Quantification of Equivalent Strut Modeling Uncertainty and Its Effects on the Seismic Performance of Masonry Infilled Reinforced Concrete Frames

ABSTRACT Quantifying aleatory and epistemic uncertainty in nonlinear structural response simulation is key to robust performance-based seismic assessments. This paper focuses on the modeling parameters that are used to describe the nonlinear force-deformation response of the equivalent infill struts used in models of reinforced concrete frames with infills. The variability in several parameters is characterized by developing empirical and theoretical multivariate probability distributions based on the deduced-to-predicted ratios derived using data from 113 physical experiments. The effect of the uncertainty in the infill strut modeling parameters on maximum story drift ratios and associated limit state fragility functions is investigated for a 3-story reinforced concrete frame building with infills. Multiple stripe analysis is performed using hazard-consistent ground motions. Relative to when only record-to-record variability is considered, modeling parameter uncertainty has a non-negligible effect on the dispersion of the maximum story drift ratios, and affects both the median and dispersion of the limit state fragilities.


Introduction
Reinforced concrete (RC) frames with masonry infills (or masonry-infilled frames) are widely used for building construction in regions with moderate-to-high seismicity.Their behavior is governed by the local and global interactions between the frames and infills.To better understand and characterize their seismic behavior, numerous experimental investigations of infilled frames have been conducted (e.g.Calvi, Bolognini, and Penna 2004;Morandi, Hak, and Magenes 2014;Stavridis, Koutromanos, and Shing 2012).Post-earthquake reconnaissance missions have also served as a valuable source of information on actual observations of the seismic vulnerability of infilled frames (e.g.Furtado et al. 2021;Li et al. 2008).The lessons learned from both controlled physical experiments and field observations have also informed the development of nonlinear structural modeling methodologies.For performance-based earthquake engineering (PBEE) assessments that require structural response simulations using large numbers of ground motions, a macromodeling strategy is usually adopted.This approach uses nonlinear flexural elements for the beams and columns and two or more compression-only diagonal axial struts to simulate the behavior of the infills.The constitutive behavior of the infill struts has been modeled using both curvilinear force-deformation (or stress-strain) relationships (e.g.Noh et al. 2017) or multilinear backbones curves (e.g.Burton and Deierlein 2014).Regardless of the modeling approach, the parameters that define the inelastic response of the infilled strut are calibrated using the results from physical experiments.When reasonable amounts of experimental data are available, empirical equations are developed to predict the model parameters (e.g.Huang, Burton, and Sattar 2020;Liberatore et al. 2017).Developed through the application of statistical regression, the equations provide central tendency estimates of the model parameters.While generally not considered in PBEE estimates, there is an implicit recognition that significant uncertainties are present in these model parameter predictions.
Uncertainty propagation is an important part of the robust implementation of the PBEE framework.In the context of nonlinear structural response estimation, two types of uncertainty can be considered.Record-to-record RTR ð Þ uncertainty is considered by performing incremental dynamic analysis IDA ð Þ, multiple stripe analysis MSA ð Þ using a suite of ground motions, or cloud analysis (Baker 2015;Vamvatsikos and Cornell 2002).The resulting structural response distribution reflects the uncertainty in the frequency content, duration, number of cycles, and other important characteristics of the ground motion.The second type of uncertainty, modeling uncertainty, is due to limitations in the data and information that inform the development of the numerical model as well as various simplifications or assumptions that are incorporated.Four categories of modeling uncertainty denoted as types I through IV are described in Bradley (2013).The lack of knowledge about the measured values of the physical quantities or "basic" parameters (e.g.compression strength of concrete, yield strength of reinforcing steel, and prism strength of masonry) give rise to Type I uncertainty.Type II uncertainty accounts for the bias in measured quantities (e.g.concrete compressive strength and elastic modulus) and the related constitutive model parameters (e.g.peak strength, initial stiffness, or ultimate deformation of the structural element).The Type III category is used for errors that originate from the specific choice of a constitutive model and its underlying assumptions in a numerical model.Finally, Type IV is used to describe uncertainties that result from the systematic errors introduced by the adoption of a less or more simplified modeling approach to represent the behavior of a structure (e.g.boundary conditions and lumped plasticity model).
The effects of Type I and Type II uncertainties on seismic performance can be generally considered by performing nonlinear response history analyses using a set of structural models with randomly sampled basic and constitutive model parameters, respectively.Type I uncertainty is easier than Type II to characterize because it primarily requires probability distributions of the material properties that are used as inputs into the nonlinear structural model (e.g.Asgarian and Ordoubadi 2016;Choudhury and Kaushik 2019;Dolšek 2009;Gokkaya, Baker, and Deierlein 2017;Haselton and Deierlein 2006;Jalayer, Iervolino, and Manfredi 2010;Jiang and Ye 2020;Kosič, Fajfar, and Dolšek 2014;Liel et al. 2009;Sattar, Weigand, and Wong 2018).Characterization of Type II uncertainty is more challenging because the probability distributions of the errors in the constitutive model parameters are needed.This requires a reasonably sized dataset such that the distribution of the difference or ratio between the predicted (by some empirical equation) and measured (from experiments) model parameters can be quantified.This partly explains why there are relatively fewer studies in the current literature on Type II model characterization and propagation within the PBEE framework (e.g.Dolšek 2009;Gokkaya, Baker, and Deierlein 2017;Haselton and Deierlein 2006;Liel et al. 2009;O'reilly and Sullivan 2018).Few recent, yet preliminary studies have also attempted to explicitly quantify the effect of Type III and Type IV uncertainties (e.g. Gaetani d'Aragona, Polese, and Prota 2021;Gokkaya, Baker, and Deierlein 2017;Mucedero, Perrone, and Monteiro 2021;Romano et al. 2021;Sattar, Liel, and Martinelli 2013;Sattar, Weigand, and Wong 2018;Wijaya et al. 2020).Within the specific context of infilled frames, Type I uncertainty has been the most considered in prior studies (e.g.Choudhury and Kaushik 2019;Wijaya et al. 2020).
This paper characterizes and propagates the effect of Type II modeling uncertainty (denoted as modeling uncertainty in the remainder of the paper) on the seismic performance of RC-infilled frames.For this purpose, the set of empirical equations developed by Huang, Burton, and Sattar (2020) is considered, where a "pinched" load-deformation hysteretic rule (Lowes and Mitra 2004) was implemented to simulate the behavior of the infill strut.A dataset comprised 113 RC infilled frame specimens (Huang and Burton 2020) was used to develop empirical equations for predicting the constitutive model parameter values and simulate the behavior of masonry infill walls.The considered dataset only included unreinforced masonry infills with no openings or retrofit measures, which were tested under monotonic or quasi-static cyclic loading.As described in Huang, Burton, and Sattar (2020), the experimental hysteresis of the entire infilled frame is considered for the regression analyses and definition of the infill strut model.In the present study, probability distributions of the errors (expressed as predicted-to-deduced ratios) in the values of the equivalent infill strut modeling parameters are derived.Both theoretical and empirical multivariate distributions are developed while considering the correlations between the model parameter errors.The error distributions are then used to propagate and quantify the effect of modeling uncertainty in the seismic performance of a prototype infilled frame building that has been used in several prior studies (including experimental).MSAs are performed on nonlinear structural model realizations of the prototype building that capture the full probability distribution of the infill strut parameters.The probability distribution of maximum story drift ratios SDRs ð Þ as well as system-level limit states are quantified while considering both record-to-record and model parameter uncertainties.The effect of the chosen distribution type (i.e.empirical vs theoretical) and correlation among the model parameter errors on the seismic response and performance outcomes is also assessed.

Overview of Huang et al. Infill Strut Model
Equivalent strut models were first used to analytically represent the in-plane behavior of masonry infills (e.g.Holmes 1961;Mainstone and Weeks 1970;Polyakov 1960;Stafford Smith and Carter 1969).More recently, the idea has been extended and refined for use in macro-element computational models (Burton and Deierlein 2014;Chrysostomou, Gergely, and Abel 2002;Crisafulli and Carr 2007;El-Dakhakhni, Elgaaly, and Hamid 2003;Sattar and Liel 2016).This has enabled more reliable analysis and seismic performance assessment of infilled frame buildings, especially within the PBEE framework.The model considered in the current study, which was developed by Huang, Burton, and Sattar (2020), uses diagonal compression-only truss elements (one in each direction) to model the behavior of the infill.The force-deformation relationship that governs the response of the struts is based on the Lowes-Mitra-Altoontash (Lowes and Mitra 2004) pinching material.As schematically depicted in Fig. 1, the in-cycle behavior is described by six model parameters: the initial stiffness K e ð Þ, the yield strength F y À � , the capping strength F c ð Þ, the associated deformation d c ð Þ, the residual strength F res ð Þ, and the ratio of post-capping-to-initial stiffness K pc =K e À � .Huang, Burton, and Sattar (2020) developed a set of equations for predicting each parameter using various geometric and material properties as the independent variables.Analytical relationships are developed from regression analysis using an experimental dataset of force-displacement responses comprised 113 infilled frame specimens, where only the response of the unreinforced masonry panel is considered.Table 1 summarizes the predictive equation for each of the six parameters including the associated coefficient of determination R 2 ð Þ and residual standard error σ ð Þ.A multiplicative functional form was adopted for K e , F c , d c and K pc =K e to be generally consistent with the mechanics-based analytical relationships between response parameters and structural properties.The independent variables used to predict these four parameters include the modulus of elasticity E m ð Þ and strength f m ð Þ of the masonry prisms, the thickness t w ð Þ, and height-to-length h w =l w ð Þ ratio of the infill panel and the length of the diagonal strut l d ð Þ.The yield F y À � and residual F res ð Þ strengths are based on a percentage of the capping strength F c ð Þ.The reported R 2 values show significant variation in the ability of each equation to capture the variance in the data.Similarly, there is also a significant variation in the σ which is more directly related to the predictive performance of each model.

Multivariate Probability Distributions of Deduced-To-Predicted Parameter Ratios
To support uncertainty quantification and propagation, a multivariate distribution of the errors associated with the Huang, Burton, and Sattar (2020) predictive equations is needed.These errors are represented as the ratio between the deduced (obtained through calibration as reported in Huang, Burton, and Sattar (2020)) value of the parameter for the i th specimen d i ð Þ and the value predicted by the relevant equation for the same specimen p i ð Þ.The d i =p i ratio is computed for the 113 specimens that are the basis of the equations.Histograms showing the empirical distribution of d i =p i for each model parameter are shown in Fig. 2, where it is observed that the deduced-to-predicted ratios for all six model parameters have skewed empirical distributions.The overwhelming majority of the ratios range from approximately 0.4 to 2.0.The minimum ratio of 0.17 is associated with the residual strength (F res ), while the maximum (9.0) occurs in the deformation at capping strength (d c ).A few outliers (i.e.values that show a clear deviation from the overall pattern) located at the right tail of the distributions occur in the initial stiffness (K e ), deformation at capping strength (d c ), and post-cappingto-initial stiffness ratio (K pc =K e ).
As described later in the paper, the empirical distribution of the d i =p i ratios can be used to sample randomized realizations of the model parameter.However, for more generalized use (especially in future studies), it is desirable to also have their theoretical probability distributions.As such, the Kolmogorov -Smirnov test (KS-test) is performed to evaluate how well the empirical distribution of the different d i =p i ratios fits the following theoretical distributions: Rayleigh, Normal, Lognormal, and Gamma.A non-parametric hypothesis test, the KS-test (Karson 1968) computes the maximum absolute difference between the cumulative distribution function CDF ð Þ of the empirical and  Huang, Burton, and Sattar (2020) to estimate the infill strut model parameters.
theoretical distributions.For this study, the KS-test is performed in MATLAB ( 2019) where the null hypothesis is that the datasets follow a specific continuous (theoretical) distribution.A p-value (i.e. a statistical value that describes how likely it is to obtain an observation if the null hypothesis is true) that is higher than the specified significance level α ð Þ indicates that the null hypothesis is accepted, and  the two datasets are indeed from the same distribution.The results for a truncated lognormal distribution, which provides the best overall fit with the empirical data, are summarized in Table 2.All of the p-values are much greater than 0.05, a value that is typically used as the significance level.Good visual matches between the empirical and log-normal CDFs for all six strut model parameters are shown in Fig. 3.One of the goals of the current study is to evaluate whether the correlation between the deduced-topredicted ratios for the six infill strut model parameters influences the seismic performance outcomes.For this reason, randomized realizations of infill strut backbone curves are generated with and without these correlations.The (Pearson) correlation coefficient across each model parameter is computed using the empirical dataset illustrated in Fig. 2, with the associated correlation matrix presented in Fig. 4. The strongest deduced-to-predicted ratio correlation is observed between F c and F y and F c and F res with coefficients of 0.93 and 0.81, respectively.This is expected because both F y and F res are computed as a function of F c .For the same reason, the deduced-to-predicted ratio correlation between F y and F res is also significant with a coefficient of 0.79.The deduced-to-predicted ratios of these three parameters F c ; F y ; F res À � are also moderately and positively correlated with K e .The deduced-to- predicted ratio of the deformation at peak strength d c has low but negative correlations with all other parameters except K pc =K e .

Sampling Distribution of Infill Strut Backbone Curves
Latin Hypercube Sampling LHS ð Þ is used to generate random realizations of the infill strut backbone curve.As discussed in several studies (Dolšek 2009;Ugurhan, Baker, and Deierlein 2014a;Vamvatsikos and Fragiadakis 2010;Vorechovsky and Novak 2003), LHS has been demonstrated to be an effective method for sampling random variables with and without correlation.By stratifying the probability function that defines the distribution of a random variable, the number of simulations is considerably reduced compared to other techniques such as Monte Carlo simulation (Dolšek 2012).An overview of the procedure used to generate the randomized infill strut backbone curves is shown in Fig. 5. Four sets of backbone curve realizations and nonlinear structural models of the prototype building (described in the next section) are generated.The first set of realizations is based on the empirical distributions of the deducted-to-predicted parameter ratios and considers the correlations shown in Fig. 4. The second set also uses the empirical distributions, but the correlations are not considered.In other words, the deducted-to-predicted parameter ratios are assumed to be independent.By comparing the performance outcomes from these two sets of model realizations, the effect of correlation between model parameters on seismic performance will be isolated.The third and fourth sets of deduced-to-predicted parameter ratio realizations are sampled from the truncated lognormal distribution.As noted earlier, using a theoretical distribution increases generalizability and will be especially useful for future studies.Comparing the performance outcomes from the first and third (or second and fourth) sets of model realizations will enable an evaluation of the effectiveness of the theoretical distribution in capturing and propagating the modeling uncertainty.
As shown in Fig. 5, LHS is used to generate realizations of the deducted-to-predicted parameter ratios of six randomized infill strut model parameters.These ratios are then multiplied by the "expected" value of the corresponding parameter obtained from the Huang, Burton, and Sattar (2020) equations.This step produces the model parameter samples that will be used in the nonlinear structural model.The deformation at yield strength d y À � for each realization is computed using the randomly sampled initial stiffness and yield strength.Each realization of randomized model parameters is checked to determine whether they are consistent with the rules of the backbone curves.For instance, although infrequent, there are some cases where the sampled F res is greater than the sampled F y .Such cases (around 10% of the sampling space) are inconsistent with the rules of the backbone curve and are therefore excluded.The impact of the excluded cases on the statistics of the response demands is assessed, and a maximum difference of approximately 4% is observed.On the contrary, when correlations are not considered, the number of rejected cases increases to approximately 60%,  emphasizing the importance of properly considering the correlations.Based on the results of a convergence analysis of the infilled framed seismic performance (i.e.effect of number of realizations on performance outcomes), 500 model realizations of the strut modeling parameters are used in the LHS method.Figure 6a shows a schematic representation of the infill strut backbone curve with the empirical distribution of the randomized parameters.For illustrative purposes, a set of 50 LHSgenerated backbone curves is presented in Fig. 6b along with the mean model curve (i.e.model constructed using the mean measured values).

Description of Prototype Building
We assess the effect of record-to-record and modeling uncertainty on the seismic response and performance of the prototype building designed and studied by Stavridis (2009).The building comprises a three-story non-ductile reinforced concrete frame structure with unreinforced triplewythe masonry infill walls.This type of building represents common construction practice in the 1920s era in California.However, masonry-infilled RC frames with similar design details as the prototype building used in this study are still often used for housing and industrial activities in many parts of the world.Furthermore, this type of structure continues to be a common construction practice in places where earthquakes are a great concern.A floor plan and elevation view of one of the longitudinal infilled frames is shown in Fig. 7.The triple-wythe masonry infill walls are located in the perimeter reinforced concrete frames.The longitudinal infilled frame without openings is considered for the current study.Additional details about the design of the prototype building are provided in Stavridis (2009).

Nonlinear Structural Modeling
As noted earlier, the seismic performance assessment is performed for one of the longitudinal perimeter infilled frames.Since the Huang, Burton, and Sattar (2020) equations for the strut model parameters are only applicable to solid infill panels, wall openings are not considered.Figure 8 shows a schematic representation of the two-dimensional nonlinear structural model of the infill frame that was constructed in the Open System of Earthquake Engineering Simulation (OpenSees) (McKenna 1997).Rayleigh damping based on the first and third modal frequencies is used for the dynamic analysis with 5% critical damping.Leaning columns are used to capture the P-∆ effects caused by half the load transmitted to the interior gravity frame.The leaning columns are connected to the RC frame system at each floor level by rigid truss elements.Rotational springs with very small stiffness are placed at the ends of the leaning columns to avoid adding flexural stiffness to the frame.
Linear elastic elements with concentrated plastic hinges are used to model the beams and columns.The semi-empirical equations calibrated by Haselton et al. (2016) are used to determine the parameters that define the hysteretic behavior of the flexural hinges and the Ibarra-Medina-Krawinkler (I-M-K) model (Ibarra, Medina, and Krawinkler 2005) is incorporated.Using the model developed by Elwood (2004), shear failure in the columns is considered by adding a shear hinge in series with the flexural hinge at each end.The centerline dimensions of the frame elements are used to model the RC frame system.The masonry infill walls are modeled as two compression-only diagonal truss elements with the Lowes-Mitra-Altoontash pinching material (Pinching4 material in OpenSees) and connected to the point at the intersection of the frame beam and column.Thus, the eccentric shear load that can be generated from the infill to the RC columns is not explicitly modeled.The model parameters that define the backbone curve for the axial behavior of the equivalent strut are based on the Huang, Burton, and Sattar (2020) empirical equations described earlier.Since the infill strut element considers compression-only action, the parameters that define the tension branch of the hysteretic curve are assumed to be almost zero to avoid numerical issues during the analysis.As indicated in Huang, Burton, and Sattar (2020) a unit strut area is used for the axial response of the equivalent strut, so that the calibrated stress is the same as the associated force.Mean values are used for the cyclic degradation and pinching parameters.For the mass and gravity loads, the measured values reported by Stavridis (2009) are used.Also, the mean measured material properties (Shing and Stavridis 2014) are used to define the modeling parameters for the reinforced concrete elements (i.e.beams and columns).Recall that the goal of the study is to isolate the effect of Type II uncertainty in the empirical strut model.Based on the geometry and measured mean values of the material properties, the computed predicted mean values for the six modeling parameters of the infill strut model are listed in Table 3.The structural model does not consider the stiffness and strength contribution provided by the concrete floor slab.The column bases are modeled as fixed without allowance for foundation uplift.The first three modal periods for the mean model are 0.119 s, 0.058 s, and 0.047 s, respectively.

Selection of Hazard-Consistent Ground Motions
A Los Angeles site (34.208 and −118.604) with short-period and one-second period spectral accelerations S s ¼ 1:5g and S 1 ¼ 0:6g, respectively, for Site Class D conditions is used for the hazardconsistent ground motion selection.Seismic hazard deaggregation is performed using the Unified Hazard Tool provided online by the United States Geological Survey (USGS) (Geological Survey 2020).The ground motion selection is performed using the fundamental period of the prototype structure in the direction of analysis at the site location.Based on Site Class D and spectral acceleration at 0.1 s Sa T1 ð Þ, eight hazard levels are selected with probability of exceedance in 50 years ranging from 20% to 0.5%.Using the disaggregation results from the USGS tool and an average of three ground motion models; the Campbell and Bozorgnia (2014), the Abrahamson, Kamai, Abrahamson, and Silva (2014), and the Chiou and Youngs (2014), a uniform hazard target spectrum is obtained for each hazard level.The conditional mean spectrum is acknowledged as an alternative to the uniform hazard spectrum target.However, the choice of target spectra (uniform vs conditional) is not expected to have a tangible effect on the findings of the current study, which is focused on the effect of equivalent strut modeling uncertainty on the seismic performance of infilled frames.Forty pairs of ground motions are then selected at each hazard level (i.e.eight record-sets) such that their mean spectra reasonably match the respective target spectra.A maximum scale factor of 3.0 is considered for the record selection.The mean magnitude, source-to-site distance and epsilon from the hazard disaggregation are shown in Table 4.The 50-year exceedance probability, the return period, and the target spectral acceleration at the fundamental period of the structure corresponding to each hazard level are also shown in Table 4. Probabilistic seismic hazard analysis PSHA ð Þ is also performed for the site and period of interest using the USGS Uniform Hazard Tool.The target spectra as well as the ones for the selected ground motions (individual and mean) and the hazard curve obtained from PSHA, are shown in Figs. 9 and Figure 10, respectively.

Effect of Record-To-Record Uncertainty on Seismic Response and Performance
MSA is performed on the nonlinear structural model constructed using the mean measured values of the basic parameters (i.e.material properties and geometry) for the infill strut parameters (mean model hereinafter) (see Table 3) to quantify the record-to-record uncertainty.Figure 11 shows the results from the MSA, where empirical responses for SDR (maximum story drift ratio) (Fig. 11a) are represented in the form of "stripes," with each corresponding to one of the hazard levels.Note that only maximum drift responses of up to 5% are shown.Each stripe (hazard level) includes a total of 80 responses and SDR � 2% is taken as the collapse threshold.The MSA responses are summarized in  higher intensities, the β SDR;RTR begins to fluctuate, which is likely due to changes in the number of collapse or near-collapse cases, where the response demands no longer follow a lognormal distribution.Three limit states are used to characterize the seismic performance of the prototype structure.A drift threshold of 0.3% is used for the first limit state LS 1 ð Þ and 0.75% and 2.0% for the second LS 2 ð Þ and third LS 3 ð Þ limit states, respectively.According to several prior studies (e.g.Basha and Kaushik 2016;Ghobarah 2004;Kalman Šipoš and Sigmund 2014;Stavridis and Shing 2010), these three limit states are associated with cracking in the masonry infill and interfaces (moderate damage), extensive   cracking of the infills and frame columns (severe damage), and shear failure and/or loss of axial capacity in the frame system (global collapse), respectively.The limit state fragilities shown in Fig. 11b are obtained from the fraction of predicted responses (at each hazard level) that causes the exceedance of a certain limit state.The maximum likelihood estimation method (Baker 2015) is applied to the empirical probabilities.The fragilities describe the probability of exceeding a defined limit state as a function of Sa T1 normalized by the spectral value corresponding to the maximum considered earthquake MCE ð Þ level, here denoted as S MT .Consistent with the observations from the SDR distribution, the logarithmic standard deviation (dispersion) generally increases with the severity of the limit states.For instance, β LS;RTR for LS 3 is 15% and 14% higher compared to LS 2 and LS 1 , respectively.

Effect of Record-To-Record and Modeling Uncertainty on Seismic Response and Performance
To account for and propagate both record-to-record and modeling uncertainty, MSAs are performed on 500 randomized nonlinear structural model realizations.For each model, one realization of the infill strut model parameters is utilized, and one of the 80 ground motions is randomly sampled at each hazard level (Ugurhan, Baker, and Deierlein 2014b).It is worth noting that an alternative to this approach would be to run the entire MSA (i.e. using all 80 ground motions at each hazard level) through each randomly generated model.However, this approach would significantly increase the computational expense of the overall procedure.In addition, the adopted approach retains the effect of record-to-record uncertainty through the random sampling of the ground motions (Baker 2015).As noted earlier, four different approaches are used to sample the infill strut model parameters.The results presented in this section are based on the empirical and correlated deduced-to-measured values of the infill strut parameters as this case is considered the "baseline" or most representative of the data.Later in the paper, we benchmark the results from the other three cases to the ones presented in this section.
Figure 12a presents the median SDR ratios for the cases with and without modeling uncertainty θ SDR;TOT =θ SDR;RTR À � at each hazard level.Similarly, the same ratio is shown for the log-standard deviation β SDR;TOT =β SDR;RTR � � in Fig. 12b.The respective values are also summarized in Table 6.As reflected in Fig. 12b, the inclusion of modeling uncertainty increases the maximum story drift dispersion in all hazard levels.On average (i.e. the mean value considering all hazard levels), the dispersion increases 22% when modeling uncertainty is included.The greatest increase in the dispersion occurs at the lowest hazard level (41%) and there is a general decrease in the relative impact of modeling uncertainty at higher hazard levels (e.g.18% increase at the 10000 years return period).The smaller demands and marginal inelastic (or elastic in some cases) response partially explains the relatively higher contribution of modeling uncertainty at lower hazard levels.The effect of modeling uncertainty on the median SDR ranges from a 17% decrease to a 22% increase at the highest and lowest hazard levels, respectively.This implies that propagating the modeling uncertainty through the record-to-record variability may decrease or increase the estimated median drift values.The effect of modeling uncertainty on limit state performance is summarized in Fig. 13 and Table 7.On average, modeling uncertainty increases the dispersion in the limit state performance by 26%.In fact, the effect of modeling uncertainty on dispersion is actually lower in the more severe limit states (i.e. at global collapse) with a 15% increase.On the other hand, the median limit state intensity, which corresponds to the intensity where the probability of exceeding a certain limit state is 50%, is increased by an average of 1% when modeling uncertainty is considered, decreasing 8% and increasing 9% for the lower and more severe limit state, respectively.The increasing trend in both the median and dispersion of the limit state intensities is consistent with the "flattening" of the fragility curve that is generally observed when modeling uncertainty is considered (FEMA 2012).Another way to interpret the effect of modeling uncertainty on the performance of reinforced concrete frames with infills is to examine how the probability of limit state exceedance is affected at the DBE and MCE hazard levels.In Fig. 13, we observe that, at the MCE level, the probability of collapse LS 3 ð Þ is reduced from 43% to 39% when  modeling uncertainty is considered.It is worth noting that both these probabilities are considered unacceptably high.At the DBE hazard level, the probability of exceeding LS 3 is minimally affected.However, the DBE exceedance probability is increased by 21% and 19% for LS 1 and LS 2 , respectively.A risk-based evaluation is also conducted to quantify the effect of modeling uncertainty on timebased performance metrics.The mean annual rate of drift-based limit state exceedance is computed by integrating the site-specific hazard curve and the fragility curves obtained for each limit state.Then, the probability of exceedance of each limit state, over a period of 50 years, is computed by assuming a Poisson distribution.Consistent with the results obtained for the dispersion in the limit states fragility curves, modeling uncertainty has a more significant impact on the lower limit state thresholds.For LS 1 , the probability of exceedance in 50 years P LS;50ys À � is increased by 67%, while for LS 3 , the same metric increased by nearly 8% when modeling uncertainty is included.Table 8 summarizes the effects of modeling uncertainty on the 50-year probability of limit state exceedance.
To identify the contribution of each random variable to modeling uncertainty, a deaggregation analysis is conducted by repeating the MSAs while considering the uncertainty in one parameter at a time while the others are assigned their mean values.The results of this analysis are shown in Table 9, where it is observed that the most significant contributors (i.e.produce a higher increase in median and dispersion) to the limit state performance of the building are the yield strength (F y ) and capping strength (F c ).

Effect of Error Distribution Type and Correlation on Seismic Response and Performance
The effect of error correlations and the implications of using a theoretical multivariate distribution is investigated.The results show that the theoretical lognormal distribution produces seismic response demands and limit state performances that are quite similar to the empirical case.Figure 14 compares the fragility curves obtained with both approaches.Up to the MCE hazard level, both approaches (i.e. using the empirical and theoretical distributions) give practically the same estimates.For higher intensity levels, the results obtained using the theoretical distributions are slightly more conservative in terms of the probability of exceedance of the limit states.More specifically, the dispersion of both the MSA drift demands and limit state fragility curves increase on average by 4% and 7%, respectively, when the theoretical distributions are assumed.The median SDR decreases by an average of 3.5%, while the median limit state intensity saw an average increase of 1.5%.In terms of the limit states, the maximum difference occurs for LS 3 , where the median limit state intensity computed based on the empirical distributions is only 2.4% greater than the theoretical distribution case.The same trend is observed for the dispersion, where the empirical case resulted in a log-standard deviation that is 1.04 times the theoretical for LS 3 .These results suggest that the use of the determined theoretical distributions (to represent the randomness of the model parameters) is reasonably adequate for quantifying the effect of modeling uncertainty on drift demands and the associated limit states.
Two types of correlation are considered when quantifying the effect of modeling uncertainty: (1) the correlation within the parameters that define the behavior of a single element (the strut model in this case) and ( 2) the correlation among the model parameters of different elements in a structure.Similar to what was observed in previous studies (Gokkaya, Baker, and Deierlein 2017;Haselton and Deierlein 2006;Ugurhan, Baker, and Deierlein 2014b), the results showed that the assumptions made about both types of correlation can substantially vary the estimated uncertainties and performance of the structure.Considering the correlation (as reported in Fig. 4) between the parameters of the same strut (but ignoring correlation across different struts) resulted in a median limit state intensity that is 6% and 12% lower for LS 1 and LS 3 , respectively.For the dispersion, an increase of 23% and a decrease of 5% are observed for the same limit states, respectively, compared to the case where the correlation is ignored i.e. zero correlation is assumed.When full correlation is assumed across different strut elements (while considering the computed correlation within each strut), the median limit state intensity decreased by 1% for LS 1 and increased by almost 2% for LS 3 , while the dispersion increased by 27% and 5%, respectively, compared to the zero-correlation case.Furthermore, for the SDR median and dispersion differ by as much as 26% depending on the correlation assumption (i.e.full, zero, or computed).

Conclusion
This study characterized the modeling uncertainty embedded in the equivalent strut model based on the Huang, Burton, and Sattar (2020) empirical equations, which were developed to represent the inplane nonlinear behavior of masonry infill walls in reinforced concrete (RC) frames.The effect of modeling uncertainty on the seismic performance of a non-ductile RC masonry-infilled frame building is then assessed.
Type II epistemic uncertainty in the modeling parameters that define the backbone of the infill strut behaviour is characterized according to Bradley (2013), where multivariate empirical and theoretical probability distributions were developed to represent the variability of each parameter.These probability distributions are based on the deduced-to-predicted ratios obtained from (1) experimental data of 113 infilled frame specimens and (2) the predictive equations developed by Huang, Burton, and Sattar (2020).Modeling uncertainty is examined in conjunction with the record-to-record uncertainty at different hazard levels.Using site-specific hazard-consistent ground motions, multiple stripe analyses MSAs ð Þ are performed on randomized nonlinear model realizations generated using Latin Hypercube Sampling (LHS).The effect of modeling uncertainty on the seismic performance of a 3-story RC infilled frame building is investigated in terms of the distributions of maximum story drift SDR ð Þ ratios and fragility curves are derived for three drift-based limit states.Additionally, the implications of different correlation assumptions and the use of a theoretical probability distribution for the deduced-to-predicted parameter ratios are also studied.
Type II epistemic uncertainty in the infill strut constitutive model parameters had a significant impact on both the median and dispersion of the SDR, and drift-based limit state fragility curves.The median SDR is shifted in a range between −17% (decrease) and+22% (increase), depending on the hazard level, when modeling uncertainty is considered.The increase in the SDR dispersion due to modeling uncertainty ranged from 9% to 41%, with an average increase of 22%.For the limit state fragility curves, modeling uncertainty primarily impacts the dispersion.The median limit state intensity is increased on average by 1%, while the dispersion increased by an average of 26%, with a maximum increase of 34% for the least severe limit state.From a risk-based perspective, the probability of limit state exceedance in 50 years increased by 67%, 42%, and 8%, for LS 1 (0.3% drift, moderate damage), LS 2 (0.75% drift, severe damage), and LS 3 (2% drift, collapse threshold), respectively, when modeling uncertainty is considered.Additionally, the results obtained from the empirical distributions of the deduced-to-predicted model parameter ratios were quite similar to those obtained when using the proposed theoretical lognormal distributions.This implies that the lognormal distribution parameters can be used in future studies on similar structures.On the other hand, the results showed that the assumptions made about the type of correlation among the random variables are critical and may alter the effect of modeling uncertainty on the performance of the structure by as much as 26% (based on the SDR as the performance metric).As such, this is an area that is worthy of further investigation.
The results show that consideration of modeling uncertainty and correlations between and within structural components has strong implications for the assessed response and performance.Future work could comparatively assess the effect of Type I (i.e.basic parameters) and II modeling uncertainty, where for the latter, the effect of other model parameters and failure modes (e.g.beam-column nonlinear modeling parameters and out-of-plane) can be incorporated.The effect of Type III modeling uncertainty can also be considered by using modeling approaches that are different from the Huang et al. model used in this study.In addition, the infill model does not account for the out-ofplane response of the infill, which can lead to greater vulnerability and therefore requires further investigation.Finally, the constitutive behavior considered in this study has been calibrated based on experimental results for fully solid infill panels (Huang, Burton, and Sattar 2020).Thus, to further generalize the findings, additional analyses need to be performed on different infilled frame structural configurations (e.g. with openings at the infill walls, with bare frames at some levels, etc.).

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
The work was supported by the National Science Foundation Award No. [1554714].

Data Availability Statement
Some or all data, models, or code that support the findings of this study are available from the corresponding author, MH, upon reasonable request.

Figure 2 .
Figure 2. Histograms showing the empirical distribution of deduced-to-predicted ratios for (a) K e , (b) F c , (c) F y , (d) F res , (e) d c , (f) K pc =K e .

Figure 3 .
Figure 3. Visual match between the empirical and log-normal CDFs of the deduced-to-predicted ratios for (a) K e , (b) F c , (c) F y , (d) F res , (e) d c , (f) K pc =K e .

Figure 4 .
Figure 4. Computed correlation matrix for the deduced-to-predicted ratios of the six model parameters.

Figure 5 .
Figure 5. Procedure used to generate randomized infill strut backbone curves.

Figure 6 .
Figure 6.(a) Schematic representation of infill strut backbone curve with empirical distribution of modeling parameters and (b) an illustrative set of randomized backbone curves generated using LHS.

Figure 7 .
Figure 7. Schematic representations of the prototype building; (a) plan and (b) elevation views.

Figure 8 .
Figure 8. Schematic representations of the of two-dimensional nonlinear structural model constructed in OpenSees.

Figure 10 .
Figure 10.Hazard curve for the Los Angeles site corresponding to the fundamental period of the infilled frame structure.

Figure 11 .
Figure 11.MSA results considering only record-to-record uncertainty: (a) empirical distribution of SDR max and (b) limit state fragility curves.

Figure 13 .
Figure 13.(a) Bar charts showing the θ LS;TOT =θ LS;RTR and β LS;TOT =β LS;RTR , respectively, and (b) change in limit state fragility curves due to the consideration of modeling uncertainty.

Figure 14 .
Figure 14.Comparison of limit state fragility curves obtained using the correlated empirical distributions (E C ) and the correlated theoretical lognormal distributions (T C ) proposed in this paper.

Table 1 .
Predictive equations developed by

Table 2 .
KS-test results for the truncated lognormal distributions.
Parameter K-Test p-value for 5% Significance Level Probability Distribution Parameters for Deduced-to-Predicted Ratios

Table 3 .
Mean values computed for the considered infill strut model.

Table 5 ,
which shows the median θ SDR;RTR À �and log-standard deviation β SDR;RTR � � for the noncollapse cases and the percentage of collapses at each hazard level.In general, β SDR;RTR increases up to the hazard level corresponding to the design base earthquake DBE ð Þ (475-year return period), which is consistent with the increase in the nonlinearity of the responses and magnitude of the demands.At

Table 4 .
Summary of considered hazard levels and parameters for the selection of ground motion sets.

Table 5 .
Median, dispersion (log-standard deviation), and percentage of collapse cases considering only record-to-record uncertainty.

Table 7 .
Effect of modeling uncertainty on MSA limit state performance in terms of θ LS;TOT =θ LS;RTR and β LS;TOT =β LS;RTR .

Table 8 .
Effect of modeling uncertainty on limit state exceedance over a period of 0 years.

Table 9 .
Effect of the deaggregated modeling uncertainty on the limit state performance.