## 4 Numerical Results and Discussion
The principle of the scanning method is that the uncertain interval \(x^{l}\) can be divided into equal-stepped samples expressed as  
\[x_{\eta}=x^{L}+\eta\frac{x^{U}-x^{L}}{\hat{\eta}}\in\left[x^{L},x^{U}\right], \eta=1,2,\ldots\hat{\eta} \tag{28}\]  
where \(\hat{\eta}\) represents the sampling number within the interval \(\left[x^{L},x^{U}\right]\). Both accuracy and convergence can be achieved through proper sampling, and it has been demonstrated that the objective can be fulfilled with more than 500 samples [22]; thus, the method sets the value of \(\hat{\eta}\) to 500. To consider the potential geometric errors that might occur due to the design, fabrication, and machining processes, the inner radius of the cross-section is regarded as an uncertain parameter with a deviation degree of 5%. The comparison of the results between the interval method and the scanning method is presented in Figure 4. In the interval method, the Chebyshev surrogate model has an expansion order of 10 and an interpolation number of 11. Firstly, it is noticeable from Figure 4 that the deterministic responses in cases 1 to 4 are all encompassed by the upper and lower bounds of the uncertain responses. Secondly, the upper and lower bounds of the uncertain response obtained through the proposed interval method and the scanning method are nearly identical. To further validate the accuracy of the proposed interval method, the notion of relative error is put forward. Supposing that the upper and lower bounds of the response interval obtained by the scanning method are the exact values, the relative error is expressed as:  
\[\varepsilon=\frac{\left|d-d^{\prime}\right|}{d}\times 100\% \tag{29}\]  
where \(d\) and \(d^{\prime}\) are the response bounds obtained by the scanning method and the interval method, respectively. The results are presented in Figure 5.  
Figure 4: The uncertain transient dynamic response of a circular arch with 5% deviation in the inner radius of cross-section under external excitation: (**a**) case 1, (**b**) case 2, (**c**) case 3, and (**d**) case 4. The maximum error range depicted in Figure 5 is within 7%, which demonstrates the suitability of the proposed interval method for computing the response interval of the arch and the excellent accuracy of the calculation results. It is worth noting that under the same external excitation, as the absolute value of the determined response increases or as time elapses, the error of the interval method becomes smaller and the results become more accurate. Meanwhile, under the harmonic load in cases 2 to 4, most of the peak errors occur at the minimum absolute value of their respective responses. Due to the small response, their impact on the structure can be disregarded. Hence, the interval method with an expansion order of 10 is accurate enough to fulfill the research objective. In addition, it is observed from Figure 4 that, although only a small uncertainty is considered here, the uncertain effects are significant. Meanwhile, the upper and lower bounds of the uncertain response increase along with the increase in the absolute value of the deterministic response. Moreover, under the harmonic load in cases 2 to 4, this phenomenon is particularly evident at the peaks and troughs of the response, signifying that the larger the absolute value of the vibration response, the more pronounced the influence of uncertain parameters. Transient Dynamic Behavior Investigation of the Generally Shaped Arches Considering Different Uncertain Parameters  
In this subsection, the uncertain transient dynamic responses of the generally shaped arches as proposed in Table 1 are presented. To facilitate comparison, the total length, mid-span height, cross-section size, and physical parameters are kept constant, while variations only occur in the curvature radius and the central angle. Regarding a specific arch shape, information has previously been provided for the circular arch in terms of its curvature radius and central angle, and others can easily be obtained through simple calculations.