## 4 Numerical Results and Discussion
For the cycloidal arch, \(a=1.6\) m, \(-\theta_{0}=-5\pi/12\) and \(\theta_{0}=5\pi/12\); for the elliptical arch, \(a=3\) m, \(b=5\) m, \(-\theta_{0}=-\pi/2\) and \(\theta_{0}=\pi/2\); for the parabolic arch, \(a=3\) m, \(x_{\mathrm{p}}=5\) m, \(y_{\mathrm{p}}=0\), \(-\theta_{0}\approx-0.21\pi\) and \(\theta_{0}\approx 0.21\pi\). For the sake of brevity and without sacrificing representativeness and distinctness, the harmonic distributed load of case 3 is selected as the external excitation. The inner radius of the cross-section is still regarded as an uncertain parameter, with a deviation degree of 5% considered. Some of these simulation results have been illustrated in the previous comparative analysis. Additionally, considering the inevitable inherent dispersion in the materials, the uncertain Young's modulus with a deviation degree of 5% is also employed, and the transient response solutions are plotted in Figure 6 and Figure 7, respectively. Figure 5: The relative error of the scanning method and the interval method in Figure 4: (**a**) case 1, (**b**) case 2, (**c**) case 3, and (**d**) case 4. By comparing the results in Figures 6 and 7 with those in Figure 4, it can be found that the various shaped arches listed in Table 1 allow the same conclusion as for the previous circular arch--that is, the greater the absolute value of the response, the more obvious the influence of uncertainty. In addition, for different shaped arches with a span of 10 m and a mid-span rise of 3 m, the deterministic and uncertain responses are not exactly the same, which is not only related to the shape type of the arch but also to its central angle \(2\theta_{0}\). Arranged in descending order of the central angles of the arch, they are the elliptical arch, cycloid arch, circular arch and parabolic arch. The deterministic response and uncertain response ranges are also positively correlated with the size of the central angle, and they are in the order of the elliptical arch, cycloid arch, circular arch and parabolic arch from large to small, and the greater the absolute value of the response, the more obvious this feature is. In addition, it can also be observed that for the same shaped arch, different uncertain parameters have different influences on the boundaries of the uncertain response, and the influence of the cross-sectional size is much greater than that of Young's modulus. Due to the intricacy of practical engineering problems, the occurrence of multiple uncertain parameters is often inevitable. Considering the scenario where both uncertainties  
Figure 6: The uncertain transient dynamic response of the general shaped arch with a 5% deviation degree in the inner radius of cross section: (**a**) circumference, (**b**) cycloid, (**c**) ellipse, and (**d**) parabola. Figure 7: The uncertain transient dynamic response of the generally shaped arch with a 5% deviation degree in [PERSON]’s modulus: (**a**) circumference, (**b**) cycloid, (**c**) ellipse, and (**d**) parabola. occur simultaneously and the deviation degree is the same, which is 5%, the uncertain response of the arch is further calculated. At this juncture, if the scanning method is employed, it is necessary to perform operations on the deterministic finite element model at each sample point due to the geometric increase in the required number of sample points (if the number of sample points for a single uncertain parameter is still set at 500, then the number of sample points for double uncertain parameters becomes 5002), thereby resulting in an unacceptable computing cost. In contrast, the Chebyshev interval method adopted in this paper can conveniently extend to higher-dimensional approximations, significantly reducing the total number of interpolation points and keeping it within an acceptable range.