## 4 Numerical Results and Discussion
The uncertain dynamic response of the generally shaped arch when considering two uncertainties simultaneously calculated by the surrogate model is illustrated in \[x_{\eta}=x^{L}+\eta\frac{x^{U}-x^{L}}{\hat{\eta}}\in\left[x^{L},x^{U}\right], \eta=1,2,\ldots\hat{\eta} \tag{28}\]0  
Upon comparing the results in Figure 8 with those in Figure 6, it is observable that the upper and lower bounds of the uncertain response are very close to the bounds of the uncertain response when only the inner radius of the cross-section is regarded as an uncertainty. This implies that even when taking into account the uncertainty of both parameters, the impact of uncertainty in Young's modulus on the response is much less than that of the cross-section size. It can be deduced that, irrespective of whether it is a single uncertainty or multiple uncertainties, the influence of the cross-section size on the uncertain response is significantly greater than that of the material properties. Transient Dynamic Responses Comparison of General Shaped Arches with the Same Total Length and Different Mid-Span Height  
The span of a particular arch bridge remains invariably constant, yet the mid-span rise may differ due to diverse usage requirements. Hence, we are highly intrigued by comparing the uncertain dynamic response ranges of arches with the same span but varying rise span ratios, taking into account the impact of uncertain parameters. Similar to the previous subsection, when the two uncertainties occur simultaneously, with the span remaining unchanged and the mid-span rise being 4 m, the interval responses are obtained through the Chebyshev interval method, as depicted in \[x_{\eta}=x^{L}+\eta\frac{x^{U}-x^{L}}{\hat{\eta}}\in\left[x^{L},x^{U}\right], \eta=1,2,\ldots\hat{\eta} \tag{28}\]1 The parameters that curvature radius needed and the central angle for each shape of arch are as follows: for the circular arch, \(R=41/8\) m, \(-\theta_{0}=-\arcsin(40/41)\) and \(\theta_{0}=\arcsin(40/41)\); for the cycloidal arch, \(a=2\) m, \(-\theta_{0}\approx-0.24\pi\) and \(\theta_{0}\approx 0.24\pi\); for the elliptical arch, \(a=4\) m, \(b=5\) m, \(-\theta_{0}=-\pi/2\)  
\[x_{\eta}=x^{L}+\eta\frac{x^{U}-x^{L}}{\hat{\eta}}\in\left[x^{L},x^{U}\right], \eta=1,2,\ldots\hat{\eta} \tag{28}\]2 The uncertain transient dynamic response of the general shaped arch with a 5% deviation degree in the inner radius of cross section and a 5% deviation degree in the Young’s modulus: (**a**) circumference, (**b**) cycloid, (**c**) ellipse, and (**d**) parabola. Firstly, as depicted in Figure 9, when the span remains unchanged and the mid-span rise increases--that is, when the rise span ratio increases--the uncertain effects brought about by uncertain parameters still remain significant. The absolute value of the response at the mid-span and the upper and lower bounds of the uncertain response at any given time are all reduced compared to the arch before the change in the rise span ratio. Secondly, it can also be observed that the degree of reduction in the response varies with the shape of the arch, and the response is no longer positively correlated with the central angle. Under the new rise span ratio, the response is still ranked from largest to smallest as the elliptical arch, cycloid arch, circular arch, and parabolic arch, while the central angle is ranked from largest to smallest as the elliptical arch, circular arch, cycloid arch, and parabolic arch. Among them, the central angle of the elliptical arch remains unchanged, the central angle of the circular arch increases by approximately one-quarter, the central angle of the parabolic arch increases slightly, and the central angle of the cycloid arch decreases by about 40%. This indicates that for arches of different shapes with different rise span ratios, the positive relationship between the central angle and the response seems not to hold. It can only be confirmed that the rise span ratio indeed has an impact on the size of the response.