## 4 Solution Methods
### Semi-Analytical Solutions of FG-GPLRC Circular Arches with Various End Conditions  
For an arch with other supporting conditions, say clamped or free end conditions, one cannot seek analytical solutions to Equation (10). [PERSON] et al. (2014); [PERSON] et al. (2014) employed the differential quadrature method (DQM) to develop semi-analytical solutions to the partial differential state equations for static and vibration problems of laminated composite structures. In this sub-section, the DQM will be applied in the free vibration analysis of FG-GPLRC circular arch with various end conditions. According to the DQM, a continuous function _f(x,y)_ and its \(n\)-th partial derivatives at a given point \(x_{i}\) are approximated as linear weighting sums of function values at all the discrete points by:  
\[\left.\frac{\partial^{n}f(x,y)}{\partial x^{n}}\right|_{x=x_{i}}=\sum_{k=1}^{ N}g_{ik}^{(n)}f(x_{k},y) \tag{26}\]where \(i=1,2,\ldots,N\); \(n=1,2,\ldots,N-1\) and \(g_{ik}^{(n)}\) are the weighted coefficients that can be thoroughly determined by the positions of all discrete points as:  
\[ \begin{split} g_{ik}^{(1)}&=\frac{\prod_{j=1,j\
eq i }^{N}(x_{i}-x_{j})}{(x_{i}-x_{k})\prod_{j=1,j\
eq k}^{Nx}(x_{k}-x_{j})},\\ g_{ik}^{(n)}&=n\Bigg{[}g_{ik}^{(n-1)}g_{ik}^{(1)}- \frac{g_{ik}^{(n-1)}}{x_{i}-x_{k}}\Bigg{]},\ (n=2,3\cdots,N-1)\end{split}  \tag{27}\]  
where \(i\), \(k=1,2,\ldots,N\), but \(i\
eq k\), \(g_{ii}^{(n)}\) are determined as:  
\[g_{ii}^{(n)}=-\sum_{k=1,k\
eq i}^{N}g_{ik}^{(n)},\ (n=1,2,\cdots,N-1;\ i=1,2, \cdots,N) \tag{28}\]  
Furthermore, it can be assumed that:  
\[ \begin{split} u_{\theta}&=HU(\theta,\zeta)\mathrm{e} ^{i\omega t}\\ \sigma_{r}&=c_{0}Y(\theta,\zeta)\mathrm{e}^{i\omega t }\\ u_{r}&=HW(\theta,\zeta)\mathrm{e}^{i\omega t}\\ \tau_{r\theta}&=c_{0}\Gamma(\theta,\zeta)\mathrm{e} ^{i\omega t}\end{split}  \tag{29}\]  
Substituting Equation (29) into Equation (10) and using the above-mentioned DQ procedure by discretizing the domain of \(\theta\), the following discretized state equations are obtained:  
\[ \begin{split}\frac{\partial U_{i}}{\partial\zeta}& =\frac{\lambda}{\eta}U_{i}-\frac{\lambda}{\eta}\!\sum_{k=1}^{N}\!g _{ik}^{(1)}W_{k}+\frac{c_{0}}{c_{55}}\Gamma_{i}\\ \frac{\partial Y_{i}}{\partial\zeta}&=\frac{\lambda ^{2}}{\eta^{2}}\frac{k_{11}}{c_{0}}\!\sum_{k=1}^{N}\!g_{ik}^{(1)}U_{k}+\frac{ \lambda}{\eta}\!\left(\!\frac{c_{13}}{c_{33}}-1\!\right)\!Y_{i}-\frac{\rho}{ \rho_{0}}\Omega^{2}W_{i}+\frac{\lambda^{2}}{\eta^{2}}\frac{k_{11}}{c_{0}}W_{i }-\frac{\lambda}{\eta}\!\sum_{k=1}^{N}\!g_{ik}^{(1)}\Gamma_{k}\\ \frac{\partial W_{i}}{\partial\zeta}&=-\frac{c_{13} }{c_{33}}\frac{\lambda}{\eta}\!\sum_{k=1}^{N}\!g_{ik}^{(1)}U_{k}+\frac{c_{0}} {c_{33}}Y_{i}-\frac{c_{13}}{c_{33}}\frac{\lambda}{\eta}W_{i}\\ \frac{\partial\Gamma_{i}}{\partial\zeta}&=-\frac{ \rho}{\rho_{0}}\Omega^{2}U_{i}-\frac{\lambda^{2}}{\eta^{2}}\frac{k_{11}}{c_{0} }\!\sum_{k=1}^{N}\!g_{ik}^{(2)}U_{k}-\frac{c_{13}}{c_{33}}\frac{\lambda}{\eta} \!\sum_{k=1}^{N}\!g_{ik}^{(1)}Y_{k}-\frac{\lambda^{2}}{\eta^{2}}\frac{k_{11}}{c _{0}}\!\sum_{k=1}^{N}\!g_{ik}^{(1)}W_{k}-\frac{2\lambda}{\eta}\Gamma_{i}\end{split}  \tag{30}\]  
where \(U_{i}=U(\theta i,\zeta)\), \(Y_{i}=Y(\theta i,\zeta)\), \(W_{i}=W(\theta i,\zeta)\), and \(\Gamma_{i}=\Gamma(\theta i,\zeta)\) are functions of the variable \(\zeta\) at a given sampling point \(\theta_{i}\). The discretized boundary conditions (\(i=1\) and \(i=N\)) should be considered in the discrete state equation Equation (30). The S-S, C-F and C-C arches are concerned herein as three representative end conditions.