Parametric sensitivity study on regional seismic damage prediction of reinforced masonry buildings based on time-history analysis Xinzheng Lu
## AbstractRegional seismic damage prediction is an important approach to discover the weakness of a city so as to effectively mitigate seismic losses. A major proportion of regional seismic losses is caused by masonry buildings. As a result, an accurate prediction of the regional seismic damage to masonry buildings has significant engineering and scientific values. Various parameters of the computational models for regional seismic damage predictions usually involve considerable uncertainty, especially for masonry buildings. Therefore, a parametric sensitivity analysis for the regional seismic damage prediction of reinforced masonry buildings is performed in some detail in this study. Damage to this kind of buildings is predicted through nonlinear time-history analysis (THA) using the multiple-degree-of-freedom (MDOF) shear model, which can better represent the features of different buildings and ground motions. In the sensitivity analysis, two widely used methods, the first-order second-moment (FOSM) method and the Monte Carlo method, are adopted and their prediction results are compared. The outcomes of this study indicate that the uncertainty of parameters has a small influence on the analysis results when the total number of regional buildings is large. However the uncertainty cannot be neglected for individual building analysis. In addition, the FOSM method, which is more time-saving, can achieve a similar level of prediction as the Monte Carlo simulation. ## KeywordsMasonry structure; Sensitivity analysis; Parameter uncertainty; Regional seismic damage prediction; Multiple-degree-of-freedom shear model; Time-history analysis |

## 1 IntroductionDestructive 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 parametersPublished 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), (1)
Calculate
the fundamental period (2)
Calculate
the actual yield strength of each story,
(3)
Calculate
the peak strength of each story,
(4)
Calculate
the strength at the softening point, (5)
Calculate
the IDR at the yield point, (6)
Determine
the IDR at the peak point,
Fig. 1 The MDOF shear model According to the procedure given
above, five variables must be determined. They include the fundamental period
This equation only requires the
total building height 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
## 2.2 Test of regression resultsTo validate the reliability of
the regression results, a test of the assumed distributions given in Eq. (4)
is necessary. As the distributions of Table 1 Results of
in
which, 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 ( ## 2.3 Determination of damage statesIn 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 ¡° Based on the work by Xiong et
al. (2017), with respect to reinforced masonry buildings, the initial cracking
point, 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
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
Substituting the mean values in Eq. (4) into Eq. (7), the
value of
Note that the |

## 3 The FOSM methodFor completeness of presentation,
a brief introduction to the FOSM method is given in this section. Assume that
the random variable S, as shown in Eqs. (9) to (11)._{X}
Suppose y, presented in Eqs. (14) and (15).
where
In this study, the gradients of
where
Note that the random variable
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 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, can be calculated. Then the
total standard deviation of the new lognormal distribution, _{MDL}b, can be obtained as Eq. (20) (FEMA 2009)._{TOT}
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, (1)
The fundamental
period of buildings is a random variable considered in this study, and the
adoption of (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
(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 methodThe 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 variablesAs mentioned in Section 2, six
random variables (i.e., 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
With the knowledge of multivariate statistical analysis, the following theorem can be used herein (Anderson 1984, He 2008, Kalpić and Hlupić 2011): If
where
the size of
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 sizeTo 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
Fig. 3 Determination of Monte Carlo sample size |

## 5 Case study## 5.1 OverviewA total of 199 reinforced masonry
buildings located at Tsinghua University campus are considered in this study.
According to the Chinese ¡°
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 buildingTornado 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 _{
} , 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; ¡°T_{1}¡± 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 (3) The peak overstrength ratio (4) The IDR at the peak point (5) The IDR at the softening point
(6) The parameters for estimating
the fundamental periods, (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 used for the FOSM method, suggesting that
the assumption for _{y}m used in the FOSM method is conservative;_{y}(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 buildingsWith 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
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
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 adopted in the Monte Carlo simulation;_{y}(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 ConclusionsThe 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. 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(in Chinese) Zhou Y, Shi W, Han R (2012) Vibration test and analysis of the fundamental period of multi-story masonry structures with large-bay. Engineering Mechanics 29(11):197-204. (in Chinese) ## 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). |