Experimental Studies of 

Reinforced Concrete 

Structures Under MultiDirectional Earthquakes

and Design Implications

Office of Nuclear Regulatory Research

NUREG/CR-7119
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p 

Experimental Studies of 

Reinforced Concrete 

Structures Under MultiDirectional Earthquakes

and Design Implications

Manuscript Completed: September 2011

Date Published: July 2013

Prepared by

N. Simos and C. H. Hofmayer

Brookhaven National Laboratory

P.O. Box 5000

Upton, NY 11973-5000

A. Murphy, NRC Program Manager

Job Code N6510

Office of Nuclear Regulatory Research

NUREG/CR-7119

iii

ABSTRACT

This report describes the results of a multi-phase study focusing on the SMART2008 (Seismic 

design and best-estimate Methods Assessment for Reinforced concrete buildings subjected to 

Torsion and non-linear effects) shaking table experiment. The aim of the project was to enable 

an international benchmarking study to take place where different methodologies, modeling,

and numerical approaches are used to study and predict the non-linear behavior and damage 

of reinforced concrete structures. Specifically, a 1/4th scale model of a nuclear reinforced 

concrete structure designed according to the French nuclear practices was tested on the 

shaking table at Commissart a' l'Energie Atomique Saclay, France using thirteen multidimensional earthquakes ranging from low seismic intensities to five times the design level. 

Three numerical prediction phases were executed which included (a) a “blind” prediction phase 

based on using best-estimate data for the structural properties and the induced seismic loads, 

(b) an “updated” prediction phase where best estimates were improved using some of the initial 

shaking table test results enabling the studies of higher loadings, and (c) a sensitivity and 

vulnerability analysis phase based on the SMART2008 specimen numerical model used in the 

prediction phase enabling the generation of damage fragility curves for the structure.

The report provides a detailed description of the SMART2008 experiments, an assessment of 

the test results, the structural response predictions using modeling and numerical techniques,

and a damage fragility assessment. The report also discusses potential design implications of 

addressing the non-linear structural behavior during seismic events exceeding the design 

values or Design Basis Earthquake (DBE). The design implications are derived primarily from 

the ability of the numerical approaches adopted in this effort to predict highly non-linear 

structural behavior under seismic loading.

While the 1/4th scale SMART2008 test structure was designed and constructed following the 

French nuclear design code, the observations and assessments stemming from the test results 

should not be considered directly applicable to the seismic behavior of US nuclear structures 

due to differences in the details in the two national codes. However, this report summarizes the 

various aspects of the benchmark and the findings that were drawn from the shaking table tests 

and the numerical analyses. Based on this study, the following specific observations were 

noted and may warrant further consideration, as consensus codes are updated and the NRC 

revises regulatory guidance:

1. For the model used in the SMART2008 experiment, the capacity of the model, built 

per the French code, was higher that the design capacity, i.e., the model was 

designed for a 0.2g peak ground acceleration and withstood shaking table induced 

ground motions of about 1.1 g with only concrete cracking and minor localized steel 

reinforcement yielding.

2. The observed reductions in stiffness at lower seismic intensities are attributed to 

micro-cracking within the concrete.

3. The use of non-linear treatment of real concrete structures should be considered by 

regulatory authorities and it may be appropriate to allow it to influence the design 

through consensus codes.

v

TABLE OF CONTENTS

ABSTRACT................................................................................................................................iii

TABLE OF CONTENTS ............................................................................................................. v

LIST OF FIGURES....................................................................................................................vii

LIST OF TABLES.....................................................................................................................xiii

EXECUTIVE SUMMARY...........................................................................................................xv

ACKNOWLEDGMENTS...........................................................................................................xix

1. INTRODUCTION .............................................................................................................1

1.1. Background .....................................................................................................................1

1.2. Objective .........................................................................................................................1

1.3. Seismic Tests and Response Predictions........................................................................2

1.4. Sensitivity Studies and Fragility Evaluation......................................................................4

1.5. Lessons Learned and Design Impact...............................................................................5

2. SMART2008 BENCHMARK.............................................................................................7

2.1. SMART2008 Test Structure.............................................................................................9

2.2. Shaking Table Characteristics .......................................................................................13

2.3. Instrumentation..............................................................................................................14

2.4. Shaking Table Seismic Tests.........................................................................................17

3. NUMERICAL ANALYSES AND TEST PREDICTIONS .................................................. 43

3.1. Structural Model.............................................................................................................44

3.2. Description of the Constitutive Relations........................................................................ 46

3.3. Shaking Table Test Predictions .....................................................................................49

3.4. Overall Assessment of Prediction Phase ....................................................................... 83

3.4.1. Damping ........................................................................................................................83

3.4.2. Torsional Effects............................................................................................................85

4. VULNERABILITY ASSESSMENT OF 3-D NUCLEAR STRUCTURES .......................... 93

4.1. Sensitivity Studies..........................................................................................................94

4.2. Vulnerability Study – Seismic Fragility Assessment ..................................................... 106

5. SUMMARY ..................................................................................................................129

5.1. SMART2008 Shaking Table Test Assessment ............................................................ 129

5.2. Numerical Prediction Assessment ............................................................................... 130

5.3. Sensitivity and Fragility Studies ................................................................................... 132

5.4. Impact on Design Approaches and Implications .......................................................... 133

6. REFERENCES ............................................................................................................ 139

vii

LIST OF FIGURES

Figure 1-1 SMART2008 test structure on the Azalee shaking table at CEA 

Saclay, France ...................................................................................................2

Figure 1-2 Details of Azalee shaking table used in the SMART2008 tests...................... 3

Figure 1-3 SMART2008 test structure design response spectrum.................................. 3

Figure 2-1 SMART2008 benchmark flowchart of the sequence of phases ..................... 8

Figure 2-2 SMART2008 test structure schematic............................................................ 10

Figure 2-3 Geometric description of the SMART2008 structure.................................... 11

Figure 2-4 Footprint of the SMART2008 structure with CG and torsion axes .............. 11

Figure 2-5 Geometric description of the base plate and foundation............................. 12

Figure 2-6 Geometric description of the AZALEE shaking table................................... 13

Figure 2-7 Shaking table hydraulic jacks/actuators orientation.................................... 14

Figure 2-8 Layout of accelerometer distribution on the shaking table, the 

foundation of the structure, and the floor slab locations ............................ 14

Figure 2-9 Acceleration sensor locations on Wall #3 and displacement 

transducers on Wall #4...................................................................................15

Figure 2-10 Schematic and actual configuration of rebar strain gauges........................ 15

Figure 2-11 Instrumented and loaded SMART2008 structure on shaking table 

prior to the seismic tests................................................................................16

Figure 2-12 Numerical estimates of the first three eigenfrequencies and modes 

of the SMART structure on the shaking table prior to the 

application of the thirteen seismic tests ....................................................... 20

Figure 2-13 Acceleration time histories of the three real earthquake records 

used in the SMART2008 shaking table experiment ...................................... 21

Figure 2-14 Acceleration time histories recorded on the shaking table during 

the SMART2008 tests RUN1, RUN2 and RUN3 and the 

corresponding response spectra................................................................... 22

Figure 2-15 Acceleration time histories recorded on the shaking table during 

the SMART2008 RUN4 test for the two horizontal directions x and 

y........................................................................................................................23

Figure 2-16 Acceleration response spectra on the shaking table during the 

SMART2008 test RUN4 indicating deviation from the design 

spectrum of Figure 2.1 near the structural modes at 7 Hz and 16 

Hz .....................................................................................................................23

Figure 2-17 Comparison of horizontal and vertical acceleration components 

recorded on the shaking table during the test.............................................. 24

Figure 2-18 Response output locations on the SMART structure................................... 25

Figure 2-19 Table horizontal accelerations recorded during test RUN3......................... 26

Figure 2-20 Recorded horizontal accelerations on the structure during test 

RUN3 ................................................................................................................26

Figure 2-21 Acceleration response spectra at level 3 and location D recorded 

during the application of the three real earthquake records (RUN1, 

RUN2 and RUN3) .............................................................................................27

Figure 2-22 Table and structure acceleration response spectra comparison for 

RUN3 test.........................................................................................................28

Figure 2-23 Relative displacement rUy between level 3 and table recorded 

during RUN3 test.............................................................................................28

Figure 2-24 Rebar strain on Wall 3 recorded during RUN3 test ...................................... 29

Figure 2-25 Recorded shaking table horizontal accelerations during the RUN4 

test ...................................................................................................................31
viii

Figure 2-26 Recorded shaking table and in-structure horizontal accelerations 

in RUN4............................................................................................................31

Figure 2-27 Recorded shaking table and in-structure horizontal accelerations 

during the sequence of RUN8 to RUN13 tests .............................................. 32

Figure 2-28 Shaking table acceleration response spectra in the two horizontal 

directions recorded during the synthetic earthquake tests (RUN4 

through RUN13)...............................................................................................33

Figure 2-29 Evolution of floor spectra and comparison of real record 

(RUN3)spectra with the design-level (RUN4) and the over-design 

RUN8 ................................................................................................................34

Figure 2-30 Comparison of shaking table acceleration response spectra in the 

two horizontal directions for design level (RUN4) and last test in 

sequence (RUN13) to identify structure-table interaction effects ............... 34

Figure 2-31 Evolution of in-structure response spectra (level 3 - location A) for 

tests RUN3 through RUN13. Location A is close to the stiffest 

section of the test structure ........................................................................... 35

Figure 2-32 Evolution of in-structure response spectra (level 3 - location D) for 

tests RUN4 through RUN1.............................................................................. 35

Figure 2-33 Evolution of in-structure response spectra (level 3 - location C) for 

tests RUN4 through RUN13............................................................................ 36

Figure 2-34 Relative displacement between 3rd and 1st level recorded during 

RUN9 ................................................................................................................36

Figure 2-35 Evolution of rebar strain in Wall 4. Comparison of RUN4, RUN9, 

and RUN13.......................................................................................................37

Figure 2-36 Identified cracking patterns at the end of the 13-run shaking table 

tests on the inside surfaces of the test structure walls............................... 38

Figure 2-37 “Pseudo” transfer functions between table accelerations and 3rd 

level of structure identifying transitions to non-linear behaviour 

Measured damping values..............................................................................39

Figure 2-38 Method schematics for estimating actual damping values in the 

SMART structure from shaking table tests. (a) logarithmic 

decrement and (b) half-power bandwidth method ....................................... 40

Figure 2-39 Free-vibration characteristics of structure during different tests 

Global Behaviour of the Structure ................................................................. 41

Figure 2-40 Identification of rotational behaviour of the test structure based on 

the displacement patterns recorded at the corners of the 3rd level............. 42

Figure 3-1 Numerical models used in the numerical analysis of the SMART 

structure ..........................................................................................................45

Figure 3-2 Numerical representation of structural details of the SMART 

structure ..........................................................................................................45

Figure 3-3 Constitutive relations governing the Winfrith concrete element 

mode used in the SMART benchmark analyses............................................ 47

Figure 3-4 Holmquist-Johnson-Cook concrete material model..................................... 48

Figure 3-5 Comparison of the two concrete constitutive models in terms of 

the SMART structure response to synthetic accelerograms ....................... 49

Figure 3-6 Stress-strain relation of steel used in the analysis ...................................... 49

Figure 3-7 Eigenfrequencies and modes of the final 3-D model used in the 

numerical predictions of the SMART2008 tests............................................ 52

Figure 3-8 Locations on the structure used for comparison of predictions 

with test data...................................................................................................52

Figure 3-9 Comparison of horizontal accelerations at the corner locations of 

the 3rd level between recorded test data and prediction from the

BNL simulation for RUN3 ...............................................................................53
ix

Figure 3-10 Comparison of horizontal acceleration response spectra between 

recorded data and predictions from BNL simulation for the real 

earthquake RUN3 ............................................................................................54

Figure 3-11 Comparison of horizontal relative displacement Uy at location C 

between recorded data and predictions from BNL simulation for 

the real earthquake RUN3 (note the offsetting of the recorded data 

in the y-axis) ....................................................................................................55

Figure 3-12 Comparison of horizontal relative displacement Ux at location D 

between recorded data and predictions from BNL simulation for 

the real earthquake RUN3............................................................................... 56

Figure 3-13 Comparison of rebar strain on Wall 3 between recorded data and 

predictions from BNL simulation for the real earthquake RUN3................. 57

Figure 3-14 Comparison of structure accelerations for the design earthquake 

level (RUN4) between the experiment and the BNL simulation................... 58

Figure 3-15 Comparison of floor response spectra for the design earthquake 

level (RUN4) between the experiment and the BNL simulation................... 59

Figure 3-16 Comparison rebar strain in wall 3 for the design earthquake level 

(RUN4) between the experiment and the BNL simulation............................ 60

Figure 3-17 Comparison of rebar strain in wall 4 for the design earthquake 

level(RUN4) between the experiment and the BNL simulation. 

Shown in (c) is the recorded strain in expanded scale depicting the 

“re-adjustment” or permanent deformation in the rebar at the 

selected location.............................................................................................60

Figure 3-18 Comparison of relative displacements at location D and along the 

x-direction for the design earthquake level (RUN4) between the 

experiment and the BNL simulation .............................................................. 61

Figure 3-19 Comparison of 3rd level (locations C and D, y-direction and xdirection respectively) accelerations between test and prediction 

for RUN6 and RUN7 inputs............................................................................. 62

Figure 3-20 Floor response spectra comparison at the 3rd level of the structure 

between test and prediction for RUN6 and RUN7 inputs............................. 63

Figure 3-21 Predicted damage in the SMART structure at the end of (a) RUN6 

and (b) RUN7. Cracking of concrete is predicted to appear during 

the RUN7 table excitation...............................................................................64

Figure 3-22 Comparison of predicted acceleration at point A of Level 3 with 

recorded accelerations during RUN8 ............................................................ 65

Figure 3-23 Comparison of predicted accelerations at point C (y-direction) and 

D (x-direction) at Level 3 of the test structure with recorded 

accelerations during RUN8............................................................................. 66

Figure 3-24 Comparison of acceleration floor response spectra at point C(ydirection) and D (x-direction) of Level 3 between recorded and 

predicted data..................................................................................................67

Figure 3-25 Predicted damage state of the SMART structure upon completion 

of the RUN8 shaking table excitation ............................................................ 68

Figure 3-26 Comparison of predicted rebar strain on Wall #4 of the structure 

(location J1V04 near the foundation) with the one recorded by the 

installed strain gauge during RUN9............................................................... 69

Figure 3-27 Comparison of predicted rebar strain on Wall #4 of the structure 

(location J6V04 mid-height between 1st and 2nd level) with recorded 

strain during RUN9 .........................................................................................70

Figure 3-28 Comparison of predicted relative displacement rUy between the 

3rdlevel and foundation top at corner C with the recorded relative 

displacement during RUN9............................................................................. 71
x

Figure 3-29 Comparison of predicted relative displacement rUx between the 3rd 

level and foundation top at corner D with the recorded relative 

displacement during RUN9............................................................................. 72

Figure 3-30 Comparison of predicted accelerations on the 3rd level of the 

structure (corners C and D) with recorded data during the shaking 

table tests RUN10, RUN11 and RUN12 .......................................................... 73

Figure 3-31 Comparison of accelerations on the 3rd level of the structure 

(corner D along the x-direction) with recorded data during the 

RUN11 test.......................................................................................................74

Figure 3-32 Floor response spectra comparison between predicted and 

recorded data during the RUN12 test ............................................................ 75

Figure 3-33 Comparison of rebar strain mid-height of Wall #4 (level 0 and level 

1) at the edge C for tests RUN10, RUN11 and RUN12 .................................. 76

Figure 3-34 Acceleration pair recorded on the shaking table during the last test 

in series, RUN13, where PGA values were 0.75g in x and 1.13g in y .......... 77

Figure 3-35 Comparison of structure accelerations at level 3 for the RUN13 test......... 78

Figure 3-36 Acceleration floor response spectra comparison for the RUN13 

test ...................................................................................................................79

Figure 3-37 Comparison of relative displacements between level 3 and level 

0(foundation/wall interface) for the RUN13 test............................................ 80

Figure 3-38 Predicted structural damage in the SMART structure at the end of 

the 13-run test .................................................................................................81

Figure 3-39 Predicted floor response spectra evolution during the application 

of the ten synthetic earthquake pairs along the y-direction ........................ 82

Figure 3-40 Predicted floor response spectra evolution during the application 

of the ten synthetic earthquake pairs along the x-direction ........................ 82

Figure 3-41 Effect of damping ratio on structural acceleration response ...................... 84

Figure 3-42 Model of a SMART2008-like structure on the shaking table with a 

rectangular instead of trapezoid cross section ............................................ 85

Figure 3-43 Fixed-base mode comparisons between the trapezoidal 

SMART2008 structure and the one with rectangular cross section 

generated for comparison purposes ............................................................. 87

Figure 3-44 Accelerations computed at the four corners of the rectangular 

cross section SMART2008-like structure for RUN3: (a) x-direction 

acceleration and (b) y-direction acceleration showing minimal 

variation between the acceleration traces and indicating that the 

structure is not exhibiting torsional behaviour ............................................ 88

Figure 3-45 Accelerations computed at two extreme locations of the 

trapezoidal SMART2008 structure (corners C and D) for RUN3: (a) 

x-direction acceleration and (b) y-direction acceleration. Depicted 

is the significant variation in acceleration amplitude indicating a 

torsional behaviour of the structure.............................................................. 89

Figure 3-46 Damage sustained by the two different structures under the 

sameseismic load: (a) actual SMART2008 structure, and (b) 

rectangular SMART2008-like structure with minimal asymmetries............. 90

Figure 3-47 Response spectra of input accelerations signals RUN3, RUN4 and 

RUN5 along x and y directions and the torsional mode (insert) of 

the SMART2008 structure on the shaking table (ft = 16.884 Hz).................. 91

Figure 3-48 Third-level acceleration response spectra of the two structures 

(actual SMART2008 and rectangular version) for RUN3 input..................... 91

Figure 3-49 Third-level acceleration response spectra of the two structures 

(actual SMART2008 and rectangular version) for RUN6 input..................... 92
xi

Figure 4-1 Numerical model used in the sensitivity analysis of the test 

structurealso depicting locations for floor response spectra 

generation........................................................................................................93

Figure 4-2 Bi-directional base input acceleration time histories (synthetic) 

generated from a white noise power spectrum and used 

simultaneously in the sensitivity analyses for the 0.2g and 0.4g 

PGA earthquakes ............................................................................................97

Figure 4-3 Power spectral densities (PSD) of the input accelerations used in 

thesensitivity analysis ....................................................................................98

Figure 4-4 Influence of the varying structural parameters on the first three 

eigenfrequencies and modal shapes of the fixed base structure 

used in the sensitivity study .......................................................................... 99

Figure 4-5 Floor response spectra variability due to uncertainties in the 

concrete modulus computed on Level 3 (x-direction at D and ydirection at C) for the 0.2g earthquake input pair....................................... 100

Figure 4-6 Floor response spectra variability due to uncertainties in the 

concrete modulus computed on Level 3 (x-direction at D and ydirection at C) for the 0.6g earthquake input pair....................................... 101

Figure 4-7 Floor response spectra variability due to uncertainties in rebar 

yield strength computed on Level 3 (x-direction at D and ydirection at C) for the 0.6g earthquake input pair....................................... 102

Figure 4-8 Floor response spectra variability due to uncertainties in structural 

damping computed on Level 3 for the 0.6g earthquake input pair............ 102

Figure 4-9 Floor response spectra variability due to uncertainties in added 

floor mass computed on Level 3 for the 0.6g earthquake input pair......... 103

Figure 4-10 Inter-story drift transients for the 0.6g seismic input pair 

comparing the nominal concrete modulus with the lowest value in 

the parametric space .................................................................................... 104

Figure 4-11 Fixed-base model used for the nonlinear analyses in fragility 

assessment depicting the discretization of the concrete structure 

and the steel rebar distribution.................................................................... 108

Figure 4-12 Peak ground accelerations histogram of the 50 earthquake sets (x 

and y directions) used in fragility analysis ................................................. 114

Figure 4-13 Regression between different indicators of the 50 bi-directional 

synthetic records used in the fragility analysis.......................................... 115

Figure 4-14 Four acceleration time histories and four response spectra of a 

sub-set of the 50 synthetic records used in the fragility analysis 

(First two pairs of time histories and response spectra are on 

previous page – 116)..................................................................................... 117

Figure 4-15 Stratified sampling Latin Hypercube used for uncertainty 

distribution of the structural properties...................................................... 118

Figure 4-16 Cumulative distribution function of structural properties (concrete 

Ec and rebar Fy) of the 50 realizations used to log-normally 

distribute uncertainty.................................................................................... 118

Figure 4-17 Damage conditions (crack extend and pattern) reached in the 

structure at the end of four seismic inputs out of the 50 randomly 

selected earthquake-realization pairs.......................................................... 119

Figure 4-18 Damage conditions (reinforcement failure near the base of the 

structure) reached during one of the randomly selected 

earthquake-realization pairs......................................................................... 119

Figure 4-19 Regression between seismic input indicators of the 50-earthquake 

ensemble and structure response in terms of floor drifts and 

eigenfrequency degradation ........................................................................ 120
xii

Figure 4-20 Regression between non-linear analysis output variables 

monitored on the test structure for the 50 pairs of structural 

property realizations and the seismic inputs. (a) Regression 

between 1st and 2nd structural frequency drop, (b) regression 

between drifts along x and y at the same structural location and (c) 

regression between drift at a structural extremity (location D) and 

drop in 1st frequency.................................................................................... 121

Figure 4-21 Non-linear response spectra generated on level 3 of the structure 

at location D (x-direction) and location C (y-direction) from the 

analysis of the 30-set earthquake pairs....................................................... 122

Figure 4-22 Non-linear response spectra generated on level 3 of the structure 

at location D (x-direction) and location C (y-direction) from the 

analysis of the extra 20-set earthquake pairs ............................................. 123

Figure 4-23 Regression analysis between the logarithms of the PGA and 

eigenfrequency drop used to estimate median capacity and logstandard deviation ........................................................................................ 124

Figure 4-24 Fragility curves for all three damage states and all capacity 

measures using PGA as the demand indicator........................................... 125

Figure 4-25 Fragility curves for all three damage states and all capacity 

measures using PGD as the demand indicator .......................................... 125

Figure 4-26 Fragility curves for all three damage states and all capacity 

measures using CAV as the demand indicator........................................... 126

Figure 4-27 Fragility curve comparison for different drift measures - PGA ................. 127

Figure 4-28 Fragility curve comparison between drift and frequency drop - PGA

........................................................................................................................127

Figure 4-29 Fragility curve comparison between drift and frequency drop - CAV....... 128

Figure 5-1 Comparison of acceleration spectra between the direct method 

(i.e., both orthogonal excitations acting on the structure 

simultaneously) and the SRSS method where the two responses 

were computed independently..................................................................... 135

Figure 5-2 Generation of design response spectra by enveloping 

proceduresusing US NRC guidelines .......................................................... 136
xiii

LIST OF TABLES

Table 2.1 Concrete properties derived from actual casts of the SMART2008 

structure. .........................................................................................................12

Table 2.2 Shaking table recorded horizontal accelerations observed during 

tests. ................................................................................................................18

Table 2.3 Observed degradation of SMART200 test structure 

eigenfrequencies.............................................................................................20

Table 4.1 Structural properties variability matrix used in the sensitivity 

analysis of the SMART test structure ............................................................ 96

Table 4.2 Summary of computed inter-story drifts between structural levels 

at corner locations A and D for the 0.2g seismic input pair....................... 105

Table 4.3 Summary of computed inter-story drifts between structural levels 

at corner locations A and D for the 0.6g seismic input pair....................... 105

Table 4.4 Maximum relative displacements computed for the fixed-base 

SMART structure and the two seismic pairs (0.2g and 0.6g)..................... 106

Table 4.5 Summary of estimates of median capacity Am and log-normal 

deviation β. .................................................................................................... 124

xv

EXECUTIVE SUMMARY

This report presents the results of a comprehensive study of reinforced concrete structures 

exhibiting three-dimensional effects as well as non-linear response under multi-directional 

earthquakes. The study was started in 2008 and it consists of (a) shaking table tests and 

response results of a 1/4th scale structure, (b) numerical prediction results based on the test 

structure design and the shaking table multi-directional excitations, and (c) sensitivity and 

structural vulnerability assessments of the test structure leading to fragility curves for predefined limit states based on different metrics.

The primary goal of this multi-faceted effort was to understand, based on both the experimental 

results and the benchmarked numerical models, the implications of three-dimensional effects in 

reinforced concrete nuclear structures. These effects manifest themselves in the ability of these 

structures to withstand multi-directional seismic loads and the generation of more realistic 

response spectra or critical equipment loadings at elevations within the structure. Further, by 

utilizing the shaking table test results and those of a comprehensive sensitivity and vulnerability 

analysis performed on the test structure, another goal was to identify the most appropriate 

metric that captures the damageability of the overall structure and the establishment of limit 

states. 

The US Nuclear Regulatory Commission participated in this international effort through 

contractor support from the Brookhaven National Laboratory. The raw information that may

influence the nuclear design code evolution as a result of this effort stems from:

• the quantification of torsion effects, 

• the performance of a structure designed according to a nuclear code (in this case the 

French code) when subjected to seismic loads that far exceed the design basis 

earthquake, 

• the ability of the much advanced numerical modelling and analysis to predict the nonlinear response of a complex reinforced concrete structure, 

• qualitatively addressing various metrics for establishing limit states for a complex 

structure and their implementation into performance-based design approaches.

The SMART2008 project was initiated in 2008 as an international benchmark study and was 

opened to teams within the practicing nuclear engineering community as well as the earthquake 

engineering research community. It was supported by the Commissariat à l’Energie Atomique 

(CEA), Electricité de France (EDF), the Organization for Economic Cooperation and 

Development of the Nuclear Energy Agency (OECD/NEA) and the International Atomic Energy 

Agency (IAEA) Extra Budgetary Program on Seismic Safety. The primary focus was the 

response of reinforced concrete nuclear structures exhibiting torsion effects as well as nonlinear response. Of interest was the generation of more realistic floor response spectra to be 

used in the design of critical equipment. The experimental effort used a 1/4th scale 3-story 

structure tested on the shaking table capable of 100 ton capacity and 1g maximum 

acceleration. By accompanying the series of shaking table tests with a benchmark study by the 

international practicing community, the goal was expanded to enable the quantification of 

margins in different design methods accepted around the world, identify the more relevant 

methods in quantifying seismic analysis variability due to uncertainties in structural properties 

and seismic input, and generating appropriate fragility data for such structures. To accomplish 

the benchmark aspect of the SMART2008 study, two distinct phases were introduced. The first 

phase consisted of a blind prediction of a subset of the actual test results based on best
xvi

estimates of the structural properties and the intended shaking table accelerations. The blind 

prediction covered both “under design” and “over design” levels. The objectives were to (a) 

assess the different design methods for spectra generation, for both dynamic response and 

floor response, and (b) compare best-estimate methods ranging from linear to non-linear 

approaches and different levels of complexity in numerically representing the structure. During 

the second prediction phase, preliminary test results for some of the actual shaking tests were 

shared with the participants. This enabled a more realistic numerical representation of the 

structure in place of the best-estimate representation of the first phase, as well as the prediction 

of the response based on achieved shaking table excitations rather than desired load levels and 

earthquake content. Lastly, following the prediction phases and by utilizing the more realistic 

representation of the structure, sensitivity and vulnerability studies were conducted leading to 

fragility curves for pre-conceived limit states of the test structure. The primary objectives of the 

sensitivity/vulnerability phase of the SMART2008 benchmark were to (a) quantify the variability 

in the structural seismic response and identify contributions stemming from uncertainties in 

structural properties, and (b) assess and compare different methods leading to the generation 

of fragility curves associated with defined limit states and qualify the different metrics that were

used. 

While the 1/4th scale test structure was designed and constructed following the French nuclear 

design code, the observations and assessments stemming from the test results should not be 

generalized to the seismic behavior of other nuclear structures due to significant deviation of 

the tested structure from actual reinforced concrete nuclear structures especially at foundation 

conditions. However, the lessons learned from this study are very useful for numerical modeling 

and analysis of reinforced concrete structures subjected to multi-directional loading in the linear 

and nonlinear ranges.

A principal lesson to be taken away from this benchmark is that the non-linear techniques 

utilized is this benchmark indicate that non-linear techniques available for structural engineering 

models are or are becoming mature enough to warrant consideration in future editions of 

consensus codes and then in regulatory guidance, i.e., code developers should be considering 

the implementation of some of these techniques for future editions.

The report is organized in sections each of which describes in detail a particular aspect of the 

multi-faceted study. The introductory section provides a description of the background 

including the objectives and the scope of this undertaking, as well as a summary of the 

SMART2008 experimental study and the accompanying numerical prediction phases. Also 

included in the introductory section is a discussion of the aims and results of the SMART2008 

project on design approaches.

In Section 2, the SMART2008 shaking table tests, which represent the core subject of this 

effort, are described in detail. Specifically, the design specifics of the test structure and of the 

AZALEE shaking table used in the test are described along with the properties of the family of 

multi-directional earthquakes that form the seismic demand array. Shaking table results of 

Phase I, which included modal characteristics of the test structure as well as response to low 

level earthquakes, and thus mostly linear response, are presented and discussed. The results 

of the entire series of thirteen three-directional earthquakes that were used in the shaking table 

test ranging from under-design levels to as much as five times the code design level are 

discussed. Specifically, the measurements of the test structure response provided by CEA 

along with the damage progression which resulted from applying the earthquake loads in 

tandem are evaluated and discussed. Finally, in Section 2 an overall assessment of the shaking 

table results is presented including (a) the global behavior of the structure including torsional 

effects, (b) the evolution of floor response spectra with increasing input loads and (c) the 

damage onset and its evolution and damping characteristics of the structure.

Section 3 depicts the description of the numerical modeling and dynamic analysis formulation 

used to simulate the shaking table tests. Discussed in this section are the results of the “blind 
xvii

prediction” and “best estimate” phases which formed the initial approach to prediction of the 

actual test. Adjusted response predictions triggered by data provided as a result of low-intensity 

shaking table tests that reflected the actual test structure and the achieved shaking table 

excitation are presented, along with a discussion on the influence of uncertainties in the bestestimate phase. The overall performance of the prediction model against the actual shaking 

table tests at different earthquake intensity levels is discussed in detail. Discussed in Section 3 

is the observed damping of the system during the shaking table tests, its impact on the 

numerical analyses performed, and its relevance to the general design practice. Included also 

in Section 3 is a discussion on the torsional effects exhibited during the tests and captured in 

the numerical evaluation.

The sensitivity and vulnerability studies that were undertaken using the finalized prediction 

model described in Section 3 are presented in Section 4 of the report. In particular, the 

objectives and the expected value of these studies are discussed along with the results of the 

numerical studies encompassing both the sensitivity or uncertainty in structural capacity and the 

fragility analysis. Discussed in detail is the vulnerability assessment based on the generated 

fragility curves using different approaches and metrics and their connection to the structural 

limit states.

In Section 5 a discussion on the lessons learned as a result of both the shaking table tests, the 

numerical predictions of the test structure response, and the “non linear” floor response spectra 

derived from both the tests and the simulations is presented while focusing on the torsion 

effects induced on three dimensional structures. Further, based on the results of the sensitivity 

and vulnerability analyses with emphasis on the fragility curve generation, the concept of the 

limit state is discussed. Most importantly in this section, the connection of the information 

manifested in (a) the 3D effects, (b) the non-linear floor response spectra, and (c) the fragility 

assessment methodology tested against actual data of the SMART2008 structure subjected to 

several times the design level is explored and discussed.

This report summarizes the various aspects of the benchmark and the findings that were drawn 

from the shaking table tests and the numerical analyses. Based on this study, the following 

specific observations were noted and may warrant further consideration, as consensus codes 

are updated and the NRC revises regulatory guidance:

1. For the model used in the SMART2008 experiment, the capacity of the model, built 

per the French code, was higher that the design capacity, i.e., the model was 

designed for a 0.2g peak ground acceleration and withstood shaking table induced 

ground motions of about 1.1g with only concrete cracking and minor localized steel 

reinforcement yielding.

2. The observed reductions in stiffness at lower seismic intensities are attributed to 

micro-cracking within the concrete.

3. The use of non-linear treatment of real concrete structures should be considered by 

regulatory authorities and it may be appropriate to allow it to influence the design 

through consensus codes.

xix

ACKNOWLEDGMENTS

The work described in this report was sponsored by the Office of Nuclear Regulatory Research 

of the U.S. Nuclear Regulatory Commission. The authors would like to express their gratitude to 

Dr. A.J. Murphy (NRC Project Manager on this research program) for appreciating the 

importance of the involvement in the two international seismic benchmark studies whose results 

and findings are reported herein.

The authors would like to also extend their gratitude to the organizers and sponsors of the 

SMART2008 benchmark (CEA, EDF, OECD/NEA and IAEA) and in particular P. Labbe of EDF 

and C. Thierry and S. Juster-Lermitte of CEA for their invaluable contributions towards the 

organization and execution of the SMART2008 project.

1

1. INTRODUCTION

1.1. Background

The seismic response of reinforced concrete (RC) structures exhibiting three-dimensional or 

primarily torsional effects as well as non-linear behavior is a concern within the earthquake 

engineering community and more so within the regulatory agencies overseeing nuclear 

facilities. The need to understand the behavior of such structures is not only the vulnerability or 

seismic capacity of the structure itself but also the proper evaluation of floor response spectra 

based on which critical equipment are designed. Traditionally, simplified approaches have been 

adopted for the seismic design of the structures (i.e., stick model representations) where the 

torsional effects are forced out of the estimation. In these simplified approaches the multidirectional nature of the earthquake and its effect on the structure as well as non-linearities in 

the structural response are accounted for through response combination schemes, i.e., Square 

Root Sum of the Squares method, and equivalent linearization.

In order for these effects to be quantified, a dedicated seismic test was proposed where both 

the nuclear design codes in above-design level situations and the ability of the earthquake 

community to predict the complex response of an equally complex structure were evaluated. A 

seismic test accompanied by a benchmark study was initiated by the Commissariat à l’Energie 

Atomique (CEA) and Electricité de France (EDF). The project was endorsed by the 

Organization for Economic Cooperation and Development (OECD)/Nuclear Energy Agency 

(NEA) and the International Atomic Energy Agency (IAEA). The project termed SMART-2008 

project (Seismic design and best-estimate Methods Assessment for Reinforced concrete 

buildings subjected to Torsion and non-linear effects) was started in 2008 as an international 

benchmark study by inviting participation by research teams from both the practicing nuclear 

engineering and the earthquake engineering research communities.

The US NRC’s Office of Nuclear Regulatory Research, in collaboration with Brookhaven 

National Lab (BNL) joined this proposed international benchmark on seismic design of nuclear

structures. The primary goals were to study in detail, through the assessment of the shaking 

table test results and the results of the numerical analyses employed to predict the response, 

(a) the torsional effects on 3-D structures, (b) the generation of “non-linear” floor response 

spectra, and (c) fragility assessment methodologies based on the SMART structure. Utilizing 

the wealth of new information relating to the seismic response of 3D structures, collectively as a 

group of participating research teams in the benchmark, and individually as a BNL team, a 

subsequent goal was to assess the potential impact on design provisions relevant to the US 

nuclear design practices. 

1.2. Objective

The objective of the study was to independently evaluate and interpret the SMART2008 

shaking table tests and to develop a state-of-the-art numerical prediction model to be used in 

the various phases of the benchmark study. The tests consisted of blind test predictions based 

on best-estimate structural data and actual predictions aided by structural data revealed by lowintensity tests performed on the test structure. Through the benchmarking simulations and the 

test results, the aim was to assess the non-linear response of the reinforced concrete structure 

exhibiting torsional effects and in the process to evaluate the evolution of non-linear floor 

response spectra. 

A key part of the overall objective was the study of structural and seismic input uncertainties on 

3D structures by conducting sensitivity studies around key structural properties and evaluating 

the seismic fragility curves of such structures using a large family of earthquakes. 
2

Further, upon the completion of the tests and simulations and their analysis, the important 

objective was the identification of areas that are expected to influence the acceptable design 

practices. .

1.3. Seismic Tests and Response Predictions

In order to study and validate approaches that can be used in dynamic analysis and response 

of reinforced concrete structures which may exhibit three dimensional effects (typical of a 

nuclear structure) as well as non-linear behavior, a 3-story test structure was built according to 

the French seismic code and was seismically tested on the AZALEE shaking table at the CEA 

Saclay, France. The test structure shown in Figure 1.1 reproduced a nuclear building at 1/4th

scale and was designed according to the RCC-G nuclear code (Regles de Conception et de 

Construction du Genie civil). Its seismic design was based on a 5% damped design spectrum 

anchored at 0.2g acceleration. The test structure was a three story trapezoid made of 

reinforced concrete and consisted of 4 walls supported on a continuous footing and mounted on 

the AZALEE shaking table via a steel plate. The design of the structure enabled the mass 

centre and the effective torsion centre to be along different axis and thus amplify the torsional 

effects resulting from the seismic loads. 

The shaking table supporting the SMART structure is shown in Figure 1.2. It has a capacity of 

100 ton and can produce1.0 g maximum acceleration. It has a total mass of 25 tons and is fixed 

to eight hydraulic jacks (4 in the horizontal directions and 4 in the vertical direction) which are 

controlled during the experiment. A more detailed description of the SMART structure, the 

shaking table and their interface is provided later in the report.

The 5% damping design spectrum anchored to 0.2g peak ground acceleration used in 

designing the test structure is shown in Figure 1.3.

Figure 1-1 SMART2008 test structure on the Azalee shaking table at CEA 

Saclay, France
3

Figure 1-2 Details of Azalee shaking table used in the SMART2008 tests

Figure 1-3 SMART2008 test structure design response spectrum

A total of thirteen (13) pairs of earthquakes (termed RUN1 thru RUN13) acting in the two 

horizontal directions simultaneously formed the basis of the seismic tests. Three of the 

earthquake pairs (RUN1, RUN2 and RUN3) were real, low-intensity records while the remaining 

ten (RUN4 through RUN13) were synthetic and were derived from the design spectrum starting 

at 0.1g and increasing to 1.0g for RUN13. The shaking table testing during which the thirteen 

pairs were applied in sequence thus ensuring the progression of damage consisted of two 

phases. In the first phase, the structure was subjected to earthquakes that represented the 

under-design, the design, and the over design levels (RUN1 to RUN8). Since it was intended by 

the designers that the test structure remains within the elastic regime during the design basis 

earthquake (RUN4), it can be assumed that this level earthquake would be equal to or less than 

an Operational Basis Earthquake (OBE) in the US. 

These tests helped reveal uncertainties in the structural properties of the specimen as well as 

the controllability of the actuators in transferring the desired acceleration record to the shaking 
4

table. Following these initial set of tests, the entire suite of thirteen earthquake pairs was 

completed. During the first phase, the over-design level, designated as RUN8 table 

accelerations of 0.41g in the x-direction and 0.55g in the y-direction, were recorded and the 

onset of non-linear structural response was observed.

As an integral part of the SMART2008 benchmark, an international numerical prediction contest 

was conducted to: 

• assess the state-of-the-art of capabilities to numerically analyze non-linear structures 

and compare the different best-estimate methods for floor response spectra evaluation, 

and

• evaluate the different methodologies and regulatory guidelines on nuclear design.

By introducing the participation of the international practicing community, the aim was to enable 

the quantification of margins in different design methodologies accepted around the world and 

the identification of more relevant methods in quantifying seismic analysis variability due to 

uncertainties in structural properties and seismic input and generating appropriate fragility data 

for such structures. 

The benchmark study consisted of two distinct phases. The first phase was conducted prior to 

the fabrication of the specimen and consisted of two stages. First, static, modal and dynamic 

analyses were performed on the structure using conventional methods of choice (i.e., spectral, 

pushover, linear and/or nonlinear time history analyses). This was followed by a “blindprediction” of the shaking table test where two seismic pairs (0.2g and 0.4g PGA) were applied 

to the structure using best-estimate values of the structural properties. These studies were 

meant to provide a measure of the variability that results from the choice and complexity of the 

numerical model, the hypotheses, and the type analysis used. Following the construction of the 

specimen and the completion of the first phase of the shaking table tests, a best-estimate 

prediction was performed. This prediction phase was based on specific information that 

became available after the construction and initial testing of the specimen. The information 

pertinent to the shaking table test set-up, the as-built structure and achieved shaking table 

excitations enabled the updating of the numerical models that were then used for the bestestimate predictions. Specifically, information on

• experimental set-up including the mass of the structure and the actual added loads 

• concrete and reinforcing steel properties deduced 

• initial frequency and damping ratio prior to the tests 

• seismic input accelerations observed on the table 

• selected test results for a low-intensity signal and the design level

was provided and used in analyzing the thirteen earthquake pairs in series. 

A comprehensive set of prediction results generated using the BNL numerical model and a fully 

non-linear analysis reflected in the constitutive relations of the concrete and the reinforcing 

steel are presented in the report. The prediction results are compared with the actual test 

results and are accompanied by assessment and discussion.

1.4. Sensitivity Studies and Fragility Evaluation

Following the completion of the shaking table tests and the best-estimate predictions, the 

variability in the response of structures stemming from (a) structural property uncertainties, 

(b) random character of the seismic input and (c) numerical representation of the structural 

system sensitivity and vulnerability analyses were conducted. The vulnerability analyses sought 

to develop fragility curves. 
5

The primary objectives of the sensitivity/vulnerability phase of the benchmark were to: 

(a) quantify the variability in the structure seismic response and identify the contribution 

stemming from uncertainties in structural properties, and 

(b) assess and compare different methods leading to the generation of fragility curves 

associated with defined limit states and to evaluate the qualification of the different 

metrics that are used. 

Results derived from the sensitivity and vulnerability analysis in terms of floor response spectra 

and fragility curves for different combinations of demand and capacity indicators are presented 

in Chapter 4 of the report. Also included are discussions on the methodologies used to develop 

fragility curves, as well as discussions on the findings and the needs for further studies. 

1.5. Lessons Learned and Design Impact

The results and assessment derived from the distinct phases of the benchmark are 

summarized in last section of the report. 

First, the shaking table tests and the recorded structural response are discussed in terms of 

(a) challenges in conducting such a test and in particular the ability to control the key elements 

comprising the test such as the shaking table and structure interaction and its influence on the 

observed in-structure response, (b) the evolution of the “damage” observed in the structure as a 

result of the sequential application of seismic loads with increasing intensity, (c) the in-structure 

response in terms of accelerations, inter-story drifts and most importantly floor response 

spectra, and (d) qualification of the overall response and exhibited structural capacity as 

compared to the design levels. Safety-significant observations were made regarding the overall 

response and the changes that occur in the concrete material, such as softening in the 

concrete, at much lower than design level intensity values.

The performance of the numerical model generated and implemented in the non-linear analysis 

and prediction of the test is summarized and the potential implications in the design practice are 

drawn and presented in the summary section. Specifically, based on the demonstrated ability of 

the computationally costly non-linear model to trace the structural response even when the 

structure entered the non-linear regime, arguments are presented for the potential use of these 

types of analyses in the design practice of reinforced concrete structures. Observations on 

studies that will further enhance the understanding of the nonlinearities in reinforced concrete, 

and subsequently the adaptation into numerical models and design analyses, are also made.

The assessments from the dedicated studies performed on the sensitivity of the structural 

response due to uncertainties in structural parameters and the vulnerability study which 

resulted in seismic fragility curves are summarized along with considerations for further study. 

The need for enhanced approaches or alternative methods that may lead to seismic fragility 

assessment with higher confidence levels is discussed.

Finally, by taking into account the results and lessons learned from the three main thrusts of the 

SMART2008 benchmark summarized above, design implications are drawn and presented. In 

particular the performance of the actual structure in terms of capacity, evolution of its dynamic 

characteristics at below design level and the exhibited damping are put into context with the 

current practice and design provisions. Further, the potential of using the non-linear type 

analysis in design provisions and especially in validating current practice and methodologies to 

account for nonlinearities in the reinforced concrete at levels which currently ensure the linearelastic behaviour is discussed.

7

2. SMART2008 BENCHMARK

To address the issue of multi-directional seismic loading on nuclear structures where torsion is 

induced while it is influenced by the non-linear behaviour of reinforced concrete, an 

international benchmark study under the term SMART-2008 (Seismic design and best-estimate 

Methods Assessment for Reinforced concrete buildings subjected to Torsion and non-linear 

effects) was initiated. The benchmark that was coordinated by CEA Saclay, France and 

supported by Electricite de France (EDF) was established around a scaled nuclear structure 

that was to be tested on the Azalee shaking table at CEA under multi-directional seismic 

loadings. In order for the study to be relevant to the nuclear sector, the scaled structure was to 

be designed according to the French nuclear code RCCG and used a 5% damped design 

spectrum anchored at 0.2g acceleration (see Figure 1.3). Following the construction of the 

scaled nuclear structure, a sequence of selected multi-directional earthquake inputs consisting 

of real seismic records and synthetic ones derived from scaled design spectra has performed 

on the shaking table to assess the vulnerability of the structure and its damage progression. 

Shown in Figure 2.1 is a flowchart of the various phases of the SMART2008 benchmark which 

addressed different aspects of the study. 

In Phase-Ia, preliminary analyses were performed on the SMART2008 structure based on the 

per-design model and the expected structural and material properties of the structure. Static 

analyses for a fixed-base structure under (a) gravity loads which included the additional masses 

on the floor slabs and (b) horizontal loads applied at the 3rd level were performed. A series of 

modal analyses were also performed which included (a) the fixed-base and free-free structure 

without the added masses and prior to its placement on the shaking table and (b) the massloaded structure under fixed-base conditions and on the shaking table. The primary goal of the 

preliminary static and modal analyses was to assess the variability of solutions stemming from 

the different numerical methods the participating teams were utilizing in the SMART2008 

benchmark (i.e. stick model representation, shell or solid element, rebar consideration, etc.). 

The dynamic analysis or blind prediction of Phase-Ia consisted of a conventional dynamic 

analysis and a best-estimate analysis subtask. For the conventional analysis the participating 

teams were provided with synthetic time histories and corresponding response spectra of a twodirectional synthetic earthquake of 0.2g peak acceleration to analyse the structure using either

time history analysis or response spectra analysis to predict the response of the structure which 

exhibits the per-design properties. Upon completion of the conventional dynamic analysis, a 

best-estimate prediction of the response of the structure based on shaking table inputs (thirteen 

pairs of horizontal accelerations consisting of real and synthetic earthquakes with increasing 

peak acceleration to 1.0g) was performed and a series of structural response data (resultant 

forces, peak displacements, peak accelerations, floor response spectra, stresses and strains in 

the concrete and rebar, etc.) were computed and provided to the SMART2008 organizers. The 

choice of analysis for the blind prediction (spectral, time-history, equivalent static, etc.) was left 

up to the participating teams. Non-linear time history analyses were adopted and used in the 

US NRC/BNL study.

Following the fabrication of the test structure and the completion of the shaking table tests, the 

as-built structural parameters were provided to the participants along with the actual shaking 

table accelerations achieved during the tests. During Phase-Ib, which followed the shaking 

table tests, the structural model was updated to reflect the “as-built” conditions and the series of 

the thirteen (13) pairs of horizontal accelerations recorded on the shaking table during the tests 

were used to predict the response and damage of the structure and make direct comparisons 

with the test results.

Upon completion of the shaking table test phase, a vulnerability study defined as Phase II of the 

SMART2008 benchmark was performed. The primary objectives of the study were to (a) 

quantify the sensitivity of the seismic response stemming from uncertainties surrounding 

structural properties and the random character of the seismic input and (b) assess the 
8

vulnerability of the structure and generate fragility curves by accounting for uncertainties in the 

structural modelling, in concrete constitutive relations, and in damage progression. The report 

focuses on the evaluation of the actual test data, the numerical prediction based on the as-built 

structure and the actual shaking table inputs (Phase-Ib) and on the vulnerability study (Phase 

II).

This chapter focuses on the experimental part of the SMART-2008 benchmark and presents 

details relevant to the actual test. This includes (a) the description of the scaled structure and 

the material properties that are influential in the seismic response, (b) the description of the 

shaking table infrastructure and the interface conditions between the test structure and the 

table, (c) the description of the input earthquake suite and discussion on its dynamic 

characteristics, and (d) the presentation of experimental campaign results and assessment.

Figure 2-1 SMART2008 benchmark flowchart of the sequence of phases
9

2.1. SMART2008 Test Structure

The SMART2008 test structure shown schematically in Figure 2.2 is a reproduction of a nuclear 

building at 1/4th scale and was designed according to the French nuclear code RCCG with its 

seismic design based on a 5% damped design spectrum anchored at 0.2g acceleration. The 

test structure is a three-story trapezoid made of reinforced concrete. IT consisted of 4 walls 

supported on a continuous footing and mounted on the AZALEE shaking table via a steel plate. 

Key provision in the design of the structure is to enable for the mass centre and the effective 

torsion centre to be along different axis and thus accentuate the torsional effects of the seismic

loads on the structure. 

Figure 2.3 illustrates all the dimensional details of the test structure. The 3-story structure is a 

1/4th scale trapezoidal shape model of a typical nuclear support structure consisting of three 

walls forming a U shape. Two of the walls have openings in all three levels to reproduce 

realistic design configurations of shear walls. A reinforced concrete column connected to all 

three level slabs and was situated in the geometrical middle of the structure. Reinforced 

concrete beams supporting the floor slabs span the distance between one of the walls and the 

column. Based on the 1/4th scale model the mass scaling is 1/16 and, therefore, the wall 

thicknesses were selected accordingly. All three walls are 20 cm thick. The three beams had a 

cross section of 15cm x 32.5 cm and the column a cross section of 20cm x 20cm. The wider of 

the tree walls (normal to the beams) was designated as Wall #V01 and Wall #V02 (beam midsection representing the separating line) while the remaining two walls were designated as Wall 

#V03 (wall with opening) and Wall #V04.

The structure was supported on a continuous 38cm wide by 15 cm high reinforced concrete 

footing (see Figure 2.5) that was built on a continuous 62cm wide steel plate of 2cm thickness. 

The steel plate was the means by which the whole structure was mounted on the top surface of 

the shaking table. The reinforced concrete column, on the other hand, was directly anchored on 

the 62cm by 62cm steel plate which in turn was mounted on the shaking table with screws.

The centre of mass of the SMART2008 structure is at xcg =1.28 m and ycg = 0.98 m based on 

the coordinate system shown in Figure 2.3c.

The steel reinforcement in the test structure was designed according to the European design 

code EC2. An elaborate reinforcing scheme was implemented using a set of different bars. 

These include HA10 (Ǿ=10mm; Rm=654-661 MPa), HA8 (Ǿ=8mm; Rm=577-591 MPa), HA6 

(Ǿ=6mm; Rm=617-637 MPa), HA4 (Ǿ=4mm; Rm=565 MPa). HA10 rebars were used in the 

foundation and the reinforced column. The reinforcement in the wall consisted of primarily 4mm 

diameter bars at 100 mm spacing which was further supplemented by HA10, HA8 and HA6 

rebars near the wall openings. The nominal yield stress for the reinforcing steel was Fe = 500 

MPa.

Nominal concrete properties were initially established in order for the preliminary estimates of 

the structural properties to be derived and also for the numerical side of the benchmark to be 

started. These basic concrete properties are listed below.

fc = 30 MPa Compressive strength

ft = 2.4 MPa Tensile strength

E = 32,000 MPa Concrete Young modulus

 = 0.20 Poisson’s ratio

Following the initial estimates, results of concrete cylinder tests from actual casts used to 

construct the structure became available. The cylinder tests showed that the actual concrete 

modulus was lower than the best estimate listed above and varied between casts. The results 

of seven (7) different casts are listed in Table 2.1. Listed in the table are both concrete tangent 
10

and secant modules with the latter derived from the first point on the stress-strain curve and the 

1/3rd of the ultimate (failure) loading.

The concrete density, based on the test cylinders was approximately 2372 kg/m3

. The average 

density of the reinforced concrete of the structure was approximately 2460 kg/m3

.

Since gravity loads and stresses do not undergo scaling, to reproduce the state of stress in the 

structure walls and column anticipated in typical nuclear structures, additional loads were 

applied on each level using lead and steel blocks placed directly on each slab. The total weight 

of the structure, following the addition of floor masses, was estimated at about 46.81 T and is 

distributed as follows:

weight of structure ~ 9.31 T

weight of foundations ~ 2.11 T

added load at level 1 ~ 12.06 T 

added load at level 2 ~ 12.70 T 

added load at level 3 ~ 10.63 T 

Figure 2-2 SMART2008 test structure schematic
11

Figure 2-3 Geometric description of the SMART2008 structure

Figure 2-4 Footprint of the SMART2008 structure with CG and torsion 

axes
12

Figure 2-5 Geometric description of the base plate and foundation

Table 2.1 Concrete properties derived from actual casts of the 

SMART2008 structure.
13

2.2. Shaking Table Characteristics

The Azalée shaking table, shown in Figure 2.6 with the schematic of the SMART2008 structure 

placed on it for reference, has a total mass of 25 tons and for simplicity it may be considered as 

a rigid block. It is fixed to eight hydraulic jacks (4 in the horizontal direction and 4 in the vertical 

direction) with their orientation shown in Figure 2.7. As indicated in Figure 2.7, the distance 

between two vertical actuators is 4 meters. The actuators controlling the horizontal motion of

the table are located at ~1.02 m below the upper face of the shaking table (where the 

SMART2008 structure is mounted via the steel base plate), while the centre of gravity is 0.60 m 

below the table top. All the jacks can be described as active systems, meaning that they are 

controlled during the experiment. The implication, however is that it is extremely difficult to 

extract dynamic properties which are expected to influence the dynamic response of the 

shaking table system that in turn will affect the response of the supported structure. 

Given the geometry of the test model, rocking motions may be experienced at the shaking 

table’s level during the seismic tests. Due to gaps in the understanding of the dynamic behavior 

of the overall table system stemming from the active nature of the actuators, the structure/table 

interaction is very difficult to predict. Approximate data regarding the effective stiffness of the 

vertical actuators have been used in past shaking table tests where the vertical actuators were 

represented by 215 MN/m springs. Based on the position of the test structure on the shaking 

table and the coordinate system used, the table center of mass is estimated to be at XCG

TABLE = 

1.50 m and YCG

TABLE = 0.92 m (1.28 m and 0.92m respectively for the test structure).

Figure 2-6 Geometric description of the AZALEE shaking table
14

Figure 2-7 Shaking table hydraulic jacks/actuators orientation

2.3. Instrumentation

An extensive array of 3-D accelerometers, displacement transducers and rebar strain gauges 

were introduced onto the test structure in an effort to monitor and capture the complex 

response. As shown in Figure 2.8, accelerometers capturing the horizontal components of the 

shaking table accelerations (AXTAB and AYTAB) were installed on the top surface at table 

mass centre. Four vertical accelerometers (AZTAB1-4) placed at each corner were to monitor 

the vertical motion and thus rocking of the table. Shown also in Figure 2.8a are a series of 

accelerometers capturing the horizontal and vertical components of acceleration at foundation 

level (i.e. Ac00xA, Ac00yA, Ac00zA). The distribution of accelerometers over the three levels of 

the structure and of the displacement transducers for Wall 3 and 4 respectively are shown in 

Figure 2.9. Displacements were measured by cable extension sensors with precision 

potentiometers.

Strain gauges were placed on several extreme locations of the structure and in particular on the 

reinforcing steel (as shown in Figure 2.10) near the foundation and the first story walls to 

monitor steel non-linear behaviour. Figure 2.10b depicts an actual strain gauge placed on one 

of the rebars of Wall 4. Shown in Figure 2.11 is the constructed SMART2008 structure on the 

shaking table with the added floor masses and the instrumentation on key locations of its walls.

(a) (b)

(b)

Figure 2-8 Layout of accelerometer distribution on the shaking table,

the foundation of the structure, and the floor slab locations
15

Figure 2-9 Acceleration sensor locations on Wall #3 and displacement

transducers on Wall #4

(a) (b)

Figure 2-10 Schematic and actual configuration of rebar strain gauges
16

Figure 2-11 Instrumented and loaded SMART2008 structure on shaking 

table prior to the seismic tests
17

2.4. Shaking Table Seismic Tests

Following the construction, instrumentation, and placement of the test structure on the Azalee 

shaking table, a suite of bi-directional horizontal earthquakes were applied and the response of 

the structure was monitored and recorded. The shaking table tests began in July of 2008 and 

were completed in the Spring of 2009. 

Seismic Input Suite

The SMART2008 benchmark consisted of thirteen pairs of seismic table excitations which were 

applied in sequence in an effort to observe the structural degradation and non-linear effects of 

the test structure. In other words damage that was induced by a given earthquake pair in the 

series would affect the response of the structure under the application of the next pair which 

was to increase in intensity. Low-intensity white noise excitations were applied between runs in 

an effort to extract key dynamic properties of the progressively damaged structure 

(i.e., eigenfrequencies). 

It was understood early in the benchmark effort that the effect of the shaking table on the 

dynamic response of the test structure may be dominant. Specifically, the eigenmodes of the 

structure will be directly linked to the dynamic properties of the table/structure system. It is to be 

expected that participation of the shaking table in the system, because of its large mass, will 

reduce the eigenfrequencies of the structure considerably. Since the response of the structure 

is the result of the convolution of its dynamic properties with the frequency content of the 

seismic excitation, it is thus expected that the structural response will differ depending on where 

its eigenfrequencies lie. Therefore, it is crucial that the dynamic properties of the system, as a 

whole, are well understood. As noted earlier, however, knowledge gaps exist regarding the 

dynamic characteristics of the shaking table stemming primarily from the active nature of the 

actuators. Recognizing the importance of this information, both the benchmark organizers and 

the participating research teams conducted sensitivity studies prior to the shaking table tests to 

quantify the effect. Based on numerical best-estimates the first three important modes (two 

flexural modes and twisting mode) differed significantly between the fixed-base configuration of 

the structure (where no shaking table is present) and the case of shaking table participation 

under best estimate dynamic property values for the table system. For example, the first 

flexural mode identified in the fixed base at frequency ~9 Hz under table participation was 

reduced to ~7 Hz. The second flexural mode from ~16 Hz in the fixed-base configuration was 

reduced to ~9 Hz and the twisting mode from ~28 Hz to 16 Hz. Hammer-induced white noise 

tests on the SMART2008 structure prior to the series of seismic loads generally confirmed the 

effect of the shaking table. Figure 2.12 depicts the numerical estimates of the eigenfrequencies 

prior to the test and the progressive damage of the structure. As a result, monitoring of the 

modal degradation of the structure was undertaken during the series of tests utilizing both white 

noise inputs between the tests or estimating structural frequencies from the tail-ends of the 

response transients.

The thirteen horizontal pairs used in sequence in the test consisted of both real and synthetic 

records. Three real records of lower intensity than the design level of 0.2g where applied first. 

The input accelerograms of these real earthquake records are depicted in Figure 2.13. These 

were scaled from the actual records by a factor between 1.03 and 1.58 in order to arrive at a 

PGA = 0.05g, a minimum level at which the shaking table can be controlled. Depicted in 

Figure 2.14 are acceleration time histories recorded on the top surface of the shaking table 

during the application of the three real records along with the corresponding response spectra.

The ten synthetic earthquake pairs were generated from the design spectrum shown in 

Figure 2.1 anchored at 0.1 g and scaled to 1.0 g in order to generate pairs with 0.1g 

incremental increase in intensity. Figure 2.15 depicts the shaking table horizontal accelerations 

recorded during the RUN4 test. Figure 2.16 shows the response spectra of the shaking table 
18

acceleration which needs to be compared to the one in Figure 2.1. Clearly seen in the actual 

table spectra are the two peaks corresponding to the first two flexural modes of the 

table/structure system.

Test Observations

Following the completion of the sequence of the thirteen horizontal acceleration pairs 

introduced on the table through the actuators and the recording of (a) accelerations on the table 

top and throughout the structure, (b) absolute displacements (measured with respect to the 

facility floor surrounding the moving table), (c) reinforcing rebar strains, and (d) damage 

progression through monitoring of the concrete crack development, the following major 

observations were made:

Shaking Table – Structure Interaction

Of primary interest in the course of the 13-run test was the qualification and quantification of the 

influence of the shaking table on the response of the SMART structure under study. As noted 

earlier in the report, controlling the shaking table and especially in the vertical direction was a 

challenge. Compounded onto the controllability of the table is the epistemic uncertainty 

surrounding its effective dynamic properties. As a result, there has been a significant variability 

between the intended motions on the shaking table and the recorded motions on the table top 

surface during the tests. Table 2.2 depicts peak ground acceleration comparisons between the 

input goals and the actual recorded motions on the table. The difference observed is assessed 

to be the combined result of both the inability to control the vertical and rocking motion of the 

table and of the structure-table interaction. As noted in Figure 2.16 where the table spectra 

during the synthetic seismic pairs for RUN4 are shown, the eigenmodes of the structure are 

influencing the shaking table response. This is clearly depicted by the appearance of two 

distinct peaks at 7 Hz and 16 Hz which in turn correspond to the fist two modes of the 

SMART2008 structure. Further, as shown in Figure 2.17 for RUN4 and RUN13 pairs, the 

vertical acceleration measured on the table (geometric corners) is approximately equal in PGA 

to the horizontal components.

Table 2.2 Shaking table recorded horizontal accelerations observed 

during tests.

Acceleration Axx (g) Acceleration Ayy (g)

Goal Test Goal Test

Run 1 0.05 0.08 0.05 0.03

Run 2 0.05 0.08 0.05 0.05

Run3 0.05 0.19 0.05 0.15

Run 4 0.1 0.20 0.1 0.24

Run 5 0.2 0.19 0.2 0.20

Run 6 0.3 0.23 0.3 0.32

Run 7 0.4 0.33 0.4 0.35

Run 8 0.5 0.41 0.5 0.55

Run 9 0.6 0.41 0.6 0.56

Run 10 0.7 0.53 0.7 0.67

Run 11 0.8 0.58 0.8 0.77

Run 12 0.9 0.70 0.9 1.06

Run 13 1.0 0.75 1.0 1.13
19

Evolution of Eigenfrequencies

To assess the progression of damage in the reinforced concrete structure resulting from the 

sequence of seismic excitations, special attention was paid in estimating the evolution of 

eigenfrequencies as a key indicator of damage. Between some of the successive tests, white 

noise low-level excitations were applied on the structure in order to arrive at an estimate of the 

condition of its dynamic properties. Estimates of the updated frequencies were derived from 

displacement response spectra that resulted from the white noise input. In addition, by utilizing 

the free-vibration of the structure that followed the end of the strong motion part of the input 

record, similar estimates were made. The two approaches resulted in similar estimates of 

eigenfrequencies. The main objective and impact of such estimation on both the shaking table 

tests assessment and the numerical evaluation of the response is a better understanding of the 

actual degradation that takes place and its influence on the floor response spectra of the 

“damaged” structure. Shifting of the frequencies due to concrete damage or rebar yield will 

indicate change in the stiffness of the structure and potential shifting away or towards 

frequency bands of the input excitation containing the bulk of its energy.

Table 2.3 lists the estimates of the first three structural eigenfrequencies (f1 = flexural along xaxis, f2 = flexural along y-axis, and f3 = twisting) upon completion of the corresponding input 

record. It should be noted that the modes identified as flexural modes are not purely flexural but 

have twisting contribution due to the asymmetry in the SMART structure. Shown in Figure 2.12 

are numerical estimates of the modes prior to the application of the thirteen seismic tests. 

Based on the eigenfrequency evolution listed in Table 2.3 the following important observations 

are made:

• A drop in eigenfrequencies is observed even after the application of very low intensity 

excitations (i.e. real records in RUN1 through RUN3) where no cracking in concrete or 

yielding in the reinforcement is expected. This has been attributed to damage in 

concrete due to micro-cracking. Similar observations have been made during similar 

shaking table tests on the seismic response of shear walls. This is significant in nuclear 

design due to the fact that the structural eigenfrequencies may experience shifting even 

at levels far below the design basis earthquake.

• A significant drop in frequency is observed following RUN5 even though the intensities 

of the input accelerations are at the level of the design basis earthquake for the 

structure. As will be discussed in more detail in the next section, this has been attributed 

to the fact that the structure during the application of the real, low intensity earthquakes 

exhibited low levels damping and as a result it experienced a significant number of 

stress cycles following the end of the strong motion input part of the RUN3 test.

• Eigenfrequency shifts of ~50% are observed for the 1st structural mode following the 

thirteen shaking table tests. Smaller shifting is observed for the next two modes. These 

observations are critical both in quantifying damage evolution but also in evaluating floor 

response spectra in structures that experience earthquakes below the design basis 

level.
20

Table 2.3 Observed degradation of SMART200 test structure 

eigenfrequencies.

Figure 2-12 Numerical estimates of the first three eigenfrequencies and 

modes of the SMART structure on the shaking table prior to 

the application of the thirteen seismic tests
21

Figure 2-13 Acceleration time histories of the three real earthquake 

records used in the SMART2008 shaking table experiment
22

 

RUN1 (real earthquake) Accelerations Applied to Shake Table

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 1 2 3 4 5 6

sec

Acceleration (g)

Run1_Table_axx

Run1_Table_ayy

RUN2 (real record) Input Accelerations

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 1 2 3 4 5 6

secs

Acceleration (g)

Run2_Table_axx

Run2_Table_ayy RUN2 (real record) Response Spectra

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 10 100

Freq (Hz)

Spectral Accel (g)

Run2_Table_RSxx

Run2_Table_Rsyy

RUN3 (real record) Input Acceleration

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5 6 7 8

secs

Acceleration (g)

Run3_Table_axx

Run3_Table_ayy

RUN3 (real record) Acceleration Response Spectra

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 10 100

Freq (Hz)

Spectral Accel (g)

Run3_Table_RSxx

Run3_Table_Rsyy

Figure 2-14 Acceleration time histories recorded on the shaking table 

during the SMART2008 tests RUN1, RUN2 and RUN3 and the 

corresponding response spectra
23

Figure 2-15 Acceleration time histories recorded on the shaking table 

during the SMART2008 RUN4 test for the two horizontal 

directions x and y

Figure 2-16 Acceleration response spectra on the shaking table during 

the SMART2008 test RUN4 indicating deviation from the 

design spectrum of Figure 2.1 near the structural modes at 7

Hz and 16 Hz
24

Figure 2-17 Comparison of horizontal and vertical acceleration 

components recorded on the shaking table during the test
25

Shaking Table Test Results

In this section, selected results from the extensive matrix of recorded data are presented and 

discussed. Shown in Figure 2.18 are the coordinate system for acceleration and displacement 

records and the designation of the key locations. Also shown are the designations and position 

of the strain gauges where rebar strains were monitored and reported herein.

Real Earthquake Records

The real earthquake records designated as RUN1, RUN2 and RUN3 were applied in the first 

phase of the shaking table tests. Of interest was to observe the response of the structure to 

realistic earthquake signals. While RUN1 and RUN2 achieved relatively low intensities (even 

though degradation in terms of eigenfrequencies was observed), RUN3 represented a more 

realistic case for analysis since it approached intensities close to design basis. Figure 2.19 

depicts the recorded horizontal table accelerations. As seen from the figure, the strong motion 

portion is approximately 3 seconds. Examining, however, the recorded acceleration data shown 

in Figure 2.20 (along the x-axis at location D and y-axis at location C) it is evident that the 

structure experiences a long-lasting free vibration. The implication of such response is that the 

structure resonated with the input, (more detailed discussion on this point is provided in 

Section 3.4.2) and that the damping of the combined system (structure and shaking table) is 

very low for this low intensity seismic excitation (measured during the test as ~1%). The 

multitude of stress cycles experienced by the structure may well explain the shifting of the 

eigenfrequencies due to micro-cracking caused by the cyclic load. 

Figures 2.21 and 2.22 depict acceleration response spectra for all three real signals used in the 

test. Relative displacements and rebar strains shown in Figures 2.23 and 2.24 respectively, 

further confirm the presence of “un-damped” free vibration of the structure during RUN3. 

Figure 2-18 Response output locations on the SMART structure
26

Figure 2-19 Table horizontal accelerations recorded during test RUN3

Figure 2-20 Recorded horizontal accelerations on the structure during 

test RUN3
27

Real Earthquakes (Run1-Run3) RS (3Dx)

0

0.5

1

1.5

2

2.5

3

3.5

4

1 10 100

Freq (Hz)

Spectral Accel (g)

RS_Run1_3Dxx

RS_Run2_3Dxx

RS_Run3_3Dxx

Figure 2-21 Acceleration response spectra at level 3 and location D 

recorded during the application of the three real earthquake 

records (RUN1, RUN2 and RUN3)
28

5% Response Spectra of Run3 Test - Shake Table & Structure

0

0.5

1

1.5

2

2.5

3

3.5

4

1 10 100

Freq (Hz)

Spectral Acceleration (g)

Run3_Table_RSxx

Run3_Table_Rsyy

RS_Run3_3Axx

RS_Run3_3Cyy

RS_Run3_3Dxx

Figure 2-22 Table and structure acceleration response spectra 

comparison for RUN3 test

Figure 2-23 Relative displacement rUy between level 3 and table recorded 

during RUN3 test

Run 3: Recorded Relative Uy Displacement [3Cy-1Cy]

1

2

3

4

5

6

7

0 2 4 6 8 10

secs

rUy (mm)

rUy [3Cy-1Cy)
29

Figure 2-24 Rebar strain on Wall 3 recorded during RUN3 test

Run 3: Recorded Rebar Strain (Wall 3 - Location J2V03) 

-80

-60

-40

-20

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

uStrain
30

Synthetic Records

The synthetic records, designated as RUN4 through RUN13, were applied following the three 

real signals of the seismic input suite. As noted previously these synthetic records were 

generated from the design spectrum of Figure 2.1 aiming to progressively reach 1.0 g intensity 

in the input acceleration. According to Table 2.2 which lists the acceleration intensities achieved 

at the table level, RUN4 with axx = 0.2g and ayy = 0.24g table accelerations ended up 

representing the design basis earthquake which governed the design of the test structure. In 

addition to the RUN4 seismic input pair, RUN8 (with axx = 0.41g and ayy = 0.55g) was 

considered in this benchmark as the “over-design” seismic level. As a result, special attention 

was given in the post-analysis as well as the predictions to these two tests.

Depicted in Figure 2.25 are the table accelerations recorded during RUN4 indicating strong 

motion duration of about 3 seconds. Figure 26 on the other hand illustrates in-structure 

accelerations induced by the RUN4 pair for both horizontal directions and compares them with 

the table accelerations. While not as dramatic as in real earthquake RUN3, the structure 

undergoes free vibration indicating low damping. Accelerations recorded on the 3rd level at 

higher intensities (RUN8 through RUN13) are shown in Figure 2.27. It is deduced from the 

acceleration transients that higher levels of damping are exhibited by the structure due to the 

absence of free vibration in the response.

Acceleration response spectra of all the synthetic earthquakes at the table top surface are 

shown in Figure 2.28. It can be deduced from the spectral traces that there exist a structuretable interaction stemming from the shifting of the frequencies which originate in the 

degradation of the structure. 

In addressing the degradation of the structure with the sequential application of the increasing 

seismic loads, floor response spectra were generated. Seen in Figure 2.29 are floor response 

spectra at the 3rd level from three tests, RUN3 (real earthquake), RUN4 (design-level) and 

RUN8 (over design level). The figure very clearly demonstrates the changes that occur in the 

stiffness of the structure which are caused by damage accumulation (through mico-cracking) in 

the concrete. To compare the table spectra with the shape of the design spectrum (Figure 2.1) 

the RUN4 and RUN13 table response spectra are compared in Figure 2.30. One notices that 

the interaction, which was more visible in the low intensity (RUN4) input has blended in and it is 

less distinguishable. In general, however, the spectra maintain the general shape of the original 

spectrum from which the input was generated.

More complete information regarding the damage evolution can be deduced from Figures 2.31-

2.33 which depict floor response spectra at the same structural location respectively and 

indicate the trend and shifting of the eigenfrequencies to lower values. As is seen in Figure 2.32 

the spectral acceleration associated with the first flexural mode decreases while the intensity of 

the input increases stemming from the fact that the structure is being progressively damaged.

Relative displacements between the 3rd and first levels of the structure are shown in Figure 2.34 

for the RUN9 test. The increased level of damping in the structure due to micro-cracking is 

evident in that the deformations of the structure cease soon after the strong motion part is over.

Interesting results of rebar strain are depicted in Figure 2.35 which traces the strain during 

different tests. For RUN4 the response is linear because of the two-sided nature of the strain 

transient. As the intensity increased in RUN9 and beyond, it is suspected that rebars at the 

extremities and near the foundation are experiencing yielding.
31

Figure 2-25 Recorded shaking table horizontal accelerations during the 

RUN4 test

Figure 2-26 Recorded shaking table and in-structure horizontal 

accelerations in RUN4
32

Figure 2-27 Recorded shaking table and in-structure horizontal 

accelerations during the sequence of RUN8 to RUN13 tests

 Test Accelerations Ayy (Table and 3C) in the OVER-DESIGN Regime (Runs 8-13)

-3

-2

-1

0

1

2

3

0 9 18 27 36 45 54

secs

Accel Ayy (g)

Ayy_3C

Table_Ayy

Test Accelerations Axx (Table and 3D) in the OVER-DESIGN Regime (Runs 8-13)

-3

-2

-1

0

1

2

3

0 9 18 27 36 45 54

secs

Accel Axx (g)

Axx_3D

Table Axx
33

Shake Table Acceleration Response Spectra 

SYNTHETIC Input Tests (RUN4 through RUN13) along X-direction 

5% Damping

0

0.5

1

1.5

2

2.5

3

1 10 100

Freq (Hz)

Spectral Acceleration (g)

Run4_Table_RSxx

Run5_Table_RSxx

Run6_Table_RSxx

Run7_Table_RSxx

Run8_Table_RSxx

Run9_Table_RSxx

Run10_Table_RSxx

Run11_Table_RSxx

Run12_Table_RSxx

Run13_Table_RSxx

Shake Table Acceleration Response Spectra 

SYNTHETIC Input Tests (RUN4 through RUN13) along Y-direction 

5% Damping

0

0.5

1

1.5

2

2.5

3

3.5

4

1 10 100

Freq (Hz)

Spectral Acceleration (g)

Run4_Table_Rsyy

Run5_Table_Rsyy

Run6_Table_Rsyy

Run7_Table_Rsyy

Run8_Table_Rsyy

Run9_Table_Rsyy

Run10_Table_Rsyy

Run11_Table_Rsyy

Run12_Table_Rsyy

Run13_Table_Rsyy

Figure 2-28 Shaking table acceleration response spectra in the two 

horizontal directions recorded during the synthetic 

earthquake tests (RUN4 through RUN13)
34

Figure 2-29 Evolution of floor spectra and comparison of real record 

(RUN3)spectra with the design-level (RUN4) and the overdesign RUN8

Recorded Shake Table 5% Acceleration Response Spectra 

SYNTHETIC Input Tests RUN4 (0.2g) and RUN13 (1.0g)

0.01

0.1

1

10

0.1 1 10 100

Freq (Hz)

Spectral Acceleration (g)

Run4_Table_Rsyy

Run13_Table_Rsyy

Run4_Table_RSxx

Run13_Table_Rxxy

Figure 2-30 Comparison of shaking table acceleration response spectra 

in the two horizontal directions for design level (RUN4) and 

last test in sequence (RUN13) to identify structure-table 

interaction effects
35

Response Spectra Evolution from Run3 (real record) to Run13 - 5% Spectral Damping

0

1

2

3

4

5

6

1 10 100

Freq (Hz)

Spectral Accel (g)

RS_Run3_3Axx

RS_Run4_3Axx

RS_Run5_3Axx

RS_Run6_3Axx

RS_Run7_3Axx

RS_Run8_3Axx

RS_Run9_3Axx

RS_Run10_3Axx

RS_Run11_3Axx

RS_Run12_3Axx

RS_Run13_3Axx

Figure 2-31 Evolution of in-structure response spectra (level 3 - location 

A) for tests RUN3 through RUN13. Location A is close to the 

stiffest section of the test structure

Response Spectra Evolution from Run3 (real record) to Run13 - 5% Spectral Damping

0

1

2

3

4

5

6

7

8

9

10

1 10 100

Freq (Hz)

Spectral Accel (g)

RS_Run4_3Dxx

RS_Run5_3Dxx

RS_Run6_3Dxx

RS_Run7_3Dxx

RS_Run8_3Dxx

RS_Run9_3Dxx

RS_Run10_3Dxx

RS_Run11_3Dxx

RS_Run12_3Dxx

RS_Run13_3Dxx

RS_Run3_3Dxx

Figure 2-32 Evolution of in-structure response spectra (level 3 - location 

D) for tests RUN4 through RUN1
36

Response Spectra Evolution from Run3 (real record) to Run13 - 5% Spectral Damping

0

1

2

3

4

5

6

7

8

9

10

1 10 100

Freq (Hz)

Spectral Accel (g)

RS_Run3_3Cyy

RS_Run4_3Cyy

RS_Run5_3Cyy

RS_Run6_3Cyy

RS_Run7_3Cyy

RS_Run8_3Cyy

RS_Run9_3Cyy

RS_Run10_3Cyy

RS_Run11_3Cyy

RS_Run12_3Cyy

RS_Run13_3Cyy

Figure 2-33 Evolution of in-structure response spectra (level 3 - location 

C) for tests RUN4 through RUN13

Figure 2-34 Relative displacement between 3rd and 1st level recorded 

during RUN9

Run 9: Recorded Relative Displacement Ux (3Dx-1Dx)

-15

-10

-5

0

5

10

15

0 1 2 3 4 5 6 7 8 9

secs

Displ. rUx (mm)
37

RUN4: Rebar Strain (Wall 4 J6V04)

-200

-150

-100

-50

0

50

100

150

200

0 2 4 6 8 10 12 14

secs

uStrain

 

RUN13: Rebar strain (Wall 4 J6V04) 

-500

0

500

1000

1500

2000

2500

0 2 4 6 8 10 12 14

secs

ustrain

Figure 2-35 Evolution of rebar strain in Wall 4. Comparison of RUN4, 

RUN9, and RUN13

Run 9: Recorded Rebar Strain (Wall 4 - J6V04) 

-500

0

500

1000

1500

2000

0 2 4 6 8 10 12

secs

uStrain
38

Observed Structural Damage

At the end of each run in the 13-run sequence the walls of the structure were thoroughly 

examined for developed cracks. Figure 2.36 depicts the crack pattern observed at the end of 

the 13-run test. They were predominantly surface cracks. Given that the structure was designed 

with the design level earthquake at 0.2g and during the shaking table tests it experienced 

seismic input in excess of 1.0 g (1.13 g during RUN13), the damage level is considered low with 

the structure maintaining its overall integrity. 

Based on the monitoring process, no cracks appeared on the structure at design level (RUN4). 

The first noticeable cracks appeared during the RUN6 test near the openings of walls 1 and 2 

(shown in red in Figure 2.36) (over design level). At the end of the 13-run tests, wall 4 exhibited 

the most damage in terms of cracking (inner and outer faces) between level 0 and level 1, while 

on wall 3 cracks emanated from the top corners of the wall openings. Small horizontal surface 

cracks could be seen in the column near the junctions with the floor slabs and the base plate. 

Yielding in the monitored rebars was first suspected to occur during RUN10 in the top corners 

of the openings in walls 1 and 2. There was no evidence of rebar rupture anywhere in the 

structure and in particular near the wall-foundation interface that is anticipated to first occur at 

loads exceeding the structural capacity. In order to shed some light on the damage 

progression, “pseudo” transfer functions were developed between the table and the structure 

based on the test records. Clearly seen in Figures 2.37 is shifting of the frequencies where the 

transfer function associated with the first frequency occurs and the reduction of the peak values 

as the structure experiences damage and exhibits non-linear behaviour (inability to transfer 

acceleration). Dramatic changes occur during RUN10 (where yielding occurred in rebars 

accompanied with cracking in all four walls) which result in the lowering of motion transfer 

capability of the structure in the subsequent and higher intensity seismic tests. Since the 

initiation of rebar yielding accompanied by concrete cracking in the US practice is associated 

with the Safe Shutdown Earthquake (SSE) then it may be assumed that the RUN10 earthquake 

of the SMART2008 test is equivalent to an SSE and therefore it can be concluded that the 

structure remained intact at 1.5 SSE which in the SMART2008 experiment is represented by 

RUN13. 

One important finding, however, was the appearance of separation gaps between the 

foundation and the steel plate that mounts the test structure to the top of the shaking table. This 

may be attributed to the structure-table interaction in which the table top is not acting as a rigid 

plate but experiences localized deformations along its interface with the structure.

Figure 2-36 Identified cracking patterns at the end of the 13-run shaking 

table tests on the inside surfaces of the test structure walls
39

Acceleration "pseudo" Transfer Function along X-direction

Table motion and 3rd Floor Location D

0

20

40

60

80

100

120

140

160

180

200

0 5 10

Freq (Hz)

"pseudo" Transfer Function

Run 2

Run 3

Run 4

Run 7

Run10

Run13

Acceleration "pseudo" Transfer Function along X-direction

Table motion and 3rd Floor Location D 

Zooming into "non-linear" transition 

0

50

100

150

200

250

1 10

Freq (Hz)

"pseudo" Transfer Function

Run 9

Run10

Run11

Run13

Figure 2-37 “Pseudo” transfer functions between table accelerations and 

3rd level of structure identifying transitions to non-linear

behaviour Measured damping values

To address the issue of real damping that is exhibited by a reinforced concrete structure during 

a dynamic event, shaking table test data were used to both quantify and identify the evolution of 

progressive damage in the structure. As noted previously, between some of the tests low-
40

intensity white noise excitation was applied to the structure in order to extract information on its 

actual dynamic properties and in particular eigenfrequencies. It was also noticed during the 

tests that during some of the runs the structure exhibited a long tail of free vibration. By utilizing 

information from both the free-vibration portion and/or response to white noise the actual 

damping of the structure was assessed through (a) the logarithmic decrement and (b) the 

power bandwidth methods. It is speculated by the authors that the energy absorption 

characteristics exhibited during the SMART2008 tests is the result of the combined system, 

reinforced concrete structure and the shaking table as well as the effect of the mounting of the 

structure on the shaking table. 

Figure 2.38 depicts the principle governing the application of the two methods. Figure 2.39 

illustrates the free-vibration tail that was observed in the structure during two of the tests (RUN3 

and RUN10). As seen in RUN3 the structure exhibits very low damping demonstrated by the 

long duration free-vibration. On the other hand, after the strong motion part of RUN10 where 

significant damage to the structure was estimated to have occurred (formation of large, visible 

cracks and rebar yielding), the free vibration of the structure is quickly damped out indicating 

overall increased energy absorption of the system prompted by the contribution provided by the 

extensive, large cracks in the concrete.

Figure 2-38 Method schematics for estimating actual damping values in 

the SMART structure from shaking table tests. (a) logarithmic 

decrement and (b) half-power bandwidth method

Based on the evaluation using the two approaches was assessed that during the low-intensity 

tests (below the design level of 0.2g) the damping in the overall system (structure and table) 

was approximately 1%. At RUN5 which was slightly over the design level the damping ratio 

increased to ~2% and gradually reached levels of 7% above Run 11.
41

Figure 2-39 Free-vibration characteristics of structure during different 

tests Global Behaviour of the Structure

The SMART2008 structure was designed to accentuate the 3-D effects during multi-directional 

seismic inputs and in particular to assess torsion in reinforced concrete structures. As a result 

the trapezoidal structure had its mass and torsion centres far apart with the torsion centre 

located closest to the stiffest corner of the structure (point A) as shown in Figure 2.40. Based 

on recorded absolute displacements the trend of the general motion of the structure can be 

deduced. Shown in Figure 2.40 is a plot of the movement of each of the four corner points of 

the structure at the 3rd level. The displacement of each corner shown has been amplified by a 

factor of 50. The displacement trend of each corner depicted in Figure 2.40 confirms the global 

rotational character of the structure with respect to the rotational axis shown. Corner A being 

the closest to the axis exhibits the smallest movement while extremities C and D the largest.
42

Figure 2-40 Identification of rotational behaviour of the test structure 

based on the displacement patterns recorded at the corners 

of the 3rd level
43

3. NUMERICAL ANALYSES AND TEST PREDICTIONS

The SMART2008 benchmark also included a series of numerical analyses in order to (a) 

assess design practices around the world and the use of analysis methods in nuclear structure 

design and (b) use the shaking table tests as the basis for assessing the capabilities of the 

different methods in predicting the response of reinforced concrete structures under multidirectional seismic inputs.

The numerical part of the benchmark consisted of two distinct phases. Phase 1 which is 

described in this chapter had the following objectives:

• Assessment of conventional design methods and structural dynamic analyses including 

floor response spectra generation 

• Comparison of best-estimate methods in predicting the response of structures which will 

be allowed to enter the non-linear regime and the generation of corresponding “nonlinear” floor response spectra

Phase 2 which is described in detail in Chapter 4 and was initiated following the completion of 

the actual shaking table tests and the corresponding analyses, aimed at (a) quantifying the 

variability in structural response resulting from uncertainties in structural properties and 

(b) generating fragility curves for structural vulnerability.

Phase 1 was divided into two parts namely Phase 1a and Phase 1b. Phase 1a was termed the 

“blind-prediction” phase where conventional and best-estimate analyses were performed to 

assess (a) the ability to analyze 3-D structures with non-linear behaviour using methodologies 

from conventional practice (which include equivalent static analysis, response spectra analysis, 

linear dynamic analysis, non-linear static or pushover analysis and non-linear time history 

analysis) and (b) the ability of an adopted best-estimate methodology using basic material and 

geometry information to predict the non-linear response of a structure resulting from a 

controlled test such as the shaking table test on the SMART structure. Phase 1a was 

conducted prior to any shaking table tests. The key objective of Phase 1a was to assess the 

variability that exists between methodologies and modelling/analysis techniques that are 

currently in use within the nuclear sector. 

Phase 1b represented the “best-estimate” analysis where specific information regarding the 

shaking table test set-up, the as-built structure and the actual excitations were available. 

Specifically, information on:

• The actual experimental set-up of the shaking table tests including the mass of the 

structure, the added loads and their distribution as well as interface conditions between 

the structure and the table

• Concrete and reinforcing steel properties of the as-built test structure which was 

deduced during construction

• The estimated initial frequency of the test structure on the shaking table and damping 

ratio prior to performing any of the 13 tests in an effort to fill the information gap on the 

dynamic properties of the table itself and the structure-table interaction

• Actual seismic input accelerations observed at the table top surface and the foundation 

level for the initial tests (actual table accelerations differed significantly from the 

intended excitation derived from scaling the design response spectrum)

• Selected structural response results for the lowest seismic input (RUN1) and the design 

level (RUN4)
44

BNL participated and contributed the results of its independent studies to all phases of the 

benchmark listed above. Results of Phase 1a (pre-test conventional analyses) generated by the 

BNL studies are included in the referenced documents and not included in the body of this 

report. This is because the objective of Phase 1a was to primarily obtain a correlated trend 

between all the participating teams and the adopted methodologies and not to quantify the 

ability of predicting the SMART shaking table test results. This overall assessment is

summarized in the summary and conclusions section. In this chapter, the results of the Phase 

1b or “best-estimate” prediction analysis will be presented and compared with the 

corresponding shaking table test results which became available after the completion of 

Phase 1b and the submission of the independent results to the SMART2008 project.

3.1. Structural Model

Various numerical representations of the SMART structure were developed and used in the 

various phases of the SMART benchmark. These were based on a three-dimensional 

description of the structure very close to the actual geometry. In addition to the structure, the 

foundation, the base plate mounting to the table, and the table itself were modelled and 

integrated. In addition, the added masses were introduced in the form of solid blocks as they 

were described in the initial phases of the study (prior to the construction of the test structure 

and the final configuration of the masses). 

Following the design drawings on reinforcing steel rebar layout, the steel rebars were discretely

modelled into the structure. The concrete walls, floor slabs and foundation were represented by 

solid elements, as well as the beams, the column and steel plates. 

The shaking table was modelled as a rigid block. However, sensitivity analyses were performed 

to observe the effect of its stiffness on the structural response. Shown in Figure 3.1 and 3.2 are 

global representations of the numerical model and modelling of local elements respectively.
45

Figure 3-1 Numerical models used in the numerical analysis of the 

SMART structure

 

Figure 3-2 Numerical representation of structural details of the SMART 

structure

The SMART structure was discretized into finite elements in a way that allowed for the

distribution and position of rebars using the capabilities of the TrueGrid software. That resulted 

in a very large numerical model which was computationally expensive (~400,000 elements in 

total were making up the numerical model). This model was used in Phase 1a of the 

benchmark which encompassed linear-static, non-linear static and modal analyses. 

For the transient analyses, the discretization was “optimized” by combining some of the 

reinforcing bars and allowing for a smaller number of finite elements in the concrete. The 

optimization was done on the basis of equivalent dynamic properties (i.e., maintaining the 

eigenfrequencies of the structure). This resulted in a smaller numerical representation 
46

(~260,000 finite elements total) which reduced the computational time significantly. Typical 

computational time for a non-linear analysis using the strong motion of the acceleration records 

was approximately 60 hours on a dedicated 8-CPU micro-station.

The structure was firmly connected to the top of the shaking table via the base plate. No local 

interaction (i.e., separation of the plate from the table between the mounting screws was 

allowed).

Because of the uncertainties regarding the shaking table dynamic properties and supports,

different options, were explored. For the equivalent static analyses and Phase 1a dynamic 

analyses, the table was assumed rigid and was restricted for both cases from vertical 

movement. For the push-over equivalent static analyses, the table was considered fixed in all

degrees of freedom. Following some additional information regarding the stiffness of the 

vertical jacks and eigenfrequency values for the actual SMART structure tested on the table, 

vertical and horizontal springs were introduced to support the rigid shaking table. This last 

configuration was used in the “best-estimate” response prediction phase (Phase 1b).

Input accelerations for the prediction phase were introduced at the interface of the horizontal 

springs (actuators) with the shaking table which was approximately at mid-height of the 

modelled table. The 25 ton mass of the table which is less uncertain as a parameter was 

introduced in the modelling of the table.

The analyses (static, modal and non-linear time history) were performed using the capabilities 

of the LS-DYNA finite element code.

3.2. Description of the Constitutive Relations

To perform the non-linear dynamic analyses required and to enable the capturing of the 

progressive damage in the concrete or the yielding or failure in the rebar, constitutive relations 

that are appropriate for such non-linear analysis were adopted for the concrete and the steel.

For the concrete, the constitutive relations implemented into the Winfrith Concrete Element of 

the LS-DYNA material library were primarily used. The volumetric compaction and yield surface 

for this material that are numerically implemented into the software are shown in Figure 3.3. In 

particular, the analysis allowed for strain rate effects in the concrete and the dissipation of 

energy (fracture energy) in crack opening. Cracking is formed upon exceeding the tensile 

(splitting) strength but was allowed to close during the shaking. Parameters such as the 

fracture energy of concrete are significant and they influence the generation and distribution of 

cracking thus causing the weakening of the structure. Also influential are concrete parameters 

such as aggregate size and current state of tensile and compressive strengths. Damage in this 

material is viewed as material cracking or crushing and its inability to carry load. By allowing for 

some load-carrying capability the material can be pushed beyond the uni-axial strength values. 

When certain thresholds are reached where the material is incapable of satisfying a 

combination of requirements, the material can erode from the matrix. An important parameter in 

establishing damage in the concrete is the fracture energy, a property that was not available for 

the concrete used in the SMART structure. As a result, fracture energies of 110 J-m were used 

for all the numerical studies. Sensitivity studies for fracture energies reaching 330 J-m were 

performed.
47

Figure 3-3 Constitutive relations governing the Winfrith concrete 

element mode used in the SMART benchmark analyses

To address the progressive damage in the concrete which results in reductions of its capacity 

from micro-cracking, other constitutive relations were explored. An alternative concrete material 

model based on the research work of Holmquist, Johnson and Cook [A Computational 

Constitutive Model for Concrete Subjected to Large Strains and High Pressures, 14th 

International Symposium on Ballistics, Quebec, 1993] that is also part of the LS-DYNA library 

was studied and compared with the Winfrith counterpart. The constitutive and damage relations 

are shown in Figure 3.4. The material is formulated around an equivalent strength model, an 

accumulated damage model and an equation of state. Key features of this model are: (a) the 

accumulated damage and (b) the equation of state. In particular, the equation of state 

describes the relation between hydrostatic pressure and volume. The material in the impact 

zone is assumed to behave like a compressible fluid. The tensile pressure allowed is defined on 

the basis of the accumulated damage according to the relation Tnew = T•(1-D), where D is the 

damage (shown in Figure 3.4b) and which reflects both the equivalent plastic strain and plastic 

volumetric strain, and T is the tensile strength.

The comparison of the two concrete constitutive and damage models is shown in Figure 3.5. 

The structural accelerations shown were derived by subjecting the SMART structure to the 

same sequence of synthetic earthquakes and implementing the different constitutive relations in

the concrete.
48

(a) (b)

Figure 3-4 Holmquist-Johnson-Cook concrete material model
49

Figure 3-5 Comparison of the two concrete constitutive models in terms 

of the SMART structure response to synthetic accelerograms

The reinforcing steel was described using a bi-linear stress-strain model. Shown in Figure 3.6 

are actual rebar stress-strain test results provided by the SMART benchmark organizers and 

the adopted equivalent stress strain relation.

Figure 3-6 Stress-strain relation of steel used in the analysis

3.3. Shaking Table Test Predictions

Following the fabrication of the SMART specimen, the organizers provided test-specific 

information and actual material properties including:

• experimental set-up of the shaking table tests including the mass of the structure and 

final distribution of added loads 

• Concrete and reinforcing steel properties of the as-built test structure which was 

deduced during construction 
50

• estimated initial frequency of the test structure on the shaking table and damping ratio 

prior to performing any of the 13 tests 

• actual seismic input accelerations observed at the table top surface and the foundation 

level for the initial tests 

• selected structural response results for the lowest seismic input (RUN1) and the design 

level (RUN4)

The finite element model and its properties were re-adjusted to reflect the as-built properties 

and the non-linear time history analysis of the 13-run. As was described in Chapter 2, actual 

concrete cylinder tests on the structure concrete casts revealed a significant deviation from the 

nominal concrete modulus of 32,000 MPa used in the studies prior to the test. Table 2.1 which 

is repeated below is indicative of the variation.
51

Table 2.1 (Reproduced) concrete properties of as-built SMART structure.

Based on the updated material properties, the updated concrete density, and the initial 

eigenfrequencies of the combined system (structure plus shaking table), the modal properties 

of the combined system used at the start of the prediction analysis were re-assessed. Figure 

3.7 depicts the first three modes of the combined table/structure exhibited by the numerical 

model prior to the application of seismic loads. The values of the eigenfrequencies are similar 

to those observed by the white noise test (7.0, 9.0 and 16.0 Hz, respectively).

In order to capture the progression of damage, the thirteen bi-directional records were applied 

in sequence and in a way that the information is transferred between tests. Due to the 

computational burden, only the strong motion portion of the provided accelerograms were used. 

Figure 3.8 depicts the locations and orientations where test and prediction data are compared. 

As in the actual test, the real earthquake records were applied first. Special attention was paid 

in the response of the structure to RUN3 input which is both a real earthquake record and has a 

PGA close to the design level. Further, as seen from the actual experimental response, the 

structure exhibited free-vibration that lasted very long and thus it was crucial that the prediction 

model was capable of replicating this interesting behaviour. 

Figure 3.9 depicts acceleration time histories on three corners of the structure. As is clearly 

shown, the numerical model also predicts the same long-lasting free vibration while it also 

matches the peak acceleration values recorded during the test. It can also be observed that 

there exists a “structure” in the free vibration transient predicted by the numerical model 

attributed to the shaking table contribution for which all its dynamic properties are not available. 

This is more evident in the comparison of the response spectra shown in Figure 3.10 for both 

horizontal directions.

Comparison of relative displacements in the structure, depicted in Figures 3.11 and 3.12, reveal 

that the displacement range is well replicated (please note the offset of the horizontal axis in the 

recorded data). Examination of the recorded and predicted strains in a rebar of wall 4 shown in 

Figure 3.13 reveals that the simulation model is capable of capturing the strain in the rebars 

and that both (test and prediction) show that it is the whole structure and not the top section 

that undergoes free vibration with very little damping in the system.
52

Figure 3-7 Eigenfrequencies and modes of the final 3-D model used in 

the numerical predictions of the SMART2008 tests

Figure 3-8 Locations on the structure used for comparison of 

predictions with test data
53

Figure 3-9 Comparison of horizontal accelerations at the corner 

locations of the 3rd level between recorded test data and 

prediction from the BNL simulation for RUN3
54

Acceleration Response Spectra - Experiment vs. Prediction (3A x-dir)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70 80 90 100

Freq (Hz)

Spectral Accel (g)

Run3_Table_RSxx

Test_Run3_3A_xx

Simul_Run3_3A_xx

Acceleration Response Spectra - Experiment vs. Prediction (3C y-dir)

0

0.5

1

1.5

2

2.5

3

1 10 100

Freq (hz)

Spectral Accel (g)

Run3_Table_Rsyy

TEST_Run3_3C_yy

Simul_Run3_3C_yy

Figure 3-10 Comparison of horizontal acceleration response spectra 

between recorded data and predictions from BNL simulation 

for the real earthquake RUN3
55

Figure 3-11 Comparison of horizontal relative displacement Uy at 

location C between recorded data and predictions from BNL 

simulation for the real earthquake RUN3 (note the offsetting 

of the recorded data in the y-axis)
56

 

Figure 3-12 Comparison of horizontal relative displacement Ux at 

location D between recorded data and predictions from BNL 

simulation for the real earthquake RUN3
57

Run 3: Recorded Rebar Strain (Wall 3 - Location J2V03) 

-80

-60

-40

-20

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

secs

uStrain

Figure 3-13 Comparison of rebar strain on Wall 3 between recorded data 

and predictions from BNL simulation for the real earthquake 

RUN3

Comparison of results for the RUN4 (design level) earthquake input

The first synthetic earthquake applied to the structure was the design level earthquake (RUN4). 

The analysis continued using the initial damping of 1% and completed 7 seconds of excitation. 

Similarly as in RUN3, response data of acceleration, relative displacement, and rebar strain in 

the four walls were post-processed. In addition, the appearance of cracks was monitored 

throughout the RUN4 simulation. 

Figure 3.14 depicts the comparison of acceleration in the two horizontal directions at locations 

C (y-direction) and D (x-direction) at the 3rd level. While the acceleration at D is slightly 

underestimated, the acceleration in location C is well reproduced. As in RUN3 it is suspected 

that the table is playing a role as can be seen in the floor response spectra comparison shown 

in Figure 3.15. 
58

Rebar strain comparison is shown in Figures 3.16-17 and relative displacements in Figure 3.18. 

From the strain comparison in Figure 3.17 it is concluded that the simulation is able to predict 

the permanent deformation or slight adjustment experienced at wall #4.

Consistent with post-test observations, the simulation showed no cracks in the concrete at the 

end of RUN4.

Figure 3-14 Comparison of structure accelerations for the design 

earthquake level (RUN4) between the experiment and the 

BNL simulation
59

Response Spectra Comparison for RUN4

0

1

2

3

4

5

6

1 10 100

Freq (Hz)

Spectral Accel (g)

RS_3Dx_test

RS_3Dx_simul (1.5% damp)

Response Spectra Comparison for RUN4

0

1

2

3

4

5

6

1 10 100

Freq (Hz)

Spectral Accel (g)

RS_3Cy_test

RS_3Cy_simul (1.5% damp)

Figure 3-15 Comparison of floor response spectra for the design 

earthquake level (RUN4) between the experiment and the 

BNL simulation
60

Run 4: Recorded rebar strain (Wall 3 - J2V03)

-150

-100

-50

0

50

100

150

0 1 2 3 4 5 6 7

secs

uStrain

Figure 3-16 Comparison rebar strain in wall 3 for the design earthquake 

level (RUN4) between the experiment and the BNL simulation

Run4: Recorded rebar Strain (Wall 4 - J2V04)

-300

-200

-100

0

100

200

300

0 1 2 3 4 5 6 7

secs

uStrain

(a) (b)

Run4: Recorded rebar strain (Wall 4 -J2V04)

-300

-200

-100

0

100

200

300

400

0 2 4 6 8 10 12 14 16 18 20

secs

uStrain

(c) (d)

Figure 3-17 Comparison of rebar strain in wall 4 for the design 

earthquake level(RUN4) between the experiment and the BNL 

simulation. Shown in (c) is the recorded strain in expanded 

scale depicting the “re-adjustment” or permanent 

deformation in the rebar at the selected location.
61

Relative Displacement from Shaking Table Test rUx_[3Dx-1Dx]

-6

-5

-4

-3

-2

-1

0

1

2

3

4

0 1 2 3 4 5 6 7

secs

Ux (mm)

TEST_rUx_[3Dx-1Dx]

Figure 3-18 Comparison of relative displacements at location D and 

along the x-direction for the design earthquake level (RUN4) 

between the experiment and the BNL simulation

According to Table 2.3, the achieved PGA for RUN5, which was the test following the design 

level earthquake was lower than RUN4. As a result, the effects of tests beyond RUN5 are 

presented in the following figures. During the test a significant drop in the frequency associated 

with the torsional mode (3rd eigen mode) was measured during RUN5 without any correlation to 

any visible damage in the structure. 
62

Shown in Figure 3.19 are comparisons of predicted accelerations and experimental 

measurements. While the predictions generally agree with the experiment, they seem to under 

predict the accelerations. This indicates that (a) the increased damping introduced into the 

analysis following the completion of RUN5 (2% damping ratio from 1.0% used in the below 

design level seismic intensity tests and 1.5% for RUN4) combined with (b) the effect of visible 

cracking on the acceleration response of the structure observed during the simulation of RUN6 

(isolated small cracks) and RUN7 (permanent cracks) are the cause of the under prediction.

Figure 3.20 depicts the comparison of floor response spectra. Figure 3.21 illustrates the spatial 

distribution of effective stress along with predicted cracking at the end of RUN6 and RUN7 tests 

respectively. As anticipated, elevated levels of stress are predicted near the corners of the wall 

openings that result from the torsional response of the structure.

Figure 3-19 Comparison of 3rd level (locations C and D, y-direction and xdirection respectively) accelerations between test and 

prediction for RUN6 and RUN7 inputs
63

 

Figure 3-20 Floor response spectra comparison at the 3rd level of the 

structure between test and prediction for RUN6 and RUN7 

inputs
64

(a)

(b)

Figure 3-21 Predicted damage in the SMART structure at the end of (a) 

RUN6 and (b) RUN7. Cracking of concrete is predicted to 

appear during the RUN7 table excitation
65

The completion of RUN7 and the observation of damage that was induced indicated that the 

structure has entered the non-linear regime of its response. The pair of accelerograms 

designated as RUN8 and termed as over-design earthquake during the benchmark induced 

shaking table PGA values of 0.41g in the x and 0.55g in the y directions. Special attention was 

paid to this level anticipating that, it being double the design level earthquake, significant levels 

of damage could be experienced. 

The analysis continued by transferring the damage information at the end of RUN7 (see 

Figure 3.21), maintaining the same level of 2% damping, and counting on the cracked concrete 

to provide additional effective damping. Figures 3.22 and 3.23 illustrate a comparison of 

accelerations predicted at the three corners of the structure at the 3rd level (A, C, and D) and 

those recorded during the test. While it slightly under predicts, the analysis reproduces the 

acceleration response very well throughout the strong motion part of the input earthquake even 

within the non-linear regime. Figure 3.24 presents the floor response spectra comparison where 

it is seen that the non-linear analysis is in excellent agreement for the first two eigenmodes. 

Figure 3.25 reflects the effective stress state of the structure during the RUN8 input earthquake 

and the damage in the form of concrete cracking.

Figures 3.26 and 3.27 illustrate the comparison of rebar strain in the Wall 4 (which appears to 

be more damaged at the end of RUN8) for the subsequent accelerograms pairs for RUN9. 

Similar to RUN8, RUN9 resulted in PGA values to the shaking table of 0.41g and 0.56g along xand y-directions respectively. The non-linear analysis reproduces both the levels and the 

transient shape of the rebar strain. Figures 3.28 and 3.29 compare relative displacements 

between the 3rd level and the foundation/wall interface and demonstrate that the displacements 

are well matched (please note the offset in the vertical axis of the recorded data). Both the test 

and the prediction results show the appearance of extra damping in the structure due to the 

damage (rapidly dampening of the free vibration at the end of the strong motion part).

Run8 (3A x-dir)

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3 3.5 4

secs

Accel (g)

Run8_3Ax_TEST

Run8_3Ax_simul

Figure 3-22 Comparison of predicted acceleration at point A of Level 3 

with recorded accelerations during RUN8
66

Run8 (3C y-dir)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3 3.5 4

secs

Accel (g)

Run8_3Cy_TEST

Run8_3Cy_simul

Run8 (3D x-direction)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3 3.5 4

secs

Accel (g)

Run8_3Dx_TEST

Run8_3Dx_simul

Figure 3-23 Comparison of predicted accelerations at point C (ydirection) and D (x-direction) at Level 3 of the test structure 

with recorded accelerations during RUN8
67

Figure 3-24 Comparison of acceleration floor response spectra at point 

C(y-direction) and D (x-direction) of Level 3 between 

recorded and predicted data
68

Figure 3-25 Predicted damage state of the SMART structure upon 

completion of the RUN8 shaking table excitation
69

Run 9: Recorded Rebar Strain (Wall 4 - J1V04) 

-600

-400

-200

0

200

400

600

800

1000

1200

0 2 4 6 8 10 12

secs

uStrain

Figure 3-26 Comparison of predicted rebar strain on Wall #4 of the 

structure (location J1V04 near the foundation) with the one 

recorded by the installed strain gauge during RUN9
70

 

Run 9: Recorded Rebar Strain (Wall 4 - J6V04) 

-500

0

500

1000

1500

2000

0 2 4 6 8 10 12

secs

uStrain

Figure 3-27 Comparison of predicted rebar strain on Wall #4 of the 

structure (location J6V04 mid-height between 1st and 2nd

level) with recorded strain during RUN9
71

 

Run 9: Recorded Relative Displacement Uy (3Cy-1Cy)

-15

-10

-5

0

5

10

0 1 2 3 4 5 6 7 8 9

secs

rUy (mm)

Figure 3-28 Comparison of predicted relative displacement rUy between 

the 3rdlevel and foundation top at corner C with the recorded 

relative displacement during RUN9
72

Run 9: Recorded Relative Displacement Ux (3Dx-1Dx)

-15

-10

-5

0

5

10

15

0 1 2 3 4 5 6 7 8 9

secs

Displ. rUx (mm)

Figure 3-29 Comparison of predicted relative displacement rUx between 

the 3rd level and foundation top at corner D with the recorded 

relative displacement during RUN9
73

The predicted response during RUN9 exhibited the effective damping in the damaged structure 

(see Figure 3.29). Originally the analysis proceeded without updating the structural damping of 

2%, anticipating that the cracked concrete will introduce additional energy absorption in the 

system. This resulted however in solution instabilities during RUN10 indicating that the structure 

is being damaged as a result of the higher acceleration demand. As a result, the damping in the 

concrete structure was increased to 4% for the non-linear analysis for the tests beyond RUN9. 

Shown in Figure 3.30 are comparisons of accelerations for the tests RUN10, RUN11 and 

RUN12. While the non-linear analysis model traces the floor accelerations quite well it is 

evident that the accelerations are under-predicted. Figure 3.31 depicts the acceleration 

comparison for RUN11, still showing a very good agreement between the test and the 

prediction results. Figure 3.32 presents the floor response spectra comparison for RUN12 at 

the corners C and D of the 3rd level of the structure Figure 3.33 depicts the rebar strain 

comparison at mid-level height (Level 0 and level 1) of Wall #4. Excellent agreement between 

the predicted and the recorded strains is observed.

Figure 3-30 Comparison of predicted accelerations on the 3rd level of the

structure (corners C and D) with recorded data during the 

shaking table tests RUN10, RUN11 and RUN12
74

Run11: Recorded Acceleration Axx (Level 3 - Location 3D)

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6

secs

Accel Axx (g)

Figure 3-31 Comparison of accelerations on the 3rd level of the structure 

(corner D along the x-direction) with recorded data during the 

RUN11 test
75

Figure 3-32 Floor response spectra comparison between predicted and 

recorded data during the RUN12 test
76

Rebar Strain Comparison between Test and Simulation (J6V04 of WAll #4)

-1000

-500

0

500

1000

1500

2000

2500

0.5 2.5 4.5 6.5 8.5 10.5 12.5 14.5

secs

uStrain

J6V04_simul

J6V04_test

Figure 3-33 Comparison of rebar strain mid-height of Wall #4 (level 0 and 

level 1) at the edge C for tests RUN10, RUN11 and RUN12

RUN13 representing the last test in the 13-run sequence induced peak ground accelerations 

higher than originally planned for the test along the y-direction and lower in the x-direction. 

Specifically, in the planning of the benchmark, the PGA values for both directions were to be 

1.0g. During the shaking table test, the PGA along the x-direction was only 0.75g and along the 

y-direction was 1.13g. The shaking table accelerations for the last test are shown in Figure 

3.34.

The analysis for the RUN13 was performed without any further increase of the assumed 

damping of 4% used in RUN12. The analysis of RUN13 relied on the continuing concrete 

cracking to provide the potential increase in the 4% damping of the system assumed at the 

start of RUN13. Figure 3.35 depicts the comparison of structural accelerations between the test 

data and the non-linear analysis. Again, while the agreement is very good, the simulation 

under-predicts the experiment especially during the free-vibration tail. This is most likely the 

result of mismatch between the actual damping exhibited by the RC structure and the 4% 

assumed in the simulation. The under-prediction is also evident when the floor response 

spectra are compared (Figure 3.36). Also seen in the spectra comparison of Figure 3.36, where 

reflected are the accumulated damage and nonlinearities from the entire 13-test series (RUN1 

through RUN13), is that the eigenfrequency drop in the test is slightly larger than what is 

predicted. This effect is more pronounced in the spectra along the x-direction. The non-linear 

analysis model reproduces the response spectra shape in the higher frequencies (~100 Hz) 

along the y-direction.

Excellent agreement in the structural displacements are seen in Figure 3.37 where relative 

displacements between the 3rd level and the level 0 (interface of vertical walls and foundation) 

at the corners C and D are presented.
77

Figure 3.38 depicts effective stress and damage state of the walls and column of the structure 

in the beginning of RUN13 (a) and at the end (b). Also shown in (c) are the crack patterns of 

the actual test structure. As seen in (a) the stress in Wall #4 is eventually relieved by the 

formation of cracks which reproduce quite well the patterns observed in the test specimen. 

Figures 3.39 and 3.40 depict the evolution of the floor response spectra as a result of the 

cumulative damage experienced by the structure due to the 13-set sequence. As compared to 

similar spectra shown in Figures 2.31 and 2.32 for the actual test, one observes the similarity in 

the trend (lowering of eigenfrequencies) and also the difference in the level of damage or 

change in the dynamic properties of the reinforced concrete structure.

Figure 3-34 Acceleration pair recorded on the shaking table during the 

last test in series, RUN13, where PGA values were 0.75g in x 

and 1.13g in y
78

 

4

Figure 3-35 Comparison of structure accelerations at level 3 for the 

RUN13 test
79

Figure 3-36 Acceleration floor response spectra comparison for the 

RUN13 test
80

Figure 3-37 Comparison of relative displacements between level 3 and 

level 0(foundation/wall interface) for the RUN13 test
81

(a)

 (b)

(c)

Figure 3-38 Predicted structural damage in the SMART structure at the 

end of the 13-run test
82

Figure 3-39 Predicted floor response spectra evolution during the 

application of the ten synthetic earthquake pairs along the ydirection

Figure 3-40 Predicted floor response spectra evolution during the 

application of the ten synthetic earthquake pairs along the xdirection
83

3.4. Overall Assessment of Prediction Phase

Based on the results of the non-linear analysis model, the general assessment is that the model 

predicted the experiment very well throughout. The numerical model was able to accurately 

predict the response of the structure on the shaking table under the action of a real earthquake 

(RUN3) which induced a very long free-vibration tail to the actual structure during the test. In 

addition, the analysis model was able to reproduce the response for structural accelerations, 

displacements and reinforcing bar strains when the structure behaved non-linearly and damage 

was accumulated in the concrete and the steel with localized yielding. 

The agreement was good even though uncertainties regarding the test configuration and thus 

its numerical implementation exist. These uncertainties that have a profound effect on the 

structural response stem primarily from the shaking table whose boundary conditions and 

dynamic properties are not well understood and are still unavailable. It was indicated by the 

benchmark organizers during the close-out workshop that prior to the upcoming second phase 

of the benchmark efforts will be made to identify the dynamic properties of the shaking 

table/actuator system. Added to the shaking table uncertainties are lack of data regarding the 

concrete material and, in particular, the fracture energy that influences its cracking and/or 

crushing. Generic values were used in the non-linear analysis that may not be representative of 

the concrete used in constructing the test structure. Further, the need to “optimize” the size of 

the computational model such that the computational cost of the demanding analysis was 

manageable most-likely contributed to the small deviations observed between test data and 

prediction data. 

3.4.1. Damping

The energy dissipation of the overall system (concrete structure, shaking table and its supports 

and of the interface conditions that exist between the bottom of the structure and the shaking 

table) is identified as the biggest uncertainty and the source of the deviations of the predicted 

data. The overall energy dissipation in the concrete structure, however, is a combination of the 

assumed initial damping and of the contribution by the progressing damage in the form of 

visible and permanent cracks. Given, however, that the accumulated damage in the concrete 

primarily takes place in micro-scale before it manifests itself into visible and permanent cracks, 

the non-linear analysis is unable to completely follow the constitutive and damage behaviour at 

this level. 

As it was observed during the SMART2008 actual tests, damage below the threshold of 

cracking occurs even at very low earthquake intensities (RUN1 through RUN5). To account for 

that, better concrete damage accumulation models are needed to support the simulation and 

prediction analyses. In an effort to account for the increased energy dissipation within the 

concrete material due to micro-cracking (this happens at a scale that the numerical processes 

are unable to capture), the damping in the concrete was artificially increased during the 13-run 

analysis as the test progressed towards higher intensity levels. However, as the comparison of 

the test and predicted results show, these values may require further adjustment. To quantify 

the effect of the material damping ratio on the structural response using the design level 

earthquake input (RUN4), different levels of damping were introduced and the structural 

response was compared. Shown in Figure 3.41 are comparisons of structural accelerations in 

y- and x-directions for damping ratios of 1%, 1.5% and 2%. As seen from the comparison, the 

differences are relatively small.
84

Figure 3-41 Effect of damping ratio on structural acceleration response
85

3.4.2. Torsional Effects

In an effort to address the torsional effects on the SMART2008 structure, a different model, 

shown in Figure 3.42 was generated and analysed under seismic loading. The structure 

depicted in Figure 3.42, has a rectangular, closed cross-section and it is made of the same 

materials as the actual SMART2008 structure. While the structure is not perfectly symmetric 

(different openings in the x- and y-walls as well as a floor support beam running only along the 

y-direction), it has eliminated the primary contributors to torsional behaviour.

In order to qualitatively assess the torsional effects the new, closed-section structure was 

analyzed for modes and frequencies and seismic excitations RUN3 and RUN6 and compared 

with the SMART2008 structural response and modal characteristics.

Figure 3-42 Model of a SMART2008-like structure on the shaking table 

with a rectangular instead of trapezoid cross section

Shown in Figures 3.43 are the fixed-base modes of the two structures. In the case of the 

SMART2008 structure the two first modes are not purely flexural but a combination of flexural 

and torsional. This, however, is not true for the rectangular, closed-section structure which 

exhibits pure flexural modes with higher eigenfrequencies indicating that the structure is stiffer.
86

The accelerations along the x- and the y-directions of the four corner locations of the third level 

of the rectangular structure are depicted in Figure 3.44. It is evident that there are minimal 

variations between the movements of the four corner locations implying the absence of 

torsional effects. For the actual SMART2008 structure, however, the accelerations at the 

extreme corners C and D shown in Figure 3.45 vary significantly indicating that the structure is 

experiencing torsion.

As shown in Figure 3.46 the actual SMART2008 structure experiences extensive cracking 

which is attributed to torsion exerted on the structure. This is evident from (a) the orientation of 

cracking in the central column (shear cracks), and (b) the cracking in the shear walls and the 

base. On the other hand, the rectangular, closed-section structure where the asymmetries are 

kept to a minimum experienced very little damage under the same loading. The slight damage 

observed is in the concrete beam which only runs along the y-direction.

Shown in Figure 3.47 are the acceleration response spectra at the shaking table level of input 

signals RUN3, RUN4 and RUN5 in the x and y directions. Also shown in the same figure is the 

torsional mode of the structure and the corresponding frequency ft = 16.884 Hz. It can be 

observed that the torsional frequency of the structure on the shaking table coincides with a 

significant peak in the response spectrum of the input signals. This leads to the conclusion that 

the torsional effects are accentuated during the action of these signals. The effect was 

observed during the actual tests where the torsional frequency dropped significantly during 

RUN5 indicating that damage in the concrete was accumulated in the form of micro-cracks at a 

higher rate due to this “torsional” resonance.

Figure 3.48 compares the response spectra at the third level between the actual SMART2008 

structure and the one of rectangular cross section where most of the asymmetries have been 

eliminated during RUN3. It can be observed that the spectral peak close to the torsional 

frequency of the structure (16.884 Hz) has disappeared for the rectangular structure which 

experiences no torsional effects. Similarly in Figure 3.49, response spectra comparison is 

depicted for RUN6.
87

(a) “flexural” mode along x-axis (for the trapezoidal SMART2008 structure the mode is a 

combination of flexural and torsional)

(b) “flexural” mode along y-axis (for SMART2008 structure the “flexural” mode is a 

combination of flexural and torsional)

(c) torsional mode

Figure 3-43 Fixed-base mode comparisons between the trapezoidal 

SMART2008 structure and the one with rectangular cross 

section generated for comparison purposes
88

(a)

(b)

Figure 3-44 Accelerations computed at the four corners of the 

rectangular cross section SMART2008-like structure for 

RUN3: (a) x-direction acceleration and (b) y-direction 

acceleration showing minimal variation between the 

acceleration traces and indicating that the structure is not 

exhibiting torsional behaviour
89

(a)

(b)

Figure 3-45 Accelerations computed at two extreme locations of the 

trapezoidal SMART2008 structure (corners C and D) for 

RUN3: (a) x-direction acceleration and (b) y-direction 

acceleration. Depicted is the significant variation in 

acceleration amplitude indicating a torsional behaviour of the 

structure
90

(a) SMART2008 structure

(b) Rectangular cross-section structure

Figure 3-46 Damage sustained by the two different structures under the 

sameseismic load: (a) actual SMART2008 structure, and (b) 

rectangular SMART2008-like structure with minimal 

asymmetries
91

Figure 3-47 Response spectra of input accelerations signals RUN3, RUN4 

and RUN5 along x and y directions and the torsional mode 

(insert) of the SMART2008 structure on the shaking table (ft = 

16.884 Hz)

Figure 3-48 Third-level acceleration response spectra of the two 

structures (actual SMART2008 and rectangular version) for 

RUN3 input
92

Figure 3-49 Third-level acceleration response spectra of the two 

structures (actual SMART2008 and rectangular version) for 

RUN6 input
93

4. VULNERABILITY ASSESSMENT OF 3-D NUCLEAR STRUCTURES

Following the shaking table test prediction effort of the SMART2008 benchmark, a 2nd phase of 

the activity was launched that was dedicated to variability, sensitivity and vulnerability analysis 

based on numerical models developed for the SMART test specimen. The objectives of the 

effort were to (a) quantify the influence of uncertainties that exist in the estimation of design 

parameters of the structure on the response and (b) investigate structural vulnerability methods 

and derive fragility curves for typical limit states. Uncertainties typically surround the generation 

of fragility curves for structural systems, where the variability of the structural demand is 

convoluted with the variability in the structural capacity. The resulting fragilities are compared 

with limit states that themselves exhibit uncertainties in their specification. Thus, the 

vulnerability part of the benchmark represented a serious attempt to streamline an important 

aspect of safety in structural design.

Given the significant uncertainties surrounding the interaction between the shaking table used 

in the experimental effort and the SMART structure, it was recommended by the benchmark 

organizers that a fixed-base representation of the SMART three-dimensional structure be 

adopted for both the sensitivity and the fragility analysis. Figure 4.1 depicts the numerical 

representation of the SMART structure used in the subsequent analyses where the shaking 

table was reduced to rigid plane and the structure is supported through the base steel plates as 

the appear in the design.

Figure 4-1 Numerical model used in the sensitivity analysis of the test 

structurealso depicting locations for floor response spectra 

generation
94

4.1. Sensitivity Studies

Though the sensitivity study, which aimed to quantify (a) the variability in the seismic response 

of the SMART2008 structure and (b) identify the contribution stemming from uncertainties in 

input design parameters such as material properties, the influence of nonlinearities was sought. 

Specifically, by establishing a range around the nominal value of structural properties that 

influence the design of the structure, the response of the structure was evaluated at (a) the 

seismic design level and (b) a seismic level significantly exceeding the design level earthquake. 

Given that the design of the test structure followed a seismic code for nuclear structures which 

only allows for linear/elastic response, this phase of the study focused on quantifying the 

potential for the appearance of nonlinearities even at the design level earthquake. This was 

achieved by considering nonlinearities in the structural materials (i.e. bilinear stress-strain 

relations in the rebar and yielding uncertainty as well as non-linear behavior in the concrete 

beyond the allowable design stress by permitting cracking to appear) and by observing and 

analyzing floor response spectra generated by the nonlinear time history analyses. Appearance 

of nonlinearities in the structural response will impact the natural structural modes by (a) 

shifting the frequencies in the floor response spectra and (b) changing the displacement 

response or floor drifts anticipated in the structure for nominal or design values of the structural 

parameters.

It is also expected that even without the appearance of non-linear behaviour of the structure, 

the overall response will be influenced by the variability of the structural properties due to the 

changes in the fundamental modes of the overall structure. This in turn, when convoluted with 

the earthquake input frequency or period content, is expected to have a significant effect on the 

floor response spectra and consequently on the design of secondary systems.

The variability in material properties, the seismic input characteristics and the results of the 

nonlinear sensitivity analyses are described in the following subsections.

Variability Matrix of Structural Properties

In an effort to cover uncertainties in estimating influential design parameters such as concrete 

strength, rebar yield, structural damping and live loads, the variability matrix shown in Table 4.1 

was established. The objective was to study and quantify the influence of each parameter on 

the structural response individually by studying its effect through the entire range of values 

while maintaining the nominal value for the remaining varying fields. Table 4.1 depicts the range

of the four design parameters evaluated in this sensitivity study and the twelve distinct cases for 

each seismic input level used. 

Each set of properties defines the dynamic characteristics of the structure prior to applying the 

base excitation. The choice of properties, regardless of the input, results in a variation of the 

eigenfrequencies of the fixed-base structure and in particular those cases where the concrete 

modulus Ec and the added mass are being varied. Therefore, even when the structure during 

the nonlinear analysis remains within the linear range, variation in the response is expected.

Seismic input

For this parametric study, two sets of horizontal synthetic accelerograms (one for each

horizontal direction) were used to study the impact of parameter uncertainty. These 

accelerograms were derived from a white noise power spectrum, a process that prevented any 

bias towards a particular frequency band. Two seismic levels were selected to study the 

variability of the response; namely,
95



 a t de s ign le ve l (with P GA e qua l to 0.2 g)

 a t ove r de s ign le ve l (with P GA e qua l to 0.6g)

The acceleration time histories used in this parametric study were provided by the SMART200 

benchmark organizers and had a duration of Tdur = 5 secs with a time step of Δt = 0.005 sec.

Figures 4.2a and 4.2b depict the pairs of horizontal accelerations used in the time integration 

analyses (nonlinear) and the response sensitivity assessment. Shown in Figure 4.3 are power 

spectral densities of the input accelerations for both the 0.2g and the 0.6g earthquake pairs. 

Based on the number of realizations per seismic level shown in Table 4.1, a total of twenty four 

(24) independent analyses were performed and post-processed.

Response Output

The parametric or sensitivity analysis was based on the following response variables:

(a) Maximum inter-story drifts (differential displacement between two floors) at each floor 

for the corner point A in the x and the y directions and point D in the x direction. The 

idea behind the selection is to compare the variability in the global response of the 

structure near the torsional center with that at a location at the extremity of the structure 

(as seen in Figure 4.1)

(b) Maximum absolute relative displacement experienced by the structure and expressed 

as the difference between the displacement at the base and any location over the 

vertical walls

(c) Floor response spectra at Level 3, and at locations D and C shown in Figure 4.1. 

Response spectra in the x-direction were to be evaluated at location D and along 

y-direction at location C.

Parametric Study Results

Shown in Figure 4.4 are the first three eigenmodes for the structure for five of the twelve 

parameter sets depicted in Table 4.1. The computed eigenfrequencies for the fixed-base 

SMART structure differ significantly from those of the actual structure that was tested on the 

shaking table. In particular, the eigenfrequencies for the second mode (flexural mode along the 

y-direction) and the third mode (twisting about the vertical axis) are much higher than the 

counterparts of the actual structure on the shaking table. As discussed previously, this is 

attributed to the interaction of the structure with the massive table. 

Figures 4.5 through 4.9 depict response spectra computed on level 3 of the structure under 

different realizations of the structural parameters and seismic input pairs. Evident are the 

“nonlinear” effects on the response spectra that are induced by the 0.6g seismic input. 

Shown in Figure 4.10 are inter-story drift transients between floors of the test structure for the 

0.6g seismic input pair. The results depicted compare the nominal concrete modulus with the 

case of the lowest concrete modulus value expected within the parametric space.

Tables 4.2 and 4.3 are summaries of computed inter-story drifts at selected locations (corners 

A and D of wall #3) on the structure for 0.2g and 0.6g PGA respectively. Table 4.4 lists the 

maximum relative displacements observed in the fixed-base SMART structure under the action 

of the two seismic pairs corresponding to 0.2g and 0.6g PGA respectively.
96

Table 4.1 Structural properties variability matrix used in the sensitivity 

analysis of the SMART test structure
97

(a)

(b)

Figure 4-2 Bi-directional base input acceleration time histories 

(synthetic) generated from a white noise power spectrum and 

used simultaneously in the sensitivity analyses for the 0.2g 

and 0.4g PGA earthquakes
98

Power Spectral Densities Input Accelerations - Sensitivity Analysis

1E-16

1E-13

1E-10

1E-07

0.0001

0.1

100

0.1 1 10 100

Hz

PSD (g**2/Hz)

PSDx_02g PSDy_02g

PSDx_06g PSDy_06g

Figure 4-3 Power spectral densities (PSD) of the input accelerations 

used in thesensitivity analysis
99

 (a) Reference (Ec = 32 GPa; Fy= 500 MPa; M = 33.85 T)

(b) Case 2 (Ec = 25.6 GPa; Fy= 500 MPa; M = 33.85 T)

(c) Case 6 (Ec = 38.4 GPa; Fy= 500 MPa; M = 33.85 T)

(d) Case 12 (Ec = 32 GPa; Fy= 500 MPa; M = 37.23 T)

(e) Case 11 (Ec = 32 GPa; Fy= 500 MPa; M = 30.46 T)

Figure 4-4 Influence of the varying structural parameters on the first 

three eigenfrequencies and modal shapes of the fixed base 

structure used in the sensitivity study
100

 

Figure 4-5 Floor response spectra variability due to uncertainties in the 

concrete modulus computed on Level 3 (x-direction at D and 

y-direction at C) for the 0.2g earthquake input pair

 
101

 

Figure 4-6 Floor response spectra variability due to uncertainties in the 

concrete modulus computed on Level 3 (x-direction at D and 

y-direction at C) for the 0.6g earthquake input pair
102

Figure 4-7 Floor response spectra variability due to uncertainties in 

rebar yield strength computed on Level 3 (x-direction at D 

and y-direction at C) for the 0.6g earthquake input pair

Figure 4-8 Floor response spectra variability due to uncertainties in 

structural damping computed on Level 3 for the 0.6g 

earthquake input pair
103

 

Figure 4-9 Floor response spectra variability due to uncertainties in 

added floor mass computed on Level 3 for the 0.6g 

earthquake input pair
104

Interstory Drift (1st floor and base) - Concrete Youngs Modulus Variation

0.6g PGA Seismic Input

 (baseline of 32 GPa vs. 25.6 GPa)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

secs

Interstory Drift (mm)

DRFT_F1-F0 (B01)

DRFT_F1-F0 (B02)

Interstory DRIFT (Second vs. First Floor at pt. A) resulting from Concrete Ec Variation

0.6g PGA Seismic Input

(Baseline 32 Gpa vs. Ec=25.6 Gpa)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

secs

Interstory DRIFT-xx (mm)

DRFT_F2-F1 (B01)

DRFT_F2-F1 (B02)

Interstory DRIFT (3rd vs. 2nd floor at pt. A) resulting from Concrete Youngs Modulus 

Variation

0.6g PGA Seismic Input

(baseline Ec=32 GPa vs Ec=25.6 GPa)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 secs

x-DRIFT (mm)

DRFT_F3-F2 (B01)

DRFT_F3-F2 (B02)

Figure 4-10 Inter-story drift transients for the 0.6g seismic input pair 

comparing the nominal concrete modulus with the lowest 

value in the parametric space
105

Table 4.2 Summary of computed inter-story drifts between structural 

levels at corner locations A and D for the 0.2g seismic input 

pair.

LEVEL 1 LEVEL 2 LEVEL 3

pt. A pt. A pt. D pt. A pt. A pt. D pt. A pt. A pt. D

Case x-dir y-dir x-dir x-dir y-dir x-dir x-dir y-dir x-dir

1-Ref 0.28 0.187 0.66 0.295 0.2 0.73 0.302 0.181 0.741

2 0.288 0.125 0.68 0.267 0.147 0.774 0.258 0.141 0.831

3 0.35 0.17 0.794 0.33 0.186 0.826 0.3 0.176 0.828

4 0.332 0.195 0.693 0.31 0.191 0.722 0.318 0.154 0.69

5 0.326 0.18 0.78 0.326 0.177 0.764 0.322 0.166 0.78

6 0.28 0.187 0.66 0.295 0.2 0.73 0.302 0.181 0.741

7 0.28 0.187 0.66 0.295 0.2 0.73 0.302 0.181 0.741

8 0.28 0.187 0.66 0.295 0.2 0.73 0.302 0.181 0.741

9 0.4 0.2 0.9 0.392 0.192 0.88 0.34 0.174 0.853

10 0.23 0.13 0.5 0.241 0.151 0.59 0.226 0.151 0.562

11 0.3 0.2 0.69 0.303 0.196 0.82 0.297 0.178 0.827

12 0.315 0.181 0.736 0.284 0.191 0.83 0.28 0.185 0.838

Table 4.3 Summary of computed inter-story drifts between structural 

levels at corner locations A and D for the 0.6g seismic input 

pair.

LEVEL 1 LEVEL 2 LEVEL 3

pt. A pt. A pt. D pt. A pt. A pt. D pt. A pt. A pt. D

Case x-dir y-dir x-dir x-dir y-dir x-dir x-dir y-dir x-dir

1-Ref. 1.62 0.8 4.62 1.35 0.57 4.26 1.29 0.552 3.94

2 1.85 0.74 4.8 1.432 0.61 4.32 1.38 0.6 3.944

3 1.817 0.77 4.93 1.41 0.595 4.454 1.38 0.568 3.94

4 1.64 0.753 4.18 1.48 0.49 3.9 1.41 0.478 3.5

5 1.47 0.8 3.14 1.1 0.53 2.93 1.12 0.5 2.7

6 1.373 0.826 4.135 1.147 0.616 3.57 1.076 0.6 3.25

7 1.908 0.785 5.16 1.516 0.604 4.6 1.486 0.58 4.24

8 1.955 0.534 4.816 1.543 0.42 4.444 1.442 0.41 4.01

9 1.333 0.78 4.33 1.165 0.736 4.25 1.128 0.716 4.06

10 2.32 0.766 5.914 1.767 0.585 5.331 1.85 0.581 4.824

11 1.533 0.74 3.51 1.124 0.488 3.17 1.035 0.461 2.986

12 2.1 0.772 5.778 1.49 0.704 5.01 1.467 0.668 4.71
106

Table 4.4 Maximum relative displacements computed for the fixed-base 

SMART structure and the two seismic pairs (0.2g and 0.6g)

0.2 g 0.6 g

Δx (mm) Δy (mm) Δx (mm) Δy (mm)

x-dir y-dir x-dir y-dir

1 –Ref. 1.96 2.1 12.7 11.13

2 2.18 2.4 12.85 11

3 2.3 1.91 13.27 11.34

4 1.984 1.93 11.24 9.75

5 2.146 2.018 8.42 7.65

6 1.96 2.1 10.95 10.08

7 1.96 2.1 13.8 11.45

8 1.96 2.1 13.01 10.17

9 2.5 2.245 12.57 14.36

10 1.574 1.25 15.87 12.7

11 2.264 2.06 9.17 7.88

12 2.35 1.86 15.25 12.46

4.2. Vulnerability Study – Seismic Fragility Assessment

An integral part of the SMART2008 benchmark was the utilization of the test structure 

numerical model to develop fragility curves in an effort to convolute uncertainties in the seismic 

input with uncertainties in the structural capacity. In the parametric study of the previous 

section, the influence of the variation of individual design or structural properties on the 

response of the global structure was quantified according to ranges that are typically introduced 

in design. In the vulnerability aspect of uncertainty treatment described in this section, a multidimensional uncertainty matrix describing the structural capacity is convoluted with 

uncertainties regarding the seismic excitation. Thus, the combined effect can be compared to 

the anticipated limit state or capacity of the structure.

Structural capacity A is defined as the limit seismic load before a pre-defined performance limit 

state (or damage state) is reached by the structure. Due to the fact that capacity is influenced 

by uncertainties surrounding structural properties that can be characterized as random, 

capacity itself can be considered and treated as a random variable. The limit seismic load is a 

function of a parameter characterizing ground motion such as PGA, PGD, PGV or CAV. If PGA 

is selected as the seismic ground motion level indicator, for example, then capacity can also be 

expressed in terms of PGA. As a result, the probability of failure (or exceedance of the 

performance limit state) of the structure due to seismic ground motion level α is expressed by 

fragility curves which represent the following relation

P (α) ≡ P( failure |α) = P(A < α) f (4.1)

Failure or exceedance of the performance state at a ground motion level α occurs when the 

actual capacity of the structure is inferior to the seismic demand at the level α. Exceedance of a 

limit state could be expressed in a variety of ways using terms such as inter-story drift, structure 

yielding, reduction of fundamental modes, etc.. To establish the relation between the demand 

and capacity, typically a computationally intensive non-linear analysis is required in which a 

randomized structure is subjected to increasing seismic input which is also random in nature. 

For the SMART structure shown in Figure 4.11, the structural elements above the shaking table 

are considered including the steel plate at the structure/table interface. The structure is 
107

subjected to horizontal pairs of earthquakes at the base and all the structural elements are 

governed by non-linear constitutive relations (i.e. concrete can crack and crush and rebar can 

yield and fail).

In order for the probability of failure or exceedance to be quantified, given that both the capacity 

and the exciting earthquake are random variables, a significant number of convolutions at any 

given seismic level α will normally be required to increase the confidence in the estimation of 

probability. Capacity is treated as a random variable due to the uncertainties surrounding its 

structural properties while the seismic input, despite specifying its ground motion level α, is still 

a random process of duration, frequency content, significant number of cycles, etc. This 

“complete” approach can be expressed according to the relation below

total a

fail a

N

N

P d C a

|

| ( ≥ ) | = (4.2)

where Ntotal represents the total number of random realizations/input pairs and Nfail is the 

number of realizations during which failure has been reached. When utilizing (4.2) to derive the 

fragility of a structure, the number of seismic inputs can be very large as one attempts to 

saturate the earthquake ensemble with duration, frequency content, magnitude, hypo-central 

distance, etc.
108

Figure 4-11 Fixed-base model used for the nonlinear analyses in fragility

assessment depicting the discretization of the concrete 

structure and the steel rebar distribution

The randomization of the SMART structure used in the fragility phase of the benchmark was 

achieved by assuming that four (4) key structural properties have a degree of uncertainty 

around a nominal or design value. Specifically, the following randomization is introduced:

• Secant concrete Young’s modulus described by a mean value of Ec=32,000 MPA and a 

standard deviation of 2,200 MPa,

• Rebar yielding stress with mean Fy = 500 MPa and standard deviation of 30 MPa,

• Structural damping with nominal (mean) value of 2% and standard deviation of 1%,

• Added floor masses with mean total of 33.85 T and standard deviation of 5%.of the 

mean value

However, while these structural parameters are expected to follow a certain distribution around 

the mean value (uniform, normal, log-normal, etc.) there is no provision for spatial variation or 
109

spatial correlation between locations around the structure. Such an added requirement will add 

additional variation in the process and result in additional computational cost. A procedure that 

may be adopted in an effort to account for the spatial variability around the structure of the key 

structural properties is discussed in the report in a later section. 

As noted previously, the randomization of the four key structural properties listed above occurs 

simultaneously rather than isolating one varying parameter at a time. This process leads to 

distinct “multi-dimensional random realizations” of the structural capacity that are in accordance 

to the adopted distribution.

Traditionally in vulnerability analysis of structures, limit or damage states have been defined 

qualitatively as light, moderate, extensive and complete. In developing fragility curves, however, 

in addition to the qualitative damage states, quantitative limit states must be established so that 

they can be compared with the seismic demand at a given intensity level. Different measures or 

indicators are used to describe the maximum state that the structure reaches as a result of a 

seismic load. The selection of these indicators may be influenced by the type of structure, 

empirical observations of performance of similar structure under real seismic loads, etc. 

No exact formulation exists in establishing limit states for a structure. Such states may be 

specified or derived using either physics-based models or judgment stemming from experience 

and observations. A physics-based approach defines some functional level to be assigned to 

the structure for various levels of damage and is based on engineering analysis. Using 

judgment to define the limit states, the goal is to define functionality levels which would be 

assigned to the structure by decision makers for different levels of observed damage. 

For the fragility analysis of the SMART structure inter-story drifts and eigenfrequency 

degradation were adopted as damage indicators. The qualitative and quantitative definition of 

the damage indicators and limit states are listed below:

Maximum inter-story drift at location A (x and y directions) and location D (x direction). The 

objective of linking the maximum drifts at the selected locations of the SMART structure is to 

evaluate the local damage effects.

Three limit states are being considered and expressed in terms of the structure story height h

(h = 1.2 m)

• Light damage (h/400)

• Controlled damage (h/200)

• Extensive damage (h/100).

Degradation of structural eigenfrequencies (first two modes) in order to account for the 

overall structural behavior and evolution of its dynamic characteristics. 

Similarly, three damage levels are considered that have been quantified as follows:

• Light damage (frequency drop of 15%)

• Controlled damage (frequency drop of 30%)

• Extensive damage (frequency drop of 50%)

The degradation of eigen-frequencies is deduced by comparing the estimated eigenfrequency 

of the structure at the end of the seismic load to the initial frequency estimated based on the 

corresponding realization of the randomized structural properties.
110

While additional uncertainty can be introduced in the damage criteria, which in turn will add 

further computational burden in an effort to capture the influence on the vulnerability 

assessment, no variability in the defined limit states have been assumed in this analysis.

Generation of Fragility Curves

Fragility curves are continuous functions reflecting the conditional probability that the response 

of a structure to increasing seismic excitations represented by a random process exceeds a 

given performance limit state defined both qualitatively and qualitatively. The structural 

response is typically defined by a pre-selected variable such as inter-story drift, frequency drop, 

yielding, etc. Theoretically, fragility represents the probability that the response of the structure 

exceeds a given threshold conditional on an earthquake intensity parameter (see Equations 4.1 

and 4.2). The conditional probability of failure of a structure can be linked to a seismic input 

motion indicator, such as PGA, CAV or PGD. This is due to the fact that both the seismic 

demand and the capacity can be expressed in terms of the selected seismic input indicator.

For a given structure, such as the SMART test structure used in the benchmark, fragility curves 

are developed from a number of seismic response data obtained using simulated ground 

motions and through a nonlinear time history analysis. Fragility curves will be affected by 

several parameters such as (a) performance limit criteria which can be both random and 

multidimensional (where a correlation of more than one damage indicator is used), (b) input 

ground motion random character, (c) uncertainties in structural properties and their proper 

randomization, and (d) appropriate modeling of the structure which captures the damage 

indicators. To establish fragility curves that incorporate uncertainties in all four groups of 

parameters, an extensive computational effort is required given that each participating 

parameter is treated as random thus requiring convolution with the remaining three groups over 

their range of uncertainty. In order to reduce the computational effort while maintaining high 

confidence levels, certain assumptions have been introduced in randomization of the influential 

parameters. These assumptions permitted the description of uncertainty in closed-form 

expressions which resulted in closed-form solutions of the fragility curves. 

For example, a two parameter lognormal distribution function has been used extensively for the 

construction of fragility curves. This was made possible by assuming that the structural 

properties defining capacity in the structure are log-normally distributed and so is capacity itself.

The assumption of a lognormal capacity distribution was introduced during the development of 

probabilistic risk assessment methodologies for nuclear power plants in the 1970’s and 1980’s. 

It was prompted by observations that the actual structural strength and the design strength are 

related through a multiplicative safety factor resulting from a multitude of factors each of which 

is associated with a specific source of uncertainty in the structure. Based on the assumption of 

lognormal distribution for each of these factors, the overall safety factor is also distributed lognormally due to the multiplicative nature of variables defined by a lognormal distribution. 

Therefore, by assuming that uncertainties in key parameters influencing capacity (i.e. strength, 

damping, etc.) are log-normally distributed, the capacity is log-normally distributed as well. This 

assumption has significant implications in that it enables the capturing of uncertainty in the 

structural capacity into a single, analytical expression in terms of either a response value 

(deformation, ductility, etc.), or better yet, a seismic input indicator, such as peak ground 

acceleration (PGA), peak ground velocity or cumulative absolute velocity (CAV). 

Evaluation of fragilities can be further simplified by developing a probability distribution for the 

demand conditioned on the seismic input indicator (i.e. PGA, PGD, CAV) and convolving it with 

the capacity distribution. If the seismic demand is assumed to also follow a log-normal 

distribution then fragilities can be deduced from the following relation 
111

] ln( / ) [ ] [ 2 2

D C

d c S S P D C

β β α

α

+

> = Φ (4.3)

where Sd is the median estimate of the demand, SC is the median estimate of the capacity, βD|α

is the dispersion or logarithmic standard deviation of the demand conditioned on the intensity 

measure α and βC is the dispersion of the capacity. Written in a different form, when the 

lognormal fragility model is chosen, the fragility curves can be defined from the relation

] ln( / ) ( ) [ β

α

α m

f

A P = Φ (4.4)

where Ф(•) is the standard Gaussian cumulative distribution function and fragility curves, are 

entirely determined by only two parameters, which are median capacity Am and lognormal 

standard deviation β. These parameters can be evaluated by means of regression analysis.

Fragility curves based on relation (4.4) were developed in this benchmark according to the 

following procedure:

• A suite of N=50 ground motions are selected that contain characteristics of earthquakes 

that are relevant to the location of the structure in question. While typically the ensemble 

consists of real earthquakes and/or a combination of real and synthetic ground motions, 

fifty synthetic records were utilized in the analysis.

• N=50 realizations of the structure were generated by considering the four random 

distributions of the structural parameters (Ec, Fy, damping and added floor mass). This 

resulted in 50 independent structural variations of the model shown in Figure 4.11.

• The 50 seismic bi-directional synthetic earthquakes were randomly paired with the 50 

structural realizations and through a full non-linear time history analysis for each pair, 

various responses were monitored and recorded (displacements, floor accelerations, 

structural frequency evolution, etc.)

• A regression analysis between responses and seismic input indicators were performed 

to derive the median capacity and the dispersion which are then used to develop the 

corresponding fragility curves.

In the following sections the details of the above adopted generic process leading to fragility 

curves is presented in more detail.

Ground Motion Suite 

An ensemble of 50 bi-directional horizontal accelerograms provided by the SMART benchmark 

organizers were used in the fragility analysis. All 50 accelerograms are synthetic and have been 

derived from a spectrum similar in shape with the one used to design the SMART2008 

specimen. Effort was made to ensure that the frequency range is saturated and that the 

duration of the strong motion in the records covers a wide range. By selecting a large ensemble 

of earthquakes, which cover the range of interest for both frequency content and duration, one 

avoids the possibility of excluding probable scenarios near the tails of the distributions 

characterizing the seismic demand and the structural capacity.

Indicators such as peak ground acceleration, displacement and velocity as well as cumulative 

absolute velocity (CAV), were provided for each of the bi-directional records. Figure 4.12 

depicts the distribution of peak ground accelerations characterizing the 50-member ensemble.
112

Figure 4.13 depicts the regression between indicators such as PGA, PGD and CAV. It is 

observed that the PGA has stronger correlation with CAV than PGD. Figure 4.14 depicts 

acceleration time histories and response spectra for a subset of the 50 earthquake pairs.

Structural Realizations

In order to perform a convolution with the N number of earthquakes selected to represent the 

demand, an equal number of structural capacity realizations need to be generated according to 

a statistical model. Having accepted the fact that uncertainties surrounding the structural 

properties follow the log-normal distribution, the main task is to enable the propagation of the 

uncertainty throughout the model. As noted earlier, the structural properties that exhibit 

uncertainty are the secant concrete modulus Ec, the yield stress in the rebar Fy, the structural 

damping and the added live load. Their distribution is defined by their mean value and the 

standard deviation. In order to propagate the uncertainty through the model without being 

biased towards the mean value of the structural parameters, the stratified Latin Hypercube 

sampling approach was employed. Figure 4.15 depicts the sampling approach used to 

construct the N=50 realizations which is based on the principle that the entire range (i.e. mean3σ – mean+3σ) is represented in the sample. This is done by selecting the random values of 

the structural property on the basis of the cumulative distribution function which is divided in 

equal intervals of cumulative probability. 

Figure 4.16 depicts two of the four randomized parameters following a log-normal distribution. 

As shown, the entire range is represented. The non-uniform distribution shown in the two traces 

is the result of the distribution of the uncertainty in two groups of 30 and 20, respectively. The 

first 30 realizations were constructed to satisfy the requirements of the benchmark, while added 

to the distribution were 20 additional earthquake-realization pairs.

Non-Linear Analysis 

The 50 realizations of the structure properties were introduced into the numerical model shown 

in Figure 4.11 and were paired with the 50 synthetic ground motions (x and y directions acting 

simultaneously at the base of the structure) leading to 50 non-linear analyses of the structure. 

Eigenfrequencies of the structure for each of the realizations were extracted prior to the 

applications of the earthquake pair, mid-way of the transient analysis, and at the end of the 

record, in an effort to observe the degradation in the structural modes resulting from damage in 

the concrete and yielding or failure in the rebars. In addition, acceleration responses were 

monitored through-out the structure and response spectra were generated. Shifting of response 

spectra peaks were monitored and were correlated with the shifting of frequencies derived 

directly from the analysis. Displacements at selected locations in the structure were also 

monitored and maximum inter-story drifts were deduced and recorded.

Figure 4.17 depicts the condition of the structure at the end of four out of the fifty analyses 

performed indicating variability in the level of damage experienced. Shown in Figure 4.18 is the 

state of the structure where reinforcing steel bars failed near the foundation. As compared to 

the shaking table test where input PGA values in excess of 1.0g were achieved leading to 

minimal progressive cracking and no rebar failure, it is evident that the convolution of seismic 

inputs with randomized structural properties has the potential of inducing higher levels of 

damage in the same structure although the fixed-base conditions may have contributed to the 

increased damage state.

Figure 4.19 depicts trends between the different seismic input indicators and the structural 

response while Fig. 4.20 shows the regression between the different response variables.

Figures 4.21 and 4.22 illustrate the variability in the response spectra observed on the 3rd level 

of the structure.
113

Estimation of Fragility Curve Parameters

Following the N=50 deterministic analyses which randomly paired the 50 seismic inputs with the 

50 property uncertainty realizations and the monitoring/estimation of the structural responses, 

the two parameters shown in equation 4.4 (median capacity Am and the dispersion β) can be 

deduced from regression analysis. 

Specifically, by denoting the numerical analysis output as Ψ (drift, displacement, frequency 

drop, etc.) and the ground motion indicator (PGA, PGD, CAV, etc.) as ς, upon completion of the 

deterministic analyses there exist N = 50 structure response outputs Ψ (Ψ1, Ψ2,,, ΨN) 

corresponding to ground motion indicator levels ς (ς1, ς2, ,, ςN). Given that Ψ is assumed to be 

log-normally distributed it implies that ln(Ψ) is a normally distributed random variable. By 

performing linear regression on ln(Ψ) and ln(ς) according to the relation

ln(Ψ) = α + b ln(ς ) + ε (4.5)

parameters α and b can be determined. ε represents the uncertainty of the regression relation 

and is a normally distributed random variable with median = 1 and logarithmic standard 

deviation σlnε, which is also equal to dispersion or log-standard deviation β. Figure 4.23 depicts 

one of the regression relations used to estimate the parameters of equation 4.5.

Following the estimation of parameters α and b in (4.5) the median seismic capacity Am can 

be evaluated from the following relation

ln( ) ln( ) ΨlimitSTATE = a + b Am (4.6)

Where ΨlimitSTATE is the response corresponding to the limit state (light, controlled, and extensive 

damage). Table 4.4 summarizes the median seismic capacity and the corresponding dispersion 

for the three damage states, damage indicators and various seismic input indicators (PGA, 

CAV, etc.). Having established the median capacity and dispersion for all cases as depicted in 

Table 4.5 fragility curves using equation 4.4 are generated.

Figures 4.24 - 4.29 depict fragility curves for three different motion indicators (PGA, PGD and 

CAV) and the three damage states based on various damage indicators (floor drift at different 

locations and degradation of first and second eigenfrequencies). As seen from the generated 

fragility curves there exists significant variability between the different input and damage 

indicators. The correlation between the structural responses seen in Figures 4.19 and 4.20 are 

a pre-cursor of the variability that would be expected in deriving the fragilities. 

To address the variation observed in fragilities when different metrics are used (either seismic 

input indicators or damage indicators) methodologies have been proposed that introduce multidimensional fragility surfaces where limit states are correlated, for instance, displacements and 

accelerations observed in a structure. For such approach, the response can be represented by 

a “bell surface” in, say, the spectral acceleration – spectral displacement plane and the limit 

states in terms of accelerations, displacements, or velocities. 

While fragility curves are difficult to defend regarding their accuracy in representing the damage 

or failure probability of a given structure, given the uncertainties in establishing limit states and 

their definition, it is important, however, for the variability between different metrics to be 

minimized. While confidence levels can be increased by increasing the number of realizations 

with a corresponding computational cost, it is the quantification of the limit states that will 

remain as the biggest challenge. Recommendations on ways to improve confidence levels in 

fragility estimates are included in the summary chapter.
114

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

1 11 21 31 41 51 61 71 81 91

Input (xx & yy)

PGA (g)

Figure 4-12 Peak ground accelerations histogram of the 50 earthquake 

sets (x and y directions) used in fragility analysis
115

 

Regression between PGA and CAV for the 50 Earthquakes

y = 14.731x + 0.8651

R2 = 0.6356

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2

PGA (g)

CAV (m/s)

Regression between PGD and CAV for the ensample of 50 earthquakes

y = 35.759x + 4.6526

R2 = 0.2466

0

5

10

15

20

25

0 0.05 0.1 0.15 0.2 0.25 0.3

PGD (m)

CAV (m/s)

Figure 4-13 Regression between different indicators of the 50 bidirectional synthetic records used in the fragility analysis
116

Acceleration Time Histories of Input Earthquake Pair #10

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 0.5 1 1.5 2 2.5 3 3.5 4

secs

Accel (g)

E10_axx E10_ayy

Response Spectra of Input Earthquake Pair #10

0

0.2

0.4

0.6

0.8

1

1.2

0.1 1 10 100

Freq (Hz)

Spectral Accel (g)

RSx_E10

RSy_E10

Aceleration Time Histories of Input Earthuquake Pair #30

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

secs

Accel (g)

E30_axx

E30_ayy

Response Spectra of Input Earthquake Pair #30

0

0.5

1

1.5

2

2.5

3

3.5

0.1 1 10 100

Freq (Hz)

Spectral Accel (g)

RSx_E30 RSy_E30
117

Acceleration Time Histories of Input Earthquake Pair #40

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2 2.5 3 3.5 4

secs

Accel (g)

E40_axx E40_ayy

Response Spectra of Input Earthquake Pair #40

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1 1 10 100

Freq (Hz)

Spectral Accel (g)

RSx_E40 RSy_E40

Acceleration Time Histories of Input Earthquake Pair #50

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3 3.5 4

secs

Accel (g)

E50_axx E50_ayy

Response Spectra of Input Earthquake #50

0

0.5

1

1.5

2

2.5

3

3.5

4

0.1 1 10 100

Freq (Hz)

Spectral Accel (g)

RSx_E50 RSy_E50

Figure 4-14 Four acceleration time histories and four response spectra of 

a sub-set of the 50 synthetic records used in the fragility 

analysis (First two pairs of time histories and response 

spectra are on previous page – 116)
118

Figure 4-15 Stratified sampling Latin Hypercube used for uncertainty 

distribution of the structural properties

Figure 4-16 Cumulative distribution function of structural properties 

(concrete Ec and rebar Fy) of the 50 realizations used to lognormally distribute uncertainty
119

Figure 4-17 Damage conditions (crack extend and pattern) reached in the

structure at the end of four seismic inputs out of the 50 

randomly selected earthquake-realization pairs

Figure 4-18 Damage conditions (reinforcement failure near the base of 

the structure) reached during one of the randomly selected 

earthquake-realization pairs
120

Regression between PGA and Drift at Level 3 Location D

y = 6.6495x + 0.7783

R2 = 0.4791

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1 1.2

PGA (g)

Drift 3Dx (mm)

Scattering of Structure Eigen-Frequency Reduction with PGA of Base Excitation

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1 1.2 1.4 PGA (g)

Eigenfrequency Drop (%)

Drop_Eigen-freq1_%

Drop_Eigen-freq2_%

Linear (Drop_Eigen-freq1_%)

Linear (Drop_Eigen-freq2_%)

Scattering of 3rd Floor Drift as a Function of Peak Ground Displacement (average)

0

2

4

6

8

10

12

14

0 0.05 0.1 0.15 0.2 0.25 0.3

PGD (m)

Drift (mm)

Drift_X_ptD

Linear (Drift_X_ptD)

Eigen-frequency reduction as function of input excitation PGD (average)

0

10

20

30

40

50

60

70

0 0.05 0.1 0.15 0.2 0.25 0.3

PGD (m)

Eigen-frequency Drop (%)

Drop_Eigen-freq1_%

Drop_Eigen-freq2_%

Linear (Drop_Eigenfreq1_%)

Linear (Drop_Eigenfreq2 %)

Scattering of 3rd Level Drift as function of CAV

0

2

4

6

8

10

12

14

0 5 10 15 20 25

CAV (m/s)

Drift (mm)

Drift_X_ptD

Linear (Drift_X_ptD)

Scattering of Eigen-frequency drop as function of input excitation CAV

0

10

20

30

40

50

60

70

0 5 10 CAV (m/s) 15 20 25

Eigen-frequency Drop (%)

Drop_Eigen-freq1_%

Drop_Eigen-freq2_%

Linear (Drop_Eigenfreq1_%)

Linear (Drop_Eigenf 2 %)

Figure 4-19 Regression between seismic input indicators of the 50-

earthquake ensemble and structure response in terms of 

floor drifts and eigenfrequency degradation
121

(a) (b)

y = 3.9124x + 22.144

R2 = 0.5348

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12

Drift at D (x-dir)

Frequency-1 Drop (%)

(c)

Figure 4-20 Regression between non-linear analysis output variables 

monitored on the test structure for the 50 pairs of structural 

property realizations and the seismic inputs. (a) Regression 

between 1st and 2nd structural frequency drop, (b) 

regression between drifts along x and y at the same 

structural location and (c) regression between drift at a 

structural extremity (location D) and drop in 1st frequency
122

Figure 4-21 Non-linear response spectra generated on level 3 of the 

structure at location D (x-direction) and location C (ydirection) from the analysis of the 30-set earthquake pairs
123

Figure 4-22 Non-linear response spectra generated on level 3 of the 

structure at location D (x-direction) and location C (ydirection) from the analysis of the extra 20-set earthquake 

pairs
124

 

Figure 4-23 Regression analysis between the logarithms of the PGA and

eigenfrequency drop used to estimate median capacity and

log-standard deviation

Table 4.5 Summary of estimates of median capacity Am and log-normal 

deviation β.

Light

damage

Controlled 

damage

Extensive

damage

metric response Am Beta Am Beta Am Beta

1 PGA max drift pt A x 0.833777 0.46724 1.581545 0.562 2.999942 0.7182

2 PGD max drift pt A x 0.264676 0.6279 0.921997 0.7013 3.211765 0.83172

3 CAV max drift pt A x 16.28560 0.57628 37.18083 0.6554 84.88570 0.79347

4 PGA max drift pt A y 1.081667 0.56126 1.792055 0.6423 2.968993 0.78263

5 PGD max drift pt A y 0.460597 0.78644 1.262226 0.8462 3.459016 0.95707

6 CAV max drift pt A y 21.74134 0.67528 40.70131 0.744 76.19571 0.86804

7 PGA max drift pt D x 0.391637 0.50586 0.780536 0.5945 1.555614 0.74391

8 PGD max drift pt D x 0.060917 0.61079 0.213835 0.686 0.750618 0.81888

9 CAV max drift pt D x 6.138204 0.61029 15.39055 0.6855 38.58932 0.81851

10 PGA 1st freq drop 0.122945 0.51906 0.347101 0.6057 0.745831 0.75294

11 PGD 1st freq drop 0.002231 0.62518 0.041942 0.6988 0.364468 0.82967

12 CAV 1st freq drop 1.532403 0.54159 5.331796 0.6252 13.36413 0.76865

13 PGA 2nd freq drop 0.026502 0.37838 0.186933 0.4906 0.788738 0.66383

14 PGD 2nd freq drop 3.678E-0 0.42019 0.007789 0.5235 0.403240 0.68852

15 CAV 2nd freq drop 0.177176 0.39605 2.319474 0.5043 15.43719 0.67406
125

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

PGA (g)

Failure Probability

Drift_Ax light

Drift_Ax controlled

Drift_Ax extended

Drift_Ay light

Drift_Ay controlled

Drift_Ay extended

Drift_Dx light

Drift_Dx controlled

Drift_Dx extended

Freq1 Drop light

Freq1 Drop controlled

Freq1 Drop extended

Freq2 Drop light

Freq2 Drop controlled

Freq2 Drop extended

Figure 4-24 Fragility curves for all three damage states and all capacity 

measures using PGA as the demand indicator

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

PGD (m)

Failure Probability

Drift_Ax light

Drift_Ax controlled

Drift_Ax extended

Drift_Ay light

Drift_Ay controlled

Drift_Ay extended

Drift_Dx light

Drift_Dx controlled

Drift_Dx extended

Freq1 Drop light

Freq1 Drop controlled

Freq1 Drop extended

Freq2 Drop light

Freq2 Drop controlled

Freq2 Drop extended

Figure 4-25 Fragility curves for all three damage states and all capacity 

measures using PGD as the demand indicator
126

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35 40

CAV (m/s)

Failure Probability

Drift_Ax light

Drift_Ax controlled

Drift_Ax extended

Drift_Ay light

Drift_Ay controlled

Drift_Ay extended

Drift_Dx light

Drift_Dx controlled

Drift_Dx extended

Freq1 Drop light

Freq1 Drop controlled

Freq1 Drop extended

Freq2 Drop light

Freq2 Drop controlled

Freq2 Drop extended

Figure 4-26 Fragility curves for all three damage states and all capacity 

measures using CAV as the demand indicator
127

Failure probabilities for drift in x and y directions 

 PGA Indicator

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

PGA (g)

Failure Probability

Drift_Ax light

Drift_Ax controlled

Drift_Ax extended

Drift_Ay light

Drift_Ay controlled

Drift_Ay extended

Figure 4-27 Fragility curve comparison for different drift measures - PGA

Comparison of fragilities using limit states of drift and 1st frequency drop 

PGA indicator

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

PGA (g)

Failure Probability

Drift_Dx light

Drift_Dx controlled

Drift_Dx extended

Freq1 Drop light

Freq1 Drop controlled

Freq1 Drop extended

Figure 4-28 Fragility curve comparison between drift and frequency drop 

- PGA
128

Fragility comparison between drift and frequency 1 drop 

CAV indicator

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20

CAV (m/s)

Failure Probability

Drift_Dx light

Drift_Dx controlled

Drift_Dx extended

Freq1 Drop light

Freq1 Drop controlled

Freq1 Drop extended

Figure 4-29 Fragility curve comparison between drift and frequency drop 

- CAV
129

5. SUMMARY

The SMART2008 benchmark addressed the complex issue of multi-directional seismic loading 

on reinforced concrete nuclear structures when subjected to torsion and nonlinearities. The 

benchmark, officially initiated in 2008, was coordinated by CEA Saclay, France and supported 

by Electricite de France (EDF). The benchmark used a quarter-scale nuclear structure 

designed according to the French nuclear code. The test structure was used as the basis for a 

series of independent numerical studies and was eventually tested on the shaking table at CEA, 

Saclay under a series of multi-directional seismic loadings.

An international numerical prediction contest that accompanied the shaking table tests aimed to 

(a) assess the state-of-the-art of numerical capabilities to analyze non-linear structures and 

compare the different best-estimate methods for floor response spectra generation and (b) 

evaluate the different methodologies and guidelines on nuclear design. 

Upon completion of the shaking table tests and the best-estimate predictions of the response, a 

vulnerability study was undertaken where seismic fragility curves were generated. The 

vulnerability study considered the variability in the response of structures stemming from (a) 

structural property uncertainties, (b) random character of the seismic input and (c) numerical 

representation of the structural systems. 

This section summarizes the findings/assessments that were drawn from the shaking table 

tests and the accompanying numerical analyses. Further, based on findings that may have 

structural design implications, discussions on specific approaches that address identified 

outstanding issues are presented.

5.1. SMART2008 Shaking Table Test Assessment

A series of shaking table tests of the SMART2008 structure were performed where real, low 

intensity earthquake signals were applied followed by a series of synthetic earthquakes 

increasing in intensity for up to 1.13g horizontal PGA in the last test. Upon completion of the 

+tests, the following observations were made:

• The SMART reinforced concrete structure was designed by precisely following the 

guidelines of the French nuclear design code with a design level earthquake of 0.2g 

PGA. In the course of the shaking table tests peak accelerations at the base of the 

structure reached PGA levels of 1.13g (a factor greater than 5 times the design level). 

Examination of the structure at the end of the tests revealed that the structure resisted 

the excessive seismic loads and suffered only minor damage. The observed damage 

was in the form of concrete cracking and minor, localized rebar yielding (observed in 

rebar recorded strain). 

• Reinforced concrete under seismic loads that are much lower than the design 

earthquake level appear to undergo stiffness changes due to micro-cracking or 

adjustment in the concrete-rebar interface. The SMART2008 tests confirmed 

observations made in a previous study of seismic behavior of reinforced concrete shear 

walls (i.e. CAMUS benchmark, D. Combescure, et al.). The common observation in the 

two shaking table experiments is that the eigenfrequencies of the structure are reduced 

following seismic loads with intensities as low as 0.1g that are attributed to a reduction 

in the reinforced concrete stiffness. As a result, floor response spectra generated from 

actual in-structure recordings undergo a shift of their peaks towards lower frequency 

values, a feature that may have implications to the design of equipment supported on 

floors.
130

• The influence of the shaking table on the dynamic response of the structure was 

assessed to be significant in that it (a) lowers the eigenfrequencies of the combined 

system and thus the response of the structure relative to the frequency content of the 

input excitation and (b) alters the shape of the floor response spectra. The role and 

participation of the shaking table can be viewed as similar to SSI where the role of the 

soil supporting the structure is significant. 

• The inability to fully control the shaking table motions, and in particular its vertical motion 

and the rotational or rocking response during the tests, introduced uncertainties 

regarding the anticipated and observed structural responses. Added to the controllability 

issue was the fact that the dynamic properties of the table supports (actuators) are not 

well understood or established, so best-estimates were introduced to explain the 

observed response. As a result, the observed motions on the table deviated from the 

excitations intended to be introduced at the different stages of the 13-run test series. 

The uncertainties surrounding the role of the shaking table (dynamic properties, 

stiffness, and interface conditions with the structure) can be said to resemble those 

associated with the interface of real structures with the supporting soil and the 

assumptions that are made to account for the SSI. 

• An extensive instrumentation array was implemented during the SMART2008 tests that 

captured a multitude of in-structure response data (multi-directional accelerations 

throughout the structure, absolute displacements, steel reinforcement strain) and 

monitored concrete damage. The collected test data (a) provide the basis for assessing 

the numerical models and their capability to predict non-linear structural response and 

(b) contain a wealth of information that can lead to important insights for the seismic 

response and capacity of reinforced concrete structures and the potential use of such 

structures in nuclear design.

5.2. Numerical Prediction Assessment

A series of numerical analyses were conducted as an integral part of the SMART2008 

benchmark. In the first phase, prior to the construction of the test structure, a blind prediction of 

the seismic response of the structure to synthetic seismic loads generated from the same 

design spectrum was performed by 35 research teams. In addition, static and modal analyses 

were performed under different loading and boundary conditions. The primary goal was for the 

benchmark to assess the different methodologies (linear and/or nonlinear time history analyses, 

modal or spectral analyses, linear static or non-linear static or push-over analyses, etc.) used in 

different parts of the world to design reinforced concrete structures when subjected to 3-D 

effects and non-linear behavior. The summary assessment of (a) the methodologies, (b) types 

of analyses and (c) result correlation or scattering was presented by the SMART2008 

organizers during the proceedings of a dedicated workshop organized in December 2010 at 

CEA, Saclay. According to the summary, the variation in the structural response using the 

different methodologies was significant and stemmed primarily from (a) assumed properties of 

materials involved such as concrete moduli and damping, and (b) the level of simplicity 

introduced in the numerical model adopted to describe the structure. Based on the identified 

sources for the wide variability, it was concluded that better guidance in the design codes is 

needed regarding the adoption of design parameters which are meant to compensate for the 

non-linear behavior (i.e. damping specification).

During the second phase which was launched after the construction and preliminary testing on 

the shaking table of the structure, predictions of its non-linear, 3D response resulting from the 

sequence of seismic loads were made and compared to the actual test data.

The integrated assessment conducted by the SMART2008 benchmark organizers regarding the 

best-estimate prediction of the shaking table tests by the participants is the following:
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• Numerical prediction models generally over-predict the in-structure accelerations while 

under-predict the displacements. This is primarily attributed to the lack of understanding 

of the evolution of damping in the reinforced concrete structure. An important parameter

is the often arbitrary selection of initial structural damping ratio for the structure under 

seismic load. Observed in the seismic tests is a reduction of stiffness in the reinforced 

concrete even at low intensity seismic loads leading to shifting of eigen-frequencies and 

increased damping. In analytical models, capturing the progression of damping or 

damage at low intensity levels is difficult and as a result the estimated stiffness of the 

structure remains high thus leading to predictions with higher accelerations and lower 

displacements as compared to the actual response. The implementation of a 

progressive increase in structural damping may help in accounting for these 

nonlinearities but the quantification is not easy and the selection may not be 

appropriate.

• The ability of the numerical models to predict the shaking table tests correlated with the 

type of analysis and the complexity of the model used to represent the structure. Nonlinear analyses based on models with nonlinearities in the constitutive models proved to 

be best suited in capturing the structural response when the structure entered the 

nonlinear regime. Numerical models which adopted an “exact” representation of the 

actual structure (concrete, rebars, shaking table, etc.) were able to capture, in spite of 

uncertainties surrounding the actual test set-up, the general behavior of the structure 

through the sequence of tests. This implies than non-linear analyses have the best 

chance in capturing the complex behavior of structures which are allowed to experience 

non-linear behavior.

As presented in the previous chapter, the BNL numerical effort was based on a threedimensional representation of the structure that included all elements of the combined system. 

This included the shaking table, the actuators and support springs (with best-estimate spring 

constant values), the concrete and its constitutive relations, and the rebars in discrete form. 

Optimization of the steel reinforcement distributions was implemented to reduce the 

computational burden. A fully non-linear scheme was adopted and used for all phases of the 

prediction (blind and best-estimate) as well as for the preliminary push-over analyses. Based on 

the comparison between the predictions made with the BNL 3-D, non-linear numerical model 

and the actual shaking table test results the following assessment is made:

• Excellent agreement, in general, with the shaking table test was obtained using the 

elaborate non-linear model developed. In particular, the model was able to reproduce 

the accelerations, displacements, and rebar strains both in terms of peak values and 

shape of the transients.

• The adoption of (a) best estimates of fracture energy in the concrete (fracture energy 

properties of normal density concrete were used) and (b) the progressive increase in 

structural damping as the concrete structure was being damaged with the seismic 

excitation sequence, resulted in often slightly under predicting both the displacements 

and the accelerations. This is attributed to the role of cracks that develop as a result of 

the fracture energy assumed and the value of structural damping assumed for the 

damaged concrete. Another potential source is the fact that no effort was made to 

adjust or update the concrete moduli (tangent and secant) with the progression of 

damage. 

• The model predicted the locations where damage was observed in the test structure 

quite accurately. Deviations that exist between the predictions and the test stem from 

the fracture energy assumed and from the simplification/optimization of the steel rebar 

distribution and layout.
132

• The model was able to accurately predict the response of the test structure under real 

earthquake signals when long-duration free-vibration tails in the resulting transients 

were observed in the test. This is important because of the damageability potential of 

such earthquakes due to the multiple significant stress cycles they induce in the 

structure even when their intensity is below the design earthquake level. Having 

available a numerical model that can predict such response induced by real earthquake 

signals can provide insights to the initiation of structural damage.

• As with all other prediction models, the BNL model/process was also influenced by the 

role of the shaking table and its dynamic properties. While sensitivity studies were 

performed regarding the modal characteristics of the combined system, research on

quantifying the interaction between table and structure, which in turn resembles SSI, 

may provide important insights and quantification of these effects.

5.3. Sensitivity and Fragility Studies

The large variability in the predicted SMART structure response observed during the numerical 

prediction phases of the benchmark, reinforced the need for a sensitivity study to quantify the 

effect of uncertainties in the structural parameters. By following a widely used approach in 

structural design where a range of values of key parameters is established, typically arbitrarily, 

the effect of individual parameters such concrete modulus, damping, steel yield stress and 

added live loads was evaluated through a multitude of analyses. The objective was to observe 

the variability of the response of the structure in terms of floor response spectra and floor drifts, 

two response parameters of importance in the design of reinforced concrete structures. The 

variability was attributed to (a) the epistemic uncertainty of structural parameters and (b) the 

sensitivity of the response to the modeling adopted and the analysis approach. To minimize the 

pool of uncertainty sources, the shaking table was eliminated, resulting in a fixed base 

structure. The variability in the input was also minimized (only two specific, white noise based 

excitations were used). Results from the study indicate that the uncertainties in the structural 

damping have the greatest influence on the response and in particular floor response spectra.

In addition to the structural parameter sensitivity studies, a vulnerability assessment was 

performed and seismic fragility curves were generated. This was accomplished by introducing 

probabilistic measures in both the seismic input and in the structural parameters, and by 

evaluating exceedance probabilities of limit states through convolution of randomized excitation 

with randomized properties. To explore the influence that the choice of demand and capacity 

indicators may have on the exceedance of pre-defined limit states, fragilities that covered the 

different demand-capacity combinations were derived. Based on the assessment of the BNL 

vulnerability study where a random earthquake family of fifty (50) records was convoluted with 

fifty (50) random realizations of the structural parameters, the following observations and 

recommendations are deduced:

• Since fragility curves are used as a tool in decision making, adopting procedures that do 

not account for uncertainties in establishing the limit states of a given structure which 

consequently deduce estimates that are non-conservative may lead to wrong decisions. 

One possible approach is to treat performance limit states as random variables instead 

of deterministic quantities, ensuring that uncertainties are accounted for and that the 

level of conservatism can be decided upon based on the sensitivity analysis results.

• The definition of the limit threshold or limit state has a significant effect on fragility 

curves, so a parametric analysis may be needed to understand the sensitivity of 

different elements such as non-linear treatment, variations in the structural model, 

number of earthquakes, etc. 
133

• Multi-dimensional performance limit states need to be explored so they can eliminate the 

large deviations that exist between fragilities derived using different demand and 

capacity indicators. As observed in the analysis, large differences exist between fragility 

curves which were established using the same demand indicator (i.e. PGA) and different 

capacity threshold (i.e. floor drift vs. frequency drop). For example, for a multidimensional performance limit state based on displacements and accelerations of a 

building story, the fragility can be derived as a convolution of the two single-threshold 

fragilities.

• Structure capacity realizations, where the distribution of strength (or its uncertainty) is 

influenced by spatial correlation conditions (rather than being described by a pre-defined 

distribution of uncertainty, i.e. log-normal), should be studied in more detail. Specifically, 

when, for the purpose of simplicity which allows fragilities to be derived in the form of 

analytical expressions, one accepts a given distribution of the uncertainty as a priori, 

influential parameters defining capacity or lack-of could be inadvertently omitted. 

Accepting, for example that the uncertainty in structural capacity is expressed in terms 

of a lognormal distribution constitutes such a simplified approach. In 3-D structures, 

however, the spatial variability or uncertainty of key structural parameters may be more 

important than the distribution governing the fluctuation of the property around the 

nominal value and so it must be captured in the fragility evaluation. Spatial variability in 

soil properties and their effect on the dynamic response of the structure they support 

has been explored in Simos et al. (2004) along with procedures through which spatial 

correlation based on limited field data can be introduced in the randomized model. 

• The number of earthquakes that form the randomized suite used to integrate with the 

random realizations of the capacity may need to be optimized both in the number of 

earthquakes and the type. In particular, the selection of the earthquake suite to reflect 

ground motions characteristic of the local site where fragilities for a structure are being 

generated should be favored over a pool of earthquakes that tend to saturate the range 

of frequencies and peak ground accelerations. Synthetic earthquakes based on power 

spectra reflecting the energy content in a given area may form a most appropriate suite.

• Efforts should be dedicated towards the development of fragilities based on static nonlinear (push-over) analyses. By relying on the demonstrated capabilities of such nonlinear methods to predict large deformations and failure in reinforced concrete 

structures, higher-confidence fragilities (in terms of the actual capacity and the 

quantification of the performance limit states) can be deduced and utilized in decisionmaking for the performance of existing structures

5.4. Impact on Design Approaches and Implications

The multi-directional shaking table tests of the 3-D SMART2008 reinforced concrete structure 

succeeded in inducing both torsion and non-linear behavior as they were intended when the 

benchmark was conceptualized. Based on the recorded test data and their assessment, as well 

as the results of the numerical analyses that accompanied the experiment, the following 

observations with potential impact on design approaches are deduced:

• The inherent capacity of the reinforced concrete structure is much higher than the “perdesign” capacity. Specifically, the SMART2008 structure was designed and constructed 

according to the French seismic design code for 0.2 PGA design level earthquake. Peak 

ground accelerations in one of the tests exceeded 1.0 g. The structure, which had been 

progressively damaged by a series of seismic loads exceeding the design level, suffered 

only minor additional damage at approximately five (5) times the input intensity. The 

available excess capacity is attributed to the fact that the design is based on linearelastic behavior of the structural materials (concrete and reinforcing steel) and does not 
134

account for stiffness reduction which in turn reduces the acceleration demand on the 

structure.

• Reinforced concrete structures are observed to undergo stiffness reduction at lower 

seismic intensities than the design level. Such observations were made in a shear wall 

test on the same shaking table (CAMUS Shear Wall Experiment, D. Combescure, et al.) 

and were reconfirmed during the SMART2008 experiment. The reduction in stiffness is 

attributed to micro-cracking that takes place within the concrete. As the test results 

indicated, changes in the dynamic properties of the reinforced concrete structure were 

noticeable even after subjecting the structure to seismic loads of 0.05g PGA. This 

observation has significant implications in the seismic design of reinforced concrete 

nuclear structures in that (a) the response of the structure could be significantly altered 

as a result of the dynamic property changes (shifting of eigenfrequencies towards lower 

values) and their correspondence to the frequency band of the input acceleration energy 

and (b) the floor response spectra estimated for the structure based on the design 

stiffness can experience large variations both in terms of spectral accelerations and 

frequency content. 

• To fully understand and quantify the stiffness reduction that takes place at low seismic 

intensities, dedicated research in this area remains. Such effort could be based on 

experimental studies which will explore the accumulated damage and its effect on the 

concrete stiffness and its implementation into the constitutive concrete models. 

Subsequently, and via the enhancement of numerical models that already account for 

damage in materials such as concrete, guidelines on concrete stiffness reduction can be 

deduced. Further, the existing guidelines on design floor response spectra 

(i.e., generation of envelope floor spectra) could be revisited to assess if the drop in 

spectral frequencies due to stiffness reduction is adequately accounted for. 

• Observed in the shaking table tests below or at design level are low energy dissipation 

characteristics of the overall system (RC structure, shaking table and mounting 

conditions). It is assumed, based on these low intensity tests, that the damping exhibited 

by the RC structure alone is still quite low. This finding is in fact in line with the generally 

accepted notion that at seismic intensity levels well below the design level (i.e., well 

below the OBE level) the damping in the concrete is typically between 1%-2%. 

Specifically, damping values as low as 1% of critical have been observed as compared 

to the 4%-7% recommended by regulatory bodies on reinforced concrete structures for 

OBE and SSE. Because the test structure and its support conditions are not typical of a 

nuclear RC structure, extrapolation of the observations on damping values from the 

SMART2008 to nuclear RC design and practice should be avoided.

• Based on the comparison of the test data with the predicted data using a fully nonlinear, 3-D analysis of the SMART structure that was implemented by BNL, it can be 

concluded that such analyses, in spite of the high computational burden, can quite 

accurately reproduce the response and the damage experienced by the structure even 

when multi-directional loads are applied on an asymmetric structure. 

• The SMART2008 benchmark organizers, who evaluated the different analyses 

approaches used by the participants to derive best-estimates of the response, 

concluded that non-linear approaches with exact representation of the geometry, 

including the shaking table which introduces SSI-like effects, were able to derive the 

most accurate estimates of the actual test accelerations and deformations. In spite of 

the computational cost, which rapidly diminishes due to parallel processing in the 

computational space, the use of non-linear treatment of a real nuclear concrete 

structure should be considered by the regulatory authorities and be allowed to influence 

the design. Confirmation of the SMART seismic tests and of other similar tests such as 
135

the CAMUS shear wall tests demonstrate that non-linear methods have matured since 

the early days of nuclear structural design. A number of other benchmarks on reinforced 

concrete under high strain loading (i.e. high velocity impact) have also shown that the 

ability of the current non-linear numerical techniques to predict the response and 

damage has improved in recent years. 

• Using the non-linear methodology that was developed and used by BNL to predict the 

seismic response of the asymmetric 3-D structure under multi-directional loading, 

conventional methods widely used in the industry can be qualified and augmented to 

include effects stemming from nonlinearities. Figure 5.1 shows a verification of the 

SRSS method using the SMART structure and a bi-directional seismic input of 0.2g 

PGA (design level). As seen in the generated floor response spectra the SRSS method, 

which was primarily applied to structures idealized as stick models, is reproducing the 

floor spectrum generated by the simultaneous action of the two-directional input. 

Further, generation of design floor spectra based on existing guidelines can be 

developed and the adequacy of the enveloping provisions can be evaluated for the 

conditions where the structure exhibits non-linear response. Figure 5.2 shows the 

deduction of “design” floor response spectra for the 3rd floor of the SMART2008 

structure using US NRC guidelines on enveloping procedures over computed spectra at 

various floor locations. Figure 5.2 also includes spectra comparison at the structural 

extremity D between the SRSS and the direct methods. 

Figure 5-1 Comparison of acceleration spectra between the direct 

method (i.e., both orthogonal excitations acting on the 

structure simultaneously) and the SRSS method where the

two responses were computed independently
136

Figure 5-2 Generation of design response spectra by enveloping 

proceduresusing US NRC guidelines

Summarized below is the overall assessment based on the comprehensive SMART benchmark 

(shaking table tests and independent analyses that were performed) drawn by the organizing 

committee and concurred by the participating teams during the closing workshop:

• There exists great variability and scattering in the results produced by the different 

approaches attempting to analyze the same structure under generally the same 

conditions (i.e. boundary conditions in the case of SMART varied depending on the 

participation of the shaking table). This prompts the question whether some of the 

conventional practice in seismically analyzing structures should be re-considered. 

• While good agreement may be observed using similar hypotheses and similar types of 

dynamic analysis, variations in the development of floor response spectra were still 

observed. These variations may have implications in design. Therefore, guidelines and 

a simplified methodology should be derived for developing floor response spectra along 

with provisions for minimum spectral values regardless of the complexity of the 

approach employed.

• More precise guidelines should be developed regarding the numerical modeling and the 

level of simplification allowed for structures using conventional analyses (i.e. equivalent 

static, response spectrum, non-linear static, linear dynamic and even non-linear 

dynamic).

• Better guidelines should be developed regarding the damping values according to the 

type of analysis that is adopted. Further research can enable a better understanding of

the damping parameter and, in particular, its evolution which greatly influences the 

structural response.
137

• Guidelines, under say the auspices of ASCE or RFS (French Fundamental Safety Rules 

RFS2001-01), should be developed in order to better account for the concrete Young’s 

modulus and how it is modified depending on the type of analysis to be performed. As 

with structural damping, further work could help to identify the degradation and thus the 

need for updating of the modulus as a function of the loading history and the structural 

damage.

• The assessment results should be considered at the industry updated related 

consensus codes and the NRC considers them for regulatory endorsement.

139

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NUREG/CR-7119 Experimental Studies of Reinforced Concrete Structures Under 

Multi-Directional Earthquakes and Design Implications