An investigation of the structural nonlinearity effects on the building seismic risk assessment under mainshock–aftershock sequences in Tehran metro city

Abstract

This research attempts to investigate the effects of neglecting the nonlinear behavior of structures on the estimated seismic risk assessment of buildings under mainshock–aftershock (MA) sequences. In this regard, the Tehran metro city is selected as a building site due to its high seismicity level. Three separate 5-, 10-, and 15-story buildings are considered and designed based on international design codes. The earthquake hazard scenarios containing mainshock–aftershock sequences are modeled randomly using a synthetic stochastic methodology for this region. Next, by implementing the Monte Carlo simulation method, buildings performances are obtained for a large number of different scenarios, and consequently, the lifetime direct losses imposed on the buildings are evaluated. To investigate the effect of structural nonlinearity, the described process is performed in two distinct scenarios: one of them assumes that the buildings behave linearly, while the other one allows the structures to respond nonlinearly. Finally, the level of dependency of calculated lifetime seismic risk to this parameter and also the contribution of different sources of losses, including physical damage, business interruption, and casualty losses, are investigated considering the aftershocks effect.

Keywords

aftershocks nonlinearity MA sequences seismic risk Tehran metro city

Introduction

Buildings located in the area surrounded by highly active faults are prone to considerable seismic risk, especially when the aftershocks’ effects are taken into account. Since the concept of aftershocks is not clearly addressed in the building design codes (European Committee for Standardization, 2004; Structural Engineering Institute, 2016), it is a big concern to know what will happen for the buildings damaged in the mainshock, and then, are supposed to withstand series of aftershocks. From a more general perspective, it is required to investigate this challenge not only from the structural aspect but also from the economic side, since the aftershocks can cause time-dependent losses to buildings and consequent disruption of businesses. One of the major causes of this high level of potential risk is the cumulative damages of buildings under sequential earthquakes, depending on the level of structures response entering the nonlinear behavior phase. The more the structural nonlinear deformations are, the higher the structural, non-structural, and economic losses level. Therefore, it is important to investigate the effect of structural nonlinear behavior in the estimated seismic risk of building.

In recent decades, different approaches are proposed for the single building seismic risk assessment throughout the life-cycle estimation concept. The damage probability matrix method (Onur et al., 2006; Yepes-Estrada et al., 2016) is one of them. In addition, the capacity spectrum method, applying the pushover analysis approach, was presented by the Federal Emergency Management Agency (2013), while many other researchers worked with the dynamic time history analysis method for the life-cycle cost assessment of buildings such as the ones performed by Porter et al. (2004), Lagaros (2007), Mitropoulou et al. (2011), Gencturk et al. (2016) Loli et al. (2017), Sharmin et al. (2018), Rabonza and Lallemant (2019), and Noureldin and Kim (2021). Besides, in another field, several other research works have been published with the focus on the application of vibration control systems in the reduction of buildings’ life-cycle cost (Park et al., 2004; Gidaris and Taflandis, 2012, 2015; Mitropoulou and Lagaros, 2016). The main disadvantage of all works, similar to the mentioned ones, is missing the aftershock effects in the evaluation of the life-cycle cost due to earthquake scenarios.

From the other side, to overcome this major shortcoming, an extensive effort has been launched to investigate the effect of aftershocks on the performance of different types of buildings (Han et al., 2014; Khansefid and Bakhshi 2019a; Luco et al., 2004; Li and Ellingwood 2007; Ruiz-García and Aguilar 2015; Zhou et al., 2013). Most of these works focused only on the changes in the structural responses of buildings taking into account the aftershocks’ effects while fewer research works dealt with the seismic loss and risk assessment of buildings under MA sequences considering the building nonlinear behavior. In this area, Raghunandan et al. (2015) evaluated the effects of severe mainshocks on the nonlinear response of buildings and the collapse capacity reduction of reinforced concrete buildings in their estimated seismic risk under following aftershocks. Shokrabadi and Burton (2018) also worked on the seismic risk of nonlinear reinforced concrete frames under MA sequences using a Markov chain approach. Zhang et al. (2019) estimated the seismic risk of a 42-story concrete building considering the nonlinear response of the structure under MA sequences. Shia et al. (2020) worked on the efficacy of shape memory alloy dampers in the reduction of seismic risk of nonlinear buildings under MA sequences. More recently, Khansefid (2021) studied the applicability of different types of passive vibration control systems on the mitigation of lifetime seismic risk of nonlinear buildings under MA scenarios.

The previous works considered the structural nonlinearity in the seismic risk assessment of an individual building. However, from the decision-makers’ point of view, the estimation of regional risk is more attractive. In this case, the consideration of nonlinearity of structures during the time history analysis of all buildings in a region can be significantly time-consuming. This fact was implied by Vargas-Alzate et al. (2020) in assessing the seismic risk of buildings in the city of Barcelona, as well as Lu et al. (2020) study where they attempted to propose a new approach for considering the effects of aftershock in the regional seismic risk assessment via nonlinear dynamic time history analysis.

As it is expected, the consideration of structural nonlinear behavior in the seismic risk estimation process would increase both accuracy and computational costs simultaneously. Therefore, some researchers did their assessment by using the simplified and/or linear structural models (Park et al., 2004; Rabonza and Lallemant, 2019). This present study is an attempt to investigate the effects of structural nonlinearity on the estimated lifetime seismic risk of buildings through a quantitative case study of buildings located in Tehran metro city by considering the mainshock–aftershock sequences. In other words, it is aimed to answer this question; whether it is accurate enough to use a simple linear structural model in the process of seismic risk assessment? Or the time history analysis of building should be done taking into account the nonlinear behavior of the structure?

In this regard, the Tehran metro city is selected as a basis for the calculation of buildings lifetime seismic risk, as an area prone to the high level of seismic hazard. Three models of 5-, 10-, and 15-story steel moment resisting frame structures are defined as a common type of residential building systems constructed in this metro city. The seismic hazard is simulated by an advanced random earthquake scenario model for the 50-year lifetime of buildings. The important feature of this hazard modeling approach is its capability to generate earthquakes with different intensity levels based on their probability of occurrence. Afterward, the buildings’ responses are evaluated in two separate cases under each random earthquake scenario. The first one considers that the building’s materials are not allowed to enter the nonlinear phase, while in the second one, the occurrence of nonlinear response is possible. This procedure is repeated as many times as required by the Monte Carlo simulation approach to generate a sufficient number of realization of evaluated lifetime seismic risk for building models.

The results show that the nonlinear behavior of buildings increases the seismic risk in comparison with the linear structure. On average, the risk level is amplified up to 25% if it is estimated only due to the mainshocks, while this change reaches 50% in the case of considering the aftershock effects. Among different types of losses, in the case of linear structural models, the physical damages of building contribute mostly to estimated loss values, while for the nonlinearly-behaved structures, the business interruption plays a more important role, which is due to the more downtime of buildings. Consideration of aftershocks also increases the lifetime seismic risk tangibly for the linear structures. It causes a 33% increase, while this value is 60% for the structures that behave nonlinearly. It highlights the importance of taking into account the aftershock effects in the lifetime seismic risk assessment of nonlinear structures. Finally, the contribution of different faults in the final estimated seismic risk is obtained.

Building models

Three separate building models are considered for this study, including 5-, 10-, and 15-story buildings with the moment resisting frame structural system. These building models are defined as representatives of typical buildings constructed in Tehran metro city with a geographical longitude and latitude of 51.39° and 35.69°. The geometrical configuration of these buildings is shown in Figure 1. In the plan, each building consists of five spans with the length 6 m in both horizontal directions. Also, the building story heights are typically 3.5 m, except for the first one, which is equal to 5.0 m. The yielding stress of steel material of all models is 240 MPa. The dead and live loads are presumed to be 6 kN/m² and 3 kN/m². The project site soil type is C, with the shear wave velocity of 450 m/s. All of the buildings are designed in accordance with the requirement of ASCE7-16 (Structural Engineering Institute, 2016). The input seismic spectrum used for the design process is developed with the short (S_DS) and 1-s period (S_D1) of buildings’ location equal to 1.13 and 0.47 for the 475-year return period (Engineering Optimization Research Group, 2011). The design of all building models lead to I-shape beam sections with the web plate of 300 × 8 mm up to 500 × 10 mm and the flange plate of 200 × 10 mm to 250 × 25 mm. The columns are also considered as a square box shape sections with a plate size of 250 × 15 mm to 500 × 30 mm.

Figure 1.

Geometry of building models.

Since it is supposed to perform a large number of analyses using the Monte Carlo simulation method, a simplified shear frame building model is used as a multi-degree-of-freedom system to calculate the response of structures with the following formula

M \ddot{x} (t) + C \dot{x} (t) + F_{R S} (t) = - M L {\ddot{x}}_{g} (t)

(1)

where

\ddot{x}

\dot{x}

, x, and

{\ddot{x}}_{g}

are structural acceleration response vector, relative velocity response vector, relative displacement response vector, and input earthquake acceleration, respectively. M, C, F_RS, and L are mass matrix, inherent damping coefficient matrix, non-linear restoring force vector, and influence vector, respectively.

To model the nonlinearity of buildings, the Bouc–Wen model (Ismail et al., 2009) is used due to its simplicity and accuracy. As it is seen in Figure 2, in this hysteretic model, four main parameters exist, including story initial stiffness (K), story yielding force (F_y), the secondary to initial stiffness ratio (α), and the maximum relative deformation capacity (D_max), which are obtained for each story of buildings via nonlinear push-over analysis approach. The values of α are 0.15, 0.17, and 0.20, and D_max is equal to 0.08 m, 0.09 m, and 0.11 m for the 5-, 10-, and 15-story buildings, respectively. The values of other two parameters for each of the building models are also depicted in Figure 3. It is important to note that when the story deformation reaches D_max, the building is collapsed and it can no more withstand the earthquake excitation. In addition, the damping matrices of all structural models are calculated using the Rayleigh method (Chopra 2011), considering 5% equivalent damping ratios for the first and third modes of vibration.

Figure 2.

Bouc–Wen nonlinear model.

Figure 3.

Stories stiffness and yielding force of building models.

Earthqauke hazard model

In accordance with the target of this research, it is required to perform a large number of nonlinear dynamic time history analyses, which is not practical to be done via using the real recorded acceleration time histories. Therefore, there are no other choices rather than adopting the earthquake models capable of simulating seismic hazard randomly. In this regard, among different models existed in this field (Dabaghi and Kiureghian, 2014; Khansefid and Bakhshi, 2019b; LeVeque et al., 2016; Rezaeian and Kiureghian, 2012; Zhuang, 2011; Zentner and Poirion, 2012), the one proposed by Khansefid and Bakhshi (2019b) is selected. Their model is developed for the Iranian plateau as a region with one of the highest seismicity levels in the world. The main advantage of this model is its capability in generating both random events, including mainshocks and aftershocks, as well as their corresponding accelerograms (Figure 4), which are elaborated in the following.

Figure 4.

Schematic of mainshock–aftershock sequences.

Sub-model for random event simulation

In the beginning, it is necessary to identify which fault is the candidate for earthquake triggering. In this regard, all the active faults in the area of Tehran metro city (Danciu et al., 2016), within a radius of 100 km around the building location, are considered (Figure 5). The seismicity properties of these faults are reported in Table 1. Next, for generating a random earthquake scenario, among all active faults, one of them is randomly selected based on their triggering probability obtained from their seismicity rate (λ). Afterward, for the selected causative fault, a random event scenario compatible with the conditions of the Iranian plateau is developed, containing mainshocks and aftershocks, for the time window of 50 years (Khansefid and Bakhshi 2019b). For the mainshock events, their number of occurrences in the considered time period is randomly determined by using the Poisson probability distribution, while the magnitude and occurrence time of each event are simulated by adopting the modified form of Gutenberg–Richter (Anagnos and Kiremidjian, 1988) and exponential distributions. In the case of aftershocks, the number of events is generated using the modified form of the Omori formula (Utsu et al., 1995). For the magnitude and occurrence time of each of these aftershocks, the proposed joint probability density function for the Iranian plateau is adopted (Khansefid and Bakhshi, 2018).

Figure 5.

Map of active faults in Tehran greater area.

Table 1.

Characteristics of active faults in Tehran greater area.

ID	Fault name	Length (km)	M_min	M_max	λ	Triggering probability
F01	North Tehran 1	72.2	5.5	7.2	0.02828	0.127
F02	North Tehran 2	46.9	5.5	7.0	0.01781	0.080
F03	Evanaki	82.1	5.0	6.9	0.01140	0.051
F04	Kahrizak	36.7	5.5	6.8	0.01802	0.081
F05	Taleghan	74.0	5.5	6.9	0.02778	0.125
F06	Mosha	155.7	5.0	7.6	0.02246	0.101
F07	Kandevan	97.2	5.5	7.3	0.01397	0.063
F08	Eshtehard	74.7	5.5	7.2	0.01958	0.088
F09	Siahkuh	86.8	5.5	7.2	0.01574	0.071
F10	Robat Karim	84.4	5.5	7.3	0.01376	0.062
F11	Garmsar	76.5	5.5	7.3	0.02164	0.097
F12	Pishva	35.3	5.5	7.5	0.01143	0.051

As a sample, in Figure 6, a randomly triggered MA earthquake scenario by the fault F03 is simulated for the probability level of 0.0006. It consists of two mainshocks with magnitudes of 5.3 and 5.9. These events occur in years 11 and 43, after the starting point of the scenario. These mainshocks are followed by 31 and 39 aftershocks, respectively. The magnitudes of aftershocks vary between 3.1 and 4.7 for all mainshocks.

Figure 6.

Sample of randomly generated event scenario.

Finally, in this research work, a total of 2000 event scenario realizations is generated using the aforementioned approach. The statistical properties of developed event scenarios are shown in Figure 7. The simulated events contain 1 or 2 mainshocks followed by the 20–57 aftershocks. The magnitudes of mainshocks are from 5 to 7, while the aftershocks’ one ranges from 3.7 to 5.4. The focal depth of events varies from 14 km to 26 km, and the site-to-source distance of all events starts from 3 km and goes up to 100 km. Moreover, the events are spread almost uniformly in all 50-year periods of simulation. In addition, the deaggregation of different magnitude, as well as site-to-source distances, due to the triggering of all faults are shown in Figure 8. It is clearly observable that, for the mainshocks, the highest contribution of magnitudes in the randomly generated scenarios are from 5 to 6, while in the case of aftershocks, magnitudes between 4 to 5 have the highest contribution. In the case of distance, the contribution of higher distance ranges (50–100 km) are less than the lower ones (0–40 km).

Figure 7.

Statistical properties of major characteristics of generated random events in all scenarios for mainshocks and aftershocks.

Figure 8.

Deaggregation of generated mainshocks and aftershocks in all random scenarios.

Sub-model for synthetic stochastic accelerogram simulation

After generating a random event scenario in the previous sub-model, it is necessary to simulate the corresponding synthetic stochastic accelerograms for each event, compatible with the seismic condition of the Iranian plateau (Khansefid et al., 2019). This sub-model is capable of simulating both far- and near-field ground motions. In this regard, for the case of far-field (broad-band) signals, first, a white noise signal is generated and then by using a time modulating function, q (t, α), and a unit-impulse response function, $h [t - τ . λ (t)]$ , the signal is modified in time and frequency domain, respectively, to achieve the final broad-band earthquake accelerogram

X (t) = q (t, α) {\frac{1}{σ_{f} (t)} \int_{- \infty}^{t} h [t - τ, λ (τ)] w (τ) d τ}

(2)

where X(t) is the earthquake acceleration signal,

λ (t)

is the time-varying parameter,

σ_{f} (t)

is the standard deviation of the unit-impulse response function filter, and

w (τ)

is the white noise signal. These functions and variables were obtained and reported by Khansefid et al. (2019) for the Iranian plateau.

An important issue in generating the random earthquake scenarios is to determine whether a simulated signal for a specific event should contain a near field velocity pulse or not. In this model, this phenomenon is considered based on the closeness of causative faults to the building location using the empirical probability distribution functions proposed by Khansefid (2020) for the observation of pulse-like signals in the Iranian plateau.

In the case of generating random pulse-like signals, first, a broad-band signal (called residual signal) is procreated by the broad-band model, and then a velocity pulse, simulated by equation (3), is added to the residual signal (Dabaghi, and Kiureghian 2014)

\begin{array}{l} v_{pul} (t) = {\frac{1}{2} V_{p} \cos [2 π (\frac{t - t_{\max, p}}{T_{P}}) + ν] - \frac{D_{r}}{γ T_{P}}} {1 + \cos [\frac{2 π}{γ} (\frac{t - t_{\max, p}}{T_{P}})]} \\ t_{0} - \frac{γ}{2} T_{P} < t \leq t_{0} + \frac{γ}{2} T_{P} \end{array}

(3)

where t_max,p, V_P, T_P , ν, γ, and D_r are the occurrence time of peak value of pulse velocity, pulse amplitude, pulse duration, phase angle, oscillatory character, and a variable eliminating the undesired residual displacement at the end of pulse time series, respectively.

In the end, by using this procedure for all events of previously described 2000 random earthquake event scenarios, the corresponding synthetic stochastic accelerograms are simulated. As an example in Figure 9, the acceleration time series for two of the previously generated random event (Figure 6) are shown: one for the first mainshock and the other one for the 15th aftershock of the second mainshock with the magnitude of 3.9.

Figure 9.

Sample of time series of generated accelerogram for mainshocks and aftershocks.

In Figure 10, the statistical properties of all generated accelerograms are depicted. It is seen that the peak ground acceleration (PGA) of mainshocks reaches 1.0 g, while the aftershock’s one remains below 0.40 g. The PGVs vary from 0.01 m/s to 0.4 m/s, in the case of mainshocks, where the aftershocks’ PGVs are in the range of 0.01–0.2 m/s. As another important characteristic of accelerograms, the duration of mainshocks and aftershocks is shown, where their maximum values are 10 s and 7 s, respectively. Finally, as a measure of frequency content, the procreated accelerograms contain signals with an approximate main period of around 0.2–1.0 s.

Figure 10.

Statistical properties of major characteristics of generated random accelerograms in all scenarios for mainshocks and aftershocks.

Seismic risk assessment

To evaluate the seismic risk of building models, the Monte Carlo simulation method is adopted. Accordingly, at the first step, a series of random seismic hazard is generated, then the responses of building models are obtained via the dynamic time history analysis in two separate cases of considering and not considering the nonlinearity of structures. Afterward, using the fragility functions, the severity of different damage types of buildings is obtained, and consequently, the total direct loss values of models are calculated.

Loss estimation

To estimate the loss values of buildings, there are different approaches, such as the ones proposed by the Hazus (FEMA, 2013) and FEMA-P58 (FEMA, 2018). Three main types of direct losses are taken into account, namely, physical damage loss, casualty loss, and business interruption loss.

First, for the physical loss, it is necessary to evaluate the overall damage state of buildings under the earthquake hazard. As of Hazus’ suggestion, four damage types are considered, including the structural elements damage (STR), non-structural drift-sensitive components damage (NSD), non-structural acceleration-sensitive components damage (NSA), and the internal contents damage (CON). Each of these damage types has four sequential states defined as slight, moderate, extensive, and complete. If the structural responses of building models are given, the probability of facing any damage states under the considered earthquake can be calculated by the fragility functions (FEMA, 2013)

P_{i} [D S_{j} | EDP] = Φ [\frac{1}{β_{DS}} \ln (\frac{EDP}{ED P_{DS}})]

(4)

where Φ is the standard normal cumulative distribution function; i is the physical damage type (STR, NSD, NSA, CON); EDP is the engineering demand parameter, either structural story acceleration or drift response; EDP_DS and β_DS are the median value and natural logarithmic standard deviation of damage states’ thresholds (given by Hazus); and DS_j is the jth damage state.

After determining the probability of occurrence of each damage level. The total direct loss value, due to physical damage of buildings and their content, is obtained by the following formula:

E_{i} (C) = A \cdot BRPC \sum_{i = 1}^{4} P_{i \cdot j} C_{i \cdot j}

where E_i(C) is the expected loss value due to damage type i, A is the total area of building in a square meter, BRPC is the building replacement cost (US$250 per square meter), P_i,j is the probability of occurring damage type i at level j, and C_i,j is the loss ratio (per BRPC) of damage type i at level j, presented in Table 2 (FEMA, 2013), that is, the repair or replacement cost per square meter for each damage type and level.

Table 2.

Damage cost of different types per percentage of building replacement value.

Damage level	Residential
Damage level	STR	NSD	NSA	CON
Slight	0.3	0.9	0.8	1.0
Moderate	1.4	4.3	4.3	5.0
Sever	6.9	21.3	13.1	25.0
Complete	13.8	42.5	43.7	50.0

STR: structural elements damage; NSD: non-structural drift-sensitive components damage; NSA: non-structural acceleration-sensitive components damage; CON: internal contents damage.

It is noteworthy to mention that to consider the cumulative damages due to the consideration of mainshock–aftershocks events, the final condition of buildings at the end of each earthquake is considered as an initial condition at the next aftershock. In addition, in order to be reassured about the capturing final condition of the building at each accelerogram, 30 s zero pad is added to the end of each acceleration time series, which allows the simulation process to take the building-free vibration effect into account.

For the casualty losses, the FEMA-P58 (FEMA, 2018) approach is used. Accordingly, the typical occupancy pattern of people in residential buildings is adopted (Figure 11). The maximum number of presented persons in a 100 m² of building area at the peak time is assumed to be equal to four. The ransom cost of a dead person is taken equal to US$100.000. Also, it should be mentioned that the collapse condition is considered separately for calculating the casualties.

Figure 11.

Occupancy pattern of residential buildings.

The physical damage of buildings will cause a temporary disruption in their operation, which may lead to considerable economic loss. In reference to the Hazus (FEMA, 2013) method, two main types of direct economic losses can be quantified after the earthquake occurrence, which are time-dependent. These losses are rental income loss and relocation cost calculating by the following formulas

E_{REC} (C) = (1 - OO) \times RT \times \sum_{i = 1}^{4} P_{i} \times TRANS + (OO) \times RT \sum_{i = 1}^{4} P_{i} \times (TRANS + RENT)

E_{RIL} (C) = (1 - OO) \times A \times RENT \times \sum_{i = 1}^{4} P_{i} \times RT

(5)

In the above equations, P_i is the occurrence probability of damage state i. OO is the owner-occupied factor considered 0.6 (Statistical Center of Iran 2017). RT and LOF are the repair time and loss of function time of the building after experiencing the damage, adopted from Hazus (FEMA, 2013), for different damage levels. RF is the recapturing factor of the business, assumed equal to be 0.9, according to the Hazus (FEMA, 2013) suggestion. RENT and TRANS are the rental rate cost per day and the required cost for the shifting and transferring to the new temporal accommodation, respectively. These parameters are considered to be US$7 per month per square meter and US$500, on average.

Monte Carlo simulation and uncertainties

There are different approaches for simulating earthquake scenarios and their consequences, including Monte Carlo sampling, importance sampling, slice sampling, and rejection sampling (Mackay 1998; Robert and Casella 2004). In this research, the first one is selected due to its simplicity and straight-forward vision. In the Monte Carlo simulation process, an important challenge is the number of generated realizations. In this research, for achieving the 0.02 coefficient of variation in estimating the target loss value, with the confidence level of 95% (Hahn 1972), more than 1225 samples are required, while in the following steps, 2000 samples are generated.

Both of the earthquake hazard as well as the buildings’ properties may cause some levels of uncertainty in the final estimated risk. The modeling procedure of the first item is described in the previous sections. However, for the latter, the uncertainties are considered by modeling the building mechanical characteristics, and parameters of loss estimation formula as lognormal random variables, with the means and coefficient of variations (COVs) described in Table 3, in accordance with the suggestions of JCSS (2001) with some modification to consider the local conditions of Tehran metro city buildings.

Table 3.

Mean values and COVs of the lognormal distributions considered for the building model parameters.

Parameter	Mean	COV
Story mass	DL = 6 kN/m², LL = 3 kN/m²	0.15
Story stiffness	Figure 3	0.15
Story yielding force	Figure 3	0.15
Building replacement value	US$250	0.25
Rental rate cost per day	US$7	0.25

COV: coefficient of variations.

Results and discussions

More than 72,000 times linear/nonlinear dynamic time history analysis is performed in this research for 2000 samples of linear and nonlinear structural models. The results of analyses are represented in two separate parts. First, the structural response of buildings under a selected mainshock–aftershock sequence is discussed, and in the second part, the obtained losses and their dependency on the structural nonlinearity are discussed.

Structural responses

In this part, the main responses of all building models under a selected earthquake mainshock–aftershock scenario are studied elaborately. This scenario includes 1 mainshock and 24 aftershocks generated by the activity of fault with the ID of F01. The magnitude of mainshock is 6.3, while the aftershocks’ magnitude varies from 4.1 to 4.9. The site-to-source distances of all events of this scenario are about 14–30 km. Figure 12 shows the variation of accelerograms’ PGA starts from 0.07 g and ends to 0.51 g. In addition, the acceleration time series of the mainshock and the aftershock with the highest PGA are also shown in this figure.

Figure 12.

Acceleration time series of mainshock and the strongest aftershock plus the PGA of all events in the selected mainshock–aftershock sequence.

In Figure 13, the structural responses of building models are illustrated. It is seen that the force–deformation relationship of the first story of all building models, in the case of nonlinear behavior, experiences considerable nonlinear hysteretic loop. The mean value for drift responses increases tangibly (about 56%) by considering the nonlinear behavior of structures, while the acceleration responses decrease about 34%, on average.

Figure 13.

Structural response of all building models under a single selected mainshock–aftershock sequence: (a) force-deflection of the first story under mainshocks event, (b) mean drift responses of all stories under all events of sequence, (c) mean acceleration responses of all stories under all events of sequence.

Loss estimation

In this part, the dependency of estimated seismic risk and loss to the structural nonlinearity is studied for all building models. In Figure 14, the probabilistic loss curves of different building models are depicted. In all models, the estimated loss values of linear structures are smaller than the nonlinear ones. The estimated loss value with the exceedance probability of 10%, called probable maximum loss (Porter 2020), is obtained 48%, 46%, and 41% for the nonlinear structure, while by neglecting the effect of building nonlinear behavior, these values are calculated 37%, 40%, and 32%, respectively. Thus, by ignoring the structural nonlinearity, the total loss is not tangibly underestimated. It is due to the higher level of damages related to the acceleration response of building (NSA and CON damage types), while the linear behavior is assumed for buildings.

Figure 14.

Estimated loss values of building models.

To investigate more about the sources of estimated loss, in Figure 15, the general trend of the contribution of main loss types, including physical, business interruption, and casualty are assessed to the level of severity of imposed losses. Interestingly, the fatality and injury losses are near zero, which implies that the target of building codes to save the life of occupants is achieved. In the linear structures, in different damage levels, the physical losses are estimated more than the business interruption one, since there is no permanent damage in the structure due to nonlinear behavior. However, in the case of considering the nonlinear structural behavior, the structural damage, and more generally, physical losses, is saturated. This high level of structural damage leads to a longer downtime of businesses, which causes more business interruption loss. This is more considerable, even in the higher seismic risk level.

Figure 15.

Contribution of different loss types in the total estimated loss value.

As another important parameter, the effect of aftershocks on the calculated seismic risk is illustrated in Figure 16. It is seen that by considering the aftershocks, the average of total seismic loss increases from 15% to 24% of building replacement value (60% increase) for the nonlinear structure, while, in the case of linear buildings, it changes from 12% to 16% of BRPC (33% increase). These results highlight that by neglecting the aftershock effects, the seismic risk is underestimated, especially when the building nonlinear behavior is possible.

Figure 16.

Effect of aftershocks on the estimated loss value.

The last but not the least important thing is the contribution of active faults in the total lifetime estimated loss. In Figure 17, this contribution is depicted for both cases of considering the building nonlinear behavior or neglecting. In both cases, the faults with the ID of 1, 4, 6, and 10 have the highest contribution. These faults are among the closest ones to the building site. In addition, the faults contribution, or better to say faults priority, does not depend considerably on the structural linear or nonlinear behavior.

Figure 17.

Contribution of different faults in the final estimated seismic risk.

Concluding remarks

This research focused on the investigation of effects of structural nonlinearity on the estimated seismic risk of an individual building located in Tehran metro city. Two separate cases were modeled with/without consideration of building nonlinear behavior. The structural performances under random mainshock–aftershock scenarios show that by considering the nonlinear behavior, the structural responses increase up to 56%, while the total seismic risk, including physical damage, business interruption, and casualty, increases in the lower range. In addition, when the nonlinear behavior of buildings is taken into account, the average estimated loss increases by about 25% and 50% without and with considering the aftershocks in comparison with the linear structure. Importantly, by ignoring the aftershocks, the total seismic risk is underestimated about 25% and 40% in the linear and nonlinear structures. Finally, it is shown that among different active faults around Tehran metro city, faults 1, 4, 6, and 10 are the most important ones. However, there are no differences in their priority by changing the building behavior from linear to nonlinear.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ali Khansefid

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