Analysis of the Effect of Lateral Collision on the Seismic Response of Bridges under Fault Misalignment

Abstract

Mutual dislocation of seismogenic faults during strong earthquakes will result in a large relative displacement on both sides of the fault. It is of great significance to explore the influence of the collision effect between the main beam and the transverse shear key on the seismic response of the bridge under fault dislocation. In this paper, a series of cross-fault ground motions with different ground permanent displacements are artificially synthesized using a hybrid simulation method. Based on the contact element theory, the Kelvin–Voigt model is used to simulate the lateral collision effect. The effect of lateral collision on the seismic response of the continuous girder bridge is compared from the two aspects of fault dislocation position and fault dislocation degree. On this basis, the analysis of lateral collision parameters is carried out with the aim of reasonably regulating the seismic response of the structure. The results show that, compared with the near-fault bridge, the influence of lateral collision on the cross-fault bridge is stronger. The amplification of the bending moment of the central pier and the limitation of the bearing displacement are five times and two times, respectively, for the near-fault bridge. When the fault has a large dislocation, the weak point of the structural damage is the bending failure of the pier bottom and the residual torsion after the earthquake. The collision parameters of conventional bridges will aggravate the bending moment demand of the pier bottom of cross-fault bridges and limit their bearing displacement too much. Therefore, by appropriately reducing the collision stiffness and increasing the initial gap, the internal force and displacement response distribution of the cross-fault bridge structure can be more reasonable. The study in this paper has reference significance for seismic analysis of cross-fault bridges with transverse shear keys.

1. Introduction

China is located between the world’s two major seismic belts: the Pacific Rim Seismic Belt and the Eurasian Seismic Belt. Under the compression of the Pacific, Indian, and Philippine plates, neotectonic activity in the western mountainous area is strong, and the fault zone is widely developed [1,2]. Large active faults can extend for hundreds or even thousands of kilometers, with high seismic hazards [3,4]. With the continuous improvement of land development intensity and the construction of basic transportation facilities along the Belt and Road, including the Sichuan–Tibet Railway [5], which has been started, more and more highways and railway bridges will inevitably have to be built next to or across large active faults.

Earthquake disasters in recent decades have shown that the relative displacement of the seismogenic fault during strong earthquakes causes different ground motions on both sides of the fault, accompanied by permanent displacement of the surface rupture [6,7]. Under the combined action of strong ground motion and surface rupture caused by the seismogenic fault, bending and shear failure of piers often occur in bridge structures adjacent to or directly crossing faults, and even beam fall or collapse of the entire bridge is severe [8]. In the 1999 Chi-Chi earthquake, the surface of the Chelongpu fault, the seismogenic fault, produced a surface rupture zone of up to 100 km, and the relative dislocation on both sides of the fault reached several meters. The huge displacement led to the failure of seven bridges near the fault, such as the Wuxi Bridge, Shiwei Bridge, and Mingzhu Bridge [9]. In the 1999 Kocaeli earthquake, the surface rupture zone of the fault passed through the piers and adjacent abutments of the Arifiye viaduct [10], and a horizontal displacement of nearly 1.5 m was generated on both sides of the ground, resulting in the collapse of the entire bridge and the complete interruption of road traffic. The 2008 Wenchuan earthquake occurred in the central fault zone of the Longmen Mountains. The Xiaoyudong Bridge and the Yingxiu Shunhe Bridge near the fault zone collapsed due to fault displacement during the earthquake. The bottom of the pier of Baihua Bridge [11] has crushing failure, and the bottom of the pier and the tie beam have severe shear failure. The above earthquake damage shows that the permanent displacement of surface fracture caused by fault dislocation is the main cause of beam fall, bearing system failure, and pier damage [12].

In order to prevent the occurrence of falling beams or to ensure the seismic safety of bearings, the measures of placing shear keys in the transverse direction of the bridge are usually adopted in bridge engineering [13,14]. Although the shear key can reduce the seismic damage of the falling beam [15], it will also collide with the main beam, causing component damage [16]. Therefore, transverse shear key damage is a very common form of damage in post-earthquake bridge structures. For example, in the 1999 Chi-Chi earthquake and the 2001 Gujarat earthquake, there are many records of lateral movement and shear key failure of bridge superstructures [9,17]. In the 2008 Wenchuan earthquake, the beam bridge damage was extremely severe, and the shear key failure rate was as high as 16.8%, while the pier failure rate was only 2.4% [18]. For the overall seismic design of the bridge, the main function of the transverse shear key is to limit the displacement and transfer force. Although the seismic design provisions of shear keys in different countries are not very clear, there are still many scholars who have discussed it in recent years. Maleki [19,20] used the linear collision model to study the collision effect between the superstructure and the transverse shear key of the simply supported beam bridge. It is shown that the collision stiffness, initial gap, and vibration period have a great influence on the seismic response of the structure. Ignoring the collision effect will underestimate the seismic demand of the substructure and lead to unsafe results. However, the problem of energy dissipation is not considered in the collision process. Jiang Hui et al. [21] established a bridge lateral collision contact element analysis model that can take into account the energy loss during the collision process and investigated the influence of collision stiffness, initial gap, and bridge pier height on the seismic collision effect of a railway RC simply supported beam bridge under shallow source strong earthquake. The results show that when the lateral collision stiffness is 1~2% of the axial stiffness of the beam, the relative balance between force and deformation can be achieved. Nie Liying et al. [22] and Deng Yulin et al. [23] studied the simulation model of lateral collision on bridge structure and conducted parameter analysis. They pointed out that the collision parameters have a significant effect on the overall seismic performance of the bridge but did not propose the specific values of the corresponding bridge collision parameters. Therefore, the above scholars only carry out correlation analysis on the transverse collision effect of conventional far-field bridges, and whether it is suitable for bridges near faults needs further study.

Compared to conventional bridges, the permanent ground displacement caused by the mutual dislocation of faults will result in different spatial position changes at the bridge support points, which need to be coordinated by the deformation of the bridge structure itself [24,25]. This often leads to a larger sliding displacement between the main girder and the pier, resulting in a more severe lateral collision with the shear key. Among these, the location of the fault relative to the bridge and the degree of fault displacement are two important factors affecting the mechanical performance of the bridge [26,27]. At present, many national codes prohibit bridges from directly crossing active faults or adopting avoidance distance measures, which leads to an insufficient understanding of the seismic response of cross-fault bridges, and their seismic design method needs to be improved [28,29]. Therefore, the study of the influence of lateral collision on the seismic response of cross-fault bridges under fault dislocation can promote the seismic strengthening of such bridges and the formulation of relevant specifications, which is of great importance to avoid or minimize seismic damage.

In this paper, based on the contact element theory, the Kelvin–Voigt model is used to establish a cross-fault bridge analysis model considering the lateral collision between the shear key and the beam. A series of cross-fault ground motions with different ground permanent displacements were artificially synthesized using the hybrid simulation method. The effect of lateral collision on the seismic response of a continuous girder bridge is compared from the position and degree of fault dislocation. Finally, the two key parameters of collision stiffness and the initial gap in the collision effect are analyzed in order to provide guidance for improving the seismic safety of cross-fault bridges.

2. Engineering Overview and Finite Element Model Establishment

2.1. Project Overview

In this paper, a typical three-span beam bridge is selected as the research object. As shown in Figure 1, the bridge is a 3 × 30 m equal-span continuous beam bridge, in which the second bridge spans vertically across a strike-slip fault surface rupture zone. According to geological structural conditions, historical earthquakes, and other seismic safety assessment data, the upper limit of potential magnitude in the bridge site area is 7, the maximum horizontal offset is 2 m, and the site category is a Class II site. The superstructure is a single-box single-cell box girder section with a beam height of 1.8 m. The lower structure adopts a rectangular single-column pier with a height of 18 m. Rectangular pile caps are used at the bottom of the pier. The pile foundation is a bored pile, and four pile foundations are arranged at the bottom of each pile cap. Two plate rubber bearings (LRB) are placed on each pier cover beam, and a reinforced concrete shear key is set on both sides of the cover beam. The transverse section of the bridge-related components is shown in Figure 2.

2.2. Bridge Finite Element Modeling

In this paper, SAP2000 v22 software is used to establish a three-dimensional finite element model of the whole bridge dynamic analysis, as shown in Figure 3. According to the principle of capacity protection, in addition to piers, bearings, and other components entering the plastic state under a strong earthquake, the main beam, cap beam, and other components are in the elastic working area. Therefore, the main beam and cap beam are simulated by a spatial linear elastic beam-column element. For the column pier used in this paper, plastic hinges are often formed at the bottom of the pier, and PMM fiber hinges are used for simulation. When defining the hinge plasticity, the section is divided into steel fiber, confined concrete fiber, and unconfined concrete fiber, as shown in Figure 4. For the material constitutive of the section fiber, the Mander model [30] is used to define the stress–strain constitutive relationship of the constrained and unconstrained concrete, as shown in Figure 5. The Giuffre–Menegotto–Pinto model [31] is used to define the stress-strain constitutive relationship of the steel fiber, as shown in Figure 6. The pile–soil interaction is simulated by adding a spring with 6 degrees of freedom at the bottom of the cap. The six spring stiffnesses are vertical stiffness, anti-push stiffness in the longitudinal and transverse directions, anti-rotation stiffness around the vertical axis, and anti-rotation stiffness around the two horizontal axes. The second stage, dead load, is considered by applying a concentrated mass to the main beam node. The damping ratio of the structure is 5%, and Rayleigh damping is assumed. The nonlinear dynamic time history analysis uses the Newmark-β direct integration method, where γ = 0.5, β = 0.25.

In Figure 5 and Figure 6, the abscissa is the strain of the material, and the ordinate is the stress of the material; f′_c and ε are the peak stress and peak strain of unconstrained concrete, respectively; f′_cc and ε_cu are the peak stress and peak strain of confined concrete, respectively; E_c and E_sce are the initial elastic modulus of concrete and the secant elastic modulus corresponding to the peak stress, respectively. E₀ and E₁ are the initial stiffness and yield stiffness of the steel bar, respectively, and ε_y is the yield strain of the steel bar.

2.2.1. Simulation of Bearing

The bearing adopts a plate rubber bearing. The test results of a large number of plate rubber bearings show that the hysteresis curve of plate rubber bearings is narrow and long, which can be processed approximately linearly [32,33]. Therefore, this study uses a linear spring element to simulate the restoring force model of the rubber plate bearing. The plane size of the plate rubber bearing is 300 mm × 400 mm, and the total thickness is 47 mm. The calculation method of shear stiffness and vertical stiffness of the bearing is as follows.

Horizontal rigidity of the bearing:

$$k=\frac{G_d{A}_r}{{\displaystyle \sum t}}$$

(1)

Vertical stiffness of bearing:

$$k_{\mathrm{v}}=\frac{E{A}_r}{{\displaystyle \sum t}}$$

(2)

where G_d is the dynamic shear modulus of the plate rubber bearing, which is generally taken as 1200 kN/m²; A_r is the shear area of the rubber bearing; ∑t is the total thickness of the rubber layer. E is the vertical compressive elastic modulus of the plate rubber bearing.

2.2.2. Simulation of Lateral Collision

Existing studies have shown that [34,35] the contact element method is an effective way to simulate the collision effect between structural components. The commonly used mechanical models for collision between structures are the linear spring model, Kelvin–Voigt model, Hertz model, and Hertz–Damp model. Among them, the linear spring model and the Hertz model cannot consider the energy consumption during the collision process when simulating the collision reaction of the structure; although the Hertz–Damp model [36] can consider the energy dissipation problem of collision, the contact element used is a highly nonlinear element, which limits its application range. The Kelvin–Voigt model [37,38] can not only consider the collision energy consumption but also the element form is concise, and the simulation effect can also meet the analysis requirements. Therefore, it is the most widely used in the study of seismic collision problems of bridge structures.

Therefore, based on the Kelvin–Voigt model, this paper considers the initial gap between the shear key and the beam. The lateral collision is simplified as the model shown in Figure 7, which consists of a spring and a damping unit in parallel and then connected in series with the initial gap. The relationship between the collision force and the displacement is shown in Figure 8.

In Figure 7 and Figure 8, m₁ and m₂ are the masses of the main girder and pier, respectively. k_k is the collision stiff-ness; g_p is the initial gap of the shear key; C_k is the damping coefficient; u₁ and u₂ are the displacement of the main beam and the shear key, respectively; F_max is the maximum contact force; u_max is the maximum displacement.

In the process of transverse collision, the collision force F_c between the shear key and the main beam can be expressed as:

$$F_{\mathrm{c}}=\left\{\begin{array}{l}{k}_{\mathrm{k}}\left(u_1-u_2-g_{\mathrm{p}}\right)+C_{\mathrm{k}}\left({\dot{u}}_1-{\dot{u}}_2\right),\hspace{1em} u_1-u_2-g_{\mathrm{p}}\ge 0\\ 0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} u_1-u_2-g_{\mathrm{p}}<0\end{array}\right.$$

(3)

where ${\dot{u}}_1$ and ${\dot{u}}_2$ is the speed of the main beam and the shear key; C_k is the damping coefficient. According to the energy conservation relationship in the collision process, the relationship between the damping coefficient C_k and the recovery coefficient e can be established as follows:

$$C_{\mathrm{k}}=2\zeta \sqrt{k_{\mathrm{k}}\frac{m_1{m}_2}{m_1+m_2}}$$

(4)

$$\zeta =-\frac{\ln e}{\sqrt{{\pi }^2+{\left(\ln e\right)}^2}}$$

(5)

where ζ represents the collision damping coefficient, which is related to the collision recovery coefficient e; when e = 1, it means that the collision is completely elastic, and when e = 0, it means that the collision is completely plastic. In this paper, the energy loss in the process of concrete collision is considered, and the collision recovery coefficient e is 0.65.

3. Synthesis and Input of Cross-Fault Ground Motions

3.1. Synthesis of Cross-Fault Ground Motion

Limited by the extreme paucity of ground motion records across active fault zones, it is a feasible solution to use artificial methods to perform a reasonable simulation of the ground motion of cross-fault bridges. Yang et al. [39] found that the seismic ground motion in the fault zone can be obtained by superimposing high-frequency records with low-frequency artificial waves containing fling-step effect pulses. Therefore, in this paper, the pulse function is used to simulate the low-frequency component, the near-field ground motion is filtered to obtain the high-frequency component, and finally, the broadband ground motion is superimposed [40].

3.1.1. Determination of Low Frequency Components

In this paper, the improved pulse model proposed by Hoseini et al. [41] is used to simulate the low-frequency pulse with a sliding effect. The analytical model of the velocity pulse is shown in Equation (6).

$$v\left(t\right)=\left\{\begin{array}{c}\mathrm{for}\left(t_0-\frac{\gamma }{4f_{\mathrm{p}}}\right)\le t\le \left(t_0+\frac{\gamma }{4f_{\mathrm{p}}}\right),\\ A{\left(\frac{4f_{\mathrm{p}}}{\gamma }\right)}^4{\left({\left(t-t_0\right)}^2-{\left(\frac{\gamma }{4f_{\mathrm{p}}}\right)}^2\right)}^2\cos \left(2\pi f_{\mathrm{p}}t+v\right),\\ \gamma \ge 1\\ \mathrm{Otherwise}, v\left(t\right)=0\end{array}\right.$$

(6)

where A is the main control of the amplitude of the velocity pulse; f_p is the pulse frequency; v is the harmonic phase; γ to define the parameters of the oscillation characteristics; t₀ is the time point that defines the peak value of the curve, which coincides with the peak time of the high-frequency bottom wave velocity.

The frequency f_p is determined using the empirical Equation (7) proposed by Kamai et al. [42]. According to the statistical results of Reference [43], the pulse parameter γ is 2. The average sliding displacement on both sides of the fault is 40, 50, 60, 70, and 80 cm. By adjusting the relevant pulse parameters, the average sliding displacement in the displacement time history can be made consistent with it, as shown in Figure 9.

$$\ln \left(1/f_{\mathrm{p}}\right)=1.16M_{\mathrm{w}}-6.42$$

(7)

where M_w is the moment magnitude.

3.1.2. Selection of High Frequency Ground Motion

For the high-frequency ground waves needed to synthesize cross-fault artificial seismic waves, this paper selects three groups of near-fault ground motions from the PEER strong earthquake database and extracts their high-frequency components.

Table 1. Three selected near-fault ground motion records.

NGA Number	Earthquake	Magnitude	Station	Rrup (km)	PGA (cm·s−2)	PGV (cm·s−1)	PGD (cm)	Tp (s)
316	Westmorland, 1981	5.9	PTS225	16.66	0.23	55.58	37.10	4.39
802	Loma Prieta, 1989	6.93	STG090	8.50	0.33	45.97	33.33	4.57
803			WVC000	9.31	0.26	42.06	24.27	5.65

lists the basic information of the corresponding ground motions.

3.1.3. Synthesis of Cross-Fault Ground Motion

The actual ground motion is a random process in which the high-frequency and low-frequency components are coupled, and cross-fault ground motion is no exception. Based on the analysis of 11 ground motions and 28 seismic recordings by Tian et al. [44], it is found that the near-fault velocity pulse is a low-frequency component of less than 1 Hz. Therefore, the Butterworth filter is used to extract the high-frequency random components with a cut-off frequency greater than 1 Hz in three ground motions. The filter frequency response formula H(f) is as follows:

$$H\left(f\right)=\frac{1}{1+{\left(\frac{f}{f_{\mathrm{c}}}\right)}^{2n}}$$

(8)

where f is the input frequency, f_c is the cut-off frequency, and n is the filter order of 4.

After filtering the near-field seismic records according to the above method, the high-frequency bottom wave and the low-frequency pulse function are superimposed to obtain the cross-fault ground motion with slip displacement. Limited to space, the time history curves of three groups of synthetic ground motions with an average slip displacement of 50 cm are given in Figure 10.

3.2. Input of Cross-Fault Ground Motion

Using the lumped mass matrix, the motion balance equation of the bridge structure in the absolute coordinate system under seismic excitation is:

$$\underset{M}{\underbrace{\left[\begin{array}{l}{M}_{aa} M_{ab}\\ M_{ba} M_{bb}\end{array}\right]}}\left\{\begin{array}{l}{\ddot{X}}_a\\ {\ddot{X}}_b\end{array}\right\}+\underset{C}{\underbrace{\left[\begin{array}{l}{C}_{aa} C_{ab}\\ C_{ba} C_{bb}\end{array}\right]}}\left\{\begin{array}{l}{\dot{X}}_a\\ {\dot{X}}_b\end{array}\right\}+\underset{K}{\underbrace{\left[\begin{array}{l}{K}_{aa} K_{ab}\\ K_{ba} K_{bb}\end{array}\right]}}\left\{\begin{array}{l}{X}_a\\ X_b\end{array}\right\}=\left\{\begin{array}{c}0\\ R_b\end{array}\right\}$$

(9)

where ${\ddot{X}}_a$, ${\dot{X}}_a$ and $X_a$ are the acceleration, velocity, and displacement vectors of the structural unsupported nodes in the absolute coordinate system; ${\ddot{X}}_b$, ${\dot{X}}_b$ and $X_b$ are the acceleration, velocity, and displacement vectors of the structural support nodes in the absolute coordinate system; $M$, $C$ and $K$ are the mass, damping, and stiffness matrices, respectively. Subscript aa denotes the degree of freedom of non-support nodes, bb denotes the degree of freedom of support nodes, and ab denotes their coupling terms. $R_b$ is the external force vector acting on the support node.

The dynamic equilibrium equations of unknown motion vectors ${\ddot{X}}_a$, ${\dot{X}}_a$ and $X_a$ can be obtained by expanding Equation (9), namely:

$${{\rm M}}_{aa}{\ddot{X}}_a+C_{aa}{\dot{X}}_a+K_{aa}{X}_a+{{\rm M}}_{ab}{\ddot{X}}_b+C_{ab}{\dot{X}}_b+K_{ab}{X}_b=0$$

(10)

Under seismic excitation, using the lumped mass model, then $M_{ab}=0$. Equation (10) can be simplified and rewritten as:

$${{\rm M}}_{aa}{\ddot{X}}_a+C_{aa}{\dot{X}}_a+K_{aa}{X}_a=-K_{ab}{X}_b-C_{ab}{\dot{X}}_b$$

(11)

Equation (11) is the dynamic equilibrium equation for solving the structural response using the displacement input model [45]. Equation (11) is solved by the direct integral method, and the absolute acceleration, absolute velocity, and absolute displacement time history of each point can be calculated.

The displacement input model is not only applicable to the uniform excitation input but also to the non-uniform excitation input, and the direct calculation result is an absolute value. The internal force of the cross-fault bridge structure depends not only on the relative displacement between the superstructure and the ground motion but also on the difference in ground motion input at the structural support. Therefore, in this paper, the multi-point excitation displacement input considering both dynamic displacement and quasi-static displacement is adopted as the reasonable ground motion input mode of this kind of bridge in SAP2000.

In order to focus on the influence of lateral collision caused by mutual dislocation of strike-slip faults on the seismic response of bridges, only the lateral ground motion input is carried out in this paper. Since the horizontal slip values of the parallel fault direction components on both sides of the strike-slip fault are equivalent and opposite in direction, they are divided into cross-fault bridges and near-fault bridges according to the fault dislocation position. The two seismic input methods are shown in Figure 11.

In order to analyze the influence of lateral collision on the seismic response of bridges under fault dislocation, in this paper, the following four working conditions are set up for comparative analysis. When setting the transverse shear key, due to the lack of research on the transverse collision of cross-fault bridges, according to the research results of the existing literature, the setting parameters of conventional bridges are selected. Combined with the basic parameters of a typical highway bridge structure [46], the initial gap between the shear key and the beam is assumed to be 6 cm. According to Jiang Hui et al. [21], the ratio of the collision stiffness to the linear stiffness of the main beam is recommended to be 0.01~0.02, and the initial collision stiffness of the transverse shear key in this paper is 1 × 10⁵ kN/m.

Condition A: consistent artificial synthetic ground motion input, no shear key is set;

Condition B: consistent artificial synthetic ground motion input, a shear key is set on both sides of the cap beam;

Condition C: non-uniform artificial synthetic ground motion input, no shear key is set;

Condition D: non-uniform artificial synthetic ground motion input, a shear key is set on both sides of the cap beam.

4. Influence of Fault Dislocation on the Collision Effect of Bridge

4.1. Fault Dislocation Location

The artificial synthetic ground motion with an average sliding displacement of 50 cm is input to the four working conditions. Due to space limitations, the influence of lateral collision on the mechanical properties of key structural members when fault dislocation is located at different positions of the bridge is discussed by taking the artificial synthetic ground motion 802 as an example.

Figure 12 is the time history curve of the torque at the bottom of the middle pier and the side pier of the bridge under various working conditions. It is easy to know that the changes of the torque time history curves of the middle pier and the side pier under various working conditions are almost the same, whether it is the difference of the fault dislocation position or the setting of the transverse shear key. The torque value of the middle pier is only slightly larger than that of the side pier. It shows that when the middle pier and the side pier structure system are the same, the difference between the two torsion effects is very small and can be ignored. Therefore, this paper only describes the change in the torque time history of the middle pier. Under the action of condition A and condition B, the change in the middle pier torque time history is almost 0; the variation range of pier bottom torque time history under conditions C and D has reached the order of 10⁴ kN·m. This is mainly because the movement direction of the piers on both sides of the fault-crossing bridge is different under the fault dislocation. The large bending stiffness in the main beam plane and the torsional stiffness of the bearing has a strong constraint effect on the torsion of the pier, which in turn causes the larger torque of the pier. At the same time, the permanent displacement of the ground also leads to a large residual torque. Therefore, each pier of the bridge across the fault bears a great torsion effect, while the torsion effect of the pier under the action of conventional near-fault ground motion is very small and can be ignored. Condition D coincides with the time history curve of condition C before the collision effect occurs. After the collision effect, the condition D curve rises as a whole, and the post-earthquake torque is on a higher horizontal line. It shows that the existence of lateral collision amplifies the torque of the pier bottom of the fault-crossing bridge, and the influence on the residual bending moment cannot be ignored.

Different from the conventional near-fault bridge, the cross-fault bridge itself has a significant torsional demand. The lateral collision between the shear key and the main beam requires piers to have higher torsion resistance. Therefore, in the torsional design of the pier column of the bridge across the fault, the adverse effect of the shear key needs to be properly considered.

Figure 13 is the time history curve of the bending moment at the bottom of the middle pier and the side pier of the bridge under various working conditions. From the perspective of fault dislocation position, the bending moment at the bottom of the pier under the action of conditions A and B is smaller than that of conditions C and D, and there is no residual bending moment. This shows that under the combined action of strong ground motion and surface rupture caused by seismogenic fault, the bending moment response of the pier bottom of the cross-fault bridge is greater than that of the conventional near-field bridge, and the pier has an obvious residual bending moment. The bending moment response of the pier adjacent to the fault is more significant than that of the pier far away from the fault.

From the perspective of the collision effect, the existence of the shear key will enlarge the maximum bending moment at the bottom of the bridge pier, but it will also reduce the residual bending moment of the cross-fault bridge. The maximum bending moment of the middle pier and the side pier of the near-fault bridge increased by 18.3% and 5.1%, respectively. The maximum bending moment of the middle pier and the side pier of the fault-crossing bridge increases by 90.6% and 129.8%, while the residual bending moment decreases by 69.0% and 80.6%. It shows that the influence of lateral collision on the bending moment of the pier bottom of the near-fault bridge is limited and only reflected at the middle pier. The amplification effect of the maximum bending moment at the bottom of the pier in the cross-fault bridge and the suppression effect on the residual bending moment are more than 90% and 60%, respectively. In contrast, the amplification effect of lateral collision on the maximum bending moment of the pier in the cross-fault bridge is about 5 times that of the near-fault bridge.

The above shows that due to the existence of mutual dislocation of faults, the bending moment response of the pier bottom of the conventional near-field bridge is not only smaller than that of the cross-fault bridge itself but also the amplification effect of the lateral collision between the shear key and the main beam is far less than that of the cross-fault bridge. Therefore, the maximum bending moment at the bottom of the middle pier is a weak point of structural damage when the shear key is set up for the cross-fault bridge (condition D), and it should be reasonably regulated.

Figure 14 shows the time history curve of bearing displacement at the bridge’s middle pier and side pier under various working conditions. From the diagram, it can be seen that the displacement degree of the bearing of the middle pier of the bridge under each working condition is greater than that of the side pier. From the position of fault dislocation, the time history of bearing displacement fluctuates up and down the zero-horizontal line under the action of conditions A and B. Under the action of condition C and condition D, the displacement time history of the bearing has a large one-way offset. It shows that when the fault is located on one side of the bridge, the bridge bearing is mainly affected by the acceleration to move back and forth in the transverse direction, and there is no residual displacement.

When the bridge passes through the fault vertically, the bridge bearing is mainly affected by the fault displacement to make a one-way movement in the transverse direction of the bridge, and there is obvious residual displacement. Therefore, it is necessary to set up a shear key to limit the bearing displacement of cross-fault bridges. When the shear key is set, the maximum displacement of the middle pier and the side pier bearing under the action of condition A is reduced by 31.0% and 31.3%, respectively. The maximum displacement of the middle pier and the side pier bearing under the action of condition C is reduced by 59.2% and 38.8%, respectively, and the residual displacement is reduced by 62.9% and 23.3%, respectively. It shows that the existence of a shear key can effectively improve the relative displacement between the pier and beam and the residual deformation after an earthquake. Among them, the limiting effect on the displacement of the pier bearing in the cross-fault bridge is about twice that of the near-fault bridge, which greatly reduces the risk of a falling beam of the cross-fault bridge itself.

In summary, compared with near-fault bridges, cross-fault bridges have unique torsional and significant bending resistance requirements. At the same time, the lateral collision between the shear key and the main beam has a greater effect on the fault-crossing bridge. It increases the internal force demand at the bottom of the pier but effectively improves the bearing deformation. This point is significant at the piers on both sides of the fault.

4.2. Degree of Fault Dislocation

Through the above correlation analysis of changing the fault dislocation position, it can be seen that when the bridge crosses the fault vertically, the effect of lateral collision on the seismic response of the bridge pier adjacent to the fault is the most significant. Therefore, the synthetic ground motion with an average slip displacement of 40~80 cm is input for condition C and condition D, respectively, and the mean value of the seismic response under the action of three ground motions is taken as the analysis result. In the following, the influence of fault dislocation degree on the lateral collision effect at the middle pier of a cross-fault bridge is discussed in detail.

Figure 15 shows the relationship between the torque at the bottom of the pier and the slip displacement of the fault under the action of conditions C and D. From the diagram, it can be seen that as the average slip displacement of the fault increases from 40 cm to 80 cm, the maximum torque and residual torque at the bottom of the pier under the action of working condition C and working condition D are in a step-like upward trend. Among them, the growth rate of the maximum torque at the bottom of the pier under the action of condition D and condition C is 94.5% and 89.2%, respectively, and the ratio of the two is 1.06. It shows that the growth rate of the maximum torque at the bottom of the pier under the action of working condition C is slightly lower than that of working condition D, resulting in an increase in the maximum torque difference between the two working conditions. Although the difference in residual torque at the bottom of the pier under the action of working condition C and working condition D is small, the pier itself has a large residual torque compared with the maximum torque.

Therefore, when the degree of fault dislocation is small, the amplification effect of a lateral collision on the maximum torque at the bottom of the pier can be considered appropriate. However, when the degree of fault dislocation is large, there is obvious residual torque after the earthquake, and relevant control measures need to be taken to prevent other factors from causing the torsional cracking of piers.

Figure 16 shows the relationship between the bending moment at the bottom of the pier and the slip displacement of the fault under the action of conditions C and D. From the perspective of the maximum bending moment at the bottom of the pier, as the average slip displacement of the fault increases from 40 cm to 80 cm, the growth trend of the maximum bending moment under the action of working condition C gradually slows down, while the working condition D increases almost linearly. This has led to a growing difference between the two. Among them, the growth rate of the maximum bending moment at the bottom of the pier under the action of condition D and condition C is 61.9% and 28.6%, respectively, and the ratio of the two is 2.16. It shows that the amplification effect of lateral collision on the maximum bending moment of the pier bottom becomes more and more prominent with the increase in fault slip displacement. From the perspective of the residual bending moment at the bottom of the pier, the residual bending moment under the action of condition C has the same trend as the maximum bending moment, but the residual bending moment under the action of condition D does not increase significantly with the increase in fault slip displacement. It shows that the limiting effect of lateral collision on the residual bending moment of the pier bottom is more effective with the increase in fault slip displacement.

With the increase in the degree of fault dislocation, the existence of transverse collision leads to the increasing probability that the bending moment demand of the pier bottom of the cross-fault bridge exceeds its own bending resistance. Therefore, when the cross-fault bridge faces a large fault dislocation, the bending effect of the pier should be considered first to avoid the bending failure of the pier due to excessive bending moment.

Figure 17 shows the relationship between the bearing displacement and the fault sliding displacement at the middle pier of the fault-crossing bridge under the action of conditions C and D. The comparison between the two shows that the maximum displacement of the bearing is consistent with the overall change law of the residual displacement. Different from the internal force response, as the average slip displacement of the fault increases from 40 cm to 80 cm, the bearing displacement under the action of working condition C shows a linear growth trend, while the growth of working condition D is relatively gentle. The growth rate of the maximum displacement of the bearing under condition D and condition C is 57.6% and 88.0%, respectively, and the ratio of the two is 65.4%. It shows that the bearing with a shear key is less sensitive to fault slip displacement and has better resistance to displacement. Under the premise of normal function, even if the fault dislocation displacement is large, the shear key can better limit the relative displacement between the pier and the beam and the residual deformation after the earthquake and greatly reduce the risk of the beam falling of the bridge across the fault.

In summary, the amplification effect of lateral collision on the local internal force of cross-fault bridges under fault dislocation with different slip displacements is inevitable. However, the lateral collision can effectively reduce the relative displacement between the pier and the beam and avoid the phenomenon of falling beams. Therefore, from the perspective of bridge functionality, it is necessary to set up a shear key for cross-fault bridges. However, the selection of conventional collision stiffness and initial gap is not suitable for the extreme case of cross-fault bridges. The larger collision stiffness and smaller initial gap under the fault dislocation will aggravate the effect of lateral collision on the seismic response of the bridge, resulting in the unreasonable distribution of internal force and displacement of the structure. In the following, the influence of the main parameters related to the lateral collision on the seismic response of the middle pier of the cross-fault bridge is analyzed in order to provide guidance for its reasonable seismic design.

5. Parametric Analysis of Lateral Collision Effects

The degree of lateral collision of bridges across faults is intense, and the collision stiffness and initial clearance are two key parameters affecting the collision effect. Therefore, this section presents a detailed analysis of the effects of collision stiffness and initial clearance on the seismic response of the middle pier of bridges across faults.

5.1. Collision Stiffness

Collision stiffness is an important parameter affecting the seismic performance of the anti-falling beam device. When it plays a role, the inertial force shared by the pier increases significantly. Now, the collision stiffness k_k is taken as 1 × 10⁴ kN/m, 5 × 10⁴ kN/m, 1 × 10⁵ kN/m, 3 × 10⁵ kN/m, 5 × 10⁵ kN/m, 7.5 × 10⁵ kN/m, and 1 × 10⁶ kN/m, respectively, and the relationship between collision stiffness and seismic response of cross-fault bridges is compared and discussed.

Figure 18 is the relationship curve between the collision stiffness with the internal force of the pier bottom and the displacement of the bearing under the action of three seismic waves. It can be seen that the trend of curve change under different seismic waves is consistent. However, the influence of collision stiffness on the internal force of the pier bottom and the displacement of the bearing is inconsistent. With the increase in collision stiffness, the torque at the bottom of the pier increases first and then decreases, and the end remains level. The variation range of the torque at the bottom of the pier is very limited, and the maximum change is only 3.3%. It shows that the influence of collision stiffness on pier bottom torque is very small, and it can be ignored when selecting parameters. However, the impact stiffness has a significant effect on the bending moment of the pier bottom, and its overall trend is obviously increasing. Because in terms of the nature of the contact element, the collision stiffness directly determines the size of the collision force; with the increase in collision stiffness, the collision effect will undoubtedly be amplified, and the increased collision force will be transmitted to the pier through the cap beam, which will eventually lead to the increase in the bending moment response at the bottom of the pier. However, the bending moment at the bottom of the pier increases slowly in the later period. The displacement of the bearing is different from the internal force of the pier bottom. Due to the increase in the collision stiffness, the restriction ability of the shear key to the displacement of the main beam is also increased, so the displacement of the bearing decreases with the increase in the collision stiffness. However, according to the slope of the bearing displacement curve, its growth trend has gradually slowed down. It shows that with the increase in collision stiffness, the limit effect of the shear key drops sharply.

The above analyses show that when the collision stiffness is large, the degree of change in the seismic response of the structure is extremely small. In the smaller range of collision stiffness, the bending moment of the pier bottom and the relative displacement of the pier beam should be the key objects to be considered. Therefore, the selected collision stiffness is in the range of k_k = 5 × 10⁴~1 × 10⁵ kN/m, and the shear key device has a better seismic effect.

5.2. Initial Gap

Usually, there is a certain initial gap between the shear key and the beam, and the width of these gaps is usually less limited by the construction requirements due to the large displacement of the bearing of the bridge across the fault. Therefore, in this paper, the maximum displacement of the bearing in condition C is 35 cm as a reference. It is assumed that the initial gap at the middle pier and the side pier is the same, and nine grades of 2 cm~36 cm are taken, respectively, of which 36 cm means that no lateral collision occurs. The remaining specific values are detailed in Figure 19.

Figure 19 is the relationship curve between the initial gap with the internal force of the pier bottom and the displacement of the bearing under the action of three seismic waves. It can be seen from the figure that the size of the initial gap has different effects on the torque and bending moment response of the pier bottom. With the increase in the initial gap, the torque at the bottom of the pier decreases first and then increases, and there is an extreme point. However, the torque at the bottom of the pier is not much different when the initial gap is small and large. The bending moment at the bottom of the pier decreases with the increase in the initial gap, and there is a certain upward fluctuation in the local range. It shows that the influence of the initial gap on the pier bottom bending moment is greater than that of the pier bottom torque. When the initial gap is smaller, the lateral collision has a greater restriction on the relative displacement of the pier beam, which leads to a significant demand for the bending moment at the bottom of the pier. Therefore, unlike conventional bridges, the shear keys of cross-fault bridges need to reserve a large initial gap to ensure the safety of related components. As far as the bearing displacement is concerned, it increases linearly with the initial gap. This is mainly because the lateral collision occurs when the displacement of the bearing exceeds the initial gap. The increase in the initial gap is equivalent to the time that the lateral collision is prolonged. Therefore, the change to the initial gap is linearly related to the change in the bearing displacement.

It can be seen that from the perspective of the internal force of the pier bottom, the reasonable initial gap should mainly consider the influence of the bending moment of the pier bottom. In addition, the growth rate of the initial gap to the bearing displacement is constant. Therefore, it is more appropriate to select the initial gap of 10~14 cm when selecting the initial gap of the case cross-fault bridge. At the same time, it is suggested that the rubber block should be filled between the shear key and the beam to dissipate the seismic energy, delay the collision, and achieve a better limit effect.

According to the influence of the above collision stiffness and initial gap on the cross-fault bridge, the collision stiffness is selected to be 5 × 10⁴ kN/m, and the initial gap is 14 cm to re-adjust the condition D, referred to as condition E. Taking the average value under the action of three seismic waves, the transverse seismic response of the cross-fault bridge under the action of condition E is compared with that of condition C and condition D, as shown in

Table 2. Comparison of the seismic response of the middle pier of the bridge across the fault before and after parameter adjustment.

	Torque/(kN·m)	Bending Moment/(kN·m)	Bearing Displacement/(cm)
Condition C	17,094	42,598	38
Condition D	19,886	75,472	14
Condition E	21,436	52,844	23

.

By comparing condition D and condition C, it can be seen that under the fault dislocation, the transverse shear key effectively controls the bearing displacement of the bridge across the fault, but the bending moment demand at the bottom of the pier is more serious. After the reasonable adjustment of the relevant parameters, the bending moment and bearing displacement of the pier bottom of the fault-crossing bridge under the action of condition E reach the moderate level of condition C and condition D, and the growth of the pier bottom torque is not obvious.

In summary, the influence of collision stiffness and initial gap on the pier bottom torque and bearing displacement of cross-fault bridges is obvious under fault dislocation. Conventional parameters lead to a large redundancy of bearing displacement, and the bending moment at the bottom of the pier becomes a weak point of structural damage. Therefore, appropriately reducing the collision stiffness and increasing the initial gap is conducive to the rational distribution of the seismic response of the structure, thereby improving the seismic performance of the cross-fault bridge.

6. Conclusions

In this paper, for continuous girder bridges vertically spanning a strike-slip fault, we take both the location and degree of fault misalignment as the starting point. Based on the quantified structural seismic response, the effect of lateral collision on bridges near/across faults is revealed. On this basis, parametric analyses of collision stiffness and initial clearance are carried out to rationally regulate the internal force and displacement responses of bridges spanning faults. The main conclusions are as follows:

• Compared with the near-fault bridge, the lateral collision has a greater effect on the cross-fault bridge. The amplification of the maximum bending moment of the middle pier and the limitation of the maximum displacement of the bearing are five times and two times that of the near-fault bridge, respectively. At the same time, the lateral collision also amplifies the torsional characteristics of the cross-fault bridge. Therefore, the risk of local failure of the middle pier of the cross-fault bridge under the action of lateral collision is much higher than that of the near-fault bridge.
• With the increase in the average slip displacement of the fault from 40 cm to 80 cm, the existence of transverse collision leads to an increase of 94.5%, 61.9%, and 57.6% in the torque at the bottom of the pier, the bending moment at the bottom of the pier, and the displacement of the bearing. Its amplification effect on the bending moment at the bottom of the pier is 2.06 times that without collision. When the bridge is faced with large fault dislocation, the bending effect of the pier should be considered first to avoid bending failure caused by an excessive bending moment. Secondly, we should also pay attention to the residual torque of the pier after the earthquake to prevent the secondary torsion cracking of the pier.
• The variation of parameters related to the collision effect shows that the influence of the transverse collision parameters on the pier bottom torque of the cross-fault bridge is not obvious, and it is only 3.3% affected by the collision stiffness. By properly reducing the collision stiffness and increasing the initial gap, the amplification effect of the transverse shear key on the bending moment of the pier bottom can be improved, and the bearing displacement also has high redundancy. Therefore, the selection of reasonable collision parameters can effectively control the seismic response of cross-fault bridges.

In short, the change in fault dislocation position and dislocation degree will greatly affect the lateral collision effect of the bridge. The improvement effect of the shear key with conventional parameters on the seismic response of the fault-crossing bridge is polarized. It is necessary to select the relevant parameters reasonably to realize the equalization of the stress state of the bridge across the fault.

However, this study simulates the restoring force model of the laminated rubber bearing in a linear form and does not consider the damage state of the shear key. Therefore, the further research focus is to consider the influence of the friction slip characteristics of the plate rubber bearing and the degradation of the mechanical properties of the transverse shear key on the transverse seismic response of the cross-fault bridge and put forward the reasonable parameters suitable for the design of the transverse shear key of the cross-fault bridge.