# Stochastic Analysis of a Large-Span Continuous
## 3 Stochastic Seismic Response of the Large-Span CGHSR Bridge
### Fully Non-Stationary Ground Acceleration Model  
Consider the ground motion \(\tilde{U}_{b}(t)\), defined as the following sigma-oscillatory Gaussian process [10]:  
\[\tilde{U}_{b}(t)=\sum_{p=1}^{q}X_{p}(t)=\sum_{p=1}^{q}A_{p}(t)Y_{p}(t), \tag{42}\]  
where the component processes \(\Big{[}X_{p}(t),p=1,2,\cdots,q\Big{]}\) are independent uniformly modulated processes with zero mean; \(A_{p}(t)\) is the deterministic time-modulating function of the \(p\)th component process admitting the following representation [10]:  
\[A_{p}(t)=\alpha_{p}\Big{(}t-\zeta_{p}\Big{)}^{\beta_{p}}e^{-\gamma_{p}(t-\zeta _{p})}H\Big{(}t-\zeta_{p}\Big{)}, \tag{43}\]  
where \(\alpha_{p}\) and \(\gamma_{p}\) are positive constants; \(\beta_{p}\) is the positive integer; \(\zeta_{p}\) is the \"arrival time\" of the \(p\)th sub-process; and \(H(t)\) is the unit step function. The auto-correlation function of the \(p\)th zero-mean stationary Gaussian process \(Y_{p}(t)\) is defined as [10]:  
\[R_{Y_{p}}(\tau)=e^{-\
u_{p}|\tau|}\cos\Big{(}\eta_{p}\tau\Big{)}, \tag{44}\]  
and its power spectral density (PSD) function is:  
\[S_{Y_{p}}(\omega)=\frac{\
u_{p}}{2\pi}\left[\frac{1}{\
u_{p}^{2}+\Big{(} \omega+\eta_{p}\Big{)}^{2}}+\frac{1}{\
u_{p}^{2}+\Big{(}\omega-\eta_{p}\Big{)} ^{2}}\right], \tag{45}\]  
where \(\
u_{p}\) is the frequency bandwidth; \(\eta_{p}\) is the predominant (or central) frequency of the process. From Equation (42), the EPSD function of \(\tilde{U}_{b}(t)\) is:  
\[S_{\tilde{U}_{b}}(t,\omega)=\sum_{p=1}^{q}\Big{|}A_{p}(t)\Big{|}^{2}S_{Y_{p}} (\omega), \tag{46}\]The parameters of the sigma-oscillatory process model are shown in Table 3. Figure 5 shows the EPSD of the fully non-stationary seismic excitation using the parameters in Table 3.  
As shown in Figure 5, the frequency content shifts towards the lower range as time elapses, indicating that the surface waves arriving last have a lower frequency than the push and shear waves, which arrive much earlier. Equation (46) reproduces both the intensity and frequency non-stationary characteristics of the real earthquake record. To analyze the structural responses under the seismic excitation that only considering the non-stationarity of the intensity, an equivalent modulated non-stationary ground acceleration model, corresponding to the fully non-stationary model, can be derived following the energy conservation principle. The modulated non-stationary ground acceleration model is represented by:  
\[S_{U_{b}}^{m}=\alpha f^{2}(t)S^{*}(\omega)=\alpha S^{\prime}(\omega,t), \tag{47}\]  
where \(\alpha\) is the parameter that ensures the energy conservation; \(f(t)\) and \(S^{*}(\omega)\) are the time-modulating function and PSD function, which can be obtained with the following steps:  
Step I: Using the MCS method to generate a large number of ground acceleration samples, according to the fully non-stationary ground acceleration model in Equation (46). Here, 500 samples are generated using the MCS method. Step II: As for each generated sample, estimate its time-modulating function using the Kernel regression method and the PSD using the Pwelch method. Step III: Calculating the ensemble average of the estimated time-modulating functions and the PSDs, which represent the \(f(t)\) and \(S^{*}(\omega)\), respectively. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(p\) & \(\alpha_{P}\) & \(\beta_{P}\) & \(\gamma_{P}\) & \(\xi_{P}\) & \(v_{P}\) & \(\eta_{P}\) \\ \hline
1 & 0.2358 & 7 & 1.3375 & 0.8672 & 1.2618 & 3.1022 \\
2 & 0.1394 & 8 & 1  
Step IV: Conducting double integration of \(S^{\prime}(\omega,t)\) in Equation (47) and the EPSD in Equation (46), which represent the total energy of the ground acceleration. Then, determine \(\alpha\) by following the energy conservation principle. The total energy of the analytical EPSD of the fully non-stationary ground acceleration is 75,281 cm\({}^{2}\)/s\({}^{2}\), while the total energy is 33,133 cm\({}^{2}\)/s\({}^{2}\) for \(S^{\prime}(\omega,t)\).