## 4 Numerical Results and Discussion  
The present section centers on the numerical simulation of generally shaped arches with uncertain parameters. In the preliminary section, a circular arch is regarded as an exemplary case, in which external forces varying with time are chosen as the external excitation to acquire the deterministic dynamic responses. Owing to the arch structure possessing a considerable span and primarily undergoing compressive loads, the radial displacement at the top of the arch is of particular concern; thus, as shown in Figure 3, the vertical excitation is selected and the radial displacement at the top is chosen as the transient response hereafter. The cross-section is regarded as a hollow circular, and \(\tau_{1}\) and \(\tau_{2}\) are, respectively, the outer and inner radii of the cross-section. Then, the Chebyshev interval method mentioned beforehand is utilized to probe the bounds of these responses, and the accuracy of the proposed approach is validated via comparison with the scanning method. In the second section, the interval method is further extended to encompass variable curvature arches, enabling a more comprehensive conclusion. For the purpose of comparison, all the arches maintain the same span and mid-span rise as the circular arch in the previous section. In the third section, the response ranges of identical arches with a constant span and different mid-span rises are presented. Figure 2: Computational flowchart of the uncertain transient dynamic response via the Chebyshev interval method. Comparison of the Chebyshev Interval Method with the Scanning Method in Transient Responses Calculation  
In this subsection, the transient response of the circular arch under various types of external excitation is initially computed. The physical and geometric parameters of the arch are presented in Table 2. The span is taken as 10 m and the mid-span rise is taken as 3 m, respectively. The curvature radius and central angle of the arch can be readily determined through straightforward calculations (R = 17/3 m, \(-\theta_{0}\) = \(-\arcsin(15/17)\), \(\theta_{0}\) = \(\arcsin(15/17)\)), respectively. The specific types of external excitation are detailed in Table 3, with a duration time of t = 2 s and a time step of 0.02 s.  
Subsequently, in order to take into account uncertain parameters, the Chebyshev interval method is utilized to obtain the uncertain response ranges. To assess the accuracy of \begin{table}
\begin{tabular}{c c} \hline
**Description** & **Value** \\ \hline Young’s modulus E & 210 Gpa \\ \hline Density of mass \(\rho\) & 7860 kg/m\({}^{3}\) \\ \hline Poisson ratio \(\upsilon\) & 0.3 \\ \hline Outer radius r\({}_{1}\) & 0.09 m \\ \hline Inner radius r\({}_{2}\) & 0.078 m \\ \hline Shear factor \(\kappa\) & 1.2 \\ \hline \end{tabular}
\end{table} 
Table 2: Physical and geometric parameters of the arch. Figure 3: (**a**) The radial displacement of the selected point, and (**b**) the cross-section of the arch. \begin{table}
\begin{tabular}{c c} \hline
**Type** & **Description** & **Value** \\ \hline Case 1 & Linear increasing concentrated load & F\({}_{1}\) = 20t (kN) \\ \hline Case 2 & Harmonic concentrated load & F\({}_{2}\) = 20 \(\times\) sin(2\(\pi\)t) (kN) \\ \hline Case 3 & Harmonic distributed load with a linear increasing variation & F\({}_{3}\) = 20t \(\times\) sin(2\(\pi\)t) (kN/m) \\ \hline Case 4 & Harmonic distributed load with negative exponential variation & F\({}_{4}\) = 20 \(\times\) e\({}^{-t/2}\)\(\times\) sin(2\(\pi\)t) (kN/m) \\ \hline \end{tabular}
\end{table} 
Table 3: Types of external excitation force. the interval procedure in predicting the response ranges of the arch, it is essential to employ certain traditional methods for quantifying uncertainty as a benchmark. The scanning method, similar to the Monte Carlo method, is frequently used as an accuracy validation for another uncertainty quantification method.