Damage risk assessment of a high-rise building against multihazard of earthquake and strong wind with recorded data

Engineering Structures 200 (2019) 109697

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Damage risk assessment of a high-rise building against multihazard of earthquake and strong wind with recorded data

T

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Xiao-Wei Zhenga, Hong-Nan Lia,b, , Yeong-Bin Yangc, Gang Lia, Lin-Sheng Huoa, Yang Liua a

State Lab. of Coastal and Offshore Engineering, Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China School of Civil Engineering, Shenyang Jianzhu University, Shenyang 110168, Liaoning, China c School of Civil Engineering, Chongqing University, Chongqing 400045, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Multihazard Probabilistic multi-hazard analysis (PMHA) High-rise building Fragility Damage risk

The high-rise buildings designed with a long lifetime may be exposed to one or more extreme hazards. Traditionally, specifications separately treated the multiple extreme hazards according to the controlling load case. Thus, the ability of high-rise buildings designed by the current codes to face the combined threats of earthquake and wind is rather vague. This paper presents a multihazard-based framework to assess the damage risk of a high-rise building subjected to earthquake and wind hazards separately and concurrently, which can be broken into three parts: the modeling of hazards, the structural fragility analysis and the damage probability computation. Firstly, based on the earthquake and wind data from 1971 to 2017 recorded in the Dali region of China, the hazard curves of single earthquake and wind, and the copula-based surface of bi-hazards are well established. Secondly, the multihazard-based fragility analysis of a high-rise building in Dali Prefecture is performed with the consideration of various load conditions. Lastly, upon completing the hazard models and fragility analyses, quantifications of the damage probabilities for the separate and concurrent hazards are determined directly. Numerical results indicate that the damage probability and contributions of each hazard circumstance are sensitive to damage severity. Furthermore, the damage probability induced by the bi-hazards dominates the total probability under most damage states conflicted with the common assumptions presented in the available researches. The comprehensive application highlights the necessity of examining the responses of high-rise buildings subjected to multihazard. The potential of the presented framework is of great help for decision-making.

1. Introduction In the past few decades, there is an unprecedented increase in the urbanization and population of China, which brings a tremendous growth of high-rise/super-tall buildings construction [1]. The high-rise buildings designed with long lifetime are commonly susceptible to multiple hazards (or referred to as multihazard), e.g. earthquakes and strong winds, which is the focus of this study. Most researches and specifications conducted on the high-rise buildings have focused on the effects of seismic and wind load separately as it respectively pertains to the strength and serviceability requirements [2]. The meteorological phenomena indicated that the atmospheric pressure varies vastly during earthquakes occurrence [3–6]. As reported in series of literature [7–10], strong winds were often accompanied by the occurrence of earthquakes. Besides, an earthquake of magnitude 6.7 attacked

Hokkaido (142.0E, 42.7N) on the Sep.6, 2018, at the same time, the super typhoon “Feiyan” (Category 14) quickly struck the west coast of Hokkaido (139.2E, 43.0N) from Sep.5 to 6. More importantly, Li et al. [10] collected the earthquake data with surface wave magnitude (Ms) ≥ 4 and wind data with speed (V) ≥ 10 m/s of Dali region with a radius of 300 km from 1971 to 2017, and there are 76 pairs of earthquake and strong wind simultaneously recorded on the same day among the total sample with a size of 3389. It is rational to speculate that there will be more extreme events of simultaneous earthquakes and strong winds (or bi-hazards for short) occurring in all the wide future. Therefore, understanding the behavior of high-rise buildings under these bi-hazards is an essential topic currently. The authors used these data to construct the joint probabilistic distribution of peak ground acceleration (PGA) and V by the Copula functions, and provided the quantitative measures of the combined-event risk [10]. The joint

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Corresponding author at: State Lab. of Coastal and Offshore Engineering, Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China. E-mail addresses: [email protected] (X.-W. Zheng), [email protected] (H.-N. Li), [email protected] (Y.-B. Yang), [email protected] (G. Li), [email protected] (L.-S. Huo). https://doi.org/10.1016/j.engstruct.2019.109697 Received 22 April 2019; Received in revised form 15 September 2019; Accepted 18 September 2019 0141-0296/ © 2019 Published by Elsevier Ltd.

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Nomenclature PMHA V V10 PGA Ms Mw CDF MCDF

JCDF SFRCT MDRA PSHA Bi-hazard MIDR IM EDP LHS FE PDF

probabilistic multi-hazard analysis strong wind speed 10-minute average wind speed at the height of 10 m above the ground peak ground acceleration surface wave magnitude moment magnitude cumulative distribution function marginal CDF

joint CDF steel frame-RC core tube multihazard-based damage risk assessment probabilistic seismic hazard analysis simultaneously having earthquake and strong wind events maximum inter-story drift ratio intensity measure engineering demand parameter Latin Hypercube Sampling finite element probability density function

the buildings with 10 and 30 stories, respectively. Venanzi et al. [11] presented a framework for the life-cycle cost estimation in tall buildings under the combination of seismic and wind excitations, which was beneficial for decision-making. However, in their work, both the earthquake and wind induced structural responses were assumed to be linear which might be unrealistic for hazards with high intensities. Other efforts have been devoted to examine the risk or optimization of medium or high-rise buildings subjected to earthquake and wind, e.g., Huang [22], Chulahwat and Mahmoud [23], Suksuwan and Spence [24], Joyner and Sasani [25] and Santos-Santiago et al. [26]. Without exception, the common defects of above mentioned studies are that the joint occurrence of these two hazards was out of consideration. To our best knowledge to the literature, the researches on the effects of simultaneous application of seismic and wind loads on the responses mainly focused on wind turbines [27,28]. The results demonstrated that the seismic and wind loads simultaneously acting on wind turbines significantly impacted the stability in comparison to the value induced by a single load. Analogously, earthquake and wind excitations acting concurrently may significantly increase the potential damage of buildings, which is imperative to be discussed in detail. In this regard, a framework for assessing the damage risk in a high-rise building under seismic and wind loads is provided in this paper. In which, the probabilistic models of individual hazard and bi-hazard are the initial step, which can be constructed based on the recorded field data by the statistical approach. Concerning the study [10], the marginal cumulative distribution functions (MCDFs) of PGA and V were respectively described by the Frechet and truncated Weibull distributions. The joint CDF (JCDF) was verified to be well characterized by Joe Archimedean copula. Fragility is functioned as a bridge between the hazard models and damage risk in the

probabilistic model was constructed by the earthquake and wind data recorded on the same day rather than the combination of arbitrary point-in-time data. The risk of the events of simultaneously occurring earthquake and strong wind may be overestimated. Nevertheless, it is still necessary to put attention on investigating the impacts of these events, which is neglected in the available literature. It will cause devastating damage to high-rise buildings and incalculable casualties, once these extreme concurrent hazards occur in the future. In recent years, the importance of ensuring an infrastructure with adequate ability in face of earthquake and wind is gaining a spike of interest. The multihazard-based assessment and optimization were investigated for different structures subjected to earthquake and wind hazards, e.g., high-rise/tall buildings [11,12], residential wood constructions [13], bridges [14], wind turbines [15] and electric power transmission systems [16]. Currently, the researches on the multihazard primarily focused on the individual earthquake or wind with an assumption of nearly nil chance of simultaneously having these two hazards. Wen and Kang [17,18] proposed the minimum life-cycle cost design criteria, including a methodology and an application, for buildings subjected to earthquake and wind. The results indicated that the design procedure was not just controlled or governed by one hazard as recommended in most specifications considering load combination [18]. Note that for their researches the structure was assumed to be restored to its original condition after occurring a hazard. In addition, some efforts were also devoted to the fragility analysis [13], life-cycle loss estimation [19] and risk assessment [14,20] of residential wood constructions and bridges subjected to earthquake and wind/hurricane. Until quite recently, Mahmoud and Guo [21] provided a probabilistic framework to assess the design alternatives against earthquake and wind hazards according to the expected lifecycle cost, and applied it to

Fig. 1. Flowchart of proposed multihazard-based damage risk assessment framework. 2

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3. Mean annual occurrence models of multiple hazards

proposed framework, which is a conditional probability of a representative engineering demand parameter (EDP) exceeding a specific damage state under the premise of hazards occurrence [29]. Separate seismic and wind fragility for the high-rise/tall buildings have been examined in many studies, e.g., Nikellis et al. [12] and Mosallam et al. [30] for seismic fragility, and Le and Caracoglia [31] and Nikellis et al. [12] for wind fragility. Yet, the simultaneous multihazard-based fragility of high-rise buildings is scarce. To remedy this gap, the fragility surface with respect to PGA and V is utilized to quantify the performance of high-rise buildings under the bi-hazards of earthquake and wind. The key aim of this paper is to develop a framework for assessing the structural damage risk under earthquake and wind hazards with the consideration of the simultaneous occurrence of these two hazards. For instance, the proposed framework is applied in a 42-story steel frameRC core tube (SFRCT) building. The next section of this paper will illustrate this proposed comprehensive framework in detail. The mean annual occurrence models of individual hazards and bi-hazards are presented in Section 3. The fragility analyses of the exemplary high-rise building under multiple hazards are discussed in Section 4. Section 5 calculates the annual damage probability of each hazard circumstance. The last section summarizes the main conclusions.

The Dali Prefecture (24.7°N ~ 26.7°N, 98.9°E ~ 101.1°E) with an altitude of about 2000 m is located on the Zhongdian-Dali seismic belt of Yun-Gui Plateau, Yunnan Province, China. Meanwhile, the Yun-Gui Plateau is near the eastern boundary between India and Eurasian plates, which leads Dali region to be one of the most seismicity and high wind hazard of China. Performing a probabilistic multi-hazard analysis (PMHA) is imperative to determine the hazard level at a given region, including the following steps: (1) identifying the multiple threat scenarios; (2) characterizing the distribution of the intensities of certain hazards, e.g., earthquake and strong wind; (3) generating the hazard curves for individual hazards and the hazard surface for bi-hazards. 3.1. Mean annual occurrence model of individual earthquake The China Strong Motion Network Centre has deployed 27 stations at the Dali region to collect the valuable earthquake records since 2007. The earthquake catalogue is employed to supplement the earthquake database. With reference to studies [39,40], for the Sichuan-Yunnan area of China, the minimum magnitude of completeness in the earthquake catalog since 1970 is 4, i.e., Mc = 4. Liu et al. [41] proposed that for the Dali region Mc in the earthquake catalogue recorded from 1970 to 2011 is 2. Herein, the earthquake data employed to construct the seismic model is with Ms ≥ 4, indicating the completeness of the earthquake catalog is well satisfied. In terms of the probabilistic seismic hazard analysis (PSHA) method presented in the studies [42,43], the mean annual exceedance rate of PGA > x can be written by

2. Framework for damage risk analysis under multiple hazards A multihazard-based damage risk assessment (MDRA) framework proposed herein is integrated of hazard model construction, fragility analysis and damage probability calculation, as shown in Fig. 1. One main assumption is introduced in the proposed framework: the source mechanism of this given area, i.e., Dali Prefecture in Southwest China, for the exemplary building located at is assumed to be the area source [32–35]. Three hazard circumstances, including the individual earthquake (e), individual wind (w) and bi-hazards (ew) of earthquake and wind, are taken into account. The flowchart illustrated in Fig. 1 demonstrates the procedure from the hazard models to computing the damage probability under earthquake and wind hazards. Ignoring the structural capacity deteriorating with time [17], the annual damage probability can be expressed as [14,36–38]:

λ (PGA > x ) = λ (m > m min )

∬ P (PGA > x|m, R, ε ) f (m) f (R) f (ε )dmdr dε (2)

where P(PGA > x|m, R, ε) denotes the conditional probability of PGA exceeding x given m, R and ε; m is the surface wave magnitude rather than the moment magnitude (Mw), because Mw was not used to describe the magnitude in the selected earthquake catalog. R is the epicentral distance in km. It is worth noting that the fault distance is not used herein due to the lack of recorded fault strike during earthquakes occurring at Dali. ε is the residual of ground motion attenuation model, which was normal distribution of (0, σlnPGA); λ(m > mmin) is the mean annual rate of exceeding the minimum magnitude, calculated by λ(m > mmin)=(AD/A) λ [44], in which AD is the area of the Dali Prefecture, A is the area of the Dali region with a 300 km radius and λ is the mean annual occurrence rate in this region. f(m) means the probability density function (PDF) of m, expressed as [42]

Pai f, j (EDP > C) = ∫ Pf,i j (EDP > C|IMi)dF (IMi > imi ) = ∫im Pf,i j (EDP > C|IMi) f (IMi )dIMi i

(1) Pf,ji(EDP

> C|IMi) is the conditional structural fragility of exwhere ceeding the j-th damage state given the occurrence of hazard circumstance i; C is the structural capacity; IMi denotes the vector intensity measure (e.g., PGA and V) for the i-th hazard circumstance; F (IMi > imi) is the mean annual exceedance probability of i-th hazard with respect to IM; f(IMi) is defined as d|1- F(IMi > imi)|dIMi, with “d (·)” meaning the differential. Definitely, the key purpose of the MDRA framework is to analyze the structural damage induced by the multiple hazards containing three separate main parts: (1) the first part is to collect the recorded hazard data, including the instrumental earthquake\wind data and earthquake catalog, for a certain given geographical region, then applying the collected data to construct the hazard models. (2) the second one is to compute the conditional fragility of a high-rise building against multiple hazards separately and concurrently. This part is the easy of explicit representing the uncertainties associated with the parameters of FE model, and the intensities of seismic and wind loads. (3) the last one is to calculate the annual damage probability by Eq. (1), which can be regarded as a crucial variable for the design of new structures and the risk assessment of existing structures.

f (m) =

b ln(10)10−b (m − mmin) m min < m < m max 1 − 10−b (mmax − mmin)

(3)

where mmin is the minimum magnitude, equaling 4; mmax is the maximum magnitude, which is treated as an uncertain variable; and b is the Gutenberg-Richter constant. Modeling as area source for the Dali Prefecture, the PDF of R can be expressed by [42]

f (R) =

d d πR2 2R F (R) = ( ) = 2 0 < R < Rt dR dR πRt2 Rt

(4)

where Rt is the threshold of R, a value of 300 km used by most researchers [45,46]. The ground motion attenuation model is commonly applied to supplement PGA values from the earthquake catalog. In order to consider the effects of different attenuation models on the results of PSHA, a benchmark model proposed by Cornell et al. [47] and the model adopted in the code [32] are employed to estimate a series of PGA. The regression analysis is used to determine the coefficients C1–C6 according to the instrumental PGA from 2007 to 2017, as listed in Table 1. The combinations of estimated and instrumental PGAs are defined as 3

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Fcumw(y) denotes the cumulative probability of V ≤ y. Li et al. [10] verified the truncated Weibull distribution (with m = 1.0, a = 10.0, b = 3.040 and c = 1.174) was the best to model the distribution of strong wind speed at the Dali Prefecture, in which, m is a parameter to ensure the integral of this distribution is 1.Then, in accordance with Eq. (7), F(V > y) can be computed and plotted in Fig. 5. According to the specification [53], the wind speeds respectively corresponding to 50- and 100-year return periods are computed as 32.4 m/s and 34.9 m/s. In contrast, the wind speeds with the same return periods calculated by the probability distribution of wind speed proposed herein are 34.5 m/s and 35.7 m/s. Comparing to the codespecified values, the errors are respective 6.0% and 2.3%, which verifies the accuracy of the presented probabilistic model.

Table 1 Summary of C1–C6 and model standard deviations. Models

C1

C2

C3

C4

C5

C6

σlnPGA

I II

0.164 −0.119

0.896 0.045

−1.980 0.067

– −1.463

– 52.32

– −0.453

0.977 0.964

the mixed PGA data.

Model I ln PGA = C1 + C2 M + C3 ln(R + 25) σln PGA

(5)

Model II ln PGA = C1 + C2 M + C3 M 2 + C4 ln[R + C5 exp(C6 M )] σln PGA

(6)

Li et al. [10] and Zheng et al. [48] applied the Gumbel, Frechet, Weibull and Lognormal distributions to describe the distribution of PGA. The authors provided quantitative evidence to justify the Frechet model was the best to match the distribution of PGA [10]. The results of regression analysis are as follows: Model I: a = 0, b = 2.683, c = 1.067, R2 = 0.9506 and RMSE = 0.0026; Model II: a = 0, b = 2.593, c = 1.165, R2 = 0.9724 and RMSE = 0.0021, in which a, b and c are respective the location, scale and shape parameters. R2 and RMSE respectively denote the coefficient of determination and the root mean square error. Then the fitting curves are shown in Fig. 2. In which, “Data 1” and “Data 2” denote the earthquake data obtained from the attenuation models I and II, respectively. With reference to previous studies [49,50], the uncertainty involved in the input variable, e.g., mmin, λ(m > mmin), b-value, mmax and attenuation relationship, has a significant impact on the results of PSHA. The logic tree has been widely used to take into account the effects of uncertain variables, and is shown in Fig. 3 with an assumption that mmin and λ(m > mmin) are deterministic. The level-1 branching in Fig. 3 deals with the uncertainty of b value-range and its weightages are determined by Roshan and Basu [49]. The second layer branching of the logic tree are formed with different mmax and the corresponding weightage is computed based on the earthquake catalog. Note that the maximum observed magnitude of an earthquake in the earthquake catalog is 7.8. Besides, the last branches, as shown in Fig. 3, are constructed to deal with the uncertainty of these two attenuation models with equal weightage of 0.5. According to Eq. (2), the seismic hazard curve derived from the logic tree is plotted in Fig. 4, and the mean annual rate will be used in the following loss analysis. It is stressed that, based on the frequency notion of probability, f (PGA) involved in Eq. (1) can be approximately determined by d|λ(PGA)|/dPGA, i.e., the slope of the seismic hazard curve [38,51]. With reference to Kiureghian [38], the error can be negligible when the considered exceedance probability is small.

3.3. Joint mean annual occurrence model of bi-hazards Referring to the analysis of individual hazard models, the joint mean annual exceedance probability of the bi-hazards is given by

F (PGA > x ∩ V > y ) = λ ew P (PGA > x ∩ V > y )

(8)

where λew is the mean annual occurrence rate of events of simultaneously having earthquake and strong wind; P(PGA > x ∩ V > y) denotes the exceedance probability. In order to construct the mean annual occurrence chance of the events of simultaneous earthquake and strong wind, the corresponding PGA and V recorded on the same day are abstracted from the established earthquake and wind databases. There are total 76 records from 1971 to 2017, as summarized in the study [10]. With this in mind, the simultaneous occurrence of earthquake and strong wind hazards refer to the data recorded on the same day, rather than emphasizing the necessity of the maximum associated with seismic and wind loads occurring simultaneously. Indeed, the assumption of maximum wind speed occurs at the same time as an earthquake over one day may lead to an overestimate of the risk of simultaneous earthquake and strong wind hazards. It may be reasonable that the arbitrary point-in-time wind speed is combined with the seismic event. However, the risk assessment under joint seismic and wind excitations may be more suitable for the high-rise or even mega-tall buildings rather than general buildings. Thus, the overestimate may be acceptable for these largesized buildings to a certain extent. Based on these 76 pairs of data, Li et al. [10] used six commonly used Archimedean copulas to construct the JCDF of PGA and V. The results indicated that the Joe Archimedean copula (with θ = 11.01) was the best to describe the JCDF. The equation of Joe copula function and Kendall’s τ are expressed as

3.2. Mean annual occurrence model of individual strong wind The China Meteorological Administration (CMA, http://data.cma. cn/) provided the daily maximum wind speed data (averaged over 10 min) recorded at the elevation of 10 m above the ground level from 1971 to 2017. The contact anemometer, composed by the sensor, indicator and recorder, with a sampling frequency of 1 Hz was applied to record the wind speed. In this subsection, only the strong wind, defined with V ≥ 10 m/s, of engineering interest is abstracted from the recorded daily wind data [52] and its sample size is 2944. It is worth remarking that the hurricanes were not included in the wind data. Then, the mean annual chance of V exceeding a given level y can be expressed by [52] w w F (V > y ) = λ w,10 Fexc (y ) = λ w,10 [1 − Fcum (y )]

(7)

where λw,10 is the mean annual occurrence rate of V ≥ 10 m/s and equals to 0.172; Fexcw(y) is the exceedance probability of V > y;

Fig. 2. Probability density function of PGA associated with Models I and II. 4

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Fig. 3. Logic tree used for considering the uncertainties in PSHA.

Fig. 5. Mean annual exceedance probability of strong wind.

Fig. 4. Uniform seismic hazard curve of PGA.

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FXY (PGA, V ) = Cθ [FX (PGA), FY (V )]

mechanics and the smeared crack model [56]. Furthermore, the P-delta effect was taken into account in modeling this exemplary building. The first three fundamental periods of this high-rise building are 3.23 s (T1), 2.61 s (T2) and 0.91 s (T3), respectively. The analysis of structural damage probability is influenced by various sources of uncertainty associated with the material property, damping ratio (ξ) and intensity of loads. The probabilistic description of material parameters and damping ratio of this SFRCT building are provided by Zheng et al. [1]. Moreover, they have investigated the wind-induced damage probability of this exemplary building considering these uncertainties. All random variables include the mean value, coefficient of variation (COV), distribution and threshold value, and are assumed to be uncorrelated [1,60]. Then, 20 samples of FE structural models are generated by the Latin Hypercube Sampling (LHS) method for the nonlinear dynamic analysis.

1/ θ

(1 − u1)θ + (1 − u2)θ ⎤ =1− ⎡ ⎢ −(1 − u1)θ (1 − u2)θ ⎥ ⎣ ⎦ τ=1+

4 θ

∫0

1

[1 − (1 − t )θ] ln{[1 − (1 − t )θ]} dt (1 − t )θ − 1

(9)

(10)

where θ is a parameter describing the dependency relationship between the variables PGA and V, which is completely reflected by the recorded data; τ can be calculated by the difference between the probabilities of concordant and discordant sample pairs of (PGA, V). The measured JCDF of PGA and V is introduced to further verify the goodness-of-fit of the proposed copula-based approach. The definition of measured JCDF is written by [54]

Pij =

tij N

1 ⩽ i, j ⩽ N

(11) 4.2. Fragility-based performance assessment

where tij is the number of pairs (x, y) falling within the interval [x(i), y(j)] with x(i), y(j) denoting the order statistics. The sample pairs of the bi-hazards are divided into 141 × 21 intervals by taking 1 as an increment. Then, the R2 and RMSE of the measured JCDF are 0.8834 and 0.0585 when the mixed PGA is obtained from Model I, while for Model II the values are respective 0.8884 and 0.0574. After determining the copula-based JCDF of PGA and V, the joint exceedance probability of PGA > x ∩ V > y can be easily given by

P (PGA > x ∩ V > y ) = 1 − FPGA (x ) − FV (y ) + C(u1, u2; θ)

In the performance assessment of high-rise buildings against multiple hazards of earthquake and wind, the maximum inter-story drift ratio (MIDR) is commonly applied to assess the structural capacity of resisting extreme hazards [11]. Six damage states according to the MIDR are adopted herein as listed in Table 3, which was also used in Wen’s work [18] to assess the minimum life-cycle cost of buildings under earthquake and wind hazards. The median EDP described by the MIDR can be expressed as a function with respect to the selected intensity measures (IMs, i.e. IM1 = PGA and IM2 = V) given by:

(12)

where C(u1,u2;θ) can be obtained by the Joe Archimedean copula; u1 and u2 respectively denote the FPGA(x) and FV(y). Then, the joint mean annual occurrence probability of PGA > x & V > y is plotted in Fig. 6. As can be observed that the occurrence probability of concurrent earthquake and strong wind seems to pertain to a weak earthquake accompanied by moderate intensities of wind. The joint exceedance chance of these simultaneous events with largest-magnitudes is small.

ln(EDP) = k 0 + k1 ln(PGA) + k2 ln(V )

4. Fragility analysis of high-rise buildings against multiple hazards

The 1-order or 2-order expansion functions can well match the structural responses with respect to the intensity of input loads [61]. However, the 2-order demand model will produce a negative value of k1 in this paper, which contradicts the fact, namely, the structural response commonly increases with the intensity of loads increasing. The K-fold cross-validation technique will be applied to determine the regression coefficients k0, k1 and k2 under multiple hazards. The basic concept of K-fold cross-validation is as follows [62]: (1) partition the sample into K separate parts; (2) randomly select (K-1) part as the training set to determine the model by a linear regression analysis, and other as the test set to verify the accuracy of the regression model; (3)

(13)

Noteworthy, the accuracy of Eq. (13) can be justified by the Taylor series expansion. Let us introduce variables x and y to represent ln (PGA) and ln(V), respectively. Then the Taylor series expansion of function f(x, y) can be expressed as

f (x , y ) = k 0 + k1 x + k2 y + k11 x 2 + k22 y 2 + k12 xy + ···

In order to examine the proposed MDRA methodology, a high-rise building is used as a case study. Fig. 7 illustrates the spatial distribution of some high-buildings in the given studied region. It is observed that Dali has relatively dense high-rise building distribution. The design general information of this site of interest is summarized in Table 2. 4.1. Description of the dynamic analysis model A benchmark high-rise building provided by [55] is redesigned by replacing the RC frames with the steel frames on the ETABS platform, i.e., the building is redesigned as a 42-story steel frame-RC core tube (SFRCT) structure, which is mainly constructed with the pipe-section steel columns, I-section steel beams and RC shear walls. The length and width of this SFRCT building are respectively 32.4 m and 30.6 m, as illustrated in Fig. 8 (a). Excepting the first floor with a height of 4.5 m, and other stories are with the height of 3.6 m. Referring to studies [1,12], the OpenSees platform is employed to model the SFRCT building with the DispBeamColumn element modeling beams and columns and the Multi-layer Shell element simulating the shear walls, as shown in Fig. 8 (b). In which, the Multi-layer Shell element is verified by various numerical simulations and experiments to be capable of simulating the coupled shear-flexural behavior of RC shear walls, as detailed in the studies of Lu et al.[56–58]. The steel columns and beams and the reinforcing bars are modeled by Steel02, meaning that the material degradation is properly considered [59]. Besides, two-dimensional concrete constitutive model is applied to capture the mechanical behavior of shear walls, which is formed based on the damage

Fig. 6. Joint mean annual exceedance probability of the bi-hazards. 6

(14)

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Table 3 Performance levels and damage states with respect to inter-story drift ratio. Performance levels

Damage states

Drift ratio (%)

Slight Light Moderate Heavy Major Destroyed

DS-I DS-II DS-III DS-IV DS-V DS-VI

> 0.2 > 0.5 > 0.7 > 1.5 > 2.5 > 5.0

aleatoric uncertainty associated with the structural capacity, which is 0.25 based on Ellingwood et. al [63]. The parameter βEDP|PGA,V is the logarithmic standard deviation of the demand conditioning on PGA and V, expressed as Fig. 7. The spatial distribution of high-rise buildings at Dali Prefecture.

βEDP|PGA, V ≅ Table 2 Basic design conditions. Parameters

Limits

Wind reference pressure Surface roughness Seismic fortification intensity Design basic acceleration Earthquake classification Site condition

0.65 kN/m2 Class B 8° 0.2 g Group II Category III

(16)

in which di is the maximum inter-story drift ratio under i-th input load; n is the size of input loads (equaling 100 herein). The nonlinear dynamic time history analyses are conducted to yield the fragility of this SFRCT building subjected to seismic and wind excitations. The effects of various load conditions, i.e., the loading angles along X-axis and Y-axis, are taken into account in the time history analyses, since the X and Y-axis are respectively the most vulnerable and safest attack angles in terms of the deformation-based fragility [1]. The fragility analysis of this SFRCT building under two-dimensional wind excitations has been carefully investigated by Zheng et al. [1]. Besides, the influences of multi-dimensional seismic loads on the responses of this building are not taken into consideration herein, which will be studied in the future work. In general, eight loading conditions are considered, as shown in Fig. 9.

repeat K times of step 2 to ensure each part to have been as a test set, and compute the predicted errors of the K models; (4) determine the optimal demand model by averaging these K models. In both earthquake and wind engineering, the EDP is assumed to be modeled by a lognormal distribution [13], thus the conditional vulnerability probability can be written by [37]

⎛ ln(C) − ln(EDP) ⎞ Pf (EDP > C|PGA,V ) = 1 − Φ ⎜ ⎟ ⎜ β2 + βc2 ⎟ ⎝ EDP|PGA, V ⎠

∑ [ln(di ) − ln(EDP)]2 n−2

4.2.1. Input ground-motion records In order to account for the uncertainty of ground-motion records, 100-acceleration series are selected from the PEER-NGA database (http://peer.berkeley.edu/nga/) based on the target spectrum corresponding to 8° (0.2 g) intensity provided by the code [64]. Fig. 10 illustrates the M-Rjb distribution of the selected records. The earthquake

(15)

where Φ(·) is the standard normal distribution, βc represents the

Fig. 8. Layout and 3D FE model of the high-rise SFRCT building. 7

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Fig. 9. The sketch of loads acting on the structure.

process. In this context, the selected ground-motion records are classified into three groups: 30-ordinary records with no long-period characteristic, 37-near-field pulse-like records and 33-far-field harmoniclike acceleration series. The comparison between the target and response spectra are shown in Fig. 11, and the thin gray lines denote the elastic response spectra of the selected ground-motion records. It can be observed from Fig. 11 that the median response spectrum marked with black solid line lies in the 0.8 and 1.2 times of target spectrum, respectively marked with green and blue dash lines, between T1, T2 and T3, which is satisfied well with the corresponding requirement of the code [64]. 4.2.2. Input wind load time-histories The wind speed along the height of the building can be decomposed into a constant mean wind speed and a fluctuating wind speed. In which, the mean wind speed is a function of height and a reference wind speed (e.g., the 10-minute average wind speed at the height of 10 m, written as V10), which can be described by the logarithmic law [67]. The fluctuating wind speed is generated based on V10 by the harmonic superposition method [68]. According to the code [53] and studies [1,27], the duration of the generated time histories of fluctuating wind speed is set to be 600 s, in which, a 20 s-wind load series from zero to the initial wind load is introduced for each simulated wind load time history to avoid the transient solution existing in dynamic analysis. The wind load acting on each floor of this SFRCT building is assumed to be fixed at only one point. Then, the along-wind load acting on the i-th floor can be determined by [53]

Fig. 10. The M-Rjb distribution of the selected ground motion records.

1 Fwi = μs ⎛ ρa Vi2 ⎞ Ai ⎝2 ⎠

(17)

where μs is the shape coefficient; the air density ρa = 1.235 kg/m ; Ai is the projected area of the i-th floor in the windward direction; Vi is the total wind speed. To account for the uncertainty associated with the intensity of wind load, the Monte-Carlo simulation is applied to generate 100 samples of V10 ranging from 0 to 40 m/s owing to the maximum wind speed of 35.7 m/s in the recorded wind data. The probability distribution of wind speed, i.e. Weibull model, is just used to calculate the occurrence chance of the generated wind speed. Since the generated wind speed sample covers a large range in the return periods, it is appropriate for developing the analytical fragility. It is worth noting that Eq. (17) is incapable of computing the effects of across and torsional wind loads. However, the torsional effects can be neglected for the symmetrical structures [1,69,70], e.g., the exemplary SFRCT building. The reliable across-wind load is determined by the 3

Fig. 11. Comparison between the median response and target spectra.

disaster reports indicated that the high-rise buildings under the longperiod ground motions would shake intensively or even occurring resonance [65,66], which was not carefully recognized during the design 8

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Table 4 Results of the coefficients k0, k1, k2 and βEDP|PGA,V. Cases Seismic Wind Seismic//Wind Seismic⊥Wind

Along Along Along Along Along Along Along Along

X-axis Y-axis X-axis Y-axis X-axis Y-axis X-axis Y-axis

Case Case Case Case Case Case Case Case

I(1) I(2) II(1) II(2) III(1) III(2) IV(1) IV(2)

k0

k1

k2

βEDP|PGA,V

PRESS

−4.5874 −4.7728 −12.436 −12.788 −6.1406 −5.8946 −5.7144 −6.3866

0.7132 0.6913 – – 0.2335 0.3073 0.2930 0.2100

– – 2.2396 2.2046 0.5042 0.3631 0.3064 0.5168

0.7311 0.6763 0.2868 0.2630 0.6096 0.5611 0.6061 0.6113

1.4e−3 9.43e−4 1.28e−4 2.92e−4 2.10e−3 1.20e−3 7.83e−4 7.94e−4

Fig. 14. Wind-based demand model. Fig. 12. Seismic-based demand model.

Fig. 15. Wind-based fragility curves. Fig. 13. Seismic-based fragility curves. Table 6 The values of V corresponding to the median conditional failure probability (units: m/s).

Table 5 The PGA values corresponding to the median conditional failure probability (units: g). Damage states

Along X-axis

Along Y-axis

Difference (%)

DS-1 DS-2 DS-3 DS-4 DS-5 DS-6

0.107 0.371 0.587 1.656 3.319 8.530

0.127 0.461 0.740 2.162 4.435 11.75

15.7 19.5 20.7 23.4 25.2 27.4

9

Damage states

Along X-axis

Along Y-axis

Difference (%)

DS-1 DS-2 DS-3 DS-4 DS-5 DS-6

16.2 24.4 28.3 39.9 50.2 68.5

19.6 29.7 34.5 48.7 61.3 83.7

17.3 17.8 18.0 18.1 18.1 18.2

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dynamic analyses are needed to be performed in the fragility analysis. However, if without the random combination, there will be 2000(=20 * 100) nonlinear analyses for this building against individual hazard, and 200,000 (=20 * 100 * 100) nonlinear analyses for concurrent hazards. Obviously, it is extremely time-consuming and ill-advised. 4.3.1. Seismic-induced fragility curves In light of the discussion in Section 4.2, the 5-fold cross-validation is used to determine the optimal demand model based on the output MIDR induced by the individual seismic load. Then, the coefficients involved in Eq. (13) and βEDP|PGA,V are listed in Table 4, in which the word “PRESS” denotes the predicted error sum of squares. Fig. 12 shows the MIDR response with respect to PGA in the logarithm coordinate. According to Eq. (15), the fragility curves of the SFRCT building against the seismic load along X and Y axis for the different damage states are plotted in Fig. 13. It can be obviously observed that the conditional failure probabilities of Case I(1) are larger than the corresponding values of Case I(2). This is due to the fact that the structural stiffness along X-axis is lower than that along Y-axis. In order to further investigate the difference between the fragility of this high-rise building under the individual seismic load along X and Yaxis, the values of PGA corresponding to the 50% conditional failure probability (i.e., the median value) are summarized in Table 5, in which, the “Difference” is defined as |PGAx-PGAy|/ PGAy. The “Difference” results indicate that there are remarkable gaps among the fragilities along X and Y axis.

Fig. 16. Random combination of PGA and V.

power spectral density (PSD) of wind pressure acting on the buildings, which is commonly obtained from a specific wind tunnel test. Yet there is no any direct wind tunnel data about this high-rise SFRCT building. Besides, the SFRCT building is assumed to be located at Dali, where there are no extreme typhoon or hurricane events. Thereby, with reference to studies [1,71–73], it may be reasonable to neglect the acrosswind effect. Nevertheless, for this exemplary high-rise building employed herein, the across-wind and torsional effects may bring a few slight impacts on the structural responses, and ignoring these effects may cause the current analysis not comprehensively sufficient. Thereby, in the future work, the authors will perform more comprehensive research to consider the influences of the probabilistic shape parameter, directionality, across-wind and torsional effects on the wind-excited high-rise buildings.

4.3.2. Wind-induced fragility curves Applying the individual wind load as input to the dynamic time domain structural analysis model, then the output MIDR can be obtained. Analogously, the 5-fold cross-validation technique is employed to calculate the coefficients of Eq. (13) as listed in Table 4. The fitted curves for the logarithm of MIDR with respect to ln(V) are shown in Fig. 14. In terms of the damage states listed in Table 3, the single wind loadinduced fragility curves along X and Y axes are demonstrated in Fig. 15. It shows similar results with the individual seismic-induced fragilities, i.e., the conditional failure probabilities of Case II(1) are larger than the corresponding values of Case II(2). The values of V corresponding to the median conditional failure probability are investigated in particular, as listed in Table 6. The definition of “Difference” is as same as mentioned above. It is evident that different wind load conditions have noticeable impacts on the conditional probability of the wind-excited high-rise

4.3. Fragility curves for individual hazards As discussed in the study [1], the LHS method is applied to consider the uncertainties associated with material properties and damping ratio, and there were 20 samples of FE structural model generated. The 100 earthquake records and the 100 wind load time histories are randomly divided into 20 groups [1,27,74]. In general, there are 20 pairs “seismic-wind loads” and 20 FE models, then the 20 pair loads and 20 FE models are matched one-to-one in random generating 20 combinations of loads and FE models in total. In brief, only 100 nonlinear

Fig. 17. Surfaces of demand model for the bi-hazards. 10

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Fig. 18. Fragility surfaces of bi-hazards under DS-3.

Fig. 19. Fragility curves for given PGA and V. Table 7 Annual damage probabilities of each hazard circumstance. Damage states

Earthquake

Wind

Earthquake & Wind

Summation

Error (%)

DS-1 DS-2 DS-3 DS-4 DS-5 DS-6

1.02e−3 2.10e−4 1.11e−4 1.93e−5 4.49e−6 3.93e−7

1.18e−2 1.06e−4 8.70e−6 5.41e−9 9.11e−12 2.19e−16

1.33e−2 2.21e−3 8.35e−4 4.56e−5 3.67e−6 5.66e−8

1.28e−2 3.16e−4 1.20e−4 1.93e−5 4.49e−6 3.93e−7

3.76 85.7 85.6 57.7 22.3 594

building. 4.4. Fragility surfaces for bi-hazards Previous work insisted that the probability of simultaneous earthquake and wind was extremely low, then, the structural responses induced by the bi-hazards were commonly ignored [2,11]. Li et al. [10] presented a copula-based approach to constructed the joint distribution of PGA and V based on the recorded data at the Dali region. As discussed above that the likelihood of simultaneously occurring earthquake and wind cannot be ignored, besides, for the high-rise buildings, even if V10 is relatively low, the value of wind speed will increase vastly at the higher building height. Thereby, it is imperative to investigate

Fig. 20. Histogram of annual damage probability.

11

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Pahf (EDP>C) =

N

∑ ∑ rkh Pahf,k,j (EDP>C) k=1 j=1

(18)

where word ‘h’ denotes the hazard circumstances, i.e., the individual earthquake, wind and bi-hazards; rkh is the chance of k-th load condition, viz., 0.5 for individual load condition and 0.25 for bi-hazards condition; Paf,k,jh(EDP > C) is the annual damage probability under a certain hazard circumstance, which is determined by Eq. (1). According to Eq. (18), the annual damage probability under each hazard circumstance is illustrated in Fig. 20 and also summarized in Table 7, in which the word “Summation” denotes the summing of the probabilities from the two individual hazards. Besides, the error is defined as |Psummation-PEarthquake&Wind|/PEarthquake&Wind. Fig. 20 shows that the annual damage probability decreases drastically with damage severity increasing. It is clear that, for the first damage state (DS-1), the wind load leads to much higher probability of damage comparing to seismic load. This is likely due to the fact that a certain conditional fragility corresponds to higher occurrence chance of wind speed than that of earthquake. This result confirms the significant effect of wind load in the design of high-rise buildings. Another observation is that, under DS-1, Pew ≈ Pe + Pw as listed in Table 7 with an error of 3.76%. This mainly because the responses of this high-rise building under the multiple hazards may be in the elastic status. Therefore, the bi-hazards induced MIDR should be sum of the responses associated with single seismic and wind loads. However, in the damage probability evaluation, the demand models are determined by the regression analysis, and the errors are inevitable. In contrast, for other damage states, the annual damage probabilities of bi-hazards differ vastly with the summation values. This may be owing to that the bihazards excited structural responses reach elastoplastic state, and a slight excitation will bring large response. In the open literature on the multiple hazards, the event of simultaneously occurring earthquake and wind was commonly assumed to be negligible groundlessly. However, it is of great interest from the results that, for the first four damage states, the bi-hazards induces the highest damage probability among these three hazard circumstances, which are contrary with the common assumptions. Because, in addition to the combination with both high intensities of seismic and wind loads, the seismic load with the lower PGA and wind load with the higher intensity simultaneously acting on the high-rise buildings also can cause larger structural responses (Fig. 17) and conditional fragility (Fig. 18). Yet, the joint occurrence probability of an earthquake with the lower PGA and wind with the higher wind speed may not be small as commonly guessed and can be comparable to the occurrence chance of an earthquake. For example, considering a certain joint event with PGA = 0.05 g and V = 18 m/s corresponding to an exceedance probability of 0.0013, the maximum inter-story drift ratio under load condition III(1) is 0.0046. In contrast, for an individual seismic event, the MIDR of 0.0046 corresponds to a PGA of 0.329 g under load condition I(1), which is associated with an exceedance chance of 0.0016. The contributions, defined as 100%×Pafh/Paf, varying with respect to damage states are depicted in Fig. 21. The total annual damage probability is the summation of all possible damage probability, given by

Fig. 21. Contribution curves of each hazard circumstance to the total probability.

the resistance of high-rise buildings, located at the multihazard-prone areas, simultaneously against seismic and wind loads. During the lifetime of a building, the combinations of different intensities of earthquake and wind may differ significantly [75]. To address this fact and evaluate the occurrence probabilities of the bi-hazards of earthquake and wind with various intensities, the selected 100-earthquake records and 100-simulated wind load time histories are used to form the 100random combined samples, as scattered in Fig. 16. In conducting the dynamic analysis of the SFRCT building against the simultaneous application of seismic and wind excitations, for the first 400 s [27], the wind load is the sole input load. Following the finish of this period, the earthquake load is combined in the simulation [27]. As mentioned in Section 4.2, the Case III(1), Case III(2), Case IV(1) and Case IV(2) are respectively considered to discuss the coupling effects of bi-hazards. Table 4 provides the regression results of the demand model as well as βEDP|PGA,V and PRESS. In which, for the demand model of bi-hazards, “Along X-axis” denotes seismic load along X-axis. It can be observed from the coefficients associated with the bi-hazards (i.e., the last four rows of Table 4) that the coefficients k2 (attributable to wind) are larger than k1, which implies that the effect of the wind load on the MIDR is significant. Besides, the values of PRESS imply that the optimal demand model closely matches the structural responses under the multiple hazards. Fig. 17 (a) and (b) respectively illustrate the MIDR with reference to PGA and V in the logarithm coordinate for Case III(1) and Case IV(1). As an example, the spatial distribution of conditional failure probability under DS-3, when the seismic load is along with X-axis, is respectively shown in Fig. 18 (a) and (b). Obviously, Fig. 18 is difficult to demonstrate the difference between these two cases clearly. For this purpose, the fragility curves for given PGA and V with the seismic load along X-axis are shown in Fig. 19, in which, ‘III(1)’ and ‘IV(1)’ respectively denote the Cases III(1) and IV(1). With reference to the individual hazard models discussed in Section 3, the given PGA and V respectively corresponding to 500 and 10-year return periods are calculated to be 0.25 g and 31.3 m/s. When the wind speed is given, the conditional failure probability with respect to PGA of Case III(1) is visibly larger than the corresponding value of Case IV(1), as shown in Fig. 19(a). The conditional failure probabilities for a certain PGA, i.e., Fig. 19(b), illustrates similar trends with figure (a).

w Paf = Paef + Paew f + Pa f

(19)

It can be observed from Fig. 21 that the contributions of earthquake increase with damage severity increasing, conversely, the contributions of wind is steadily dropping. Because, the wind load is difficult to cause large elastoplastic response compared to earthquake. The contribution curve of bi-hazards can be divided into three sections, i.e., the rapidly increasing segment (OA), slow descent (AB) and fast descending section (BC). The reason for OA is that, comparing to earthquake induced responses, the occurrence probability of bi-hazards with certain PGA and V inducing the equal inter-story drift is higher. Besides, the contribution of bi-hazards descends slowly between points A and B, and this is likely

5. Annual damage risk analysis Upon completion of the hazard models and fragility analyses, the annual damage probability can be calculated by 12

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Acknowledgement

because of the slow descent of occurrence chance of bi-hazards from DS-2 to DS-3. Furthermore, the occurrence chance of bi-hazards at both higher intensities descends fast, which may be the cause of the BC segment. It is worth stressing that, via a thorough examination of the resulting data, the bi-hazards obviously dominates the total annual damage probability under the first four damage states. The results yield greater insight into the importance of comprehensively investigating the impacts of bi-hazards on the high-rise buildings rather than blindly and groundlessly ignoring these influences.

The authors would like to express our sincere gratitude to the financial support of National Key R&D Program of China (2016YFC0701108) and National Natural Science Foundation of China (51738007). The China Meteorological Administration and China Strong Motion Network Centre at Institute of Engineering Mechanics, China Earthquake Administration, who provided the recorded wind and earthquake data, are also greatly acknowledged. References

6. Conclusions

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This paper presents a multihazard-based framework for the damage risk assessment under the earthquake and wind hazards separately and concurrently, which is applied to a 42-story steel frame-RC core tube buildings located at Dali, a region prone to the earthquake and strong wind hazards. The effects of uncertainties associate with material parameters, damping ratio, load intensity and various load conditions are taken into account. The main conclusions are summarized as follows: (1). The results of multihazard demand model indicate that the wind speed has a significant impact on structural responses. The fragility of the exemplary building under the seismic and wind excitations are vastly different against various load conditions. (2). The mean annual occurrence models of individual earthquake/ strong wind and the bi-hazards are constructed according to the recorded earthquake and wind data. Due to lacking of the arbitrary point-in-time data, the constructed joint annual occurrence rate may be overestimated. Besides, the annual damage probability is sensitive to damage severity. Furthermore, when obtaining the certain cost corresponding to each damage level, the fragility curves and surface can be directly incorporated into a loss estimation of high-rise buildings against multiple hazards of earthquake and wind. (3). The bi-hazards induced damage probability is the largest compared to individual earthquake and wind under most damage states, which is contradicted with the widely accepted assumption of ignoring the joint event of earthquake and wind. It is essential to examine the safety of high-rise buildings under simultaneously acting seismic and wind loads. The observations may not be well extended to other locations, however, the proposed framework provides a vital support for the necessity of examining the response of high-rise buildings against the combined seismic and wind loads. Its application provides insight on characterizing the damage probability of high-rise/tall buildings under the earthquake and wind hazards separately and concurrently considering the hazard models. Besides, the efforts devoted by this paper are a tentative step of Performance-based Multihazard Engineering (PBME) and have the potential to serve as an aid to decision-making and risk-based design. Note that China is a multihazard-prone country, e.g., the earthquake, wind and flood, in the world. The Yun-Gui Plateau, Tibet Plateau and Pamirs are the regions with both high seismicity and wind hazard. In contrast with the eastern region of China, the economy of these mentioned regions located in the western China is relatively backward. Once the extreme hazards (such as the event of concurrent earthquake and wind) occur, it will cause enormous economic losses and casualties. Thereby, a multihazard-based approach to assess the damage risk of high-rise buildings against separate and concurrent seismic and wind excitations should be discussed in details. Declaration of Competing Interest The authors declared that there is no conflict of interest. 13

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