Fragility curves accounting for uncertainties in material parameters and ground motion characteristics using a data driven surrogate model
Civil, Structural and Environmental Engineering, University at Buffalo, Buffalo, NY 14260, USA.
Correspondence to: Dr. Supratik Bose, Civil, Structural and Environmental Engineering, University at Buffalo, Buffalo, NY 14260, USA. E-mail:
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 three-staged 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 first-story 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 non-structural 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[5-8]), 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 shear-dominated 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 time-consuming for large and complex buildings. Nowadays, an emerging tool to assess the seismic vulnerability of structures is the use of surrogate models (among others[11-13]). 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[14-20]). 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 et al. propose a computationally efficient machine learning (ML)-based surrogate model framework for seismic fragility prediction of RC bridge piers.
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. 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 three-staged 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.
CASE STUDY BUILDING
This study focuses on the four-story school building shown in Figure 1, which is located in Sankhu, Nepal. The building consists of a masonry-infilled RC frame with seven bays in one direction and two bays in the perpendicular direction. It suffered extensive damage in the infills and RC members during the 2015 Gorkha earthquake. The damage in the first story included the shear failure of a beam-column joint in SE corner of the building, dominant cracks and spalling of the RC columns, and horizontal and diagonal cracks in the infills, while no damage was observed in the upper stories. The damaged columns revealed inadequate reinforcement detailing, with stirrups spaced at distances larger than the column dimensions. The concentration of damage in the south end of the ground story indicates that the frame exhibited a torsional response to the ground excitation. More information on the design details and the observed damage can be found in the study by Bose et al..
DEVELOPMENT OF FINITE ELEMENT MODEL
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 and utilizes displacement-based inelastic beam-column elements 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 Menegotto-Pinto and extended by Filippou et al. to include isotropic strain hardening effects. The Kent-Scott-Park material model with linear tension softening is used for concrete and masonry, as it can represent the nonlinear behavior of quasi-brittle materials.
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 nearby-recorded 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 et al..
DEVELOPMENT OF DATA-DRIVEN SURROGATE MODEL
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 (fc′, fm′); the strains at the peak strength (ε1c, ε1m); the residual strengths (frc, frm); the strains at the onset of the residual strength (ε2c, ε2m); the parameters lambda (λc, λm) defining the ratio of the unloading stiffness at the onset of the residual strength to the initial stiffness; the tensile strengths (ftc, ftm); and the tension softening stiffnesses (Etsc, Etsm). The damping ratio, ξ, of the structure is numerically modeled using the mathematically convenient Rayleigh formulation, considering 3% of the critical damping for the first two modes. To incorporate the effect of the uncertainty associated with damping, it is also considered as one of the variables.
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 (λc, λm) and the damping ratio; therefore, these are varied by 80% of the calibrated values to investigate their effects on the structural response. The calibrated properties, along with their percentage variations in the sensitivity analysis, are presented in Table 1. A total of 30 nonlinear models are created and, in each case, a nonlinear model is subjected to the two components of the ground motion recorded at the Univ Grants Comm., Sanothimi, Bhaktapur (THM) station, the closest station to the school building[33,34]. The modeling uncertainties associated with the parameters used to represent reinforcing steel are neglected in this study, as they are not expected to vary considerably.
Calibrated properties of the baseline numerical model
|Panels||Peak compressive strength||Residual strength||Tensile strength||Strain at peak strength||Strain at residual strength||Tension softening stiffness||Lambda*|
|MPa (ksi)||MPa (ksi)||MPa (ksi)||-||MPa (ksi)|
|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)|
The results of the sensitivity analysis are summarized in Figure 2. The figure demonstrates the variability in the peak first-story drift values in the two orthogonal directions for the response history analyses compared to the values obtained from the baseline model. The first story drift is selected for this comparison, as it is the most critical engineering demand parameter for these structures. The results indicate that the response is sensitive to eleven parameters, which are shaded in Figure 2. These parameters are considered as input variables in the surrogate model developed in this study.
Ground motion selection and parameters
To incorporate the ground motion uncertainties, the bi-directional ground motion time-histories 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 is used to obtain additional motions in this study. FEMA P695 provides 22 far-field pairs of ground motions representing magnitudes in the range of 6.5-7.6 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 Table 2, are employed here to run dynamic analyses to obtain data for the calibration of the surrogate model. These ground motions are selected to represent different soil types, frequency contents, amplitudes, etc.
Ground motions selected in the study
|EQ index||Earthquake||Recording station||
|Vs_30 (m/s)||PGA (g)||Scale factor|
|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|
|9||7.3||1992||Landers||Yermo Fire Station||86.0||354||0.245||0.20|
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 record-to-record 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. Figure 3 shows the comparison between the NBC 1994 design spectrum and the median spectrum of the ten selected ground motions, scaled to the design hazard level considered in Nepal.
The selected records are used here to conduct incremental dynamic analysis (IDA) to correlate a performance indicator (PI), such as the peak inter-story 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[37-39] 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 et al. in their study on the response of bridge piers concluded that it is not possible to estimate the structural response satisfactorily using a single IM. Hence, a series of statistically significant IMs are considered in this study, including the following. In all cases, the IMs are considered for the ground motions in both the orthogonal directions.
• the PGA, peak ground velocity (PGV), and peak ground displacement (PGD);
• the Arias Intensity, IA, proportional to the integral of the squared acceleration A(t) time history defined according to Equation 1;
• the spectral IMs in terms of acceleration (Sa), velocity (Sv), and displacement (Sd);
• the pseudo-spectral IMs in terms of acceleration (PSa), velocity (PSv), and displacement (PSd);
• the Housner Intensity, IH, integral of the pseudo-velocity response spectrum over the period range of 0.1 to 2.5 s according to Equation 2;
• the root-mean-square measures in terms of acceleration (ARMS), velocity (VRMS), and displacement (DRMS) as the square roots of the mean values over the duration of the ground motion record;
• the characteristic Intensity, IC, defined according to Equation 3;
• the sustained maximum measures in terms of acceleration (SMA) and velocity (SMV) defined as the third highest absolute peaks in the time histories;
• the spectrum IMs in terms of acceleration (ASI) and velocity (VSI), defined according to Equation 4 as the integral of the spectral acceleration (Sa) and velocity (Sv), respectively, over the period range of 0.1 to 0.5 s;
• the acceleration parameter, A95, defined as the level of acceleration which contains up to 95% of the Arias intensity;
• the strong motion duration or significant duration, Dsig, defined as the interval of time between the accumulations of 5% and 95% of Arias intensity;
• the Predominant Period, TP, estimated using the 5% damped acceleration response spectrum at its maximum, as long as TP > 0.2 s;
• the mean Period, Tmean, obtained from the Fourier amplitude spectrum, Ci for each frequency fi within the range of 0.25 to 20 Hz, according to Equation 5;
• the effective peak acceleration (EPA), defined as the mean of the spectral acceleration over the period range of 0.1 to 2.5 s according to Equation 6;
• the spectral acceleration measure, S*, defined in terms of pseudo-spectral accelerations PSa(T1) and PSa(T2), at the first and second mode periods, respectively, according to Equation 7;
• the cumulative Absolute Velocity, CAV, defined in terms of the minimum and maximum accelerations according to Equation 8.
where N is the number of 1-second windows in the time history, PGAi and ti are the maximum recorded acceleration (g) and start time of the window i, respectively, Amin is an acceleration threshold (user-defined, taken as 0.025 g typically) to exclude low-amplitude motions, and H(x) is the Heaviside step function;
• the standardized CAV (SCAV), defined in terms of the CAV according to Equation 9;
• the normalized energy density (ED), defined in terms of the velocity time history according to Equation 10;
• the IM, Izdefined in terms of the acceleration history and PGA, PGV (Cosenza and Manfredi, 1998) according to Equation 11;
• the uniform duration, DU, the summation of the time windows where the ground acceleration is above a certain threshold, which is considered 0.05 g in this study;
• the number of peaks, defined as the number of times the acceleration time history crosses threshold acceleration values -0.025 g (Np025), 0.05 g (Np050), and 0.1 g (Np100).
Nonlinear regression equations are developed herein to predict the seismic response of the four-story school building with functions of the material properties and ground motion characteristics. To this end, the 3SLS statistical modeling approach used by Anastasopoulos et al. to examine the effectiveness of various IMs in predicting the structural response of a bridge pier is adopted. This method correlates the selected damage indices with a number of statistically important IMs and material properties. It also takes into account the errors associated with bias due to omissions of variables and the regression properties, such as heteroscedasticity, autocorrelation, and exogeneity of the regressors, among others.
Overview of the statistical modeling approach
In nonlinear regression analysis with multiple output parameters (PIs in this case), it is necessary to allow for cross-equation 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, cross-equation 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 cross-equation 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 two-stage 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 cross-equation error correlation. Thus, when the damage indices are endogenous with cross-correlated 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.
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 et al.,)[10,21,22], the first story drift ratio is considered in this study, as discussed earlier. More specifically, the peak and the residual ISDs along the X and Y directions, i.e., PISDX, PISDY, RISDX, and RISDY, where P and R stand for the peak and residual values, respectively, are selected here. Note that the absolute values of the PIs are considered here as the direction of the story displacement is not important. These are treated as dependent variables and are modeled using the 3SLS approach. Hence, the system uses one equation to estimate each of these variables. The PIs are related to one another, especially the peak and residual drifts along each direction, and therefore, the parameters of the regression equations are estimated simultaneously so that the relations between the dependent variables can be captured accurately. Mathematically, the PIs are represented by the system of equations as follows:
where IMs and MPs are the explanatory variables in the equations to define the selected PIs, and ε1 to ε4 are the correlated error terms.
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., PISDx and PISDy, from Stage 1 also replace the endogenous variables on the right-hand side of the equations for RISDx and RISDy. Consequently, in Stage 3, the correlations of the error terms are used to improve the parameter estimates from Stage 2. Since the PIs can take only positive values, exponential relations are preferred between the dependent variables and the regressors (IMs and MPs) to avoid invalid negative values of the PIs. In addition, this exponential transformation further improves the statistical fit with increased forecasting accuracy. The relations between the PIs and the instrumental variables are nonlinear; therefore, several transformations (power forms, logarithmic relationships, etc.) are considered to obtain better statistical fits. Dummy variables are also created for each earthquake (E1 to E10) to account for the fixed effects of the panel data, which are generated as a result of the earthquake scaling.
Model estimation using 3SLS
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 Figure 2, i.e., the peak compressive strengths (fc′, fm′), the tensile strengths (ftc, ftm), the strains at the peak strength (ε1c, ε1m), the strains at the onset of the residual strength (ε2c, ε2m), the lambdas (λc, λm,) controlling the unloading for concrete and masonry, and the damping coefficient (ξ), are considered here. All eleven model parameters are varied, considering four values for each of them to develop 44 numerical models in total. The model parameters considered here are presented in Table 3. These parameters are varied one at a time in the parametric study. Hence, the possible correlations between them are neglected to limit the number of dynamic analyses and reduce the computation time. Moreover, there are no established relations to reliably describe the intra-dependencies of these parameters. The ten selected ground motions are scaled with a scale factor ranging from 0.4 to 4.0 at an increment of 0.4, resulting in ten levels of ground motions. The consideration of 44 models, ten ground motions, and ten excitation levels results in 4,400 dynamic analyses.
Model parameters considered to develop the numerical models to fit the surrogate model
|Model parameters||Variation (%)||Value-1||Value-2||Calibrated value||Value-3||Value-4|
|fc′ (ksi)||+/- 10 and 20||1.12||1.26||1.40||1.54||1.68|
|ftc (ksi)||+/- 10 and 20||0.16||0.18||0.20||0.22||0.24|
|ε1c||+/- 15 and 30||0.0021||0.00255||0.003||0.00345||0.0039|
|ε2c||+/- 15 and 30||0.0056||0.0068||0.008||0.0092||0.00104|
|λc||x1/4, 1/2, 3.75, and 7.5||0.025||0.05||0.1||0.375||0.75|
|fm′ (ksi)||+/- 10 and 20||0.40||0.45||0.50||0.55||0.60|
|ftm (ksi)||+/- 10 and 20||0.066||0.075||0.083||0.091||0.10|
|ε1m||+/- 15 and 30||0.0014||0.0017||0.0020||0.0023||0.0026|
|ε2m||+/- 15 and 30||0.0028||0.0034||0.0040||0.0046||0.0052|
|λm||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 R-squared (R2) value, which is used as a statistical measure of goodness of fit. Hence, a threshold of 90% level of confidence is selected here. The R2 value is adjusted for the number of predictors in the model. The adjusted R2 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. Tables 4 and 5 present the estimation results of the 3SLS surrogate models for the peak and residual ISDs along the orthogonal directions of the school building. In the tables, it can be observed that the signs of the coefficients might vary between equations in the 3SLS model. This indicates that the effect of an independent variable on the dependent variable could be affected by the inclusion of other independent variables, as the coefficients are calculated simultaneously through the system of equations. In addition, it can be observed that the interstory drift ratios along one direction can be dependent on the IMs along the other direction, as they are statistically significant with a high level of confidence.
Model estimation results for the peak ISDs of the building using the 3SLS approach
|Dependent variable = ln(PISDx)||Dependent variable = ln(PISDy)|
|R-square = 0.9535||R-square = 0.9549|
|Adjusted R-square = 0.9531||Adjusted R-square = 0.9546|
Model estimation results for the residual ISDs of the building using the 3SLS approach
|Dependent variable = ln(RISDx)||Dependent variable = ln(RISDy)|
|R-square = 0.9114||R-square = 0.9055|
|Adjusted R-square = 0.9109||Adjusted R-square = 0.9048|
Assessment of the surrogate model
To assess its accuracy, the statistical model is used to estimate the PIs for the models and ground motions used for its training. Figures 4 and 5 compare the interstory drift values obtained from the detailed nonlinear time history analyses and those estimated using the surrogate models. The comparisons indicate reasonably good matches between the statistical and the FE models. The fits are better for the higher drift levels, which are of interest when structural damage and/or collapse are estimated. This is particularly encouraging as the primary focus in this study is the estimation of the collapse potential of the structure, which occurs at larger drifts.
Figure 4. Observed and predicted values in terms of the peak ISD for the case-study building. (A) Peak ISD along the X direction; (B) Peak ISD along the Y direction.
Figure 5. Observed and predicted values in terms of the residual ISD for the case-study 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 Table 3 used in the training process. These models are also subjected to ground motions not considered in the development of the surrogate model. These “unseen” ground motions are taken randomly from the set of 42 ground motions, which comprises the far-field suite of FEMA P695 and motions from the 2015 Gorkha earthquake, with the exclusion of those used to train the surrogate models. The performance of the surrogate model is evaluated for ten such cases using five “unseen” ground motions and two sets of “unseen” MPs. The two sets of MPs obtained by estimating the average of Values 1 and 2 and the average of Values 3 and 4 in Table 3, respectively for all MPs. For example, the values used for fc′ are 1.19 ksi and 1.61 ksi. The five “unseen” ground motions used here are presented in Table 6. The results of these analyses, marked as blue stars in Figures 4 and 5, are rather encouraging, considering the complex nonlinear behavior of the structure.
“Unseen” ground motions selected to evaluate the surrogate model
|Earthquake||Recording station||PGA (g)|
|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||1989||Loma Prieta||Gilroy Array #3||0.555|
MONTE CARLO SIMULATIONS
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.
Uncertainties in model parameters
The statistical models summarized in Table 4 indicate that only six of the eleven sensitive parameters identified in Figure 2 are statistically significant: the peak compressive strengths and corresponding strains of concrete and masonry, the tensile strength of masonry, and the damping coefficient. Therefore, these are considered in the MC simulations. The probability density functions of these parameters are estimated based on data from material tests on concrete and masonry available in the literature[54-58]. In previous studies, the distributions are typically assumed to be lognormal[20,59]; however, to remove any possible bias due to the selection of the probability density functions and investigate which distribution is more suitable for these MPs, both Weibull and lognormal distributions are fitted here, as shown in Figure 6.
Figure 6. Fitting cumulative and probability distributions to available test data. (A) fc′ from the lab tests in Taiwan; (B) fc′ from the actual buildings in the USA[56,57]; (C) fc′ from the lab tests in UB, USA; (D) ε1cfrom the actual buildings in the USA[56,57]; (E) ε1cfrom the lab tests in UB, USA; (F) fm′ from an actual building in the USA; (G) fm′ from the lab tests in UB, USA; (H) ε1mfrom an actual building in the USA; (I) ε1mfrom the lab tests in UB, USA; (J) fm′ from the lab tests in IITK, India; (K) ε1mfrom the lab tests in IITK, India; (L) ftm from the lab tests in IITK, India.
It can be observed in Figure 6 that both distributions fit the test data equally well. Hence, a lognormal distribution is assumed in line with previous studies. Table 7 presents the median, eμ, and standard deviation, σ for a lognormal distribution adopted for all the model parameters. In the case of the damping ratio, ξ, for which test data is very limited, the parameters are taken from the literature[20,54,60-62]. The standard deviation values of the datasets in Table 7 that have mean values closest to the parameters of the calibrated model summarized in Table 1 are employed here to generate samples for the MC simulations. These are indicated with bold text in the table.
Estimated lognormal distribution statistical parameters of the experimentally obtained material properties
|Parameters||fc′ (ksi)||ε1c||fm′ (ksi)||ftm (ksi)||ε1m||ξ (%)|
|Log normal distribution fitting parameters||Yeh, 1998||Median (eµ)||4.7||-||-||-||-||
COV = 0.4
(Porter et al., 2003)
|Yousefianmoghadam et al., 2015, 2018[25,56]||eµ||4.2||0.0029||-||-||-|
|Cristofaro et al., 2011||Set 1||eµ||1.4||-||-||-||-|
|Schueremans and Dionys, 2006||Set 1||eµ||-||-||0.6||-||0.003|
|Shimizu et al., 1999||Set 1||eµ||2.7||-||-||-||-|
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 Nsamp times, generating a set of realizations consistent with the assumed lognormal distribution. Then, two approaches are considered to generate the cases analyzed with the statistical models.
Approach-1: The realizations are set up such that the first random variable is varied from 1 to Nsamp while keeping the other random variables constant. Then, the second variable is varied from 1 to Nsamp without changing the other variables, and so on. Thus, this approach can consider the interactions between the random variables. However, the number of models
Approach-2: With this approach, Nsamp sets of random variables are generated, with each set representing one realization. In this case all MP values are different between the different models. Thus, solving the problem Nsamp times gives information on the randomness in the output response of the system to each set of input variables. Thus, the number of models (Nmod) in this approach is equal to Nsamp.
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 record-to-record uncertainties. The geometric mean of the spectral acceleration at the first modal period of the building, Sa[T1] 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 post-earthquake damage assessment[63-65], the damage classification used by Brzev et al. to assess low-rise RC buildings in Nepal affected by the 2015 Gorkha earthquake is adopted here. This damage classification scheme includes five DGs ranging from DG1 to DG5 based on the EMS-98 scale, to characterize the severity of damage in structural components. This damage classification is correlated to maximum drifts reached during an earthquake by using data from laboratory specimens[59,66] and by consulting with the field engineers from the National Society of Earthquake Technology (NSET), Nepal who implemented it after the 2015 earthquake. As proposed by Hilly, the limit states, DG1 to DG5, correspond interstory drift ratios of 0.3%, 0.6%, 1.2%, 2.0%, and 3.0%, with DG5 representing severe damage and partial or total collapse of the building. The medians and standard deviations of Sa[T1] for the five DGs obtained from both approaches of MC simulations are summarized in Figures 7 and 8. It can be observed that the medians and standard deviations converge to the same values of Sa[T1], representing the five DGs for both approaches. However, the first approach needs 46,656 models for all median and standard deviation values to converge compared to the 100 analyses needed for the second approach. Therefore, the second approach is adopted here for further analyses.
Figure 7. Medians and standard deviations of the Sa[T1] at various damage grades obtained from MCS - Approach 1. (A) Medians; (B) Standard deviations.
Figure 8. Medians and standard deviations of the Sa[T1] at various damage grades obtained from MCS - Approach 2. (A) Medians; (B) Standard deviations.
To capture the uncertainties associated with record-to-record 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 Nmod = 100 are presented in Figure 9.
Figure 9. IDA curves for the calibrated model and from Monte Carlo Simulation using the surrogate model. (A) Baseline model; (B) Monte Carlo simulations Approach 2, Nmod = 100.
Table 8 presents the medians and standard deviations obtained from the MC simulations, the baseline FE model, and the baseline surrogate models. For all DGs, the methods considered here result in similar trends for the median and dispersion values, thus further validating the use of the statistical models for MC simulations. However, one can observe that for all DGs, the median values of the spectral acceleration are lower in the case of the MC simulations, which account for the variability in the MPs. Moreover, one can note that the standard deviation increases by approximately 10% in the case of the surrogate model compared to the FE model. The increase is larger in the case of the MC simulations, and especially for the larger DGs. The substantially higher scatter can be expected, considering that in the case of the MC simulations, the uncertainties in the MPs are also included. This is evident in Figure 9, which demonstrates a higher scatter in the data points.
Results of the Monte Carlo simulation compared to the base model
|Damage grades||Median Sa[T1] (g)||Standard deviation Sa[T1] (g)|
|Baseline model||Monte carlo simulations||Baseline model||Monte carlo simulations|
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 Figure 10. It can be observed that the lognormal distribution provides a good estimate of the fragility data. Therefore, it is used in this study to develop the fragility curves.
Figure 10. 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 [Figure 11]. These curves incorporate the uncertainties associated with the MPs and the ground motion characteristics. Figure 11B demonstrates the effects of the uncertainties of the MPs by comparing the fragility curve of the baseline model, considering only record-to-record uncertainty with that obtained from the MC simulations for DG5. The difference between the two curves illustrates the importance of incorporating modeling uncertainties in vulnerability analysis. The baseline model, when not considering the modeling uncertainties, can be unconservative and underestimate the probability of collapse for spectral accelerations below 0.6 g.
EFFECT OF CORRELATIONS BETWEEN MATERIAL PARAMETERS
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 Set-1 the concrete properties are assumed to be correlated, but there are no correlations between the masonry properties. In Set-2 the masonry parameters are assumed to be correlated, while the concrete properties are not. Finally, in Set-3 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 Set-3 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 Nmod = 1,000 are summarized in Table 9. It can be observed that the correlations between the parameters have no noticeable effects on the median Sa[T1], particularly for the first two sets. In fact, for Sets 1 and 2, correlation effects tend to be conservative if neglected. However, the standard deviations change slightly with the correlations between the variables.
Parametric study on the effects of the correlation assumptions on the fragility curves
|Correlation assumptions||Sa[T1] for Set-1||Sa[T1] for set-2||Sa[T1] for set-3|
|Median||Std. dev.||Median||Std. dev.||Median||Std. dev.|
In this study, a reliable data-driven 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 cross-equation 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 performance-based 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 physics-based 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 KAvailability of data and materials
The data and materials pertinent to this paper can be provided upon request.Financial support and sponsorship
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.Conflicts of interest
All authors declared that there are no conflicts of interest.Ethical approval and consent to participate
Not applicable.Consent for publication
© The Author(s) 2023.
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Cite This Article
Bose S, Stavridis A, Anastasopoulos PC, Sett K. Fragility curves accounting for uncertainties in material parameters and ground motion characteristics using a data driven surrogate model. Dis Prev Res 2023;2:17. http://dx.doi.org/10.20517/dpr.2023.20
Bose S, Stavridis A, Anastasopoulos PC, Sett K. Fragility curves accounting for uncertainties in material parameters and ground motion characteristics using a data driven surrogate model. Disaster Prevention and Resilience. 2023; 2(3): 17. http://dx.doi.org/10.20517/dpr.2023.20
Bose, Supratik, Andreas Stavridis, Panagiotis Ch. Anastasopoulos, Kallol Sett. 2023. "Fragility curves accounting for uncertainties in material parameters and ground motion characteristics using a data driven surrogate model" Disaster Prevention and Resilience. 2, no.3: 17. http://dx.doi.org/10.20517/dpr.2023.20
Bose, S.; Stavridis A.; Anastasopoulos PC.; Sett K. Fragility curves accounting for uncertainties in material parameters and ground motion characteristics using a data driven surrogate model. Dis. Prev. Res. 2023, 2, 17. http://dx.doi.org/10.20517/dpr.2023.20
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