# Stochastic Analysis of a Large-Span Continuous
## 2 Analysis Methodology
### Stochastic Analysis Using the PEM
Denote \(\mathbf{V}(\omega)\) as [27]:  
\[\mathbf{V}(\omega)=diag\Big{[}e^{i\theta_{1x}},e^{i\theta_{1y}},e^{i\theta_{1z }},e^{i\theta_{2x}},e^{i\theta_{2y}},e^{i\theta_{2z}},\cdots,e^{i\theta_{mx}},e^{i\theta_{my}},e^{i\theta_{mz}}\Big{]}, \tag{14}\]  
According to Equation (12), the coherence matrix can be rewritten as:  
\[\mathbf{\Gamma}(\omega)=\sum_{l=1}^{r}\lambda_{l}\mathbf{V}^{*}(\omega)\{ \beta(\omega)\}_{l}\{\beta(\omega)\}_{l}^{T}\mathbf{V}^{T}(\omega), \tag{15}\]Thus, the Equation (9) can be rewritten as:  
\[\mathbf{S}(\omega,t)=\sum_{l=1}^{r}\lambda_{l}\mathbf{D}(\omega,t)\mathbf{V}^{*}( \omega)\{\beta(\omega)\}_{l}\{\beta(\omega)\}_{l}^{T}\mathbf{V}^{T}(\omega) \mathbf{D}^{T}(\omega,t), \tag{16}\]  
The Hariichandran-Vanmarche model is selected as the lagged coherence function here [35]:  
\[\left|\gamma_{jk}(\omega)\right|=a\exp\left[-\frac{2d_{jk}}{a\rho(\omega)}(1-a +\alpha a)\right]+(1-a)\exp\left[-\frac{2d_{jk}}{\rho(\omega)}(1-a+\alpha a) \right], \tag{17}\]  
\[\rho(\omega)=K\left[1+(\omega/\omega_{0})^{b}\right]^{-1/2}, \tag{18}\]  
where \(a=0.736\), \(\alpha=0.147\), \(K=5210\), \(\omega_{0}=6.845\) and \(b=2.78\) are the empirical model parameters given in [26; 35], which are adopted here just as an example. To solve Equation (4), the ground motion \(\left\{\widetilde{\mathbf{X}}_{b}\right\}\) at the right-hand side can be replaced by the following pseudo excitation:  
\[\widetilde{\mathbf{F}}(\omega,t)=\sqrt{\mathbf{S}(\omega,t)}e^{i\omega t}=\sum _{l=1}^{r}\sqrt{\lambda_{l}}\mathbf{D}(\omega,t)\mathbf{V}(\omega)\{\beta( \omega)\}_{l}e^{i\omega t}, \tag{19}\]  
where the \(l\)th pseudo excitation vector can be represented by:  
\[\widetilde{\mathbf{F}}_{l}(\omega,t)=\sqrt{\lambda_{l}}\mathbf{D}(\omega,t) \mathbf{V}(\omega)\{\beta(\omega)\}_{l}e^{i\omega t}, \tag{20}\]  
To ease the usage of the PEM, Equation (4) can be decomposed and rewritten as state-space representation under \(l\)th pseudo excitation:  
\[\dot{\mathbf{Z}}=\mathbf{H}\mathbf{Z}+\mathbf{Q}(t)= \begin{bmatrix}0&\mathbf{ I}\\ -\mathbf{\lambda}^{2}&\widetilde{\mathbf{C}}\end{bmatrix} \mathbf{Z}+ \begin{bmatrix} 0\\ -\mathbf{\Phi}^{T}\mathbf{M}_{\xi}\mathbf{R}_{\xi}\widetilde{\mathbf{F}}_{l}( \omega,t)\end{bmatrix} , \tag{21}\]  
\[\text{where}\,\mathbf{Z}=\left\{\widetilde{\mathbf{X}}_{s}^{d},\widetilde{ \mathbf{X}}_{s}^{d}\right\}_{l}^{T};\lambda=diag\big{(}\omega_{1},\omega_{2}, \cdots,\omega_{n}\big{)}\text{ and }\widetilde{\mathbf{C}}=diag\big{[}-2 \xi_{1}^{2}\omega_{1},-2\xi_{2}\omega_{2},\cdots,-2\xi_{n}^{2}\omega_{n} \big{]};\]  
\(\omega_{i}\) and \(\xi_{i}\) represent the \(i\)th modal frequency and damping ratio, respectively; \(\mathbf{\Phi}\) represents the modal matrix. The numerical solution of Equation (21) is:  
\[\mathbf{Z}\Big{(}t_{\widetilde{k}+1}\Big{)}=\exp(\mathbf{H}\Delta t)\mathbf{Z }\Big{(}t_{\widetilde{k}}\Big{)}+\int_{t_{\widetilde{k}}}^{t_{\widetilde{k}+1 }}\exp\Big{(}\mathbf{H}\Big{(}t_{\widetilde{k}+1}-\tau\Big{)}\Big{)}\widetilde {\mathbf{F}}_{l}(\omega,\tau)d\tau, \tag{22}\]  
In Equation (22), the first part, at the right-hand side, can be solved by the precise integration method, while the second part is solved by second-order Newton-Cotes integration methods to improve the accuracy [36; 37]. \[\begin{array}{l}\int_{t_{\widetilde{k}+1}}^{t_{\widetilde{k}+1}}\exp\Big{(} \mathbf{H}\Big{(}t_{\widetilde{k}+1}-\tau\Big{)  
Consider that the ground motion is determined by the following modulated EPSD [27]:  
\[\mathbf{S}(\omega,t)=g(t)\mathbf{S}(\omega), \tag{24}\]  
where \(g(t)\) is a slowly varying modulation function; \(\mathbf{S}(\omega)\) is the power spectral density of a stationary stochastic process.