1 Introduction

Destructive earthquakes around the world in recent years have resulted in severe building damage and economic losses. Regional seismic damage prediction is an important approach to discover the weakness of a city so as to effectively mitigate seismic losses. For most cities, masonry buildings account for a large proportion of urban constructions. Historical earthquakes indicate that the earthquake-induced damage to masonry buildings contributes significantly to the total regional seismic losses (He et al. 2011, Ural et al. 2012, Lin et al. 2015, Bayraktar et al. 2016). Therefore, an accurate prediction of the regional seismic damage to masonry buildings has significant engineering and scientific values.

The existing approaches for regional seismic damage predictions mainly include: the damage probability matrix method (ATC 1985), the capacity-demand spectrum curve method (D¡¯Ayala et al. 2004, MAE 2006, FEMA 2012a), and the time-history analysis (THA)-based method (D¡¯Ayala et al. 2004, Hori 2006, Lu et al. 2014). The damage probability matrix method is especially convenient and relatively accurate for areas with sufficient statistical damage data (Kircher et al. 1997). However, this method does not take full consideration of the characteristics of individual buildings or ground motions (Xiong et al. 2017). The capacity-demand spectrum curve method used for the regional seismic damage prediction is similar to the capacity spectrum method for individual buildings (FEMA 2012a). It can, to some extent, consider the seismic resistance of individual buildings and the spectral characteristics of different ground motions. In addition, this method can also be used to generate the record-to-record fragility curves of structures under earthquakes (D¡¯Ayala et al. 2004, Vamvatsikos and Cornell 2006, Rossetto et al. 2016). However the duration and pulse effect of the ground motions, or the high-order vibration modes of buildings, cannot be fully considered by this method (Lu et al. 2014). To overcome the limitations of the abovementioned two methods, a THA-based seismic damage prediction method was developed based on the multiple-degree-of-freedom (MDOF) models (Hori and Ichimura 2008, Lu et al. 2014, Lu and Guan, 2017). This method takes a full account of the features of different buildings and ground motions. In addition, realistic dynamic scenarios of seismic damage, which can be easily understood by non-professional users, can be generated based on the time-history responses of buildings obtained through this method (Xiong et al. 2015). For this reason, THA-based seismic damage prediction method will be adopted in this work.

The nonlinear THA has already been widely used to predict the seismic damage of individual buildings (Poon et al. 2011, Jiang et al. 2014, Lu et al. 2013a, 2016, Lu et al. 2013b, 2015a, Xie et al. 2015, Tian et al. 2016). Some researchers (Hori and Ichimura 2008, Yamashita et al. 2011, Xu et al. 2014, Lu et al. 2014, Xiong et al. 2016, 2017, Yepes-Estrada et al. 2016) have also applied nonlinear THA to regional buildings. Nevertheless two critical issues must be tackled before performing the THA-based regional seismic damage prediction:

(1) Enormous computing workload associated with the large number of regional buildings.

Different from the nonlinear THA of an individual building, the number of buildings in an urban area can be excessive, which demands massive computing workload. Various solutions to improving the computational efficiency have been proposed on the basis of latest computer science. Yamashita et al. (2011), Lu et al. (2014, 2015b), and Xu et al. (2016) proposed the solution using the supercomputers, the CPU/GPU coarse-grained parallel computing method, the cloud computing method, and the distributed computing method, respectively, to meet the huge amount of computing demand.

(2) Parameter determination of the computational models for buildings.

To date, it is almost impossible to obtain detailed structural data of each building in an urban area. Therefore, various parameters of the computational models must be determined according to the accessible macroscopic data of the buildings. For example, Xiong et al. (2016, 2017) proposed a parameter determination method for reinforced concrete (RC) frames, tall buildings and masonry structures, based on a simulated design procedure and the statistics of extensive experimental and analytical results. Through this method, the hysteretic behavior of each building at each story can be obtained. The reliability of this model has also been validated through the comparison with experimental results and earthquake site investigations. Note that Xiong et al. (2016, 2017) performed only the deterministic analysis without considering the uncertainty of the model parameters. As evident in the parameter determination process reported by Xiong et al. (2016, 2017), there exists an inherent uncertainty in the model parameters (e.g., the randomness of the peak drift ratio). In consequence, such uncertainty must be properly considered for an accurate seismic performance prediction.

Sensitivity analysis for the seismic damage prediction of an individual building has been widely performed (Lee and Mosalam 2005, Liel et al. 2009, Shin and Kim 2014), and several relatively mature approaches have also been developed. However, the study on the sensitivity of the seismic damage prediction of regional buildings is very limited (Homma et al. 2014, Michel et al. 2016, Remo and Pinter 2012). Research on the regional seismic damage prediction based on the MDOF models and nonlinear THA method is particularly inadequate. Although a Monte Carlo simulation of regional buildings was conducted by Homma et al. (2014), their work mainly focused on the computing methodologies, rather than the influence of parameter uncertainty on the seismic performance of buildings. Therefore, it is of great importance to fill this research gap by evaluating the seismic damage to regional buildings influenced by the parameter uncertainty.

The first-order second-moment (FOSM) method (Melchers 1999) and the Monte Carlo method (Rubinstein 1981) are two widely used methods for sensitivity analysis, which have been used by many researchers investigating various aspects of sensitivity analysis (Porter et al. 2002, Lee and Mosalam 2005, Na et al. 2008, Fellin et al. 2010, Shin and Kim 2014). The Monte Carlo method is considered to be the most accurate method, and is frequently used to benchmark the accuracy of other simplified methods. However, this method is computationally time-consuming and requires the distribution of each random variable. By contrast, the FOSM method is relative simple and requires much less computational workload. For many cases, the FOSM method can also produce similar results to the Monte Carlo method, whereas the variable distribution form is not required to be specified. Given the special features of the two methods, this work will employ both methods to perform necessary calculation and comparison.

The main focus of this work relates to masonry buildings as they suffer the most severe damage during earthquakes. Due to the absence of necessary engineering design process, the seismic performance of unreinforced masonry buildings has great dispersion, and the corresponding statistical data are far from sufficient. Therefore, a sensitivity analysis of reinforced masonry buildings will be performed in this work. Note that some researchers have also studied the uncertainty problems associated with masonry structures (Milani and Benasciutti 2010, Atamturktur et al. 2012, Parisi and Augenti 2012, Parisi and Galasso 2014, Rota et al. 2014); however, their research mainly focused on the performance of one specific masonry structure. Pagnini et al. (2008) provided an easy-to-use mechanical method for the fragility assessment of masonry structures, but it was basically a capacity spectrum method. In this study, the sensitivity analysis of the seismic performance of both individual and regional reinforced masonry structures will be discussed based on the time history analysis. The outcomes of this study may provide useful references to similar research projects.

2 Determination and test of model parameters

2.1 Determination of key model parameters

Published literature (Hori 2006, Lu et al. 2014, Xiong et al. 2017) shows that the MDOF shear model performs well in describing the nonlinear characteristics and failure modes of multi-story buildings. Therefore, the MDOF shear model is adopted herein to simulate reinforced masonry buildings. A typical MDOF shear model is shown in Fig. 1(a). The backbone of the inter-story force-displacement relation is represented by a tri-linear curve as shown in Fig. 1(b), where Points A, B, C and D denote the design point, yield point, peak point and softening point, respectively. The notations of the inter-story drift ratio (IDR), d, and the corresponding strength of each point, V, are also given in Fig. 1(b) and will be used in the subsequent discussions. The parameter determination procedure proposed by Xiong et al. (2017) is detailed as follows:

(1)     Calculate the fundamental period T₁ of the building based on the empirical equations, and perform an equivalent lateral force analysis to obtain the design shear force of each story, V_design,i, according to the design codes (MOHURD 2010), where i is the story number;

(2)     Calculate the actual yield strength of each story, V_yield,i, based on V_design,i, and the yield overstrength ratio W_y defined in Eq. (1);

(3)     Calculate the peak strength of each story, V_peak,i, based on V_yield,i, and the peak overstrength ratio W_p defined in Eq. (2);

(4)     Calculate the strength at the softening point, V_soft,i, as 85% of V_peak,i;

(5)     Calculate the IDR at the yield point, d_y, based on the corresponding inter-story stiffness k and the yield strength V_yield,i;

(6)     Determine the IDR at the peak point, d_p, and the IDR at the softening point, d_s, based on the statistics of the available experimental data.

Fig. 1 The MDOF shear model

According to the procedure given above, five variables must be determined. They include the fundamental period T₁, the yield overstrength ratio W_y, the peak overstrength ratio W_p, the IDR at the peak point d_p, and the IDR at the softening point d_s. Among them, the fundamental period T₁ should be determined firstly. Considering the limited accessible building information for regional seismic damage prediction, the empirical equation proposed by Zhou et al. (2012) is used to estimate the fundamental period of reinforced masonry buildings, as shown in Eq. (3).

This equation only requires the total building height h, which can be readily obtained for regional buildings from the available geographic information system (GIS). Two parameters in Eq. (3), i.e. a and b, can be determined statistically. According to the regression statistics of 39 reinforced masonry buildings, Zhou et al. (2012) reported the mean and standard deviation of a being 0.09094 and 0.02784 respectively, with the corresponding values of b being 0.01395 and 0.00183. A normal distribution is adopted for a and b when performing the Monte Carlo simulation in this work.

In this study, the experimental data of 135 reinforced masonry walls (Liu et al. 1981, Yan 1985, Shi and Yi 2000, Yang et al. 2000, Wang et al. 2003, Yu 2003, Wang et al. 2004, Ye et al. 2004, Zhou 2004, Zhang 2005, Huang 2006, Sun et al. 2006, Zhou et al. 2006, Zhang 2007a, 2007b, Gong 2008, Hao et al. 2008, Yang 2008a, Yang et al. 2008b, Fang 2009, Han 2009, Gu et al. 2010, Weng 2010, Zhang 2010, Zheng 2010, Liu et al. 2011, Wu et al. 2012a, Wu 2012b, Xiao et al. 2012, Guo et al. 2014, Wang et al. 2014, Zhang 2014) are adopted to obtain the regression results of the yield overstrength ratio W_y, the peak overstrength ratio W_p, the IDR at the peak point d_p, and the IDR at the softening point d_s. A lognormal distribution is used for these parameters. Note that the yield overstrength ratio W_y and the peak overstrength ratio W_p should be larger than 1. In consequence,  and  are used for regression. Combined with the parameters a and b, the distributions of these six parameters are presented in Eq. (4). More details can be found in Appendices A and B.

2.2 Test of regression results

To validate the reliability of the regression results, a test of the assumed distributions given in Eq. (4) is necessary. As the distributions of a and b are taken from Zhou et al. (2012), only four parameters (W_y, W_p, d_p, and d_s) that are obtained from the regression statistics of the experimental data of 135 reinforced masonry walls, are tested herein, through the widely-used hypothesis test method, i.e., the Kolmogorov-Smirnov test (K-S test) (Massey 1951, Yi 1996). The K-S test results of the four parameters are shown in Table 1.

Table 1 Results of K-S test

in which, a denotes the significance level; n refers to the sample size; D_n is the absolute difference between the distribution of regression results and that of the actual samples; and D(n, a) is the threshold of the distribution difference under the significance level of a. A higher significance level will lead to a stricter threshold.

It should be noted that the four parameters (W_y, W_p, d_p, and d_s) may not be obtained simultaneously by each test, therefore the sample sizes n and the corresponding threshold values D(n, a) are not identical for different parameters. Table 1 indicates that the regression results are reliable even under higher significance level.

2.3 Determination of damage states

In order to predict the level of seismic damage to buildings, it is also necessary to set different criteria for different damage states. According to the ¡°Classification of Earthquake Damage to Buildings and Special Structures¡± (SAC 2009) in China and the Hazus report (FEMA 2012a), the seismic damage states of buildings can be described as: None, Slight, Moderate, Extensive, and Complete.

Based on the work by Xiong et al. (2017), with respect to reinforced masonry buildings, the initial cracking point, V_{initialcrack,i}, and the yield point, V_yield,i, can be used as the criteria for ¡°Slight damage¡± and ¡°Moderate damage¡± as shown in Fig. 2(a); the displacement-based criteria for ¡°Extensive damage¡±, d_extensive, and ¡°Complete damage¡±, d_complete, proposed in the Hazus report (FEMA 2012a) can be used to determine the corresponding damage states of reinforced masonry buildings. This implies that, the criteria recommended by Xiong et al. (2017) for ¡°Slight damage¡± and ¡°Moderate damage¡± can be changed simultaneously with different backbone curves of buildings. But the criteria for ¡°Extensive damage¡± and ¡°Complete damage¡± are assumed to be deterministic (i.e., constant values) for the same type of buildings according to the Hazus report (FEMA 2012a), which means that these criteria will not be influenced by the parameters mentioned in Eq. (4). Note that, such deterministic criteria can be problematic for carrying out the sensitivity analysis in this work. Especially, when the Monte Carlo simulation is performed, the random change of backbone curves will lead to unreasonable analysis results. For example, the randomly generated zero-resistance point (i.e., Point E in Fig. 2(b)) may be smaller than the criterion for ¡°Complete damage¡±, or even the criterion for ¡°Extensive damage¡±, which is obviously irrational. Therefore, the deterministic criteria for ¡°Extensive damage¡± and ¡°Complete damage¡± of Xiong et al. (2017) must be modified in this study.

With respect to the state of ¡°Complete damage¡±, many researchers (Ibarra and Krawinkler 2004, Ibarra et al. 2005, Haselton et al. 2008, Del Gaudio et al. 2016) recommended the zero-resistance point (Point E in Fig. 2(b)) as the criterion point. Therefore, the zero-resistance point, which varies simultaneously with the backbone curve, is adopted herein to represent the ¡°Complete damage¡± point.

Fig. 2 Determination of damage state criteria

In order to develop a more rational criterion for ¡°Extensive damage¡±, the criteria proposed in the Hazus report (FEMA 2012a) are studied, firstly. Through the analysis of different damage state criteria for reinforced masonry buildings designed to different seismic design codes (pre-code, low-code, moderate-code, high-code) recommended in the Hazus report (FEMA 2012a), the interrelationship among the criteria for ¡°Moderate damage¡±, ¡°Extensive damage¡± and ¡°Complete damage¡± can be approximately expressed in Eq. (5), where q is very close to a deterministic value (with a mean value of 0.259 and a standard deviation of 0.001).

Furthermore, if the criterion for ¡°Extensive damage¡± proposed in the Hazus report (FEMA 2012a) is plot in Fig. 2(b), it will be located between Point C (i.e., peak strength point) and Point E (i.e., ¡°Complete Damage¡± point). Thus, similar to Eq. (5), an interrelationship among the ¡°Extensive damage¡± point, the peak point, and the ¡°Complete damage¡± point, can be developed in the form of Eq. (6),

Eq. (6) performs better than Eq. (5) for the Monte Carlo simulation, because the criterion for ¡°Extensive damage¡± given in Eq. (6) is always located between Point C and Point E regardless of any randomness of the parameters, which strictly follows the fundamental definition of the criterion for ¡°Extensive damage¡±.

When the damage criteria given in Eq. (6) are represented by the corresponding definitions in Fig. 2(b), the value of g  can be given as:

Substituting the mean values in Eq. (4) into Eq. (7), the value of g equals 0.214. Thus, the criterion for ¡°Extensive damage¡± used in this work is defined as Eq. (8), which varies simultaneously with different backbone curves.

Note that the d_extensive predicted by Eq. (8) using the mean values in Eq. (4) is very close to the criterion proposed by Hazus report (d_{Hazus,extensive}) (FEMA 2012a), which further validates the rationality of this method.

3 The FOSM method

For completeness of presentation, a brief introduction to the FOSM method is given in this section. Assume that the random variable X has a mean of m_X and a covariance matrix of S_X, as shown in Eqs. (9) to (11).

Suppose y is a function of X, as given in Eq. (12). If the derivatives of f(X) with respect to X exist, the first-order approximation of f(X) using Taylor series expansion evaluated at m_X can be expressed in Eq. (13). This gives the approximations of the mean and standard deviation value of y, presented in Eqs. (14) and (15).

where

In this study, the gradients of f are numerically obtained using the finite difference approach as shown in Eq. (17)

where

Note that the random variable X is taken as Eq. (19).

The incremental dynamic analysis (IDA) is performed by using the 22 far-field ground motion records recommended by FEMA P695 (FEMA 2009), and the ground motion intensity is recorded when a certain damage state (i.e. None, Slight, Moderate, Extensive, and Complete damage) is reached for the first time. Previous research (FEMA 2012a, 2012b) demonstrated that, the fragility function of structures under earthquakes can be assumed to be a lognormal distribution. Therefore, for an individual building, the variable z is defined as the mean ground motion intensity when a certain damage state is reached, and y is defined as ln(z). For regional buildings, the mean fragility curves are obtained by evaluating the mean probability of exceeding each damage state for all buildings under a certain level of ground motion intensity. Note that this fragility curve is essentially not in a precise lognormal distribution form. Nevertheless the lognormal distribution can still be used as an approximation of this curve, which will be illustrated in Section 5.3. For simplicity, the mean value of y for each individual building is also used as an approximation of the mean value of the distribution.

When no parameter uncertainty is considered, the record-to-record fragility function of structures under earthquakes usually follows a lognormal distribution with a standard deviation of b_RTR (FEMA 2009). When the abovementioned FOSM method is employed, the standard deviation contributed by parameter uncertainty, b_MDL, can be calculated. Then the total standard deviation of the new lognormal distribution, b_TOT, can be obtained as Eq. (20) (FEMA 2009).

In this study, the peak ground acceleration (PGA) is used as the intensity measure of the ground motion, instead of the response spectrum of fundamental period, S_a(T₁), due to the following reasons:

(1)     The fundamental period of buildings is a random variable considered in this study, and the adoption of S_a(T₁) will further complicate the analysis;

(2)     When regional buildings are considered, the THA needs to be performed for hundreds of buildings with different fundamental periods. Thus the use of the same S_a(T₁) level is not appropriate;

(3)     Many related studies were also performed based on the PGA (Xiong et al. 2017, Xu et al. 2014, Zeng et al. 2016). Therefore, using the PGA in this study enables the current analysis outcomes to be easily compared to those of the existing studies;

(4)     As the PGA is adopted in the Chinese code as the intensity measure for design of building structures (MOHURD 2010), the analysis outcomes based on the PGA can be better compared against the relevant contents in the code.

4 The Monte Carlo method

The Monte Carlo method is a powerful and widely used method in sensitivity analysis (Lee and Mosalam 2005, Fellin et al. 2010, Shin and Kim 2014). In this study, the conditional distribution of random variables will be firstly introduced, and subsequently the sample size of the Monte Carlo simulation will be determined.

4.1 Conditional distribution of random variables

As mentioned in Section 2, six random variables (i.e., W_y, W_p, d_p, d_s, a, and b) are considered herein. These random variables, theoretically, are not independent from each other, and their interactions are still unknown. Thus, the conditional distribution of these variables should be introduced here to take the interdependency into consideration (Fellin et al. 2010, Galasso et al. 2014, Galasso et al. 2015)..

To calculate the conditional distribution of the random variables, the joint distribution form of these variables should be determined. Considering the distribution form of each variable and the simplification of the analysis, the following two assumptions are adopted:

(1)   The joint distribution of the six random variables in Eq. (19) is assumed to be a 6-dimensional normal distribution, as shown in Eq. (21), and the covariance matrix is calculated based on the collected experimental data discussed before.

(2)     Due to the different sources of data, the variables a and b are thought to be independent from any other variables, hence the covariance values corresponding to a and b are 0 as shown in Eq. (22).

With the knowledge of multivariate statistical analysis, the following theorem can be used herein (Anderson 1984, He 2008, Kalpić and Hlupić 2011):

If X is a p-dimensional random variable, and follows a p-dimensional normal distribution, with a mean of m and a covariance matrix of S (i.e. , , ), then X, m, S can be decomposed as follows:

where the size of X⁽¹⁾ and m⁽¹⁾ is q¡Á1, and that of S₁₁ is q¡Áq. Then the conditional distribution of X⁽¹⁾ given X⁽²⁾ is a normal distribution with a mean of m_1•2 and a covariance matrix of S_11•2 as follows:

where

Based on this theorem, the conditional distribution of random variables can be obtained when other variables are given, and further adopted in the Monte Carlo sampling.

4.2 Monte Carlo sample size

To ensure the accuracy of the Monte Carlo simulation, sufficient samples are required. Fig. 3 shows the typical results of normalized mean and standard deviation obtained from the analysis presented in Section 5. The curves represent the results of RM-1 (a 1-story reinforced masonry building in Section 5.1) with respect to the parameter d_s under the ¡°Complete damage¡± state. When the sample size exceeds 500, the results begin to remain stable. Note that through a number of numerical studies, the convergence rate of the results shown in Fig. 3 is found to be the slowest among all different cases of parameters and damage states. Thus, the sample size of 500 is adopted in this study.

Fig. 3 Determination of Monte Carlo sample size

5 Case study

5.1 Overview

A total of 199 reinforced masonry buildings located at Tsinghua University campus are considered in this study. According to the Chinese ¡°Code for seismic design of buildings¡± (MOHURD 2010), these buildings have a 8-degree seismic design intensity (with the PGA of 0.20g at the 10% probability of exceedance in 50-year hazard level), with the site condition of Class ¢ò specified in the Chinese code, which approximately corresponds to site classes C and D in ASCE 7-10 of the United States (Luo and Wang 2012, ASCE 2010). Fig. 4 illustrates the percentage of different story numbers and construction years. Table 2 provides the mean of story numbers, story heights, and floor areas of the buildings. In this study, three typical buildings of 1-story, 3-story, and 6-story (denoted as RM-1, RM-3, and RM-6) are analyzed before the regional seismic damage prediction. During the analysis, the parameters of different buildings are considered to be independent from each other. In addition, the interaction between different buildings is not considered for simplification.

Fig. 4 Composition of regional reinforced masonry buildings

Table 2 Mean of typical parameters of regional reinforced masonry buildings

5.2 Sensitivity analysis results of individual reinforced masonry building

Tornado diagram is a widely used tool to illustrate the sensitivity analysis results (Porter et al. 2002, Lee and Mosalam 2005, Na et al. 2008, Shin and Kim 2014). Figs. 5 to 7 show the sensitivity analysis results obtained for RM-1, RM-3, and RM-6, considering 4 damage states. The gray vertical line in each figure denotes m_y defined in Eq. (14). The blue solid lines denote the results obtained using the FOSM method, and the red lines symbolize the results of the Monte Carlo simulation. Here, the hollow square refers to , and the solid dot represents , which are expressed in Eq. (18). For the Monte Carlo simulation outcomes, the difference between the left/right triangles and the middle one indicates the standard deviation of the results. Furthermore, ¡°all-C¡± implies that all the random variables change simultaneously with the consideration of the conditional distribution; ¡°all-N¡± denotes the simultaneous change of all the random variables without the consideration of the conditional distribution; ¡°T₁¡± stands for the results being obtained when a and b are changed simultaneously; other symbols denote the results obtained when only the specific variable is varied during the analysis.

Fig. 5 Tornado diagrams of the uncertainty analysis results for RM-1

Fig. 6 Tornado diagrams of the uncertainty analysis results for RM-3

Fig. 7 Tornado diagrams of the uncertainty analysis results for RM-6

From the results displayed in Figs. 5 to 7, the following observations with regard to the seismic damage sensitivity of reinforced masonry buildings can be drawn:

(1)     The relative sensitivity of each parameter is similar for buildings with different story numbers;

(2)     The yield overstrength ratio W_y is important for the ¡°Slight¡±, ¡°Moderate¡± and ¡°Extensive¡± damage states, and its influence cannot be neglected even for the ¡°Complete¡± damage state;

(3)     The peak overstrength ratio W_p is important for the ¡°Extensive¡± damage state, but has little or no influence to the ¡°Slight¡± and ¡°Moderate¡± damage states, and its influence to the ¡°Complete¡± damage state is small;

(4)     The IDR at the peak point d_p is important for the ¡°Complete¡± damage state, whereas no influence is found to the ¡°Slight¡± and ¡°Moderate¡± damage states;

(5)     The IDR at the softening point d_s is an important parameter for the ¡°Extensive¡± and ¡°Complete¡± damage states; however no influence is observed to the ¡°Slight¡± and ¡°Moderate¡± damage states;

(6)     The parameters for estimating the fundamental periods, a and b, have a minor influence to all the damage states;

(7)     Based on the observations provided above, the accuracy of the individual building analysis can be further improved by using more accurate critical parameters which have greater influence on the seismic performance of the individual building.

Based on the results obtained from the FOSM method and the Monte Carlo simulation, the probability density curves or the fragility curves can also be determined as shown in Figs. 8 to 10. Note that the density curves can illustrate the sensitivity results more clearly than the cumulative fragility curves. In these figures, ¡°IDA¡± denotes the results obtained when only the uncertainty of ground motions is considered; ¡°FOSM¡± denotes the results due to the FOSM method; ¡°MCS-C¡± refers to the results of the Monte Carlo simulation considering the conditional distribution; and ¡°MCS-N¡± represents the results of the Monte Carlo simulation without consideration of the conditional distribution.

Fig. 8 Probability density results for RM-1

Fig. 9 Probability density results for RM-3

Fig. 10 Probability density results for RM-6

Results shown in Figs.8-10 lead to the following additional observations:

(1)     The FOSM method, with significantly reduced workload, can produce similar results as the Monte Carlo simulation;

(2)     When all the random variables change simultaneously in the Monte Carlo simulation, a larger value of m_y is obtained compared to the assumed value of m_y used for the FOSM method, suggesting that the assumption for m_y used in the FOSM method is conservative;

(3)     The consideration of conditional distribution yields a smaller standard deviation of results, especially for the ¡°Complete¡± damage state, and is closer to the results obtained from the FOSM method.

(4)     When analyzing the seismic performance of an individual building, the influence of uncertainty of model parameters is significant and cannot be neglected.

5.3 Sensitivity analysis results of regional reinforced masonry buildings

With respect to the regional buildings, the mean fragility curves (or the probability density curves) are obtained by evaluating the mean probability (or probability density) of exceeding each damage state for all the buildings under a certain level of PGA, as shown in Fig. 11 (the solid lines are the fragility curves, and the dashed lines are corresponding probability density curves). Note that these fragility curves are not in a precise lognormal distribution form. Notwithstanding, the lognormal distribution can still be used as an approximation of these curves. Similar to Section 5.2, the tornado diagram results are illustrated in Fig. 12.

Fig. 11 Mean fragility curves of regional buildings

Fig. 12 Tornado diagram of results for regional reinforced masonry buildings

Comparing the results presented in Figs. 5 to 7 and Fig. 12, it can be found that the sensitivity relationship of each parameter for the regional buildings is similar to the results of individual buildings. The influence by parameter uncertainty of the regional buildings, however, becomes much smaller than that of individual buildings.

To quantify the relationship of sensitivity between the regional building analysis and the individual building analysis, the ratios of b_MDL,RM-Urban (the standard deviation of regional analysis induced by parameter uncertainty) over b_MDL,RM-i (the standard deviation of individual building analysis induced by parameter uncertainty) for each case are provided in Fig. 13.

Fig. 13 Comparison of sensitivity between regional and individual building analyses

In Fig. 13, each point represents the ratio obtained only when the specific variable (denoted along the horizontal axis) changes during the analysis. The sensitivity of the regional buildings decreases when the number of buildings (denoted as n_b) increases, and it approaches approximately  of the corresponding sensitivity of an individual building. Note that, if a group of variables follows independent and identical normal distribution of N(m, s), then the mean of these variables will follow a normal distribution of N(m, s/ ). Therefore the value of  can be mathematically obtained when the region consists of n_b independent and identical buildings. This is particularly true when all the parameters are changing concurrently (i.e. ¡°all-N¡± and ¡°all-C¡± in Fig. 13), which may reduce the dispersion contributed by each parameter alone.

Fig. 14 Probability density results for regional reinforced masonry buildings

The probability density curves considering the model parameter uncertainty are illustrated in Fig. 14, from which several observations can be obtained:

(1)     The influence of uncertainty of model parameters for the regional seismic damage prediction is very small.

(2)     The FOSM method, with significantly reduced workload, can achieve similar results as the Monte Carlo simulation for the regional seismic damage prediction. Therefore, the FOSM method can be used as an approximation of the Monte Carlo simulation;

(3)     The assumption for m_y used in the FOSM method is conservative compared to the m_y adopted in the Monte Carlo simulation;

(4)     The consideration of conditional distribution can lead to a smaller standard deviation, especially for the ¡°Complete¡± damage state, and is closer to the results obtained by the FOSM method.

6 Conclusions

The FOSM and Monte Carlo methods are adopted in this study to perform the parametric sensitivity study for the regional seismic damage prediction of reinforced masonry buildings. Several important conclusions are drawn as follows:

(1)   The relative sensitivity of each parameter is similar for individual building and regional building analyses;

(2)   The uncertainty of model parameters should be considered for the seismic damage prediction of an individual building. This is however not necessary for regional buildings, when the interaction between buildings is not considered and the parameters of different buildings are assumed to be independent from each other;

(3)   The conditional distribution is necessary to be considered for the Monte Carlo simulation which reduces the variance of the results;

(4)   The FOSM method, with significantly reduced workload, can produce similar results as the Monte Carlo simulation, and can therefore be used as an approximation of the Monte Carlo simulation.

The seismic damage prediction method used herein can also be implemented to other types of structures. It is worth noting that certain limitations still exist in this study, such as the ignorance of the interaction of different buildings with respect to their dynamic responses. Such dynamic effects will be investigated by further developing an improved computation model to simulate the interactions between the buildings and the sites.

Acknowledgement

The authors would like to acknowledge the financial supports of the National Natural Science Foundation of China (No. 51578320, 51378299) and the National Key Technology R&D Program (No. 2015BAK14B02). The authors would like also to acknowledge Professor Quanwang Li and Mr. Xiang Zeng for their contributions to this work.

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Appendix A Typical parameters of reinforced masonry test specimens considered in this study

Appendix B Parameters obtained for reinforced masonry test specimens considered in this study

*This value is based on the average value provided by Shi and Yi (2000).