# Seismic Response and Damage Analysis of an Isolated Bridge Subjected to Near-Fault Ground Motion
## 2 Finite Element Model of Bridge
### Pier Analysis Model
#### 2.2.1 Constitutive Model for Concrete Material  
[PERSON]'s uniaxial concrete model [23] is implemented in the analysis (Figure 5). The model has high simulation accuracy and computational efficiency. The stirrup constraint effect can be simulated by modifying the characteristic parameters of concrete materials. To consider the damage behavior in the uniaxial constitutive model of concrete, [PERSON] [19] used the tensile and compressive damage indices of concrete to describe the tensile and compressive damage of concrete, respectively. The calculation method of compression damage index \(D_{\text{c}}\) is as follows:  
\[D_{\text{c}}= \begin{cases}0&\varepsilon_{\text{cm}}^{\text{c}}<\varepsilon_{ \text{cd0}}^{\text{c}}\\ \frac{\varepsilon_{\text{cd0}}-\text{Em}}{\varepsilon_{\text{cd0}}-\text{E} _{\text{c20}}}&\varepsilon_{\text{cd0}}^{\text{c}}\leq\varepsilon_{\text{cm}} ^{\text{c}}\leq\varepsilon_{\text{20}}\\ 1&\varepsilon_{20}<\varepsilon_{\text{cm}}^{\text{c}}.\end{cases}  \tag{1}\]  
Figure 4: The cross-section fiber discretization method. Figure 5: [PERSON]’s concrete constitutive model. where, \(E_{\text{cd0}}=\left(\sigma_{\text{cd0}}^{\text{c}}-\sigma_{r}\right)/\left(\varepsilon _{\text{cd0}}^{\text{c}}-\varepsilon_{r}\right)\) is the initial compression damage modulus, determined according to the compression damage starting point \(D_{0}\big{(}\varepsilon_{\text{cd0}}^{\text{c}},\sigma_{\text{cd0}}^{\text{c}} \big{)}\) and the point \(R(\varepsilon_{r},\sigma_{r})\); \(E_{\text{cm}}=\left(\sigma_{\text{cm}}^{\text{c}}-\sigma_{r}\right)/( \varepsilon_{\text{cm}}^{\text{c}}-\varepsilon_{r})\) is the modulus of the current unloading point, determined according to the unloading point \(D(\varepsilon_{\text{cm}}^{\text{c}},\sigma_{\text{cm}}^{\text{c}})\) and the point \(R(\varepsilon_{r},\sigma_{r})\). The tensile damage index \(D_{\text{t}}\) is calculated as follows:  
\[D_{\text{t}}= \begin{cases}0&\varepsilon_{\text{cm}}^{\text{t}}<\varepsilon_{ \text{cr}}\\ \frac{E_{\text{cd0}}-E_{\text{cm}}}{E_{\text{cd0}}}&\varepsilon_{\text{cr}} \leq\varepsilon_{\text{cm}}^{\text{t}}\leq\varepsilon_{\text{ut}}\\ 1&\varepsilon_{\text{ut}}<\varepsilon_{\text{cm}}^{\text{t}}.\end{cases}  \tag{2}\]  
where, \(\varepsilon_{\text{cm}}^{\text{t}}\) is the strain at the current unloading point (shown in Figure 6); \(\varepsilon_{\text{cr}}\) is the strain corresponding to the peak tensile stress; \(\varepsilon_{\text{ut}}\) is the ultimate tensile strain; \(E_{\text{cd0}}\) is the initial tangent modulus; \(E_{\text{tm}}\) is the secant modulus of the current unloading point. The parameters of the concrete material are shown in Table 1.  
In the analysis, the modified Menegotto-Pinto constitutive model [24] was used for modeling the stress-strain relationship between steel bars. The model was proposed by [PERSON] and [PERSON] and modified by [PERSON] et al. [25] to consider the isotropic strengthening effect. The Bauschinger effect, under cyclic loads, is considered in the model. The modified model is in good agreement with experimental results and has high solution efficiency, so it has been widely used. [PERSON] [19] introduced the Bonora damage model [26] modified by [PERSON] and [PERSON] [27] into the modified Menegotto-Pinto constitutive model. The modified Bonora damage model can simulate the damage behavior of steel under cyclic loads and has a high simulation accuracy. The damage index in the Bonora damage model is calculated as follows:  
\[\dot{D}=\alpha\frac{\left(D_{\text{cr}}-D_{0}\right)^{1/\alpha}}{\ln\varepsilon _{\text{cr}}-\ln\varepsilon_{\text{th}}}f(\frac{\sigma_{m}}{\varepsilon_{ \text{eq}}})(D_{\text{cr}}-D)^{\alpha-1/\alpha}\frac{dp}{p}. \tag{3}\]  
where, \(\dot{D}\) is the damage increment; \(D\) is the cumulative damage value; \(D_{0}\) is the initial damage value; \(D_{cr}\) is the critical damage value; \(\varepsilon_{\text{cr}}\) is the critical strain corresponding to the