## 4 Calculation and Discussion for Eccentrically Compressed Steel Piers
### Out-of-Plane Pseudo-Static Analysis  
Figure 2 shows that, under the action of the eccentric compression \(N\) and the out-of-plane eccentric horizontal cyclic load \(H_{\text{Y}}\), the steel piers are subjected to the torsional  
Figure 9: Load–displacement curves of the in-plane pseudo-static analysis: (**a**) b2; (**b**) b3; (**c**) b4; (**d**) b13–10; and (**e**) b3–d. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Pier**} & \multirow{2}{*}{**Model**} & **Elastic** & \multirow{2}{*}{\(H_{\text{m}}\)} & \multirow{2}{*}{\(\delta_{\text{m}}\)} & \multirow{2}{*}{\(\delta_{\text{80}}\)} & **Total** \\  & & **Stiffness** & & & & **Hysteretic** \\ \hline \multirow{2}{*}{b2} & Fiber model & 0.966 & 0.998 & 2.65 & 3.65 & 21.36 \\ \cline{2-7}  & Hybrid model & 0.987 & 1.019 & 2.65 & 3.26 & 20.97 \\ \hline \multirow{2}{*}{b3} & Fiber model & 0.967 & 0.997 & 2.65 & 3.60 & 21.38 \\ \cline{2-7}  & Hybrid model & 0.989 & 1.014 & 2.65 & 3.21 & 20.31 \\ \hline \multirow{2}{*}{b4} & Fiber model & 0.962 & 1.009 & 2.65 & 4.15 & 32.11 \\ \cline{2-7}  & Hybrid model & 0.981 & 1.019 & 2.65 & 4.02 & 30.29 \\ \hline \multirow{2}{*}{b13–10} & Fiber model & 0.997 & 1.256 & 4.15 & 7.05 & 135.87 \\ \cline{2-7}  & Hybrid model & 0.969 & 1.288 & 4.15 & 6.24 & 123.29 \\ \hline \multirow{2}{*}{b3–d} & Fiber model & 0.968 & 0.666 & 2.80 & 5.51 & 21.32 \\ \cline{2-7}  & Hybrid model & 0.989 & 0.699 & 2.80 & 5.45 & 20.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of the in-plane calculation results. moment \(H_{\text{Y}}e\) and the bending moment \(Ne\). [PERSON] et al. [25] confirmed that the restraint of the upper bearings should be considered when studying a single steel pier, i.e., the torsional deformation should be restrained. Therefore, this study restrained the torsional deformation of the steel pier under eccentric compression. Although the recommended formula of the scale coefficient for the bearing capacity and ductility performance index between the eccentric and centrally loaded rectangular-section steel piers under out-of-plane horizontal earthquakes was given in a previous study [27; 28], the relationship between the hysteresis curves of these two kinds of structures remains unclear. Therefore, each pier's hybrid-element calculation models (Figure 2) were established to perform the out-of-plane numerical simulation analysis in this paper. When performing the out-of-plane pseudo-static analysis for the eccentrically compressed steel piers, the adopted displacement loading form is the same as that used in the pseudo-static analysis of the centrally loaded steel piers (Figure 10) [11; 18; 19]. Out-of-plane horizontal pseudo-static numerical simulations were conducted through the improved-fiber and hybrid-element calculation models for piers, as shown in Table 1. Figure 11 shows the calculation results of the out-of-plane load-displacement curve of each pier, as shown in bold in Table 1. The hysteresis curve results of the corresponding steel piers without eccentricity are also plotted in the figure to reflect the influence of eccentric compression. The calculation results show that, due to the eccentric bending moment effect, the maximum structural bearing capacity is worse than that of the steel piers without eccentricity, and the ductility is also reduced. Therefore, eccentrically compressed piers will attain the limiting state in advance. Because the torsional deformation is restrained, the stiffness does not change in an obvious manner. Furthermore, each pier's load-displacement curve in Figure 11, calculated according to the improved fiber model, is quite close to that obtained using the hybrid-element calculation model, and other piers in Table 1 also have similar comparison results.