# Stochastic Analysis of a Large-Span Continuous
## 2 Analysis Methodology
### Stochastic Analysis Using the PEM
Thus, the pseudo static displacement \(\left\{\widetilde{\mathbf{X}}_{s}^{s}\right\}_{l}\) under the \(l\)th pseudo excitation can be simplified as [27]:  
\[\left\{\widetilde{\mathbf{X}}_{s}^{s}\right\}_{l}=-\frac{1}{\omega^{2}} \mathbf{R}_{s}\sqrt{\lambda_{l}}\mathbf{D}(\omega,t)\mathbf{V}(\omega)\left\{ \beta(\omega)\right\}_{l}e^{i\omega t}, \tag{25}\]  
However, as for fully non-stationary seismic excitation, the frequency content has obvious time-varying characteristics. The pseudo static displacement \(\left\{\widetilde{\mathbf{X}}_{s}^{s}\right\}_{l}\) should be reconsidered and calculated with direct integral using Equation (20):  
\[\left\{\widetilde{\mathbf{X}}_{s}^{s}\right\}_{l}=\mathbf{R}_{s}\int_{0}^{t} \int_{0}^{\tau_{1}}\sqrt{\lambda_{l}}\mathbf{D}(\omega,\tau)\mathbf{V}(\omega )\left\{\beta(\omega)\right\}_{l}e^{i\omega\tau}d\tau d\tau_{1}, \tag{26}\]  
With the aid of POD, the non-separable matrix \(\mathbf{D}(\omega,t)\) can be decomposed as [38]:  
\[\mathbf{D}(\omega,t)=\sum_{p=1}^{N_{p}}\Lambda_{p}(t)\Psi_{p}(\omega), \tag{27}\]  
where \(\Lambda_{p}(t)\) is the real time function and \(\Psi_{p}(\omega)\) represents the orthonormal bases. To give the analytical solution of Equation (26), the following sum of sine function is considered to fit the time function \(\Lambda_{p}(t)\):  
\[\Lambda_{p}(t)\approx\sum_{q=1}^{N_{q}}a_{q}\sin\Bigl{(}b_{q}t+c_{q}\Bigr{)}, \tag{28}\]  
Substituting Equations (27) and (28) to Equation (26), the pseudo static displacement \(\left\{\widetilde{\mathbf{X}}_{s}^{s}\right\}_{l}\) can be represented as:  
\[\left\{\widetilde{\mathbf{X}}_{s}^{s}\right\}_{l}=\sqrt{\lambda_{l}}\mathbf{R }_{s}\mathbf{V}(\omega)\left\{\beta(\omega)\right\}_{l}\sum_{p=1}^{N_{p}}\sum _{q=1}^{N_{q}}a_{q}Ans_{q}(t)\Psi_{p}(\omega), \tag{29}\]  
\[ \begin{array}{l}Ans_{q}(t)=\int_{0}^{t}\int_{0}^{\tau_{1}}\sin\Bigl{(}b_{q} \tau+c_{q}\Bigr{)}e^{i\omega\tau}d\tau d\tau_{1}\\ =\frac{1}{\left(b_{q}^{2}-\omega^{2}\right)}\left(b_{q}\Bigl{(}b_{q}^{2}t+ \omega(2i-\omega t)\Bigr{)}\cos\Bigl{(}c_{q}\Bigr{)}+\Bigl{(}b_{q}^{2}-ib_{q}^ {2}\omega t+\omega^{2}+i\omega^{3}t\Bigr{)}\sin\Bigl{(}c_{q}\Bigr{)}-\\ e^{i\omega t}(2 ib_{q}\omega\cos\Bigl{(}c_{q}+b_{q}t\Bigr{)}+\Bigl{(}b_{q}^{2}+ \omega^{2}\Bigr{)}\sin\Bigl{(}c_{q}+b_{q}t\Bigr{)}))\end{array}  \tag{30}\]  
As the integral \(Ans_{q}(t)\) in Equation (29) has the analytical solution, the pseudo static displacement \(\left\{\widetilde{\mathbf{X}}_{s}^{s}\right\}_{l}\) for fully non-stationary seismic excitation can also have a similar form as Equation (25). Note that \(N_{p}=3\) and \(N_{q}=8\) is selected here to reach a compromise between the computational efficiency and the accuracy. This POD-based pseudo static displacement determination framework is summarized in Table 1.  
Finally, the pseudo nodal absolute displacement \(\left\{\widetilde{\mathbf{X}}_{\text{s}}\right\}_{l}\) under the \(l\)th pseudo excitation can be represented as the summation of the \(\left\{\widetilde{\mathbf{X}}_{\text{s}}^{s}\right\}_{l}\) and \(\left\{\widetilde{\mathbf{X}}_{\text{s}}^{d}\right\}_{l}\). Then, the EPSD of the nodal absolute displacement \(\mathbf{X}_{\text{s}}\) can be calculated by:  
\[\mathbf{S}_{\mathbf{X}}(\omega,t)=\sum_{l=1}^{r}\left\{\widetilde{\mathbf{X}}_ {\text{s}}^{*}(\omega,t)\right\}_{l}\left\{\widetilde{\mathbf{X}}_{\text{s}}^{ T}(\omega,t)\right\}_{l}, \tag{31}\]  
The flow chart of the stochastic analysis of the CGHSR bridge under fully non-stationary seismic excitation using PEM is shown in Figure 1.