## 4 Response Characteristics of Stabilizing Piles under Seismic Load
### Bending Moment  
The typical response time-history curve of bending moments in the centrifuge model test (EQ4, S3) is shown in Figure 14. The residual bending moment increment \(\Delta\)M\({}_{\rm r}\) in each shaking event is defined as the difference between the beginning and the end of the response time-history curve of the bending moment. During the seismic loading process, the maximum dynamic bending moment M\({}_{\rm max}\) and the minimum dynamic bending moment M\({}_{\rm min}\) in each shaking event are taken as the difference between the beginning and the peak and trough, respectively, of the time-history curve of the bending moment. Figure 15 shows the time-history curves of bending moments measured by strain gauges on a stabilizing pile under five continuous earthquake excitations (only 6 s of the time-history are taken in each case to show the variation in the bending moment). It can be seen from Figure 15 that the bending moment measured at different points on the stabilizing piles does not reach the maximum value at the same time. Taking EQ3 as an example (Figure 15c), the times corresponding to the peak signal amplitude are 109.91 s, 109.91 s, 109.91 s, 109.92 s, 109.93 s, 109.95 s, and 110.03 s (S1-S7), and the corresponding times to the wave trough are 109.35 s, 109.35 s, 109.36 s, 109.37 s, 109.38 s, 109.37 s, and 109.48 s (S1-S7) at each strain gauge. It can be seen that the bending moment on the stabilizing piles embedded in the bedrock (S1, S2, and S3) exhibits no phase difference,  
Figure 12: Pile top horizontal displacement under seismic excitations in the 2# model. Figure 13: The amplitude ratios of the transfer function of the acceleration measured at A-1079 (pile top) and A-0 (bottom of the model box) in the 2# model during centrifuge testing. while there is a phase difference in the bending moment on the pile above the bedrock (S4, S5, S6, and S7), especially near the top of the pile (S7) where the time difference is about 0.1 s. This is mainly due to the interaction between the stabilizing piles and the deposit, and the onset of the effect of viscous damping, which will be beneficial to the dissipation of earthquake energy. The bending moment corresponding to the characteristic time shown in Figure 15 is extracted. Meanwhile, the maximum and minimum bending moments generated during the earthquake (due to the phase difference, the bending moment at each measurement point on the stabilizing pile cannot reach a maximum or minimum at the same time, which is called the maximum or minimum bending moment envelope) and the residual bending moment after the earthquake areextracted. After deducting the initial static bending moment generated by centripetal acceleration before the earthquake, the bending moment diagram along the pile's height is plotted in Figure 16.  
Figure 14: Typical response time–history curves of bending moments in the centrifuge test (EQ4, S3). Figure 15: _Cont._  
It can be seen from Figure 16 that the dynamic bending moment corresponding to each characteristic time changes within the envelopes of the maximum and minimum dynamic bending moments. Therefore, the envelopes of maximum and minimum dynamic moments can be used as a basis to determine the load in the seismic design of a stabilizing pile. When the input seismic load is relatively small (EQ1, PGA = 0.055 g; EQ2, PGA = 0.125), the positive and negative dynamic bending moments of the pile are quasi-symmetrically distributed, and the residual bending moment after the earthquake is small (and even negligible in EQ1) (Figure 16a). Figure 15: Time–history curves of bending moments in centrifuge tests: (**a**) EQ1; (**b**) EQ2; (**c**) EQ3; (**d**) EQ4; (**e**) EQ5. Figure 16: Maximum and minimum bending moment envelopes, residual bending moment after an earthquake, and the dynamic bending moment on a stabilizing pile: (**a**) EQ1; (**b**) EQ2; (**c**) EQ3; (**d**) EQ4; (**e**) EQ5.