Seismic plus energy upgrading of masonry buildings using advanced materials : analysis of a combined structural and thermal retrofitting.
Date: 2024-09-25
Institutions: Joint Research Centre
Subjects: advanced materials, building insulation, building materials, building safety, building technique, earthquake, energy efficiency, research report, seismic monitoring

ISSN 1831-9424 

Seismic plus energy upgrading of masonry 
buildings using advanced materials 

Analysis of a combined structural and thermal retrofitting 

Kouris, L., Bournas, D., Pohoryles, D., Valsamos, G., 
Savvas, T., Maduta, C. 

2024 

 EUR 30563 EN 

 
  
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Contact information 
Name: Dionysios Bournas 
Address: Via E. Fermi, 2749, I-21027 Ispra (VA), Italy 
Email: Dionysios.BOURNAS@ec.europa.eu  

EU Science Hub 
https://joint-research-centre.ec.europa.eu 

JRC123617 

EUR 30563 EN 

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ISBN 978-92-76-28603-5   ISSN 1831-9424   doi:10.2760/66556 

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Luxembourg: Publications Office of the European Union, 2024 

© European Union, 2024  

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How to cite this report: European Commission, Joint Research Centre, Kouris, L., Bournas, D., Pohoryles, D., Valsamos, G., Savvas, T. and 
Maduta, C., Seismic plus energy upgrading of masonry buildings using advanced materials, Publications Office of the European Union, 
Luxembourg, 2024, https://data.europa.eu/doi/10.2760/66556, JRC123617. 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
Contents 

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

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

Acknowledgements .......................................................................................................................................................................................................................................... 3

1 

Introduction..................................................................................................................................................................................................................................................... 4

1.1  Problem statement and Review of Integrated Retrofitting in the Literature .................................................................... 4

1.2  Context of relevant EU policies ...................................................................................................................................................................................... 6

2  State-of-the-art on the experimental performance of the textile-reinforced mortar (TRM) ........................................... 7

2.1  Performance of TRM ................................................................................................................................................................................................................ 7

2.2 

In-plane performance of the hybrid system ..................................................................................................................................................... 8

2.3  Out-of-plane performance of the hybrid system ......................................................................................................................................... 9

3  Homogenisation of masonry properties........................................................................................................................................................................... 11

3.1  Overview of homogenisation ........................................................................................................................................................................................ 11

3.2  Gradient elastic theory ........................................................................................................................................................................................................ 12

3.3  Gradient homogenisation model (GREHM) ....................................................................................................................................................... 14

3.3.1

Homogenised masonry elastic modulus Ev perpendicular to mortar bed joints (vertical) ............... 15

3.3.2

Homogenised masonry elastic modulus Eh parallel to mortar bed joints (horizontal) ........................ 15

3.3.3

Homogenised masonry shear elastic modulus G .................................................................................................................... 16

3.3.4

Homogenised masonry Poisson’s ratios vxy and vyx ............................................................................................................... 16

3.3.5

Normalised graphs for the estimation of the homogenised properties ............................................................ 17

3.4  Verification of the proposed homogenisation model ............................................................................................................................ 19

3.4.1

Comparison with numerical homogenisation models ......................................................................................................... 19

3.4.2

Comparison with analytical homogenisation models ......................................................................................................... 25

4  Numerical simulations and analyses of reinforced masonry with TRM ............................................................................................ 29

4.1 

Implicit simulation of TRM reinforced masonry .......................................................................................................................................... 29

4.1.1

Description of the model ................................................................................................................................................................................ 29

4.1.1.1

Parameters of TRM layers ............................................................................................................................................................... 29

4.1.1.2

Parameters of masonry ..................................................................................................................................................................... 31

4.1.2

Calibration of the model for out-of-plane loads ..................................................................................................................... 33

4.1.2.1

Calibration of masonry properties ........................................................................................................................................... 33

4.1.2.2

Calibration of the textile bond behaviour.......................................................................................................................... 34

4.1.3

Empirical equations for the reduction coefficients................................................................................................................ 35

4.1.4

Analyses for out-of-plane quasi-static loads ............................................................................................................................. 38

4.2  Explicit simulation of TRM reinforced masonry ........................................................................................................................................... 40

4.2.1

Parameters of the model ............................................................................................................................................................................... 40

4.2.2

Calibration of the model ................................................................................................................................................................................. 41

4.2.3

Analysis for dynamic loading from explosions .......................................................................................................................... 43

5  Assessment of the combined retrofitting in city scale ...................................................................................................................................... 45

i 

5.1  Studied cities and buildings’ population ............................................................................................................................................................. 45 

5.2  Energy performance .............................................................................................................................................................................................................. 49 

5.3  Seismic performance ............................................................................................................................................................................................................ 52 

5.4  Combined performance ...................................................................................................................................................................................................... 55 

6  Conclusions .................................................................................................................................................................................................................................................. 58 

References ............................................................................................................................................................................................................................................................. 59 

List of abbreviations and definitions ........................................................................................................................................................................................... 68 

List of boxes......................................................................................................................................................................................................................................................... 69 

List of figures ..................................................................................................................................................................................................................................................... 70 

List of tables ........................................................................................................................................................................................................................................................ 72 

ii 

 
 
 
 
 
 
 
 
 
 
 
 
 
Abstract 

Upgrading  existing  masonry  buildings  in  the  EU  has  become  very  important  due  to  their  poor  performance 
during  earthquakes  resulting  in  significant  human  and  economic  losses,  and  on  the  other  hand,  their  low 
energy  performance  which  significantly  increases  their  energy  consumption.  Therefore,  a  holistic  approach 
with a concurrent seismic and energy retrofitting is necessary and has been explored in this project. Namely, 
SPEctRUM  investigated  a  hybrid  structural-plus-energy  retrofitting  solution  to  achieve  cost  effectiveness, 
which  combines  inorganic  textile-based  composites  with  thermal  insulation  systems  for  masonry  building 
envelopes. Specifically, textile reinforced mortar (TRM) composites are used to reinforce unreinforced masonry 
(URM) buildings enhancing their seismic capacity, while thermal insulation is added to improve the buildings’ 
energy  performance.  The  EU  policy  context  of  energy  efficiency  and  sustainability  and  safeguarding  of 
cultural heritage is benefitted by the project as discussed in chapter 1. 

Firstly in chapter 2, a thorough review of the experimental investigations applying textiles in inorganic mortars 
is carried out and conclusions are drawn regarding its efficiency. A substantial increase in the in-plane and the 
out-of-plane strength and displacement capacity can be seen in all the test campaigns despite the different 
experintal techniques and strengthening materials. Moreover, the hybrid composite material made of layers of 
TRM and thermal insulation was found advantageous and capable to fulfil its structural and energy targets if 
the various layers are carefully bonded. 

Then, in chapter 3, numerical models and design solutions were developed to facilitate the application of the 
proposed system. The investigation in the first place focused on a new homogenisation model for masonry to 
reduce  the  complexity  of  the  simulation.  Indeed,  in  a  retrofitted  masonry  wall  there  exist  at  least  four 
different  materials  (masonry  units,  masonry  mortar,  strengthening  mortar  and  textile)  built  in-situ  with  an 
internal  structure  which  has  many  imperfections.  Therefore,  there  are  a  lot  of  interaction  levels  and 
parameters  which  are  not  easily  to  predict  in  a  micro-modelling  approach.  The  homogenisation  of  masonry 
offers a viable solution to simplify masonry in one material. To this end, the proposed homogenisation model 
includes  a  gradient  term  to  represent  the  internal  structure  of  masonry.  The  proposed  gradient  elastic 
homogenisation model (GREHM) is furnished with easy-to-apply charts to simplify its application on masonry 
walls. 

In the following chapter 4 the numerical modelling of the strengthened masonary with TRM is attempted. Two 
approaches  for  a  detailed  modelling  are  followed:  (i)  an  implicit  model  for  quasi-static  loading  and,  (ii)  an 
explicit  model  for  dynamic  loading.  The  former  model  includes  the  homogenised  properties  of  masonry  to 
limit the numerical cost of the analysis. Moreover, the interaction of the textile with the inorganic  matrix is 
accounted  for  introducing  in  the  model  effective  material  properties  with  the  aid  of  an  empirical  equation 
specifically  designed  for  this  aim.  This  empirical  equation  has  been  calibrated  using  an  experimental 
campaign  on  the  out-of-plane  performance  of  masonry  walls  strengthened  with  TRM.  It  can  distinguish 
between  coated  and  uncoated  textiles.  Parametric  investigations  show  that  the  capacity  can  be  increased 
substantially even when less strong textiles from natural fibres are used. The explicit model on the contrary, is 
able to treat simulations with higher non-linearities and in this regard, a distinct simulation of masonry units 
and mortar joints was adopted. Based on this latter model the response of strengthened walls in blast loads 
has been investigated and it was found that TRM layers offer high safety from explosions for the occupants.  

In  Chapter  5,  the  cost  effictiveness  of  the  proposed  retrofitting  system  has  been  explored  in  a  large-scale 
investigation  across  various  locations  in  Europe.  A  number  of  cities  with  different  seismic  risk  and  climatic 
conditions were selected and their building stock was investigated. The seismic hazard varies from low to high 
risk characterised by the expected PGA and the climatic conditions were classified into four zones (warm to 
cold)  characterised  by  the  heating  degree-days.  Then,  archetype  buildings,  i.e.  buildings  with  typical 
architectural and structural characteristics were defined and the building stock was classified in the respective 
categories.  The  renovation  of  the  building  population  of  the  selected  cities  was  examined  assuming  target 
annual renovation rates from 1 to 3%. Then, the payoff period was estimated. Relativley short payoff times 
have  been  found  generally  which  decrease  when  high  seismic  risk  combines  with  old  masonry  structures. 
Finally, the last chapter summarises the conclusions. 

1 

Foreword 

This is the final report of the SPEctRUM project funded by Marie Skłodowska-Curie Actions concluding its main 
findings. This project has been inspired by the always more demanding need to upgrade the existing masonry 
EU buildings due to their poor seismic performance resulting in severe human and economic losses and their 
low energy  performance which significantly increases their energy  consumption. The issue of  upgrading the 
unreinforced  masonry  (URM)  buildings  is  of  great  importance  since  they  are  unengineered  vernacular 
structures  and  far  from  the  levels  of  the  current  standards  for  seismic  capacity  and  energy  consumption. 
Moreover,  the  latter  combined  with  deterioration  due  to  ageing  of  materials,  environmental  degradation, 
experience  of  several  earthquakes  and  lack  of  maintenance,  yield  to  even  highers  structural  and  energy 
deficiencies.  Every  recent  moderate  to  high  seismic  shaking  has  caused  damage  ranging  from  cracks  to 
partial  or  total  collapse  with  a  high  death  toll  and  economic  loss.  Therefore,  this  project  aimed  to  confront 
both  deficiencies  in  an  integrated  manner  by  exploring  innovative  techniques  and  advanced  materials.  The 
synergy of this project with the JRC institutional project iRESIST+ which focused on the upgrading of old RC 
buildings, tried to address the problem for the most common building typologies providing viable solutions in 
the context of EU regulations for energy savings of buildings and protection of cultural heritage. 

A  simultaneous  seismic  and  energy  retrofitting  of  the  masonry  building  stock  can  be  realised  integrating 
advanced  strengthing  and  insulating  materials.  In  masonry  buildings  however,  there  are  some  more 
requirements for the set targets of the retrofitting apart from the structural and energy performance related 
to their traditional nature; these can be summarised in the reversibility and compatibility of the strengthening 
materials. Moreover, cost effectiveness remains an important aspect which should be added to them.  

SPEctRUM  explored  a  hybrid  structural-plus-energy  retrofitting  solution  which  combines  inorganic  textile-
based  composites  with  thermal  insulation  systems  for  masonry  building  envelopes  to  meet  all  the 
aforementioned  requirements.  Analytical  models  and  design  solutions  were  developed  to  facilitate  the 
investigation and the application of the proposed system. Parametric analyses showed that the capacity can 
be  increased  substantially  with  the  use  of  various  strength  textiles.  The  cost  efficiency  of  the  combined 
retrofitting system was explored in a large-scale investigation accross Europe and rather short payoff times 
were found especially in regions of moderate to high seismicity as those in the Mediterranean region.  

2 

Acknowledgements 

The fund received from the Marie Skłodowska-Curie Actions - Individual Fellowships of the Call H2020-MSCA-
IF for the project SPEctRUM (No. 799593) is gratefully acknowledged.  

The kind help of Dr A Anthoine and Dr FA Kariou is thankfully acknowledged.  

Authors 

Leonidas Alexandros S. Kouris, Dionysios A. Bournas, Daniel A. Pohoryles 

Georgios N. Valsamos, Savvas P. Triantafyllou, Carmen Maduta 

3 

 
 
 
1 

Introduction 

1.1  Problem statement and Review of Integrated Retrofitting in the Literature 

Large parts of the built infrastructure need repair or strengthening due to ageing, rehabilitation, or obsolete 
design. The issue of upgrading the existing unreinforced masonry (URM) structures is of great importance and 
priority  for  all  EU  countries  as  most  of  them  are  unengineered  vernacular  structures.  URM  buildings  suffer 
from  low  seismic  capacity  and  high  energy  consumption.  Moreover,  the  deterioration  due  to  ageing  of 
materials, environmental degradation, experience of several earthquakes and lack of maintenance, contribute 
to  an  evolving  higher  vulnerability  and  lower  energy  efficiency.  Every  moderate  to  high  seismic  shaking 
causes  damage  ranging  from  cracks  to  partial  or  total  collapse  with  a  high  death  toll  and  economic  loss 
(Indirli et al., 2013; Penna et al., 2014b). In the Mediterranean region the existing masonry building stock is 
susceptible to a significant seismic hazard. Given that part of these URM buildings constitutes the famous EU 
built  cultural  heritage  which  is  at  a  high  stake  and  whole  traditional  settlements  have  been  devastated  in 
strong  earthquakes  (Kouris  et  al.,  2010),  their  structural  health  monitoring  and  conservation  (Rossi  and 
Bournas, 2023) comprises priority for the EU. 

On  the  other  hand,  the  low  energy  performance  of  masonry  buildings  increases  their  energy  consumption 
which  is  mainly  attributed  to  the  no  or  low  thermal  insulation  of  their  envelopes.  At  EU  level,  buildings 
consume  40%  of  energy  and  release  36%  of  related  greenhouse  gas  (GHG)  emissions,  with  most  existing 
buildings  built  before  energy  efficiency  standards  were  established.  The  Energy  Performance  of  Buildings 
Directive (EPBD) fosters requirements to improve the energy performance of both new and existing buildings. 
The 2024 recast EPBD introduces minimum energy performance standards for non-residential buildings and 
mandates progressive renovation pathways for residential buildings, aiming for a zero-emission building stock 
by 2050 (European Commission, 2024). This goal can be particular challenging for older masonry buildings, as 
it may require significant investments in energy efficiency and on-site renewable technologies, which can be 
at risk due to natural hazards like earthquakes. While demolition and new construction might be considered 
sometimes, they are not fully in line with the EU’s emphasis on resource efficiency and sufficiency principles 
(Maduta  et  al.,  2022).  Therefore,  focusing  on  renovation  is  crucial  for  reducing  vulnerability  to  hazards, 
reducing environmental impact, and improving indoor environmental quality in existing buildings. Renovation 
also supports job creation, boosts industry competitiveness, and strengthens energy security (Zangheri et al., 
2020).  

Extensive  research  has  been  carried  out  to  upgrade  URM  structures  in  terms  of  strength  and  deformation 
capacity  using  composites  as  they  cause  minimum  invasion.  Numerous  structures  have  been  efficiently 
strengthened  with  fibre-reinforce  polymers  (FRP),  which  achieve  sufficient  bond  with  masonry  substrates 
through  an  organic,  commonly  epoxy,  agent  e.g.  (Bakis  et  al.,  2002;  Carozzi  et  al.,  2017;  Krevaikas  and 
Triantafillou, 2005; Triantafillou, 1998). Despite the fact that force transfer is probably sufficiently secured if 
surfaces are adequately prepared, during actual application there are a series of disadvantages related to the 
epoxy adhesives: 

—  Irreversibility: polymeric resins cannot be detached without damaging masonry.  

—  Damp incompatibility: it is not possible to apply epoxies on humid surfaces whereas freeze problems may 

appear during application from isolated water.  

—  Lack  of  vapour  permeability:  organic  resins  are  not  vapour  permeable,  hence  in  a  case  of  frescoes 

existing at the opposite side of the walls problems is possible to arise.  

—  Thermal  compatibility  problems:  a  different  thermal  coefficient  with  masonry  substrate  cause  strain 

incompatibility; in addition, it is hard to employ epoxies at low temperatures. 

—  Lack of fire resistance: polymer composites cannot resist high temperatures.  

—  Working and  environmental  hazard: toxicity of epoxy resin can harm workers or, the  environment  when 

uncautiously worked.  

—  High cost: the cost of epoxies is relatively high.  

—  Poor  bond  with  rough  masonry  surfaces:  the  substrate  roughness  and  irregularity  increase  the 
workmanship cost and can cause a premature delamination and reduce the quality of the intervention.  

4 

The  replacement  of  organic  matrices  with  inorganic  ones,  that  is  the  use  of  textile  reinforced  mortar  (TRM) 
instead  of  FRP,  offers  an  effective  solution  to  the  shortcomings  listed  above,  especially  for  heritage 
structures, where the limited invasiveness and compatibility aspects are extremely important. 

TRM  jacketing  has  minimal,  if  any,  thermal  insulation  capacity.  Therefore,  to  tackle  the  problem  of  a 
concurrent seismic plus energy retrofitting, TRM can be combined with thermal insulation to form a structural-
plus-thermal upgrading of existing buildings (Figure 1, (Bournas, 2018)). A wide range of internal and external 
insulations  can  achieve  high  energy  savings  together  with  a  quick  and  easy  application  and  low  cost 
(Anastaselos et al., 2009). Various types of thermal insulation boards can be combined with TRM layers and 
used  externally  and  internally,  for  instance  made  from  mineral  wool  or  expanded  polystyrene/polyurethane. 
The finishing involves placing continuous insulation boards using mineral renders and coatings, although some 
kinds of discontinuous tile finishing could also be applied (Jelle, 2011).  

Figure 1. Seismic plus energy retrofitting system: (a) TRM Jacket + insulation material on a masonry wall; (b) TRM + 
heating system on the masonry infill of RC framed structure. 

(a) 

(b) 

Source: (Bournas, 2018) 

Box 1. Advantage of TRM over FRP for retrofitting masonry buildings 

The  replacement  of  organic  matrices  with  inorganic  ones,  that  is  the  use  of  textile  reinforced  mortar  (TRM) 
instead of FRP, offers an effective solution, especially for heritage structures, where the limited invasiveness 
and compatibility aspects are extremely important. 

Integrated  seismic  and  energy  retrofitting  of  buildings  has  been  increasingly  investigated  in  the  scientific 
literature from both, a theoretical and an experimental, point of view. A recent study provided a state-of-the-
art review and analysis of integrated retrofitting interventions (Pohoryles et al. 2022), dividing them into four 
different  broad  groups:  (1)  integrated  exoskeleton  solutions;  (2)  integrated  interventions  on  the  existing 
building  envelope;  (3)  replacement  of  the  existing  envelope  with  better  performing  materials;  and  (4) 
interventions on horizontal elements.  

Research into exoskeleton solutions includes BRB-braces with solar shading and diagrids with thermal panels. 
Auxiliary structures feature insulated RC walls and RC frames with thermal panels or masonry infills. Shear-
wall  solutions  support  energy  retrofits  with  external  steel  structures,  accommodating  green  facades, 
photovoltaics,  or  shading  devices,  and  can  expand  living  space.  These  solutions  are  highly  invasive, 
significantly altering the building's exterior. For more details, see in Di Lorenzo et al. 2020.  

Instead  of  adding  external  structures,  existing  buildings  can  be  upgraded  by  strengthening  walls  with 
composite materials like TRM and adding thermal insulation (as described in Fig. 1a). TRM-based retrofitting is 
less disruptive, cost-effective, and environmentally friendly. Prefabricated TRM or TRC panels with insulation 
or capillary tubes (Baek et al. 2022, 2024), and Cross-laminated timber (CLT) panels with insulation (Smiroldo 
et al. 2023, Kallioras et al. 2024), offer reduced construction time and modularity. Implementing low-carbon 
retrofits,  such  as  mass  timber,  is  key  to  decarbonizing  buildings  and  achieving  climate  neutrality  by  2050. 
Finally, strengthening URM walls with new windows and structural frames improves both structural integrity 
and thermal performance. 

Another  approach  is  replacing  external  walls  of  masonry-infilled  RC  buildings  with  stronger,  thermally 
insulating  bricks  or  deformable  infill  walls  decoupled  from  the  RC  frame.  Additional  thermal  insulation  is 

5 

 
 
usually  needed,  but  sustainable  or  insulated  bricks  can  enhance  energy  efficiency.  Though  invasive  and 
disruptive, this method is typically low-cost (Pohoryles et al. 2022). 

Finally, a study by Gkatzogias et al. 2022 analyzed the potential impacts of integrated renovation across EU 
regions,  considering  cost-benefit  ratios  and  reductions  in  losses  and  energy  use.  Integrated  renovation 
showed significant economic and safety benefits, reducing both fatalities and energy consumption at EU level. 

The  following  chapters  present  a  a  systematic  investigation  of  the  concurrent  retrofitting  of  masonry 
structures. This includes a review comparing TRM structural properties, model development through analytical 
studies, and a final analysis of loss mitigation benefits across the EU. 

1.2  Context of relevant EU policies 

The ageing European building stock requires remarkable renovation efforts to improve its energy performance 
and ensure structural integrity during extreme loadings and natural disasters such as earthquakes. The impact 
of the existing buildings stock on the environment is a topic of growing importance at international level. At 
EU level, buildings consume about 40% of the final energy consumption and are resposable for about 36% 
related greenhouse gas (GHG) emissions (European Commission, 2023). These large GHG emissions and high 
energy consumption are largerly linked to the old stock of buildings in Europe, with a third of buildings being 
over 50 years (Filippidou and Jimenez Navaro, 2019). Currently, only a small portion of the existing building 
stock is renovated annually, with the rate estimated at 1% in 2020 (European Commission, 2020). To achieve 
the EU’s ambitious goals of a zero-emission building stock and a climate-neutral society by 2050, there is a 
pressing need to significantly accelerate the rate of building renovations. 

Box 2. Boost EU policies implementation 

Integrating  energy  efficiency  upgrading  with  seismic  strengthening  will  incentivise  renovation  further  and 
boost EU policies implementation. 

Renovation  and  retrofit  strategies  are  required  to  upgrade  the  ageing  building  stock  in  the  developed 
countries  of  the  world  since  demolition  and  rebuilding  are  neither  an  economically  viable  nor  an 
environmentally  friendly  solution  at  large  scale.  That  is  why  comprehensive  energy  and  climate  policy 
packages  have  been  introduced  to  progressively  improve  the  energy  performance  of  EU  buildings 
(Economidou et al., 2020). The Green Deal strategy sets the ambitious target of transforming the EU into the 
first climate-neutral continent by 2050, emphasizing the importance of a cross-sectoral approach (European 
Commission,  2019).  The  Renovation  Wave  strategy  aims  to  at  least  double  the  annual  building  renovation 
rate by 2030 and promotes deep renovation. In 2021, the Commission introduced the Fit for 55 package to 
align EU laws with the target of reducing GHG emissions by 55% by 2030, compared to 1990 levels, which is 
a stepping stone toward the 2050 climate neutrality goal (European Parliament, 2021).  

One of the key legislative instruments is the Energy Performance of Buildings Directive (EPBD), the main legal 
instrument for boosting energy efficiency in EU buildings. First introduced in 2002, the EPBD has been revised 
several times to align with the more ambitious EU energy and climate targets. The 2018 amendment placed a 
strong  emphasis  on  an  integrated  renovation  approach,  introducing  strengthened  provisions  to  support 
building renovations (European Commission, 2018). The directive encouraged Member States to adopt highly 
energy-efficient  measures,  while  also  addressing  risks  related  to  seismic  activity,  which  can  affect  energy 
efficiency renovations and the lifespan of buildings. In response, several Member States incorporated seismic 
risk mitigation into their Long-Term Renovation Strategies (LTRSs). An evaluation of the LTRSs by (Castellazzi 
et  al.,  2022)  revealed  that  many  countries  positively  received  the  recommendation  to  integrate  seismic 
strengthening  with  energy  efficiency  upgrades.  Although  not  a  mandatory  requirement,  several  national 
strategies  addressed  this  issue,  with  a  few  countries  providing  specific  measures  (e.g.,  Croatia,  Cyprus, 
Hungary, Italy, Romania, Slovenia, and Spain). 

The  2024  recast  EPBD  introduces  a  formal  definition  for  deep  renovation  and  recognizes  it  as  a  prime 
opportunity  to  increase  building  resilience  against  disaster  risks,  including  earthquakes,  create  better  indoor 
conditions,  and  increase  climate  resilience  besides  reducing  energy  consumption  (European  Commission, 
2024). The importance of this cross-sectoral approach is highlighed by a growing body of literature over the 
past decade, which highlights its potential for delivering holistic solutions to complex challenges (Pohoryles et 
al., 2022, Kakoulaki et al., 2023). This trend is expected to further evolve as the focus on integrated policies 
continues to expand. Further research is  needed to explore how renovation strategies can address safe and 
healthy indoor environments, energy and resource efficiency with a view of minimizing the life-cycle impacts 
of buildings (Maduta et al., 2023).  

6 

2  State-of-the-art on the experimental performance of the textile-

reinforced mortar (TRM) 

2.1  Performance of TRM 

TRM is a composite material consisting of textiles embedded in an inorganic matrix as a replacement of the 
organic  resins.  Commonly,  the  fibres  are  made  of  high  strength  industrial  materials  such  as  glass,  carbon, 
basalt,  aramid,  polypropylene  (PP),  polyparaphenyle  benzobisoxazole  (PBO),  steel  etc.  (Brameshuber,  2016; 
Gries et al., 2016; T.C. Triantafillou, 2016) but also less strong natural materials from plant or animal fibres 
such as abaca, coir, flax, hemp, jute, sisal, hemp etc. (Cevallos et al., 2015; Gkournelos et al., 2022; Habibi et 
al., 2019; Olivito et al., 2016, 2014; Valluzzi et al., 2014), can be applied. The most common matrix is cement-
based, but also hydraulic lime mortars which are more compatible with traditional masonries can be used, e.g. 
(Mechtcherine et al., 2016). Several experimental campaigns have investigated the performance of TRM as a 
strengthening  material.  The  effectiveness  and  appropriateness  of  TRM  as  a  strengthening  and  retrofitting 
material  for  unreinforced  masonry  structures  is  due  to  the  high  strength-to-weight  ratio,  minimal 
invasiveness,  resistance  at  high  temperatures,  applicability  at  low  temperatures,  applicability  on  high 
dampness  and  irregular  surfaces,  vapour  permeability,  easy  and  safe  application  for  workers  and  the 
environment,  chemical  and  mechanical  compatibility  with  various  masonry  substrates,  and  a  high  level  of 
reversibility  which  is  an  important  issue  for  cultural  heritage  structures.  As  it  will  be  shown  below,  the 
effective  combination  of  a  TRM  jacket  with  thermal  insulation  boards  without  side  effects  on  their 
performance  is  a  top  advantage  as  it  achieves  cost  savings  due  to  the  same  nature  of  workmanship 
(scaffolding, preparation of the substrate etc.) and minimum invasiveness during application.  

Generally, the binder system of TRM, being of inorganic and mineral nature, present a looser adhesion to the 
substrate in respect to organic resins of FRPs provided that the textile is properly bonded and depending on 
the  surface  preparation  (Bellini  and  Mazzotti,  2017)  no  matter  what  the  fibre  chemical  and  mechanical 
properties are. The inorganic matrix is a fresh material and cannot achieve the same level of bonding of an 
epoxy.  However,  the  bond  mechanism  between  textile  and  matrix  is  complicated  in  the  case  of  TRMs  and 
presents a pseudo-ductile behaviour due to gradual cracking and textile pull-out phenomena which is actually 
preferable in the case of seismic performance as it can absorb energy.  

The bond at the textile-mortar interface is a macroscopic mechanism which at a micro level is responsible for 
the stress transferred between the filaments within a yarn and between the yarn and the matrix. Simulating 
the bonding mechanism explicitly in a model is a very challenging task (Düreth et al., 2020) since most of the 
interactions  have  a  random  nature.  To  this  end,  in  this  report  an  empirical  expression  is  derived  based  on 
experimental and numerical  data (Kouris et al.,  2023). The key element during the  bond  mechanism is that 
the  inorganic  matrix  is  not  able  to  fully  penetrate  between  the  filaments  that  constitute  the  fabric  yarns, 
because the cement grain dimensions are too large compared to the space among the filaments. Therefore, a 
stochastic penetration appear which leads a different yarn inner and outer bond. The bond failure mechanism 
is finally a “telescopic” one, involving several local failures in the micro level, including slippage and unequal 
stress levels among filaments (Donnini et al., 2016). The outer filaments carry a higher stress level compared 
to  the  inner  filaments  of  the  yarn  and  cause  the  telescopic  failure.  Hence,  the  achieved  ultimate  stress  is 
lower than the nominal value of the material. Impregnation of the yarns into organic resin before embedment 
in  the  inorganic  matrix,  along  with  coating  of  the  rovings  with  sand  grains  has  been  proposed  in  order  to 
smooth the stress variations and improve bonding e.g. (Donnini et al., 2016), albeit at the expense of limiting 
the ability to apply the materials on elements with high curvature. Studies have shown that the more layers 
of textiles the higher the actual bond of the fibers achieved (Askouni and Papanicolaou, 2019; Makashev et 
al., 2023). 

To derive the basic material mechanical properties of TRM, rectangular coupon specimens should be tested is 
tension monotonically, e.g. (ICC Evaluation Service, 2013). A tensile coupon is made in a flat mold by applying 
successive  layers  of  mortar  (3-4  mm  thick)  and  textiles  in  sandwich-type  manner.  A  typical  stress  –  strain 
curve  is  illustrated  in  Figure  2  and  an  idealised  trilinear  form  can  be  defined,  where  E1,  E2  and  E3  are  the 
elastic moduli in phases I, IIa and IIb, respectively e.g. (De Santis et al., 2017; Kouris and Triantafillou, 2018; 
T.C.  Triantafillou,  2016;  Thanasis  C.  Triantafillou,  2016).  The  strains  are  measured  either  by  digital  image 
correlation which can capture a high resolution and nonuniformly distributed strain field (Bilotta et al., 2017) 
or applying conventional measurement systems (i.e. strain gauges and LVDTs).  

7 

 
Figure 2.  Response of TRM in tension: experimental behaviour and trilinear idealisation. 

s
s
e
r
t
s

IIa

I

E1

IIb

E3

strain

2.2 

In-plane performance of the hybrid system 

In-plane  failure  of  masonry  members  is  the  result  of  actions  parallel  to  the  walls.  Typical  seismic  actions 
include bending and shear, with or without axial force. Diagonal in-plane shear failure in masonry is typical in 
shear  wall-type  elements.  On  the  contrary,  a  flexural  failure  appears  in  slender  piers.  Strengthening  can  be 
provided by applying TRM jackets so that a “truss” mechanism may be activated (in analogy to the way shear 
is carried by reinforced concrete members).   

TRM-based strengthening of masonry walls subjected to in-plane cyclic loads has been studied experimentally 
by Papanicolaou et al. (Papanicolaou et al., 2007).  In this study it was shown that TRM overlays provide an 
extremely  high  gain  in  strength  and  deformation  capacity,  comparable -  if  not  higher  - to  that  provided  by 
equivalent  FRP  systems.  Namely,  slender  and  squat  piers  and  spandrel  beams  have  been  tested  and  their 
strength and deformation capacity has been found to increase several times. 

In-plane  diagonal  compression  tests  were  performed  in  various  types  of  specimens,  such  as:  two-leaf  brick 
walls (Benedetti, 2019; Ismail and Ingham, 2016, 2014),  one-leaf brick walls (Babaeidarabad et al.,  2014b; 
Koutas  et  al.,  2014),  tuff  volcanic  masonry  walls  (Del  Zoppo  et  al.,  2019;  Marcari  et  al.,  2017;  Prota  et  al., 
2006), brick masonry walls (Sagar et al., 2017). In-plane tests on cantilever piers and spandrel beams have 
also been performed by Papanicolaou et al. (Papanicolaou et al., 2007), who tested one-leaf brickwork walls 
with various aspect ratios and axial load levels. The structural integrity during loading was achieved with TRM. 
The  increase  of  the  capacity  was  substantial  in  all  the  cases  which  reached  even  more  than  100%  of  the 
unreinforced  capacity  limited  by  the  compressive  strength  of  masonry.  Double-sided  jackets  offer  a  higher 
strength increase albeit at the expense of a lower ductility (Del Zoppo et al., 2019).  

Moreover,  pier-spandrel  assemblages  have  been  tested  to  study  the  critical  interaction  between  reinforced 
spandrels  and  unreinforced  piers  (Augenti  et  al.,  2011;  Ismail  and  Ingham,  2016;  Parisi  et  al.,  2011).  In  all 
cases,  the  improved  behaviour  of  URM  with  TRM  in  terms  of  strength  and  deformation  capacity  has  been 
demonstrated. Moreover, the better performance of TRM versus FRP with regards to deformation capacity has 
been  reported  (Papanicolaou  et  al.,  2011).  In  cyclic  loading  the  improved  deformation  capacity  was 
accompanied by reduced strength degradation and high energy dissipation (Augenti et al., 2011). 

8 

 
Box 3. TRM strengthened masonry elements 

In  all  the  tests  carried  out  on  masonry  elements  strengthened  with  TRM  and  loaded  in-plane,  an  increased 
strength and deformation capacity was noticed with improved performance in terms of strength degradation 
which can add significant ductility and energy dissipation during seismic oscillations. 

The application of lime mortars which present a higher compatibility with old masonry walls has been tested 
and the strength capacity was increased twice as much (Benedetti, 2019). The external layer in this case is 
thicker than the one made of cementitious matrices. 

The  hybrid  thermo-structural  retrofitting  has  also  been  tested  in  in-plane  cyclic  loads  (Baek  et  al.,  2022; 
Gkournelos et al., 2020; Triantafillou et al., 2018). The position of the TRM layer  has been investigated and 
interestingly, it was found that even if the thermal insulation is attached on the wall in between an external 
TRM layer, the capacity of the wall maintained similar strengths when a careful bonding of the various layers 
is  realised.  In  this  direction,  the  use  of  lightweight  mortars  with  a  higher  thermal  resistance  can  contribute 
also to the insulation of the wall and use of thinner thermal boards (Longo et al., 2020) 

2.3  Out-of-plane performance of the hybrid system 

TRM-based strengthening of masonry walls subjected to out-of-plane cyclic loads showed an extremely high 
gain  in  strength  and  deformation  capacity,  comparable  -  if  not  higher  - to  that  provided  by  equivalent  FRP 
systems  (Harajli  et  al.,  2010;  Papanicolaou  et  al.,  2008).  The  observed  failure  is  due  to  either  masonry 
cracking or textile failure (usually debonding or tensile breakage). 

Out-of-plane  four-point  bending  tests  were  carried  out  to  investigate  the  improvement  of  the  masonry 
performance  in  terms  of  strength  and  deformation  capacity,  due  to  the  application  of  this  strengthening 
technique  (Gattesco  and  Boem,  2017).  Three  point  bending  (Kariou  et  al.,  2018;  Papanicolaou  et  al.,  2008; 
Shermi and Dubey, 2017) or four point bending (Babaeidarabad et al., 2014a; Harajli et al., 2010; Kadam et 
al., 2015; Martins et al., 2015; Padalu et al., 2019; Sagar et al., 2017; Valluzzi et al., 2014) tests have been 
performed  in  various  types  of  masonry  walls  made  of  natural,  concrete  or  brick  units.  In  all  the  cases,  the 
capacity of the strengthened wall was several times larger than that of the URM wall. 

A  test  campaign  investigating  plenty  of  parameters  including  the  thickness  of  the  masonry  wall,  three 
different textile materials, various number of layers, the coating of the textiles etc. carried out by (Kariou et 
al., 2018) and it is used in the next chapters as a benchmark to calibrate analytical models. The test campaign 
involved  eighteen  specimens  in  three-point  bending  including  two  control  specimens  (unstrengthened).  The 
load was increasing monotonically at the mid-span of the masonry piers which had 18 courses of bricks units 
in a stretcher bond. Mortar joints were approximately 10 mm thick, and the specimens’ dimensions were 1.34 
m  long  ×  0.44  m  wide.  Wall  thickness  was  a  parameter  under  investigation  and  therefore,  the  eight  single 
wythe  piers  (S)  were  0.1025  m  thick,  while  the  eight  double  wythe  piers  (D)  were  0.215  m  thick  as  they 
include one head mortar joint. Three different fibres were used: (i) carbon (C), (ii) glass (G) and (iii) basalt (B). 
In  this  study,  only  the  carbon  textile  is  investigated.  The  basic  material  properties  details  are  presented  in 
Table 1. Details about the epoxy resin and the impregnation process can be found in (Kariou et al., 2018). The 
thickness of the coat is 3 mm for the one-layer carbon textile. 

Shaking table tests of a full scale U-formed specimen, sub-assemblage of a real structure, were performed to 
study the out-of-plane high efficiency of a TRM system comprising of steel cords embedded in lime mortar 
(De Santis et al., 2016). 

The  hybrid  system  of  the  simultaneous  retrofitting  has  been  also  tested  in  out-of-plane  loads  where  the 
thermal insulation has been  either attached on the layer  of the  structural reinforcement or  placed  between 
the wall and the TRM (Triantafillou et al., 2017). In the former case the insulation can protect TRM from fire 
(Triantafillou  et  al.,  2017)  whereas  in  the  latter  case  there  is  a  strength  and  deformation  gain  due  to  an 
increased lever arm provided that the bonding is sufficiently ensured (Gkournelos et al., 2020). However, in-
plane cracking of the wall can reduce the capacity of the out-of-plane performance by 10% on in comparison 
to an intact member (Gkournelos et al., 2020). In such a case it was found that the hybrid system would not 
have the latter extra gain in respect to the merely TRM reinforced wall despite the increased lever arm due to 
a premature shear and debonding failure. 

9 

 
Table 1. Material properties: elastic modulus [GPa] and strength [MPa]. 

Material 

Masonry 

Reinforcing mortar (matrix) 

Carbon 

Glass 

Basalt (coated) 

Source: (Kariou et al., 2018). 

Strength [MPa] 

Elastic modulus [GPa] 

9.7 

6.0 

3800 

1400 

1351 

2.5 

0.8 

225 

74 

89 

Box 4. The thermo-structural hybrid retrofitting of masonry elements 

The  hybrid  system  of  the  simultaneous  retrofitting  apart  from  a  cost-effective  solution  which  tackles  two 
problems  in  one  solution,  offers  a  thermal  protection  to  the  TRM  layer(s)  in  elevated  temperatures  and  no 
matter the position of the layers the strength and deformation capacity is at least the same if not higher in 
respect to a wall with only structural reinforcement. 

As it can be deduced from the previous, the simulation of masonry walls strengthened with jackets of TRM is 
a  challenging  task  due  to  complex  interactions  of  the  various  materials.  A  first  simplification  is  the 
homogenisation of the properties of mortar joints and brick units which is described in the next chapter. 

10 

3  Homogenisation of masonry properties 

3.1  Overview of homogenisation  

The  mechanical  properties  of  vernacular  masonry  structures  present  a  large  discrepancy  depending  on  the 
variety  of  materials,  bonding  properties,  geometry  of  the  layout,  and  other  arrangements  related  to  the 
craftsmanship  of  masons.  Masonry  material  is,  in  fact,  a  multiphase  continuum  comprising  variations, 
discontinuities,  cracks  and  interfaces.  In  view  of  the  real  nature  of  the  material  during  service,  a  detailed 
discrete micro-model should consider the exact arrangement of masonry units and mortar joints along with 
their  distinct  properties  and  those  of  the  interfaces  (Figure  3a).  Due  to  the  inherent  complexity  of  such  an 
approach its use is only limited to element-wise (local) analysis and they are neither feasible, nor applicable 
especially  for  the  design  of  buildings  (Lagomarsino  et  al.,  2013)  or,  building  aggregates  (Formisano  et  al., 
2015;  Maio  et  al.,  2015)  since  the  additional  computational  effort  is  very  high  and  the  exact  units  and 
mortars geometry difficult to be accurately defined and meshed.  

Therefore, the commonly applied design methods are based on homogenised macro-mechanical properties of 
masonry  utilising either shell or beam elements i.e. the well-known equivalent frame  model to describe the 
global behaviour of masonry structures (Belmouden and Lestuzzi, 2009; Chen et al., 2008; Kouris et al., 2014; 
Kouris and Kappos, 2012; Lagomarsino et al., 2013; Penelis, 2006; Penna et al., 2014a).  

In the first place, the elastic moduli are needed for modelling masonry walls. The current codes however do 
not  include  physics  or  theoretical  estimates.  Instead,  estimates  of  the  elastic  moduli  of  masonry  walls  are 
used  based  on  either  wallette  experiments  or,  on  empirical  formulas  involving  the  wallette  compressive 
strength  which  in  turn  is  also  given  by  an  empirical  formula.  Such  an  approach  typically  results  in  large 
discrepancies (Tomazevic, 1999). Clearly the consideration of a perfectly homogeneous and isotropic material 
with a single adjustment of the elastic modulus is far from being a realistic assumption. To this end, a new 
homogenization  model  was  deemed  necessary  and  developed  here  with  a  dual  aim:  (i)  to  achieve  a  higher 
accuracy in respect to the existing models, but also (ii) to be as practical and easy-to-apply as possible. For a 
more detailed description the reader is referred to (L. A. S. Kouris et al., 2020). 

Box 5. Homogenisation model for masonry walls 

The  homogenised  properties  are  commonly  used  in  the  simulation  of  masonry  buildings.  A  new 
homogenization method is developed here aiming to achieve an accurate and a practical and easy-to-apply 
model. 

Traditional  masonry  comprises  brick  or  stone  units  bound  together  with  mortar.  When  a  masonry  wall  is 
loaded can fail in either constituent material or their interfaces. Therefore, a meticulous modelling approach 
of  masonry  is  a  distinct  model  of  both  brick  units  and  mortar  joints  and  all  their  interfaces.  A  first 
simplification can be to homogenise the mortar joints with the brick units while keeping their interaction. This 
proposed approach is a balance between a simplified smeared cracked approach with a continuum model and 
on  the  other  hand,  a  detailed  model  with  several  interaction  interfaces,  thus,  keeping  the  model  affordable 
numerically.  

In this modelling approach a representative volume element (RVE) is considered (Figure 3), i.e. the brick unit 
together with half mortar joint around it. The NL behaviour is attributed to the stiffness degradation based on 
two material models: (i) plasticity and, (ii) damage. The former is related to the post-yield response up to the 
maximum capacity and the latter to the material softening. 

11 

 
 
 
Figure 3. The homogenised modelling approach for masonry using the representative volume element; in light blue the 
interaction interfaces. 

R V E

RV E

Source: (L. A. S. Kouris et al., 2020). 

3.2  Gradient elastic theory 

Under  the  assumption  of  periodicity,  the  RVE  elastic  properties  can  be  determined  experimentally  from 
masonry  prisms  or,  from  the  properties  of  the  constituents  and  the  application  of  a  homogenisation  rule 
which ideally should consider the orthotropic elastic properties (L. A. S. Kouris et al., 2020). Classical elasticity 
theory is extensively used in a wide range of design problems and applications in civil engineering. However, it 
fails  to  represent  local  heterogeneity  phenomena.  The  incorporation  into  the  classical  linear  elastic 
constitutive  relation  of  strain  εkl  and  stress  σij  of  the  contimuum  media  of  the  Laplacian  of  the  strain  can 
remedy this shortcoming of the elastic theory (Aifantis, 2003, 1992). Assuming a general elastic material the 
enhanced constitutive elasticity law between the stress σij and the strain εkl tensors becomes as follows: 

σ
ij

=

C

ijkl

(
ε
kl

2
− ∇
l

2
ε
kl

)

(1) 

where Cijkl is the material elastic moduli fourth order tensor and ℓ is the internal non-local length. The larger 
the ℓ is the wider is the area affected by the local developed strain gradients.  

Next, an axial single degree of freedom (SDOF) gradient elastic spring is assumed with cross-section area Aξ 
and length Lξ along the longitudinal axis ξ (Figure 4). 

The considered element is assumed fixed at the one side (ξ=0) and free at the end (ξ=Lξ) where the axial load 
Nξ is applied. 

12 

 
 
 
Figure 4. Axial 1-D gradient elastic element. 

E, Aξ , l

Νξ

Lξ

Source: (L. A. S. Kouris et al., 2020). 

Kinematic  consistent  boundary  conditions  (Eqs.  (2))  are  considered  to  estimate  the  constants;  one  classical 
condition  for  the  displacement  at  the  fixed  end;  two  non-classical  ones  for  the  second  derivative  of  the 
displacement at the fixed end and the other based on its vanishing of the derivative at the free end (Akintayo 
et al., 2012; Askes et al., 2008; Askes and Aifantis, 2011; Mousavi and Aifantis, 2016; Polizzotto, 2018, 2003; 
Ru and Aifantis, 1993; Tsepoura et al., 2002): 

u

0

=

0;

2
d u
0
2
ξ
d

=

0;

ξ

=Ldu
ξ
d

0

(2a,b,c) 

By  substituting  Eqs  (2)  in  the  constitutive  Eq  (1)  and  solving  the  third  order  ordinary  differential  equation 
(ODE) the displacement and axial force at the free end (ξ=Lξ) are expressed as: 

u
ξ

=

N L
ξ ξ
EA
ξ


−
1





L
ξ

tanh





L
ξ










−
1

Ν =
ξ

EA
ξ

L
ξ

−


1






L
ξ

tanh





L
ξ






Thus, the axial stiffness k of the gradient elastic spring is given: 






(3a,b) 

⋅

u
ξ

k
ξ

=

iEA
*
L
ξ

*
L
ξ

=

L
ξ


−
1





L
ξ

tanh

L
ξ















(4a,b) 

* is the effective ‘gradient’ quantity of the length Lξ of the spring given by Eq. (4b). Obviously, the 
In Eq. (4a) Lξ
classical elastic spring and the gradient one differ only in the estimation of the spring length, i.e. the former’s 
*  is  given  by  a  simple  expression 
length  L  is  the  actual  length  of  the  spring,  while  the  effective  length  Lξ
involving the ratio of the internal length ℓ to the geometric length Lξ. 

Following the same procedure, a gradient elastic shear spring can be established starting from Eqs. (1) and 
(2) and considering only shear stresses and strains for a shear element with dimensions L×H, with y being the 

13 

 
 
 
 
 
 
 
vertical  axis.  Denoting  τ  the  shear  stress,  γ  the  shear  strain,  As  the  shear  cross  section  and  G  the  shear 
modulus, the respective equations for the gradient elastic shear strain are: 

τ

=

G

k

s

=


γ
−

GA
s
*
H

2



2
γ
2

d
dy





(5a,b,c) 

*

H

=

H


1



−



tanh





H










where the shear strain is as usual equal to γ = du/dy. 

3.3  Gradient homogenisation model (GREHM) 

GREHM is developed for a  running or a stack  bond masonry but the methodology can be easily adapted to 
other  masonry  bonds.  The  representative  element  of  the  periodic  masonry  wall,  which  is  the  smallest 
repetitive  cell,  for  these  two  bonds  consists  of  one  brick  unit  bounded  by  half  thickness  mortar  joints  as 
shown in Figure 5. This element is repeated throughout a masonry wall excluding boundaries.  

Figure 5. The representative surface element (RSE) of running bond brickwork: (a) on the wall, (b) a close-up. 

y
,
m

t

+
y
,
b
t

Y

y
,
m
t

y
,

b
t

2
/
y
,
m
t

T2

R S E

tb,x + tm,x

T1

R S E

(a) 

(b) 

X

tm,x/2

tb,x

tm,x

Source: (L. A. S. Kouris et al., 2020). 

The periods of the spatial variation of the medium T in the 2D dimensional space are equal to the respective 
dimensions of the  representative surface element (RSE),  namely the length and  height of the brick and the 
mortar  joint,  respectively,  i.e.  T  =  [(tm,x  +  tb,x)  (tm,y  +  tb,y)]T  (see  also  Figure  5).  The  geometric  and  mechanical 
properties  of  masonry  represented  by  the  function  Ƒ  fulfil  the  regular  periodicity  rule  for  composite  media 
(Hassani and Hinton, 1998): 

14 

 
 
 
 
F

+
x N




( )
b

⋅

T

=




F

( )
x

where

x

=





x
i
y
i





and

N

( )
b

=












1

2
0

0

1






,

b

=

1

1 0

0 1





,

b

=

2

(6) 

In Eq. (6) x is the coordinates vector of a certain i point, T is the period vector and N matrix depends on the 
bond b ={1 for running bond, 2 for stack bond}. It should be noted that the geometry of the RSE presents a 
double symmetry in the horizontal and vertical axes. 

On  the  boundary  of  the  RSE  a  uniform  stress  distribution  is  assumed  which  is  valid  for  gravity  loads.  For 
dynamic loads the spatial stress  variation is assumed  very small and neglected in the RSE as normally the 
vector T will be very small to the total dimensions of the wall. For the elastic response of the RSE the mortar 
joints are considered fully interconnected to the brick unit and no sliding is taken into account. Moreover, the 
thickness  of  the  wall  is  assumed  very  thin  compared  to  the  other  dimensions  and  the  wall  as  a  plane  2-D 
element. In the following elastic analysis the assumption that plane sections remain plane after deformation 
is also made and the Poisson effect is neglected. Based on gradient elasticity and these assumptions, the RSE 
is equivalent to a set of springs for the constitutive components respectively; brick units, mortar head and bed 
joints. The dimensions of the RSE are shown in Figure 5b. The thickness of mortar joints is denoted as tm,x for 
head  joints  and  tm,y  for  bed  joints.  The  brick  units  have  dimensions  tb,x  and  tb,y  in  the  horizontal  and  vertical 
direction,  respectively.  A  raised  asterisk  (∙)*  is  used  in  the  following  to  denote  the  gradient  quantity  of  the 
respective dimension according to Eq. (4b). 

The elastic modulus of mortar is denoted as Em whereas hollow brick units generally present an orthotropic 
behaviour  having  two  elastic  moduli  Eb,x  and  Eb,y  in  the  two  directions  (solid  units  are  isotropic).  The  longer 
dimension is typically on the horizontal x axis. The Poisson’s ratios are for mortar vm and for the orthotropic 
bricks vb,xy = vb,yxEb,x/Eb,y. The gradient elastic stiffness of the axial and shear springs representing the masonry 
elements is estimated according to Eqs. (4a) and (5a). The same notation applies to the horizontal and vertical 
stiffnesses of mortar km,x|y and bricks kb,x|y, respectively. 

3.3.1  Homogenised  masonry  elastic  modulus  Ev  perpendicular  to  mortar  bed  joints 

(vertical) 

For the estimation of the homogenised elastic modulus Ev in the vertical direction perpendicular to mortar bed 
joints a  simplification on RSE (Figure 5) is possible: it is  sufficient to  consider only the bed joints assuming 
that  the  load  is  uniformly  distributed  and  tm,x<  tb,x/10.  The  homogenised  modulus  of  elasticity  Ev  after 
substitution of the spring stiffnesses in Eq. (4) and some algebraic calculation can be found: 

E
v

=

t
tot y
,

t

m,

y

2





2

E E
,
b y m
*




E

b y
,

+

*
t E
y m
b,

(7) 

where ttot,y = tb,y + tm,y is the total thickness of the representative volume in the respective direction.  

3.3.2  Homogenised  masonry  elastic  modulus  Eh  parallel  to  mortar  bed 

joints 

(horizontal) 

For the estimation of the  homogenised elastic modulus Eh in the  horizontal direction  parallel to mortar bed 
joints the full RSE (Figure 5) should be taken into consideration as the omission of the latter results in larger 
difference. Therefore, a uniform stress is applied both on the mortar head and bed joints are loaded. Due to 
strain compatibility and the interaction between mortar bed joints and the brick units the stress field becomes 
more  complicated.  A  simplification  is  possible  due  to  the  small  thickness  of  the  mortar  joints:  the  stress 
distribution in the bed joints can be assumed uniform. The closer to unity is the ratio Eb/Em and the smaller the 

15 

 
 
 
 
mortar  thickness  the  more  accurate  is  this  assumption.  By  analogy  with  Eq.  (7)  the  homogenised  elastic 
modulus in the horizontal direction Eh will be: 

E

h

=

t

tot x
,

E E
c m
*




E
c

t

m x
,

2





2

+

*
t E
x m
b,

(8) 

where ttot,x = tb,x + tm,x is the total thickness of the representative volume in the horizontal direction. 

3.3.3  Homogenised masonry shear elastic modulus G 

The shear elastic modulus of the homogenised masonry is estimated using the gradient elasticity law of Eq. 
(5a).  When  the  simplified  RSE  (i.e.  without  the  head  joints,  see  Figure  5)  is  loaded  with  a  horizontal  shear 
force then in all the layers the same stress τ is developed. Obviously, the total shear strain of the RSE is equal 
to the addition of the three elements. It is noted that the shear springs are in row loaded by the same shear 
force. As a result, Eq. (7) are here transformed as follows: 

=
G t

tot y
,

t

m y
,

2





2

G G
b m
*




*
+
G t G
y m
b,

b

(9) 

where Gb and Gm are the shear elastic moduli of the brick unit and the mortar joints respectively. 

3.3.4  Homogenised masonry Poisson’s ratios vxy and vyx 

The  Poisson’s  ratio  vyx  is  defined  as  the  ratio  of  the  horizontal  dilatation  to  the  vertical  deformation  for  a 
vertical stress σyy of the RSE (Figure 5) and can be expressed as: 

v

yx

σ
= − x
σ
y

E
v
E

h

(10) 

The  sign  at  the  right-hand  side  of  Eq.  (10)  is  minus  because  the  developed  stresses  for  classical  materials 
have  opposite  signs.  The  horizontal  Eh  and  the  vertical  Ev  elastic  moduli  are  given  by  Eqs.  (8)  and  (7), 
respectively, whereas the vertical stress is σy = Fy /Ay = Fy /tb,x and the horizontal stress σx = Fx /Ax = Fx /(tb,y + 
tm,y). The vertical springs are in series and thus, they carry an equal force: Fy = Fby = Fmy. In the case that the 
brick  unit  is  a  hollow  brick  with  an  orthotropic  behaviour  it  will  have  two  Poisson’s  ratios  vb,xy  and  vb,yx. 
Therefore, combining the latter with Eqs. (7), (8) and (10) it is concluded that: 

*
v t
b b y
,

+

2

v
m

t

m,

y





*
t
b
,x

*
v t
b b y
,

+

v
2
m

t

m,

y





*




*




2

2

*
t
b
,x

v

yx

=

v

xy

=

t
b x
,
+
t

t

b y
,

m y
,

E
v
E

h

t
b x
,
+
t

t

b y
,

m y
,

(11a,b) 

16 

 
 
 
 
 
 
 
 
3.3.5  Normalised graphs for the estimation of the homogenised properties  

The  influence  of  the  most  important  parameters  on  the  orthotropic  behaviour  of  masonry  is  investigated 
herein; these are (see Figure 5): (i) the internal length which is an internal non-local parameter of the model, 
(ii) the mortar joints and especially their thickness and, (iii) the comparative stiffness of the brick units to the 
one of the mortar which are the main physical parameters. Normalised plots for the elastic moduli are then 
provided. 

—  The thicker the joints are, the more affected is the macro-behaviour of masonry.  

—  The level of the influence depends on the mortar to brick modulus of elasticity ratio, Em/Eb. 

—  The internal length is to limit this role. For example, a value of internal length ℓ equal to the mortar joint 

thickness leads to a much lower impact of the ratio Em/Eb on the homogenised elastic moduli. 

The  influence  of  mortar  joints  in  the  vertical  homogenised  modulus  is  presented  in  Figure  6  where  the 
thickness is represented in the horizontal axis, the ratio of Em/Eb in various lines (see legend of Figure 6), and 
the internal length in successive plots. The normalised plots of the homogenised horizontal elastic modulus Eh 
and the homogenised shear modulus G are presented in Figure 7 and Figure 8 respectively. It is interesting to 
note that a soft mortar joint (Em/Eb = 0.25) will affect the vertical elastic modulus up to a thickness tmy = tby. 
Higher  values  of  tmy  will  not  have  a  significant  impact  on  the  mechanical  properties  which  reach  a 
stabilisation.  This  stabilisation  thickness  increases  as  the  ratio  Em/Eb  increases.  Certainly,  the  ratio  of  Em/Eb 
most predominantly rules the homogenised vertical modulus.  

Figure 6. The (normalised) vertical elastic modulus Ev of masonry in respect to tmy / tby for various ratios of Em/Eb and 
ℓ/tmy. 

b

E

/

v

E

b

E

/

v

E

   l

 / t

m y

 =0.1

   l

 / t

m y

 =0.2

1.3

1.2

1.1

1

0.9

0.8

0.7

0.6

0.5

1.4

1.3

1.2

1.1

1

0.9

0.8

0.7

0.6

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

2

1.8

1.6

1.4

1.2

1
0.05

t

m y

 / t

b y

   l

 / t

m y

 =0.5

t

m y

 / t

b y

   l

 / t

m y

 =1

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

m y

 / t

b y

t

m y

 / t

b y

E

E

m

m

 / E

 / E

b

b

 =0.25

 =0.5

E

E

m

m

 / E

 / E

b

b

 =0.75

 =1

E

E

m

m

 / E

 / E

b

b

 =1.25

 =1.5

Source: (L. A. S. Kouris et al., 2020). 

3.5

3

2.5

2

1.5

1
0.55

b

E

/

v

E

b

E

/

v

E

17 

 
 
 
   
 
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
Figure 7. The (normalised) horizontal elastic modulus Eh of masonry in respect to tmy / tby for various ratios of 
Em/Eb and ℓ/tmy. 

   l

 / t

m y

 =0.1

   l

 / t

m y

 =0.2

b

E

/

h

E

1.3

1.2

1.1

1

0.9

0.8

0.7

0.6

1.4

1.3

1.2

1.1

1

0.9

0.8

0.7

0.6

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

t

m y

 / t

b y

   l

 / t

m y

 =0.5

t

m y

 / t

b y

   l

 / t

m y

 =1

1.5

1.4

1.3

b

1.2

E

/

h

E

1.1

1

0.9

0.8

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1

0.9

b

E

/

h

E

b

E

/

h

E

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

t

m y

 / t

b y

t

m y

 / t

b y

E

E

m

m

 / E

 / E

b

b

 =0.25

 =0.5

E

E

m

m

 / E

 / E

b

b

 =0.75

 =1

E

E

m

m

 / E

 / E

b

b

 =1.25

 =1.5

Source: (L. A. S. Kouris et al., 2020). 

Figure 8. The (normalised) shear elastic modulus G of masonry in respect to tmy / tby for various ratios of Em/Eb and ℓ/tmy. 

   l

 / t

m y

 =0.1

   l

 / t

m y

 =0.2

0.55

0.5

0.45

0.4

b

E

/

G

0.35

0.3

0.25

0.2

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

t

m y

 / t

b y

   l

 / t

m y

 =0.5

t

m y

 / t

b y

   l

 / t

m y

 =1

b

E

/

G

0.8

0.7

0.6

0.5

0.4

1.4

1.2

1

0.8

0.6

0.4

b

E

/

G

b

E

/

G

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

t

m y

 / t

b y

t

m y

 / t

b y

E

E

m

m

 / E

 / E

b

b

 =0.25

 =0.5

E

E

m

m

 / E

 / E

b

b

 =0.75

 =1

E

E

m

m

 / E

 / E

b

b

 =1.25

 =1.5

18 

 
 
   
 
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
   
 
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
Figure 9. The Poisson’s ratio vyx of masonry in respect to tmy / tby for various ratios of Em/Eb and ℓ/tmy. 

   l

 / t

m y

 =0.1

   l

 / t

m y

 =0.2

0.2

0.18

x
y

0.16

v
o
i
t
a
r

n
o
s
s
o
P

i

0.14

0.12

0.1

0.2

0.18

0.16

0.14

0.12

0.1

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

t

m y

 / t

b y

   l

 / t

m y

 =0.5

t

m y

 / t

b y

   l

 / t

m y

 =1

0.2

0.18

x
y

0.16

v

o
i
t
a
r

n
o
s
s
o
P

i

0.14

0.12

0.1

0.2

0.18

0.16

0.14

0.12

0.1

x
y

v
o
i
t
a
r

n
o
s
s
o
P

i

x
y

v

o
i
t
a
r

n
o
s
s
o
P

i

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

t

m y

 / t

b y

t

m y

 / t

b y

E

E

m

m

 / E

 / E

b

b

 =0.25

 =0.5

E

E

m

m

 / E

 / E

b

b

 =0.75

 =1

E

E

m

m

 / E

 / E

b

b

 =1.25

 =1.5

Source: (L. A. S. Kouris et al., 2020). 

Moreover, standard Poisson’s ratios were assumed vb = 0.2 for bricks and vm = 0.15 for mortar and a ratio of 
the  brick  dimensions  tbx/tby  =  3.25  to  plot  the  normalised  graphs  (which  are  the  only  fixed  values  that  are 
included). The homogenised Poisson ratio νyx is presented in Figure 9 for various joint thicknesses and various 
internal lengths. It is noted that for the above assumed values of Poisson ratios for the bricks and mortar the 
typical value 0.25 of the codes is here the upper bound limit. An increasing trend is observed for the Poisson 
ratio as the internal length increases. This increasing trend is stronger when the ratio of mechanical properties 
of the two materials (Em/Eb) is lower and for smaller thicknesses of the joints (which is the common case). The 
homogenised  Poisson’s  ratio  νxy  can  then  be  easily  estimated  using  the  formula  νxy  =  νyx∙Eh/Ev  (Equation 
(11a,b)). 

3.4  Verification of the proposed homogenisation model 

3.4.1  Comparison with numerical homogenisation models 

The proposed model is compared with a finite-element (FE) based homogenisation model (Anthoine, 1995) to 
verify the  proposed equations and  confirm the assumptions of the model. Apparently, the internal length in 
the FE  comparisons is not  playing a major role as the  models are  perfectly bonded and linear elastic apart 
from one case where the load creates a local singularity. The actual advantage of GREHM with the inclusion 
of an internal length will be shown in the next paragraph where the model is compared to experiments. 

The  wall  used  as  a  benchmark  problem  represents  a  specimen  tested  at  the  University  of  Patras,  Greece 
(Papanicolaou et al., 2011). The wall is 1.17 m long, 1.20 m high and 0.095 m thick with an aspect ratio close 
to unity. The in-plane dimensions of the homogeneous solid bricks are 0.195×0.060 m (tb,x×tb,y) and the wall is 
single-leaf (i.e. the bricks are 0.095 m thick). The mortar thickness varies slightly in the two directions: tm,x = 
0.0195 m and tm,y = 0.0160 m. The mechanical properties of the constituent materials needed for GREHM are 
summarised  in  Table  2.  The  shear  moduli  of  brick  units  and  mortar  can  be  found  610  MPa  and  290  MPa 
respectively. 

19 

 
   
 
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
 
 
 
   
 
   
 
   
 
   
 
   
 
   
 
Table 2. Properties of brick units and mortar joints for GREHM. 

Size (tx×ty) 

Elastic modulus (E) 

Poisson ratio (v) 

[m] 

[GPa] 

Brick units 

0.195×0.060 

1.4 

Mortar joints 

0.0195×0.0160 

0.7 

- 

0.15 

0.20 

The application of GREHM is carried out as follows: (i) for the estimation of the homogenised elastic moduli in 
the  vertical  Ev  and  the  horizontal  Eh  directions  Equations  (7)  and  (8)  are  applied  respectively;  (ii)  for  the 
estimation of the homogenised shear modulus G Equation (9b) is applied; (iii) finally for the estimation of the 
Poisson’s  ratio  νyx  and  νxy  Equations  (11a,b)  are  applied.  As  already  mentioned  in  these  equations  the 
‘gradient’ dimension (∙)* is given by Equation (4b). A number of internal lengths are assumed ranging from 0 
to  tmy.  The  application  of  these  equations  and  the  influence  of  the  internal  length  is  depicted  in  Figure  10. 
Obviously, it is not a linear influence, with an increasing trend for the elastic moduli and a decreasing one for 
the Poisson ratio. 

Figure 10. The influence of the internal length normalised against the thickness of mortar joints on (a) the horizontal and 
vertical elastic moduli and (b) the shear modulus (normalised against the brick elastic modulus) and, (c) the Poisson’s ratio 
of the homogenised model. 

1.6

1.5

1.4

1.3

1.2

1.1

1

0.9

0.8

]

%

[

b

E

/

v

l

E
s
u
u
d
o
m

l

a
c
i
t
r
e
V

1.2

1.15

1.1

1.05

1

0.95

0.9

0.85

]

%

[

b

E

/

h

l

E
s
u
u
d
o
m

l

a
t
n
o
z
i
r
o
H

E

E

v

h

0

0.2

0.4
  l

 / t

0.6

0.8

0.8

1

m y  [%]

y x

x y

(a) 

(b) 

(c) 

Source: (L. A. S. Kouris et al., 2020). 

The estimation of the elastic properties of the considered wall is then, carried out using FE models (Anthoine, 
1995) and compared with the previous results of the homogenised properties. In all the analyses the CPS4R 
four  node  plane  stress  element  is  used.  The  FEs  representing  the  mortar  joints  and  the  bricks  are  fully 
connected between them (using nodal constraints) not allowing for any sliding at the interfaces. To this end a 
RSE with a rectangular pattern for T (see also Figure 5) is selected and is presented in Figure 11 where red 
are  the  elements  of  brick  units  and  orange  those  of  the  mortar.  The  mesh  of  the  RSE  is  performed  using 
rectangle  (close  to  square)  elements  of  size  8×8  mm  approximately.  The  selection  of  this  RSE  essentially 
simplifies the periodic conditions on the boundaries (PCBs) around the RSE to the following ones: 

20 

 
 
 
 
 
 
 
 
 
 
  
 
   
 
 
 
 
  
 
   
 
 
 
 
 
 
 
δ

x right

,

−

δ

δ

δ

,

y bottom
−

y right

,

δ

δ

x bottom

,

−

x left
,
δ

−

y top
,

R

= ∆ ∈
x
= ∆ ∈
y

y left
,
δ

x top
,

= ∆ ∈
y
= ∆ ∈
x

&

δ

=

0

x right
,
δ

,

y bottom
=

y right
,
δ

x bottom

,

R

&

R

&

δ

R

&

=

0

0

=

0

(12a,b,c,d) 

where δx, δx are the displacements in x and y axis at the right, left, bottom and top boundaries of the RSE. The 
displacements  Δx  =  0.01  mm  and  Δy  =  0.01  mm  are  applied  to  estimate  the  horizontal  and  vertical  elastic 
modulus Eh and Ev  and the Poisson’s ratio νxy and νyx respectively using the PBCs given in Equations (12a,b). 
For the shear homogenised modulus G the PBCs expressed in Equations (12c,d) on the boundaries are applied 
with a horizontal displacement Δx = 0.01 mm. The deformations and the strain and stress states for this latter 
case are presented in Figure 11. The results in terms of reaction forces (ΣF), stresses (σ, τ) and strains (ε, γ) of 
the RSE for all the loading cases and the respective homogenised resulted orthotropic moduli of Eh, Ev and G 
and Poisson ratios vxy, vyx are summarised in Table 3. 

Figure 11. The FE model of the RSE: (a) model, (b) shear displacements, (c) shear strain, and (d) shear stress. 

(a)                                                                                                             

(b) 

 (c) 

(d) 

Source: (L. A. S. Kouris et al., 2020). 

21 

 
 
 
 
 
 
 
 
 
Table 3. Results from the FE analysis of the RSE and homogenised mechanical properties. 

Results  &  Homogenised 
properties 

Units 

Load direction 

Horizontal 

Vertical 

Shear 

A
E
F
m
o
r
f

s
t
l
u
s
e
R

ΣF 

ε | γ 

ε 

(transversal) 

σ | τ 

  Ε | G 

d
e
s
i
n
e
g
o
m
o
H

s
e
i
t
r
e
p
o
r
p

(Ε | G)/Eb 

v 

kN 

-787.34 

1499.59 

628.61 

- 

- 

GPa 

GPa 

- 

- 

4.66E-05 

6.58E-05 

6.58E-05 

7.61E-06 

1.03E-05 

- 

-0.05 

0.07 

0.03 

1169.55 

1118.58 

468.90 

0.84 

0.16 

0.80 

0.16 

0.33 

- 

In the FE analysis of RSE the inclusion of the internal length ℓ does not improve the accuracy. Comparing the 
results  of  Table  3  with  those  of  GREHM  presented  in  Figure  10  it  can  be  seen  that  for  ℓ  =  0  there  is  an 
excellent  match:  Ev  =  1156.5  GPa,  Eh  =  1168.8  GPa,  G  =  495.34  GPa,  νxy  =  0.16  and  vyx  =  0.16  and  the 
relevant differences 3%, 0%, 5%, 0% and 0%.  

Figure 12. FEM of the wall: (a) the wall configuration, (b) mesh of the wall. 

(a) 

(b) 

Source: (L. A. S. Kouris et al., 2020). 

The  influence  of  the  internal  length  is  also  examined  in  a  global  analysis  of  a  FE  wall  model.  The  wall  is 
shown in Figure 12a. The brick pattern of the wall is not symmetric in the horizontal and the vertical direction: 
there are 5 ½ bricks in the horizontal courses and 16 vertical courses in total (Figure 12a).  

22 

 
 
 
 
 
 
 
 
Figure 13. Comparison between the (Mises) stress field in kPa of the distinct-properties model Γd (on the left) and GREHM 
model Γh (on the right) for a vertical (a,b), horizontal (c,d) and moment (e,f) load. 

(a)

(b)

(c)

(d)

(e)

(f)

Source: (L. A. S. Kouris et al., 2020). 

A first FE model, called the ‘distinct’ one Γd, comprises different (classical) elastic plane elements for brick and 
mortar bed and head joints shown in Figure 12b. The bricks have been meshed using 6×20 elements whereas 
the  mortar  using  2×20  and  8×2  elements  for  the  bed  and  the  head  joints  respectively  using  a  mesh 
dimension approximately equal to ½ of the mortar joint thickness. The elements are fully connected. The wall 
is  restrained  at  its  base  to  represent  the  experimental  conditions.  The  previously  defined  homogenised 
properties are used throughout the homogenised FE model Γh(ℓ), using various internal lengths ℓ to compare 
its  influence  on  the  analysis.The  stress  and  deformed  state  of  Γd  FE  model  is  presented  in  the  left  side  of 

23 

 
Figure 13 for a number of different external loadings p: (i) an evenly distributed vertical load 100kPa on the 
top  (p1),  (ii)  a  lateral  load  of  100kPa  applied  on  the  vertical  left  side  of  the  wall  (p2)  and,  (iii)  an  in-plane 
moment produced by a vertical distributed load inverting sign at the middle of the wall, i.e. 100 kN/m and -
100  kN/m  at  the  top  (p3),  all  applied  statically.  The  Von  Mises  stress  state  is  presented  in  kPa.  It  can  be 
observed that the brick units being the stiffer material attract higher stresses. This is more dominant in the 
first loading case. 

Next, a second FE model Γh is run but in this case applying GREHM: all plane elements of Γh have the same 
homogenised properties. A number of internal lengths ℓ ranging from 0 to 20 mm are assumed and hence, 
the respective models are generated Γh(ℓ). The same loads as previously are applied to all {Γh(ℓ), lϵ[0,20mm]}. 
The  stress  and  deformed  state  of  Γh(ℓ=0)  are  presented  on  the  right  of  Figure  13  where  it  is  seen  that 
comparing the two stress states the homogenised model smooths out the distinct one. 

A displacement approach is followed to compare the two models: the displacements of the model Γd with the 
distinct  properties  for  the  materials  are  assumed  the  correct  values,  and  the  error  is  estimated  from  the 
deviation of the respective displacements of the models with the GREHM properties Γh(ℓ) for the same loading 
conditions.  

The error e(ℓ,p)  between the nodal displacements j (j =1:n) on the free boundaries of the distinct (δj
the GREHM (δj

h(ℓ,p)) models is defined as follows: 

d(p)) and 

(



,

e

p

)

n

1
= ∑
n

=

1

j

d
δ
j

(

)
p
d
δ
j

h
−
δ
j
(
p

(
)



,

p

)

(13) 

Therefore,  the  error  e(ℓ,p)    is  defined  as  the  averaged  summation  of  the  errors  between  the  respective 
displacements of the two models, i.e. e(ℓ,p) = Σej for ℓ,p and εjϵΓ where j  is a node on the free boundary of 
the wall S=∂Γε≠0.  

In Figure 14 the error e(ℓ,p)  vs. the assumed internal length of each analysis for the three external loads p1, 
p2  and  p3  is  plotted.  Interestingly  the  inclusion  of  the  non-local  length  can  increase  the  accuracy  of  the 
analysis for the moment load. In fact, the application of a vertical load changing sign in the middle of the top 
side of the wall creates a local singularity. An internal length ℓ around 4 mm can improve the estimation of 
the  displacements in this  case. For the other loading  conditions the non-local  parameter does not  decrease 
the error. This would be expected for classical linear elastic FE models as has been already mentioned. For ℓ = 
0 the maximum error for all the loading conditions does not exceed 0.03%. 

Figure 14. The influence of the internal length on the accuracy of the model for three different loading conditions. 

0.14

0.12

0.1

0.08

0.06

]

%

[

r
o
r
r

E

0.04

0.02

0

0

Vertical load

Horizontal load

Moment

2

4

6

8

10

12

14

16

18

20

 Internal length

  l

 [mm]

24 

 
 
 
 
 
 
3.4.2  Comparison with analytical homogenisation models 

A review of the closed-form homogenisation models can be found in (Taliercio, 2014). Here a comparison is 
made between the proposed GREHM and three other models: the (Pande et al., 1989) model, and two other 
recent models (Di Nino and Luongo, 2019; Taliercio, 2014). The expression proposed in codes for the elastic 
modulus will be shown in a following section and compared with GREHM.  

Table 4. Experimental testing of masonry walls: main parameters of brick units and mortar joints (continued). 

Exp.  

Campaign 

Eb 

tb,y 

tb,x 

w* 

Em 

tm,y 

tm,x 

ℓ 

Eexp 

Ev 

fexp 

[MPa] 

[mm] 

[mm] 

[mm] 

[MPa] 

[mm] 

[mm] 

[mm] 

[MPa] 

[MPa] 

[MPa] 

1BRE1 

1530 

55 

250 

110 

1545 

10 

1BRE2 

1530 

55 

250 

110 

1365 

10 

2BOF1 

2BOF2 

2BOF3 

2BOF4 

2BOF5 

1122 

61 

257 

126 

149 

13 

1122 

61 

254 

125 

149 

13 

1122 

61 

258 

126 

149 

13 

1224 

49 

265 

129 

143 

13 

1224 

49 

259 

125 

143 

13 

2BOF6 

2215 

46 

251 

131 

143 

13 

3DRO1 

4200 

45 

290 

140 

125 

10 

3DRO2 

4200 

45 

290 

140 

225 

10 

4FRT1 

7500 

40 

4FRT2 

7500 

40 

65 

65 

40 

40 

220 

20 

220 

10 

10 

10 

13 

13 

13 

13 

13 

13 

10 

10 

20 

10 

5GUM1 

3372 

75 

230 

105 

5450 

12.5 

14 

5GUM2 

3372 

75 

230 

105 

7083 

12.5 

14 

5GUM3 

3372 

75 

230 

105 

8568 

12.5 

14 

8 

2035 

2067 

13.5 

12 

2310 

2280 

13.2 

3 

743 

743 

2.9 

3.2 

767 

762 

3.1 

3.4 

785 

780 

3.0 

6.7 

1120 

1121 

2.5 

6.7 

1091 

1121 

2.8 

5.1 

1225 

1220 

2.7 

1 

1 

3 

3 

15 

19 

17 

729 

738 

12.0 

1181 

1195 

13.7 

878 

874 

9.0 

1938 

2000 

3.9 

4824 

4883 

8 

5232 

5248 

9.4 

5024 

5068 

12.2 

6HOS 

7LLO1 

7LLO2 

7LLO3 

7LLO4 

12930 

36 

123 

60 

3270 

10 

13595 

140 

290 

70 

7976 

10 

13595 

140 

290 

70 

5102 

10 

4003 

140 

290 

45 

7976 

10 

4003 

140 

290 

45 

5102 

10 

10 

10 

10 

10 

10 

0.2 

8000 

8064 

18.2 

7 

15047  15060  20.4 

0.1 

10958  12284  12.3 

40 

40 

6042 

5999 

14.4 

5964 

5998 

11.1 

1(Brencich et al., 2008), 2(Boffill et al., 2019), 3(Drougkas et al., 2016), 4(Furtmüller and Adam, 2011), 5(Gumaste et al., 2007), 6(Hossain et 

al., 1997), 7(Llorens et al., 2019) * w is the wall thickness. 

Experimental campaigns of masonry prisms and wallettes tested in vertical loading were collected from the 
literature.  Herein  those  relevant  tests  that  presented  the  material  characterisation  data  needed  for  the 
application of GREHM, namely the mechanical properties of both the brick units and mortar, were selected and 

25 

the  relative  experimental  values  of  the  vertical  elastic  modulus  Eexp  and  the  compressive  strength  fexp  are 
presented  in  Table  4  and  Table  5  (ten  different  experimental  campaigns  resulted  in  23  different  masonry 
types). The calibrated value of ℓ to achieve the highest possible consistency with the experimental values and 
the vertical homogenised modulus Ev according to GREHM are also provided. 

Table 5. Experimental testing of masonry walls: main parameters of brick units and mortar joints. 

8NWO1 

8490 

8NWO2 

8490 

8NWO3 

8490 

72 

72 

72 

224 

106 

3900 

10 

224 

106 

3450 

10 

224 

106 

750 

10 

9OLI 

12000 

45 

285 

130 

4200 

10 

10RED1 

8000 

100 

305 

143 

6600 

6 

10RED2 

8000 

100 

305 

143 

6600 

12 

10RED3 

8000 

100 

305 

143 

6600 

20 

10RED4 

8000 

100 

305 

143 

6600 

30 

11VER 

16700 

52 

210 

100 

2100 

13 

10 

10 

10 

10 

6 

12 

20 

30 

13 

2 

0 

4 

8410 

8370 

13.5 

7210 

7206 

11.5 

6610 

6670 

10.6 

1.1 

10000  9972 

28.6 

0.1 

7800 

7930 

3.2 

0.1 

7200 

7846 

3.3 

0.1 

7100 

7748 

3.5 

0.1 

6900 

7646 

3.6 

0.1 

6800 

7063 

- 

8(Nwofor, 2012), 9(Oliveira et al., 2007), 10(Reddy et al., 2009), 11(Vermeltfoort et al., 2007).  
* w is the wall thickness. 

Figure 15. Comparison of vertical elastic modulus Ev between the proposed GREHM and the models of (Taliercio, 2014), 
(Di Nino and Luongo, 2019), and (Pande et al., 1989) for the considered experimental values Eexp: the red dashed lines at 
±10%. 

GREHM

TALIERCIO

DiNINO & LUONGO

PANDE ET AL.

p
x
e

E

/

v

E

1.5

1.4

1.3

1.2

1.1

1

0.9

0.8

0.7

0.6

0.5

1
E
R
B

2
E
R
B

1
F
O
B

2
F
O
B

3
F
O
B

4
F
O
B

5
F
O
B

6
F
O
B

1
O
R
D

2
O
R
D

1
T
R
F

2
T
R
F

1
M
U
G

2
M
U
G

3
M
U
G

S
O
H

1
O
L
L

2
O
L
L

3
O
L
L

4
O
L
L

1
O
W
N

2
O
W
N

3
O
W
N

I
L
O

1
D
E
R

2
D
E
R

3
D
E
R

4
D
E
R

R
E
V

The  comparison  of  the  models  is  presented  by  means  of  the  ratio  between  the  analytical  Ev  to  the 
experimental Eexp elastic modulus in the vertical direction (data given in Table 4 and Table 5), as illustrated in 
Figure 15. The homogenised properties applying GREHM can be estimated fast using the normalised graphs of 

26 

 
Figure 6-Figure 8. It is noted that the limits of the vertical axis of Figure 12 have been kept between 50% and 
150%  for  better  readability  although  a  higher  difference  appeared  in  some  cases.  In  general  the  proposed 
GREHM is more precise while the other models estimate lower values of the elastic modulus. The total error 
(i.e. root of the sum of squares over the total number of samples) across the 23 experimental tests is 0.74% 
for GREHM, 5.70% for (Di Nino and Luongo, 2019), 5.47% for (Taliercio, 2014) and 5.22% for (Pande et al., 
1989). 

Figure 16. A comparison between GREHM and the models of Taliercio (Taliercio, 2014), Di Nino and Luongo (Di Nino and 
Luongo, 2019), and Pande et al. (Pande et al., 1989) for the elastic modulus in the horizontal direction Eh, the shear 
modulus G and the Poisson ratio vxy. 

b

E

/

v

E

b

/

E
G

1

0.8

0.6

0.4

0.2

0

0.6

0.5

0.4

0.3

0.2

0.1

0

b

E

/

h

E

1

0.8

0.6

0.4

0.2

0

S
O
H

2
O
W
N

1
D
E
R

R
E
V

S
O
H

2
O
W
N

1
D
E
R

R
E
V

0.25

0.2

0.15

0.1

0.05

y

x

v

S
O
H

2
O
W
N

1
D
E
R

R
E
V

0

S
O
H

2
O
W
N

1
D
E
R

R
E
V

GREHM

TALIERCIO

DiNINO & LUONGO

PANDE ET AL.

Source: (L. A. S. Kouris et al., 2020). 

As can be seen in Figure 12 for the specimens HOS (Hossain et al., 1997), NWO2 (Nwofor, 2012), RED1 (Reddy 
et al., 2009) and VER (Vermeltfoort et al., 2007) (see Table 4 and Table 4 for the details of the tests) all the 
models  estimate  very  similar  values.  In  particular  for  those  tests  the  estimation  of  the  internal  length  ℓ 
becomes  minimum  and  a  further  comparison  between  the  models  is  shown  in  Figure  16  for  the  elastic 
modulus  in  the  horizontal  direction  Eh,  the  shear  modulus  G  and  the  Poisson  ratio  vxy.  This  is  a  purely 

27 

 
analytical  comparison  of  the  homogenised  values  as  these  quantities  had  not  been  measured  in  relevant 
tests. As can be seen GREHM appears to give slightly higher values for Ev for the first three tests and for the 
fourth one it is (Pande et al., 1989) to be higher. This tendency remains generally also for the other quantities 
apart  from  the  shear  modulus  G  where  (Taliercio,  2014)  increases.  This  comparison  shows  a  general 
consistency between the models for these experiments. 

28 

 
4  Numerical simulations and analyses of reinforced masonry with TRM 

Traditional  and  cultural  heritage  buildings  need  advanced  models  for  seismic  analysis  and  strengthening.  A 
model for reinforced masonry with TRM should consider all the constituents and their interactions. URM piers 
in general can fail due to crushing or rupture of the units and/or the mortar or, due to failure of their cohesion. 
The  external  coat  of  the  TRM  interacts  with  masonry  and  complicated  bonding  and  interaction  phenomena 
among  them  appear  and  numerous  damage  modes  can  occur.  An  FE  model  to  be  able  to  describe  them 
converts the partial differential equations (PDEs) into matrix analyses. Two approaches there are to solve the 
matrices: (i) the implicit method based on the inversion of stiffness matrices to get the displacements and, (ii) 
the explicit method based on the inversion of mass matrices to get the accelerations. The former method is 
an  iterative  solution  where  each  step  solution  depends  on  the  previous  one  and  suits  for  static  or,  time-
independent problems. The latter method is also an iterative method but contrary to the previous, the solution 
of each step is used to define the displacements of the next step. Therefore, such an approach needs smaller 
time steps and suits well for dynamic loads, or time-dependent problems. The former method has the merit 
of being unconditionally stable while, the latter demands less computational effort as the mass matrix being 
diagonal is easily invertible and suits for highly non-linear problems. In this simulation approach two models 
are developed: an implicit model for quasi static problems and an explicit model for dynamic loading. Below 
the developed methods are presented and analyses are carried out. 

Box 6. Implicit and explicit models 

Two simulation approaches are proposed here for modelling traditional and cultural heritage buildings: (i) an 
implicit one for quasi-static loads, and (ii) an explicit one for dynamic loads. The former method has the merit 
of being unconditionally stable while, the latter demands less computational effort as the mass matrix being 
diagonal is easily invertible and suits for highly non-linear problems. 

4.1 

Implicit simulation of TRM reinforced masonry   

4.1.1  Description of the model 

In this modelling approach some simplifications are necessary to reduce the complexity of the simulation to 
an affordable cost. First of all, the masonry is treated using its homogenised properties taking advantage of 
the  model  of  the  previous  chapter.  Then,  the  interaction  between  the  textile  and  the  reinforcing  mortar  is 
macroscopically simulated using macro-indices to reflect the bonding behaviour of the textile in the TRM. This 
is a deliberate choice to keep the numerical problem in an affordable implicit analysis cost and to avoid the 
exact  description  of  properties  which  otherwise  would  need  profound  investigation.  For  a  more  detailed 
description the reader is referred to the paper (Kouris et al., 2023). 

4.1.1.1  Parameters of TRM layers 

The reinforcing layer of TRM is applied externally on either or both sides of a wall. The response a TRM coat 
defined  in  coupon  tests    is  generally  assumed  to  be  trilinear  (Kouris  and  Triantafillou,  2018)  as  shown  in 
Figure 2, where the three subsequent phases I, IIa and IIb are characterised by the moduli E1, E2 (=0) and E3.  

The textile in the mortar is simulated using rebar sub-elements. These sub-element stiffnesses are overlaid 
onto the  parent  mortar elements in which they  reside, but do not  have separated degrees of freedom, and 
therefore  have  strain  compatibility  with  the  mortar.    In  reality  however,  this  is  not  the  case,  as  a  slip  bond 
occurs  between  yarns  and  the  matrix  and  in  the  macro-scale  this  local-scale  loss  of  bonds  appears  as 
reduced  stiffness  and  strength  of  the  textile.  Although  the  bond  behaviour  has  been  the  focus  of  several 
recent experimental and analytical studies e.g. (Carozzi et al., 2017; Larrinaga et al., 2014; Raoof et al., 2016; 
Veljkovic  et  al.,  2020)  the  topic  still  remains  open  and  the  applications  of  an  implicit  model  would  be 
dramatically  limited  by  the  increased  complexity  of  all  the  interfacial  properties,  let  alone  the  testing 
procedures  needed  to  define  them.  Hence,  to  take  into  account  the  non-directly  simulated  local  slip 
phenomena macro-factors are introduced limiting the elastic modulus and the strength of the textile. These 
macro-indices are defined in three steps (Figure 17): first, the textile stress-strain behaviour is simplified into 
a linear behaviour (Figure 17a). The actual and complex interaction process between filaments, yarns, and the 
matrix  takes  place  together  with  an  uneven  stress  distribution  of  the  fibres  microscopically;  all  these 
interactions result in several micro-cracking and micro-debonding phenomena (Kouris and Triantafillou, 2018) 
which  in  the  macro-scale  appear  as  a  telescopic  debonding.  Then,  the  macroscopic  slip  is  simulated  as  a 

29 

bilinear curve (Askouni and Papanicolaou, 2019, 2017) as shown in Figure 17b. A thorough description of the 
several steps that the debonding undergoes with increasing load can be found in (D’Antino et al., 2014). 

Figure 17. The modelling approach for bond slip phenomena in TRM: (a) a linear textile response, (b) a bilinear bonding 
response, and (c) the effective linear textile response. 

fd

s
s
e
r
t
s

E

τb

s
s
e
r
t
s
d
n
o
b

strain

εult

(a) 

δ0

δbond,ult

slip

(b) 

s
s
e
r
t
s

feff

Eeff

(c) 

εeff,ult

strain

Source: (Kouris et al., 2023). 

In the third step, the macroscopic slip δbond is turned into a strain εbond using the fibre length L: εbond = δbond / L. 
The  effective  strain  of  the  fibre  εeff  is  the  sum  of  the  linear  one  and  εbond.  Therefore,  the  effective  elastic 
modulus and the effective strength of the textile can be defined as follows: 

𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒 = 𝜀𝜀 +   𝜀𝜀𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 

𝐸𝐸𝑒𝑒𝑒𝑒𝑒𝑒 =

𝜎𝜎
𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒

In Equation (14) the ultimate strain is denoted as εeff,ult and is the minimum of the sums:   

𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒 = 𝐸𝐸𝑒𝑒𝑒𝑒𝑒𝑒 ∙ 𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒,𝑢𝑢𝑢𝑢𝑢𝑢

(14). 

(15) 

𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒,𝑢𝑢𝑢𝑢𝑢𝑢 = min {𝜀𝜀𝑢𝑢𝑢𝑢𝑢𝑢 +   𝜀𝜀𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏|𝜀𝜀 +   𝜀𝜀𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏,𝑢𝑢𝑢𝑢𝑢𝑢}

where the  ultimate strains εult and εbond,ult = δbond,ult /L are  defined in Figure  17a and  b  respectively.  Applying 
these  empirical  rule  the  loss  of  the  cohesion  and  the  consequent  slip  of  the  textile  inside  the  reinforcing 
mortar  is  included  within  an  effective  elastic  modulus  and  strength  of  the  textile  representing  both 
phenomena: this schematically is shown in Figure 17c. 

30 

 
 
 
 
 
 
 
 
The Drucker-Prager plasticity model is used to simulate the compressive non-linear behaviour of the mortar. 
The model allows an isotropic hardening and softening of materials under compression. The evolution of the 
yield  surface  with  non-linear  deformation  is  defined  based  on  the  uniaxial  compression  yield  stress,  of  the 
masonry  assemblage.  In  this  criterion,  flow  stress  ratio  R,  dilation  angle  ψ,  and  friction  angle  β,  need  to  be 
defined. The linear yield surface in the DP plasticity model is expressed as 

𝐹𝐹 = 𝑡𝑡 − 𝜎𝜎𝐻𝐻 tan𝛽𝛽 − 𝑑𝑑

3

(16a,b) 

where in Eq. (16) σH stand for the hydrostatic pressure equal to 1/3 of the sum of the normal stresses in the 3 
directions and q for the von Mises equivalent stress defined as follows: 

𝑡𝑡 =

𝑞𝑞 �1 +

− �1 −

� �

�

�

1
2

1
𝑅𝑅

1
𝑅𝑅

𝑟𝑟
𝑄𝑄

3
𝑞𝑞 = �
2

(𝛴𝛴: 𝛴𝛴)

𝛴𝛴 = 𝜎𝜎 + 𝑝𝑝 𝐼𝐼

𝑟𝑟 = �

𝛴𝛴 ∙ 𝛴𝛴: 𝛴𝛴�

9
2

(17) 

1
3

Finally, in Eq. (17) Σ is the deviatoric stress tensor and p the hydrostatic stresses.   

4.1.1.2  Parameters of masonry 

The FEM of masonry comprises masonry blocks representing the RVEs interconnected through interfaces. Each 
RVE is modelled using the homogenised properties: the elastic one derived using the GREHM and a concrete 
plastic-damage  model  (Lubliner  et  al.,  1989).  The  latter  plasticity  law  despite  the  fact  that  it  has  been 
developed  specifically  for  concrete,  it  can  be  also  used  well  for  masonry  or  other  brittle  materials,  with 
different  plasticity  and  damage  behaviour  in  compression  and  tension.    For  the  uniaxial  compression 
behaviour a law proposed for masonry specifically (Kaushik et al., 2007) is adopted and introduced in pairs of 
axial plastic strain εp vs. compressive stress σ. as shown in Figure 18.  

Figure 18. The stress-plastic strain curve of masonry. 

]

a
P
M

[

s
s
e
r
t

S

12

10

8

6

4

2

0

0

0.002

0.004

0.006

0.008

0.01

Plastic strain [%]

Source: (Kouris et al., 2023). 

Therefore,  the  key  parameters  needed  to  define  in  the  model  are:  (i)  the  elastic  moduli  and  the  Poisson’s 
ratios, (ii) the dilation angle, ψ, in the p–q plane (estimated as 15 degrees), (iii) the flow potential eccentricity, 
ϵ  (assumed  equal  to  0),  (iv)  the  ratio  of  initial  equibiaxial  compressive  yield  stress  to  initial  uniaxial 

31 

 
 
 
 
 
 
 
 
 
compressive  yield  stress  (estimated  equal  to  1.16  i.e.  the  default  value),  (v)  the  ratio  of  the  second  stress 
invariant on the tensile meridian to that on the compressive meridian (assumed as 2/3), and (vi) the viscosity 
parameter (assumed equal to 10-6). Moreover, the post-cracking behaviour in tension and the tensile fracture 
energy  is  estimated  from  a  sensitivity  analysis  of  a  URM  pier.  The  FEM  comprises  masonry  blocks 
representing  the  RVE  interconnected  through  interfaces.  The  cohesive  elements  in  the  interfaces  are 
introduced  using  the  command  *Contact  Pair  where  the  surfaces  in  contact  are  defined  and  their 
properties.  Details  about  the  properties  of  the  contact  surfaces  are  spedified  below  using  the  command 
*Surface Interaction followed by the out-of-plane thickness (defined as 1 m).  
The dilatancy angle of masonry is the ratio of the perpendicular displacement to the one parallel to the joints 
for a wall in shear load. It controls the amount of plastic volumetric strain developed during plastic shearing 
and  is  assumed  constant  during  plastic  yielding.  However,  the  dilation  phenomena  in  masonry  are  complex 
related  to  the  asperity  and  roughness  of  the  developed  cracks  in  mortar  joints  due  to  shearing  loads 
(Stupkiewicz  and  Mróz,  2001).  The  dilatancy  angle  is  not  constant  during  loading  (Stupkiewicz  and  Mróz, 
2001) and it has been shown that for extensive and cyclic shear loads to tend to zero (Lourenço and Ramos, 
2004). Therefore, the angle of dilation ψ can be intuitively related to the angle of internal friction φ. For non-
cohesive soils (i.e. sand, gravel) with the angle of internal friction φ > 30° the value of dilation angle can be 
estimated as ψ = φ - 30°.  A negative  value of  dilation angle is acceptable only for rather loose  sands.  The 
value of ψ = 0 corresponds to the volume preserving deformation while in shear. 

The Drucker-Prager plasticity model is used to simulate the compressive non-linear behaviour of the mortar 
of the reinforcement. The model allows the isotropic hardening and softening of materials under compression. 
The adopted modelling approach requires the definition of the properties of the interface between the RVEs. 
The friction between the masonry prisms is defined equal to 0.7 kPa introduced and the slip tolerance is the 
default  value  0.005.  The  behaviour  of  interfaces  when  overlapped  is  determined  as  hard  (*Surface 
Behavior). 
The cohesive behaviour is ruled through the traction separation stiffness: the elastic part of the response is 
governed  by the  stiffness in  the normal  plane Knn  and the stiffnesses in two perpendicular  shear  planes Kss 
and Ktt. The respective values have been estimated from the experimental results and are equal to 4MPa/m, 
40MPa/m and 40MPa/m. The response is before any damage is linear and after reaching the damage criterion 
is linearly decreased related to the scalar variable d as follows: 

𝒕𝒕 = �

𝑡𝑡𝑏𝑏
𝑡𝑡𝑠𝑠
𝑡𝑡𝑢𝑢

� = (1 − 𝑑𝑑) �

𝐾𝐾𝑏𝑏𝑏𝑏
0
0

0
𝐾𝐾𝑠𝑠𝑠𝑠
0

0
0
𝐾𝐾𝑢𝑢𝑢𝑢

� �

𝛿𝛿𝑏𝑏
𝛿𝛿𝑠𝑠
𝛿𝛿𝑢𝑢

� = (1 − 𝑑𝑑)𝑲𝑲 ∙ 𝜹𝜹

The maximum stress criterion is adopted to characterise damage initiation: 

(18). 

(19). 

𝑡𝑡𝑠𝑠
0 ,
𝑡𝑡𝑠𝑠
In Eq. (19) the tractions do not regard the compressive stresses (negative sign) on the fracture behaviour of 
the  joints  in  the  normal  direction.  Regarding  the  properties  of  the  stiffness  matrix  of  Eq.  (18)  these  should 
correspond to the RVE’s elastic properties as during the elastic response there is not relative movement at the 
interfaces.  Using  the  RVE  and  given  that  the  interfaces  have  no  thickness  the  following  simple 
homogenisation rule can be applied:  

𝑡𝑡𝑢𝑢
0� = 1
𝑡𝑡𝑢𝑢

𝑡𝑡𝑏𝑏
0 ,
𝑡𝑡𝑏𝑏

𝑚𝑚𝑚𝑚𝑚𝑚 �

where Eu, Em are the elastic moduli of the units and the mortar and tu is the thickness of the mortar joints. 

𝐾𝐾𝑏𝑏𝑏𝑏 =

𝐸𝐸𝑢𝑢𝐸𝐸𝑚𝑚
𝑡𝑡𝑚𝑚(𝐸𝐸𝑢𝑢−𝐸𝐸𝑚𝑚)

(20) 

32 

 
 
 
 
 
 
4.1.2  Calibration of the model for out-of-plane loads 

Applying the proposed FE model the specimens of the experimental campaign made by (Kariou et al., 2018) 
are modelled and a calibration of the slippage and bond loss of the textile is carried out. An 8-node reduced 
integration 3D FE element, namely the C3D8R element is used to model masonry elements, while the TRM is 
modelled  using  3D  shell  elements  with  4  nodes,  namely  the  S4  element.  Masonry  blocks  representing  the 
RVEs are interconnected through interfaces. Attention is paid to match the location of nodes to improve the 
performance of interfaces. 

The  specimens  are  discretised  refinedly  with  FEs  of  close  to  unity  aspect  ratios  to  ensure  accurate 
representation of stresses and strains. The specimens were supported by steel plates and rollers, with "zero" 
resistance to uplift and very high compression stiffness.  The model does not consider the subgrade stiffness 
and  instead  a  full  retrain  of  the  vertical  DOFs  along  the  support  line  is  applied.  Finally,  a  displacement-
controlled load is applied directly to the nodes of the specimen instead of the steel plate. 

4.1.2.1  Calibration of masonry properties 

The  parameters  of  the  proposed  model  which  were  not  experimentally  found  are  fine-tuned  carrying  out  a 
sensitivity  analysis  with  the  out-of-plane  response  of  the  control  double-wythe  specimen  made  of 
unreinforced masonry.  The control specimen of single-wythe unreinforced masonry failed in its own weight 
(Kariou et al., 2018).  

The post-cracking behaviour in tension is assumed to be related to the tensile strength equal to 0.5MPa while 
the tensile fracture energy equal to 0.25kN/m (Lourenço, 2000) was found precise. Furthermore, however, a 
dilation angle ψ = 150 was found to correlate well with the experimental results. Standard values for the other 
parameters  are  found  alright:  (i)  the  flow  potential  eccentricity,  ϵ  =  0,  (ii)  the  ratio  of  initial  equibiaxial 
compressive yield stress to initial uniaxial compressive yield stress equal to 1.16, (iii) the ratio of the second 
stress  invariant  on  the  tensile  meridian  to  that  on  the  compressive  equal  to  2/3,  and  (iv)  the  viscosity 
parameter equal to 10-6. 

Figure 19. Comparison of the out-of-plane response of the unreinforced double wythe specimen with the analytical model. 

3.5

3

2.5

2

1.5

1

0.5

]

N
k
[

e
c
r
o
F

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Displacement [mm]

Model

Experiment

Source: (Kouris et al., 2023). 

The comparison with the experimental response shown a very good agreement both in terms of strength and 
displacements. The softening behaviour is also captured sufficiently well (Figure 19). 

33 

 
 
 
 
 
4.1.2.2  Calibration of the textile bond behaviour 

The elastic modulus ET and the strength fy,T of TRM are calibrated using the experimental results. The effective 
properties of TRM reflect the simplifications of the simulation regarding the failure modelling; the failure of 
the  bond  between  the  textile  and  the  matrix  and  the  failure  of  the  cohesion  of  the  matrix  have  not  been 
modelled. The effective properties of TRM can account for these two failure modes. 

Figure 20. Comparison of the out-of-plane response of the reinforced single wythe specimen with the analytical model. 

Single wythe specimens

S_C1

S_C1 _(Co)

25

20

15

10

]

N
k
[

e
c
r
o
F

5

0

0

5

10

15

0

2

4

6

8

10

12

14

16

18

20

Displacement [mm]

S_G3

Displacement [mm]

S_G3_(Co)

40

35

30

25

20

15

10

5

0

30

25

20

15

10

5

0

]

N
k
[
e
c
r
o
F

]

N
k
[

e
c
r
o
F

45

40

35

30

25

20

15

10

5

0

]

N
k
[

e
c
r
o
F

45

40

35

30

25

20

15

10

5

0

]

N
k
[

e
c
r
o
F

16

14

12

10

8

6

4

2

0

35

30

25

20

15

10

5

0

25

20

15

10

]

N
k
[

e
c
r
o
F

]

N
k
[

e
c
r
o
F

]

N
k
[

e
c
r
o
F

0

2

4

6

8

10

12

0

5

10

15

20

25

30

Displacement [mm]

S_G7

Displacement [mm]

S_G7 _(Co)

0

2

4

6

8

10

12

14

0

5

10

15

20

25

30

35

40

45

50

Displacement [mm]

S_B3

Displacement [mm]

S_B7

5

0

0

5

10

15

20

0

5

10

15

20

25

30

Displacement [mm]

Displacement [mm]

Model

Experiment

Source: (Kouris et al., 2023). 

34 

 
 
 
 
 
 
 
 
 
 
 
 
Figure 21. Comparison of the out-of-plane response of the reinforced double wythe specimen with the analytical model. 

60

50

40

30

20

10

]

N
k
[

e
c
r
o
F

0

0

45

40

35

30

25

20

15

10

5

0

0

70

60

50

40

30

20

10

0

0

]

N
k
[

e
c
r
o
F

]

N
k
[

e
c
r
o
F

Double wythe specimens

D_C1_(Co)

D_C1

5

10

15

20

25

0

1

2

3

4

5

6

7

8

9

Displacement [mm]

D_G3_(Co)

Displacement [mm]

D_G3

5

10

15

20

0

1

2

3

4

5

Displacement [mm]

D_G7_(Co)

Displacement [mm]

D_G7

2

4

6

8

10

12

0

2

4

6

8

10

12

Displacement [mm]

Displacement [mm]

D_B3

45

40

35

30

25

20

15

10

5

0

35

30

25

20

15

10

5

0

70

60

50

40

30

20

10

0

45

40

35

30

25

20

15

10

5

0

]

N
k
[

e
c
r
o
F

]

N
k
[

e
c
r
o
F

]

N
k
[

e
c
r
o
F

]

N
k
[

e
c
r
o
F

0

2

4

6

8

10

12

14

16

18

20

Displacement [mm]

Model

Experiment

Source: (Kouris et al., 2023). 

4.1.3  Empirical equations for the reduction coefficients 

Using the proposed model, the elastic modulus and the textile strength should be treated with their effective 
values  depending on the bonding slip and the effective total strain (Eq. (14)).  The imperfect bonding allows 
the  textile  to  slip  in  the  matrix  up  to  the  total  loss  of  cohesion.  This  telescopic  slippage  (Banholzer  et  al., 
2006)  is  mainly  influenced  by  the  external  treatment  of  the  textiles  (coating)  and  the  impregnation  of  the 
textile yarn with the surrounding inorganic matrix (Alecci et al., 2016). In addition to the latter the placement 

35 

 
 
 
 
 
 
 
 
 
 
of the textile can influence the bonding (Bertolesi et al., 2014). The impregnation as well as the placement of 
the textile are factors with a random nature. It  should be noted that a  slippage between the  substrate and 
masonry is not commonly observed (D’Antino et al., 2014).  

Apart  from  the  external  coating  of  the  yarns  the  geometric  and  material  parameters  of  the  textile  and  the 
matrix  which  control  the  bonding  are:  (i)  the  radius  r  of  the  yarns,  (ii)  the  distance  w  between  the  yarns 
number of layers, (iii) the thickness of the layer, (iv) the perpendicular distance between the yarns h, (v) the 
elastic moduli of the textile Et and the matrix Emtx, (vi) the tensile strength of the textile and the matrix, (vii) 
the number of layers, (viii) the volumetric ratio of the textile in matrix and (ix) the stress of the textile σt. The 
correlation  coefficients  of  these  parameters  with  the  effective  ratios  of  the  elastic  modulus  and  tensile 
strength  of  the  textile  reveal  the  most  important  ones  which  are  included  in  the  following  unitless  three 
coefficients: 

𝛼𝛼 =

𝜎𝜎𝑢𝑢
𝑓𝑓𝑢𝑢

2

𝜋𝜋 ∙ 𝑟𝑟
𝑤𝑤 ∙ ℎ𝐿𝐿 =

𝑡𝑡𝑏𝑏𝑏𝑏𝑚𝑚 ∙ 𝑁𝑁𝑁𝑁
𝑡𝑡𝑐𝑐𝑏𝑏𝑐𝑐𝑢𝑢

𝛽𝛽 =  

𝛾𝛾 =

𝐸𝐸𝑚𝑚𝑢𝑢𝑚𝑚
𝐸𝐸𝑢𝑢

(21a,b,c) 

where No is the number of layers and tcoat is the total thickness of the TRM jacket. The relationship defining 
the  ratio  of  the  effective  textile  elastic  modulus  Et,eff  and  bonding  slip  follows  the  shape  of  the  bond-slip 
model of embedded steel reinforcement in reinforced concrete structures (CEB-FIB, 2011): 

, 

𝛦𝛦𝑡𝑡,𝑒𝑒𝑒𝑒𝑒𝑒

𝑒𝑒𝑡𝑡,𝑒𝑒𝑒𝑒𝑒𝑒

𝛺𝛺𝐸𝐸 =

𝐸𝐸𝑡𝑡

𝛺𝛺𝑒𝑒 =

𝑒𝑒𝑡𝑡

𝛺𝛺𝐸𝐸,𝑒𝑒 = 1 − �1 − 𝑒𝑒

−𝑝𝑝2𝛽𝛽

−𝑝𝑝1𝛼𝛼

−𝑝𝑝3

� 𝑙𝑙𝑙𝑙𝛾𝛾

(22a,b) 

where Ω is the reduction coefficient and p1, p2 and p3 coefficients given in. The comparison of the empirical 
model  and  the  calibrated  experimental  values  is  shown  in  Figure  22  where  a  good  performance  of  the 
selected reduction function is noted. 

The coefficients of the empirical equations are given in Table 6. In this table the goodness-of-fit metrics, i.e. 
the  R2  and  the  p-values  are  provided.  The  empirical  model  is  found  to  converge  effectively  with  the  data 
achieving robust determination values (R2>0.8 and p<0.01).  

The  variation  of  the  empirical  equation  is  easier  interpreted  in  the  2D  plots  vs.  the  α,  β  coefficients.  The 
empirical equation is bounded and for the experimental carbon and matrix properties this variation is between 
0.3 and 0.9 as presented in Figure 24a. Obviously, the higher the tension that a fibre can sustain the higher 
would  ΩΕ  be  and  consequently,  as  α  increases  the  ΩΕ  coefficient  increases.  Moreover,  the  thinner  nominal 
thickness  (and  respectively  the  yarn)  the  less  influence  has  the  tensile  stress.  This  can  be  attributed  to  the 
degree of impregnation of the yarn; if the roving is thick then, it is made of several fibres and the degree of 
impregnation of the surrounding matrix in the inner fibres is low resulting in an uneven stress state. Due to 
this stress distribution, in Figure 24b it is seen that low stress receives a smoother variation of ΩΕ in regard to 
the radius of the yarn whereas for high stresses the ΩΕ coefficient degrades quickly. 

36 

 
 
 
 
 
 
 
]

%

[

f

o
n
o

i
t

a
m

i
t
s
E

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

Figure 22. Experimental vs. estimated values of Ω. 

(uncoated)

(coated)

E

E

(uncoated)

(coated)

f

f

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Experimental 

 [%]

Source: (Kouris et al., 2023). 

Table 6.Calibrated coefficients 

,

, 

and regression metrics of the empirical Eq. (22) (Kouris et al., 2023).   

2
1
 p
p
p1 

3 

p

p2 

p3 

R2 

p-value 

ΩΕ 
(uncoated) 

0.63 

559 

0.12 

0.85 

3∙10-3 

ΩΕ (coated) 

0.15 

1557 

0.12 

0.84 

2∙10-3 

Ωf 
(uncoated) 

2.34 

2000 

0.13 

0.93 

2∙10-3 

Ωf (coated) 

3.35 

277 

0.06 

0.94 

1∙10-3 

Source: (Kouris et al., 2023) 

Figure 23. The variation of ΩΕ vs. α and β coefficients.  

)
d
e
t
a
o
c
(

E

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

)
d
e
t
a
o
c
(

E

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0

0.2

0.4

0.6

0.8

1

0

0.002

0.004

0.006

0.008

0.01

37 

 
 
 
 
 
 
Source: (Kouris et al., 2023). 

4.1.4  Analyses for out-of-plane quasi-static loads 

The parametric analysis investigates the effectiveness of a large range of textiles and matrices as reinforcing 
materials  for  out-of-plane  loads  which  was  not  possible  to  consider  in  the  experimental  campaign.  To  this 
end,  textiles  with  a  varying  strength  from  as  low  as  100  MPa  to  2500  MPa  (effective  strength  values)  are 
applied to the masonry piers. The key parameter is the mechanical reinforcement ratio ω which varies largely 
(from 0.05 to 1) whereas in the experimental campaign varied from 0.03 to 0.43 as shown in Table 7.  

Table 7. Experimental parameters. 

Jacket thickness 

[mm] 

ρw 

[%] 

ωt 

Max Force 

Max displacement 

[kN] 

[mm] 

Specimen 

S_C1 

S_C1_(Co) 

S_G3 

S_G3_(Co) 

S_G7 

S_G7_(Co) 

S_B3_(Co) 

3 

5 

4 

7 

8 

9 

9 

0.095% 

0.35 

23.4 

0.095% 

0.35 

35.3 

0.129% 

0.19 

14.3 

0.129% 

0.19 

25.8 

0.300% 

0.43 

30.6 

0.300% 

0.43 

42.5 

0.108% 

0.15 

23.2 

S_B7_(Co) 

13 

0.253% 

0.35 

44.5 

D_C1 

D_C1_(Co) 

D_G3 

D_G3_(Co) 

D_G7 

D_G7_(Co) 

D_B3_(Co) 

3 

5 

4 

7 

8 

9 

9 

D_B7_(Co) 

13 

Source: (Kariou et al., 2018). 

0.045% 

0.17 

40.1 

0.045% 

0.17 

58.8 

0.061% 

0.09 

32 

0.061% 

0.09 

40.4 

0.143% 

0.03 

67.1 

0.143% 

0.21 

63.8 

0.052% 

0.07 

43.1 

0.120% 

0.17 

66.2 

15.6 

20.8 

12.2 

30 

14.7 

55 

24.7 

32.1 

9.7 

14.9 

5.8 

28.8 

12.8 

16.2 

16.7 

13.6 

The textile volumetric ratio varies also up to 0.49% whereas in the experimental campaign the corresponding 
values  ranged  up  to  0.3  %.  Each  textile  has  its  own  texture  with  varying  filament,  roving  and  yarn  areas, 
distances  etc.  The  textile  tensile  force  per  metre  versus  the  strengthening  mechanical  ratios  is  plotted  in 
Figure 24.  The reinforcement mechanical ratio ω is realised applying 20 different combinations of: (i) textile 
fibre strength fTRM (as well as elastic modulus), (ii) yarn areas, and (iii) number of layers as shown in Figure 24. 

38 

 
 
Obviously,  lower  strength  textiles  resemble  those  made  from  natural  fibres  and  have  thicker  yarns  while, 
higher number of layers (e.g. 3 or more) are more common for composite textiles.  

Figure 24. Textile properties for parametric analyses with varying mechanical ratios combining a range of textile 
strengths, yarn area (area of the points), and number of layers (colour of the points, see colourbar).  

1

0.9

]

%

[

0.8

o
i
t
a
r

l

i

a
c
n
a
h
c
e
m
g
n
n
e
h
t
g
n
e
r
t

i

S

0.7

0.6

0.5

0.4

0.3

0.2

0.1

7

1

0.9

0.8
6

0.7

0.6

5
0.5

0.4

0.3
4

0.2

0.1

3

7

6

5

4

3

]

%

[

o
i
t
a
r

l

i

a
c
n
a
h
c
e
m
g
n
n
e
h
t
g
n
e
r
t

i

S

0

10

20

30

40

50

60

70

500

1000

1500

2000

2500

Textile tensile force [MN/m]

Textile tensile strength f

 [MPa]

T R M

Source: (Kouris et al., 2023). 

Moreover,  the  axial  load  is  included  in  the  investigation.  The  axial  load  has  not  been  considered  in  the 
experiments  due  to  increased  set-up  complexity,  but  real  piers  carry  gravity  loads.  Therefore,  two  levels  of 
axial  loads  are  considered:  (i)  10%  of  the  axial  pier  capacity  and,  (ii)  20%  of  the  axial  pier  capacity  as 
common masonry buildings can exhibit. In this parametric analysis the piers configuration remains as the one 
of the experimental campaign to serve as reference for validation. 

Figure 25. Maximum vertical load of piers out-of-plane loaded for 3 levels of vertical load: (a) N = 0, (b) N = 10% fm, and 
(c) N = 20% fm. 

80

60

40

20

]

N
k
[

e
c
r
o
F

0

0

0.5

1

1.5

2

2.5

3

3.5

N 

 10% fm

N = 20% fm

80

60

40

20

]

N
k
[

e
c
r
o
F

0

0

0.5

1

1.5

2

2.5

3

3.5

80

60

40

20

]

N
k
[

e
c
r
o
F

0

0

0.5

1

1.5

2

2.5

3

3.5

Omega

Omega

Omega

Single-wythe & coated

Double-wythe & coated

Single-wythe & uncoated

Double-wythe & uncoated

Source: (Kouris et al., 2023). 

For the previous parameters and employing the proposed FE model, a parametric analysis is carried out. To 
estimate the nominal elastic modulus and the textile strength from their effective values Eq. (22) is applied. 
The results of the analyses  are summarised in Figure 25: small variations for the three levels of axial load 
can  be  seen.  For  the  single-wythe  piers  the  axial  load  can  lead  to  a  slightly  higher  capacity  for  heavily 
strengthened  jackets  with  ω  higher  than  1.  Obviously,  uncoated  textiles  add  lower  capacity  to  the  piers. 
Generally, even low strength textiles (e.g. made from natural fibres) resulting to as low values of ω as 0.05 to 

39 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
0.1  approximately  can  add  a  substantial  increase  of  the  out-of-plane  pier  capacity  which  can  be  around  5 
times the capacity of the unstrengthened pier without axial load. The out-of-plane capacity increases with ω 
but then, it reaches a plateau value which does not alter substantially. This value for the single-wythe piers is 
0.25 approximately for the uncoated textiles and 0.5 for the coated ones and for the double-wythe piers the 
respective values are 0.5 and 1 approximately. 

4.2  Explicit simulation of TRM reinforced masonry   

4.2.1  Parameters of the model 

An  explicit  model  is  also  developed  for  the  cases  where  the  nature  of  the  load  is  dynamic.  The  reader  is 
referred to (L. A. Kouris et al., 2020) for more details on this topic. The FE simulations were carried out using 
the  explicit  finite  element  solver  EUROPLEXUS  (EUROPLEXUS,  2019)  jointly  developed  by  the  French 
Commissariat à l’Energie Atomique et aux Energies Alternatives (CEA Saclay) and the European Commission’s 
Joint Research Centre (JRC Ispra). The main application domain of the software is numerical simulation of fast 
transient  phenomena  including  non-linear  effects,  such  as  explosions  and  impacts  in  complex  three-
dimensional fluid structure systems and therefore, suits perfectly for the masonry blast analysis applied here. 

The  numerical  model  of  the  masonry  wall  was  discretized  by  8-node  linear  brick  elements  with  reduced 
integration  to  avoid  locking  phenomena.  A  uniform  element  size  of  1cm  was  opted  for  to  also  resolve  the 
mortar joints, see Figure 26. The TRM layer has been modelled with 4-node shell elements with 6 DOFs per 
node, 4 integration points in the plane and 5 integration points through the thickness. A similar element size 
of 1cm has been employed for the TRM layer which is sharing common nodes with the masonry wall in order 
to take into account the bonding between the two different components. The specimen was constrained along 
the  vertical  direction  (parallel  to  the  load)  at  the  support  locations.  For  the  quasi-static  tests  the  load  has 
been implemented somewhat faster (constant velocity of 0.05m/s), using time scaling which implies that the 
simulation  ran  during  a  shorter  time  interval  than  the  duration  of  the  physical  process  of  the  test.  Time 
scaling  increases  the  strain  rates  in  the  structure,  therefore  rate-sensitivity  has  been  deactivated  from  the 
constitutive model. 

Figure 26. Numerical model of the masonry wall. 

Source: (L. A. Kouris et al., 2020). 

The  mortar  and  the  bricks  exhibit  a  brittle  behaviour,  therefore  a  three-invariant  cap  model  with  mixed 
hardening has been used (Kristoffersen et al., 2018). It includes strain rate sensitivity and isotropic damage, 
and  has  been  successfully  employed  in  previous  studies  representing  concrete  structures  (Guilbaud,  2015). 
For the TRM layer a Von Mises material law has been used, where elasto-plasticity with isotropic hardening is 
implemented via a radial return algorithm. 

40 

 
 
 
Failure was modelled using element erosion (element excluded from the calculation for the remaining period) 
that is triggered when all the integration points in the respective element reach a critical value related with 
the  constitutive  law.  For  the  brittle  material  law,  the  volumetric  plastic  strain  has  been  used  as  a  failure 
criterion, while the threshold has been defined after appropriate parametric studies. For the TRM, a criterion 
based  upon  the  principal  strains  has  been  engaged.  The  criterion  is  active  when  the  hydrostatic  stress  is 
positive (traction), while when the hydrostatic stress is negative (compression) failure is inactive. 

4.2.2  Calibration of the model 

The  numerical  model  and  the  assumptions  described  in  Section  3  are  applied  to  the  masonry  single  and 
double  wythe  specimens  of  the  experimental  campaign  presented  in  Section  2  to  calibrate  the  effective 
material properties. The calibrated material properties for both specimens retrofitted with a carbon TRM layer 
are  presented  in  Table  8  and  Table  9.  It  is  noted  that  a  uniform  weight  is  applied  to  all  materials  as  the 
analysis  is  explicit.  Masonry  and  mortar  joints  are  modelled  using  a  concrete  NL  material  law  applying  a 
Drucker-Prager strain threshold (EUROPLEXUS, 2019). 

Table 8. Material parameters for brittle constitutive law.  

Material 

Brick 

Mortar 

Density 

[kg/m³] 

2000 

1800 

Young modulus 

Poison  

[GPa] 

6 

0.4 

ratio 

0.3 

0.2 

Compressive 
strength 

[MPA] 

15 

5 

Tensile 
strength 

[MPa] 

1.5 

0.5 

Volumetric 
strain 
threshold 

0.2 

0.1 

Table 9. Effective material parameters for the carbon TRM layer. 

Density 

[kg/m³] 

3650 

Young modulus 

Poison ratio 

Elastic limit 

Ultimate stress 

[GPa] 

225 

[-] 

0.2 

[MPA] 

250 

[MPa] 

1600 

Ultimate 
strain 

[%] 

1.5 

The  calibration  of  the  effective  material  parameters  for  the  carbon  textile  is  shown  in  Table  9  and  the 
comparison  between  analytical  and  experimental  results  are  presented  in  Figure  27  where  an  excellent 
matching is observed. Failure is calibrated with high accuracy in both cases as well as the initial stiffness. A 
reasonably small variation of the stiffness is observed for the double wythe masonry specimen which can be 
attributed to local aleatory bonding imperfections. 

41 

Figure 27. Comparison between experimental results of single (S_C1) and double wythe (D_C1) masonry and 
numerical model (EPX). 

Source: (L. A. Kouris et al., 2020) 

Moreover, comparing the failure mode and the crack propagation among the experimental specimens and the 
numerical models show a reasonably accurate coincidence as depicted in Figure 28.  

Figure 28. Failure mode of experimental specimens and analytical representation. 

42 

 
 
Source: (L. A. Kouris et al., 2020) 

4.2.3  Analysis for dynamic loading from explosions 

An idealized form of a blast pressure-time curve at a certain distance from the detonation point and without 
intermediate obstacles is presented in Figure 3. The shock wave first reaches the point under consideration at 
the arrival time ta (which includes the time of the detonation itself). The pressure reaches its maximum value 
pmax  (peak-overpressure)  almost  instantly  and  starts  decaying  until  it  reaches  the  ambient  pressure  p0  (in 
most  cases  the  atmospheric  pressure)  within  time  td,  called  the  positive  phase  duration.  The  pressure 
continues to drop below the ambient pressure reaching its minimum value pmin, and then starts to rise again 
reaching the ambient pressure within time tn, known as the negative phase duration. The exponential decay of 
the  pressure-time  history  is  represented  through  the  modified  Friedlander  equation  (Baker,  1983),  that 
describes the rate of decrease of the blast pressure values. Depending on the distance of the obstacle from 
the detonation point and the mass of the explosive charge, the desired parameters (time of arrival, pressure 
magnitudes, duration and impulses) can be defined using the graphs included in (Kingery and Bulmash, 1984). 

Figure 29. Idealized form of pressure-time history due to explosion (Friedlander equation (Baker, 1983)). 

Source: (L. A. Kouris et al., 2020) 

 A  hemispherical  blast  wave  produced  from  a  80  kg  TNT  charge  at  a  standoff  distance  of  5m  has  been 
considered to impinge the upper surface of the wall. This scenario implies a Hopkinson-Cranz scaled distance 
of 1.16 

 according to the following Equation:  

1/3

𝑚𝑚/𝑘𝑘𝑘𝑘

(23) 

3
𝑍𝑍 = 𝑅𝑅
�
√𝑊𝑊

43 

 
 
 
 
where 

 is the standoff distance and 

 is the charge mass. 

𝑅𝑅

𝑊𝑊

Figure 30. Damage states for: (a) unreinforced masonry, and (b) retrofitted masonry with TRM. 

(a) 

(b) 

Source: (L. A. Kouris et al., 2020). 

The  materials  comprising  the  masonry  wall  behave  differently  under  impulsive  loading  when  compared  to 
static  or  quasi-static  loading,  due  to  the  effect  of  the  increased  strain-rates.  Figure  30  presents  the 
comparison  of  the  masonry  wall  response  under  blast  loading  with  and  without  TRM  enhancement.  The 
magenta  coloured  zones  indicate  the  eroded  elements  after  reaching  their  maximum  limit.  More  accurate 
representation of the crack propagation needs finer discretisation, but the current one is sufficient to extract 
qualitative conclusions.  

The additional TRM layer appears to prevent the extensive cracking of the mortar. In fact, the opening of the 
mortar  bed  joints  in  the  middle  zone  due  to  tensile  failure  is  prevented  allowing  for  an  additional  masonry 
arch  mechanism  to  develop  which,  however,  is  still  far  from  high  plastic  strains.  This  additional  arch 
mechanism appears  with a  moderate diagonal  cracking  and a compression zone in the middle mortar head 
joints. Moreover, this mechanism relieves part of the stress and strain tension at the supports. 

Monitoring of the number of the failed elements for each case, reveals that the TRM enhanced wall has 20% 
less  eroded  elements  that  is  a  clear  indication  of  damage  reduction.  The  TRM  is  not  exfoliated,  keeping 
together the individual components of the wall, a characteristic that can contribute towards the reduction of 
the  potential  flying  debris  and  their  lethal  consequences,  and  thus,  increase  substantially  the  safety  of  the 
structure. 

44 

 
 
 
 
 
5  Assessment of the combined retrofitting in city scale 

This  chapter  investigates  the  application  of  the  combined  energy  and  seismic  retrofitting  on  masonry 
buildings at a city level across different locations in Europe with a range of combinations of seismic hazard 
and  climatic  conditions  and  assesses  its  potential  benefits.  For  a  broader  analysis  including  other  types  of 
buildings  the reader is referred to (Pohoryles et al., 2020) and for RC buildings specifically to (Pohoryles and 
Bournas, 2021). Indeed, in the following study the focus is only on masonry buildings but when the losses are 
estimated at a city level the total building stock is considered. Representative existing masonry buildings from 
various construction periods have been selected reflecting typical building characteristics from a thermal and 
structural point of view and the targets for energy and seismic retrofitting were determined for four climatic 
and  seismic  zones.  The  retrofitting  scheme  combining  thermal  and  structural  upgrade  for  the  building 
envelope  is  applied  to  reach  the  respective  performance  targets  close  to  those  set  by  modern  codes.  The 
buildings’  energy  demand  and  seismic  safety  before  and  after  retrofitting  are  assessed  for  the  different 
climatic and  seismic scenarios, leading to a large-scale  assessment of the building stock  using a  combined 
metric, namely the expected annual loss.  

5.1  Studied cities and buildings’ population 

Twenty European cities located in areas of different seismic hazard and climatic conditions were selected as 
case study locations to investigate the impact of the combined energy and seismic retrofitting. Those cities 
are grouped in four climatic and seismic zones. As shown in Table 10, the climatic zones were classified by 
heating degree-days (HDD), a weather-based index representing the difference between a base temperature 
and a day’s mean temperature. A base temperature of 18°C was chosen to calculate the HDD values for this 
study.  The  outdoor  air  temperature  was  adopted  from  EnergyPlus  (US  Department  of  Energy,  2010) 
generated from a period of climate record of 30 years. 

Table 10. Matrix of case study cities by seismic risk (PGA) and climatic condition (HDD). 

HDD 

PGA 

HDD≤1200 

1200<HDD≤2200 

2200<HDD≤3000  3000<HDD≤4000 

PGA ≤ 0.10 

Seville  

Porto  

Milan/Geneva  

Bratislava/Warsaw/ 
Groningen  

0.10 < PGA ≤ 0.25 

Faro  

Bari  

Belgrade  

Bucharest 

0.25 < PGA ≤ 0.40 

Lisbon/Athens 

Rome/Thessaloniki 

Perugia 

Sofia 

PGA >0.40 

Larnaca 

Istanbul 

Potenza 

The  seismic  zones  were  defined  on  the  basis  of  the  expected  peak  ground  acceleration  (PGA)  for  a  return 
period  of  475  years  (10%  exceedance  in  50  years).  Gas  extraction  induced  earthquakes  in  Groningen  area 
were also considered assuming a PGA value of 0.1g (Dost et al., 2017). The values vary from as low as 0.05g 
to  0.65g  (Pagani  et  al.,  2018)  divided  in  four  zones  from  low  to  high  seismicity  as  illustrated  in  .  For  each 
climatic zone there is at least one city in each seismic zone apart from the coldest zone (3000<HDD≤4000) 
where no city exists in Europe with high seismic hazard (PGA >0.40).  

45 

 
 
 
Figure 31. The seismic hazard in PGA values (%g) per city. 

5
6
0

.

4
0

.

1
4
0

.

Climatic zone 1

Climatic zone 2

Climatic zone 3

Climatic zone 4

7
2
0

.

5
2
0

.

5
1
0

.

2
0

.

1
0

.

5
1
0

.

1
0

.

5
0
0

.

Seismic zone 4

Seismic zone 3

Seismic zone 2

Seismic zone 1

1
4
0

.

5
2
0

.

2
0

.

5
3
0

.

5
3
0

.

3
0

.

5
2
0

.

1
0

.

5
0
0

.

]
g
%

[
A
G
P

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

The residential sector constitutes the major share (75%) of the EU building stock. In many European countries 
over half of the residential building stock was built before the 1970s and hence does not comply with modern 
energy  efficiency  and  seismic  safety  requirements.  Low  to  mid-rise  stone  and  brick  masonry  buildings 
constitute the typical masonry traditional buildings widespread in historical areas of Europe, with very limited 
construction of reinforced concrete (RC) structures before 1945. Most of RC buildings were constructed after 
the 1960s. Between  1945-1960 there is a transitional period for constructions  moving from the traditional 
masonry to the new RC unti the overwhelming prevail of the latter after 60s. During this period brick masonry 
and RC buildings is assumed to be evenly proportioned.  

Representative masonry building typologies need to be defined to assess the impact of combined seismic and 
energy retrofitting for the European building stock. In the literature, the classification of structures is typically 
different for  describing the  building stock based on either its energy or  seismic performance.  Generally, the 
key parameters for both energy and seismic performance of buildings are: (i) the construction period, (ii) the 
main structural system or material (e.g. RC, masonry, steel, wood etc.) and, (iii) the geometric dimensions (e.g. 
number of storeys). Here, as the focus is only masonry buildings, the building stock is classified into low and 
medium-rise according to their number of storeys, grouped typically into 1-2 and 2-4. High-rise buildings with 
more than 5-6 storeys  have  a very small proportion and in this analysis are neglected.  In terms of energy 
performance, the U-values of envelope elements varies with the material. Regarding the material of masonry 
structures, typically a distinction between stone-masonry and brick-masonry is made, which can classify the 
buildings into different age groups necessary to gather information from building inventories. Fired bricks are 
known since the antiquity; however, its mass production has been initiated during the industrialisation era. In 
this regard, the majority of the existing brick-masonry buildings were erected after the beginning of the 20th 
century. In this regard the year 1900 is assumed as the margin between the two age groups, assuming that 
the  exisiting  stone-masonry  buildings  were  built  before  this  time.  In  reality,  however,  there  can  be  found 
mixed  types  of  masonry  buildings  (e.g.  stone-masonry  ground  floor  and  upper  storeys  from  brick-masonry) 
and exceptions from the age assumption should be expected. 

Data on the building stock for each case study location is obtained by two main sources. On the one hand the 
number  of  dwellings  or  average  useful  floor  space  constructed  by  decade  is  obtained  from  the  respective 
Census  data  of  2011.  Cities  with  historical  centres  have  a  large  proportion  between  20-30%  of  masonry 
structures (e.g. Perugia, Potenza, but also Groningen and Geneva), classified as stone- and low and mid-rise 
brick masonry buildings (Figure 32).  

46 

 
 
Figure 32. Relative proportion of masonry and RC building per location. 

Masonry

RC

100

80

60

40

20

0

]

%

[

e
p
y
t

s
g
n
d

i

l
i

u
B

Faro
Larnaca

Lisbon

Seville

Athens

Porto

Perugia

Bari

Rome
Istanbul
Thessalonniki

Milan
Potenza
Belgrade

Geneva
Bucharest

Sofia
Warsaw
Bratislava
Groningen

On  the  other  hand  NERA  project  provides  data  on  building  height  which  allowed  for  a  discretization  of 
buildings into low to mid-rise masonry buildings. As shown in Table 11, for the masonry building categories, 
low-rise and mid-rise buildings made of stone or clay brick masonry are defined.  

The floor area of 8 x 18 m is chosen to be globally representative, as the selected buildings are theoretical in 
nature  and  their  geometrical  characteristics  represent  broadly  the  average  of  the  building  stock,  similar  to 
other large-scale studies on the building stock.  Typical traditional masonry buildings do not have a constant 
storey-height and respectively the size of openings decreases along their height. Lower storeys tend to have a 
higher  inter-storey  height  than  typical  RC  structures.  An  average  height  of  4  m  for  the  ground  floor  is 
assumed here and window dimensions equal to 1.0 × 1.90 m, resulting in glazing 18.3% for low-rise and mid-
rise masonry buildings. 

Table 11. Parameters of the case study masonry buildings. 

Elevation 

Sketch 

Low-Rise stone or brick 

Mid-Rise brick 

Plan 

Tot. floor area 

No. of storeys 

Storey height 

Total height 

m2 

m2 

m 

m 

Source: (Pohoryles et al., 2020) 

8 x 18 

288 

2 

8 x 18 

576 

4 

4 (first) / 3.30 

4 (first) / 3.10 

9.60 

15.90 

The relative proportion of the masonry building types in terms of their building height or material (stone or 
brick  masonry)  aggregated  from  the  NERA  project  is  shown  in  Figure  33.  Among  the  building  envelope 
components,  the  walls  and  windows  typically  represent  the  major  share  of  the  overall  energy  loss,  so  this 
study  focuses  on  these  elements.  Considering  the  extensive  variations  over  the  years  and  from  country  to 

47 

 
 
 
 
 
 
country,  five  different  typical  external  wall  types  were  selected  representing  the  different  time  periods,  as 
illustrated in Table 13.  

Figure 33. Relative proportion of masonry building types per location. 

Stone

Brick LR

Brick MR

100

80

60

40

20

0

]

%

[

e
p
y
t

s
g
n
d

i

l
i

u
b

y
r
n
o
s
a
M

Faro
Larnaca

Lisbon

Seville

Athens

Porto

Perugia

Bari

Rome
Istanbul
Thessalonniki

Milan
Potenza
Belgrade

Geneva
Bucharest

Sofia
Warsaw
Bratislava
Groningen

The characteristics of the external walls of masonry buildings have been based on seismic vulnerability and 
energy studies of existing masonry buildings. While typically the thickness of the walls in masonry buildings 
varies  along  their  height  especially  for  mid-rise  structures,  here,  average  values  have  been  considered, 
actually 30 cm for solid brick walls and 45 cm for stone walls. Stone walls are generally multi-leaf and an 
assumption of 60% stone and 40% mortar is made. The U-values in Table 12 are compared to the range of 
U-values of external wall elements in EU residential buildings based on the data collected in for the respective 
time periods.  

A  key  parameter  of  the  thermal  performance  of  buildings  is  the  thermal  transmittance  of  its  envelope 
elements (U-value in W/m2/K). In Table 12, the U’-values, i.e. the U-value considering any thermal bridges, are 
shown  for  the  selected  envelope  types.  These  were  calculated  using  Eq.  (24)  after  evaluating  the  effect  of 
different thermal bridges in HEAT2, a two-dimensional heat transfer simulation software following ISO 10211 
and ISO 10077-2 standards:   

 (24) 

′

𝑈𝑈

= 𝑈𝑈 +

∑ 𝜓𝜓∙𝑢𝑢

2
𝐴𝐴  [W/(m

K)]

where  U  is  the  thermal  transmittance  of  the  building  envelope  element  (exterior  walls),  A  the  area  of  the 
building  envelope  element  (in  m2)  and 
  the  sum  of  all  products  of  thermal  bridge  ψ-values  (thermal 
transmittance  of  the  linear  thermal  bridge  in  W/(mK))  and  the  lengths  of  thermal  bridges.  The  ψ-values 
corresponding to the internal dimensions of the thermal bridge cross-section were considered.  For masonry 
buildings,  the  majority  of  linear  thermal  bridges  are  generated  by  the  geometry  of  the  buildings,  such  as 
external masonry walls intersections (corners), internal masonry wall intersections with external masonry wall, 
while  only  a  few  are  generated  by  combining  different  thermal  conductive  materials  such  as  wooden  floor 
intersection with external masonry wall and wooden window frame and masonry wall connection.  

∑ 𝜓𝜓 ∙ 𝑙𝑙

It  can  be  observed  that  higher  ψ-values  are  generated  by  wall-roof  and  wall-floor  intersection  and  in  the 
upper  part  of  the  windows,  but  generally  the  highest  impact  in  the  recalculation  of  the  walls  U-value  is 
generated by the wall - floor intersection due to the length of this thermal bridge type, especially in the case 
of mid and high-rise buildings. 

48 

 
 
  
 
 
 
 
 
 
 
 
 
Table 12. Wall characteristics of the masonry buildings.   

Period 

Material 

Type 

Section 

U (W/m2K) 

EU-range   

1900-1960 

Brick 
masonry  

Signle-leaf  

1.30 - 2.24 

0.9 - 2.5 

45

Pre-1945 

Stone 
masonry 

Multi-leaf  

2.15 

0.9 - 2.4 

5.2  Energy performance 

Normally, an energy renovation project would encompass a variety of interventions including changes in HVAC 
system, adding renewable energy sources or shading devices; however, here the renovation targets are set in 
terms of the U-value of the envelope elements in order to reduce transmission heat losses. National energy 
performance  legislations  typically  recommend  U-values  for  different  envelope  elements  (exterior  walls, 
windows, roofs) to be achieved by newly built or renovated structures. The values for all case study locations 
are shown in Table 13 and are used to define U-value targets for the retrofit solution. In countries were U-
value  requirements  for  retrofitting  of  existing  buildings  are  not  available,  the  requirements  for  new  built 
structures  were  taken,  as  illustrated  in  the  last  column  of  Table  13.  Average  U-values  target  values  were 
considered in each climatic zone of the twenty climate zones selected. In a very recent study (Veljkovic et al., 
2023),  a  hybrid  tool  combining  artificial  neural  networks  and  building  physics  approach  was  developed  to 
calculate the space heating energy demand was calculated for the whole EU residential sector. 

The windows were also upgraded by means of airtight frames and glass with low solar heat gain coefficient 
(SHGC)  of  0.55.  Replacing  the  existing  windows  has  substantial  effects  on  decreasing  the  infiltration  heat 
losses  by  increasing  the  overall  airtightness  of  the  building.  Therefore,  the  number  of  air  changes  was 
decreased from 1.0 to 0.60 h-1 corresponding to moderate shielded buildings with medium airtightness. The 
energy  demand  for  space  heating  (HED)  and  cooling  (CED)  per  conditioned  floor  area  (kWh/m2)  across  the 
different  locations  of  the  case-study  buildings  was  estimated  by  means  of  dynamic  energy  simulations  in 
EnergyPlus  (US  Department  of  Energy,  2010).  Details  of  the  simulations  can  be  found  in  (Pohoryles  et  al., 
2020). It is noted that, system losses, auxiliary energy, hot water, and lighting energy use are conservatively 
assumed not to change between the existing and retrofitted states. The primary energy use and associated 
equivalent-CO2  emissions  are  calculated  from  heating  and  cooling  demands.  The  heating  and  cooling 
demands  are  multiplied  by  the  respective  total  primary  energy  factors  (fPtot)  and  converted  in  equivalent 
operational CO2 emissions (kCO2e in g/kWh). These are obtained by multiplying the primary energy factors and 
equivalent  operational  CO2  emission  factors  of  different  energy  sources  with  the  relative  proportions  of 
different energy sources used for heating and cooling based on the energy mix of the relevant countries from 
the  2015  residential  profiles  (ISO,  2017).  The  energy  costs  were  transformed  into  estimated  annual  losses 
(EALE) using the next equation:  

49 

 
 
Zone  

City 

Walls 

Roof 

Floor on ground 

Windows 

U-values [W/m²K] 

New 
existing* 

build 

/  

Table 13. Target U-values in each case-study location. 

1 

2 

3 

4 

Larnaca 

Lisbon 

Athens 

Seville 

Faro 

Bari 

Porto 

Rome 

Salonniki 

Istanbul 

Perugia 

Milan 

Potenza 

Belgrade 

Geneva 

Bucharest 

Groningen 

Sofia 

Bratislava 

Warsaw 

0.85 

0.80 

0.50 

0.82 

0.80 

0.40 

0.70 

0.36 

0.45 

0.60 

0.34 

0.34 

0.36 

0.40 

N/A 

0.56 

0.20 

0.35 

0.32 

0.23 

0.75 

0.50 

0.40 

0.35 

0.75 

0.70 

0.45 

0.45 

0.70 

0.38 

0.60 

0.32 

0.40 

0.40 

0.30 

0.30 

0.32 

0.40 

N/A 

0.20 

0.20 

0.28 

0.20 

0.20 

*N- requirements for new buildings 
N+R – same requirements for both new and retrofitted buildings 

Source: (Pohoryles et al., 2020) 

0.68 

0.46 

0.39 

0.30 

3.80 

2.80 

3.00 

3.10 

2.80 

2.60 

2.40 

2.40 

2.80 

2.40 

2.20 

2.20 

2.40 

1.50 

N/A 

1.30 

1.65 

1.70 

1.40 

1.10 

3.10 

2.52 

2.07 

1.40 

N+R 

N 

N 

N+R 

N 

N 

N 

N 

N 

N+R 

N 

N 

N 

N+R 

- 

N 

N+R 

N 

N 

N+R 

0.75 

0.70 

0.45 

0.82 

0.70 

0.42 

0.50 

0.36 

0.40 

0.60 

0.33 

0.33 

0.36 

0.40 

N/A 

0.22 

0.29 

0.40 

0.40 

0.25 

0.61 

0.42 

0.33 

0.22 

50 

annual heating+cooling cost 

𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 =

total building value

 [%]

 (25). 

The  EALE  is  obtained  by  dividing  the  heating  and  cooling  energy  costs  per  m²  of  each  building  type  by  the 
current average building prices per m² at each location. The energy costs are calculated based on the annual 
heating and cooling energy consumptions multiplied by the energy prices in the selected case study locations 
from (Eurostat, 2020a, 2020b).  The current building prices per m² are evaluated from (NUMBEO, 2019). The 
annual  energy  savings  due  to  retrofitting  are  then  expressed  as  the  difference  in  EALE  before  and  after 
retrofitting, again as a percentage of the total building value. Note that the building value would potentially 
increase because of retrofitting. As this is difficult to precisely quantify, it is conservatively ignored from the 
analysis. The EALE ‘s are averaged over the considered energy zones and presented in Figure 34. As expected 
the savings are higher for the colder zones.  

The savings due to the energy retrofit are given as the difference of the initial annual losses (EALj,i and j = E) 
and the losses after retrofit application (EALj,r and j = E) as in Eq. 27: 

∆𝐸𝐸𝐸𝐸𝐸𝐸𝑗𝑗 = 𝐸𝐸𝐸𝐸𝐸𝐸𝑗𝑗,𝑖𝑖 − 𝐸𝐸𝐸𝐸𝐸𝐸𝑗𝑗,𝑟𝑟 [%]

where j stands for type of retrofit (E for energy retrofit, S for seismic and C for a combined one). The ΔEALE ‘s 
are averaged over the considered cities and presented in Figure 34 per energy zone. It is seen that the energy 
savings do not vary remarkably as different targets have been set for the U values of each zone (Table 13). 

 (27) 

Figure 34. Expected annual savings (%) after retrofitting per energy zone: (i) energy, (ii) seismic, and (iii) combined. 

Energy Zone

]

%

l

[
e
u
a
v
g
n
d

i

l
i

u
b
r
e
p
s
g
n
i
v
a
s

l

a
u
n
n
A

Energy         Seismic        Combined

Energy         Seismic        Combined

Energy         Seismic        Combined

Energy         Seismic        Combined

STONE

LR-BR

MR-BR

51 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
5.3  Seismic performance 

Strengthening  structures  using  TRM  has  been  shown  to  improve  their  seismic  performance  considerably  in 
various experimental studies on masonry walls or masonry-infilled RC frames, e.g. summarised by the authors 
in  (Kouris  and  Triantafillou,  2018;  Koutas  et  al.,  2019;  Pohoryles  and  Bournas,  2020).  Despite  this  clear 
evidence the retrofitted buildings were not assumed to achieve the behaviour of modern high-code reinforced 
masonry  buildings  (RMB),  but  instead  a  conservative  improvement  in  performance  of  a  lower  category  in 
terms  of  seismic  design  level  is  assumed  after  receiving  seismic  retrofitting  (Table  14).  Therefore,  URM 
buildings  achieve  a  performance  after  the  TRM-retrofitting  which  is  assumed  to  be  that  of  low-  (for  stone 
walls) or medium-code (for brick walls) RBM buildings. 

Table 14. Target seismic design level of the seismic retrofit scheme. 

Material 

Current design level 

Target design level 

Stone  

Brick  

URM (stone) 

RBM (low) 

URM (brick) 

RBM (medium) 

The  expected  annual  loss  due  to  seismic  events  (EALS)  was  evaluated  applying  the  performance  based 
methodology  (Ellingwood  et  al.,  2007;  Kappos  et  al.,  2007;  Kouris  and  Kappos,  2015).  Rather  than  using  a 
deterministic approach with  specific  mechanical characteristics of the building materials, the next  steps are 
followed to consider the typical spread of material characteristics for the different construction periods: 

—  First, fragility functions are used to determine the probability of reaching a certain level of damage 
(or  damage  state  DS)  for  a  certain  level  of  earthquake  intensity,  presented  in  terms  of  PGA  to  be 
compatible with the intensity of seismic hazard at each location (Table 10).  

—  The defined damage levels are associated to a monetary loss as a fraction of the existing building 

value, by means of a damage-to-loss function specific for masonry and RC structures.  

—  The fragility curves and damage-to-loss functions are then combined to give the vulnerability curve 
of  the  specific  building  type,  i.e.  the  monetary  loss  (as  a  percentage  of  the  building  value)  vs.  the 
selected intensity measure (PGA). 

—  The annual probability of exceedance (PE) in PGA values is defined based on the seismic hazard at 

each location. 

—  The annual probability of exceedance of loss (in % of building value) is the combination of the annual 

probability of exceedance of a certain PGA with the intensity to loss curve.  

—  The expected annual seismic loss (EALS) is finally the integral of the latter function. 

Typical  fragility  functions  for  the  european  building  stock  with  four  DS  (ranging  from  ‘no  damage’  to 
‘collapse’)  for  the  chosen  stone  (Ahmad  et  al.,  2011)  and  brick  (FEMA,  2003)  unreinforced  masonry  were 
selected according to their age, number of storeys (low and mid-rise) and building material  (stone, brick). The 
proposed  fragility  curves  of  the  RBM  buildings  from  HAZUS  (FEMA,  2003)  were  adopted  to  represent  the 
target  performance.  The  fragility  curves  for  the  four  damage  states  of  stone  masonry  low  rise  before  and 
after retrofitting based on the aforementioned assumptions are illustrated in Figure 35. 

52 

Figure 35. Fragility curves for four damage states of stone URM and RMB (dashed lines) buildings. 

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

]

%

[

e
c
n
a
d
e
e
c
x
E

f

o

y
t
i
l
i

b
a
b
o
r
P

0.1

0.2

0.3

0.4

0.5

0.6

PGA [%g]

The damage-to-loss functions were selected from the literature (Kappos et al., 2006).  Note that this loss is 
based  on  the  cost  of  repair  for  the  specific  damage  states  and  is  always  less  than  or  equal  to  the 
replacement cost, i.e. the value of the building. Other losses that can be associated to earthquakes, such as 
human losses and the social impact are however not considered, as the scope of this study focuses only on 
buildings and the assumption of repair costs alone renders more conservative results. The vulnerability curves 
of the buildings types (unreinforced and strengthened) according to the previous assumptions are shown in 
Figure 36 where a ceiling value equal to 0.75 has been assumed for the losses. 

53 

 
 
 
 
Figure 36. Vulnerability curves for stone and low/medium rise brick masonry buildings and their strengthened 
counterparts. 

0.8

0.7

0.6

0.5

]

%

[

s
s
o
L

0.4

i

c
m
s
e
S

i

0.3

0.2

0.1

0

0

Stone Masonry

Stone Masonry-Retr

LR Brick Masonry

LR Brick Masonry-Retr

MR Brick Masonry

MR Brick Masonry-Retr

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PGA [%g]

 The annual rate of exceedance H of a reference PGA can be assumed to vary according to (CEN, 1998): 

−𝑘𝑘

 (26) 

𝐻𝐻(𝑚𝑚𝑔𝑔𝑔𝑔) ≅ 𝑘𝑘0 ∙ 𝑚𝑚𝑔𝑔𝑔𝑔

where  the  proposed  value  for  k  is  3  and  k0  can  be  computed  based  on  the  agR  that  corresponds  to  a 
probability  of  exceedance  of  10%  in  50  years,  i.e.  the  values  of  PGAs  in  Table  10.  For  example  the  annual 
probability of exceedance for Milan, Italy of the PGA = 0.15g (10% probability of exceedance in 50 years) is 
0.002 approximately which is equal to 1/475 i.e. the inverse of the return period (Figure 37). The savings are 
higher  for  the  zone  with  higher  hazard.  The  EALS  ‘s  are  averaged  over  the  considered  energy  zones  and 
presented in Figure 34. Seismic savings for colder climatic zones are lower due to a lower seismic hazard (). 

54 

 
 
 
 
 
 
 
 
 
Figure 37. Annual probability of exceedance vs. ground acceleration for Milan. 

-1

10

-2

10

-3

10

-4

10

-5

10

e
c
n
a
d
e
e
c
x
e
f
o

y
t
i
l
i

b
a
b
o
r
p

l

a
u
n
n
A

0.1

0.2

0.3

0.4

0.5

0.6

Ground Acceleration [%g]

 The EALS ‘s are averaged over the considered energy zones and presented in Figure 34.  

5.4  Combined performance 

The combined losses are defined as the sum of the energy and the seismic ones EALC = EALE + EALS (Bournas, 
2018). The impact of the renovation strategy in terms of cost benefits for each seismic zone is evaluated by 
comparing the savings in terms of energy costs and seismic losses with the initial investment costs by means 
of assessing the payback periods, calculated using Eq. 28.  

The indicative payback period of the retrofitting interventions can be calculated as the ratio of the assumed 
costs of the retrofit (with respect to the non-retrofitted building) to the annual cost savings: 

⁄
Retrofit costk building value

𝑡𝑡𝑝𝑝𝑐𝑐𝑝𝑝𝑏𝑏𝑐𝑐𝑐𝑐𝑘𝑘 =  

∆𝐸𝐸𝐴𝐴𝐿𝐿𝑘𝑘

[years]

 (28) 

where k corresponds to the  respective values of the energy, seismic or  combined retrofit, depending on the 
evaluation. Details about the assumptions of the building values can be found in (Pohoryles et al., 2020). It is 
worth noting that the cost estimations do not precisely consider the spatial variation in the case study cities 
but reflect gross current mean trends. The payback periods are estimated for the energy, the seismic and the 
combined retrofitting and averaged over the seismic zone in Figure 38. The exact values of payback periods 
should be considered with caution, as average gross cost estimations have been made and may considerably 
change depending on the global economy crisis and other financial conditions. 

As mentioned initially, the effect of combined seismic and energy retrofitting is evaluated in this study and 
seismic risk mitigation should be considered according to the new Energy Performance of Buildings Directive 
(EU Council and EU Parliament, 2010). The plots in Figure 38 allow to assess whether a combined retrofit is 
worthwhile compared to an energy retrofit alone for each building type and seismic zone. The payback periods 
of separate seismic and energy interventions are also indicated. The results shown herein indicate important 
trends  regarding  the  benefits  of  combined  retrofit  interventions.  Obviously,  for  vulnerable  buildings  in  high 
seismicity locations the payoff period for the seimic retrofitting shortens considerably. The combined retrofit 
achieves a shorter payback period, less than 10 years, for the cities with higer seismicity and extends to the 
same level with the energy retrofit for low seismicity zones. It is interesting to note that a combined retrofit is 
beneficial  for  masonry  buildings  already  from  zone  2,  that  is  in  cities  with  low  to  medium  seismic  hazard 
(0.10g < PGA ≤ 0.25g). 

55 

 
  
 
 
 
 
 
Figure 38. Payback periods in years per seismic zone: (i) energy, (ii) seismic, and (iii) combined. 

Seismic Zone

* 

]
s
r
y
[
d
o
i
r
e
p
f
f
o
y
a
P

Energy        Seismic       Combined

Energy         Seismic        Combined

Energy        Seismic       Combined

Energy         Seismic        Combined

STONE

LR-BR

MR-BR

* The payoff periods for stone, low-rise and mid-rise masonry buildings in zone 1 are 73, 176 and 166 years respectively. 

Therefore, for locations of low seismicity (seismic zones 1 and 2), the benefit of adding seismic retrofitting is 
generally limited (only for stone-masonry buildings noticeable) and the payback periods are rather similar for 
energy  retrofitting  alone  and  the  combined  retrofitting  scenario  for  all  masonry  structures.  However,  when 
assessing  locations  in  zones  of  medium  to  high  seismicity  (seismic  zones  3  to  4),  the  benefits  of  providing 
simultaneously the seismic and energy retrofitting are illustrated, as in these zones the importance of seismic 
loss reduction  becomes  more critical than the energy savings alone. The benefits are particularly important 
for  stone-masonry  buildings,  as  losses  due  to  energy  costs  and  due  to  seismic  events  can  be  significantly 
reduced. Even for seismic zone 2, with low seismic hazard, the combined retrofit payback period is reduced for 
the oldest masonry buildings. It should be highlighted that for zones in which seismic retrofit is essential (i.e. 
above  seismic  zone  2),  implementing  the  seismic  intervention  at  separate  times  from  the  energy  retrofit, 
would lead to longer payback periods.  

The  benefits  of  a  simultaneous  seismic  and  energy  retrofitting  of  the  building  stocks  in  all  case  study 
locations are compared to a mere energy retrofitting in their 2030 state assuming a renovation rate 3% per 
year  only  for  masonry  structures.  By  this  year  and  with  this  rate  all  masonry  buildings  will  have  been 
renovated in the considered cities. In this study the total building population is considered for the estimation 
of losses and the reader is referred to (Pohoryles et al., 2020) regarding the current RC building stock which is 
assumed  not  to  be  involved  in  the  retrofitting  process.  The  classification  from  A+  to  F  is  done  in  terms  of 
expected savings ΔEAL (in % of the building value) by 2030 in respect to the 2011 state and allows an insight 
onto  the  effect  of  different  retrofitting  scenarios  in  various  seismic  risk  and  climatic  conditions.  Values  of 
ΔEALC  above  0.35%  correspond  to  a  category  A+,  while  a  value  below  0.075%  of  expected  savings 
corresponds to an F rating with intermediate categories A, B, C, D and E (Calvi et al., 2016) (Figure 39). 

56 

 
 
 
 
 
 
 
Figure 39. Average classification in terms of ΔEAL of the building stock across all seismic and climatic zones by 2030 
assuming a renovation rate of masonry buildings 3% for a merely energy or a combined retrofitting. 

Energy

Combined

EAL (%)

1

0.25%

0.11%

0.08%

0.19%

0.27%

0.12%

0.12%

0.21%

2

0.12%

0.20%

0.15%

0.14%

0.28%

0.51%

0.25%

0.25%

3

0.10%

0.12%

0.43%

0.15%

0.37%

0.43%

1.36%

0.31%

4

0.22%

0.06%

0.47%

0.63%

0.46%

1.58%

A+

A

0.35

B

0.3

C

D

E

0.25

0.2

0.15

0.1

F

0.05

0

NaN

e
n
o
Z
c
m
s
e
S

i

i

1

2

3

4

1

2

3

4

Climatic Zone

Climatic Zone

As shown in Figure 39, the increase in seismic risk (going from seismic zone 1 to 4) within the same climatic 
zone, leads to an increase of the ΔEAL due to the corresponding increase in the seismic ΔEALS. In spite of the 
relatively small percentage of masonry buildings in the cities (Figure 32) the benefit from a combined retrofit 
of masonry buildings can be remarkable as can be noticed from the change of savings category for several 
combinations  of  seismic  and  climatic  zones  in  respect  to  the  energy  retrofit.  Carrying  out  a  simultaneous 
upgrading of masonry buildings, cities with moderate to high seismicity (seismic zones 3 and 4) achieve an A+ 
or A category of savings.  

Box 7. Implementation of a simultaneous seismic + energy retrofitting of buildings in Europe 

The  investigation  of  the  implementation  of  a  simultaneous  seismic  and  energy  retrofitting  of  buildings  in 
cities across Europe yielded the EAL’s and payback periods. The trend that this investigation reveals, is that 
cities with old masonry buildings in high seismicity areas will be substantially benefitted from a simultaneous 
retrofitting rather than a merely energy retrofitting. The payback period of such a refurbishment investment 
for URM buildings will be remarkably shorter. 

57 

 
 
 
 
6  Conclusions 

The  emerging  need  to  upgrade  and  protect  traditional  masonry  buildings  both  in  terms  of  structural  and 
thermal retrofit, required also by currect EU and/or national legislation, is addressed in this report by exploring 
the  potential  of  integrating  seismic  and  energy  retrofitting  for  masonry  buildings.  More  specifically  seismic 
retrofitting  in  the  form  of  TRM  jacketing  is  simultaneously  with  termal  insulation  materials.  To  allow  for  a 
wider  investigation  of  such  an  integrated  system,  analytical  models  were  developed  and  showed  that  the 
hybrid system can highly benefit cities with high seismicity. 

Initially, a homogenisation model, in the form of a set of closed-form expressions, was proposed to estimate 
the  in-plane  elastic  and  shear  moduli  and  the  Poisson’s  ratios  in  two  orthogonal  directions.  Gradient 
homogenisation  model  (GREHM)  was  developed  introducing  an  internal  non-local  length  to  account  for  the 
micro-structure of multiphase masonry and the respective local phenomena that take place in reality in the 
identified RSEs for the horizontal and the vertical directions (i.e. parallel and perpendicular to the bed joints). 
The gradient elasticity was opted for it can address phenomena such as discontinuities and cracks appearing 
in masonry in local scale.  The internal length was found a significant parameter that can affect the accuracy 
of the model when compared with experimental results. A value for the internal length is proposed to be used 
where there are no other data calibrated on the available experimental data. 

Moreover, an implicit  model  for simulating the out-of-plane behaviour of TRM-strengthened  walls  has  been 
presented, which by including an empirical rule for the debonding of the textile reinforcement avoids complex 
further  simulations  in  micro  and  macro  scale.  Using  this  simplified  modelling  methodology,  a  parametric 
analysis was carried out where a variation of textiles has been used. Textiles with low strength capacity (e.g. 
natural fibres) to rather strong ones (e.g. composite fibres) with varied number of layers and yarn dimensions 
were applied to investigate their potential in reinforcing masonry for out-of-plane loads. The analysis shows 
that  TRM  jackets  can  effectively  enhance  the  out-of-plane  capacity  of  walls.  Even  when  natural  fibres  are 
used  (e.g.  low  capacity  fibres)  a  substantial  out-of-plane  capacity  is  recorded  which  becomes  much  higher 
increasing  the  fibres  strength.  The  axial  load  increases  substantially  the  out-of-plane  capacity  of  the  walls 
when the prevailing failure mode is the textile damage. 

Furthermore, a model for masonry strengthened  walls with TRM  has  been  developed in an explicit FE  code. 
The  material  models  of  the  simulation  have  been  calibrated  using  static  out-of-plane  experiments. 
Unreinforced and retrofitted masonry piers using one layer of carbon textile in inorganic cement-based matrix 
(TRM)  have  been  exposed  to  blast  loads.  Application  of  one  layer  of  TRM  (using  carbon  textile  in  inorganic 
matrix) can substantially increase the capacity against impact and blast loads. 

The  current  low  renovation  rates  of  buildings  make  it  challenging  to  meet  the  ambitious  EU  climate  and 
energy targets at  2030 and 2050. The effect of a combined retrofit of the European building stock allow for 
an implementation of an increased renovation rate which was shown not only to meet the emission reduction 
targets, but also to be economically feasible, as reduced losses from energy costs and seismic damage make 
the  renovation  strategies  more  viable.  Looking  at  payback  periods  for  the  combined  energy  and  seismic 
retrofit  investment  of  old  traditional  stone-masonry  buildings,  integrated  interventions  showed  financial 
benefits  over  energy  retrofitting  alone  for  high  seismicity  areas.  The  growing  emphasis  on  cross-sectoral 
approaches  shows  their  potential  to  provide  comprehensive  solutions  to  complex  challenges.  As  integrated 
approaches  evolve,  further  research  is  needed  to  ensure  renovation  strategies  increase  seismic  resilience, 
improve  indoor  environments,  enhance  energy  efficiency,  and  minimize  life-cycle  impacts.  Developing  and 
using  low-carbon  technologies  will  be  crucial  for  achieving  decarbonized  buildings  and  advancing  climate 
neutrality by 2050. 

58 

References 

Ahmad, N., Crowley, H., Pinho, R., 2011. Analytical Fragility Functions for Reinforced Concrete and 
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List of abbreviations and definitions 

DOF 

EAL 

Degree of Freedom 

Expected Annual Losses 

EPBD 

2010 Energy Performance of Buildings Directive 

FE 

FRP 

Finite Element  

Fiber Reinforced Polymer 

GREHM  GRadiEnt Homogenisation Model 

HDD 

Heating Degree-Days 

NZEB 

Nearly Zero-Energy Buildings 

ODE 

PDE 

PGA 

RSE 

RVE 

TRM 

URM 

Ordinary Differential Equation 

Partial Differential Equation 

Peak Ground Acceleration 

Representative Shell Element 

Representative Volume Element 

Textile Reinforced Mortar 

UnReinforced Masonry 

68 

List of boxes 

Box 1. Advantage of TRM over FRP for retrofitting masonry buildings ............................................ 5 

Box 2. Boost EU policies implementation ............................................................................ 6 

Box 3. TRM strengthened masonry elements ........................................................................ 9 

Box 4. The thermo-structural hybrid retrofitting of masonry elements ..........................................10 

Box 5. Homogenisation model for masonry walls ..................................................................11 

Box 6. Implicit and explicit models ..................................................................................29 

Box 7. Implementation of a simultaneous seismic + energy retrofitting of buildings in Europe ...............57 

69 

 
List of figures 

Figure 1. Seismic plus energy retrofitting system: (a) TRM Jacket + insulation material on a masonry wall; (b) 
TRM + heating system on the masonry infill of RC framed structure. .............................................. 5 

Figure 2.  Response of TRM in tension: experimental behaviour and trilinear idealisation. ...................... 8 

Figure 3. The homogenised modelling approach for masonry using the representative volume element; in 
blue the interaction interfaces. .......................................................................................12 

Figure 4. Axial 1-D gradient elastic element. .......................................................................13 

Figure 5. The representative surface element (RSE) of running bond brickwork: (a) on the wall, (b) a close-up.
 ..........................................................................................................................14 

Figure 6. The (normalised) vertical elastic modulus Ev of masonry in respect to tmy / tby for various ratios of 
Em/Eb and ℓ/tmy. ........................................................................................................17 

Figure 8. The (normalised) shear elastic modulus G of masonry in respect to tmy / tby for various ratios of 
Em/Eb and ℓ/tmy. .........................................................................................................18 

Figure 9. The Poisson’s ratio vyx of masonry in respect to tmy / tby for various ratios of Em/Eb and ℓ/tmy. .......19 

Figure 10. The influence of the internal length normalised against the thickness of mortar joints on (a) the 
horizontal and vertical elastic moduli and (b) the shear modulus (normalised against the brick elastic 
modulus) and, (c) the Poisson’s ratio of the homogenised model. .................................................20 

Figure 11. The FE model of the RSE: (a) model, (b) shear displacements, (c) shear strain, and (d) shear stress.
 ..........................................................................................................................21 

Figure 12. FEM of the wall: (a) the wall configuration, (b) mesh of the wall. ....................................22 

Figure 13. Comparison between the (Mises) stress field in kPa of the distinct-properties model Γd (on the left) 
and GREHM model Γh (on the right) for a vertical (a,b), horizontal (c,d) and moment (e,f) load. ................23 

Figure 14. The influence of the internal length on the accuracy of the model for three different loading 
conditions. ..............................................................................................................24 

Figure 15. Comparison of vertical elastic modulus Ev between the proposed GREHM and the models of 
(Taliercio, 2014), (Di Nino and Luongo, 2019), and (Pande, Liang, and Middleton, 1989) for the considered 
experimental values Eexp: the red dashed lines at ±10%. ...........................................................26 

Figure 16. A comparison between GREHM and the models of Taliercio (Taliercio, 2014), Di Nino and Luongo 
(Di Nino and Luongo, 2019), and Pande et al. (Pande, Liang, and Middleton, 1989) for the elastic modulus in 
the horizontal direction Eh, the shear modulus G and the Poisson ratio vxy. .......................................27 

Figure 17. The modelling approach for bond slip phenomena in TRM: (a) a linear textile response, (b) a 
bilinear bonding response, and (c) the effective linear textile response. ..........................................30 

Figure 18. The stress-plastic strain curve of masonry. ............................................................31 

Figure 19. Comparison of the out-of-plane response of the unreinforced double wythe specimen with the 
analytical model........................................................................................................33 

Figure 20. Comparison of the out-of-plane response of the reinforced single wythe specimen with the 
analytical model........................................................................................................34 

Figure 21. Comparison of the out-of-plane response of the reinforced double wythe specimen with the 
analytical model........................................................................................................35 

Figure 22. Experimental vs. estimated values of Ω. ...............................................................37 

Figure 23. The variation of ΩΕ vs. α and β coefficients.............................................................37 

Figure 24. Textile properties for parametric analyses with varying mechanical ratios combining a range of 
textile strengths, yarn area (area of the points), and number of layers (colour of the points, see colourbar). .39 

70 

Figure 25. Maximum vertical load of piers out-of-plane loaded for 3 levels of vertical load: (a) N = 0, (b) N = 
10% fm, and (c) N = 20% fm. ..........................................................................................39 

Figure 27. Comparison between experimental results of single (S_C1) and double wythe (D_C1) masonry and 
numerical model (EPX). ................................................................................................42 

Figure 28. Failure mode of experimental specimens and analytical representation. ............................42 

Figure 30. Damage states for: (a) unreinforced masonry, and (b) retrofitted masonry with TRM. .............44 

Figure 33. Expected annual savings (%) after retrofitting per energy zone: (i) energy, (ii) seismic, and (iii) 
combined. ..............................................................................................................51 

Figure 34. Fragility curves for stone URM and RMB buildings. ....................................................53 

Figure 35. Vulnerability curves for stone and low/medium rise brick masonry buildings and their strengthened 
counterparts. ...........................................................................................................54 

Figure 36. Annual probability of exceedance vs. ground acceleration for Faro. .................................55 

Figure 37. Payback periods in years per seismic zone: (i) energy, (ii) seismic, and (iii) combined. .............56 

Figure 38. Average combined classification in terms of EAL of the building stock across all seismic and 
climatic zones. .........................................................................................................57 

71 

 
List of tables 

Table 1. Material properties: elastic modulus [GPa] and strength [MPa]. .........................................10 

Table 2. Properties of brick units and mortar joints for GREHM. ..................................................20 

Table 3. Results from the FE analysis of the RSE and homogenised mechanical properties. ...................22 

Table 4. Experimental testing of masonry walls: main parameters of brick units and mortar joints (continued).
 ..........................................................................................................................25 

Table 5. Experimental testing of masonry walls: main parameters of brick units and mortar joints. ..........26 

Table 6.Calibrated coefficients 

,

, 

and regression metrics of the empirical Eq. (22). ....................37 

Table 7. Experimental parameters. ...................................................................................38 

1
p

2
 p

3 

p

Table 8. Material parameters for brittle constitutive law. .........................................................41 

Table 9. Effective material parameters for the carbon TRM layer. ...............................................41 

Table 10. Matrix of case study cities by seismic risk (PGA) and climatic condition (HDD). ......................45 

Figure 31. Relative proportion of masonry and RC building per location. ........................................47 

Figure 32. Relative proportion of masonry building types per location. ..........................................48 

Table 12. Wall characteristics of the masonry buildings. ..........................................................49 

Table 13. Target U-values in each case-study location. ...........................................................50 

Table 14. Target seismic design level of the seismic retrofit scheme. ...........................................52 

72 

 
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