## 3 Review of the Improved Fiber Model  
To establish a high-precision calculation model for centrally compressed square-section steel piers that can be used in horizontal bidirectional seismic response analysis, previous studies [18; 19] proposed a kind of fiber model that can accurately consider the local instability of steel plates. Figure 4 shows a schematic of the basic principle of this improved fiber model. It can be seen from Figure 4 that the core idea of the model is to use an equivalent hysteresis model as the constitutive relationship of the fiber element in the length of the effective seismic damaged zone \(L_{\text{ed}}\) at the bottom of the pier to reflect the local instability effect of the steel plates. Since the local instability effect of steel plates distorts the section, the traditional fiber model based on the plane cross-section assumption and the material elastoplastic constitutive relationship becomes inapplicable. Therefore, in the improved fiber model that can consider local instability effect, it is firstly necessary to determine the length of the effective seismic damaged zone, \(L_{\text{ed}}\), where the steel plates may undergo local instability deformation and the plane cross-section assumption is no longer applicable. Only one fiber  
Figure 4: Basic principle of the improved fiber model [18; 19] for the centrally compressed square-section steel pier. Figure 3: Sensitivity analysis results of the shell element mesh size. element will be established in this area. The empirical value formula for \(L_{\text{ed}}\) given in [18,19] is shown below. \[L_{\text{ed}}=L_{\text{w}}\approx\min[0.7B_{0},a_{0}] \tag{2}\]  
where \(L_{\text{w}}\) is the half length of a buckling wave of steel plates. Furthermore, it is necessary to establish the equivalent stress-strain hysteresis model of the fiber element within \(L_{\text{ed}}\). Figure 4 shows that this hysteretic model was established by fitting the axial average-stress-average-strain relationship within \(L_{\text{ed}}\) obtained by the hybrid-element calculation model. To unify the constitutive relationship of the fiber elements in other regions, the modified two-surface uniaxial constitutive steel model was selected as the initial form of the equivalent hysteretic model. By comparing the average-stress-average-strain relationship and the stress-strain relationship of the steel modified two-surface model, it can be seen that, because the steel plates undergo out-of-plane geometric deformation after local instability, the geometric deformation of the steel plates is restored when reverse tension is applied. Thus, the axial stiffness of the fiber element within \(L_{\text{ed}}\) must be reduced. According to the characteristics of the average-stress-average-strain curve, the fiber element in the effective seismic damaged zone adopted the following equivalent elastic modulus \(E^{\prime}\):  
\[E^{\prime}=\left\{ \begin{array}{l}E,\qquad\overline{\varepsilon}\geq \overline{\varepsilon}_{\text{m}}\\ E\cdot k\cdot e^{m\overline{\varepsilon}},\qquad\overline{\varepsilon}< \overline{\varepsilon}_{\text{m}}\end{array} \right. \tag{3}\]  
where \(\overline{\varepsilon}\) is the current strain, \(\overline{\varepsilon}_{m}\) is the average strain value corresponding to the peak value of the horizontal displacement load, \(E\) is the elastic modulus value of the steel alloy, and \(k\) and \(m\) are equivalent hysteretic model parameters. Through the parameter analysis of several centrally compressed steel piers, the fitting formulas were obtained in the literature [18], as follows:  
\[\left\{ \begin{array}{l}k=0.951R_{R}^{-0.981}\lambda^{0.186}\alpha^{0.333}(N/N_ {y})^{0.286}L_{\text{ed}}^{0.322}(\gamma/\gamma^{*})^{-0.080}\\ m=28.578R_{R}^{1.635}\lambda^{-0.345}\alpha^{0.178}(N/N_{y})^{0.260}L_{\text{ed }}^{-0.260}(\gamma/\gamma^{*})^{-0.244}k^{2.016}\end{array} \right.