Identifying Optimal Intensity Measures for Predicting Damage Potential of Mainshock–Aftershock Sequences

Abstract

Large earthquakes are followed by a sequence of aftershocks. Therefore, a reasonable prediction of damage potential caused by mainshock (MS)–aftershock (AS) sequences is important in seismic risk assessment. This paper comprehensively examines the interdependence between earthquake intensity measures (IMs) and structural damage under MS–AS sequences to identify optimal IMs for predicting the MS–AS damage potential. To do this, four categories of IMs are considered to represent the characteristics of a specific MS–AS sequence, including mainshock IMs, aftershock IMs (i.e., IM^MS and IM^AS, respectively), and two newly proposed IMs through taking an entire MS–AS sequence as one nominal ground motion (i.e., IM₁^MS–AS), or determining the ratio of IM^AS to IM^MS (i.e., IM₂^MS–AS), respectively. The single-degree-of-freedom systems with varying hysteretic behaviors are subjected to 662 real MS–AS sequences to estimate structural damage in terms the Park–Ang damage index. The intensities in terms of IM^MS, IM^AS, and IM₁^MS–AS are correlated with the accumulative damage of structures (i.e., DI₁^MS–AS). Moreover, the ratio (i.e., DI₂^MS–AS) of the AS-induced damage increment to the MS-induced damage is related to IM₂^MS–AS. The results show that IM₂^MS–AS exhibits significantly better performance than IM^MS, IM^AS, and IM₁^MS–AS for predicting the MS–AS damage potential, due to its high interdependence with DI₂^MS–AS. Among the considered 22 classic IMs, Arias intensity, root-square velocity, and peak ground displacement are respectively the optimal acceleration-, velocity-, and displacement-related IMs to formulate IM₂^MS–AS. Finally, two empirical equations are proposed to predict the correlations between IM₂^MS–AS and DI₂^MS–AS in the entire structural period range.

1. Introduction

Past large earthquake events, such as the 1999 Chi-Chi earthquake [1], the 2008 Wenchuan earthquake [2], and the 2011 Tohoku earthquake [3] have shown that structures located in seismically active regions are commonly exposed to sequential-type excitations during a strong earthquake event. The cascading effects of mainshock (MS) and the following aftershocks (AS) could lead to a significant accumulation of structural damage, which tends to make a structure vulnerable to severe damage and even collapse. Nevertheless, current seismic design codes usually consider the design earthquake as a single event only, yet neglecting the detrimental effects caused by MS–AS sequences. Therefore, it is important to estimate the cumulative damage of structures under MS–AS sequences in an appropriate way, for accurately evaluating their dynamic performance subjected to a major earthquake event.

In recent years, numerous studies have been conducted to evaluate structural damage caused by MS–AS sequences using single-degree-of-freedom (SDOF) systems. These studies could be traced back to Mahin [4] who first examined the inelastic response of a SDOF system under a specific MS–AS sequence consisting of one MS and two ASs. It was reported that the ASs could lead to a significant increase in structural inelastic deformation and energy dissipation. After that, various structural performance indicators, e.g., peak displacement response [5,6,7], behavior factor [8,9], ductility demand [8,10,11,12,13], input energy [14], damage index [15], and collapse capacity [16], have been adopted to examine the AS effect by subjecting SDOF systems to a suite of real [11,12] or artificial [8,9,10,11,15,16] MS–AS sequences. Recently, multiple degree of free dom systems (MDOFs) have been used to investigate the structural damage caused by MS–AS sequences; the studied structures include steel frames [17] bare and infilled reinforced concrete frames [18,19,20], wood structures [21], equipped structures [22], bridges [23] and other kind of structure [24]. The aforementioned studies compare the structural damage under MSs alone and MS–AS sequences, showing the significant effect of aftershocks on structural performance under earthquakes.

Accurate prediction of the damage potential of an earthquake has always been a well concerned issue in earthquake engineering [25]. To estimate the earthquake damage potential, it is necessary to introduce two intermediate variables, one representing the strength or severity of the earthquake and the other describing the actual damage or destructiveness due to the earthquake. Therefore, investigation of the interdependence between the earthquake intensity measures (IMs) and the structural damages plays a core role in the prediction of earthquake damage potential. An IM with a close dependence on structural damage can well reflect the damage potential of an earthquake. In recent years, different IMs have been linked with various structural damage indices to examine their correlation levels [26,27,28,29,30,31,32]. These studies have reported that the correlation between IMs and structural damage is affected by many factors, including the near- or far-fault ground motions [33,34], dimensionality of earthquake inputs [35,36], structural properties [37], and structural damage indicators adopted [38]. Due to the complex characteristics of ground motions, no IM alone can be regarded as the optimal one for assessing earthquake damage potential. Recently, several advanced IMs have been developed by combining classic IMs together to improve the capability of predicting damage potential [39,40,41].

In the studies stated above, however, the focus is mostly attached on the damage potential of single earthquake events without accounting for the effect of sequential earthquakes. In other words, the damage potential for the MS–AS sequences has not been paid adequate attention as that for the MSs alone. Recently, Elenas et al. [42] and Kavvadias et al. [43] examined the damage potential of MS–AS sequences by relating MS and AS intensities to the accumulative damage of reinforcement concrete (RC) frame structures under MS–AS sequences, respectively. Yet, only one RC frame structure was considered in both studies, leading to the limitation of the results to specific structural property. Moreover, only MS [42] or AS [43] IMs are considered to represent MS–AS sequences. This is unfavorable to reflect the real damage potential of a MS–AS sequence since the MS–AS-induced accumulative damage of structures is, to a large extent, dependent on the intensities of both MS and AS. Therefore, there is an urgent need to make a comprehensive investigation on the prediction of damage potential for MS–AS sequences.

The main purpose of this study is to identify a set of optimal IMs of MS–AS sequences which exhibit a high correlation to structural damages. To cover a wide range of general structural properties, several representative SDOF systems with varying hysteretic behaviors are taken as the study cases. The considered SDOF systems are then subjected to a suite of 662 sequential earthquake inputs selected from 13 earthquake events. The structural accumulative damage caused by the MS–AS sequences is obtained using the damage index proposed by Park et al. [44] and modified by Kunnath et al. [45]. Based on the obtained results, a comprehensive correlation analysis is conducted to search for the optimal IMs for predicting damage potential of MS–AS sequences.

2. Intensity Measures for MS–AS Sequences

In this study, four categories of IMs were used to represent MS–AS sequences based on 22 classic IMs. They are PGA [46], I_A [47], P_a [48], E_a [49], a_rms [50], a_rs [51], I_c [44], I_a [52], PGV [46], P_v [48], P_D [53], E_v [49], v_rms [50], v_rs [51], I_v [52], I_F [54], PGD [46], P_d [48], E_d [49], d_rms [50], d_rs [51], I_d [52]. The 22 classic IMs have also been used in [38], which can be divided into three groups related to acceleration (see Figure 1), velocity (see Figure 2), and displacement (see Figure 3), respectively. It is noteworthy that the structure-specific IMs are not included in this study since they are commonly related to the structural fundamental period that would be elongated due to the MS-induced damage. In other words, two different structural periods are needed to define a structure-specific IM for the MS and the following AS. Thus, this study only adopts IMs associated with the basic properties of ground motions.

The considered four MS–AS IM categories include the mainshock IM, i.e., IM^MS, and aftershock IM, i.e., IM^AS, which are taken as the important intensity indicators for predicting the damage potential of MS–AS sequences according to Elenas et al. [42] and Kavvadias et al. [43], respectively. Besides IM^MS and IM^AS, two new types of IMs for MS–AS sequences were proposed herein, which are briefly represented by IM₁^MS–AS and IM₂^MS–AS. Using the considered 22 classic IMs, IM₁^MS–AS is defined by taking the entire MS–AS sequence as one nominal time history of ground motion acceleration. In particular, the peak amplitude-related IMs, including PGA, PGV, and PGD, are used to determine IM₁^MS–AS by max (IM^MS, IM^AS). As for the other IMs related to ground motion duration, they are used to define IM₁^MS–AS using the combined duration of the MS and the associated AS ground motions. As for IM₂^MS–AS, it is defined by the ratio of IM^AS to IM^MS, i.e., IM₂^MS–AS = IM^AS/IM^MS. It is clear that IM₂^MS–AS is a dimensionless and compound parameter by combing IM^AS and IM^MS together.

Table 1. Acceleration-related intensity measures (IMs) and the corresponding MS–AS IMs.

IMs	$\mathit{I}{\mathit{M}}^{\mathbf{M}\mathbf{S}}$	$\mathit{I}{\mathit{M}}^{\mathbf{A}\mathbf{S}}$	$\mathit{I}{\mathit{M}}_1^{\mathbf{M}\mathbf{S}–\mathbf{A}\mathbf{S}}$	$\mathit{I}{\mathit{M}}_2^{\mathbf{M}\mathbf{S}–\mathbf{A}\mathbf{S}}$
$PGA$	$PG{A}^{\mathrm{MS}}=Max\left[a{\left(t\right)}^{\mathrm{MS}}\right]$	$PG{A}^{\mathrm{AS}}=Max\left[a{\left(t\right)}^{\mathrm{AS}}\right]$	$PG{A}_1^{\mathrm{MS}–\mathrm{AS}}=Max\left(PG{A}^{\mathrm{MS}},PG{A}^{\mathrm{AS}}\right)$	$PG{A}_2^{\mathrm{MS}–\mathrm{AS}}=\frac{PG{A}^{\mathrm{AS}}}{PG{A}^{\mathrm{MS}}}$
$I_A$	$I_A^{\mathrm{MS}}=\frac{\pi }{2g}{{\displaystyle \int }}_0^{t_f^{\mathrm{MS}}}{\left[a{\left(t\right)}^{\mathrm{MS}}\right]}^{2}dt$	$I_A^{\mathrm{AS}}=\frac{\pi }{2g}{{\displaystyle \int }}_0^{t_f^{\mathrm{AS}}}{\left[a{\left(t\right)}^{\mathrm{AS}}\right]}^{2}dt$	$I_{A,1}^{\mathrm{MS}–\mathrm{AS}}=\frac{\pi }{2g}{{\displaystyle \int }}_0^{t_f^{\mathrm{MS}-\mathrm{AS}}}{\left[a{\left(t\right)}^{\mathrm{MS}-\mathrm{AS}}\right]}^{2}dt$	$I_{A,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{I_A^{\mathrm{AS}}}{I_A^{\mathrm{MS}}}$
$P_a$	$P_a^{\mathrm{MS}}=\frac{1}{t_{90}^{\mathrm{MS}}}{{\displaystyle \int }}_{t_5^{\mathrm{MS}}}^{t_{95}^{\mathrm{MS}}}{\left[a{\left(t\right)}^{\mathrm{MS}}\right]}^{2}dt$	$P_a^{\mathrm{AS}}=\frac{1}{t_{90}^{\mathrm{AS}}}{{\displaystyle \int }}_{t_5^{\mathrm{AS}}}^{t_{95}^{\mathrm{AS}}}{\left[a{\left(t\right)}^{\mathrm{AS}}\right]}^{2}dt$	$P_{a,1}^{\mathrm{MS}–\mathrm{AS}}=\frac{1}{t_{90}^{\mathrm{MS}-\mathrm{AS}}}{{\displaystyle \int }}_{t_5^{\mathrm{MS}-\mathrm{AS}}}^{t_{95}^{\mathrm{MS}-\mathrm{AS}}}{\left[a{\left(t\right)}^{\mathrm{MS}-\mathrm{AS}}\right]}^{2}dt$	$P_{a,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{P_a^{\mathrm{AS}}}{P_a^{\mathrm{MS}}}$
$E_a$	$E_a^{\mathrm{MS}}={{\displaystyle \int }}_0^{t_f^{\mathrm{MS}}}{\left[a{\left(t\right)}^{\mathrm{MS}}\right]}^{2}dt$	$E_a^{\mathrm{AS}}={{\displaystyle \int }}_0^{t_f^{\mathrm{AS}}}{\left[a{\left(t\right)}^{\mathrm{AS}}\right]}^{2}dt$	$E_{a,1}^{\mathrm{MS}–\mathrm{AS}}={{\displaystyle \int }}_0^{t_f^{\mathrm{MS}-\mathrm{AS}}}{\left[a{\left(t\right)}^{\mathrm{MS}-\mathrm{AS}}\right]}^{2}dt$	$E_{a,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{E_a^{\mathrm{AS}}}{E_a^{\mathrm{MS}}}$
$a_{r\mathrm{MS}}$	$a_{r\mathrm{MS}}^{\mathrm{MS}}=\sqrt{\frac{1}{t_{90}^{\mathrm{MS}}}{{\displaystyle \int }}_{t_5^{\mathrm{MS}}}^{t_{95}^{\mathrm{MS}}}{\left[a{\left(t\right)}^{\mathrm{MS}}\right]}^{2}dt}$	$a_{r\mathrm{MS}}^{\mathrm{AS}}=\sqrt{\frac{1}{t_{90}^{\mathrm{MS}}}{{\displaystyle \int }}_{t_5^{\mathrm{AS}}}^{t_{95}^{\mathrm{AS}}}{\left[a{\left(t\right)}^{\mathrm{AS}}\right]}^{2}dt}$	$a_{r\mathrm{MS},1}^{\mathrm{MS}–\mathrm{AS}}=\sqrt{\frac{1}{t_{90}^{\mathrm{MS}}}{{\displaystyle \int }}_{t_5^{\mathrm{MS}-\mathrm{AS}}}^{t_{95}^{\mathrm{MS}-\mathrm{AS}}}{\left[a{\left(t\right)}^{\mathrm{MS}-\mathrm{AS}}\right]}^{2}dt}$	$a_{r\mathrm{MS},2}^{\mathrm{MS}–\mathrm{AS}}=\frac{a_{r\mathrm{MS}}^{\mathrm{AS}}}{a_{r\mathrm{MS}}^{\mathrm{MS}}}$
$a_{rs}$	$a_{rs}^{\mathrm{MS}}=\sqrt{{{\displaystyle \int }}_0^{t_f^{\mathrm{MS}}}{\left[a{\left(t\right)}^{\mathrm{MS}}\right]}^{2}dt}$	$a_{rs}^{\mathrm{AS}}=\sqrt{{{\displaystyle \int }}_0^{t_f^{\mathrm{AS}}}{\left[a{\left(t\right)}^{\mathrm{AS}}\right]}^{2}dt}$	$a_{rs,1}^{\mathrm{MS}–\mathrm{AS}}=\sqrt{{{\displaystyle \int }}_0^{t_f^{\mathrm{MS}-\mathrm{AS}}}{\left[a{\left(t\right)}^{\mathrm{MS}-\mathrm{AS}}\right]}^{2}dt}$	$a_{rs,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{a_{rs}^{\mathrm{AS}}}{a_{rs}^{\mathrm{MS}}}$
$I_c$	$I_c^{\mathrm{MS}}={\left(a_{r\mathrm{MS}}^{\mathrm{MS}}\right)}^{1.5}{\left(t_{90}^{\mathrm{MS}}\right)}^{0.5}$	$I_c^{\mathrm{AS}}={\left(a_{r\mathrm{MS}}^{\mathrm{AS}}\right)}^{1.5}{\left(t_{90}^{\mathrm{AS}}\right)}^{0.5}$	$I_{c,1}^{\mathrm{MS}–\mathrm{AS}}={\left(a_{r\mathrm{MS}}^{\mathrm{MS}-\mathrm{AS}}\right)}^{1.5}{\left(t_{90}^{\mathrm{MS}-\mathrm{AS}}\right)}^{0.5}$	$I_{c,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{I_c^{\mathrm{AS}}}{I_c^{\mathrm{MS}}}$
$I_a$	$I_a^{\mathrm{MS}}=PG{A}^{\mathrm{MS}}·{\left(t_{90}^{\mathrm{MS}}\right)}^{1/3}$	$I_a^{\mathrm{AS}}=PG{A}^{\mathrm{AS}}·{\left(t_{90}^{\mathrm{AS}}\right)}^{1/3}$	$I_{a,1}^{\mathrm{MS}–\mathrm{AS}}=PG{A}^{\mathrm{MS}–\mathrm{AS}}·{\left(t_{90}^{\mathrm{MS}-\mathrm{AS}}\right)}^{1/3}$	$I_{a,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{I_a^{\mathrm{AS}}}{I_a^{\mathrm{MS}}}$

Notes: $I{M}^{\mathrm{MS}}$, $I{M}^{\mathrm{AS}}$, and $I{M}^{\mathrm{MS}–\mathrm{AS}}$ are the intensities for the MS, AS, and MS–AS sequence, respectively; $a{\left(t\right)}^{\mathrm{MS}}$, $a{\left(t\right)}^{\mathrm{AS}}$, and $a{\left(t\right)}^{\mathrm{MS}–\mathrm{AS}}$ are the acceleration time-histories for the MS, AS, and MS–AS sequence, respectively; $t_f^{\mathrm{MS}}$, $t_f^{\mathrm{AS}}$, and $t_f^{\mathrm{MS}–\mathrm{AS}}$ are the duration of the MS, AS, and MS–AS sequence, respectively; $t_{90}^{\mathrm{MS}}=t_{95}^{\mathrm{MS}}-t_5^{\mathrm{MS}}$, $t_{90}^{\mathrm{AS}}=t_{95}^{\mathrm{AS}}-t_5^{\mathrm{AS}}$, and $t_{90}^{\mathrm{MS}–\mathrm{AS}}=t_{95}^{\mathrm{MS}–\mathrm{AS}}-t_5^{\mathrm{MS}–\mathrm{AS}}$ are the duration for the MS, AS, and MS–AS sequence, respectively, between the corresponding times in terms of t₅ and t₉₅ corresponding to 5% and 95% of total I_A, respectively.

,

Table 2. Velocity-related IMs and the corresponding MS–AS IMs.

IMs	$\mathit{I}{\mathit{M}}^{\mathbf{M}\mathbf{S}}$	$\mathit{I}{\mathit{M}}^{\mathbf{A}\mathbf{S}}$	$\mathit{I}{\mathit{M}}_1^{\mathbf{M}\mathbf{S}–\mathbf{A}\mathbf{S}}$	$\mathit{I}{\mathit{M}}_2^{\mathbf{M}\mathbf{S}–\mathbf{A}\mathbf{S}}$
$P_v$	$P_v^{\mathrm{MS}}=\frac{1}{t_{90}^{\mathrm{MS}}}{{\displaystyle \int }}_{t_5^{\mathrm{AS}}}^{t_{95}^{\mathrm{MS}}}{\left[v{\left(t\right)}^{\mathrm{MS}}\right]}^{2}dt$	$P_v^{\mathrm{AS}}=\frac{1}{t_{90}^{\mathrm{AS}}}{{\displaystyle \int }}_{t_5^{\mathrm{AS}}}^{t_{95}^{\mathrm{AS}}}{\left[v{\left(t\right)}^{\mathrm{AS}}\right]}^{2}dt$	$P_{v,1}^{\mathrm{MS}–\mathrm{AS}}=\frac{1}{t_{90}^{\mathrm{MS}-\mathrm{AS}}}{{\displaystyle \int }}_{t_5^{\mathrm{MS}-\mathrm{AS}}}^{t_{95}^{\mathrm{MS}-\mathrm{AS}}}{\left[v{\left(t\right)}^{\mathrm{MS}-\mathrm{AS}}\right]}^{2}dt$	$P_{v,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{P_v^{\mathrm{AS}}}{P_v^{\mathrm{MS}}}$
$P_D$	$P_D^{\mathrm{MS}}=\frac{I_A^{\mathrm{MS}}}{{\left(v_0^{\mathrm{MS}}\right)}^2}$	$P_D^{\mathrm{AS}}=\frac{I_A^{\mathrm{AS}}}{{\left(v_0^{\mathrm{AS}}\right)}^2}$	$P_{D,1}^{\mathrm{MS}–\mathrm{AS}}=\frac{I_A^{\mathrm{MS}-\mathrm{AS}}}{{\left(v_0^{\mathrm{MS}-\mathrm{AS}}\right)}^2}$	$P_{D,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{P_D^{\mathrm{AS}}}{P_D^{\mathrm{MS}}}$
$E_v$	$E_v^{\mathrm{MS}}={{\displaystyle \int }}_0^{t_f^{\mathrm{MS}}}{\left[v{\left(t\right)}^{\mathrm{MS}}\right]}^{2}dt$	$E_v^{\mathrm{AS}}={{\displaystyle \int }}_0^{t_f^{\mathrm{AS}}}{\left[v{\left(t\right)}^{\mathrm{AS}}\right]}^{2}dt$	$E_{v,1}^{\mathrm{MS}–\mathrm{AS}}={{\displaystyle \int }}_0^{t_f^{\mathrm{MS}-\mathrm{AS}}}{\left[v{\left(t\right)}^{\mathrm{MS}-\mathrm{AS}}\right]}^{2}dt$	$E_{v,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{E_v^{\mathrm{AS}}}{E_v^{\mathrm{MS}}}$
$v_{r\mathrm{MS}}$	$v_{r\mathrm{MS}}^{\mathrm{MS}}=\sqrt{\frac{1}{t_{90}^{\mathrm{MS}}}{{\displaystyle \int }}_{t_5^{\mathrm{MS}}}^{t_{95}^{\mathrm{MS}}}{\left[v{\left(t\right)}^{\mathrm{MS}}\right]}^{2}dt}$	$v_{r\mathrm{MS}}^{\mathrm{AS}}=\sqrt{\frac{1}{t_{90}^{\mathrm{AS}}}{{\displaystyle \int }}_{t_5^{\mathrm{AS}}}^{t_{95}^{\mathrm{AS}}}{\left[v{\left(t\right)}^{\mathrm{AS}}\right]}^{2}dt}$	$v_{r\mathrm{MS},1}^{\mathrm{MS}–\mathrm{AS}}=\sqrt{\frac{1}{t_{90}^{\mathrm{MS}-\mathrm{AS}}}{{\displaystyle \int }}_{t_5^{\mathrm{MS}-\mathrm{AS}}}^{t_{95}^{\mathrm{MS}-\mathrm{AS}}}{\left[v{\left(t\right)}^{\mathrm{MS}-\mathrm{AS}}\right]}^{2}dt}$	$v_{r\mathrm{MS},2}^{\mathrm{MS}–\mathrm{AS}}=\frac{v_{r\mathrm{MS}}^{\mathrm{AS}}}{v_{r\mathrm{MS}}^{\mathrm{MS}}}$
$v_{rs}$	$v_{rs}^{\mathrm{MS}}=\sqrt{{{\displaystyle \int }}_0^{t_f^{\mathrm{MS}}}{v}^2\left(t\right)dt}$	$v_{rs}^{\mathrm{AS}}=\sqrt{{{\displaystyle \int }}_0^{t_f^{\mathrm{AS}}}{\left[v{\left(t\right)}^{\mathrm{AS}}\right]}^{2}dt}$	$v_{rs,1}^{\mathrm{MS}–\mathrm{AS}}=\sqrt{{{\displaystyle \int }}_0^{t_f^{\mathrm{MS}-\mathrm{AS}}}{\left[v{\left(t\right)}^{\mathrm{MS}-\mathrm{AS}}\right]}^{2}dt}$	$v_{rs,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{v_{rs}^{\mathrm{AS}}}{v_{rs}^{\mathrm{MS}}}$
$I_v$	$I_v^{\mathrm{MS}}={\left(PG{V}^{\mathrm{MS}}\right)}^{2/3}·{\left(t_{90}^{\mathrm{MS}}\right)}^{1/3}$	$I_v^{\mathrm{AS}}={\left(PG{V}^{\mathrm{AS}}\right)}^{2/3}·{\left(t_{90}^{\mathrm{AS}}\right)}^{1/3}$	$I_{v,1}^{\mathrm{MS}–\mathrm{AS}}={\left(PG{V}^{\mathrm{MS}-\mathrm{AS}}\right)}^{2/3}·{\left(t_{90}^{\mathrm{MS}-\mathrm{AS}}\right)}^{1/3}$	$I_{v,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{I_v^{\mathrm{AS}}}{I_v^{\mathrm{MS}}}$
$I_F$	$I_F^{\mathrm{MS}}=PG{V}^{\mathrm{MS}}·{\left(t_{90}^{\mathrm{MS}}\right)}^{0.25}$	$I_F^{\mathrm{AS}}=PG{V}^{\mathrm{AS}}·{\left(t_{90}^{\mathrm{AS}}\right)}^{0.25}$	$I_{F,1}^{\mathrm{MS}–\mathrm{AS}}=PG{V}^{\mathrm{MS}-\mathrm{AS}}·{\left(t_{90}^{\mathrm{MS}-\mathrm{AS}}\right)}^{0.25}$	$I_{F,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{I_F^{\mathrm{AS}}}{I_F^{\mathrm{MS}}}$

Note: $v{\left(t\right)}^{\mathrm{MS}}$, $v{\left(t\right)}^{\mathrm{AS}}$, and $v{\left(t\right)}^{\mathrm{MS}–\mathrm{AS}}$ are the velocity time-histories for the MS, AS, and MS–AS sequence, respectively; $v_0^{\mathrm{MS}}$, $v_0^{\mathrm{MS}}$, and $v_0^{\mathrm{MS}–\mathrm{AS}}$ are the numbers of zero-crossing per unit time of the acceleration time-histories corresponding to the MS, AS, and MS–AS sequence, respectively.

and

Table 3. Displacement-related IMs and the corresponding MS–AS IMs.

IMs	$\mathit{I}{\mathit{M}}^{\mathbf{M}\mathbf{S}}$	$\mathit{I}{\mathit{M}}^{\mathbf{A}\mathbf{S}}$	$\mathit{I}{\mathit{M}}_1^{\mathbf{M}\mathbf{S}–\mathbf{A}\mathbf{S}}$	$\mathit{I}{\mathit{M}}_2^{\mathbf{M}\mathbf{S}–\mathbf{A}\mathbf{S}}$
$PGD$	$PG{D}^{\mathrm{MS}}=Max\left[d{\left(t\right)}^{\mathrm{MS}}\right]$	$PG{D}^{\mathrm{AS}}=Max\left[d{\left(t\right)}^{\mathrm{AS}}\right]$	$PG{D}_1^{\mathrm{MS}–\mathrm{AS}}=Max\left(PG{D}^{\mathrm{MS}},PG{D}^{\mathrm{AS}}\right)$	$PG{D}_2^{\mathrm{MS}–\mathrm{AS}}=\frac{PG{D}^{\mathrm{AS}}}{PG{D}^{\mathrm{MS}}}$
$P_d$	$P_d^{\mathrm{MS}}=\frac{1}{t_{90}^{\mathrm{MS}}}{{\displaystyle \int }}_{t_5^{\mathrm{AS}}}^{t_{95}^{\mathrm{MS}}}{\left[d{\left(t\right)}^{\mathrm{MS}}\right]}^{2}dt$	$P_d^{\mathrm{AS}}=\frac{1}{t_{90}^{\mathrm{AS}}}{{\displaystyle \int }}_{t_5^{\mathrm{AS}}}^{t_{95}^{\mathrm{AS}}}{d}^2\left(t\right)dt$	$P_{d,1}^{\mathrm{MS}–\mathrm{AS}}=\frac{1}{t_{90}^{\mathrm{MS}-\mathrm{AS}}}{{\displaystyle \int }}_{t_5^{\mathrm{AS}}}^{t_{95}^{\mathrm{MS}-\mathrm{AS}}}{d}^2\left(t\right)dt$	$P_{d,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{P_d^{\mathrm{AS}}}{P_d^{\mathrm{MS}}}$
Ed	$E_d^{\mathrm{MS}}={{\displaystyle \int }}_0^{t_f^{\mathrm{MS}}}d{\left(t\right)}^{\mathrm{MS}}dt$	$E_d^{\mathrm{AS}}={{\displaystyle \int }}_0^{t_f^{\mathrm{AS}}}d{\left(t\right)}^{\mathrm{MS}}dt$	$E_{d,1}^{\mathrm{MS}–\mathrm{AS}}={{\displaystyle \int }}_0^{t_f^{\mathrm{MS}-\mathrm{AS}}}d{\left(t\right)}^{\mathrm{MS}}dt$	$E_{d,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{E_d^{\mathrm{AS}}}{E_d^{\mathrm{MS}}}$
drms	$d_{r\mathrm{MS}}^{\mathrm{MS}}=\sqrt{\frac{1}{t_{90}^{\mathrm{MS}}}{{\displaystyle \int }}_{t_5^{\mathrm{MS}}}^{t_{95}^{\mathrm{MS}}}{\left[d{\left(t\right)}^{\mathrm{MS}}\right]}^{2}dt}$	$d_{r\mathrm{MS}}^{\mathrm{AS}}=\sqrt{\frac{1}{t_{90}^{\mathrm{MS}}}{{\displaystyle \int }}_{t_5^{\mathrm{AS}}}^{t_{95}^{\mathrm{AS}}}{\left[d{\left(t\right)}^{\mathrm{AS}}\right]}^{2}dt}$	$d_{rms,1}^{\mathrm{MS}-\mathrm{AS}}=\sqrt{\frac{1}{t_{90}^{\mathrm{MS}}}{\int }_{t_5^{\mathrm{MS}-\mathrm{AS}}}^{t_{95}^{\mathrm{MS}-\mathrm{AS}}}{\left[d{\left(t\right)}^{\mathrm{MS}-\mathrm{AS}}\right]}^{2}dt}$	$d_{r\mathrm{MS},2}^{\mathrm{MS}–\mathrm{AS}}=\frac{d_{r\mathrm{MS}}^{\mathrm{AS}}}{d_{r\mathrm{MS}}^{\mathrm{MS}}}$
drs	$d_{rs}^{\mathrm{MS}}=\sqrt{{{\displaystyle \int }}_0^{t_f^{\mathrm{MS}}}{\left[d{\left(t\right)}^{\mathrm{MS}}\right]}^{2}dt}$	$d_{rs}^{\mathrm{AS}}=\sqrt{{{\displaystyle \int }}_0^{t_f^{\mathrm{AS}}}{\left[d{\left(t\right)}^{\mathrm{MS}}\right]}^{2}dt}$	$d_{rs,1}^{\mathrm{MS}-\mathrm{AS}}=\sqrt{{\int }_0^{t_f^{\mathrm{MS}-\mathrm{AS}}}{\left[d{\left(t\right)}^{\mathrm{MS}-\mathrm{AS}}\right]}^{2}dt}$	$d_{rs,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{d_{rs}^{\mathrm{AS}}}{d_{rs}^{\mathrm{MS}}}$
Id	$I_d^{\mathrm{MS}}=PG{D}^{\mathrm{MS}}·{\left(t_{90}^{\mathrm{MS}}\right)}^{1/3}$	$I_d^{\mathrm{AS}}=PG{D}^{\mathrm{AS}}·{\left(t_{90}^A\right)}^{1/3}$	$I_{d,1}^{\mathrm{MS}–\mathrm{AS}}=PG{D}^{\mathrm{MS}–\mathrm{AS}}·{\left(t_{90}^{\mathrm{MS}-\mathrm{AS}}\right)}^{1/3}$	$I_{d,2}^{\mathrm{MS}–\mathrm{AS}}=\frac{I_d^{\mathrm{AS}}}{I_d^{\mathrm{MS}}}$

Note: $d{\left(t\right)}^{\mathrm{MS}}$, $d{\left(t\right)}^{\mathrm{AS}}$, and $d{\left(t\right)}^{\mathrm{MS}-\mathrm{AS}}$ are the displacement time-histories for the MS, AS, and MS–AS sequence, respectively.

provide the considered IMs related to acceleration, velocity, and displacement, respectively, and the corresponding definitions for MS–AS ground motions.

Figure 1, Figure 2 and Figure 3 show the obtained values of the 22 classic IM^MS and IM^AS by using the acceleration-, velocity-, and displacement-related IMs, respectively. It is found that most of the selected MS ground motions have larger intensities than the corresponding AS ground motions regardless of the IMs adopted, indicating significantly higher hazard caused by MSs compared to ASs. As observed, approximately over 90% (except only 89.32% of PGA^MS values is greater than the corresponding PGA^AS values, as shown in Figure 1a) of the considered MS ground motions show larger IMs than the corresponding AS ground motions. This result shows the larger damage potential of MSs than the corresponding ASs. Since the I_A, E_a and a_rs have similar definition equations (see Table 1), their percentages corresponding to $I_A^{\mathrm{MS}}>I_A^{\mathrm{AS}}$ (Figure 1b), $E_a^{\mathrm{MS}}>E_a^{\mathrm{AS}}$ (Figure 1c), and $a_{rs}^{\mathrm{MS}}> a_{rs}^{\mathrm{AS}}$ (Figure 1d), respectively, are almost the same. Similar results can be also observed between E_v (Figure 2c) and v_rs (Figure 2e) and that between E_d (Figure 3c) and d_rs (Figure 3e). Moreover, among the considered acceleration, velocity, and displacement IMs, the integral-formed IMs show slightly larger percentage with IM^MS > IM^AS than the other IMs.

3. Structural Damage under MS–AS Sequences

The damage index (DI) is used to simulate structural damage. In this study, four kinds of structural damage are considered, including the mainshock-induced damage for the initial system, i.e., DI^MS, the AS-induced incremental damage for the mainshock-damaged system, i.e., DI^AS|MS, the accumulative damage caused by MS–AS sequences, i.e., DI₁^MS–AS, and the ratio of between DI^AS|MS and DI^MS, i.e., DI₂^MS–AS = DI^AS|MS/DI^MS. In the following studies, DI₁^MS–AS is related to IM^MS, IM^AS, and IM₁^MS–AS, and DI₂^MS–AS is related to IM₂^MS–AS.

To determine DI₁^MS–AS and DI₂^MS–AS, the structural damage index proposed by Park and Ang [44] is adopted herein. The Park–Ang damage index is a linear combination of maximum deformation response and hysteretic energy dissipation. Therefore, it increases monotonically so that it is beneficial to quantify the accumulative damage caused by MS–AS sequences [15]. The Park–Ang damage index has been modified by Kunnath et al. [45], and the modified damage index, DI, is expressed by

$$DI=\frac{u_{max}-u_y}{u_u-u_y}+\beta \frac{E_H}{F_y{u}_u}$$

(1)

where u_y and u_u are the yielding and ultimate displacements of a specific SDOF system under monotonic loading, respectively; u_max and E_H are the maximum displacement and absorbed energy of the system under earthquake loading, respectively; and β is the energy consumption factor, which is taken as β = 0.15 herein following Zhai et al. [15].

4. SDOF Systems

A total of eight different hysteretic models are considered herein to simulate the nonlinear behavior of SDOF systems under MS–AS sequences. Similar SDOF model assumptions and considerations have been used in [41]. Figure 4 shows the hysteretic rules of the considered models, where the force and displacement are both normalized by the yield force and yield displacement, respectively. A polygonal type of hysteretic model is adopted in this study for illustration. Compared to the smooth type of hysteretic models (e.g., Bouc–Wen models [55,56,57,58], which have the advantage of using a set of differential or algebraic equations), the polygonal type of hysteretic model requires less parameters, which can also include both strength and stiffness deterioration. The SDOF systems partly or totally w/wo considering P-Δ effect, pinching, and damage-induced deterioration are used to account for the varying hysteretic behaviors of the SDOF system.

Table 4. Detailed information of the considered SDOF systems.

	Delta	PinchX	PinchY	Damage1	Damage2
Model-1	0	1.0	1.0	0	0
Model-2	0.03	1.0	1.0	0	0
Model-3	0	0.8	0.2	0	0
Model-4	0	1.0	1.0	0.05	0.02
Model-5	0.03	0.8	0.2	0	0
Model-6	0.03	1.0	1.0	0.05	0.02
Model-7	0	0.8	0.2	0.05	0.02
Model-8	0.03	0.8	0.2	0.05	0.02

provides the considered eight SDOF models, where Delta is the parameter controlling the P-Δ effect; PinchX and PinchY are the pinching factors for strain (or deformation) and stress (or force) during reloading, respectively; and Damage1 and Damage2 denote the damage due to ductility and energy, respectively, which are used to account for the deterioration due to damage.

Among the considered models, the basic model, which is numbered as Model-1, is an elastic-plastic-with-hardening model (see Figure 4a). Based on the basic model, three important factors, including the P-Δ effect, pinching effect, and unloading degradation, are partly or totally incorporated in Models 2–8 to reflect more complex nonlinear behaviors of structures under seismic loading. In particular, Models 2–4 represent the bilinear hysteretic models by individually considering the P-Δ effect, pinching effect, and unloading degradation, respectively (see Figure 4b–d). For Models 5–7, two of the above-stated influence factors are accounted for in the hysteretic behavior, as shown in Figure 4e–g, respectively. Besides, Model-8 has the most complex hysteretic behavior since it collectively incorporates the P-Δ effect, pinching effect, and unloading degradation (see Figure 4h). It is noteworthy that the considered Models 2–8 are used for parametric study to examine the effect of hysteretic models of SDOF systems on the obtained results.

A total of 44 fundamental vibration periods ranging from 0.2 s to 6.0 s are considered for each SDOF system, consisting of 29 periods between 0.2 s and 3.0 s (with an increment of 0.1 s), and 15 periods between 3.2 s and 6.0 s (with an interval of 0.2 s). Four yield strength reduction factors, i.e., R = 2, 3, 4, and 5, are considered to account for the varying ductility levels of structures. Moreover, a constant damping coefficient ξ = 0.05 is utilized for all the considered SDOF systems according the past available studies, such as [15,21]. It is noteworthy that the analysis process of this paper is also suitable for the cases with other ξ values. However, the discussion on the effect of damping ratios is beyond this study and will be discussed in the future work. Consequently, a total of 1408 (8 hysteretic models × 44 vibration periods × 4 ductility levels) SDOF systems are used in subsequent calculations.

5. Correlation Analysis

5.1. Selection of MS–AS Sequences

A set of 662 real MS–AS earthquake sequences were selected from 13 earthquake events based on the NGA-West2 database (https://peer.berkeley.edu/research/nga-west-2) (Ancheta et al [58]). The NGA-West2 database has been used to develop the latest generation of ground motion prediction equations for shallow crustal earthquakes (e.g., Campbell and Bozorgni [59], Du et al. [32] The detailed information of the selected ground motion records and the related earthquake events can be found in Zhu et al. [60]. It is noteworthy that, among the aftershock sequences triggered by a specific mainshock, only aftershock record with the largest magnitude was adopted. Moreover, the selected mainshock and associated aftershock ground motions should be recorded at the same station.

Figure 5 shows the distribution of moment magnitudes M_w and rupture distances R_rup for the selected ground motions of MSs and ASs. The magnitude of MSs ranges from M_w = 5.8 to M_w = 7.62, while the magnitude of the corresponding ASs ranges from M_w = 5.01 to M_w = 7.14. As for rupture distances, the range of R_rup for MSs is between 0.32 km and 208.47 km and that for ASs is between 1.98 km and 218.16 km, respectively. As seen from Figure 5b, the M_w values of MSs are always larger than that of AS. This is compatible with the Bath’s rule [61]. Figure 6 compares the elastic response spectra for the selected ground motions of MSs and ASs. It is found that the median response spectrum of MS ground motions is significantly larger than that of AS ground motions, which is expected due to the greater shaking intensities of MSs than ASs.

For a given ground motion recorder, it could only detect seismic signals in a certain range of frequency due to its settings. This makes the recorded ground motions have a specific lowest usage frequency, i.e., f_low. Therefore, for dynamic analysis, a ground motion can only be used for nonlinear SDOF systems with their vibration periods no larger than T = 1/f_low. As for a specific MS–AS sequence, the f_low values for the MS and AS ground motions are probably different. For simplicity, the representative f_low value for a MS–AS sequence is taken as max (f_low^MS, f_low^AS), where f_low^MS and f_low^AS are the f_low values for the MS and AS ground motions, respectively. In this study, the highest structural period considered for the nonlinear SDOF systems is 6.0 s, and thus, the usable f_low values of the selected MS–AS sequences should be no less than 0.166 Hz. Figure 7 shows the number of usable earthquake sequences at different periods with the consideration of f_low requirement, where the number of usable sequences decreases with increasing periods of SDOF systems.

5.2. Correlation Coefficient

The Pearson correlation coefficient ρ is calculated for four IM-DI pairs in the logarithm space, including IM^MS vs. DI₁^MS–AS, IM^AS vs. DI₁^MS–AS, IM₁^MS–AS vs. DI₁^MS–AS, and IM₂^MS–AS vs. DI₂^MS–AS, respectively. The basic system with R = 2 at T = 0.2 s, 1.0 s, and 5.0 s are taken herein for illustration to represent the structure with short (acceleration-sensitive), medium (velocity-sensitive), and long (displacement-sensitive) vibration periods, respectively (Riddell [38]).

Figure 8 shows the scatters in terms of lnPGA^MS vs. lnDI₁^MS–AS, lnPGA^AS vs. lnDI₁^MS–AS, lnPGA₁^MS–AS vs. lnDI₁^MS–AS, and lnPGA₂^MS–AS vs. lnDI₂^MS–AS. It is clear that the correlation between the MS (or AS) intensity alone (i.e., PGA^MS or PGA^AS) and the structural cumulative damage (i.e., DI₁^MS–AS) is insignificant with the ρ values less than 0.3. Moreover, there is no clear improvement in the correlation between lnPGA₁^MS–AS and lnDI₁^MS–AS in which the entire earthquake sequence is regarded as one nominal ground motion. Yet, PGA₂^MS–AS shows a much stronger interdependence with DI₂^MS–AS, with correlation coefficients exceeding 0.5. Besides PGA, similar observation can also be obtained from the other IM-related data, which are summarized in Figure 9. As observed, the correlations between lnIM₂^MS–AS and lnDI₂^MS–AS for different considered IMs are always much larger than those of the other pairs (e.g., between lnIM^MS and lnIM₁^MS–AS). These results confirm that IM₂^MS–AS performs much better than the other three types of IMs, i.e., IM^M, IM^AS, and IM₁^MS–AS, for the prediction of the damage potential of earthquake sequences.

Figure 10 shows the correlations associated with the considered classic IMs between IM₂^MS–AS and DI₂^MS–AS. Among the acceleration-related IMs, I_A is the most optimal one to generate IM₂^MS–AS since I_A,2^MS–AS has higher correlation with DI₂^MS–AS than E_a,2^MS–AS, PGA^MS–AS, I_a,2^MS–AS, a_rms,2^MS–AS, I_c,2^MS–AS, a_rs,2^MS–AS, and P_a,2^MS–AS. As for the velocity-related and displacement-related IM groups, the corresponding optimal IMs are v_rs and PGD, respectively.

According to the results of Figure 10, I_A, v_rs and PGD are found as the optimal IMs to formulate IM₂^MS–AS in the acceleration-, velocity-, and displacement-based IM group, respectively. Then the ρ values between the three pairs, including I_A,2^MS–AS vs. DI₂^MS–AS, v_rs,2^MS–AS vs. DI₂^MS–AS, and PGD₂^MS–AS vs. DI₂^MS–AS, are plotted together at the entire period range, as shown in Figure 11. It is noteworthy that only the results related to the basic SDOF model (i.e., Model-1) are used. It is clear that there is no single IM exhibiting the best correlation throughout the entire period range, and thus, a transition period (i.e., T_c) exists. In fact, before and after T_c, I_A and v_rs are the most optimal IMs (with the highest correlation to DI₂^MS–AS) to formulate IM₂^MS–AS, respectively.

6. Effect of Hysteretic Models

Two aspects of investigations are conducted on the effect of hysteretic models, including hysteretic model types and R levels. Figure 12 shows the calculated ρ values between the concerned four IM-DI pairs, including lnIM^MS vs. lnDI₁^MS–AS, lnIM^AS vs. lnDI₁^MS–AS, lnIM₁^MS–AS vs. lnDI₁^MS–AS, and lnIM₂^MS–AS vs. lnDI₂^MS–AS, using all the considered eight types of models with a constant R value of 2. As seen from this figure, the correlation between lnIM₂^MS–AS and lnDI₂^MS–AS is significantly higher than those of the other three IM-DI pairs regardless of the hysteretic model types. Among the acceleration-, velocity-, and displacement-related IMs, I_A, v_rs, and PGD are the most optimal IMs to formulate IM₂^MS–AS, respectively, due to the associated high correlation with DI₂^MS–AS. The observations stated above are consistent with those obtained in Figure 9. Moreover, varying hysteretic models does not cause significant differences in the obtained ρ values. Therefore, the effect of hysteretic model types on the obtained correlation results is limited and has no significant effect over the response for the current purposes.

To examine the effect of strength reduction ratio, the systems simulated by the basic model with varying R values, i.e., R = 2~5, are adopted for illustration. Figure 13 compares the calculated ρ values between lnIM^MS vs. lnDI₁^MS–AS, lnIM^AS vs. lnDI₁^MS–AS, lnIM₁^MS–AS vs. lnDI₁^MS–AS, and lnIM₂^MS–AS vs. lnDI₂^MS–AS. It is clear that varying R values does not change the observations obtained in Figure 10 and Figure 11, including: (1) lnIM₂^MS–AS shows better performance than IM^MS, IM^AS, and IM₁^MS–AS due to its higher correlation with DI₂^MS–AS; and (2) I_A, v_rs, and PGD are the most optimal IMs to formulate IM₂^MS–AS among the acceleration-related, velocity-related and displacement-related IMs, respectively. Figure 14 illustrates the effect of changing R on the ρ values between I_A,2^MS–AS and DI₂^MS–AS, v_rs,2^MS–AS and DI₂^MS–AS, and PGD₂^MS–AS and DI₂^MS–AS at the entire period range. As observed, the change of R values yields a slightly larger difference on the ρ values than that of varying hysteretic models, especially for the two IM-DI pairs in terms of v_rs,2^MS–AS and DI₂^MS–AS, and PGD₂^MS–AS and DI₂^MS–AS. This is because a system with larger R value is easier to suffer nonlinear behavior under a MS excitation, leading to a more complex effect of the AS on the structural damage.

7. Empirical Prediction Equations

In this section, empirical equations are developed to predict the correlation coefficients, ρ, between the most optimal IM₂^MS–AS and the resulting DI₂^MS–AS. As seen from Figure 11, I_A,2^MS–AS and v_rs,2^MS–AS are separately the most optimal IM to formulate IM₂^MS–AS before and after the transition period T_c. This conclusion is verified with considering different hysteretic models and varying R levels in this section. Therefore, a step-wise relationship is feasible to represent the relationship between ρ and T. Figure 15 collects the calculated ρ values between the most optimal IM₂^MS–AS and DI₂^MS–AS corresponding to different hysteretic models with varying R values. From a viewpoint of median prediction, the median ρ values, i.e., ρ_50%, at different T values are obtained by statistics; a stepwise equation consisting of a linear and a nonlinear section is regressed by Equation (2). It is clear that Equation (2) can appropriately fit the calculated ρ_50% values with a coefficient of determination as R² = 0.9.

$${\rho }_{50\%}=\{\begin{array}{c}-0.143\times T+0.87 T\le T_c=0.45 \mathrm{s}\\ 0.92\times exp\left[-{\left(\frac{T-4}{10}\right)}^2\right] T_c=0.45 \mathrm{s}\le T\le 6.0 \mathrm{s}\end{array}$$

(2)

where the transition period T_c is taken as 0.45 s. When T ≤ T_c, I_A is recommended as the best candidate to formulate I_A,2^MS–AS = I_A^AS/I_A^MS with the corresponding correlation with DI₂^MS–AS approximated as ρ_50% = −0.143 × T + 0.87. For the case of T < T_c < 6.0 s, v_rs,2^MS–AS = v_rs^AS/v_rs^MS is taken as the best IM for MS–AS sequences, whose median correlation with DI₂^MS–AS can be evaluated as ρ_50% = 0.92 × exp{−[(T − 4)/10]²}. Based on Equation (2), the predicted correlation ρ_50% is generally higher than 0.8, showing a significant interdependence between I_A,2^MS–AS (or v_rs,2^MS–AS) and DI₂^MS–AS. This result provides a basis that it is feasible to develop a linear relationship between I_A,2^MS–AS (or v_rs,2^MS–AS) and DI₂^MS–AS in logarithm space. This log-linear relationship is valuable for the quick assessment of structural damage caused by MS–AS sequences.

The variability of ρ in terms of the standard deviation σ_ρ is further calculated based on the obtained ρ values. Figure 16 shows the σ_ρ values along with the period. It is clear that a stepwise type equation is suitable to fit the σ_ρ values. Then the stepwise function shown in Equation (3) is developed, which shows an acceptable goodness-of-fitting with a value of R² equal to 0.84.

$${\sigma }_{\rho }=\{\begin{array}{c}0.024\times T+0.021\\ 0.003\times T^2-0.0254\times T+0.021\end{array} \begin{array}{c}T\le 1.0 \mathrm{s}\\ 1.0 \mathrm{s}\le T\le 6.0 \mathrm{s}\end{array}$$

(3)

8. Discussions

In this study, a new type of IM for MS–AS sequences, i.e., IM₂^MS–AS = IM^AS/IM^MS, is verified to be an desirable IM to predict MS–AS damage potential due to its high correlation with DI₂^MS–AS = DI^AS|MS/DI^MS. Based on this observation, reliable log-linear relationships can be established between IM₂^MS–AS and DI₂^MS–AS to predict the structural damage due to MS–AS sequences. Take the cases of T = 0.4 s and T = 3.0 s as examples; both periods are selected according to Equation (2) in the ranges of T ≤ T_c and T < T_c < 6.0 s, respectively. According to the results of Section 5 and Section 6, I_A,2^MS–AS and v_rs,2^MS–AS are proven to be the optimal IMs, respectively. The linear regressions between I_A,2^MS–AS and DI₂^MS–AS and between v_rs,2^MS–AS and DI₂^MS–AS are also provided at log-log axis in Figure 17.

It is clear that clear linear relationships have been developed in these two figures. Based on the developed log-linear relationships, once the IMs of the mainshock and the following aftershock are determined, the damage ratio DI₁^MS–AS can be predicted directly. If the initial damage of structures due to mainshock (i.e., DI^MS) has been estimated, the aftershock-induced damage increment (i.e., DI^AS|MS) and the accumulative damages (i.e., DI₁^MS–AS) due to the MS–AS sequence can be further calculated. Besides, a reliable log-linear relationship between IM₂^MS–AS and DI₂^MS–AS is also useful for fragility assessment under MS–AS sequences since it commonly assumes a log-linear relationship between earthquake IMs and engineering demand parameters (EDPs). To sum up, there are three folds of applications of the developed log-linear relationship between IM₂^MS–AS and DI₂^MS–AS, as: (1) to predict DI₂^MS–AS given IM₂^MS–AS; (2) to predict DI₁^MS–AS and DI^AS|MS given IM₂^MS–AS and DI^MS; and (3) to represent the relationship between IM and EDP for fragility assessment under MS–AS sequences.

9. Conclusions

The damage potential for MS–AS sequences was comprehensively conducted by examining the correlation between the intensities of MS–AS sequences and the corresponding structural damage. The following general conclusions have been drawn:

(1)

The correlations between IM^MS and DI₁^MS–AS, IM^AS and DI₁^MS–AS, IM₁^MS–AS and DI₁^MS–AS, and IM₂^MS–AS and DI₂^MS–AS were examined in the logarithm space for the SDOF systems with varying periods. The results show that the IM-DI pair in terms of IM₂^MS–AS and DI₂^MS–AS has the highest correlation among the considered four IM-DI pairs. In other words, the proposed IM, which is defined as IM^AS/IM^MS, has a better capability to predict damage potential caused by earthquake sequences in comparison with IM^MS, IM^AS, and IM₁^MS–AS.

(2)

A total of 22 classic IMs were considered as the candidates to define the intensities of MS–AS sequences. Amongst these IMs, I_A, v_rs, and PGD are the most optimal ones to formulate IM₂^MS–AS for the acceleration-related, velocity-related, and displacement-related IM groups, respectively, due to the high correlations with DI₂^MS–AS. There is no single IM to best formulate IM₂^MS–AS in the entire structural period range. I_A,2^MS–AS and v_rs,2^MS–AS are the best IMs before and after a transition period T_c, separately.

(3)

A comprehensive parametric study was conducted by considering various types of hysteretic models and different yield reduction factors. The results show that the effects of varying hysteretic models and yielding reduction factors are limited on the correlation between IM₂^MS–AS and DI₂^MS–AS. This result reveals that the obtained conclusion in this study is in a general prospect.

(4)

A step-wise equation consisting of linear and nonlinear parts was regressed as the median prediction of ρ between the best IM₂^MS–AS and the caused DI₂^MS–AS. Moreover, through regression analysis, another step-wise function was developed to denote the standard deviation of the ρ values along with T. The empirical equations can fit the obtained data reasonably well, providing an opportunity for the further studies to conduct a quick prediction of damage potential for a specific MS–AS earthquake sequence.