## 2 Seismic Response Calculation Method for Soil Layers Based on the Modal Superposition Method
### Required Modes for Acceleration Response Analysis  
Equation governing the motion of a linear single DoF system subjected to the ground acceleration \(\ddot{u}_{g}(t)\) can be expressed as  
\[\ddot{d}_{i}+2\zeta_{i}\omega_{i}d_{i}+\omega_{i}^{2}d_{i}=-\ddot{u}_{g}(t) \tag{8}\]  
Figure 1: Two-dimensional irregular site.  
The contribution of the \(i\)-th mode to the displacement is  
\[\{u_{i}(t)\}=\{\phi_{i}\}q_{i}(t)=\gamma_{i}\{\phi_{i}\}d_{i}(t) \tag{9}\]  
The equivalent static force for the \(i\)-th mode is  
\[\{f_{i}(t)\}=[K]\{u_{i}(t)\} \tag{10}\]  
Substituting Equations (3) and (9) into Equation (10) gives  
\[\{f_{i}(t)\}=\gamma_{i}[M]\{\phi_{i}\}\Big{[}\omega_{i}^{2}d(t)\Big{]} \tag{11}\]  
The contribution of the \(i\)-th mode to the structural displacement under the excitation of static force \(\{f_{i}(t)\}\) is  
\[\{u_{i}\}=\left\{u_{i}^{st}\right\}\Big{[}\omega_{i}^{2}d(t)\Big{]} \tag{12}\]  
The peak values of the contribution of the \(i\)-th mode to the structural displacement and acceleration are  
\[\{u_{i,max}\}=\{u_{i}^{st}\}S_{a}(\zeta_{i},\omega_{i}) \tag{13}\]  
\[\{\bar{u}_{i,max}\}=\{A_{i}^{st}\}S_{a}(\zeta_{i},\omega_{i}) \tag{14}\]  
\[\left\{u_{i}^{st}\right\}=\gamma_{i}\{\phi_{i}\}/\omega_{i}^{2} \tag{15}\]  
\[\left\{A_{i}^{st}\right\}=\omega_{i}^{2}\{u_{i}^{st}\}=\gamma_{i}\{\phi_{i}\} \tag{16}\]  
where \(\{u_{i}^{st}\}\) is the static displacement response of \(i\)-th mode, and \(S_{a}(\zeta_{i},\omega_{i})\) is the pseudo-acceleration response spectrum value of the \(i\)-th mode.  
[PERSON] [25] proposed that the modal contribution factor \(\overline{r}_{ji}^{st}\) based on \(\left\{u_{i}^{st}\right\}\) can be expressed as  
\[\overline{r}_{ji}^{st}=u_{ji}^{st}/u_{j}^{st} \tag{17}\]  
where \(u_{j}^{st}\) is the static displacement response for the \(j\)-th degree of freedom, and \(u_{ji}^{st}\) is the static displacement response for the \(j\)-th degree of freedom of the \(i\)-th mode.  
The modal contribution factor is based on displacement response. Comparing Equations (15) and (16), it can be seen that \(\left\{A_{i}^{st}\right\}\) is \(\{u_{i}^{st}\}\) multiplied by \(\omega_{i}\)2. Therefore, the contribution of higher modes to acceleration will be greater than that to displacement. To estimate the structural acceleration response error caused by modal truncation, the mode contribution factor of acceleration for the \(j\)-th degree of freedom of the \(i\)-th mode is proposed and can be expressed as  
\[a_{ji}=\gamma_{i}\phi_{ji} \tag{18}\]  
Since \(\sum\limits_{i=1}^{n}\gamma_{i}\{\phi_{i}\}=\{I\}\), the degrees of freedom in the direction of the seismic input are  
\[\sum\limits_{i=1}^{n}a_{ji}=1 \tag{19}\]  
The error of cumulative mode contribution factor of acceleration is defined as  
\[e_{aj}=1-\sum\limits_{i=1}^{N}a_{ji} \tag{20}\]  
Typically, finite element software can extract modal participation factors and mode shapes. Therefore, the mode contribution factor of acceleration can be readily obtained through a simple calculation using these two quantities. As comparison, the error of cumulative modal participation mass \(e_{m}\) and the error of cumulative modal contribution factor \(e_{uj}\) are expressed as:  
\[e_{m}=1-\sum_{i=1}^{N}\gamma_{i}^{2}/m_{t} \tag{21}\]  
\[e_{uj}=1-\sum_{i=1}^{N}\overline{\tau_{ji}^{st}} \tag{22}\]  
where \(m_{t}\) is total mass of the system.