Vulnerability-Based Economic Loss Rate Assessment of a Frame Structure Under Stochastic Sequence Ground Motions

Abstract

Modeling mainshock–aftershock ground motions is essential for seismic risk assessment, especially in regions experiencing frequent earthquakes. Recent studies have often employed Copula-based joint distributions or machine learning techniques to simulate the statistical dependency between mainshock and aftershock parameters. While effective at capturing nonlinear correlations, these methods are typically black box in nature, data-dependent, and difficult to generalize across tectonic settings. More importantly, they tend to focus solely on marginal or joint parameter correlations, which implicitly treat mainshocks and aftershocks as independent stochastic processes, thereby overlooking their inherent spectral interaction. To address these limitations, this study proposes an explicit and parameterized modeling framework based on the evolutionary power spectral density (EPSD) of random ground motions. Using the magnitude difference between a mainshock and an aftershock as the control variable, we derive attenuation relationships for the amplitude, frequency content, and duration. A coherence function model is further developed from real seismic records, treating the mainshock–aftershock pair as a vector-valued stochastic process and thus enabling a more accurate representation of their spectral dependence. Coherence analysis shows that the function remains relatively stable between 0.3 and 0.6 across the 0–30 Rad/s frequency range. Validation results indicate that the simulated response spectra align closely with recorded spectra, achieving R² values exceeding 0.90 and 0.91. To demonstrate the model’s applicability, a case study is conducted on a representative frame structure to evaluate seismic vulnerability and economic loss. As the mainshock PGA increases from 0.2 g to 1.2 g, the structure progresses from slight damage to complete collapse, with loss rates saturating near 1.0 g. These findings underscore the engineering importance of incorporating mainshock–aftershock spectral interaction in seismic damage and risk modeling, offering a transparent and transferable tool for future seismic resilience assessments.

1. Introduction

Earthquakes are one of the most destructive natural disasters, necessitating a comprehensive understanding of their engineering properties. As seismic design theory and priorities evolve, engineers have increasingly recognized the importance of considering the sequence of seismic characteristics in structural performance. Historical research and seismic damage statistics indicate that large earthquakes are often followed by secondary ground motions, known as “aftershocks”. These events, though frequently overlooked, can significantly impact building performance and safety [1,2].

In recent years, the modeling of mainshock–aftershock ground motion sequences has advanced beyond simplified assumptions—such as treating aftershocks as amplitude-modulated versions of the mainshock or as randomly selected strong motion records [3,4]—and toward a growing emphasis on data-driven approaches. Notably, Copula-based joint distributions and machine learning techniques have been increasingly adopted to characterize the statistical dependencies between mainshock and aftershock parameters, such as the peak ground acceleration (PGA), duration, spectral content, and energy measures. These methods have shown considerable promise in capturing nonlinear relationships that traditional regression-based models often fail to represent. For example, Copula models enable flexible dependence structures that account for asymmetric tail behavior between variables, such as in Shen and Wang et al.’s research [5,6,7,8,9], while machine learning algorithms—particularly neural networks and decision-tree ensembles—are capable of learning complex patterns from large seismic databases, such as in Fayaz and Yu et al.’s research [10,11,12,13]. However, despite their technical sophistication, these approaches face significant limitations when applied to engineering practice. First, both Copula-based and machine learning methods typically function as “black box” models, offering limited interpretability of the underlying physical mechanisms. This makes it difficult for engineers to integrate the results into design-oriented frameworks where transparency and traceability are crucial. Second, these models are highly dependent on the quality and representativeness of the training data. Their performance often deteriorates when applied to regions or site conditions not well represented in the original dataset, reducing their generalizability across different tectonic environments. Finally, the lack of explicit parametric relationships in these methods hinders their incorporation into standard performance-based seismic assessment procedures, where engineers require clearly define input–output functions. Moreover, a fundamental limitation lies in the fact that most existing models, whether statistical or data-driven, fail to capture the spectral coherence between mainshock and aftershock ground motions. By focusing primarily on marginal or joint parameter distributions, these methods essentially treat mainshocks and aftershocks as conditionally independent processes, neglecting their underlying frequency–domain interdependence. This oversight can lead to incomplete representation of real seismic sequences, especially in simulations that aim to reflect realistic energy transfer, phase alignment, or site-specific spectral behavior [14,15,16].

In recent years, increasing attention has been paid to the seismic fragility of building structures under mainshock–aftershock (MS–AS) sequences. Traditional fragility analyses often assume earthquakes to be single events. However, in seismically active regions, aftershocks following a mainshock can cause secondary damage or even trigger structural collapse. To address this, researchers have developed various assessment methods specifically tailored for MS–AS sequences. Yu et al. proposed fragility surface models for reinforced concrete (RC) buildings based on dual intensity measures (IMs), enabling a more comprehensive representation of structural failure probabilities under MS–AS excitations [17]. Di Sarno and Pugliese investigated the effect of aging in RC structures, finding that aftershocks significantly increase structural vulnerability, particularly after weaker mainshocks [18]. Wang et al. examined a novel “mega-sub” structural system and showed that an increasing aftershock intensity substantially reduced the collapse margin capacity [19]. Additionally, Zhang et al. analyzed masonry structures by generating over 36 million earthquake-structure samples to quantify the probabilistic responses of various failure modes under MS–AS loading [20]. Shafaei and Naderpour focused on fiber-reinforced polymer (FRP) retrofitted RC frames, demonstrating the effectiveness of retrofitting in maintaining structural integrity under sequential earthquakes [21]. Salami et al. emphasized the influence of advanced structural modeling techniques and ground motion types on the seismic fragility of low-rise RC structures [22]. Overall, MS–AS sequences have a cumulative and amplifying effect on structural damage.

Given the increasing need for reliable seismic risk assessment in engineering practice, this study aims to provide a modeling framework that captures both the physical realism and statistical coherence of mainshock–aftershock ground motion sequences. Existing approaches often fail to reflect the spectral interdependence between events or deliver explicit, adaptable formulations suitable for integration into performance-based seismic design. To fill this gap, we propose a spectrum-based stochastic modeling framework with clear engineering applicability, which enables accurate simulation of mainshock–aftershock sequences and supports downstream analyses such as structural vulnerability and economic loss estimation. The model utilizes recorded data from strong seismic events and integrates the evolution power spectral density (EPSD) function with parameter attenuation relationships to simplify modeling. A coherence function model is proposed to analyze the relationship between mainshock and aftershock ground motions. The proper orthogonal decomposition (POD) model, as described by Di Paola [23,24,25], is employed to accurately and efficiently simulate ground motions in mainshock–aftershock sequences. This study then evaluates the seismic performance of a simplified nonlinear frame structure under simulated mainshock–aftershock scenarios using incremental dynamic analysis (IDA). Finally, this study estimates seismic economic losses based on the proposed methodology. The remainder of this document is organized as follows. Section 2 discusses the selection concept and classification foundation of measured data. Section 3 introduces the EPSD function of ground motions and presents a recommended approach for parameter attenuation relationships. Section 4 describes the proposed model for analyzing the relationship between mainshock and aftershock ground motions using a coherence function. Section 5 details the application of the POD model for simulating ground motions in mainshock–aftershock sequences. Section 6 focuses on the seismic performance analysis of a nonlinear frame structure under simulated ground motions, and presents the seismic economic loss estimation based on the proposed methodology. Finally, Section 7 concludes the study with key remarks. This study contributes an explicit, engineer-friendly spectral modeling framework for mainshock–aftershock ground motions. By incorporating physically interpretable parametric relationships for amplitude attenuation and spectral coherence, the proposed method offers a transparent and practical alternative to black box models. It is especially suitable for engineering applications such as structural fragility assessment and seismic loss estimation, thereby addressing a critical gap in current seismic modeling practices.

2. Selection Principle of Measured Mainshock–Aftershock Ground Motions

Using the following guidelines, 468 mainshock–aftershock ground motion data samples were meticulously chosen from the Pacific Earthquake Engineering Research (PEER) Centre for a complete assessment of their seismic characteristics:

(1) It is recommended that the same station record both the mainshock and its matching aftershock.

(2) The only aftershock data taken into account was the one with the mainshock record’s maximum moment magnitude scale.

(3) To completely eliminate the influence of near-fault impacts, the fault distance ought to exceed 10 km.

(4) To exclude ground motions whose structural effects are negligible, events with a moment magnitude Mw < 5 are omitted [26,27].

In addition, measured records were divided into three main groups based on site soil conditions using the average shear wave velocity at the top 30 m ($V_{\mathrm{S},30}$). These groups correspond to ground motions that occurred on stiff or soft rock soil, medium-hard soil, and medium soft soil. This classification was performed for the purpose of conducting detailed research on the characteristics of multi-dimensional ground motions.

Table 1. Correspondence between site classes and $V_{\mathrm{S},30}$.

Measured Record	Site Classes
	1	2	3
$V_{\mathrm{S}30}$ (m/s)	>450	300~450	<300
Number of groups	184	139	145

displays the record counts for each site class and how they correspond to one another.

(1) This class includes rock or extremely stiff soil conditions. In the Chinese Code for Seismic Design of Buildings (GB50011-2010) [1], this corresponds to Site Classes I0 and I1, which typically represent weathered rock or hard soil layers with minimal amplification effects. In Eurocode 8 [2], it is equivalent to Site Class A, which indicates rock or stiff deposits of sand, gravel, or overconsolidated clays. Despite being generally referred to as “rock”, this class may include both soft and stiff rock materials with high seismic wave transmission efficiency.

(2) This class corresponds to medium-stiff soil layers, such as dense sand or stiff clay. According to the Chinese code, this aligns with Site Class II. In Eurocode 8, it overlaps with the transition zone between Site Class B (softer range) and Site Class C (stiffer range), depending on the local stiffness and damping conditions. These soils exhibit moderate amplification effects during seismic shaking and are typical of urban subsoils with intermediate stiffness.

(3) This class represents soft soil, including loose sand, silty clay, or organic deposits with high seismic amplification potential. In the Chinese code, this corresponds to Site Classes III and IV, which are associated with significant amplification and longer-period ground motion effects. In Eurocode 8, this class matches Site Class C, which includes deep deposits of soft clays or silts with poor load-bearing capacities. Such soils are of particular concern in seismic design due to resonance risk and increased structural demand.

The 468 sets of ground motion records from mainshocks to aftershocks are detailed in

Table 2. Detailed information of selected mainshock–aftershock earthquake records.

NO.	Earthquake Name	Number of Stations	Moment Magnitude	Range of Fault Distance (km)	Range of ${\mathit{V}}_{\mathbf{S},30}$ (m/s)
1	Chi-Chi_Taiwan_China	256	7.62	10.48~116.57	160.67~1525.85
	Chi-Chi_Taiwan-06_China	256	6.3	29.33~145.62	160.67~1525.85
2	Friuli_Italy-01	1	6.5	33.4	249.28
	Friuli_Italy-02	1	5.91	41.39	249.28
3	Hollister-01	1	5.6	19.56	198.77
	Hollister-02	1	5.5	18.08	198.77
4	L’Aquila(aftershock1)_Italy	29	6.3	20.46~210.9	199~649.67
	L’Aquila_Italy	29	5.6	11.12~208.47	199~649.67
5	Livermore-01	5	5.8	15.19~34.66	367.57~517.06
	Livermore-02	5	5.42	15.17~30	367.57~517.06
6	Mammoth Lakes-01	1	6.06	15.46	537.16
	Mammoth Lakes-06	1	5.94	16.03	537.16
7	Northern Calif-01	1	6.4	44.68	219.31
	Northern Calif-04	1	5.7	57.21	219.31
8	Whittier Narrows-01	56	5.99	14.66~62.07	160.58~1222.52
	Whittier Narrows-02	56	5.27	11.06~64.27	160.58~1222.52
9	Imperial Valley-07	6	5.01	13.32~49.93	162.94~242.05
	Imperial Valley-06	6	6.53	11.17~49.4	162.94~242.05
10	Chalfant Valley-01	4	5.77	17.17~24.47	303.47~585.12
	Chalfant Valley-02	4	6.19	15.13~24.45	303.47~585.12
11	Irpinia_Italy-01	7	6.9	10.84~13.49	356.39~574.88
	Irpinia_Italy-02	7	6.2	14.74~64.37	356.39~574.88
12	Duzce_Turkey	7	7.14	131.45~188.7	175~523
	Kocaeli_Turkey	7	7.51	13.49~145.06	175~523
13	Darfield_New Zealand	56	7	11.86~382.64	247.5~649.67
	Christchurch_New_Zealand	56	6.2	11.25~431.73	247.5~649.67
14	Nenana Mountain_Alaska	19	6.7	104.73~276.74	212.48~456.75
	Denali_Alaska	19	7.9	54.78~275.9	212.48~456.75
15	Northridge-01	17	6.99	20.72~46.74	269.14~602.1
	Northridge-02	17	6.05	19.53~43.79	269.14~602.1
16	Superstition Hills-01	1	6.22	17.59	179
	Superstition Hills-02	1	6.54	23.85	179
17	Coalinga-01	1	6.36	12.11	257.38
	Coalinga-05	1	5.77	16.05	257.38

. This study also used an energy cut-off range of 1–99% of the total energy to thoroughly examine the effective intensity component of the recorded data.

3. EPSD of the Mainshock–Aftershock Ground Motions and Its Parameter Attenuation Relationships

3.1. EPSD of the Non-Stationary Mainshock–Aftershock Ground Motions

Generally, the EPSD function of mainshock–aftershock ground motions can be regarded as a stochastic process $U^i(t)$, which can be described by the Priestley theory of non-stationary stochastic process. This can be written as follows [28]:

$$S(\omega ,t;{\mathit{\lambda }}_S^i)={\left|f(t;{\mathit{\lambda }}_f^i)\right|}^2\overline{S}(\omega ;{\mathit{\lambda }}_{\overline{S}}^i)$$

(1)

where $S(\omega ,t;{\mathit{\lambda }}_S^i)$ indicates the two-sided EPSD function of $U^i(t)$; $f(t;{\mathit{\lambda }}_f^i)$ indicates the intensity of the non-stationary modulating function; and $\overline{S}(\omega ;{\mathit{\lambda }}_{\overline{S}}^i)$ denotes the two-sided power spectral density (PSD) function. Aside from this, $i=1,2$, corresponding to the mainshock and aftershock ground motions, respectively.

It is well known that the ground motions are provided with a significant non-stationarity intensity. To this end, the model which can reflect the non-stationary characteristics of ground motions is employed in this study, which was proposed by Wang [29]:

$$f_p(t)={[{\displaystyle \frac{t}{c^i}}\exp (1-{\displaystyle \frac{t}{c^i}})]}^2$$

(2)

In Equation (2), parameter $c^i$ represents the arrival time of the earthquake, which has a certain relationship with the simulation duration; that is, $c^i=0.2T_{\mathrm{d}}^i+1$. $T_{\mathrm{d}}^i$ is the strong seismic duration of the mainshock and aftershock. It can be clearly acknowledged that the aforementioned model can describe the ground motion change over time well, and the parameter vector ${\mathit{\lambda }}_f^i$ can be concluded to be ${\mathit{\lambda }}_f^i=(c^i)$.

For the PSD function of the corresponding stationary ground motions, the Kanai-Tajimi model is adopted in this paper [30]:

$$\overline{S}(\omega ;{\mathit{\lambda }}_{\overline{S}}^i)={\displaystyle \frac{{({\omega }_{\mathrm{g}}^i)}^4+4{({\xi }_{\mathrm{g}}^i{\omega }_{\mathrm{g}}^i\omega )}^2}{{[{\omega }^2-{({\omega }_{\mathrm{g}}^i)}^2]}^2+4{({\xi }_{\mathrm{g}}^i{\omega }_{\mathrm{g}}^i\omega )}^2}}{\displaystyle \frac{{\omega }^4}{{[{\omega }^2-{({\omega }_{\mathrm{f}}^i)}^2]}^2+4{({\xi }_{\mathrm{f}}^i{\omega }_{\mathrm{f}}^i\omega )}^2}}{S}_0^i$$

(3)

In Equation (3), the parameters ${\omega }_{\mathrm{g}}^i$ and ${\xi }_{\mathrm{g}}^i$ indicate the dominant frequency and critical damping of the soil layer, respectively, while ${\omega }_{\mathrm{f}}^i$ and ${\xi }_{\mathrm{f}}^i$ are the dominant frequency and critical damping of the rock bed, respectively, which ensure the rationality of the seismic spectrum energy and have a certain function relation with ${\omega }_{\mathrm{g}}^i$ and ${\xi }_{\mathrm{g}}^i$ (${\omega }_{\mathrm{f}}^i=0.1{\omega }_{\mathrm{g}}^i$ and ${\xi }_{\mathrm{f}}^i={\xi }_{\mathrm{g}}^i$). $S_0^i$ is a measure of the spectral intensity factor, which represents the intensity of the white noise bedrock acceleration process. This can be mathematically stated in the following form [31]:

$$S_0^i={({\displaystyle \frac{A_{\max }^i}{r^i}})}^2{\displaystyle \frac{1}{{\omega }_{\mathrm{e}}}};{\omega }_{\mathrm{e}}={\displaystyle \frac{{\displaystyle {\int }_{-\infty }^{\infty }\overline{S}(\omega ;{\mathit{\lambda }}_{\overline{S}}^i)\mathrm{d}\omega }}{S_0^i}}$$

(4)

where $A_{\max }^i$ and $r^i$ indicate the peak ground acceleration and peak factor, respectively. To this end, the parameter vector of the PSD function can be written as ${\mathit{\lambda }}_{\overline{S}}=({\omega }_{\mathrm{g}}^i,{\xi }_{\mathrm{g}}^i,A_{\max }^i,r^i)$.

According to Equations (1–4), the parameter vector of the EPSD function of the ith direction component ground motions $U^i(t)$ can defined as follows:

$${\mathit{\lambda }}_S^i=({\mathit{\lambda }}_f^i,{\mathit{\lambda }}_{\overline{S}}^i)=(c^i,{\omega }_{\mathrm{g}}^i,{\xi }_{\mathrm{g}}^i,A_{\max }^i,r^i)$$

(5)

3.2. Attenuation Relationships for Mainshock–Aftershock Parameters

Determining the parametric relationships between the mainshock and aftershocks is a prerequisite for modeling them. In engineering, the focus is on the engineering characteristics of ground motion, i.e., the peak ground acceleration, duration, and dominant frequency. Therefore, this paper focuses on the relationship between the engineering characteristics of them.

PGA represents the intensity of the ground motion and is the main basis for the seismic design of structures. According to the empirical formula of Joyner–Boore, the PGA can be expressed as follows [32]:

$$\lg (A_{\max })=0.49+0.23(M-6)-\lg \sqrt{R^2+8^2}-0.0027\sqrt{R^2+8^2}$$

(6)

where R is the fault distance in km and M is the magnitude.

Gutenberg–Richter obtained the relationship between the magnitude M occurring at an arbitrary time and region and their number of occurrences N from a large amount of measured data [33]:

$$N={10}^{A-BM}$$

(7)

where M is the magnitude of earthquakes occurring in the time domain or region; N is the number of M-magnitude earthquakes; A is the frequency of M-magnitude earthquakes occurring; and B is usually one. Since the formula has been explicitly stated to be applicable to the time, it can be used to predict the magnitude of aftershocks at the same location and moment.

Assuming that earthquakes of magnitudes $M_1$ and $M_2$ occur, obeying the same frequency-magnitude law in Equation (7), the relationship between the occurrence of $N_1$ earthquakes of magnitude $M_1$ and the occurrence of $N_2$ earthquakes of magnitude $M_2$ is given by

$$M_1+\lg \left(N_1\right)=M_2+\lg \left(N_2\right)$$

(8)

Substituting the magnitude relationship of Equation (8) into Equation (6) gives the ratio of peak accelerations of the mainshock and aftershocks:

$${\displaystyle \frac{A_{\max }^{(2)}}{A_{\max }^{(1)}}}={10}^{-0.23\Delta M}$$

(9)

From Equation (9), when a mainshock with PGA $A_{\max }^{(1)}$ occurs, K aftershocks with a peak acceleration of ${10}^{-0.23\Delta M}{A}_{\max }^{(2)}$ will follow, where K denotes the total number of aftershocks triggered by a given mainshock.

Duration represents the experience time from the beginning to the end of the ground motion, but because there are more low-intensity parts at the beginning and end of the measured ground motion, the energy share of these parts is extremely small, and the impact on the engineering structure is negligible. Thus, only the main energy interval of the ground shaking is retained in the engineering, and the duration of this interval is defined as the strong seismic duration $T_{\mathrm{d}}$, which is used as the simulation of the total duration of the ground motion in this paper. Zhang et al. [34] proposed a strong seismic duration linear fitting formula for the mainshock and aftershock based on the magnitude difference:

$$T_{\mathrm{d}}^{(2)}=k{T}_{\mathrm{d}}^{(1)}+d$$

(10a)

$$k=-1.934\times {10}^{-5}{\mathrm{e}}^{{\displaystyle \frac{\Delta M}{0.157}}}+0.622$$

(10b)

$$d=0.664{\mathrm{e}}^{{\displaystyle \frac{\Delta M}{0.534}}}+1.653$$

(10c)

In Equation (10), $T_{\mathrm{d}}^{(2)}$ and $T_{\mathrm{d}}^{(1)}$ denote the strong seismic duration of the mainshock and aftershock, respectively, in seconds, and $\Delta M$ is the magnitude difference between the mainshock and aftershock, which can be calculated using Equation (7).

Furthermore, the spectral characteristics reflect the dynamic properties of the ground motion, which have a direct and important influence on the structural response. Yuan suggested the following relationship between the PGA with a domain period $T_g$ and seismic intensity I for a strong earthquake region [35]:

$$A_{\max }=1.6\times T_{\mathrm{g}}^{-1.3}\times {10}^{0.18I}$$

(11)

Meanwhile, Ou suggested a statistical regression relationship between the seismic intensity I and magnitude M based on the measured data [36]:

$$M=0.66I+0.98$$

(12)

The intensity $I_{\mathrm{aft}}$ of the aftershock can be obtained by substituting the magnitude difference $\Delta M$ of the main aftershock according to Equation (8) into Equation (12):

$$I_{\mathrm{aft}}={\displaystyle \frac{M_{\mathrm{main}}-\Delta M-0.98}{0.66}}$$

(13)

By substituting Equations (9) and (13) into Equation (11), the ratio of the dominant period of the mainshock to the aftershock can be obtained:

$${\displaystyle \frac{T_{\mathrm{g}}^{(1)}}{T_{\mathrm{g}}^{(2)}}}={\displaystyle \frac{{\omega }_{\mathrm{g}}^{(2)}}{{\omega }_{\mathrm{g}}^{(1)}}}={({\displaystyle \frac{A_{\max }^{(1)}\times {10}^{0.18I_{\mathrm{aft}}}}{A_{\max }^{(2)}\times {10}^{0.18I_{\mathrm{main}}}}})}^{\frac{-1}{1.3}}={10}^{0.03\Delta M}$$

(14)

As a result, the relationship between mainshock ground motions and aftershock ground motions in terms of amplitude, duration, and domain frequency can be more comprehensively grasped. Furthermore, the EPSD for each of them can be modeled on the basis of the above relationships.

4. Correlation Analysis of Mainshock–Aftershock Ground Motions

In practical signal analysis, coherence is typically calculated based on spectral estimates derived from time series data. This involves evaluating the cross-spectral and auto-spectral densities of two signals using techniques such as the Fourier transform, Welch’s method, or wavelet-based spectral decomposition. These approaches allow researchers to capture how two signals relate to each other at different frequency bands over time. Depending on the characteristics of the signals—such as the stationarity, noise level, or resolution requirements—various smoothing or windowing techniques may also be applied to ensure robust estimation of coherence values. Moreover, coherence analysis can be extended to time-frequency representations for studying nonstationary signals, where the frequency content evolves over time, as is often the case in earthquake ground motions. Coherence is a fundamental concept in the field of signal analysis, being used to describe the degree of correlation or consistency between two signals in the frequency domain. Unlike traditional statistical correlation, coherence focuses on the coupling of signals in the frequency domain, rather than simply their temporal trends or magnitude similarities. This frequency-based perspective provides deeper insights into the hidden structure of signal interactions, especially when dealing with nonstationary, nonlinear, or stochastic systems. In summary, coherence is not just a mathematical tool; it is a vital concept for understanding and characterizing the complexity of natural and engineered systems. As modeling in engineering increasingly involves multi-variable, large-scale, and multi-resolution data, coherence analysis is becoming an essential component in modern scientific and engineering methodologies. One recent study demonstrated that soil conditions exert a significant influence on the seismic characteristics of ground motions. Ultimately, it is important to establish an association between the various elements of mainshock–aftershock ground motion, as evidenced by the explicit manifestation within the EPSD functions. This section primarily focuses on investigation of the spectrum correlation (coherence) in mainshock–aftershock ground motions.

Generally, the coherence function of the mainshock ground motion and aftershock ground motion of the kth group of a measured record can be expressed as follows [37]:

$${\overline{\eta }}_k^{ij}(\omega )={\displaystyle \frac{\left|E_k^{ij}(\omega )\right|}{\sqrt{E_k^i(\omega )\cdot E_k^j(\omega )}}}$$

(15)

where $E_k^{ij}(\omega )$ represents the estimated cross-PSD between the measured mainshock ground motion record and measured aftershock ground motion record, which can be obtained with the Matlab toolbox function “cpsd”. $E_k^i(\omega )$ and $E_k^j(\omega )$ represent the estimated PSDs of the measured mainshock ground motion record and measured aftershock ground motion record, respectively, and can be obtained with the Matlab toolbox function “pwlech”, while $\left|\cdot \right|$ represents the modular function.

In order to fit the coherence function, a three-order Fourier series model is adopted in this paper, which can be represented as follows:

$$\eta (\omega ;{\mathit{\lambda }}_{\eta })=A+{\displaystyle \sum _{l=1}^3{B}_l\sin (l\times D\times \omega )+C_l\cos (l\times D\times \omega )}$$

(16)

where l represents the order of the Fourier series model and ${\mathit{\lambda }}_{\eta }=(A,B_l,C_l,D)$ is the parameter vector.

Following this, taking the average coherence function of all the measured records as the target, the parameter vector of the Fourier series model can be identified according to the best square approximation criterion. Figure 1 illustrates the outcome of the fitting process, indicating that the coherence function remained relatively constant between 0.3 and 0.6 across the frequency range from 0 $\mathrm{rad}/\mathrm{s}$ to 300 $\mathrm{rad}/\mathrm{s}$. The parameters of the Fourier series model that were identified are presented in

Table 3. The identified parameters of the proposed Fourier series model.

Order	A	Bl	Cl	D
1	$-3.78\times {10}^7$	$5.66\times {10}^7$	$4.23\times {10}^6$	$4.32\times {10}^{-4}$
2		$-2.24\times {10}^7$	$-3.38\times {10}^6$
3		$3.69\times {10}^6$	$8.41\times {10}^5$

.

5. POD Representation of Vector Process of Mainshock–Aftershock Ground Motion

5.1. The POD Method

Generally speaking, the mainshock–aftershock ground motions can be considered a 1D-2V non-stationary vector process $\mathit{U}(t)=[U^1(t),U^2(t)]$, and these can be presented by the two-sided EPSD function matrix, which is as follows:

$$\mathit{S}(\omega ,t)=\left[\begin{array}{cc}S(\omega ,t;{\mathit{\lambda }}_S^{11})& S(\omega ,t;{\mathit{\lambda }}_S^{12})\\ S(\omega ,t;{\mathit{\lambda }}_S^{21})& S(\omega ,t;{\mathit{\lambda }}_S^{22})\end{array}\right]$$

(17)

where $\mathit{S}(\omega ,t)$ denotes the two-sided EPSD function matrix and $i=1,2$, corresponding to the mainshock and aftershock ground motions, respectively. The diagonal and off-diagonal elements represent the EPSD function matrix of mainshock–aftershock ground motions. The elements of $\mathit{S}(\omega ,t)$ can be defined as follows:

$$S(\omega ,t;{\mathit{\lambda }}_S^{ij})=\eta (\omega ;{\mathit{\lambda }}_{\eta }^{ij})\sqrt{S(\omega ,t;{\mathit{\lambda }}_S^i)\cdot S(\omega ,t;{\mathit{\lambda }}_S^j)}i,j=1,2$$

(18)

where $S(\omega ,t;{\mathit{\lambda }}_S^i)$ denotes the auto-EPSD function of the ith component of mainshock–aftershock ground motion and the parameter vector ${\mathit{\lambda }}_S^i$, which are related to the soil classes. $S(\omega ,t;{\mathit{\lambda }}_S^{ij})$ denotes the cross-EPSD between the mainshock ground motions and aftershock ground motions, and $\eta (\omega ;{\mathit{\lambda }}_{\eta }^{ij})$ denotes the coherence function between the mainshock ground motions and aftershock ground motions, which can be referred to in Equations (6–14).

Generally speaking, ${\mathit{S}}_{\mathit{U}}(\omega ,t)$ can be further decomposed as follows:

$$\mathit{S}(\omega ,t)=\mathit{D}(\omega ,t){\rm I}(\omega ){\mathit{D}}^{\ast \mathrm{T}}(\omega ,t)$$

(19)

in which

$${\rm I}(\omega )=\left[\begin{array}{cc}1& \eta (\omega ;{\mathit{\lambda }}_{\eta }^{12})\\ \eta (\omega ;{\mathit{\lambda }}_{\eta }^{21})& 1\end{array}\right]$$

(20)

where * and T indicate the conjugate and matrix transpose, respectively. The diagonal matrix of EPSD is $\mathit{D}(\omega ,t)=\mathrm{diag}[S(\omega ,t;{\mathit{\lambda }}_S^{11}),S(\omega ,t;{\mathit{\lambda }}_S^{22})]$. ${\rm I}(\omega )$ indicates the coherence function matrix of the mainshock–aftershock ground motion.

Obviously, ${\rm I}(\omega )$ can be decomposed by adopting eigendecomposition into the following product since it is a nonnegative Hermitian matrix:

$$\left\{\begin{array}{l}{\rm I}(\omega )=\mathit{\Psi }(\omega )\mathbf{\Lambda }(\omega ){\mathit{\Psi }}^{\ast \mathrm{T}}(\omega )\hfill \\ {\mathit{\Psi }}^{\ast \mathrm{T}}(\omega )\mathit{\Psi }(\omega )=\mathbf{I}\hfill \end{array}\right.$$

(21)

where $\mathit{\Psi }(\omega )=[{\mathit{\psi }}_1(\omega ),{\mathit{\psi }}_2(\omega )]$ and its element ${\mathit{\psi }}_r(\omega )={[{\varphi }_{1r}(\omega ),{\varphi }_{2r}(\omega )]}^{\mathrm{T}}$ represents the shape of the rth $(r=1,2)$ eigenmode. $\mathbf{\Lambda }(\omega )=\mathrm{diag}[{\Lambda }_1(\omega ),{\Lambda }_2(\omega )]$, and its element ${\Lambda }_r$ indicates the diagonal eigenvalue matrix, while I indicates the identity matrix of a $2\times 2$ order. Generally, ${\mathit{\psi }}_r(\omega )$ can be further expressed in the following form:

$${\mathit{\psi }}_r(\omega )={\mathit{\chi }}_r({\omega }_k)+\mathrm{i}{\mathcal{Z}}_r({\omega }_k)$$

(22)

Furthermore, suppose that $\mathit{U}(t)=[U^1(t),U^2(t)]$ is a real-valued, zero-mean stochastic vector process. Then, the mainshock–aftershock ground motions can be written as follows [23,24,25]:

$$\begin{array}{l}{U}^i(t)\approx 2{\displaystyle \sum _{r=1}^2{\displaystyle \sum _{k=1}^N\sqrt{S(\omega ,t;{\mathit{\lambda }}_S^i){\Lambda }_r({\omega }_k)\Delta \omega }}}\times \left[{\chi }_{ir}({\omega }_k)\right.\left(R_{rk}\cos {\omega }_{k}t-\right.\left.Q_{rk}\sin {\omega }_{k}t\right)\\ -{\mathcal{Z}}_{ir}({\omega }_k)\left.\left(R_{rk}\sin {\omega }_{k}t+Q_{rk}\cos {\omega }_{k}t\right)\right]\end{array}$$

(23)

where ${\omega }_k$ denotes the discrete frequency series, $\Delta \omega ={\omega }_{\mathrm{u}}/N$ indicates the frequency increment, and N indicates the number of frequency intervals, respectively, while i indicates the ith composition process of $\mathit{U}(t)$ and $\{R_{rk},Q_{rk}\}$ refers to a set of real-valued, zero-mean standard orthogonal random variables which satisfies the following basic conditions:

$$\begin{array}{l}E\left[R_{rk}\right]=E\left[Q_{rk}\right]=0,E\left[R_{rk}{Q}_{sl}\right]=0,\\ E\left[R_{rk}{R}_{sl}\right]=E\left[Q_{rk}{Q}_{sl}\right]={\displaystyle \frac{1}{2}}{\delta }_{rs}{\delta }_{kl},\\ r,s=1,2,\dots ,M;k,l=1,2,\dots ,N\end{array}$$

(24)

To this end, the orthogonal random variables $R_{rk}$ and $I_{rk}$ can be further defined in the following form:

$$R_{rk}=\cos {\varphi }_{rk},I_{rk}=\sin {\varphi }_{rk}$$

(25)

where ${\varphi }_{rk}$ is a sequence of independent random phase angles distributed uniformly over the interval $[0,2\mathsf{\pi })$ and the defined $\{R_{rk},I_{rk}\}$ are obviously satisfied with the basic conditions defined in Equation (24). Then, the conventional POD scheme can be obtained as follows:

$$U^i(t)\approx 2{\displaystyle \sum _{r=1}^2{\displaystyle \sum _{k=1}^N\sqrt{S(\omega ,t;{\mathit{\lambda }}_S^i){\Lambda }_r({\omega }_k)\Delta \omega }\left|{\mathit{\psi }}_r({\omega }_k)\right|\times \cos [{\omega }_{k}t+{\phi }_r({\omega }_k)+{\varphi }_{rk}]}}$$

(26)

5.2. The Sample Realization Procedures of Mainshock–Aftershock Ground Motions

The representative sample realization for mainshock–aftershock ground motions can be implemented using the following procedures:

(1) Construct the EPSD function $S(\omega ,t;{\mathit{\lambda }}_S^i)$. Substitute the given mainshock EPSD parameters and the aftershock parameters obtained from Equations (6)–(14) into Equation (1), and then the EPSD function $S(\omega ,t;{\mathit{\lambda }}_S^i)$ can be obtained.

(2) Determine the coherence function matrix ${\rm I}(\omega )$. The elements in ${\rm I}(\omega )$ can be calculated from the mainshock–aftershock ground motions model $\eta (\omega ;{\mathit{\lambda }}_{\eta })$ proposed in Section 4.

(3) Generate the independent random phase angles ${\varphi }_{rk}$ distributed uniformly over the interval $[0,2\mathsf{\pi })$ with the Monte Carlo method. Then, substitute $S(\omega ,t;{\mathit{\lambda }}_S^i)$, ${\rm I}(\omega )$, and ${\varphi }_{rk}$ into Equation (26) to generate the representative samples of mainshock–aftershock ground motions.

The main procedures are illustrated following flowchart in Figure 2.

5.3. Numerical Examples

In the numerical example in this paper, it is assumed that two aftershocks occur after a mainshock, and the magnitude difference is $\Delta M=0.301$ according to Equation (7). Then, the EPSD parameters of the mainshock can be obtained with the Code for Seismic Design of Buildings for site 2, and the corresponding parameters of the aftershocks are shown in

Table 4. Mainshock and aftershock parameters.

Parameters	$\mathbf{Mainshock} (\mathit{i}=1)$	$\mathbf{Aftershock} (\mathit{i}=2)$
${\overline{\omega }}_{\mathrm{g},p} (\mathrm{rad}/\mathrm{s})$	15.71	16.03
${\overline{\xi }}_{\mathrm{g},p}$	0.72	0.72
${\overline{A}}_{\max ,p} (\mathrm{cm}/{\mathrm{s}}^2)$	200	170.52
$T_{\mathrm{d},p}(\mathrm{s})$	25	18.36
$c_p$	6	4.8
r	3	2.75
Mw	6.8	6.5

according to Equations (6)–(14). The other simulation parameters were follows: number of frequency truncation terms $N=1600$; truncation frequency ${\omega }_{\mathrm{u}}=240 \mathrm{rad}/\mathrm{s}$; time step $\Delta t=0.01 \mathrm{s}$; and number of representative samples $n_{\mathrm{sel}}=100$.

Figure 3 illustrates the generated representative samples of the mainshock and aftershock ground motions of site 2. It should be noted that representative samples exhibited a non-zero intensity. It is evident that the representative samples effectively demonstrated the seismic characteristics. Specifically, there were clear distinctions in the frequency component, duration, and amplitude observed in the typical samples of the mainshocks and aftershocks occurring under the soil conditions. Therefore, it accurately demonstrated the seismic attributes of a sequence. Specifically, in comparison with the ground motions of the mainshock, the aftershock ground motions exhibit a shorter length, lower amplitude, and greater number of frequency components.

A comparison of the average acceleration response spectrum between the simulated outcomes and the recorded data for soil 2 is shown in Figure 3. This comparison was made in order to improve the knowledge of the engineering feasibility of the stochastic mainshock–aftershock model described in this study. It is essential to recognize that the amplitudes of the recorded data were modified in order to correspond with the appropriate peak ground acceleration (PGA) of the mainshocks and aftershocks that were simulated. In terms of the average acceleration response spectrum, it is clear that the simulated samples had a extremely excellent agreement with the measured records, as is visible. Specifically, it is of the utmost importance to ensure that optimal consistency is maintained in the geographic region surrounding the principal peak of the acceleration response spectrum. The research that was mentioned before revealed that there is a significant association between the statistical features of simulated stochastic mainshock–aftershock ground motions and the strong motion records that were measured. This research provides evidence that the simulation method that was proposed is capable of being successfully implemented in engineering through practical application.

The accuracy of the proposed method is demonstrated by the coefficient of determination (R²) between the simulated and recorded response spectra in the numerical example shown in Figure 4, with values of 0.90 for the mainshock and 0.91 for the aftershock.

The comparison of the coherence function between the representative samples that were generated and the targets that corresponded to them are shown in Figure 5. Again, the success of the suggested simulation approach for mainshock–aftershock ground motions is demonstrated by the fact that the simulated coherence function between both the mainshock and the aftershock were highly consistent with the related goals. This is demonstrated in the figures.

6. Seismic Economic Loss Assessment of Frame Structures Based on Vulnerability Theory

This study shows how the proposed model can be applied in engineering using a standard nonlinear frame construction. Using the aforementioned stochastic model to generate synthetic ground motion, this study delves into the evaluation of seismic economic damage.

6.1. Seismic Vulnerability Theory Based on the IDA Method

The incremental dynamic analysis (IDA) approach is a way of conducting analysis that uses dynamic time history analysis to look at how a structure’s sustained response varies over time. Modulating the strength of seismic motion input achieves this. The objective of this study is to improve understanding of the correlation between structural performance variations by creating a quantitative relationship between structure reaction and seismic intensity. Moreover, it meticulously considers the building’s seismic requirements and structural capacities. A high-level overview of the IDA method follows, as described in [38,39,40].

In this study, the seismic intensity index, also known as IM, is typically defined as the PGA, and the structural damage index, often known as DM, is determined to be the maximum story drift ratio ${\theta }_{\max }$. It is possible to accept, through the use of the IDA approach, that there is an exponential correlation between the structural damage index (DM) and the seismic intensity index (IM), and this correlation can be given as follows:

$$DM=a\cdot I{M}^b$$

(27)

Meanwhile, Equation (27) can be further expressed by logarithmizing both sides:

$$\ln DM=\ln (a)+b\cdot \ln (IM)=A+B\ln IM$$

(28)

where A and B are constants.

In the event, it is assumed that the seismic resist ability of a structure at a particular level of performance is denoted by the letter C, and the structural reaction capacity of a structure under a particular seismic intensity is denoted by the letter D. The following is an example of how the chance of structural failure might be expressed in this manner:

$$P_{\mathrm{f}}=P(R\le 0)=P(C/D\le 1)$$

(29)

Suppose that C and D correspond to normal distribution; that is, we have

$$C~N({\mu }_C,{\sigma }_C)$$

(30)

$$D~N({\mu }_D,{\sigma }_D)$$

(31)

where $\mu$ and $\sigma$ indicate the mean and standard deviation, respectively. In this end, $R$ also corresponds to a normal distribution, and we convert R to a standard normal distribution:

$$P_{\mathrm{f}}=P(R\le 0)=P(T\le {\displaystyle \frac{\mu }{\sigma }})=\mathit{\Phi }({\displaystyle \frac{-\ln ({\mu }_C/{\mu }_D)}{\sqrt{{\beta }_C^2+{\beta }_D^2}}})$$

(32)

where $\beta$ denotes the logarithmic standard deviation. By combining Equations (27) and (32) and letting $\sqrt{{\beta }_C^2+{\beta }_D^2}=0.5$, the formula for the failure probability can be written as follows [41]:

$$P_{\mathrm{f}}=\mathit{\Phi }({\displaystyle \frac{\ln (\exp (A)\cdot I{M}^B/{\mu }_D)}{0.5}})$$

(33)

6.2. Engineering Background and Vulnerability Analysis

In order to explore the dynamic reliability evaluation of a nonlinear frame structure that is subjected to explosive ground vibrations, the engineering object being studied here was a 10-story frame structure that took into consideration the nonlinearity of the material. A schematic diagram of the nonlinear frame construction can be found in Figure 6. The height of the first level is $h_1=4\mathrm{m}$, and the height of the remaining stories is $h=3\mathrm{m}$. The numbers $750\times 750{\mathrm{mm}}^2$ and $600\times 600{\mathrm{mm}}^2$ correlate to the sections of the bottom and remainder columns, respectively. For more details regarding particular structural parameters, please refer to [41].

A numerical example of how to simulate the average seismic characteristics of site 2 by stimulating ground motions during mainshocks and aftershocks from outside sources is given here. The simulated samples exhibited seismic engineering features that were congruent with the measured records as shown in Figure 3, which compares the acceleration response spectra of the two sets of data. The mainshock PGA was 200 cm/s², and there were 100 simulated samples.

In Figure 7, we can observe the relationship between the story drift ratio, the restoring force, and the mainshock–aftershock ground motion representative sample with an intensity of 0.2 g for the eighth story. This frame structure model incorporated considerable nonlinearity, as shown in Figure 6.

In this numerical simulation, simulated mainshock–aftershock ground motions with correlation were involved, according to Equation (28). Figure 8 shows the fitted results for the logarithmic PGA with the logarithmic maximum story drift ratio, which demonstrated that there was a strong linear relationship between them, and the correlation coefficient was 0.91. The results of linear fit were A = 1.20 and B = −4.75. Here, A and B are the intercept and slope, respectively, of the log-linear regression between the intensity measure (IM) and damage measure (DM) used in the vulnerability analysis.

Additionally, to assess the condition of ultimate failure of the structure, it was required to determine the boundaries of the quantitative indicators for the different stages of the structural reaction that were deemed ultimate failure states. The following levels of failure states can be used frame structures nonlinearly according to the Code for Seismic Design of Buildings (GB 50011-2022) [1]. Each ultimate failure state level hass the level-specific description in

Table 5. The descriptions and quantitative indicators of each degree of structural damage.

Degree of Structural Damage	Descriptions	Quantitative Indicators
Completely undamaged	The primary load-bearing components are in good condition, and there are only a few load-bearing components that have been damaged. Therefore, there is no need for repairs.	<1/500
Minimally impaired	There are some load-bearing components that have tiny cracks, and there are also some non-load-bearing components that are damaged, but they do not require immediate replacement or only require modest repairs.	1/500
Partially impaired	As opposed to some non-load-bearing components, which are significantly damaged and need to be repaired, the majority of load-bearing components have rather minor damage.	1/200
Extensively impaired	The majority of load-bearing components suffered serious damage or even collapse because of the load.	1/100
Collapsed	There was a collapse of the majority of the primary load-bearing components.	1/50

.

The exceeding probability of a nonlinear frame structure under different degrees of structural damage is depicted in Figure 9. It is clear that as the PGA of the mainshock ground motions reached 0.2 g, 0.4 g, 0.7 g, and 1.2 g, the frame structure would be slightly damaged, moderately damaged, severely damaged, and collapsed, respectively. This is indicated by the fact that the frame structure would be capable of collapse.

6.3. Analysis of Economic Loss Rate

In general, the replacement value and economic losses of a structure can be described as the seismic economic loss rate of the structure, and the definition of this rate is as follows:

$$DF={\displaystyle \frac{L}{R}}$$

(34)

in which DF stands for the damage factor, L is the economic losses, and R is the replacement value. In addition, the following table illustrates the relationship between the degree of structural damage and the economic loss rate.

According to

Table 6. The relationship between the degree of structural damage and the range of the loss rate.

Degree of Structural Damage	Range of Economic Loss Rate (%)	Median Value of Economic Loss Rate
Essentially unaltered	0	0
Minimally impaired	1~10	5
Partially impaired	10~40	25
Extensively impaired	40~80	60
Collapsed	80~100	90

and the results of the vulnerability analysis, the loss rate can be represented by a vulnerability curve [42,43]:

$$\begin{array}{ll}DF=& 0\times (1-P_1)+0.05\times (1-P_1-P_2)+0.25\times (1-P_1-P_2-P_3)+\\ & 0.6\times (1-P_1-P_2-P_3-P_4)+0.9\times P_4\end{array}$$

(35)

where $P_1,P_2,P_3,P_4$ indicate the exceeding probability.

To this end, the economic loss of frame structure under the mainshock–aftershock ground motions is displayed in Figure 10, which shows that the economic loss was relatively gentle between 0 g and about 0.3 g. Then the economic loss rapidly rose as the PGA became larger than 0.3 g. The findings of this research give enhanced theoretical substantiation for the precise assessment of economic losses incurred by frame structures when subjected to mainshock–aftershock ground motions and provide evidence of the engineering applicability of proposed stochastic model for mainshock–aftershock ground motions.

7. Conclusions

In this paper, a coherence function model of mainshock–aftershock ground motions was proposed to describe the correlation between mainshock ground motions and aftershock ground motions, and we realized the simulation of mainshock–aftershock ground motions based on the POD simulation method via measured records. Based on this, the purpose of IDA method-based vulnerability analysis and economic assessment is achieving a nonlinear frame structure to evaluate the degree of structural damage caused by mainshock–aftershock ground motions. The specific conclusions are as follows:

(1) The analysis of strong mainshock–aftershock motion records showed that the coherence between mainshocks and aftershocks was 0.3~0.6, and the three-order Fourier series model recommended by this paper can describe the variation trend with the frequency of coherence better.

(2) The numerical examples presented in this study illustrate that the seismic characteristics of the simulated samples could effectively represent the average seismic properties observed in the measured records of various soil types. This finding offers a valuable theoretical foundation for conducting more precise analyses of structural dynamic responses.

(3) This study examined the seismic economic evaluation of a nonlinear frame structure using vulnerability theory in the context of mainshock–aftershock ground motions. The findings indicate that mainshocks and aftershocks might lead to significant economic losses when the PGA exceeds 0.3 g.

Additionally, future work will involve a systematic benchmarking study to compare the proposed method with existing empirical and data-driven ground motion models. Such comparisons will focus not only on the predictive accuracy for response spectra and structural demand but also evaluating computational efficiency, interpretability, and usability in engineering design workflows. This will help further clarify the relative advantages and application scenarios of the proposed approach. Therefore, while the current results demonstrate the potential and feasibility of the method, additional studies are still required to fully validate and generalize its use across different seismic environments and structural types.