## 7 Conclusions  
This study further verifies that the MSRS method without the contribution of the structural support degrees of freedom in the seismic response of the multi-support structure can result in a large deviation of the structural response, particularly when near to the structural supports. Hence, three methods are proposed to improve the MSRS method. Using a model of a five-span reinforced concrete continuous rigid frame bridge under incoherent ground displacement excitation, it was verifies that the modified MSRS method could reflect the seismic responses of the structure more accurately than does the MSRS method.  
The collective response spectrum method is proposed for the seismic analysis of structures under coherent and incoherent ground displacement excitation. For the generalized response spectrum method, the generalized peak response factors are introduced based on the fact that the structural response peak factor mainly depends on structural characteristics rather than structural response. The generalized response spectrum method greatly promotes computational efficiency.  
In addition, it should be remarked that this study does not directly relate the response spectrum with the power spectrum. The future work will continue to address this issue by establishing the transformation relationship between the response and power spectrum.  
**Author Contributions:** [PERSON]. ([PERSON]) and [PERSON] ([PERSON]) designed and coordinated the study. [PERSON]. ([PERSON]) and [PERSON] performed theoretical study under the supervision of [PERSON]. ([PERSON]). [PERSON]. ([PERSON]) and [PERSON] checked the results of this study.  
**Funding:** This research was funded by National Natural Science Foundation of China, grant number 51608069.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Structural Response**} & \multicolumn{2}{c}{**The MSRS Method**} & \multicolumn{2}{c}{**The Power**} & \multicolumn{2}{c}{**The Simplified Power**} & \multicolumn{2}{c}{**The Collective Response**} \\  & & & **Spectrum Method** & **Spectrum Method** & **Spectrum Method** \\ \cline{2-6}
**(mm)** & \(\sigma_{\text{b,mm}}\) & \(\mu_{\text{b,mm}}\) & \(\sigma_{\text{b,mm}}\) & \(\mu_{\text{b,mm}}\) & \(\mu_{\text{b,mm}}\) & \(\mu_{\text{b,mm}}\) \\ \hline VD\({}^{1}\) at Point 2 & 6.46 & 16.66 & 6.46 & 18.02 & 17.76 & 17.93 \\ VD at Point 3 & 108.3 & 27.94 & 10.83 & 30.19 & 29.77 & 30.26 \\ VD at Point 4 & 118.0 & 30.48 & 118.0 & 32.86 & 32.44 & 33.38 \\ VD at Point 5 & 9.33 & 24.15 & 9.35 & 25.98 & 25.71 & 26.95 \\ VD at Point 6 & 4.68 & 122.23 & 4.76 & 13.17 & 13.10 & 14.09 \\ VD at Point 8 & 2.90 & 7.93 & 3.07 & 8.44 & 8.43 & 9.01 \\ VD at Point 9 & 3.96 & 11.00 & 4.05 & 11.27 & 11.14 & 11.33 \\ VD at Point 10 & 3.68 & 10.54 & 3.68 & 10.37 & 10.12 & 9.41 \\ VD at Point 11 & 2.69 & 8.01 & 2.80 & 7.92 & 7.70 & 6.53 \\ VD at Point 12 & 1.57 & 4.72 & 1.86 & 5.18 & 5.13 & 4.63 \\ \hline \hline \end{tabular}  
* VD indicates vertical displacement.  
\end{table}
Table 6: The comparison of displacements calculated by the four methods.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Structural Response**} & \multicolumn{2}{c}{**The MSRS Method**} & \multicolumn{2}{c}{**The Power**} & \multicolumn{2}{c}{**The Simplified Power**} & \multicolumn{2}{c}{**The Collective**} \\  & & & **Spectrum Method** & **Spectrum Method** & **Response Spectrum Method** \\ \cline{2-6}  & \(\sigma_{\text{b,mm}}\)\({}^{2}\) & \(\mu_{\text{b,mm}}\) & \(\sigma_{\text{b,mm}}\) & \(\mu_{\text{b,mm}}\) & \(\mu_{\text{b,mm}}\) & \(\mu_{\text{b,mm}}\) \\ \hline BBM \({}^{1}\) at Point 02 & 0.61 & 1.81 & 0.62 & 1.54 & 1.71 & 1.76 \\ BBM at Point 0  
**Acknowledgments:** This research has been financially supported by National Natural Science Foundation of China (51608069). The authors would like to thank the members of the HIT 504 office for their selfless help and useful suggestions.  
**Conflicts of Interest:** The authors declare no conflict of interest