## 3 The MSRS Method under Incoherent Ground Excitations
### Solution of Equation (10)  
As mentioned above, \(z(t)\) denotes a structural seismic response, such as nodal displacement or internal force in members. Decomposing the structural absolute displacement, \(\mathbf{X}\), into the pseudo-static displacement, \(\mathbf{X}^{s}\), and the dynamic displacement, \(\mathbf{X}^{d}\), the structural seismic response can be written as Equation (11)  
\[z(t)=\mathbf{q}^{T}\mathbf{X}=\mathbf{q}^{T}\Big{(}\mathbf{X}^{s}+\mathbf{X}^{ d}\Big{)}. \tag{11}\]  
By using the mode decomposition method, the mean value and mean square deviation of the structural peak response, \(\mu_{\parallel,\max|}\) and \(\sigma_{\parallel,\max|}\), can be calculated as Equations (12) and (13)  
\[\begin{array  
and the ground acceleration, \(\ddot{x}_{gl}\) at support \(l\); \(\rho_{\ddot{x}_{gk}\ddot{x}_{gl}}\) denotes the cross-correlation coefficient between the ground accelerations, \(\ddot{x}_{gk}\) and \(\ddot{x}_{gl}\) at supports, \(k\) and \(l\); \(\sigma_{x_{gk}}\) and \(\sigma_{x_{gl}}\) represent the mean square deviations of displacements at supports, \(k\) and \(l\); \(S_{x_{gk}x_{gl}}(i\omega)\) is the cross-power spectral density function between the ground displacements \(x_{gk}\) at support \(k\), and \(x_{gl}\) at support \(l\) for the \(i\)th mode; \(\sigma_{\ddot{x}_{gk}}\) and \(\sigma_{\ddot{x}_{gj}}\) are, respectively, the mean square deviations of displacements of the ground accelerations, \(\ddot{x}_{gk}\) at support \(k\) and, \(\ddot{x}_{gl}\) at support \(l\); \(H_{i}(i\omega)\) is the complex frequency response function for \(i\)th mode and \(H_{j}(-i\omega)\) is the conjugate one; \(S_{x_{gk}\ddot{x}_{gl}}(i\omega)\) is the cross-power spectral density function between the ground displacements, \(x_{gk}\) at support \(k\) and the accelerations, \(\ddot{x}_{gl}\) at support \(l\); \(S_{\ddot{x}_{gk}\ddot{x}_{gl}}\) is the cross-power spectral density function of the accelerations between \(\ddot{x}_{gk}\) at support \(k\) and at the \(\ddot{x}_{gl}\) support; \(\sigma_{x_{gk}}\) and \(\sigma_{\ddot{x}_{gk}}\) can be obtained by Equation (17)  
\[\sigma_{x_{gl}}^{2}=\int_{-\infty}^{\infty}S_{x_{gk}x_{gl}}(\omega)d\omega, \ \sigma_{\ddot{x}_{gk}}^{2}=\int_{-\infty}^{\infty}\bigl{|}H(i\omega)\bigr{|}^{2} S_{x_{gk}\ddot{x}_{gk}}(\omega)d\omega. \tag{17}\]  
And the coefficients, \(a_{k}\) and \(b_{jk}\) are expressed by Equation (18),  
\[a_{k}=\mathbf{q}^{T}\mathbf{r}_{k},\ b_{jk}=\mathbf{q}^{T}\mathbf{\varphi}_{j }\beta_{jk}k=1,2,\cdots,m;j=1,2,\cdots,n, \tag{18}\]  
where \(\mathbf{q}\) is the transfer vector, which can transform the displacement response into other structural responses, such as the bending moment, shear force, and so on; \(\mathbf{\varphi}_{j}\) is the column vector of the \(j\)th mode; \(\mathbf{r}_{k}\) is the \(k\)th column in the pseudo-static influence matrix, \(\mathbf{R}=-\mathbf{K}^{-1}\mathbf{K}_{c}\); \(\beta_{jk}\) denotes the \(j\)th mode participation factor under the ground motion acceleration at support \(k\), which can be calculated by the following formula  
\[\beta_{jk}=\frac{-\mathbf{\varphi}_{j}^{T}\mathbf{M}\mathbf{r}_{k}}{\mathbf{ \varphi}_{j}^{T}\mathbf{M}\mathbf{\varphi}_{j}}k=1,2,\cdots m;j=1,2,\cdots n. \tag{19}\]