# Error Caused by Damping Formulating in Multiple Support Excitation Problems
## 5 Numerical Simulation
### Long Span Cable-Stayed Bridge
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Forces** & **Symbol** & \(\mathbf{e^{Rx}}\) **(\%)** & \(\mathbf{e^{LLr}}\) **(\%)** & \(\mathbf{e^{Am}}\) **(\%)** & \(\mathbf{e^{Ar}}\) **(\%)** \\ \hline \multicolumn{6}{c}{**Pier 2**} \\ \hline Base shear & \(F_{b,2}\) & \(-3.64\) & \(-4.05\) & \(-8.63\) & \(473.91\) \\ Shear at deck level & \(F_{d,2}\) & \(-9.09\) & \(-8.88\) & \(-0.07\) & \(-9.15\) \\ Base Moment & \(M_{b,2}\) & \(-1.44\) & \(-2.14\) & \(-4.54\) & \(-4.40\) \\ Moment at deck level & \(M_{d,2}\) & \(-3.31\) & \(-3.08\) & \(-0.12\) & \(-3.22\) \\ \hline \multicolumn{6}{c}{**Pier 3**} \\ \hline Base shear & \(F_{b,3}\) & \(-3.80\) & \(-3.01\) & \(18.35\) & \(503.38\) \\ Shear at deck level & \(F_{d,3}\) & \(-16.15\) & \(-15.93\) & \(0.91\) & \(-16.23\) \\ Base Moment & \(M_{b,3}\) & \(-1.66\) & \(-1.04\) & \(5.21\) & \(56.43\) \\ Moment at deck level & \(M_{d,3}\) & \(-5.16\) & \(-4.68\) & \(1.46\) & \(-4.83\) \\ \hline \multicolumn{6}{c}{**Cable forces**} \\ \hline Largest Cable force & \(F_{c}\) & \(-3.42\) & \(-3.64\) & \(-0.28\) & \(-3.73\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Error ratios of the mean peak responses of the responses of cable-stayed bridge. In Figure 14, compared with the PSD of UN-M model, due to the influence of MSE, some of the resonance peaks in the PSD of RMM-M are increased and some are decreased. The errors in the UN-R model and the RMM-R model originated from different modes. Therefore, it is hard to decide whether the effect of MSE would increase the damping truncation error or not. In Table 8, the error ratio of ADM-M and ADM-R have large values when base forces are calculated. The largest error ratio reaches 503%, which is unacceptable showing that the neglect of support damping could cause huge errors. Though the error ratios in ADM of the other responses are acceptable, ADM is not recommended in the applications of the MSE problem. In this paper, the influence of the apparent propagation velocity of the ground motion is also studied. The influence of the incident angle has a similar effect with the apparent propagation velocity. Figure 15 shows the variation of the mean peak responses of the mid-tower displacement of Pier 2, \(D_{m,2}\), with respect to the variation of \(1/v_{p}\). As is shown in Figure 15, the results of RMM-M and ILMM-M match well with each other. \(v_{p}\) mainly influence the vibration frequencies of the ripples caused by the wave passage effect. The error ratios of the other methods either increase or decrease with the variation of \(1/v_{p}\). \(v^{method}\) is defined as follows to measure the error sensitivity of \(v_{p}\). \[v^{method}=\max_{v_{p}}\left|e^{Method}\right|-\min_{v_{p}}\left|e^{Method}\right|, \tag{44}\]  
Table 9 shows the maximum changes of the error ratios when \(v_{p}\) changes from 800 m/s to infinity. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Forces** & \(v^{RF}\) **(\%)** & \(v^{ILr}\) **(\%)** \\ \hline \multicolumn{3}{c}{**Pier 2**} \\ \hline Base shear & 0.94 & 1.36 \\ Shear at deck level & 0.95 & 1.09 \\ Base Moment & 0.34 & 0.49 \\ Moment at deck level & 0.62 & 0.80 \\ \hline \multicolumn{3}{c}{**Pier 3**} \\ \hline Base shear & 1.54 & 0.93 \\ Shear at deck level & 1.03 & 0.89 \\ Base Moment & 1.16 & 0.69 \\ Moment at deck level & 0.69 & 0.91 \\
**Largest Cable force** & 3.53 & 3.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Maximum variation of the error ratio for different apparent propagation velocity. Figure 15: The variation of the mean peak response of \(D_{m,2}\) with respect to \(1/v_{p}\), when different modelling methods are used. As is shown in Table 9, the value of \(v^{Rr}\) and \(v^{ILr}\) are mainly around 1%. The largest value is observed from the largest cable force, which is around 3.5%. The influence of MSE on the error ratio is also studied. Three variables are defined as follows.