# Stochastic Analysis of a Large-Span Continuous
## 3 Stochastic Seismic Response of the Large-Span CGHSR Bridge
### Structural Stochastic Seismic Response  
The structural seismic responses are analyzed using both the PEM and MCS method. The obtained results are cross-checked against each other. To utilize the MCS method, 500 samples of the fully non-stationary tri-directional seismic excitations are first generated with the fully non-stationary EPSD in Equation (46). Figure 7 shows one of the generated seismic excitation samples. Figure 6: Estimated modulated non-stationary ground acceleration model. Figure 7: \(Cont.\)  
The auto-/cross-correlation functions are estimated using the generated samples. Meanwhile, the estimated correlation functions are compared with the target ones to validate the reliability of the generated samples, as shown in Figure 8.  
Figure 8: _Cont_. Figure 7: A sample of the fully non-stationary tri-directional seismic excitations: (**a**) Seismic excitations in \(x\) direction; (**b**) Seismic excitations in \(y\) direction; (**c**) Seismic excitations in \(z\) direction. From Figure 8, it is seen that the estimated correlation functions generally agree well with the target ones, indicating the reliability of the generated fully non-stationary tri-directional seismic excitations. Then, the structural stochastic response (quasi-static and dynamic displacement) can be calculated using the MCS method in time domain. To facilitate the comparison of the structural stochastic response obtained using the MCS and PEM methods, the time-dependent response root mean square (RMS) \(\eta(t)\) is derived as [52]:  
\[\eta(t)=\sqrt{E\Big{[}\mathbf{X}^{2}(t)\Big{]}}=\sqrt{\frac{1}{2\pi}\int_{- \infty}^{+\infty}\mathbf{S}_{\mathbf{X}}(\omega,t)d\omega}, \tag{48}\]  
where the EPSD of the structural response in Equation (31) can be transformed into the time-dependent response RMS; the response RMS is utilized for the following illustrations of the structural stochastic analysis. Using the PEM and MCS method, the nodal absolute displacement RMS of the bridge girder is obtained, respectively, and shown in Figures 9 and 10. Figure 8: The estimated correlation functions with the target ones: (**a**) \(x\) direction (Auto-correlation); (**b**) \(x\) direction (Cross-correlation); (**c**) \(y\) direction (Auto-correlation); (**d**) \(y\) direction (Cross-correlation); (**e**) \(z\) direction (Auto-correlation); (**f**) \(z\) direction (Cross-correlation). Figure 9: Time-dependent absolute displacement RMS of the girder in \(z\) direction (Vertical): (**a**) MCS; (**b**) PEM (Unit: m). As shown in Figures 9 and 10, the time-dependent absolute displacement RMS obtained by the MCS method generally agree well with that by PEM, validating the reliability of the estimated results. It is evident that the \(z\) direction value is relatively larger than that in the \(y\) direction. It can also be observed that the nodal absolute displacement RMS is time-varying due to the non-stationary effects of the ground motion. Furthermore, the time-dependent absolute displacement RMS at the mid-span is shown in Figure 11.  
As shown in Figure 11, the maximum displacement RMS at the mid-span is 0.092 m in the \(z\) direction and decreases to 0.073 m in the \(y\) direction. It is easy to see that the maximum displacement RMS occurs at \(t=12.1\) s, which roughly coincides with the occurrence time of the maximum value in the fully non-stationary EPSD model in Figure 5. Furthermore, the displacement RMS has a noticeable increase after \(t=16.8\) s and reaches the second peak value, 0.078 m, in the \(z\) direction and 0.061 m in the \(y\) direction, respectively, at \(t=18.6\) s. This phenomenon is caused by the arrived surface waves from \(t=16\) s to 20 s in Figure 7:0 Note that the second peak values of the absolute displacement RMS are more than 80 percent of the maximum in both directions, indicating that the structural seismic response caused by the last arrived surface waves should be carefully considered.