Reliability background of the Eurocodes : support to the implementation, harmonization and further development of the Eurocodes.
Date: 2024-10-31
Institutions: Joint Research Centre
Subjects: EFTA countries, EU Member State, European standard, building industry, building regulations, harmonisation of standards, report, technical standard

ISSN 1831-9424 

Reliability background of the Eurocodes 

Vrouwenvelder, T., Dimova, S., Sousa, M.L., Marková, J., 
Mancini, G., Kuhlmann, U., Taras, A., Jockwer, R., Jäger, 
W., Schweckendiek, T., Franchin, P., Skejić, D., Sørensen, 
J.D., Spehl, P., Stacy, M., Feldmann, M., Croce, P., Zhao, 
B., Tanner, P., Knippers, J., Mollaert, M., André, J., 
Steenbergen, R., Kohler, J., Bakeer, T., Maljaars, J., Allaix, 
D., Spross, J. 

2024 

 EUR 40072 

 Support to the implementation, harmonization and further development of the Eurocodes  
 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
  
 
 
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JRC139110 

EUR 40072 

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ISBN 978-92-68-20980-6   ISSN 1831-9424   doi:10.2760/9482837  

KJ-01-24-076-EN-N 

Luxembourg: Publications Office of the European Union, 2024  

© European Union / Vrouwenvelder, T. / Marková, J. / Mancini, G. / Kuhlmann, U. / Taras, A. / Jockwer, R. / Jäger, W. / 
Schweckendiek, T. / Franchin, P. / Skejić, D. / Sørensen, J.D. / Spehl, P. / Stacy, M. / Feldmann, M. / Croce, P. / Zhao, B. / 
Tanner, P. / Knippers, J. / Mollaert, M. / André, J. / Steenbergen, R. / Kohler, J. / Bakeer, T. / Maljaars, J. / Allaix, D. / Spross, J., 
2024 

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How to cite this report: European Commission, Joint Research Centre, Vrouwenvelder, T., Dimova, S., Sousa, M.L., Marková, J., Mancini, G., 
Kuhlmann, U., Taras, A., Jockwer, R., Jäger, W., Schweckendiek, T., Franchin, P., Skejić, D., Sørensen, J.D., Spehl, P., Stacy, M., Feldmann, M., 
Croce, P., Zhao, B., Tanner, P., Knippers, J., Mollaert, M., André, J., Steenbergen, R., Kohler, J., Bakeer, T., Maljaars, J., Allaix, D., and Spross, J., 
Reliability background of the Eurocodes, Publications Office of the European Union, Luxembourg, 2024, 
https://data.europa.eu/doi/10.2760/9482837, JRC139110 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
CONTENTS 

Abstract ................................................................................................................................................................................................................................... 1 

Foreword ................................................................................................................................................................................................................................. 2 

Acknowledgements .......................................................................................................................................................................................................... 4 

1  Introduction ................................................................................................................................................................................................................... 5 

2  Principles of structural reliability ..................................................................................................................................................................... 6 

2.1  Steps in structural design and assessment .................................................................................................................................. 6 

2.2  Uncertainty, probability and frequency............................................................................................................................................ 6 

2.2.1 

Types of uncertainty .................................................................................................................................................................... 6 

2.2.2  Model uncertainty .......................................................................................................................................................................... 7 

2.2.3  Observed failure frequency in practice, notion of hidden safety .................................................................... 8 

2.3  Approaches to structural decision making.................................................................................................................................. 10 

2.3.1 

Safety and Economy ................................................................................................................................................................. 10 

2.3.2 

Risk based approach ................................................................................................................................................................. 10 

2.3.3 

Reliability-based approach ................................................................................................................................................... 11 

2.3.4 

Semi-probabilistic approach ................................................................................................................................................ 12 

2.3.5  Member versus system behaviour, robustness ....................................................................................................... 13 

2.4  Time effects ................................................................................................................................................................................................... 14 

2.4.1 

Bath tub curve .............................................................................................................................................................................. 14 

2.4.2 

Inspection, monitoring and maintenance .................................................................................................................... 16 

2.5  Target Reliability ......................................................................................................................................................................................... 17 

2.5.1 

Safety and economy ................................................................................................................................................................. 17 

2.5.2 

Reliability differentiation ....................................................................................................................................................... 17 

2.5.3 

Annual versus life time targets, influence of design life ................................................................................... 18 

2.5.4  Minimum versus average reliability ................................................................................................................................ 18 

2.5.5  Member versus system ........................................................................................................................................................... 18 

2.5.6 

Brittle versus ductile failure ................................................................................................................................................. 18 

2.5.7 

Role of Quality Assurance ..................................................................................................................................................... 19 

2.6  Derivation of partial factors for semi-probabilistic design .............................................................................................. 19 

2.6.1  Design value method ............................................................................................................................................................... 19 

2.6.2  Minimizing the deviation from the target ................................................................................................................... 20 

2.6.3 

Closing remark .............................................................................................................................................................................. 22 

3  The Eurocode semi-probabilistic verification method ..................................................................................................................... 23 

3.1  Historical background .............................................................................................................................................................................. 23 

3.1.1 

Chronological overview ........................................................................................................................................................... 23 

3.1.2 

Reliability methods applied in the Eurocodes. .......................................................................................................... 23 

3.1.3 

Values for key parameters ................................................................................................................................................... 25 

3.1.4 

Conclusion ....................................................................................................................................................................................... 27 

i 

 
3.2  The verification formats according to EN 1990:2023 ........................................................................................................ 27 

3.3  Target Reliability ......................................................................................................................................................................................... 29 

3.3.1 

Target for consequence class CC2 and reference period 50 year............................................................... 29 

3.3.2  Differentiation due to consequences of failure ....................................................................................................... 30 

3.3.3 

Brittle versus ductile behaviour ......................................................................................................................................... 30 

3.3.4  Differentiation on structure type ...................................................................................................................................... 30 

3.3.5  Other types of limit states and design situations .................................................................................................. 30 

3.3.6 

Influence of time: design life, temporary structures, transient design situations, execution ..... 31 

3.3.7  Material deterioration (fatigue, corrosion, etc.) ....................................................................................................... 31 

3.4  Background of partial factors for permanent and variable loads ............................................................................... 32 

3.4.1 

General introduction.................................................................................................................................................................. 32 

3.4.2 

Portfolio, Limit state function and Weights ............................................................................................................... 32 

3.4.3 

Probabilistic models for partial factor calibration ................................................................................................. 35 

3.4.4 

Resulting reliabilities for design according to the Eurocodes ......................................................................... 39 

3.4.5 

Reflection ......................................................................................................................................................................................... 40 

3.5  Further issues related to action aspects ...................................................................................................................................... 41 

3.5.1 

Load combination factors ..................................................................................................................................................... 41 

3.5.2 

Free and fixed actions, single source rule ................................................................................................................... 41 

3.5.3 

The limit state of static equilibrium ................................................................................................................................ 44 

3.5.4 

Prestressing .................................................................................................................................................................................... 44 

3.5.5 

Accidental actions ....................................................................................................................................................................... 48 

3.5.6 

Fire ....................................................................................................................................................................................................... 49 

3.5.7 

Seismic design situation ......................................................................................................................................................... 50 

3.6  Issues related to specific materials ................................................................................................................................................ 57 

3.6.1 

Geotechnics .................................................................................................................................................................................... 57 

3.6.2 

Glass ................................................................................................................................................................................................... 59 

3.6.3 

Fibre-Polymer Composites .................................................................................................................................................... 64 

3.6.4  Membrane structures ............................................................................................................................................................... 67 

3.7  Checking deformation capacity / brittle behaviour ............................................................................................................... 69 

3.8  Nonlinear analysis / FEM ........................................................................................................................................................................ 73 

3.8.1 

Types of nonlinearity ................................................................................................................................................................ 73 

3.8.2 

Use of the non-linear finite element method (NL-FEM) in assessment and design: ....................... 74 

3.8.3 

Application of the partial factors on the action side ............................................................................................ 74 

3.8.4 

Application of the partial factors on the resistance side ................................................................................... 75 

3.8.5 

Checks on deformations and deformation capacity ............................................................................................. 77 

3.8.6 

Reflection on the choice of upper and lower design values of structural properties ...................... 77 

3.8.7 

Adjustment of model uncertainty and corresponding partial factors under FEM calculation .... 77 

3.8.8 

Requirements for the quality of nonlinear numerical methods used ........................................................ 78 

3.9  Calibration  in case of degradation ................................................................................................................................................. 78 

ii 

 
3.9.1 

Fatigue using SN-curves ........................................................................................................................................................ 78 

3.9.2  Durability Verification Concrete ......................................................................................................................................... 81 

3.10 Robustness...................................................................................................................................................................................................... 84 

3.10.1 

Introduction .................................................................................................................................................................................... 84 

3.10.2  Design strategies to prevent unacceptable damage............................................................................................ 84 

3.10.3  Design methods for robustness ........................................................................................................................................ 85 

3.10.4  Reflection ......................................................................................................................................................................................... 87 

3.11 Observational methods, design assisted by testing and monitoring ......................................................................... 87 

4  Assessment of Existing Structures .............................................................................................................................................................. 89 

4.1  Motivation........................................................................................................................................................................................................ 89 

4.2  Scope .................................................................................................................................................................................................................. 89 

4.3  Target reliability levels ............................................................................................................................................................................ 90 

4.4  Updating information ............................................................................................................................................................................... 91 

4.4.1 

General .............................................................................................................................................................................................. 91 

4.4.2 

Updating the failure probabilities ..................................................................................................................................... 91 

4.4.3 

Updating the basic variables ............................................................................................................................................... 92 

4.5  Verification...................................................................................................................................................................................................... 92 

4.5.1  Overview ........................................................................................................................................................................................... 92 

4.5.2 

Partial factor format ................................................................................................................................................................. 93 

4.5.3 

Assessment values of the basic variables .................................................................................................................. 93 

4.6  Deterioration .................................................................................................................................................................................................. 95 

4.6.1 

General .............................................................................................................................................................................................. 95 

4.6.2 

Remaining structural capacity and verification ....................................................................................................... 95 

4.7  Closure ............................................................................................................................................................................................................... 95 

5  Guideline for probabilistic and risk-based decision making ........................................................................................................ 97 

5.1  Introduction .................................................................................................................................................................................................... 97 

5.2  Reliability based design procedures ............................................................................................................................................... 97 

5.3  Risk based decision procedures ......................................................................................................................................................... 98 

5.3.1 

Step 1 Qualitative Risk analysis ........................................................................................................................................ 99 

5.3.2 

Step 2 Quantitative Risk Analysis (QRA) ................................................................................................................... 100 

5.3.3 

Step 3 Decision making, optimization, risk acceptance and communication .................................... 102 

6  Concluding remarks ........................................................................................................................................................................................... 105 

References ....................................................................................................................................................................................................................... 106 

List of abbreviations and definitions .............................................................................................................................................................. 114 

List of boxes ................................................................................................................................................................................................................... 115 

List of figures ................................................................................................................................................................................................................ 116 

List of tables .................................................................................................................................................................................................................. 118 

Annexes ............................................................................................................................................................................................................................. 119 

Annex A. Probabilistic Models ...................................................................................................................................................................... 119 

iii 

 
Annex B. Applications and Examples ....................................................................................................................................................... 150 

iv 

 
 
 
Abstract 

EN 1990 “Eurocode: Basis of structural design” defines structural reliability as the ability of a structure to fulfil 
the  specified  requirements  during  the  service  life  for  which  it  has  been  designed;  the  notion  covers  safety, 
serviceability and durability. According to this definition reliability is the concern of all parts of the whole suite 
of Eurocodes. In a more narrow sense of the word, reliability refers to the way uncertainties are dealt with in 
design and assessment by the use of partial factors or more advanced probabilistic methods. This document 
deals  primarily  with  this  more  narrow  definition,  it  describes  both  the  probabilistic  and  semi-probabilistic 
approach  and  gives  special  attention  to  their  underlying  relationship.  Chapter  2  should  be  considered  as  a 
general introduction to this report and introduces relevant basic notions such as uncertainty, probability, risk, 
member  and  system failure,  time  dependency, effects of  deterioration  and inspection,  etc.  In  chapter  3  the 
focus is on the basic reliability requirements in EN 1990 related to the persistent as well as the accidental and 
seismic design situations. Some special attention is given to the new materials: glass, fibre reinforced polymers 
and membrane structures included in the second generation of the Eurocodes. In chapter 4, attention is given 
to  the  special  problems  that  may  arise  in  assessment  of  existing  structures.  Important  is  the  treatment  of 
several types of observations (inspections, monitoring, proof loads etc), the various decision options (further 
inspection, repair, and demolishment) and of course typical reliability issues (updating, reduced targets). The 
explicit risk and full probability methods described in Chapter 5 are intended to support these approaches where 
allowed  and  meaningful.  It  also  gives  background  for  code  calibrations.  Annex  A  gives  an  overview  of 
probabilistic  models  for  actions,  material  properties  and  model  uncertainties.  It  may  be  considered  as  a 
reflection  of  the  present  state  of  knowledge  in  the  various  Eurocode  subcommittees,  working  groups  and 
horizontal groups. Annex B shows a number of example applications related to various parts of the theory. 

1 

 
Foreword 

In  building  and  civil  engineering,  most  structures  are  designed  and  assessed  on  the  basis  of  codes  and 
standards. These documents describe the structural calculation models, the values for the load (action, demand) 
and  resistance  (strength,  capacity)  parameters,  the  safety  margins,  the  way  to  cope  with  accidental  and 
extraordinary design situation, etc. The intention is to arrive at structures that are both safe and economic and 
serve implicitly or explicitly sustainability goals like limiting emissions and consumption of natural resources. In 
several countries the codes may be mandatory in a legal sense on the basis of a building act. 

The  suite of Eurocodes  (EN  1990 – EN 1999)  has been  written with the above intentions to  harmonize the 
various national rules within Europe. Its field of application are all relevant structures like buildings, bridges, 
towers, etc. including their foundations for all kind of materials, loads, geometry and models. The various parts 
of the Eurocodes deal with one or more of those topics. Within this spectrum of codes the aim of EN 1990 Basis 
of Structural Design is to harmonize the reliability concepts over the various structures and design situations. 
It should be emphasized that as far as safety levels are concerned, every member state within CEN has the 
right to make individual choices and modifications. Actually, that is the role and duty of every national code 
committee as explained in document CEN/TC 250 N1250. 

The present report is a Background Document to the EN 1990:2023 standard mentioned above. It has been 
written  by  a  special  Ad  Hoc  Group  that  was  installed  on  the  basis  of  the  following  formal  decision  by 
Subcommittee 10 of CEN/TC250. 

DECISION 13 taken by CEN/TC 250/SC 10 on 2018-12-06  

Subject:  Formation  of  a  CEN/TC  250/SC  10  Ad-Hoc  Group  to  prepare  a  JRC  Science  and  Policy  Report  on 
‘Reliability Background in the Eurocodes’  

The CEN/ TC 250/SC 10 EN 1990 – Basis of structural design, considering the requirement to engage efficiently 
with progress of SC10 related outputs from Mandate M/515  

−  agrees  to  the  formation  of  an  Ad-Hoc  Group  to  prepare  a  JRC  Science  and  Policy  Report  on  ‘Reliability 
Background  in  the  Eurocodes’  with  a  convenor  (TBC),  maximum  1  representative  from  each  of  CEN/TC  250 
subordinate group (SCs, WGs, HGs), 3 members from SC 10 as key members and 2 members from JRC, noting 
the possibility for the SC Chairman to enlarge the group in the future as needed;  

− agrees the following terms of reference for the Ad-Hoc Group:  

  Draft  an  explanatory  report  on  the  assumptions  in  EN  1990  for  the  most  representative  set  of 
partial factors so as to avoid doubts and possible confusion both at national level and for practical 
users,  

 

 

 

 

 

 

To provide guidance to NSBs for their decision on partial factors in Annex A.1 of EN 1990 in each 
country,  illustrating  common  methods,  procedures,  etc.  also  by  means  of  examples,  taking  into 
account the work of SC 10/WG 1,  

Provide guidance on stochastic modelling on the most representative failure modes for different 
materials (e.g. concrete, steel, timber, masonry) and for different loads (e.g. imposed loads, wind 
actions, snow loads),  

Provide  basic  assumptions,  methods  and  provide  examples  to  serve  as  a  reference  for  the 
implementation of full probabilistic methods and, potentially, of risk-based methods, in line with 
EN 1990 guidance given in its informative Annex C,  

To report regularly on the progress of the work to SC 10, thus enabling SC 10 to report to TC 250,  

To deliver within the time frame for the publication plan for EN 1990.  

SC 10 requests TC 250 to encourage each subordinate group to appoint a member to the Ad-Hoc 
Group.  

The decision was taken by unanimity. The appointed members of the Ad Hoc Group can be found on page 8 of 
this report. The group started their activities in the summer of the year 2019 and has met about  12 times, 
mostly in the form of electronic meetings. Subcommittee SC10 was informed on a  regular basis and in the 
summer of 2021 a request for feedback under all relevant subcommittees and workgroups was organised. A 
summary of the main comments received and the reactions of the Ad Hoc Group can be found in chapter 6.1. 

2 

 
It should be noted that the report is intended to support those engaged in the application of reliability concepts 
for the calibration of National Annexes to the Eurocodes and/or for specific projects. The report has therefore 
been drafted for a target audience that is familiar with this technical field and it also assumes an understanding 
of  the  fundamental  concepts  of  reliability  analysis.  Such  'text  book'  material  is  not  included  in  this  report. 
References is made to e.g. Melchers and Beck 2007, Madsen et al. 2006, Schneider and Vrouwenvelder 2017. 
It is also assumed that the reader is familiar with methods and terminology in EN 1990.  

It  is  helpful  to  acknowledge  that  CEN/TC  250  unanimously  agreed  the  approach  to  be  taken  in  seeking  to 
enhance ease of use of the second generation of the Eurocodes.  Two aspects of this approach are of particular 
relevance to this report. 

First is the aim defined for the primary target user group of competent practitioners which states: ‘We will aim 
to  produce  standards  that  are  suitable  and  clear  for  all  common  design  cases  without  demanding 
disproportionate levels of effort to apply them’.  This aim inevitably results in some degree of simplification 
and conservatism in certain design situations.   

Second is the principle that that Eurocodes should include ‘state-of-the-art material the use of which is based 
on  commonly  accepted  results  of  research  and  has  been  validated  through  sufficient  practical  experience’. 
Standards such as the Eurocode are reviewed on a regular basis and it is expected that they will be periodically 
updated. Whilst this report focusses on the background to EN 1990, it also discusses some matters that have 
not been incorporated into the Eurocodes because they were not yet felt to sufficiently satisfy this principle. 
However, it is valuable to include them in anticipation that, with further research and application, they could 
potentially be included in the future and therefore to encourage that such research is undertaken.  

Finally, it is appropriate to recognise that during the development of the Eurocodes there has been significant 
debate on some of the matters covered in this report before arriving at a consensus position. Inevitably, such a 
consensus does not necessarily represent the first preference of all parties.  

Steve Denton 

Chair, CEN/TC 250 “Structural Eurocodes” 

Paolo Formichi 

Chair, CEN/TC 250/SC 10 “EN 1990 Basis of structural design” 

3 

 
 
 
 
 
Acknowledgements 

The  authors  would  like  to  thank  Steve  Denton,  chair  of  CEN/TC  250  ‘Structural  Eurocodes’,  for  his  detailed 
technical and editorial review, and his full support to the publication of this report. 

The authors would also like to thank Paolo Formichi, chair of CEN/TC 250/SC 10 ‘EN 1990 - Basis of structural 
and geotechnical design’ for his careful supervision of the project. 

Authors 

This  report  it  the  result  of  the  work  performed  by  the  CEN/TC  250/SC  10  Ad-Hoc  Group  on  the  Reliability 
Background in the Eurocodes. The group had the following members: 

Ton Vrouwenvelder 

Convenor CEN/TC 250/SC 10 Ad Hoc Group Reliability  

Silvia Dimova  

Luisa Sousa   

formerly Joint Research Centre of the European Commission, Ispra-Italy 

formerly Joint Research Centre of the European Commission, Ispra-Italy 

Jana Marková 

CEN/TC 250/SC 1  

(Actions on structures) 

Giuseppe Mancini  

CEN/TC 250/SC 2  

(Concrete structures) 

Ulrike Kuhlmann   

CEN/TC 250/SC 3  

(Steel structures) 

Andreas Taras 

CEN/TC 250/SC 4  

(Composed structures)  

Robert Jockwer 

CEN/TC 250/SC 5  

(Timber structures) 

Wolfram Jäger 

CEN/TC 250/SC 6  

(Masonry structures) 

Timo Schweckendiek 

CEN/TC 250/SC 7  

(Geotechnical structures)  

Paolo Franchin 

CEN/TC 250/SC 8  

(Earthquake resistance) 

Davor Skejić   

CEN/TC 250/SC 9  

(Aluminium structures) 

John Dalsgaard Sørensen CEN/TC 250/SC 10  

(Basis of structural and geotechnical design)  

Pierre Spehl   

Mungo Stacy   

CEN/TC 250/SC 10  

(Basis of structural and geotechnical design) 

CEN/TC 250/SC 10 

(Basis of structural and geotechnical design) 

Markus Feldmann 

CEN/TC 250/SC 11  

(Structural Glass) 

Pietro Croce   

CEN/TC 250 HGB   

(Horizontal Group Bridges) 

Bin Zhao 

Peter Tanner   

Jan Knippers   

CEN/TC 250 HGF   

(Horizontal Group Fire) 

CEN/TC 250 WG2  

(Existing Structures) 

CEN/TC 250 WG4  

(Fibre reinforced polymer structures) 

Marijke Mollaert   

CEN/TC 250 WG5  

(Membrane structures) 

João André       

CEN/TC 250 WG6  

(Robustness) 

The AHG would like to thank the following experts for their contributions as co-authors to some specific parts 
of this document: 

Raphael Steenbergen   TNO, Delft, the Netherlands  Section 3.4 and annexes B1 and B2 on partial factors 

Jochen Kohler 

         NTNU, Trondheim, Norway  Section 3.4 and annexes B1, B2 and B3 on partial factors 

Tammam Bakeer          TU Dresden, Germany         Section 3.18 and annex B4 on numerical analysis 

Johan Maljaars 

         TNO, Delft, the Netherlands  Annex B6 on fatigue 

Diego Allaix 

         TNO, Delft, the Netherlands   Annex B7 on deterioration of concrete 

Johan Spross  

         KTH, Stockholm, Sweden  

Annex B8 on risk analysis and design by observation 

4 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1 

Introduction 

Society expects that buildings and other structures are safe for the people who use them or are near them. 
They  should  also  protect  against  adverse  economic  consequences.  The  failure  of  a  building  or  structure  is 
expected to be an extremely rare event.  

Structural engineers devote their effort to meeting society’s expectations efficiently. Engineers and scientists 
work together to develop solutions to structural problems. Given that nothing is absolutely and eternally safe, 
the goal is to attain an acceptably small probability of failure for a structure. Reliability analysis is part of the 
science  and  practice  of  engineering  today,  not  only  with  respect  to  the  safety  of  structures,  but  also  for 
questions of serviceability and other requirements of technical systems that might be impacted by uncertainties. 

The  elaborations  of  these  tasks  may  take  place  on  several  levels  often  referred  to  (ISO  2394)  as  (1)  the 
risk-based approach, (2) the reliability-based approach and (3) the semi-probabilistic or partial factor-based 
approach.  Appropriate calibration procedures should lead  to the various levels giving  consistent results (see 
section 2.3). In addition to this internal calibration of the various design approaches there is also a need to 
compare the results externally to design and assessment methods used in the  past (see Figure 1.1). Where 
these  methods  were  successful,  they  should  not  be  changed  substantially.  On  the  other  hand,  unsafe  or 
uneconomic situations should be corrected when encountered, as has been done systematically at every code 
revision in the past decades. 

On the highest level of abstraction, the design as well as the assessment of a structure (or structural system 
or  part  of  it)  comes  down  to  a  decision  analysis.  The  question  is  which  lay  out,  which  material  and  what 
dimensions are needed to give a structure that satisfies all preferences, taking into account possible constraints 
of several types. Only in special cases the total optimization is part of the individual design process; in most 
cases  this  process  is  solved  in  several  steps.  Authorities  often  prescribe  the  minimal  required  reliability  by 
balancing implicitly or explicitly on the one hand the use of resources needed and on the other the consequences 
of  unsatisfactory  behaviour.  This  includes,  of  course,  also  aspects  of  human  safety.  The  resulting  required 
reliability  may  be  presented  in  terms  of  target  failure  probabilities,  but  in  daily  practice  use  is  made  of  a 
translation of probabilities into demands on the level of characteristic values and partial (safety) factors, the 
so-called partial factor method or semi- probabilistic approach. In this document the full risk and reliability-
based design and assessment procedures are treated in Chapter 4, the semi-probabilistic method is treated in 
Chapter 3. Chapter 2 treats the information that is general for all levels. 

Figure 1.1. Basic scheme of calibration procedure: the first step is to determine a target reliability for an appropriate 
class of structures based on a combination of several criteria, the second step is to find corresponding partial factors. 

5 

 
 
2  Principles of structural reliability 

2.1  Steps in structural design and assessment 

The  design  and  assessment  of  structural  systems  comprise  many  aspects.  In  this  document  we  focus  on 
structural reliability related issues. In this context the word structural includes geotechnical and other relevant 
disciplines. This chapter 2 deals, by way of introduction,  with a number of the  general aspects of structural 
design  and  assessment  and  does  not  give  specific  background  information  to  clauses  and  formulas  the 
Eurocode. This will be done in chapter 3. 

The basic question is, whether for a given structure (planned or existing) the level of reliability for an intended 
(remaining) service life is optimal or at least sufficient. This is called a reliability verification procedure. This 
verification procedure may formally be subdivided into a number of steps either explicitly in the case of risk 
based approaches or implicitly using the partial factor approach:  

1. 

In  the  first  stage  the  system  to  be  considered  is  defined  and  an  inventory  of  all  relevant  external 
threats and hazards is made; the threats can be from any source of mechanical, physical, chemical or 
biological origin. Note that even so called “unforeseen” actions may need attention.  

2.  Next the internal structural processes that may lead to failure need to be described, usually referred 
to as mechanisms or failure scenarios, including the planned or achieved effects of quality assurance, 
inspection,  monitoring  and  maintenance.  The  combination  of  hazard  and  failure  scenario  is  often 
referred to as a design situation. 

3.  The third step describes the possible suite of consequences of the resulting structural behaviour.  

In all the above steps “models” are used. The models reflect our scientific and empirical knowledge, usually in 
the form of mathematical expressions and algorithms like for instance a structural Finite Element Model. Models 
can be defined as relations between variables describing the state of input and output of physical processes. 
Because of the uncertainties involved, the outcome of the structural behaviour as well as the net benefit of all 
design and assessment decisions are uncertain. The best we can do is to find the decision that optimizes the 
expectation of the benefit, using the concepts of probability theory. Actually, that brings us to the next steps in 
our procedure: 

4.  Quantification of relevant uncertainties in probabilistic terms. 

5.  Finding solutions leading to an optimal expectation of the benefit (or a minimum of costs), meeting at 
the same time all societal preferences and demands related to human safety, environmental and other 
intangibles.  

Given the fundamental role of the notions uncertainty and probability it makes sense to discuss these issues 
first before going into further elaboration in structural engineering.  

2.2  Uncertainty, probability and frequency 

2.2.1  Types of uncertainty 

Like  in  everyday  life,  uncertainties  of  multiple  different  nature  are  encountered  in  decisions  concerning  the 
design and assessment of structures. The most obvious uncertainties are related to the natural variability or 
randomness of actions, material properties and geometric data. However, as soon as we want to establish the 
statistical properties of those variables, we have to deal with many unclear sets of subpopulations and often a 
substantial shortage of data. This leads to the important class of so-called statistical uncertainties. In addition, 
we  have  to  admit  that  our  theories  and  models  about  environmental  and  structural  behaviour  are 
approximations  of  reality  and  make  predictions  with  some  higher  or  lower  degree  of  accuracy.  The  related 
uncertainties are referred to as model uncertainties. Additionally we have to deal with uncertainties present in 
measurements and observations. Finally, in practice uncertainties can arise from human errors of several kind 
as well as unforeseen (or even unforeseeable) events.  

Notes:  

Different kinds of uncertainty may require different means of counteracting; for instance, partial factors 
may not be very effective against gross errors.  

6 

 
Uncertainty is not a (constant) property of a structure, but depends on the state of knowledge, which can 
change over time, e.g. during the process of design, building and use; it is one of the reasons that existing 
structures can have to be judged in a slightly different way (see chapter 4). 

In  reliability literature  uncertainties are  often  subdivided  into  aleatory  uncertainties resulting  primarily  from 
(natural) variability and epistemic uncertainties resulting from incomplete knowledge. The idea is that epistemic 
uncertainties can be reduced by investigations while aleatory cannot, although the exact subdivision to some 
extent is arbitrary. It is of course useful to know whether further investigations or research activities can reduce 
uncertainty  or  not.  Sometimes  reduction  of  uncertainty  is  possible  in  theory  but  not  in  practice.  The  basic 
question, however, is whether, in structural engineering, we want to make a distinction in treatment between 
the two types of uncertainty if they exist. It turns out that there are two schools of thought.  

In  the  classical  (frequentist)  school  of  statisticians  a  clear  distinction  in  the  treatment  is  made.  The  basic 
philosophy is that the quantification of epistemic uncertainties in principle requires subjective estimates and 
should be kept carefully separated from the objective scientific part. In the alternative so called Bayesian school, 
the notion of probability is understood as a “degree of belief” which makes an equal treatment of objective and 
subjective uncertainties a logical next step. In applications in structural engineering the Bayesian approach has 
been adapted from the beginning as the most appropriate one. In short, the idea is that as subjective estimates 
of likelihood simply cannot be completely excluded anyhow, the best we can do is to deal with them and make 
sure that these estimates reflect the view of a large group of professionals, thereby making them as objective 
as possible.  This  will  not  make  every  individual  estimate  fully  reliable.  The  only  claim is that,  if  carried  out 
properly, the probabilities will be correctly averaged over a large number of decision situations. The requirement 
to fulfil that claim, is that the purely intuitive part is neither systematically too optimistic nor systematically too 
pessimistic. Checking whether the theoretical results match reality can only be done by long term observations 
over a wide variety of structures. 

A pronounced difference in application is visible in the statistical evaluation procedures for design by testing. 
For example, in the frequentist formulas for the 5% characteristic values we first find the best guess for the 
5% fractile (natural uncertainty) and an additional say 10% confidence interval (statistical uncertainty). In the 
Bayesian approach (see EN 1990:2023 Annex D) the 5% fractile is found including the statistical uncertainty 
and no confidence interval is given. 

2.2.2  Model uncertainty 

Uncertainties are not only  present in  variables like loads  and material properties but  also in the action and 
resistance models used in structural analysis. The models usually are based on a combination of first principles 
and  (semi)  empirical  relations.  Depending  on  the  material  and  the  type  of  mechanism  (e.g.  bending,  shear, 
buckling, corrosion) models will predict the real behaviour with lower or greater accuracy. The main sources of 
model  errors  are  related  to  influences  or  parameters  neglected  in  the  models  and  approximations, 
simplifications or errors in the sets of equations. To incorporate these uncertainties, the outcome of the model 
Ymodel can be represented as:  

Often reduced to: 

Ymodel = f (X1,…Xn, a1,…am , θ1,…θk) 

Ymodel =θ1 f(X1,…Xn; a ) 

(2.1) 

(2.2) 

The variables Xi … Xn describe the relevant physical (loading, material or geometrical) parameters; the additional 
random variable  accounts  for the  model  uncertainty,  a is a vector containing the parameters that can be 
assumed as deterministic. 

As far as possible models are compared with and calibrated on measurements and experimental tests using a 
best  fitting  criterion  and  estimates  of  the  degree  of  uncertainty,  usually  respectively  the  bias  ()  and  the 
coefficient of variation V(). Measurement errors and statistical uncertainty due to a limited data set should be 
taken  into  account.  Note  that  well  calibrated  advanced  models  sometimes  are  used  to  calibrate  simplified 

7 

 
 
 
engineering models. In those cases it is important to realize that model and statistical uncertainties are present 
on both levels and should be taken into account simultaneously.  

In the absence of experimental evidence expert judgement (Bedfort and Cooke, 2003) is sometimes considered 
as an option to estimate the model uncertainty parameters, although utmost care is needed. 

2.2.3  Observed failure frequency in practice, notion of hidden safety 

In the light of the above discussion the question is often raised whether or not the theoretical probability of 
failure corresponds or should correspond to the frequency of failed structures in practice. In this discussion the 
notion of hidden safety plays an important role, among other factors such as gross/human errors or negligence 
during design, execution or operation. Gross/human errors are supposed to be covered by quality control and 
operational  requirements  (e.g.  execution  standards);  they  cannot  be  adequately  covered  by  partial  factors 
calibrated based on engineering models and physical and model uncertainties related to uncertain parameters. 
Excluding these type of errors, the observed failure rates (e.g. ULS exceedance) appear in general to be small 
compared to the estimations with engineering models. 

This section addresses different types of hidden safety, illustrated by examples, and the implications for the 
safety philosophy in the Eurocodes. It is noteworthy that there are also situations where unconservative choices 
or assumptions are made, however this section focusses on the more common safe-sided elements. 

Generally  speaking,  assumptions  and  choices  in  design  and  the  related  engineering  modelling  tend  to  be 
conservative, or in other words, on the safe side. In most cases such conservative choices are very rational, 
taking into account the asymmetry in consequences from errors or deviations on the safe and unsafe side. The 
list below contains several  (maybe interconnected) examples of hidden safety that can arise in construction 
design: 

1.  Resistance modelling: We may opt for elastic analysis instead of plastic analysis. Model choices are 
often  based  on  the  wish  to  make  simplifications  (ease  of  use).  The  simplified  models  are  often 
relatively safe-sided and robust approximations of more advanced models. 

2.  Load  modelling:  The  load  patterns  assumed  for  modelling  the  spatial  distribution  of  imposed  and 
climatic loads are often chosen with very safe assumptions, or the seismic action is described through 
a uniform hazard spectrum, rather than a less conservative conditional spectrum (See.3.5.7). 

3.  Design type or structural system: In many design situations, a structure is designed according to its 
critical section or member, while the rest is overdesigned. Sometimes this is done in order to enforce 
a desired favourable plastic mechanism (Žižmond et al, 2016). Also, it is quite common that ULS does 
not govern the structural dimensions, but rather SLS does. In these cases, the safety margin can be 
considerable. Finally there may be robustness consideration, for example, anchors of retaining walls 
are designed such that neighbouring anchors could cope with the failure of an in-between anchor.  

4.  Structural behaviour modelling: The structural behaviour is simplified using conservative assumptions. 
For example, a  hinge is assumed where in fact an  encasement providing at least partial  continuity 
exists. Such system redundancies are often not explicitly exploited. 

5.  Material strength modelling: For example, the usual steel design is for yielding, while strain hardening 
and the margin to the ultimate strength is typically neglected (unless covered by bias in model factors 
in calibration of partial factors); another example are the conservative assumptions with respect to 
geometrical imperfections.  

6.  Constructional type: For many designs there are minimum values for dimensions and additional often 
prescriptive safety rules. For example, in steel construction, there is a minimum weld thickness given 
even if not needed. Besides there are explicit robustness rules and rules like “one bolt is no bolt”, also 
providing  extra  safety.  Minimum  values  for  member  sizes  and  detailing  in  Eurocodes  compliant 
structures have been shown to lead to reliability, with respect to seismic action, higher than the target 
one by orders of magnitude in low-seismicity areas (Iervolino, et al. 2018). 

7.  Practical issues  rounding off structural dimensions to the safe side and producers aiming at better 

quality then required to avoid a negative acceptance test. 

8.  The limited number of safety elements in the safety format intended for a large set of applications. 
This  may  lead  to  a  subset  of  applications  where  already  the  theoretical  reliability  level  is  not  in 
accordance with the target.  

8 

 
The list above is not exhaustive but meant to provide examples of types of hidden safety, many of which are 
related to modelling choices. It should be clear that the estimation of actual, observable failure rates in reality 
using engineering models is difficult, given the unavoidable bias introduced by hidden safety elements, even 
though some types can be covered by taking model bias explicitly into account. 

When considering hidden safety, it is meaningful to make a distinction between the following two categories: 

  Hidden safety which is actually ‘hidden’ for the normal user of a code but can be modelled as model 
uncertainties based on comparisons between physical tests and the engineering models in the codes. 
Hence, this type can be accounted for in calibration of the partial factors, though it can be difficult to 
determine model  uncertainties accurately. If physical test results are available, model uncertainties 
can be quantified using the method in EN 1990:2023 Annex D.  

9.  An  important  implicit  conclusion  of  this  type  of  hidden  safety  poses  limitations  to  using  advanced 
analysis methods other than those the code was calibrated to. For example, a user may think that an 
increase of the characteristic resistance can be obtained using an advanced numerical method (e.g. 
FEM) instead of the engineering model in the code but the conservative bias of the engineering model 
in the code may already be accounted for in the choice of the material factor. A reduction of the scatter, 
however, still may be possible. 

  Hidden  safety  related  to  aspects  like  conservative  assumptions  in  design.  These  aspects  can  be 
substantial but there is no guarantee that they always are present. Some structures are optimized just 
to the limit of the requirements in the code. 

Related  to  the  first  category  the  transparency  is  of  great  importance:  if  hidden  safety  is  made  explicit  by 
comparing model predictions with experiments or real behaviour, the implication for design mostly depends on 
the starting point for the reliability target, which has been and probably will be treated differently in the different 
Eurocode member states. The basic options, as stated earlier, are: 

  Design rules for new structures  are  calibrated to  similar existing structures as long as they exhibit 
satisfactory  performance  (essentially  ensuring  that  roughly  the  same  structural  dimensions  are 
obtained by new design rules). 

  Designs  of  new  structures  meet  a  reliability  target  based  on  considerations  of  human  safety  and 

economy of design, verified by reliability analysis. 

In the first case there is almost no chance to remove a (useful) hidden safety, in the second case there is with 
possible unintended consequences. For completely new types of structures and materials the second option is 
the only one available. 

In addition to hidden safety there might also be hidden unsafety. Sometimes models may be based on incorrect 
assumptions that are not detected in underlying experiments and tests. As an example, one might think of a 
change of failure mode from ductile to brittle when reaching the tail of the distribution. One observes always 
ductile modes in even a large number of tests, but at the relevant low fractiles the mode may be brittle having 
large  consequences  for  the  system  behaviour  and  consequences  to  human  safety  because  of  the  lack  of 
warning. 

However, even if we could fully exclude the issues of hidden (un)safety, the nature of probability itself would 
still give rise some careful considerations. The failure probability calculated is always the result of the chosen 
physical models for the relevant mechanisms and the appropriate corresponding statistical properties for the 
random variables. These choices are based on a particular state of information. As a consequence, the failure 
probability for a certain building may change if we get/obtain more information  e.g., on the concrete quality 
actually delivered instead of the concrete quality in general. The state of information is to some extent a choice 
in itself that can be optimized. 

This  ambiguity  is  also  one  of  the  reasons  that  we  should  not  interpret  a  failure  probability  as  an  absolute 
attribute of the physical structure or structural system. This would only be the case if no further information 
could be obtained at all and in structural engineering that can hardly  ever be the case. In the context of risk 
informed decision making, the failure probability is mainly a comparative measure and should always be seen 
and communicated conditional to that.  

However, the question of the relation between the outcome of a reliability analysis and the failure frequency in 
the real world will remain. Notions like optimization and updating are meaningful only if the statistical numbers 

9 

 
have a meaning in the world where optimization and updating is performed. So the need to have numbers as 
realistic and up to date as possible for that aspect is there.  

2.3  Approaches to structural decision making  

2.3.1  Safety and Economy 

In section 2.1 the general decision-making approach with respect to the verification of structural reliability has 
been  presented  and  in  chapter  1  the  possible  levels  of  risk  based,  reliability  based  and  semi-probabilistic 
methods were introduced. In this section these three levels and in particular their step wise coherence will be 
discussed. For all levels the result should be a sufficiently safe as well as economic structure. The notions safe 
and economic are not fully clear in themselves and need some further elaboration.  

“Safe” has primarily to do with the safety of people who use the structure or are in the vicinity of the structure. 
Attention should be given to the risk of individuals, to the so-called societal risk (the possibility that a number 
of persons gets killed or hurt) and the fact that people should feel safe in a built environment. To some extent 
“safe” is also understood as a requirement of the structure to protect economical value. The central attention, 
however, is usually given to the notion of “individual risk”. The individual risk follows from the combination of 
the probability of structural failure and the probability that a person is killed in the event of structural failure. 
The socially accepted value for the individual risk level related to structural failure level is usually considered 
to be at least one order lower than for other accidents. Maximum acceptable individual risk values are generally 
in the order of 10-6 to 10-5 (often industrial) per year (see also Vrijling et al, 2005). In many countries values 
for the maximum individual risk are also often depending on regulations for specific types of hazards such as 
industrial  accidents,  flood  defence,  traffic  safety,  etc.  Usually,  they  are  expressed  in  terms  of  annual 
probabilities.  

“Economy” should be interpreted as “socio-economic”, including all types of direct and indirect costs related to 
building,  use,  damage and demolition of  a structure.  It includes  monetary  values as  well as intangibles like 
environment issues and sustainability. In other words: one should not build structures that require large amounts 
of resources that could be used in better ways. On the other hand, where with little effort a higher reliability 
level can be achieved this should be done  to avoid unnecessary  socio-economic costs of failure. So, we are 
looking  for  a  balance  between  safety,  sustainability  and  economy.  This  is  a  responsibility  for  every  code 
committee when developing new codes or when rewriting existing ones. The past in this context is a guidance, 
not a must. The aim must be to combine new developments with proven technology. 

2.3.2  Risk based approach 

In  the  full  risk-based  approach  the  above  decision  process  is  carried  out  explicitly,  including  the  safety 
constraints mentioned. In its most simplified way, the optimization can be written down as a minimization of 
the sum of the direct costs of the design decision and the risk due to failure. That is, leaving out for the time 
being  discounting  and  other  complicating  factors  like  maintenance,  we  want  to  minimize  the  total  cost 
expectation Ctot: 

Under the safety constraint: 

min Ctot = Ccon + Pf Cf 

Pf < Pft 

(2.3) 

(2.4) 

where Ctot are the expected total costs, Ccon the cost of the (design) decision, Cf the costs of the consequences 
of unsatisfactory performance and Pf the probability of unsatisfactory behaviour during the design service life. 
The extra condition Pf < Pft is based on ethical and societal aspects of human safety where relevant. In many 
cases the so called ALARP  principle is used (As Low  As Reasonably Practicable).  The  risk-based approach is 
generally used for individual structures, not for a portfolio.  

The  approach  implicitly  assumes  that  the  structural  behaviour  can  be  subdivided  into  a  satisfactory 
performance  and  an  unsatisfactory  performance  or  failure.  The  border  line  between  the  two  possibilities  is 

10 

 
 
 
called the limit state. To enable a further numerical approach a limit state function g(X) is defined, where X is 
the  vector  of  all  random  variables  (actions,  material  properties,  etc).  By  definition  the  domain  g(X)<0 
corresponds to failure and g(X)>0 to non-failure. In order to formulate the function g(X) use is made of well-
known action and structural models. This means that the function g(..) might be much more complex than it 
looks at first sight. The failure probability can now formally be calculated from: 

Pf = P(g(X) <0) = ∫

g(x)<0

f(x)dx

(2.5) 

where f(x) is the joint probability density function of all the random variables, which means that for all variables 
involved we need estimates of their mean values, their uncertainty and also their mutual dependencies. Note 
that X in equation refers to the random variables themselves and x to particular realisations of X.  

Notes: 

The term “failure” in reliability theory is not restricted to collapse but can refer to any form of unintended 
behaviour. It can refer to the system level as well as to the level of members or (sub) systems. In some 
cases, the binary subdivision “failure versus nonfailure” is of course a simplification as there can also be a 
gradual transition from satisfactory to non-satisfactory behaviour like in service-ability limit states. In other 
cases, the real limit state can often be difficult to model and use is made of approximate or condition limit 
states.  

Random entities (like for instance the wind velocity) can vary continuously in in time and space; in those 
cases they are usually referred to as random processes and random fields respectively 

2.3.3  Reliability-based approach 

In the reliability-based approach the optimization (2.3) to (2.5) reduces to: 

Under the condition  

min C = Ccon 

Pf < Pft  

(2.6) 

(2.7) 

The minimal required  reliability  Pft now includes (next to immaterial constraints like human safety) also the 
economic considerations that were left to an explicit optimization procedure in the risk-based approach. This in 
fact  is  the  first  step  in  Figure  1.1.  Note  that  this  way  minimum  human  safety  requirements  and  economic 
average targets are mixed up. It is the price of simplification. 

The required minimum or target reliability levels (in fact the maximum acceptable failure probability) usually 
are proposed by the engineering community (in particular code committees) but are in the end the responsibility 
of the relevant authority and given in the National Annex. Required reliability levels are generally presented as 
a  classification  system  of  the  consequences  of  failure  expressed  in  terms  of  human  safety,  economy  and 
environment. Recourses needed to achieve a higher safety level may play a role, explicitly or implicitly. In many 
code calibrations also a comparison to the design procedures applied in the in the past is made. The idea is that 
the design rules in the past on the whole turned out to be quite satisfactory and should only be modified to 
achieve more consistency or to enable a wider field of application. There is, in other words, no need for major 
shifts.  For  completely  new  types  of  structures  or  materials  the  challenge  is  somewhat  different.  Also  new 
requirements, for instance relating to sustainability matters, can lead to the need for reconsideration of target 
reliabilities.. The expectation is that this will be even more in the future. 

Note that (2.7) may refer to the whole system or to individual members of failure modes (say ductile versus 
brittle). If used for a portfolio the safety constraint could in (2.7) refer to the highest failure probability or to an 
average value. See also sections 2.3.5 and 2.5. 

11 

 
 
   
 
 
2.3.4  Semi-probabilistic approach 

For the semi-probabilistic approach, used in daily practice, the requirements (2.6) and (2.7) are transferred to: 

Under the condition:  

min C = Ccon 

g(Xd) > 0 

(2.8) 

(2.9) 

The Xd are the so-called design values of the random variables. In terms of reliability theory, the design values 
represent  the  values  of  X  for  which  the  likelihood  density  of  failure  (after  a  transformation  to  a  set  of 
independent standard Gaussian variables) is maximal. If X has a normal distribution the value Xd can (se also 
the Design Value Format method, chapter 2.6) be found from: 

Xd = µX - X t X 

(2.10) 

where µX and X are the mean and standard deviation of X respectively, X is the (FORM) importance factor of 
the  variable  X  with  0,0    X   1,0  for  resistance  properties and  -1,0    X   0,0  for  actions, t  is the  target 
reliability index corresponding to the target failure probability Pft via Pft = (-t), where ( ) is the distribution 
function of the zero-mean normal distribution with unit standard deviation, see further Table 2.1. For other than 
normal distributions the value of Xd may be found from F(Xd) = (Xt) with F(..) being the cumulative distribution 
function of the variable X. As the result must be conservative for a large group of structures, the selected  X 
values will usually be conservative (uneconomic) in individual cases. 

Table 2.1. Relationship between the failure probability Pf and the reliability index . 

Pf 

 

Source: Excel 

10-1 

10-2 

10-3 

10-4 

10-5 

10-6 

10-7 

1,3 

2,3 

3,1 

3,7 

4,2 

4,7 

5,2 

In the Eurocodes the design values are derived by multiplying (or dividing by) the characteristic values Xk with 
some appropriate partial or combination factor.  

Xd = Xk /     

for resistance  

 (2.11a) 

Xd =  Xk of  𝜓 Xk 

for actions 

 (2.11b) 

These  partial  factors  have  been  calibrated  (second  step  in  Figure  1.1)  in  such  a  way  that  the  design  or 
assessment decision for a large class of structures fulfils with some acceptable level of accuracy the reliability 
requirements (see Chapter 3 for more details). The basic problem is that the alpha-values X vary from design 
situation to design situation. For that reason, the semi-probabilistic method is implicitly or explicitly based on a 
conservative (weighted) choice of the alpha-values. This is not only the case for the Design Value Format (DVF) 
method  (section  2.6)  but  implicitly  also  holds  for  the  procedure  where  partial  factors  are  derived  using  an 
optimization procedure. 

12 

 
 
 
 
 
 
Note:  both  in  EN 1990:2002  and  EN 1990:2023  the  DVM  method  is  recommended  and  often  used  on  the 
resistance side; the proposed partial factors for variable loads, however, are often lower than would follow from 
the DVM method. See section 3.4 and Annex B.2 for further information.  

2.3.5  Member versus system behaviour, robustness 

The  analysis of  reliability  and  risk should  in  principle  be  applied  to  the  total  structural  system,  including  its 
interactions with the environment. In practice however the verification of the reliability is often performed on 
the level of single members (beams, columns, connections), single modes (shear, bending) or even on the level 
of a single cross section or individual point. The verification rules in the Eurocodes are in most cases formulated 
at the member level and also the target reliability is used for the individual failure mode at member level.. This 
might be considered as the outcome of a trade-off between correctness and complexity of the analysis.. We 
shall discuss it below in somewhat more detail. 

Even if collapse modes on the structural level are considered (like in non-linear (e.g. buckling) analysis), the 
verification is actually carried out on an individual  failure mode basis; the fact that the presence of several 
(partially dependent) potential modes may lead to an increase of the failure probability is usually neglected. 
The  JCSS  probabilistic  model  code,  however,  recommends  a  more  system-based  approach  and  a  relatively 
simple application in practice may be found in (Manen et al., 1998).  

In general, however, the extension of member to actual system reliability is a very challenging process. System 
effects may be load path and load history dependent (including the initial conditions). System effects involve 
concepts such as failure modes, damage progression, robustness, fragility and system reliability. As a result, 
the level of uncertainties that influence the determination of the reliability of a system are larger than the one 
applicable to members. 

The general belief is that due to a combination of redundancy and ductility most structures have a (substantial) 
higher reliability than the individual members. However, ductility requirements are not always linked explicitly 
to target reliabilities as mentioned above and for the possible effects of low ductility (brittle behaviour) there 
is little attention from a reliability point of view. Also, in cases of low redundancy (e.g. a statically determinate 
structure)  comprising  multiple  element  the  failure  probability  of  the  system  is  typically  higher  than  the 
maximum member failure probability.. 

Typically,  in  design  codes,  system  effects  are  dealt  with  predominantly  by  indirect  design  methods  (i.e. 
prescriptive design and detailing rules), envisaged as a second layer of safety, additional to the direct design 
methods  used  to  check  the  reliability  of  members.  For  example,  interstorey  drift  limits  are  used  to  define 
performance-based levels in seismic design while tying rules and ductility requirements are specified to provide 
baseline levels of robustness performance. 

The details of the indirect methods depend on the severity level of the (direct and indirect) consequences due 
to failure of a structural member, structural part or the entire system. Usually, the severity is not considered as 
a gradual scale but as a rough classification of Consequence Classes, actually the same as the classification 
used for differentiation of partial factors. It should be noted here that controlling the member reliability, does 
not, in general, cause a change of the potential (direct and indirect) consequences that can occur in the event 
of failure, but rather concern the control of the expected probability of occurrence of such events (and this way 
the expected risk level).  

The supplementary indirect design provisions for system behaviour (also referred to as design for robustness) 
are intended for the uncertain and unknown adverse triggering events where the failure consequences are not 
negligible. The rules for robustness are usually presented as a set of rules that cover indirect and direct design 
methods: 

Indirect design methods are usually provided in terms of minimum tying forces, which attempt to indirectly 
simulate the structural effects caused by a loss of a column of foundation support, 

Direct  design  methods  may  involve  alternative  load  paths,  identification  and  design  of  key  members, 
structural segmentation. 

While indirect design methods apply primarily to hazard independent scenarios (i.e. only the structural effects 
of a set of unidentified hazards are simulated, most commonly by notionally removing of a key member), direct 
design  methods  apply  to  both  hazard  independent  and  dependent  scenarios.  Indirect  design  methods  and 
alternative load paths often use catenary action effects to mobilise additional resistance. Reference is made to 
the work done by (WG6.T2). 

13 

 
Another type of relevant system behaviour is the micro system behaviour inside a member. This may result in 
volume effects as for instance are relevant for timber (Rouger, F. 2017) and ground (see chapter 3.6.1). For 
brittle materials the resistance of a member may also change with the volume (size effect in concrete) due to 
energy balance effects  

A very special cases in this respect is the material glass. Given its very brittle behaviour it is very sensitive to 
all kinds of influences that cause extra stresses normally not accounted for. It is not always possible to design 
for adequate reliability of all individual elements which makes it essential that fail safe system behaviour is 
considered, given the failure of a single element. See also chapter 3. 

2.4  Time effects 

2.4.1  Bath tub curve 

Time effects are almost always present. Even in a standard design we have to deal with a reliability requirement 
 =3,8 for 50 years and a time varying load for which the characteristic value is defined on a 2% exceedance 
probability for the period of 1 year. The given time periods are essential. The point is that the fluctuations in 
time of the load leads to a dependency of the failure probability on the period of time under consideration. In 
addition, a structure may lose resistance due to time dependant degradation processes like creep, corrosion, 
surface or internal damage, fatigue, etc. Both effects should be taken into consideration regardless of whether 
we adopt a probabilistic or semi-probabilistic approach. 

If we include the time, the random variables X as well as the g function itself may become time dependent. If 
we subdivide the total period tL in a number of subintervals (for instance periods of one year) we may write: 

Pf (tL) = P1 + P2 + P3 +  + Pn 

(2.12) 

where  Pi  is the  probability  of  failure  exactly  in  year number i, which  may  be  written  in  terms of  limit  state 
functions as: 

P1= P(g1 < 0) 
P2= P(g2 < 0 and g1 > 0) 
P3 =P(g3 < 0 and g1 > 0 and g2 > 0) 

Etc. 

(2.13) 

where gi is the (minimum) value of the limit state function in year i. The logic is that failure in the second year 
is feasible only if the structure has survived first year. 

Using the definition of conditional probability, we may also write (2.13) as: 

P1= P(g1<0) 
P2= P(g2<0 | g1>0) P(g1>0) 
P3 =P(g3<0 | g1>0 and g2>0) P(g1>0 and g2>0) 

(2.14) 

etc. 

The vertical bar is the symbol for “conditional on” or “given”. Note that: 

Pi =P(gi<0 and g1>0 and g2>0..) are referred to as the unconditional annual failure probabilities. 

Pi =P(gi<0 | g1>0 and g2>0) are referred to as the conditional annual first failure probabilities.  

14 

 
 
 
 
 
 
 
 
The conditional failure probabilities are larger than the unconditional ones. As long as the failure probabilities 
are small like in most structural reliability applications, the difference between the two is negligible, as in that 
case all survival probabilities P(g1>0 and g2>0..) are almost equal to 1,0.  

For the sake of completeness:  

Pi =P(gi<0) is called the failure probability for year i (irrespective of other points in time). 

Example 2.1. Time varying load, no deterioration 

As  an  example,  take  a  member  with  time  independent  resistance  R  loaded  by  self-weight  G  and  a  yearly 
independently varying stationary climatic load Q. In that case: 

g1 = R-G-Q1 

g2 = R-G-Q2 

g3 = R-G-Q3 

Let now R and G (for the case of simplicity) be deterministic; in that case: 

P1 = P(R-G-Q1<0)=P 

P2 = P(R-G-Q2< 0 | R-G-Q1 0) P(R-G-Q1> 0) = P(R-G-Q2< 0) P(R-G-Q1 0)= P(1-P) 

Similarly:  

P3 = P(1-P)2 

In that case the annual probabilities Pi will decrease although very slowly in the course of time; if R and G are 
random the decrease will go faster, depending of course on the ratio of the standard  deviations of the time 
dependent part (Q) and the time independent parts (R and G). 

Example 2.2. Deterioration 

As another example, consider the same case as above, but now the resistance R is subject to mechanisms like 
corrosion or fatigue: 

Ri = R0 –  Rj    (j=1..i-1)  where Rj is assumed to be larger than 0.   

For an unprotected steel element all Rj may be a set of (correlated) random variables with constant statistical 
properties each year 

For a corrosion protected steel element (e.g. bars in a reinforced concrete structure) Rj, may be zero for some 
random time and non-zero later on. 

In  the  case  of  fatigue,  the  resistance  at  year i  is a  function  of  all  the  load  cycles that  have  occurred  in  all 
previous intervals from 0 to i-1.  

Figure 2.1 shows a possible resulting course of the conditional annual failure probabilities. The curve first goes 
downwards  because  of  the  correlation  between  the  subsequent  years.  Later  is  goes  upwards  because  of 
degradation processes, if present. The total curve is called the bathtub curve. The sum of all conditional failure 
probabilities may be larger than 1,0. 

The  unconditional annual probabilities are slightly smaller in the beginning  but the  difference  may increase 
later. The sum of the unconditional annual failure probabilities to a certain point in time is equal to the total 
failure probability of the period of time under consideration (see equation (2.12). The sum of all unconditional 
probabilities (from one to infinity) is equal to 1,0.  

Note that the failure probability for the total period from year 1 to year n is given by (2.12) and can be found 
by adding the unconditional failure probabilities. It is an increasing (strictly speaking a  non-decreasing) function 
of time. 

In practical applications the use of (2.13) and (2.14) is quite cumbersome. The calculation can be simplified by 
using the so-called outcrossing approach: 

15 

 
 
Pi =P(g i< 0 and (g i-1 > 0 and g i-2 > 0… and g1 > 0))  P(g i< 0 and g i-1 > 0 ) 

(2.15) 

So instead of taking all past survivals from 1 to i-1 on board, we limit ourselves to only the previous year i-1 
(or event). Whether this is accurate or not depends on the correlation  properties. In literature refinements of 
this method and elaborations for continuous cases can be found. 

Figure 2.1. Bathtub curve (black line) showing the conditional failure probability as a function of time, with zones of (I) 
increasing reliability (dominant time independent uncertainties like resistance and self-weight), (II) constant reliability 
(dominating time dependent uncertainties, usually loads) and (III) decreasing reliability (dominating deterioration). The 
blue line shows the unconditional failure probabilities. Note that the annual reliability index has just the opposite profile: it 
first goes up, is constant and then goes down. 

In  present  practice  (also  in  the  Eurocodes)  the  combination  of  cumulative  deterioration  and  failure  of  the 
reduced  resistance  under  some  kind  of  peak  load  is  often  neglected.  Both  aspects  are  often  considered  as 
separate phenomena with independent safety criteria without considering that the reliability requirement (say 
a total life time beta of 3,8 or an annual beta of 4,4 or 4,7 for every year) holds for the combination of effects. 
In chapter 3 this item will be discussed further.  

2.4.2 

Inspection, monitoring and maintenance 

As  counterweight  to  the  loss  of  reliability  due  to  deterioration  there  is  usually  a  program  of  monitoring, 
inspection and maintenance. Given for instance the result of an inspection at a certain point of time t we might 
update the probability of failure: 

Pf = P(g(X) <0 | h(X) = 0) 

(2.16) 

The event h=0 represents the outcome of the inspection; depending on the type of inspection this can also be 
in the form of h>0 or h<0. If inspection took place at time t, the updated failure probability is in principle derived 
for some period following this point in time.  

Example 2.3 Inspection for fatigue  

As an example, we can rewrite the above general equation in the case of fatigue, assuming no crack was found 
in year k: 

Pfi = P( ai > acrit,i | ak < adet) 

16 

 
 
 
 
where Pfi is the failure probability in one of the following years i, ai is the length of the fatigue crack in year i, 
acrit,i is the critical crack length in year i, ak is the length of the fatigue crack in the year of inspection k and adet 
is the detectable crack given the method of inspection. Note that the critical crack length depends among others 
on the height of the extreme load in year i. Further examples may be found in Annex B.  

Important notions to be included in the analysis are the statistical properties of the inspection method like the 
probability of detecting a defect and the accuracy with respect to location, nature and size.  

Inspection planning should in fact already be taken into account in the design stage. The first inspection should 
be scheduled at latest at the point in time where the (usually annual) failure probability reaches its maximum 
acceptable level. Given the result of the inspection a new point in time for inspection can be derived or direct 
maintenance or repair can be necessary or optimal. The other way around is to start from a given inspection 
plan and then verify that it the combination of design and inspection meets the target. 

In equation. (2.14) we might also use the observation that the structure has survived until the point of inspection 
(proven strength) or even more data if available. In geotechnics often data already gathered during the building 
process are included (observational method) enabling sometimes a more economic design. 

2.5  Target Reliability  

2.5.1  Safety and economy 

For the reliability-based approach and the partial factor approach the target reliability is a prescribed quantity.  

Usually, the target reliability level is proposed by the engineering community (often the code committee) and 
endorsed by the National Authorities who finally are responsible. The common starting point is that a structure 
should be safe as well as economic during the design or remaining service time. The notions safe and economic 
have been discussed in section 2.3. 

In general, one may state that the accepted reliability level with respect to ULS-failure depends on: 

The consequences of failure which may depend on the failure mode (see 2.5.6)  

The resources needed for structural dimension, in particular the costs related to additional safety 

The anticipated service life, 

The variability and uncertainty in resistance and load, including the likelihood of mistakes (as far as not 
covered by quality control). 

In most codes only the first factor is used for explicit reliability differentiation in terms of different levels of 
target  reliability  levels.  It  is  actually  the  connection  between  reliability-based  design  and  risk  analysis.  The 
consequences of failure, however, are not elaborated in detail, but only a rough classification of structural types 
(dwellings, high rise buildings, grand stands, etc.) is given. There usually is also a link to robustness requirements 
and QA-procedures. 

2.5.2  Reliability differentiation  

Reliability differentiation depends on the consequences of failure (related to economy losses, environmental 
effects and losses of human lives) but may also be used as a tool to apply the available resources of society 
in an optimal way. As such it is one of the means to contribute to more sustainability in building industry.  

For  various  reasons  it  does  not  seem  appropriate  to  distinguish  in  reliability  requirements  between  various 
materials. All structures or structural elements within a certain class of use and failure consequences should 
have the same reliability, whatever material present.  

However, there is no reason why the same value (as for instance the in the Eurocode commonly accepted value 
of 3,8 for 50 years as derived over 30 years ago for CC2 buildings under variable loads only. see chapter 3.1) 
should automatically hold for other limit states like deflection, for other structures like masts, towers, silos or 
for loads with a higher degree of uncertainty (e.g. fire, impact, earth quakes). The wish for transparency in the 
field of reliability requires primarily the same target for comparable consequences. Differentiation, however 
can  be  accepted,  provided  a  careful  consideration  is  given  to  all  aspects  discussed  before  (life  safety, 
consequences of failure, economical and historical aspects). The topic in relation to the Eurocode will be dealt 
with further in chapter 3.3.  

17 

 
A very special application of the reliability differentiation concerns the field of assessment of existing structures. 
Strengthening an existing structure in general requires a lot more efforts than strengthening a structure that 
only exists on paper. Based on these considerations it can make sense to lower the reliability targets for existing 
structures. Have in mind that when assessing an existing structure also the (remaining) design service life might 
be (much) shorter than 50 years. The further elaboration in the Eurocode is given attention in chapter 4. 

2.5.3  Annual versus life time targets, influence of design life  

Reliability targets always have to be  specified for a  certain event or a  certain period of time (the so-called 
reference period). Annual values are commonly in use when dealing with human safety; in that case the value 
is usually independent of the design life time. The life time option is convenient if one only considers economic 
optimization. In practice the life time option was connected directly to a design service life of 50 years. Rules 
for structures for a different design service life (temporary structures), large civil engineering works) were never 
very clear. It seems logical to make a difference between economy and human safety dominated targets.  

Confusing in this respect is also what is meant by an annual value criterion. Does it mean that during the design 
service life the average or the maximum yearly failure probability is decisive?  As the operational difference 
between the life time approach and the annual average is small it makes sense to set a target for the maximum 
annual value during the design or remaining service life. This also simplifies calculations.  Usually only the first 
and the last year need to be checked. 

If failure events in various years are independent and have small similar probabilities, the relation between an 
annual and a 50-year target is simply approximated as (see also example 2.1): 

P50 = 50 x P1 

(2.17) 

The  formula  still  holds  for  moderate  correlation  but  complex  relation  arises  if  there  is  a  large  degree  of 
correlation (for instance due to dominating permanent loads or large resistance uncertainties or the presence 
of degradation processes giving rise to the increasing failure probability at the end of the life time. This also 
means that there is no one to one correspondence for a large class of different structures. One can only claim 
an equivalence for instance between  = 3,8 for 50 years and (depending on the degree of correlation)  = 
between 3,8 and 4,7 for one year. Some structures however will have higher dimensions in the one system and 
lower in the other. These differences may also be present in corresponding -values and/or -factors. 

Note by the way that there is no need to make a choice between the two approaches: one can also adapt the 
life time target for design and add constraints for annual failure probabilities. 

2.5.4  Minimum versus average reliability 

As already pointed out in chapter 2.3.3, the target reliability can be formulated for a portfolio of structures (say 
all bridges). In that case it may be questioned whether the target refers to the average reliability of the portfolio 
or to the minimum reliability (average or maximum failure probability respectively). If aiming at an economic 
optimum the average value makes sense. However in order to fulfil requirements related to human safety the 
minimum reliability level seems to be the most appropriate.  Of course, such differentiation would make the 
derivation of design rules more complicated. A compromise could be to take the average but to add a limit to 
the scatter of the reliability levels in each portfolio. 

2.5.5  Member versus system  

This issue was already discussed in chapter 2.3.5. In most practical applications target reliabilities are used for 
designing individual members (including connections) for separate failure modes. However, consequences are 
usually related to system failures and the system failure probability may be the result of contribution of a set 
of system failure modes. For a part this has to be considered as the state of the art. However, in some cases 
we encounter exceptions as in the design of laminated timber beams or multiple pile foundations. 

2.5.6  Brittle versus ductile failure 

Brittle  failures  on  the  member  level  have  in  many  situations  more  serious  consequences  for  the  structural 
system  integrity  than  ductile  failure.  This  might  be  a  reason  to  use  a  higher  consequence  class  and  higher 

18 

 
 
reliability targets for brittle failure modes or to force the structure into ductile modes such as in capacity design 
used in seismic design situations. An additional aspect is the degree of warning: ductile failure modes may give 
some kind of pre-warning and thus time for people to leave the building in time (see also section 3.7). 

2.5.7  Role of Quality Assurance 

Quality Assurance (QA) procedures (for more information see EN 1990:2023 Annex B) may be carried out for 
design, fabrication and execution. The intention of those procedures is to ensure that the reality is in accordance 
with the assumptions in the verification (like schematisation and properties of materials and geometry). As far 
as the  overall  design  is  concerned  the  best  way  to  reach  this goal  is to  make  use  of  independent  checking 
procedures.  

There obviously is a direct relation between QA and reliability verification. One often heard opinion is that the 
use of a higher level of QA leads to a lower uncertainty in the material properties and  therefore provides a 
justification  for  a  reduction  of  partial  factors.  Although  there  is  some  logic  in  this  reasoning  there  is  no 
recommendation (also not in the Eurocodes) to do so for new structures. For existing structures this may be 
different.  QA  in  that  case  is  very  close  here  to  the  actual  measuring  the  actual  properties  of  the  existing 
structure ; one may then follow the corresponding rules from statistical inference 

2.6  Derivation of partial factors for semi-probabilistic design  

2.6.1  Design value method 

The  theory  of  structural  reliability  offers  several  methods  to  find  partial  factors  given  a  prescribed  target 
reliability t level. The simplest one is the method using standardised importance (sensitivity, weight) factors  
as already introduced in formula (2.10):  

Xd = µX - X t X 

(2.10) 

This formula holds for the normal distribution. Given the design values the partial factors can be derived using 
(2.11).  

In EN 1990:2002 the choice was made that for t =3,8 and a reference period of 50 years, the value X = 0,8 
for the dominant resistance variable and -0,7 for the dominant load variable; for non-dominant variables the 
values are 0,4x0,8=0,32 and -0,4x0,7=-0,28. (See also section 3.1.3). These choices are also the basis for the 
load combination rules. Where relevant the various loads have to be considered as dominant one by one. The 
practical advantage of the design value method is that all load and resistance design values can be derived 
individually, assuming a consistent application on both load and resistance side and taking care of limits as 
mentioned in EN 1990:2023 Annex C.  

The  proposed  -values,  of  course,  can  be  varied  to  improve  the  result  in  a  certain  application.  Usually,  the 
results of a specific choice are checked by considering a set of structural members and load combinations and 
calculating  the  corresponding  -values (like  for  instance  as presented  in  Figure  2.2  taken  from  Meinen  and 
Steenbergen, 2018.)  The result should give -values close to the target over the whole range of application. 
Close in this respect may refer to the average  or to the minimum value (see also section 2.6.3). For example, 
for the seismic design situation the values used in driving the partial factors in the second generation Eurocode 
8 are 0,42 for the resistance and -0,91 for the seismic action effect (Franchin and Noto, 2023). 

19 

 
 
 
 
 
 
 
 
Figure 2.2. Resulting reliability index for steel members, designed according to the Eurocodes,, for the limit state of 
bending under self-weight, permanent and imposed loads for a 1 (red) and 50 (blue) year reference period as a function 
of Q which represents the ratio of the variable and total load; the dashed and dotted lines refer to a zero and maximum 
contribution of permanent loads respectively; the full lines show the average value in the aria of interest (illustrative 
example only)  

Source: Meinen and Steenbergen, 2018  

When the combination of loads is considered, it would be too conservative to assume that the maxima of two 
or more-time dependent loads in the reference period will occur at the same point in time. Where this is taken 
into account the so-called FBC models (Ferry Borges and Castanheta, 1971) are recommended (Figure 2.3). 
Note that time dependency is also important for the safety formats for degrading systems and finding load 
factors for fatigue loads and/or environmental actions.  

Figure 2.3. Ferry Borges and Castanheta load model (FBC) for imposed loads (sustained and intermittent components) 
(illustrative example)  

Source: Faber, Koehler & Sorensen, 2018 

2.6.2  Minimizing the deviation from the target 

More advanced methods aim at minimizing the averaged difference between the target and the reliability levels 
obtained by a certain set of partial factors  for a certain field of application (see EN 1990:2023 Annex C):  

    min  wi (i() - t)2   

                                                                            (2.18 a) 

or min  wi (Pfi () – Pft)2 

                                                                            (2.18 b) 

or min  wi [(Ri() -Ri(t)) / Ri(t)]2 

 (2.18 c) 

20 

 
 
 
 
 
 
 
 
The summation is over the selected set of structures. 

The first option mentioned (2.18 a) is the one most commonly used. See (Baravalle, 2017) for further discussion. 
The method will further be referred to as the optimization method. The weight factors wi may reflect the number 
of structures of type i or even economic considerations. In addition, there may be or should be constraints like 
a minimum beta or a maximum deviation. Other constraints may relate to having material independent load 
factors or minimum partial factors for permanent loads.  

One should keep in mind that a minimization problem does not need to have a unique solution. There may be 
a set of different solutions for the vector  leading to the same structural dimensions (Madsen et al, 1986). For 
instance, in the simple case of example 2.1 one may increase the load factor and reduce the resistance factor 
accordingly, leading to the same structure. So, in order to have a technically meaningful answer one has to 
make a proper choice or introduce an additional constraint. One might for instance fix the partial factor for the 
self-weight and study the results for several values. Alternatively one may fix the values on the resistance side; 
actually the example calculations in Annex B2 of this report have been performed along that line, using the 
gamma values according to EN 1992-EN 1999). 

Example 2.4 Simple code calibration example 

This example is only intended to show the procedure with minimum mathematics. It is not intended as a useful 
example as far as the numerical values are concerned. 

In this example the following steps will be taken: 

1. Choose a portfolio of structures and design situations with corresponding limit state fucntions and weights 
wi ; set the target reliablity 

2. Define the set of statistical properties  

3. Define a verification code format, estimate a set of partial factors and calculate for each design situation 
(exactly) the minimal required structural dimensions. 

4.  Calculate  for  design  situation  i  the  corresponding  reliability  index  I and  for  the  whole  set  the  weighted 
average  = (wi I ) / wi  

If the (weighted) average value of i over the set of  design situations (step 4) is about equal to the target, we 
have a reasonable proposition for the partial factors; other wise we have to repeat steps 3 and 4.  

One might also  go further and try to find a  minimum for the  relative deviation of the taget, which may  be 
defiened as: D=  wi { (i -t ) / t }2 . But that is outside the scope of this simple example. 

Step 1: Definition of portfolio and choice of target 

Let us consider a portfolio of two materials and two loads. The limit state function is given by: 

g = R - F1  - F2  

The ratio of the variable loads to the total load is governed by the parameter , defined as:  

 = F2k / {F1k+F2k } 

where  ranges from 0,2 to 0,8 with intervals of 0,2. So actually we will take the values  = 0,3,  = 0,5 and 
 = 0,7. The two materials, the two loads and the three ratio values  will have the same weight w. In total we 
have 2 x 3 = 6 design situations. The target reliability t is set as 3,8 for the reference period under consideration. 

Step2: Statistical properties  

The statistical properties are presented as follows: 

distribution type  mean value 

coefficient of variation V  characteristic value 

material 1  Normal 

material 2  Normal 

load 1  

Normal 

load 2  

Normal 

m1 

m2 

1 

 / (1-) 

0,10 

0,20 

0,10 

0,20 

21 

m1 (1-1,64V1) 

m2 (1-1,64V2) 

1.0 

 / (1-) 

 
 
The characteristic values of the loads are taken equal to their mean value; the mean value of the first load has 
been set equal to 1,0. The mean for the resistance follows for each design from the design procedure in steps 
3 and 4. 

Step 3: Design format and partial factors 

As design format we choose: 

Rk / R >  F1 F1k + F2 F2k  

Let the estimate for a set of partial factors be: R1=1,2, R2 =1,5, F1 = 1,2 and  F2=1,4.  

Step 4: Resulting reliability level 

The first design situation where  = 0,3 leads to: 

R1d = 1,2 x 1 + 1,4 x 0,43 = 1,8;     R1k = 1,8 x 1,2 = 2,16    and    m1 = 2,16 / (1-1,65 x 0,1) =2,6 

Using FORM this leads to  = 3,83. If this analysis is carried out for all 6 design situations we find beta values 
ranging from 2,8 to 4,5 with 3,5 as an average. As this is below the target a second trial is performed: 

Steps 3 and 4: second trial: 

The  second  choice  for  the  partial  factors  are:  R1=1,3,  R2=1,8,  F1  =  1,2,  F2   =1,4.  This leads to  beta-values 
ranging from 3,3 to 4,3 with an average of 3,8. This average is okay but the range may, from an ecomomical 
or poitical point of view, be considered as quite wide. This can be improved by further optimizing, but, as stated, 
this is not in the scope of this example. In some applications there is also a requirement on a minumum reliability 
level. 

In the above example the presence of two variable loads in one limit state verification is not considered. If we 
want to do so, we have to model the variable loads in more detail (e.g. using FBC models as indicated in Figure 
2.3) and use time dependent reliability calculation procedures as described in section 2.4. As an approximation 
one  could  simply  add  the  maxima  of  variable  loads  in  the  reference  period,  but  that  would  lead  to  an 
overestimation of the failure probability as the maxima of two or more variable loads need not to occur on the 
same point in time. 

2.6.3  Closing remark 

Whatever  method  is  chosen  to  define  reliability  targets  and  establishing  partial  factors,  a  general 
recommendation will be that abrupt large changes in the successful design procedures from the past should be 
avoided, unless there are very convincing reasons. The behaviour of the structures designed according to the 
rules  of  the  past  present  important  empirical  evidence  from  a  major  experiment  that  should  be  taken  into 
account. On the other hand: 

Periods of 20 or even 50 years may be too short to check safety against e.g.  design values of extreme 
winds having return periods of over thousand years.  

Some structures may have been uneconomically overdesigned (should be avoided of course). 

Results from research (hidden/unhidden safety; more refined models) should be incorporated. 

The conclusion again has to be that we are looking for the balanced combination of practical experience, theory 
and research developments. 

For more information see also (Gayton et al., 2004) and (Silva et al, 2014).  

22 

 
 
3  The Eurocode semi-probabilistic verification method 

3.1  Historical background 

3.1.1  Chronological overview 

CEN’s decision to move the EN Eurocodes into a second generation in 2015 following and with the important 
support of the European Commission and EFTA through mandate M/515,  was, preceded by a long history of 
European standardisation of design standards for structural engineering. Based on the Rome Treaties of 1957, 
the Coal and Steel Commission of the European Community began drawing up seven Eurocodes with rules for 
the serviceability and structural safety of building structures from 1975 onwards. For this purpose, the European 
associations of the corresponding construction methods were asked for support.  

After the decision of the European and EFTA member states in 1989 the development of the so-called ENV 
Eurocode versions for structural design began, the first of which were published in 1992. Due to difficulties in 
harmonizing  all  the  aspects  of  the  calculation  methods,  the  ENV  Eurocodes  included  "boxed  values"  which 
allowed member states to adapt the safety level for use on their territory and for which the partial safety factor 
concept was very much suited. 

As  the  basic  document  of  the  ENV  series  establishing  design  principles  and  requirements  as  well  as  the 
approaches for safety, serviceability and durability of structures, the Eurocode 1, ENV 1991-1 “Basis of design 
and action on structures – Part 1: Basis of design”, was published in 1994 (CEN, 1994). 

After 1998 the conversion of the ENVs to ENs under the full normative leadership of CEN began. In the course 
of this, it was decided to separate the standards for basis of design and actions on structures, i.e. to dedicate 
an EN 1990 to basis of design and to leave the regulations for the load approaches in EN 1991. 

After the conversion of ENV 1991-part 1 into EN 1990 (CEN, 2005/2010) in 2002, in 2007 the publication of 
EN Eurocodes was concluded. The implementation programme entered the coexistence period, during which the 
EN Eurocodes were used in parallel with National Standards that had the same scope. 

By  the  end  of  this  coexistence  in  2010  the  Eurocodes  are  introduced  in  the  EU  countries  including  the 
corresponding national documents, which gave the details of how to apply EN- Eurocodes in Member States. 

Over the past decade the evolution of the Eurocode suite has been progressed to develop the second generation 
of EN Eurocodes. This work has included the extension and updating of the principles and fundamentals of the 
basis of structural and geotechnical design in EN 1990. 

In  the  period  until  the  1990,  for  the  purpose  of  practical  design,  there  was  a  mix  of  national  and  material 
oriented design standards, partly based on the classical allowable stress approach, partly on load (or load & 
resistance) factor limit state concepts. However, in many research centres as well in pre-codification committees 
and some code committees both probabilistic and semi-probabilistic methods (as presented in chapter 2 of this 
document) were well developed. The Eurocodes were committed to the semi-probabilistic partial factor design 
concept from the very beginning (Sörensen, 2004). At all times of development, the Eurocodes reflected the 
current  international  state  of  the  art  of  safety  concepts  in  civil  engineering.  Some  of  the  most  important 
elements are briefly described hereafter. 

3.1.2  Reliability methods applied in the Eurocodes.  

Design  of  structures  can  be  performed  either  based  on  a  reliability  analysis or  on  the  design  value  format 
(Sörensen, 2004). Within the Eurocodes suite of the ENV 1991 (CEN, 1994) and EN 1990 (CEN, 2005/2010), 
the  design  values  of  basic  variables  are  determined  from  their  characteristic  values  using  partial  (γ)  and 
combination (ψ) factors (partial factor design). 

It  has  been  agreed  that  numerical  values  of  these  factors  can  be  determined  using  several  methods 
(Gulvanessian et al, 2012): 

a)  calibration to past experience 

b)  application of probabilistic reliability theory and calibration 

For evaluation at a linearized limit state, method (b) can also be expressed as:  

the use of the probabilistic FORM (see hereafter) level II method and after calibration derivation of design 
values for action and resistance. 

23 

 
According to (Gulvanessian et al, 2012) explaining EN 1990 (CEN, 2005/2010): “… as a rule, both procedures 
(calibration to past experience and application of probabilistic reliability theory) for the determination of γ and 
ψ factors (level 1) are combined in order to increase the credibility of obtained results.” This is visualised by 
Figure 3.1. 

Figure 3.1. Overview of reliability methods 

Source: EN 1990 (CEN, 2005/2010), Annex C. 

The design with partial factors should be the usual method for practice in structural engineering. As far as we 
talk  about  probabilistic  methods  in  EN 1990,  we  can  say  that  EN 1990,  especially  its  Annex  C  is  somehow 
oriented  to  level  II  methods,  but  not  only:  it  also  allows  safety  evaluation  according  to  full  or  quasi  full 
probabilistic (level III) methods without further ado. 

As mentioned EN 1990 (CEN 2005/2010), especially Annex C is based on concepts of “First Order Reliability 
Methods” (FORM, an advanced level II method [DIN, 1981]. Consider for the sake of further discussion the basic 
case where the limit state function is given by Z = R – E with R and E independent normally distributed variables 
representing resistance and action. In that case we have: 

μZ= μR- μE  ;     σZ

2 = σR

2+σE

2;      Pf = Φ(- β) 

(3.1) 

where    is  the  safety  or  reliability  index,  named  as  such  by  Cornell  in  1969  (Cornell,  1969)  that  can  be 
considered as invariant for mechanically equivalent problem formulations (Kjerengtroen et al 1996). For the 
more  general  non-linear  limit  state  function  Z  =  g  (X1,  X2,  …  ,  Xn)  with  non-normal  variables  a  suitable 
transformation from the X-variables to the standard zero mean unit standard deviation Gaussian u-variables 
was  proposed  by  Hasofer  and  Lind  in  1974  (Hasofer  et  al,  1974).  For  independent  variables  Xi  the 
transformation is given by: 

 (ui) = FXi (Xi) or simply for normal variables: ui = 

Xi- μi
σi

(3.2) 

Alternatively, one could choose the tail approximation according to (Rackwitz and Fiessler, 1978). 

Given this transformation the reliability index β could be conceived as the minimum distance from the origin of 
the normalized coordinate system to the failure surface g(u) = 0 (the so-called design point): 

24 

 
 
 
 
 
β=√u1

 2+u2

 2+…+un

 2  = √(α1β)2+(α2β)2+…+(αnβ)2 

(3.3) 

The sensitivity or weighting factors “αi“ are equal to the partial derivatives ∂g/∂ui at the design point and could 
be interpreted as direction cosines of the stretch from the origin to the design point. It can easily be proven that 
the sum of the squares should be equal to 1. 

3.1.3  Values for key parameters 

The choice of a value for the target reliability βt was subject to some evolution process. In the various national 
studies at that time (around 1980) there was not much uniformity in the presentation of the results with respect 
to the reference period. In some studies, the one-year period was chosen, in other studies the 50-year period. 
The transition between the two results were (neglecting correlation) simply presented as  

P50 = 1 –(1-P1)50  50 P1 

or in terms of reliability index and n representing the number or years: 

βn= Φ-1 [Φn(β1)] 

(3.4) 

(3.5) 

The scatter between the various studies was quite large (see Table 3.1); the scatter between various materials, 
types of elements and loading situation was even larger, but showed similarities between the various studies.  

In the German  national regulation  (DIN, 1981)  basically, an (annual) failure probability of  pf = 1x10-6 being 
equivalent with β = 4,7 for ULS and 3,0 for SLS was chosen. In the light of predominantly permanent loads this 
failure probability was  considered as relatively high: as (real) permanent loads are  not time  dependent this 
value holds for a reference period of 1 year as well as for any time. Therefore, it was proposed to define β = 
4,7 as anchor value for a limit state function considering variable loads only, when referring to one year, for 
normal failure consequences (CC=2). Other documents that were on the table at that time and pointing more 
or less in the same direction are presented in Table 3.1. The choice for a life time reliability measure is also 
already present in the CEB-bulletins 116 (Part 2) and 124/125 (Vol 1, appendix 1). It is there stated that the 
reference period for the reliability should be chosen “identical the anticipated time of use of the structure”. As 
a central value (medium number of endangered persons and medium economic consequences) 10-5 (=4,3) 
was tentatively proposed. In addition, a tolerable annual individual risk was set at 10-5. As the probability that 
a  user  of  a  structure  is  killed  due  to  a  collapse  (depending  on  the  situation)  is  less  than  1,0  the  collapse 
probability of the structure may be larger. The reduction from 4,3 to 3,8 was probably the result of a number 
of national studies. It has never been an important point of discussion during the drafting of the first EN 1990.  

Note:  

It should be realized that the relevant parts of the CEB-bulletins 116 and 124 were not mere products of 
the concrete industry but the joint product of all professional organisations of the Liaison Committee (CEB, 
fib, ECCS, RILEM, IASS, CIB) and the JCSS. The title was in both cases “Common unified rules for different 
types of construction and material” 

Anyhow, already in ENV 1991-1 (CEN, 1994), the target reliability index for the ULS was chosen equal to β = 
3,8 (CEN, 1994) (CEN, 2005/2010) (Gulvanessian et al, 2012) (DIN, 1981) related to a period of 50 years (for 
SLS β = 1,5). In current EN  1990  (CEN, 2005/2010)  different target reliabilities can be  found  depending on 
various  reference  periods  and  consequence  classes  (CCs).  Reliability  differentiation  was  earlier  a  subject  of 
discussion in for instance CEB 124 and NKB, mentioned in Table 3.1. 

The Design Value Method also asks for choices for the sensitivity factors αX. For the Eurocodes, in line with e.g. 
(DIN, 1981), the sensitivity factors αE (action side) and αR (resistance side) were predefined as to αE = -0,7 and 
αR = 0,8, thus allowing to treat action and resistance side independently in the design, and where the action 

25 

 
 
 
 
side was given some higher importance by increasing the αE value’s amount from 0,6 to 0,7 (deviating from 
the unity condition, see above). This gave a certain flexibility for cases where the numeric values of αE and αR 
are not completely satisfied. One can allow for an undercut or exceed the target reliability βt by 5% if the range 
of the standard deviation ratio is in between 0,16 < σE/σR < 7,6 (CEN, 2005/2010) (König et al, 1982). 

In case of more than one load or resistance variable it was on the basis of theoretical considerations decided 
to use 0,4 αX for the non-dominant parameters. On the loading side this formed the basis for the derivation of 
the  and  factors (Ferry-Borges et al, 1971) (Turkstra et al, 1980) (Vrouwenvelder, 2002) (Vrouwenvelder et 
al, 1996). 

Characteristic values for loads and resistances were in principle defined as 5% fractile for strength parameters, 
50 % fractile for low variable permanent loads and 2% fractile for annual maxima of variable loads. Given 
these choices, the partial factors could be defined in the frame work of the Design Value Method. This leads 
however to different values than the well-known G = 1,35 and Q = 1,5 which were also chosen on historical 
grounds (see Figure C1). For more information see Chapter 3.4 and Annex B2.  The resulting ψ-values of EN 
1990 (CEN, 2005/2010) are conservative (Vrouwenvelder, 2002) (Vrouwenvelder et al, 1996). 

Table 3.1. Overview main results of historical documents. 

Country 

Study 

Ref 
[year] 

t  
1 yr 

t 
50 yr 

R 

G 

Gco
m 

Q 

0 

kF1 

kF3 

3,0 

3,8 

3,3 
3,6d 

4,3 
4,3 

4,7 
4,7 
4,2 

>4 

? 

0,8 
0,8 

0,8 

1,3a 
1,3 
1,0c 
1,35 
1,3 

1,2 
1,2 

1,35 

3,1f 

0,8 

1,2 

1,3a,b 
1,3 
1,3c 
1,5 
1,5 

1,4 
1,4 

1,5 

x 
x 

x 

x 

x 

x 

0,9 

1,2 

0,9e 

1,1e 

x 

x 

x 

x 

Galambos 
DIN 
Koenig/Hosser 

USA 
Germany 
Germany 
Scandinavia  NKB 
Belgium 
SECO 
Netherlands  NEN 
CIRIA 
UK 

1982 
1981 
1981 
1978 
1984 
1987 
1977 

1976 
1976 
1978 
1980 
1984 
1998 

International  CEB-112 
International  CEB-116 
International  CEB-124/125 
International  CEB-127/128 
Eurocode 1 
Euro Com 
International 
ISO 
a here is also a sys =1,1;  
b char value defined as Qk = Q120 yr  
c  in combination with high material factor  
d t = 2,6 for wind 
e variable actions only 
f economic optimum 
x several values, most in the order of 0,4-0,7 

Background studies mentioned  in Table 3.1: 
USA 
Galambos T., B. Ellingwood, J. G. MacGregor, C. Allin Cornell, Probability Based Load Criteria: Assessment of Current Design Practice, Journal 
of the Structural Division, ASCE, Vol. 108, No. ST5, May, 1982 
Germany: 
Koenig, Hosser, The simplifieds level II metnod and its application on the derivation of safety elements for level 1, Technische Hochshule 
Darmstadt (1981) 
Koenig Hosser, Sicherheitsanforderungen fur die Bemessung der bauliche Anlagen nach den mpfehlungen des NABau, Bauingenieur 57 
(1982) 
NaBau, Grundlagen zur Festlegung von Sicherheitsanforderungen für bauliche Anlagen, DIN Deutsches Institut für Normung, Berlin, Köln, 
Beuth Verlag, 1981 
DIN – Deutsches Institut für Normung e.V. (1981). General Principles on the Specification of Safety Requirements for Structures. 1st edition 
in English, 1981. 
Belgium:  
R. D'Havé and P. Spehl, Principes généraux de détermination de la sécurité et de l'aptitude au service des structures, Annales des travaux 
publics de Belgique n°1 of 1984. 
The Netherlands: 
Vrouwenvelder, A.C.W.M., Siemes, A.J.M.. Probabilistic calibration procedure for the derivation of partial safety factors for the Netherlands 
building codes. HERON, 32 (4), 9-31, 1987 

26 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Scandanavia: 
NKB – The Nordic Committee on Building Regulations, Recommendation for Loading- and Safety Regulations for Structural Design. NKB-
Report No. 36. (1978)  
NKB – Nordic Committee and Work Reports (1999:01 E). Basis of Design of Structures. Proposals for Modification of Partial Safety Factors 
in Eurocodes. (1999) 
UK:  
Ciria, Rationalisation of safety and serviceability factors in Structural codes, Ciria report 63, UK, 1977 
EC: 
Rowe et al., Eurocode 1, Common unified rules for different types of construction and material; Commission of the European communities, 
DG Internal market and industrial affairs, Report 8847 (1984) 
CEB:: 
Bulletins 112, 116, 124, 125,127,128 
Mathieu H. General overview of structural reliability problems, Updated comprehensive abridgement, of the CEB manual “ Structural safety 
CEREMA, 1995 

3.1.4  Conclusion 

EN 1990, Basis of structural design, as head document in the Eurocodes suite establishes for all the structural 
Eurocodes  the  principles  and  requirements  for  safety,  serviceability  and  durability  of  structures.  It  further 
describes the basis of design and verification and provides guidelines for related aspects of structural reliability. 
It  gives  all  the  operative  material  independent  rules  (e.g.  partial  factors  for  actions,  load  combination 
expressions for ultimate and serviceability limit states), and therefore EN 1992 to EN 1999, which do not provide 
material independent guidance, cannot be used without EN 1990. 

EN  1990,  Basis  of  structural  design,  as  a  worldwide  recognized  standard,  has  undergone  a  progressing 
development in the past. Care was always taken to ensure that this and its predecessor documents were kept 
up to date with the state of the art. Thus, it allows not only an implementation of the safety level based on 
simple empirical values or on level II methods, but also fully probabilistically oriented evaluations can easily be 
considered if allowed by the National Annex. 

The development of this document also shows its flexibility in terms of application to the most diverse issues, 
starting with those of structural engineering, through those of earthquakes, up to geotechnics. 

Nevertheless, EN 1990 must be further developed as a part of the ongoing need for standards to be updated, 
partly because of presently still existing shortcomings, partly because of the new challenges that will come. 
(See also chapter 6.} 

3.2  The verification formats according to EN 1990:2023 

The  basic  code  format  for  the  verification  of  the  ULS  in  the  Eurocode  is  given  by  (inequality  (8.1)  in  EN 
1990:2023):  

Ed ≤ Rd 

(3.6a) 

where Ed is the design value of an action effect in a certain member, cross section, stress concentration point 
or connection in the structure and Rd the design value of the corresponding resistance. The verification is carried 
out for each specific design situation and for all limit states, load combinations and load arrangements.  

For  the  ultimate  limit  state  of  excessive  deformation  there  is  a  similar  formulation  (inequality  (8.2)  in  EN 
1990:2023): 

Ed ≤ Cd,ULS 

(3.6b) 

where Ed is the relevant displacement or deformation and Cd,ULS a corresponding limit value; numerical values 
may be given in material related codes; no recommendations are given for the partial factors in EN 1990:2023. 
In EN 1998 (seismic analysis) the deformation criterion is used as a mean to prevent brittle failures.  

For the resistance Rd the design value is calculated from: 

27 

 
 
 
Rd = R{𝜂 Xk/𝛾m; ad; ΣFEd} /𝛾dR 

(3.7a) 

In this expression a distinction is made between a factor covering the uncertainties in the resistance model and 
a factor covering the uncertainties in the material properties. A variant formulation is given in EN 1990:2023, 
equation (8.21): 

Rd = R{𝜂1 X k1 ; 𝜂i X ki 𝛾m1/𝛾mi ; ad; ΣFEd} /𝛾Rd1 

(3.7b) 

In most design situations the expression is simplified to:  

Rd = R{𝜂 Xk/𝛾M; ad; ΣFEd}  

(3.8) 

It may be surprising to find loads or load effects in this expression for the resistance. This, however is necessary 
to cover all possible limit state expressions. For instance, if the failure domain is governed by the limit state 
expression M/Mp  + (N/Np)2 >  1,0 (or more advanced models as often present in  EN 1992 to EN  1999)), it is 
otherwise impossible to use equation (3.6)  as verification format.  

The  design  value  of  the  effect  of  actions  Ed  for  a  specific  combination  of  actions  is  calculated  from 
EN1990:2023 eq (8.3):  

Ed = 𝛾Sd E {Σ(𝛾f 𝜓 Fk); ad; XRd} 

(3.9) 

In  this  expression  a  distinction  is  made  between  a  factor  𝛾Sd  covering  the  uncertainties  in  the  load  effect 
calculation and a factor 𝛾f covering the uncertainties in the load. Note that the uncertainty in the load model as 
such is also included in 𝛾f. In most design situations this may be simplified to EN1990:2023 eq (8.4):  

Ed = E {Σ(𝛾F 𝜓 Fk); ad; XRd}    and  𝛾F = 𝛾fSd 𝛾f 

 (3.10a) 

This simplification is correct if the relation between load and load effect is linear or if the model uncertainty in 
the load effect calculation is absent or relatively small.  The simplification may be unsafe in the non-linear case 
where the load model  uncertainty is relatively  large and the  application of  partial factors on the effects of 
actions is more adverse than the application of partial factors on actions. This type of load effect functions E(F) 
is often referred to as ”under linear”, ”under proportional” or “convex”.  The code refers to certain limit states in 
cable and membrane structures as well as some geotechnical structures where this might be the case. The best 
option is of course to use the basic equation (8.3) in EN1990:2023, presented above as (3.9). Another option is 
to use EN 1990:2023 eq (8.5): 

Ed = 𝛾E E {Σ(𝜓 Fk); ad; XRd}    and  𝛾E = 𝛾fSd 𝛾f 

 (3.11b) 

Some remarks (see also section 3.8): 

  use of the most adverse case of (8.4) and (8.5) leads to over conservative and uneconomic results. 

  The  load  factors  derived  for  the  linear  case  are  not  always  appropriate  for  cases  of  strong  non-linear 
behaviour. We  may leave the field of application and end  up with too large  deviations from the  desired 
reliability level. In case of doubt advanced full probabilistic verifications are recommended. 

28 

 
 
 
 
 
 
 
  The situation becomes even  more complex in the case of load combinations of  permanent, variable and 

prestressing loads 

It should further be noted that load effects may also depend on the structural properties. In case of a linear 
elastic analysis this may be the stiffness but also strength parameters can be involved. It is not always obvious 
whether, to be on the unfavourable side, in a certain design situation upper or lower design values for ad and 
XRd should be taken. A famous example is the case of the shear force in a girder. By using the lower design 
values for the bending moment capacity, the maximum value of the shear force is limited and the possible 
brittle mechanism of a shear failure may be missed. In most cases this is not taken into account explicitly but 
where  important  additional  requirements  are  formulated  to  guarantee  a  ductile  behaviour,  like  in  capacity 
design in seismic engineering. However, these requirements are seldom calibrated to a safety target (see also 
section 3.7). 

The summation symbol in the above equations stands for a load combination, not necessarily a mere addition. 
The 𝜓 can be either “1,0”,  , 𝜓0, 𝜓1 or 𝜓2. As an example, consider the permanent and transient design situation 
in EN 1990:2023 equation (8.12). In that case we have: 

Σ(F 𝜓 Fk ) = Σ𝛾G,i Gk,i + 𝛾Q,1 Qk,1+Σ𝛾Q,j 𝜓0,j Qk,j+(𝛾PPk) 

(3.12) 

This formula actually means that for the verification of the ULS reliability we have to consider the combination 
of all factored permanent and variable loads, but that we cany include the combination factor 𝜓0,j for all but 
the  leading  variable  action.  All  actions  should  be  considered  as  leading  one  by  one.  The  more  refined  load 
combination rules (8.13) and (8.14) serve the same purpose. In a similar way EN 1990:2023 shows combination 
rules for the limit states for the accidental and seismic design situation, for fatigue and for serviceability. 

Not all actions or action components have an unfavourable effect. This is relevant both for (3.7) and (3.9). For 
instance, within certain bounds, the presence of a normal compression force in a concrete or masonry cross 
section may increase the bending resistance. Other examples are overturning or uplift. If that is the case it may 
be unsafe to consider the upper design values in the load combinations.  

For variable actions one simply can ignore the load if it acts favourable and take 𝛾F equal to zero. However, one 
should keep in mind that loads sometimes follow patterns. For wind for instance, physics may dictate that a 
whole part of a façade is under suction, regardless of the influence on an action effect. The instructions for 
those type of load arrangements is taken care of in EN 1991. 

In case of favourable permanent loads one should replace 𝛾G by 𝛾G,fav or 𝛾G . The first one is used if the net 
total contribution of a single permanent load source is relevant and gives a favourable load effect. The second 
one is applied for cases where it is relevant to distinguish between parts of a single source permanent action 
that  act  favourably  and  other  parts  that  do  not.    It  is  then  important  to  consider  the  degree  of  correlation 
between load effects resulting from different parts of the same structure. This is particularly important for the 
limit states of static equilibrium with no resistance term being active. In those cases, the terms stabilizing and 
un-stabilizing are used. In structural limit state verification, the effect of partially favourable and unfavourable 
contributions of permanent actions usually is small. As it is quite cumbersome to take these effects fully into 
account, EN 1990:2023 has introduced the so-called single source rule. Reference is made to section 3.5 for 
discussion on the accuracy of this approximation.  

3.3  Target Reliability  

3.3.1  Target for consequence class CC2 and reference period 50 year 

According to annex C (informative), the basic target for design of new structures is a reliability index t =3,8 for 
a reference period of 50 years and consequence class CC2. As an alternative, EN 1990:2023 Annex C offers 
annual values with a conversion based on independent failures in subsequent years. In this chapter we will not 
consider this annual option once more. Reference is made to chapters 2 and 4.  

Actually,  the  beta  value  of  3,8  (or  one-year  values  in  the  range  4,2  to  4,7)  was  derived  on  the  basis  of 
compromises as well as calibration studies (see section 3.1) at the end of the last century. The calibration was 
mainly intended to derive partial factors leading on average to the same structural dimensions as before, but 

29 

 
 
with less scatter in reliability. To some extent one should be careful to use the same target in combination with 
other (improved) probabilistic models for the random variables (see discussion on hidden safety). 

Note: See also Annex B 

3.3.2  Differentiation due to consequences of failure 

The CC2 reliability level of 3,8 may be adjusted in the case of larger or smaller consequences of failure. In 
chapter 4.3 of EN 1990:2023 five consequence classes (CC0 to CC4) are defined on the basis of human safety 
and socio-economic considerations. According to Annex C the corresponding reliability indices changes to 4,3 
and 3,3 respectively for CC3 and CC1.This indicates a difference of a factor 3 to 10 in consequences. Note that 
EN 1990:2023 translate the change in beta into a change in some partial load factors by a factor kFI. This can 
easily be incorporated in a calibration procedure if needed. In the current 2002 version of the Eurocode actually 
the numerical values for kF have been chosen quite intuitively although some background was available (see 
chapter 3.1).  

The consequences in case of structural failure may depend heavily on the type of structure. For that reason, 
the  Annexes  A1  to  A4  give  specific  examples for the  detailed  categorisation  for  respectively  buildings (A1), 
bridges  (A2),  towers  and  masts  (A3)  and  silos  and  tanks  (A4).  In  principle  the  consequences  for  the  same 
subcategory are intended to be in the same order regardless the type of structure. 

Note  finally  that  from theoretical  considerations  the  target  should  also  depend  on  the  recourses  needed  to 
improve the level.EN1990:2023 does not explicitly accommodate for this aspect, but can be implicitly accounted 
for, for instance in accidental and seismic design.    

3.3.3  Brittle versus ductile behaviour 

Sometimes the presence of brittle behaviour is considered as a reason to choose higher safety levels. Of course, 
brittle behaviour can lead to less redistribution in case of overloading, and due to the lack of warning possibly 
more victims. Actually, it is primarily a matter of higher consequences in case of failure and may be settled that 
way (see also 2.5). EN 1990:2023 does not give further guidance. To some extent the task to provide adequate 
rules for brittle failure is left to the material related codes and the National Annexes (for instance via Table EN 
1990, A1.1). Some specific guidance is given in EN 1998 (capacity design rules for seismic loading). Note by the 
way that sudden failure without  pre-warning also  sometimes  can occur in structures  with  ductile materials. 
Further information can be found in section 3.7. 

The question of safety under brittle failure is in a pronounced way present in the design and assessment of 
glass structures as glass is an ideal brittle material. Thoughtful design and construction methods  can help to 
reduce the warning and redistribution problems, otherwise the use of adjusted target reliability levels should 
be considered (see also chapter 3.6.2). 

3.3.4  Differentiation on structure type 

The central target value of 3,8 in EN 1990:2002 for CC2 has mainly been derived on the basis of studies for 
buildings only. It has never been shown that this value can be justified for other types of structures like bridges, 
silo’s,  towers,  masts,  geotechnical  structures,  etc.  As  long  as  there  is  no  proof  that  the  value  is  highly 
uneconomic or unsafe there seems however no reason to change it. In EN 1990:2023 no explicit distinction of 
this type for that reason has been made. Implicitly there is a higher (annual) reliability level for many types of 
bridges  as  the  design  life  time  is  100  years.  This  is  on  top  of  an  often-higher  reliability  when  bridges  are 
classified as CC3-structures. 

3.3.5  Other types of limit states and design situations 

Note that the background of the standard ULS target is limited to permanent and transient design situations 
only.  

For the accidental design situations EN 1990 and EN 1991-1-7 do not give any specific reliability target. The 
background document to EN 1991-1-7:2002 refers to ISO2394 where for the loading an annual exceedance 
probability of the load of 10-4 is recommended. This is, by and large, in line with the findings in (Hingorani et al, 
2019)  and  leads  to  lower  values for  the  design  load  than  would  follow  from    =  3,8  for  50  years.  On  the 
resistance side, partial factors equal to 1,0 are usually chosen. Also, this indicates that (for economic reasons) 

30 

 
a lower reliability level is considered as appropriate.  In a  qualitative way this is a rational way of  spending 
resources, but more scientific research is needed to make quantitative. 

In the case of unforeseen actions usually a damage scenario is specified and a certain reliability level for the 
damaged structure for some specific (usually short) period of time is implicitly or explicitly required. The actual 
reliability is then a combination of the likelihoods of the damage and later failure. No specific background for 
the Eurocode rules on this point are known. 

The new generation of EN 1998 for seismic design sets a reliability target in terms of annual probability of 
exceedance  of  collapse  (which  is  close  to  the  formal  limit  state  NC  in  seismic  analysis)  equal  to  2x10-4 
corresponding to  = 3,5. This is close to the ISO2394:2015 recommendation  = 3,3 in Table G.4 for the central 
category of consequences, relatively large costs of safety measures and a high variability. Note also that this 
target is meant for the system level where Eurocode design is often limited to the member level.  

In  fire  design  a  procedure  aiming  at  the  level  of  a  reliability-index  of  3,8  for  50  years  is  now  present  in 
EN 1991-1-2 The reliability for the traditional classification system based on a minimum time to collapse if 
exposed to the ISO standard fire curve is not clear.  

For the limit state of EQU a target reliability differentiation similar to STR is possible and a similar set of life 
time or annual reliability targets. However, the effect on the design in general will be less. 

In  the  case  of  fatigue  the  target  is  not  different  from  the  target  for  other  failure  mechanisms  if  similar 
consequences are expected; relevant in this respect is the distinction between damage tolerant and safe life 
(see also 3.9.1).  

For  the  verification  of  serviceability  limit  states  a  distinction  should  be  made  between  reversible  and  non-
reversible limit states. The value of target reliability index  = 0 is considered in Eurocodes for checking the 
reversible serviceability limit state and  = 1,5 for irreversible serviceability limit state considering the reference 
period of 50 years. For selection of the serviceability reliability level, see also ISO 2394 and JCSS PMC. 

Influence  of  time:  design  life,  temporary  structures,  transient  design  situations, 

3.3.6 
execution 

See 2.5.3 for a general technical introduction  

The standard design service life in Eurocode is 50 years. Other options for choice for the design service life, 
ranging from 10 to 100 years, are given. Consequences for the design are relatively clear for the treatment of 
deterioration mechanisms, but whether partial factors (or characteristic values) should be changed is not clear. 
The following observations can be made: 

It makes sense to aim for a higher target annual reliability level for a longer design life time if the effects 
of  damage  in  case  of  collapse  are  dominated  by  economic  consequences;  investments  in  structural 
reliability are more cost efficient if you can profit for a longer time;  

For human  safety  dominated  failure  mechanisms an  annual  target  independent  of  the  design  life-time 
seems to be more appropriate;  

For temporary structures that are re-used several times (apart from other safety considerations) the total 
time of use should be considered; 

Also, transient design situations should be considered for the total time of their duration. 

The present EN 1990:2023 does not have operational rules. The above considerations, however, should give 
sufficient guidance for additional rules in the National Annexes. As an example: for CC2 the Dutch code keeps 
the reliability index of 3,8 as a target for all (accumulated) design life time larger than 15 years. For shorter 
periods an annual limit is (-3,8) / 15= 2 x 10-5. 

3.3.7  Material deterioration (fatigue, corrosion, etc.) 

Although  = 3,8 was derived for non-deteriorating structures, it is reasonable to include deterioration simply 
as  one  of  the  mechanisms  contributing  to  failure.  One  may  include  here  inspection  and  repair.  If  annual 
probability  is  used  the  Table  C2  standard  present  annual  reliability  index  of  4,7  in  EN 1990:2023  Annex  C 
definitely is too high; a value of say 4,2 seems to be more appropriate due to corelation and time dependency 
(see section 3.9 and Annex B).  

31 

 
3.4  Background of partial factors for permanent and variable loads   

3.4.1  General introduction 

The  objective  of  this  chapter  is  to  provide  further  background  for  the  recommended  partial  load  factors  in 
Annex A of EN  1990:2023. According to the generally accepted  basic principles of the Eurocode, the  partial 
factors for actions in ULS-verifications are material and structure independent, at least within a wide category 
of structures like buildings, bridges and so on. As a result of that principle, one has to accept that the factors 
cannot serve all design situations in an equally adequate way. Instead, they may for instance be calibrated as 
a  weighted  average  over  a  set  of  design  equations  that  represent  the  field  of  their  application.  This  is  the 
consequence of the chosen level of simplification and generalization implied by the partial factor format of the 
Eurocode. The aim is to avoid unfavourable deviations in design situations that either occur very frequently or 
can have a major impact. On the other hand, one should be careful not to create very large deviations, even for 
a  small  niche  of  applications.  For  that  reason,  results  for  individual  cases  count  and  should  be  checked; 
sensitivity studies are essential. 

It should also be kept in mind that material related Eurocodes EN 1992-EN 1999 can choose material resistance 
partial factors for a specific material (or even a subcategory of structures) in order to reach the chosen target 
reliability (e.g. 3,8 for 50 year) given overall and fixed values for the partial load factors in EN 1990. At that 
stage it is also possible to give tailor made attention to nonlinear effects. 

Notes:  

- A differentiation of the limit state equations for all the resistance models and all materials is not possible 
within the scope for this report. Such a refinement, however, is also believed to be less relevant for the goal of 
this sub-chapter that is a global reflection on the recommended partial action factors. See also the general note 
above. 

- The methods and elaborations presented in this sub-chapter as well as in Annex B should be considered as 
example  applications  of  EN  1990:2023  Annex  C  clause  C.4.4;  possible  considerations  outside  this  reliability 
based approach will be mentioned where appropriate (see also Figure 3.1).  

-  In  this  subchapter  the  same  steps  are  followed  as  in  the  simple  example  2.4,  but  probabilistic  models, 
characteristic values and partial factors .are here in accordance with the Eurocode approach. 

3.4.2  Portfolio, Limit state function and Weights  

The limit state considered in this sub-chapter 3.4 is for buildings according to the column DC1 in EN 1990:2023 
Table A1.8. We will primarily consider: 

●  6 different materials; 

●  1-2 limit state models per material; 

●  1 self-weight and 1 permanent load;  

●  3 different variable loads, but no combinations of variable loads;  

●  STR, permanent design situation; 

●  a reference period (and design service time) of 50 years; 

●  a target reliability of 3,8; 

In addition there are some calculations to study sensitivities.  

3.4.2.1  The limit state function 

The load effects are linear and unfavourable; no details of the resistance model are included. For most materials 
a simply supported beam is considered, starting from the following limit state function: 

g(X) = Z f - (1/8) [g + q] L2 

(3.13a) 

32 

 
 
where: 

g(X)  limit state function;  

X 

vector of random variables; 

f  material strength [kN/m2]; 

g 

q 

Z 

L 

self-weight (including permanent load) [kN/m]; 

imposed load [kN/m]; 

effective section modulus [m3]; 

span [m]. 

After dividing by (1/8)(gm+qm)L2 , introducing Q=q/qm, G=g/gm and R = f/fm and adding the model uncertainties 
for the resistance (R), the load calculation (e) and the variable loads (Q) we arrive at: 

g(X) = R p R - e [ (1-aQ) G + aQ Q Q] 

where  aQ  =  qm  /(  qm+gm )  and  p  =  Z  fm  /  (1/8)(gm+qm)L2  is  a  deterministic  standardized  resistance.  Index  m 
indicates the mean value. Next we split up the weight term into a self-weight part and a permanent load: 

G = aG GS + (1- aG) GP with aG = GSm /(GSm+GPm ) 

So finally a generic standardised linear limit state function is obtained as:  

g (X,pij) =pijΘR,iRi-e [(1-aQ) {aGGS,U+(1-aG)GP,U}+aQΘQ,jQj ] 

(3.12b) 

R,  GS,  GP  and  Q  represent  the  standardised  (mean  1,0)  random  variables  representing  resistance,  self-
weight, permanent and variable load; 

The  - variables represent the model uncertainties for the resistance, load effect and load model; 

p is the resistance factor following from the Eurocode design requirement (see 3.15); 

aQ represents the ratio between variable and permanent loads (aQ = 1 for variable load only);  

aG represents the ratio between permanent load and self-weight aG =1 for self-weight only); 

i and j refer to material and load cases. 

Notes: 

Geometrical uncertainties are not explicitly taken into account but may be part of the model uncertainty. 

In other publications often the parameter  = Qk/(Gk+Qk) is used. For snow loads the values of  and aQ are 
similar.  For  wind  and  office  load    relation  between    and  aQ  (for  a  50  year  reference  period)  may  be 
estimated as  = 1,4 aQ / (1+0,4 aQ)..  

The load model uncertainty for the self-weight model is neglected.  

3.4.2.2  The design equation 

Given the span, the characteristic actions, the characteristic resistance and the partial factors we may find the 
appropriate dimensions of the structure (actually the Z-parameter) from: 

Z fk / R = (1/8) [ G gk+Q qk ] L2 

(3.13) 

where R = Rd m (covering the resistance model and the material) and G and Q are equal to the corresponding 
products Sd f .(covering the model effect calculation and the combination of action model and action itself). In 
the standardised representation this is modified into: 

33 

 
 
 
p Rk / R = G (1-aQ ) Gk+ Q aQ Qk 

(3.14) 

After separating the self-weight part and permanent load: 

p Rk / R = (1-aQ) [G {aG GSk+(1-aG) GPk } + Q aQ Qk] 

(3.15) 

Using this equation one finds the parameter p in equation (3.12b). The factors R on the resistance side will be 
chosen in accordance with present material related Eurocodes, see Table 3.2. The partial factors on the loading 
side (to be investigated in this sub-chapter) are G=1,35 and Q=1,5 in accordance with EN1990:2023, Annex A.. 
The procedure is performed for a range of values for aG and aQ, , each value receiving equal weight in this sub-
chapter. Table 3.3 gives information over the applied ranges and the weights w (see section 2.6.2).. These values 
are best guesses intended to give more weight to what is believed the most frequent or important applications.  

It should be noted that in the design equation no model uncertainties are explicitly considered; implicitly this 
means  that  the  characteristic  values  are  equal  to  1,0  (irrespective  of  the  actual  mean  and  coefficient  of 
variation) and are actually omitted in the design equation above and the input tables. 

Table 3.2. Partial factors for the resistance according to related Eurocodes. 

i 

1 

2 

3 

4 

5 

6 

7 

8 

9 

i 

1a 

2a 

2b 

3a 

4a 

5 

6a 

7a 

7b 

Material and Limit state 

R     M 

Structural steel yielding strength (bending) 

Concrete compressive strength  

Re-bar yield strength (bending) 

Glulam timber (bending) 

Solid timber (bending)  

Masonry compression strength 

Aluminium bending 

Ground pile foundation 

Shallow foundation 

1,00 

1,50 

1,15 

1,25 

1,30 

1,50 

1,10 

1,50 

1,40 

Source: Eurocodes 1992-1999 

Table 3.3. Weights and ranges of variations of aG and aQ. 

i 

1 

2 

3 

4 

Material and Limit state 

wR,i (weight) 

aQ ranges 

aG ranges 

Structural steel yielding strength 
(bending) 

17 % 

Concrete compressive strength  

12 % 

Re-bar yield strength (bending) 

40 % 

Glulam timber (bending) 

3,5 % 

[0,3- 0,8] 

[0,1- 0,7] 

[0,1- 0,7] 

[0,2- 0,8] 

[0,6; 1,0] 

34 

 
 
 
[0,2- 0,8] 

[0,1- 0,7] 

[0,3-0,8 ] 

[0,1-0,5] 

[0,1-0,5] 

5 

6 

7 

8 

9 

1 

2 

3 

Solid timber (bending)  

1,7 % 

Masonry compression strength 

12 % 

Aluminium bending 

1,7 % 

Ground pile foundation 

Shallow foundation 

6% 

6% 

Variable Load 

wR,i (weight) 

Imposed load 

Snow 

Wind 

50% 

10% 

40% 

3.4.3  Probabilistic models for partial factor calibration 

Table 3.7 (at the end of this section) shows probabilistic models and corresponding characteristic values for 
calibration purposes used in this sub-chapter. This set may be considered as a simplified version of Annex A. 
The models are based on minimum code requirements.  

The mean values for resistance and loads in Table 3.7 do not refer to the actual physical mean values (like a 
mean strength of 120 MPa or a load of 45 kN). The point is that the mean values of the actions in 3.4.2 have 
been  normalized  to  1,0.  This  is  meaningful  as  the  absolute  values  of  the  loads  and  resistances  are  of  no 
importance for the determination of the reliability in this example. Only the ratios are relevant. The mean value 
of the resistance is also taken as 1,0. For an assumed or given set of partial factors the value of p follows from 
equation 3.15. Next the reliability index given the required value of p may be found using (3.12b).  

As long as the models provided in the codes are unbiased also the mean values of the model uncertainties are 
equal to 1,0. Consequently mean values unequal to 1,0 show that the model has a bias.  The coefficients of 
variation in Table 3.7 follow from Annex A.  

The columns with characteristic values show the  values that are actually used in the  design or assessment 
calculations.  

Further detailed information is given below of each category. 

3.4.3.1  Materials 

According  to  EN  1990:2023  clause  6.2  characteristic  values  for  materials  are  (unless  specified  otherwise) 
defined  as  the  5%  fractiles  of  the  underlying  probability  distribution  function  for  materials.  So,  the  typical 
situation in Table 3.7 would be: 

Table 3.4. Typical statistical information for materials in this sub-chapter 

Random 
variable  

Material A 

X 

R4 

Distribution 

Mean  

V 

Xk  (fractile) 

Lognormal 

1,00 

0,10 

(0,05) 

In  this  case  the  characteristic  value  in  design  equation  3.15  follows  (as  a  good  approximation,  the  exact 
expression can be found in EN1990:2023, Annex C.4.4.2) from:  

Xk = µ exp(-1,65V) = 0,83 

(3.16) 

35 

 
 
 
 
 
 
 
 
 
 
 
A deviating fractile in the column of Xk requires some formal background, that is, the number should be specified 
in codes or material product standards. In Table 3.7 this is (making reference to Annex A) the case for steel and 
aluminium. 

3.4.3.2  Model uncertainties resistance side. 

Model uncertainties for resistance models are often  based on comparison  between a  theoretical model and 
experiments where all other variables in the model (yield stress, geometry etc.) have been measured. According 
to EN 1990:2023 Annex D the experiments lead to a scatter V and a bias b. A positive bias indicates that the 
model is on the safe side (e.g. lower bound model or models implicitly taking care of the scatter) (see Figure 
3.2). 

Figure 3.2. Definition of Bias according to EN1990:2023 Annex D 

The options that could have been followed for the bias in the various material related Eurocodes after the tests 
are: 

( 1 )  No change to the model and no change in the partial factor, i.e. Rd = 1,0 

( 2 )  The model was left unchanged and the bias was taken into the partial factor Rd  

( 3 )  The model was changed by a factor b making it unbiased. 

The consequences for mean and partial factor are shown in Table 3.5: 

Table 3.5. Options for treating the bias of the model uncertainty  

Option 

 () 

(1)  No changes  

(2)  bias in Rd 

(3)  bias in model  

b 

b 

1 

V 

V 

V 

V 

Rd (bias part only) 

1 

1/b 

1 

Note that, as mentioned before, the model uncertainty in the design equations 3.15 (just like in the Eurocodes) 
is not considered explicitly and so implicitly the characteristic value is equal to 1,0, as also stated in Table 3.7).  

Options (2) and (3) in the Table 3.5 lead to the same results, it is only a different manner of presentation. Option 
2 is in line with a rd as defined in EN 1990:2023 Annex D eq (D28). Option (3) uses the test results as a correction 
tool to obtain an unbiased model. The values () and Rd (bias part only) are changed accordingly.  

Option (1) is used when the model and the partial factor equal 1,0 already have been fixed and are used in the 
design  stage  of  the  calibration.  The  tests  and  the  model  uncertainty  parameters  are  used  only  to  find  the 
corresponding reliability index for a given design. The design will be different from options 2 and 3, but in the 
end this will be compensated by different values for the partial load factors in order to reach the same target 
beta. 

So finally the same target reliability level is being reached in all three cases: the option type has influence on 
the  intermediate  results,  but  not  on  the  result  of  the  calibration.  One  should  only  make  sure  that  in  the 
calibration procedure the same definitions are followed as in the material related Eurocodes. In particular one 

36 

 
 
has to be careful if a Eurocode makes adjustments in the model in order to get “nice values” like 1.0 for the 
partial factors. 

For the scatter part of the model uncertainty (so b=1) only two options are used (see Table 3.6): 

( 1 )  No change to the model and no change in the partial factor, i.e. Rd = 1,0. 

( 2 ) The scatter was accounted for via EN 1990:2023, equations (D.21) for rk and (D.28) for rd. where 
the expression for rk refers to the 5% fractile of the product R R and a similar statement holds for rd . 

Table 3.6. Options for treating the scatter part of the model uncertainty (lognormal distribution) 

Option 

(1)  no change 

Mean 

1     

(2)  bias in model (rk  and rd ) 

 exp(V) 

V 

V 

V 

Rd  

1  

rk / rd  

where  may follow from Annex D. Also here the characteristic values implicitly are taken as 1,0. Note that V 
of course follows from the tests and is not changed. 

In Table 3.7 the mean values of the model uncertainty on the resistance are equal to 1 or b. For the calculations 
in this report it is assumed that Rd is included in the R  or  M  values presented in Table 3.2; for that  reason the 
Rd values are not present in Table 3.7. 

Note finally that we should carefully consider how the bias and scatter in the model uncertainty is obtained. 
According  to  Annex  D  all  random  variables  present  in  the  model  should  be  measured  carefully.  In  practice, 
however, sometimes nominal values are used. That way the bias may be incorrect and a part of the scatter in 
resistance parameters may enter the coefficient of the variation of the model. In order to remove the double 
counting of uncertainties one should base m on the difference between the scatter in the laboratoy and the 
real world only. There may also be measurement errors disturbing the procedure. These can be accounted for 
if the standard deviation of the error is known. 

3.4.3.3  Model uncertainty in load effect calculations  

According to Annex A the model uncertainty for the load effect calculation depends on the type of structure and 
type  of  load  effect.  For  statically  determinate  structures  the  uncertainty  is  smaller  (say  V  =  0,05)  than  for 
statically indeterminate structures (say V = 0,10 – 0,20). In this subchapter the value of 0,10 has been chosen, 
knowing that higher values are possible. But, in particular for (ductile) indeterminate structures there usually is 
a redistribution of stresses possible that is not taken into account. Although the Eurocode in principle is limited 
to member level design (and not the system level) it can be argued that a value V=0,10 is high enough. Note 
further that no characteristic value is specified: the uncertainty is believed to be covered by the partial load 
factors and the characteristic value is not present in equation 3.15. 

3.4.3.4  Self-weight and permanent loads 

For  self-weight  in  general,  the  50%  fractiles  are  taken  as  representative  values  as  in  agreement  with  EN 
1990:2023. The scatter in the table includes the variation in volume, specific weight and the model uncertainty 
in  the  self-weight  model.  Characteristic  values for  permanent  loads are  mean  values  (or  50%  fractiles)  for 
V=0,1 and 95% fractiles for V=0,2. 

3.4.3.5  Climatic actions 

A time variant load is usually the product of a time variant part and a time invariant part. According to clause 
6.1.2.3 of EN 1990:2023 the characteristic value for the time variant part is chosen as the 98% fractile of the 
annual maximum. The characteristic value for the time invariant part is tuned in such a way that for the product 
of the two characteristic values the exceedance probability for a one-year period is 2%. As a simple upper bound 
value k = (exp V ) can be used, however, calculating the exact value is to be preferred.   

In this model no attention is paid to climate change.  

37 

 
3.4.3.6 

Imposed load (office building) 

For imposed loads actually no specific statement is present in EN 1990:2023 for the fractile corresponding to 
the characteristic value. In this study we take a slightly modified JCSS model as starting point (see Annex A) 
which  leads  to  mean  of  2,22  kN/m2  and  a  V=  0,26  for  the  50-year  maximum.  According  to  EN  1991  the 
representative  value  is  3,0  kN/m2.  The  quotient  3,0  /  2,22  =1,35  explains  the  characteristic  value  in  the 
standardized load model in Table 3.7.  

Table 3.7. Stochastic models for partial factor calibration (based on Annex A). 

Random variable 

X 

Distr, 
type 

Mean 
) 
(

V 

Char value 
(Fractile) 

Logn.  1,15 
Logn.  0,97 
Logn.  1,03 
Logn.  1,00 
Logn.  1,00 
Logn.  1,16 
Logn.  1,27 
Logn.  1,00 
Logn.  1,00 

Logn.  1,00 
 R1 
Logn.  1,00 
 R2 
Logn.  1,00 
 R3 
Logn.  1,00 
 R4 
Logn.  1,00 
 R5 
Logn.  1,00 
 R6 
Logn.  1,00 
 R7 
 R9 
Logn.  1,00 
 R10  Logn.  1,00 

Resistance model uncertainty   
R 
Steel bending 
R 
Concrete compression 
R 
Concrete bending 
R 
Glulam bending 
R 
Solid timber bending 
R 
Masonry (flexure) 
R 
Aluminium bending 
R 
Pile foundation 
Shallow foundation/slope instab  R 
Material strenght 
Steel yielding strength 
Concrete compr strength 
Rebar yielding strength 
Glulam bending strength 
Solid timber bending strength 
Masonry compr. Strength 
Aluminium 0,2% limit 
Cone penetration test value 
Undrainred shear strength 
Model Unc Load effect calc 
statically indetermiate frames  
Self-weight 
Self-weight (steel) 
Gs.1  Normal  1,00 
Self-weight (reinforced concrete)  Gs.2  Normal  1,00 
Gs.4  Normal  1,00 
Self-weight (glulam) 
Gs.5  Normal  1,00 
Self-weight (solid timber) 
Gs.6  Normal  1,00 
Self-weight (masonry) 
Normal  1,00 
 Gs.7
Self-weight (auminium) 
Normal  1,00 
 Gs.8
Self-weight (ground) 
Permanent load 
Permanent load (small V) 
Permanent load (large V) 
Variable loads 
Wind time-invariant part  
Wind velocity pressure 
Snow time-invariant part  
Snow load on ground   
Imposed load model  
Imposed load (office) 

Logn.  0,97 
Gumbel  1,00 
Logn.  0,81 
Gumbel  1,00 
Logn.  1,00 
Gumbel  1,00 

Normal  1,00 
GP 
GP*  Normal  1,00 

Q1 
Q1 
Q2 
Q1 
Q2 
Q3 

Logn.  1,00 

c.l 

0,05 
0,14 
0,07 
0,10 
0,10 
0,18 
0,15 
0,20 
0,15 

0,05 
0,10 
0,045 
0,15 
0,20 
0,16 
0,05 
0,12 
0,20 

Averaged over several applications 
All variability not in concrete lab strength  

1,00 
1,00 
1,00 
1,00 
1,00 
1,00 
1,00 
1,00 
1,00 

0,83 
(0,05)  Not including lab – in situ variation. 
(0,05) 
(0,05) 
(0,05) 
(0,05) 
0,85 
(0,05) 
(0,05) 

0,10 

none 

0,025 
0,05 
0,10 
0,10 
0,07 
0,04 
0,05 

0,10 
0,20 

0,26 
0,14 
0,26 
0,20 
0,10 
0,26 

Load model uncertainty  included  

For Aluminium: see Annex A 

1,00 
0,98 
0,95 
0,95 
1,00 
1,00 
1,00 

1,00 
1,33 

1,08 

Extreme for period of 50 a   

0,82 

Extreme for period of 50 a 

1,35 

Extreme for period of 50 a 

Note: Mean values of the strength and load variables are all equal to 1.0 as a consequence of the standardization of the limit state function. 
Only annual action maxima are different form 1.00 to keep them consistent with the distribution of the 50 year maximum. No values are 
presented for the characteristic values of the model uncertainties as they are not explicitly present in the design equations. Mean values 
of the model uncertainties different from 1.00 account for bias in the models. 

38 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
3.4.4  Resulting reliabilities for design according to the Eurocodes  

The resulting reliability indices corresponding to EN 1990:2023 format (8.10) are presented in Table 3.8. For 
the design all partial factors are in line with the Eurocodes: the values for the actions are 1,35 for permanent 
and 1,5 for variable loads, the values for the resistance side are according to Table 3.2. 

The results in Table 3.8 are presented for all considered combinations of the materials and variable actions. 
Each value is averaged over the considered range of the aQ parameter, which is the ratio between the variable 
and total load (see Table 3.3). The overall average result can be characterized by  = 3,74. In order to check the 
sensitivity to the choice of the weight values the analysis has been repeated for equal weights. In that case the 
result was  = 3,65. 

Table 3.8. Resulting 50 year reliability indices for each combination of material and action (averaged over all aQ and aG 
values)  

Wind 

Snow 

Imposed 

Steel bending  

Concrete compression  

Concrete bending  

Glulam bending  

Solid timber bending  

Masonry  

Aluminium bending  

Pile foundation  

Shallow foundation 

3,52 

3,84 

3,42 

3,34 

3,41 

4,10 

3,61 

3,55 

3,73 

3,17 

3,62 

3,15 

3,08 

3,17 

3,92 

3,31 

3,42 

3,59 

3,94 

4,15 

3,78 

3,71 

3,74 

4,37 

4,00 

3,79 

3,98 

Figure  3.3  shows  graphically  the  scatter  in  beta-values  for  the  various  loads  (first  3  graphs,  including  all 
materials)  and  materials  (last  9  graphs,  including  all  three  loads).  Figure  3.4  shows  the  dependence  of  the 
reliability index on the aQ parameter for snow. Each line gives the result for one material. The results for wind 
snow and live load show quite similar trends. 

To some extent the relatively small scatter for foundations may be surprising, given the large uncertainty in 
resistance. But actually that fact makes the result relatively less sensitive to the scatter in the climatic loads 
as for instance compared to steel. Note also that the partial factors for foundations are large and the range of 
aQ is rather narrow. 

39 

 
 
 
 
 
Figure 3.3. Results of reliability indices for 50 years, for each loading type and each material; calculations based on 
design format prEN 1990:2023 (8.12) using partial factors according to the Eurocodes.  

Figure 3.4 Reliability index β as function of αQ for snow load, each line representing a specific material (steel 
concrete. etc).  

4.5

4

3.5

3

2.5

β

2

0

0.2

0.4

0.6

0.8

1

αQ

3.4.5  Reflection 

Given the result of 3.4.4 the main conclusion is that on average a design according to the rules of the Eurocode 
(including the values 1,35 and 1,50 for the partial factors for permanent and variable actions) meets the target 
reliability level  = 3,8 for a design life time of 50 years with sufficient accuracy. Of course, we should keep in 
mind  that  the  above  analysis  is  limited  in  its  set  up  and  contains  a  number  of  quite  arbitrary  choices. 
Nevertheless, it may be considered as a reasonable indication.  

At the same time, we may also conclude from Table 3.8 and Figure 3.4 that the scatter in calculated reliability 
levels  is  large.  Numbers  run  roughly  from  2,5  to  4,5,  the  larger  values  corresponding  to  weight  dominated 
design  situations  and  the  smaller reliability  levels are  obtained  for  design  situations  dominated  by  climatic 

40 

 
 
 
 
actions. The background reason for this scatter has partly to do with the historical development of the present 
partial factors for permanent and variable loads. The familiar values of G = 1,35 and Q = 1,50 were not derived 
on the basis of reliability theory only but on a mix of theoretical considerations and traditional values. Also, for 
reasons outside the domain of the theory of structural reliability the values were considered as a good and 
acceptable compromise. 

For additional information and reflection on this issue, reference is made to Annex B where also results from 
some sensitivity studies are shown. 

3.5  Further issues related to action aspects 

3.5.1  Load combination factors 

The load combination factors 𝜓0 and  are intended to reduce the non-dominant or non-leading actions in a 
load combination (see section 3.2). Having all actions in the load combination with their unreduced design value 
would be over conservative if the actions are statistically independent. The combination values can be included 
in a standard calibration procedure, but this can make the procedure more extensive and is often omitted.  

An approximation is to assume that the maxima in time of all variable loads occur at the same point in time. 
This leads  to  upper  bound  values for  𝜓0  values  (although  still  less  than  1,0).  The  error in  case  of  a  1-year 
reference period is less than for the 50-year reference period. 

It is worth to point out that the 𝜓0 values for (8.12) are theoretically different from the 𝜓0 values for (8.13). 
The reason is that in (8.12) the leading variable action has no reduction factor and so only all other factors may 
profit from the fact that maxima do not occur at the same point in time. In (8.13a), however, also the leading 
variable load is multiplied with 𝜓0 . The reduction results from the combination with the permanent load and in 
that case the maxima always occur at the same point in time by definition. Note further that in (8.13b) the   
factor actually is also a kind of 𝜓0 -values but with a different name. 

If we neglect the time variability, the design value for the accompanying action QdA = 𝛾Q,j 𝜓0,j 𝑄k,j follows from 
(see section 3.1.3):  

F(QdA) = (0,4 x0,7 )= (0,28) 

(3.17a) 

However, if we want to include the time factor we have to apply a more complex formula using the FBC model 
(Ferry Borges and Castanheta M. 1971): 

F(QdA) = {(-0,4 -1((-0,7)/n)}n 

(3.17b) 

Here n is the number of intervals in the reference period for the slowest fluctuating load in the combination. 
This may lead to the unpleasant but correct result that the 𝜓0 - for snow is different if combined with wind or 
combined with live load. Taking the highest value is a practical solution. 

See also example B3 in Annex B. 

3.5.2  Free and fixed actions, single source rule 

According to clause (6.1.1) of EN 1990:2023 actions can be classified as free or fixed. The distribution of a 
fixed action is unambiguously defined for the whole structure or structural member. For free actions various 
spatial distributions over the structure are possible and should be accounted for. In some cases the analysis 
should be based on the most unfavourable combination of loaded and unloaded areas. This holds for instance 
for imposed loads in buildings (including partition walls) and traffic loads on bridges. In other cases (like snow 
and  wind)  the  situation  is  mixed  and  a  set  of  prescribed  fixed  load  arrangements  have  to  be  considered. 
Construction loads (cranes, auxiliary construction loads) can be considered as fixed or free depending on their 
use. 

41 

 
 
 
Special attention in this respect is needed for permanent loads. In order to avoid cumbersome calculations for 
the free component of permanent loads in practice, EN 1990:2023 introduces in 6.1.1 the notion of a ‘single 
source” simplification. We will discuss here the rule for the structural part of the permanent load..  

The structural self-weight is a random field with in general relatively well known means and coefficients of 
variations. The difficulty is in the correlation  pattern, as there is relatively little data. Parts coming from one 
source (one material, one manufacture) are believed to be highly correlated. According to the JCSS model code 
the correlation within one member is almost complete and for the correlation between members in one building 
a value of 0,7 is recommended. This is probably meant as an average for different situations, but that is not 
mentioned explicitly. 

The  simplifying  one  source  (full  correlation)  assumption  is  on  the  conservative  side  as  long  as  all  parts 
contributing to a certain load effect are all favourable or all unfavourable. If parts of the permanent load are 
unfavourable and other parts are favourable, the one source assumption is no longer conservative. We may 
arrive at the famous problem of subtracting two numbers leading to an almost zero load effect. Using the one 
source assumption we may arrive at a design load effect of zero. See also section 3.5.3.  

We should however also keep in mind that the partial factor G takes care of the uncertainty in the load itself 
as well as in the load effect calculation uncertainty . The statistical correlation for the weight has to do with 
method of fabrication. Clearly in cases where the “single source” simplification is applicable (same material, 
same  manufacture,  same  time  of  fabrication)  the  dependency  will  be  close  to  1,0  in  most  cases.  For  “two 
sources” it will be closer to 0,0. The uncertainty in  has nothing to do with the source of the material. In the 
draft update for the model uncertainty chapter of the JCSS model code the following general statement can be 
found:  

In literature, until now, little attention has been given to correlation patterns in model uncertainty. As a general 
indicator one may say that uncertainties related to influences or parameters neglected in the models may leave 
room to independence between various parts of the structure and between structures when the neglect leads 
to  unbiased  predictions  over  the  structure.  If  we  neglect  for  instance  temperature  effects  in  a  statically 
indeterminate structure, this will lead to tensile stresses in some parts and compression in other ones. In case 
of a large number of members, the correlation between two arbitrary members is judged to be small. However, 
if there are only two members, they may have a strong (negative) correlation. Simplifications on the other hand 
tend to correlation over the entire population and will partly be observed as a bias in case of testing.  

An important group of random variables neglected in the structural analysis are the effects due to deformation 
constraints  in  statically  indeterminate  structures  as  may  follow  from  temperature  variations,  humidity 
variations, settlements and stresses due to imperfect mounting. It is well known that for ductile structures this 
is not very important, but for brittle elements this can be quite different. Also local errors in the stress calculation 
model, normally not relevant, can become very important in case of brittle behaviour. Their determination can 
be  more  complicated  as  for  ductile  behaviour  as  structural  behaviour  of  brittle  structures  can  show 
discontinuities. 

In general the model uncertainties may be considered as dominant with respect to the weight (see e.g. Annex 
A). So even in a case when there is clearly only one source in the origin of the structural weight, there still may 
a  non-full-correlation,  assuming  that  the  model  uncertainties  are  not  fully  correlated  between  different 
elements. Let us consider two specific cases: 

42 

 
 
 
 
 
 
 
 
 
 
 
Figure 3.5 Considered cases for the single source discussion. 

Case 1 

Given two floors on a continuous beam as in Figure 3.5 left. The floors (neglecting the beams) have self-weights 
g1 and g2 with equal mean values. The bending moment in the centre of the left beam is equal to (including 
the model uncertainty factors ): 

M = (5/48) 1 g1 a2.- (1/48) 2 g2 a2 

Note that this may be slightly different from the maximum moment, but that is considered as less important 
here. If considered as one source we have for the design load effect M (using (EN 1990:2023 equation 8.12): 

Md = (4/48) G (g) a2.= (4/48) 1,35 G (g) a2

 = 5,4/48 (g) a2 

If we consider the loads as two (independent) sources we have to distinguish between the favourable and the 
unfavourable contributions: 

Md = (5/48) G (g) a2 - (1/48) Gfav (g) a2 = (5x1,35-1x1,0) /48 (g) a2 =5,75/48 (g) a2 

Given a mean for M of 4/48 (g) a2 (assuming the mean value being equal to the characteristic values) the 
difference is considered as too small to bother, so we may still consider it as one source even if it is not. 

Case 2 

For the case of Figure 3.5 right, we find for the design load effect Md in the left beam (using (8.12) and the 
single source rule): 

Md = (1/8) G (g) a2.- (1/4) G (g) (0,6a)2

 = 0,047 (g) a2 

If we consider the terms as coming from two (independent) sources we have: 

Md = (1/8) G (g) a2 - (1/4) Gfav (g) (0,6a)2 = 0,079 (g) a2 

Note that the mean value of M is equal to (1/8) (g) a2 - (1/4) (g) (0,6a)2 = 0,035 (g) a2. This means that the 
difference between the two design values is large. This is a typical case where the sensitivity is large. In order 
to get more insight, let us redo the calculation for the second design load value of the bending moment, but 
using reliability theory. If we define the limit state function and statistical data as:  

g(X) = 0,067 – { 0,125 1 g1 - 0,09 2 g2} 

(g1)=(g2)=1,0, V(gi)=0,05, V(i)=0,10, (gi gj)=0,7, (i j) = 0,5  

we find that S = 3,8*0,7 = 2,66, in line with the required reliability. So, it follows that Md = 0,067 (g) a2 which 
still is high compared to the “one source” approach.  

The difference reduces if we add imposed load: taking qk = 0,3 gk the error of the one source approximation 
becomes in the order of 20 % if compared to the probabilistically derived value. This is still quite substantial 
and we might even think of worse examples. EN 1990:2023 Clause 8.3.3.1(3) states that the single source rule 
may be applied provided that the design is not sensitive to spatial variation of those permanent actions. The 
basic question is whether the designer will recognize this criterion in practice or not. The safe way out is of 
course by making both assumptions and accept the most conservative result. 

Finally, a remark is made about the self-weight from building elements other than structural. These loads can 
either be time independent or change a number of times during the working life of the structure. They can be 
fixed or quite random in space. As a consequence, no large correlation can be assumed and the single source 

43 

 
 
rule is not recommended. The variability is usually not large, so the partial factor for self-weight seems to be 
sufficient. If cases where the variability is large the 95% fractile is used. Detailed rules are left to the National 
Annex. 

3.5.3  The limit state of static equilibrium 

Closely related to the single source problem is the limit state of static equilibrium. Actually, it is the extreme 
example where the  single  source rule  is not applied in order to ensure the safety of  structures with a  high 
sensitivity to an out of balance between action effects arising from permanent actions from in most cases a 
single source. Also, the factors in EN 1990:2023 Table A1.8 need calibration. The specific point is that in the 
verification procedure no resistance term is present. For instance, the limit state function could be: 

g(X)=

1
4

θ1g1a2-

1
4

z a2θ2g2 

(3.18) 

The first term represents the de-stabilizing part and the second term stabilizing part. The factor z is chosen 
such that reliability index is equal to the CC2 target reliability value of 3,8. Let us take:  

V(gi) = 0,05, V(i) = 0,05, (gi gj)=0,7 and (i j) = 0,5 

(3.19) 

The difference with the statistics used in the single source example is that the coefficient of variation for the 
model uncertainty has been reduced from of the V(i) = 0,10 to 0,05. The background is that most calculation 
modes in static equilibrium verifications will be less complex than in structural load effect calculations. Using 
these numbers, it follows that (for z=0,79 and =3,8): 

1dg1d / 2dg2d = 0,86 / 1,09 = 0,78 

(3.20) 

So according to the theory we should take  a partial factor 0,86 for the stabilizing part and 1,09 for the de-
stabilizing part of the weight in order to achieve the required reliability level. In EN 1990:2023 Table A1.8 the 
values for the partial factors are 1,15 and 1,35 leading to a ratio = 0,85 instead of 0,78. Looking from the 
theory this ratio is slightly  on the unsafe side.. It is also quite against the spirit of the partial factor approach 
to increase a load that acts favourable. 

Note 1: For the case of static equilibrium EN 1990:2023 does not specify kF-factors for G stb, making the margin 
with G destb very small for CC1.  

Note 2: See also example B4 in Annex B which is discussed in more detail.  

3.5.4  Prestressing 

In the context of this note prestressing is conceived as an active measure to influence the stress distribution in 
the structure by prestressed cables (tendons), bars or other elements. The purpose of prestressing could be to 
avoid certain limit states (less deflection, cracking) or mechanisms to occur. In particular one might think of: 

 

Internal prestressing in girders (bonded and unbonded tendons). 

  Cables supporting towers and masts 

  Cable stayed bridges (or buildings) 

 

 

 

Extradosed bridges (or buildings) 

Positioning of supports (e.g. by jacking);  

Tension grid (membrane) structures 

44 

 
 
 
 
Note  that  already  the  order  of  the  constructive  adding/mounting  building  elements  have  influence  on  the 
internal stress distribution, but that is not discussed in this note. 

A general aspect of prestressing is that the prestressing action is not important for the Ultimate Limit state 
capacity as long as the behaviour is geometrically linear and full ductility is present. If not, or if other criteria 
are important (deflection, cracking) limit state functions have to be formulated. In most cases the force in the 
prestressing element will appear as a design value (see EN 1990:2023 eq (8.9):  

Pd=𝛾P Pk 

(3.21) 

Pk  is  the  characteristic  value  of  the  prestressing  force  (usually  a  single  value)  and  γP  is  the  partial  factor 
according to EN 1990:2023 Annex A or as given in the relevant Eurocodes. General recommendations, assuming 
no (cor)relation with self-weight, are:𝛾Pinf = 0,9 and :𝛾Psup =1,1 for the standard favourable and unfavourable 
effects. For internal  prestressing in  concrete beams the value 1.0 is taken; for local effects (crunching) and 
stability limit states the values are 1,2 and 1,3. For membrane systems we have: 𝛾Psup =1,25. 

For possible (cor)relations with self-weight, see section on cable stayed bridge. 

Some codes allow factors closer to 1,0 if there is sufficient control on the action force. It should be mentioned 
that  a  reliability-based  background  for  partial  factors  on  prestressing  is  lacking.  Most  factors are  based  on 
experience and direct expert judgement only. 

As in many cases the prestressing action is actually the result of a deformation type of action, it could be more 
reasonable to let the partial factor act on the deformation. However, that is usually not done. 

3.5.4.1 

Internal prestressing in concrete beams 

In case of concrete elements with internal bonded prestressing tendons the reaction force of the prestressing 
action is found in the compressive stress in the concrete. The  main purpose is to avoid early cracking or to 
reduce deformations. The upper value of the prestressing force is well controlled (so low uncertainty) and  

𝛾P = 1,0 

(3.22) 

seems therefore to be appropriate. The lower bound value is more uncertain as it depends on stiffness ratios 
of steel and concrete, relaxation and creep. This can influence serviceability limit states related requirements 
but  does  not  have  a  significant  effect  on  ultimate  limit  states.  EN  1990:2023  allows  National  Annexes  to 
introduce  correction  factors  rsub  and  rinf  where  relevant.  In  the  case  of  unbonded  prestressing  systems  the 
concrete  element  is  ready  first  (post-tensioned)  and  the  prestressing  force  can  even  be  calibrated  better 
assuming the friction forces are small. Long term loss effects may be compensated in the course of time if 
wanted and physically possible. 

As far as the ULS is concerned one might even neglect the pretension and consider the tendons as ordinary 
reinforcement to understand the sensitivity to prestressing force, assuming that sufficient deformation capacity 
(also for concrete crushing) is available. The contribution is different in case of bonded or unbonded tendons 
because the unbonded tendons are able to develop only a limited increase of elongation at ULS. This may also 
influence the stresses at the anchorage zone 

3.5.4.2  External prestressing 

In the case of external prestressing, we may distinguish between cable stayed structures and guyed towers and 
masts. 

 - Cable stay structures, 

The main goal of the prestressing is to control the actual deflections. The reacting forces come partly from the 
self-weight of the bridge deck and partly from the stiffness of the deck structure. At the other side we may 
have pylons and or anchors. For extremely flexible systems of deck structure and pylon there is only the weight 
and actually there is no real prestressing any more. For those cases the load combination: 

45 

 
 
 
𝛾G Gk + 𝛾P Pk = 1,35 Gk + 1,00 Pk 

(3.23) 

is over conservative. 

The  point  is  that  if  the  self-weight  in  a  practical  case  is  higher than  the  predicted  value  during  design,  the 
prestressing force will also be higher as it will be tuned to the combined effect for instance by controlling a 
deflection at a key location. However, taking 𝛾P exactly equal to 𝛾G might be too optimistic due to uncertain 
short- and long-term changes in stiffness properties; also model uncertainties and possible measurement errors 
may disturb the accuracy of the calculation. Note that partial load factors also keep track of the uncertainties 
in the calculation model. The factors should actually be calibrated to a reliability analysis, taking into account 
all these uncertainties. In that case one could make use of the fact that the uncertainty in the actual weight is 
very  small  and  only  the  uncertainty  in  the  load  effects  is  relevant.  Further,  the  actual  uncertainty  in  the 
prestressing control system (e.g. based on measurements of deformations or displacements) could be taken 
into account. A possible outcome could be 

self-weight unfavourable 

1,35 Gk + 1,30 Pk and 1,35 Gk + 1,40 Pk 

(3.24a) 

self-weight favourable 

1,00 Gk + 0,95 Pk and 1,00 Gk + 1,05 Pk 

(3.24b) 

However, this is not used in the present Eurocode. In EN 1993-1-11, for the case where the prestressing force 
and the self-weight are carefully tuned, the above set 3.24a and 3.24b is simplified to: 

1,35 Gk + 1,35 Pk 

1,00 Gk + 1,00 Pk 

(3.24c) 

(3.24d) 

In particular for locations where the load effects of weight and prestressing exactly or almost  exactly cancel 
each other, the differences might be noticeable. Again, if sufficient deformation capacity is available, the ULS 
reliability will hardly be affected. 

In  deriving  the  partial  factors  we  should  bear  in  mind  that  also  the  strength  of  the  prestressing  elements 
themselves is limited. Depending on the properties of the prestressing elements a value of 𝛾P = 1,40 (or even 
already 1,35) may already be unrealistically high and exceeding the actual resistance.  

- Guyed towers and masts  

Towers and masts need to resist wind loads and offshore structures even waves (coming from all directions) 
and  tying  them  by  cables  is  a  very  efficient  way  to  do  so.  The  prestressing  actions  in  the  cables  will  be 
counteracted by the prestressing actions in other cables and a compression force in the mast. At the lower end 
the cables are anchored to a foundation. The purpose of prestressing is to avoid compression in cables and 
reduction in stiffness of the whole system. Guy ropes in guyed masts are normally relative long and non-linear 
effect of the ropes are more important. In those cases the permanent load is usually less relevant.. 

3.5.4.3  Lightweight structures with load reversal 

To ensure adequate structural behaviour for all relevant design situations, prestressing is required in structures 
where load reversal is possible for different combinations of actions or load arrangements. This is particularly 
the  case  with  lightweight  structures.  Steel  or  composite  bow-string  bridges  are  considered  as  an  example. 
Hangers made of prestressing tendons, prestressing bars or even steel profiles are often used for load transfer 
from the deck to the arch. Vertical or inclined hangers can be provided. 

46 

 
 
 
 
 
 
 Figure 3.6 Bow-string bridge with inclined, prestressed hangers.  

Source: CESMA Ingenieros SL – Madrid; photo: MEGADRONE S.A.C. 

The overall structural behaviour depends on numerous parameters, such as the relative stiffness of the deck 
and the arch, the arrangement and separation of the hangers, as well as their stiffness, among others. Non-
symmetrical  load  arrangements  cause  the  system  to  bend  and  can  introduce  compression  forces  into  the 
hangers. In order to avoid possible damage to these elements, in particular their anchorages, and changes to 
the structural system, during the construction process the pre-elongation in the hangers must be adjusted. This 
adjusting is a very delicate operation since internally the system is statically indeterminate. Generally, it should 
be  realized  in  several  runs  with  a  particular  tensioning  sequence.  However,  deviations  from  the  nominal 
prestressing forces are possible. The limit state is that the internal force in the hangers is negative. Sometimes, 
a minimum tensile force is assumed as a limit, taking into account possible losses in prestressing forces.  

In practice, it is not realistic to check all possible combinations of favourable and unfavourable deviations from 
the  specified  prestressing  forces  in  the  different  hangers  by  applying  the  rules  for  combining  actions  for 
persistent and transient design situations (see EN 1990:2023 8.3.4.2). There are also no specific provisions or 
simplified  rules  in  the  various  Eurocodes,  such  as  EN  1993-1-11,  for  this  type  of  prestressing  application. 
However, when verifying Ultimate Limit States, including those related to instabilities of the arch, it should be 
taken  into  account  that  the  prestressing  forces  can  be  higher  or  lower  than  the  specified  values.  The 
corresponding partial factors P should also cover measurement uncertainties. Considering  the tolerances in 
conventional  stressing  procedures  and  the  uncertainties  associated  with  converting  the  applied  prestressing 
forces into their effects, an upper and lower value of, respectively P = 1,1 and P = 0,9 seem to be appropriate. 

3.5.4.4  Prestressing in membrane structures 

Tensile surface structures, being either membrane structures or cable nets (see section 3.6.4), are prestressed 
systems that can carry loads perpendicular to the surface. The spatial shape is mostly doubly curved. A tensile 
surface structure can  carry loads by activating only tensile stresses. The form can  be anticlastic (in general 
mechanically prestressed like saddle shaped structures) or synclastic (in general pneumatically prestressed by 
air inflation) or even monoclastic or flat. 

Usually, membrane structures are mixed structures. The supporting structure for the membrane can be made 
of steel, timber or other material. The membrane itself is carrying the external loads to the supporting structure. 

For membrane structures the relationship between action and effect is not proportional. Membrane structures 
require a geometric non-linear analysis. 

47 

 
 
 
 
Figure 3.7 The membrane structure Palais Thermal Bad Wildbad.  

© Form TL.  

Prestress is applied mechanically by membrane compensation or pneumatically by inner pressure. To ensure 
the biaxial loadbearing behaviour and to avoid slack areas as well as large deflections, membrane structures 
are prestressed biaxially. The prestress level is in accordance with the structural analysis. The minimum design 
prestress should be maintained throughout the design service life.  

The  recommended  lower and  upper partial  safety  factor  values for  prestress  are  γPinf  =  0,9  and  γPsup  =1,25 
respectively (these values are based on expert judgement).  

Note: Tolerances in the supporting structure might affect the prestress level, therefore special consideration 
should be given to the sensitivity of membrane structures to these tolerances. The acceptable tolerances of the 
supporting structure to the interface to the membrane should be agreed in the design phase. 

3.5.4.5 

Jacking systems at supports 

Prestressing  without explicit tension elements like cables etc. may be realized by imposing deformations by 
jacking at the supports at convenient times during the construction  process (Handbuch Brücken, 2014). The 
counteracting force is delivered by the foundation on one side and the structure on the other. Using this method, 
one  may  for  instance  achieve  a  compression  layer  against  the  cracking  that  usually  appears  in  continuous 
composite bridge girders at support (EN 1994-2). One way to deal with this type of prestressing is by treating 
the jack force as an independent permanent action. As the jack force is relatively well known the partial factor 
may be lowered. If the jacking process is tuned to some measurable deflection or deformation of the structure 
the relevant uncertainties should be considered. And like the situation in cable stayed bridges there may be a 
dependency with the gravity loads. 

3.5.5  Accidental actions  

In  the  accidental  design  situations  attention  is  given  to  exceptional  conditions.  The  load  combination  to  be 
considered is (EN1990:2023, clause 8.3.4.3):: 

Ed = Σ Gk,i + Ad + Σ { 𝜓1,1 or 𝜓2,1 } Qk,j+ Σ 𝜓2,j Qk,j + (Pk) 

(3.25) 

where Ad is the design value for the accidental action which is specified directly. Operational information on Ad 
is  presented  in  EN  1990:2023  (related  to  flooding,  notional  loading  on  key  elements  and  local  damage 
scenario’s like the notional removal of a column) and in EN 1991 (related to fire, impact and explosion). For 
specific structures more types of accidental actions may need to be taken into account. Typical for accidental 
actions is that some degree of structural damage is explicitly accepted. EN 1990:2023 Annex E specifies also 
design strategies related to unforeseen or unforeseeable actions in the form of tying and segmentation. 

For water related accidental actions a return period of 1000 year is recommended (see clause 6.13.2); for other 
accidental actions no recommended fractiles or reliability targets are specified. Guidance may be found in the 
general principles for setting target reliabilities as discussed in ISO 2394, Annex G. For accidental actions usually 

48 

 
 
 
the relative cost of safety measure should be classified as high, leading to lower reliability targets  compared 
with permanent design situations. According to ISO 10252, target values for the annual failure probability in 
case of accidental design situations are  usually set between 10-6 and 10-4 . For the failure probability given 
some prescribed damage (e.g. removed column), depending on the circumstances, values between 0,001 and 
0,1 are considered as appropriate for a short reference period following the event. 

3.5.6  Fire 

The fire resistance assessment of Eurocodes is considered as a structural design at Ultimate Limit State in fire 
situation.  In  consequence,  as  it  is  the  case  for  room  temperature  design,  the  fire  resistance  assessment  of 
Eurocodes  comprises  several  parts  related  to  the  reliability  issue,  such  as  mechanical  and  thermal  actions, 
material properties and calculation models. The difference compared to normal room temperature design is 
that the design actions for fire resistance assessment are composed of both thermal and mechanical actions. 
Moreover, the former part, i.e. thermal action, occupies a very important place in fire resistance assessment. As 
a result, thermal effects also play an important role in the response of the structure. 

The following paragraphs will provide a summary about the reliability background of the fire parts of current 
Eurocodes. 

3.5.6.1  Mechanical actions 

The mechanical actions for fire resistance assessment according to Eurocodes need to be determined with help 
of a specific combination method taking account of the fact that fire is an accidental situation. Consequently, 
the design values for various combination coefficients are smaller than those used for room temperature ULS 
design. According to EN 1990:2023, Clause 8.3.4.3, we have for the accidental fire design situation to take into 
account, similar as for the first generation of Eurocodes: 

ΣGk,i+ [𝜓1,1 or 𝜓12,] Qk,1+Σ 𝜓2,j Qk,j 

(3.26) 

where  the  choice  between    𝜓1,1  and  𝜓2,1  is  up  to  the  National  Annex.  However,  as  there  is  no  background 
document, an estimate for the resulting reliability index on annual or life time is not known. 

3.5.6.2  Thermal actions 

The thermal actions for fire resistance assessment of Eurocodes are classified into two families: nominal fire 
temperature curves and physically-based thermal actions. The first family of thermal actions, in particular, ISO-
standard  fire  curve,  has  been  applied  for  several  decades  and  their  reliability  depends  in  fact  on  the  fire 
resistance requirement imposed by the regulatory authorities. From a scientific point of view, it is actually a 
classification system with unclear reliability target, but very largely based on real fire feedback. Note also that 
for instance one positive test is considered as sufficient, neglecting the rules for design by testing as given in 
prEN 1990:2023 Annex D. 

As far as the second family of thermal actions is concerned, the reliability background of corresponding design 
parameters adopted in current EN 1991-1-2 are based on the outcomes of the research works conducted in 
Europe (CE, 2002). In the scope of these research works, one of the key parameters for physically-based thermal 
actions,  the  fire  load,  is  dealt  with  in  detail  according  to  the  semi-probabilistic  approach  of  Annex  A  of 
ENV1991-1 (CEN, 1994) in order to get a specific partial factor for this action. The starting point of this approach 
is to take the failure probability value of 7,23 x 10-5 per working life of the building corresponding to the safety 
index β=3,8 and adjust it by taking account of the occurrence probability of fully developed compartment fire 
so that the partial factor for the fire load can be derived. The adopted occurrence probability of a fully developed 
compartment fire is from German data (Bub, 1979). As several parameters can influence either positively or 
negatively the occurrence probability of a fully developed compartment fire, they are introduced separately for 
a more accurate design of fire load. This design approach has been incorporated in EN 1991-1-2. In addition, 
the characteristic values of design fire load recommended in current EN 1991-1-2 are based on a 80% fractile 
value which is about 22% higher than the mean value.  

In the current draft of EN 1991-1-2, new design values of the partial factor for the fire load are incorporated 
to take account of two other reliability classes, namely β=4,3 and β=3,3. The same methodology to reach a 

49 

 
 
safety index of 3,8 has been applied in the determination of additional design values of safety factor for the 
fire load.                

3.5.6.3  Material properties 

For fire design, two types of material properties are needed, the first one is relative to thermal properties and 
the second one concerns mechanical properties, both of them at elevated temperatures. 

For  fire  resistance  assessment  according  to  Eurocodes,  the  design  values  for  partial  factors  (thermal  and 
mechanical) of all the materials are brought to 1,0 whatever their design values for room temperature design 
are. The characteristic values of all the thermal properties are taken as the mean values. The characteristic 
values for mechanical resistance are the following: 

Steel: 

nominal value [CEN, 2020] (0,12% fractile) 

Concrete: 

characteristic value (5% fractile) 

Timber:  

20% fractile (5% fractile for room temperature design) 

Masonry:  

characteristic value (5% fractile in principle) 

Aluminium:   nominal value (not exactly but similar to steel). 

It can be found that the basic design values are quite different from one material to another. However, the 
reliability background of using only one partial factor value of 1,0 for all materials remains  still a matter of 
ongoing research., similar as in the first generation of Eurocodes. 

3.5.6.4  Calculation models  

As  far  as  the  calculation  models  (in  particular  for  finding  the  temperatures  in  the  compartment  and  the 
structure) of Eurocodes for fire resistance assessment are concerned, the majority of them has no reliability 
background. However, certain design rules recently developed for the second generation of Eurocodes are based 
on following validation criteria according to testing standard of passive fire protection materials [CEN, 2002]: 

The results predicted with  design rules shall not be on the unsafe side by  more than  15% of the reference 

result; 

Individual  calculation  results  from  design  rules  on  the  unsafe  side  compared  to  reference  results  shall  not 

exceed 20%; 

The mean value of all differences between calculation results from design rules shall be on the safe side. 

One  can  conclude  simply  that  a  probability  of  85%  is taken  to  validate  the  design  rules for  fire  resistance 
assessment.  

It is necessary to point out that all the design models using tabulated data are derived from currently used 
basic combination coefficients for room temperature ULS design, namely 1,35 for permanent loads and 1,5 for 
imposed loads. 

3.5.6.5  Conclusions 

The review of the design procedures of current Eurocodes with respect to fire resistance assessment shows 
that on the one hand, the reliability of the design rules is not dealt with in a global way and on the other hand, 
in a lot of cases, the reliability background is not available. Note also that in many countries the fire safety level 
is governed by requirements specified in minutes fire resistance using ISO 834. These requirements are not 
part of the Eurocode. 

3.5.7  Seismic design situation   

3.5.7.1  Characterization of the hazard 

Earthquakes result from strain energy accumulation in the earth crust due to tectonic movements, followed by 
sudden release at the attainment of the crust material strength. The energy released at the source (fault) results 
in a local kinetic energy density that decreases with distance 𝑅 from the source and therefore in ground motion 
of decreasing amplitude (geometric attenuation). 

50 

 
From a structural point of view earthquakes are rapidly varying support motions, rather than applied loads. The 
motion of the structure (its response) is influenced by its dynamic properties, i.e. mass and stiffness, as well as 
its strength and ductility. The relation between the maximum response of a simple structure and its natural 
period of vibration T and damping ratio  is described by the (absolute) acceleration spectrum and the (relative) 
displacement spectrum, shown for an individual earthquake event (i.e. the motion recorded at one site during 
one event of magnitude M on a source rupturing at distance R from the site.  

Figure 3.8  Acceleration (a) and displacement (b) response spectra for an earthquake event (a single M, R pair).  

Given the high degree of unpredictability in both loading and response use is made of a probabilistic approach 
(PSHA or Probabilistic Seismic Hazard Analysis). In most PSHA’s the occurrence of seismic events is modelled 
as a stationary Poisson process. The interval time between two successive events is random with an exponential 
distribution.  The  mean  arrival  time  is  equal  to  the  so-called  return  period  TR.  Equivalently  one  can  use  the 
exceedance rate λSa(T)(s) of a given ground motion level s, which is the reverse of the return period. 
The basic equations are given by: 

λSa(T)(s)=λ0[1-FSa(T)(s)] 

(3.27) 

FSa(T) (s) = P [Sa(T) < s] =  P[Sa(T) < s | M=m,R=r] fM.R (m,r) dm dr 

(3.28) 

where: 

λSa(T)(s) 
𝜆0  

Sa(T)  

M 

R 

frequency of exceeding any given ground motion value; 

rate of relevant seismic events generated by all considered seismic sources;  

spectral acceleration at any given vibration period;  

  magnitude of the earthquake; 

source to site distance. 

By repeating the analysis for different values of the natural vibration period T a set of 𝜆-curves (seismic hazard 
*  values of Sa at different natural vibration 
curves) is obtained. By setting a value of the return period TR
periods, characterized by the same probability of exceedance are obtained and can be plotted versus T. The 
spectrum obtained by this procedure is called the Uniform Hazard Spectrum (UHS), and is shown in Figure 3.9. 

* =1/λSa

51 

 
 
 
 
 
 
 
 
 
 
 
Figure 3.9  Seismic hazard curves for spectral acceleration at different vibration periods (a), uniform hazard spectrum for 
a fixed return period TR (b) and analytical approximation of the uniform hazard spectrum in the code (c).  

The  UHS  is  at  the  basis  of  the  definition  of  the  seismic  action  in  all  modern  codes  for  seismic  design  and 
assessment,  including  of  course  EN  1998.  Usually,  the  UHS  resulting  from  PSHA  is  approximated  with  an 
analytical shape that can be described with a reduced set of parameters, as illustrated in Figure 3.10.  

The scatter in ground motion is described within the attenuation law (or ground motion prediction equation, 
GMPE), which is a key component of the PSHA. These laws, obtained by regression on empirical data (spectra 
(M,R)+ ln Sa, essentially 
like those in Figure 3.8 for events of known M and R) usually in the form ln Sa(T) =μln Sa
describe  Sa(T)  as  a  lognormal  random  variable  and  its  (logarithmic)  standard  deviation  ln Sa  (the  so-called 
record-to-record variability) that accounts for the scatter and is stochastically independent of M and R;  is a 
standardised  normal  variable  indicating  the  number  of  standard  deviations  that  the  ground  motion  at  the 
building site is above or below the median predicted value. In practice: 

P[Sa(T)<s|M=m,R=r]=Φ (

(m,r)

ln s -μln Sa
σln Sa

) 

(3.29) 

EN 1998-1:2004 contained two different spectral shapes, denoted as Type I and II, associated with different 
magnitude ranges and to be used in regions of lower and higher seismicity, respectively. These shapes were 
than "anchored" to a TR-dependent value of the peak ground acceleration ag. In the current draft of Eurocode 8, 
prEN 1998-1-1:2020, the spectrum is defined in terms of two TR-dependent ordinates, Sα and Sβ, respectively 
at  maximum  acceleration  and  at  Tβ=1s,  a  choice  that  allows  more  refined  description  of  the  geographical 
variation of the seismic action in Europe. 

Note: PSHA studies have been carried out Europe wide and are being continuously updated in interaction with 
SC8 in order to support an harmonized cross-border definition of Sα and Sβ within EN 1998.  

3.5.7.2  Limit states, return period and load combinations 

A total of four limit states are considered in EN 1998: 

- operational limit and damage limitation (both SLS);  

- significant damage and near collapse (both ULS).  

In EN 1998-1:2004 the return period for action related to the SD (significant damage state) for CC2 is set equal 
to 475 years (actually 10% exceedance probability in 50 years). In the current draft this value is confirmed. If 
calculations are based on the limit state NC (near collapse) a return period of 1600 years is recommended. It 
should be noted that EN 1998 also gives attention to the collapse of non-structural elements. They are very 
sensitive to the seismic load and can cause relatively large damage. 

The load combination to take into account for the seismic situation is given by EN 1990:2023 equation (8.16): 

52 

 
 
 
Ed = Σ G k,i + AEd + Σ 𝜓2,j Qk,j + (𝑃k) 

(3.30) 

In seismic engineering an extra factor 𝜓E is present to indicate the amount of mass related to imposed load live 
that contributes to the inertia in the dynamic calculation. No reliability-based background is given. 

When use is made of response-history analysis (RHA, EN 1998-1-1:2020) the main selection criterion for the 
sets of recorded ground motions is a good match between the response spectrum obtained as an average of 
the individual motion spectra and the target spectrum. The latter is the UHS for the site but, notably, the latest 
EN  1998  allows  for  a  more  realistic  conditional  spectrum  as  a  target  (see  Annex  A).  The  code  requires  a 
minimum of 7 records and there is no compensation for statistical uncertainty resulting from this low number. 
The resulting average deformation value serves as the demand for the structural deformation capacity. Since 
usually for structural properties mean values are taken and strain hardening is considered in the model, capacity 
design requirements checks are checked differently from linear analysis. 

While  NC  is  the  limit  state  of  real  interest  for  safety,  the  SD  limit  state  is  the  one  that  marks  the 
technical/economic limit between reparability of damage and need for demolition after the earthquake. It should 
be noted that, according to the current design philosophy, unless other seismic protection strategies like, e.g., 
seismic isolation, are adopted, damage is expected as a consequence of relying on ductility in the design. For 
this reason, SD is taken as the reference limit state for the design of new structures. The detailing for ductility 
implies satisfaction of NC verifications. For the very same reason, NC is the reference limit state in EN 1998-
3:2020,  i.e.  existing  structures  are  explicitly  checked  for  near  collapse  since  the  global  ductility  cannot  be 
"implied”. 

3.5.7.3  Structural analysis  

Figure 3.10  Deformation resulting from support motion (a), deformation resulting from (uniformly distributed) 
equivalent lateral forces (b), deformation resulting from more "realistic" lateral force distribution (also in terms of 
intensity) (c).  

Effects  of  earthquakes  (imposed  displacements,  as  in  Figure  3.10a  have  been  traditionally  translated  into 
equivalent horizontal (lateral) forces, initially uniform Figure 3.10b and then more realistically distributed Figure 
3.10c. In the past, however, design procedures of this type did not provide the structure with the necessary 
displacement capacity (Figure 3.11a) due to the development of unfavourable mechanisms, as for instance the 
"soft story" Figure 3.11b. It was only in the 1960s (Blume et al, 1961) that the concept of "capacity design" was 
introduced, i.e. the idea of proportioning member strengths so as to prevent brittle mechanisms, both at the 
local  level,  by  requiring  brittle  shear  resistance  to  be  larger  than  the  plastic  shear,  and  at  the  global  level, 
favouring beams' yielding over that of columns Figure 3.11c. 

53 

 
 
 
 
 
 
 
 
Figure 3.11. Base shear vs top-displacement curve of a structure with inadequate ductility (a), soft-storey plastic 
mechanism (b) and beam-sway plastic mechanism (c).  

Like other codes worldwide, the first generation EN 1998(2005) also contained the concept of a force-reduction 
factor, called "behaviour factor" and denoted by q. This factor allows design based on reduced equivalent forces 
and  linear  structural  analysis  (static  or  dynamic,  depending  on  the  relevance  of  higher  modes),  relying  on 
ductility and over-strength provided by capacity design and redundancy, as shown in Figure 3.12. 

Figure 3.12  Reduction of design base shear as a function of ductility, redundancy and overstrength.  

The  second  generation  EN  1998  maintains  this  previous  well-established  procedure  referred  to  as  "force-
based", but it also introduces a procedure referred to as "displacement-based". In this procedure the checks 
of verification are on the comparison of the displacement demands versus their corresponding capacities. This 
procedure,  present  in  the  EN  1998  first  generation  primarily  for  assessment  of  existing  structures,  is  now 
codified also for design. 

In  the  displacement-based  approach  the  verification  of  ductile  mechanisms  is  carried  out  in  terms  of 
deformations (i.e., according to the format of inequality 8.2 in EN 1990:2023). For example, the chord rotation 
θ (Figure 3.13) is used for frames. For this reason, models for the inelastic rotation capacity of members are 
provided for reinforced concrete and steel members in Clause 7 of Part 1-1 of EN 1998-1-1:2020. These models 
are the updated versions of those already used in EN 1998-3:2005 (see Annex A for further information on 
these models, and their model uncertainty). 

Figure 3.13. Definition of chord rotation for a cantilever column (a), a fixed-end column (b) and beams and columns in a 
frame (c).  

54 

 
 
 
 
 
Figure 3.14. Moment-rotation and definition of ultimate rotation θu.  

In terms of rotations, the design values for the capacities related to the limit states SD and NC are expressed 
as (Figure 3.14): 

θSD = 1
γRd,SD

(θy + αSDθu

pl)  

θNC=

1
γRd,NC

(θy+θu

pl)   

(3.31) 

(3.32) 

where: 

αSD 

γRd,SD  

γRd,NC  

is the part of the plastic rotation  u
of q used in force-based design (αSD is between 0,3 and 0,5); 

 pl associated with a state of damage consistent with the values 

is partial factor on the resistance (deformation capacity)at SD; 

is partial factor on resistance at NC; 

θy and θu

pl are the median values for the rotation at yield and plastic part of the rotation at ultimate level. 

It is noted how the γRd factors are to be applied to the member resistance (generalized deformation capacity 
for  ductile  failure  modes  and  generalized  stress  for  brittle  or  force-controlled  failure  modes),  not  to  the 
individual input variables like material strengths (fc, fy, etc). They are derived to account for the variability of 
input  variables  describing  the  material  properties,  geometry  and  construction  details,  as  well  as  the  model 
uncertainty θ in each resistance model. For existing structures they are functions of the amount and quality of 
the collected information on  geometry,  details and materials, described with the  so-called knowledge levels 
(KLG, KLD and KLM, respectively), which can take three possible discrete levels: minimum, average and high 
(Franchin and Pagnoni, 2018 and Background report CEN/TC250/SC8/N1076). 

Verification of brittle mechanisms in the displacement method, such as shear in members or joints, is carried 
out in terms of forces (generalised stresses). It is important to note that the shear strength models 𝑉𝑅𝑑, however, 
are modified with respect to those used for other actions, since the shear resistance of members developing 
plastic hinges, i.e. undergoing cyclic inelastic deformations is obviously different and needs to be reduced as a 
function of maximum demand ductility (Biskinis, 2004, 2020). 

Note: The models used for the concrete shear resistance in EN 1998-1-1:2020 resolves a long-standing issue. 
In the previous edition the models used for the seismic analysis were not in line with those in EN 1992. In the 
present version of Eurocode 8, the models (in particular for columns) can be considered as just a modification 
of  the  Eurocode  2  approach.  The  modification  is in  particular of  importance  for  the  assessment  of  existing 
structures. 

55 

 
 
 
 
 
3.5.7.4  Material properties 

As far as fractiles for material properties are concerned, materials are divided in "new", "existing" and "added". 
For  all  materials,  mean  properties  enter  the  analysis  models  for  the  purpose  of  establishing  action  effects 
(demand) through analysis. 

For the member-by-member verifications: 

material of new members (either part of a new design or of the retrofit of an existing structure) are represented 
as  usual  with  their  characteristic  values  and  enter  into  the  resistance  models  from  the  corresponding 
material-dependent Eurocode  

materials in existing members are represented in terms of mean values, determined from all available sources 
of information (EN 1998-3:2020,5), i.e., integrating original design documents, like specifications or better 
the test reports, with in-situ destructive and non-destructive tests carried out at the time of assessment.  

3.5.7.5  Target reliability 

In the background of seismic engineering in the first-generation Eurocode there was no clear target reliability. 
Given the set of rules in the former EN 1998 calculations (Iervolino et al, 2018) indicate that annual reliability 
indices may range from (see Figure 3.15) about 2.9 to 4,3. The average would be in the order of  = 3,5 which 
is close to the ISO2394 recommendation of 3,3 in Table G.4 for the category CC2 and relatively large costs of 
safety measures. The second generation EN 1998 finally sets a reliability target in terms of annual probability 
of exceedance of collapse (which is close to NC) equal to 2 × 10−4 ( =  3,5).  

Figure 3.15. Range of annual failure probabilities  

Source: Iervolino et al. 2018 

3.5.7.6  Seismic protection strategies  

The  previous  paragraphs  describe  the  action  characterization,  load  combinations,  material  properties  and 
analysis methods, limit  states and target  reliability  in  the  seismic  design  or assessment  situation,  implicitly 
referring to conventional moment-resisting frames. The same notions apply to reinforced concrete as well as 
steel, composite or timber construction, and of course to other similar structural types, like shear wall buildings 
or dual systems (where frames and walls share the lateral inertia forces arising from the seismic action). In all 
cases,  the  underlying  seismic  protection  strategy  relies  on  capacity-design  to  provide  the  global  ductility 
necessary for the structure to undergo safely inelastic deformation as a result of the seismic energy input. 

The traditional ductility-based strategy, however, is definitely not the only one employed, even though it is the 
most used one, mainly for its similarity to conventional construction practices. The use of mechanical devices 
as  a  means  to  control  response  to  seismic  action  (control  of  vibrations)  is  widespread  and  it  is  usually  the 
preferred  solution  for  structures  in  higher  consequence  classes,  due  to  the  superior  performance.  EN  1998 
provides  the  basis  of  design  of  structures  equipped  with  anti-seismic  devices  (which  are  regulated  by  the 
product standard EN 15129 in Part 1-1, chapter 6) and additional indications on their use in buildings in EN 
1998-1-2, chapter 8 (Base isolated buildings) and chapter 9 (Buildings with energy dissipation systems), and 

56 

 
 
bridges in EN 1998-2, chapter 8 (Specific rules for bridges equipped with anti-seismic devices). These devices 
belong to two great categories, often used jointly. 

Seismic isolators are devices able to reduce the seismic forces within the structures and are characterized by 
high stiffness in the vertical plane and low stiffness in the horizontal plane, so that the structural behaviour 
between structure and ground be uncoupled. The following typologies of isolators are currently used (metallic 
elastic-plastic isolators were more in use in the past): 

Elastomeric isolators, with or without lead core, denoted respectively rubber bearings and lead-rubber bearings; 

Sliding isolators, with either plane or curved sliding surfaces, treated for controlled friction. 

Supplemental  damping  is  provided  with  devices  of  viscous  (solid,  fluid)  or  hysteretic  (steel,  shape-memory 
alloys) behaviour, working in parallel with the isolation system in the relative displacements of the structure 
with  respect  to  the  foundation  (buildings)  or  the  piers  (bridges,  commonly  isolated  at  the  deck  level). 
Supplemental damping is also provided with devices working in the relative displacements between floors of 
buildings, in the form of braces. Buckling-restrained braces (BRB) gained widespread use. 

For the numerical simulation of structures equipped with anti-seismic devices it is of fundamental importance 
to have knowledge of main parameters governing the behaviour of such devices, like damping ratio, friction 
coefficient, viscosity coefficient, horizontal stiffness, etc. All those parameters should be described including 
their  uncertainty  and  time-dependency;  actually,  the  mean  values  and  the  coefficient  of  variation  of  those 
parameters can vary more than 50% with the ageing process. A probabilistic analysis of the isolation system 
should then account for the above variability, describing each isolator as uncoupled by the other ones of the 
same system, both at short and long-life time. 

3.6 

Issues related to specific materials  

3.6.1  Geotechnics 

3.6.1.1  Representative values of ground properties 

In geotechnical engineering the representative value of a ground property is determined either: 

using a nominal value based on -engineering judgement as a ‘cautious estimate of the value affecting the 
occurrence of the limit state’, or by  

determining a characteristic value using statistical methods ‘such that the calculated probability of a worse 
value […] is not greater than 5%’.  

Evidently, also judgement-based cautious estimates of nominal values should consider to have the same degree 
of  confidence  in  order  to  ensure  consistency  between  the  statistical  and  judgement-based  approach  in  a 
reliability sense. 

Besides the degree of confidence, the other important element in the definition of representative values for 
ground properties refers to the notion of the ‘value affecting the occurrence of the limit state’. The meaning of 
the statement deals (apart from the adequate strain level) with the influence of spatial variability on the ground 
property value to be used in design, as explained below.  

Broadly  speaking,  most  geotechnical  failure  modes  involve  large  failure  surfaces  or  volumes  of  ground, 
compared with the size of ground specimens used in laboratory testing or the ground volumes involved in in-
situ  testing  (both  in  the  order  50-500  mm,  depending  on  the  type  of  test).  Common  examples  are  failure 
surfaces in slopes or under shallow foundations, or the volumes of ground involved exerting active and passive 
earth  pressures  on  retaining  walls.  In  such  conditions,  the  performance  of  a  geotechnical  structure,  or  the 
occurrence of the limit state, is influenced rather by the average ground property value in the affected ground  
volume than by the lowest value present (i.e. the ‘weakest spot’). However, there are also limit states where 
spatial averaging is not applicable. Common examples are basal heave of deep excavations or internal erosion 
under cut-off walls or dams. That is the reason why EN 1997-1:202x considers both cases, with and without 
spatial averaging, as explained below.  

57 

 
 
 
Figure 3.16. Examples of failure modes with and without spatial averaging. The slope failure surface (left) cuts through 
the entire subsoil layer underneath the embankment, hence, the average shear strength along the failure surface is to be 
used for the characteristic value. On the other hand, internal erosion under a dam (right) is dominated by the weakest 
path (right hand side source: Internet).  

As a consequence, EN 1997-1: 202x considers the use of two (limit) cases for determining the representative 
value of a ground property: 

 

 

a ‘mean value’ (spatial average) when the occurrence of the limit state is ‘insensitive to the spatial 
variability’ of the ground property in question (i.e. full spatial averaging is applicable);  

a ‘point value’ when the occurrence of the limit state is sensitive to spatial variability (e.g. the weakest 
spot governs the behaviour). 

When statistical methods are used, in both cases the characteristic value is to be determined such  that the 
probability of a more unfavourable value should not be greater than 5%. When cautious estimates are based 
on expert judgement (i.e. nominal values), the same degree of confidence should be targeted. The equations in 
EN 1997-1: 202x are fully in line with Annex D of EN 1990:2023.  

When regional or global data sets are used, the equations to be extended by accounting for inter-site variability 
(i.e. variability of the mean values between sites) ought to be extended by accounting for inter-site variability 
(i.e. variability of the mean values between sites).  Similarly, additional terms can be included to account for 
partial averaging (i.e. between no averaging and full averaging), where relevant. For background information 
and  derivations  reference  is  made  to  ISO2394  (2015),  (Prästings  et  al,  2018)  and  (Deltares  2020).  The 
references (Prästings et al., 2018) and (Bond & Harris, 2008) also provide guidance for deriving characteristic 
values for shear strength parameters using regression analysis. 

The future version of EN 1997 also addresses a form of using prior knowledge in determining the characteristic 
value,  more  specifically  by  using  recommended,  literature-based  values  for  the  coefficient  of  variation 
(Vx=σx/μx; see Annex A of EN 1997-1; 2021). When using these Vx, the statistical uncertainty reduces and the 
Student-t factor for a virtual unlimited test series should be used (i.e. t∞

0,95=1,645). 

Recent work involving random field simulation (e.g. Hicks, 2019) has provided insight into the ‘effective property 
distributions’, i.e. the probability distributions of equivalent ground properties to be used in limit state analyses 
when using uniform ground layer properties as opposed to modelling random fields explicitly. Such approaches 
are  superior to  the  solely  statistical-based  approximations  as  described  above  in  so  far that  they  allow  for 
quantifying the effective mean (usually lower than the field mean due to failure surfaces being attracted to 
weaker zones) and the effective variance (lower than the field variance due to partial averaging) without making 
strong assumptions. 

The considerations above mainly address the  situation with direct  measurements of the ground  property or 
parameter  of  interest,  for  example  the  undrained  shear  strength  or  the  weight  density  as  determined  in 
laboratory  tests.  Often  the  estimation  of  ground  properties  is  based  on  indirect  measurements  by  using 
transformation models (in geotechnical engineering often called ‘correlations’, not to be confused with the more 
narrow definition of statistical  correlation). An example would be the estimation of the undrained shear strength 
from  the  results  of  cone  penetration  tests  (CPT).  When  determining  the  representative  value  using  indirect 
measurements and transformation models, it is important to account for the model error (bias and variance) in 
the transformation model. 

Finally,  for  the  purpose  of  this  document  the  technical  committee  ‘TC304  Risk  Management  in  Engineering 
Practice’ of the International Society of Soil Mechanics and Geotechnical Engineering (ISSMGE) has compiled a 

58 

 
 
state-of-the-art review  document of  uncertainty and  variability in geotechnical  parameters (ISSMGE-TC304, 
2020), including approaches to determine characteristic values and transformation uncertainties. 

3.6.1.2  Partial factors 

The calibration of partial factors as described in section 3.4 involves a generic limit state function that was 
developed to  represent typical structural elements with  self-weight, permanent and variable loads, the load 
effects of which are compared to the corresponding resistance. While this approach should work reasonably 
well for foundations (e.g. piles or shallow foundations), other geotechnical structures such as embankments or 
retaining walls do not fit into this format very well. The main reasons is that the ground is involved in (effects 
of) actions as well as in the resistance for many geotechnical structures at the same time. Hence, the distinction 
between actions and resistance is not that straightforward, and they are often correlated. Another point is the 
uncertainty in ground properties in combination with model uncertainty frequently outweighs the uncertainties 
in the (external) actions, whereas in structural engineering the load uncertainty is typically at least as important 
as the resistance uncertainty, if not dominant.  

These  differences  between  geotechnical  structures  with  other structural  elements  (and  among  geotechnical 
structures themselves) are reflected in the various design combinations (DC 1-4) (Arnold P et al.) as detailed in 
EN 1990:2023 Table A.1.8, and the different sets of partial factors attributed to them. For many geotechnical 
design situations, the partial factors on permanent actions (G) are equal to 1 or not applicable at all; and the 
partial factors on variable actions (Q) are lower than for structural design in some cases. For the remainder, 
the differences are dealt with in EN 1997  by choosing the resistance and or material factors such that the 
desired level of reliability is achieved.  

Notably, the choice of partial resistance and material factors was based on different grounds in the different 
countries in the first version of the EN 1997, and it probably will be for the implementation of the new version, 
too. Some countries rely on reliability-based calibration of partial factors as described  in section 3.4 of this 
document. The majority, however, chooses the partial resistance and material factors such that the resulting 
designs match the designs obtained with earlier safety concepts as much as possible, such that the achieved 
level  of  reliability  is  at  least  comparable  to  pre-Eurocode  design  concepts.  (Arnold  et  al.  (2013))  provides 
information on how this issue was approached for the first version of EN 1997 in different countries. 

3.6.1.3  Factor of safety, a confusing term in geotechnics  

In geotechnical engineering often the ratio  FS = Rd/Ed. (or Rk/Ek) is often referred to as the ‘factor of safety’. 
Clearly this is not a safety factor like the partial factors in  EN 1990:2023. Actually, it is an indicator of the 
degree  of  overdesign.  To  avoid  confusion  FS  nowadays  is  also  referred  to  as  the  “factor  of  stability”.  Note 
further that the inverse (1/FS) corresponds to the well-known unity check UC =Ed/Rd), traditionally often used by 
structural engineers.  

3.6.2  Glass 

3.6.2.1 

Introduction 

Glass is increasingly used as structural  building  material as transparency and brightness become  more and 
more important in modern architecture. For that reason, TC250 has decided that in addition to the existing suite 
of Eurocodes there is a need for a Eurocode on the design of Glass structures. As first steps a Technical Report 
on the state of the art in Europe [Feldmann et al, 2014] and the pre standard (EN/TS19100) have been realised. 
At the moment the EN itself is under development. The background for the reliability and robustness aspects 
can be found in (Feldmann et al. 2014; Pisano et al 2015a; Ballarini et al. 2016; Pisano et al. 2019a;  Sedlacek 
et al. 1999). This section 3.6.2 is mainly based on that document. Some additional comments are given. 

For the material glass there are certain special aspects which have to be considered within the structural design. 
Firstly, glass is extremely brittle. Cracks in glass propagate at a velocity of up to 1500 m/s (Feldmann et al, 
2014) (Nielsen et al. 2009). Consequently, the physical integrity of a monolithic glass pane may be lost in a 
split  second,  without  any  prior  plastic  deformation  or  other  prior  warning  of  impending  failure.  This  may 
seriously affect the load-bearing capacity and the residual stiffness of the structural glass member or system 
of which the glass pane is part. Moreover, depending on the specific glass type and composition, there may be 
a risk of detached glass fragments or shards detaching from the pane that could cause human injury. Secondly, 
glass fracture may be caused by actions which are of no consequence in a design with other structural materials. 
Some examples: i) while an accidental drop of a hammer on a concrete slab is a non-event, the same may have 

59 

 
serious consequences on a glass floor or roof, when no corresponding design provisions are preventing failure 
of the glass element; ii) partial shading of glass exposed to solar radiation may be sufficient to cause thermal 
fracture if no corresponding design provisions are preventing this kind of failure; iii) the presence of a nickel 
sulphide inclusion of the order of  magnitude of only  0,1  mm may  cause spontaneous fracture in thermally 
toughened glass even years after installation. Also, here provisions are required, so that the required reliability 
can be achieved. 

Given the above considerations, in addition to the limit states under normal service conditions (SLS and ULS), 
for structural glass also specific limit state scenarios like  Fracture Limit State (FLS) and Post Fracture Limit 
State (PFLS) are considered. An overview is presented in Table 3.9. Safety verifications are therefore required 
for LSS-1, LSS-2 and LSS-3 not only for the intact state, but also at the time of fracture (FLS) and also in the 
post-fracture state (PFLS). FLS and PFLS verifications are required for reasons of robustness, because partial 
glass breakage must always be expected, even with the best structural design of the intact configuration. FLS 
and PFLS are actually special ULS verifications, but they are not called such because of the risk of confusion 
with "normal" ULS verifications.  

Table 3.9. Limit State Scenarios (LSS) from prCEN/TS 19100-1 

Design for the unfractured glass state  

Design for the glass fractured state (safe glass fracture) 

Limit State Scenario (LSS) 

LSS-0 

LSS-1 

LSS-2 

LSS-3 

SLS 

ULS 

SLS 

ULS 

FLS 

SLS 

ULS 

SLS 

ULS 

FLS 

Design  for  the  post-fractured  state  (residual 
capacity) 

load 

PFLS 

PFLS 

3.6.2.2  Strength characteristics of glass (according to prCEN/TS19100-1) 

Physically seen, glass has an extremely high strength. It may go up to ten thousand of N/mm². However, for 
real products such as e.g. flat glass, it is strongly influenced by the size of its surface flaws. The typical sizes 
of  flaws  (scratches)  of  marketed  glass  are  in  the  range  of  0,01  to  0,001  mm.  Therefore,  technically  seen, 
annealed glass provides strengths of about 50 to 100 N/mm². 

A highly reliable measure to increase the effective strength of glass is the  tempering process where surface 
compression stresses, see Figure 3.17, are induced by rapidly cooling down the surfaces, when coming out of 
the  furnace.  These  compressive  stresses  are  in  equilibrium  with  tensile  stresses  in  the  middle  of  the  cross 
section. As these regions are not in contact with surface cracks, the material here is able to withstand very high 
tensile  stresses.  The  pre-stress  can  be  induced  also  by  other  techniques,  e.g.  by  a  chemical  toughening 
procedure. 

60 

 
 
 
 
 
 
 
Figure 3.17. Pre-stress distribution across thickness in the centre of a glass pane  

Source: Laufs 2000 

Glass breaks when a dominant surface flaw under positive opening stress reaches the critical condition of fast 
propagation. To model the failure distribution, in the glass world (scientists and industry), the Weibull statistics, 
based upon the weakest link-in-the-chain concept, has been the most widely used approach. Especially the two 
parameter Weibull (2PW) distribution, for the probabilistic characterization of the mechanical resistance of glass 
has been a preference for a long time. 

However, for annealed float glass there is evidence (see Figure 3.18) that the 2PW is not accurate in the left-
hand-side  tail  (Danzer,  2006)  (Yankelevsky,  2014)  (Pisano  et  al,  2015).  In  addition,  it  was  found  that  the 
lognormal approach, as used for other structural materials, is very suitable to describe the failure distribution 
of glass. (Wellershof, 2006) (Fink, 2000) (Schula, 2015) (Datsiou et al, 2017).  

Figure 3.18. Normal distribution, lognormal distribution, Weibull distribution (from left to right)  

Source: Wellershoff 2006 

The samples for the surveys came from single suppliers and care was taken in the investigation of the specimen 
used. The testing setup was in accordance to EN 1288-2 and -3 (EN, 2000). The studied glass types were: float, 
chemically toughened, thermally toughened and heat strengthened glass. The samples of each population form 
a  homogeneous  population  from  the  same  distributors.  The  use  of  the  basic  strength  of  f=45 MPa  for  the 
characteristic strength and the 𝛾M factor of 1,8 can be justified from these tests. Statistical uncertainties were 
included. Moreover, there is other independent research existing (Wellershoff, 2006) in which the glass strength 
and its corresponding safety were evaluated according to Annex D of EN 1990 (CEN, 2005/2010). Results from 
these  investigations  are  given  in  Figure  3.19.  It  should  be  noted  that  uncertainties  reflecting  the  model 
uncertainties  of  e.g.  a  glass  beam  or  column  is  not  explicitly  accounted  for  in  in 𝛾M.  There  are,  however, 
indications (Wellershoff, 2006) that the value of 𝛾M.nevertheless, is large enough. 

61 

 
 
 
Figure 3.19. (left) strength data of annealed glass surfaces: aged with grit paper (aged (a)), aged with falling abrasives 
(aged (b)) and as-received; (right) strength data of annealed glass edges: strong axis bending cut edge (a), strong axis 
bending polished edge (b) and weak axis bending 

Source: Wellershoff 2006 

The strength of  heat-treated glass results from the capacity of pristine float glass and the residual surface 
compression  induced  by  the  thermal  treatment,  which  has  to  be  overcome  before  cracks  open.  A  micro-
mechanically  motivated  approach  (Pisano  et  al,  2017)  (Bonati  et  al,  2018)  indicates  that  the  probability  of 
failure is obtained as the statistical convolution of the distributions of pristine annealed glass strengths and 
heat induced pre-stresses. The beneficial effect of the residual stresses is even more relevant when the state 
of  stress  from  external  actions  is  not  equi-biaxial.  Another  favourable  contribution  is  represented  by  crack 
healing due to synergistic effect of the thermal process and the consequent surface compressions, as discussed 
in (Pisano et al, 2017) (Zaccaria et al, 2016)  

Figure 3.20. Measured results for heat strengthened glass.  

Source: Feldmann et al. 2014 

In the formula for the design strength of glass the pre-stress is given a separate partial factor γP . On the basis 
of the published literature, in which different distribution functions for the thermal pre-stress were used, it is 

62 

 
 
 
shown  that  a  Gaussian  distribution  is  well  suited  to  represent  the  scatter  of  the  pre-stress    (Schula, 2015) 
(Pisano et al, 2019 and Pisano et al, 2017). 

In Figure 3.20, the results obtained by four-point bending and optical measurements, for heat strengthened 
glass show that the experimentally determined strength values and optically measured values of pre-stress are 
subject to different scattering. It is easy to see from Figure 3.20, that the values of pre-stress and strength 
values of heat strengthened glass are statistically independent variables.  

If one considers the annealed glass, its strength depends only on the surface or edge condition (micro cracks). 
However, the thermally induced residual stress state within the glass plate after the plate has been thermally 
toughened depends on its chemical material  composition, its physical properties, the roler conditions of the 
toughening furnace and its quench section parameters (e.g. speed of the air as it exits the nozzle). 

3.6.2.3  Design bending strength formulae and calibration of safety factors 

In the current version of prCEN/TS 19100-1 two formulae for calculating the bending strength resistance of 
glass are given. In both of them, the pre-stress (index p) is taken into account in the design strength fg,d via the 
difference between the strengths of pre-stressed glass (index  b) and annealed glass (index g), taken as the 
corresponding  5%  lower  fractiles  of  the  corresponding  statistical  distribution,  in  agreement  with  product 
standards.  

fg,d = ke • ksp • λA • λl • kmod • 

fg,k 
γM

 + ke • ke,p • 

fb,k-fg,k 
γP

fg,d = ke • ksp • λA • λl • kmod • 

fg,k 
γM

 + ke • ke,p • 

fb,k-fg,k 
kiγP

(3.33) 

(3.34) 

The factor ki is the interference factor, which specifically accounts for this statistical property. Whereas formula 
3.33 follows the classical rules reported by most of the existing standards, according to which heat-treated 
glass is a material per se, formula 3.34 paves the way for a new approach. The presence of two formulae will 
permit a gradual transition to industries and engineers of the practice. 

The  partial  safety  factors γM  and γP  (Table  3.10)  are  borrowed  from  the  established  practice,  but  are  also 
statistically  justified  to  be  in  agreement  with  the  target  in  EN  1990:2023.  The  factor γP  results  from  the 
statistical  analysis  of  non-destructive  pre-stress  measurements,  but  does  not  consider  the  aforementioned 
statistical  interference.  This  is  instead  recovered  through  the  coefficient  ki.  Note  that  this  reliability 
differentiation is on top of the differentiation factor KF according to prEN 1990:2023-Annex A.1.7.    

Table 3.10. Partial safety factors for material and pre-stress copied from prCEN/TS 19100-1 

Design situations 

Class of consequences 

Persistent & Transient 
(fundamental combination) 

Basic material M 

Surface pre-stress P 

Accidental 

Basic material M 

Surface pre-stress P 

CC1 

CC2 

CC3 

1,6 

1,1 

1,0 

1,0 

1,8 

1,2 

1,1 

1,0 

2,0 

1,3 

1,2 

1,0 

The choice of the partial safety factor for CC1 and CC3 is due to the fact, that higher or lower β-values of CC1 
and CC3 respectively compared to CC2 render safety factors being 10% higher or lower. This is also confirmed 
by level III analyses by (Pisano et al, 2017) (Pisano et al, 2019). The 𝛾-factors for the surface prestress are 
already explained above. For the 𝛾-factors of the accidental load combination usually γM = 1,0 for CC2 is used 

63 

 
 
  
 
  
 
for ordinary material. In glass design, these values are increased by 10% on the safe side. This is not based on 
an explicit or formal reliability background. For the time being this is left to the individual member states. For 
the sake of completeness it should be mentioned that (Pisano et al. 2019) is based on the traditional Weibull 
statistics for glass.  

Furthermore, the strength of annealed glass (fg,k) certainly depends on several factors, such as the waviness of 
the surface (caused by the float process), the size of the glass plate, the type of applied stress and the duration 
of  the  load.  The  size-  and  stress-effect  are  properly  considered  in  the  Weibull  statistics,  whereas  the 
phenomenon  of  static  fatigue,  due  to  the  subcritical  crack  propagation,  renders  the  design  strength  time-
dependent.  This  is  due  to  subcritical  crack-growth  and  is  described  as  standard  in  different  literature  via  a 
micromechanical crack propagation approach: 

1

kmod = ( 1
n+1

)

n (R)

1

n(n+1)(σ̇ )− 

1
n+1(tL)

1
n 

n: Material parameter 
(n=16 soda-lime glass)     

(3.35) 

where  n=16  is  the  value  chosen  for  humid  conditions  as  worst  case.  Therefore,  this  kmod  accounts  for  the 
reduction due to subcritical crack growth, or in other words intercrystallite corrosion, formerly known as “static 
fatigue”. Note that in glass design this is relevant and classical cyclic fatigue is of minor importance, see below. 

If the design/verification is carried out in proximity of the glass edge, the influence of the edge finishing (e.g. 
cut, strong defects; ground, lighter defects; polished, very little defects) has not to be considered. The level of 
pre-stress in the glass is affected additionally by the thermal toughening process (horizontal, vertical) itself. 
These factors are taken into account in the design formulae by introducing the coefficients ke, ksp, λA, λl, kmod, kp 
and ke,p. The values of those factors are believed to chosen in such a way that they can accommodate the bias 
and scatter involved. 

Ageing  was  considered  via  the  research  on  abrasive  actions  during  the  service  life  of  glass,  as  well  as  the 
implementation of those actions into destructive testing in (Datsiou, 2017) (Datsiou et al, 2017) (Datsiou et al 
2017). In the process of researching work cyclic fatigue of glass has also been studied (Hilken). Therein the 
influence of fatigue during the service life was found to be substantial. However, when considering the cyclic 
fatigue effect of fluctuating wind, it turns out, that the consideration of the  kmod -factor coming from static 
fatigue is sufficient. This is due to the fact, besides others, that a threshold below which no failure occurs was 
found (Hilken)  Using common crack propagation laws, the cyclic fatigue can  be predicted sufficiently. When 
using pre-stressed glass, almost no cyclic fatigue effects could be observed, except of the case, where all of 
the peaks of the cyclic amplitudes significantly reached beyond the surface prestress, which normally does not 
occur. Note that CEN TS does only refer to structures, whose design assumes quasi-static loading only including 
wind, which means that other fatigue actions are out of scope of the CEN TS 19100. Possibly, the Eurocode to 
be developed will consider this later on. 

3.6.2.4  The bending strength safety of annealed glass in national codes on design 

In the frame of establishing the safety level of the European design code of glass structures, it is also of interest 
to know the corresponding levels of the current national codes. Many of the current national codes, are using 
the semi-probabilistic design approach, including the actions as specified in EN 1991. Therefore, the γM values 
on  the  resistance  side,  can  more  or  less  directly  be  compared.  The  literature  survey  shows,  that  the 
differentiation of the glass strength in terms of reduction factors, as given in CEN/TS 19100 Annex A and Annex 
B cannot be found in the national codes. However, most of the national codes are also using γM in the range of 
1,8, just as CEN/TS 19100 does. Moreover, the newly introduced reduction factors (k- and 𝜆-factors) of CEN TS 
19100 effectively result into additional safety margins, compared to national codework. 

3.6.3  Fibre-Polymer Composites  

The  CEN  Technical  Specification  (TS  19101)  ‘Design  of  Fibre-Polymer  Composite  Structures’  applies  to  the 
design  of  buildings,  bridges  and  other  civil  engineering  structures,  including  permanent  and  temporary 
structures,  composed  of  Fibre-Polymer  Composite  materials  (also  known  as  Fibre-Reinforced  Polymer 
structures) or combinations of Fibre-Polymer Composite materials and other structural materials (for hybrid- 
Fibre-Polymer  Composite  structures).  The  scope  includes,  but  is  not  limited  to,  Fibre-Polymer  Composite 
laminates, profiles, sandwich panels (including web-core panels), bolted and bonded joints, composed of glass, 
carbon, basalt or aramid fibres, thermoset resins and adhesives, and polymeric foam or balsa wood cores. Fibre-

64 

 
 
Polymer Composite cables, concrete rebars, strengthening components and honeycomb cores are not covered 
by  the  TS.  The  main  manufacturing  processes  included  are  pultrusion,  filament  winding,  hand  layup,  resin 
transfer moulding (RTM), resin infusion moulding (RIM), and vacuum-assisted resin transfer moulding (VARTM). 
In contrast to most other building materials the manufacturing process has a major impact on strength and 
stiffness of the Fibre-Polymer Composite structure as it defines the fibre lay-up.  

The TS applies to Fibre-Polymer Composite structures in which the temperatures inside members, joints and 
components, under service conditions, are higher than -40°C and lower than Tg - 20°C, where Tg is the lowest 
glass transition temperature of either the Fibre-Polymer Composite material, the core or adhesive materials. Tg 
is  specified  as  the  onset  value  of  the  storage  modulus  decay,  obtained  from  dynamic  mechanical  analysis 
(DMA), and should be at least 60°C. 

3.6.3.1  Basis of design  

The standard equation for the design value of the resistance is given by: 

Rd = 

1
Rd

 R {c,i 

Xki
mi

 ; ad ; Σ FEd}  

(3.36) 

where: 

Rd 

is  a  partial  factor  accounting  for  the  uncertainty  in  the  resistance  model,  and  for  geometrical 
deviations, if these are not modelled explicitly; for values see prCEN/TS 19101:2020. 

R{…}  denotes the output of the resistance model; 

c,i 

Xki 

mi 

is the  conversion  factor accounting  for  effects  of  moisture  and temperature,  effects of  ageing  of 
materials, including the uncertainty of those effects; 

represents the characteristic values of material or product properties (defined as 5% fractiles); 

is a partial factor for a material or product property accounting for unfavourable deviations of the 
material or product properties from their characteristic values and the random part of the conversion 
factor; 

denotes the design values of geometrical parameters; 

FEd 

denotes the design values of actions used in the assessment of the design value of the effect of 
actions, Ed; 

I 

is for the ith material or product property. 

For the design of creep rupture, fatigue, adhesive connections, and fire, the product  Rdm is replaced by M, 
which  is  a  single  partial  material  factor  taking  into  account  all  the  uncertainties  and  deviations  mentioned 
above. 

The partial factor for a material or product property, m, depends only on their coefficient of variation and shall 
be determined from testing. 

For the partial factor for the resistance model, Rd, values are given for (i) profiles and laminates (for material 
failure  and  global  and  local  buckling);  (ii)  sandwich  panels  (for  material  failure,  global  and  local  buckling, 
laminate wrinkling, core indentation and punching failure); (iii) bolted joints (for net-tension, pin-bearing, shear-
out, block shear, and pull-out failure). 

Where possible, Rd values were derived from detailed statistical analyses of a significant number of resistance 
models,  based  on  the  assumptions  of  EN  1990:2023  and  ISO  2394:2014;  some  of  the  values  were  also 
validated through explicit reliability analyses of design equations based on the Eurocode format. (Sá et al, 2021) 

65 

da 
 
and (Pacheco et al, 2021) present probability distributions for the model uncertainties of the resistance models 
based on trustworthy experiments collected from the literature. 

The conversion factor, c,i, is composed of two conversion factors for temperature and moisture effects and can 
be expressed as c = ct cm. Nominal values are given for specific Fibre-Polymer Composites, core and adhesive 
materials,  and  for  specific  environmental  conditions,  for  a  50-year  period  (which  may  be  extended,  if 
appropriate maintenance and inspection procedures are implemented). These nominal values were derived from 
experimental  results  collected  from  the  literature.  As  an  alternative  to  the  nominal  conversion  factors,  the 
temperature-  and moisture-dependent  material  properties  can  be  determined  by  testing  in  accordance  with 
Annex D in EN 1990:2023. The same applies for properties affected by environmental conditions not covered 
by the nominal factors.  

The conversion factor for temperature, ct, takes into account changes of material properties due to material 
temperatures under service conditions (see above), excluding effects of long-term exposure. Nominal values 
are  provided  for  Fibre-Polymer  Composite  materials;  these  depend  on  whether  the  properties  are  fibre-  or 
matrix-dominated and are functions of Tg. Further values are listed for sandwich foam and balsa wood core 
materials, with the former also depending on Tg and the latter on density.  

The  conversion  factor  for  moisture,  cm,  takes  into  account  changes  of  material  properties due  to  moisture 
absorption over time, including ageing effects resulting from long-term exposure. Nominal values were derived 
for unprotected Fibre-Polymer Composite materials and epoxy adhesives, for three exposure classes.  

A sub-clause  concerning creep effects is further included, complemented by an annex. Creep effects on the 
strength  of  Fibre-Polymer  Composite  structures  are  considered  by  limiting  the  stress  levels  for  the  quasi-
permanent  combinations  of  actions.  Fibre-type-specific  stress  limits  are  provided  in  the  clause  concerning 
Ultimate Limit States, see below. Creep effects on the deformations are taken into account by creep coefficients, 
which reduce the relevant moduli of elasticity of the materials. Values are specified for different moduli for 
pultruded Fibre-Polymer Composite profiles, Fibre-Polymer Composite laminates/plies and core materials, for a 
50-year period; they were derived from experimental results collected from the literature.  Annex  A provides 
further values for additional periods of design service lives. 

3.6.3.2  Materials 

Strength and stiffness of Fibre-Polymer Composite components depend not only on the chosen type of fibre 
and resin as well as on the specific fibre lay-up. Standard test methods are specified to obtain the Fibre-Polymer 
Composite, core and adhesive material properties normally required in the design of Fibre-Polymer Composite 
structures.  Instructions  for  how  to  extract  or  manufacture  test  coupons  are  also  provided, in  order to  avoid 
damaging a Fibre-Polymer Composite member and for when the laminates are relatively thick. For a preliminary 
design, indicative (mean) values of selected material properties are listed in an Annex, which can be applied, 
provided that appropriate values of coefficients of variation are assumed to derive design values.  

3.6.3.3  Connections and joints 

In  Fibre-Polymer  Composite  structures  bolted,  adhesive  and  hybrid  bolted-adhesive  connections,  and  joints 
(whereby a joint is composed of one or more individual connections) occur. Concerning bolted connections under 
in-plane forces, the resistances in the case of laminate failure modes to be verified are net-tension, pin-bearing, 
shear-out,  and  block-shear  failure.  Subjected  to  out-of-plane  tensile  forces,  the  failure  modes  at  the  bolt 
locations are pull-out failure through the laminate, bolt failure in tension, and failure of the laminate at a bolt 
location  owing  to  interaction  between  in-  and  out-of-plane  forces.  At  the  joint  level,  shear  and  tying  force 
failures of web cleat joints are considered. 

A Fibre-Polymer Composite structure comprising adhesive joints shall be designed as fail-safe, i.e. joint failure 
shall not result in failure of the structure or critical parts thereof. Failure of an adhesive joint is considered as 
an accidental situation according to EN 1990:2023. The failure mode of adhesive connections shall be either 
cohesive  failure  in  the  adhesive  or  fibre-tear  failure  in  the  adherend;  adhesive  failure,  i.e.  failure  in  the 
adherend-adhesive interface, shall be prevented by appropriate material selection and surface preparation.  

The  resistance  verification  of  adhesive  connections  is  performed  according  to  Eq.  1,  using  a  single  partial 
material factor for the adhesive connection resistance, M,ac, as mentioned above. This partial factor depends on 
(i) the type of inspection and maintenance and accessibility of the adhesive connection, and (ii) the application 
conditions,  either  with  fully  controlled,  i.e.  reproducible  process  parameters,  or  only  partially  controlled 
parameters. Owing to insufficient experimental results, the range of values specified is adopted from literature 

66 

 
(EUROCOMP, 5.1.10, P(4), i.e. from the products of the partial factors m,1m,2). The background to these values 
could not be traced and they thus require further validation. Three methods are described for the design of 
adhesive joints and connections: (i) design assisted by testing; (ii) design based on stress analysis; (iii) design 
based on fracture mechanics. Stress-based and fracture mechanics-based failure criteria are provided for the 
latter two methods and experimental validation is mandatory in both cases. 

Provisions are included for the design of hybrid bolted-adhesive connections. Depending on the relative stiffness 
of the adhesive layer, the individual resistances of the bolted and adhesive connections can be summed or not. 

3.6.4  Membrane structures  

3.6.4.1  Analysis 

The membrane surface is form found by numerical methods to determine an equilibrium state. The form found 
state shall result in acceptable levels of stress, strain, and geometry under design loads that are determined in 
accordance with the form found geometry. This means that a geometrically non-linear analysis is mandatory. 

Uncertainties regarding the effects of the action due to uncertainties of the structural analysis model and the 
analysis type should be considered. 

The supporting structure shall also be stable without the membrane or membrane parts. The stability of the 
whole  structure  depends  on  the  integrity  of  each  component.  A  local  failure  can  cause  a  general  collapse. 
Complementary cables and redundant devices should be envisaged to increase the safety at critical points, such 
as mast tops, arches or frames, in order to prevent failure which can cause great damage or personal injuries. 

3.6.4.2  Material properties, characteristic values 

Tensioned membrane structures can be fabricated from fabrics like polyvinyl chloride (PVC)-coated polyethylene 
terephthalate (PET)-fabrics (PES-PVC), polytetrafluoroethylene (PTFE)-coated glass fibre fabrics (glass-PTFE) or 
foils made from ethylene-tetrafluoroethylene (ETFE), or other materials. 

Characteristic (or nominal) values shall be determined in accordance to Table 3.11. 

Table 3.11. Relevant standards for different materials and properties 

Material 

Property 

Relevant standards or rules 

fabrics 

tensile strength 

EN ISO 1421 or EN ISO 13934-1 and EN 1990 :2023 Annex D 

fabrics 

tensile stiffness 

EN 17117-1 or any other appropriate rule 

foils 

tensile strength  

5% fractile from tests at 23 °C according to EN ISO 527-3 

base material 

minimal 5 specimen with kn according to EN 1990:2023 Annex D 

The coefficient of variation Vx shall not exceed 0,05 

Recommended for ETFE: 40 MPa 

foils 

tensile strength 

Similar to base material 

seam or edge 

The coefficient of variation Vx shall not exceed 0,08 

Recommended for EFTE: 30 MPa 

foils 

tensile stiffness 

biaxial testing 

3.6.4.3  Material properties, design values 

The design resistance for a specific design situation is calculated according to equation (8.18) of EN 1990:2023. 

For textile structures, the design value Rd of a resistance is calculated as  

67 

 
Rd = 

1 
γM

Rk (kage; kbiax; kdur,*; ktemp,*; ksize; kx) 

                    (3.37a) 

For foil structures, the design value Rd (ULS) or R (SLS) of a resistance is calculated as  

Rd (or R) = 

1 
γM

Rk (kage; kbiax; kdur,*; ktemp,*;ksingle,* kQL,*kx) 

 (3.38) 

where  

Rk  
values for the material properties and dimensions;  

is the characteristic value of the particular resistance determined with characteristic or nominal 

γM  
combined into a single partial material factor (γM = γm · γRd);  

for simplicity, the  partial factors  γm and γRd  given in Formula (8.18) of EN 1990:2023  may be 

kage  

kbiax  

kdur,*  

ktemp,*  

ksize  

ksingle  

kQL 

kx  

is the modification factor for environmental (aging, deterioration) effects; 

is the modification factor for biaxial effects;  

is the modification factor for various load durations (permanent, long-term, medium-term, short-
term 

 is the modification factor for various temperature effects ;  

is the modification factor linked to the panel size; 

is the modification factor for single layer ETFE structures;  

is the modification factor considering the manufacturers quality level; 

is a variable for a modification factor for not yet specified effects; 

Modification factors kbiax, kage, kdur,*, ktemp,* can be determined with project specific tests. If compliant tests from 
other projects are available, these values can be used.  

The recommended value for the partial factors M are given in Table 3.12: 

Table 3.12. The recommended value for the partial factors M. 

Application 

Resistance parameter 

Partial factor 

structures from  

material  

PES-PVC or glass-PTFE fabrics  

connections  

material  

connections  

ETFE foils  

Note: 

γM0 =1,4 

γM1 = 1,5 

γM0 = 1,1 

Source: prCEN-TS19102-MasterDoc-2021-11-20 

The partial factor for the resistance of connections γM1 may be determined according to formula 3.38 taking 
into account the production quality of the executing production facility with Quality Level 3 (corresponding to 
EXC3, High execution quality level). 

68 

 
 
 
 
 
 
At least 30 series of minimum 5 tests per series from the last 6 month shall be available on connections of the 
base  material  under  consideration.  Weld  seams  and  edge  connections  should  be  considered,  as  well  as  all 
welding machines and welders, for which this partial factor should be applied. 

γM1 = (1-1.64*Vx) / (1-0.8*ß*Vx) ≥ γM0 

(3.38) 

where: 

Vx  is the mean value of the coefficient of variation of at least 30 series of minimum 5 tests per series from 
the production of the last 6 months [Vx = (standard deviation)/(mean value)]; Vx shall be validated by the tests 
carried out in the specific project on the specific detail. The coefficient of variation Vx shall not exceed 0.08 for 
the mean value and in the specific project. 

ß 

is the reliability index, ß = 3.8; other values may be chosen depending on the reliability class of the project 

3.7  Checking deformation capacity / brittle behaviour  

Ductility and global/local deformation capacity are important structural properties and can have large influence 
on the behaviour of structures. The following aspects are worth mentioning: 

Sufficient ductility  or deformation capacity  contributes to a  redistribution of loads  when locally  internal 
forces and stresses have reached the limit value; 

As a result, local peak stresses or locally weak strength areas can be neglected or averaged; 

In  a  similar  way  it  makes  the  structure  less  sensitive  to  some  types  of  modelling  and  calculation 
inaccuracies; 

Temperature effects, settlements, eigen stresses (e.g. resulting from inhomogeneous cooling), etc are of 
less or no importance; 

Ductility can help to survive dynamic actions for instance in case of seismic or wind actions; 

Under certain circumstances (e.g. gradual load increase, observable deformation) ductility cangive a timely 
warning for collapse and thus a reduction of the failure consequences.  

Note  that  ductility  may  be  defined  on  the  material  level,  the  member  level  and  the  system level.  A  ductile 
material  does not  guaranty  automatically  a  ductile  structure:  detailing  rules are  equally  important.  Material 
related codes contain many rules implicitly dealing with this matter. Examples are the minimum reinforcement 
requirements and limitations on redistribution in concrete or the fu/fy ration in the case of steel structures. 

It is important to realise that also materials classified as ductile in the end have a finite deformation capacity 
and that  we  have  to  deal  with  many  types of  hardening  and softening  behaviour.  For  this  reason  we  need 
models for e.g. the moment-rotation-relationships of joints or other potential critical zones. To some extent they 
are available in the literature and also in codes (e.g. 8.2 in EN 1990:2023), however, from a reliability point of 
view the deformation or strain properties are not treated on the same level as the resistance side. The distinction 
between  characteristic  values  and  design  values  is  not  always  made  and  mean  values  and  coefficients  of 
variation are not codified. It is also not always clear how to use ductility information in the nonlinear (FEM) 
analysis and the background of some rules are only documented in a limited way. Given the increased use of 
nonlinear numerical methods of analysis an explicit modelling of post yield behaviour (including fracture energy 
and strains at rupture) are however becoming more and more important. For advanced non-linear calculations 
for structures with large failure consequences it can, for that reason, be recommended to take results from 
recent research projects into consideration. This section gives an impression. 

Example: Plastic design of steel structures 

Typical  situations  where  deformation  capacity  or  rotation  capacity  is  checked  (at  least  indirectly)  is  when 
deriving the conditions for the application of the plastic hinge theory. There it makes a difference whether the 
plastic hinge forms in the beam or in the joint. For beams the difference of rotation b,redist is considered, whereas 
for joints the total rotation j,tot counts, see Figure 3.21.  

69 

 
 
 
 
 
 
 
 
 
Figure 3.21. Comparison of moment-rotation curve of (a) beams und (b) joints according to (Kühnemund, 2003) 

Source: Kuhlmann et al. 2014 

Regarding the reliability concept it should not be neglected that there is an interaction between resistance and 
deformation,  If  the  rotation  capacity  is  taken  belonging  to  the  reduced  resistance  Mj,pl,Rd  it  is  automatically 
increased. So may be the reference value should be taken from Mj,pl,Rk and in a second step reduced by an own 
partial factor (Figure 3.21). A better approach is a combined checking as realised by the so-called beamline 
theory. 

Figure 3.22. Definition of the available rotation capacity of joints according to Kühnemund, 2003  

Source: Kuhlmann et al. 2014 

In the recent years, (Schäfer, 2005) introduced the concept of the so-called beamline method. The beamline 
method is an integrative design approach regarding the design of the structural system and the choice of the 
joints as part of it, also mentioned in (Demonceau et al, 2021). It provides in a single diagram and compares 
with each other (i) the requirements of the  global structural system, in terms of joint  strength and rotation 
capacity (system line), and (ii) with respect to the local system the available strength and rotation capacity of 
a joint (joint line), see Figure 3.23b. For the derivation of the system beamlines extensive analyses varying both 
the system´s characteristics as well the joint characteristics should be done. The global system analyses are 
performed  on  beam  models  and  the  mechanical  characteristics  of  the  joints  are  simulated  through  the 
application of spring elements. The simplest, representative for the derivation of a beamline structural system 
is depicted in Figure 3.23a.  

70 

 
 
The semi-continuous joints are represented by springs, the stiffness Si of which is varied. In such way beamlines 
can be derived for one type of structural system, one type of load case and for different values of the joints´ 
mechanical characteristics. A further variation of the other two parameters, structural system and load case 
leads  to  the  derivation  of  system  beamlines  for  certain  value  ranges  of  structural  systems´  type  and 
geometrical characteristics, load cases and mechanical characteristics of joints. On the other hand, the beamline 
for the joint can be derived dependent only on the M-φ curve of the joint configuration. The beamline method 
of (Schäfer, 2005) was developed for sway-frames. 

Figure 3.23. (a) Example of beam model for the derivation of beamlines (b) derived beamline. 

Source : Huber et al. 2004 

Later, [Keller, 2019] using the results of ROBUSTIMPACT (Demonceau et al, 2021) evaluated the influence of 
the system on the required joint rotation capacity and expanded the “beamline method” for non-sway frames 
and for the load case “loss of a column”.  

For the  load  case  “loss  of  a  column”  the  directly  affected  joint  is  under  positive  bending  and  the  indirectly 
affected  joint  under  negative  bending  moment.  Therefore,  two  beamlines  for  the  system  are  needed,  one 
providing the requirements of the system in terms of rotational capacity of joints under positive bending and 
one providing the requirements in terms of the rotational capacity of joints under negative bending, together 
with the M-φ curves of the joint for positive and negative bending, see Figure 3.24.  

Figure 3.24. Check of sufficient rotation capacity for composite joints.  

Source: Keller, 2019  

The  topic  of  deformation  capacity  and  brittle  behaviour  is  very  relevant  for  the  resistance  against  many 
accidental  loads  and  the  design  for  robustness.  Recommendations  like  prescriptive  tying  rules  in  the  code 
usually require the fulfilment of a set of underlying assumptions and performance requirements like a large 
deformation capacity in joints and prevention of brittle failures (e.g. shear failures). In O’Connor et al (2021), 
minimum levels of ductility are given for a specific level of stiffness from the surrounding structure of common 

71 

 
 
 
buildings. Nonlinear structural analysis offer a useful tool in the assessment of robustness rules. Depending on 
the sophistication of the adopted models, nonlinear phenomena preceding component failure/structural collapse 
can be represented to various degrees of accuracy, including the flexural strength, compressive arching action, 
tensile catenary/membrane action, diaphragm action of slabs/walls, etc.. In this respect, the use of nonlinear 
analysis in robustness assessment allows the consideration of the structure at relatively large deformations 
thus exceeding conventional strength‐based norms, provided the deformation capacities at which components 
fail by fracture are not exceeded (Izzuddin et al., 2008). It would be helpful if codes would give more guidance 
on models and reliability based design values.  

Most explicit attention to the ductility aspects and avoidance of brittle failure until now has been given in the 
seismic design situation. Economic solutions for a high seismicity are possible only by making use of the energy 
dissipation  in  the  plastic  ranges  of  the  load  deflection  diagrams  (Freeman,  1978;  Fajfar,  2000).  To  avoid 
unintended premature brittle behaviour the notion of capacity design was introduced, making sure that brittle 
elements were sufficiently stronger than the ductile ones. Eurocode 8 (EN 1998-1-1,7:2020) now also includes 
criteria  on  inelastic  deformation  and  strength  criteria  for  reinforced  concrete,  steel,  timber  and  masonry 
construction.  The  increasing  use  of  nonlinear  FEM  in  non-seismic  design  situation  will  likely  require  similar 
developments. 

Next to the mechanical and statistical modelling of post yield behaviour on the material, member and structural 
level there is the question of the reliability level required. The consequences of ductile (member) failure are 
often less serious than for brittle failure, partly because of the load distribution effects and partly because of 
the warning effect. As a result it may make sense to uses a higher consequence class in case of brittle behaviour. 
According  to  EN  1990:2023  this  leads  to  increased  levels  of  the  partial  load  factors.  It  may  however  be 
considered as more convenient to keep the load factors constant and have an increased partial factor on the 
resistance side. For the moment this is up to the National Annex. 

Example from Eurocode 8 

For the sake of illustration, the general form of these models is described with reference to frame members, 
exemplified in Figure 3.25 with a cantilever scheme,  where the chosen deformation parameter is the chord 
rotation 𝜃 (see also 3.5.7). The cantilever scheme is also the prevalent one in the experimental campaign that 
has produced over the years the data supporting the models now included in the code, by now amounting to 
hundreds of inelastic cyclic tests. 

Figure 3.25. Deformed shape, top displacement and chord rotation for a cantilever column: a) at yield, b) at ultimate.  

As shown inFigure 3.25a, until yield, the response is elastic, curvature and moment are proportional and linear 
(this is an approximation for concrete members where stiffness changes in the portion where the moment is 
larger than the cracking one). After yield, due to  hardening, the  moment can  still increase until a maximum 
value  and  an  area  undergoes  plastic  deformations:  this  is  the  plastic  hinge  of  length  lp.  Deformations  are 
concentrated in this area and the integral of curvature of the plastic hinge length can be considered a rotation 
around the hinge centroid, that causes a plastic increase of the top displacement δp. The simplest mechanical 
models of the chord rotation at yield and ultimate are thus: 

(3.39) 

θy =

δy
LV

=

ϕy LV
3

72 

 
 
 
 
 
θu=

δu
LV

=

δy+δp
LV

=θy+θu

pl=θy+

1
LV

lp(ϕu-ϕy) (LV-

lp
2

) 

(3.40) 

where the yield and ultimate curvature are obtained from a moment-curvature section analysis, (or predictive 
analytical  equations  for  the  most  common  cross  section  shapes  like  rectangular  and  circular),  and  
pl = θu-θy is the plastic part of the chord rotation  
θu

Even this so-called mechanical model, however, has an element of empiricism, in the model for the plastic hinge 
length. It is included in corrected form in the Eurocode 8 to cover those elements whose section shape is not 
common.  For  typical  elements  in  frames,  with  rectangular  sections  in  RC  buildings  or  wide  flange/I-beam 
sections in structural steel, empirical formulas are preferred, due to their better agreement with experimental 
results. 

Specific models for the yield and plastic rotations included in Eurocode 8 are based on research by (Biskinis and 
Farids 2010a, b) for new and existing RC members (evolution of those in EN 1998-3:2005, due to Panagiotakis 
and Fardis, 2001), by Grammatikou et al (2018a, b) for retrofitted members, by Lignos and Krawinkler (2010, 
2013) for steel members. 

All these models are developed to be unbiased, i.e. have a median of the ratio of experimental to  predicted 
rotation equal to one, and report the coefficient of variation of this ratio, which describes the model uncertainty 
(something between 20% and 40%, indicatively). Examples are shown in Figure 3.26. In particular, Figure 3.26a 
and  b  refer  to  the  ultimate  rotation  of  concrete  members  (Grammatikou  et  al,  2018a),  and  two  different 
statistical techniques to fit the same model and leading to markedly different coefficients of variation (42% 
versus 18%). Figure 3.26c refers to the plastic rotation of steel beams (Lignos & Krawinkler, 2011). 

Figure 3.26. Comparison of predicted and experimental values of ultimate chord rotations for: a) and b) concrete, c) steel 
beams.  

Source: Grammatikou et al. 2018b and Lignos and Krawinkler 2011  

Final note: the models and rules  presented here may differ from the solutions that have been developed in 
EN1998 for the seismic design situation. However, in both cases the intention is to reach a certain minimum 
safety level. For more discussion see also chapter 3.8. 

3.8  Nonlinear analysis / FEM 

3.8.1  Types of nonlinearity 

The majority of design calculations and partial factor calibration is performed on the basis of Linear structural 
behaviour that is defined by a linear relation between strains and displacements, a formulation of equilibrium 
which is independent of deformations and a linear dependency of stresses and strains. 

Within EN 1990:2023 the following types of nonlinearity are defined 

73 

 
 
 
 
 
geometric non-linearity, caused by changes in geometry, in most cases sufficiently accurate approximated 
by using the second order theory; 

material non-linearity, caused by non-linear stress-strain relationship in the material; 

contact non-linearity,  caused  by  non-linear behaviour at  the  contact  boundary  between  structural  parts 
during introduction of actions. 

Additionally, we may have non-linearity of the limit state function if the local resistance is a non-linear function 
of the various influencing variables. This, of course, is highly relevant for a proper application of reliability theory 
(in particular for simplified methods like FORM and PFM) and is dominantly present in the literature on reliability 
theory. In this chapter, however, we will concentrate on the classical types of structural non-linearities as defined 
in EN 1990:2023. 

Note: Additional to the general statements in EN 1990:2023, chapter, 8 a special Annex to EN1990:2023 for 
this topic is under preparation;  unfortunately this annex  was not ready  when writing  the background in this 
report. 

3.8.2  Use of the non-linear finite element method (NL-FEM) in assessment and design: 

Two different types of procedures are to be distinguished: 

(a)  Conventional member by member checking, calculating load effects by applying linear or nonlinear FEM 
to be used as input to local resistance models. 

(b)  Checking the overall failure of the whole structural system (non-linear resistance) using nonlinear FEM. 

The linear version of (a) is already a standard for a long time. When using the non-linear option, the load effects 
are not only depending on elastic properties, but also on other material properties.  

In both cases (a) and (b) the consideration of non-linearities (in particular second order effects) are mandatory 
if they are unfavourable and relevant for the overall failure. 

Notes: 

Some computer programs generate automatically the actions from self-weight. The user has to take care 
himself that partial factors for favourable and unfavourable acting self- weight are introduced correctly 
Note that these automatic generations of the self-weight can also influence the load path. 

As superposition is not valid in nonlinear analysis, the start of the analysis is the formulation of a relevant 
load combination including a (time dependent) specific load arrangement. As such it is to be recommended 
to start with a classical elastic analysis to find out which load combinations and arrangements are relevant 
to be considered in the non-linear case. 

FEM/NM on a small local scale (part of a wall or beam) can also be used to derive equivalent averaged 
material parameters for use in the overall analysis.  

In non-linear limit state functions the theoretical values of the partial factors depend on the degree of non-
linearity; for instance: for a steel column the coefficient of variation (and thus the partial factor) of the load 
bearing capacity may depend on the slenderness ratio. If nevertheless the same partial factor is used over 
the total range of column types it should be checked that deviations from the target reliability are within 
the accepted limits (see EN 1990:2023, Annex C). 

3.8.3  Application of the partial factors on the action side  

As discussed in section  3.2,  EN  1990;2023 offers in 8.3.2.2 three options to calculate the load effects in a 
structure. In general (see section 3.2)  

Ed = 𝛾Sd E {Σ (𝛾f 𝜓 Fk); 𝑎d; X Rd}      

EN1990:2023 (8.3)  
(3.41a) 

To simplify matters, Sd and f can be combined in one factor F 

74 

 
 
 
Ed = E {Σ (𝛾F 𝜓 Fk); 𝑎d; X Rd}  

EN1990:2023 (8.4) 
(3.41b) 

When the problem is linear the partial factor also can be applied on the effects of actions:  

Ed = 𝛾E E {Σ ( 𝜓 Fk); 𝑎d; X Rd  

EN1990:2023 (8.5)  
(3.41c) 

The first one is theoretically to be preferred, but from a practical point of view the other two are used mostly 
in practice. In most FEM software only the option (3.41b) is operational, which is usually on the safe side. 

In  equation  (3.41c)  the  design  values  of  the  action  effect  will  be  based  on  the  highest  partial  factor  (For 
example, a  factor  1,5  for  self-weight)..  This  may  lead  to  a  very  uneconomic  result.  In  the  linear  case  when 
superposition is allowed it may be better to use eq. (3.41d).  

Ed = Σ {𝛾E,i Ei{( 𝜓 Fk,i); 𝑎d; X Rd}} 

(3.41d) 

3.8.4  Application of the partial factors on the resistance side 

For the resistance side we may choose as starting points the EN 1990:2023 formula (8.18) or the simplified 
formulas (8.19) and (8.20).  

When applying factors to material properties, the design value of resistance should be calculated from:  

Rd= R ( 

Xrep
M

; ad ;  FEd)   

EN 1990:2023 (8.19) 

When applying partial factors to the resistance, its design value should be calculated from:  

Rd= 

R (Xrep; ad ;  ∑ FEd  )
R

EN 1990:2023 (8.20) 

Formula (8.19) is valid if the model uncertainty factor is close to 1.0 and (8.20) if the material factor is close 
to 1.0. Both expressions are correct if the relation is linear. When choosing an option in the nonlinear case, one 
should consider both reliability as well as economic aspects. Material  related Eurocode have made their own 
choices (see EN 1990:2023  eq 8.3.5.1).  Note that  some  types of non-linearity may  work out  differently for 
(8.19) and (8,20).  

Notes:  

For application of the formulas for  Rd in NL-FEM on the system level, the resistance parameter R is usually 
defined as an overall load amplifier that in the incremental analysis runs from zero up to the value where the 
structure  is  no  longer  stable.  The  design  value  of  the  resistance  then  corresponds  to  the  value  of  the  load 
amplifier where  instability  occurs.  Of  course  one  may  also  stop  the  analysis earlier as soon  as  a  sufficient 

75 

 
 
 
 
 
 
 
 
 
 
 
 
degree  of  reliability  has  been  proven.  This  way,  however,  one  does  not  get  information  on  how  much  the 
structure is overdesigned. For more information see sections 3.8.4.1 and 3.8.4.2. 

The factor R includes the effects of uncertainty in all materials, local resistance models and overall system 
behaviour (see also the text on the adjustment of model uncertainty under FEM calculation further below). 

In both equations 8.19 and 8.20 the material factor m and the model uncertainty resistance factor Rd have to 
be combined into one factor, M or R which of course may lead to some inaccuracies, for instance in the case 
of nonlinear resistance models. Part of this type of inaccuracy, however, also is present outside the application 
of non-linear FEM. In cases of multiple failure modes maybe conservative approximations have to be made. 
(see EN 1990:2023, eq. 8.21). 

3.8.4.1  Application of EN1990:2023 (8.19) 

The input to the FEM analysis is given by:  

design values of the loads Fd;  

design values of the material properties: Xd=Xrep/ M;  

geometrical properties ad. 

After (in steps) applying the loads Fd and assuming the structure is still stable there are members in the linear 
state and members in the non-linear (or yielded) state. In this approach, in principle, the local load effects in all 
members can be checked for all standard local limit state requirements according to the relevant Eurocodes in 
the same way as for linear FEM. However, if all necessary checks as well as the corresponding post limit state 
behaviour have been implemented correctly in the non-linear FEM this is not necessary as it is already carried 
out implicitly (actually we have arrived at case b). 

It may occur that different limit states (local yield or rupture conditions) require different M values for the same 
basic variable X. Most FEM-software offer no option to deal automatically with that situation. One may of course 
make a conservative choice by taking the highest value. Otherwise, one has to make several runs and suppress 
where relevant (artificially) the limit state requiring the low M value. 

Note: See also the section on avoidance of brittle behaviour. 

3.8.4.2  Application of EN1990:2023 (8.20) 

The input to the FEM analysis is given by:  

design values of the loads Fd;  

representative / characteristic values of the material properties: Xrep; 

geometrical properties anom (or ad where appropriate). 

FEM programs do not always allow to introduce the partial factor R on the level of local (member) resistance. 
A possible solution is to transfer the R to the loading side and raise the loads up to R Fd: We may formulate 
this as:  

Rd= R (Xd ; ad ;  R FEd)   

EN EN 1990:2023 (8.20a) 

Actually, this comes down to a kind of load factor design). In case of material softening or geometrical stability 
problems it is conservative (but possibly very uneconomic). Next all members can be checked in the conventional 
way, leaving out of course R. If the FEM contains all required limit state checks (and corresponding post line 
state behaviour) there is no need for further checking. Note finally that using (8.20a) in an incremental non-
linear analysis the total loads R Fed should be applied proportionally in steps. 

Remarks for the application of (8.20): 

Depending on the type of resistance criteria favourable acting permanent actions should be multiplied by 
a factor 1.0 instead of R (see also Annexes B.6.1 and B.6.2). 

If several materials are present, one may choose one material as dominant (or decisive) and adjust the 
representative values of others (like in (8.21)). Note that the choice may have influence on the final result. 

76 

 
 
 
If in (8.20) mean values are used as representative values for the material properties the above procedures 
should be adjusted accordingly. 

In  general  non-linear  analysis  is  more  complicated  than  linear  analysis  because  of  the  fact  that  the 
superposition rule is not valid anymore. One has to concentrate on the most critical load combinations. It 
may be helpful to start from a survey in the elastic domain.      

For further discussion and background, see the FEM-shear wall example in Annex B.5 

3.8.5  Checks on deformations and deformation capacity  

Next to (or as part of) the load bearing capacity assessment of the structure there is a need to consider the 
related deformations. This may be part of the checking of service-ability limit states with respect to deflection 
criteria. However, (see section 3.7) it is also relevant for the checking of local and global ductility requirements. 
In incremental FEM-analysis where proper models for post-elastic behaviour of both members and connections 
are  incorporated  (as  done  for  instance  in  chapter  7  of  EN  1998-1-1)  this  may  occur  partly  implicitly.  If 
calculations are based on the upper or lower bound theorems of plasticity theory no stepwise increase of the 
loads is performed and a separate check that the deformations and deformation capacities are sufficient maybe 
necessary. Note finally that a force controlled incremental FE analysis can only detect the ultimate capacity of 
the structure but can’t detect the degree of post critical ductility in the global behavior. 

3.8.6  Reflection on the choice of upper and lower design values of structural properties 

In addition to the direct strength parameters f, also values for other structural parameters have to be chosen, 
like the modulus of elasticity, hardening or softening models, strain at rupture, imperfections, bond, fracture 
energy, etc. A usually conservative approximation is to use standard upper and lower design values according 
to C4.4.2 in EN 1990:2023 Annex C. More refined methods are possible but at the cost  of more calculation 
work. It is for instance pessimistic and not always necessary to take the lowest design values for the (ductile) 
resistances of all members. For less dominant members or parameters the use of say characteristic values (5% 
fractiles) could be safe enough. This depends of course on the degree of correlation between members, which 
however usually is only scarcely known. 

One should also be aware that in general the use of lower design values for local resistance parameters is not 
always on the safe side. In some cases, a high value of a local resistance parameter may be unfavourable. A 
well-known example is that brittle (or strongly softening) mechanisms like shear or connection failure may be 
overlooked due to ductile yielding in nearby cross sections or members. To detect these brittle mechanisms on 
member level one should perform simulations using high (design) values for resistance parameters in ductile 
mechanisms.  Another  option  is  to  prevent  these  mechanisms  by  applying  capacity  checks  as  in  seismic 
engineering. Missing these mechanisms may have far stretching consequence for redistribution of loads and 
mobilizing  reserve  capacity  under  dynamic  loadings.  This  issue  is  in  particular  necessary  to  consider  when 
capacity design, seismic design or even full plastic design is done. 

3.8.7  Adjustment  of  model  uncertainty  and  corresponding  partial  factors  under  FEM 
calculation 

When using FEM, it might be that the model uncertainties related to the FEM model are not consistent with the 
model uncertainties for classical design using design equations in the material Eurocodes (which may be biased 
for  several  reasons).  In  those  cases,  it  is  possible  to  check  the  FEM-calculation  model  directly  with  a 
representative set of experiments and add model uncertainties using EN 1990:2023 Annex D. This may give 
rise to the need for additional general or operational rules in chapter 8 of EN 1990:2023 of the following type: 

-Design for non-linear problems assisted by numerical models 

-Design for non-linear problems assisted by use of computational methods shall achieve the level of reliability 
required  for the  relevant  design  situation  taking  into  account  the  model  uncertainty  of  the  computational 
method.  

-Adjustment of material partial factors to account for the model uncertainty related to the numerical model. 

Note: Further information about method validation and model uncertainty is given in Annex D following the 
same principles as for design assisted by experimental testing.  

In this respect the status of Annex D in a specific CEN member state may be important. 

77 

 
Note: a revision of Annex D to include design assisted by numerical methods is under preparation in SC10/WG3.  

3.8.8  Requirements for the quality of nonlinear numerical methods used 

According  to  EN  1990:2023,  clause  7.2.2(4)  non-linear  numerical  models  should  be  validated  (or  rather 
justified), to establish whether the numerical model correctly reproduces the necessary physical phenomena. It 
means that the numerical models used should be compared to numerical, experimental or benchmark results 
representing the main failure modes of the elements. These covers: 

General aspects: 

Checks on the correctness of the material modelling 

Tests in order to check the characterization of relevant structural failure modes 

Note: Comparing the result of a model of a structure with an experiment may also be misleading because the 
sources of differences cannot always be determined and errors from various sources can cancel each other. 

Verification for the specific project: 

Mesh sensitivity 

Sensitivity to action input values 

Adequacy and type of the solution method. 

The loading process adopted by non-linear simulation in order to evaluate the structural resistance shall be 
defined according to engineering judgement consistently with the considered combination of actions at ultimate 
limit  state.  The  loading  process  should  where  relevant  account  for  the  loading  sequence  due  to  staged 
construction methods. 

Note: If methods go beyond the calculation models described in the Eurocodes, e.g. a more advanced nonlinear 
numerical model, then the partial factors cannot be used with calibration as described above.  It is up to the 
National Annex to set up further guidance in order to achieve the required reliability level. 

3.9  Calibration  in case of degradation 

3.9.1  Fatigue using SN-curves  

3.9.1.1  Limit state formulation  

Fatigue is mechanism where cyclic actions or action inducing repeated stress cycles cause effects like crack 
growth that can lead to failure. The actual failure can be caused by a relative large cycle or by stresses casued 
by another type of action. In some cases dynamics can play an important role. Given the nature of fatigue, not 
only the intensity of the action cycles but also the number of stress cycles is of great importance.  

A common approach to deal with the fatigue mechanism is the so called  S-N-curve-method. This mehtod is 
based on experimental S-N-curves for constant amplitude loading, relating the (constant) stress range S = Δ.  
to the number of stress cycles to failure N. Above the fatigue limit (if present) and in its most simple form the 
failure criterion is given by 

N Δ -m = K 

(3.42) 

The parameters K and m are resistance parameters depending on material specific aspects and local structural 
geometry. Note that for practical applications in steel and aluminium structures a two branch version for the S-
N-model is used having parameters K1, K2, m1 and m2. More details are discussed in Annex B.6 of this document. 
For other applications the model can also depend on the (cycle) mean stress level (as for instance in concrete, 
timber and FRP). 

To account for variable amplitude (VA) loading, the Palmgren-Miner damage accumulation rule (hereafter called 
“Miner sum”) is often used which leads to the following limit state function for a given period of time t (design 
service life or time of inspection): 

78 

 
 
g(t) = Dcr – Dn 

(3.43) 

where Dcr is the Miner sum at failure (with expectation often assumed to be close to 1,0) and Dn is the damage 
based on equation due to n applied load cycles present in the period t. 

n
Dn= ∑
i=1

1
Ni

(3.44) 

with Ni following from equation (3.42) for an applied stress range Δi. To evaluate Dn we thus need the stress 
ranges Δi, usually presented either as a stress range histogram or a spectrum and a total number of cycles n 
for the design life time (or other period) under consideration. The histogram or spectrum may be derived on the 
basis of load effect simulations in combination with rain flow counting or reservoir counting methods. In some 
cases  the  load  model  presents  an  equivalent  constant  amplitude  value.  In  general  the  model  for  the 
determination of the load effect Δ contains several (partly uncertain) global and local model factors. 

Due to the fact that according to Miner’s rule (although not really true in reality) the order of the loads is not 
relevant, we may evaluate (3.44) by an ordering in magnitude. This means that we can divide the spectrum into 
a number of stress intervals, calculate the damage Di for each stress interval and multiply with the number of 
cycles in the interval; Dn follows by adding the contributions of all stress intervals. It is clear that the model 
uncertainty in the fatigue model should be accounted for where in fact the standard rules from EN 1990:2023 
Annex D apply. 

3.9.1.2  Target reliability  

For structures in reliability class RC 2, the European standard EN 1990:2023 recommends a target reliability 
index for the design life time. It ranges from 1,5 to 3,8, depending on the degree of inspectability, repairability 
and damage tolerance.  

Note that one should be careful here: the correct way to calculate the failure probability for the life time is by 
explicitly including the effect of inspection. This will lead to an increase of the reliability index and the target 
should still be 3,8. The reduction of the target to 1,5 is intended to compensate for the case where inspection 
is carried out in reality but not included in the calculation. It is in fact a strong simplification. One should never 
combine the two approaches: that is incorporating the positive effect of inspection on the calculated reliability 
level and taking the reduced 1,5 value as a reliability requirement. Actually there is even a need for a better 
specification of the inspection requirements and corresponding repair that would justify this reduction.  

3.9.1.3  Probabilistic verification 

The failure probability for a total period of time from 0 to t is given by: 

Pf (t) = P(g(t)<0) 

(3.45a) 

where  g(t) is  given by (3.58). The stress distribution or spectrum in the expression for  g(t) follows from the 
application and the structural properties. One might also calculate the  last year of the design service life or, 
where relevant, the intended inspection or repair period (see also section 2.4). For the annual evaluation holds: 

P(failure in year t) = P(g(t)<0)  -  P(g(t-1)<0) 

(3.45b) 

(or, alternatively, the conditional failure rate, see chapter 2). This probability should be less than the annual 
target, which according to Table C.2 in EN 1990:2023 for reliability class 2 corresponds to  = 4,7. This value 

79 

 
 
 
 
 
 
however, is an upperbound and a lower value may be more appropriate because of the correlations in resistance 
and model uncertainties in time. 

Note:  the  approach  using  an  annual  reliability  target  is  considered  to  be  more  suitable  for  the  fracture 
mechanics approach. 

3.9.1.4  Semi-probabilistic approach   

According to EN 1990:2023 formula (8.11) the design value of the spectral stress range level Δσi,d should be 
derived from: 

Ffat,d = 𝛾Ff Ffat 

(3.46) 

where Ffat is the (nominal) value of relevant fatigue action and  𝛾Ff is a partial factor for fatigue actions and 
covers  the  model  uncertainties  in  the  actions  and  the  stress  calculation.  The  precise  meaning  of  Ffat 
(characteristic, mean or maximum loads/stresses) may be material specific. The same holds for the choice of 
the verification is in the elastic or plastic region. The value of 𝛾Ff depends on the models for fatigue actions and 
on the models for the effects of the fatigue actions. The recommended value of 𝛾Ff is 1.0 when the models for 
fatigue actions are safe sided such, that the uncertainties of the action effects, which are approximately equal 
to 10%, are covered. Otherwise, the recommended value is 1,1. Models for fatigue loads can be found in EN 
1991 for e.g. wind (prEN 1991-1-4, Annexes F and H), traffic and waves (prEN 1991-1-8). These models offer 
(or should offer) the basis for the fatigue calculation for all materials. However, the further elaboration from 
the external loads to the relevant internal load effect that is suitable for a fatigue damage are highly material 
specific.  As  a  result  also  the  systematic  deviations,  the  sources  and  degrees  of  uncertainty  and  thus  the 
appropriate  partial  factor  𝛾Ff  depend  both  the  type  of  loading,  the  type  of  structure  and  the  material. 
Uncertainties can be related to the simplified set of vehicles, the accuracy of the influence lines, the local stress 
model, the dynamic effects and in some cases the presence of the permanent load. Note that also the inspection 
regime and the consequences of the fatigue damage (e.g. damage tolerant) are relevant for the value of partial 
factors. 

Accordingly, in EN 1992 and EN 1993 factors (referred to as λ factors at the moment) have been introduced 
for a number of load models in order to avoid damage accumulation. . Presently in the SC3-WG9, SC3-WG13 
and  AHG  Fatigue, attempts  have  started  to  work  on  more  realistic  traffic  assumptions  considering  also  the 
influence  of  uncertainties.  For  the  use  of  FRP  in  wind  turbine  blades  technical  specifications  are  under 
development. It is important that similar approaches are followed for other materials as well as for wind and 
wave actions. 

On  the  resistance  side  the  (characteristic)  S-N  curves  are  usually  based  on  the  95%  survival  probability 
(including  statistical  uncertainty  in  accordance  with  EN  1990:2023-Annex  D).  The  design  values  for  the 
resistance  (to  be  defined  in  the  same  dimensions  as  the  fatigue  load  F)  may  be  found  according  to  EN 
1990:2023 (8.22): 

Rdfat,d = Rfat / 𝛾Mf 

(3.47) 

where Rfat is the representative fatigue resistance and 𝛾Mf  the partial factor for fatigue resistance accounting 
for  the  consequence  of  fatigue  failure  and  the  application  of    inspections  (and  ease  of  repair)  of  fatigue-
sensitive members. 

It should be noted, that there are certain preconditions for the fatigue verification which are hidden in additional 
serviceability limit state requirements. Examples are that the steel structure should stay in the elastic range for 
the frequent load combination or that for the headed studs in EN 1994 there is a restriction on the upper loading 
for the characteristic load combination  

Fatigue can be caused by more than one fluctuating action. In that case the distribution (spectrum) should be 
based on the combined history. When the mean stress level needs to be considered in the damage rule, the 
mean values may be based on EN 1990:2023 (8.17): 

80 

 
 
 
ΣFd=Σ Gk,i+Σ 𝜓2,j Qk,j + (Pk) + Ffat,d 

(3.48) 

Variable actions of the same type as the fatigue action should not be considered in the part Σψ2,j Qkj. 

This applies if the absolute value of the load effect or if the mean load effect is of importance for the fatigue 
verification. Depending on the type of sensitivity of the model with respect to the mean value of the stress 
cycle it may be necessary to consider the mean value of fluctuation load per cycle and not the life time quasi 
static value. 

3.9.1.5  Reflection 

Fatigue in welded steel or aluminium structures or reinforcement is a process where initial defects at the micro 
level develop into visible cracks that may reduce the load bearing capacity of the joint. During a moment or 
period of heavy loading the weakened structure may come to yielding and/or to instable fracture. In that sense, 
splitting up the reliability verification into a standard ULS verification and a fatigue verification using the SN-
approach  is  a  simplification  that  might  be  on  the  unsafe  side.  On  the  other  hand,  it  is  believed  that  the 
parameters in the SN-model based on small specimen are on the safe side. Since the majority of the life is in 
the initiation and growth of small cracks, the crack size at final fracture is relatively unimportant for the total 
life, provided that the material satisfies the toughness rules of EN 1993-1-10. This situation becomes different 
when evaluating fatigue inspections. 

The  probabilistic  S-N  curve  as  given  in  Eq.  3.42)  is  a  simplified  approximation.  More  accurate  but  more 
complicated descriptions are given in literature, e.g. (D.Leonetti et al. 2017, 2020). Using such descriptions often 
results into lower partial factors (Maljaars, 2022). The partial factors in EN 1993-1-9 are based on these more 
accurate and complicated descriptions. 

Models  that  describe  the  process  of  crack  growths  and  subsequent  fracture  are  available  and  should  be 
preferred from a reliability point of view, in case fatigue inspections apply. A problem in practical application is 
that keeping track of all local geometrical details and spatial variability is an almost impossible exercise.  

See Annex B for further information on crack growth models and examples, also for other materials. 

3.9.2  Durability Verification Concrete 

3.9.2.1  Limit state formulation and target reliability 

Under  normal  conditions  concrete  protects  embedded  reinforcing  steel  against  corrosion.  However, 
environmental actions such as carbonation or penetration of chloride ions disrupt the protective properties of 
the passive layer and may lead, over time, to corrosion of reinforcing steel. This stage until corrosion is active 
is called the initiation period.  

Figure 3.27. Schematic presentation of degradation with time.  

sls 

uls 

After  the  start  of  corrosion  (the  so-called  propagation  period)  its  effects  on  a  RC  structure  usually  include 
cracking  of  the  concrete  cover,  reduction  and  eventually  loss  of  bond  between  concrete  and  corroding 

81 

 
 
 
reinforcement  and  finally  reduction  of  cross-sectional  area  of  reinforcing  steel.  So,  it  may  lead  to  the 
exceedance of SLS (loss of appearance) or in combination with mechanical actions to exceedance of the ULS.  

As identifying the onset of corrosion is quite difficult, because very early corrosion does not induce any external 
sign and the concrete properties are not affected, proxy limit states or condition limit states (CLS) (see also ISO 
2394, fib-bulletin 34 (2006), fib-model-code 2010, Brite EuRam III, 2000) are introduced like for instance the 
“durability/deterioration  limit  state”  (DLS)  which  considers  the  corrosion  penetration  attack  until  50  μm  for 
homogeneous corrosion and 500 µm for pit corrosion as not impairing the structural reliability assumed in the 
design. Note that if a condition limit state verification is used, it does not replace the more complex serviceability 
of ultimate limit state verification, the targets should be chosen accordingly.  

Having  this  in  mind  the  following  typical  limit  states  associated  with  corrosion  of  reinforcement  may  be 
distinguished: 

  De-passivation and onset of corrosion; 

  Development of 50 µm-homogeneous attack or 500 µm pit depth which may result into a micro crack 
(say < 0,1 mm) parallel to the bar in concrete surface (referred to as Durability or Deterioration Limit 
State); 

  Crack associated with appearance, say 0,3 mm in width (Serviceability Limit State);  

 

Loss of bond; 

  Actual loss of (local) load bearing capacity (Ultimate Limit State). 

The  target  reliability  required  for  each  of  these  limit  states  in  general  will  be  stricter  when  going  from 
depassivation to the ultimate limit state, roughly climbing from for instance  = 1,3 for de-passivation and 1,5 
for the DLS to 3,8 for the ULS (design working life values). 

3.9.2.2 

Initiation and propagation models  

Corrosion due to carbonation may start when the depth of the carbonation front is larger than the cover depth. 
So the LSF for the start of corrosion may be defined as: 

g(t) = c- xc(t) 

(3.49) 

where: 

c 

is the concrete cover [mm];  

xc(t)  is the carbonation depth [mm] at the time t;  

t  

is the period of time [year] under consideration.  

The  value  of  xc  depends  on  a  diffusion  process  of  CO2  and  depends  on  many  geometrical  and  material 
parameters as well as the outside environment (fib, JCSS, EN 1992). Note that given (3.47) we may calculate 
the probability of depassivation for every point in time. Taking t=td (where td is the design working life) we may 
find the failure probability P{g(td)<0} for passing this limit state for the design service life.  

The limit state function for the initiation of corrosion due to penetration of chloride ions can be formulated as: 

g(t) = Ccrit – C(t) 

(3.50) 

where Ccrit is the critical chloride content with respect to the binder to achieve depassivation of the reinforcement 
and C(c,t) is the chloride content at the level of the reinforcement and time t.  

The propagation model is based on onset, rate and accumulation of damage. Propagation models include the 
effects of corrosion on structural behaviour, like cracking, spalling and loss of mechanical strength. Important 
is the spatial distribution of cross-sectional loss. This is crucial for modelling pitting corrosion. In general, the 
limit state function at time t can be expressed as (fib-model-code 2010, Val et al 2019): 

82 

 
 
 
g(t) = pcrit – Vcorr (t – t dep)   for  t > tdep 

(3.51) 

where  pcrit is the  critical corrosion penetration for the limit state under consideration (e.g. spalling or loss of 
bending strength, Vcorr is the speed of corrosion, t is time and tdep is the starting time of the depassivation state. 
Note that P{g(td ) < 0} is the probability of reaching the limit state within the design service life. 

All models include quite a number of parameters (concrete quality, cover exposure conditions, etc.),  

3.9.2.3  Reliability verification for the ULS 

For the ULS of a beam under bending the Limit state function may be given by: 

gULS (t)=θMRMR

(t)-θMEME 

(3.52) 

where θMR is the model uncertainty of the resisting bending moment; MR(t) is the resisting bending moment at 
time t; θMR is the model uncertainty of the load-induced bending moment and ME is the bending moment induced 
by the applied loads: 

ME=0,08 (g+q)L2   

(3.53) 

For simplicity we take conservatively the load effect M as the maximum value in the life time. In case of possible 
degradation  on  the  desistance  side  the  resisting  bending  moment  MR(t)  includes  the  reduction  in  the 
reinforcement area:  

MR (t) = n As(t)  fy  heff 

(3.54) 

where As (t) = ( /4) d(t) 2 with d(t) the time dependent diameter of the reinforcement bars at time t given by 
(assuming uniform corrosion, no pitting): 

d(t) = d0 - 2d(t) =  d0 – 2 (t-tint) Vcorr 

(3.55) 

where d0 is the initial diameter and d(t) is the corrosion loss, Vcorr is the time average corrosion rate; tinit is the 
time to initiation, which follows from g (tinit )= 0 but of course keeping d(t) > 0. A further numerical elaboration 
can be found in Annex B. 

In present practice such an analysis is usually not carried out, not even in a semi-probabilistic approach. The 
above  combined  analysis  of  chemical  and  mechanical  hazards  is  normally  split  up  into  two  separate 
verifications:  one  for  the  uncorroded  structure  and  one  for  the  corrosion  initiation.  The  verification  of  the 
uncorroded structure is based on beta=3,8 for 50 years or some equivalent annual target, For the corrosion 
initiation target the target is not specified (deemed to satisfy rules are used) or taken quite loosely as  =1,3 
or 1,5 for 50 years. 

In cases having an aggressive environment and little possibilities for inspection this might be quite dangerous. 
Note also that in addition to the direct effect of rebar reduction we may have to deal with loss of steel ductility, 
loss of bond and reduction in concrete cross section. In those cases, in design as well as assessment, extra 
attention compared to standard procedures may be justified.    

83 

 
 
 
 
 
 
3.10  Robustness 

3.10.1  Introduction 

Robustness  is  defined  as  the  ability  of  a  structure  to  withstand  unforeseen  adverse  events  without  being 
damaged to an extent disproportionate to the original (unforeseen) cause. The term unforeseen refers to actions 
and influences not defined in quantitative terms. The basic requirement for robustness is formulated in clause 
(4.4) of EN 1990:2023 and some guidance on design strategies and design methods is given in EN 1990:2023 
Annex E (informative). Further elaborations can be found in EN 1991-1-7 and the material related Eurocode’s. 
These elaborations however still give rise to much discussions due to different opinions on a general design 
philosophy.  This  section  in  combination  with  Chapter  5  is  intended  to  give  some  further  guidance  to  the 
developments in this field. 

Design for robustness should be an integral part of the conceptual design phase of the structure, taking into 
account different strategies that can be applied. In general, the design of a structure for robustness can involve 
the – explicit or implicit – identification of: 

  Hazards H; 
 
 
  Direct and indirect consequences. 

Local (direct) damage D; 
Systemic damage S (follow-up/indirect damage), encompassing progressive collapse; 

In  accordance  with  EN  1990:2023,  robustness  is  primarily  intended  to  be  used  with  respect  to  unforeseen 
events, actions and influences (unspecified or really unknown hazard) but may also prove to be an adequate 
design  strategy  for  actions  which  the  code  does  define  in  quantitative  terms  (usually  termed  ”identified 
actions”). 

According to EN 1990:2023 the occurrence of (gross) human errors should be counteracted by adequate Quality 
Assurance procedures as specified in Annex B. Nevertheless, still human errors can occur and a proper degree 
of robustness is then helpful in mitigating the consequences. At the moment, however, no numerical operational 
methods are available.  

3.10.2  Design strategies to prevent unacceptable damage 

It is appropriate to underline that unacceptable damage can be prevented by adopting design strategies which 
do not necessarily increase the robustness of the system. This can be clearly be concluded by analysing how 
(structural) risk can be determined. 

In a risk-based context, the expected value of risk can be determined based on the following equation according 
to Annex B in EN 1991-1-7 (see also chapter 5): 

Risk = Σ Σ Σ CijkP[Sk|Dj∩Hi]P[Dj|Hi]P[Hi] 

(3.56) 

where: 

P[Hi]  

represents the probability of occurrence of hazard Hi; 

P[Dj|Hi]  

represents the probability of initial damage Dj conditional on hazard Hi; 

P[Sk |Dj∩Hi]   represents the probability of follow-up damage Sk conditional on damage Dj and hazard Hi; 

C  

represents the consequences. 

Reducing any of the four factors can be considered as a design strategy to control risk levels. 

The design strategies preventing unacceptable damage have been largely investigated in literature (Ellingwood 
&  Leyendecker  1978,  Gross  &  McGuire  1983,  Dusenberry  &  Juneja  2002)  and  they  can  be  categorised  as 
follows: 

Design  strategies  which  prevent  the  occurrence  of  abnormal  and  dangerous  events  (i.e.  influencing  P[Hi]  in 
equation 3.56). This strategy aims at reducing the exposure of the structure to hazards, by reducing the 
probability  of  occurrence  and/or  the  intensity  of  abnormal  and  hazardous  events.  Examples  of  such 
strategies include the use of safety  barriers against impact or base isolators to limit the intensity of a 

84 

 
 
seismic  action..  Event  control  strategies  commonly  do  not  lead  to  a  change  in  structural  robustness,  in 
particular in cases when the structure’s tolerance to damage (or progression thereof) is uncorrelated with 
the event control strategies used, e.g. non-structural measures such as impact barriers  

Design strategies which prevent the occurrence of an initial damage (i.e. influencing P[Dj|Hi] in equation 3.56. In 
general, this strategy aims at identifying and designing key members by improving the member behaviour 
through sufficient resistance and energy dissipation capacity. It is to be noted, however, the key member 
design approach does not necessary enhance the robustness of the system, unless the member changes 
are  correlated  with  the  system  robustness  behaviour.  In  addition,  problems  may  arise  if  unidentified 
adverse events or actions occur, which are not covered by the design of key members, neither against the 
identified, variable or accidental, nor the notional actions. Further weak points of the  strategy lie in the 
process used to identify the key members themselves and the reference to a tolerable damage limit. When 
key members are being "identified as those whose absence would result in damage that is greater than a 
tolerable  limit"  [EN  1990:2023  E.4(4)],  individual  member  removal  seems  to  be  assumed,  with  no 
consideration of scenarios with more than one member being damaged under an extreme event. In addition, 
the  term  “tolerable  limit”  is  vague  and  does  not  imply  guidance  for  the  practicing  engineer  [CEN/TC 
250/WG6.T2]. 

Design strategies which prevent unacceptable spreading of damage (i.e. influencing P[Sk |Dj ∩ Hi] in equation 
3.56).  This  strategy  is  aimed  at  the  global  system  behaviour  so  that  the  spread  of  the  initial  damage 
remains limited. Two alternatives are typically considered: 

The  alternative  load  path  approach  which  redistributes  the  forces  originally  carried  by  the  failed 
members. However, if the initial damage produces an overloading in the remaining structure, the 
alternative load path procedure may promote progressive collapse and reduce the robustness of 
the system. Thus, robustness design by providing alternative load paths may require a higher load 
bearing  capacity  for  the  remaining  structure  than  that  needed  for  persistent  and  identified 
accidental design situations. Since, in principle, any member may fail due to unidentified accidental 
actions, an uneconomic design is a possible outcome of such a strategy. In this regard, as with the 
key member design strategy, open questions also relate to the need to consider or not consider 
scenarios where more than one member is damaged under an extreme event. 

Structural segmentation through which the spreading of failure is prevented or limited by isolating the 
initially damaged portion of the system. This strategy has advantages of being able to define by 
design the extent and location of the failure regions, but this comes also with some disadvantages. 
Structural  segmentation  typically  involves  designing  structural  fuses,  e.g.  weak  interfaces,  and 
favours dividing the system in statically determinate sub-systems. When exposed to persistent, 
transient and identified accidental situations these sub-systems will exhibit increased risks when 
compared  with  adequately  designed  and  detailed 
indeterminate  solutions.  Structural 
segmentation aims to reduce risks globally, but in the process may increase risks locally. A key 
issue is therefore how to fine tune the design details so that adequate robustness and risks levels 
are attained. Note that indirect design methods are currently missing for segmentation, and that 
design of statically determinate sub-systems may involve increased requirements for structural 
safety and serviceability and thus can lead to a higher material consumption compared to other 
strategies  

3.10.3  Design methods for robustness 

Design for robustness should be based on the Consequence Class (CC) of the structure, as given for example in 
Table 3.13. The table is intended as a possible inspiration for the NBS’s.  

The specificities of each design approach can be as follows: 

For CC1 structures no particular robustness provisions are required if the structure is designed according to the 
partial  factor  design  philosophy  specified  in  the  Eurocodes  for  persistent,  transient  and  seismic  design 
situations  and  the  failure  consequences  are  acceptable.  Note  that  the  latter  should  result  from  using 
methods given in Chapter 5. 

For CC2  structures two  groups  can  be  defined,  respectively  CC2a  and CC2b  according  to  different  levels of 
consequences. Both refer to similar groups of structures but exhibit different levels of magnitude of failure 
consequences. Therefore, different levels of sophistication of the analysis and design methods should be 
applied. A possible discrimination between the two classes can relate to: 

85 

 
When using indirect design methods for CC2a, typically only horizontal tying is involved whereas for 
CC2b vertical tying is required in addition.  

When using direct design methods for CC2a typically simpler analysis requirements are involved, for 
example, linear elastic models may still be appropriate, whereas for CC2b non-linear analyses are often 
deemed necessary. 

Note that although it should be required that the reliability of the designs achieved using either direct 
or indirect design methods should be comparable, the current robustness provisions are not generally 
associated with a specific target level of reliability. Alternatively, indirect design methods may be used 
as baseline (minimum) for robustness requirements. 

For  CC3  structures,  additional  explicit  identification  and  enhanced  quantitative  characterisation  of  adverse 
events are typically considered necessary, involving, in particular, for existing structures the use of more 
complex reliability-based methods. Member states may provide rules for the additional use of qualitative 
or quantitative risk analysis in special cases. 

Table 3.13. Relationship between Consequences Classes and Robustness design approaches (example)  

Consequences 
Classes 

CC1 

CC2a 

Lower Risk Group 

Robustness design requirements 

 Analysis method 

Design method 

No particular provisions are necessary with respect to robustness. 

Analyses can be based on simplified models of 
loads and structural behaviour. 

Design  for  robustness  using  direct 
and/or indirect design methods. 

CC2b 

Simplified models still possible. 

Higher Risk Group 

Dynamic  and/or  non-linear  analysis  models 
may  become  relevant.  Analyses  may  however 
be  based  on  simplified  models  of  loads  and 
structural behaviour. 

CC3 

Additional explicit identification and enhanced characterisation of adverse events. 

Level  II  to  IV  reliability  methods  including  qualitative/quantitative  consideration  of 
possible risks where appropriate and (as stated in EN 1990:2023, Table E2) specified by 
the relevant authority or agreed upon for the specific project by the relevant parties  

It is highlighted that design for robustness consists of the identification and assessment of design scenarios. 
Herein, each design scenario relating to a set of events or conditions occurring during the execution or lifetime 
of  the  structure  leading  to  a  state  of  the  system  for  which  the  potential  for  attaining  unacceptable 
consequences should be assessed. Design scenarios depend on the type of design method to be considered, 
ranging from realistic event sequences to notional damage scenarios.  

In general, the rules for accidental design situations are applicable; however, in some cases persistent design 
situations should be considered, in particular for cases where permanent actions control the design.  

In addition, it is noted that the design solution obtained using direct and indirect design methods is not the 
same. The former may involve a design focused on the system scale whereas the latter focuses, in general, 
only on the local scale. One  does not replace the other. The  same applies to  hazard  dependent and  hazard 
independent scenarios (e.g. identified and unforeseen hazards, respectively). 

Therefore,  in  many  cases,  a  suitable  way  for  economically  meeting  all  structural  safety  and  robustness 
requirements should be based on a combination of different design strategies, as appropriate. For this purpose, 
a pragmatic approach can be adopted implying continuous structural systems with a ductile behaviour;  

86 

 
 
-For  hazard  independent  scenarios,  provision  of  either  alternative  load  paths  via  direct  or  indirect  design 
methods, or, predefined collapse mechanisms for the event of local failure of selected key members, simulated 
by notional scenarios (as noted above, indirect design methods are currently missing for segmentation);  

-For hazard dependent scenarios, provision of alternative load paths via direct or indirect design methods 

When using the alternative load path strategy, check if the associated failed area can be deemed acceptable. 
If not, repeat until the latter is verified. Research was performed by (Tanner and Hingorani, 2015) which can be 
used to define a acceptance criteria based on the failed area. 

3.10.4  Reflection 

Even  before  the  well-known  Ronan  Point  tower  building  accident  in  1968,  structural  integrity  design 
requirements  for  ties  were  included  in  CEB  bulletin  60  concerning  precast  panel  constructions.  A  complete 
historical review of prescriptive design rules for robustness  can be found in (Russell et al, 2019). The “tying 
force” rules included in many codes are inspired in these provisions. There is still ongoing debate on whether 
the objective of these tying forces was just to aid in keeping the structural shape of precast panel constructions 
or to prevent local failure and maintaining integrity after a member loss. What is clear that there is no available 
evidence  that  rules  developed  for  precast  panels  construction  still  apply  to  modern  construction  methods. 
Furthermore, although the original intention of the ties was to prevent brittle failure, this is not explicitly checked 
and therefore it is not certain that the tie, or secondary mechanism can form without further failures occurring. 
Advanced  structural  analysis  in  combination  with  experimental  evidence  may  help  to  come  to  a  better 
understanding. 

Finally  note  that  current  design  loads  to  design  key  members  (i.e.  pressure  load  of  34  kPa)  have  not  been 
derived from case studies which allow for their generalised application. It makes sense to keep this in mind. In 
any case, it appears reasonable to include the consequences of key member failures in the considerations.. 

3.11  Observational methods, design assisted by testing and monitoring  

Ideally all structural design models, both on the action and on the resistance side, are based on the laws of 
mechanics  and,  where  relevant,  on  other  laws  from  the  world  of  natural  sciences.  These  laws  have  been 
validated in the past on the basis of numerous experiments, both on a fundamental level (first principles) and 
on an applied level (buckling curves, shear models). The same holds for the values adopted for the variables in 
the models. This whole set of experiments and observations is the basis for the models and design values in 
the codes.  

In some cases, the available set of data may be considered as insufficient, for instance for innovative materials) 
and a new set of experiments is set up. Annex D of EN 1990:2023 (Design by Testing) gives guidance to that 
process and gives formulas on how to deal with the test results compatible with the Eurocode design philosophy. 
A main topic in this respect is the treatment of the scatter in the test results, in particular in the situation of a 
small  number  of  tests  (statistical  uncertainty).  The  procedure  in  Annex  D  is,  in  line  with  the  rest  of  EN 
1990:2023,  based  on  the  Bayesian  interpretation  of  probability  where  no  distinction  is  made  between  the 
various sources of uncertainty. In practice this means for instance that there is no need to choose, in addition 
to the target reliability, separate confidence intervals for the estimates of means and standard deviations as 
has to be done in classical frequentist approaches.  

The observational method is a design approach with testing/observations not only before but also during the 
execution or early stage of use. It is already used for quite a long time in geotechnical engineering (Peck (1969) 
and later modified as it formally entered EN 1997. However, the basic idea can be and are also applied to other 
types of structures. An example is the application of vortex limiting devices: they are only applied if practical 
behaviour show the need.  

The essence of the approach is to design a structure based on an estimate of the structural or ground properties 
less cautious than the design values in partial factor design (e.g. best estimate values), monitor the behaviour 
of the structure  during  or after construction and to have prepared contingency measures to  be taken if the 
encountered  properties  deviate  negatively  from  the  expectation.  A  key  aspect  of  the  approach  is  to  assign 
trigger values to the observed parameters for the implementation of contingency measures. Such decision rules 
establishing the triggering of prepared contingency measures have to produce a safe margin to structural failure 
during the execution phase and ultimately have to ensure sufficient reliability of the finalized structure. 

When the method is applied the reliability level in the final state would be insufficient if based on the large 
uncertainties present  at  the  beginning.  However,  during  the  process of  execution  the  reliability  level  can  be 

87 

 
updated using Bayesian procedures. (e.g. Spross & Johansson, 2017). The effect of updating and, hence, the 
effectiveness  of  the  approach  will  depend  largely  on  the  degree  to  which  uncertainties  in  time-invariant 
variables (e.g. ground properties or permanent loads) dominate the design problem. The objective is to meet 
the formal reliability requirements in the final  state of the  structure, using the information obtained during 
execution (and early use) to update (reduce) uncertainties in the structural or ground properties. Contingency 
measures should be applied when necessary. In order to achieve this a risk analysis at the start of the project 
is recommended. 

Next to the reliability of the final product also the reliability in the various phases of construction (for instance 
related to the safety of human lives) can be important. In principle such requirements can be treated using the 
similar methods as discussed above.  

Note that standard inspection and monitoring activities during the entire life time of the structure are based on 
the same principle as design by observations. The basic issue, however, in that case is not the almost immediate 
behavior but the long term behavior, including all kinds of deteriorating influences.  

Finally  it  is important  to  realize  that  it  may  be  important  to  have  in  advance  a  clear  understanding  on  the 
procedure with the client and the relevant authorities.  

88 

 
 
4  Assessment of Existing Structures 

4.1 

 Motivation 

Current and future activities related to the development of cities, industrial areas and infrastructures should be 
determined by sustainability goals (JRC 2015). New needs are therefore not simply answered by adding new 
buildings and infrastructures to the existing built environment or by replacing existing with new engineering 
works. Rather, ways are being explored to modify existing systems to meet new demands, or simply to extend 
their service life. For this purpose, the reliability of existing structures must be verified, activity that is usually 
denominated as assessment CEN/TS 17440 2020].  

In general, an assessment of an existing structure may be required in the case of (ISO 2394 2015): 

a change in the purpose of the structure compared to that for which it was originally designed or previously 
assessed;  

deviations in the properties of the structure from those adopted in the original design or in the previous 
assessment.  

Examples of changes in the purpose of a structure include: 

change of use;  

increase of actions; 

extension of the originally intended design service life; 

increase in reliability requirements (e.g. due to higher consequences in case of failure).  

Deviations in the properties of a structure can be due to the following causes, among others:  

modifications of the structural system (e.g. extension, upgrading, repair);  

deterioration induced by time-dependent actions and influences (e.g. corrosion, fatigue); 

damage  induced  by  accidental  events,  overloads,  changes  of  the  boundary  conditions  (e.g.  differential 
settlements); 

defects resulting from previously undetected errors during design, construction or use.  

The general principles of structural reliability laid down in the Eurocode for structural design (EN 1990; 2023) 
and of course also in chapter 3 of this document also apply to the assessment of existing structures. However, 
there  are  also  important  differences  between  assessing  existing  and  designing  new  structures.  The  use  of 
design-oriented methods to assess existing structures therefore often leads to a high degree of conservatism 
with serious economic, ecological and socio-political consequences if resources are invested in the unnecessary 
strengthening or replacement of existing structures with the associated disruptions  (JRC 2015). This chapter 
briefly  discusses  relevant  differences  between  assessment  and  design.  In  the  Eurocode  system,  these 
differences are addressed in the Technical Specification (CEN/TS 17440 2020). The conversion of this document 
into an additional Eurocode part “prEN1990-2 Basis of structural and geotechnical design – Part 2: Assessment 
of existing structures” is under development. 

4.2  Scope  

The main difference between assessing the performance in existing and design phase structures is that many 
characteristics whose values are merely anticipated in the latter can be measured in the former, taking also 
into account the effects of the construction process and subsequent life of the structure, during which it may 
have undergone alteration, deterioration, misuse and other changes to its as-built state. The accuracy of the 
assessment results obtained by applying load and strength models can usually be improved by collecting more 
data about the structure under analysis and about the actions and influences to which it is exposed (see 4.4 
and 4.5). This does not mean, however, that the uncertainties can be completely resolved: in-service inspection 
and testing are also associated with  uncertainties  (ISO 2394  2015).  Therefore, assessment is conducted  by 
stages (CEN/TS 17440 2020), raising the quality of the information available from stage to stage where needed. 
Note also that at the start of an assessment the uncertainties may be even greater than in design for instance 
as the consequence of (unexpected) damage or deterioration. 

89 

 
Another relevant difference between the design of new structures and the assessment of existing ones concerns 
the  level  of  reliability  required  (see  4.3).  The  choice  of  this level  depends,  among  other parameters, on  the 
amount and expense of effort needed to reduce the risks associated with a structure. The cost of achieving a 
higher level of performance (i.e. increased safety) is usually high in existing structures compared to the cost of 
improving the same performance in the  design  phase of  a new structure. This aspect  should be considered 
when choosing the level of reliability required for an existing structure (JRC 2015), along with the influence of 
the remaining service life. Note that in principle a minimum reliability level related to human safety resulting 
from ethic reasons should not be lowered.   

Deterioration due to environmental influences, repeated actions or use-induced wear, is typically a cumulative 
process that can adversely affect structural reliability. Differences between existing and new structures also 
evolve  from  the  treatment  (not  in  principle  but  in  practice)  of  the  possible  interactions  of  cumulative 
deterioration and extreme action effects (see 4.6).  

Although the detailed design of any engineering structure requires multiple iterative steps, the ultimate result 
is a clear definition of the type, layout and dimensions of the load bearing system and its individual elements 
and details, as well as an appropriate selection of construction materials. This can be fundamentally different 
in  an  existing  structure  in  which  different  approaches  may  be  advisable,  depending  on  the  results  and 
conclusions from the assessment with regard to the relevant requirements for the system (see 4.7). Appropriate 
recommendations  should  be  formulated  to  the  owner,  taking  into  account  the  conditions  for  possible 
interventions  and  future  operation  of  the  construction  work.  Such  recommendations  can  therefore  include 
constructional or operational intervention measures that belong to different categories.   

4.3  Target reliability levels 

Significant aspects to consider when determining the target value of the reliability index for the assessment 
situations relevant to an existing structure include (CEN/TS 17440 2020):  

the possible cause and the mode of reaching a limit state (as far as they are relevant for the consequences 
of failure);  

the  possible  direct  and  indirect  consequences  of  failure  in  terms  of  risk  to  life,  injury,  potential 
environmental and economic losses, social and political consequences, loss of cultural heritage value, etc.;  

the relative cost of safety measures to increase reliability (as long as the level is not governed by minimum 
human safety aspects); 

the reference period. 

The acceptable reliability levels for existing structures can be different from those required for new structures. 
The following types of considerations can justify lower levels for existing structures:  

economic: the relative cost of safety measures to increase the reliability of an existing structure can be 
very high, while the additional cost of increasing the reliability in the design phase of new structures is 
generally low; 

societal: the strengthening or replacement of existing structures can lead to the resettlement of residents, 
the interruption of activities or may influence the values of cultural heritage, circumstances that normally 
do not play a role in the design of new structures; 

sustainability: sustainability goals are of fundamental importance when deciding to extend the service life 
of existing buildings and infrastructures, or when adapting such systems to new needs, as this implies a 
reduction in resource consumption compared to replacement with new structures or structural elements, 
likewise, the rehabilitation of existing structures usually allows the selection of the most suitable solutions 
and materials from the point of view of sustainability. 

Target reliability levels can be derived based on explicit risk analysis (see chapter 5) or economic optimization, 
meeting acceptable human safety levels, e.g. with regard to current best practice (orEN 1990-2:2021). When 
selecting target reliability levels, it should be taken into account that the intended remaining service life of an 
existing  structure  often  may  be  shorter  than  the  design  service  life  of  new  structures.  This  is  in  particular 
relevant for cases where consequences of failure are dominated by economical losses. Note that the reference 
period to which the target reliability is related can be selected regardless the anticipated remaining service life. 
If annual target reliabilities are used, the corresponding structural performance should be achieved in each year 

90 

 
of the remaining service life. This is particularly relevant in the case of deteriorating structures  in which the 
final year of the remaining service life is decisive (see 4.6). 

The target reliability levels for new structures can in most cases be considered conservative, when used for the 
assessment of existing structures due to the aspects discussed above. Note that the  final responsibility for 
setting  target  reliability  levels  and  corresponding  safety  elements  is  a  on  the  national  level  and  should  be 
mentioned in the relevant National Annex. Chapter 5 given some guidance in making appropriate choices.    

4.4  Updating information    

4.4.1  General 

The acquisition of new data  about an existing structure  by means of inspections, measurements or tests is 
intended to supplement the available prior information, which may often be quite vague, with respect to aspects 
such as geometrical properties, actions and influences, construction material and geotechnical properties, as 
well as the actual condition of the structure, its behaviour or deformation capacity (CEN/TS 17440 2020). When 
new information becomes available, all relevant data need to be evaluated, taking into account the  existing 
prior information.  This  process  is known  as  updating  of  information,  which  is one  of  the  main  tasks of  any 
assessment. In addition to the rules given in the Eurocode on basis of assessment (CEN/TS 17440 2020; prEN 
1990-2,  2021),  the  updating  of  information  is  also  covered  in  the  assessment  and  retrofitting  part  of  the 
Eurocode  on  earthquake  resistance  (prEN  1998-3  2021),  in  particular  in  its  chapter  5  on  information  for 
structural assessment.  

Two complementary approaches can be considered to update information about the properties of a structure 
and its performance under the actions and influences to which it is exposed (ISO 2394 2015):  

the updating of the probability of structural failure by using information from load testing or about the 
performance of the structure in the past; 

the collection of data on individual basic variables by performing on-site inspections to update previously 
available uncertain information.  

4.4.2  Updating the failure probabilities   

Formally, the direct update of the probability of failure can be carried out using fundamental relationships from 
probability theory (ISO 2394 2015; prEN 1990-2 2021), together with the information that an existing structure 
or structural component has withstood an applied load effect. In practice, load effects are known in the case of 
load tests or, to a lesser extent, if the maximum loads applied over the past service life can be estimated. 

In this context it should be noted that the observation that a structure or structural component has withstood 
a  load  test  does  not  directly  reveal  its  actual  resistance,  nor  does  it  provide  a  direct  measure  of  structural 
reliability. Such an observation only indicates that the minimum resistance of the structure at the time of the 
test is greater than the effect of the applied load, allowing for the uncertainties associated with the conversion 
from loads to load effects. Therefore, to update the probability of failure of a structure on the basis of a known 
load, the original cumulative distribution function for the structural resistance used for this purpose can be cut 
off at the level of the known deterministic level. In the more general case a more advanced procedure may be 
needed,  for  instance  ,  allowing  for  the  relevant  model  uncertainties  including  those  associated  with  the 
conversion  from  an  action  to  its  effects  (i.e.  depending  on  measurement  uncertainties  or  other  less  known 
effects the cut off may be sharp or gradual). If no prior knowledge on the resistance is available, the probability 
of  failure  may  (conservatively)  be  assessed  based  on  the  effects  of  the  known  action  and  its  statistical 
distribution.  

Similarly, satisfactory performance of a structure over a period  T of service indicates that in the absence of 
significant degradation, its minimum resistance is greater than the maximum (random) load effect applied over 
that period. To estimate the probability of failure, the distribution function for the structural resistance can be 
updated taking into account the estimated random action effect over the same period T. Note that if there is 
deterioration the analysis becomes more cumbersome. 

Additional  measurements  and  inspections  may  reveal  more  information  of  the  structural  properties.  This 
information,  of  course,  can  be  taken  on  board  in  the  updating  procedure.  If  there  is  for  instance  a  well-
established degree of correlation between stiffness and strength, information on the stiffness may be used to 
update the distribution of the resistance. 

91 

 
4.4.3  Updating the basic variables   

When  updating  the  probability  distribution  function  of  a  basic  variable,  Xi,  its  parameters  (e.g.,  mean  value, 
standard deviation, coefficient of skewness, lower bound, etc.) may be considered as random variables (JCSS 
2000;  ISO  2394  2015;  prEN 1990-2  2021).  Prior  distribution  functions  for  the  unknown  parameters  of  the 
investigated variable should reflect all the information available before the in-service data acquisition is carried 
out. For example, distribution parameters that represent the state of uncertainty associated with the Eurocode 
design rules, compiled in Section 3.4 and Annex A, can often be used as a starting point for updating. Given 
such prior distributions and statistical data from new observations, posterior distributions can be derived, e.g. 
by (repeatedly) applying a Bayesian method (JCSS 2000). In general, the following assumptions are appropriate 
for the type of distribution in most applications (no difference with the design situation):  

for  dimensions  and  material  properties,  respectively  a  Gaussian  and  a  log-normal  distribution  may  be 
adopted;  

a Gaussian distribution may be appropriate for permanent action effects;  

an extreme value distribution may be convenient if it is intended to represent a maximum value within a 
chosen reference time (e.g., effects due to variable or accidental actions).  

The  choice  of  probability  distribution  functions  should  be  made  with  caution,  considering  possible  bias  and 
skewness. The coefficients of asymmetry and kurtosis may provide valuable information for determining the 
appropriate  theoretical  model  (i.e.,  probability  distribution  function).  Note  also  that  variables  originally 
uncorrelated may get correlated after updating; in that case updating the marginals is not sufficient.. It is also 
important  to  note  that  any  method  used  for  inspection,  measurement  or  testing  has  a  limited  resolution. 
Uncertainties associated with such methods include (ISO 2394 2015):   

measurement error; 

inherent variability of a measured parameter; 

model uncertainty when a parameter cannot be measured directly so that a relationship is needed between 
it and the corresponding measured parameter; 

statistical uncertainty due to a limited number of measurements.  

These  uncertainties  should  be  taken  into  account  when  determining  the  statistical  data  that  describe  the 
measured  parameter.  This  can  be  achieved,  for  example,  by  increasing  the  coefficient  of  variation  of  the 
corresponding  parameter (prEN 1990-2  2021).  In this context, it should also be taken into account that the 
uncertainties  associated  with  non-destructive  testing  methods  are  generally  higher  than  with  destructive 
methods.  For  assessing  the  earthquake  resistance  of  existing  structures,  EN  1998-3:2021  accounts  for  the 
above uncertainties by using partial factors 𝛾
Rd that depend on the amount of information gathered during the 
assessment, as described in discrete terms by so-called Knowledge levels (KLs), and are derived considering 
the  inherent  variability,  the  statistical  uncertainty  as  well  as  the  model  uncertainty  for  each  resistance  or 
deformation capacity formula in the code. 

4.5  Verification  

4.5.1  Overview 

Similar to the design of new structures, when assessing existing structures, it should be verified that with an 
appropriate reliability level no limit state is exceeded for all relevant assessment situations (prEN 1990-2 2021), 
verifications can be performed using different methods: 

 

 

 

quantitative assessment based on calculations; 

qualitative assessment based on past performance; 

combination of both types of methods. 

If qualitative methods are used to assess a structure at the relevant limit states based on satisfactory past 
performance over a sufficiently long period of time, a detailed structural investigation should be carried out to 
show that a number of specific conditions (e.g., no damage nor deterioration, no changes to the structure or the 
environment, etc.) are met. Note that there is no reliability background for qualitative assessment methods. On 
the other hand (see 4.4.2), the available information about the past performance of a structure can also be 

92 

 
used in a  quantitative assessment, for example, for  direct updating of failure probabilities or for  resistance 
updating based on action effect histories.  

The  most  accurate  way  of  assessment  would  be  to  explicitly  consider  updated  load  and  strength  variables 
through  the  use  of  reliability  methods  or  risk-based  decision  procedures.  However,  such  methods  and 
procedures are time-consuming, calling for a specific operational knowledge of probabilistic methods, and are 
only used in special cases. For example: 

 

if, using the partial factor format , it cannot be demonstrated that the structure or element achieves 
the required reliability; 

  when uncertainties are outside the usual ranges; 

 

 

in cases with severe failure consequences or insufficient robustness; 

for decisions regarding a whole group of similar structures (e.g. calibration of partial factors); 

  when evaluating the efficiency of different options for intervention (e.g. monitoring and maintenance 

strategies).   

To verify whether an existing structure meets the relevant reliability requirements for all assessment situations, 
the partial factor format is normally used, equivalent to the method specified for structural design (prEN 1990 
2020). The difference is that the relevant parameters, including the characteristic values of the basic variables 
and partial factors, can be modified based on updated information (prEN 1990-2 2021). A particular form of 
additional information is a proof loading (see 4.4.2). Such tests, as well as study of track records, may reveal 
implicitly a part of the  hidden safety (see 2.2.3). When elaborating the results it is important to distinguish 
between conclusions related to the structure (or structural member) that was actually proof loaded and other 
(similar) structures or members. 

4.5.2  Partial factor format  

When using the partial factor format, the assessment values for the action effects, Ea, are compared to the 
assessment values for the respective strengths, Ra. The so-called assessment values (prEN 1990-2 2021) are 
equivalent to the design values for new structures  (prEN 1990 2020). Structural reliability requirements are 
met if, similar as in design, inequality 4.1 is fulfilled for all relevant assessment situations: 

Ea≤Ra 

(4.1) 

where Ea and Ra can be expressed as functions of the assessment values of the basic variables like in design.  

Resistance models that cannot be used for the assessment of all existing structures should be adjusted so that 
the  effects  of  structural  condition,  including  deterioration  on  the  load  bearing  capacity  can  be  modelled 
explicitly. Such assessment-specific provisions should be developed for future editions of the material-oriented 
Eurocodes (CEN/TC 250/WG2/WG2.T2 2021). If resistance models for the design are adjusted or expanded in 
their scope, in order to enable the assessment of existing structures, partial factors for resistance,  Ra, should 
cover the degree of uncertainty associated with these models.  

Note  that  the  observation  of  a  certain  degree  of  deterioration  should  not  only  result  in  a  modification  of 
structural dimensions or properties. It may also be necessary to adjust other parameters as for instance mean 
or extreme load values. The mere fact that the structure was affected more than foreseen could be an indication 
that some actions (or environmental influences) were estimated too low. Similar adaptations may be necessary 
for structural or geotechnical parameters. 

4.5.3  Assessment values of the basic variables   

Provided that the basic variables relevant for the reliability verification of an existing structure are specified by 
suitable  probability  distribution  functions  (see  4.4.3),  their  characteristic  values,  xki,  the  associated  partial 
factors,  xi,  and the  assessment  values,  xai,  can  be  determined.  By  way  of  example, consider a  log-normally 
distributed resistance variable xi, associated with a non-deteriorated structure, with updated mean value, mxi, 
standard deviation, xi and coefficient of variation, Vxi. The updated characteristic value, xki, is obtained as the 

93 

 
 
 
lower 5% fractile of the updated probability distribution function of xi. If we use, the Design Value Method, we 
may derive for coefficients of variation, Vxi ≤ 0,25, the updated partial factor, xi, from:  

γxi=

xki
μxi

⋅eαR⋅βt⋅Vxi 

(4.2) 

where t is the target reliability index and aR the FORM sensitivity factor for resistance. For the latter, the value 
recommended  in  the  Eurocode  (prEN  1990  2021)  may  be  used  as  a  first  approximation  if  the  respective 
application limits are met. If not, as the uncertainties and relative importance of variables for existing structures 
may differ from those in the design phase, R could be determined by reliability analysis or could be adopted 
conservatively (e.g., aR = 1). Finally, the assessment value of the same variable, xai, to be applied in the reliability 
verification of the existing structure analysed, is obtained from: 

xai=

xki
γxi

(4.3) 

Similarly, for a normally distributed  action variable,  xi, with  updated  parameters  xi, xi and  Vxi, the updated 
partial factor, gxi, can be obtained from Formula (4.4). As in the previous case, the value recommended in the 
Eurocode (prEN 1990 2021) for the FORM sensitivity factor for effects of actions, E, may be used as a first 
approximation (but the rules for application should be checked):  

γxi=

μxi
xki

⋅(1-αE⋅βt⋅Vxi) 

(4.4) 

The  updated  assessment  value  of  the  action  variable,  xai,  can  be  derived  as  follows  from  the  updated 
characteristic value, xki: 

xai=γxi⋅xki 

(4.5) 

For  permanent  actions,  the  updated  characteristic  value,  xki,  is  normally  considered  to  be  equivalent  to  the 
updated mean value, mxi.  

For the effects of variable actions with a Gumbel distribution (e.g. imposed loads, climatic actions),  assuming 
that the uncertainties associated with the time-invariant part of the action model (see 3.4) are not relevant, the 
updated  partial  factor,  xi,  may  be  determined  on  the  basis  of  formula  (4.6),  where    is  the  cumulative 
distribution function of the standardised normal distribution, xi represents the updated standard deviation and 
the other parameters correspond to those of formula (4.4). As already mentioned (see 4.3), all these parameters 
should  relate  to  the  chosen  reference  period.  In  the  absence  of  a  specific  reliability  analysis,  the  sensitivity 
factor,  E,  may  again  be  approximated  by  using  recommended  values  (prEN  1990  2021),  if  the  application 
limits are met. Finally, the updated assessment value of the action variable, xai, is obtained by applying formula 
(4.5). The latter also applies to the general case, if the uncertainties associated with the time-invariant part of 
the action are relevant, although the updated partial factor cannot be represented with a closed formula.   

γxi=

1
xki

⋅ (μxi-σxi⋅

√6
π

⋅ (0,577+ ln (- ln ϕ (-αE⋅βt)))) 

(4.6) 

94 

 
 
 
 
 
 
 
 
 
 
The above examples of updated partial factors and assessment values take no account of additional effects, 
e.g. due to model uncertainties. Further information on general procedures for taking such uncertainties into 
account can be found in the Eurocode (prEN 1990 2021). 

For what concerns seismic assessment, EN 1998-3 adopts nonlinear analysis methods as the default choice. 
These methods employ nonlinear finite element models of the whole structural system (see 3.8). To avoid using 
different sets of material properties for  respectively the model and the verifications (a choice that with the 
extensive use of the corresponding first generation Eurocode Part, has led to considerable complications in its 
practical application), the assessment values of material properties are prescribed to be the mean, i.e., the same 
as in the nonlinear model. Partial factors are then calibrated to consider this fact. 

4.6  Deterioration   

4.6.1  General 

Deterioration due to environmental influences, repeated actions or use-induced wear, is typically a cumulative 
process that can adversely affect the reliability of existing structures. When designing new structures according 
to  the  Eurocodes,  the  possible  combination  of  cumulative  deterioration  and  extreme  action  effects  is  often 
neglected: durability design is usually considered separately from the design for ultimate and serviceability limit 
states (see chapter 3). In some material-oriented Eurocodes, the design to prevent deterioration is based on 
condition or proxy limit state verifications. Such simplifications are certainly not appropriate for the assessment 
of existing structures that are already affected by deterioration mechanisms: 

reliability requirements should be verified for the combined effects of cumulative deterioration and the 
relevant actions and influences likely to occur during the remaining service life; 

condition limit states intended to prevent deterioration from affecting the performance of a new structure 
may not apply to existing structures that are affected by ongoing deterioration; 

indicators that are not based on measurable quantities cannot be used for inspection and maintenance 
planning. 

In those cases adjusted project specific risk based criteria may be needed. 

4.6.2  Remaining structural capacity and verification 

Resistance models are usually based on a combination of mechanical principles and empirical relationships. 
Models  that  explicitly  take  into  account  the  effects  of  deterioration  on  the  resistance,  adjusted  or  newly 
developed  should  preferably  be  based  on  physical  principles.  The  scope  of  such  models  should  include  all 
relevant material-specific deterioration mechanisms and the associated uncertainties should be quantified. 

Models should also be developed to describe the propagation of deterioration as a function of time, with the 
aim of predicting the condition of an existing structure over the remaining service life, going out from its actual 
condition  at  the  time  of  assessment.  Depending  on  the  conditions  to  which  the  structure  is  exposed  (e.g. 
environmental  influences,  repeated  actions),  these  models  should  describe  the  onset  and  the  rate  of  the 
cumulative processes that affect the parameters influencing the remaining structural resistance. The spatial 
distribution of the processes should be accounted for if relevant.  

In deteriorating structures, the reliability requirements should be based on a reference period of one year (see 
4.3) and verifications should be carried out for the final year of the intended remaining service life (see 2.4 for 
time effects and 2.5.3 for annual versus (remaining) lifetime target reliabilities). The uncertainties associated 
with  the  models  that  describe  the  propagation  of  deterioration  as  a  function  of  time  should  be  taken  into 
account,  along  with  those  associated  with  the  resistance  models  for  deteriorated  structures.  However,  the 
former may be reduced by implementing structural health monitoring techniques to provide information about 
environmental influences on the structure, degradation processes or structural performance and their variation 
over time 

4.7  Closure  

The staged assessment process of an existing structure can be completed if clear conclusions can be drawn 
from the findings regarding the assessment objectives, or if an additional assessment step is unlikely to provide 
relevant new knowledge. Depending on the assessment findings, the structure or a  structural member may, 
within the scope of the assessment: 

95 

 
achieve  the  reliability  required,  assuming  adequate  inspection  and  maintenance  during  the  remaining 
service life;   

achieve the reliability required at the time of the assessment, but not for the complete period of time during 
which  the  existing  structure  is  intended  to  remain  operational,  taking  into  account  the  anticipated 
development of its condition and the planned level of maintenance;  

fail to achieve the reliability required; 

need immediate correction of the existing condition by means of urgent risk mitigation measures. 

If the required degree of reliability is not achieved with regard to structural safety, robustness, serviceability or 
durability, intervention is needed. Appropriate interventions should be defined case-specifically, taking account 
of the following: 

the type and importance of the structure; 

the type of basic requirement (prEN 1990 2021) that is not met; 

possible cause and mode of attaining a limit state;  

expected consequences of failure; 

options of interventions that are available.  

96 

 
5  Guideline for probabilistic and risk-based decision making 

5.1 

Introduction 

Note on terminology:. In this document risk is understood as measure for the combination (product) of likelihood 
and consequences of an adverse event (see also chapter 2.3). Although this definition is widely accepted in the 
European construction sector, it is different to the one in other publications where a risk analysis is limited to 
the  probabilistic  aspects  like  in  “seismic  risk  analysis.  Sometimes  even,  the  terms  risk  and  probability  are 
considered as synonyms.  

The main application of risk assessment and reliability analysis is to serve as an indicative background for the 
calibration  of  partial  factor  design  procedures,  being  the  mainstream  of  the  Eurocodes.  This  makes  the 
information in this chapter meaningful for all NSB’s that are responsible for formats and numerical values in 
the National Annexes. However, the NSB may also define conditions and procedures for direct application of 
risk and reliability-based design on a project level. The most common condition for such allowances is that the 
Eurocode gives no solution for certain types of structures or certain types of design situations. However, also in 
cases where Eurocodes are believed to be less adequate from a safety or economy point of view, the National 
Annex may allow, recommend or require a more advanced risk or reliably based approach. Examples are: 

as  an  alternative  to  the  prescriptive  robustness  rules  for  structures  with  high  consequences  in  case  of 
failure; 

within  procedures  described  in  EN  1991-1-8  (for  design  purposes)  and  EN  1990-2  2021  (for  existing 
structures);  

for unconventional structures (e.g. very large structures, new design concepts, new materials). 

Note that risk and reliability-based approaches may be used for special aspects of the structure, while most of 
the design is using conventional partial factor design procedures based on standardised consequence classes 
and target reliability classes. 

5.2  Reliability based design procedures  

This approach is used if the standardised target reliability according to EN 1990:2023 Annex C is accepted as 
it is, but no satisfactory corresponding set of partial factors is available. 

The goal might be to check for adequate reliability only, but one might also aim at minimization of costs and/or 
resources.  It  is  strongly  recommended  to  use  similar  probabilistic  models  as  presented  in  Annex  A  of  this 
document or (if not possible) to use the same level of schematisation.  

In order to calculate the annual or lifetime failure probabilities one needs: 

Limit state functions that represent underlying scenarios; 

Probabilistic models for the random quantities; 

Reliability calculation procedures. 

Limit state functions and the underlying failure scenarios should in principle not be different from those used 
in partial factor design and they correspond where possible to the mechanisms described in the material related 
Eurocodes.  Of  course,  it  is  possible  that  refinements  of  those  models  are  intended  or  that  no  limit  state 
formulations  are  presented  in  the  Eurocodes.  Note  that  when  deriving  a  mechanism  formulation  for  use  in 
reliability analysis from existing design rule (either in Eurocode or elsewhere) one should: 

take care of application limits; 

investigate the model uncertainties; 

avoid introduction of hidden safety elements or make them transparent and public available to practitioners 
and code-writers (see also chapter 2.2.3). 

Probabilistic properties for reliability-based design should follow if available from Annex A of this background 
document.  If  not  possible  a  combination  of  data  and  engineering  judgement  should  be  used.  Engineering 
judgement should be treated with care and in particular the influence of persons having somehow an interest 
in the outcome of the calculations should be avoided. Sensitivities to arbitrary choices should be assessed and 
distributions with fixed upper or lower limits should not be used unless there are convincing reasons. Updating 

97 

 
of information on the basis of tests and observations shall be taken into account where appropriate (e.g. when 
assessing existing structures). 

The calculation of the failure probability can be performed by (a combination of) standard calculation techniques 
such  as  FORM/SORM,  mode  combination  rules,  response  surface  techniques,  the  outcrossing  approach  and 
various  Monte  Carlo  Simulation  techniques.  In  principle,  the  outcome  of  a  reliability  analysis  is  the  failure 
probability for the reference period under consideration and there is no need to check a set of specific design 
situations.  However,  for  reasons  of  computational  manageability  it  may  be  necessary  to  consider  a  set  of 
individual  design  situations  (transient,  accidental,  member,  yielding,  brittle  fracture,  etc)  and  combine  them 
afterwards. For examples reference is made to Annex B. 

In the seismic context, EN 1998-1-1 introduces with Annex E (Simplified reliability-based verification format) 
the possibility of checking reliability in the seismic design/assessment situation with a simplified formulation 
based  on  nonlinear  analysis,  using  the  total  probability  theorem  to  evaluate  the  annual  probability  of 
exceedance of the (global, structural system level) limit state according to: 

∞

PF= ∫ P(LS|Se) |

0

dH(Se)
dSe

| dSe

(5.1) 

where P(LS|Se) is the fragility function and H(Se) the hazard function (mean annual probability of exceedance 
of a given value of the local intensity Se, see 3.5.7. The method is simplified because under the assumption that 
the fragility is lognormal and the hazard is linear in the log-log space PF can be obtained in closed-form, using 
a nonlinear static analysis and tabulated values of the uncertainty in Se (Cornell et al, 2003; Dolšek and Fajfar, 
2007). 

The choice of the computational tool including the setting of possible steering parameters is of course essential 
for the accuracy of the result. The suitability of a method depends on the shape of the failure domain (defined 
by the set of limit state conditions) and the number and type of the (independent) random variables. Problems 
resulting in linear or almost linear limit state formulations are usually easy to solve (e.g. using FORM) while 
difficulties may arise in highly nonlinear and time dependent structural problems, in particular if the decisive 
limit state is reached after a sequence of plastic and brittle local failures. Also, the presence of spatial variability 
may be a complicating and time consuming factor. 

In order to find out whether a method will lead to sufficiently accurate results or not, the characteristics of both 
the computational method and the reliability problem at hand should carefully be investigated. In this respect 
it is difficult to formulate generic rules. It may be helpful to apply several methods and compare the results 
and convergence behavior. The depth of such endeavours may depend on the decision making context.. 

Crude Monte Carlo is generally speaking always reliable and will converge to the exact result, but is of course 
time consuming. An interesting feature however is that the (statistical) uncertainty can be quantified explicitly 
on the basis of probability theory. This uncertainty may the also be incorporated in the final result in a similar 
way as described in EN1990:2023 Annex D for physical experiments.  

5.3  Risk based decision procedures  

In risk management usually the following steps are distinguished: 

Step 1 

Qualitative Risk analysis  

-Description of the system and processes 

-Identification of hazard scenarios, failure modes and consequences 

-Selection of decisions to be taken 

Step 2 

Quantitative Risk Analysis (QRA)  

-Definition of models, quantification of probabilities 

-Quantitative definition of consequences 

98 

 
 
 
-Effects of mitigation measures 

Step 3 

Risk evaluation, optimization, acceptance and communication  

-Human safety, cost efficiency, environmental and sustainability issues  

-Presentation of results and communication with stakeholders 

The essence of a risk based or risk informed design (assessment) is that the decisions are taken with respect 
to  a  set  of  possible  alternatives in  such  a  way  that  the  benefits  minus the  use  of  the  resources  needed  is 
optimal. All models used should not be looked upon as absolute, but depend on the context of this decision 
making space. The steps will be discussed in more detail in the next section. For more information see also (ISO 
2394, IS031000, JCSS Risk analysis). 

5.3.1  Step 1 Qualitative Risk analysis 

As a first step in a risk informed decision making the system under consideration shall be identified, including 
the  environment,  all  its  components  and  processes,  the  limitations  in  space  and  time,  the  relevant  hazard 
scenarios  and  the  possible  direct  and  indirect  consequences.  Special  considerations  shall  be  given  to  the 
interaction  between  the  structure  and  its  environment  which  may  include  natural  constituents,  possible 
mechanical  systems  as  well  as  human  and  organizational  procedures.  Examples  are  operational  and 
environmental loads, aggressive chemicals, human errors, poor quality of design, materials and construction 
etc., individually as well as in combination. 

The  inventory  of  hazard  scenarios  is  in  principle  a  creative  process.  In  order  to  stimulate  these  processes 
systematic approaches like PHA, HAZOP, FMEA, fault and event trees etc. have been invented. A typical approach 
is to consider all relevant variables in the system under more or less normal conditions and answer the question 
what could be the origin of deviations of those values (or range) as well as the direct and indirect consequences. 
Very helpful may also be to use check lists like the examples in Table 5.1. One should however keep in mind 
that these lists always can be incomplete. 

Table 5.1. Overview of hazards. 

Normal loads  Accidental 

/natural 

Accidental 
/manmade 

Human influences  Human Errors 

Self-weight 

Earthquake a) 

Internal explosion 

Mining subsidence 

Design error 

Imposed loads 

Landslide 

External explosion 

Demonstrations 

Material error 

Car park loads 

Hurricane 

Internal fire 

Public disorder 

Construction error 

Traffic  

Tornado 

External fire 

Vandalism 

Misuse 

Snow 

Avalanche 

Impact by vehicle etc   Terrorist attack  

Lack of 

maintenance 

Wind 

Rock fall 

Mining subsidence 

Miscommunication. 

Hydraulic  

High groundwater 

Temperature 

Flood/storm surge 

“Unforeseen” 

Physical attack 

Volcanic eruption 

Chemical attack  Tsunami 

Earthquake a)  

a)  depending on the region it is normal or accidental 

99 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Some comments with respect to this table might be given: 

Generally, the human error is considered as the main cause of accidents. Most researchers give values in the 
order  of  60-90  %.  Errors  may  be  made  during  the  design  (conceptual  errors,  misinterpretations  of  rules, 
calculating  errors,  software  errors,  drawing  errors),  during  execution  (misreading  of  specifications,  bad 
workmanship, inferior materials) and use (operation, inspection, maintenance, refurbishment). The elaboration 
is closely related to EN 1990:2023 Annex B 

The  category  of  vandalism  and  terrorist  attack  is a  type  of  action  where  there  is  typically  a  deliberate 
aspect  involved.  This  complicates  the  modelling  task.  The  intention  of  the  action  is  destruction  and  for  the 
destroyer the strength of a structure is the starting point. To some extent it does not help to make the structure 
stronger, as it can provoke more action on the loading side. Of course, by proper design and robustness, one 
can make it more difficult for persons who want to destroy the structure as a whole.  

A very special category is the so-called unforeseen hazards. Even in the most careful QRA some hazards may 
be  underestimated,  unrecognized  or  objectively  unforeseeable  (sometimes  referred  to  as  black  swans).  By 
definition no specific information is available. The category "unrecognised or ignored" is a kind of human error, 
the “black swan category” is a shortcoming of the whole profession. Due to the fact that unforeseen hazards 
cannot be modelled directly another approach has to be followed. The usual way is to model the possible effect 
of the hazard in the form of a reduction in strength or complete (notional) removal of one or more structural 
members.  Some  of  the  approaches  in  EN  1990:2023  and  EN  1991-1-7  and  corresponding  robustness 
requirements are related to this category. 

Elaborating all hazards in combination with  possible mechanisms in the structure (yielding, rupture, fatigue, 
chemical/physical mechanisms, etc) may lead to undesired behaviour and in the end to failures with serious 
consequences. The local structural damage may be referred to as the direct consequences, the further global 
effects on the structure, the environment, the users as well as the indirect consequences. See Figure 5.1 for 
illustration. Table 5.2 gives an overview of possible final consequences to be considered. 

Table 5.2. Types of the possible consequences to be considered. 

Health 

Cost of damage 

Business 
Continuity 

Environmental  Social and Political 

Reversible 
damage to 
nature 

Irreversible 
damage to 
nature 

Fatalities 

structures 

Loss of income 

surrounding 
properties 

contents 

Injuries 

Damaging vital 
facilities  

(e.g. hospitals),  

Spread of diseases 

Delayed and long-
term health effects 

Psychological 

Loss of customers 

Inability to 
provide vital 
services  

detours and 
delays 

economy of a 
region 

Loss of reputation 

Increase of public 
fears 

Loss of political 
support/. 

stringent new 
measures 

“Blight”/long-term 
evacuation 

5.3.2  Step 2 Quantitative Risk Analysis (QRA) 

In this step the hazards, the behaviour of the structural system and corresponding consequences are modelled 
and all uncertainties are quantified. For the modelling of hazards reference is made to Annex A of this report, 

100 

 
 
 
 
 
 
 
 
 
 
 
although for a number of models a direct further reference to Eurocodes EN 1991, EN 1997 and EN 1998 is 
made. Structural behaviour should be modelled as according to the material related model codes, although in 
some  cases  more  advanced  models  like  present  in  EN  1998  would  be  required.  For  models  related  to 
consequences  (in  terms  of  the  expected  number  of  casualties)  reference  is  made  to  (Tanner  et  al.  2015, 
Hingorani et al. 2019). 

Given all information, the risk R for a certain time reference interval can be assessed as the expected value of 
the  sum  of  the  consequences,  where  the  expectation  operation  is taken  over all  uncertainties affecting  the 
performance of the system: 

nE
R= ∑ Pi Ci
i=1

(5.2) 

where Pi and Ci are the probability and the consequence associated with event i respectively, all events being 
mutually  exclusive.  Both  the  number  of  possible  events,  their  probabilities  as  well  as  the  associated 
consequence generally depend on the decision (design) alternative chosen.  

According to EN 1991-1-7, Annex B, the equation can be developed further to (see also Figure 5.1): 

NS

NH

ND
R= ∑ p(Hi) ∑ ∑ p(Dj|Hi)
j

k=1

i=1

p(Sk|Di)C(Sk) 

(5.3) 

where: 

Hi 

Dj 

hazard scenario (or hazard combination i); 

structural damage state j; 

Sk  

the exceeding of adverse (limit) state k; 

C(Sk) expected consequences related to Sk. 

Additionally, NH is the number of different hazards considered, ND the number of selected damage states and 
NS  the  number  of  adverse  states.  In  principle  all  hazards  Hi  according  to  Table  5.1  and  all  consequences 
according to Table 5.2 should be considered, although in most cases a risk screening process will lead to a focus 
on the most relevant scenarios and consequences. 

Figure 5.1. Illustration of factors risk analysis of structures subject to accidental actions. (a) Identification and modelling 
of relevant accidental hazards. (b) Direct structural damage (often damaged column or floor). (c) Further performance of 
the damaged structure and final indirect consequences. (source Eurocode 1991-1-7) 

101 

 
 
 
 
 
 
In case of human safety, the consequences 5.3 should be read as the expected number of casualties conditional 
upon the exceedance of the  limit state Sk. The total risk R corresponds in that case to expected  number of 
people being killed in the reference period (usually one year). In some cases, there may be a consideration of 
corresponding economic aspects (see Step 3). 

For indicative statistical data, depending for instance on the role and characteristics of the structure, reference 
is made to annex A; some examples may be found in Annex B.  

The  goal in the end is to find an optimal solution, at least one that is feasible. For that reason, one  has to 
evaluate  the  risk  for  a  number  of  design  alternatives.  In  particular  one  has  to  consider  several  levels  and 
methods of risk mitigating measures (see Table 5.3). 

Table 5.3. Examples of risk mitigating measures. 

Elimination or reduction of the hazard 

Protection of the structure, provision of sprinkler system, etc. 

Controlling the hazard, for example, warning systems or monitoring. 

Increased reserves of strength 

Alternative load paths through structural redundancy 

Controlled collapse of where the probability of injury or fatality may be reduced 

5.3.3  Step 3 Decision making, optimization, risk acceptance and communication 

In risk-based decision making one seeks the decision alternative that maximizes the utility of the decision maker 
AND is acceptable in regard to human safety from a societal perspective. In order to assess both criteria one 
has to (see ISO 2394, Annex G): 

 

Identify an optimal decision alternative. The optimisation considers the utility of the decision maker. 
The  utility  might  be  expressed  in  monetary  terms  and  intangible  assets  might  be  translated  to 
monetary  terms  reflecting  their  economic  implication  for  the  decision  maker.  Interest  rates  for 
discounting future costs and benefits should be chosen as representative for the type of economic 
endeavour. 

  Check the optimal decision alternative for societal acceptance. The marginal lifesaving cost principle 
is employed and the cost surrogate for the representation of the societal preference for the prevention 
of the loss of human lives is obtained by the LQI (JCSS, 2008). Interest rates for discounting future 
costs should be chosen as representative for societal economic growth. 

 

Further criteria reflecting codes, standards and legislation might be added. 

As already discussed in section 2.3.2 the utility of the decision maker should include: 

all the relevant benefits; 

all expected resources related to building activities, future maintenance (inspection/ monitoring/repair) and 
dismantling;  

the cost (utility) expectation over all other possible random consequences of structural behaviour.  

As far as human safety is concerned (and maybe also other intangible assets) there is the choice to include 
nominal compensation costs or costs based on a marginal lifesaving cost principle. The first option seems to be 
appropriate for the private decision maker, the second for the public decision maker. In the end of course the 
private decision maker is also bound to public constraints. 

It  can  be  concluded  that  the  marginal  lifesaving  cost  principle  is  applied  in  the  optimization  (by  the  public 
decision  maker)  and  in  the  societal  acceptance  criterion  (usually  proposed  by  NSB’s  and  endorsed  by  the 
relevant authorities). Typical notions related to the LQI (Life Quality Index) are: 

The Societal Willingness To Pay (SWTP) and  

102 

 
The Societal Value of a Statistical Life (SVSL).  

The results of the societal acceptance criteria are often presented in the form of minimum reliability levels (e.g. 
ISO2394, Table G3). Criteria concerning life safety risk acceptance are often specified in terms of the As Low 
As Reasonably Possible/Practicable (ALARP) format. The idea is that risks depending on their magnitude can be 
evaluated  to  be  negligible,  tolerable  or  non-acceptable,  as  shown  in  Figure  5.2  where  the  ALARP  format  is 
illustrated. 

Figure 5.2. Illustration of the ALARP principle in life safety risk regulation.  

Table 5.4 shows an example of corresponding numbers for individual human life safety. In between the upper 
and lower limits, the cost optimization method is acceptable. 

Table 5.4. Overview of risk criteria in terms of annual individual fatality rates. 

Application 

Fatality rate [a-1] 

Upper limit extra ordinary cases 

10-4 

Upper ALARP limit for existing structures and installations  10-5 

Upper ALARP limit for new structures and installations 

3 x 10-6 

Lower ALARP value 

10-6 

It should  be kept in  mind that criteria following from the above notions  refer to averages. This means that 
within a group under consideration there may be persons with a less and a higher fatality rate than the average. 
From an  ethical  point  of  view  it  makes  sense  to  have  an  additional  check  for the  highest  risk  levels  on  an 
individual basis. An individual may be defined either as a real person (going all the time from one location to 
a next one) or as a hypothetical person spending all the time on one place. The second option may be considered 
as the most operational in regulatory concepts (Faber et al 2015). 

In addition to the ALARP the so-called precautionary principle deserves mentioning. This principle has been 
formulated  in  the  context  of  European  risk  regulation.  The  principle  can  be  interpreted  as  a  measure  to 
safeguard society against  (technical) developments  having  disproportionally large consequences for society. 
The precautionary principle finds application in relation to the assessment of activities for which there  is no 
experience and for which there is suspicion that they might have unacceptable adverse consequences. For such 
situations it is required that more knowledge must be collected to ensure that such activities can be sufficiently 
controlled and managed using known best practice technologies.  

An important issue is the scale of an accident. In general, one large accident attacks more attention in the 
news media and the public than a large number of small accidents. These perceptions may be called irrational 
but the effects on the general acceptance cannot be ignored and some extra weight to the consequence of 
large accidents may be needed to reflect the public risk aversion. The most common approach is to take the 
damage not proportional to N but to Nk, where k is a number between 1 and 2, reflecting increasing risk aversion.  

103 

RiskisintolerableRiskistolerable isALARPRiskisbroadlyacceptableThe activitycannotbeacceptedexceptin extraordinarycasesThe activityistolerable only if further reduction of risk is impracticable or its cost is grossly disproportionate to the improvement gainedThe activityisacceptableanditisnot requiredtodemonstrateALARP 
 
Usually and for several reasons a structure is planned for some specified period of time. The principles in this 
chapter may hold for every possible period of anticipated design (working) life. One should however keep in 
mind: 

The actual lifetime is a random variable, also depending on future events and decisions. 

There may be a need for the structure after that period and that makes the design service life as such a 
design parameter in itself. 

In the case of temporary structures, parts of it may be re-used for a next period of time; the continuous 
processes of mounting and demounting may involve special risks. 

There is a limit to the time a structure can serve society in a certain set up; this may lead to demolishing 
(parts of) the structure after a certain period. When designing a structure, options for reuse should be kept 
in mind. 

An essential aspect of risk management is risk communication before, during and after the undesired events. 
One  should  have  in  mind  the  various  parties  involved  who  may  all  have  different  interests  and  different 
perceptions.  The  degree  of  acceptance  normally  depends  on  the  degree  of  exposure,  influence  and 
involuntariness. A particular aspect of risk communication is responsibility. Risk acceptance and correspondence 
is always the responsibility of the authority. The QRA-specialists are responsible only for presenting information 
in a transparent way on the basis of traceable up to date knowledge. 

Finally, it should always be realized and also clearly mentioned that calculated risks are estimates of reality 
based on a given state of information. So, later, it might prove to be incorrect to some extent, but nevertheless 
it should now be accepted as the best basis for making decisions.  

104 

 
 
 
6  Concluding remarks 

Achieving an adequate level of reliability is an essential part of design and assessment of structural systems. 
It  requires  a  thorough  knowledge  of  structural  behaviour  under  the  various  possible  loading  conditions.  In 
addition,  all  disciplines  in  the  field  of  structural  engineering  have  to  deal  with  uncertainties,.  It  is  of  great 
importance  that  in  all  action  and  material  related  Eurocodes  the  treatment  of  uncertainty  is  done  in  a 
transparent and standardized way. Otherwise, misunderstandings between experts working in different fields 
could occur and lead to dangerous or uneconomic situations.  

The basis for the way to deal with uncertainties in structural engineering is formulated in EN 1990:2023, Basis 
of Structural and Geotechnical Design. The present report has been written to give background information to 
the various clauses and annexes of EN 1990:2023 as well as to present other state of the art material that 
might  inform  future  developments.  The  aim  has  been  to  show  how  the  contents  of  EN  1990:2023  are 
compatible with the relevant parts of other Eurocodes and to contribute to a common understanding of basic 
notions like uncertainty, reliability, safety, risk, code calibration and so on. 

It turned out not to be an easy job and in some cases the report can only mention that discrepancies do exists 
between present practice and present knowledge or that present knowledge simply has no adequate answers 
with reliance therefore often placed on continuity of past practices. These of course are the topics for future 
research which hopefully will continue now and in due course support further future evolution of the Eurocode. 
Subject areas where further research could be helpful include: 

Annual versus lifetime reliability requirements; 

Target reliability: required resources versus risk reductions for all types of limit states; 

Use of nonlinear FEM methods; 

Partial Factors for prestressing; 

Background of load combinations; 

Risk based inspection and monitoring; 

More guidance on risk analysis and robustness; 

More background for requirements for existing structures; 

An important (but not new) observation concerns the discrepancies between target and calculated reliabilities 
and real failure frequencies. To some extent these discrepancies are related to so-called “hidden safeties” in 
the design formula and the design process (see chapter 2). It is important to realise these hidden safeties, in 
particular when accepting design methods which may decrease the degree of conservatism like for instance 
certain wind tunnel tests. In those cases one might easily throw away a part of the hidden safety margin, and 
without realising, making the structure less safe than desired. 

The calculations carried out in the process of writing this document gave indications that the recommended 
partial factors indeed on average lead to a reliability index of 3,8 for a 50 year reference period, as was taken 
as a standard. In Annex B example calculations are presented for the derivation of partial factors using the 
methods mentioned in Annex C.4.4 of EN1990:2023. The main conclusion is that NBS’s should only aim for this 
approach  if  a  more  extensive  analysis  is  performed.  However,  also  on  the  level  of  CEN  TC250/SC10  a 
continuation of the work started  in this document is to be recommended. Possible issues for research could 
include  reduction  of  uncertainties  in  climatic  actions,  performing  combined  optimization  on  loading  and 
resistance side, setting new targets related to use of materials and modifying the system of reliability formats.  

An important remark to be made is that he calculations in this report have been carried out using the exact 
definition  of  the  characteristic  values  according  to  EN1990:2023.  In  some  member  states,  however,  values 
specified  in  National  Annexes,  for  example  environmental  actions,  appear  to  have  been  established  on  a 
different basis. This should be taken into account carefully when calibrating corresponding partial factors for 
inclusion in National Annexes. An example is given in Annex B.2.5. 

In order to give guidance to reliability and risk based calculations, whether code oriented (as in Annex B) or for 
specific  applications  (as  in  chapter  5),  a  set  of  probabilistic  models  is  presented  in  Annex  A.  The  models 
presented reflect the state of the art in 2021. It is recommended that research in this field is continued and 
new data be  gathered to  update Annex  A  say every 5 years  in collaboration  with the  JRC. Cooperation with 
organisations like ISO and JCSS could be helpful.  

105 

 
 
References 

Foreword 

Melchers, R.E. and Beck, A. (2007), Structural Reliability Analysis and Prediction, Wiley 

Madsen, H.O, Krenk, S. and Lind, N.C. (2006), Methods of Structural Safety, Dover Publications 

Schneider, J. and Vrouwenvelder, A. (2017), Introduction to Safety and Reliability of structures, IABSE Publication 

2.2 Basics 

Bedford, T., Cooke, R. (2003), Probability risk analysis foundations and methods, Cambridge University Press 

Žižmond,  J.,  and  Dolšek,  M.  (2016),  ‘Evaluation  of  factors  influencing  the  earthquake-resistant  design  of 
reinforced  concrete frames according to Eurocode 8’, Structure and  Infrastructure Engineering: Maintenance, 
Management, Life-Cycle Design and Performance, Vol. 12, Issue 10, pp. 1323–1341.  

Iervolino, I.,  Spillatura,  A.,  and  Bazzurro,  P.  (2018),  ‘Seismic  Reliability  of  Code-Conforming  Italian  Buildings.’ 
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2.3 Decision making 

Vrijling, J. K., Van Gelder, P. H. A. J. M. and Ouwerkerk, S. J. (2005), ‘Criteria for acceptable risk in the 
Netherlands’, in: Taylor, C. and Van Marcke, E. (eds.), Infrastructure Risk Management Processes: Natural, 
Accidental, and Deliberate Hazards, American  Society of Civil Engineers, pp. 143-158.  

van Manen, S.E., Vrouwenvelder, A. and van Foeken, R.J. (1998), ‘Probabilistic Design of the Steel Structure of 
the  New  Waterway  Storm  Surge  Barrier’,  in:  P.  Thoft-Christensen  (ed),  Industrial  Application  of  Structural 
Reliability Theory, ESREDA - European Safety Reliability & Data Association, Det Norske Veritas, pp. 187-204 

WG6, T2 (2019), Robustness Rules for material related Eurocode parts, CEN/TC 250/WG 2/T6 

Rouger,  F.  (2017),  ‘Volume  and  stress  distribution  effects’,  in:  Blaß,  H.  J.,  and  Sandhaas,  C.  (eds),  Timber 
engineering - Principles for design, KIT Scientific Publishing 

2.6 Partial factors 

Ferry Borges, J. and Castanheta, M. (1971), Structural Safety, Course 101, 2nd edition, LNEC, Lisbon 

Faber, M, Koehler J. and Sorensen J.D. (2018), Framework for the reliability based assessment of load partial 
safety factors in EC CEN TC250/SC10 

Meinen, M.E and Steenbergen, R.D.J.M. (2018), ‘Reliability levels obtained by Eurocode partial factor design; a 
discussion on current and future reliability levels’, Heron, Vol. 63, Issue 4, TNO/TU Delft 

Franchin, P. and Noto, F. (2023), ‘Reliability-based partial factors for seismic design and assessment consistent 
with second-generation Eurocode 8’, Earthquake Engineering and Structural Dynamics, Vol. 52,  pp. 4026–4047  

Madsen, H.O., Krenk, S. and Lind, N.C. (1986), Methods of structural safety, Prentice Hall 

Gayton,  N.,  Chateuneuf,  A.,  Sorensen  J.D.,  Pendola,  A  M.  and  Lemaire,  M.  (2004),  ‘Calibration  methods  for 
reliability-based design codes’, Structural Safety, Vol. 26, Issue 1, pp. 91-121 

Silva, R.C., Cremona, Ch. (2014), ‘Safety factor calibration for bridge concrete girders’, Structural Engineering 
and Mechanics, Vol. 49, Issue 2, pp.163-182 

Baravalle, M. (2017), Risk and reliability based decision making of structural design codes, Thesis, Norwegian 
University of Science and Technology, Trondheim 

3.1 History 

CEN - European Committee for Standardisation (1994), Eurocode 1- Basis of design and actions on structures 
– Part 1: Basis of design, ENV 1991-1 

CEN - European Committee for Standardisation (2010), Eurocode – Basis of structural design, EN 1990: 2002 
+ A1: 2005 + A1: 2005/AC: 2010 

106 

 
 
Sörensen, J. D. (2004), Notes in structural reliability Theory and risk analysis, Aalborg University, Aalborg  

Gulvanessian, H., Calgaro, J-A., Holicky, M. (2012), Designers guide to Eurocode basis-of-structural design 2nd 
edition, ICE – Institution of Civil Engineers, London 

DIN  -  Deutsches 
Sicherheitsanforderungen für bauliche Anlagen, Beuth Verlag, Berlin 

Institut  für  Normung  e.V.,  GruSibau 

(1981),  Grundlagen  zur  Festlegung  von 

Cornell, C. A. (1969), ‘A probability based structural code’, Journal of the American Concrete Institute, Vol. 66, 
Issue 12, pp. 974-985 

Kjerengtroen. L. and Comer, J. (1996), ‘Probabilistic methods in design – an Overview of current technologies’, 
Journal of Commercial Vehicles, Vol. 105, Section 2, pp 244 – 251 

Hasofer,  A.  M.  and  Lind,  N.  C.  (1974),  ‘Exact  and  invariant  second  moment  code  format’,  Journal  of  the 
Engineering Mechanics Division of American Society of Civil Engineers, Vol. 100, Issue 1, pp. 111-121 

König,  G.,  Hosser,  D.  and  Schobbe,  W.  (1982),  ‘Sicherheitsanforderungen  für  die  Bemessung  von  baulichen 
Anlagen nach den Empfehlungen des NA Bau – eine Erläuterung’, Bauingenieur, vol. 57 pp. 69-78 

Rackwitz, R. and Fiessler, B. (1978), ‘Structural Reliability under combined random load sequences’, Computers 
and structures, Vol. 9, Issue 5, pp. 489-494  

Ferry Borges, J. and Castanheta, M. (1971), Structural Safety, Course 101, 2nd edition, LNEC, Lisbon 

Turkstra, C. J. and Madsen, H. (1980), ‘Load combinations in codified structural design’, Journal of the Structural 
Division of American Society of Civil Engineers, Vol. 106, Issue 12, pp. 2527-2543 

Vrouwenvelder, T. (2002), ‘Reliability based code calibration the use of the JCSS probabilistic model code’, Proc. 
of JCSS Workshop on Code Calibration, Zürich 

Vrouwenvelder,  T.  and  Gulvanessian  H.  (1996),  Basis  of  Design  and  Actions  on  Structures:  Background  and 
Application of Eurocode 1, Delft  

ISO – International Organisation for Standardisation (2015), General principles on reliability for structures, ISO 
2394  

JCSS  (2001),  JCSS  Probabilistic  Model  Code,  JCSS  web  site,  https://www.jcss-lc.org/jcss-probabilistic-model-
code/  

3.3 Target 

Hingorani, R., Tanner, P. and Zanuy, C. (2019), ‘Life safety risk-based requirements for concrete structures in 
accidental situations caused by gas explosions’, Structural Safety, Vol. 76, pp. 184-96. 

3.4 Code calibration 

Köhler, J. and Baravalle, M. (2019), ‘Risk-based decision making and the calibration of structural design codes 
– prospects and challenges’, Civil Engineering and Environmental Systems, Vol. 36, Issue 1, pp. 55-72.  

3.5.4 Prestress 

Ferry Borges, J. and Castanheta, M. (1971), Structural Safety, Course 101, 2nd edition, LNEC, Lisbon 

Mehlhorn, G. and Curbach, M. (2014), Handbuch Brücken, Springer Verlag 

3.5.6 Fire 

European  Commission:  Directorate-General  for Research  and Innovation,  Cajot,  L.,  Schleich, J.  and  Pierre,  M. 
(2002), Competitive steel buildings through natural fire safety concepts – Final report, Publications Office 

CEN - European Committee for Standardisation (1994), Eurocode 1- Basis of design and actions on structures 
– Part 1: Basis of design, ENV 1991-1 

Bub, H. (1979), Baulicher Brandschutz im Industriebau, Beuth Verlag, Berlin 

CEN - European Committee for Standardisation (2002), Fire tests on elements of building construction, Part 4 
Test method for determining the contribution to the fire resistance of structural members by applied protection 
to steel structural elements, CEN ENV 13381-4 

107 

 
CEN  -  European  Committee  for  Standardisation  (2020a),  Eurocode  3  –  Design  of  steel  structures  -  Part1-1 
General rules and rules for buildings, prEN 1993-1-1 

3.5.7 Seismic design situation 

Biskinis, D., Fardis, M., and Roupakias G. (2004), ‘Degradation of Shear Strength of Reinforced Concrete’, ACI 
Structural Journal, Vol. 101, Issue 6, pp. 773-783 

Biskinis, D. and Fardis, M. (2020), ‘Cyclic Shear Resistance Model for Eurocode 8 Consistent with the Second-
Generation Eurocode 2’, Bulletin of Earthquake Engineering, Vol. 18, Issue 6, pp. 2891–2915. 

Fardis, M. (2018), ‘Capacity Design: Early History’, Earthquake Engineering & Structural Dynamics, Vol. 47, Issue 
14, pp. 2887–96. 

Fajfar, P. (2018), ‘Analysis in Seismic Provisions for Buildings: Past, Present and Future.’ Bulletin of Earthquake 
Engineering, Vol. 16, Issue 7, pp. 2567–2608. 

Blume, J.A., Newmark, N.M. and Corning, L.  H. (1961), Design of Multistory Reinforced  Concrete Buildings for 
Earthquake Motions. Portland Cement Association, Skokie 

Franchin, P., and Pagnoni, T. (2018). ‘A general model of resistance partial factors for seismic assessment and 
retrofit’, in Proceedings of the 16th European conference on earthquake engineering, Thessaloniki, Greece. 

3.6.1 Geotechnics 

Bond, A. and Harris, A. (2008), Decoding Eurocode 7, Taylor & Francis, London 

Calle, E.O.F, Schweckendiek, T. and Kanning, W. (2020), ‘Background of characteristic values for geotechnical 
parameters, Deltares report number TBD (document in progress). 

Hicks, M. A., Varkey, D., van der Eijnden A. de Gast, T. and Vardon, Ph. J. (2019), ’On characteristic values and the 
reliability-based assessment of dykes’, Georisk: Assessment and Management of Risk for Engineered Systems 
and Geohazards, Vol. 13, Issue 4, pp. 1-7 

ISO – International Organisation for Standardisation (2015), General principles on reliability for structures, ISO 
2394  

International Society of Soil Mechanics and Geotechnical Engineering (ISSMGE) - Technical Committee TC 304 
‘Engineering  Practice  of  Risk  Assessment  and  Management’  (2020),  State-of-the-art  review  of  inherent 
Download: 
variability 
http://140.112.12.21/issmge/2021/SOA_Review_on_geotechnical_property_variablity_and_model_uncertainty.
pdf 

geotechnical 

uncertainty 

properties 

models, 

and 

and 

in 

Prästings, A., Spross, J. and Larsson, S. (2018), ‘Characteristic values of geotechnical parameters in Eurocode 
7’, Geotechnical Engineering, Vol. 172, Issue 4, pp. 1-21 

Arnold, P., Fenton, G.A., Hicks, M.A., Schweckendiek, T. and Simpson, B. (2013), Modern Geotechnical Design Codes 
of Practice - Implementation, Application and Development, IOS Press 

3.6.2 Glass 

CEN  -  European  Committee  for  Standardisation  (2021a),  Design  of  Glass  Structures  -  Report  of 
CEN/TC250/SC11, prCEN/TS 19100  

Feldmann  M,  Kaspar  R,  Abeln  B,  Gessler  A,  Langosch  K,  Beyer  J,  Schneider  J,  Schula  S,  Siebert  G,  Haese  A, 
Wellershoff F, Cruz P, Belis J, Colvin J, Morgan T, Ensslen F, Eliasova M, Šulcová Z, Royer-Carfagni G, Galuppi L, 
Grenier  C,  Hoegner  H,  Kruijs  R,  Louter  C,  Manara  G,  Neugebauer  J,  Rajcic  V,  Zarnic  R.  (2014),  Guidance  for 
European  Structural  Design  of  Glass  Components.  EUR  26439.  Publications  Office  of  the  European  Union, 
Luxembourg, JRC86637  

Schneider,  J.  (2001),  Festigkeit  und  Bemessung  punktgelagerter  Gläser  und  stossbeanspruchter  Gläser,  PhD 
Dissertation, Technische Universität Darmstadt 

Nielsen,  J.,  Olesen,  J.  and  Stang  H.  (2009),  ‘The  fracture  process  of  tempered  soda-lime-silica  glass’, 
Experimental Mechanics, Vol 49, Issue 6, pp. 855-870 

108 

 
CEN  -  European  Committee  for  Standardisation  (2000a),  Glass  in  building  -  Determination  of  the  bending 
strength of glass - Part 2: Coaxial double ring test on flat specimens with large test surface areas, EN 1288-
2:2000-09 

CEN  -  European  Committee  for  Standardisation  (2000b),  Glass  in  building  -  Determination  of  the  bending 
strength of glass - Part 3: Test with specimen supported at two points (four point bending), EN 1288-3:2000-
09 

Danzer, R. (2006), ‘Some notes on the correlation between fracture and defect statistics: Are Weibull statistics 
valid for very small specimens?’, Journal of the European Ceramic Society, Vol. 26, Issue 15, pp. 3043–3049 

Yankelevsky, D. (2014), ‘Strength prediction of annealed glass plates—A new model’, Engineering Structures, 
Vol. 79, pp. 244–255  

Pisano,  G.  and  Royer  Carfagni,  G.  (2015),  ‘The  statistical  interpretation  of  the  strength  of  float  glass  for 
structural applications’, Construction and Building Materials, Vol. 98, pp. 741–756  

CEN  -  European  Committee  for  Standardisation  (2006),  An  overview  of  prEN-13474  and  the  work  of 
CEN/TC129/WG8 from which it was developed’, Brussels 

Ballarini, R., Pisano, G. and Royer Carfagni, G. (2016), ‘The Lower Bound for Glass Strength and Its Interpretation 
with Generalized Weibull Statistics for Structural Applications’, Journal of the Engineering Mechanics Division of 
American Society of Civil Engineers, Vol. 142, Issue 12 

CEN  -  European  Committee  for  Standardisation  (2012),  Glass  in  building  –  Basic  soda  lime  silicate  glass 
products – Part 2: Float glass, EN 572-2:2012 

Madjoubi,  M.,  Bousbaa,  C.,  Hamidouche,  M.,  and  Bouaouadja,  N.  (1999),  ‘Weibull  statistical  analysis  of  the 
mechanical strength of a glass eroded by sand blasting’, Journal of the European Ceramic Society, Vol. 19, Issue 
16, pp. 2957–2962 

Wellershoff, F. (2006), Nutzung der  Verglasung zur  Aussteifung von Gebäudehüllen, PhD  Dissertation, RWTH 
Aachen 

Feldmann, M. (2020), Glass in Structures – Resistance & Safety (Presentation on AHG – Reliability Meeting 5th 
February 2020) 

Ballarini, R., Pisano, G., Royer Carfagni, G. (2016), ‘New calibration of partial material factors for the structural 
design of float glass. Comparison of bounded and unbounded statistics for glass strength’, Construction and 
Building Materials, Vol. 121, pp. 69-80 

Fink,  A.  (2000),  Ein  Beitrag  zum  Einsatz  von  Floatglas  als  dauerhaft  tragender  Konstruktionswerkstoff  im 
Bauwesen, PhD Dissertation, TU Darmstadt 

Schula,  S.  (2015),  Charakterisierung  der  Kratzanfälligkeit  von  Gläsern  im  Bauwesen,  PhD  Dissertation,  TU 
Darmstadt  

CEN - European Committee for Standardisation (2010), Eurocode – Basis of structural design, EN 1990: 2002 
+ A1: 2005 + A1: 2005/AC: 2010 

Laufs,  W.  (2000),  Ein  Bemessungskonzept  zur  Festigkeit  thermisch  vorgespannter  Gläser,  PhD  Dissertation, 
RWTH Aachen 

Pisano, G., Royer Carfagni, G., Schneider, J. (2019), ‘Statistical interference of crack healing on the strength of 
thermally-treated glass’, Engineering Fracture Mechanics, Vol. 205, pp. 511-531 

Sedlacek, G., Blank, K., Güsgen, J., and Laufs, W. (1999), Glas im konstruktiven Ingenieurbau. Ernst & Sohn, Berlin 

Zaccaria, M. and Overend, M. (2020), ‘Nondestructive Safety Evaluation of Thermally Tempered Glass’, Journal 
of Materials in Civil Engineering, Vol. 32, Issue 4  

Pisano, G. and Royer Carfagni, G. (2017), ‘Statistical interference of material strength and surface prestress in 
heat-treated glass’, Journal of the American Ceramic Society, Vol. 100, pp. 954–967 

Bonati, A., Occhiuzzi, A., Pisano, G., Royer Carfagni, G. (2018), ‘A micro-mechanically motivated model for the 
strength of heat-treated glass’, Glass Structures and Engineering, Vol. 3, Issue 2, pp. 153-166 

Zaccaria,  M.,  Overend,  M.  (2016),  ‘Thermal  healing  of  realistic  flaws  in  glass’,  Journal  of  Materials  in  Civil 
Engineering, Vol. 28, Issue 2, 04015127/1-9 

109 

 
Datsiou, K. (2017), Design and Performance of Cold Bent Glass, PhD Dissertation, University of Cambridge 

Datsiou K., Overend M. (2017a), ‘The Strength of Aged Glass’, Glass Structures and Engineering, Vol. 2, pp. 105–
120 

Datsiou K., Overend M. (2017b), ‘Ageing of glass with sand abrasion’, Construction and Building Materials, Vol. 
142, pp. 536–551. 

Hilcken, J. (2015), Zyklische Ermüdung von thermisch entspanntem und thermisch vorgespanntem Kalk-Natron-
Silikatglas (Cyclic fatigue of annealed and tempered soda-lime glass), Dissertation, TU Darmstadt 

Pisano, G., Royer-Carfagni, G., Schneider, J. (2019), ‘Open issues in the calibration of partial safety factors for 
heat-treated glass’, Structural Safety, Vol. 79, pp. 1-11 

3.6.3 PFC Fibre-Polymer Composites  

Sá, M., Pacheco, J., Correia J. R., Silvestre, N. and Sørensen, J. D. (2021a), ’Structural safety of pultruded FRP 
profiles  for  global  buckling.  Part  1:  Approach  to  material  uncertainty,  resistance  models,  and  model 
uncertainties’, Composite Structures, Volume 257, 2021 

Pacheco, J., Sá, M., Correia J. R., Silvestre, N. and Sørensen, J. D. (2021b),  ’Structural safety of pultruded FRP 
profiles  for  global  buckling.  Part  Part  2:  Reliability-based  evaluation  of  safety  formats  and  partial  factor 
calibration’, Composite Structures, Volume 257, 2021 

3.7 Checking deformation capacity / brittle behaviour 

Kühnemund,  F.  (2003),  Zum  Rotationsnachweis  nachgiebiger  Knoten  im  Stahlbau,  Dissertation,  Institut  für 
Konstruktion und Entwurf, Universität Stuttgart, Mitteilungen Nr. 2003-1 

U.  Kuhlmann,  M.  Feldmann,  J.  Lindner,  C.  Müller  and  R.  Stroetmann  (2014),  Eurocode  3:  Bemessung  und 
Konstruktion  von  Stahlbauten,  DIN  EN  1993-1-1  mit  Nationalem  Anhang,  Kommentar  und  Beispiele,  vol.  1, 
Beuth Verlag, Berlin 

Schäfer, M. (2005), Zum Rotationsnachweis teiltragfähiger Verbundknoten in verschieblichen Verbundrahmen, 
PhD Dissertation, University of Stuttgart 

Demonceau, F., Marginean, I., Golea, T., Jaspart, J. P., Santiago, A., Santos, A. F., Simoes da Silva, L., Elghazouli, 
A., Khalil, Z, Kuhlmann, U., Skarmoutsos, G., Baldassino, N., Zandonini, R., Zordan, M., Dubina, D., Dinu, F., Obiala, 
R. and Candeias, M. (2021), FAILNOMORE - Mitigation of the risk of progressive collapse in steel and composite 
building frames under exceptional events, Deliverable 1-2 - Background document (not published),’ European 
Commission, Brussels 

Huber,  G.,  Kuhlmann,  U.  and  Schäfer,  M.  (2004),  ‘Rotation  capacity  of  joints’,  in:  Improvement  of  Buildings´ 
Structural Quality by New Technologies, Final report, COST Action C12 

Keller, N. (2019). Robustheit von Stahl-und Verbundrahmen durch gezielte Knotenausbildung, PhD Dissertation, 
University of Stuttgart 

Biskinis, D. and Fardis, M. (2010a) ‘Deformations at flexural yielding of members with continuous or lap-spliced 
bar’, Structural Concrete, Vol. 11, Issue 3, pp. 127-138 

Biskinis, D. and Fardis, M. (2010b), ‘Flexure-Controlled Ultimate Deformations of Members with Continuous or 
Lap-Spliced Bars’, Structural Concrete, Vol. 11, Issue 2, pp. 93–108. 

Fajfar, P. (2000) ‘A Nonlinear Analysis Method for Performance-Based Seismic Design,’ Earthquake Spectra, Vol. 
16, Issue 3, pp. 573–92. 

Freeman,  S.A.  (1978).  ‘Prediction  of  Response  of  Concrete  Buildings  to  Severe  Earthquake  Motion’,  in: 
Proceedings of Douglas McHenry International Symposium on Concrete and Concrete Structures, Publication SP-
55, American Concrete Institute, Detroit, Michigan, U.S.A. 

Grammatikou, S., Biskinis D., and Fardis, M. (2018a), ‘Flexural Rotation Capacity Models Fitted to Test Results 
Using Different Statistical Approaches’, Structural Concrete, vol. 19, Issue 2, pp. 608–24. 

Grammatikou, S., Biskinis D., and Fardis, M. (2018b), ‘Models of the Flexure-Controlled Strength, Stiffness and 
Cyclic  Deformation  Capacity  of  Rectangular  RC  Columns  with  Smooth  Bars,  Including  Lap-Splicing  and  FRP 
Jackets’, Bulletin of Earthquake Engineering, Vol. 16, Issue 1, pp. 341–75. 

110 

 
Lignos,  D.  and  Krawinkler,  H.  (2011),  ‘Deterioration  Modeling  of  Steel  Components  in  Support  of  Collapse 
Prediction of Steel Moment Frames under Earthquake Loading’, Journal of Structural Engineering, Vol. 137, Issue 
11, pp. 1291–1302. 

Lignos,  D.  and  Krawinkler,  H.  (2012),  ‘Development  and  Utilization  of  Structural  Component  Databases  for 
Performance-Based Earthquake Engineering’, Journal of Structural Engineering, Vol. 139, Issue 8, pp. 1382–94. 

Panagiotakos,  T.  B.,  and  Fardis,  M.  (2001),  ‘Deformations  of  RC  Members of  Infilled  Frames  at  Yielding  and 
Ultimate’, ACI Structural Journal, Vol. 98, Issue 2, pp. 135–48. 

Izzuddin, B., Vlassis, A., Elghazouli, A. and Nethercot, D. (2008), ‘Progressive Collapse of Multi‐Storey Buildings 
due to Sudden Column Loss — Part I: Simplified Assessment Framework’, Engineering Structures, Vol. 30, Issue 
5, pp. 1308‐1318. 

O’Connor, A., Izzuddin, B., Molkens, T., Belletti, B. and Martinelli, P. (2021), Final Report of Project Team WG6.T2 
Robustness Rules in Material Related Eurocode Parts’ 

Hoffmann,  N.,  Kuhlmann,  U.  and  Skarmoutsos,  G.  (2018),  ‘Design  of  steel  and  composite  structures  for 
robustness’, in: IALCCE. The Sixth International Symposium On Life-Cycle Civil Engineering, Ghent, Belgium 

Bigaj van Vliet, A. J. (1999), Structural dependence of rotation capacity of plastic hinges in RC beams and slabs, 
PhD Dissertation, TU Delft 

3.9.1 Fatigue  

DNVGL-RP-C210  (2015),  Probabilistic  methods  for  planning  of  inspection  for  fatigue  cracks  in  offshore 
structures, DNV GL AS, Recommended Practice 

DNVGL-RP-C203 (2016), Fatigue design of offshore steel structure, DNV GL AS, Recommended Practice 

Zhang, Y-H. and Maddox S.J. (2009), ‘Investigation of fatigue damage to welded joints under variable amplitude 
loading spectra’, International Journal of Fatigue, Vol. 31, pp. 138–152 

D’Angelo, L. and Nussbaumer,  A. (2017),  ‘Estimation of  fatigue S-N  curves of welded joints using advanced 
probabilistic approach’, International Journal of Fatigue, Vol. 97  

Leonetti,  D.,  Maljaars,  J.  and  Snijder,  H.  H.  (2017),  ‘Fitting  fatigue  test  data  with  a  novel  SN  curve  using 
frequentist and Bayesian inference’, International Journal of Fatigue, Vol. 105 

Leonetti, D., Maljaars, J. and Snijder, H. H. (2020), ‘Probabilistic fatigue resistance model for steel welded details 
under variable amplitude loading – Inference and uncertainty estimation’, International Journal of Fatigue, Vol. 
135 

Baptista, C., Reis, A. and Nussbaumer, A. (2017), ‘Probabilistic S-N curves for constant and variable amplitude’, 
International Journal of Fatigue, Vol. 101, pp. 312–327 

Maljaars,  J.  (2020),  ‘Evaluation  of  traffic  load  models  for  fatigue  verification  of  European  road  bridges’, 
Engineering Structures, Vol. 225 (2020) 111326 

Maljaars, J., Leonetti, D., Hashemi, B. and Snijder, H. H. (2022), ‘Systematic derivation of safety factors for the 
fatigue design of steel bridges’, Structural Safety, Vol. 97, 102229J.  

3.9.2 Durability Concrete 

Brite EuRam III (2000a), Probabilistic calculations. Duracrete – Probabilistic performance based durability design 
of concrete structures, BE95-1347/R12-13 

Brite EuRam III (2000b), Benchmarking of the conventional design methodology, BE95-1347/R10, 2000 

Val,  D.,  Andrade,  C.  and  Vrouwenvelder,  A.  (2019),  Section  2.21  ‘Environmental  exposure’,  JCSS  Probabilistic 
Model Code, draft 12 January 2019.  

fib (2016), Partial factor methods for existing concrete structures, fib Bulletin, Vol. 80 

fib (2010), fib model code, Ernst and Sohn 

3. 10 Robustness 

111 

 
Ellingwood, B. and Leyendecker, E.V. (1978), ‘Approaches for design against progressive collapse’, Journal of 
the Structural Division, vol 104, Issue 3, pp. 413-423.  

Gross, J.L. and McGuire, W. (1983), ‘Progressive collapse resistant design’, Journal of Structural Engineering, Vol. 
109, Issue 1, pp. 1-15.  

Dusenberry,  D.O.  and  Juneja,  G.  (2002),  ‘Review  of  existing  guidelines and  provisions  related  to  progressive 
collapse’, in: Workshop on prevention of progressive collapse (Vol. 92), National Institute of Building Sciences, 
Washington, USA 

Tanner, P. and Hingorani, R. (2015), ‘Acceptable risks to persons associated with building structures’, Structural 
Concrete, vol. 16, Issue 3, pp. 314-322.  

Russell, J.M., Sagaseta, J., Cormie, D. and Jones, A.E.K. (2019), ‘Historical Review of Prescriptive Design Rules for 
Robustness After the Collapse of Ronan Point’, Structures, Vol. 20, pp. 365-373.  

3.11 Observational method 

Peck,  R.B.  (1969),  ‘Advantages  and  limitations  of  the  observational  method  in  applied  soil  mechanics’, 
Géotechnique, Vol. 19, p. 171–87. 

Spross, J., Johansson, F. (2017), ‘When is the observational method in geotechnical engineering favourable?’, 
Structural Safety, Vol. 66, pp. 17-26. 

International Society of Soil Mechanics and Geotechnical Engineering (ISSMGE) - Technical Committee TC 304 
‘Engineering  Practice  of  Risk  Assessment  and  Management’  (2020),  State-of-the-art  review  of  inherent 
variability 
Download: 
http://140.112.12.21/issmge/2021/SOA_Review_on_geotechnical_property_variablity_and_model_uncertainty.
pdf 

geotechnical 

uncertainty 

properties 

models, 

and 

and 

in 

4. Existing structures 

CEN  -  European  Committee  for  Standardisation  (2021b),  Report  on  requirements  and  guidance  for  the 
development of material-specific assessment and retrofit provisions, CEN TC 250/WG2/WG2.T2 Project Team 
Version, Final Document, CEN, Brussels 

CEN - European Committee for Standardisation (2021c), Eurocode – Basis of assessment and retrofitting of 
existing  structures:  general  rules  and  actions,  prEN  1990-2  2021,  CEN  TC  250/WG2/WG2.T2  Project  Team 
Version, Final Document, CEN, Brussels 

CEN  -  European  Committee  for  Standardisation  (2021d),  Eurocode  8  –  Design  of  Structures  for  earthquake 
resistance, Part 3: Assessment and retrofitting of buildings, prEN 1998-3 2021, CEN, Brussels 

CEN  -  European  Committee  for  Standardisation  (2020b),  Technical  Specification  –  Assessment  of  existing 
structures, CEN/TS 17440, CEN, Brussels 

CEN - European Committee for Standardisation (2020c), Eurocode  – Basis of structural and geotechnical design, 
prEN 1990:2020 (E), CEN TC 250 N2555 Enquiry Draft, CEN, Brussels 

ISO – International Organization for Standardisation (2015), General principles on reliability for structures, ISO 
2394:2015, ISO, Geneva 

JCSS  –  Joint  Committee  on  Structural  Safety  (2000)  Assessment  of  Existing  Structures,  RILEM  Publications 
S.A.R.L. 

JRC 2015, JRC Science and Policy Report, New European technical rules for the assessment and retrofitting of 
existing structures, European Commission, Joint Research Centre, JRC, Ispra, 2015, ISBN 978-92-79-46022-7, 
doi: 10.2788/052881. 

Luechinger, P., Fischer, J., Chrysostomou, C., Dieteren, G., Landon, F., Leivestad, S., Malakatas, N., Mancini, G., 
Markova, J., Matthews, S., Nolan, T., Nuti, C., Osmani, E., Ronnow, G., Schnell, J., Tanner, P. (2015), New European 
Technical Rules for the Assessment and Retrofitting of Existing Structures, EUR 27128, Publications Office of 
the European Union, Luxembourg, JRC94918 

5. Risk analysis 

ISO – International Organization for Standardisation (2018), Risk management, ISO 31000, ISO, Geneva 

112 

 
JCSS  –  Joint  Committee  on  Structural  Safety  (2008),  ‘Risk  Assessment  in  Engineering  -  Principles,  System 
Representation 
https://www.jcss-
lc.org/publications/raie/01_jcss_riskassessment.pdf  

Criteria’, 

website, 

2008, 

JCSS 

June 

Risk 

& 

Schneider,  J.  and  Vrouwenvelder,  A.,  (2017),  Introduction  to  Safety  and  Reliability  of  structures,  IABSE 
Publications, Zürich 

Faber  M.  H.,  Sørensen,  J.  D.  and  Vrouwenvelder,  T.  (2015),  ‘On  the  Regulation  of  Life  Safety  Risk’,  in:  12th 
International Conference on Applications of Statistics and Probability in Civil Engineering - ICASP12, Vancouver, 
Canada 

Tanner, P. and Hingorani, R. (2015), ‘Acceptable risks to persons associated with building structures’, Structural 
Concrete, Vol. 16, Issue 3, pp. 314-322 

Hingorani,  R.,  Tanner,  P.,  Prieto,  M.  and  Lara,  C.  (2020),  ‘Consequence  classes  and  associated  models  for 
predicting loss of life in collapse of building structures’, Structural Safety, Vol. 85, Issue 1, 101910 

Allin Cornell, C., Jalayer, F, Hamburger, R.O., and Foutch, D.A. (2002), ‘Probabilistic Basis for 2000 SAC Federal 
Emergency Management Agency Steel Moment Frame Guidelines’, Journal of Structural Engineering, Vol. 128, 
Issue 4, pp. 526-533.  

Dolšek, M. and Fajfar, P. (2007), ‘Simplified probabilistic seismic performance assessment of plan-asymmetric 
buildings’, Earthquake Engineering and Structural Dynamics, Vol. 36, Issue 13, pp. 2021-2041 

113 

 
 
 
 
 
 
List of abbreviations and definitions 

AHG 

Ad Hoc Group 

ALARP 

As low as reasonable possible/practicable 

CC 

CEN 

EN 

Consequence class 

Comité Européen de Normalisation (European Committee for Standardization) 

European Standard 

ENV  

European Preliminary Standard 

FORM 

First Order Reliability Method 

ISO 

MC 

International Standard Organisation 

Monte Carlo 

NBS 

National Bureau of Standards 

QA 

SC 

SLS 

TC 

UC 

ULS 

WG 

Quality Assurance 

Subcommittee 

Serviceability Limit State   

Technical Committee 

Unity Check 

Ultimate Limit State 

Working Group 

114 

 
 
List of boxes 

Example 2.1. Time varying load, no deterioration ...................................................................15 

Example 2.2. Deterioration ............................................................................................15 

Example 2.3 Inspection for fatigue...................................................................................16 

Example 2.4 Simple code calibration example ......................................................................21 

Example: Plastic design of steel structures ..........................................................................69 

Example from Eurocode 8 ............................................................................................72 

115 

 
 
List of figures 

Figure 1.1. Basic scheme of calibration procedure: the first step is to determine a target reliability for an 
appropriate class of structures based on a combination of several criteria, the second step is to find 
corresponding partial factors. ......................................................................................... 5 

Figure 2.1. Bathtub curve (black line) showing the conditional failure probability as a function of time, with 
zones of (I) increasing reliability (dominant time independent uncertainties like resistance and self-weight), (II) 
constant reliability (dominating time dependent uncertainties, usually loads) and (III) decreasing reliability 
(dominating deterioration). The blue line shows the unconditional failure probabilities. Note that the annual 
reliability index has just the opposite profile: it first goes up, is constant and then goes down. ................16 

Figure 2.2. Resulting reliability index for steel members, designed according to the Eurocodes,, for the limit 
state of bending under self-weight, permanent and imposed loads for a 1 (red) and 50 (blue) year reference 
period as a function of Q which represents the ratio of the variable and total load; the dashed and dotted 
lines refer to a zero and maximum contribution of permanent loads respectively; the full lines show the 
average value in the aria of interest (illustrative example only) ...................................................20 

Figure 2.3. Ferry Borges and Castanheta load model (FBC) for imposed loads (sustained and intermittent 
components) (illustrative example) ..................................................................................20 

Figure 3.1. Overview of reliability methods ........................................................................24 

Figure 3.2. Definition of Bias according to EN1990:2023 Annex D ..............................................36 

Figure 3.3. Results of reliability indices for 50 years, for each loading type and each material; calculations 
based on design format prEN 1990:2023 (8.12) using partial factors according to the Eurocodes. ...........40 

Figure 3.4 Reliability index β as function of αQ for snow load, each line representing a specific material (steel 
concrete. etc). ..........................................................................................................40 

Figure 3.5 Considered cases for the single source discussion. ....................................................43 

Figure 3.6 Bow-string bridge with inclined, prestressed hangers. ................................................47 

Figure 3.7 The membrane structure Palais Thermal Bad Wildbad. ...............................................48 

Figure 3.8  Acceleration (a) and displacement (b) response spectra for an earthquake event (a single M, R 
pair). ....................................................................................................................51 

Figure 3.9  Seismic hazard curves for spectral acceleration at different vibration periods (a), uniform hazard 
spectrum for a fixed return period TR (b) and analytical approximation of the uniform hazard spectrum in the 
code (c). ................................................................................................................52 

Figure 3.10  Deformation resulting from support motion (a), deformation resulting from (uniformly 
distributed) equivalent lateral forces (b), deformation resulting from more "realistic" lateral force distribution 
(also in terms of intensity) (c). ........................................................................................53 

Figure 3.11  . Base shear vs top-displacement curve of a structure with inadequate ductility (a), soft-storey 
plastic mechanism (b) and beam-sway plastic mechanism (c). ....................................................54 

Figure 3.12  Reduction of design base shear as a function of ductility, redundancy and overstrength. ......54 

Figure 3.13. Definition of chord rotation for a cantilever column (a), a fixed-end column (b) and beams and 
columns in a frame (c). ................................................................................................54 

Figure 3.14. Moment-rotation and definition of ultimate rotation θu. ...........................................55 

Figure 3.15. Range of annual failure probabilities .................................................................56 

Figure 3.16. Examples of failure modes with and without spatial averaging. The slope failure surface (left) 
cuts through the entire subsoil layer underneath the embankment, hence, the average shear strength along 
the failure surface is to be used for the characteristic value. On the other hand, internal erosion under a dam 
(right) is dominated by the weakest path (right hand side source: Internet). .....................................58 

116 

 
Figure 3.17. Pre-stress distribution across thickness in the centre of a glass pane ............................61 

Figure 3.18. Normal distribution, lognormal distribution, Weibull distribution (from left to right) .............61 

Figure 3.19. (left) strength data of annealed glass surfaces: aged with grit paper (aged (a)), aged with falling 
abrasives (aged (b)) and as-received; (right) strength data of annealed glass edges: strong axis bending cut 
edge (a), strong axis bending polished edge (b) and weak axis bending ..........................................62 

Figure 3.20. Measured results for heat strengthened glass. ......................................................62 

Figure 3.21. Comparison of moment-rotation curve of (a) beams und (b) joints according to (Kühnemund, 
2003) ...................................................................................................................70 

Figure 3.22. Definition of the available rotation capacity of joints according to (Kühnemund, 2003). ........70 

Figure 3.23. (a) Example of beam model for the derivation of beamlines (b) derived beamline. ..............71 

Figure 3.24. Check of sufficient rotation capacity for composite joints. .........................................71 

Figure 3.25. Deformed shape, top displacement and chord rotation for a cantilever column: a) at yield, b) at 
ultimate. ................................................................................................................72 

Figure 3.26. Comparison of predicted and experimental values of ultimate chord rotations for: a) and b) 
concrete, c) steel beams. ..............................................................................................73 

Figure 3.27. Schematic presentation of degradation with time. ..................................................81 

Figure 5.1. Illustration of factors risk analysis of structures subject to accidental actions. (a) Identification 
and modelling of relevant accidental hazards. (b) Direct structural damage (often damaged column or floor). 
(c) Further performance of the damaged structure and final indirect consequences. (source Eurocode 1991-1-
7) ..................................................................................................................... 101 

Figure 5.2. Illustration of the ALARP principle in life safety risk regulation. ................................... 103 

117 

 
 
 
List of tables 

Table 2.1. Relationship between the failure probability Pf and the reliability index . ..........................12 

Table 3.1. Overview main results of historical documents. .......................................................26 

Table 3.2. Partial factors for the resistance according to related Eurocodes. ...................................34 

Table 3.3. Weights and ranges of variations of aG and aQ. ........................................................34 

Table 3.4. Typical statistical information for materials in this sub-chapter .....................................35 

Table 3.5. Options for treating the bias of the model uncertainty ................................................36 

Table 3.6. Options for treating the scatter part of the model uncertainty (lognormal distribution) ............37 

Table 3.7. Stochastic models for partial factor calibration (based on Annex A). ................................38 

Table 3.8. Resulting 50 year reliability indices for each combination of material and action (averaged over all 
aQ and aG values) ......................................................................................................39 

Table 3.9. Limit State Scenarios (LSS) from prCEN/TS 19100-1 .................................................60 

Table 3.10. Partial safety factors for material and pre-stress copied from prCEN/TS 19100-1 ...............63 

Table 3.11. Relevant standards for different materials and properties ..........................................67 

Table 3.12. The recommended value for the partial factors M. ..................................................68 

Table 3.13. Relationship between Consequences Classes and Robustness design approaches (example) ....86 

Table 5.1. Overview of hazards. .....................................................................................99 

Table 5.2. Types of the possible consequences to be considered. .............................................. 100 

Table 5.3. Examples of risk mitigating measures. ............................................................... 102 

Table 5.4. Overview of risk criteria in terms of annual individual fatality rates. .............................. 103 

118 

 
 
 
Annexes 

Annex A. Probabilistic Models 

A.1 Introduction 

This annex contains recommended statistical (probabilistic) models to be used in structural reliability and risk 
calculations. The models may also be used as basis for the calibration of partial factors. The models are the 
basis for most of the reliability calculations in chapter 4 and Annex B of this document. 

For all models the background to the information is presented. Where possible the models given in this annex 
correspond  to  the  standardized  models  given  in  Eurocodes  EN 1991  to  EN 1999  or  in  the  corresponding 
background documents. If no such information is available other sources like the JCSS Probabilistic Model code 
have been used. Some data models are partly based on engineering judgement. See also chapter 3.2.1. 

The  models  represent  the  state  of  knowledge  of  2021  and  refer  to  the  situation  that  no  project-specific 
information is available. The model uncertainties are intended to reflect the uncertainty in normal engineering, 
numerical or experimental models. If in special cases more information is available the models in this Annex 
can be used as priors in a Bayesian updating procedure (see text below for detailed information). 

Note further that this annex does not address the issue of climate change and the possible consequences for 
design  and  assessment.  Some  information  can  be  found  in  chapter  4  of  CEN-TC250-SC1  N1735 
Interdependence of climatic actions (2021). However, at the moment there is no translation to the corresponding 
Eurocode sections, but this of course may be different in the future. 

Further general technical introductory remarks: 

Data  for  geometrical  properties  (thickness  etc.)  are  given  in  the  chapter  on  self-weight  but  may  also  be 
important for the resistance part. 

The mean or bias in the tables is defined as the ratio of the actual mean value and the mean that follows 
theoretically from the characteristic value and the coefficient of variation. 

Range  in  Tables  may  refer  to  different  references,  locations  or  circumstances;  without  further  information 
medium values are recommended. 

In some cases the user may have some local or material specific data, In that case the models described in this 
annex may be used as kind of prior. Typical values for the equivalent sample sizes of the priors are: 

n = 1 to 5 

for the mean value  

 = 3 to 10   for the scatter  

In  most  tables  also  values  for  the  weight  factors  wk  are  given;  they  should  be  considered  as  based  on 
engineering judgement only. They reflect the relative importance of the load or material, both with respect to 
the presence in design as with respect to the possible consequences in case of failure. 

A.2 Permanent actions 

A.2.1 Self-weight 

The self-weight of structural elements or structures of material 𝑖 may be modelled as (JCSS PMC 2001): 

GS,i= ∫ ρidV
Vol

(A.4) 

where: 

ρi  

is the weight density which is a random field; 

Vol  
material-dependent. 

is the volume of the structure which is affected by deviations from the nominal values which are 

119 

 
 
  
The geometrical dimensions (thickness, cross-sectional area etc.) are modelled as d=dnom+δ where dnom is the 
nominal dimension (deterministic) and  δ is the  deviation which is represented  by a normal distribution  with 
parameters μδ,δ. It follows that d is normally distributed with parameters: 

μd=dnom+μδ and σd=σδ. 

(A.5) 

Notes: 

To cover (unsafe) conservative estimations of self-weight a bias of 1,05 is sometimes proposed. 

Spatial correlation maybe important when self-weight has both large positive and negative contributions to a 
load effect. The JCSS PMC recommends =0,7 for two different members. 

Values for distribution parameters depend on the material and type of structural member. 

A.2.1.1 Self-weight concrete structures 

The self-weight of a box shaped concrete element with dimensions d1, d2 and l may be modelled as 

GS=d1d2l ρ 

(A.6) 

Reinforcement is not considered, usually around 1-2% of cross section and accounted for by increasing density 
of concrete. 

The  dimensions  usually  are  considered  as  uncorrelated.  The  properties  for  the  structural  dimensions  are 
presented in Table A.1 (JCCS- PMC, 2001). 

Table A.1. Concrete elements self-weight models 

wk 

Weight density  

Large  elements  casted 
1000 mm<dnom<2000 mm  

in  situ 

30 % 

Small elements casted in situ 

100 mm<dnom<1000 mm  

70 % 

X 

 

d 

d 

μ 

nominal 

dnom+3mm 

dnom(1+0.003)  

V 

0,04 

10mm
dnom+3mm

4+0,006dnom
dnom(1+0,003)

The resulting GS,i for concrete elements may be considered to have a (on average) coefficient of variation V = 
0,046. 

References 

JCSS (2001), JCSS Probabilistic Model Code, Joint Committee on Structural Safety, https://www.jcss-lc.org/jcss-
probabilistic-model-code/ (periodically updated, online publication) 

A.2.1.2 Self-weight steel structures 

The self-weight for steel members may be modelled as  

Gs=Alρ  

Gs=l1l2tρ  

for profiles,  

for plates. 

If items like welds, bolts and stiffeners are not dealt with explicitly, one may on average take them into account 
by an additional 2 %. 

The structural steel weight density may be taken as normal with mean equal to the nominal value and V=0,01. 

Parameters for geometrical properties A and t may be taken as given in Table A2 (JCSS PMC, 2001). 

120 

 
 
 
 
 
 
 
 
 
 
Table A.2 Steel elements self-weight models 

gross section area  

X 

A 

µ 

0,995 Anom 

net section area (bolt holes etc.) 

Anet 

0,99 Anet,nom 

section modulus (for bending capacity) 

W 

0,995 Wnom 

CoV 

2,0% 

2,5% 

2,0% 

The resulting GS,i steel elements may be considered to have a coefficient of variation V=0,025 to 0,033.  

References 

JCSS (2001), JCSS Probabilistic Model Code, Joint Committee on Structural Safety, https://www.jcss-lc.org/jcss-
probabilistic-model-code/ (periodically updated, online publication  

A.2.1.3 Self-weight timber structures 

Timber element self-weight may be modelled as (steel joints and plates not accounted for); 

Gs=d1d2lρ 

(A.7) 

with (nominally): 

0,05m<(d1,d2)<0,45m for sawn elements, and  

0,05m<d1<1,55m and 0,05m<d2<0,45m for glulam elements. 

The timber weight density may be taken as normal with mean equal to the nominal value and V=0,10; where 
relevant  a  more  detailed  model  taking  into  account  means  and  fluctuations  of  moisture  content  may  be 
appropriate. 

Geometrical parameters may be taken from Table A.3. 

Table A.3 Timber elements self-weight models 

Sawn  

Glulam  

wk 

10 % 

90 % 

Geometrical 
dimension 

μ 

V 

di 

di 

dnom(1+0,05) 

2mm
dnom(1+0,05)

dnom 

1mm
dnom

The resulting GS,i timber elements may be considered to have on average a coefficient of variation 0,105. 

In some cases it may be necessary to add a certain weight (say 2 or 5 %) to take into account the weight of 
steel elements used in connections. 

References 

CIB-W81 (1989), Actions on structures Self-Weight Loads, Publication no. 115, CIB, Rotterdam 

A.2.1.4 Self-weight aluminium structures 

It  may  be  assumed  that  the  structural  aluminium  alloys  weight  density  is  normal  with  mean  equal  to  the 
nominal value and V=0,01. Because deviation in density is non-existent for practical purposes, the self-weight 
of aluminium structural members is mainly influenced by their geometrical properties. 

The member self-weight may be modelled as: 

121 

 
 
 
 
 
 
GS=A l ρ  

for profiles, 

GS=l1 l2 t ρ   for plates. 

Statistical  parameters,  mean  value  (μ),  coefficient  of  variation  (V)  and  distribution  types  for  available 
dimensional parameters (geometrical properties) may be taken from Table A.4 For dimensional parameters not 
specifically mentioned in Table A.4, it may be assumed that the mean values are equal to the nominal values 
and the standard deviations are equal to half of the interval between the nominal value and the lower bound 
of the applicable tolerance interval in EN 1090-3 or other relevant product standards. 

Table A.4 Variability of available aluminium members’ dimensional properties 

Dimension type 

Parameter / weight factor 
wk 

Xm 

Mean value 

Coefficient  of 
variation 

Distribution 
type 

Extruded profiles / 75% 

0,95 tnom 

5,0% 

Lognormal 

Thickness 

Welded profiles / 5% 

1,00 tnom 

4,0% 

Normal 

Plates / 20% 

1,00 tnom 

2,0% 

Normal 

Depth h 

1,00 hnom 

0,3 % 

Normal 

Outer  dimensions  of 
cross-section 

Width b 

1,00 bnom 

0,3 % 

Normal 

Outer  diameter  d  of  circular 
hollow section 

1,00 dnom 

0,5 % 

Normal 

Nominal dimensions according to the applicable product standard or specification. 

References 

Dokšanović, T. (2017), Lateral-torsional buckling of aluminium elements with open unsymmetrical cross section 
PhD thesis, University of Zagreb (in Croatian) 

A.2.1.5 Self-weight masonry 

In general the mean value is taken as the nominal value and a coefficient of variation of 5% is assumed. 

References 

Rackwitz,  R. 
Tragwerkszuverlässigkeit, Einwirkungen. Ernst & Sohn Verlag, Berlin, Germany (in German) 

(1996),  Einwirkungen  auf  Bauwerke. 

In:  Mehlhorn,  G. 

(Ed),  Der 

Ingenieurbau: 

A.2.1.6 Self-weight ground  

The weight of ground is usually not relevant for the self-weight of a structural element (Gs) but for the stress 
conditions in the ground surrounding foundations elements. It is also important for determining the load effects 
and resistances in other geotechnical structures such as embankments or retaining walls. 

The coefficient of variation for the weight density of soil may be considered in the range of 2 to 5 % for both, 
clay and sand, considering the typical amount of site investigation and spatial averaging within the zone of 
influence. (5 to 10% CoV total variability according to prEN1997, i.e. without spatial averaging). 

A.2.2 Non-structural permanent loads 

Next to the permanent load caused by the weight of structural elements there may be other loads of (almost) 
permanent character, originating for instance from: 

manufactured elements such as flooring systems, surfacing, facades and ceilings; 

122 

 
finishing elements connected with the structure, including hand rails, safety barriers, parapets, weight of 
cables, pipes and service ducts; 

additional provisions like lifts; 

machinery fixed permanently to, or within, the structure; 

ventilation and colling equipment; 

loads due to ballast systems; 

earth loads on roofs and terraces. 

Some of these loads may have a quite large uncertainty, this however may also depend on the type of analysis 
(generic or specific, long or short period). Table A.5 gives some guidance for numerical values to be taken into 
account. 

Table A.5 Numerical values for non-structural permanent loads 

Random variable 

X 

Distr. type  Mean(μ) 

V 

Char value 

(Fractile) 

Permanent load 

GP 

Normal 

1,00 

0,10 

0,50 

Permanent load (large V) 

GP* 

Normal 

1,00 

0,20 

0,95 

Some of the non-structural elements may be exchanged or removed during the design lifetime, e.g. ventilation 
and cooling equipment and could therefore be considered as a free permanent load not present when it acts 
favourable for the load effect considered. 

A.3 Loads varying in space and time 

A.3.1 General 

The uncertainties on the action side in structural analysis are associated with:  

The intrinsic variations in time and space; 

The  possible  deviation  of  the  adopted  and  simplified  action  model  from  the  real  action  (load  model 
uncertainties). 

The time variability is accounted by considering maxima over a reference period T which in general is considered 
to be one year. If one has to change from one year to a period T= n years, it is necessary to make the distinction 
between the varying in time and constant in time random variables: 

FQT (q) =  FQ1  (q;c) n fC (c) dc 

(A.8) 

where:  

QT  

Q1  

C  

is the maximum load in period T;  

the annual maximum for the time varying variable and  

represents the time invariant variable(s). 

The time variant part of most loads is dependent on the geographical location and the use of the building. Even 
if wind- or snow zones are distinguished there still may be an important systematic (but often randomized) 
variation within the zone. 

Note finally that according to EN 1990:2023 the characteristics values for variable loads are defined as values 
corresponding  to  an  annual  exceedance  probability  of  0,02,  including  both  time  variant  and  invariant 
uncertainties.  One  should  be  aware  of  the  fact  that  code  values  in  EN 1991  and/or  National  Annexes  may 
deviate from that definition. 

123 

 
 
A.3.2 Imposed load 

In calculations the space variability of the loads is accounted by using an equivalent uniformly distributed load 
(EUDL) as described in the JCSS PMC. Live load is modelled as the sum of equivalent  uniformly distributed 
sustained q and intermittent loads p where (see Figure A.1). 

The sustained load q may be modelled as an FBC process with random time intervals; the mean occurrence rate 
of a load change is given by q (that is, a mean interval duration equal to 1/ q). The distribution of the arbitrary 
point in time load may begiven by: 

q∼Gamma (μq=mq;σq=√σV

2+σU

2 A0
A

κ) 

(A.9) 

A is the tributary area and A0 is the reference area usually taken as 10 m2 (although depending on the type of 
use), keeping A0/A at a maximum of 1,0; the value of  follows from the shape of the influence factor. As an 
alternative one may choose the Gumbel instead of the Gamma distribution having a less “fat” tail. 

The intermittent load intensity may be modelled as a Poisson pulse process having a mean occurrence rate p 
(so the mean time between the occurrence of pulses is 1/p ); the non-zero load realization may be modelled 
as:  

p ∼ exponential(μp=σp=mp) 

(A.10) 

The duration of the non-zero loading period is dp 

Figure A.1. Arbitrary realization of the sustained intermittent load for an office space [N/m2] 

p 

q 

An approximation of the cumulative distribution function representing the maxima of the sum process q+p over 
a period T is given by: 

F(q+p)maxT(r)≅ exp {-λqT (1-Fq(r)) - λpT (1-Fq+p(r))} 

∞

Fq+p(r)= ∫ fq(x)Fp(r-x)dx

0

(A.11) 

(A.12) 

The functions F and f represent, respectively, the cumulative distribution functions and the probability density 
functions for the various variables.  

The upper tail of the cumulative density in equation A.11 may, for convenience, be approximated by a Gumbel 
and/or a Lognormal distribution with the parameters presented in Table A.6. This simplifies the code calibration 
work.  Although  the  Gumbel  distribution  is  best  for  representing  extremes  in  general,  the  lognormal  has  the 
advantage of being defined on the positive real space only. 

124 

 
 
 
 
 
 
 
The  model  uncertainty  for  the  live  load  model  θO,mod  is  not  known  and  probably  dependent  on  the  type  of 
application; the present recommendation is a lognormal distribution with unit mean and with 0,10 coefficient 
of variation. 

Table A.6 Parameters of the sustained and intermittent loads 

q-sustained load 

p-intermittent load 

Mean 
value 

Mean  duration 
of 
non-zero 
part 

  Mean 
value 

Mean  duration  of 
non-zero part 

Mean 
between 
zero part 

time 
non 

mq 

V 

U 

d,q=1/q 

  mp 

d,p 

N/m2 

N/m2  N/m2 

years 

  N/m2 

days 

200 

400 

300 

200 

2 

2 

2 

2 

Office 

500 

300 

600 

5 

Lobby 

200 

150 

300 

10 

Residence  300 

150 

300 

7 

300 

50 

100 

10 

Hotel 
room 

Patient 
room 

400 

300 

600 

8 

200 

2 

School 

600 

150 

400 

10 

Merchant  900 

600 

1600  5 

500 

400 

3 

7 

1/p 

Yr 

0,3 

1 

1 

0,1 

1 

0,3 

1 

For office loading the maximum load in 50 years has been calculated which resulted in a Gumbel distribution 
with mean 2,22 kN/m2 and V=0,26. 

References 

JCSS (2001), JCSS Probabilistic Model Code, Joint Committee on Structural Safety, https://www.jcss-lc.org/jcss-
probabilistic-model-code/ (periodically updated, online publication) 

Gulvanessian,  H.  and  Holicky,  M.  (2005),  ‘Eurocodes:  using  reliability  analysis  to  combine  action  effects’, 
Structures & Buildings, Vol. 158, Issue 4,  pp. 243-252. 

fib COM3 TG3.1 (2016), Partial Factor Methods for Existing Structures (technical recommendation), fib 

Melchers, R. E. (2001) Structural Reliability Analysis and Prediction, 2nd ed., John Wiley & Sons Ltd., Chichester, 
England 

Choi, E. C. C. (1992), ‘Live Load in Office Buildings. Lifetime Maximum Load and the Influence of Room Use’, 
Proceedings of the Institution of Civil Engineers - Structures and Buildings, Vol. 94, Issue 3, pp. 307-314. 

Honfi, D. (2014), ‘Serviceability floor loads’, Structural Safety, Vol. 50, pp. 27-38. 

Sedlacek, G. and Gulvanessian, H. (1996), ‘Basis of design and actions on structures: part 2.1: densities, self 
weight, imposed loads’, IABSE reports, Vol. 74, pp. 61-70. 

Rackwitz,  R. 
Tragwerkszuverlässigkeit, Einwirkungen. Ernst & Sohn Verlag, Berlin, Germany (in German) 

(1996),  Einwirkungen  auf  Bauwerke. 

In:  Mehlhorn,  G. 

(Ed),  Der 

Ingenieurbau: 

Glowienka, S. (2007), Zuverlässigkeit von Mauerwerkswänden aus großformatigen Steinen, PhD thesis, Institut 
für Massivbau, Technische Universität Darmstadt, Darmstadt (in German) 

CIB W81 (1989), Actions on Structures, Live Load in Buildings, Report No. 116, Rotterdam 

125 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
European Commission: Joint Research Centre, Markova, J., Pinto, A., Sousa, M., Dimova, S. et al. (2018),  Reliability 
of  structural  members  designed  with  the  Eurocodes  NDPs  selected  by  EU  and  EFTA  Member  States,  Pinto, 
A.(editor) 
2018, 
Publications  Office 
https://data.europa.eu/doi/10.2760/426349  

European  Union, 

and  Dimova, 

S.(editor), 

the 

of 

A.3.3 Wind load 

Eurocode 1 Part 1-4 (CEN, 2005) model for wind load is derived from the A. G. Davenport wind load chain. The 
characteristic  peak  velocity  pressure  is calculated  in  equation  A.13  where,  cg, co, cdir, cseason, cr  represent  the 
gust, orography, direction, season, altitude and roughness coefficients, respectively; k stands for characteristic 
value and ρ for the air density. Here co = cdir = cseason = 1 are considered deterministic. 

qp,k=

1
2

ρ⋅cg,k⋅(co,kcdir,kcseason,kcalt,kcr,kū max, 1yr,k)

2

(A.13) 

Two methods are proposed in the Eurocode for calculating the characteristic wind action on a structure. The 
first, named surface pressure method, calculates the total force acting on a structure as the vectoral summation 
of the external, internal and frictional forces (included in order in equation A.14): 

Fw,k=Fw,e,k+Fw,i,k+Ffr,k=csd,k ∑ cpe,k

i

qp,k

i

i + ∑ cpi,k
Aref
i

i qp,k

i +cfrqp,kAfr 
Aref

(A.14) 

where: 

cpe, cpi, cfr  

are the external pressure, internal pressure and friction coefficients, respectively; 

csd  
effects and have small dimensions.  

is the structural factor which is equal to unity for structures which are not sensitive to dynamic 

Friction forces are not considered further since cfr in most cases is one or two orders of magnitude smaller than 
the other pressure (force) coefficients.  

The second method named force coefficient method evaluates the characteristic force acting on a structure by 
means of an overall force coefficient cf, see equation A.15. 

Fw,k=csd,k ∑ cf,k

j qp,kAref

j

(A.15) 

j

The summations in equation A.14 are performed over different areas which the structure surface is divided in; 
the one in equation A.15 is performed over different elements composing the structure. The structural factor 
may be written as Csd = CsCd and takes into account the non-simultaneous occurrence of peak wind pressures 
on the surface (Cs) and the dynamic effects (Cd). 

According to equation (5.3.1) in Davenport, 1987, the annual maximum wind pressure is obtained as 

q= cp (cgcrco)2 qb1 

(A.16) 

in which: 

cp 

cg 

is the external or internal coefficient of pressure; 

is the gust factor equal to 1+ku Iu with ku expected peak factor and I the turbulence intensity; 

cr 
is the roughness factor accounting for the variability of the mean wind velocity at the site of the 
structure due to the height above ground level and the ground roughness of the terrain upwind of the structure 
for the wind direction considered; 

co  

is the orography factor which takes into account the orography effect (e.g. hills, cliffs, etc.); 

qb1 
function of wind direction and time of the year at 10 m above ground of terrain category II. 

is  the  basic  velocity  pressure  associated  with  the  basic  wind  velocity  ū max, 1yr,k,  defined  as  a 

126 

 
 
 
 
 
 
 
The air density is modelled as a deterministic variable since its scatter at large wind speeds is small, see e.g. 
Kasperski, 2009 for an example of real measurements. The roughness factor Cr describes the variation of the 
pressure with height. The gust factor Cg is the ratio between the peak velocity pressure to the mean velocity 
pressure. By approximating the longitudinal velocity pressure fluctuation as a linear function of the velocity 
fluctuation the mean and standard deviation of Cg are obtained as equal to the ones of the gust velocity, see 
e.g.  JCSS,  2001.  The  aerodynamic  shape  factor  Ca  is  standing  for  Cpe, Cpi, Cfr  or  Cf  depending  on  the  force 
component under consideration. 

Cpe and Cf are best represented by a Fisher-Tippett type I distribution, see e.g. Cook and Mayne, 1979. Holmes 
and  Cochran,  2003,  propose  a  GEV  distribution  with  slightly  positive  ξ  (sub-exponential  tail).  Table  A.7 
summarizes the basic random variables representation, characteristic fractiles and partial safety factors. 

Table A.7 Basic variables properties from and partial safety factors   

Description 

Distribution  Bias 

V 

(cr)2   Roughness factor 

Lognormal 

0,8(9) 

10 - 20% (5)(9) 

(cg)2   Gust factor 

Lognormal 

1,00(9) 

10 - 15% (5)(9) 

Characteristic 
fractile 

FCr (μCr

) 

FCg (μCg

) 

Cpe 

External pressure coefficient 

Gumbel 

1,00(9) 

10-30% (6)  

(14% 
tunnel test) 

for  wind 

80 % (7) 

Cpi 

Cfr 

Cf 

Internal pressure coefficient 

Gumbel 

10-30% (5) 

Friction coefficient 

Force coefficient 

Gumbel 

1,00 

1,00 (9) 

10 - 15 % (5)(9) 

78 % (5) 

78 % 

78 % 

2

Umax, 1yr

  Mean wind pressure (1yr max)  Gumbel 

1,00 

20 - 40% 

0,98 

According to [6] the bias in the cpe and cf -coefficients are size dependent and given by: 

20% underestimation for cpe1 

(bias of approx. 1,20 for 1 m2) 

20% overestimation for cpe10  

(bias of approx. 0,80 for 10 m2 and larger) 

According  to  EN  1991-1-4  the  coefficient  of  variation  of  Umax, 1yr
  may  be  taken  as  0.23  if  no  site  specific 
information is available. Further, the stochastic model for the wind action is calibrated such that the annual 
probability of exceedance of the calculated characteristic value does not exceed 0.02. 

2

A simplification of the model may be obtained by combining all model uncertainties in one model uncertainty 
parameter θ: 

q=θ   (

1
2

2
ρŪ max, 1yr

)   

(A.17) 

where, for uncorrelated random variables (or weakly correlated),  is lognormal distributed for the central limit 
theorem with mean equal to the product of the mean values, characteristic value equal to the product of the 
characteristic values and the coefficient of variation satisfying the following equation: 

127 

 
 
 
 
 
 
 
 
 
 
(1 + 𝑉𝜃

2) = (1 + 𝑉𝐶𝑎

2 ) (1 + 𝑉(𝐶𝑟)2

2

) (1 + 𝑉

2
2
(𝐶𝑔)

) 

(A.18) 

This gives Vq between 0,17 (with all minimum coefficients of variation) and 0,40 (with all maximum coefficients 
of variation). The expected coefficient of variation may be taken around 0,28.  

Note:  the  terrain  category  should  also  be  introduced,  variation  of  mean  wind  with  height  and  intensity  of 
turbulence, see JCSS PMC. 

Model uncertainty does not cover uncertainty in thermal effects for special cases (buildings where temperature 
is intentionally kept below 0 °C, roofs with high thermal transmittance etc.) which should be noted. 

References 

JCSS (2001), JCSS Probabilistic Model Code - Part 1 - Basis of Design, Joint Committee on Structural Safety, 
https://www.jcss-lc.org/jcss-probabilistic-model-code/ (periodically updated, online publication 

CEN – European Committee for Standardisation (2005), Eurocode 1 - Actions on structures - Part 1-4: General 
actions - Wind actions, EN 1991-1-4:2005 

Kasperski, M. (2009), ‘Specification of the design wind load — A critical review of code concepts’, Journal of 
Wind Engineering and Industrial Aerodynamics, Vol. 97, Issues 7–8, pp. 335-357. 

Cook, N.J., and Mayne, J.R. (1979), ‘A novel working approach to the assessment of wind loads for equivalent 
static design’, Journal of Wind Engineering and Industrial Aerodynamics, Vol. 4, Issue 2, pp. 149-164. 

Holmes, J.D., and Cochran, L.S. (2003), ‘Probability distributions of extreme pressure coefficients’, Journal of 
Wind Engineering and Industrial Aerodynamics, Vol. 91, Issue 7, pp. 893-901. 

ISO – International Organisation for Standardisation (2009), Wind Actions on Structures, ISO 4354:2009 

Hansen, S.O., Pedersen, M.L. and Sørensen, J.D. (2015), Probability based calibration of pressure coefficients, in: 
14th International Conference on Wind Engineering, Porto Alegre, Brazil. 

CEN – European Committee for Standardisation (2021), Interdependence of climatic actions, CEN-TC250-SC1 
N1735 

Davenport A.G., (1987) Proposed new international (ISO) wind load standard. High winds and building codes, in: 
Proceedings of WERC/NSF Wind Engineering Symposium, Kansas City, USA 

A.3.4 Snow load 

The model in accordance with prEN 1991-1-3:2020 (CEN, 2020a-b) is given by: 

sr = θ μi(Ce) Ct sg 

(A.19) 

where: 

sg  

θ 

snow load on the ground; 

model uncertainty 

μi  
exposure of the site (through the exposure coefficient Ce)  

conversion factor or snow load shape coefficient, depending on the roof’s geometry and the wind 

Ct  

is the thermal coefficient, to take account of the effects of heat flux through the roof.  

The  model  uncertainty  does  not  cover  uncertainty  in  thermal  effects  for  special  cases  (buildings  where 
temperature is intentionally kept below 0°C, roofs with high thermal transmittance etc.) which should be noted. 

Values for use in design calculations can be found in prEN 1991-1-3:2020 and or in the corresponding NAs. The 
following tables show the corresponding statistical properties. All values may be adjusted based on local data. 
Reference  is  made  to  the  Background  documents  of  SC1.T2  (CEN,  2020c)  and  to  the  Final  Reports  of  the 
European Snow Load Research Project (Sanpaolesi et al, 1998 and 1999). Figure A.2 gives an impression of the 

128 

 
 
 
 
 
distribution  of  the  load  intensity  over  Europe.  Note  that  here  the  characteristic  values  are  presented  which 
depend on the definition and rounding off rules used per country or region. 

Figure A.2. European Ground snow load map (indicative characteristic values) defined by EN 1991-1-3, 2005 (source: 
Croce et al., 2019).  

Source: Croce at al. 2019 

The model for the time-invariant part of the action, i.e. the shape coefficient, which can be calibrated on the 
basis of  available  data  for  roof  snow  loads,  is  significantly  affected  by  the  shape  of  the  roof  and  its  wind 
exposure conditions and the climatic region. Therefore, uncertainties in the snow load model, which are related 
to the time-invariant part of the action, will be evaluated in two different ways depending on the level of detail: 

CASE A:  

By means of a single random variable, which covers all the uncertainties in the time-invariant part and will be 
represented by the model uncertainty  assuming deterministic the shape coefficients. Statistical parameters 
of this random variable are evaluated considering the ratio between measured and modelled data of snow load 
on roofs for all available measurement collected in Thiis and O’Rourke, 2015. In the table below the statistical 
data for snow load model are illustrated. 

Table A.8 Statistical data for snow load (CASE A) 

Variable 

X 

Distrib. 

Bias 

V 

Xk 

Comments 

Snow  load  on 
the ground 

(annual 
maximum) 

sg1 

Gumbel 

1,00 

See  
A.10 

Table 

X0,98 

Table A.10 

See Thiis and O’Rourke, 2015 for 
in 
ground  snow 
Europe according to EN 1991-1-
3 and National Annexes. 

load  maps 

129 

 
 
model 
uncertainty  

Θ 

lognormal 

0,81 

0,26 

+ 

See  N.71  "NewCEN  model  for 
predicting  snow  load  on  gable 
roofs" (Croce at al. 2019). 

It  is  estimated  considering  the 
ratio  between  measured  and 
modelled data of snow load on 
gabled  roofs  in  the  Norwegian 
Case  History  Database  (Thiis 
and  O’Rourke,  2015)  under 
different exposure conditions. 

conversion 
from ground to 
roof snow load 

μi(Ce) 

det 

EN 1991-1-
3:2020 

Ct 

det 

1,00 

-- 

thermal 
coefficient  

CASE B:  

Considering  the  specific  statistical  distribution  of  shape  coefficients  for  the  investigated  roof  shape,  wind 
exposure condition, and climatic region, derived from available measurements. In the table below, statistical 
data are given for flat roofs (measurements taken from Sanpaolesi et al, 1998 and 1999, or CEN, 2020d) and 
gabled roofs in case of balanced load and unbalanced load (measurements taken from CEN, 2020d). 

Table A.9 Statistical data for snow load (CASE B) 

Variable 

X 

Distrib.  bias 

V 

Xk 

Comments 

Snow load on ground(annual 
maximum) 

sg1 

Gumbel  1,00 

See 
Table 
A.10  

Table 
A.10 

X0,98 

See (Croce at al., 2019) for ground snow 
load  maps 
in  Europe  according  to 
EN 1991-1-3 and National Annexes. 

model uncertainty  

Θ 

- 

- 

- 

- 

Sheltered 

0,94 

0,09 

X0,58 

conversion  from 
ground  to  roof 
snow  load  (flat 
roofs)* 

Normal 

μi(Ce)  LN 

0,92 

0,39 

X0,76 

Exposed 

0,91 

0,32 

X0,73 

conversion  from 
ground  to  roof 
snow  load  (flat 
roofs)** 

Sheltered 

0,78 

0,16 

X0,63 

μi(Ce)  LN 

Normal 

0,83 

0,16 

X0,63 

130 

Model uncertainties in the time-invariant 
part of the snow load are directly taken 
into  account  through  the  statistical 
distribution of shape coefficients. 

roofs  according 

*Flat 
the  data 
collected during the ESLRP (Sanpaolesi et 
al, 1998 and 1999). 

to 

SC1.T2  DOC_087  (CEN,  2020b)  shows 
the  dependence  of  design  conversion 
factors  with  the  coefficient  of  variation 
of ground snow loads (CoVg). 

Xk are evaluated considering a CoVg = 0,6.  

**Flat  roofs  according  to  the  data 
collected in the Norwegian Case History 
Database  (Thiis  and  O’Rourke,  2015). 
Data are available only for sheltered and 
normal  exposure  and  are  characterized 

 
 
 
 
 
 
by lower CoV for normal exposure than 
that 
the  ESLRP 
(Sanpaolesi et al, 1998 and 1999). 

collected  during 

Design  conversion  factors  is  almost 
constant  for  both  sheltered  and normal 
conditions (CEN, 2020b). 

***Gabled  roofs  according  to  the  data 
collected in the Norwegian Case History 
Database (Thiis and O’Rourke, 2015).  

***Gabled  roofs  according  to  the  data 
collected in the Norwegian Case History 
Database (Thiis and O’Rourke, 2015).  

Exposed 

- 

- 

- 

conversion  from 
ground  to  roof 
snow load  

Sheltered 

0,71 

0,24 

X0,69 

(gabled 
25°<α<30°) 

roofs 

Normal 

μi(Ce)  LN 

0,73 

0,29 

X0,71 

Balanced load  

(case (i) 
prEN 1991-1-3:
2020)*** 

conversion  from 
ground  to  roof 
snow load  

Exposed 

0,78 

0,37 

X0,75 

Sheltered 

0,73 

0,25 

X0,68 

(gabled 
25°<α<30°) 

roofs 

Normal 

μi(Ce)  LN 

0,70 

0,27 

X0,70 

Unbalanced load 

(case (ii) and (iii) 
prEN 1991-1-
3:2020)*** 

Exposed 

0,70 

0,35 

X0,74 

thermal coefficient  

Ct 

det 

1,00 

- 

LN stands for lognormal distribution 

131 

 
 
 
 
 
Table A.10 Coefficient of variation for yearly maximum snow load on ground (from Annex 3 in Sanpaolesi et al. 1998) 

Region 

min CoVSG  

m CoVSG  

max CoVSG  

min 

mean 

max 

Switzerland 

0,4 

(seldom 

0,2 
below 0,3) 

Italy 

0,5 

0,4 

Germany 

(seldom 

0,3 
below 0,4) 

Austria 

France 

UK and EIRE 

Iberian 
Peninsula 

0,3 

0,4 

1,0 

0,7 

0,6 

0,5 

1,0 

0,8 

1,0 

0,7 

0,7 

0,8 

0,45 

1,8 

1,6 

0,8 

6,0 

Comments 

Number  of 
repetitions 
in a year 

2 

1 

<1000m 

>1000m  

1<r<2 

<1000m 

1<r<2 

>1000m 

1 

No altitude dependency 

1<r<2 

Weak altitude dependency 

1<r<3 

Only two stations analysed 

1 

2 

>3 

1 

Mean value of V is equal to 
2;  majority  of  stations 
have CoV between 1,0 and 
2,0 

Mediterranean 
region 

0,1 
for 
(0,4 
most stations) 

1,5 
for 
(1,1 
most stations) 

1<r<3 

Do not depend on altitude 

Central East 

0,5 

1,2 

<100m 

0,3/0,4 

0,5/0,6 

1<r<6 

>100m 

Central West 

for 
(0,7 
0,6 
most stations) 

for 
(1,0 
1,2 
most stations) 

1<r<2 

Greece 

Norway 

0,4 

0,2 

Finland Sweden 

0,2 

Iceland 

Croatia* 

0,3 

0,5 

0,4 

0,67 

1,2 

0,6 

0,7 

1,3 

0,7 

0,7 

132 

1<r<2 

1 

1<r<2 

Weak altitude dependency 

1 

1 

1 

No altitude dependency 

Pannonian region 

Dinaric Alps region 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
0,4 

Czech Republic 

0,3 

0,6 

1,0 

1,2 

1 

Mediterranean region 

1<r<2 

Altitude dependent 

* Croatian Meteorological and Hydrological Service (DHMZ)    ** Czech Hydrometeorological Service (CHMI) 

Sanpaolesi et al., 1998 concluded that, although there is a wide deviation, the value of 0,6 is calculated as the 
average from all the data. Also Soukhov, 1998 seems to support 0,6 as a representative value of CoV of  annual 
ground snow load maxima for Germany. Aldea et al, 2016 indicate CoV between 0,35 and 1,38 for Romania, 
with an average value of 0,59. See also Figure A.3 for information. 

Figure A.3. Coefficient of variation of annual maximum of snow load versus altitude  

Source: Soukhov, 1998 

References 

CEN – European Committee for Standardization (2020a), Eurocode 1: Actions on structures - Part 1-3: General 
actions - Snow loads, EN 1991-1-3:2020 (Final draft) 

CEN  –  European  Committee  for  Standardization  (2020b),  Background  on  design  values  of  roof  snow  loads, 
SC1.T2 DOC_087 

CEN  –  European  Committee  for  Standardization  (2020c),  Background  document to  prEN 1991-1-3  (Final  PT 
Document - April 2020), SC1.T2 DOC_095 

CEN – European Committee for Standardization (2020d), New CEN model for predicting snow load on gable 
roofs, SC1.T2 DOC_071 

Sanpaolesi L., Del Corso R., Formichi P., Currie D., Sims P., Sacré C., Stiefel U., Lozza S., Eiselt B., Peckham R., 
Solomos G., Holand I., Sandvik R., Gränzer M., König G., Sukohv D. (1998), Phase 1 Final Report to the European 
Commission, Scientific Support Activity in the Field of Structural Stability of Civil Engineering Works: Snow Loads, 
Department of Structural Engineering, University of Pisa, Italy 

Sanpaolesi L., Del Corso R., Formichi P., Brettle M., Currie D., Dillon P., Sims P., Delpech P., Dufresne M., Sacré C., 
Stiefel U., Lozza S., Eiselt B., Peckham R., Solomos G., Holand I., Leira B., Sandvik R., Gränzer M., König G., Sukohv 
D. (1999), Phase 2 Final Report to the European Commission, Scientific Support Activity in the Field of Structural 
Stability of Civil Engineering Works: Snow Loads, Department of Structural Engineering, University of Pisa, Italy 

133 

 
 
 
Thiis T. K., O’Rourke M. (2015), ‘Model for Snow Loading on Gable Roofs’, Journal of Structural Engineering, Vol. 
141, Issue 12, pp. 1-9. 

Croce P., Formichi P., Landi, F. Marsili F. (2019), ‘Harmonized European ground snow load map: Analysis and 
comparison of national provisions’, Cold Regions Science and Technology, Vol. 168, 102875. 

Soukhov, D. (1998), "The Probability Distribution Function for Snow Load in Germany", LACER, Volume 3 

Aldea, A., S. Demetriu, D. Lungu, C. Neagu, R. Vacareanu and C. Arion, (2016), The snow loads code in Romania, 
in: Proceedings of ICSE 2016, CSTB - Centre scientifique et technique du bâtiment, Nantes, France 

A.3.5 Traffic loads 

For information reference is made to the JCSS Probabilistic Model Code.  

References 

JCSS (2001), JCSS Probabilistic Model Code - Part 2 (under development), Joint Committee on Structural Safety, 
https://www.jcss-lc.org/jcss-probabilistic-model-code/ (periodically updated, online publication 

A.3.6 Accidental actions 

Probabilistic models for accidental actions, in particular impact from vehicles, trains, shops and aircrafts and 
loads due to internal and external explosions may be found in Eurocode EN 1991-1-7 and ISO 10252. Relevant 
Background information is presented in (Vrouwenvelder and Stiefel, 2012). Table A.11 and Table A.12 give a 
summarized and indicative information on models for vehicle impact and internal natural gas explosions. 

Table A.11 Indicative data for car impact 

Random variable 

X 

Distr. type 

Mean  

V 

Vehicle velocity - highway 

vo 

Lognormal 

Vehicle velocity - urban area 

Vehicle velocity - courtyard 

Lognormal 

Lognormal 

80 km/h 

40 km/h 

15 km/h 

Vehicle velocity - parking garage 

Lognormal 

5 km/h 

0,12 

0,20 

0,33 

1,00 

Deceleration  after  leaving  the 
track 

a 

Lognormal 

4,0 m/s2 

0,33 

Vehicle mass – lorry  

m 

Normal 

20 000 kg 

0,6 

Vehicle mass – car 

Deterministic 

1 500 kg 

Vehicle stiffness 

Angle of leaving the track 

Vehicle failure intensity 

k 

 

 

Deterministic 

300 kN/m 

-- 

Raleigh 

10˚ 

1,0˚ 

Poisson 

10-7/ m 

Table A.12 Indicative data for internal Natural Gas explosion 

Random variable 

X 

Distr. type 

Mean  

V 

Explosion pressure in buildings 

p 

Lognormal 

EC-Model  

0,7 

Explosion rate 

 

Poisson 

2x10-6 /a 

134 

 
 
 
 
 
References 

Vrouwenvelder A. and Stiefel U. (2012), Eurocode 1, Part .1-7, Accidental Actions Background Document, TNO, 
the Netherlands  

A.3.7 Seismic actions 

Seismic action is described as a response spectrum (usually spectral acceleration Sa(T)). This may be a so-called 
uniform  hazard spectrum, or, more recently, a conditional spectrum. The former is the collection of spectral 
ordinates  characterized  by  the  same  probability  of  exceedance  (collection  of  fractile  from  their  respective 
marginal  distributions),  while  the  latter  is  obtained  from  the  joint  distribution  of  spectral  ordinates  (i.e.,  it 
requires a correlation model) and represents more closely the spectrum of real motions conditional on a spectral 
ordinate taking a specific value. The distribution of spectral acceleration at a given vibration period T at a chosen 
site is obtained through probabilistic hazard analysis (Cornell, 1968, McGuire, 1995, Araya and Der Kiureghian, 
1988,  Bazzurro  and  Cornell,  1999,  Woessner  et  al.,  2015,  GEM  2019),  which  employs  the  total  probability 
theorem (Benjamin and Cornell, 1970) to obtain the probability of Sa(T), being exceeded. 

The  classic  probabilistic  seismic  hazard  analysis  (PSHA)  describes  the  earthquake  occurrence  in  time  as  a 
Poisson process, and the interarrival time  as an exponential distribution, both having the parameter λ0 when 
considering all seismic sources. The return period is the inverse of the mean annual rate of exceedance of a 
given ground motion level at a site, product of  λ0 times the probability of exceedance given an event of any 
magnitude on any of the sources affecting the site. The return periods for the significant damage (SD) and near 
collapse (NC) limit states are Nationally Determined Parameters in EN 1998, with recommended values equal 
to 475 years and 1600 years, respectively.  

Table A.13 presents the relevant distributions to model earthquake occurrence in time, magnitude and space 
and the conditional distribution of Sa(T), given a seismic event, describing the attenuation laws or ground motion 
prediction equations. 

Table A.13 Seismic actions 

Random variable  X 

Distribution type 

Number  of  events 
in t 

Nt 

Poisson: 

P(Nt=n)=

e-λ0t(λ0t)n
n!

, n=0,1,2,.. 

Time  till  the  first 
event 

 

Exponential: 

P(  t)=1-e-λ0t 

Distribution 
parameters1 

Xk 

λ0 - annual mean rate 
of 
earthquakes 
occurrence 

Magnitude 

M 

Gutenberg-Richter  magnitude  recurrence 
model for a seismic source 

mmin – lower bound 
magnitude. 

fM(m)=βe-β(m-mmin) 

(Gutenberg and Richter, 1942), 

β- parameter 
governing the relative 
of 
likelihood 
earthquakes 
of 
various magnitudes  

Site-to-source 
distance 

R 

Uniform  distribution  for  area  seismic 
sources (simplified models) 

135 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
Spectral 
acceleration  for  a 
given event 

Sa(T)  Lognormal  distribution  depending  on  the 
natural period T, for a given magnitude m 
and site-to-source distance r 

Mean - μln Sa

P[Sa(T)<s|m,r]=Φ (

(m,r)

ln s -μln Sa
σln Sa

) 

Standard deviation 
 - σln Sa 

Spectral 
acceleration due to 
all 
sources 
affecting the site 

Sa(T)  Mixture 

of 
obtained through PSHA 

lognormal 

distributions 

Return 
period 
[year] 

SD: 475 

NC: 1600 

1 Seismic action parameters are site dependent. 

References 

Araya,  R.,  and  Der  Kiureghian,  A.  (1988),  Seismic  hazard  analysis  improved  models.  Uncertainties  and 
sensitivities, Rep. UCB/EERC 90/11. California University, Berkeley, USA 

Bazzurro, P. and Cornell, C.A. (1999), ‘Disaggregation of seismic hazard’, BSSA, Vol. 89, pp. 501-520. 

Benjamin, J.R. and Cornell C., (1970), Probability, statistics, and decision for civil engineers, McGraw-Hill, New 
York, USA 

Cornell, C.A. (1968), ‘Engineering seismic risk analysis’, BSSA, Vol. 8, pp. 1503-1606. 

Global Earthquake Model Foundation – GEM (2019), The OpenQuake-engine User Manual, version 3.7.1., Global 
Earthquake Model Foundation, Pavia, Italy, doi: 10.13117/GEM.OPENQUAKE.MAN.ENGINE.3.7.1.  

Gutenberg, B. and Richter, C.F. (1942), ‘Earthquake magnitude, intensity, energy, and acceleration’, BSSA, Vol. 3, 
pp. 163-191. 

McGuire, R.K. (1995), ‘Probabilistic seismic hazard analysis and design earthquakes: closing the loop’, BSSA, Vol. 
85, pp. 1275-1284. 

Woessner J, Danciu L, Giardini D, Crowley H, Cotton F, Grunthal G, Valensise G, Arvidsson R, Basili R., Demircioglu 
M,  Hiemer 
(2015), 
‘The 2013 European seismic hazard model: key components and  results’, Bulletin of Earthquake Engineering 
Vol. 13, pp. 3553–3596 

Stucchi  M. 

C,  Musson 

S,  Meletti 

Sesetyan 

Rovida 

K, 

A, 

R, 

A.4 Material Properties 

The  statistical  properties  in  this  section  are  as  much  as  possible  in  accordance  with  the  data  presented  in 
material  related  Eurocodes  or  their  background  documents  and  may  be  considered  as  appropriate  for 
performing reliability analysis in combination with the corresponding structural models. 

A.4.1 Concrete 

Table A.14 Concrete properties 

X 

description 

Mean or bias factor  

CoV V 

Concrete bending 

fC 

Compressive strength  

fcm/fck = exp(1,645Vfc)  

0,100 

Concrete shear 

136 

 
 
 
 
 
 
 
 
 
 
 
 
 
S 

Model uncertainty (without shear reinforcement.) 

 = 1,10 

Residual uncertainties (??) 

- 

0,107 

0,046 b) 

In-situ factorS accounts for the difference between the actual in-situ concrete strength in the structure fc,ais and the strength 

a)  
of the control specimen fc. 
b)  

The residual uncertainties (??) refer to aggregate size, reinforcement area and spacing and column size. 

All Model Uncertainties for concrete bending have been removed and put together in A6.1 

The higher qualities have usually a lower coefficient of variation. 

References 

CEN – European Committee for Standardization (2021), prEN 1992-1:2021 Part1, Annex A, Table A,2 

A.4.2 Reinforcement Steel 

Table A.15 Basic variables reinforcement 

X 

fy 

Description 

Distr 

Bias 

V 

yield strength 

LN 

fym/fyk = exp(1,645Vfy) 

0,045 

References 

prEN 1992-1 Part1, Annex A, Table A,2 (2021) 

A.4.3 Structural Steel 

Table A.16 Basic variables steel 

X 

fy 

Description 

Mean or bias factor 

CoV  -  V 

yield stress – steel grades S235 to S420 

fym/fyk a) = 1 + 4,00 Vfy 

yield stress – steel grades S460 and above b) 

fym/fyk a) = 1 + 3,00 Vfy 

fu 

tensile strength – steel grades S235 to S420 

fum/fuk a) = 1 + 4,00 Vfu 

tensile strength – steel grades  S460b) 

fum/fuk a) = 1 + 3,00 Vfu 

5,0% 

4,5% 

5,0% 

4,5% 

a) The design and the assessment of steel structures according to EC3 or EC4 do not make use of the 5% fractile as characteristic value of 
material strengths. The minimum guaranteed values are used instead, which are checked after production through specific testing. Due to 
quality control measures aimed at minimizing the sorting out of material, the minimum guaranteed value typically corresponds to a very 
low fractile of a PDF, below the 0,1% fractile.   
b) The current market share of steels of this strength in construction is below 10%. 

For the E modulus the usual value is 210 000 N/mm2. 

References 

CEN – European Committee for Standardization (2022), EN 1993-1-1:2022 Annex E (small differences possible) 

A.4.4 Timber 

The strength and stiffness properties of structural timber products are specified in the product standards EN 
338  for  solid  timber  and  EN  14080  for  glued  laminated  timber.  Structural  timber  is  graded  according  to 
different  indicators  and  assigned  to  the  different  strength  classes  in  the  product  standards  based  on  the 
reference  properties bending strength, bending  MOE and  density. The coefficient of  variation differs for the 

137 

 
 
different timber products, strength grades and is dependent on the grading method. The strength classes C24 
for solid timber and GL24h for glulam are typically taken as a reference. 

Table A.17 Timber Strength models and parameters 

Mechanical property 

Distribution 
type 

μ  V 

Comment 

Solid timber (EN 338) 

Bending strength 

fm  

Lognormal 

1,0  0,25 

Tension parallel to grain 

ft,0  

Lognormal 

1,0  0,30 

Tension perp. to grain 

ft,90  

2-p Weibull 

1,0  0,25 

Compression parallel 

fc,0  

Lognormal 

1,0  0,20 

Compression perp. 

fc,90   Normal 

1,0  0,10 

Shear (parallel) 

fv  

Lognormal 

1,0  0,25 

MOE parallel (bending) 

Em  

Lognormal 

1,0  0,13 

MOE parallel (tension) 

E0  

Lognormal 

1,0  0,13 

MOE perpendicular 

E90  

Lognormal 

1,0  0,13 

Shear modulus 

Gv  

Lognormal 

1,0  0,13 

Density 

ρ  

Normal 

1,0  0,10 

Glulam (EN 14080) 

Bending strength 

fm  

Lognormal 

1,0  0,15 

Tension parallel to grain 

ft,0  

Lognormal 

1,0  0,10-0,15 

(Frese 2016), JCSS 

MOE (bending) 

Em  

Lognormal 

1,0  0,05-0,10 

(Fink 2014; Frese 2016)  

Density 

ρ  

Normal 

1,0  0,10 

Source: JCSS PMC Section 3.5.3.2 Table 2, if not otherwise stated 

The coefficients of variation for engineered wood products are typically lower due to homogenization effects. 
Ranta-Maunus at al., 2001 report V=0,10 for softwood LVL in edgewise bending and V=0,14 for softwood LVL 
in flatwise bending. 

References  

Fink,  G.  (2014),  Influence  of  Varying  Material  Properties  on  the  Load-Bearing  Capacity  of  Glued  Laminated 
Timber, PhD Thesis Nr. 21746, ETH, Zurich, Switzerland 

Frese,  M.  (2016),  Computergestützte  Verfahren  zur  pragmatischen  Beurteilung  der  Tragwiderstände  von 
Brettschichtholz:  Zusammenfassung  exemplarischer  Simulationsstudien,  KIT  Scientific  Publishing, 
https://doi.org/10.5445/KSP/1000052710 (in German) 

Ranta-Maunus, A., Fonselius, M., Kurkela, J., and Toratti, T. (2001), Reliability Analysis of Timber Structures - VTT 
Research Notes, 2109, Technical research centre (VTT) of Finland, Espoo, Finland 

138 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
A.4.5 Aluminium  

Wrought aluminium alloys for structural applications according to EN 1999-1-1:2007 are shown in Table A.18. 

Table A.18 Wrought aluminium alloys to be used in structural applications according to EN 1999-1-1:2007 

Aluminium Alloys 

Common tempers 

Structural use in Europe 

EN AW-3004 

EN AW-3005 

EN AW-3103 

EN AW-5005/5005A 

EN AW-5049 

EN AW-5052 

- 

- 

- 

- 

- 

- 

EN AW-5083 

O/H111, H24, H36, H116/H321 

EN AW-5383 

H12|H22/H34 

EN AW-5454 

- 

EN AW-5754 

O/H111, H12|H22/H34 

EN AW-6060 

EN AW-6061 

EN AW-6063 

EN AW-6005A 

EN AW-6082 

EN AW-6106 

EN AW-7020 

EN AW-8011A 

T6 

T6 

T6, T66 

T6 

T6, T4 

T6 

- 

- 

Not common 

Common 

Not common 

Statistical parameters: mean value (μ), coefficient of variation (V) and distribution types for material strength 
models of available aluminium alloys that are commonly  used for  structures in Europe may be taken  from 
Table A.19. These statistical parameters have been obtained from a recent study performed within SC9 on more 
than 12 000 tension test results collected from producers, manufactures and design offices all over Europe. 

Table A.19 Variability of available material properties 

Parameter 

Alloy 

EN AW 

Temper 

Mean value 

Coefficient 

of variation 

Distribution type 

5754 

H111 

1,56 Rp0,2,min 

11 % 

Lognormal 

139 

 
 
6005A 

T6 

1,18 Rp0,2,min 

4,7 % 

Normal 

6060 

T6 

1,25 Rp0,2,min 

5,0 % 

Lognormal 

Y  be  taken  from  2%  proof 
strength, fo 

6060 

6063 

T66 

T6 

1,26 Rp0,2,min 

4,4 % 

1,28 Rp0,2,min 

8,8 % 

Lognormal 

6063 

T66 

1,19 Rp0,2,min 

4,1 % 

6082 

T6 

1,20 Rp0,2,min 

6,2 % 

Normal 

Normal 

Normal 

5754 

H111 

1,14 Rm,min 

3,4 % 

Lognormal 

6005A 

T6 

1,07 Rm,min 

2,8 % 

Lognormal 

6060 

T6 

1,15 Rm,min 

5,3 % 

Lognormal 

Ultimate tensile strength, fu 

6060 

T66 

1,07 Rm,min 

6,5 % 

Lognormal 

6063 

T6 

1,09 Rm,min 

3,0 % 

Lognormal 

6063 

T66 

1,07 Rm,min 

3,3 % 

Lognormal 

6082 

T6 

1,12 Rm,min 

5,0 % 

Lognormal 

Modulus of elasticity, E 

All 

Poisson ratio,  

All 

All 

All 

69943 MPa 

5,5 % 

Lognormal 

0,33 

3,3 % 

Normal 

Rp0,2,min and Rm,min are the minimum yield strength Rp0,2 and the lower bound of the ultimate tensile strength Rm, 
according to the applicable product standard, e.g. of the EN 755 or EN 485 series. 

References 

Skejić, D.; Dokšanović, T.; Čudina, I.; Mazzolani, F.M. (2021), ‘The Basis for Reliability-Based Mechanical Properties 
of Structural Aluminium Alloys’, Applied Sciences, Vol. 11, 4485, https://doi.org/10.3390/app11104485  

A.4.6 Masonry 

For the analysis of masonry structures the data in Table A.20 may be used. 

Table A.20 Probabilistic model for generic Masonry 

X 

fC 

References 

description 

Mean/char value 

Masonry Compressive strength 

fcm/fck = exp (1,645 Vfc)  

CoV 

0,16 

Jäger, W. and Pech, A. (2015), ‘Compressive strength of masonry – statistical evaluation of test data by power 
functions as basis of EC6, in: Jäger, W. (ed), Mauerwerk-Kalender 40, Ernst & Sohn, Berlin, Germany pp. 423 - 
448 

A.4.7 Ground 

The prEN1997 provides ranges of the coefficient of variation for various ground properties, see Table A.21. 

140 

 
 
 
Table A.21 Indicative ranges of coefficient of variation of ground properties from EN1997-1 Annex A (representing 
spatial variability, i.e. no spatial averaging is included yet)  

Ground/rock 
type 

Ground property 

Fine-grained soils  Shear strength in total stress analysis 

All soils and rocks  Peak or residual effective cohesion 

All soils and rocks  Angle of friction 

All soils and rocks  Shear strength at failure 

All soils and rocks  Unconfined compressive strength 

Symbol 

cu 

c’p, c’r 

 

zf 

qu 

CoV (X)  

30-50 

30-50 

5-15 

15-25 

20-80 

All soils 

Modulus of deformability  

E or G 

20-70 

Fine-grained soils  Vertical or horizontal consolidation coefficient 

cv or ch 

30-70 

All soils 

Hydraulic conductivity  

k 

70-250 

These indicative CoV values) have recently been supported in broad literature study by ISSMGE-TC304 (2020) 
to reasonably represent the point-to-point variability of individual sites. It is important to notice, however, that 
a geotechnical design parameter refers to larger volumes of ground involved with the occurrence of a limit 
state.  Since  spatial  averaging  takes  place,  the  geotechnical  design  parameter  typically  refers  to  the  spatial 
mean of a larger volume (with a few exceptions), including the related statistical uncertainty. Notice that the 
amount  of  averaging  is  therefore  application  dependent,  depending  on  the  volume  of  ground  involved.  See 
section 3.5.6 in the report for more details.  

Table A.21a presents recommended values for coefficients of variation of the most important ground strength 
parameters  for  calibration  purposes,  taking  into  account  that  the  variables of  interest  in  calibration  studies 
more often than not refer to spatial averages. That is the reason why the recommended CoV values in the third 
column are sometimes lower than the ranges of the values reported in Table A.21a.  

Table A.21a Typical coefficients of variation representing the uncertainty in spatial average ground strength parameters, 
recommended for calibration of partial factors  

Distribution 

CoV* 

Angle of friction φ’ 

Lognormal 

0,10 

Effective Cohesion c’ 

Lognormal 

0,20 

Undrained shear strength cu  Lognormal 

0,20 

* The CoV refers to the uncertainty in the geotechnical parameter, meaning that spatial averaging and statistical uncertainty are considered. 

It is important to notice that there is a large variety of (strength-related) parameters involved with geotechnical 
design, and that the uncertainty is related to the type and amount of site investigation (e.g. laboratory versus 
in-situ  testing).  Table  A.21  merely  reflects  typical  values  of  the  most  important  properties.  More  detailed 
information on statistics of soil properties can be found in the literature; the literature review by ISSMGE-TC304 
(2020) provides a useful overview. 

References 

International Society of Soil Mechanics and Geotechnical Engineering (ISSMGE) - Technical Committee TC 304 
‘Engineering  Practice  of  Risk  Assessment  and  Management’  (2020),  State-of-the-art  review  of  inherent 
Download: 
variability 

geotechnical 

uncertainty 

properties 

models, 

and 

and 

in 

141 

 
 
 
 
http://140.112.12.21/issmge/2021/SOA_Review_on_geotechnical_property_variablity_and_model_uncertainty.
pdf 

A.5 Imperfections and eccentricities 

For the time being calculations will be based on current design values in the codes. 

A.6 Model Uncertainties for load effect calculation 

The model uncertainty for the load effect calculation  θE is primarily depending on the type of internal force 
(axial forces, shear forces etc.) considered. 

As far as the global load effects are concerned and assuming the use of the computer codes according to the 
state of the art (common practice, in principle Eurocode models), the values according to Table A.22 may be 
considered as recommended. 

Table A.22 Model uncertainties for the global analysis; the correlation  is between members 

Structural type 

Load effect type   Bias  CoV 

Frames statically determinate  

N, M, D 

1,0 

0,05 

Frames statically indeterminate  

N, M, D 

1,0 

0,10 

Plate structures   

Plate structures 

Shells  

2D solids  

3D solids  

m 

n 

m, n 

 

 

1,0 

0,20 

1,0 

0,10 

1,0 

0,20 

1,0 

0,10 

1,0 

0,15 

 

0,8 

0,8 

0,8 

0,8 

0,8 

0,8 

0,8 

All model uncertainties may be assumed to have a lognormal distribution. 

It  is  assumed  here  that  applied  FEM  models  have  appropriate  geometrical  and  physical  linear  or  nonlinear 
features. In case of geometrical nonlinear analysis relevant initial imperfections should be included. In case of 
physical  nonlinearities  care  is  taken  of  deformation  capacity  and  possible  path  dependence.  Usually  the 
fulfilment of code requirements avoids buckling and/or brittle behaviour with sufficient reliability. 

References 

JCSS (2001), JCSS Probabilistic Model Code, Joint Committee on Structural Safety, https://www.jcss-lc.org/jcss-
probabilistic-model-code/ (periodically updated, online publication) 

A.7 Model Uncertainties for (local) resistance models 

Model uncertainties are defined according to: (not including parameter uncertainty in principle) 

ӨR = Rexperiment / Rmodel 

The statistical properties of a model uncertainty are always connected to a specific model and a well-defined 
population of structures for which it has been derived and will be used. For the determination of the statistical 
properties reference is made to EN 1990:2023, Annex D. 

Note  that  theoretically  model  uncertainties  do  not  include  the  uncertainties  or  scatter  in  the  other  basic 
variables. However, in practice this is not always possible resulting in some double counting of uncertainties. 

Uncertainties  regarding  resistance,  θR,  are  dependent  on  both  the  material  and  the  type  of  internal  force 
considered. All model uncertainties in this chapter are assumed to have a lognormal distribution. 

142 

 
 
 
 
 
 
 
 
 
 
 
A.7.1 Reinforced concrete 

Table A.23 Model uncertainty related to the resistance model for concrete 

Relative 
frequency 

Random 
variable 

Mean 

CoV 

wk 

μθ 

Vθ 

Bending moment 

Shear 

Axial force 

Bending in plates 

Buckling 

30% 

30% 

10% 

10% 

20% 

*Including position of the reinforcement 

θR, mod, ik 

1,035 

7% * 

θR, mod, ik 

θR, mod, ik 

θR, mod, ik 

θR, mod, ik 

1,40 

1,00 

1,20 

1,40 

25,0% 

5,0% 

15,0%* 

25,0% 

More in detail, for bending use has been made of (see EN 1992-1, Annex A): 

Table A.24 Detailed background information for resistance model for concrete 

X 

Description 

Distr 

Bias 

CoV 

bending 

d 

d 

 

Effective depth 

Effective depth d 

Model uncertainty 

Compression 

S 

Ac 

C 

In-situ factor (=

fc,als
a)  ) 
fc

Concrete area 

Model uncertainty steel 

LN 

LN 

LN 

LN 

LN 

LN 

md = 1 -0,05(200/d)2/3. 

Vd = 0,05(200/d)2/3 

 = 0,95  

S= 1,09 

 = 0,95 

 = 1,00 

 = 1,02 

0,050  

0,045  

0,120 

0,040 

0,060) 

In total resulting in m = 1,07 en CoV=0,14, as indicated in the first table 

References 

CEN – European Committee for Standardization, EN 1992-1, Annex A 

CEN – European Committee for Standardization, Background document FprEN 1992-1-1 (final version), 2023-
04-05), CEN-TC 250-SC 2_N2087 

A.7.2 Steel 

Parameters for the model uncertainty in steel structures designed to EC3 or (where pertinent) EC4 are presented 
in Table A.25; the values are based on data found e.g. in the reference list. 

143 

 
 
 
 
 
Table A.25 Model uncertainty related to the resistance model for steel 

Load 
or ULS limit state 

effect  

Relative 
frequency 

Random 
variable 

Mean 

CoV 

 a) 

wk

µ b) 

 b) 

V

Cross-sectional resistances – no buckling, associated with fy and  M0  

Bending – plastic, compact, elastic sections 

Shear  

Axial force - yielding 

Torsion 

15% 

3% 

15% 

2% 

R,mod,ik 

1,225 

7,5% 

1,40 

10% 

1,00 d) 

- d) 

1,40 

10% 

Cross-sectional resistances – slender sections, local buckling, associated with fy and  M1  

Bending moment and shear 

Axial compression  

7,5% 

7,5% 

R,mod,ik 

1,15 

1,10 

5% 

4% 

Member resistances -  global buckling, associated with fy and  M1  

Column or compression member buckling 

Lateral-torsional buckling 

15% 

15% 

R,mod,ik 

1,15 

1,20 

5% 

7,5% 

Local section or connector resistances -  associated with fu and  M2  

Net section and bolt bearing failure 

Welds 

7,5% 

7,5% 

1,10 

R,mod,ik 

1,10 

Bolts, studs and other mechanical connectors 

5% 

1,10 

5% 

5% 

5% 

a) Weights are based on best estimates  of dominant ultimate limit states in common building, bridge and 
industrial structures 

b) The mean value of the model uncertainty is defined as the average of the ratio between available test 
results and strength predictions given by the corresponding Eurocode design model , the corresponding CoV 
as the coefficient of variation of this ratio  

c)  Recommended  values  for  the  partial  factors  are:  γM0=1,0  ;  γM1=1,0  or  1,1  (depending  on  application); 
γM2=1,25    

d) The model uncertainties associated with axial force yielding are covered by the uncertainties for material 
and geometry 

The resulting weighted R,mod,ik for structural steel has a mean of ca. µ=1.15 and a CoV of 5%. 

144 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Note: Model uncertainties for buckling may depend on the eccentricities used in the model: a large value for the 
eccentricity will lead to a higher average of the model uncertainty.  

References  

Braconi,  A.,  et  al.  (2013),  Optimizing  the  seismic  performance  of  steel  and  steel-concrete  structures  by 
standardizing material quality control, OPUS, European Commission, Technical  steel research, Steel products 
and applications  for  building  construction  and  industry,  Final  report,  EUR  25893,  EC  Directorate  General  for 
Research, Brussels, 2013 

Byfield, M.P., Nethercot, D.A. (1997), ‘Material and geometric properties of structural steel for use in design’, The 
Structural Engineer, Vol. 75, pp. 363-367 

Byfield, M.P., and Nethercot, D.A. (1998), ‘An analysis of the true bending strength of steel beams’, Proceedings 
of the Institution of Civil Engineers. Structures and Building,. Vol. 128, pp. 188-197 

Commission of the European Community - CEC (1988), ENV background doc. 5.03 

Francis P. et al. (2016),  High Strength Long Span Structures  - Final Report, HILONG (RFCS-CT-2012-00028): 
Final Report, EC Directorate General for Research, Brussels 

Greiner R. et al (2011), SEMI-COMP+: Valorisation Action of Plastic Member Capacity of Semi-Compact Steel 
Sections – a more Economic Design, RFS2-CT-2010-00023, Background Documentation, Research Programme 
of the Research Fund for Coal and Steel – RTD, Final report, EUR 25893, EC Directorate General for Research, 
Brussels, 2011 

Greiner, R., Kaim, P. (2001), Comparison of LT-buckling design curves with test results, ECCS TC8 – Report No. 
TC8-2000-014, European Convention for Constructional Steelwork, Brussels 

Kleiner,  A.  (2014),  Beurteilung  des  Tragverhaltens  von  Flankenkehlnahtverbindungen  aus  normal-  und 
höherfestem Baustahl unter Berücksichtigung statistischer Kriterien, Institut für Konstruktion und Entwurf, PhD 
Dissertation, Universität Stuttgart (in German) 

Može. P., Sinur. F. (2014), Statistical evaluation of net cross-section strength function, Research Report, Faculty 
of Civil and Geodetic Engineering, University of Ljubljana, Slovenia 

Müller, C. (2003), Zum Nachweis ebener Tragwerke aus Stahl gegen seitliches Ausweichen, PhD Dissertation, 
RWTH Aachen  

Schillo, N., Taras, A., Feldmann, M. (2018), ‘Assessing the reliability of local buckling of plates for mild and high 
86-98, 
strength 
https://doi.org/10.1016/j.jcsr.2017.12.001. 

Constructional 

Research, 

Journal 

steels’, 

Steel 

142, 

Vol. 

pp. 

of 

Simões  da  Silva,  L.,  et  al.  (2016),  Standardization  of  safety  assessment  procedures  across  brittle  to  ductile 
failure modes (SAFEBRICTILE) – Final Report, EC Directorate General for Research, RFSR-CT-2023-00023, KI-
NA-28906-EN-C, ISSN 978-9 -79-76999-3. DOI 10.2777/76892, Brussels 

Simões da Silva, L., Rebelo C., Nethercot D. et al. (2008), ‘Statistical Evaluation of the Lateral-Torsional Buckling 
Resistance of Steel I-beams, Part 2: Variability of steel properties’, Journal of Constructional Steel Research., 
Vol. 64, pp. 832-849 

Tankova, R., Simoes da Silva, L., Marques, L., Rebelo, C., Taras, A. (2014), ‘Towards a standardized procedure for 
the safety assessment of stability design rules’, Journal of Constructional Steel Research, Vol. 103, pp. 290-
302, https://doi.org/10.1016/j.jcsr.2014.09.010. 

Taras, A. (2010), Contribution to the development of consistent stability design rules for steel members, PhD 
Dissertation, TU Graz 

Taras, A., Greiner, R. (2010), New design curves for lateral–torsional buckling—Proposal based on a consistent 
pp.  648-663, 
derivation, 
https://doi.org/10.1016/j.jcsr.2010.01.011. 

Research,  Vol.  66, 

of  Constructional 

Issue  5, 

Journal 

Steel 

145 

 
 
 
 
A.7.3 Timber 

(Sørensen,  Svensson,  and  Stang  2005)  report  values  CoVθ=0,05  for  bending  and  short  term  load  duration. 
(Kohler 2006) reports CoVθ=0,05-0,10 for bending and short term load in additon CoVθ=0,10 for bending and 
long term load duration including damage accumulation. 

Table A.26 Model uncertainty related to the resistance model associated with the load effect for timber 

Bending moment 

Shear 

Axial force 

Buckling 

Joints  with  laterally  loaded  dowel  type 
fasteners / European yield model: 

-embedment failure   

-failure  with  two  plastic  hinges  in  the 
fasteners per shear plane   

wk 

variable 

μθ 

CoVθ 

30 % 

30 % 

10 % 

30 % 

θR, mod, ik 

θR, mod, ik 

θR, mod, ik 

θR, mod, ik 

θR, mod, ik 

θR, mod, ik 

1,00 

1,00 

1,00 

1,00 

0,8 

1,25 

10,0% 

15,0% 

5,0% 

15,0% 

10,0% 

10,0% 

The resulting θR,i for timber has mean 1,00 and coefficient of variation 0,15. 

A considerable model uncertainty is reported for connections including a bias (μθ≈0,8-1,3 for single fasteners 
and μθ≈1,25 for multiple fasteners) and variability (CoVθ≈0,07-0,10  for single fasteners and CoVθ≈0,20 for 
multiple fasteners), e.g. Kohler, 2006 and Jockwer, Fink, Köhler, 2018. 

Other examples are the  design of notched beams with fracture in tension  perpendicular to grain and shear 
(Jockwer et al. 2011) with μθ≈0,8 and CoVθ≈0,25. 
References 

Jockwer, R., Steiger, R., Frangi A., J. Köhler, J. (2011), ‘Impact of Material Properties on the Fracture Mechanics 
Design Approach for Notched Beams in Eurocode 5’, in: Proceedings of the 44th CIB-W18 Meeting, Paper 44-6-
1, Alghero, Italy. 

Jockwer,  R.,  Fink,  G.,  and  Köhler  J.  (2018)  “Assessment  of  the  Failure  Behaviour  and  Reliability  of  Timber 
Connections  with  Multiple  Dowel-Type  Fasteners”,  Engineering  Structures,  Vol.  172,  pp.  76–84. 
https://doi.org/10.1016/j.engstruct.2018.05.081. 

Kohler, J. (2006.), Reliability of Timber Structures, PhD Thesis Nr. 16378, Zurich, Switzerland: IBK ETH Zurich. 

Sørensen, J. D., Svensson, S. and Stang B. D. (2005), ‘Reliability-Based Calibration of Load Duration Factors for 
153–69 
Timber 
https://doi.org/10.1016/j.strusafe.2004.10.001  

Structures’, 

Structural 

Safety, 

Issue 

Vol. 

27, 

pp. 

2, 

A.7.4 Masonry 

Information on the model uncertainty for masonry under compression and buckling can be found in Bakeer & 
Salehi, 2019 and Brehm, 2011; a summary is presented in Table A.27. 

146 

 
 
 
 
 
 
 
 
 
 
Table A.27 Summary of the stochastic models of the model uncertainties for masonry 

Failure mode 

Unit 

Model 

Dist. 

m 

σ 

CoV 

Compression θfC 

Bakeer & Salehi 2019 

0,97 

16% 

Diagonal tension θdt 

Sliding shear θs 

Flexure θf 

AAC 

CB 

CS 

CB 

CS 

DIN EN 1996-1-1/NA 

1,14 

0,20 

17% 

1,03 

0,18 

17% 

DIN EN 1996-1-1/NA 

1,15 

0,24 

21% 

Lognormal 

DIN EN 1996-1-1/NA 

1,24 

0,24 

19% 

1,00 

0,18 

18% 

AAC 

Fully-plastica 

1,05 

0,16 

15% 

DIN EN 1996-1-1/NA 

1,00 

0,20 

20% 

1,10 

0,20 

18% 

Shear crushing θc 

CB 

all 

astress-strain relationship 

a See also (Salehi 2018). 

References 

Bakeer, T. and Salehi, H. (2019), ‘Determination of a model partial safety factor for unreinforced masonry walls 
under buckling’, European Masonry Journal, Vol. 23, Issue 1, pp. 32 – 39 

German  Institute  for  Standardisation  –  DIN  (2012),  National  Annex  –  Nationally  determined  parameters. 
Eurocode 6: Design of masonry structures – Part 1-1: General rules for reinforced and unreinforced masonry 
structures, DIN EN 1996-1-1/NA.  

Brehm  E.  (2011),  Reliability  of  Unreinforced  Masonry  Bracing  Walls  -  Probabilistic  Approach  and  Optimized 
Target Values, in: Dissertationsreihe Prof. Dr. C.-A. Graubner, TU Darmstadt  

Salehi. H. (2018), ‘Determination of partial factor for model uncertainty for unreinforced masonry sheer wall’, 
European Masonry Journal, Vol. 22, Issue 2, pp. 103 -112 

A.7.5 Aluminium 

A  list  of  model  uncertainty  statistics  related  to  the  resistance  models  associated  with  the  load  effects,  i.e., 
typical failure modes of aluminium members is given in Table A.28. Relative frequency values are assumed, 
and they should be obtained from real design situation when considering relevant structural member. 

Table A.28. Model uncertainty related to the resistance model associated with the typical failure modes of aluminium 
member 

Resistance / failure mode 

Axial resistance of cross-section 

Random 
variable 

Mean 

CoV 

μθ 

Vθ 

θR, mod, ik 

1,00 

0,0% 

Relative frequency 

wk

1 

10% 

147 

 
 
 
 
 
 
 
 
 
 
 
 
 
Bending resistance of cross-section2 

Shear resistance of cross-section 

30% 

5% 

θR, mod, ik 

1,27  

14,9% 

θR, mod, ik 

1,00 

5,0% 

Buckling resistance of members in compression3  20% 

θR, mod, ik 

1,18  

14,6% 

Buckling resistance of members in bending4 

25% 

θR, mod, ik

4 

1,00 

10,0% 

1 Each load effect is sampled with a different relative frequency, wk. The values are assumed. 
2 Including the effects of normal and shear forces. The values are obtained from tests related to bending of extruded tubes (EN AW 
6061-T6 and 6063-T5). 
 3 The values are obtained from tests related to buckling of extruded tubes and I-sections (EN AW 6061-T6, 6063-T5 and 6082-T6). 
4Including the effects of axial compression. 
5 For lateral-torsional buckling of unsymmetrical extruded profiles, for resistance model, θR, mod, ik, the mean of 1,19 and coefficient of 
variation of 0,08 are obtained (Doksanovic, PhD). The distribution is lognormal. When results are separated by relative slenderness 
areas, the values are 1,27/0.06 (less than 1,2) and 1,13/0.03 (higher than 1,2). 

References 

Zhu, J.H. and Young,  B. (2009), ‘Design of Aluminum  Alloy Flexural Members  Using Direct Strength  Method’, 
Journal of structural engineering Vol. 135, pp. 558–566. 

Zhu, J.H., Young, B. and Gardner, L. (2014), ‘Deformation-based design of aluminium alloy beams’, Engineering 
Structures, Vol. 80, pp. 339-349. 

Zhu, J.H., Young, B. and Gardner, L. (2016), ‘Flexural response of aluminium alloy SHS and RHS with internal 
stiffeners’, Engineering Structures, Vol. 121, pp. 170-180. 

Zhu,  J.H.  and  Young,  B.  (2006a),  ‘Tests  and  design  of  aluminum  alloy  compression  members’,  Journal  of 
structural engineering, Vol. 132, Issue 7, pp. 1096–1107. 

Zhu, J.H. and Young, B. (2006b), ‘Experimental investigation of aluminum alloy circular hollow section columns’, 
Engineering Structures, Vol.  28, pp. 207–215. 

Adeoti, G. O., Fan, F., Wang, Y. and Zhai, X. (2015), ‘Stability of 6082-T6 aluminium alloy columns with H-section 
and rectangular hollow section’, Thin-walled structures, Vol. 89, pp. 1-16. 

Wang, Y., Fan, F. and Lin, S. (2015), ‘Experimental investigation on the stability of aluminium alloy 6082 circular 
tubes in axial compression’, Thin-walled structures, Vol. 89, pp. 54-66. 

Wang,  Y.  Q.,  Yuan,  H.  X.,  Chang,  T.,  Du,  X.  X.  and  Yu,  M.  (2017),  ‘Compressive  buckling  strength  of  extruded 
aluminum alloy I-section columns with fixed-pinned end conditions’, Thin-walled structures, Vol. 119, pp. 396-
403. 

Wang,  Z.  X.,  Wang,  Y.  Q.,  Sojeong,  J.,  and  Ouyang,  Y.  W.  (2018),  ‘Experimental  investigation  and  parametric 
analysis on  overall  buckling  behavior of  large-section  aluminum  alloy  columns.’,  Thin-walled  structures, Vol. 
122, pp. 585–596. 

Dokšanović, T. (2017), Lateral-torsional buckling of aluminium elements with open unsymetrical cross section 
(in Croatian). PhD Thesis, University of Zagreb, Slovenia 

A.7.8 Ground 

Resistance models for geotechnical structures refer to a large variety of structures, some of which are part of 
larger  structures  (e.g.  foundations  of  buildings  and  bridges),  while  others  are  structures  on  their  own  (e.g. 
embankments, retaining structures). For the main purpose of this annex, namely calibration of partial factors 
for typical structural members, Table A.29 contains a list of model uncertainty statistics for foundation elements 
(expert judgement). More information on  model  uncertainty for  geotechnical applications  for broader  use in 
reliability and risk analyses can be found in the literature; ISSMGE-TC304 (2020) contains a valuable literature 
review. 

148 

 
 
 
 
Table A.29 Model uncertainty related to the resistance model associated with the load effect for ground 

wk 

Random 
variable 

μθ 

COVθ 

Spread foundation – bearing capacity  

Piled Foundation – vertical bearing capacity (* 

θR, mod, ik 

1,00 

θR, mod, ik 

1,00 

0,15 

0,20 

Reference 

International Society of Soil Mechanics and Geotechnical Engineering (ISSMGE) - Technical Committee TC 304 
‘Engineering  Practice  of  Risk  Assessment  and  Management’  (2020),  State-of-the-art  review  of  inherent 
Download: 
variability 
http://140.112.12.21/issmge/2021/SOA_Review_on_geotechnical_property_variablity_and_model_uncertainty.
pdf 

geotechnical 

uncertainty 

properties 

models, 

and 

and 

in 

Annex A List of boxes 

CASE A: 

CASE B: 

129 

130 

Annex A List of figures 

Figure A.1. Arbitrary realization of the sustained intermittent load for an office space [N/m2] 

124 

Figure A.4. European Ground snow load map  

Figure A.3. Coefficient of variation of annual maximum of snow load versus altitude  133 

Annex A List of tables 

Table A.1. Concrete elements self-weight models  120 

Table A.2 Steel elements self-weight models 

Table A.3 Timber elements self-weight models 

121 

121 

Table A.4 Variability of available aluminium members’ dimensional properties 122 

Table A.5 Numerical values for non-structural permanent loads 

Table A.6 Parameters of the sustained and intermittent loads 

123 

125 

Table A.7 Basic variables properties from and partial safety factors  127 

Table A.8 Statistical data for snow load (CASE A)  129 

Table A.9 Statistical data for snow load (CASE B)  130 

Table A.10 Coefficient of variation for yearly maximum snow load on ground (from Annex 3 in [4]) 

132 

Table A.11 Indicative data for car impact  134 

Table A.12 Indicative data for internal Natural Gas explosion 

134 

Table A.13 Seismic actions 

135 

Table A.14 Concrete properties  136 

149 

 
 
 
 
 
 
 
 
 
Table A.15 Basic variables reinforcement  137 

Table A.16 Basic variables steel  137 

Table A.17 Timber Strength models and parameters 

138 

Table A.18 Wrought aluminium alloys to be used in structural applications according to EN 1999-1-1:2007

139 

Table A.19 Variability of available material properties 

139 

Table A.20 Probabilistic model for generic Masonry 140 

Table  A.21  Indicative  ranges  of  coefficient  of  variation  of  ground  properties  from  EN1997-1  Annex  A 
(representing spatial variability, i.e. no spatial averaging is included yet) 

141 

Table A.21a Typical coefficients of variation representing the uncertainty in spatial average ground strength 
parameters, recommended for calibration of partial factors  141 

Table A.22 Model uncertainties for the global analysis; the correlation  is between members  142 

Table A.23 Model uncertainty related to the resistance model for concrete 

143 

Table A.24 Detailed background information for resistance model for concrete 

143 

Table A.25 Model uncertainty related to the resistance model for steel 

144 

Table A.26 Model uncertainty related to the resistance model associated with the load effect for timber 146 

Table A.27 Summary of the stochastic models of the model uncertainties for masonry 147 

Table A.28.  Model uncertainty related to the resistance model associated with the typical failure modes of 
aluminium member 

147 

Table A.29 Model uncertainty related to the resistance model associated with the load effect for ground 149 

Annex B. Applications and Examples 

B.1 Introduction 

This annex contains examples and applications intended to clarify parts of the main report. It is intended to be 
helpful in checking or modifying reliability related National Determined Parameters by NSB’s. Some examples 
might  also  prove  useful  when  advanced  reliability  methods  are  directly  applied  in  practical  design  and 
assessment procedures. 

B.2 Reliability based background information on partial factors  

B.2.1 Results for design format EN 1990:2023-(8.12) 

B.2.1.1 Optimization method 

In chapter 3.4 of the main text it was concluded that using the partial factors currently proposed in Eurocode 
EN1990 the average reliability level for a reference period of 50 year is close to t = 3,8 which is formulated 
as  the  target  value.  However,  the  scatter  around  this  value  is  quite  large  and  the  question  arises  whether 
another choice of partial factors could lead to the same average but less scatter, that is could lead to less error 
in a number of individual cases. 

In this section an example of that type is presented for the derivation of partial factors for the permanent and 
variable loads, on the basis of chapter 2.6.2 in the main part of this document and introduced in EN1990 C.4.4.3. 
The object of the procedure is to minimize the weighted difference of squared deviations::  

150 

 
 
 
 
  
min  wi (i() - t) 2 

(B.1) 

The data for this example are the same as presented in the Tables 3.2, 3.3 and 3.7 of the main text and used 
for the calculations in chapter 3.4. That is: 

Design format EN 1990 equation (8.12) 

Partial resistance factors M according to the current Eurocodes as presented in Table 3.2; 

Load ratios aQ, aG and weight factors w as presented in Table 3.3; 

Probabilistic models according to Table 3.7. 

The target reliability is t = 3,8 for a period of 50 year as is the basis in the Eurocode. The resulting partial 
factors on the basis of equation B.1 and the data from chapter 3.4 mentioned above are:  

G = 1,20 for the self-weight and  

Q = 1,94, 2,35 and 1,60 for respectively wind, snow and imposed load.  

Figure B.1 shows the violin plots for the calibrated values (right hand side and green bars) and the standard 
Eurocode values G = 1,35 and Q = 1,50 (left hand side and blue bars). As could be expected, the scatter for 
the calibrated values is lower than for the classical Eurocode values because for the assumptions inherent in 
this example, the partial factors have been optimised to minimise the scatter. 

If a system with only one common partial factor for variable loads is chosen, the optimal values G = 1,20 for 
the  self-weight  and  Q  =  1,75  for  wind  snow  and  imposed  load  are  obtained.  In  that  case  the  scatter  is  in 
between two results shown in Figure B.1. 

Figure B.1. Violin plots for the scatter in resulting β-values 

Figure  B.2  shows  the  corresponding  β  -values  as  function  of  αQ  for  wind,  snow  and  live  load  for  the  non-
optimized case. Each line corresponds to one of the materials.  

151 

 
 
 
 
 
 
 
Figure B.2. Non-optimized Reliability index β as function of αQ for wind, snow and live load ranging from 2.5 to 4.5 

0.2

0.4

0.6

0.8

1

αQ

0.2

0.4

0.6

0.8

1

αQ

4.5

4

3.5

3

2.5

β

2

0

wind 

4.5

4

3.5

3

2.5

β

2

0

snow 

5

4.5

4

β

3.5

3

2.5

2

0

0.2

0.4

0.6

0.8

1

αQ

Imposed load 

152 

steel bending

concrete bend

concrete comp

glulam bend

timber bend

masonry

aluminium

pile found

shallow found

steel bending

concrete bend

concrete comp

glulam bend

timber bend

masonry

aluminium

pile found

shallow found

steel bending

concrete bend

concrete comp

glulam bend

timber bend

masonry

aluminium

pile found

shallow found

 
 
 
 
 
B.2.1.1 Design Value Method 

As mentioned in chapter 2.6.1 of the main text and as introduced in EN 1990 C.4.4.2 partial factors can also be 
obtained using the Design Value method (DVM) which in the Eurocode is commonly used to derive the partial 
factors on the resistance side. Based on the same probabilistic models as above the following results for the 
partial factors for variable loads are obtained (the values in parenthesis are with the load effect uncertainty c.l 
included): 

Wind load: 

Characteristic value Qk = 1,08, Design value Qd = 2,05 implying a partial factor γQ = 1,89 (1,96) 

Snow load: 

Characteristic value Qk = 0,82, Design value Qd = 1,86 implying a partial factor γQ = 2,27 (2,31) 

Imposed load: 

Characteristic value Qk = 1,35, Design value Qd = 2,08 implying a partial factor γQ = 1,54 (1,60) 

where: 

The  characteristic  value  Qc  for  the  wind  and  snow  loads  are  based  on  P(θQ Q1a≤Qc)=0,98  i.e.  the  annual 
probability of exceedance of the load [θQ Q1a] is 0,02, where Q1a is the annual maximum of the time-variant 
part of the load and  Q is the time independent load model uncertainty.  

The design value Qd for loads is based on P(θQ Q50a≤Qd)=Ф(0,7 βt), where the 50 year target reliability index 
βt=3,8. 
It can be concluded that the results are close to the result of the optimization method. For further comments 
reference is made to  section B.2.2.5. Note also that the results are similar to the results in the Background 
document  to  the  earlier  version  of  the  Eurocode  (PT  Eurocode  1.1  and  JCSS,  Background  Documentation, 
Eurocode 1 (ENV1), Part1 Basis of Design, ECCS No 94, March 1996). 

B.2.2 Results for design formats EN1990:2023 (8.13a/b) 

Using  the  same  models  as  in  section  B.2.1  we  may  derive  reliability  levels  for  designs  based  on  EN  1990 
(8.13a/b) with = 0,89 (which corresponds to  G = 1,20, as used in some counties),  Q =1,5 and 01 values 
equal to 0,6, 0,5 and 0,7 for respectively wind, snow and imposed loads.  

Table B.1 Average reliability indices per material and action 

wind 

snow 

imposed 

Steel bending  

Concrete compression  

Concrete bending  

Glulam bending  

Solid timber bending  

Masonry  

Aluminium bending  

Pile foundation  

Shallow foundation 

3,26 

3,55 

3,04 

3,08 

3,17 

3,86 

3,38 

3,24 

3,41 

2,88 

3,31 

2,75 

2,80 

2,91 

3,65 

3,05 

3,09 

3,24 

153 

3,71 

3,90 

3,46 

3,48 

3,54 

4,15 

3,79 

3,52 

3,69 

 
 
The  overall  result  can  be  characterized  by    =  3,45  (crude  average)  and  =  3,37  (using  the  weights  wi as 
specified in the Table 3.3). As expected the result is lower than for design based on (8.12). 

Calibrating the partial factors in this case leads to: 

G = 1,30 for the self-weight and  

Q = 2,05, 2,50 and 1,65 for respectively wind, snow and imposed load.  

When imposing a common partial factor for all variable loads one obtains: 

G = 1,33 for the self-weight and  

Q = 1,78 for wind, snow and imposed load.  

The target reliability is t = 3,8 for a period of 50 year as is the basis in the Eurocode.  

B.2.3 Results for annual reliability and design formats EN1990:2023-(8.12) 

In section B.2.1 the reliability index for the 50 year period was calculated. As explained in EN 1990-Annex C the 
β = 3,8 for this period is the anchor point for the Eurocode reliability target. However, in some applications and 
also as a choice in some member states, the annual reliability level is preferred. Using the same models as in 
section B.2.1 (but of course adjusted to annual probabilities where relevant) we may calculate for the annual 
reliability levels according to Table B.2. 

Table B.2 Average reliability indices per material and action 

wind 

snow 

imposed 

Steel bending  

Concrete compression  

Concrete bending  

Glulam bending  

Solid timber bending  

Masonry  

Aluminium bending  

Pile foundation  

Shallow foundation 

4,36 

4,50 

4,25 

4,10 

4,11 

4,67 

4,42 

4,07 

4,27 

4,10 

4,37 

4,08 

3,95 

3,98 

4,57 

4,22 

4,02 

4,22 

4,44 

4,60 

4,33 

4,21 

4,23 

4,77 

4,50 

4,20 

4,39 

The overall result can be characterized by  = 4,36 (crude average) and = 4,29 (average using the weights wi 
as specified in the Table 3.3 of the main text). The average annual reliability index is close to 4,3, i.e. much 
lower than the upper bound value 4,7 mentioned in EN 1990 Annex C. As to be expected the lowest values close 
to 4,0 occur for foundations and the highest (masonry under imposed loads is close to the upper bound). 

B.2.4 Results sensitivity studies 

In order to get a feeling for the sensitivity of the calibration results to some of the input assumptions, a number 
of additional analyses has been made based on modifications in section 3.4. The results have been summarized 
in table B.3: 

154 

 
 
 
 
Table B.3 Overview of sensitivity studies 

COV 
wind 

𝛼𝑄 
distribution 

Procedure 

Gamma  

Gamma 

perm 

variable  

loads  

loads 

WEIGHTED 
AVERAGE 
BETA   

AVERAGE 
BETA 

All materials/all loads 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

0,14 

uniform 

fixed  

0,12 

uniform 

fixed 

0,17 

uniform 

fixed 

1,35 

1,35 

1,35 

0,14 

uniform 

optimized a  

1,20 

0,12 

uniform 

optimized 

1,20 

0,17 

uniform 

optimized  

1,2 

1,50 

1,50 

1,50 

1,75 

1,70 

1,85 

0.14 

uniform 

optimized 

>1,25 

1,70 

0.14 

uniform 

optimized 

>1,30 

1,65 

0.14 

triangular 

optimized 

1,20 

linear 

optimized  

1,25 

Combination ”Steel-wind” only 

11 

12 

0,12 

uniform 

fixed 

1,35 

0,12 

Uniform 

Optimized  

1,10 

a optimized is always for the weighted average 

1,75 

1,65 

1,50 

1,75 

3.74 

3,79 

3,59 

3,78 

3,78 

3,81 

3,80 

3,83 

3,81 

3,82 

3,71 

3,78 

3.64 

3,68 

3,67 

3,68 

3,67 

3,73 

3,70 

3,73 

3,68 

3,65 

3,71 

3,78 

Calculation 1 (C1 for short) on the first line is the standard calculation as presented in Chapter 3.4. It shows 
that the Eurocode partial load factors 1,35 and 1,50 lead to an average reliability index close to the target 3,8, 
both for the assumed and for the constant w-factors (last column). This shows the relative moderate influence 
of the weight factors. The lines C2 and C3 show the sensitivity to the uncertainty in the wind model by changing 
the CoV from 0,14 (standard) respectively down to 0,12 and up to 0,17. As expected the beta-values go up for 
a lower wind uncertainty and down for a larger one. However, the result is not dramatic. 

The lines 4-6 give the result of the partial factor optimization procedure for the same cases  with optimized 
partial factors rounded off to a discretization with 0,05. The factors for snow, wind and imposed load are forced 
be the same like in the Eurocode EN1990:2021. The resulting beta-values are a bit closer to the target of 3,8, 
but also the scatter in the results is less. This can be seen in the violin plots in figure B1 where the scatter of 
Beta of the EN1990 factors is on the left side and the optimized values on the right hand side, although here 
all partial factors for the variable loads are optimized individually.  

155 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Calculations C7 and C8 show the results if in the optimization procedure the factor for the permanent load is 
constrained to have values larger than 1,25 respectively 1,30. The optimization procedure then leads to the 
minimum allowed values for the permanent loads and to 1,70 and 1,65 for the variable loads. Of course the 
scatter in the results increases if these higher values for the permanent load is imposed. 

In the basic calculations the aQ values (governing the ratio of permanent and variable load) for all have been 
given an equal weight. In C9 and C10 the shape of the aQ distribution is changed from uniform to triangular 
and linear (see Figure B.3). Only assuming a full linear distribution has a substantial influence and makes the 
partial  factor  for  self-weight  rise.  Note  that  this  distribution  gives  a  high  weight  to  a  very  low  relative 
contribution of the total load and is only valid for soil and a specific subset of concrete structures. 

Figure B.3. Examples of aQ distribution (uniform, triangular symmetric, triangular 0,4, linear) 

0.250

0.200

0.150

0.100

0.050

0.000

0

2

4

6

8

10

12

Finally, the calculations C11 and C12 show the results for the standard case (CoV wind =0,14 and uniform aQ 
values) for the case of only one material (steel) and one variable load (wind). The result is that the partial factor 
for permanent load should be lowered and the factor for the variable loads should be increased compared to 
the overall average.  

B.2.5 Effect of bias in characteristic value on partial factors 

The example calculations performed in Chapter 3.8 of the main text as well as in Annex B2 are always based 
on the definitions of the characteristic values according to EN1990 clause 6.1.2.3. However, one should keep in 
mind  that  National  Annexes  to  the  relevant  Eurocode  parts  may  (and  often  do)  provide  slightly  different 
characteristic  values  if  compared  to  the  results  of  extreme  value  analysis  of  measured  data.  This  holds  in 
particular for the characteristic values of climatic loads and it is mainly due to the needs of mapping procedures 
in the derivation of climatic maps, as well as to past practice. This section will show the possible effects on the 
partial factors. 

As an example we will consider a bias in wind and snow loads. Starting from data collected and made available 
by the German Meteorological Office, (Formichi and Landi, 2022) calculated a preliminary estimation of the 
mean  value  of  the  bias for  ground  snow  loads and  wind  pressure  at  approximately  20  weather  stations  in 
Germany, spread over the country as illustrated in Figure B.4, where the location of the selected weather station 
is plotted against the corresponding climatic zone, for both snow and wind actions as given in the German NAs 
to the pertinent Eurocode parts.  

156 

 
 
 
 
 
 
 
Figure B.4. Selected weather stations and snow zones (left) and wind zones (right) (with courtesy of the University of 
Pisa)  (Formichi et al, 2023) 

It turned out that the mean value and standard deviation of the bias (ratio between the characteristic value 
resulting from EV analysis and the corresponding NA’s value) are given by:  

-for wind pressure 

: mean(b) = 0.83 and st dev(b)=0.23   

-for snow load      

: mean(b) = 0.72 and st dev(b)=0.12   

where b = Qk/Qk* with Qk the characteristic value according to EN1990 and Qk* the characteristic value provided 
by the regional code. 

Note:: the data were extracted from the website of the German Meteorological Office: Deutscher Wetterdienst 
(DWD) 
2022 
(:https://opendata.dwd.de/climate_environment/CDC/observations_germany/climate/).  Comparisons  with  NAs 
data were made against: DIN EN 1991-1-3/NA. Eurocode 1 – Actions on Structures – Part 1–3: General Actions 
– Snow Loads –National Annex (in German). and  DIN EN 1991-1-4/NA. Eurocode 1 – Actions on Structures – 
Part 1–4: General Actions – Wind Actions –National Annex (in German). 

observations. 

Climate 

If the bias b has no scatter (just single values over the whole area) and a value lower than 1.0, it means that 
the design load used in the design equation (see chapter 3) using the partial factor derived according to the 
procedure in chapter 3 is higher than required. So, in that case we might actually multiply the partial factor by 
the  bias  b  and  end  up  with  the  desired  result.  Characteristic  values  and  partial  factors  are  communicating 
vessels: higher characteristic values for loads mean lower partial factors. 

If the bias is random (meaning that we do not know the exact value in every location within a zone or neglect 
it for simplicity) we can split up the bias into a mean part and a fluctuating part. The mean part may be treated 
as before. The zero mean fluctuating part does not affect the dimensions but leads to a larger uncertainty. It 
has to be taken care of as an additional random factor in the LSF in the reliability calculation. If the effects of 
the fluctuating part is significant (i.e. high C.O.V.), this will reduce beta which has to be compensated by a larger 
partial factor.  

In the present example case of climatic loads in Germany we find the following results for the optimized partial 
factors (see Formichi et al, 2023) 

Table B.4 Calibrated partial factors for different actions and for different treatment of bias in climatic actions 

 G 

 wind   snow   imposed 

No bias, no scatter, as in Annex B2                              

1,2 

1,9 

With bias, no scatter                

With bias and scatter              

1,2 

1,2 

1.6 

2,1 

157 

2,35 

1,65 

1,85 

1,6 

1,6 

1,6 

 
  
 
 
 
The values presented in the line “with bias, no scatter” have no direct meaning at regional scale, but make sense 
at local scale where it can be assumed that the characteristic value is known from available measurements. 
The values are also intended to illustrate the contribution of the various effects. The reason for deviating from 
the formal definition of the characteristic value in the NA could be that codes keep definitions and corresponding 
values from previous codes. In those cases one should be aware of the fact that the characteristic value and 
the partial factor are communicating vessels: high characteristic values may be compensated by lowered partial 
factors and vice  versa. But it is essential to keep track of the difference  between the  average part and the 
fluctuating part. 

From this example it follows a final recommendation to the National Standard Bodies in charge of the definition 
of  National  Annexes  to  the  Eurocodes  to  be  used  in  a  country,:  the  calibration  of  partial  factors  is  highly 
dependent on the choices taken in the definition of the characteristic values of actions (mainly climatic actions) 
in the NA’s to the relevant EN1991 parts.  This aspect should be carefully dealt with in the calibration process 
since, as it is shown in the example, overlooking at it could introduce overestimation of the actions’ effects. So 
it is important to have the background of action models available in accessible documents. 

B.2.6 Summary, conclusion and reflection 

In  this  annex  the  derivation  of  partial  factors  for  actions  was  demonstrated  on  the  basis  of  both  the  DVM 
method and the Code Optimization Method (minimizing deviations from the target reliability) as described in 
Annex  C.4.4,2  respectively  C.4.4.3  of  EN1990:2022.  The  calculations  should  primarily  be  considered  as 
“examples” showing how the methods work for a given input. It should be realized that for the Code Optimization 
Method the calculations were based on simple limit state functions for bending of beams only with mean values 
and coefficients of variation estimated to be representative for a large class of member verifications. These 
limitations can be considered as acceptable for the demonstration purposes of this Annex, but need further 
development to reach final conclusions. 

Member  states  that  want  to  apply  the  methods  for  evaluating  Eurocode  recommendations  or  developing 
national determined parameters in their National Annexes are advised to review the various inputs for relevance, 
and  how  representative  they  are,  for  their  applications  and  balance  their  consideration  of  the  findings  of 
reliability analysis with past performance and practice.  

The DVM was already presented in the first editions of the Eurocode EN1990:2002 as well as in the preceding 
ENV1991 version. In the EN1990:2002 the method was described in the informative Annex C and served as the 
basis for Annex  D (Design by Testing).  As such it  has been used  widely to derive the partial factors  for the 
resistance side of the design equations in the material related Eurocodes. In the Background Document to the 
EN1990:2002 [REF] the method was for reasons of consistency and demonstration also used to derive partial 
factors on the loading side. The numerical basis of the method for use in the Eurocode can be described as: 

 

 

 

 

a target reliability  =3,8 for a reference period of 50 year. 

a sensitivity factor Y = 0,8 for dominating resistance parameters and 0,4 x 0,8=0,32 for others 

a sensitivity factor Y = 0,7 for dominating and 0,4 x 0,7=0,28 for accompanying actions 

probabilistic models for actions and structural properties (resistances). 

The split -values for load and resistance have the advantage that they allow action and resistance side to be 
treated  independently.  This  allows  also  an  optimisation  for  each  material  individually  based  on  refined 
calculations concerning the different failure modes assuming adapted resistance function. Within the revisions 
for the second generation of Eurocodes a systematic review of the chosen partial factors and the belonging 
resistance functions has been done tuning those where deficiencies were recognised and allowing also for a 
differentiation of “more critical” failure modes in comparison to each other. Such a procedure is much more 
cumbersome when using the optimization method.  

The DVM also has the advantage that no explicit assumptions need to be done on the choice of a portfolio and 
corresponding weight factors. One may of course claim that these assumptions implicitly are reflected in the 
above -values. The -present values have been determined and agreed long time ago and form the basis of 
the  existing  derivation  of  most  partial  resistance  factors,  see  3.1.3.  Also,  many  proposals  dealing  with  the 
loading side have been published. However, these papers do not always correspond to the current definition of 
the characteristic values in EN1990 and EN1991 and do not make use of up-to-date probabilistic models. 

158 

 
The Code Optimization Method [Kohler] is an alternative to the DVM and has been added in the EN1990:2022. 
The code optimisation in principle deals with all materials and all actions in one straight forward optimization 
procedure, although it also may be used to optimize a subset (say for one material, like the lines 11 and 12 in 
Table B.3). The explicit aim is to reach at a set of partial factors that on average reaches the target level and 
at the same time minimizes the sum of individually squared deviations. In EN1990:2022, C.4.4.3, the deviations 
in terms of the reliability index have been chosen. A discussion on other options like the deviation in terms of 
failure probabilities can be found in [Baravalle, 2017]. The idea to avoid structures with very large reliability 
(uneconomic) and too small reliability (unsafe) was already the leading thought behind the transfer from “load 
factor design” to “load and resistance factor design”. One may also find the idea reflected in the requirement 
0,16 < E/R < 7.6 in C.7(3) of EN1990:2002 intended to set limits to the use of the DVM split alpha values 0,7 
and 0,8. 

The summarised basis for the optimization method for the mainstream Eurocode is: 

 

 

 

 

a target reliability  =3,8 for a reference period of 50 year. 

a chosen portfolio of applications and a corresponding set of limit state functions  

a corresponding set of weight factors 

probabilistic models for actions and structural properties. 

The weight factors are intended to represent the portfolio of design equations for which the partial factors are 
supposed  to  be  applied  and  over this domain  to  achieve  minimum  deviations  from  the  target  reliability.  Of 
course, it is difficult to find these weight factors in an objective way and so some sensitivity study to the choice 
is relevant. In B.2.4 results are shown for uniform weights and some alternatives. 

For a good understanding of the optimization method, it is necessary to realize that when all partial factors are 
kept  as  “free  to  optimize  parameters”,  one  should  be  aware  of  the  fact  that  the  outcome  of  the  Code 
Optimization  is  mathematically  undetermined.  An  infinite  large  set  of  factors  may  be  obtained,  all  leading 
however to the same minimum deviation with the same or similar structural dimensions.. An illustrative example 
is the case when we have only one load and one resistance term. The reliability requirement is: 

Rk / R > F Sk 

All  partial  factor  choices  leading  to  the  same  value  of  the  product  (R  F)  are  fully  equivalent  from  a 
mathematical point of view and may all be equally optimal. However, the values like R = 0,5 and F = 3 would 
look very strange to the designer and is not in line with the basic idea behind the partial factor method. So, if 
all partial factors are free at least one value should be selected to be fixed. This however does not affect the 
final result in an essential way.  

In the current example study reported in B.2.1 this issue is not relevant since the partial factors on the resistance 
side have been fixed. The values chosen are the ones derived in the various material related Eurocodes, based 
on the DMV-method and so automatically taking on board the optimizations already carried on the structural 
level.  

Another (related) issue is the fact that the objective function may  be relatively flat around the optimum or 
shows multiple minima, allowing a set of slightly or more widely different solutions, also in terms of structural 
dimensions. These different solutions are equivalent from the mathematical optimization point of view, but of 
course may be preferred or not from other considerations (e.g. use of material consumption). It is important to 
be aware of this possible effect. 

As can be seen also from the sensitivity study in Table B.3 completely different partial factors (corresponding 
to different structural dimensions) come to the same averaged beta values. One should bear in mind here that 
the optimization is not intended to change the average but to reduce the extreme deviations from the scatter. 
Information on those aspects can be found in the violin plots in B.2.1, an effect that is of course less pronounced 
if the “modified values” of 1.75 are assumed for all variable loads, see violin plots in Figure B.1. 

Discussion 

Having the  results of the calculations in this annex in mind, the AHG would like to emphasize the following 
points: 

159 

 
 

 

 

 

 

 

The present partial load factors 1.35 for permanent and 1.50 for variable loads in EN1990 may be 
considered as correct in the sense that they meet on average the target value (see chapter 3.4); also, 
the use in practice (although over a limited period of time compared to desired return periods) has not 
shown  any  obvious  shortcomings.  Overconservative  and  possibly  uneconomic  solutions  of  course 
cannot be observed.  

The optimization method in the example calculations in this Annex leads to partial load factors having 
almost the same average value for the reliability index as the partial factors recommended in EN1990, 
Annex A., but show less scatter especially in view of the influence of wind and snow actions. In view of 
the  dominating influence of  these  climatic actions the conclusion is that optimisation  of resistance 
functions for different failure modes of a single material leading to a reduction of the corresponding 
model uncertainties has nearly no influence at all. A better approach would be to try to reduce the 
scatter by improved (but also often complex) loading functions. We should have mind, however, that 
we cannot reduce the natural variability. Maybe a reduction in the load models (pressure coefficients 
etc) is possible by additional research, but this takes time. We can only profit form that once the results 
are on the table. Anyhow, it is recommendable to perform coordinated studies on this topic, possibly 
initiated on the level of TC250 and SC1. 

In this annex the characteristic values for snow and wind have been based on the definition given in 
chapter 6 of EN1990, that is the value with an annual 2% exceedance probability; if member states 
use values corresponding to a lower exceedance probability this should be taken into account, either 
by lowering the partial factor or (preferably) by adapting the characteristic value. One should also be 
aware of the fact that the example calculations in this annex are based on an average geographical 
location and that results could be different for other parts of Europe. Some indication of the influence 
of this effect can be found in Annex B2.5. 

For the calculated example in Annex B.2,1 the result of the DVM and the result of the Code Optimization 
method are roughly pointing in the same direction. The result is also similar to the result presented in 
the  Background  Document  to  the  ENV  and  implicitly  to  EN1990:2002  {REF}.  The  result  can  be 
summarised as a drastic increase of the partial factor of the variable loads and some decrease of the 
partial factors for the permanent loads. It should be noted that as a consequence of the given results 
(1,20 for the self-weight and 1,94, 2,35 and 1,60 for respectively wind, snow and imposed load). For 
lighter materials as steel there would be an increase of the dimensions between 25% and 50% if used 
for building roofs or facades. Of course, there will be gains for other parts of the application field. But 
for several reasons one should be careful with such a jump in dimensions in a new code: (1) one may 
question  the  theoretical  basis;  (2)  there  seems  to  be  no  urgent  need  from  an  unsatisfactory  track 
record and (3) one should also consider possibly far stretching economic consequences 

The statistical models used in this Annex correspond to the present stage of knowledge and is as far 
as possible in agreement with models mentioned in the Eurocodes or the relevant Backgrounds. The 
weight factors are to some extent intuitive and may need improvement, and should be reconsidered 
by national code committees to reflect country specific building traditions. 

Further/other refinements are possible by adopting the EN1990 design formats (8.13) or (8.14); also, 
partial factors on the resistance side could be fine-tuned (given the partial factors on the loading side) 
to achieve a more homogeneous reliability. 

  Once the basis of a reliability optimization procedure has been accepted it could in future be extended 
to include explicitly additional issues like risks or be a basis to compare current design procedure with 
to new developments. 

  One should be aware of the fact that some other Eurocodes the NDP values for the partial load factors 
have already been fixed on 1,35 and 1,50 to arrive at the value of some other parameters. Examples 
are the documents on fire safety and the value kmod in the timber code The same warning of course 
holds for all members states using (814) and (8.15) instead of (8.12). 

 

In the ideal situation, from a scientific point of view, all partial factors should be based on the same 
principles and methods. As present factors on the resistance side were found using DVM, the use of 
the load factors found in B.2.1.2 would be the most appropriate.  

Finally, it is important to bring to the attention that in 2018 the Coordination Group of CEN TC250 decided that 
the recommended partial load factors in the EN1990 ENQ-version of the year 2020 should not be changed 
which has also been confirmed by TC250. The arguments used at that time were that: 

160 

 
refined  partial  factors  have  been  derived  for  the  new  generation  of  Eurocodes  on  the  resistance  side 
assuming the current values on the loading side. The interdependencies would have needed an extensive 
new research to achieve the same level of accuracy and detailing without too much of  conservatism or 
reduced safety. 

the background of this new method of code optimisation was considered as relatively immature,  

the ease of use of the practise was not believed to be served by a change,  

a possible disturbance of the market balance, 

the fact that the values were NDP’s anyway.  

Note also that at the time of the development of the ENV and subsequent EN-2002-version, the partial factors 
for variable loads following from the DVM were also considered as being too high to be codified.  

It is now of course up to the member states to deal with the above arguments and look for optimal solutions. 

References 

Project Team Eurocode 1.1 and JCSS (1996), Background documentation Eurocode 1 (ENV 1991), Part 1: Basis 
of Design. Service document, no. 94, ECCS- European Convention for Constructional Steelwork., First Edition  

Rowe, R. E. (1969), ‘Safety concepts, with particular emphasis on reinforced and prestressed concrete’, in: IABSE 
Symposium: Symposium on Concepts of Safety of Structures and Methods of Design, London, United Kingdom, 
https://doi.org/10.5169/seals-5005 

Baravalle,  M.  (2017),  Risk  and  Reliability  Based  Calibration  of  Structural  Design  Codes,  Dissertation  NTNU, 
Trondheim 

Formichi, P., Landi, F., Croce, P. (2023), ‘A consistent approach to the calibration of partial factors for permanent 
and variable  loads  in  the  2nd  generation  of  the  Eurocodes’,  Structure  and  infrastructure  engineering,   1-15, 
https://doi.org/10.1080/15732479.2023.2290704, 2023 

B.3 Column examples / Euler Column 

B.3.1 Introduction 

For nonlinear limit state functions the procedure is slightly more complicated than the one shown in sections 
3.4 and B2. However, we can still follow the general procedure as explained in example 2.4 in Chapter 2.6.3, 
summarized here for the sake of convenience: 

1. 

2. 

3. 

4. 

5. 

Choose a portfolio of structures and design situations with corresponding limit state functions 

Define the corresponding weights wi   

Set the target reliability  

Define the set of statistical properties and characteristic values  

Define a verification code format 

6. 
summed squares of the deviations from the target. 

Find  iteratively  the  set  of  partial  factors  that  leads  to  a  minimum  of  D,  being  the  (weighted) 

In order to find the -value for each individual design situation we first have to design the structural dimensions 
using  the  characteristic  values  and  the  chosen  set  of  partial  factors;  next  a  probabilistic  analysis  (FORM  or 
Monte Carlo) is used to find the corresponding probability of failure or reliability index. If the minimum value of 
D is too large to whatever standard, there may be a need to split up the portfolio in a set of smaller ones. 

In  the  beam  example  in  section  3.4  it  was  possible  to  divide  resistance  and  load  by  the  total  load  effect 
(1/8)(gm+qm) L2 leading to a generalized limit state function (3.7) and a generalized design equation (3.8). The 
advantage was that the length L was not important any more as a design parameter and only the ratio of the 
characteristic  values  over  the  mean  was  important.  In  the  general  non-linear  case  this is  not  possible.  In  a 
column analysis the CoV of the “reduced resistance” r will depend on the slenderness. As a consequence, we 

161 

 
 
have to consider several individual values of the slenderness as separate parts of the portfolio, and give them 
a weight in correspondence to market share or importance. 

However, the next paragraphs demonstrate the analysis for just a single slenderness ratio, respectively for steel 
and concrete. 

B.3.2 Steel column  

In this section the reliability level for a steel column designed according to the Eurocode is analysed. For the 
analysis the following LSF and Design Equation have been chosen: 

Limit state function:  

g(X) = R Ncr –E N = R Ncr –E [ G + Q Q] 

Design equation:   

Ncr k / R = G Gk+Q Qk 

The loading parameters have the same CoV’s and partial factors as in the bending case (see chapter 3.4 of the 
main text) and are presented in Table B.5. 

In EN 1993 the critical load is given by: 

Ncr =  A fynom with  a function of the slenderness ̅ = (A fynom /FE)) and FE is the Euler load; the definition of 
the slenderness is not of importance for this analysis. In this model   includes the randomness in the yield 
stress,  the  modulus  of  elasticity,  the  dimensions,  the  eccentricities  and  the  model.  Consider  an  IPE-160-
column.with ̅ = 1.0. The curves ( ̅ ) for this column are shown in Figure B.5 (SAFEBRICTILE) for the design 
value (red line), the mean value and the mean plus/minus 2 standard deviations. For ̅ equal to 1,0 the design 
value of  is 0,6 leading to a design resistance exactly equal to the design value of the total load G Gk+Q Qk..  
(note that R = 1.0). Note that the SAFEBRICTILE project was carried out on the resistance side only and contains 
many details not discussed here. Only the results of the probabilistic resistance model are used for combination 
with the loading models.  

All data is presented in Table B.5. No values are presented for the CoV of the cross sectional area and the yield 
stress as their uncertainty is taken care of by the buckling coefficient . Mean and scatter for  given in Table 
B.5  are  corresponding  to  the  diagram  shown  in  Figure  B.5.  It  follows  that  for  slenderness  1,0  the  mean 
resistance is 0,74 and the 4-sigma-range is about 0,32. This leads to the mean and CoV for  given in Table 
B.5. In reality there will be some uncertainty in the value of ̅  which has been translated into a model uncertainty 
R 

Table B.5 Overview of variables and numerical input values; last column shows resulting FORM -values for steel column 

X 

designation 

unit 

mean  CoV 

Xnom Xk 

X 

α 

Anom  Cross sectional area (IPE 160) 

mm2 

2000 

2000 

fynom  Nominal yield stress (S235) 

MPa 

283 

235 

1,00 

̅ 

 

G 

Q 

R 

E 

Q 

Slenderness (dimensionless) 

Buckling coefficient 

Structural weight 

Variable load (wind 50a) 

Resistance model  

Load effect model 

Wind force model 

- 

- 

kN 

kN 

- 

- 

- 

1 

- 

1,00 

0,74 

0,12 

0,60 

 0,63 

170 

0,05 

170 

1,35 

-0,20 

31,5 

0,14 

34 

1,50 

-0,16 

1,00 

0,05 

1,0 

0,10 

0,97 

0,26 

- 

- 

- 

1,00 

  0,38 

 -0,52 

-0,29 

162 

 
 
 
 
 
 
 
 
 
 
 
 
Figure B.5. Data and α results of reliability c-values for steel column IPE-160 and with ̅ = 1.0.  

Having all data we perform a FORM-analysis. The resulting reliability index is 4.12 (so larger than the target 
3.8) and the α-vector is shown in Table B.5, last column. 

The resulting reliability index is in line with the result for a steel beam with the same ratio of permanent and 
variable loads. For the above column the aQ parameter is around 0,25. From Figure B.2, we find that for aQ=0,3 
the reliability index for the beam is around 4.2. 

B.3.3 Concrete column  

In this section the reliability level for a concrete designed according to the Eurocode is analysed. For the analysis 
the following LSF and Design Equation have been chosen: 

Limit state function:  

g(X) = R Ncr –E N = R Ncr –E [G + Q Q] 

Design equation:   

Ncr k / R = G Gk+Q Qk 

The loading parameters have the same CoV’s and partial factors as in the bending case. 

The external bending moment (load effect) due to the total eccentricity e = eo + u (where eo + u is the initial 
eccentricity and u corresponds to the displacement) is given by: 

ME = N e 

An upper bound for the eccentricity is the given by (max being the ultimate strain): 

e = e0 + e2max = eo + Lk

2 max / (2

 h / 2) 

For a square column (area A = h2), the bending moment capacity is given by: 

MR= 0,8  A fy h + 0,5 N h [ 1- (N/A fc)] 

where: 

0,8 is to take care of the cover and the steel dimensions. 

 A is the area of steel on one side of the column only 

The axial load bearing capacity Ncr can be found by putting ME = MR: 

N e = 0,8  A fy h + 0,5 N h [ 1- (N/A fc)] 

Which can be elaborated to: 

163 

(B.2) 

(B.3) 

(B.4) 

(B.5) 

 
 
 
 
 
 
 
0,5 N2 h / A fc +N e – 0,5 N h - 0,8  A fy h = 0 

(B.6) 

Which is a standard quadratic equation. 

The values for the input parameters are given in Table B.6. 

Table B.6 Overview of variables, numerical input values and resulting FORM-values concrete column 

X 

H 

Column width (square) 

Lk 

Buckling length 

max  Maximum strain (compression) 

eo/L 

Imperfection (design value) 

Reinforcement ratio 

 

fy 

unit 

mean 

CoV 

Xnom Xk 

X 

α 

m 

m 

- 

- 

- 

0,4 

10 

0,0035 

2/1000 

0,4 

10 

0,0035 

1,2 

2/1000 

0,02 

- 

0,02 

yield stress reinforcement FE500 

MPa 

540 

0,05 

500 

1,15  

-0,16 

Fc 

Concrete strength C45 

MPa 

60 

0,17 

45 

1,50 

-0,41 

G 

Q 

R 

E 

Q 

Structural weight  

Variable load (wind 50a) 

kN 

kN 

1980 

0,05 

1750 

1,35 

0,21 

396 

0,14 

350 

1,50 

0,18 

Resistance model  

Load effect model 

Wind force model 

1,1 

1,0 

0,10 

0,10 

0,97 

0,26 

- 

- 

- 

-0,56 

0,56 

0,31 

Note: The values for fy and fc have to be confirmed. 

Design calculation ‘leads to a design value for the resistance Ncd =:2897 kN and an axial design load Nd =2888. 
So, the design is sufficient. The resulting reliability index is 4.16 (above the target 3,8) and the α-vector is shown 
in Table B.6. 

For  more  information  on  the  background  of  partial  factors  for  concrete  colukmns  the  reader  is  referred  to 
Chateuaneuf et al, 2001. 

References 

Chateuneuf  A.,  Soars  R.  and  Venturin  W.  S.  (2001),  ‘Partial  safety  factors  for  homogeneous  reliability  of 
nonlinear reinforced concrete columns’, Structural Safety, Vol. 23, Issue 2, pp. 137-156 

B.4 Partial factors in the case of static equilibrium 

A girder (see Figure B.6) is positioned on a block without structural connection, so it is a case of static equilibrium 
and factors according to case DC3 of EN 1990 Table A.1.8 should be used. In that case we use the load factors 
1,35 and 1,15 for the two forces leading to moment  M  = 0,2Ga and a vertical load  of 2,5  G. The  resulting 
eccentricity is 0,08 a. So, if the block in total is wider than 0,16a the structure fulfils the Eurocode requirement. 

164 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure B.6. Girder on a block, case for EQU example  

According to the theory of reliability one would expect load factors in the order of 1,1 and 0,9. This leads to 
M=0,2Ga and a vertical force equal to 2G and an eccentricity of 0,10a. So a block of 0,16a would be considered 
as unsafe.  

Let  us  consider  the  results  of  a  reliability  analysis  based  on  the  limit  state  function  using  the  minimum 
dimensions following from the Eurocode factors: 

g(X) = 0,08 a (N1 G1 + N2 G2) – a (M1 G1 - M2 G2) 

with statistical properties (see section 3.5.3): 

V(G) = 0,05, V() = 0,05, (G1G2)=0,7 and (i j) = 0,5 

(B.7) 

(B.8) 

The values of (G) and a are not relevant and can be taken as unity. Note that the correlations for the chosen 
four model uncertainties (N1, N2, M1 and  M2) negative values are possible in this specific case. If the girder is 
positioned a bit out of the centre then G1 is larger and G2 is smaller; also the lever on one side is larger and 
smaller on the other. So maybe uncorrelated values for the model uncertainties are better. On the other hand, 
the correlation of G1 and G2 may be very high (closer to 0,9) as this is one continuous plate (in fact a single 
source case). An overview of results for different correlations maybe found in Table B.7. The key values for 
(G1G2)=0,9 and 0<(i j)<0,6 are highlighted in bold. 

Table B.7 Resulting reliability indices for as a function of (i j) 

(i j)  Block designed according to 

Block designed according to 

-values 1,35 and 1,15 

-values 1,1 and 0,9 

(G1G2)=0,7 

(G1G2)=0,9 

(G1G2)=0,7 

(G1G2)=0,9 

-0,4 

1,7 

-0,2 

1,8 

0,0 

0,2 

0,4 

0,6 

0,8 

2,0 

2,2 

2,4 

2,7 

3,2 

1,8 

2,0 

2,2 

2,4 

2,7 

3,2 

4,1 

2,2 

2,3 

2,5 

2,7 

3,0 

3,4 

4,0 

165 

2,3 

2,5 

2,7 

3,0 

3,4 

4,0 

5,1 

                                                 G                                      G                                                              a                     a              
 
 
 
 
 
So -values “1,35 and 1,15” give rise to β-values in the range 2,2 to 3,20 and -values “1,1 and 0,9” to β is 2,7 
to 4,0. So from this perspective the values 0,9 and 1,1 are to be preferred, as they are closer to the EN1990 
target value 3.8., 

Note finally that the single source distinction is not used in the Eurocode for the limit state of static equilibrium. 

B.5 Load combinations 

B.5.1 Load combination probabilistic analysis 

The combination of two independent variable loads Q1 and Q2 is considered. The extremes of variable loads per 
reference period T are represented by their corresponding extreme value distributions Q1;max;T and Q 2;max;T. The 
load processes are represented as sequences of rectangular pulses of  uniform (and  deterministic) length  i, 
such  that  the  reference  period  T  contains  ni  =  T/i,  load  pulses,  see  Figure  B.7.  Furthermore,  are  the  loads 
represented such that 1 > 2 and that n2/n1 is always an integer. 

Figure B.7. Loads Q1 and Q2 presented as FBC models 

The  value  of  T  may  be  specified  as  1  year  so  that  Q1;max;T  and  Q2;max;T  represent  the  annual  extreme  value 
distributions or as 50 year, being the design service life according to prEN 1990:2021 

The  extreme  value  distribution  Q2;max;T  can  be  scaled  down  to  a  smaller  reference  period,  e.g.  1  =  T/n1  by 
considering the realisations of Q2 in time as independent pulses: 

FQ2,max,1

1
n1⁄

(q)= FQ2,max,T

(q). 

(B.9) 

Similarly, the distribution of Q1;max;T can be scaled down to Q1;1. 

According to the combination procedure as proposed by (Ferry-Borges and Castanheta 1968) the  maximum 
combination Q1+2;max;T can be approximated as: 

Q1+2;max;T = max [Q1+2;max;1 ] 

Q1+2;max;T=max[Q1+2;max;1] 

Q1+2;max;1 = Q1;1 + Q2;max;1 

166 

(B.10) 

(B.11) 

 
 
 
 
 
 
 
The distribution for Q1+2;max;T can easily be evaluated in a reliability analysis. In this note we will concentrate 
however on the treatment in semi-probabilistic design.  

B.5.2 Load combination in semi-probabilistic design 

B.5.2.1 Introduction 

In semi-probabilistic design random variables are represented as Design Values, i.e. for Qi;max;T this means that 
the design value Qid follows from: 

FQi,max,T

(Qid)=  (-Qi max Tt) 

(B.12) 

Where (..) is the cumulative standard normal distribution function, Qi;max;T is the FORM sensitivity factor of the 
variable Q i;max;T and t is the annual or life time target reliability index. The sensitivity factors  may follow from 
a reliability analysis on a specific design case or taken as standardised values according to ISO2394. It should 
be noted that the standardised values may lead to usually unknown inaccuracies. 

In order to explore the problem first the load combination for two variables constant in time will be treated. The 
formulas derived are also valid for the combination of tow loads, one constant in time and the other fluctuating. 

B.5.2.2 Combination of two independent loads, at least one load constant in time. 

For two time independent loads (or one time dependent and one time independent load) the combination rule 
is as follows: 

Combination 1, Q1 is dominant and Q2 is accompanying;  

Combination 2, Q2 is dominant and Q1 is accompanying.  

The design values for the dominant and the accompanying or combination load in each combination can be 
found  by  solving  (using  the  subscript  “dom”  and  “com”  to  indicate  the  design  values  of  the  dominant  and 
combination design value for Q respectively): 

FQ (Qdom) =  (- S t) 

FQ (Qcom) =  (- 0,4 S t) 

(B.13) 

(B.14) 

where FQ (..) is the probability distribution function for  Q if the load is time independent or of the maximum 
value in the reference period if the load is time variant. According to prEN 1990 we take S = - 0,7 and t = 3,8 
for a reference period T=50 year and consequence class CC2. The formulas hold for both Q1 and Q2. 

The combination factor 0 now follows from: 

0 = Qcom/ Qdom 

(B.15) 

Table B.8 shows the elaboration for the 0 values for the general case, the normal and the Gumbel distribution. 

Table B.8 Formulas for 0 time independent case 

General 

Normal 

Gumbel 

F-1{(0,28 )}
F-1{(0,70 )}

1+0,28 V
1+0,70 V

1+0,78 {0,58+ln(- ln   (0,28  ))} V
1+0,78 {0,58+ln(- ln   (0,70  ))} V

167 

 
 
 
 
 
 
 
 
 
The target reliability t for T=50 year is here simply denoted as  and V is the coefficient of variation for the 
maximum load in the reference period. In EN 1990 the 0 for the time independent (permanent) load is denoted 
as . 

B.5.2.3 Combination of two in time varying, statistically independent loads 

In the case of two time varying loads the maxima of the loads do not need to occur at the same point in time. 
Let  both  loads  be  presented  by  FBC  process  as  defined  in  Figure  B.7.  The  relation  between  the  probability 
distributions for the maxima Qmax of Q(t) in  and T is given by equation B.9. 

The appropriate procedure in this case is to apply the combination rule of B.5.2.2 within the time interval   
where the maxima of the two loads actually will occur on the same point in time. This leads to: 

 = max(1, 2) 

(B.16) 

For  convenience  we  will  use  the  notation  PT  for  the  probability  that  the  design  value  Qd  of  a  load  Q  is  not 
exceeded in the reference period T: 

PT =  (- S t) 

In the interval  that probability (using n = T/ ) is given by: 

P = {  (-S t) } 1/n 

Now define: 

 = -1( P ) 

(B.17) 

(B.18) 

(B.19) 

Which enable us to define the design values for the dominant and the accompanying loads in terms of the 
distributions for the period : 

Qdom = F-1

Qmax  {Pdom} 

with Pdom = {} 

Qcom = F-1

Qmax  {Pcom} 

with Pcom = {0,4 } 

Of course, we may equivalently express Qd and Qcom in terms of the distribution of QmaxT: 

Qdom = F-1

QmaxT { Pdom n } 

Qcom = F-1

QmaxT { Pcom n } 

Elaborating Qdom we arrive (of course) at: 

Qdom = FQ

-1 {  (- S t) } 

There is no difference with the time independent case treated in B.5.2.2 

Elaborating Qcom we arrive at: 

Qcom = F-1

QmaxT {{(0,4 )}n} 

with  = -1 ({(-St)}1/n) 

As a simplification the following approximation is proposed: 

Qcom = F-1

QmaxT {{(-0,4St)}n} 

(B.20) 

(B.21) 

(B.22) 

(B.23) 

(B.24) 

(B.25) 

(B.26) 

This  formula  is  also  presented  in  EN  1990,  Annex  C.  This  approximation  is  close  to  the  classical  Turkstra 
approximation  (Turkstra  and  Mdsen,  1980)  for  load  combinations.  Section  B.5.3  shows  the  accuracy  of  the 
approximaton for a specific example.  

168 

 
 
 
 
 
 
 
 
 
 
 
 
Table B.9 gives an overview of the formulas for this case, including further approximations for the normal and 
the Gumbel case; the factors 0,7 in front of ln N have been tuned to get results very close to the exact solution. 
Note that exact means: given the FBC modelling of the action and given the -factors 0,7 and 0,28 according 
to EN 1990 and ISO 2394. 

Table B.9 Formulas for 0 time dependent case 

General 

Exact 

General 

Approximation 

Normal 

calibrated to “exact” (N<100) 

Gumbel 

calibrated to “exact” 

 =  -1 [  (0,7) 1/n ] 

n

F-1{ (0,4)
F-1{(0,7 )}

}

F-1{( (0,28 ))n}
F-1{ (0,70 )}

1+(0,28 -0,7 ln n) V
1+0,70 V

1-0,78 V {0,58+ln(- ln   (0,28  ))+0,7 ln n  }
1-0,78 V {0,58+ln(- ln   (0,70  ))}

B.5.2.4 Loads including a time dependent and a time independent part. 

Consider the load combination given by equation B.27 on the interval (0, T):  

S(t) = S1 + a S2 = C1 Q1(t) + a C2 Q2(t) 

(B.27) 

The coefficients Ci represent the time independent random part of the load model as for example load model 
uncertainties  and  pressure  coefficients  for  generic  building  types.  The  factor  a  runs  from  0  to  infinity  to 
represent all possible ratios of combinations of the two loads in the load effect. 

The code representation of the design value Sd for combined load effect S(t) is given by: 

Sd = max [ S1d + a 02 S2d ; 01 S1d + a S2d ] 

(B.28) 

The design values Sdi for the individual actions are defined for the reference period T and can be found from: 

Sid = FSi

-1 {  (- S t) } 

FSi (s) =  FQi max T (s/c) fCi (c) dc 

(B.29a) 

(B.29b) 

The parameters S and t are defined as before. The task is to find values 01 and 02 in such a way that for 
all values of a (or at least on average): 

P(max S(t) > Sd) =  (S t) 

(B.30) 

One might use here a minimum distance principle comparable to the calibration processes for partial factors. 
The  exact  determination  this  way,  however,  is  quite  cumbersome  and  it  may  be  convenient  to  have  some 
approximations. 

First of all, instead of the integral expression B.29 for the individual design values Sdi we may use the standard 
DVM approximation for a function of two random quantities: 

Sid = max [ Ci com Qi dom , Ci dom Qi com ] 

(B.31) 

where (dropping the index i for clarity): 

169 

 
 
 
 
 
 
 
 
 
 
 
 
Qdom = FQmaxT

-1 {  (- S t) } 

Qcom = FQmaxT

-1 {  (- 0,4 S t) } 

Cdom = FC

-1 {  (- S t) } 

Ccom = FC

-1 {  (- 0,4 S t) } 

(B.32a) 

(B.32b) 

(B.32c) 

(B.32d) 

As by definition C is constant in time, the combination value Qi com in (B.32d) is calculated using N=1. 

A simple approximation for the combination value Si com = 0i Sid may follow if we continue to neglect the time 
variability in the loads and use again N=1: 

Si com = [Ci com Qi com ] 

(B.33) 

So, in all formulas B.31-B.33 the number N is absent and the time variability of Qi does not play a role, which 
makes the approach conservative.  

An alternative formulation may be found when the variability (that is the CoV) in Q (for both loads) is dominant 
compared to the variability in C. In that case we have: 

Sid = max [ Ci com Qi dom , Ci dom Qi com ] = Ci com Qi dom 

And the load combination (1) may be written as: 

S(t) = C1com Q1(t) + a C2com Q2(t) 

(B.34) 

(B.35) 

This way the variables C1 and C2 are reduced to deterministic quantities and Q1(t) and Q2(t) are two random 
time varying processes. This means that we can use the same 0-values as in the standard case without the 
time  independent  variables  C1  and  C2.  If  the  condition  is  not  fulfilled  the  exact  approach  B.27-B.30  or  the 
conservative approximation B.31-B.33 has to be used.  

Example  calculations  showing  the  accuracy  (and  so  the  usefulness)  of  approximations  B.33  and  B.35  by 
comparing with the exact approach may be found in section B5.3.2. 

B.5.3 Numerical example 

As an example consider a load Q having a Gumbel-distribution with mean value 1 and coefficient of variation 
V=0,2. Let the reference period T be 50 years and the target reliability 3.8 according to EN 1990:2021. Figure 
B.8 shows the value of 0 as a function of n. The values have been calculated using the exact formula B.25 
and the approximation B.26.  

Figure B.8. Calculated PSI-0 value for a Gumbel distributed with V=0,2 

170 

 
 
 
 
 
 
 
 
 
So, for N=1 we get the combination factor to be applied if the variable load Q is combined with a permanent 
load  (as  for  instance  in  EN1990  (8.13a).  Values  of  N  between  50  and  300  seem  to  be  reasonable  for 
combination of for instance wind and snow. Very high values of N may apply to combinations with accidental 
loads. Note that N should never be chosen higher than the number of -intervals present in T for the load Q 
itself, irrespective of the other load. 

So actually, a PSI-0 values should be defined for combination with every other load. The Eurocode neglects 
however this differentiation, except for the combination with accidental loads where PSI-1 and PSI-2 values 
can be used. 

References 

Ferry Borges J. and Castanheta M. (1968), Structural Safety, Laboratório Nacional de Engenharia Civil, course 
101, Lisbon. 

Turkstra  C.  J.,  and  Madsen  H.  O.  (1980),  ‘Load  combinations  in  codified  structural  design’,  ASCE  Journal  of 
Structural Engineering, Vol. 106, Issue 12, https://doi.org/10.1061/JSDEAG.0005599  

B.6 Nonlinear FEM (probabilistic and semi-probabilistic) 

B.6.1 Shear wall example 

Figure B.9. Shear wall and main data 

Gk=60 kN 

Hk=11,5 kN 

3 m 

2 m 

Reinforced concrete element  

to distribute the forces 

Contact elements 

Friction coefficient =0.6 

Cohesion =0 

fk=2,9 MPa  

Thickness =115 mm 

Figure B.9 shows a masonry shear wall loaded by a permanent vertical load and a variable horizontal load. The 
example illustrates how applying partial safety factors to FEM analysis can change the estimation of safety 
and the economy. The safety check has been calculated for the following cases: 

—  Case 1: the vertical action is applied as G Gk, the horizontal action is applied as Q Qk, and the material 

parameters are applied as fk/M 

—  Case 2: the vertical action is applied as G Gk, the horizontal action is applied as M Q Qk, (compare 3.8.4.2 

in the main report) and the material parameters are applied as fk 

The characteristic values of the input parameters are: 

—  The characteristic value of horizontal action (unfavorable)  

Hk = 11, 5 kN 

171 

 
 
 
—  The characteristic value of vertical action (favorable)  

Gk = 60 kN 

—  The characteristic compression strength 

fk = 2,9 MPa 

—  Partial factors  

M = Q = 1,5; G = 1,0 (favorable) 

—  Corresponding other strength parameters (friction and shear strength) 

In the FEM analysis the vertical load was applied first, and the horizontal load was raised in steps from 0 up to 
the failure load. The failure load has been recorded for each case from the finite element analysis. This result 
was compared to a hand calculation using a simplified rectangular vertical stress distribution at the right end 
corner and a separate checking of the normal and shear stresses at the support. A summary of the result is 
given in Table B.10. 

Table B.10 Summary of the calculation results 

Application of partial safety factors for: 

Case 1 

Case 2 

Vertical action (favorable) 

Horizontal action (unfavorable) 

Material parameters 

Gk 

Q Qk 

fk/M 

Input parameters in FEM  

Vertical action (favorable)  

[kN] 

60,0 

Horizontal action (unfavorable) 

[kN] 

17,25 

Compressive strength masonry  

[MPa] 

1,93 

Calculated Horizontal resistance using 

Hand calculation VRd  

[kN] 

17,30 

FEM calculation VRd  

[kN] 

16,14 

Unity Check 

Hand calculation 

FEM calculation 

0,997 

1,07 

Permissable charteristic horizontal action 

Hand calculation 

[kN] 

11,53 

FEM calculation 

[kN] 

10,76 

Gk 

M Q Qk 

fk 

60,0 

25,88 

2,90 

18,20 

17,55 

1,422 

1,474 

 8,09 

 7,80 

Reference case 

uneconomic 

The unity check for case 1 in the finite element analysis was 1,07. In this case, the partial factors were applied 
in accordance with EN1990. In cases 2, where the unfactored characteristic values were used as input for the 
material parameters and for compensation the partial factor of the material was applied additionally to the 
horizontal action, the design was safe but very uneconomic.  

172 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Consequently,  it  is  advised  to  use  the  design  values  of  the  action  and  design  values  of  material  as  input 
parameters for the finite element calculation to avoid uneconomic results. 

B.6.2 Example numerical analysis using characteristic or mean values for resistance 

As explained in section 3.8 and demonstrated in B.6.1 it is to be preferred to use design values for resistance 
parameters in FEM (both options are allowed by EN1990). However, sometimes (e.g. in seismic engineering) it 
is  a  custom  to  use  characteristic  or  even  mean  values.  Arguments  are  that  (in  particular)  mean  values 
corresponds better to average test results and that having design values all over the structure is too pessimistic. 
However, the starting point is that the reliability level using FEM should not deviate from standard analysis, and 
so a compensation is required. This annex discusses a simple example in more detail. 

In the most common situation we have the reliability requirement: 

Rk/R > G Gk + Q Qk 

(B.36) 

If we neglect the partial factor on the resistance side we may compensate by adding a factor R on the loading 
side: 

Rk > R G Gk +R Q Qk 

This also holds when G acts as a favourable action (taking G = 1,0): 

Rk / R > - Gk+ Q Qk 

Note that Gk itself here is a positive value. This leads to: 

Rk >  R ( - Gk +Q Qk) 

(B.37) 

(B.38) 

(B.39) 

So, in this case, we may even enlarge the load Gk even when it acts favourably. However, in a more complex 
LSF this may be unconservative. Consider the limit state where friction forces are involved: 

Rk/R + Gk (k /R) > Q Qk 

Where  is a friction coefficient. Then we should have: 

Rk + Gk k > R Q Qk 

(B.40) 

(B.41) 

A factor R on G should be on the unsafe side in this case. To be always on the safe side one should not increase 
favourable acting loads in these cases.  

Final note: even that correction may not be enough in all cases consider the slightly different failure criterion 
G tan() > Q and let the partial factor be applied on the friction angle . Let the mean value of  be given by 
45o and M = 1,5. In that case the design is okay if:  

Qd < Gd tan(d) = Gd tan(m/M) = 0,58 Gd 

(B.42) 

Now use the mean value for  and adjust the partial factor for Q. The factor for G is left unchanged equal to 
1,0. In that case the design is considered okay if:  

M Qd < Gd tan(m)  or  Qd < Gd tan(45) /1,5 = 0,67 Gd 

(B.43) 

Which makes the approximation unconservative. 

173 

 
 
 
 
 
 
 
 
 
 
 
B.7 (Semi) Probabilistic fatigue analysis using advanced models 

B.7.1 Fatigue modelling and limit states 

In this annex three different approaches to the formulation of the fatigue limit state verification for welded 
connections will be considered, respectively based on: 

1.  S-N curves in combination with Miner’s Damage accumulation rule; 

the fatigue crack growth model; 

the fatigue crack growth model in combination with fracture resistance. 

B.7.1.1 S-N curves 

The  S-N  curve  approach  for  fatigue  is  based  on  the  standard  experimental  curves  for  constant  amplitude 
loading, relating the number of stress cycles to failure N to the stress the range Δ. For practical applications 
in  steel  structures  subject  to  variable  amplitude  (VA)  loading,  a  bi-linear  log-log  S-N-model  is  used  having 
parameters K1, K2, m1 and m2. The two linear parts are considered fully correlated, with an intersection at either 
ND = 5,106 or ND = 107 cycles, depending on the type of detail (see prEN 1993-1-9): 

log(N)= max  [log(K1)-m1log(Δσ), log(K2)-m2log(Δσ)] 

(B.44) 

where 

log(K2)=

1
m1

log(K1)+ (

1
m2

-

1
m1

) log(ND) 

(B.45) 

and “log” is the 10-th base logarithm. In some models a third horizontal line is added (cut off value) which may 
simplify calculations but the physical meaning is questionable in variable amplitude loading. The value of 𝑚1is 
usually taken as a fixed prior at a certain reasonable value (e.g. 𝑚1 = 3 or 5) where the other values (K1 and 
ND) are determined to get the best agreement with test results. The value of 𝑚2 is again a prior with, usually, 
a value of 𝑚2 = 2𝑚1 − 1. 

To account for variable amplitude (VA) loading, the Palmgren-Miner damage accumulation rule (hereafter called 
“Miner sum”) is often used which leads to the following limit state function for a given period of time t (design 
service life): 

g(t) = Dcr – Dn 

(B.46) 

where Dcr is the Miner sum at failure and Dn is the damage based on (3.1) due to n applied load cycles present 
in the period t. 

n
Dn= ∑
i=1

1
Ni

(B.47) 

To evaluate Dn we need the stress ranges Δ corresponding to n, usually presented as a stress range histogram 
or spectrum and a total number of cycles n for the design life time (or other period) under consideration. The 
histogram or spectrum may be derived on the basis of load simulations in combination with rain flow counting 
or  reservoir  counting  methods. In some cases the load  standard  presents an equivalent constant amplitude 
value. 

Due to the fact that according to the damage accumulation of Palmgren-Miner rule (not in reality!) the order of 
the loads is not relevant, we may evaluate B.2.4 by an ordering in magnitude. This means that we may divide 
the spectrum into a number of stress range intervals, calculate the damage  Di for each stress range interval 
and multiply with the number of cycles in the interval; Dn follows by adding the contributions of all stress range 
intervals.  

Additionally the model contains global and local model factors for the determination of the load effect Δ: 

174 

 
 
 
 
 
 
Δσ=θlmθglobΔσmodel 

Δσ=θlmθglob

[(Kscf-1)θscf+1] Δσmodel 

(B.48) 

(B.49) 

if based on the nominal stress concept respectively on the hot-spot or effective notch stress concept, where 
Δσmodel is the nominal stress, Kscf is the stress concentration factor, θlm is the uncertainty factor with respect to 
the representation of loads through a model, θglob is the uncertainty factor with respect to the (approximations 
and  simplifications)  of  the  structural  model,  and  θscf  is  the  uncertainty  factor  with  respect  to  the  stress 
concentration. 

Fatigue may be caused by more than one fluctuating action. In that case the distribution (spectrum) should be 
based on the combined history. When the mean stress level needs to be considered in the  damage rule, the 
mean values may be based on the on EN 1990 (8.17): 

Σ Fd = Σ Gk,i+Σ 𝜓2,j Qk,j + (Pk) + FfQfat 

(B.50) 

With j>1 and Qfat is the characteristic is the relevant fatigue action, and Qk,j refer to the variable actions excluding 
the action type(s) to which Qfat belongs. Depending on the sensitivity of the model with respect to the mean 
value of the stress cycle it may be necessary to consider the mean value of fluctuation load per cycle and not 
the life time quasi static value. In equation B.50 it is implicitly assumed that uncertainties and sensitivities with 
respect to the first three terms of the right-hand side of the equation are sufficiently low so that no partial 
factors are required.  

B.7.1.2 Crack Growth model, (partly) crack through criterion 

For welded joints initially present micro-cracks at the surface tend to coalesce to form a single, dominant fatigue 
crack of roughly semi-elliptical shape. According to the pioneer work of Paris & Erdogan (1963) the crack growth 
description is given by: 

da
dN

=A(∆𝐾)𝑝 

(B.51) 

Where: 

a 

da/dN  

K  

is the crack depth; 

is the instantaneous crack propagation rate (or crack extension per cycle number); 

is the stress intensity factor range at the crack tip;  

A and p  

are material constants.  

The stress intensity factor K for a given applied stress  is given as: 

K=σ Ya√πa 

(B.52) 

The stress  is the applied, nominal stress in the vicinity of the crack location, see Figure B.10. The factor Ya 
depends on the crack size normalised with respect to other relevant local dimensions (e.g. plate thickness), the 
geometry of the weld (if any) and the nature of stress distribution (membrane or out-of-plane bending). As in 
case of the S-N model, rainflow or reservoir counting methods can be applied to determine the ranges ΔK from 
the history of K. More complicated than for the S-N model, the crack size a increases in time equation B.51 and, 
through equation B.52, this means that K and ΔK are determined by combining equations B.51 and B.52. In 
most practical cases, it is sufficient to apply the counting procedure to the stress history and the ΔK histogram 
is subsequently determined by multiplication with Ya√πa. 

175 

 
 
 
 
 
 
 
 
 
 
 
 
Figure B.10. A semi-elliptical crack in a steel plate at/near the weld toe 

For practical cases, Equation B.51 is often too simplistic and a more accurate model that accounts for the near 
threshold condition is used, such as: 

da
dN

= {

0
min(A1[ΔK]p1,A2[ΔK]p2)

if ΔK≤ΔKth
if ΔK>ΔKth

(B.53) 

where A1, A2, p1 and p2 are material constants and the threshold of the stress intensity factor,  ΔKth, can be 
considered as a material constant (BS 7910:2013 + A1:2015) or as a material-specific function of the crack 
size (McEvily et al, 2003) (Chapetti, 2003) (El Haddad et al, 1979) (Zerbst et al, 2015) (Leonetti et al, 2021).  

The limit state function can be formulated as 

g(t) = acr - a(t) 

(B.54) 

where acr is a limiting crack depth (for example 75% of the plate thickness) and a(t) is the crack depth after a 
service exposure of time t. At t = 0 the crack size has an initial value ao. 

The calculation of the crack size at time t is not trivial as the stress is a random process. Assuming for  a 
sufficiently mixing (ergodic) process we may use the expectation of the stress range to find the expectation of 
da/dN: 

E(

da
dN

)=A1E[Δσp1]Δσ0

Δσtr(θglobθsifYa√πa)p1+A2E[Δσp2]Δσtr

Δσmax(θglobθsifYa√πa)p2 

E(

da
dN

)=A1E[Δσp1]Δσ0

Δσtr(θglobθsif[(Kscf-1)θscf+1]Ya√πa)p1+.. 

(B.55a) 

(B.57b) 

for a nominal and a hot-spot stress based evaluation, respectively. 

Flaws  or  fatigue  cracks  in  civil  engineering  structures  often  manifest  as  surface  cracks,  which  are  usually 
approximated as semi-elliptical (Figure B.11). In such a case two values of K and ∆K are considered, namely, 
one for the deepest point of the crack (A), for which the previous equations apply, and one for the surface point 
of the crack (C), for which similar equations apply where Ya is replaced by Yc. A similar differential equation as 
equation B.51 applies for the crack growth at the surface point. As both Ya and Yc are functions on the crack 
depth,  a,  and  the  semi  crack  with,  c,  the  two  differential  equations  are  coupled  and  should  be  solved 
simultaneously. 

B.7.1.3 Crack Growth model, fracture based failure 

Failure occurs when: 

a compression stress value arrives that causes buckling of the section weakened by the fatigue crack; 

a tensile stress value arrives that causes instable crack growth, caused by an interaction of (brittle) fracture 
and plastic failure of the remaining cross section to be exceeded. The plastic and the brittle fracture failure 
modes are a function of the crack size (a and c). 

The  compression  failure  mode  can  be  evaluated  with  the  finite  element  method.  For  tension,  a  failure 
assessment  diagram  can  be  used, which  gives  Lr  –  a  measure  of  the  proximity  to  plastic  collapse  –  on  the 
abscissa and Kr – the fracture ratio – on the ordinate. 

176 

AC 
 
 
 
 
 
 
Kr=

K+Ks
Kmat,T

+ρ 

Lr=

σnet
σyT

(B.56) 

(B.57) 

Where Kmat,T is the fracture toughness at temperature T, σyT is the yield stress at temperature T, net is stress in 
the  remaining  cross  section  and  ρ  is  an  interaction  term  which  depends  on  Lr  ,  K  and  Ks,  the  latter  factor 
depending  on  the  residual  stresses.  The  residual  stress  profile  depends on  the  joint  geometry  and  restraint 
conditions.  Reference  is  made  to  (BS  7910:2013  +  A1:2015,  2012)  for  approximate  equations  of  all  these 
factors. 

A flaw or crack is acceptable if the assessment point (Kr, Lr) is within the bounds of the (stress-strain relation 
dependent) curve in the diagram, for which reference is made to (BS 7910:2013 + A1:2015, 2012). 

The limit state function of the tension failure mode can be formulated as:  

g(t)=rFAD-rF+θFAD-1 

(B.58) 

2, rFAD is the polar coordinate that gives 
where rF is the polar coordinate of the assessment point, i.e. rF=√Kr
the point on the curve in the diagram for the same polar angle (see Figure B.11a), and θFAD is a model factor 
evaluated through experiments. Several options are available to evaluate Kmat,T and θFAD is depending on it. See 
Figure B.11b for the case that Kmat,T is evaluated from three KJC tests, from which the average is considered 
and  this  average  is  subsequently  corrected  for  the  crack  tip  constraint  through  the  triaxial  stress,  using 
procedures in (BS 7910:2013 + A1:2015, 2012). 

2+Lr

Figure B.11. Failure assessment diagram: (a) with definitions of rF and rFAD; (b) with data from large-scale wide plate and 
tubular joint tests (Maljaars et al, 2019)  

To evaluate the failure probability on the basis of (B.58) a time variant reliability method like the out-crossing 
approach is needed, e.g. (Maljaars et al, 2012). 

B.7.2 Verification 

Verification  can  be  performed  using  a  probabilistic  and  a  semi-probabilistic  method.  In  the  case  of  semi-
probabilistic analysis one should derive partial factors on the basis of calibration to probabilistic methods. So, 
in both cases, one needs to state a reliability requirement.  

For structures in reliability class 2, the European standard EN 1990 recommends a target reliability index for 
the entire fatigue life (actually the design lifetime of 50 year) ranging from 1,5 to 3,8, depending on the degree 
of inspectability, repairability and damage tolerance.  

177 

00.20.40.60.811.200.20.40.60.811.2Kr [-]Lr [-]UnacceptableAcceptableAssessmentpoint00.20.40.60.811.21.41.61.800.20.40.60.811.21.41.61.8Kr [-]Lr [-] 
 
 
 
 
 
 
One should be careful here: the correct way to calculate the failure probability for the lifetime is by explicitly 
including the effect of inspection. This will lead to an increase of the reliability index and the target should still 
be 3,8. The reduction of the target to 1,5 is intended to compensate for the case where inspection is carried in 
reality but included in the calculation. It is in fact a strong simplification. One should never combine the two 
approaches: that is incorporating the positive effect of inspection on the calculated reliability level and taking 
the reduced 1,5 value as a reliability requirement. Actually, there is even a need for a better specification of the 
requirements and corresponding repair for this reduction.  

B.7.2.1 Probabilistic method using the S-N model 

The failure probability for a total period of time from 0 to t is given by: 

Pf (t) = P(g(t)<0) 

(B.59) 

Where  g(t)  is  given  by  B.46.  The  stress  distribution  or  spectrum  in  the  expression  for  g(t)  follows  from  the 
application and the structural properties. Data for the random variables for welded steel connections may be 
taken from a table like Table B.11. 

Table B.11 Input for fatigue reliability analysis for welded steel connections using S-N curves 

Variable 

Units 

Distribution  Mean 

Standard deviation 

Log(K1) 

Parameter  

N and mm 

normal 

Slope value S-N curve 

- 

- 

b) 

a) 

- 

m1 

m2 

Dcr 

lm 

glob 

scf 

Slope value S-N curve 

2m1 -1 

- 

Miner's sum at failure 

- 

Lognormal 

1,0 

0,30 e) 

MU load model c) 

Lognormal 

d) 

MU global stress model c) 

Lognormal 

1,0 

0,10 

MU stress concentration c) 

- 

Lognormal 

1,0 

0,20 

a) Detail dependent. In absence of test data, it is often assumed as 0,2 

b) Detail dependent, and usually selected as m1 = 3 or m1 = 5 for most steel welded details. See prEN 1993-
1-9, prEN 1993-1-11 and prEN 1999-2 for m1 of the detail considered. 

c) MU = model uncertainty (correlated for all cycles).  

d) Load model dependent. Can be obtained by comparing with measurements.  

e) Original source: Wirsching (1984) 

In case of a probabilistic calculation there is a choice to evaluate the failure probability for the total design 
service life or of the last year of the design service life. For the annual evaluation holds: 

P(failure in year t) = P(g(t) <0) - P(g(t-1)<0) 

(B.60) 

(or, alternatively, the conditional failure rate, see chapter 2). This probability should be less than the annual 
target, which according to Table C.2 in EN 1990 for reliability class 2 corresponds to  = 4,7. This annual value, 
however, is an upper bound and a lower value may be more appropriate because of the correlations in resistance 
and model uncertainties in time. 

It should be noted that the probabilistic resistance model proposed here is a simplified representation of reality. 

178 

 
 
 
 
 
 
 
 
 
 
 
B.7.2.2 Semi-probabilistic verification using the S-N model 

According to EN 1990 formula (8.11) the design value of the spectral stress range level Δσi,d should be derived 
from: 

Δσi,d = γF,f Δσi,m 

(B.61) 

where Δσi,m is the (nominal) value of the i-th stress range of the stress range spectrum following from the action 
and  structural  model  used  and  γF,f  is  a  partial  factor  for  fatigue  actions.  In  EN  1990  a  value  γF,f  =  1,0  is 
recommended,  which  would  mean  that  safety  margin  should  be  present  in  the  action  models,  such  as  the 
-values in the equivalent load models. At present the -values need more elaboration. 

The relevant fatigue load models may be found in EN 1991 for e.g. wind, traffic and waves. Some of these 
models contain simplifications which  probably have been  calibrated for a  particular application but  have no 
general validity. It is important to keep this in mind as some constructional details are more sensitive to high 
stress ranges than others. 

On the resistance side the (characteristic) S-N curves in the Eurocodes are based on the 95% survival probability. 

The design values may be found by dividing by the partial factor 𝛾Mf: According to EN 1990 (8.22): 

ΔσC,d = ΔσC / γM 

(B.62) 

where ΔσC is the characteristic fatigue resistance corresponding to NC cycles (NC=2 106). Assuming that Δ𝜎C is 
based  on  a  large  number  of  tests,  the  mean  value  of  the  10-base  logarithm  of  variable  K1  in  Table  B.11, 
μ(log(K1)) can be obtained as: 

μ(log(K1))=log(NC)+1,645∙s(log(K1))+m1log(ΔσC) 

(B.63) 

where s(log(K1)) is the standard deviation of the 10-base logarithm of variable K1. Since the two branches of 
the S-N curve are considered to be fully correlated, log(K2) can subsequently be obtained from equation B.45. 

The partial factor for fatigue resistance should account for the consequence of fatigue failure and the ease of 
inspection  and  repair  of  fatigue-sensitive  members.  The  partial  factors  should  be  calibrated  to  the  same 
reliability  levels  as  for  the  probabilistic  approach.  As  mentioned,  the  probabilistic  resistance  model  of  the 
previous section is a simplified representation of reality. Adopting such a model generally yields heavier designs 
or shorter fatigue lives than the semi-probabilistic verification with the partial factors given in EN 1990 and EN 
1993. This is because these partial factors are based on a more advanced probabilistic resistance model (see 
(Maljaars  et  al.,  2021),  the  connected  presentations  at  WG9  (Maljaars,  2019),  (Maljaars,  2020)  and  related 
publications and (Leonetti et al, 2020). It may be recommended to perform the probabilistic analysis also on 
more advance models. 

It is noted that the partial factors are not calibrated for wind loads (with fatigue load in prEN1991-1-4 Annex 
F.8), and for other  

EN 1991 also gives models to be used to check infinite life. If adopted, there is a need to carefully select the 
reliability target and corresponding partial factors. 

B.7.2.3 Probabilistic verification using crack growth models 

The failure probability for a total period of time from 0 to t is given by: 

Pf (t) = P(g(t)<0) 

(B.64) 

Where g(t) is given by B.54 or B.58. The stress distribution or spectrum in the expression for g(t) follows from 
the application and the structural properties. Data for the random variables for welded steel connections may 
be taken from a table like Table B.12. 

179 

 
 
 
 
 
 
Table B.12 Input for fatigue reliability analysis for welded steel connections using fracture mechanics 

Variable 

Units 

Distribution  Mean 

St. dev. 

A1 

A2 

m1 

m2 

Crack growth constant (in air) 

N and mm  Lognormal 

4,8e-18 

8,16e-18 

Crack growth constant (in air) 

N and mm  Lognormal 

5,86e-13 

3,52e-13 

Crack growth exponent (air) 

- 

- 

Crack growth exponent (air) 

5,10 

2,88 

- 

- 

ΔKth 

Threshold value (in air) a) 

N and mm  Lognormal 

140 

56 

ai 

Initial crack depth (as-welded) 

mm 

Lognormal 

0,15 

0,10 

ai / ci 

Initial aspect ratio (as-welded) 

mm 

Lognormal 

0,62 

0,25 

FAD 

glob 

scf 

sif 

MU failure assessment diagram c) 

Lognormal 

1,09 

0,14 b) 

MU global stress model c) 

Lognormal 

1,0 

0,10 

MU stress concentration c) 

MU stress intensity factor c) 

- 

- 

Lognormal 

1,0 

0,20 

Lognormal 

1,0 

d) 

a) In its most simplistic form, i.e. no consideration of the geometry and crack size 

b) Values applies if Kmat,T is evaluated from three KJC tests, from which the average is considered and this 
average is subsequently corrected for the crack tip constraint through the triaxial stress, using procedures in 
(BS 7910:2013 + A1:2015, 2012) 

c) MU = model uncertainty (correlated for all cycles).  

d) Depending on the source. Reasonable values: 0,03 in case of very often applied (i.e. lots of experience) 
parametric equations or 0,07 in case of evaluation through the finite element model 

The values for the initial crack sizes are valid for an initial imperfection at a weld to in as-welded state. 

The approach using an annual reliability target is considered to be more suitable for the fracture mechanics 
approach, as this approach is usually adopted to determine the remaining life or the inspection interval. 

B.7.3 Worked example 

Let us consider a motorway bridge with a single span of 20 m. A cover plate is welded to the bottom flange of 
the  main  girder  at  midspan  (Figure  B.12).  The  main  girder  is  loaded  by  one  (slow)  lane  only.  The  bridge  is 
designed with the S-N curve of prEN 1993-1-9 and Fatigue Load Model 4 (FLM4) of prEN 1991-2. The S-N 
curve has a characteristic reference fatigue resistance of ∆𝜎𝐶 = 50 MPa and a knee-point at 𝑁𝐷 = 107 cycles. 
A ‘safe life’ design period of 100 years is adopted, and the corresponding values of the partial factors are 𝛾𝐹𝑓 =
1.0 or 1.1 (depending on the NA) and 𝛾𝐹𝑚 = 1.35 (large consequences of failure) according to prEN 1990 and 
prEN  1993-1-9,  respectively.  Using  the  damage  accumulation  rule  of  Palmgren-Minder,  the  required  elastic 
section modulus is determined for this bridge such, that the design value of the damage is equal to 𝐷𝑑 = 1.0. 

180 

 
 
 
 
 
 
 
 
Figure B.12.  Structure considered in the fatigue numerical example 

20 m 

The  probabilistic S-N curve is evaluated  from constant amplitude tests described in the literature regarding 
beams with cover plates and with the dimensions corresponding with ∆𝜎𝐶 = 50 MPa as in prEN 1993-1-9, see 
Figure B.13. Using the value 𝑚1 = 3 as a prior, the mean value of log(𝐾1) = 11.65 and the standard deviation 
of log(𝐾1) = 0.15. This gives a characteristic fatigue reference strength at 2 million cycles of ∆𝜎𝐶 = 50 MPa, 
i.e. exactly the value that prEN 1993-1-9 prescribes. Full correlation is assumed between 𝐾1 and 𝐾2. The knee-
point of the curve is considered as a fixed prior in agreement with prEN 1993-1-9 at 𝑁𝐷 = 107 cycles. The 
second slope is also considered as a prior as 𝑚2 = 2𝑚1 − 1 = 5. The critical value of the damage accumulation 
of Palmgren-Miner is lognormal distributed with a mean of 1 and a standard deviation of 0.3. 

Figure B.13. Typical constant amplitude fatigue test data of cover plate beams 

A Weigh-In-Motion (WIM) database recorded during 10 months in 2018 on motorway E19/A16 near Moerdijk 
(the Netherlands) is considered as the basis of the actual traffic load. The vehicles with their recorded axle 
loads and distanced between axles are run over the influence line and the load effect is determined. The 
bending moment ranges ∆𝑀 are subsequently determined using the rainflow counting method.  

Figure B.14 gives the resulting histogram, adjusted for a full year (i.e. multiplying the recorded number of cycles 
with 12/10). The figure also gives the histogram of the 5 vehicles that comprise FLM4 and that is used to design 
the bridge. 

181 

101001.E+41.E+51.E+61.E+71.E+8Δσ[MPa]N[-]failedrun-outmeanchar 
 
 
 
 
 
 
 
 
 
 
 
Figure B.14. Typical bending moment range histograms of the fatigue example. 

Variable scf does not apply since the nominal stress method is adopted. The variable glob (mean=1, st.dev=0.1) 
is applied together with variable lm (mean=1.1, st.dev.=0.05). The first variable accounts for approximations 
and  simplifications  in  the  derivation  of  the  influence  line.  The  second  variable  takes  into  consideration  an 
expected increase in the total traffic load in the coming 100 years, as well as its uncertainty. It should be noted 
that this is an estimate (experts’ opinions). Dynamic amplification due to bridge-vehicle interaction is usually 
small compared to the above-mentioned uncertainty factors for the relevant (relatively heavy) vehicles (Kalin 
et al, 2021) and it is hence not considered. 

Using the designed elastic section modulus and the probabilistic model as described above, the reliability index 
is estimated with crude Monte Carlo. The estimate for a 100 year life gives 𝛽 = 4.1 or 𝛽 = 4.6 for  𝛾𝐹𝑓 = 1.0 
or 1.1 respectively but same model. This value should be compared with the target reliability of 𝛽 = 4.3 (CC3, 
life time) for this case. A few notes should be made: 

 

 

 

The reliability varies per type of detail, since the standard deviation of the fatigue tests varies per type of 
detail and the slope parameter may also vary. 

The reliability also varies per road. Some motorways in Europe have a more aggressive traffic composition, 
others a less aggressive one as compared to the considered database, (Maljaars, 2020). 

Especially the long-term trend in traffic is highly uncertain. In that respect, the standard deviation used for 
lm may be too optimistic. 

  A more refined fatigue resistance model may be adopted as compared to the one given here. In particular, 
random fatigue limit models are able to consider the uncertainty in the constant amplitude fatigue limit 
(Pascual et al, 1999) and variations thereof). Models are also available that consider a gradual change of 
the  constant  to  the  variable  amplitude  S-N  curve,  depending  on  the  magnitude  and  number  of  ranges 
exceeding  the  constant  amplitude  fatigue  limit  (Leonetti  et  al,  2020).  Adopting  these  models  generally 
increases the estimated reliability. 

 

If the hot-spot stress method is used and hence scf applies, a lower reliability index or a higher partial 
factor would follow from the analysis. 

References 

British Standards Institute – BSI (2012), Guide on methods for assessing the acceptability of flaws in metallic 
structures, BS 7910:2013 + A1:2015. 

McEvily A., Endo M., Murakami Y. (2003), ‘On the relationship and the short fatigue crack threshold’, Fatigue and 
Fracture of Engineering Materials and Structures, Vol. 26, Issue 3, pp. 269–78. 

182 

10010001.E+01.E+11.E+21.E+31.E+41.E+51.E+61.E+7ΔM[kNm]n[-]WIM E19/A16FLM4 
 
Chapetti  M.  D.  (2003),  ‘Fatigue  propagation  threshold  of  short  cracks  under  constant  amplitude  loading’, 
International Journal of Fatigue, Vol. 25, Issue 12, pp. 1319–26. 

El  Haddad  M.,  Topper  T.  and  Smith  K.  (1979),  ‘Prediction  of  non  propagating  cracks’,  Engineering  Fracture 
Mechanics, Vol. 11, Issue 3, pp. 573–84. 

Zerbst U., Madia M. (2015), ‘Fracture mechanics based assessment of the fatigue strength: approach for the 
determination of the initial crack size’, Fatigue and Fracture of Engineering Materials and Structures, Vol.  38, 
Issue 9, pp. 1066–75. 

Leonetti D., Maljaars J., Snijder H.H. (2021), ‘Fracture mechanics based fatigue life prediction for a weld toe 
crack  under  constant  and  variable  amplitude  random  block  loading—Modeling  and  uncertainty  estimation’, 
Engineering Fracture Mechanics, Vol. 242, 107487. 

Maljaars J., Steenbergen  H.,  and  Vrouwenvelder  A.  (2012), ‘Probabilistic  model for fatigue crack growth and 
fracture of welded joints in civil engineering structures’, International Journal of Fatigue, Vol. 38, pp. 108–17. 

Leonetti D., Maljaars J., Snijder H.H. (2020), ‘Probabilistic fatigue resistance model for steel welded details under 
variable amplitude loading – Inference and uncertainty estimation’, International Journal of Fatigue, Vol. 135, 
105515.  

Kalin J., Znidaric A., Anzlin A., Kreslin, M. (2021), ‘Measurements of bridge dynamic amplification factor using 
bridge  weigh-in-motion  data’,  Structure  and  Infrastructure  Engineering,  Vol.  18,  Issue  8,  pp.  1164–1176. 
https://doi.org/10.1080/15732479.2021.1887291  

Maljaars J., ‘Evaluation of traffic load models for fatigue verification of European road bridges’, Engineering 
Structures, Vol. 225, 111326. 

Pascual  F  G,  and  Meeker,  W.  Q.  (1999),  ‘Estimating  fatigue  curves  with  the  random  fatigue-limit  model’, 
Technometrics, Vol. 41, pp. 277-289. 

Maljaars,  J.,  Rózsás,  Á.,.  Walters,  C.  L.,  Slot,  H.  (2019),  ‘Uncertainty  quantification  of  the  failure  assessment 
diagram for flawed steel components in BS 7910:2019’, Engineering Fracture Mechanics, Vol. 268, 108446 

Wirsching, P.H. (1984), ‘Fatigue reliability for ofshore structures’, Journal of Structural Engineering, Vol. 110, 
Issue 10, pp. 2340-2356 

B.8 Failure probability of deteriorated concrete structure 

B.8.1 Introduction 

The goal of this section is to present the results of the calculations of the annual probability of failure of a 
deteriorated  bridge  due  to  chloride-induced  corrosion  for  a  period  of  100  years.  In  the  following,  the 
methodology used for the reliability analysis, the structure under investigation, the models of the structural 
performance and the probabilistic models are described. 

B.8.2 Methodology 

The annual probability of failure in year i is calculated by: 

Pf,i=P(Fi∩S1∩S2∩…∩Si-1) 

(B.65) 

where: 

Fi  

denotes the failure event during year i; 

S1, S2, Si-1  

are the survival events during the first year, the second year and year i-1.  

In the simple case where only one failure mode is considered, the annual probability of failure can be expressed 
as follows: 

Pf,i=P[g(Xi)≤0∩ g(X1)>0  ∩  g(X2)>0∩…∩g(Xi-1)>0] 

(B.66) 

where: 

183 

 
 
 
g  

Xi  

is the limit state function; 

is the vector of the uncertain parameters considered in the limit state function. 

The  vector  X  contains  both  time-invariant  and  time-variant  components.  Examples  of  time-invariant 
components are the  dead weight and the  uncertainties of the  structural  models.  Variable loads, the loss of 
reinforcement area due to corrosion are examples of time-variant uncertain parameters. Due to the presence 
of  these  time-variant  parameters  in  the  limit  state  function,  vector  X  is  different  from year-to-year.  In  the 
following, the annual probability of failure is approximated by using the outcrossing approach: 

Pf,i≈P[g(Xi)≤0  ∩ g(Xi-1)>0] 

(B.67) 

This  failure  probability  is  estimated  by  using  the  First  Order  System  Reliability  Method  (Hohenbichler  and  
Rackwitz, 1983): 

P[g(Xi)≤0 ∩g(Xi-1)>0]= 

=P[αiU+βi≤0∩ αi-1U+βi-1>0]= 
=P[Zi≤-βi∩ Zi-1>βi-1]=Φ2(β;ρ) 

(B.68) 

where:  

U  

i  

i  

is the vector of independent random variables with standard normal distribution; 

is the vector of the sensitivity factors of the components of vector X; 

is the reliability index estimated using a FORM reliability calculation considering the limit state function 
g(Xi); 

Zi  

is the standard normal random variable; 





is the vector of the reliability indices [-i, i+1]; 

is the 2 x 2 correlation matrix, whose off-diagonal elements are: 

ρ12=ρ21=-αi-1⋅αi 

(B.69) 

2is the CDF function of the bi-variate normal distribution.  

Only the components of the sensitivity vector  of the time-invariants random variables are considered in the 
calculation of the correlation matrix. 

Description of the case 

The structure considered in this section is a simply supported slab bridge Holicky and Sykora, 2009). The slab 
is 11 m long, 15,6 m wide and 0,6 m thick. The concrete cover is equal to 50 mm. The concrete class is C30 
and the steel grade is S500.  

The bridge is subjected to the permanent load g and the traffic load q. The characteristic values of the bending 
moment E(Gk) and E(Qk) in the midspan section are, per meter of width, equal to 340 kNm/m and 330 kNm/m. 
The longitudinal reinforcement consists of bars with diameter of 25 mm and the reinforcement area is equal 
to 4310 mm2/m (per meter of width), leading to the unity check value (UC = Ed / Rd) of 1: 

UC = 

Ed
Rd

 =

= 

γGE(Gk)+γQE(Qk)

As

fyk
γS

h- (c+

d
2

) -

[

As

2α

fyk
γS
fck
γC ]

184 

(B.70) 

 
 
 
 
 
 
 
 
 
 
 
=

1,35⋅340+1,35⋅330

10-6⋅4310⋅

500
1,15

⋅ [600- (50+

25
2

) -

4310⋅

500
1,15
2⋅103⋅0,85⋅

30
1,5

=

904
904

=1 

]

where: 

s (=1,15)  

is the partial factor for steel reinforcement; 

c (=1,5)  

is the partial factor for concrete compressive strength;  

G (=1,35)  

is the partial factor for permanent action; 

Q (=1,35)  

partial factor for traffic load; 

 (=0,85)  

is the factor accounting for the long-term effects on the concrete compressive strength. 

B.8.3 Limit state modelling 

Only the limit state of flexural failure is considered. It is assumed that the resistance of the structure decreases 
in time due to corrosion of the reinforcement caused by the use of de-icing salts. The limit state function is 
expressed as: 

g(X,t)=θMRMR

(t)-EG -θEQEQ 

(B.71) 

where: 

θMR  

MR(t)  

is the model uncertainty of the resisting bending moment; 

is the resisting bending moment at time t; 

EG and EQ 

are the bending moments induced in the midspan section by the permanent and traffic loads; 

θEQ  

is the model uncertainty of the traffic load-induced bending moment. 

The resisting bending moment MR(t) at time t is evaluated as follows  

MR(t)=Asg(t)fy [h - (c+

d
2

) -

Asfy
2αfc

] 

The degradation function g(t) is defined as: 

 1                                                                              if t≤ Tinit

g(t)= {

[1-

2⋅0,0116⋅1,2⋅icorr0(t-Tinit)0,71
d0

]

2

                              if  t> Tinit

(B.72) 

(B.73) 

where: 

icorr0  

is the initial corrosion rate in A/cm2; 

Tinit  

d0  

is time to initiation of corrosion in years; 

is the initial diameter of the bars. 

B.8.4 Probabilistic model 

The probabilistic model of the parameters involved in the limit state functions (Equations  (B.71), (B.72) and 
(B.73)) are listed in Table B.13.  

185 

 
 
 
 
 
 
 
 
Table B.13 Input for case under consideration 

Variable  Description 

Units 

Distribution  Mean 
value 

CoV 

Note 

h 

c 

As 

fy 

fc 

E(g) 

E(q) 

EQ 

MR 

icorr0 

Tinit 

Cross-section height 

m 

Normal 

0,60 

0,017  Time-invariant 

Concrete cover 

mm 

Gamma 

60 

0,17 

Time-invariant 

Initial reinforcement area 

mm2 

Normal 

4310 

0,03 

Time-invariant 

Yield stress S500 

N/mm2  Lognormal 

560 

0,054  Time-invariant 

Concrete  compressive  strength 
C30 

N/mm2  Lognormal 

37,5 

0,13 

Time-invariant 

Permanent load 

kNm/m  Nornal 

340 

0,10 

Time-invariant 

Traffic load (1-year max) 

kNm/m  Gumbel 

290,4 

0,02 

Time-variant 

Mod.  unc. bending  moment  (load 
effect) 

Mod.  unc.  bending  moment 
(resistance) 

- 

- 

Lognormal 

1,0 

0,15 

Time-invariant 

Normal 

1,08 

0,1 

Time-invariant 

initial corrosion rate 

A/cm2  Normal 

2,7 

0,63 

Time-invariant 

Time to initiation of corrosion 

y 

Exponential 

7,4 

1,0 

Time-invariant 

B.8.5 Results 

The annual probability of failure and the annual reliability index are presented in Figure B.15 and Figure B.16 
for a period of 100 years. 

Figure B.15. Annual probability of failure 

186 

 
 
 
Figure B.16. Annual reliability index 

References 

Hohenbichler, M., and Rackwitz, R. (1983), ‘First-order concepts in system reliability’, Structural Safety; Vol. 1, 
pp. 177-188 

Holicky, M., Sykora, M. (2009), ‘Assessment of deteriorating reinforced concrete road bridge’, in: Proceedings of 
the ESREL2009 conference 

187 

 
 
 
 
B.9 Example risk based approach: Observational method in geotechnics 

B.9.1 Decision-theoretical aspects of the observational method 

As discussed in section 3.9 of the main text an inherent issue in geotechnical engineering is the considerable 
lack of knowledge (epistemic uncertainty) of the ground properties at the outset of a new project. Although 
geotechnical investigations can be used to increase the knowledge about the ground, the investigations may at 
times turn out too difficult or expensive, to determine the ground properties with adequate precision. In such 
cases,  the  observational  method  offers  a  design  framework  to  increase  the  knowledge  about  the  ground 
properties and how they affect the structural behaviour. The observational method is defined in EN 1997-1, 
clause 4.7.  

The observational method can best be demonstrated by considering the decision tree as shown in Figure B.17. 
In the general case there exists a number of design alternatives E = {e1, e2, …} as branches extending from the 
leftmost node, but here only two alternatives are shown for illustrative purposes, the observational method and 
a  conventional  design  alternative.  If  the  observational  method  is  applied,  structural  behaviour  is  monitored 
during construction. The outcome Z determines whether or not to implement one of the prepared contingency 
measures A = {a1, a2, …}, which would modify the design from the initial variant into another prepared design 
variant  .  In  other  words,  the  decision  to  modify  the  design  into  another  variant  is  not  free,  but  dictated  by 
passing monitoring thresholds that have been established in the monitoring plan. In the last branching ϴ, it is 
revealed whether limit state violation occurs with probability pF.  

Each scenario is associated with some cost, the base cost C(eOM) including development of all design variants 
and implementation of one of them at the outset, as well as the cost to prepare and execute the monitoring; 
the cost C(a) to change into another design variant; and the cost C(F) for violated limit state. The conventional 
design only has two costs: the base cost C(eCON) for the implementation of the design, and C(F) for a violated 
limit  state.  The  optimal  design  alternative  is  the  one  with  the  smallest  expected  cost,  conditionally  that  it 
satisfies  the  target  failure  probability,  pF,T.  This  can  be  determined  straightforwardly  with  a  pre-posterior 
analysis  (see  e.g.  the  textbook  by  Benjamin  and  Cornell,  1970).  An  example  is  presented  in  section  B10.3 
(summarised from Spross and Johansson, 2017). 

B.9.2 Reliability-based monitoring thresholds 

An important safety aspect of the observational method is how monitoring thresholds are determined, as they 
ensure  sufficient  safety  margin  to  the  limit  state.  On  the  other  hand,  the  thresholds  cannot  be  set  too 
conservatively, as they then will trigger expensive contingency measures that are not really warranted. Applying 
reliability-based design principles, Spross and Johansson (2017) showed how reliability-based thresholds could 
be established when applying the observational method. Based on the target failure probability, pF,T, a threshold 
xalarm for a monitored parameter X1 can be determined by solving the following equality: 

P(G(X) < 0 | X1 < xalarm) = pF,T 

(B.74) 

where G(X) = 0 is the considered limit state with associated random variables in the vector X. Equation B.74 is 
derived from the definition of conditional probability. It can be interpreted such that the probability of limit state 
violation is acceptable, conditionally on the measurement readings falling below the threshold. Equation B.74 
assumes a time-invariant analysis; Spross and Gasch (2019) discussed the implications of having to account 
explicitly  also  for  an  unavoidable,  triggered  lead  time  between  threshold  violation  and  an  unacceptable 
structural behaviour. 

188 

 
 
 
 
Figure  B.17. Observational  method compared against a  conventional design in a  decision tree (Spross and 
Johansson, 2017), CC-BY, http://creativecommons.org/licenses/by/4.0/, minor changes) 

Figure  B.18.  Section  of  the  analysed  rib  pillar.  Should  rock  anchors be  installed  immediately, or is it  more 
favourable to first observe the pillar deformation during excavation (making the pillar thinner and thinner) and 
then install rock anchors only if measurements show that they are needed. 

Figure  B.19.  For  the  analysed  rib  pillar,  the  observational  method  is  found  more  favourable,  because  the 
expected cost of that design alternative (95 MU) is less than that of the conventional design (150 MU) (Spross 
and Johansson, 2017), CC-BY, http://creativecommons.org/licenses/by/4.0/, minor changes) 

189 

  
 
 
 
Application example: vertical deformation of a rock pillar 

A very long rib pillar is to be excavated in rock (Figure B.18). The limit state concerns its vertical strain ε1, which 
must  not  exceed  a  maximum  strain  ε1,max,  i.e.  we  have  that  G(X)=ε1,max-ε1,  where  X  collects  a  number  of 
underlying rock mass properties as random variables (the reader is referred to Spross and Johansson (2017) 
for the complete geomechanical model). In this case pF,T is set equal to 0,001. Performing a reliability-based 
analysis (e.g. using Monte Carlo simulation) it is found that for an unsupported pillar, the probability of limit 
state violation at the end of the excavation process is pF = 0,0046, which exceeds pF,T. However, if four horizontal 
rock anchors are added in the model to provide confinement, pF becomes negligible.  

The decision maker now stands before two alternatives:  

1. 

2. 

to go directly with the more expensive design using the rock anchors (i.e. a conventional design), or 

to  apply  the  observational  method  to  reduce  the  uncertainty  about  the  rock  mass  properties  by 
monitoring the deformation of the unreinforced pillar during the excavation process.  

The excavation is assumed to be carried out stepwise, making the pillar thinner and thinner with each step. 
Measurements  are  made  after  each  excavation  step,  providing  a  number  of  measurements  from  which  a 
trendline is derived to predict the final deformation, including the scatter.  

If  the  actual  ground  conditions  turn  out  to  be  favourable,  rock  anchors  will  not  be  needed  after  all.  The 
alternatives are illustrated as a decision tree in Figure B.19 together with their corresponding costs. The cost 
components are given by (MU = monetary unit):  

anchor system 150 MU is applied directly or 170 MU if applied later,  

advanced monitoring system: 20 MU 

cost increase due to more careful excavation using the observational method: 30 MU 

failure 5000 MU 

The threshold for the vertical strain without anchors can be found from application of Equation. (B8.1), in this 
case  P(G(X)<0|ε1≤ε1, alarm)=pF,T leading to  ε1, alarm = 0,33⸱10-3 (corresponding to 2,7  mm  deformation). This 
allows evaluation of the probability of the unsupported design variant being successful: P(ε1≤ε1, alarm) = 76%. 
Hence, threshold violation that triggers the rock anchor installation occurs with 24% probability. 

A so-called pre-posterior analysis shows that the expected cost of applying the observational method is 95 MU, 
while the expected cost of the conventional design is 150 MU. The analysis is performed by weighing the costs 
with respect to their probability of occurrence in the decision tree. Based on the result, a risk-neutral decision 
maker should choose the observational method. Note however that if the observations trigger the contingency 
measure to install rock anchors, the final cost becomes larger (220 MU) than that of the conventional design 
alternative. A risk-avoiding decision maker may therefore consider the conventional design better, as it comes 
with a known cost. 

Note that in this case the cheapest solution is “to do nothing” and accept the high probability of failure leading 
to a cost expectation of 0,0046 ⸱ 5000 MU = 23 MU. Whether this is legally allowed may depend on the type of 
limit state (SLS or ULS) and the relevant clauses in the National Annex, as pF = 0,0046 exceeds pF,T. 

References 

Benjamin, J.R. and Cornell, A.C. (1970), Probability, statistics, and decision for civil engineers, McGraw-Hill, New 
York.  

Spross,  J.  and  Gasch,  T.  (2019),  ‘Reliability-based  alarm  thresholds  for  structures  analysed  with  the  finite 
element method’, Structural Safety, Vol. 76, pp. 174-183. DOI: 10.1016/j.strusafe.2018.09.004 

Spross, J. and Johansson, F. (2017), ‘When is the observational method in geotechnical engineering favourable?’, 
Structural Safety, Vol. 66, pp. 17-26. DOI: 10.1016/j.strusafe.2017.01.006  

190 

 
 
 
 
Annex B: List of boxes 

Description of the case  184 

Application example: vertical deformation of a rock pillar 

190 

Annex B: List of figures 

Figure B.1. Violin plots for the scatter in resulting β-values 

151 

Figure B.2. Non-optimized Reliability index β as function of αQ for wind, snow and live load ranging from 2.5 
152 
to 4.5 

Figure B.3. Examples of aQ distribution (uniform, triangular symmetric, triangular 0,4, linear) 

Figure B.4. Selected weather stations and snow zones (left) and wind zones (right) (with courtesy of the 
University of Pisa)  (Formichi et al, 2023) 

Figure B.5. Data and α results of reliability c-values for steel column IPE-160 and with  = 1.0. 

Figure B.6. Girder on a block, case for EQU example 

Figure B.7. Loads Q1 and Q2 presented as FBC models 

Figure B.8. Calculated PSI-0 value for a Gumbel distributed with V=0,2 

Figure B.9. Shear wall and main data 

Figure B.10. A semi-elliptical crack in a steel plate at/near the weld toe 

156 

157 

163 

165 

166 

170 

171 

176 

Figure B.11. Failure assessment diagram: (a) with definitions of rF and rFAD; (b) with data from large-scale 
wide plate and tubular joint tests (Maljaars et al, 2019) 

177 

Figure B.12.  Structure considered in the fatigue numerical example 

Figure B.13. Typical constant amplitude fatigue test data of cover plate beams 

181 

181 

A Weigh-In-Motion (WIM) database recorded during 10 months in 2018 on motorway E19/A16 near Moerdijk 
(the Netherlands) is considered as the basis of the actual traffic load. The vehicles with their recorded axle 
loads and distanced between axles are run over the influence line and the load effect is determined. The 
bending moment ranges ∆𝑀 are subsequently determined using the rainflow counting method. 

181 

Figure B.14. Typical bending moment range histograms of the fatigue example. 

182 

The annual probability of failure and the annual reliability index are presented in Figure B.15 and Figure B.16 
for a period of 100 years. 

186 

Figure B.15. Annual probability of failure 

Figure B.16. Annual reliability index 

Annex B: List of tables 

Table B.1 Average reliability indices per material and action 

Table B.2 Average reliability indices per material and action 

Table B.3 Overview of sensitivity studies 

Table B.4 Calibrated partial factors for different actions and for different treatment of bias in climatic 
actions 

186 

187 

153 

154 

155 

157 

Table B.5 Overview of variables and numerical input values; last column shows resulting FORM -values for 
steel column 

162 

191 

 
 
 
 
Table B.6 Overview of variables, numerical input values and resulting FORM-values concrete column 

Table B.7 Resulting reliability indices for as a function of (i j) 

Table B.8 Formulas for 0 time independent case 

Table B.9 Formulas for 0 time dependent case 

Table B.10 Summary of the calculation results 

Table B.11 Input for fatigue reliability analysis for welded steel connections using S-N curves 

X
X
-
N
A
-
X
X
X
X
X
-
E
N
-
N

164 

165 

167 

169 

172 

178 

Table B.12 Input for fatigue reliability analysis for welded steel connections using fracture mechanics  180 

Table B.13 Input for case under consideration 

186 

192 

 
 
 
 
 
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