## 3 Review of the Improved Fiber Model
\tag{4}\]  
After determining the equivalent elastic modulus, the average plastic strain can be calculated as follows and the stress-strain relationship can be translated into the stress-plastic strain relationship. \[\overline{\varepsilon}^{\text{P}}=\varepsilon-\frac{\overline{\sigma}}{E^{ \prime}} \tag{5}\]  
Furthermore, Figure 4 shows that, due to the local buckling of the steel plates, the average-stress-plastic strain curve produces a negative stiffness in the compression section. To describe the characteristics of the deformation history, the compression section of the stress-plastic strain path ABCDE (blue solid line in Figure 5) was introduced, based on the initial stress-plastic strain curve ABDF' (blue dotted line in Figure 5) of the material two-surface model. The initial value of the compressive skeleton curve M\({}_{1}\)M\({}_{2}\)N\({}_{1}\)N\({}_{2}\) (red solid line in Figure 5) is \(\sigma_{\text{y}}\). When the plastic strain \(\overline{\varepsilon}^{\text{P}}\) lower than the peak average plastic strain \(\overline{\varepsilon}^{\text{P}}_{\text{m}}\), the skeleton curve decreases with the slope \(AE\), and remains unchanged until reaching to 0.25 times the \(\sigma_{\text{y}}\). It can be seen from Formula (3) that, when the strain is equal to \(\overline{\varepsilon}_{\text{m}}\), the elastic modulus does not need to be reduced, so the peak average plastic strain \(\overline{\varepsilon}^{\text{P}}_{\text{m}}\) in the figure can be calculated as follows:  
\[\overline{\varepsilon}^{\text{P}}_{\text{m}}=\overline{\varepsilon}_{\text{ m}}-\frac{\sigma_{\text{y}}}{E^{\prime}} \tag{6}\]Figure 5 also shows that the bounding surface YY' of the modified two-surface model must be removed to \(Y_{1}Y_{2}\) to obtain a reasonable result. Therefore, it is necessary to define the bounding-surface reduction law of the equivalent hysteretic model after local instability deformation occurs, and the radius of the elastic range should also be reduced. According to the trial calculation, the initial radius of the bounding surfaces in the equivalent hysteresis model was modified to  
\[\overline{\kappa}_{0}^{1}=B\cdot\overline{\kappa}_{0} \tag{7}\]  
where \(B\) is the model parameter. The modified bounding surface radius and elastic range radius can be calculated by  
\[\overline{\kappa}_{1}=\overline{\kappa}-\left(\overline{\varepsilon}_{\text{m }}^{\text{P}}-\overline{\varepsilon}_{1}^{\text{P}}\right)C\cdot E \tag{8}\]  
\[\kappa_{1}=\kappa-\frac{\left[E_{0}^{\text{P}}+\left(A+C\right)\cdot E\right] \left(\overline{\varepsilon}_{\text{m}}^{\text{P}}-\overline{\varepsilon}_{ 1}^{\text{P}}\right)}{2} \tag{9}\]  
where \(C\) is the model parameter and \(\overline{\varepsilon}_{1}^{\text{P}}\) is the plastic strain from the previous compression-to-tension turning point in the strain history. \(\overline{\kappa}\) and \(\kappa\) are the radii of the elastic range and the bounding surface of the material two-surface model, respectively. The remaining parameters are the same as previously used. The fitting formulas of model parameters \(\overline{\varepsilon}_{\text{m}}^{\text{P}}\), \(A\), \(B\), and \(C\) are also obtained through the statistics of the many parameter-analysis results of steel piers. The results are given as follows, fitted in the literature [18]. \[\left\{ \begin{array}{l}\bar{\varepsilon}_{\text{m}}^{\text{P}}=0.00163R_{R}^ {3,00}\lambda^{0.208}a_{0}^{0.203}(N/N_{y})^{0.305}L_{\text{ed}}^{0.053}(\gamma /\gamma^{*})^{0.435}\\ A=0.445R_{R}^{1.387}\lambda^{0.744}a_{0}^{-0.284}(N/N_{y})^{0.693}L_{\text{ed}} ^{0.312}(\gamma/\gamma^{*})^{-0.504}\\ B=1.095R_{R}^{0.122}\lambda^{0.182}a_{0}^{-0.091}(N/N_{y})^{-0.104}L_{\text{ed}} ^{-0.029}(\gamma/\gamma^{*})^{-0.006}\\ C=1.370R_{R}^{0.592}\lambda^{1.889}a_{0}^{-0.299}(N/N_{y})^{0.985}L_{\text{ed}} ^{0.432}(\gamma/\gamma^{*})^{-0.197}\end{array} \right.