This study uses a statistical surrogate model to develop fragility curves for an infilled reinforced concrete frame building, considering uncertainties in both material properties and ground motion parameters. The focal point of this study is a school building in Nepal damaged during the 2015 Gorkha earthquake. The school was instrumented, and its seismic response was simulated using a nonlinear numerical model. The model, developed following a recently proposed framework and extensively validated with the field data, is used in a parametric study conducted to identify the most influential material parameters (MPs). The model is then used in incremental dynamic analyses conducted to provide data for the calibration of a surrogate model. The threestaged least square statistical modeling approach is adopted to relate the influential MPs and ground motion intensity measures with important response quantities related to the peak and residual firststory drift ratios. The surrogate model is employed to generate fragility curves accounting for the two sources of uncertainty. The results indicate that accounting for uncertainties associated with the MPs can alter the fragility curves, causing a shift in the prediction of the median and dispersion of intensity measures.
Reinforced concrete (RC) frames with masonry infills are commonly found in seismically active regions worldwide. The infills are generally considered nonstructural elements and are typically ignored in analysis and design despite their interaction with the bounding RC frame under seismic loads. In fact, their presence can drastically increase the lateral stiffness and strength of the structure, but it can lead to catastrophic brittle failure once the strength is reached. Past earthquakes have demonstrated the vulnerability of this type of structures to even moderate ground excitations, causing casualties and high economic losses^{[1,2]}. Therefore, predicting the seismic behavior and assessing the vulnerability of these structures are necessary, yet challenging, tasks for practicing engineers. The variability in material properties and the complexities involved with the nonlinear behaviors and interactions between brick, mortar, and the bounding RC frame add to the uncertainties involved with the infilled RC frames.
The tools often used to address this challenge include limit analysis methods (among others^{[3,4]}), strut models (among others^{[58]}), and detailed finite element (FE) models (among others^{[9,10]}). The limit analysis methods, though efficient, require the prediction of the failure pattern, which may not always be possible, given the number of possible failure patterns of actual structures. The strut models, which are commonly used in practice, may not capture the sheardominated failure patterns and, hence, may lead to inaccurate results. The FE method, on the other hand, can provide the most accurate results and capture the failure mechanisms. However, an FE model is computationally expensive and timeconsuming for large and complex buildings. Nowadays, an emerging tool to assess the seismic vulnerability of structures is the use of surrogate models (among others^{[1113]}). Such models, once calibrated, can account for the important structural features and excitation characteristics, and efficiently predict key features of the structural response.
Besides the numerical or statistical models, a significant effort has been made recently to develop fragility curves for classes of buildings and other structures. These curves can provide the probability of a structure reaching or exceeding a limit state or level of response when a ground motion parameter reaches a certain value. This is a quick and efficient tool for the prediction of the state of a structure as a function of the intensity of a potential earthquake. The accuracy of fragility curves depends on the information used to derive them, as they can be derived based on test, field, and/or numerical data (among others^{[1420]}). In most cases, these curves are developed based on numerical analyses of the behavior of a structure under different seismic excitations. Hence, they account for the uncertainty of the potential ground motion, but the possible inaccuracies and uncertainties related to the numerical models used are typically not accounted for.
To account for the modeling uncertainties, Narinder
The objective of this study is to develop fragility curves for a school building in Nepal consisting of infilled RC frames, accounting for the uncertainties associated with the material parameters (MPs) and the ground motion records. For that purpose, an extensively validated nonlinear numerical model of the structure is first used in a sensitivity study to identify the influential model parameters^{[21]}. In a second parametric study, incremental dynamic analyses are conducted to estimate the influence of the MPs and ground motion characteristics on the seismic response of the building. The generated data is used for the calibration of a surrogate model using the threestaged least square (3SLS) approach. The surrogate model is cross validated and then used in Monte Carlo (MC) simulations, which facilitate the development of fragility curves accounting for the uncertainties in the modeling parameters and the ground motion characteristics.
This study focuses on the fourstory school building shown in
A fourstory school building at Sankhu. (A) west elevation view from the entrance; (B) plan view (dimensions in m).
The numerical model of the school building is developed based on the methodology proposed by
The numerical model of the building, consisting of 428 elements, is developed in OpenSEES^{[26]} and utilizes displacementbased inelastic beamcolumn elements^{[27]} for the RC members and diagonal truss elements for the struts. The RC members are divided into fibers to simulate the potential development of plastic hinges. The steel reinforcement is represented by the uniaxial material model proposed by MenegottoPinto^{[28]} and extended by Filippou
The comparison with the observations and data from the field indicates that the model can accurately simulate the torsional response of the actual structure and the concentration of damage in the south end of the first story. Moreover, the modal frequencies and shapes for the first three modes of the numerical model, after it is subjected to the nearbyrecorded horizontal ground motions, match those extracted from the ambient vibration recordings obtained during the reconnaissance trip. More details on the numerical model and the calibrated properties can be found in the study by Bose
The validated FE model is used here to train a statistical surrogate model to efficiently represent the seismic response of the building. To this end, a sensitivity analysis is conducted with the FE model to investigate the effects of MP selection, as well as the ground motion parameters on the seismic response of the structure.
The material model adopted here for the RC members and diagonal masonry struts is defined by seven parameters for concrete and masonry: the peak compressive strengths (
Sensitivity analyses are conducted to quantify the effect of each of these 15 parameters on the structural response. In the sensitivity study, one parameter is varied at a time; once to a lower and once to a higher value. The variation range of each variable is selected to reflect the level of confidence in the selected value. The strength, deformation, and stiffness parameters are perturbed by 20%, 30%, and 40%, respectively, from the calibrated values of the numerical model, as those values are selected based on information available from material tests in Nepal^{[31,32]}. However, no information is available on the calibration of the parameter lambda (λ
Calibrated properties of the baseline numerical model^{[17]}
















Concrete  9.71 (1.40)  0.97 (0.14)  1.38 (0.20)  0.0030  0.0080  276 (40)  0.1 
Panel A  3.44 (0.50)  0.35 (0.05)  0.55 (0.08)  0.0020  0.0040  117 (17)  0.1 
Panel B  0.55 (0.08)  0.0017  0.0042  117 (17)  
Panel C  1.72 (0.25)  0.0012  0.0025  345 (50)  
Panel D  1.72 (0.25)  0.0012  0.0025  345 (50)  
Panel E  1.72 (0.25)  0.0012  0.0024  345 (50)  
Panel F  0.55 (0.08)  0.0016  0.0029  117 (17)  
Panel G  0.55 (0.08)  0.0016  0.0029  117 (17) 
*Ratio of the unloading slope to the initial slope at the onset of the residual strength.
The results of the sensitivity analysis are summarized in
Variations in the peak ISD model in two orthogonal directions as compared to the values obtained from the calibrated model during the sensitivity analysis.
To incorporate the ground motion uncertainties, the bidirectional ground motion timehistories from the mainshock and the three major aftershocks of the 2015 Gorkha earthquake^{[28,29]} recorded at five stations in the Kathmandu valley [KATNP (Kanti Path, Kathmandu, Nepal), PTN (Pulchowk Campus, Tribhuvan Univ., Patan), THM, KTP (Municipality Office, Kirtipur), and TVU (Dept. Geology, Tribhuvan Univ., Kirtipur)] are considered to obtain a set of 20 ground motions. Due to the lack of other recordings from Nepal, FEMA P695^{[35]} is used to obtain additional motions in this study. FEMA P695 provides 22 farfield pairs of ground motions representing magnitudes in the range of 6.57.6 recorded on firm soil. Five of these 22 pairs of recorded motions and five from the set of 20 ground motions recorded in Nepal, summarized in
Ground motions selected in the study












1  7.9  2015  Gorkha  Municipality Office, Kirtipur (KTP)  75.8    0.260  0.24 
2  Dept. Geology, Tribhuvan Univ., Kirtipur (TVU)  77.1    0.234  0.18  
3  Kanti Path, Kathmandu, Nepal (KATNP)  59.9  305  0.163  0.26  
4  Pulchowk Campus, Tribhuvan Univ., Patan (PTN)  79.3    0.154  0.30  
5  Univ Grants Comm., Sanothimi, Bhaktapur (THM)  83.7  205  0.154  0.18  
6  6.7  1994  Northridge  Beverly Hills  Mulhol  13.3  356  0.516  0.07 
7  7.1  1999  Duzce, Turkey  Bolu  41.3  326  0.822  0.08 
8  6.5  1979  Imperial Valley  Delta  33.7  275  0.351  0.17 
9  7.3  1992  Landers  Yermo Fire Station  86.0  354  0.245  0.20 
10  7.6  1999  ChiChi, Taiwan  CHY101  32.0  259  0.440  0.11 
The individual ground motions are first normalized by their peak ground velocities to remove unwarranted variability between records due to inherent differences in event magnitude, distance to the source, source type, and site conditions, while still maintaining the inherent recordtorecord variability. To achieve statistical stability, the records are scaled to minimize the difference with the response spectrum used in Nepal (NBC 1994). The selected scale factors also ensure that the median of the geometric mean response spectra of all ground motions matches the response spectrum used in Nepal.
Acceleration response spectra of the selected ground motions.
The selected records are used here to conduct incremental dynamic analysis (IDA)^{[36]} to correlate a performance indicator (PI), such as the peak interstory drift (ISD), with an intensity measure (IM) of the seismic excitation. The peak first story drift ratios in the two orthogonal dimensions are selected as PIs, as discussed in the sensitivity study. A number of IMs have been used in the literature^{[3739]} besides the commonly used peak ground acceleration (PGA). Each of these IMs considers different characteristics of the ground motion with varying levels of influence on the structural response. Anastasopoulos
• the PGA, peak ground velocity (PGV), and peak ground displacement (PGD);
• the Arias Intensity,
• the spectral IMs in terms of acceleration (
• the pseudospectral IMs in terms of acceleration (
• the Housner Intensity,
• the rootmeansquare measures in terms of acceleration (
• the characteristic Intensity,
• the sustained maximum measures in terms of acceleration (
• the spectrum IMs in terms of acceleration (
• the acceleration parameter, A95, defined as the level of acceleration which contains up to 95% of the Arias intensity^{[44]};
• the strong motion duration or significant duration,
• the Predominant Period,
• the mean Period,
• the effective peak acceleration (
• the spectral acceleration measure,
• the cumulative Absolute Velocity,
where
• the standardized CAV (
• the normalized energy density (
• the IM,
• the uniform duration,
• the number of peaks, defined as the number of times the acceleration time history crosses threshold acceleration values 0.025 g (
Nonlinear regression equations are developed herein to predict the seismic response of the fourstory school building with functions of the material properties and ground motion characteristics. To this end, the 3SLS statistical modeling approach used by Anastasopoulos
In nonlinear regression analysis with multiple output parameters (PIs in this case), it is necessary to allow for crossequation error correlation and endogeneity across the PIs that serve as dependent variables. The PIs would possibly be affected by unobserved characteristics that may be highly correlated with each other; therefore, crossequation error correlation is anticipated. In addition, the endogeneity is underlying as the PIs are expected to be directly or indirectly affected by at least one or more of the other PIs. The 3SLS approach adopted here simultaneously accounts for both crossequation error correlation and endogeneity across the PIs. More information on this econometric modeling approach can be found in the literature^{[40,52]}.
When the dependent variables (PIs) are endogenous, one PI is defined by a set of explanatory variables (IMs or MPs) that influence the variation of another PI and so on, and then the twostage least squares (2SLS) approach is used for parameter estimation of the equations simultaneously. Nevertheless, the dependent variables (PIs) are influenced by similar unobserved factors because of which the random error terms are correlated. In those cases, the parameters are obtained using the seemingly unrelated regression equation (SURE) approach to account for the crossequation error correlation. Thus, when the damage indices are endogenous with crosscorrelated error terms, as in this study, the 3SLS approach, which combines the 2SLS and SURE methods, can be used to estimate the parameters of the equations simultaneously^{[40]}.
A number of response quantities can be used to represent the response of the structure. Since the damage in these structures tends to concentrate in the first story (Stavridis and Shing; Bose
where IMs and MPs are the explanatory variables in the equations to define the selected PIs, and
The parameters of the simultaneous equations are estimated using the least square approach in Stage 1, where the instrumental variables are used as regressors to project the dependent variable. The correlation matrix of the error terms is calculated using the residuals of each equation in Stage 2. The projected values of the dependent endogenous variables, i.e.,
To train the surrogate model, the results of IDAs performed using the numerical model of the school building are used. In these analyses, the modeling parameters and ground motions are varied. The eleven sensitive MPs for concrete and masonry identified in
Model parameters considered to develop the numerical models to fit the surrogate model








+/ 10 and 20  1.12  1.26  1.40  1.54  1.68 

+/ 10 and 20  0.16  0.18  0.20  0.22  0.24 

+/ 15 and 30  0.0021  0.00255  0.003  0.00345  0.0039 

+/ 15 and 30  0.0056  0.0068  0.008  0.0092  0.00104 
λ 
x1/4, 1/2, 3.75, and 7.5  0.025  0.05  0.1  0.375  0.75 
+/ 10 and 20  0.40  0.45  0.50  0.55  0.60  

+/ 10 and 20  0.066  0.075  0.083  0.091  0.10 

+/ 15 and 30  0.0014  0.0017  0.0020  0.0023  0.0026 

+/ 15 and 30  0.0028  0.0034  0.0040  0.0046  0.0052 
x1/4, 1/2, 3.75, and 7.5  0.025  0.05  0.1  0.375  0.75  
ξ  +/ 33 and 67  0.01  0.02  0.03  0.04  0.05 
The results of the 4,400 dynamic analyses are used to train the surrogate model. The criterion for the inclusion of the input parameters in the models is their statistical significance and the statistically significant improvement of the overall model fit. In the calibration process, it was found that disregarding the variables with a level of confidence lower than 90% has no effect on the Rsquared (R^{2}) value, which is used as a statistical measure of goodness of fit. Hence, a threshold of 90% level of confidence is selected here. The R^{2} value is adjusted for the number of predictors in the model. The adjusted R^{2} values are greater than 0.90 for the four PIs, indicating good overall statistical fits, as the models account for at least 90% of the variance in the data.
Model estimation results for the peak ISDs of the building using the 3SLS approach









Constant  12.4621  6.17  ln(PGA_{2})  1.10084  8.96 
PGA_{1}  18.2248  11.02  (PGA_{2})^{2}  40.0635  10.18 
ln(PGA_{1})  2.0090  10.57  (PGV_{1})^{2}  0.00016  5.20 
PGA_{2}  31.4898  9.49  (PGV_{2})^{2}  0.00072  12.26 
ln(PGA_{2})  3.49719  6.46  PGD_{1}  0.00832  11.68 
(PGA_{2})^{2}  46.6106  1.29  (PGD_{1})^{2}  2.98E05  7.19 
PGV_{1}  0.07481  8.83  PGD_{2}  0.01914  11.62 
ln(PGV_{1})  3.63489  1.62  1/PGD_{2}  0.91117  9.71 
(PGV_{1})^{2}  0.00073  9.25  ln(PGD_{2})  0.49799  4.07 
ln(PGV_{2})  2.49558  7.83  I_{A1}  0.02132  5.50 
PGD_{1}  0.01837  14.77  (I_{A1})^{2}  1.74E05  5.85 
(PGD_{1})^{2}  2.85E05  8.40  ln(I_{A2})  0.43301  4.34 
PGD_{2}  0.00755  4.19  (S_{a1})^{2}  11.9568  6.11 
(PGD_{2})^{2}  1.03393  8.52  S_{a2}  9.73939  19.46 
ln(PGD_{2})  1.59E05  2.92  I_{C1}  0.00694  13.54 
I_{A1}  0.05540  3.30  (I_{C1})^{2}  4.37E07  8.81 
(I_{A1})^{2}  2.65080  11.14  I_{C2}  0.00534  7.65 
ln(I_{A1})  2.18E05  6.78  D_{sig2}  0.00032  6.50 
I_{A2}  0.07171  12.84  N_{p0252}  0.01181  11.45 
(I_{A2})^{2}  4.10723  11.27  (N_{p0252})^{2}  2.54E05  4.50 
ln(I_{A2})  2.75E04  8.90  ( 
0.48764  2.15 
(S_{a1})^{2}  11.1860  11.81  123.708  5.37  
I_{C1}  0.00250  6.94 

1.76156  11.88 
(I_{C1})^{2}  2.14E07  2.74  0.99478  9.09  
T_{mean1}  4.14491  4.92 

382.273  13.07 
SCAV_{1}  0.00136  6.36  10.2329  9.04  
N_{p0251}  0.00233  5.76  E1  1.10084  7.86 

0.14210  10.18  

63.8062  2.63  

1.74919  17.06  

2.20036  6.72  
305.414  10.02  
11.0669  9.73  
E7  0.26461  2.13  
Rsquare = 0.9535  Rsquare = 0.9549  
Adjusted Rsquare = 0.9531  Adjusted Rsquare = 0.9546 
Indices 1 and 2 for the intensity measures represent the two components of the ground motions.
Model estimation results for the residual ISDs of the building using the 3SLS approach









Constant  1.24975  5.93  Constant  9.73778  1.28 
ln(PGA_{1})  0.38759  8.01  PGA_{2}  7.29838  6.02 
(PGA_{1})^{2}  9.62633  15.45  ln(PGA_{2})  2.22195  13.17 
PGV_{1}  0.0181  9.39  (PGA_{2})^{2}  22.6409  12.47 
ln(PGV_{1})  0.51379  7.74  PGV_{1}  0.0601  6.31 
ln(PGV_{2})  0.75928  12.71  ln(PGV_{1})  1.95624  11.23 
(PGD_{1})^{2}  8.00E05  7.77  PGV_{2}^{2}  0.00031  4.50 
I_{A1}  0.00402  4.54  ln(PGV_{2})  0.14066  2.27 
(I_{A1})^{2}  0.87201  17.38  PGD_{1}  0.00477  4.00 
ln(I_{A1})  7.43E06  6.08  ln(PGD_{2})  0.90549  16.86 
I_{A2}  0.01232  12.25  (PGD_{2})^{2}  1.94E05  6.62 
(I_{A2})^{2}  1.34569  19.98  I_{A1}  0.01196  9.79 
ln(I_{A2})  6.86E06  4.79  (I_{A1})^{2}  2.82E05  12.08 
(S_{a1})^{2}  5.27862  17.21  I_{A2}  0.02003  15.50 
0.40747  19.61  (I_{A2})^{2}  3.41E05  9.52  

1.35843  4.87  (S_{a2})^{2}  6.39184  10.86 

128.111  5.05 

0.25552  10.56 

1.40116  15.64 

1.4873  4.67 

564.748  12.44 

487.725  9.35 
ln(PISD_{x})  1.17262  11.16  0.1844  3.21  
E1  0.32877  8.82  3.09035  2.20  
E2  0.46238  10.25  ln(PISD_{y})  1.16615  18.63 
E3  1.12909  15.80  E2  1.24972  16.41 
E5  0.35982  8.70  
E8  0.84758  9.74  
Rsquare = 0.9114  Rsquare = 0.9055  
Adjusted Rsquare = 0.9109  Adjusted Rsquare = 0.9048 
To assess its accuracy, the statistical model is used to estimate the PIs for the models and ground motions used for its training.
Observed and predicted values in terms of the peak ISD for the casestudy building. (A) Peak ISD along the X direction; (B) Peak ISD along the Y direction.
Observed and predicted values in terms of the residual ISD for the casestudy building. (A) Residual ISD along the X direction; (B) Residual ISD along the Y direction.
To further assess the confidence in the surrogate model, it is used to predict the dynamic responses of numerical models with all 11 MP values different than those presented in
“Unseen” ground motions selected to evaluate the surrogate model








7.3  2015  Gorkha  Municipality Office, Kirtipur (KTP)  0.069 
6.7  Univ Grants Comm., Sanothimi, Bhaktapur (THM)  0.097  
6.7  1994  Northridge  Canyon County  WLC  0.482 
6.9  1995  Kobe  ShinOsaka  0.243 
6.9  1989  Loma Prieta  Gilroy Array #3  0.555 
The statistical models for the school building are used here to perform additional simulations needed to develop fragility curves incorporating modeling uncertainties. The MPs of the models for these simulations are selected with the MC method.
The statistical models summarized in
Fitting cumulative and probability distributions to available test data. (A)
It can be observed in
Estimated lognormal distribution statistical parameters of the experimentally obtained material properties





ξ 

Yeh, 1998^{[54]}  Median (e 
4.7          
Dispersion (σ)  0.55          
Yousefianmoghadam 
e 
4.2        
σ  0.33        
Gao, 2021^{[58]}  e 
4.1  0.003  3.9    0.0062  
σ  0.18  0.3  0.17    0.18  
Bose, 2013^{[55]}  e 
   




σ     




Cristofaro 
Set 1  e 
        
σ          
Set 2  e 
1.9          
σ  0.57          
Set 3  e 
2.4          
σ  0.46          
Schueremans and Dionys, 2006^{[71]}  Set 1  e 
    0.6    0.003  
σ      0.23    0.55  
Set 2  e 
    0.7    0.0035  
σ      0.18    0.4  
Set 3  e 
    0.9      
σ      0.11      
Shimizu 
Set 1  e 
2.7          
σ  0.3          
Set 2  e 
3.4          
σ  0.32          
Set 3  e 
3.3          
σ  0.23          
Set 4  e 
1.6          
σ  0.38         
MC simulations are conducted here using the statistical models to investigate the effects of the uncertainties associated with the random variables (IMs and MPs) on the fragility curves. To generate the models for these analyses, the MPs are sampled
All the material properties considered here for developing the set of structural models are assumed to be uncorrelated. For each set of realizations of MPs, the statistical surrogate model is used to calculate the peak ISDs for the ten selected ground motions. The IMs are increased incrementally to perform the additional IDAs incorporating both modeling and recordtorecord uncertainties. The geometric mean of the spectral acceleration at the first modal period of the building, S_{a}[T_{1}] of the two components of each ground motion is selected as the IM, while the maximum drift ratio is selected as the PI to develop the IDA curves. The periods of the structure can differ from one model to another as some MPs can affect the stiffness; however, for consistency, the period of the calibrated model is used in all cases.
To examine the effects of modeling uncertainties at different levels of the structural behavior, distinct limit states in the form of damage grades (DGs) are considered for the case study building. For the damage classification, a critical aspect of postearthquake damage assessment^{[6365]}, the damage classification used by Brzev
Medians and standard deviations of the S_{a}[T_{1}] at various damage grades obtained from MCS  Approach 1. (A) Medians; (B) Standard deviations.
Medians and standard deviations of the S_{a}[T_{1}] at various damage grades obtained from MCS  Approach 2. (A) Medians; (B) Standard deviations.
To capture the uncertainties associated with recordtorecord randomness but not the MPs, IDA is also performed for the baseline model and the ten ground motions considered here. The IDA curves of the deterministic model and the extended IDA plot for
IDA curves for the calibrated model and from Monte Carlo Simulation using the surrogate model. (A) Baseline model; (B) Monte Carlo simulations Approach 2,
Results of the Monte Carlo simulation compared to the base model


















DG1  0.227  0.215  0.220  0.219  0.131  0.147  0.164  0.165 
DG2  0.312  0.304  0.295  0.296  0.202  0.183  0.266  0.268 
DG3  0.391  0.384  0.361  0.364  0.297  0.342  0.329  0.328 
DG4  0.419  0.408  0.411  0.409  0.211  0.243  0.338  0.339 
DG5  0.477  0.462  0.445  0.443  0.228  0.249  0.346  0.345 
The fragility curves represent the expected probability of exceeding a DG as a function of an IM. In this study, the fragility curves are defined in terms of the spectral acceleration at the period of the first mode of the structure. Similar to previous studies^{[68,69]}, the fragility function parameters are obtained from the IDA plots for computing the median and standard deviation of the selected IM at each damage state, assuming a lognormal distribution.
To assess the efficiency of this fitting approach for the fragility curves, IDA plots are generated for the baseline model using the entire set of 42 ground motions. The empirical cumulative distribution function for exceeding 1.0% drift obtained from the IDA data is compared to the fragility function fitted to the data generated by assuming lognormal distribution in
Incremental dynamic analysis results used to fit fragility curves assuming lognormal distribution. (A) IDA results of FE model; (B) Fitted fragility function.
Following this approach, the fragility curves are obtained for the five DGs from the MC simulations [
Computed collapse fragilities of the fourstory case study building. (A) Fragility curves for all damage grades; (B) Collapse fragility for DG5.
Possible correlations between the MPs are very difficult to quantify, particularly in the absence of field or test data for the material properties. However, it is important to identify the impact of the correlation assumptions on the effects of uncertainties associated with material properties. Three sets of MPs are considered here to examine the implications of correlation assumptions. In Set1 the concrete properties are assumed to be correlated, but there are no correlations between the masonry properties. In Set2 the masonry parameters are assumed to be correlated, while the concrete properties are not. Finally, in Set3 all the random variables, including concrete and masonry parameters, are assumed to be correlated. The damping coefficient in all cases is assumed to have no correlation with the MPs. The correlations considered in Sets 1 and 2 are highly likely as the properties of concrete and masonry can be correlated among themselves, but the Set3 correlation is considered here more for illustration purposes, and it does not correspond to a realistic or practical application. For each set of assumptions, the correlation coefficients are considered to be 0.2, 0.5, and 1.0 between the random variables. These correlation assumptions only affect the MC simulations for the generation of the input MPs for the statistical surrogate model. The statistical surrogate model is used here since varying the correlation assumptions using the FE model would require significant computational effort.
The results for DG5 using
Parametric study on the effects of the correlation assumptions on the fragility curves












0  0.443  0.346  0.443  0.346  0.443  0.346 
0.2  0.443  0.339  0.444  0.339  0.443  0.340 
0.5  0.444  0.338  0.444  0.336  0.445  0.331 
1.0  0.445  0.338  0.446  0.331  0.449  0.327 
In this study, a reliable datadriven surrogate model for a school building is obtained and used to generate fragility curves that consider uncertainties associated with the MPs and the ground motion characteristics. The surrogate model is developed using the 3SLS statistical approach to capture the relations among the modeling parameters and IMs with selected structural response quantities. The 3SLS approach is employed as it accounts for two important misspecification issues when concurrently statistically modeling PIs as functions of IMs and MPs: namely, endogeneity and crossequation error correlation. The obtained model is verified with results from nonlinear analyses not considered in the training process. The comparisons between the results obtained from the detailed nonlinear model and the surrogate statistical model indicate that the statistical model can accurately predict the first story drift ratios. Hence, it is further used in MC simulations.
The extended incremental dynamic analyses performed using the surrogate models lead to the development of fragility curves, which incorporate both the uncertainties associated with structural modeling parameters and the ground motions. Incorporating modeling uncertainties increases the dispersion and decreases the median in the response fragility. This indicates that neglecting the modeling uncertainties is unconservative and may lead to unsafe conclusions. Hence, it is important to incorporate the modeling uncertainties in performancebased earthquake engineering for the probabilistic seismic performance assessment of structures. Additional analyses indicate that the impact of correlations between the MPs on uncertainty quantification is insignificant, and it is conservative to neglect those. These valuable findings are based on the study of one school building. Additional studies on other buildings and structural systems can be conducted to further validate these findings. With the framework introduced here, alternative ML tools, possibly fused with physicsbased constraints, can be considered for the prediction of the structural response if they are proven to be more accurate than the statistical surrogate model employed here.
Conceptualization, implementation, numerical analysis, model assessment, visualization: Bose S
Conceptualization, methodology, supervision, model assessment, funding acquisition: Stavridis A
Conceptualization, model assessment: Anastasopoulos PC
Conceptualization, supervision: Sett K
The data and materials pertinent to this paper can be provided upon request.
Partial support of this study by the National Science Foundation Grants 1254338 and 1545595 and USGS Grant G17AC00249 is gratefully acknowledged. The first author of the paper is grateful for the financial support provided by the University at Buffalo during his doctoral studies. However, the opinions expressed in this paper are those of the authors and do not necessarily represent those of the sponsor or the collaborators.
All authors declared that there are no conflicts of interest.
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© The Author(s) 2023.