Multi-parameter assessment method for seismic resilience of subway stations based on economic losses and recovery paths

Abstract

The recurring occurrence of seismic hazards constitutes a significant and imminent threat to subway stations. Consequently, a meticulous assessment of the seismic resilience of subway stations becomes imperative for enhancing urban safety and ensuring sustained functionality. This study strives to introduce a probabilistic framework tailored to assess the seismic resilience of stations when confronted with seismic hazards. The framework aims to precisely quantify station resilience by determining the integral ratio between the station performance curve and the corresponding station recovery time. To achieve this goal, a series of finite element models of the soil-station system were developed and employed to investigate the impact of site type, seismic intensity, and station structural type on the dynamic response of the station. Then, the seismic fragility functions were generated by developing the relationships between seismic intensity and damage index, taking into account multidimensional uncertainties encompassing factors such as earthquake characteristics and construction quality. The resilience assessment was subsequently conducted based on the station’s fragility and the corresponding economic loss, while also considering the recovery path and recoverability. Additionally, the impacts of diverse factors, including structural characteristics, site types, functional recovery models, and peak ground acceleration (PGA) intensities, on the resilience of stations with distinct structural forms were also discussed. This work contributes to the resilience-based design and management of metro networks to support adaptation to seismic hazards, thereby facilitating the efficient allocation of resources by relevant decision makers.

Graphical abstract

Keywords

subway stations / seismic resilience / seismic fragility / recovery models / parameter analysis / site types

1 Introduction

The subway station stands as an indispensable component within the contemporary urban transportation network, playing a pivotal role in the city’s economic development and fostering interpersonal exchanges. Responding to the escalating needs of the city, the ongoing construction and expansion of the subway transport system are proceeding at an unprecedented pace. Historically, engineers held the belief that seismic design considerations for underground structures were superfluous, based on the assumption that they incurred less damage during seismic events compared to their above-ground counterparts. However, the catastrophic repercussions of recent seismic hazards have profoundly challenged and dispelled this confidence. Recent instances of intense seismic activity have unequivocally demonstrated that underground structures are susceptible to severe damage or even collapse in the wake of earthquakes [1,2]. A poignant example is the extensive destruction of underground tunnels following Japan’s 1995 Great Kobe Earthquake, culminating in the complete collapse of the Daikai station [3]. In stark contrast to above-ground edifices, even minor impairments to underground stations during seismic events can yield substantial economic losses. This is primarily attributable to the intricate maintenance challenges associated with subterranean structures, often compounded by an augmented risk of casualties [4,5].

In response to the aforementioned challenges, the seismic performance assessment of underground structures has garnered substantial attention from the research community. Le et al. [6] and Huh et al. [7] introduced the Ground Response Acceleration Method for Subsurface Structures, incorporating Soil-Structure Interaction. They subsequently derived seismic fragility curves for box-shaped underground tunnels based on this method. Building on this foundation, Nguyen et al. [8] extended the scope by determining seismic fragility curves for single, double, and triple-box underground tunnels. Their investigation delved into the nuanced effects of diverse seismic intensity indexes and site types on the seismic risk profile of these tunnels. Exploring alternative facets of this issue, Avanaki et al. [9] scrutinized the impact of steel fiber concrete on the seismic fragility curves of tunnels. Their study further involved a detailed analysis of the probability of damage to tunnels under seismic hazards. Extending this line of inquiry, Zhong et al. [10,11] contributed to the field by establishing seismic fragility curves for distinct underground structural configurations. This encompassed structures such as the Dakai underground station, a two-story and three-span underground station, and a three-story and three-span underground station, achieved through the implementation of incremental dynamic analysis (IDA) and cloud analysis methods, respectively. Adding granularity to the understanding of seismic vulnerability, Huang et al. [12] conducted an investigation into the impact of site types, tunnel depth, and seismic intensity index on the seismic fragility of tunnels situated in soft soils within the Shanghai area. Their approach involved a thorough utilization of the seismic fragility analysis method. The preceding literature review underscored the predominant emphasis in subsurface seismic hazard research, primarily centered on analyzing influencing factors and seismic fragility, specifically targeting pre-disaster hazard analysis. However, a discernible gap in the scholarly discourse becomes evident-a deficiency lies in the absence of comprehensive analyses regarding station performance losses, robustness, and post-disaster recoverability from a macroscopic viewpoint. This deficiency compromises the establishment of a reliable foundation for the seismic analysis of underground structures.

In response to this gap and with the aim of enhancing the safety of construction and operation within subway transport systems, seismic resilience theory has undergone substantial development in recent years [13]. Seismic resilience, in essence, is defined as the capacity of building structures to withstand, adapt to, and swiftly recover from seismic hazards [14]. Specifically applied to subway stations, it manifests as the explicit seismic design objective of minimizing economic losses and infrastructure system damage while ensuring rapid recovery from disasters. This objective has progressively integrated into the temporal and operational frameworks of associated transport infrastructure design and management processes. In general, performance indicators related to the damaged state of engineered structures serve as crucial metrics for evaluating the resilience of infrastructure—measuring its capacity to absorb damage, recover from disasters, and adapt to evolving conditions [15,16]. These performance indicators are intricately linked to resilience curves, providing a comprehensive depiction of the engineered structure’s robustness and the velocity of functional recovery following a disaster. This perspective incorporates the full-cycle resilience of the structure, considering both the construction and operation phases of the infrastructure [17,18]. Numerous resilience assessment frameworks tailored for single or multiple hazards have been proposed and applied across a diverse spectrum of infrastructures [19,20], encompassing critical components such as bridges [21,22], transport networks [23,24], and various other essential engineering structures [25]. In the realm of seismic resilience for subterranean structures, Huang et al. [26] introduced a comprehensive framework dedicated to the assessment of inter-district tunnels, aimed at evaluating their seismic performance in the face of seismic hazards. Building on this foundation, Chen et al. [27] delved into an extensive analysis of the seismic performance of entire subway systems. This analysis utilized methodologies such as fragility analysis and weighted topology maps assessment. The overall seismic performance of the subway system was quantified employing network indicators and Monte Carlo methods. In a related study, Du et al. [28] scrutinized the existing metro system evaluation indicator system. They based their analysis on the five fundamental dimensions of the physical elements of the metro system, dissecting these into four essential characteristics of resilience over time. Subsequently, they established an evaluation method for the resilience target of the metro system. Expanding the focus to shield tunnel structures, Chen et al. [29] incorporated the concept of resilience into the assessment of seismicity. Their approach was underpinned by a meticulous review methodology, addressing seismic design, construction, operation, and post-earthquake rehabilitation. While the majority of previous investigations predominantly concentrated on the evaluation of non-seismic hazards, particularly fire, on subterranean structures, studies specifically addressing seismic resilience in underground structures have been limited. Within this limited scope, the emphasis has primarily rested on quantifying the resistance of inter-district tunnels. Notably, studies pertaining to stations have been conspicuously scarce. This paucity is further accentuated by the absence of a comprehensive analysis from a macroscopic perspective. Such an analysis should encompass economic losses, robustness, and the trajectory of recovery for stations. Consequently, a critical gap in the existing body of knowledge has been identified, underscoring the imperative for a more holistic examination of seismic resilience in underground stations.

This research presents a practical framework to evaluate the seismic resilience of subway station, considering the effect of economic losses and recovery pathways. The theoretical methodology for constructing the station seismic resilience assessment framework was introduced in Section 2. Additionally, it delineates the development of a numerical model using ABAQUS software in Section 3 to conduct dynamic response analysis for the pertinent station. Section 4 delves into fragility analyses conducted on specific seismic waves. These analyses establish the correlation between seismic intensity and resultant damage, encompassing diverse uncertainties like seismic wave characteristics and building quality. This process aids in formulating station-specific fragility functions for peak ground acceleration (PGA). Connecting station fragility with economic losses completes the seismic resilience assessment, accounting for recovery paths and recoverability. Furthermore, the resilience of a station is significantly influenced by factors such as station structural, site type, functional recovery model, and PGA intensity. In light of these considerations, this study scrutinizes the evolution of station seismic resilience across single-span, two-span, and three-span stations under three pivotal factors: differing site types, functional recovery models, and varying PGA intensities. This comprehensive investigation aims to unveil the impact of these key factors on station performance. Finally, the study critically analyzes the limitations inherent in existing resilience assessment frameworks and proposes recommendations for further research. This exhaustive approach ensures a robust exploration of the proposed framework and its implications for understanding and enhancing station seismic resilience.

2 Seismic resilience assessment framework

2.1 Overview

The station resilience assessment framework introduced in this paper relies on the transverse complete dynamic numerical analysis method applied to the station. This method allows for a comprehensive consideration of key factors influencing seismic resilience, including the structural characteristics of the station, functional recovery models, site type, and seismic intensity. These parameters collectively play a pivotal role in shaping the seismic resilience of the station. The schematic representation of the assessment framework, illustrating the process for determining the seismic resilience of stations, is presented in Fig.1. The proposed methodology is systematically structured, comprising eight distinct steps.

(a) Scrutinize and compile the structural attributes of the resilience assessment subject, the specific characteristics of the site where it is situated, and the seismic inputs.

(b) A numerical model for dynamic analysis is built using the station structural characteristics and site type physical information collected from step (a). At the same time, the determined ground seismic is input into the numerical model to complete the finite element model IDA.

(c) By characterizing station damage in response to seismic hazards, the inter-story drift ratio (IDR) is identified as the station’s damage measure (DM). Subsequently, the station’s damage state (DS) is intricately categorized, drawing upon the DM as a pivotal criterion.

(d) Based on the results of the analysis in step (b) and the DM and DS determined in step (c), the logarithmic, linear relationship between the seismic intensity index (PGA) and the DM is determined. The critical parameter of seismic intensity IM_Si corresponding to different DS is obtained.

(e) Based on the relationship equation of IM–DM obtained in step (d), obtain the logarithmic standard deviation of ground shaking β₁, and associate the standard deviation of construction quality β₂ and the standard deviation of the analysis model β₃ to obtain the total standard deviation (β_tot), which is a crucial parameter.

(f) Based on the two critical parameters obtained in steps (d) and (e), a selected seismic intensity index (PGA) is used to generate station fragility curves.

(g) According to the fragility curves generated based on step (f), the probability of each damage stage of the station under different seismic intensities is obtained, the loss coefficients and the recovery time are associated with generating the loss function and recovery function, and the station function is further determined.

(h) The assessment of station resilience involves the integration of the station function over time in response to seismic hazard disturbances.

This resilience framework facilitates the evaluation of the station’s capacity to withstand earthquake-induced damage and promptly recover from such disruptions. This assessment is achieved through a comprehensive analysis of fragility, coupled with the construction of functional functions. The intricate components of these functional functions are elucidated in detail in the subsequent sections.

2.2 Definition of seismic resilience index R

In this paper, the term “seismic resilience” serves as a precise technical descriptor, indicating the ability of underground stations to withstand seismic events and promptly recover from their impacts. Fig.2 presents a conceptual diagram defining the seismic resilience of subway station, and the corresponding resilience index can be defined as follows:

(1)R=\frac{{\int }_{T_0}^{T_1}Q(t)\mathrm{d}t}{T_2-T_0}.

In Eq. (1): R represents the station resilience, Q(t) represents the functionality function of the station during the seismic hazard restoration period, T₀ is the time of the earthquake, and T₂ is the time of the end of the station restoration. A notable contribution to this field comes from Huang and Zhang [30], who conducted an examination of resilience in the context of tunnel structures following disasters. Their findings revealed that immediate post-disaster recovery efforts are impractical, necessitating a dedicated period for the mobilization of rescue teams and preparation for recovery activities. This temporal gap, referred to as the “evolutionary phase,” represents a critical aspect identified by Huang and Zhang [30]. Expanding upon the insights gained from Huang and Zhang [30], our study acknowledges the existence of an evolutionary phase in underground stations preceding the actual recovery process after a seismic disaster. We posit that, in accordance with a fundamental principle, stations experiencing more severe damage necessitate a more prolonged preparatory phase. Throughout this evolutionary phase, station performance is presumed to remain unchanged, with the duration of this phase directly correlating to the subsequent recovery time. Within this diagram, ‘T₁’ represents the point in time at which the station initiates its recovery efforts, as calculated using the equations provided [26,31–33]:

2.3 Quantification of station performance function Q(t)

The station function plays a pivotal role in delineating the performance degradation and subsequent recovery of a subway station in the wake of a seismic hazard. It stands as a fundamental component in the comprehensive assessment of station resilience. It is crucial to highlight that, in the formulation of the station function, a common assumption prevails: the absence of any performance loss before the occurrence of a seismic hazard. Moreover, the functional decline of the station attributed to aging is typically overlooked in this context, with the initial performance of the station being assumed to be at its optimal level of 100%. This simplifying assumption is made to streamline the analysis of seismic impacts on station performance, focusing primarily on the effects triggered by seismic events rather than considering pre-existing factors such as aging-induced degradation.

The station function constitutes a composite expression encompassing both the loss and recovery functions of a subway station following a seismic hazard. Illustrated in Fig.2, the occurrence of a seismic hazard inflicts damage upon the subway station, leading to a discernible loss of function. Following a preparatory interval, the subway station initiates diverse restoration measures, guiding the trajectory of the station’s functional recovery along a distinct path, ultimately reinstating it to its initial 100% performance level. The formula representing the station function is presented below [34]:

(2)

In Eq. (2): L(I) and frec(t, T₁, T₂–T₁) denote the station loss function and recovery function, respectively, which will be described in detail in Eqs. (4)–(7) below; H(x) is the Heaviside step function [34], as follows:

(3)

In the computation of the building function under seismic hazard, Cimellaro et al. [34] proposed that the building’s function could be derived by assessing the direct economic losses incurred. This conceptual framework has been widely employed in the realm of seismic resilience assessment, as evident in the works of various scholars [26,31–33,35]. In alignment with this prevalent approach, our study adopts this methodology to ascertain the structural loss function specifically tailored for subway stations, as illustrated below:

(4)L(I)=\sum _{j=1}^N{p}_j\cdot u_j.

In Eq. (4): L(I) denotes the station loss function; P_j represents the probability of the station transitioning to DS_j, and u_j denotes the repair coefficient required when the station reaches the DS_j. To ascertain the value of P_j, it is crucial to conduct a thorough station fragility analysis, the intricacies of which will be elaborated upon in Subsection 2.4. The repair coefficient, a pivotal parameter in this context, is defined as the ratio between the station’s repair cost and reconstruction cost. The precise correlation coefficients for this calculation are delineated in Tab.1.

The recovery function serves as a vital tool for delineating the dynamics and pace of a station’s restoration during the recovery phase subsequent to a seismic hazard. Both the recovery duration and the representation of the recovery function are pivotal elements in the assessment of seismic resilience [35]. In this section, drawing upon the works of Huang et al. [32,33] and Kassem and Nazri [35], we opt for the implementation of corresponding exponential, linear, and trigonometric recovery models. These models are selected to capture the nuances of progress and speed in a station’s recovery following a seismic event, considering three distinct types of station recovery: rapid emergency repair, uniform repair, and the inability to repair quickly. The equation for the linear recovery model is presented as follows:

(5)f_{\text{mex}}\left(t,T_1,T_2-T_1\right)=1-\frac{t-T_1}{T_2-T_1}.

The equation for the trigonometric recovery model is shown in the following:

(6)f_{\mathrm{mas}}\left(t,T_1,T_2-T_1\right)=\frac{1}{2}\cdot \left(1+\cos \left(\frac{\text{π}\left(t-T_1\right)}{T_2-T_1}\right)\right).

The equation for the exponential recovery model is shown in the following:

(7)f_{\mathrm{tas}}\left(t,T_1,T_2-T_1\right)=\exp \left(-\left(t-T_1\right)\frac{\ln 200}{T_2-T_1}\right).

In Eqs. (5)–(7), T₁ is the station recovery start time, and T₂ is the end time. The total time required for the recovery process is defined as shown in the following equation [31]:

(8)T_{\text{total}}=T_1=T_2-T_1=\sum _{j=1}^N{t}_j\cdot p_j.

In Eq. (8): t_j is the time required to repair the station completely when its DS is j. Detailed insights into the various station DSs and their corresponding probabilities, denoted as P_j, will be systematically expounded upon in Subsection 2.4 through a meticulous exploration of station fragility. Guided by the stipulations outlined in the Chinese specification “RISN-TG041-2022” [36], the repair time necessary for each station DS is established, as delineated in Tab.2. We posit the assumption of employing a uniform recovery time, regardless of variations in the type and location of the station structure, as long as the station’s DS remains constant.

2.4 Definition of seismic fragility curves

The foundation of resilience analysis lies in the thorough examination of station fragility. The station fragility analysis formula, is instrumental in probing the likelihood of surpassing thresholds for each DS within the station under varying seismic intensities. This process entails establishing a probabilistic correlation between Seismic Intensity Measures (IM) and DM. Both the seismic IM and the structural DM stand out as pivotal parameters in the seismic fragility analysis of subway stations. Seismic IMs are employed to elucidate the seismic capacity to induce damage, with considerations centered on efficiency, practicality, and adequacy [37–40]. In their study, Jiang et al. [37,40] scrutinized 15 seismic intensity indicators (IMs), encompassing peak ground acceleration (PGA), peak ground velocity (PGV), and permanent ground displacements (PGD). Their findings underscored PGA as the most efficient and practical IM for structural fragility analysis of shallow stations. Consequently, PGA is designated as the seismic IM in this paper.

The seismic fragility of stations is conventionally obtained through IDA, employing a numerical simulation approach, as established in prior studies [31,41]. Once the probability of each damaged state of the station is determined through the station fragility analysis, the subsequent step involves defining the loss function and recovery function of the station through coupled equations (Eqs. (4)–(7)).

In seismic fragility analyses, a widely adopted convention is to posit that the structural dynamic response to seismic forces conforms to a lognormal distribution [26,31–33], as depicted in Eq. (9):

(9)P\left(s\ge s_j/M\right)=\mathrm{\Phi }\left[\frac{1}{\sqrt{{\beta }_1^2+{\beta }_2^2+{\beta }_3^2}}\cdot \ln \left(\frac{IM}{I{M}_j}\right)\right].

In Eq. (9): S is the engineering demand parameter of the station after a seismic hazard, S_j is the threshold value of the jth DS corresponding to the station, IM is denoted as the intensity of the seismic wave, P denotes the exceedance probability of a station reaching a particular DS for a given seismic intensity IM; Φ is a standard normal distribution; IM_j is the average value of seismic intensity corresponding to different DS thresholds of the station; β₁ is the standard deviation characterizing the seismic uncertainty; β₂ is the standard deviation characterizing the uncertainty of the construction quality; β₃ is the standard deviation characterizing the uncertainty introduced by the uncertainty of the material and the analysis model standard deviation. In this paper, according to FEMA-P695 [42], β₂ and β₃ are 0.3 and 0.2, respectively.

DM are instrumental in quantifying the extent of structural damage resulting from seismic events, encompassing parameters such as IDR, shear force, and structural acceleration. Recent investigations [37,40–44] have elucidated that the prevailing form of damage sustained by the station during seismic activity is interlayer drift deformation. Furthermore, the structural damage level is intricately linked to this deformation. Therefore, building upon the successful research and application conducted by previous scholars [37,43,44], this paper adopts the IDR as both the DM and the engineering demand parameter. The calculation formula for IDR is presented in the following equation:

(10)\theta =\frac{W}{H}.

In Eq. (10): θ is the IDR, W is the horizontal displacement of the station top plate relative to the bottom plate, and H is the station floor height. Furthermore, Du et al. [44] conducted a comprehensive analysis of 18 underground stations with frame sections, systematically determining the IDR across various structural performance levels. Their classification of subway station damage status involves five distinct levels, delineated by four limits: no damage, minor damage, moderate damage, severe damage, and complete collapse, as detailed in Tab.3. This study adopts the outcomes of their research as the criteria for discriminating the DSs of the stations.

3 Numerical modeling of soil-station system

3.1 Station properties

This study conducts a resilience assessment on three distinct types of stations drawn from actual projects in China. The selected station structures encompass typical single-span, double-span, and triple-span configurations, each designed to suit varying site types and operational environments. Notably, the single-span and two-spans differ in both dimensions and spans, whereas the double-span and three-span stations share identical dimensions but exhibit differing spans. In the numerical simulations, all three stations are modeled using plane-strain solid units, with the center columns and side walls constructed from C50 concrete with uniform material properties. To enhance computational efficiency and cost-effectiveness, the numerical modeling exclusively incorporates primary reinforcement, with a protective layer of concrete set at 5 cm. The reinforcement rate is established at 0.6%. For clarity, the detailed structural dimensions and section reinforcement of the stations are presented in Fig.3–Fig.5.

3.2 Soil profiles

This study analyzes soil profiles based on three distinct site types, as stipulated by the Chinese seismic code [45]: Site types I, II, and III. The objective is to investigate how varying site types impact the seismic resilience of stations. The cumulative thickness of the three site types is 74 m, and the equivalent shear wave velocities for Site types I, II, and III are 478, 278, and 248 m/s, respectively. Fig.6 illustrates the variations in shear wave velocity (V_S), density (ρ), cohesion (c), and friction angle (φ) with burial depth for each of the three site types.

3.3 Input motions

To precisely evaluate station resilience against seismic effects, it’s crucial to determine an adequate number of seismic waves for analysis. Typically, employing 10 seismic waves suffices for the necessary calculations [46]. In this study, ten distinct seismic wave records from the Pacific Earthquake Engineering Research database [47] in the United States were chosen as the seismic loads for analysis. The selection process was based on matching Acceleration Response Spectra, a widely used method [48]. Subsequently, these chosen ground vibration records underwent scaling from 0.1g to 1.0g using amplitude modulation with a 0.1g gradient. This scaling process generated 100 seismic waves that were then incorporated into the finite element model to conduct IDA. Detailed seismic wave data are outlined in Tab.4, while the seismic acceleration response spectra of the selected ground motions are depicted in Fig.7.

3.4 Development of the soil-station numerical models

The finite element numerical model (2D), developed using ABAQUS software, meticulously considers the intricate interplay between the soil and station dynamics, enabling a comprehensive analysis of seismic response through full-time dynamic computations. To mitigate potential boundary effects, the model’s width spans 300 m, while its height stands at 74 m. The accurate selection of a constitutive relationship is pivotal in faithfully capturing the material stress–strain behavior. Within this study, the Moore Cullen constitutive model is employed to simulate the nonlinear dynamic response of the station-soil coupled numerical model under seismic loading. This model effectively accounts for inelastic material deformation, stiffness degradation due to damage, and the interplay between concrete damage evolution and plasticity by introducing pertinent damage variables [49]. The primary station structure is composed of C50-strength concrete. For the simulation of the lining, this paper adopts the Concrete Plastic Damage constitutive model. The concrete material properties align with the Chinese specification [50] and are meticulously detailed in Tab.5. The evolution of the concrete damage factor and the stress–strain relationship are visually represented in Fig.8(a) and Fig.8(b), providing a comprehensive overview of the material behavior under varying conditions. This approach ensures a robust understanding of the structural response to seismic forces, emphasizing the significance of accurately defined constitutive relationships in the analytical process.

The primary reinforcing bars employed in the station structure predominantly consist of HRB335 steel. In accordance with the stipulations outlined in the Chinese specification [50] and drawing upon relevant findings from prior research [49,51], an elasticity constitutive simulation is applied to model the behavior of these reinforcing bars. The intricacies of the material properties specific to the reinforcing bars are meticulously outlined in Tab.6. Additionally, the stress–strain relationship for these reinforcing bars is visually depicted in Fig.8(c), providing a comprehensive representation of their mechanical response under varying loads.

The transverse boundaries at both ends of the finite element model are configured with multi-point constraint (MPC) kinematic coupling, ensuring synchronized movement of the two end boundaries. This kinematic connection constraint finds widespread application in the seismic response analysis of subterranean structures, as evidenced by its utilization in various studies [12, 26,32,33,41]. To model the bottom boundary of the finite element, we adopt the methodology proposed by Lysmer and Kuhlemeye [41], simulating it as “elastic bedrock.” The seismic loading of the finite element model is achieved by appropriately incorporating dampers. Fig.9 provides a comprehensive illustration of the model’s dimensions and boundary conditions. The damper coefficient, denoted as C, is derived from the product of the mass density (ρ), shear wave velocity (Vs), and the “control area” of each damper (A), as expressed in Eq. (11). Specific values for parameters such as mass density (ρ) and shear wave velocity (Vs) are obtainable by referencing Fig.6.

(11)C=\rho \times V/s\times A.

The finite element model of both the station and the soil employed a plane strain cell model with the specific cell type CPE4R. For the reinforcement cells, a two-dimensional truss cell model with cell type T3D2 was implemented. The mesh size adhered to the criteria outlined in the study by Lysmer and Kuhlemeye [41], ensuring optimal mesh sensitivity. In terms of interaction relationships, the reinforcement is intricately integrated into the station lining through a built-in embedding technique. The contact dynamics between the lining and the surrounding soil are categorized into normal and tangential behaviors. A robust hard contact relationship governs the normal behavior, while a penalty function relationship governs the tangential behavior. The friction coefficient for the tangential behavior is meticulously set to 0.6, as stipulated in previous research [26,32,33].

3.5 Typical dynamic responses of soil-station system

3.5.1 Inter-story drift ratio (θ) of stations

As a representative case study, we have chosen to focus on the seismic analysis and characterization of the response of a shallowly buried station located in Site type I when subjected to Eq. (6) seismic waves. Fig.10 illustrates the curves of IDR for three distinct station types, namely single-span, double-span, and triple-span, at PGA values of 0.1g, 0.5g, and 1.0g. Examining Fig.10, it becomes evident that the IDR for all station types exhibits an increasing trend with escalating seismic intensity, as measured by PGA. For instance, considering the single-span station, depicted in Fig.10(a)–Fig.10(c), the maximum IDR rises from 0.63‱ at 0.1gPGA to 48.6‱ at 1.0gPGA. This observed trend underscores the direct correlation between seismic intensity and the extent of station deformation. A higher PGA corresponds to a more pronounced deformation of the strata, consequently leading to a more significant structural deformation in the station. Notably, when the input PGA is relatively small (0.1g), the station’s IDR oscillates around the 0‱ mark, suggesting that the station deformation at this seismic intensity is entirely recoverable. However, as the input PGA increases to 0.5g and 1.0g, residual IDR values emerge 1.8‱ and 30.5‱, respectively-indicating that the station’s deformation becomes partially irrecoverable at these higher seismic intensities. Similar trends are observed for both double-span and three-span stations, where the IDR increases with rising PGA levels, and residual IDR values manifest at certain PGA thresholds. Notably, at equivalent PGA levels, the IDR values are consistently smaller in two-span stations and smallest in three-span stations compared to their single-span counterparts. This discrepancy can be attributed to the multifactorial interplay of factors such as decreasing station span-to-height ratios, station dimensions, material properties, and surrounding stratigraphy. These factors collectively contribute to a higher relative stiffness in two-span stations and the most substantial relative stiffness in three-span stations, influencing the observed differences in IDR among the station types.

3.5.2 Seismic damage distribution at stations

The seismic analysis in this study focuses on the dynamic response of stations under the representative working condition of seismic wave Eq. (6) and Class I site types. Fig.11 illustrates the damage cloud maps of three station types at different PGA values: 0.1g, 0.5g, and 1.0g. Tensile damage, characterized by the DAMAGET parameter, is employed to analyze crack development in concrete for various station structures [49]. This parameter quantifies the extent of damage, ranging from 0 (undamaged) to 1 (completely damaged). Examining Fig.11 reveals a progressive increase in concrete damage with rising PGA. At 0.1g, the single-span, double-span, and three-span stations exhibit minimal concrete damage, evolving to partial damage at 0.5g, and extensive damage at 1.0g. Notably, single-span stations display a more pronounced area of concrete damage than double-span and three-span stations at the same PGA intensity, indicating higher susceptibility to damage. This trend aligns with the IDR depicted in Fig.10. Furthermore, the concrete damage primarily concentrates at the intersection of walls and floor slabs, with less damage observed at the floor span location. This observation underscores that the predominant damage mechanism involves inter-story misalignment. The selection of IDR as the DM is justified in this context. The consistent patterns in Fig.10 and Fig.11 underscore the significant influence of seismic input intensity, soil-station interaction, and station type on the seismic response of the soil-station system. This aligns with findings from prior studies on the seismic response of underground structures [26,32,33].

4 Evaluation of seismic resilience of subway stations

4.1 Development of fragility curves

In accordance with the discussions presented in Subsections 2.2, 2.3, and 2.4, a comprehensive characterization of the structural performance of the station under seismic conditions necessitates the introduction of two intermediary variables: the Station DM and the IM. Additionally, two pivotal parameters are indispensable for the formulation of station fragility curves: first, the mean value of seismic intensity (IM_j) corresponding to distinct DS thresholds, and secondly, the standard deviation denoted as β₁, which serves to quantify the seismic uncertainty. This paper adopts a log-linear regression analysis methodology, employing the natural logarithm of the DM as the dependent variable and the IM as the independent variable. Through the application of the regression model to the numerical analysis results and in alignment with the DSs defined in Tab.3, we determine the mean values of seismic intensities (IM_j) corresponding to the thresholds of varying DSs. Simultaneously, the standard deviation β₁, a crucial factor characterizing seismic uncertainty, is further ascertained.

4.1.1 Evolution of the damage measure with intensity measure

Fig.12−Fig.13 delineate the progression of the relationship between the logarithm of PGA and log IDR across three prototypical station structures under varying site types. Each data point within these figures signifies the IDR associated with a specific station configuration, site type, and diverse seismic waves and PGA values. The red solid line within these figures illustrates the regression fitting curve derived from the log IDR data. Following the establishment of the logarithmic regression correlation between the IM and DM, the mean seismic intensity values (IM_j) corresponding to distinct DS thresholds are computed, leveraging the regression equation and the delineations of diverse DSs outlined in Tab.3. Simultaneously, the standard deviation β₁, encapsulating seismic uncertainty, is calculated, and further amalgamation of β₂ and β₃ yields the standard deviation representing the total uncertainty. Notably, within Fig.12−Fig.13, the correlation coefficient R² surpasses 0.70 subsequent to the linear regression fitting. This substantial value signifies a robust linear relationship between log DM and IM, thereby underpinning the validity and reliability of subsequent probabilistic fragility analyses. Such a correlation reinforces the credibility and feasibility of these analyses, ensuring a dependable basis for further investigation and inference.

4.1.2 Proposed fragility curves

The average seismic intensity values (IM_j) corresponding to the DM across different DSs in Fig.12 have been compiled. The standard deviation β₁, reflecting the seismic uncertainty, is then calculated to establish the station fragility data, as delineated in Tab.7. Utilizing this station fragility database, in conjunction with the information detailed in Subsection 2.4, enables the construction of fragility curves for the three stations under the three specified site types. These curves serve as a foundational framework for conducting a comprehensive assessment of station resilience. Furthermore, Tab.7 illustrates a noteworthy trend: as site types degrade from I to III, the mean seismic intensity (IM_j) diminishes across various station types. Conversely, both the standard deviation β₁ and the total standard deviation exhibit an increment. This indicates a reduction in the seismic performance of the stations with worsening site types, accompanied by an augmentation in the dispersion of seismic performance levels. This observation underscores the correlation between site types and station seismic performance, emphasizing the need for a thorough understanding of these dynamics in resilience assessments.

Based on the data presented in Tab.7 and as elucidated in Subsection 2.4, fragility curves were generated for the three stations under three distinct site types, as illustrated in Fig.15–Fig.17, which visually depicts the variations in fragility curves corresponding to different site types and station structural types. Notably, it is evident that both site types and station forms exert a substantial influence on the seismic fragility of the station. It is imperative to acknowledge regional disparities in seismic protection standards across China. In the Shanghai region, for instance, a PGA ranging from 0.10g to 0.20g aligns with the design ground acceleration possessing a 10% probability of exceedance over a 50-year period. This specifically caters to the seismic protection requirements of levels 7 and 8. Remarkably, at a PGA of 0.1g, the probability of experiencing moderate damage across all three station types under various site types is deemed negligible. The highest likelihood of surpassing moderate damage for single-span, double-span, and three-span stations across diverse sites is recorded at 1.86%, 1.49%, and 1.01%, respectively. This suggests that, under these site types, the stations can endure seismic intensities of this magnitude without incurring any significant damage. Upon reaching a PGA of 0.2g, there is a marginal increase in the probability of exceeding moderate damage for the three station types under different site types. However, the maximum probabilities recorded are only 9.82%, 8.62%, and 6.99%, respectively. This implies that, although the station’s fundamental performance is maintained under these site types, there is a possibility of minor damage. The stations, however, remain resilient and are likely to withstand this level of seismic intensity without suffering substantial harm.

As the PGA escalates to a maximum intensity of 1.0g, the maximum probabilities of exceeding moderate damage surge to 70.66%, 69.51%, and 68.99% for the three station types. Concurrently, the maximum probabilities of experiencing collapse reach 34.57%, 28.15%, and 27.93%, respectively. These findings underscore the susceptibility of the stations to collapse under conditions of exceptionally strong seismic activity for the specified site scenarios. It is noteworthy that the computational outcomes reveal a distinct disparity in damage probabilities under equivalent seismic intensities. Specifically, the damage probability for a single-span station surpasses that of a two-span and three-span station significantly. Moreover, the damage probability associated with a Class III site type markedly exceeds that of a Class I site type. This implies that, under identical seismic intensities, single-span stations face a notably higher risk of damage compared to their multi-span counterparts, and locations characterized by Class III site type exhibit a substantially elevated risk compared to Class I site type.

4.1.3 Comparisons with existing fragility curves

The American Lifeline Alliance (ALA) [52] has introduced empirical fragility curves for underground structures, accounting for variations in construction conditions ranging from suboptimal to excellent. Within hazards united states (HAZUS) [53], fragility curves specific to underground structures have been empirically derived, leveraging PGA and PGD from existing seismic databases. Nguyen et al. [8] has produced fragility curves based on single-span, double-span, and triple-span underground structures implemented in a subway system. Zhang et al. [38,39] has delved into the determination of the optimal seismic IM for the probabilistic seismic demand model of stations and subsequently formulated seismic fragility curves based on this optimal IM. In this study, we undertake a comparative analysis of the fragility curves for the proposed single-span station situated in a Class III site with those derived from prior investigations, as illustrated in Fig.18. It is crucial to emphasize that the majority of published fragility curves for underground structures primarily rely on PGA [8]. Consequently, our comparison will specifically focus on seismic fragility curves derived from PGA.

As depicted in Fig.18, the overall trend of the fragility curves proposed in this study, alongside the two empirical fragility curves and the numerical fragility curves investigated by previous researchers, exhibits a remarkable similarity. However, nuanced distinctions exist. Notably, for a given PGA, the fragility curves presented in this paper align more closely with those put forth by Nguyen et al. [8] and Zhang et al. [38,39]. Moreover, the associated damage probabilities are generally higher than the empirical fragility curves of ALA [52] and HAZUS [53]. This convergence can be attributed to the shared geological considerations between this study, Zhang et al. [38,39], and Nguyen et al. [8], equivalent to the European standard for Class C or D site types. Conversely, the empirical fragility curves of ALA [52] and HAZUS [53], when averaged across all considered site types, tend to outperform those considered in this paper. Consequently, the damage probabilities presented in this study, when compared with Nguyen et al. [8] and Zhang et al. [38,39], exhibit higher values than those derived from the empirical brittleness curves. It is essential to underscore that discrepancies between this study and Nguyen et al. [8] and Zhang et al. [38,39], arising from factors such as the structural form of the underground facilities and the definition of DSs, contribute to variations in the fragility curves. Nonetheless, the overarching developmental pattern remains consistent.

4.2 Seismic resilience assessment of subway stations

In the computation of building functionality under seismic hazards, Cimellaro et al. [34] posited that the assessment of building function can be derived from the evaluation of direct economic losses. This concept has found substantial application in seismic resilience assessment research [26,31–33,35]. Leveraging this conceptual framework, the present study extends its exploration into the seismic resilience of underground stations. This exploration is based on fragility data compiled in Subsection 4.1, focusing on three representative stations situated in distinct site types. The analysis is coupled with the resilience assessment methodology outlined in Subsection 2.1.

4.2.1 Resilience assessment of the single-span station

1) Development of resilience curves

This paper explores the influence of different recovery models, site types, and PGA on the resilience curve of a single-span station, visually depicted in Fig.19. Extending the work of Cimellaro et al. [34] and Kassem and Nazri [35], our study integrates three distinct recovery models: exponential, linear, and triangular. These models are associated with three unique station recovery scenarios, namely rapid emergency recovery, uniform recovery, and no rapid recovery. In this section, we present a detailed case study of a typical single-span station, examining its response under three distinct site types and two varying levels of PGA. This case serves as a representative example, allowing us to calculate the functional curve of the station and assess how different recovery models influence its resilience capacity. As observed in Fig.19, the resilience values obtained using the exponential recovery model for rapid repair notably surpass those determined through the linear and triangular recovery models. Furthermore, the resilience values derived from the linear recovery model are slightly higher than those yielded by the triangular recovery model under identical operational conditions. For instance, when we focus on Site Type III with a PGA of 1.0g, as depicted in Fig.19(f), the resilience values (denoted as R) calculated using the three recovery models are as follows: 0.688 for the exponential model, 0.606 for the linear model, and 0.602 for the triangular model. In addition, upon a meticulous comparison of Fig.19(a) and Fig.19(d), it becomes evident that the resilience values (R) derived from the exponential recovery model for the single-span station are 0.954 and 0.854 under identical site types. This reveals a discernible diminishing trend as the PGA escalates from 0.5g to 1.0g. Subsequently, a comprehensive analysis of Fig.19(d)–Fig.19(f) indicates that the resilience values (R) computed through the exponential recovery model for the single-span station are 0.854, 0.744, and 0.688, respectively, when subjected to the same 1.0gPGA. This highlights a consistent decline in resilience as the soil type transitions from Type I to Type III. In practical seismic scenarios, the judicious selection of expedited repair methodologies and optimal site types can substantially enhance the operational efficacy of underground stations.

2) Development of resilience index R

Based on the fragility data obtained for the single-span station under diverse site types, as meticulously compiled in Subsection 4.1 and in conjunction with the resilience assessment method delineated in Subsection 2.1, we have systematically gathered and computed the resilience values (denoted as R) for the single-span station across varying PGA conditions, as illustrated in Fig.20. A careful examination of Fig.20 reveals a discernible diminishing trend in the resilience index (R) as the PGA escalates, with this reduction becoming particularly pronounced in instances of softening site types. To elucidate further, let us consider the scenario of the single-span station adopting a linear recovery model. At a PGA of 0.1g, the resilience of the station stands at 0.999, 0.992, and 0.985 under site types I, II, and III, respectively. In stark contrast, when the PGA is elevated to 1.0g, the resilience undergoes a notable decline, registering values of 0.816, 0.677, and 0.605 under site types I, II, and III, respectively. This substantial divergence in resilience among the three site types underscores the discernible impact of increased PGA on the overall resilience of the single-span station.

Drawing upon Huang’s classification of underground structures [26,32,33], we conducted a meticulous assessment of seismic resilience (denoted as R) for single-span stations. The analysis, illustrated in Fig.20, specifically focuses on diverse site types and recovery models preceding a PGA of 0.3g, revealing a notably high level of resilience. Extending our investigation to scenarios involving a higher PGA of 1.0g, the seismic resilience R, while slightly diminished, consistently maintains a commendable level, characterized as medium resilience under varying site types and recovery models. Furthermore, aligning with the stipulations outlined in the Chinese code [45], it is noteworthy that the observed PGA of 0.3g has attained a seismic protection intensity equivalent to level 8. This signifies compliance with seismic protection standards in the majority of regions across China. The implication is that the seismic resilience of single-span stations, particularly noted for its high resilience under specific conditions, attests to their adherence to rigorous seismic protection criteria.

4.2.2 Resilience assessment of the two-span station

1) Development of resilience curves

This paper systematically investigates the impact of various recovery models, site types, and PGA on the resilience curve of a two-span station, as delineated in Fig.21. Within this section, we meticulously examine a prototypical scenario involving a two-span station subjected to three distinct site types and two different PGA levels. In congruence with findings related to single-span stations, the rapid repair resilience values (denoted as R) obtained from the exponential recovery model for the two-span stations exhibit a marked increase compared to those derived from the linear and triangular recovery models. Noteworthy is the observation that the resilience values R, calculated by the linear recovery model, slightly surpass those computed by the triangular recovery model. As an instance, under site type III and 1.0gPGA, as depicted in Fig.21(f), the resilience values R derived from the three recovery models are 0.712, 0.640, and 0.636, respectively. Moreover, an analysis of Fig.17 reveals a consistent pattern: akin to their single-span counterparts, the seismic resilience value R of the two-span stations demonstrates a diminishing trend concurrent with an escalation in PGA levels and the softness of site types. This trend underscores the inherent sensitivity of the station’s resilience to variations in both seismic intensity and site characteristics.

2) Development of resilience index R

In accordance with the fragility data pertaining to the two-span station under diverse site types, as meticulously compiled in Subsection 4.1, and leveraging the resilience assessment methodology delineated in Subsection 2.1, the resilience values (designated as R) for the two-span station are systematically gathered and computed across varying PGA scenarios, as depicted in Fig.22. Analysis of Fig.22 reveals, analogous to the trends observed in single-span stations, a diminishing trajectory in the two-span station’s resilience index R with an escalating PGA, a trend that becomes more pronounced as site types soften. Illustrating this point with the specific case of a two-span station employing a linear recovery model, when the PGA is 0.1g, the resilience values for the station are 0.999, 0.991, and 0.986 under site types I, II, and III, respectively. However, as the PGA intensifies to 1.0g, the corresponding resilience values decrease to 0.898, 0.703, and 0.636 under site types I, II, and III, respectively. This widening gap in resilience among the three site types underscores the heightened vulnerability of the two-span station under more severe seismic loading, accentuated by exacerbated site types.

4.2.3 Resilience assessment of the three-span station

1) Development of resilience curves

This paper investigates the impact of diverse recovery models, site types, and PGA on the resilience curve of a three-span station, as illustrated in Fig.23. In this section, we delve into a representative case study of a three-span station under three distinct site types and two varying PGA levels. This case serves as an illustrative example to compute the functional curve and analyze the effects of different recovery models on the station’s recovery capability. Consistent with observations in single-span and two-span stations, the rapid repair resilience values (denoted as R) calculated using the exponential recovery model for the three-span station are significantly larger than those derived from the linear and triangular recovery models. Notably, the resilience values R obtained through the linear recovery model are marginally greater than those calculated by the triangular recovery model. For instance, under site type III and a 1.0gPGA, as depicted in Fig.23(f), the resilience values R, computed using the three recovery models, are 0.715, 0.642, and 0.639, respectively. Furthermore, an observation from Fig.23 reveals a consistent trend observed in single-span and two-span stations. The seismic resilience value R of the three-span station exhibits a decreasing pattern with the escalation of PGA and softness in site types.

2) Development of resilience index R

Based on the fragility data presented in Subsection 4.1 for the three-span station across diverse site types and utilizing the resilience assessment method outlined in Subsection 2.1, the resilience values (denoted as R) of the three-span station are systematically gathered and computed under varying PGA conditions, as illustrated in Fig.24. The graphical representation in Fig.24 reveals a pattern akin to that observed in single-span and two-span stations, wherein the resilience index R of the three-span station diminishes with escalating PGA values, and this decline becomes more pronounced as site types soften. To illustrate this trend further, consider the case of a two-span station employing a linear recovery model. When the PGA is 0.1g, the station’s resilience stands at 0.999, 0.994, and 0.989 under site types I, II, and III, respectively. However, with an increase in PGA to 1.0g, the resilience of the station undergoes a noticeable reduction to 0.904, 0.701, and 0.639 under site types I, II, and III, respectively. This disparity underscores the amplified impact of higher PGA values on the resilience of the station, especially under adverse site types.

4.2.4 Discussions

In this section, we investigate the impact of varying PGA levels on the resilience of a single-span station. The resilience analysis focuses on a representative case, specifically a station characterized by Site type III and employing the exponential recovery model. The ensuing exploration aims to discern the nuanced effects of PGA intensities at 0.4g, 0.7g, and 1.0g on the station’s overall resilience, with the results graphically depicted in Fig.25. Upon scrutinizing Fig.25, a clear pattern emerges. The resilience values computed at 0.4g for PGA exhibit a noteworthy superiority over those determined at 0.7 and 1.0g for PGA under identical operational conditions. Furthermore, the resilience values calculated at 0.7g for PGA surpass those obtained at 1.0g for PGA. This discernible trend underscores the pronounced influence of PGA intensity on the station’s resilience. Quantitatively, the resilience values corresponding to the three PGA intensities are delineated as 0.880, 0.772, and 0.688, respectively. These numerical outcomes provide a precise metric for evaluating the station’s ability to recover under different seismic intensities. This rigorous analysis contributes valuable insights into the intricate relationship between PGA levels and the resilience of the single-span station.

To investigate the influence of diverse structural forms on the resilience of a station, this section conducts a resilience analysis using a representative scenario with Site Type III, a PGA of 1.0g, and an exponential recovery model. The impact of single-span, two-span, and three-span structural configurations on the station’s resilience is comprehensively examined and graphically presented in Fig.26. Fig.26 reveals a notable disparity in the calculated resilience values of the station structure under identical operational conditions. Specifically, the resilience values for double-deck double-span and double-deck triple-span configurations are markedly higher than those for the double-deck single-span configuration. The quantified resilience values for the three station structure forms are 0.688, 0.712, and 0.715, respectively. It is imperative to highlight that while the seismic resilience of double-span and three-span stations appears superior in this context, this outcome should not be generalized across all scenarios. Numerous influencing factors, such as the structural span-height ratio, the relative stiffness of the station, and the surrounding strata, necessitate consideration. Therefore, in practical applications, the selection of the station’s structural form must be judiciously made, taking into account these influencing factors to enhance the station’s resilience and ensure the uninterrupted functionality of the station.

5 Conclusions

This study proposes a comprehensive framework for the seismic resilience evaluation of subway stations. The impact of various factors, including station structural design, site types, functional recovery models, and varying PGA intensities on the seismic resilience of these stations were assessed and quantified. The relevant conclusions are presented below.

1) A comprehensive framework for assessing the seismic resilience of stations was developed by integrating the effects of seismic fragility functions, economic losses and recovery models.

2) Seismic fragility functions specific to the examined stations have been formulated. These functions allow for the quantification of the influence of site type softness and escalating seismic intensity on the heightened probability of station damage. Additionally, the impact of transitioning between different station types on seismic fragility has been thoroughly examined and discussed. To validate the credibility of the proposed susceptibility curves, a meticulous comparison has been conducted against established susceptibility curves.

3) The investigation revealed a noteworthy reduction in the seismic resilience of the station as the site type experienced softness and seismic intensity escalated. It is imperative to underscore that, when exposed to identical conditions, the seismic resilience of two-span and three-span stations experiences a noteworthy augmentation in comparison to that of single-span stations. Furthermore, it is discerned that three-span stations demonstrate a slightly superior seismic resilience when juxtaposed with their two-span counterparts. However, exercising caution in extending this conclusion is paramount, given the presence of various influential factors. These factors include, but are not limited to, the structural span-to-height ratio, the relative stiffness of the station, and the characteristics of the site types.

4) The effect of various recovery models on the seismic resilience of subway stations were revealed, it was found that the resilience values computed using the exponential recovery model with rapid repair markedly surpass those derived from both the linear and trigonometric recovery models under identical operational scenarios,. Furthermore, the resilience values obtained from the linear recovery model slightly exceed those from the trigonometric recovery model. This discrepancy emphasizes the substantial impact of the expeditious post-disaster repair efforts on enhancing the seismic resilience of the stations. The observed trend underscores the significance of the speed at which post-disaster repair activities are conducted as a pivotal factor in augmenting the overall seismic resilience of the stations.

In the context of seismic station design, the efficacy of various parameters, such as diverse structural systems and soil types, can be quantified through the framework of resilience assessment presented in this study. This assessment plays a pivotal role in aiding decision-making and prioritization, particularly in the context of mitigating seismic risks in station environments. It is worth highlighting that the station loss assessment and all the functional recovery models proposed in this paper draw upon the seminal research conducted by Cimellaro et al. [34]. These research findings have found extensive application in the evaluation of seismic performance, not only in above-ground structures but also in various underground constructions [21,22,34,35]. Looking forward, further research in the realm of performance restoration modeling is imperative. This entails a comprehensive consideration of key parameters influencing station functional restoration, including the nature and extent of damage, resource availability, and the methodologies employed in construction and maintenance management. This extended research horizon aims to enhance the restoration process, transitioning from component-level restoration to the holistic restoration of the entire station system. Such an evolution in research methodology is pivotal for achieving a more comprehensive assessment of the seismic resilience of stations.

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