## 2 FE Model of Steel Piers
### Hybrid-Element Model  
To study the seismic performance and practical calculation method of the seismic response of eccentrically compressed steel piers, the hybrid-element models, which are combined with fiber elements and shell elements for multiple steel piers, were established in the common FE software ABAQUS 6.14 to perform the pseudo-static numerical simulation analysis. Figure 2 shows the horizontal bidirectional pseudo-static analysis model of a typical eccentrically compressed steel pier with a square-stiffened section. The height of the pier is \(h\), the diaphragm spacing is \(a_{0}\), and the fixed boundary condition is applied at the bottom of the pier. The vertical constant axial force \(N\) and horizontal cyclic loads \(H_{\mathrm{X}}\) and \(H_{\mathrm{Y}}\) were applied at a distance of \(e\) (i.e., the eccentricity) from the center of the section. In this case, note that the subscript \(X\) represents the horizontal direction in the eccentric plane, i.e., the transverse direction of the actual bridge, whereas \(Y\) represents the out-of-plane horizontal direction perpendicular to the X-direction, i.e., the longitudinal direction of the actual bridge. For the material elastoplastic constitutive model, the bilinear kinematic and isotropic hardening models are traditional nonlinear hysteretic constitutive steel models. However, these two models can only consider the Bauschinger effect or the strain-strengthening effect separately. Actual steel has these two performances simultaneously, so these two constitutive models have limited accuracy when applied to steel piers with significant plastic strain history and local instability behavior [32]. To improve the calculation accuracy, this study adopts the modified two-surface hysteretic constitutive model modified by [PERSON] et al. [33; 34], which has been widely used in the seismic response analysis of steel piers [4; 12; 13; 18; 19; 27; 28]. [PERSON] et al. [35] further modified this constitutive model locally to avoid the defect of unreasonable hysteresis history under slight cyclic loading. The modified two-surface model was used, which is realized by programing and calling the UMAT subroutine in ABAQUS. The steel type used in this paper was Q345 qC, a kind of structural steel alloy widely used in bridge engineering in China. The material parameters of the modified two-surface model constitutive model were calibrated by [PERSON] et al. [35] (Table 2). The detailed meaning of each material parameter can be found in the literature [33; 34; 35]. Figure 2: Hybrid-element model of the eccentrically compressed steel pier. To reflect the local instability effect of steel plates that may occur at the bottom of piers, shell elements were established within the range of thrice the spacing of the transverse diaphragms, which can cover the area where local buckling deformation occurs [31]. Fiber elements were established in the remaining part (Figure 2). The shell element type is a four-node reduced integral isoparametric element (S4R) in Abaqus, and there are five integration points in the element's thickness direction. The fiber-element type is a two-node linear spatial beam element (B31). In order to determine the reasonable shell element size to model the local instability behavior, the mesh size sensitivity analysis was carried out by comparing the calculation results and pseudo-static test results of a square-section steel pier in the literature [10]. The material and structural parameters of the tested pier are shown in Tables 3 and 4, respectively. Within the length of diaphragm spacing, 15, 20 and 25 layers of shell elements were divided in three different models. The comparison between the calculation results of the load-displacement curve at the top of the pier and the test results in the literature [10] is shown in Figure 3. In the figure, \(H\) and \(H_{y}\) are the horizontal force and the cross-section yield force, respectively.