Vulnerability Analysis of Ancient Timber Architecture by Considering the Correlation of Different Failure Modes

The paper proposes an innovative method of analyzing the seismic fragility of ancient timber architecture.Themethod is based on the Copulas, in which correlation between different failuremodes is considered.Thismethod is applied to assess the vulnerability of ancient timber architecture in Ming dynasty. The assessment includes four steps. In the first step, the incremental dynamic analysis is employed to establish seismic vulnerability curves of different failuremodes for the structure.After that, Copula joint distribution function is used to analyze the correlation among different failure modes. In the third step, fragility curves considering correlation among different failure modes are established. In the last step, the fragility curves are compared with those obtained by first-order bound method.The results show that seismic vulnerability of ancient timber architecture based on the Copulas is greater than that of any single failure modes. Moreover, the occurrence probability ranges between the upper and lower boundaries of the first-order bound method, but is close to the lower boundary.


Introduction
Many buildings of ancient timber architecture are listed as cultural heritage sites for their historical, artistic, and cultural value.Carved beams, painted rafters, painting, and colored drawings are valuable parts of ancient timber architecture [1].However, according to [2][3][4] many precious ancient buildings have been damaged and even destroyed in earthquakes alone, causing great economic and cultural loss.Therefore, it is essential to perform a seismic fragility analysis for these structures.
Seismic fragility analysis is a quantitative method that has been widely used in antiseismic analysis of buildings.Fragility curves describe seismic response of buildings from the perspective of probability.Some researchers [5,6] used this method to analyze vulnerability of RC structures.Macabuag et al. [7] built tsunami fragility curves for buildings.A simplified model of ancient timber architecture was built by Kouris et al. [8].Seismic fragility curves of the architecture were established by nonlinear finite element analysis.The analysis showed that the traditional ancient architecture had good deformation and seismic capacities.Faye et al. [9] established seismic fragility curves of a traditional timberframe wall by nonlinear spectrum analysis, and it showed the wall had a good mechanical property.Ahmad et al. [10] analyzed the vulnerability of three traditional timber-frame masonry structures, and the results showed that this kind of structure was suitable in the limit of peak ground acceleration (PGA) < 0.7 g.
Interstory drift ratio of building is usually selected as the only seismic demand parameter to build seismic fragility curves.Under this situation, this method could examine only one failure mode.Given the complex structure and mechanical properties [11], ancient timber architecture could have more than one failure model.Therefore, different seismic demand parameters [12] should be adopted to construct the seismic fragility curves.In the seismic vulnerability analysis, effects caused by correlation among different failure modes on structure vulnerability need to be considered.
Sklar's theorem [13] is the theoretical foundation for the Copula usage, which can describe the linear and nonlinear correlation between different failure modes.Copulas have been widely used in a variety of applications.Fengler and Okhrin [14] used Copulas to capture salient features of multivariate dependence for risk-management purposes.Peng et al. [15] used Copulas to improve flood risk for confluence flooding control downstream of a reservoir.Yang et al. [16] applied Copulas to assess fatigue reliability of Yangtze River Bridge.
In this study, a vulnerability analysis model of Chinese traditional timber architecture is built.Based on Gaussian Copula, t Copula, and Frank Copula, the correlation between different failure modes is taken into account.The model is applied to evaluate the vulnerability of Chinese ancient timber architecture, so as to provide a reference for practical engineering.

Definition of Seismic Fragility of Ancient Timber Architecture
Seismic fragility of an ancient timber architecture is the occurrence probability of a certain state of the structure under different forcing intensities.The fragility is defined as a probability of the structural seismic demand   exceeding capacity   under the condition of a given level of intensity measure of the hazard [17].It is given by the following.
It is generally assumed that the structural demand parameter follows the lognormal distribution [18] and   = ln( S ,   ),   = ln( S ,   ).S is the average ductility, S is the average value of the seismic demand,   is the logarithmic standard deviation of   , and   is the logarithmic standard deviation of   .
Cornell et al. [19] revealed that there is an exponential relation between S and ground motion parameter , which can be expressed as where  is a ground motion parameter;  and  are constant terms.
Taking logarithms on both sides of (2), we get the following.
The fragility curve for a specific state of damage is as follows: where Φ[•] is the standardized normal distribution.The fragility curves under different failure modes can be constructed by (4).
In this study, a new advanced methodology for obtaining the analytical fragility curves of ancient timber architecture is proposed.The method is based on the Copulas, in which correlation between different failure modes is considered.

Failure Modes of Ancient Timber
Architecture Based on Copulas
Rank correlations are Copula-based dependence measures, and the Kendall rank correlation coefficient can be taken as a measure of concordance for bivariate random variables.In this study, Kendall rank correlation coefficient  was used to calculate the Copula parameter .The study carried out by Nelsen [20] shows that relation between  and  is as follows.

Joint Bivariate Collapse Probability Using Copula Approach.
In a bivariate series system, if  1 and  2 are the structural performance functions of ancient timber architecture for different collapse modes, then, the probability of the simultaneous occurrence of two collapse modes is given as follows.
The collapse probability of the bivariate series system is given by where  1 ,  2 are the collapse probabilities of two collapse modes, respectively.
t Copula, Gaussian Copula, and Frank Copula are used to compute the fragility curves of the ancient timber structure architecture in this study, which are as follows: (1) Bivariate t Copula (2) Bivariate Gaussian Copula (3) Bivariate Frank Copula where  −1  is the inverse function of the  distribution with degree and Φ −1 is the inverse function of the standardized normal distribution function.

Seismic Fragility Analysis Process of Ancient Timber Architecture Based on Copulas
A new methodology for obtaining fragility curves of ancient timber architecture is proposed in this study.This method is based on Copulas, in which effect of correlation between different failure modes on the vulnerability of the whole structure is considered.The process of analysis is as follows: First, two different seismic demand parameters are selected.Then, incremental dynamic analysis is employed to establish seismic vulnerability curves of different failure modes for the structure.Next, Copula parameters are calculated by (9).Forth, Copula joint distribution function is used to analyze the correlation among different failure modes.Finally, fragility curves considering correlation among different failure modes are established.

Description of the Architecture.
The case in study is the rear hall of Huashi Fire God Temple which is located in Chongwenmen, Dongcheng District, Beijing.It was built during the rule of the Ming Dynasty and has a history of almost 450 years.The structure was damaged over the years and was renovated during the reign of Emperor Qianlong of the Qing dynasty.In 2004, the building was renovated again by the government.The total area of the Huashi Fire God Temple is 543.44  2 and that of the rear hall is 132.2  2 .The rear hall is a single storied structure with a post and lintel wood frame and a Yingshan Roof.The heights of eaves, ridge, and stylobate are 3.7 , 7.0 , and 0.7 , respectively.Details of the size of the building can be obtained from [21].Elevation and plane graphs of Huashi Fire God Temple are shown in Figures 1 and 2. Yin et al. [22] proposed a method to estimate the volume of historical timber-frame buildings in China.Referring to this research, an on-site test was carried out for the rear hall.The results show no obvious decay on the surface of timber components.There are local damages on wall.Some of the wood components are cracked and several of the plinths are damaged.The damaged appearance of the building is shown in Figure 3.

Ground Motion Selection.
A set of 100 ground motions are selected from the Pacific Earthquake Engineering Research (PEER) Center and used for earthquake input loading.According to the seismic fragility analysis carried out by Rota [23], Vafaei [24], and Hosseinpour [25], PGA, PGV, and Sa (T) can be used as indicators to describe earthquake intensity.Therefore, PGA is selected in this study as the indicator of the intensity of ground motion.

Finite Element Model of the Components of the Rear
Hall of the Huashi Fire God Temple.A three-dimensional finite element model is built by using the ANSYS software to evaluate the seismic global response of the rear hall of the Huashi Fire God Temple.Zhang [26] used beam element to simulate columns and beams of ancient timber buildings.So, in this paper beam element is used for the simulation of beams and columns, and spring element and shell element are selected for the simulation of mortise-tenon joints and roof of the ancient timber architecture.Reference [27] illustrated the roof structure of ancient timber architecture and offered bulk density of materials.Based on [27] and Chinese Code [28] load of the roof was calculated, which is 4030 / 2 .Study [29] showed that mechanical properties of historic timber structure would decrease along with time and environment.Mechanical properties of the materials are appropriately reduced in the process of simulation, because the building has been in service for hundreds of years.The simplified finite element model is shown in Figure 4.

Mechanical Definition of Damage States.
Wooden frames are main load-bearing system of ancient timber architecture.Mortise-tenon joints are unique connections between the components in ancient timber architecture.Seismic data [30], shaking table test carried out by Zhang et al. [31] and research carried out by Keita Ogawa [32] showed mortise-tenon joints are key points for ancient timber architectures.Li et al. [12] carried out a series of on-site test for Yingxian wood tower.They evaluated damages, bearing capacities, and reliability levels of the tower.And, a 1/10 scale shaking table test and nonlinear analysis of the wood tower were carried out.At last, a classification for seismic damaged levels and corresponding interstory drift ratio limitations based on shaking table test and nonlinear analysis was proposed in [12].The classification is shown in Table 1.Zhang et al. [33] carried out a series of cyclic loading test of timber frames with mortise-tenon joints.Stiffness, energy dissipation, and bearing capacities of the frames were studied.They also described the pull-out length of tenon at different rotation angle of mortise-tenon joints and damage state.They found that the bigger the pull-out length of tenon, the more serious the damage of the frames.
The study carried out by Institute of Earthquake Resistance and Disaster Reduction (IERDR) [34] proposed a damage classification levels for mortise-tenon joints.In the reference, damage levels are measured by pull-out length of tenon and damage degree.Based on [33,34], the connection between the damage states and cosine rotation angle of mortise-tenon joints is shown in Table 1.
According to the results of the incremental dynamic analysis, a linear regression analysis is carried out based on least square method.Then, seismic demand models for different failure modes are built (Table 2).Copula parameters  of different seismic demand parameters are used in this method to describe the correlations among different failure modes, which can be calculated by (9).For example, Table 3 shows the Gaussian Copula parameters of different failure modes.Table 3 shows the Gaussian Copula parameters range from 0.8 to 1.This indicates that the failure modes are not completely linearly correlated, but the correlation effect is significant.

Fragility Curves for Single Failure Modes.
By substituting the values given in Table 3 in (11)∼(13), the fragility curves for different failure modes are obtained and shown in Figure 5. √   2 +   2 = 1 as suggested by Chen [35]. Figure 5 shows the comparison of the fragility curves in X and Y directions for different seismic demand parameters.It is observed that the rear hall extends more in the X direction.In particular, the failure probability in four damage states of the rear hall with the interstory drift ratio as seismic parameter is obviously greater than that with the rotation angle of mortise-tenon as seismic parameter.This indicates that consideration of the rotation angle of the mortise-tenon as the seismic parameter to assess the vulnerability of ancient timber architecture might underestimate the vulnerability of the structure.

Fragility Curves with
where  and  constitute the number of failure modes.The upper and lower boundaries of the failure probability of different damage states of the rear hall of the Huashi Fire God Temple are obtained by using (15), which are shown in Figure 6.
It can be seen from Figure 6 that there is a marked deviation between the curves related to the upper and lower boundaries.The seismic vulnerability of the rear hall based on Gaussian Copula, Frank Copula, and t Copula lies between the curves related to the upper and lower boundaries of that based on the first-order bound method and is close to lower boundary.

Conclusions
This study investigates and analyzes the effect of correlation among the different failure modes on the seismic fragility of ancient timber architecture.A vulnerability assessment model for ancient timber architecture is established based on the finite element analysis theory and Copulas.In this model, different seismic demand parameters are adopted to obtain the fragility curves for different failure modes.This method takes into consideration the effect of correlation among the modes on the seismic fragility of buildings.This vulnerability assessment model is applied into a case study.The findings are as follows: (1) When rotation angle of mortise-tenon and interstory drift ratio are taken as the seismic parameters to study the damage rates of the rear hall, the occurrence probability is different.It is obvious that the

Figure 3 :
Figure 3: External damage of the building.
(a) Displacement in X with the excitation of EL Centro (b) Displacement in Y with the excitation of EL Centro

Table 1 :
Limit states of components of Huashi Fire God Temple.

Table 2 :
Seismic demand models of components of Huashi Fire God Temple.

Table 3 :
Gaussian Copula parameters.Δ  / and Δ  / are interstory drift ratio in X and Y direction, respectively.
.5.2.First-Order Bound Method.In order to verify the accuracy of the Copulas, first-order boundary method was carried out to obtain the fragility curves of the rear hall.The formula for the calculation is given by similar, and the occurrence probability is greater than any other failure modes.The highest vulnerability obtained is that calculated by Frank Copula function.With increasing PGA, the gap between the Copulas curves and others is increased.Therefore, when the earthquake intensity is high, the fragility curves based on a single seismic parameter might underestimate the vulnerability of ancient timber architecture.5max =1 [  ] ≤   ≤ 1 −  ∏ =1 [1 −   ]