Fragility analysis of a subway station structure by incremental dynamic analysis

Abstract

Fragility analysis constitutes the basis in seismic risk assessment and performance-based earthquake engineering during which the probability of a structure response exceeding a certain limit state at a given seismic intensity is sought to relate seismic intensity and structural vulnerability. In this article, the seismic vulnerability assessment of a subway station structure is investigated using a probabilistic method. The Daikai subway station was selected as an example structure and its seismic responses are modeled according to the nonlinear incremental dynamic analysis procedure. The limit states are defined in terms of the deformation and waterproof performance of the subway station structure based on the central column drift angle and the structural tension damage distribution obtained from the incremental dynamic analysis. Fragility curves were developed at those limit states and the probability of exceedance at the limit states of operational, slight damage, life safety, and collapse prevention was determined for the two seismic hazard levels. Results reveal that the proposed fragility analysis implementation procedure to the subway station structure provides an effective and reliable seismic vulnerability analysis method, which is essential for these underground structural systems considering their high potential risk during seismic events.

Keywords

damage measure fragility curves incremental dynamic analysis limit states subway station structure

Introduction

Lifelines and civil infrastructural systems, such as subways, tunnels, and underground stores built in densely populated earthquake regions, require a proper seismic risk assessment to foreseen their potential failures and ensure earthquake preparedness. Although underground structures generally perform better than surface structures during earthquakes, significant damages were reported in underground facilities subjected to strong ground motions (Hashash et al., 2001). This is especially true for shallow subway station structures that may experience high bending moments as well as axial and transversal loads during an earthquake (Huo et al., 2005). Hence, it is crucial to conduct seismic vulnerability assessment for subway station structures.

Studies have been carried out to understand the seismic behavior and proper design approaches of subway station structures. Kawashima (1994) proposed the response deformation method, which imposed the ground deformations as a static load on the underground structure through the foundation of soil springs. The method has been widely used in seismic design of underground structure due to its less computational efforts. Hashash et al. (2001) proposed a relatively simple free-field ground deformation approach during which the interaction of the underground structure and the surrounding soil ground is ignored. This approach is suitable when low intensities of earthquake are considered or the underground facility is in a stiff such as rock. Liu et al. (2014) extend the pushover analysis approach which is a nonlinear static analysis to underground structure. In pushover analysis approach, the monotonically increasing forces that replace the seismic loads are applied to a soil-structure system with an additional free field until a predetermined target displacement is reached and then the structural peak responses as well as internal forces under a given seismic intensity can be obtained. However, force distributions provided by Liu et al. cannot be available for all kinds of underground structures and site conditions. Besides, pushover analysis is a deterministic approach, which cannot consider the uncertainty of ground motions. The above-mentioned approaches are all pseudo-static analysis approaches, contrary to which are dynamic analysis approaches. Huo et al. (2005) studied the seismic performance of the Daikai Station using dynamic numerical analyses during which two key factors (i.e. the relative stiffness between the structure and the degraded surrounding soil and the fractional characteristics of the interface) that determine the response of the subway station structure are determined. In addition, the input ground motions are specified in the study of Huo et al., which cannot take the probabilistic ground motions into consideration.

Most of the previous studies have been deterministic, rather than probabilistic. Due to the random nature of earthquakes and structural seismic performance, it would be beneficial to study this problem from a probabilistic point of view. Fragility function quantifies the probability of selected system responses exceeding a given damage measure (DM) at a specified ground motion intensity and it is a probabilistic approach to conduct the vulnerability and risk assessment of subway station structures. Traditionally, fragility curves have been established for buildings (Brunesi et al., 2015b; Kiani et al., 2016; Park and Kim, 2010), and improved fragility analysis methods are also proposed for surface structure (Banerjee et al., 2016; Karapetrou et al., 2016; Mandal et al., 2016). In this article, fragility analysis is extended to the lifeline system of the subway station structure during which a probabilistic framework is developed. For this purpose, the Daikai subway station was selected and modeled and an incremental dynamic analysis (IDA) was performed utilizing 12 ground motions. Limit states of the subway station structure are obtained according to the intensity measure (IM), central column drift angle, and structural damage distribution. Finally, fragility curves are derived at different limit states. The fragility analysis of the subway station structure presented herein partially fills the knowledge gap required for performing risk assessment of subway station structures.

Numerical modeling

According to the general profile of the Daikai Station (Huo et al. 2005), the main section of the station that collapsed during the Kobe earthquake is considered as the structure under investigation (Figure 1). This section consists of a rectangular reinforced concrete box structure with dimensions of 17 m wide and 7.17 m high and central columns that are spaced at 3.5 m in the longitudinal direction. The columns are about 5 m high and have a rectangular reinforced concrete cross section of 0.4 m by 1.0 m. The reinforcement details of central column in Daikai Station are illustrated in Figure 1. Due to limited space, for more reinforcement layout information, please refer to Parra-Montesinos et al. (2006).

Figure 1.

(a) Cross section of the Daikai Station and (b) reinforcement details of central column, dimension in (mm) (Huo et al., 2005; Parra-Montesinos et al., 2006).

A detailed two-dimensional (2D) numerical model of the structure and the surrounding soil is developed using the finite element software package ABAQUS (2010). The software package has the ability to accurately determine the dynamic behavior from elastic to plastic, while taking into account both material inelasticity and geometric nonlinearities. The subway station structural model is 1000 m long and 58 m high (Figure 2). For the selection of structural element type, solid elements based on classical principles of nonlinear fracture mechanics are more accurate in predicting stress or strain concentrations that occur in shear-sensitive members (Brunesi et al., 2015a; Brunesi and Nascimbene, 2015). However, considering the calculation efficiency, beam element (B21), which can also simulate the structural force and deformation very well (Chen et al., 2014), is adopted for the reinforced concrete structural members here. The unit weight and the Young’s modulus of the concrete are set as 25 kN/m³ and 24 GPa, respectively. The concrete plastic damage model proposed by Lubliner et al. (1989) and Lee and Fenves (1998) was adopted. This model uses two damage variables (i.e. the tensile damage and the compressive damage) and a yield function to account for different damage states. The reinforcement of the 2D frame (Huo et al., 2005) is attained through the rebar command. Constitutive behavior of the reinforcement steel (yield strength: 312 MPa and Young’s modulus: 200 GPa) was modeled using the Menegotto-Pinto steel model (Menegotto and Pinto, 1973) modified by Filippou et al. (1983). This model assumes a bilinear backbone curve with isotropic strain hardening of 1% (Elnashai and Sarno, 2010).

Figure 2.

Numerical model of Daikai Station and the surrounding soil.

The four-node plane strain element (CPE4R) and the quadrilateral plane strain infinite element (CINPE4) were adopted for soil element, and the Mohr–Coulomb model was used to simulate the soil’s constitutive characteristics. The soil parameters are shown in Table 1. The interface between the soil and the structure is modeled as a frictional surface with a coefficient of friction µ of 0.4 and a friction angle of 22°. There is no cohesion between the structure and the soil.

Table 1.

Soil parameters of the Daikai Station.

Sequence	Name	Depth (m)	Unit weight (kN/m³)	Cohesion yield stress (kPa)	Friction angle (°)	Poisson’s ratio	Elastic modulus (MPa)
1	Fill	0–1	19	20	15	0.33	101.308
2	Holocene clay	1–2	19	30	20	0.32	101.308
3	Holocene sand	2–4.8	19	1	40	0.32	147.840
4	Pleistocene sand	4.8–8	19	1	40	0.40	195.972
5	Pleistocene clay	8–17	19	30	20	0.30	290.342
6	Pleistocene gravel	17–	20	1	40	0.26	560.045

The boundary conditions of the model are as follows: the horizontal and vertical displacements are fixed at the bottom surface while the top of the structure is free. Infinite elements were applied at the lateral boundaries and the ground motions are imposed at the bottom of the model.

The analysis results of this subway station numerical model are compared to those obtained by Huo et al. (2005) and they are reasonably close, which validated the accuracy of this numerical model that would be utilized in the IDA as discussed next. The detailed verification process and identity for the numerical model are provided by Chen et al. (2014).

Incremental dynamic analysis

During IDA, a series of earthquake ground motions are applied to the structural model. By increasing the ground motion intensity, the seismic performance of the structural system from the linear elastic phase through yielding to collapse is captured. Thus, the seismic performance and its progressive limit states can be identified until collapse. Results of IDA are utilized to generate fragility curves (Vamvatsikos and Cornell, 2002).

An IDA curve is a curve whose horizontal axis is the DM that shows the structural responses, and the vertical axis is the IM that shows the ground motion intensity (Vamvatsikos and Cornell, 2002). To perform IDA of the subway station structure, the ground motion records were gradually scaled up according to a specific proportion that made the peak ground velocity (PGV) equivalent to 5, 10, 20, 30, 40, 50, 60, 70, and 80 cm/s, respectively. Then, time history analysis was conducted at each intensity level during which the ground motions are horizontally applied at the bottom of the model (Figure 2). The detailed procedure is illustrated in Figure 3.

Figure 3.

Flow chart of IDA.

Considering the seismic performance characteristics of subway station structures, the selection of ground motions, IM as well as DM are discussed in detail in the following sections.

Selection of ground motion records

The ground motion records are generally selected based on the soil type of the site where the structure is located (FEMA-351, 2000). The sites are divided into Classes A, B, C, D, E, and F, which correspond to hard rock, rock, very dense soil and soft rock, stiff soil, soil or any profile with more than 3 m of soft clay, and soils requiring site-specific evaluations, respectively (Building Seismic Safety Council (BSSC), 2004). The average shear wave velocity at small shear strains in the top 30 m $ν_{s} 30$ is used to determine the site type. The $ν_{s} 30$ of the site of the Daikai Station is 241 m/s and it refers to site Class D.

The number of ground motions required for IDA depends on the research objectives and structural characteristics. In total, 10–20 records are considered to be enough to capture the randomness of ground motions and provide sufficient accuracy in the estimation of seismic demands (Shome and Cornell, 1999). Therefore, in this study, 12 earthquake records were selected from the Pacific Earthquake Engineering Research (PEER) Center Strong Motion Database (2000), listed in Table 2. The site type of the station is Class D with the corresponding $ν_{s} 30$ of 180–360 m/s.

Table 2.

Twelve selected ground motion records.

No.	Event	Station	Component	PGA (g)	PGV (cm/s)
1	Kocaeli Turkey, 1999	ERD 99999 Duzce	DZC180	0.312	58.86
2	Kocaeli Turkey, 1999	ERD 99999 Duzce	DZC270	0.358	46.40
3	Kocaeli Turkey, 1999	KOERI 99999 Yarimca	YPT060	0.268	65.75
4	Kocaeli Turkey, 1999	KOERI 99999 Yarimca	YPT330	0.349	62.18
5	Chi-Chi, Taiwan, 1999	CWB 99999 CHY101	CHY101-E	0.353	70.64
6	Chi-Chi, Taiwan, 1999	CWB 99999 CHY101	CHY101-N	0.439	115.0
7	Chi-Chi, Taiwan, 1999	CWB 99999 TCU050	TCU050-E	0.147	36.91
8	Chi-Chi, Taiwan, 1999	CWB 99999 TCU050	TCU050-N	0.130	42.36
9	Chi-Chi, Taiwan, 1999	TCU065	TCU065-E	0.79	125.35
10	Chi-Chi, Taiwan, 1999	TCU065	TCU065-N	0.575	92.13
11	Chi-Chi, Taiwan, 1999	CHY025	CHY025-E	0.162	50.92
12	Chi-Chi, Taiwan, 1999	CHY025	CHY025-N	0.155	32.88

PGA: peak ground acceleration; PGV: peak ground velocity.

Selection of IM and DM

Rational selections of IM and DM are the primary basis of correctly and effectively conducting fragility analysis. IM characterizes the intensity of ground motion and should be monotonic and scalable (Vamvatsikos and Cornell, 2002). In the field of earthquake engineering, peak ground acceleration (PGA) directly specifies the peak ground motion acceleration and it is often used as an IM. However, seismic damage of underground structure has been demonstrated to have a strong correlation with PGV (Chen and Wei, 2013). Therefore, both PGA and PGV are adopted as the IMs for IDA in this study.

DM is a parameter that characterizes the seismic damage of structures. The selection of DM depends on the application and structure itself (Vamvatsikos and Cornell, 2002). In this study, maximum story drift angle (θ_max), central column damage (CCD), and energy ratio (R_E) are selected as DM.

The reasons for the selection of above-mentioned DMs are as follows. First, till now research on DM selection for underground structure is relatively few and the indicators used to measure seismic response in seismic design of underground structures and the corresponding codes still follow those used for surface structures (GB50011, 2010), such as maximum story drift angle (θ_max). However, it was observed from the damage of underground structures due to earthquake excitation that central column is the weakest component of the underground structure. For example, in the Kobe earthquake, the Daikai Station collapsed due to the compression-shear failure of the central column, which resulted in the side walls collapsed inward (Huo et al., 2005). Therefore, in this study, the CCD is chosen as one of the indicators to measure the extent of seismic damage which is determined using the ABAQUS results as described below. The average compression damage (DC) index of element section for each element is calculated first. Then, the total compression damage indicator of the central column is computed by weighting of dissipation energy summation DC of each element as is done in equation (1)

CCD = \sum_{i} \frac{E_{i}^{e}}{\sum_{i} E_{i}^{e}} {DC}_{i}^{e}

(1)

where ${DC}_{i}^{e}$ is the average compression damage index of element section for element i and $E_{i}^{e}$ is the dissipation energy of element i.

In terms of energy conservation, the destruction of underground structures under seismic load can be considered as a transformation, transmission, and consumption process of seismic energy into the structure. Consequently, it is possible to evaluate the extent of seismic damage in terms of structural dissipated energy. In this study, the ratio of the summation of plastic energy dissipation $E_{p}$ and the damage energy dissipation $E_{DMD}$ to the summation of the plastic energy dissipation $E_{p}$ , the damage energy dissipation $E_{DMD}$ , the kinetic energy $E_{KE}$ , and the elastic strain energy $E_{E}$ is calculated and selected as the third DM, as shown in equation (2)

R_{E} = \frac{E_{p} + E_{DMD}}{E_{KE} + E_{P} + E_{E} + E_{DMD}}

(2)

where $R_{E}$ is the energy ratio.

IDA curves and their characteristics

Figure 4 depicts the IDA curves of the three DMs (i.e. θ_max, CCD, and R_E) with respect to the two IMs being PGV and PGA. As can be seen, although seismic responses of the same structural model and the same surrounding soil condition have discrepancies when subject to different ground motions, they share similar patterns.

Figure 4.

IDA curves: (a) PGV-θ_max, (b) PGA-θ_max, (c) PGV-CCD, (d) PGA-CCD, (e) PGV-R_E, and (f) PGA-R_E.

Unlike surface structures, the IDA curves of the subway station with θ_max as the DM have no noticeable flat period mainly due to the restraint of the surrounding soil (Figure 4(a) and (b)). θ_max increases with the increase in ground motion intensity and the slope of the curves decreases gradually, but the decreasing rate is very limited.

On examining the IDA curves of CCD, it is found that CCDs are close to zeroes during the elastic response stage. When the input ground motion intensity increases, the structure enters into the elastic-plastic response stage as CCD gradually increases. When CCD values approach to 0.85, the slopes of the IDA curves see a rapid increase. The structure is considered to collapse as CCD reaches its maximum value in the IDA curves (Figure 4(c) and (d)).

The IDA curves of R_E as the DM have similar shape as those of CCD IDA curves. R_E closes to zero at the beginning of the curves under low ground motion input. When the value of R_E tends to 0.85, the structure tends to collapse (Figure 4(e) and (f)).

Besides, it can be observed that the trends of IDA curves when CCD and R_E are selected as DMs are different from that of θ_max as DM (Figure 4). θ_max increases gradually with the increase in ground motion intensity (Figure 4(a)). However, CCD and R_E change rapidly before PGV = 40 cm/s while after that the two DMs have small variations (Figure 4(c) and (e)), which mean that θ_max is more sensitive to reflect structural dynamic response than CCD and R_E. Therefore, we applied θ_max during the fragility analysis in the following part. What is more, the IDA curves of CCD and R_E as DMs can be used for the determination of structural limit states in the following section, which is an important preparation for fragility analysis.

Comparison of PGA and PGV

Generally, a proper IM will result in low dispersion in IDA curves for a given DM. Considering the massive data represented by the IDA curves and the discrepancies of IDA curves corresponding to different ground motions, it is essential to summarize IDA curves by defining the 16%, 50%, and 84% fractile IDA curves for further analysis (Vamvatsikos and Cornell, 2002). Assuming that DMs follow lognormal distributions for the condition distribution of IMs (Jalayer and Cornell, 2004), fractile IDA curves of DMs with respective to PGV and PGA as the IM were attained (Figure 5). These figures demonstrate that throughout the entire seismic response stages, the dispersion of the curves when using PGV as the IM is lower than that when using PGA. To quantify the dispersion, average of the ln(DM)’s standard deviation can be computed and compared where smaller value represents lower dispersion and the associated IM is more suitable for IDA curves. Table 3 lists the average of ln(DM)’s standard deviation of the three DMs with respective PGV and PGA as the IM. It was found that the average values of PGV are all smaller than those of PGA. Consequently, it was concluded that PGV is better than PGA for the IDA of the subway station structure presented herein. Thus, PGV was selected as the IM in the fragility analysis to further evaluate the seismic vulnerability of the subway station structure.

Figure 5.

Summarized IDA curves: (a) PGV-θ_max, (b) PGA-θ_max, (c) PGV-CCD, (d) PGA-CCD, (e) PGV-R_E, and (f) PGA-R_E.

Table 3.

Average of $σ_{\ln (DM)}$ .

IM\DM	$θ_{max}$	CCD	R_E
PGV	0.427	0.274	0.179
PGA	0.536	0.351	0.222

Fragility analysis

Considering IDA results, the probability of exceeding a limit state under a given ground motion intensity for the subway station structure can be obtained by fragility analysis. For this purpose, in this subsection, limit states for subway station structure are defined first. Then, seismic vulnerability assessment for the subway station structure is conducted through fragility analysis.

Definition of limit states

Limit state describes the anticipated maximum extent damage of a structure under a predetermined design seismic intensity level, and the properly defining limit states based on the structural characteristics and research objectives are essential in seismic vulnerability analysis. In this subsection, limit states for the subway station structure are defined first. Then, these limit states are identified from the IDA curves from which the thresholds of DMs are obtained.

Limit states for subway station structure

Limit states of subway station structures define the anticipated damage of the structure under a specified design seismic level. The four damage limit states defined by FEMA-273 (1997) are as follows: operational (OP; structure maintains normally used), slight damage (SD; structure ensures its availability with simple repair), life safety (LS; structure suffers from serious damage and needs a lot of repair to return to normal service), and collapse prevention (CP; structure is difficult to restore the function, but does not collapse). These four limit states were adopted in this study to describe the damage levels of the subway station structure.

Seismic performance of structures can be generally assessed from three aspects, which are safety, serviceability, and durability. Similar to buildings, safety of subway station structures can be indicated by their deformations. However, for serviceability and durability of subway station structures, waterproof performance has a direct impact that cannot be ignored as is done for building structures. Therefore, the waterproof performance of concrete was also adopted as a factor in the definition of limit states for subway station structure.

Table 4 describes the four limit states considered in this study that are related to damage levels as indicated by structural deformation and waterproof performances of roof, floor, and side walls. Each limit state defined herein indicates a particular level of structure functionality, from which the fragility can be directly estimated.

Table 4.

Division of limit state for subway station structure.

Limit state	Description of limit states	Qualitative description
Operational	Structure components do not suffer serious damage. Structure can be normally used	Invisible micro-cracks appear on the roof, floor, and side walls, and central columns are operational
Slight damage	Structure maintains the same strength and stiffness before and after the earthquake. Structure can ensure its availability with minor repair	Visible cracks appear on the roof, floor, side walls, and central columns
Life safety	Structure components suffer serious damage and stiffness degeneration, but are not collapsed. The waterproof performances of roof and floor, as well as the side wall are failure. Structure needs a lot of maintenance to come back to normal service	Macro-cracks that cross the protective layer appear on the roof, floor, and side walls. Concrete at some plastic hinge areas spalls greatly
Collapse prevention	Structure components destroy and their strength and stiffness greatly degenerate. Structure nearly collapses. The waterproof performances of roof and floor, as well as the side wall are failure	Inclined cracks of concrete are widened continuously and become a breakage band

Determining limit states on IDA curves

Determining limit states of building structures using IDA curves are relatively mature. For example, immediate occupancy (IO) and CP limit states of buildings are identified based on IDA curve’s slope (FEMA-351, 2000). However, limited slope variations are observed from the IDA curves of the subway station structure (Figure 4(a)), making it difficult to define limit states in the same way as is done for buildings. In the relevant studies, on the other hand, IM and structural damage distribution are utilized to determine the limit states of dam structures (Alembagheri and Ghaemian, 2013, 2016) instead of using the slope of IDA curves. Considering the similar structural characteristics between dam and subway station structures, it is promising to adopt the same method for dam when defining limit states for the station under investigation.

As an important component for the subway station structure, the damage state of central column can be determined using the threshold of column drift angle (θ_c) stipulated in GB50011 (2010) (Table 5), which reflects the structural deformation. As for the waterproof performance, crack width is a direct indicator and the thresholds for subway station structure are stipulated in GB50157 (2013). In addition, based on the concrete plastic damage model which establishes the relationship between tension damage (DT) and tension strain and the DT results obtained from ABAQUS, tension strain of concrete can be first estimated. Then, concrete crack width can be calculated using the relationship between tension strain and crack width obtained from concrete axial tension experiments (Guo and Zhang, 1988). Hence, the limit state can be determined in terms of the central column drift angle (θ_c) and the tension damage distribution at each IM. Note that the maximum crack width is computed based on the maximum structural tension damage (DT_max).

Table 5.

Thresholds of θ_c at each damage state of column (GB50011, 2010).

Damage state	Basically intact	Minor damage	Moderate damage	Serious damage
θ_c	0.0018	0.004	0.0083	0.0167

An investigation on the PGV-CCD IDA curves shows that there is no damage up to PGV = 5 cm/s, at which the first compression damage appears at the plastic hinge areas of the central column and the tension damage appears at roof (Table 6). It is observed from Figure 4(a) that the IDA curves remain in the elastic region until PGV = 10 cm/s. When PGV attains 20 cm/s, the IDA curves deviate from the elastic slope and structural tensile damage starts to propagate from the roof to the central column, side walls, and the floor (Table 6). When PGV increases to 40 cm/s, knee point appears on the IDA curves and the discrepancy of different ground motions enhances. Structural compression damage is now propagated from the central column to the plastic hinge areas of side walls, roof, and floor, while the tensile damage is throughout the structure with a value greater than 0.95 (Table 6). When PGV approaches to 70 cm/s, the CCD and R_E attain their respective maximum values of 1. Compression damage is fully propagated to the plastic hinge areas of all structural components.

Table 6.

Development of damage distribution for subway station structure.

PGV (cm/s)	Compression	Tension
5
10
20
40
70

Based on the calculated results mentioned above, four limit states (i.e. OP, SD, LS, and CP) are determined, respectively, when the corresponding IMs are equal to 10, 20, 40, and 70 cm/s, as shown in Table 7. The crack width threshold for subway station structure is 0.2 mm (GB50157, 2013), from which a crack width larger than 0.2 mm can be considered as a waterproof failure. Meanwhile, the structural limit states are mainly controlled by the damage state of central column after a waterproof failure.

Table 7.

Indicators for the determination of limit states.

PGV(cm/s)	θ_c	DT_max	Crack width (mm)
10	0.0018	0.55	0.04–0.08
20	0.004	0.92	0.1–0.2
40	0.008	>0.95	>0.2
70	0.016	>0.95	>0.2

From these analyses, DMs at each limit state are determined as listed in Table 8.

Table 8.

Thresholds of each limit state for the subway station structure.

Limit state	$θ_{max}$	CCD	R_E
Operational	0.0011	0.07	0.25
Slight damage	0.0025	0.35	0.5
Life safety	0.0059	0.8	0.75
Collapse prevention	0.013	0.95	0.9

Fragility curves

Fragility curve depicts the probability of a selected structure system response exceeding specified limit states as a function of IM of ground motion. It is a description of probability distribution of all the limit states.

To construct fragility curves, it is common to assume that DM follows a lognormal distribution for the condition distribution of IM (Jalayer and Cornell, 2004) with a lognormal cumulative distribution function as follows

P (L S_{i} | IM = im) = P (DM > d m_{i} | IM = im) = 1 - P (DM < d m_{i} | IM = im) = 1 - Φ (\frac{\ln d m_{i} - μ_{\ln DM | IM = im}}{σ_{\ln DM | IM = im}})

(3)

where the limit state LS_i is quantified by DM and expressed as dm_i. Equation (3) calculates the probability of DM exceeding dm_i when IM = im. $μ_{\ln DM | IM = im}$ and $σ_{\ln DM | IM = im}$ are the mean and the standard deviation of ln(DM), respectively. And Φ is the standard cumulative normal distribution function.

Based on the limit states and fragility function mentioned above, for a certain IM, the probability of DM exceeding a specific limit state can be calculated, from which a series points of P (LS_i) versus IM for all the IMs considered can be obtained. Then, a lognormal cumulative distribution function was fitted to the P (LS_i) versus IM points through nonlinear regression analysis of MATLAB, in order to provide the probability of exceeding each limit state in a continuous fashion. What is more, the fragility curves have a good match with fragility points for each limit state (i.e. OP, SD, LS, and CP), with R² in the range of 0.9995–0.9999. Fragility curves of the four limit states, for example Daikai subway station structure, are shown in Figure 6.

Figure 6.

Seismic fragility curves corresponding to each limit state.

Fragility analysis

Fragility analysis, which means an evaluation of fragility curves, is conducted in this subsection to offer a valuable insight into the seismic risk assessment of subway station structure.

Based on the fragility curves shown in Figure 6, the probabilities of exceeding four considered limit states are calculated and listed in Table 9 at the two hazard levels: the frequent seismic intensity and the design seismic intensity both at 8° (GB50011, 2010). The PGV in Table 9 is obtained through average conversion from the PGA for the given seismic intensity. The average conversion process is shown as follows: taken the frequent seismic intensity at 8° as an example, the PGA of which is 0.07 g at the 63.2% in a 50-year probability level, first scale each selected ground motion to 0.07 g in terms of PGA, and calculate PGV of each scaled ground motion, then take the average of PGV calculated from all the ground motions, as shown in Table 9. After the determination of PGV for each seismic intensity level, the corresponding vertical line can be drawn as shown in Figure 6, from which the vertical coordinate of the cross point on a certain fragility curve means the probability of exceeding the certain limit state. The process of generating the exceeding probability for the design seismic intensity at 8°, the PGA of which is 0.2 g at the 10% in a 50-year probability level, is the same as the procedure mentioned above.

Table 9.

Exceeding probabilities corresponding to each limit state.

Seismic intensity level	PGA (g)	PGV (cm/s)	Limit state
Seismic intensity level	PGA (g)	PGV (cm/s)	Operational (%)	Slight damage (%)	Life safety (%)	Collapse prevention (%)
Frequent seismic intensity at 8°	0.07	20	97	48	2	0
Design seismic intensity at 8°	0.2	45	100	99	65	11

As shown in Table 9, the probability of exceeding the OP, SD, LS, and CP limit states is decreasing in sequence at a certain seismic intensity level. The high probability of exceeding OP and SD limit states under frequent seismic intensity at 8° indicates that the subway station structure may suffer SD but not collapse under frequent seismic intensity at 8° in future. Besides, the probability of exceeding all four limit states under the design seismic intensity at 8° reflects the high potential risk of this subway station structure during earthquakes. The subway station structure more likely attains LS or even CP limit states under the design seismic intensity at 8° than that of frequent seismic intensity at 8°.

The fragility analysis results of this subway station structure also reveal that the subway station structure will suffer high risks of serious damage during an earthquake. Therefore, it is imperative to carry on seismic design for subway station structure and conduct seismic vulnerability assessment for this kind of structure. The reasonable fragility analysis results also demonstrate that the analysis approach developed in this article for the subway station structures is effective and reliable.

Conclusion

This article presented a seismic vulnerability assessment for a subway station structure. The Daikai subway station was selected as an example structure and a validated nonlinear numerical model was established. IDA was conducted and the fragility functions and curves at the four limit states (i.e. OP, SD, LS, and CP) were developed based on the IDA results to be utilized in the seismic vulnerability assessment of the subway station structure.

From the IDA results, it is shown that using PGV as an IM is better than PGA for the subway station structure under investigation due to lower dispersion of IDA curves. Thresholds of DMs (maximum story drift ratio, CCD, and R_E) provided in this article are good references for the further study on seismic performance of subway station structure.

Limit states associated with the structural deformation and waterproof performance of the subway station structure are determined using the central column drift angle (θc) and the tension damage distribution, respectively, from which fragility functions and curves were developed.

Reasonable fragility results obtained using the proposed procedure demonstrate the reliability and effectiveness of the seismic vulnerability assessment method implemented for the subway station structure. The fragility results reveal that the subway station structure investigated is not safe for CP and does not satisfy other limit states in high seismicity sites.

Fragility analysis for a subway station structure based on IDA approach is presented in this article. Considering the fundamental limitations of IDA, such as inherent limitation of scaling of records (Banerjee et al., 2016), the future studies should focus on improved methodologies of fragility analysis for underground structures to eliminate these limitations. Besides, the conclusions drawn herein are available for the subway station structure investigated. To make the conclusions more general, different kinds of subway station structures should be considered in our future study.

Footnotes

Declaration of Conflicting Interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (grant no. 51278524), Innovation Program of Shanghai Municipal Education Commission (14ZZ034), and State Key Laboratory of Disaster Reduction in Civil Engineering (SLDRCE12-MB-02). All supports are gratefully acknowledged.

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