Abstract

Currently, it is a challenge to effectively assess the seismic performance of the high-speed railway bridge line. To figure it out, this paper discussed the applicability of the Pushover analysis in the seismic fragility of the high-speed railway bridge. As the piers are the core components to resist the earthquakes, a typical high-speed railway bridge line consisting of 22 piers was established by the finite element software OpenSees. The influences of the different pier height and sites on the fragility analysis of the pies were investigated. From the component level, the seismic performance of the high-speed railway bridge line was evaluated by the Pushover analysis. The results show that the seismic responses of the piers by the Pushover analysis are agreeable with those by the incremental dynamical analysis when the peak ground acceleration is less than 0.4g. The high piers have better seismic performance than the lower piers. The high-speed railway bridge line exhibits good seismic performance under the 7-degree design earthquake (0.15g) and the 8-degree low-level earthquake (0.10g) but may be severely damaged under the 9-degree low-level earthquake (0.40g).

1. Introduction

The high-speed railway (HSR) in China is rapidly developing due to its high velocity, safety, and comfort [1, 2]. The total length of the HSR lines will reach 30,000 kilometres by the end of 2020. With the effect of the mountainous terrain and rivers [3, 4], the HSR bridge is mainly adopted to support the railway line because of its advantages of the high smoothness, standardized design, and easy construction [5, 6]. For example, the simply supported bridge accounts for 70% of the Beijing-Shanghai HSR line. However, amounts of bridges are likely to be damaged in the high seismic risk region [79]. The Wenchuan earthquake with 8.0 magnitude hit the western area of China on May 12, 2008, causing 6410 bridges to be severely damaged and hard to be repaired [10]. Therefore, the seismic performance evaluation of the HSR bridge is of significance to ensure the normal operation of the HSR bridge line.

Piers, as the core components of the HSR bridge, are designed based on the stiffness. They have distinct characteristics with the large cross section so as to satisfy the requirements of the line smoothness and moving train safety. However, the hysteric curves of the HSR piers in the quasi-static tests [11] show that piers exhibit the low ductility due to the less longitudinal reinforcement ratio. This means that the HSR piers are likely to be damaged under the strong earthquakes. Some studies compared the dynamic responses of the HSR bridge under the vehicle load and the earthquakes [1214]. It is pointed out that the strong earthquakes have more distinct effect on the responses of the HSR piers [13]. The elastic-plastic deformation may occur at the bottom of the HSR piers under the high-level earthquake [14]. However, these studies only discussed the seismic responses of the HSR bridge under several specific seismic records. The seismic performance of the HSR bridge is not carefully investigated.

The fragility analysis is a common method for the seismic performance evaluation of the structures. It can reflect the probability of the structures exceeding a critical value under a specific seismic intensity [15]. The different methodologies, such as the nonlinear time history analysis and the Pushover analysis, are widely adopted for the seismic fragility analysis of the highway bridge [1621]. The nonlinear time history analysis is more reliable and accurate, while the Pushover analysis has the distinct advantages of the conceptual simplicity and the computational effectiveness [22, 23]. Moreover, the seismic fragility of the highway bridges by the Pushover analysis is aggregable with those obtained from the nonlinear time history analysis [24]. The Pushover analysis provides a more practical-oriented alternative for the seismic performance evaluation of the highway bridges [25]. However, the HSR bridge is designed by the stiffness, which is different from the design of the highway bridge. The applicability of the Pushover analysis in the seismic analysis of the HSR bridge is not clear. To figure it out, this paper will extend the application of the Pushover analysis into the seismic evaluation of the HSR piers.

In the current seismic code of China [26], the simply supported bridge is recommendable to be established as the simply pier or the bridge model for the seismic analysis. In this paper, the 22 HSR pier models with the different height and four types of sites were established by the finite element software OpenSees. The application of the Pushover analysis into the seismic evaluation of the HSR piers will be carefully investigated. The seismic performance of the HSR piers and the whole bridge line will be evaluated using the Pushover analysis.

2. Model Description of the HSR Bridge Piers

For the simply supported bridge system, the total length of the girder for each span is 32.6 meters, and the clear spacing between two adjacent spans is 0.1 meters. Four bearings are located on the pier to support the girder. They are the fixed bearing, longitudinal moveable bearing, transverse moveable bearing, and bidirectional moveable bearing. The girder usually keeps elastic under the earthquakes. According to the seismic code in China [26], the simply supported bridge is recommended to be established as the single pier model for the seismic analysis. For the seismic analysis in the longitudinal direction of the simply supported bridge, the girder is simplified as the lumped mass at the top of the numerical model of piers, as shown in Figure 1. Based on this, the simply supported bridge is established as the single pier model by the Open System for Earthquake Engineering Simulation (OpenSees).

The HSR bridge piers with the round-shaped cross section are one of the most common piers adopted in the simply supported bridge. Different from the highway bridge [27], the HSR piers are designed by the stiffness with the characteristics of the large cross section. The dimensions of the HSR pier are shown in Figures 1(a) and 1(b). The design strength of the concrete is 23.4MPa. The longitudinal reinforcement ratio and the axial compression ratio are 0.75% and 10%, respectively. Concrete02 and Steel02 material are used to simulate the concrete and longitudinal reinforcement, respectively. The total weight of the girder for each span is 1360 tonnes, which is regarded as the lumped mass at the top node of piers. In this paper, the 22 HSR piers of the simply supported bridge system are established to compose the HSR bridge line, as shown in Figure 1(c). The influence of the different heights is considered in the pier’s finite element model. The effect of each site is included by adopting the different response spectrum associated with the characteristic period of the soil, as shown in Table 1 and Figure 1.

3. Definition of the Pier’s Damage Limit

The hysteric curves of the HSR piers with 8m, 16m, and 24m were tested by the low cyclic load [11, 28]. The bending damage was the failure mode of these HSR piers. The experiment showed that the first crack occurred at the bottom of the pier, and then more cracks extended as the reinforcement began to yield. Based on this, the different damage limits of the HSR piers are descripted by the displacement ductility coefficient , which is expressed as [11]where d (unit: m) is the maximum displacement at the top of the HSR pier relative to the ground; (unit: m) represents the relative displacement of piers corresponding to the first yield of the longitudinal reinforcement. According to literatures [29, 30], the different damage state of the pier is shown in Table 2.

In Table 2, represents the displacement ductility coefficient when the steel bars begin to yield, which is the upper limit of the slight damage state; is the yield displacement ductility coefficient of the piers, representing the upper limit of the moderate damage state; corresponds to the displacement ductility coefficient when the concrete is cracked at the strain of 0.002 and is the upper limit of the severe damage state; is the maximum displacement ductility ratio and is the complete damage upper limit.

The damage state limits of the piers with the different heights can be calculated according to the literature [27, 29, 30], as shown in Table 3. The ductility coefficient of the tested HSR piers ranges from 3 to 6 [11]. In Table 3, the displacement ductility coefficient of the HSR piers with the different heights is less than 3.5 in the severe damage state, which indicates that the HSR piers have the low ductility.

4. Comparison of the Performance Evaluation Methods

4.1. The Procedure of the Pushover and IDA Methods

In this section, the computational efficiency and accuracy of the Pushover analysis and the incremental dynamic analysis (IDA) are compared in the seismic analysis of the HSR piers. In the Pushover methodology, it is important to select the lateral load pattern, such as the suggested inverted triangle distribution (modal pattern), uniform distribution, and the adaptive distribution [31, 32]. The selected load pattern is used for the nonlinear static analysis step by step until the capacity curve of the structures is obtained. In this paper, it is assumed that the seismic response of the HSR pier is controlled by the first mode vibration. The load pattern with the first mode vibration remains constant throughout the Pushover analysis [26], specified as (2). To identify the strength and displacement quantities, the simplified bilinear base shear-roof displacement curves of the HSR piers need to define the properties of the HSR piers. The transformation from the simplified bilinear curve into the capacity curve is calculated by (3).where (unit: kN) is the horizontal seismic load of the ith node at the first model; (unit: g) is the horizontal acceleration; is the dynamic amplification factor of the HSR pier at the first mode; is the mode participation coefficient at the first model; is the modal coordinate of the ith node at the first model; (unit: kg) is the mass of the ith node; (unit: g) is the pseudo acceleration; (unit: m) is the pseudo displacement; (unit: kN) is the maximum base shear of the pier; (unit: m) is the roof displacement of the pier; (unit: kg) is the effective mass of the HSR pier at the first model; is the displacement of the ith node at the first mode; and is the roof displacement of the piers at the first modal.

The target displacement of the HSR piers is determined by the intersection of the inelastic demand spectra and the capacity curve. The inelastic demand spectra for each site are given by the reduction of the elastic response spectrum related to the characteristic period of the soil. It reflects changes in the stiffness and damping associated with the maximum displacement response of the structure. The reduction of the structural stiffness is described by the secant stiffness [30] related to the equivalent period, defined as (4a). The equivalent viscous damping ratio is used to describe the changes of the structural damping. It represents the energy dissipation of structures in one cycle of the vibration [33]. Considering the pitching phenomenon of structures, the modified equivalent damping ratio [34, 35] is expressed as (4b).where (unit: s) is the fundamental period of structures; is the ratio of the postyield stiffness to the preyield stiffness; and is the displacement ductility coefficient, which represents the ratio of the maximum displacement to the postyield displacement of structures; is the viscous damping ratio of the structures in the elastic state; and 0.05 is adopted for the reinforced concrete structure; is the damping modification factor, which is the effective ratio of the energy dissipation of the real hysteresis of the structure to that of the simplified bilinear model.

The incremental dynamic analysis (IDA) [36] is commonly used for the seismic analysis of structures by employing the nonlinear time history analysis. This methodology can reflect the probabilistic distribution of the structural response under a suite of ground motion records, as shown in Figure 3.

In the IDA method, each ground motion is scaled to the different level of intensity in the nonlinear dynamic analysis. In this paper, the influence of four types of sites is included in the seismic analysis of the HSR pier by the IDA. Based on the seismic code in China [26], the ten seismic records are selected to match the design response spectrum associated with the characteristic period of each site. The deviations of the mean spectra of the selected records and the design spectra are less than 20% in the critical periods in Figure 4, which satisfies the requirements of the seismic record selection in the seismic code [26]. All seismic records are obtained from the database of the Pacific Earthquake Engineering Research Center (PEER) [37], as shown in Table 4.

4.2. Efficiency Comparison

According to the flowcharts shown in Figures 2 and 3, the seismic responses of the HSR piers can be calculated by the Pushover analysis and IDA. The used computer information is shown in Table 5. The cost time of two methods is shown in Table 6. The cost time ratio by the Pushover analysis and IDA is 1:88, which indicates that the Pushover analysis is more computationally effective than the IDA.

4.3. Accuracy Comparison

The IDA can accurately reflect the seismic responses of the HSR piers under the earthquakes by employing the nonlinear history analysis. However, the Pushover analysis can only approximately describe the seismic responses of the HSR piers. Therefore, the calculation accuracy is worth attention. The seismic responses of the HSR pier model by Pushover analysis are compared with those by IDA in the following part.

4.3.1. The Fragility Analysis

The fragility analysis is widely used for the seismic vulnerability assessment of structures [18]. The fragility curves represent the damage probability of structures at a given earthquake intensity, which is described by the lognormal cumulative distribution function [38, 39]. The fragility probability of the structures is given byin which P is the damage probability of the HSR piers at a given earthquake intensity. denotes the peak ground acceleration (PGA) corresponding to a 50% damage probability, and describes the uncertainties related to the structural properties and seismic characteristics.

The fragility curves of the HSR piers with the different heights in the different damage states can be obtained by (5). The fragility curves of the HSR piers on site II (medium-hard soil) are shown in Figure 5.

The fragility curves of the HSR piers by the Pushover analysis are different from those by the IDA. For most of the HSR piers, the damage probability error by the Pushover analysis is increasing as PGA increases. However, the calculation error induced by the Pushover analysis is less than 20%. When PGA is not greater than 0.4 g, the fragility curves obtained from the Pushover analysis are agreeable with those by the IDA. The damage probability calculated by the Pushover analysis is generally larger than that by the IDA. This indicates that the seismic performance of the HSR piers is underestimated by the Pushover analysis. In other words, when the Pushover analysis is applied to analyse the seismic fragility of the HSR piers with PGA less than 0.4g, the safety of the HSR piers is adequate from the perspective of the practical engineering.

4.3.2. The Effective Height Ratio Analysis

The time history analysis can accurately reflect the seismic responses of the HSR piers. Both the moment and shear force time history curves are shown to investigate the force distribution of the HSR piers under earthquakes. Take 10 m and 22 m piers as examples, the time history responses of 10 m pier on site I under the seismic record (Taiwan) and 22 m pier on site IV under seismic record (Loma Prieta) are shown in Figure 6.

In the Figures 6(a) and 6(b), the moment and shear force of the HSR pier with 10 m reach the maximum values at the same time of 12.17 sec when the PGA is 0.1 g. When the PGA increases to 1.0 g, the time difference between the maximum moment and shear force is as small as 0.01 sec. Figures 6(c) and 6(d) show that the moment and shear of 22 m pier on site IV reach the maximum values at different time. The maximum moment emerges at 9.51 sec, while the maximum shear force occurs at 9.50 sec at 0.1g PGA. With PGA increasing to 1.0 g, the time of the maximum moment and shear force is 12.06 sec and 8.19 sec, respectively. The changing time demonstrates that the seismic force distribution of the HSR piers is changing as PGA increases. This is because the plastic deformation of the HSR pier has been formed under the strong earthquake.

The Pushover analysis is based on the assumption of the constant seismic force distribution obtained from the first vibration mode of the HSR pier. This means that the Pushover analysis is inapplicable in the seismic analysis of the HSR piers when piers occur with large plastic deformation, because, in this case, it cannot capture the accurate seismic force distribution of the piers.

To better investigate the applicability of Pushover analysis in the seismic analysis of the HSR piers, the effective height , which is defined as the ratio of the maximum moment to the maximum base shear force calculated by the IDA, is adopted to analyse the change of the seismic force distribution. The effective height ratio C is defined as the ratio of the effective height by IDA to that calculated by the Pushover analysis. They are defined aswhere C is the ratio of the maximum moment to the maximum base shear force calculated through the IDA and Pushover analysis, respectively. Mmax and Fmax are the maximum moment and base shear force of the HSR pier, respectively. The effective heights Heff and effective height ratios C of the HSR piers on the four sites are shown in Figures 7 and 8, respectively.

Figure 7 shows that the effective height Heff of the HSR piers by the IDA is reduced with PGA increasing. This demonstrates that the assumption of the constant seismic load distribution in the Pushover analysis is inconsistent with that obtained from the IDA. The inconsistency will lead to the calculation error by the Pushover analysis. In Figure 8, the effective height ratio C is near to 1.0 when PGA is small. However, it decreases as the PGA increases, which has decreased to 0.7 when the PGA is more than 0.4 g. The seismic intensity and pier height have distinct effect on the effective height ratio C of the HSR piers. The effective height ratio C of the lower piers decreases more slowly than that of the high piers. It indicates that the Pushover analysis is more applicable for the lower piers than the high piers. This is because the higher-order vibration modes have distinct effect on the seismic responses of the high pier, which are not considered in the Pushover analysis.

As discussed above, the application of the Pushover analysis into the seismic analysis of the HSR piers is limited. The calculation error by the Pushover analysis will increase as PGA and the height of the HSR piers increases. This is because the seismic load distribution assumed in the Pushover analysis, in fact, is changing in the time history analysis. On the other hand, the influence of higher-order vibration modes of the higher piers are not considered in the Pushover analysis. However, the calculation error by the Pushover analysis is acceptable when the PGA is less than 0.4 g. The fragility probability error of the HSR piers by the Pushover analysis is less than 20%, and the effective height ratio is above 0.7 compared with that obtained by the IDA. Therefore, the Pushover analysis will be employed as an effective approach to evaluate the seismic performance of the HSR piers and the HSR bridge line in the following part.

5. Seismic Performance Evaluation of the HSR Piers by the Pushover Analysis

5.1. Fragility Analysis of the HSR Piers

Several factors, including the pier height and the site, influence the seismic performance of the HSR piers. The fragility curves calculated by (5) are used to describe the seismic performance of the HSR piers with different heights located. Figure 9 shows the fragility probability curves of the HSR piers with 10 m, 15 m, 21 m, and 23m on site II.

Figures 9(a) and 9(b) show that the slight damage and moderate damage probabilities of 10 m, 15 m, 21 m, and 23 m piers are similar. The height of the HSR piers has slight impact on the slight damage and moderate damage probabilities. The 15 m and 21 m piers exhibit better seismic performance than the 10 m and 23 m piers. Figures 9(c) and 9(d) demonstrate that the HSR piers have low damage probabilities at the severe damage and complete damage states. The 10 m and 15 m piers are prone to suffer severe and complete damage compared with the 21 m and 23 m piers, which indicates the high piers have a better seismic performance. Moreover, the seismic performance of the high piers has been improved with PGA increasing. It is likely that the high piers have larger stiffness, which makes it less fragile to be damaged compared with the lower piers.

The effect of the sites on the seismic fragility of the HSR piers is discussed here. Four types of sites, namely, the rock or hard soil (site I), medium-hard soil (site II), medium-soft soil (site III), and soft soil (site IV), are involved here. The 15 m pier is taken as a representative for the discussion, as shown in Figure 10.

The four types of sites have the same influence on the seismic performance of the 15 m pier. The 15 m pier is the most fragile on soft soil (site IV) but exhibits excellent seismic performance on medium-hard soil (site II). The damage probability of 15 m pier on hard soil (site I) is lower than that on the medium-soft soil (site III).

5.2. Seismic Performance Evaluation of the HSR Bridge Line

The seismic performance of the HSR bridge line plays a vital role in ensuring the normal operation and the safety of the moving train. The HSR pier, as the core component of the HSR bridge, is adopted to evaluate the seismic performance of the HSR bridge line from the component level. In the Chinese seismic code [26], the bridges are usually designed according to three principles: no structural damage under the low-level earthquakes (63% excessive probability over 50 years), being structurally repairable under the design earthquake (10% excessive probability over 50 years), and no collapse during the high-level earthquakes (2~3% excessive probability over 50 years). To evaluate the seismic performance of the HSR bridge line, the different seismic intensities are involved. They include 0.15 g PGA for a 7-degree design earthquake, 0.32 g PGA for the 7-degree high-level earthquake, 0.1 g PGA for the 8-degree low-level earthquake, and 0.4 g PGA for the 9-degree low-level earthquake [26]. Based on the displacement ductility coefficient index, the seismic performance of the HSR bridge line is evaluated by the Pushover analysis from the component level, as shown in Figure 11.

Figure 11(a) shows that the HSR piers on sites I, II, and III are not damaged, while most of the piers on site IV are moderately damaged under the 7-degree design earthquake (PGA=0.15 g). The HSR piers on sites I, II, and III almost suffer the moderate damage, while the piers on site IV are severely damaged under the 7-degree rare earthquake (PGA=0.32 g), as shown in Figure 11(b). Figure 11(c) demonstrates that all of the HSR piers under the 8-degree low-level earthquake (PGA=0.10 g) are not damaged. In Figure 11(d), the piers on sites I and II are moderately damaged under the 9-degree low-level earthquake (PGA=0.40 g), and the HSR piers on sites III and IV suffer severe damage. The 10 m pier on site IV is completely damaged. The seismic performance of the HSR bridge line, designed by the 7-degree design earthquake (PGA=0.15 g), can satisfy the seismic requirements in Chinese seismic code. The HSR bridge line has good seismic performance under the 7-degree design earthquake (0.15g) and 8-degree low-level earthquake (0.10g) from the perspective of the structural safety. However, the HSR bridge line is likely to be severely damaged even under the 9-degree low-level earthquake (0.40g). Therefore, the seismic design of the HSR piers should be improved.

6. Discussion

This paper extends the application of the Pushover analysis into the seismic performance evaluation of the HSR piers. In the previous literatures, the applicability of the Pushover analysis in the seismic evaluation of the structures such as the ordinary bridge [40], underground structures [23], and buildings [41] has been well investigated. However, there is limited research on the applicability of the Pushover analysis on the seismic performance evaluation for HSR. The Pushover analysis presents the computational attractiveness and effectiveness for the seismic analysis of the HSR piers. This paper provides a practical-oriented approach for the seismic performance evaluation of the HSR bridge line.

The HSR piers are the resisting components, which is recommendable for the seismic analysis in the current HSR seismic code [26]. The high HSR pier exhibits better seismic performance than the lower pier. This is consistent with the experiment of the HSR piers by the low cycle test [11].

The Pushover analysis was adopted to evaluate the seismic performance of the HSR bridge line from the component level. The seismic performance of the five-span HSR bridge [42] demonstrates that the HSR bridge has a good seismic performance with sufficient ductility ability under the strong earthquake. However, the HSR bridge line exhibits inadequate antiseismic ability under the 9-degree low-level earthquake (0.40g).

7. Conclusions

In this paper, the finite element models of the HSR piers are established by OpenSees. The Pushover analysis and IDA are applied into the seismic analysis of the HSR pier. The calculation error by the Pushover analysis is discussed. Based on the displacement ductility coefficient index, the Pushover analysis is adopted to evaluate the fragility of the HSR piers and the seismic performance of the HSR bridge line from the component level. Some conclusions are given as follows:

(1) The Pushover analysis is an attractive approach for the seismic analysis of the HSR piers. the Pushover analysis is computationally effective within the acceptable calculation error for the seismic performance of the HSR piers when PGA is less than 0.4 g.

(2) The damage probabilities of piers are slightly different under slight damage and moderate damage states. The high piers (e.g., 21 m pier) have better seismic performance than the low piers (e.g., 10-m pier) under severe damage and complete damage states.

(3) The fragility analysis of the HSR piers on the four types of site indicates that the piers on site II have the lowest damage probability while piers on site IV are the most seismically fragile.

(4) From the component level, the HSR bridge line can satisfy the seismic demands of the Chinese seismic code under the 7-degree design earthquake (PGA=0.15 g) and 8-degree low-level earthquake (PGA=0.10 g). However, the HSR bridge line is likely to be severely damaged under the 9-degree low-level earthquake (PGA=0.40 g). The seismic design of HSR bridge line should be improved.

Data Availability

The data used to support the findings of this study may be released upon application to the authors, who can be contacted at guowei@csu.edu.cn.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research is supported by the Fundamental Research Funds for the Central Universities of Central South University [Project No. 502221804], the National Natural Science Foundation of China [Project Nos. 51878674, 51878563], the Foundation for Key Youth Scholars in Hunan Province [Project No. 150220077], and the Project of Yuying Plan in Central South University [Project No. 502034002].