Parameter determination and damage assessment for THA-based regional seismic damage prediction of multi-story buildings
Chen Xiong 1, Xinzheng Lu 1, *, Xuchuan Lin 2, 3, Zhen Xu 4, and Lieping Ye 1
1 Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Department of Civil Engineering, Tsinghua University, Beijing 100084, China
2 Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, China
3 Key Laboratory of Earthquake Engineering and Engineering Vibration, China Earthquake Administration, Harbin 150080, China
4 School of Civil and Environmental Engineering, University of Science and Technology Beijing, Beijing 100083, China
Regional seismic damage prediction based on multiple-degree-of-freedom shear model and nonlinear time-history analysis can comprehensively consider the characteristics of buildings and ground motions. Two major challenges of applying such methodology, namely (1) parameter determination and (2) damage assessment of buildings in urban scale, are addressed in this study. The reliability of the proposed methods are validated using the tests of three individual buildings and the observed seismic damages of Longtoushan Town in 2014 Ludian earthquakes of China. Finally, a regional seismic damage prediction is performed for a large urban region, which demonstrates the applicability and scalability of the proposed methods.
Keywords: Regional seismic damage prediction; Parameter determination; Damage assessment; Multiple-degree-of-freedom shear model; Time-history firstname.lastname@example.org
Modern cities are becoming integrated systems that consist of a high density of population and buildings. Once they are hit by earthquakes, the damage or collapse of buildings will result in huge economic losses and casualties. Being one of the major structural systems in cities, multi-story buildings occupy a large proportion of seismic damage and collapse in previous earthquakes, e.g., 1994 Northridge earthquake in the United States (US) [Trifunac and Todorovska, 1997], 1995 Kobe earthquake in Japan [Miyakoshi et al., 1997] and 2008 Wenchuan earthquake in China [Wang, 2008; Lu et al., 2012]. Therefore, an accurate and efficient regional seismic damage prediction method is required to assess the seismic damage of multi-story buildings in order to mitigate the earthquake disasters in modern cities.
Existing regional seismic damage prediction methods include the following: (1) the damage probability matrix method [ATC, 1985], (2) the capacity spectrum method [MAE, 2006; FEMA, 2012a] and (3) the time-history analysis (THA)-based method [Hori, 2006; Lu et al., 2014]. The damage probability matrix method is developed based on the statistical damage of different types of structures in previous earthquake events. However, this method is sometimes not reliable for some earthquakes (e.g., extremely strong earthquakes) or regions for which limited historical building damage statistics are available. The capacity spectrum method adopts single-degree-of-freedom (SDOF) building models and pushover analyses to predict seismic damage. The capacity spectrum method can well represent the global strength and ductility of buildings with moderate computational workload, and has been widely used in previous studies [Tantala et al., 2008; Remo and Pinter, 2012]. Nevertheless, this method cannot easily represent the concentration of damage to different stories (e.g., soft-story failure mode) and the time-domain properties of ground motions (e.g., the velocity impulse of ground motions) [Lu et al., 2014]. In contrast, the THA-based method adopts multi-degree-of-freedom (MDOF) building models and nonlinear THA to predict seismic damage, which can fully represent the time/spectral domain characteristics of ground motions and the nonlinear characteristics of buildings [Hori, 2006; Lu et al., 2014]. Thus, the THA-based method is more accurate in theory and can be a very good option for regional seismic damage prediction.
Despite its larger computational workload, the THA-based regional seismic damage prediction is feasible due to recent advances in computer science. For example, Yamashita et al.  performed the regional seismic damage prediction for Tokyo urban area using the THA-based regional seismic damage prediction based on a super computer. To avoid the high maintenance costs of super computers, Lu et al.  simulated a medium-sized urban area of 4000 buildings in China using the THA-based regional seismic damage prediction powered by Graphic Processing Units (GPUs). However, parameters determination and damage criteria of MDOF building models remain challenges when using the THA-based regional seismic damage prediction [Yamashita et al., 2011]. The MDOF building models require a nonlinear inter-story hysteretic model for each story of each building [Hori, 2006; Lu et al., 2014]. The seismic performances of buildings differ with structural type or year of construction. Because it is impractical to collect the drawings of each building in a city region, a nonlinear inter-story hysteretic model must be established based on macro-scale building attribute data, such as the structural height, structural type, year of construction etc. Lu et al.  proposed a method to generate the parameters and determine the damage states of MDOF building models based on the capacity curves and damage criteria proposed by Hazus [FEMA, 2012a]. However, the capacity curves in Hazus are based on the statistical results of buildings in the US, and hence cannot be easily applied to other regions (for example, China or Japan) where different building codes are used. Therefore, a more adaptive parameter determination methodology is required.
On August 3rd, 2014, a M6.5 earthquake occurred in Ludian County, China. Fortunately, the accelerograph at the epicenter, Longtoushan Town, recorded the ground motion of the main shock. In addition, reconnaissance teams collected the detailed seismic damage and attribute data of buildings in the Longtoushan Town [Lin et al., 2015]. These data can be used to validate the MDOF building models for regional seismic damage prediction.
Based on the above background, this work proposes a parameter determination method and the corresponding damage assessment method for the MDOF building models of widely used multi-story structures, including reinforced concrete (RC) frames, reinforced masonry (RM) structures and unreinforced masonry (URM) structures. The parameter determination method, based on a simulated design procedure and the statistics of extensive experimental and analytical results, can be easily applied to different regions with different design requirements. The damage assessment method adopts a combination of force and deformation-based criteria, which is superior to the conventional deformation-based criteria. The proposed parameter determination method is examined by comparing its results with the experimental results of three individual buildings. Subsequently, the seismic damage to Longtoushan Town is predicted using the THA-based regional seismic damage prediction with the proposed parameter determination and damage assessment method. Finally, the seismic damage to a large city in China is predicted using the proposed method, which demonstrates the scalability and feasibility of the method for large cities. This research will provide a reference for the seismic damage prediction of large urban areas.
The framework and methodologies of the THA-based regional seismic damage prediction are illustrated in Figure 1. The framework mainly consists of three modules (shown as the blue boxes in Figure 1): (1) Parameter determination: The building attribute data (e.g., year of construction, number of stories, height, design intensity, structural type, etc.) obtained from the geographic information system (GIS) are used at this stage to generate the MDOF model of each building and calibrate the parameters of the inter-story hysteretic models, through a simulated design procedure and extensive statistical analyses; (2) Nonlinear THA: Selected ground motions are input into the MDOF shear models to implement the THA, and the engineering demand parameters (EDPs) (e.g. inter-story drifts and peak floor accelerations) are generated; (3) Damage assessment: Based on the corresponding damage criteria of each structural type and the computed inter-story drifts, the damage states for different stories of each building are determined. Stages (1) and (3) will be presented in detail in this study, and a detailed description of Stage (2) can be found in the previous work of the authors [Lu et al., 2014].
FIGURE 1: The framework and methodologies of the THA-based regional seismic damage prediction
2.2. MDOF shear model
According to the “Chinese code for design of civil buildings” [MOHURD, 2005], multi-story buildings denote buildings less than 7 stories or 24 m. In China, most of such buildings are masonry structures or RC frame structures, which are widely used as residential or office buildings. Therefore, this work mainly focuses on these two types of multi-story buildings.
Considering the attribute data of each building is limited and the number of building in urban area is huge, the seismic response prediction model of buildings should be relatively simple. In this study, the MDOF shear model (as shown in Figure 2) is adopted. Existing researches proved that the MDOF shear model can well capture the nonlinear properties of multi-story buildings, predict the EDPs on each story and consider the damage concentration on different stories [Lu et al., 2014; Xu et al. 2014]. The MDOF shear model has the following assumptions:
(1) The model assumes that the mass of each story is concentrated on its elevation and represented by a mass point.
(2) The seismic responses of multi-story structures are dominated by the inter-story shear deformation.
(3) The model is most suitable for multi-story buildings with regular planar layout, which is the majority of buildings in urban areas. Special considerations are needed for buildings with unconventional irregular planar layouts.
(4) Because multi-story buildings usually have regular layout along the height, the stiffnesses and masses of different stories are assumed to be identical [Hori, 2006].
FIGURE 2: The MDOF shear model
A tri-linear backbone curve is used herein to simulate the inter-story nonlinear properties (Figure 2b) [FEMA, 2012a]. Many previous studies showed that the tri-linear backbone curve model can accurately represent the inter-story behavior of a structure [Vamvatsikos and Cornell, 2005; Shi et al. 2014] with acceptable modeling complexity and computational accuracy. Due to limited amount of detailed building information, a single parameter hysteretic model proposed by Steelman and Hajjar  is adopted herein (Figure 2c).
3. Parameter determination
The parameter determination method includes the determination of elastic parameters, backbone curve parameters and hysteretic parameter, as illustrated in Figure 3.
FIGURE 3: Schematic view of the parameter determination method
Due to the limited information of the building attribute data, the simulated design procedure and statistical analyses are used. Specifically, the elastic parameters, design strengths and hysteretic parameters are determined based on the simulated design procedures. The overstrength factors (i.e. the ratios of yield/peak strengths to the design strength) and deformation parameters are determined according to the statistics of extensive experimental and analytical results (Figure 3).
Although the proposed parameter determination method is conducted for the multi-story buildings of China, the methodology can be easily applied to other regions using the local design codes and statistical data. Thus, it is more adaptive than the previous work of Lu et al. , which fully replies on the Hazus data.
As illustrated in Section 2.2 the stiffness and mass of each story are assumed to be constant along the height [Hori, 2006]. The elastic parameters of a building can be represented by the inter-story shear stiffness, k0, and the mass, m, of each story. Equations (1) and (2) show the global stiffness and mass matrices of a MDOF model using k0 and m [Lu et al., 2014].
The mass of each story, m, in Equation (2) can be determined based on the area of each story, A1, and the mass per unit area, m1 (Equation (3)) [Sobhaninejad et al., 2011]; m1 can be estimated according to the occupancy of each story.
The relationship between the stiffness, mass and first vibration period, T1, can be expressed using Equation (4), by which k0 can be determined.
where [Φ1] is the first mode vector. It is independent of the value of k0 and can be computed using the generalized eigenvalue analysis.
As shown in Equation (4), m and T1 are required to obtain k0. The vibration periods of different types of structures can be estimated using empirical equations. For example, the fundamental period of an RC frame can be calculated using the empirical equation (Equation (5)) specified in the Chinese Code [MOHURD, 2012], which is proposed based on the field test of 160 RC frames.
where H and B are the height and width of a RC frame, respectively, both of which can be obtained from the GIS data. For a structure whose width differs along its two directions, the two translational vibration periods should be determined according to the corresponding width.
Similarly, Zhou et al.  performed the field tests of 110 masonry structures in China (which is the largest data set in literature) and proposed their empirical equations. The fundamental periods of URM and RM structures can be obtained using Equations (6) and (7), respectively.
for URM structures (6)
for RM structures (7)
3.2. Determination of the backbone curve parameters for RC frames
The tri-linear backbone curve (Figure 2b) features three key points: (1) the yield point, which is the turning point between the linear behavior and the nonlinear behavior and after which the stiffness is significantly reduced; (2) the peak point, which is the point where the peak strength is reached; and (3) the ultimate point, after which the story is deemed collapsed or completely damaged. The determination of strength and deformation parameters of each key point will be discussed in the following sections.
3.2.1 Strength parameters
(a) Design strength
Consider that RC frames are usually engineering designed, the design strength of which can be estimated according to the simulated design procedures with small uncertainty and therefore be adopted as the basis to determine other strengths.
Because the seismic responses of multi-story buildings are usually dominated by their first vibration modes, an equivalent lateral force analysis can be used to calculate the design shear force, Vdesign, i, of each story, where i is the story number [ASCE, 2010; MOHURD, 2010a].
(b) Yield strength
The actual yield strength of different stories, Vyield, i, can be calculated using the equation below:
where W1 is the yield overstrength ratio of RC frames. One of the main causes of yield overstrength is the design redundancy of material [FEMA, 2012a]. The yield strengths of RC frames are more sensitive to the strength of steel reinforcement than the strength of concrete. Consequently, W1 is determined according to the partial factor of steel reinforcement, i.e., W1 = gs = 1.1 [MOHURD, 2010b]. Note that this value is also adopted in the Hazus report [FEMA, 2012a] for RC frames.
(c) Peak strength
Considering the hardening of reinforcement and concrete, the variation of neutral axis and the influences of construction measures, the peak strengths of structures should exceed the yield strengths to certain extent. Therefore, the peak strength, Vpeak, i, can be calculated with a peak overstrength ratio, W2, as follows:
To determine W2, the statistical analysis is performed by collecting 155 pushover results of RC frames designed following the Chinese seismic design code [Liu, 2006; Li, 2006; Zhai and Xie, 2007; Zhao, 2008; Zhang, 2009; Li, 2013; Shi et al., 2014]. These collected RC frames are dominated by soft-story failure modes, which agree with the actual failure modes of Chinese RC frames in earthquakes. Based on the pushover results, the relationship between W2 and the building properties (i.e., the design intensity, the number of stories) is obtained via curve fitting, as shown in the following equations:
where DI is the design intensity (ranging from 6 to 9), and n is the number of stories.
The fitted W2 values are compared to those published in the literature, as shown in Figure 4. Considering the dispersion of W2 in the literature, the curve fitting in Figure 3 shows an acceptable accuracy.
FIGURE 4: Comparison of W2 values obtained from the literature and Equation (10)
(d) Ultimate strength
RC frames usually have relatively good ductility. After reaching their peak strength, they can sustain their lateral strength to a larger deformation. According to the previous work by FEMA [2012a] and Lu et al. , this study also assumes that the ultimate strength of RC frames equals the peak strength, as shown in Equation (13).
3.2.2 Deformation parameters
The deformation parameters of RC frames include the yield, peak and ultimate deformations.
(a) Yield deformation
Structures remain linear before the yield point. Therefore, the yield deformation, dyield, i, can be computed according to k0 and Vyield, i, as shown in Equation (14).
(b) Peak deformation
The deformation at the peak strength of a structure can be calculated using its corresponding secant stiffness, as shown in Equation (15).
The secant stiffness can be determined using Equation (16),
where h is the stiffness reduction factor. Note that the design philosophy of RC structures in the Chinese code is similar to that specified in the US codes [Song and Ye, 2007; MOHURD, 2010b]. Note also that the stiffness reduction factor in ACI 318-11 [ACI, 2011] has experienced a number of validations and received a wide acceptance [Tran and Li, 2012; Avşar et al., 2014]. Therefore, the h provided in the Provision 10.10.4.1 of ACI 318-11 [ACI, 2011] can be used.
(c) Ultimate deformation
The lateral resistance of RC frames is abruptly reduced after reaching its ultimate deformation due to the rupture of rebar or the crushing of concrete. The ultimate deformation can be determined according to the inter-story drift limit of the “complete damage” state, which will be presented in Section 4
3.3. Determination of backbone curve parameters for masonry structures
The backbone curves of masonry structures also feature three key points. The difference between masonry structures and RC frames is that the yield point of masonry structures represents the significant inclined cracking of masonry walls rather than the yield of the steel reinforcement. The following methods are used to determine the backbone curve parameters of RM structures and URM structures:
3.3.1 Strength parameters
The fundamental principles used to determine the strength parameters of masonry structures and RC frames are similar. Specifically, a strength that is relatively reliable and easy to determine is first selected as the reference strength. Subsequently, the strengths of the other turning points on the backbone curve can be determined by multiplying or dividing overstrength ratios to the reference strength. For RC frames, the design strength is selected as the reference strength. The yield and peak strengths of RC frames are determined by multiplying W1 and W2 to the design strength.
Because RM structures are usually well designed, the design lateral strength can be used as the reference strength based on the above principle. In contrast, URM structures lack a design strength to be used as the reference strength. Yin  proposed a distribution curve of the peak strength per unit area for URM structures based on the statistical results of 1000 URM buildings in China, as shown in Figure 5. Thus, the peak strengths of URM structures can be selected as the reference strength to determine the other strengths. The detailed parameter determination processes for URM and RM structures are presented below.
FIGURE 5: The distribution of the peak strength per unit area of URM structures
(a) URM structures
Yin  proposed a distribution curve of the peak strength per unit area for URM structures based on the statistical results of 1000 URM buildings in China, as shown in Figure 5. Thus, the peak strengths of URM structures are selected as the reference strength to determine the other strengths. The peak strength of URM structures, Vpeak, i, can be calculated using Equation (17),
where R is the strength per unit area that can be determined according to Figure 5. Ai is the area of the ith story.
The yield strength can be obtained using Equation (18),
where W3 is the overstrength ratio between the peak strength and the yield strength. To obtain a reliable value of W3, the experimental data of 98 URM walls [Yang et al., 2000; Zhang, 2007a; Gong, 2008; Yang, 2008; Yang et al., 2008; Li and Wang, 2009; Han, 2009; Gu et al., 2010; Weng, 2010; Zhao et al., 2010; Zheng, 2010; Wu et al., 2012; Zheng, 2012; Lei, 2013; Zhang, 2014] are used to determine the distribution of W3 as shown in Figure 6. The median value of W3 = 1.40. The failure modes of these URM walls are diagonal shear failures.
FIGURE 6: Statistical results of W3 for URM structures
(b) RM structures
Similar to RC frames, the design strengths of RM structures are determined using an equivalent lateral force analysis [MOHURD, 2010a]. Subsequently, the yield and peak strengths are calculated according to Equations (19) and (20),
where W4 is the overstrength ratio between the yield strength and the design strength. W5 is the overstrength ratio between the peak strength and the yield strength.
The experimental results of 137 RM walls [Shi and Yi, 2000; Yang et al., 2000; Wang et al., 2003; Yu, 2003; Ye et al., 2004; Zhou, 2004; Zhang, 2005; Zhang, 2007a; Zhang, 2007b; Gong, 2008; Yang et al., 2008; Yang, 2008; Huang and Wang, 2009; Han, 2009; Fang, 2009; Zheng, 2010; Gu et al., 2010; Weng, 2010; Zhang, 2010; Liu et al., 2011; Xiao et al., 2012; Wu, 2012; Wu et al., 2012; Guo et al., 2014; Zhang, 2014] are used to determine W4 and W5, and the statistical results are shown in Figures 7 and 8. The median value of W4 = 2.33 and the median value of W5 = 1.41. The RM walls fail when diagonal shear cracks propagate through the entire component and rebars at both ends of the constructional column yield.
FIGURE 7: Statistical results of W4 for RM structures
FIGURE 8: Statistical results of W5 for RM structures
3.3.2 Deformation parameters
Masonry structures are assumed to be linear before the yield point. Therefore, the yield deformation can also be determined using Equation (14).
The experimental data of 98 URM walls and 137 RM walls are also used to estimate the peak deformation. The statistical data show that the drift ratios of the peak deformation of URM, dURM,peak, and RM, dRM,peak, also follow a lognormal distribution, as shown in Figures 9a and 9b. The median value of dURM,peak = 0.00268, and the median value of dRM,peak = 0.00317.
FIGURE 9: Statistical results of masonry deformation parameters
The deterioration of masonry structures after the peak point is more significant than that of RC frames. Therefore, the softening stiffness cannot be ignored. The softening stiffness is estimated using the experimental data. In pseudo-static experiments, most tests are finished when the strength decreases by 15%. However, this state does not represent the complete damage of the structure because the structure still has sufficient lateral resistance. Therefore, the point at which the strength decreases by 15% is named the “softening point” and is used to determine the softening stiffness. The strength at the softening point, Vsoftening, i, can be computed using Equation (21).
According to the experimental data of 97 URM and 137 RM walls, the drift ratios at the softening points of URM structures, dURM,softening, and RM structures, dRM,softening, also follow a lognormal distribution (Figures 9c and 9d). The median value of dURM,softening = 0.00507, and the median value of dRM,softening = 0.00960. After the softening point, assuming the backbone curve maintains the same softening stiffness [Shi et al., 2012], the ultimate displacement, dultimate, is determined based on the complete damage states of masonry structures, which is shown in Section 4.
3.4. Calibration of the hysteretic parameter
Considering the information of each building is limited, the single-parameter hysteretic model proposed by Steelman and Hajjar  is adopted in this study. The only parametert is defined by Equation (22)
where Ap and Ab are, respectively, the areas enclosed by the pinching envelope and that under the full bi-linear envelope (see Figure 2c). Given the structural type, design information and the type of ground motion used, the value t can be easily calculated using the degradation factor κ in Table 5.18 of the Hazus report [FEMA 2012a] together with the work of Steelman and Hajjar .
4. Damage assessment
According to the Hazus report [FEMA, 2012a] and the Chinese code (i.e., Classification of Earthquake Damage to Buildings and Special Structures [GAQSIQ, 2009]), seismic damage to buildings can be classified into five levels (none, slight, moderate, extensive and complete damages). To assess the seismic damages of buildings, there are two sets of criteria in existing literatures: (1) the force-based damage criteria [Yin et al., 2003] and (2) the deformation-based criteria [FEMA, 2012a]. The force-based criteria define the damage states according to the inter-story shear force. In contrast, the deformation-based criteria define the damage states according to the inter-story deformation.
Both the force-based damage criteria and the deformation-based damage criteria have their advantages and limitations. At the earlier stage of seismic damage, the stiffness of a structure is high, and a small variation in deformation will lead to a significant change in the internal force; thus, the force-based damage criteria are more reliable. In contrast, when approaching the peak strength, the tangent stiffness of a structure is quite small, and a small variation in force will induce a significant deformation change; thus, the deformation-based damage criteria are more suitable.
This study defines the damage states by taking the advantages of both the force-based and deformation-based damage criteria. The force-based damage criteria are used for the “slight” and “moderate” damage states, whereas the deformation-based criteria are used for the “extensive” and “complete” damage states. The details for each type of structure are as follows.
(1) RC frames
As proposed by Yin et al. , the RC frames reach the “slight damage” and “moderate damage” states when the internal force exceeds Vyield,i and (Vyield,i +Vpeak,i)/2, respectively, as shown in Figure 10a and Table 1. The criteria of “extensive damage”, δextensive, and “complete damage”, δcomplete, are defined according to the deformation-based method proposed by the Hazus report [FEMA, 2012a] (Table 1). Specifically, the Hazus report provides δextensive and δcomplete for RC frames with different numbers of stories and different design code levels (Table 2). According to the work of Lin et al. , the criteria in Table 2 can be successfully applied to Chinese buildings following the mapping rules of Table 3. For example, a 5-story Chinese RC frame built in 2000 with a seismic design intensity of VII will be mapped to the Low-Code C1M building in the Hazus report following the mapping rules in Table 3. Consequently, δextensive = 0.0133 and δcomplete = 0.0333 according to Table 2.
FIGURE 10: Inter-story backbone curves and damage limits
TABLE 1: Damage criteria of different types of structures
(2) Masonry structures
The yield point of masonry structures corresponds to the moment at which significant cracking occurs and the lateral stiffness abruptly declines [Shi and Yi, 2000]. Note that many smaller cracks have already formed in masonry walls prior to the yield point. According to the Classification of Earthquake Damage to Buildings and Special Structures [GAQSIQ, 2009], the initial cracking point, Vinitialcrack,i, is the “slight damage” criterion, and the cross-sectional cracking point, Vyield,i, is the “moderate damage” criterion, as shown in Figure 10b and Table 1. To determine the initial cracking force, Vinitialcrack,i, the test results of masonry buildings by Zhao (1993), Miao et al. , Zhou et al.  and Wang et al.  are collected, and the average ratio between the peak strengths and the initial cracking strengths is 2.455. Therefore, Vinitialcrack,i can be determined as follows:
Similar to RC frames, the criteria of δextensive and δcomplete proposed by the Hazus report [FEMA, 2012a] can be used to determine the “extensive” and “complete” damage states of masonry structures (Table 1). Also following the work of Lin et al. , the values of δextensive and δcomplete can be determined using Tables 2 and 3.
TABLE 2: Damage criteria of RC frames and Masonry structures from Hazus
where C1L/C1M, RM2L/RM2M, URML/URMM represent the RC frames, RM and URM structures in the Chinese code, respectively; 1F-3F indicates the number of stories of this type of structures ranging from 1-3.
TABLE 3: Divisions of seismic design levels for Chinese buildings
The pseudo-static tests of three individual buildings and the seismic damage results of Longtoushan Town are used to validate the proposed parameter determination method and damage assessment method.
5.1. Validation for RC frames
Two experiments of RC frames [Xie et al., 2015; Zhou and Zhou, 2005] are used to validate the proposed method. Both of the two frames are designed according to Chinese codes [MOHURD, 2010a; MOHURD, 2010b]. Frame 1 is a 1:2 scale test while Frame 2 is a 1:2.5 scale test. The prototype of Frame 1 has 6 stories. Considering the damage of RC frames is generally concentrated on the lower stories, the bottom three stories of the prototype RC frame were tested. The experiment is shown in Figure 11a. The prototype of Frame 2 has 3 stories and it was tested with only one actuator on the third story, as shown in Figure 11b.
FIGURE 11: Test setup of the RC frames (unit: mm)
According to the experimental results, these two RC frames experienced different failure modes (i.e., soft-story failure mode for Frame 1 and “strong-column-weak-beam” failure mode for Frame 2). The two RC frames are simulated with the MDOF shear models (Figure 2a). The inter-story nonlinear parameters are determined using the method presented in Section 3.2. The capacity curves of the simulation and experimental results are shown in Figure 12. As evident in the results, both simulations show good agreement, which demonstrates that the proposed parameter determination method can estimate the backbone curves of RC frames reasonably well.
FIGURE 12: Capacity curves of the RC frames
5.2. Validation for a RM structure
Wang et al.  have performed a full-scale, pseudo-static test of an RM structure. The prototype structure had six stories. Because of the height limitation of laboratory, only 1-5 stories of the prototype building were built and tested. The weight of the 6th story was added to the 5th story. The layout of the masonry structure is shown in Figure 13. The inter-story parameters of this masonry structures are estimated according to the method proposed in Section 3.3. The predicted results and the test results are compared in Figure 14, and the two curves have a good agreement, which demonstrates the reliability of the parameter determination method for the RM structure.
FIGURE 13: Layout of the RM structure (unit: mm)
FIGURE 14: Comparison for the RM structure
5.3. Validation of the Seismic Damage of Longtoushan Town
The seismic damage to Longtoushan Town during the Ludian earthquake is studied to validate the proposed method for regional buildings. Both the conventional damage probability matrix method and the proposed method are used for the validation.
5.3.1 Comparison with field investigation results
The post-earthquake field investigation of Longtoushan Town has collected the attribute data and damage information of 56 buildings. The buildings are simulated with the MDOF shear models in Figure 2a and the parameters of each building are determined using the method proposed in Section 3.
The ground motions recorded in Longtoushan Town are shown in Figure 15. Because the peak ground acceleration (PGA) of this earthquake attenuated quickly [Lin et al., 2015], the ground motions recorded at 16 different stations are collected, and the fitted attenuation relationship is shown in Figure 16. The ground motion is scaled according to the distance to the epicenter using the attenuation relationship in Figure 16, and input to each of the 56 buildings.
FIGURE 15: Ground motions recorded at Longtoushan Town station
FIGURE 16: The attenuation function of Ludian earthquake
The simulated damage states are compared with the damage states obtained from the field investigation, as shown in Figure 17. The comparison indicates that for half of the buildings in Longtoushan Town, the simulated damage states are identical to those of the field investigation. The differences in the remaining damage states are within one damage state level. Given the complexity of the actual situation of buildings and the variation of ground motions, the result shown in Figure 17 is deemed to be acceptable. The proposed method can predict the regional seismic damage with reasonable accuracy and reliability.
FIGURE 17: The comparison between the predicted damage states and actual damage states
5.3.2 Comparison with damage probability matrix method
Yin  proposed the damage probability matrices of different types of buildings based on the building damage data of several large earthquakes in China, which have been widely used [Lin et al., 2010]. Therefore, the damage probability matrices of Yin are used, and the prediction is compared with that of the proposed method.
According to the data from the China Earthquake Administration [CEA, 2014], the modified Mercalli intensity of the Ludian earthquake is Level IX in Longtoushan Town. The seismic design intensity of buildings in this region is VII. Therefore, the damage probability matrices of Classes A, B and C with seismic design intensity VII proposed by Yin are adopted for the RC frames, RM structures and URM structures in Longtoushan Town, respectively [Yin, 1996]. The seismic damage predicted by the damage probability matrix method and the proposed method are compared in Figure 18. Figure 18 shows that Yin’s damage probability matrix method significantly underestimates the damage to Longtoushan Town, because the ground motion recoded in Longtoushan Town was proven to be destructive despite the relatively low magnitude (M6.5) of the Ludian earthquake [Lin et al., 2015]. Therefore, the THA-based regional seismic damage prediction can more accurately consider the effects of local ground motions and building conditions and yields more reasonable results.
FIGURE 18: Comparison of predicted seismic damage and actual damage
6. Regional seismic damage prediction for a large urban area
To demonstrate the scalability of the proposed method, the seismic damage of a large Chinese urban area consisting 53,877 multi-story buildings is predicted using the THA-based regional seismic damage prediction. First, the site spectra are obtained from the local seismic hazard report. Second, note that SIMQKE [Gasparini and Vanmarcke, 1976] is a widely used and easy-to-implement program which can generate artificial acceleration time-histories from user specified response spectra [Bommer and Acevedo, 2004; Özhendekci and Özhendekci, 2012]. Consequently, it is used here to generate the local ground motions. Third, the nonlinear parameters of each building are determined according to Section 3. The nonlinear THA of each building is performed using the GPU-powered high-performance computing method proposed by Lu et al. . Consequently, the EDPs on each story of each building are used to predict the damage states. Using the visualization technology proposed by Xu et al.  and Xiong et al. , the damage states on each story are clearly presented in Figure 19. The detailed inter-story drifts and floor accelerations generated by the THA-based regional seismic damage prediction can provide a much more accurate regional loss estimation according to the fragility function and consequence function of FEMA P-58 report [FEMA, 2012b] and the work of DeBock and Liel . Such outcomes are important to determine the disaster prevention strategy of a city [Xu et al., 2016].
FIGURE 19: Regional seismic scenario of a large urban area
The parameter determination and damage assessment methods for the THA-based regional seismic damage prediction of multi-story buildings are proposed in this work. The parameter determination method, based on a simulated design procedure and the statistics of extensive experimental and analytical results, can be easily applied to different regions with different design requirements. The damage assessment method adopts a combination of force and deformation-based criteria, which is more reasonable than the previous work of Lu et al. .
The parameter determination method and the damage assessment method are validated using the pseudo-static tests of three individual buildings and the field investigation of Longtoushan Town after the Ludian earthquake. The predicted results based on the proposed method agree reasonably well with the actual damage and are more accurate than the results obtained by the damage probability matrix method.
Finally, the proposed method is applied to a large urban area consisting of 53,877 multi-story buildings. This application demonstrates that the proposed method can be used for large scale urban prediction. Furthermore, the THA-based regional seismic damage prediction provides more detailed building response results than conventional methods, which may facilitate future EDP-based regional seismic loss estimation.
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