# Free Vibration Analysis of Arches with Interval-Uncertain Parameters
###### Abstract
Even a slight parametric variation may lead to a drastic change in the natural frequency. Due to the inherent parametric uncertainties associated with arch structures, occasional variations occur in their structural dynamic characteristics which can potentially impact both the robustness and the accuracy of structural diagnoses and assessments. Therefore, it is important to investigate the effect of uncertain parameters on the dynamic characteristics of arch structures. Generally, the probabilistic model [20; 21] and non-probabilistic models such as the fuzzy set model [22; 23] and the interval model [24; 25; 26; 27; 28; 29; 30; 31; 32] are employed to characterize the uncertainty inherent in structural systems. The probabilistic model requires the probability distribution function, whereas the fuzzy set model necessitates the membership function. The aforementioned approaches are frequently high-cost and occasionally challenging to accomplish. To address this issue, the interval model has been proposed as a valuable supplement. The interval model only requires the boundaries that define the extreme values of uncertainties, rather than their accuracy distribution. Additionally, there is no need to clarify the evolution in varying ranges, making the model easy to implement. To date, there have been limited studies about the free-vibration characteristics of arches while considering the interval uncertainty. [PERSON] et al. used a probability field model, including interval fields, to represent spatially correlated uncertain parameters and analyze the non-deterministic free vibration of CFST arch structures [33]. [PERSON] et al. considered the uncertain material properties and creep effect of CFST arch structures, established a unified interval field model, and determined the limit range of the natural frequency of an arch using the interval finite element method [34]. The aforementioned methodologies are considered intrusive and require modification of the original deterministic models, resulting in complex processes. In contrast, non-intrusive interval methods utilize orthogonal polynomial approximation theory combined with interval operations to overcome the limitation of interval solutions having a wide range [27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43]. In this paper, the free-vibration differential equations of a linear elastic arch are initially established using the traditional approach, and the natural frequencies are determined by employing the DQ method, which has been shown to be more efficient and accurate. Then, the uncertain parameters are treated as interval variables, and the non-intrusive Chebyshev polynomial surrogate (CPS) model is for the first time employed to address the natural frequencies of arches with uncertainties. The natural frequencies are regarded as the objective functions of the surrogate model, and the CPS model is established based on the original deterministic value of these frequencies. Finally, the model is utilized to obtain the bounds of natural frequencies. In addition, the Monte Carlo simulation (MCS) is employed as a means of validating the accuracy of the CPS model. The remainder of this paper is organized as follows. The derivation of the free-vibration equation and the determination of the natural frequency are elaborated upon in Section 2. The resolution of the natural frequency boundary through the CPS model is addressed in Section 3. Several numerical examples are analyzed in Section 4. In Section 5, the conclusions are summarized.