## 2 Generally Shaped Arch Model and Equations of Motion
\(\eta_{ij}^{(n)}\) can be expressed as follows:first order: \(\eta_{ij}^{(1)}=\frac{L^{(1)}(\theta_{i})}{(\theta_{i}-\theta_{j})L^{(1)}(\theta_{ j})}\), \(i,j=1,2,\cdots N\), \(i\
eq j\)  
\[\eta_{ii}^{(1)}=-\sum\limits_{j=1,j\
eq i}^{N}\eta_{ij}^{(1)},\ i,j=1,2,\cdots N \tag{8}\]  
where \(L^{(1)}\big{(}\theta_{j}\big{)}=\prod\limits_{i=1,i\
eq j}^{N}(\theta_{j}- \theta_{i})\)  
higher order: \(\eta_{ij}^{(n)}=n\Bigg{(}\eta_{ii}^{(n-1)}\eta_{ij}^{(1)}-\frac{\eta_{ij}^{(n- 1)}}{\theta_{i}-\theta_{j}}\Bigg{)}\), \(i,j=1,2,\cdots N\),\(i\
eq j\),\(n=2,3,\cdots N-1\)  
\[\eta_{ii}^{(n)}=-\sum\limits_{j=1,j\
eq i}^{N}\eta_{ij}^{(n)},i,j=1,2,\cdots N,i\
eq j,n=2,3,\cdots N-1 \tag{9}\]  
The selection of sample points is an essential step in the DQ method. The Chebyshev-Gauss-Lobatto (C-G-L) sampling rule is employed here. According to the rule, the sample point \(a_{i}\) in the standard interval [0,1] is expressed as  
\[a_{i}=\frac{1-\cos\Bigl{(}\frac{i-1}{N-1}\pi\Bigr{)}}{2},i=1,2,\cdots N \tag{10}\]  
The sample point \(\theta_{i}\) in the real domain can be obtained by mapping to \(a_{i}\)  
\[\theta_{i}=-\theta_{0}+a_{i}\cdot 2\theta_{0},\ i=1,2,\cdots N \tag{11}\]  
Then, Equation (6) can be rewritten as:  
\[\left\{\begin{array}{c}\frac{EA}{R_{i}^{2}}\sum\limits_{j=1}^{N} \eta_{ij}^{(2)}u_{j}(t)-\frac{EA}{R_{i}^{3}}\sum\limits_{j=1}^{N}\eta_{ij}^{(1 )}R_{j}\sum\limits_{j=1}^{N}\eta_{ij}^{(1  
The curvature radius term \(R_{i}\), which is contained within \(K\) in Equation (13), remains constant in the case of a circular arch; however, for a variable curvature arch, it is necessary to derive the analytical expression of \(R_{i}\) to determine its value at each sample point. According to Riemannian geometry, while the analytical expression of a curve is expressed as \(y=f(x)\) or \(\left\{ \begin{matrix}x=\varphi(t)\\ y=\eta(t)\end{matrix} \right.\), which has the second derivative, the corresponding curvature radius \(R\) can be derived as \(R=\frac{\left(1+\left(y^{\prime}\right)^{2}\right)^{3/2}}{\left[y^{\prime \prime}\right]}\), or \(R=\frac{\left[\varphi^{\prime 2}(t)+\eta^{\prime 2}(t)\right]^{3/2}}{\left[ \varphi^{\prime}(t)\eta^{\prime\prime}(t)-\eta^{\prime}(t)\varphi^{\prime \prime}(t)\right]}\). The curvature radii \(R\) concerning \(\theta\) of some common shapes for arch structure applications are listed in Table 1. It should be noted that there is a parametric transformation between the global polar system and the orthogonal system. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type of Arch** & **Geometry Description** & **Parametric Representation** & **Curvature Radius Concerning \(\theta\)** \\ \hline Circumference & & \\ \(\left\{\begin{matrix}X=R\cos\theta\\ Y=\sqrt{R^{2}-X^{2}}\end{matrix}\right.\) & & \(R(\theta)=R\) \\ Cycloid & & \\ \(\left\{\begin{matrix}X=a(2\theta+\sin(2\theta))\\ Y=a(1+\cos(2\theta))\end{matrix}\right.\) & & \(R(\theta)=4a\text{cos}\theta\) \\ Ellipse & & \\ \(\left\{\begin{matrix}X=\frac{\rho^{2}\tan\theta}{b\sqrt{\frac{b^{2}(a-X^{2})}{ a^{2}}}}\\ Y=\sqrt{\frac{b^{2}(a-X^{2})}{a^{2}}}\end{matrix}\right.\) & & \\ \(\left.\begin{matrix}R(\theta)=\frac{b^{2}}{a\cos^{\beta}\sqrt{\left(1+b^{2}\tan^{ 2}\theta/a^{2}\right)^{3}}}\\ Y=\sqrt{\frac{b^{2}(a-X^{2})}{a^{2}}}\end{matrix}\right.\) & & \\ \(\left.\begin{matrix}R(\theta)=\frac{b^{2}}{a\cos^{\beta}\sqrt{\left(1+b^{2}\tan^{ 2}\theta/a^{2}\right)^{3}}}\\ Y=a-\frac{X^{2}(a-Y_{2})}{X_{2}^{2}}\end{matrix}\right.\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Analytical expression of the curvature radius of arches. Figure 1: Generally shaped arch with (**a**) the global and local reference configurations; (**b**) the displacement and internal force components; and (**c**) the calculation diagram with the span \(L\) and mid-span rise \(H\). Then, the structural damping is assumed by using Rayleigh damping theory as  
\[\mathbf{C}=\alpha\mathbf{M}+\beta\mathbf{M} \tag{14}\]  
where \(\alpha\) and \(\beta\) are the parameters of Rayleigh damping, which can be obtained from  
\[\left\{