## 3 Synthesis and Input of Cross-Fault Ground Motions
### 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:  
\[\underbrace{\left[ \begin{array}{cc}\mathbf{M}_{aa}&\mathbf{M}_{ab}\\ \mathbf{M}_{ba}&\mathbf{M}_{bb}\end{array} \right]}_{\mathbf{M}}\left\{ \begin{array}{cc} \ddot{\mathbf{X}}_{a}\\ \ddot{\mathbf{X}}_{b}\end{array} \right\}+\underbrace{\left[ \begin{array}{cc} \mathbf{C}_{aa}&\mathbf{C}_{ab}\\ \mathbf{C}_{ba}&\mathbf{C}_{bb}\end{array} \right]}_{\mathbf{C}}\left\{ \begin{array}{cc} \dot{\mathbf{X}}_{a}\\ \dot{\mathbf{X}}_{b}\end{array} \right\}+\underbrace{\left[ \begin{array}{cc}\mathbf{ K}_{aa}&\mathbf{K}_{ab}\\ \mathbf{K}_{ba}&\mathbf{K}_{bb}\end{array} \right]}_{\mathbf{K}}\left\{ \begin{array}{cc} \mathbf{X}_{a}\\ \mathbf{X}_{b}\end{array} \right\}=\left\{ \begin{array}{cc}0\\ \mathbf{R}_{b}\end{array} \right\} \tag{9}\]  
where \(\ddot{\mathbf{X}}_{a},\dot{\mathbf{X}}_{a}\) and \(\mathbf{X}_{a}\) are the acceleration, velocity, and displacement vectors of the structural unsupported nodes in the absolute coordinate system; \(\ddot{\mathbf{X}}_{b},\dot{\mathbf{X}}_{b}\) and \(\mathbf{X}_{b}\) are the acceleration, velocity, and displacement vectors of the structural support nodes in the absolute coordinate system; \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) 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. \(\mathbf{R}_{b}\) is the external force vector acting on the support node. The dynamic equilibrium equations of unknown motion vectors \(\ddot{\mathbf{X}}_{a},\dot{\mathbf{X}}_{a}\) and \(\mathbf{X}_{a}\) can be obtained by expanding Equation (9), namely:  
\[M_{aa}\ddot{\mathbf{X}}_{a}+\mathbf{C}_{aa}\dot{\mathbf{X}}_{a}+\mathbf{K}_{aa}\mathbf{X}_{a}+M_{ ab}\ddot{\mathbf{X}}_{b}+\mathbf{C}_{ab}\dot{\mathbf{X}}_{b}+\mathbf{K}_{ab}\mathbf{X}_{b}=0 \tag{10}\]  
Figure 10: Acceleration, velocity, and displacement time histories of three synthetic seismic waves: (a) synthetic seismic wave 316, (b) synthetic seismic wave 802, and (c) synthetic seismic wave 803. Under seismic excitation, using the lumped mass model, then \(\mathbf{M}_{ab}=0\). Equation (10) can be simplified and rewritten as:  
\[\mathbf{M}_{aa}\ddot{\mathbf{X}}_{a}+\mathbf{C}_{aa}\dot{\mathbf{X}}_{a}+\mathbf{K}_{aa}\mathbf{X}_{a}=- \mathbf{K}_{ab}\mathbf{X}_{b}-\mathbf{C}_{ab}\dot{\mathbf{X}}_{b} \tag{11}\]  
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.