## 4 Calculation and Discussion for Eccentrically Compressed Steel Piers
### Seismic Response Analysis  
To further verify the applicability of the improved fiber model in seismic response analysis for eccentrically compressed steel piers, considering the five groups of piers marked in bold font in Table 1 as an example, improved fiber models were established to calculate the structural seismic response under horizontal bidirectional earthquakes. Simultaneously, hybrid-element calculation models were also established for comparison. Figure 12 shows the schematics of the two models used in the seismic response analysis. The axial force in the pseudo-static analysis model was converted into the pier top mass \(m_{0}\). To verify the structural applicability under actual horizontal bi-directional earthquakes, ground motions were input simultaneously along the two main axis directions at the bottom of the pier. NS waves were input along the X-direction (in-plane direction), whereas the EW waves were input along the Y-direction (out-of-plane direction). The EW and NS ground motion inputs use the acceleration waves from the Niigata earthquake in Japan recorded by JMA (2004) [37]. The earthquake record is not related with the bridge design. Figure 13 shows the acceleration waves and response spectra of the EW and NS inputs, where a represents the acceleration and \(SA\) represents the acceleration response spectrum. \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}{*}{\(\xi_{\text{80}}\)} & \multicolumn{2}{c}{**Total**} \\  & & **Stiffness** & & & & **Hysteretic** \\ \hline \multirow{2}{*}{b2} & Fiber model & 0.967 & 1.248 & 2.00 & 2.708 & 24.89 \\ \cline{2-7}  & Hybrid model & 0.997 & 1.263 & 2.00 & 2.852 & 24.57 \\ \hline \multirow{2}{*}{b3} & Fiber model & 0.967 & 1.243 & 2.00 & 2.695 & 25.21 \\ \cline{2-7}  & Hybrid model & 1.099 & 1.241 & 2.00 & 2.510 & 23.66 \\ \hline \multirow{2}{*}{b4} & Fiber model & 0.962 & 1.259 & 2.50 & 3.21 & 31.62 \\ \cline{2-7}  & Hybrid model & 0.983 & 1.276 & 2.50 & 2.64 & 30.69 \\ \hline \multirow{2}{*}{b13\(-\)10} & Fiber model & 0.968 & 1.362 & 3.50 & 5.15 & 98.89 \\ \cline{2-7}  & Hybrid model & 0.997 & 1.410 & 4.00 & 4.93 & 96.26 \\ \hline \multirow{2}{*}{b3\(-\)d} & Fiber model & 0.973 & 1.042 & 1.50 & 1.77 & 10.86 \\ \cline{2-7}  & Hybrid model & 1.103 & 1.026 & 1.50 & 1.78 & 10.62 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of the out-of-plane calculation results. Figure 14 shows the calculation results of the displacement trajectory at the top of the eccentrically compressed steel piers. The parameters of these piers are shown in bold in Table 1. Taking the b3 pier as an example, the shear force-displacement curves in the in-plane (X) and out-of-plane (Y) directions are given in Figure 15. The calculation results of the horizontal bidirectional displacement trajectory and shear force-displacement curves obtained using the improved fiber model are basically consistent with those obtained using the hybrid-element calculation model. Under the action of horizontal bidirectional earthquakes, the overall deformation in the in-plane direction >> that in the out-of-plane direction for the eccentrically compressed steel piers. This trend is due to the eccentric compression causing the piers to move toward the eccentric side in the plane. Combined with Figure 11, it is evident that the deformation in the in-plane direction adversely affects the structural bearing capacity and stiffness in the out-of-plane direction, which deteriorates the seismic performance of the structures. Figure 12: Calculation models for the seismic response of eccentrically compressed steel piers: (**a**) hybrid-element calculation; (**b**) improved fiber models. Figure 13: Niigata earthquake record: (**a**) acceleration curves; (**b**) response spectrum.