# Stochastic Analysis of a Large-Span Continuous
## 2 Analysis Methodology
### Stochastic Analysis Using the PEM  
As for the structural system with \(m\) supports, the cross-EPSD matrix of the fully non-stationary seismic excitations can be represented as:  
\[\mathbf{S}(\omega,t)= \begin{bmatrix}\mathbf{S}_{11}(\omega,t)&\mathbf{S}_{12 }(\omega,t)&\cdots&\mathbf{S}_{1m}(\omega,t)\\ \mathbf{S}_{21}(\omega,t)&\mathbf{S}_{22}(\omega,t)&\cdots&\mathbf{S}_{2m}( \omega,t)\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{S}_{m1}(\omega,t)&\mathbf{S}_{m2}(\omega,t)&\cdots&\mathbf{S}_{mm}( \omega,t)\end{bmatrix} , \tag{5}\]  
where \(\omega\) represents the circular frequency; \(\mathbf{S}_{ii}(\omega,t)\) is the auto-EPSD matrix at \(i\)th support; \(\mathbf{S}_{ij}(\omega,t)\)\((i,j=1,2,\cdots,m)\) is the cross-EPSD matrix between \(i\)th and \(j\)th supports and can be obtained by [33]:  
\[\mathbf{S}_{ij}(\omega,t)=\tau_{ij}(\omega)\sqrt{\mathbf{S}_{ii}(\omega,t) \mathbf{S}_{jj}(\omega,t)}, \tag{6}\]where \(\gamma_{ij}(\omega)\) denotes the coherence function between \(i\)th and \(j\)th spatial supports. The sub-matrix element \(\mathbf{S}_{ij}(\omega,t)\) is determined by two horizontal and a vertical component of the ground motion. \[\mathbf{S}_{jk}(\omega,t)= \begin{bmatrix}S_{xx}&S_{yy}&S_{xz}\\ S_{yx}&S_{xy}&S_{yz}\\ S_{xx}&S_{zy}&S_{zz}\end{bmatrix} _{jk}, \tag{7}\]  
As for tri-directional ground motions, the horizontal components \(S_{xx}(\omega,t)=S_{yy}(\omega,t)=S_{xy}(\omega,t)\) are assumed to be the same when there exists strong correlation between the horizontal ground motions. Here, \(S_{xz}(\omega,t)\) and \(S_{yz}(\omega,t)\) is given by [32]:  
\[S_{xz}(\omega,t)=S_{yz}(\omega,t)=0.6\sqrt{S_{xx}(\omega,t)S_{zz}(\omega,t)}, \tag{8}\]  
According to the Equations (5) and (6), the EPSD matrix can be rewritten as:  
\[\mathbf{S}(\omega,t)=\mathbf{D}(\omega,t)\mathbf{\Gamma}(\omega)\mathbf{D}^{T }(\omega,t), \tag{9}\]  
\[\mathbf{D}(\omega,t)=diag\Big{[}\sqrt{S_{kkx}(\omega,t)},\sqrt{S_{kky}(\omega,t)},\sqrt{S_{kkz}(\omega,t)}\Big{]}\ (k=1,2,\cdots,m), \tag{10}\]  
\[\mathbf{\Gamma}(\omega)= \begin{bmatrix}1&\gamma_{12}(\omega)&\cdots&\gamma_{1 m}(\omega)\\ \gamma_{21}(\omega)&1&\cdots&\gamma_{2m}(\omega)\\ \vdots&\vdots&\ddots&\vdots\\ \gamma_{m1}(\omega)&\gamma_{m2}(\omega)&\cdots&1\end{bmatrix} , \tag{11}\]  
The elements in \(\mathbf{\Gamma}(\omega)\) can be represented by the product of the lagged coherence part \(\left|\gamma_{ij}(\omega)\right|\) and the wave passage effect \(\exp\left[i\omega d_{ij}/\gamma_{app}\right]\), \(d_{ij}\) represents the spatial distance between \(i\)th and \(j\)th supports and \(v_{app}\) is the apparent wave velocity and taken as 50 m/s in this work [34]. Thus, the coherence matrix can be represented by:  
\[\mathbf{\Gamma}(\omega)=\left|\mathbf{\Gamma}(\omega)\right|\otimes\Theta( \omega), \tag{12}\]  
where \(\left|\mathbf{\Gamma}(\omega)\right|\) is the lagged coherence matrix; \(\Theta(\omega)\) is the coherence phase matrix; \(\otimes\) denotes the element-wise production. Taking the first support along the direction of propagation as the reference node, the time delay of the motion at \(j\)th support to the reference node can be represented as \(T_{j}=\left(d_{j}-d_{1}\right)/v_{app}\). Then, the phase angle caused by the wave passage effect can be denoted as \(\theta_{j}(\omega)=-T_{j}\omega\)[27]. As the lagged coherence matrix is the diagonally dominant non-negative definite matrix, the eigenvalue decomposition (EVD) of the \(\left|\mathbf{\Gamma}(\omega)\right|\) can be shown as:  
\[\left|\mathbf{\Gamma}(\omega)\right|=\sum_{l=1}^{r}\lambda_{l}\{\beta(\omega) \}_{l}\{\beta(\omega)\}_{l}^{T}, \tag{13}\]  
where \(r\) is the rank of the lagged coherence matrix; \(\lambda_{l}\) represents the \(l\)th eigenvalue; \(\{\beta(\omega)\}_{l}\) represents the \(l\)th normalized eigenvector.