# Stochastic Analysis of a Large-Span Continuous
## 2 Analysis Methodology
### Stochastic Analysis Using the MCS Method  
To obtain the statistical moments and the mean peak values of the structural responses, a sufficient number of ground motions should be generated, first using the specific fully non-stationary EPSD model and the coherence function through numerical simulation [19]. Once the ground motion samples are obtained, the structural responses in Equations (3) and (4) can be obtained using the numerical integral method. The spectral representation method (SRM) is widely used in engineering practice due to its accuracy and simplicity and is therefore adopted here [39; 40]. Several efforts have been made to generate a multivariate non-stationary process via FFT [39; 41; 42; 43; 44]. Here, the POD-based method proposed by [PERSON] [41] is adopted to generate the fully non-stationary seismic excitation due to its performance in terms of the accuracy, easy implementation and fewer matching items. Consider the ground motion characterized by the cross-EPSD in Equation (5) with the coherence matrix shown in Equation (11), then the cross-EPSD matrix can be rewritten as Equation (9). As the \(\mathbf{\Gamma}(\omega)\) is the Hermitian matrix with non-negative definite property, the coherence matrix can be decomposed as [42]:  
\[\mathbf{\Gamma}(\omega)=\mathbf{U}(\omega)\mathbf{U}^{T*}(\omega), \tag{32}\] \begin{table}
\begin{tabular}{l} \hline \hline Algorithm: determine the pseudo static displacement under \(l\)th pseudo excitation \\ \hline
**for**\(p=1:N_{p}\) \\
1. Calculating \(\Lambda_{p}(t)\) and \(\Psi_{p}(\omega)\) using POD. \\
2. Conducting curve fitting of \(\Lambda_{p}(t)\) with Equation (28). \\
**for**\(q=1:N_{q}\) \\
3. Solving the analytical solution \(Ans_{q}(t)\) with Equation (30). \\
4. Conducting multiplication of \(Ans_{q}(t)\) with remainder in Equation (29). \\
**end** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The POD-based pseudo static displacement determination framework. Figure 1: The calculation flow chart using the PEM. \[\mathbf{U}(\omega)= \begin{bmatrix}u_{11}(\omega)&0&\cdots&0\\ u_{21}(\omega)&u_{22}(\omega)&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ u_{n1}(\omega)&u_{n2}(\omega)&\cdots&u_{nn}(\omega)\end{bmatrix} , \tag{33}\]  
According to the cross-EPSD and coherence matrices, the cross-correlation matrix \(R(t,t+\tau)\) can be given by [45]:  
\[R(t,t+\tau)= \begin{bmatrix}R_{11}(t,t+\tau)&R_{12}(t,t+\tau)&\cdots&R_{1n}(t,t+ \tau)\\ R_{21}(t,t+\tau)&R_{22}(t,t+\tau)&\cdots&R_{2n}(t,t+\tau)\\ \vdots&\vdots&\ddots&\vdots\\ R_{n1}(t,t+\tau)&R_{n2}(t,t+\tau)&\cdots&R_{nn}(t,t+\tau)\end{bmatrix} , \tag{34}\]  
\[R_{ij}(t,t+\tau)=\int_{-\infty}^{+\infty}\sqrt{S_{ii}(\omega,t)S_{jj}(\omega,t+\tau)}\gamma_{ij}(\omega)e^{i\omega\tau}d\omega, \tag{35}\]  
where \(\tau\) is the time lag. Taking only the correlation in spatial distributions of the ground motion into account, the fully non-stationary seismic excitation can therefore be divided into three independent 1D components in tri-directions. With the aid of the POD, the EPSD matrix can be factorized as [40]:  
\[\sqrt{S_{kk}(\omega,t)}\approx\sum_{c=1}^{N_{c}^{kk}}\mu_{c}^{kk}(t)\phi_{c}^{ kk}(\omega), \tag{36}\]  
The final expression of the generated samples for structural stochastic analysis using MCS method can be give as [40]:  
\[y_{k}(t)=\text{Re}\left[2\sqrt{\Delta\omega}\sum_{c=1}^{N_{c}^{kk}}\mu_{c}^{kk} (t)\sum_{l=1}^{N}\phi_{c}^{kk}(\omega_{l})e^{-i\omega_{l}t}\sum_{p=1}^{k}u_{kp }(\omega_{l})e^{-i\varphi_{nl}}\right], \tag{37}\]  
where \(\Delta\omega=\omega_{z}/N\) is the frequency resolution; \(\omega_{z}\) represents the cutoff frequency; \(N\) denotes the number of discretized frequencies; \(\omega_{l}=l\Delta\omega\); \(\varphi_{pl}\) are random phase angles subject to uniform distribution on \([0,2\pi]\). Based on the generated acceleration samples, the dynamic displacement \(\mathbf{X}_{s}^{d}\) in Equation (4) can be solved using numerical integral methods.