# A Simplified Analysis Method for Seismic Response of
## 4 Example Verification
### Comparison of Shear Stress Distribution at Pile Top of Pile Group  
Figure 7 is the envelope diagram of the peak shear stress at the pile top of the pile group, in which the time history analysis results are taken from the peak value on the time history curve of the shear stress at the top of a single foundation pile. Similar to the conclusion in Section 4.2, the time history analysis results of different seismic wave inputs are pretty different from the results of the EBSM-RAM in some cases. However, when the average action results of three seismic inputs (Figure 7d) are taken, the results of quasi-static calculation and time history analysis are very close.  
The results for the peak value of shear stress in the pile group of the quasi-static calculation and the time history analysis are compared in Table 3. It can be seen that with the increase in the reduction coefficient, the quasi-static calculation results gradually increase, and the gap with the time history analysis results gradually narrows until it grows in the opposite direction. When considering the average action of the three seismic waves, the quasi-static result corresponding to the reduction factor of 0.35 is the closest to the peak value of the time history analysis, with an error of only 3.41%; when it is 0.4, the quasi-static result exceeds the peak value of the time history analysis. It shows that the  
Figure 6: Comparison of shear stress along the pile length between central pile and angle pile: (**a**) artificial earthquake; (**b**) EI earthquake; (**c**) Northridge earthquake; (**d**) avg.  
optimal reduction coefficient that can make the quasi-static calculation results and the peak value results of time history analysis most consistent is between 0.35 and 0.4.  
Table 4 shows the envelope relationship between the quasi-static calculation results of all piles under the action of the average value of the three seismic waves and the peak results of the time history analysis. When the reduction factor is 0.25, the quasi-static pile top shear stress of 97% of the piles does not cover the peak results of the time history analysis. With the increase of the reduction factor, until it reaches 0.4, the probability of no envelope is only 11%, and the maximum error of the shear stress at the top of the pile without envelope is 9%. It can be considered that when the reduction factor is set to 0.4, the obtained quasi-static results envelop the peak results of the time history analysis.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Reduction Factor** & **0.25** & **0.3** & **0.35** & **0.4** \\ \hline A & 97\% & 86\% & 53\% & 11\% \\ B & \(-\)40\% & \(-\)30\% & \(-\)19\% & \(-\)9\% \\ \hline \end{tabular} Note: A represents the probability that for all the peak shear stresses at the top of the pile, the quasi-static calculation results cannot envelop the time history analysis results; B represents the maximum error percentage of the part of the peak shear stress that is not enveloped in A.  
\end{table}
Table 4: Quasi-static results are less than the peak value of time history analysis probability and the percentage of error between them.  
Figure 7: _Cont._  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Reduction Factor** & **Artificial Seismic Wave** & **Northridge** & **EI** & **avg** \\ \hline
0.25 & 51.48\% & 21.86\% & 15.04\% & 29.91\% \\
0.3 & 42.39\% & \(-\)2.22\% & \(-\)0.97\% & 16.71\% \\
0.35 & 33.22\% & \(-\)7.43\% & \(-\)17.11\% & 3.41\% \\
0.4 & 24.00\% & \(-\)22.22\% & \(-\)33.33\% & \(-\)9.95\% \\ \hline \end{tabular} Note: Error percentage = (time history analysis result\(-\)quasi-static result)/time history analysis result.  
\end{table}
Table 3: The percentage error of maximum shear stress of the pile top was calculated by the EBSM-RAM and the time history analysis method.  
Figure 7: Comparison of shear stress envelope of pile top of pile group: (**a**) artificial earthquake; (**b**) EI earthquake; (**c**) Northridge earthquake; (**d**) avg.