## 5 Conclusions  
This paper investigates the characteristics of the transient uncertain response of a generally shaped arch with uncertain interval parameters under vertical external excitation. The transient dynamic response of the generally shaped arch is obtained using the DQ method and the Newmark-\(\beta\) method, and the uncertain dynamic response is determined by employing an interval method based on the Chebyshev polynomials surrogate model. Through comparison with the scanning method, the accuracy of the proposed method in calculating the uncertain responses is demonstrated. Notably, under multiple uncertainties, the interval method requires less sample points and exhibits higher computational efficiency compared to the scanning method. Based on the numerical simulation results of several examples in the previous sections, the following conclusions can be drawn:  
1. Different uncertain parameters exert varying influences on the responses. The cross-sectional geometric parameters (the inner radius of the circular tube cross-section) have a more significant influence on the uncertain responses than Young's modulus. Moreover, the greater the absolute value of the deterministic response, the more pronounced this influence becomes. When considering these two uncertain parameters concurrently, the influence of the cross-sectional geometric parameter is predominant. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Size** & **Response Type** & **Circumference** & **Cycloid** & **Ellipse** & **Parabola** \\ \hline \multirow{6}{*}{H = 3 m} & \multirow{2}{*}{IRCS *} & \multirow{2}{*}{Lover} & 1.156 & 1.396 & 5.073 & 0.271 \\ \cline{3-6}  & & & Lower & 0.893 & 1.069 & 3.959 & 0.197 \\ \cline{3-6}  & & & Upper & 1.673 & 2.027 & 7.267 & 0.416 \\ \cline{2-6}  & & & Lower & 1.101 & 1.330 & 4.831 & 0.258 \\ \cline{3-6}  & & & Upper & 1.217 & 1.470 & 5.340 & 0.286 \\ \cline{2-6}  & & & Lower & 0.851 & 1.018 & 3.770 & 0.187 \\ \cline{2-6}  & & & Upper & 1.761 & 2.133 & 7.650 & 0.438 \\ \hline \multirow{6}{*}{H = 4 m} & \multirow{2}{*}{Both *} & \multirow{2}{*}{Deterministic} & 0.953 & 1.240 & 1.681 & 0.098 \\ \cline{3-6}  & & & Lower & 0.701 & 0.895 & 1.240 & 0.064 \\ \cline{1-1} \cline{3-6}  & & & Upper & 1.452 & 1.926 & 2.557 & 0.168 \\ \hline \hline \end{tabular}  
* IRCS means the uncertain inner radius of the cross-section, YM means the uncertain Young’s modulus, Both means two uncertain parameters.  
\end{table} 
Table 4: The maximum positive responses and corresponding uncertain responses (\(\times 10^{-3}\) m). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Size** & **Response Type** & **Circumference** & **Cycloid** & **Ellipse** & **Parabola** \\ \hline \multirow{6}{*}{H = 3 m} & \multirow{2}{*}{IRCS *} & \multirow{2}{*}{Lover} & \(-1.611\) & \(-1.948\) & \(-7.068\) & \(-0.379\) \\ \cline{3-6}  & & & Lower & \(-2.333\) & \(-2.825\) & \(-10.125\) & \(-0.581\) \\ \cline{3-6}  & & & Upper & \(-1.245\) & \(-1.491\) & \(-5.516\) & \(-0.275\) \\ \cline{2-6}  & & & Lower & \(-1.696\) & \(-2.050\) & \(-7.440\) & \(-0.399\) \\ \cline{2-6}  & & & Upper & \(-1.535\) & \(-1.855\) & \(-6.731\) & \(-0.361\) \\ \cline{2-6}  & & & Lower & \(-2.455\) & \(-2.974\) & \(-10.658\) & \(-0.612\) \\ \cline{2-6}  & & & Upper & \(-1.186\) & \(-1.421\) & \(-5.253\) & \(-0.261\) \\ \hline \multirow{6}{*}{H = 4 m} & \multirow{2}{*}{Deterministic} & \(-1.328\) & \(-1.730\) & \(-2.343\) & \(-0.137\) \\ \cline{3-6}  & & & Lower & \(-2.023\) & \(-2.686\) & \(-3.562\) & \(-0.234\) \\ \cline{1-1} \cline{3-6}  & & & Upper & \(-0.977\) & \(-1.248\) & \(-1.728\) & \(-0.089\) \\ \hline \hline \end{tabular}  
* IRCS means the uncertain inner radius of the cross-section, YM means the uncertain Young’s modulus, Both means two uncertain parameters.  
\end{table} 
Table 5: The maximum negative responses and corresponding uncertain responses (\(\times 10^{-3}\) m). 2. Arches of different shapes with the same rise span ratio exhibit distinct deterministic responses and corresponding uncertain responses under the same external excitation.