Three-dimensional poroelasticity solution of sandwich, cylindrical, open, functionally graded composite panels under multi-directional initial stress: semi-numerical modeling

Abstract

Up to now, no studies have been yet reported to study the mechanical behaviors of three-dimensional functionally graded graphene platelets reinforced composite (FG-GPLRC) open-type panel. In this paper, the free vibration of FG-GPLRC open-type panel under multi-directional initially stressed using three-dimensional poroelasticity theory is investigated for the first time. Weight fraction of graphene open-type panel is assumed to be distributed either uniformly or functionally graded (FG) along the radial direction. Modified Halpin–Tsai model is used to compute effective Young’s modulus, whereas effective Poisson’s ratio and mass density are computed using the rule of mixture. State-space differential equations are derived from the governing equation of motion and constitutive relations in cylindrical co-ordinates. The accuracy of the obtained formulation is validated by comparing the numerical results with those reported in the available literature as well as with the finite-element modeling. The influences of several importance parameters, such as various directional initial stress, compressibility coefficient, porosity, and various type of sandwich open-type cylindrical panel, are investigated on the frequency of the structures. The results of the present study can be served as benchmarks for future mechanical analysis of cylindrical FG-GPLRC structures.

Appendix

The general form of the matrix G is written as follows and it can be conveniently specified for each section of the sandwich panel such as the top face-sheet ${{\varvec{G}}}_{\mathrm{ft}}$, the core section ${{\varvec{G}}}_{\mathrm{c}}$, and the bottom face-sheet ${{\varvec{G}}}_{\mathrm{fb}}$

$$ {\mathbf{G}} = \left[ {\begin{array}{*{20}c} {\frac{1}{r}\left( {\frac{{{\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }} - 1} \right)} & {a_{12} } & {a_{13} } & {a_{14} } & {a_{15} } & {a_{16} } \\ 0 & 0 & 0 & { - \frac{\partial }{\partial z}} & {\frac{1}{{{\mathbb{Q}}_{55} }}} & 0 \\ 0 & 0 & \frac{1}{r} & { - \frac{1}{r}\frac{\partial }{\partial \theta }} & 0 & {\frac{1}{{{\mathbb{Q}}_{66} }}} \\ {\frac{1}{{{\mathbb{Q}}_{33} }}} & { - \frac{{{\mathbb{Q}}_{13} }}{{{\mathbb{Q}}_{33} }}\frac{\partial }{\partial z}} & { - \frac{{{\mathbb{Q}}_{23} }}{{r{\mathbb{Q}}_{33} }}\frac{\partial }{\partial \theta }} & { - \frac{{{\mathbb{Q}}_{23} }}{{r{\mathbb{Q}}_{33} }}} & 0 & 0 \\ {a_{51} } & {a_{52} } & {a_{53} } & {a_{54} } & { - \frac{1}{r}} & 0 \\ {a_{61} } & {a_{62} } & {a_{63} } & {a_{64} } & 0 & {a_{66} } \\ \end{array} } \right] $$

$$ a_{12} = \frac{1}{r}\left( {{\mathbb{Q}}_{12} - \frac{{{\mathbb{Q}}_{13} {\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{\partial }{\partial z} + \frac{{\sigma_{r}^{0} }}{r}\left( {\frac{{{\mathbb{Q}}_{12} - {\mathbb{Q}}_{13} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{\partial }{\partial z},\;a_{13} = \frac{1}{{r^{2} }}\left( {{\mathbb{Q}}_{22} - \frac{{{\mathbb{Q}}_{23}^{2} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{\partial }{\partial \theta } - \frac{{2\sigma_{\theta }^{0} }}{{r^{2} }}\frac{\partial }{\partial \theta } + \frac{{\sigma_{r}^{0} }}{{r^{2} }}\left( {\frac{{{\mathbb{Q}}_{22} - {\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{\partial }{\partial \theta } - \frac{2}{{r^{2} {\mathbb{Q}}_{33} }}\sigma_{r}^{0} \sigma_{\theta }^{0} \frac{\partial }{\partial \theta }, $$

$$ \begin{aligned} a_{14} & = \frac{1}{{r^{2} }}\left( {{\mathbb{Q}}_{22} - \frac{{{\mathbb{Q}}_{23}^{2} }}{{{\mathbb{Q}}_{33} }}} \right) + \sigma_{z}^{0} \frac{{\partial^{2} }}{{\partial z^{2} }} + \frac{{\sigma_{\theta }^{0} }}{{r^{2} }}\left( {\frac{{\partial^{2} }}{{\partial \theta^{2} }} - 1} \right) + \rho \frac{{\partial^{2} }}{{\partial t^{2} }} + \frac{{\sigma_{r}^{0} }}{{r^{2} }}\left( {\frac{{{\mathbb{Q}}_{22} - {\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}} \right) + \frac{{\sigma_{r}^{0} \sigma_{z}^{0} }}{{{\mathbb{Q}}_{33} }}\frac{{\partial^{2} }}{{\partial z^{2} }} \\ & + \frac{{\sigma_{r}^{0} \sigma_{\theta }^{0} }}{{r^{2} {\mathbb{Q}}_{33} }}\left( {\frac{{\partial^{2} }}{{\partial \theta^{2} }} - 1} \right) + \frac{{\rho \sigma_{r}^{0} }}{{{\mathbb{Q}}_{33} }}\frac{{\partial^{2} }}{{\partial t^{2} }} + \sigma_{r}^{0} \frac{{{\mathbb{Q}}_{13} }}{{{\mathbb{Q}}_{33} }}\frac{{\partial^{2} }}{{\partial z^{2} }} + \frac{{\sigma_{r}^{0} }}{{r^{2} }}\frac{{{\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}\frac{{\partial^{2} }}{{\partial \theta^{2} }} + \frac{{\sigma_{r}^{0} }}{{r^{2} }}\frac{{{\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}, \\ \end{aligned} $$

$$ a_{15} = - \frac{\partial }{\partial z} - \frac{{\sigma_{r}^{0} }}{{{\mathbb{Q}}_{33} }}\frac{\partial }{\partial z} - \sigma_{r}^{0} \frac{{{\mathbb{Q}}_{13} {\mathbb{Q}}_{33} }}{{{\mathbb{Q}}_{55} }}\frac{\partial }{\partial z},\;\,a_{16} = - \frac{1}{r}\frac{\partial }{\partial \theta } - \frac{{\sigma_{r}^{0} }}{{{\mathbb{Q}}_{33} }}\frac{\partial }{\partial \theta } - \frac{{\sigma_{r}^{0} }}{r}\frac{{{\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{66} {\mathbb{Q}}_{33} }}\frac{\partial }{\partial \theta },\;a_{51} = - \frac{{{\mathbb{Q}}_{13} }}{{{\mathbb{Q}}_{33} }}\frac{\partial }{\partial z} - \frac{{\sigma_{r}^{0} }}{{{\mathbb{Q}}_{33} }}\left( {1 + \frac{{{\mathbb{Q}}_{13} }}{{{\mathbb{Q}}_{55} }}} \right)\frac{\partial }{\partial z} $$

$$ \begin{gathered} a_{52} = \sigma_{z}^{0} \frac{{\partial^{2} }}{{\partial z^{2} }} + \frac{{\sigma_{\theta }^{0} }}{{r^{2} }}\frac{{\partial^{2} }}{{\partial \theta^{2} }} + \sigma_{r}^{0} \frac{{{\mathbb{Q}}_{13} }}{{{\mathbb{Q}}_{33} }}\frac{{\partial^{2} }}{{\partial z^{2} }} + \frac{{\sigma_{r}^{0} \sigma_{z}^{0} }}{{{\mathbb{Q}}_{55} }}\frac{{\partial^{2} }}{{\partial z^{2} }} + \frac{{\sigma_{r}^{0} }}{{{\mathbb{Q}}_{55} }}\rho \frac{{\partial^{2} }}{{\partial t^{2} }} - \frac{{\sigma_{r}^{0} {\mathbb{Q}}_{66} }}{{r^{2} {\mathbb{Q}}_{55} }}\frac{{\partial^{2} }}{{\partial \theta^{2} }} \hfill \\ - \frac{{\sigma_{r}^{0} }}{{C_{55} }}\left( {{\mathbb{Q}}_{11} - \frac{{{\mathbb{Q}}_{13}^{2} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{{\partial^{2} }}{{\partial z^{2} }} + \frac{{\sigma_{r}^{0} \sigma_{\theta }^{0} }}{{r^{2} {\mathbb{Q}}_{55} }}\frac{{\partial^{2} }}{{\partial \theta^{2} }} + \rho \frac{{\partial^{2} }}{{\partial t^{2} }} - \frac{{{\mathbb{Q}}_{44} }}{{r^{2} }}\frac{{\partial^{2} }}{{\partial \theta^{2} }} - \left( {{\mathbb{Q}}_{11} - \frac{{{\mathbb{Q}}_{13}^{2} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{{\partial^{2} }}{{\partial z^{2} }}, \hfill \\ \end{gathered} $$

$$ \begin{gathered} a_{53} = \frac{{\sigma_{r}^{0} {\mathbb{Q}}_{23} }}{{r{\mathbb{Q}}_{33} }}\frac{{\partial^{2} }}{\partial \theta \partial z} - \frac{{\sigma_{r}^{0} }}{{r{\mathbb{Q}}_{55} }}\left( {{\mathbb{Q}}_{12} + {\mathbb{Q}}_{44} - \frac{{{\mathbb{Q}}_{13} {\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{{\partial^{2} }}{\partial \theta \partial z} - \frac{1}{r}\left( {{\mathbb{Q}}_{12} + {\mathbb{Q}}_{44} - \frac{{{\mathbb{Q}}_{13} {\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{{\partial^{2} }}{\partial \theta \partial z}, \hfill \\ a_{54} = \frac{{\sigma_{r}^{0} {\mathbb{Q}}_{23} }}{{r{\mathbb{Q}}_{33} }}\frac{\partial }{\partial z} - \frac{{\sigma_{r}^{0} }}{{r{\mathbb{Q}}_{55} }}\left( {{\mathbb{Q}}_{12} - \frac{{{\mathbb{Q}}_{13} {\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{\partial }{\partial z} - \frac{{\sigma_{r}^{0} }}{r}\frac{\partial }{\partial z} - \frac{1}{r}\left( {{\mathbb{Q}}_{12} - \frac{{{\mathbb{Q}}_{13} {\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{\partial }{\partial z}, \hfill \\ \end{gathered} $$

$$ \begin{gathered} a_{61} = - \frac{1}{r}\frac{{{\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}\frac{\partial }{\partial \theta } - \frac{{\sigma_{r}^{0} }}{{r{\mathbb{Q}}_{33} }}\left( {1 + \frac{{{\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{66} }}} \right)\frac{\partial }{\partial \theta }, \hfill \\ a_{62} = \frac{{\sigma_{r}^{0} }}{r}\frac{{{\mathbb{Q}}_{13} }}{{{\mathbb{Q}}_{33} }}\frac{{\partial^{2} }}{\partial \theta \partial z} - \frac{{\sigma_{r}^{0} }}{{r{\mathbb{Q}}_{66} }}\left( {{\mathbb{Q}}_{12} + {\mathbb{Q}}_{44} - \frac{{{\mathbb{Q}}_{13} {\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{{\partial^{2} }}{\partial \theta \partial z} - \frac{1}{r}\left( {{\mathbb{Q}}_{12} + {\mathbb{Q}}_{44} - \frac{{{\mathbb{Q}}_{13} {\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{{\partial^{2} }}{\partial \theta \partial z}, \hfill \\ \end{gathered} $$

$$ \begin{gathered} a_{63} = \sigma_{z}^{0} \frac{{\partial^{2} }}{{\partial z^{2} }} + \frac{{\sigma_{\theta }^{0} }}{{r^{2} }}\left( {\frac{{\partial^{2} }}{{\partial \theta^{2} }} - 1} \right) + \frac{{\sigma_{r}^{0} }}{{r^{2} }}\frac{{{\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}\frac{{\partial^{2} }}{{\partial \theta^{2} }} + \frac{{\sigma_{r}^{0} \sigma_{z}^{0} }}{{{\mathbb{Q}}_{66} }}\frac{{\partial^{2} }}{{\partial z^{2} }} + \frac{{\sigma_{r}^{0} \sigma_{z}^{0} }}{{r^{2} {\mathbb{Q}}_{66} }}\left( {\frac{{\partial^{2} }}{{\partial \theta^{2} }} - 1} \right)\, + \frac{{\sigma_{r}^{0} }}{{{\mathbb{Q}}_{66} }}\rho \frac{{\partial^{2} }}{{\partial t^{2} }} \hfill \\ \,\,\,\,\,\,\,\,\,\,\, - \frac{{\sigma_{r}^{0} }}{{r^{2} {\mathbb{Q}}_{66} }}\left( {{\mathbb{Q}}_{22} - \frac{{{\mathbb{Q}}_{23}^{2} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{{\partial^{2} }}{{\partial \theta^{2} }} - \sigma_{r}^{0} \frac{{{\mathbb{Q}}_{44} }}{{{\mathbb{Q}}_{66} }}\frac{{\partial^{2} }}{{\partial z^{2} }} + \frac{{\sigma_{r}^{0} }}{{r^{2} }} + \rho \frac{{\partial^{2} }}{{\partial t^{2} }}\, - \frac{1}{{r^{2} }}\left( {{\mathbb{Q}}_{22} - \frac{{{\mathbb{Q}}_{23}^{2} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{{\partial^{2} }}{{\partial \theta^{2} }} - {\mathbb{Q}}_{44} \frac{{\partial^{2} }}{{\partial z^{2} }}, \hfill \\ \end{gathered} $$

$$ a_{64} = \frac{{2\sigma_{\theta }^{0} }}{{r^{2} }}\frac{{\partial^{2} }}{{\partial \theta^{2} }} - \frac{{\sigma_{r}^{0} }}{{r^{2} {\mathbb{Q}}_{66} }}\left( {{\mathbb{Q}}_{22} - \frac{{{\mathbb{Q}}_{23}^{2} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{\partial }{\partial \theta } + \frac{{2\sigma_{r}^{0} \sigma_{z}^{0} }}{{r^{2} {\mathbb{Q}}_{66} }}\frac{\partial }{\partial \theta } - \frac{{\sigma_{r}^{0} }}{{r^{2} }}\frac{\partial }{\partial \theta } + \frac{{\sigma_{r}^{0} }}{{r^{2} }}\frac{{{\mathbb{Q}}_{23} }}{{{\mathbb{Q}}_{33} }}\frac{\partial }{\partial \theta } - \frac{1}{{r^{2} }}\left( {{\mathbb{Q}}_{22} - \frac{{{\mathbb{Q}}_{23}^{2} }}{{{\mathbb{Q}}_{33} }}} \right)\frac{\partial }{\partial \theta },\;a_{66} = - \frac{{\sigma_{r}^{0} }}{{r{\mathbb{Q}}_{66} }} - \frac{2}{r}. $$

$$ {\overline{\mathbf{G}}} = \left[ {\begin{array}{*{20}c} {\frac{1}{{\overline{r}}}\left( {\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }} - 1} \right)} & {\overline{a}_{12} } & {\overline{a}_{13} } & {\overline{a}_{14} } & {\overline{a}_{15} } & {\overline{a}_{16} } \\ 0 & 0 & 0 & { - \overline{p}_{n} \frac{{R_{m} }}{L}} & {\frac{{R_{m} }}{h}\frac{1}{{\overline{\mathbb{Q}}_{55} }}} & 0 \\ 0 & 0 & {\frac{1}{{\overline{r}}}} & { - \frac{{\overline{p}_{m} }}{{\theta_{m} \overline{r}}}} & 0 & {\frac{{R_{m} }}{h}\frac{1}{{\overline{\mathbb{Q}}_{66} }}} \\ {\frac{{R_{m} }}{h}\frac{1}{{\overline{\mathbb{Q}}_{33} }}} & {\overline{p}_{n} \frac{{R_{m} }}{L}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}} & {\frac{{\overline{p}_{m} }}{{\theta_{m} \overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} & { - \frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}} & 0 & 0 \\ {\overline{a}_{51} } & {\overline{a}_{52} } & {\overline{a}_{53} } & {\overline{a}_{54} } & { - \frac{1}{{\overline{r}}}} & 0 \\ {\overline{a}_{61} } & {\overline{a}_{62} } & {\overline{a}_{63} } & {\overline{a}_{64} } & 0 & {\overline{a}_{66} } \\ \end{array} } \right] $$

$$ \overline{a}_{12} = - \frac{h}{L}\frac{{\overline{p}_{n} }}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{z} - \frac{h}{L}\overline{p}_{n} \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\left( {\frac{{\overline{\mathbb{Q}}_{12} - \overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{z} , $$

$$ \overline{a}_{13} = - \frac{h}{{R_{m} \theta_{m} }}\overline{p}_{m} \frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{\theta } + \frac{h}{{R_{m} \theta_{m} }}\frac{{2\overline{p}_{m} \overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\overline{u}_{\theta } - \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{p}_{m} \sigma_{r}^{0} }}{{\overline{r}^{2} }}\left( {\frac{{\overline{\mathbb{Q}}_{22} - \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{\theta } \, + \frac{h}{{R_{m} \theta_{m} }}\frac{{2\overline{p}_{m} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{33} }}\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} \overline{u}_{\theta } , $$

$$ \begin{gathered} \overline{a}_{14} = \frac{h}{{R_{m} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{r} - \frac{{R_{m} h}}{{L^{2} }}\overline{p}_{n}^{2} \overline{\sigma }_{z}^{0} \overline{u}_{r} - \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\left( {\frac{1}{{\theta_{m}^{2} }}\overline{p}_{m}^{2} + 1} \right)\overline{u}_{r} - \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \overline{u}_{r} \hfill \\ \,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\left( {\frac{{\overline{\mathbb{Q}}_{22} - \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{r} - \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{p}_{n}^{2} \overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{r} - \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{33} }}\left( {\frac{1}{{\theta_{m}^{2} }}\overline{p}_{m}^{2} + 1} \right)\overline{u}_{r} \hfill \\ \,\,\,\,\,\,\,\,\,\,\, - \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{r} - \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{p}_{n}^{2} \overline{\mathbb{Q}}_{13} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{r} - \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{\overline{p}_{m}^{2} \overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{r} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{r} , \hfill \\ \end{gathered} $$

$$ \overline{a}_{15} = \frac{{R_{m} }}{L}\overline{p}_{n} \overline{\tau }_{rz} + \frac{{R_{m} }}{L}\frac{{\overline{p}_{n} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\overline{\tau }_{rz} + \frac{{R_{m} }}{L}\overline{p}_{n} \overline{\sigma }_{r}^{0} \frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} \overline{\mathbb{Q}}_{55} }}\overline{\tau }_{rz} , $$

$$ \overline{a}_{16} = \frac{1}{{\theta_{m} }}\frac{{\overline{p}_{m} }}{{\overline{r}}}\overline{\tau }_{r\theta } + \frac{{R_{m} }}{{\theta_{m} }}\frac{{\overline{p}_{m} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\overline{\tau }_{r\theta } + \frac{1}{{\theta_{m} }}\frac{{\overline{p}_{m} \overline{\sigma }_{r}^{0} }}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{66} \overline{\mathbb{Q}}_{33} }}\overline{\tau }_{r\theta } , $$

$$ \overline{a}_{51} = - \frac{{R_{m} }}{L}\frac{{\overline{p}_{n} \overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\overline{\sigma }_{r} - \frac{{R_{m} }}{L}\frac{{\overline{p}_{n} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\left( {1 + \frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{55} }}} \right)\overline{\sigma }_{r} , $$

$$ \begin{gathered} \overline{a}_{52} = - \frac{{R_{m} h}}{{L^{2} }}\overline{p}_{n}^{2} \overline{\sigma }_{z}^{0} \overline{u}_{z} - \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{p}_{m}^{2} \overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\overline{u}_{z} - \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{p}_{n}^{2} \overline{\sigma }_{r}^{0} \overline{C}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{z} - \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{p}_{n}^{2} \overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\overline{u}_{z} - \left( {\frac{h}{{R_{m} }}} \right)^{3} \frac{{\overline{\omega }^{2} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\overline{u}_{z} \hfill \\ \,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{p}_{m}^{2} \overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{66} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{55} }}\overline{u}_{z} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{p}_{n}^{2} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{11} - \frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{z} - \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{p}_{m}^{2} \overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{55} }}\overline{u}_{z} - \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \overline{u}_{z} \hfill \\ \,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{p}_{m}^{2} \overline{\mathbb{Q}}_{44} }}{{\overline{r}^{2} }}\overline{u}_{z} + \frac{{R_{m} h}}{{L^{2} }}\overline{p}_{n}^{2} \left( {\overline{\mathbb{Q}}_{11} - \frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{z} , \hfill \\ \end{gathered} $$

$$ \overline{a}_{53} = - \frac{h}{{L\theta_{m} }}\frac{{\overline{p}_{n} \overline{p}_{m} \overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{23} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\overline{u}_{\theta } + \frac{h}{{L\theta_{m} }}\frac{{\overline{p}_{n} \overline{p}_{m} \overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{C}_{33} }}} \right)\overline{u}_{\theta } + \frac{h}{{L\theta_{m} }}\frac{{\overline{p}_{m} \overline{p}_{n} }}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{\theta } $$

$$ \overline{a}_{54} = \frac{h}{L}\frac{{\overline{p}_{n} \overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{23} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\overline{u}_{r} - \frac{h}{L}\frac{{\overline{p}_{n} \overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{r} - \frac{h}{L}\frac{{\overline{p}_{n} \overline{\sigma }_{r}^{0} }}{{\overline{r}}}\overline{u}_{r} - \frac{h}{L}\frac{{\overline{p}_{n} }}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{r} , $$

$$ \overline{a}_{61} = - \frac{1}{{\theta_{m} }}\frac{{\overline{p}_{m} }}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\overline{\sigma }_{r} - \frac{1}{{\theta_{m} }}\frac{{\overline{p}_{m} \overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\left( {1 + \frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{66} }}} \right)\overline{\sigma }_{r} , $$

$$ \overline{a}_{62} = - \frac{h}{{L\theta_{m} }}\overline{p}_{m} \overline{p}_{n} \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{z} + \frac{h}{{L\theta_{m} }}\overline{p}_{m} \overline{p}_{n} \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{66} }}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{z} + \frac{h}{{L\theta_{m} }}\frac{{\overline{p}_{m} \overline{p}_{n} }}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{z} $$

$$ \begin{gathered} \overline{a}_{63} = - \frac{{R_{m} h}}{{L^{2} }}\overline{p}_{n}^{2} \overline{\sigma }_{z}^{0} \overline{u}_{\theta } - \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\left( {\overline{p}_{m}^{2} + 1} \right)\overline{u}_{\theta } - \frac{h}{{R_{m} \theta_{m} }}\overline{p}_{m}^{2} \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{\theta } - \frac{{R_{m} h}}{{L^{2} }}\overline{p}_{n}^{2} \frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{66} }}\overline{u}_{\theta } , \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{{R_{m} h}}{{L^{2} }}\overline{p}_{n}^{2} \overline{\sigma }_{r}^{0} \frac{{\overline{\mathbb{Q}}_{44} }}{{\overline{\mathbb{Q}}_{66} }}\overline{u}_{\theta } + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\overline{u}_{\theta } - \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \overline{u}_{\theta } + \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{\overline{p}_{m}^{2} }}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{\theta } + \frac{{R_{m} h}}{{L^{2} }}\overline{p}_{n}^{2} \overline{\mathbb{Q}}_{44} \overline{u}_{\theta } \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{64} = - \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{2\overline{p}_{m}^{2} \sigma_{\theta }^{0} }}{{\overline{r}^{2} }}\overline{u}_{r} - \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{p}_{m} \overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{r} + \frac{h}{{R_{m} \theta_{m} }}\frac{{2\overline{p}_{m} \overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\overline{u}_{r} - \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{p}_{m} \overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\overline{u}_{r} \hfill \\ \,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{p}_{m} \overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{r} - \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{p}_{m} }}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{r} ,\;\,\overline{a}_{66} = - \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{66} }}\overline{\tau }_{r\theta } - \frac{{2\overline{\tau }_{r\theta } }}{{\overline{r}}}. \hfill \\ \end{gathered} $$

1.1 Simply–simply–simply–simply

$$ {\overline{\mathbf{G}}}_{SSSS} = \left[ {\begin{array}{*{20}c} {\overline{a}_{11}^{ssss} } & {\overline{a}_{12}^{ssss} } & {\overline{a}_{13}^{ssss} } & {\overline{a}_{14}^{ssss} } & {\overline{a}_{15}^{ssss} } & {\overline{a}_{16}^{ssss} } \\ {\overline{a}_{21}^{ssss} } & {\overline{a}_{22}^{ssss} } & {\overline{a}_{23}^{ssss} } & {\overline{a}_{24}^{ssss} } & {\overline{a}_{26}^{ssss} } & {\overline{a}_{26}^{ssss} } \\ {\overline{a}_{31}^{ssss} } & {\overline{a}_{32}^{ssss} } & {\overline{a}_{33}^{ssss} } & {\overline{a}_{34}^{ssss} } & {\overline{a}_{35}^{ssss} } & {\overline{a}_{36}^{ssss} } \\ {\overline{a}_{41}^{ssss} } & {\overline{a}_{42}^{ssss} } & {\overline{a}_{43}^{ssss} } & {\overline{a}_{44}^{ssss} } & {\overline{a}_{45}^{ssss} } & {\overline{a}_{46}^{ssss} } \\ {\overline{a}_{51}^{ssss} } & {\overline{a}_{52}^{ssss} } & {\overline{a}_{53}^{ssss} } & {\overline{a}_{54}^{ssss} } & {\overline{a}_{55}^{ssss} } & {\overline{a}_{56}^{ssss} } \\ {\overline{a}_{61}^{ssss} } & {\overline{a}_{62}^{ssss} } & {\overline{a}_{63}^{ssss} } & {\overline{a}_{64}^{ssss} } & {\overline{a}_{65}^{ssss} } & {\overline{a}_{66}^{ssss} } \\ \end{array} } \right] $$

$$ \overline{a}_{11}^{ssss} = \frac{1}{{\overline{r}}}\left( {\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }} - 1} \right)\overline{\sigma }_{rjn} ;\,\,\,\,\,\left( {\,j,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right) $$

$$ \overline{a}_{12}^{ssss} = \frac{h}{L}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{zjn} + \frac{h}{L}\frac{{\overline{\sigma }_{r}^{o} }}{{\overline{r}}}\left( {\frac{{\overline{\mathbb{Q}}_{12} - \overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{zjn} ;\,\,\,\,\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right) $$

$$ \begin{gathered} \overline{a}_{13}^{ssss} = \frac{h}{{R_{m} \theta_{m} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} - \frac{h}{{R_{m} \theta_{m} }}\frac{{2\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} + \frac{h}{{R_{m} \theta_{m} }}\frac{{\sigma_{r}^{0} }}{{\overline{r}^{2} }}\left( {\frac{{\overline{\mathbb{Q}}_{22} - \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m} }}\frac{2}{{\overline{r}^{2} \overline{\mathbb{Q}}_{33} }}\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} ;\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{14}^{ssss} = \frac{h}{{R_{m} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{rjn} + \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{z}^{0} \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{rjn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\left( {\frac{1}{{\theta_{m}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{rjn} - \overline{u}_{rjn} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \overline{u}_{rjn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\left( {\frac{{\overline{\mathbb{Q}}_{22} - \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{rjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{33} }}\left( {\frac{1}{{\theta_{m}^{2} }}\overline{P}_{m}^{2} \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{rjn} - \overline{u}_{rjn} } \right) + \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{rjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\mathbb{Q}}_{13} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{rjn} ;\,\,\,\,\,\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \overline{a}_{15}^{ssss} = - \frac{{R_{m} }}{L}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\tau }_{rzjn} - \frac{{R_{m} }}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\tau }_{rzjn} - \frac{{R_{m} }}{L}\overline{\sigma }_{r}^{0} \frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} \overline{\mathbb{Q}}_{55} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\tau }_{rzjn} ;\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) $$

$$ \overline{a}_{16}^{ssss} = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\tau }_{r\theta n} - \frac{{R_{m} }}{{\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\tau }_{r\theta n} - \frac{1}{{\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{66} \overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\tau }_{r\theta n} ;\left( \begin{gathered} \,i = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right) $$

$$ \overline{a}_{21}^{ssss} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{22}^{ssss} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{23}^{ssss} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right)} \right)}} ,\;\overline{a}_{24}^{ssss} = - \frac{{R_{m} }}{L}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{25}^{ssss} = \frac{{R_{m} }}{h}\frac{{\overline{\tau }_{rzjn} }}{{\overline{\mathbb{Q}}_{55} }}\,;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right),\;\overline{a}_{26}^{ssss} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right)} \right)}} , $$

$$ \overline{a}_{31}^{ssss} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{32}^{ssss} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right),\,\,n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{33}^{ssss} = \frac{{\overline{u}_{\theta j} }}{{\overline{r}}}\,\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right), \hfill \\ \,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right),\;\overline{a}_{34}^{ssss} = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{35}^{ssss} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right),\,\,n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{36}^{ssss} = \frac{{R_{m} }}{h}\frac{{\overline{\tau }_{r\theta jn} }}{{\overline{\mathbb{Q}}_{66} }};\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{41}^{ssss} = \frac{{R_{m} }}{h}\frac{1}{{\overline{\mathbb{Q}}_{33} }}\overline{\sigma }_{rjn} ;\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{42}^{ssss} = - \frac{{R_{m} }}{L}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{zjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{43}^{ssss} = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} ;\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{44}^{ssss} = - \frac{1}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{rjn} ;\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right),\overline{a}_{45}^{ssss} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{45}^{ssss} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{46}^{ssss} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right)} \right)}} , $$

$$ \overline{a}_{51}^{ssss} = - \frac{{R_{m} }}{L}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\sigma }_{rjn} - \frac{{R_{m} }}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\left( {1 + \frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{55} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\sigma }_{rjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \begin{gathered} \overline{a}_{52}^{ssss} = \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{z}^{0} \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} + \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {\frac{h}{{R_{m} }}} \right)^{3} \frac{{\overline{\omega }^{2} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\overline{u}_{zjn} - \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{66} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{55} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} - \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{11} - \frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{55} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} - \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \overline{u}_{zjn} - \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\mathbb{Q}}_{44} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} - \frac{{R_{m} h}}{{L^{2} }}\left( {\overline{\mathbb{Q}}_{11} - \frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{{R_{m} h}}{{L^{2} }}\left( {\overline{\mathbb{Q}}_{11} - \frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\left( {\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{i1} \overline{g}_{1j} \overline{u}_{zjn} + \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{{iN_{z} }} \overline{g}_{{N_{z} j}} \overline{u}_{zjn} } } } \right);\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{53}^{ssss} = \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{23} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{\theta jn} - \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{L\theta_{m} }}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{\theta jn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{54}^{ssss} = \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{23} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} - \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} - \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{L}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \overline{a}_{55}^{ssss} = - \frac{{\overline{\tau }_{rzjn} }}{{\overline{r}}};\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right),\;\overline{a}_{56}^{ssss} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right),\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right)} \right)}} $$

$$ \overline{a}_{61}^{ssss} = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\sigma }_{rjn} - \frac{1}{{\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\left( {1 + \frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{66} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\sigma }_{rjn} ;\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \begin{gathered} \overline{a}_{62}^{ssss} = \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{zjn} - \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{66} }}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{L\theta_{m} }}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{zjn} ;\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right), \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{63\,}^{ssss} = \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{z}^{0} \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{\theta jn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\left( {\frac{1}{{\theta_{m}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} - \overline{u}_{\theta jn} } \right) + \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{66} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{\theta jn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\left( {\frac{1}{{\theta_{m}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} - \overline{u}_{\theta jn} } \right) - \left( {\frac{h}{{R_{m} }}} \right)^{3} \frac{{\overline{\omega }^{2} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{66} }}\overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} - \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{r}^{0} \frac{{\overline{\mathbb{Q}}_{44} }}{{\overline{\mathbb{Q}}_{66} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{\theta jn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\left( {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{m1} \overline{h}_{1n} } \overline{u}_{\theta jn} + \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{{mN_{\theta } }} \overline{h}_{{N_{\theta } n}} } \overline{u}_{\theta jn} } \right)\,;\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{64}^{ssss} = \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{2\sigma_{\theta }^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{rjn} - \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} + \frac{h}{{R_{m} \theta_{m} }}\frac{{2\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} + \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} \hfill \\ \end{gathered} $$

$$ \overline{a}_{65}^{ssss} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right),n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{66}^{ssss} = - \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{66} }}\overline{\tau }_{r\theta jn} - \frac{{2\overline{\tau }_{r\theta jn} }}{{\overline{r}}};\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right) $$

1.2 Clamped–clamped–clamped–clamped

$$ {\overline{\mathbf{G}}}_{CCCC} = \left[ {\begin{array}{*{20}c} {\overline{a}_{11}^{cccc} } & {\overline{a}_{12}^{cccc} } & {\overline{a}_{13}^{cccc} } & {\overline{a}_{14}^{cccc} } & {\overline{a}_{15}^{cccc} } & {\overline{a}_{16}^{cccc} } \\ {\overline{a}_{21}^{cccc} } & {\overline{a}_{22}^{cccc} } & {\overline{a}_{23}^{cccc} } & {\overline{a}_{24}^{cccc} } & {\overline{a}_{26}^{cccc} } & {\overline{a}_{26}^{cccc} } \\ {\overline{a}_{31}^{cccc} } & {\overline{a}_{32}^{cccc} } & {\overline{a}_{33}^{cccc} } & {\overline{a}_{34}^{cccc} } & {\overline{a}_{35}^{cccc} } & {\overline{a}_{36}^{cccc} } \\ {\overline{a}_{41}^{cccc} } & {\overline{a}_{42}^{cccc} } & {\overline{a}_{43}^{cccc} } & {\overline{a}_{44}^{cccc} } & {\overline{a}_{45}^{cccc} } & {\overline{a}_{46}^{cccc} } \\ {\overline{a}_{51}^{cccc} } & {\overline{a}_{52}^{cccc} } & {\overline{a}_{53}^{cccc} } & {\overline{a}_{54}^{cccc} } & {\overline{a}_{55}^{cccc} } & {\overline{a}_{56}^{cccc} } \\ {\overline{a}_{61}^{cccc} } & {\overline{a}_{62}^{cccc} } & {\overline{a}_{63}^{cccc} } & {\overline{a}_{64}^{cccc} } & {\overline{a}_{65}^{cccc} } & {\overline{a}_{66}^{cccc} } \\ \end{array} } \right] $$

$$ \overline{a}_{11}^{cccc} = \frac{1}{{\overline{r}}}\left( {\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }} - 1} \right)\overline{\sigma }_{rjn} ;\,\,\,\,\,\left( {\,j,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right), $$

$$ \overline{a}_{12}^{cccc} = \frac{h}{L}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{zjn} + \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\left( {\frac{{\overline{\mathbb{Q}}_{12} - \overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{zjn} ;\,\,\,\,\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) $$

$$ \begin{gathered} \overline{a}_{13}^{cccc} = \frac{h}{{R_{m} \theta_{m} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} - \frac{h}{{R_{m} \theta_{m} }}\frac{{2\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} \theta_{m} }}\frac{{\sigma_{r}^{0} }}{{\overline{r}^{2} }}\left( {\frac{{\overline{\mathbb{Q}}_{22} - \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} - \frac{h}{{R_{m} \theta_{m} }}\frac{2}{{\overline{r}^{2} \overline{\mathbb{Q}}_{33} }}\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} ;\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{14}^{cccc} = \frac{h}{{R_{m} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{rjn} + \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{z}^{0} \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{rjn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\left( {\frac{1}{{\theta_{m}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{rjn} - \overline{u}_{rjn} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \overline{u}_{rjn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\left( {\frac{{\overline{\mathbb{Q}}_{22} - \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{rjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{33} }}\left( {\frac{1}{{\theta_{m}^{2} }}\overline{P}_{m}^{2} \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{rjn} - \overline{u}_{rjn} } \right) + \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{rjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\mathbb{Q}}_{13} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{\overline{\mathbb{Q}}_{66} }}{{\overline{r}^{2} }}\left( {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{m1} \overline{h}_{1n} } \overline{u}_{\theta jn} + \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{{mN_{\theta } }} \overline{h}_{{N_{\theta } n}} } \overline{u}_{\theta jn} } \right)\,;\,\,\,\,\,\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \overline{a}_{15}^{cccc} = - \frac{{R_{m} }}{L}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\tau }_{rzjn} - \frac{{R_{m} }}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\tau }_{rzjn} - \frac{{R_{m} }}{L}\overline{\sigma }_{r}^{0} \frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} \overline{\mathbb{Q}}_{55} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\tau }_{rzjn} ;\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) $$

$$ \overline{a}_{16}^{cccc} = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\tau }_{r\theta n} - \frac{{R_{m} }}{{\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\tau }_{r\theta n} - \frac{1}{{\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{66} \overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\tau }_{r\theta n} ;\left( \begin{gathered} \,i = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) $$

$$ \overline{a}_{21}^{cccc} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{22}^{cccc} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{23}^{cccc} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{24}^{cccc} = - \frac{{R_{m} }}{L}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{25}^{cccc} = \frac{{R_{m} }}{h}\frac{{\overline{\tau }_{rzjn} }}{{\overline{\mathbb{Q}}_{55} }}\,;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right),\;\overline{a}_{26}^{cccc} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{31}^{cccc} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{32}^{cccc} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{33}^{cccc} = \frac{{\overline{u}_{\theta j} }}{{\overline{r}}}\,\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{34}^{cccc} = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{35}^{cccc} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\,\overline{a}_{36}^{cccc} = \frac{{R_{m} }}{h}\frac{{\overline{\tau }_{r\theta jn} }}{{\overline{\mathbb{Q}}_{66} }};\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{41}^{cccc} = \frac{{R_{m} }}{h}\frac{1}{{\overline{\mathbb{Q}}_{33} }}\overline{\sigma }_{rjn} ;\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{42}^{cccc} = - \frac{{R_{m} }}{L}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{zjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{43}^{cccc} = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} ;\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{44}^{cccc} = - \frac{1}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{rjn} ;\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right),\;\overline{a}_{45}^{cccc} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{46}^{cccc} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{51}^{cccc} = - \frac{{R_{m} }}{L}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\sigma }_{rjn} - \frac{{R_{m} }}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\left( {1 + \frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{55} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\sigma }_{rjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \begin{gathered} \overline{a}_{52}^{cccc} = \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{z}^{0} \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} + \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {\frac{h}{{R_{m} }}} \right)^{3} \frac{{\overline{\omega }^{2} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\overline{u}_{zjn} - \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{66} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{55} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} - \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{11} - \frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{55} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} - \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \overline{u}_{zjn} - \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\mathbb{Q}}_{44} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} - \frac{{R_{m} h}}{{L^{2} }}\left( {\overline{\mathbb{Q}}_{11} - \frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}\left( {\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{i1} \overline{g}_{1j} \overline{u}_{zjn} + \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{{iN_{z} }} \overline{g}_{{N_{z} j}} \overline{u}_{zjn} } } } \right);\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{53}^{cccc} = \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{23} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{\theta jn} - \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{L\theta_{m} }}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{\theta jn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{54}^{cccc} = \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{23} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} - \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} - \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{L}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \overline{a}_{55}^{cccc} = - \frac{{\overline{\tau }_{rzjn} }}{{\overline{r}}};\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right),\;\overline{a}_{56}^{cccc} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} $$

$$ \overline{a}_{61}^{cccc} = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\sigma }_{rjn} - \frac{1}{{\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\left( {1 + \frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{66} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\sigma }_{rjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \begin{gathered} \overline{a}_{62}^{cccc} = \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{zjn} - \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{66} }}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{L\theta_{m} }}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{zjn} ;\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{63\,}^{cccc} = \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{z}^{0} \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{\theta jn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\left( {\frac{1}{{\theta_{m}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} - \overline{u}_{\theta jn} } \right) + \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{66} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{\theta jn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\left( {\frac{1}{{\theta_{m}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} - \overline{u}_{\theta jn} } \right) - \left( {\frac{h}{{R_{m} }}} \right)^{3} \frac{{\overline{\omega }^{2} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{66} }}\overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} \overline{C}_{66} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} - \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{r}^{0} \frac{{\overline{\mathbb{Q}}_{44} }}{{\overline{\mathbb{Q}}_{66} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{\theta jn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \overline{u}_{\theta jn} - \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} - \frac{{R_{m} h}}{{L^{2} }}\overline{\mathbb{Q}}_{44} \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{1}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}\left( {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{m1} \overline{h}_{1n} } \overline{u}_{\theta jn} + \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{{mN_{\theta } }} \overline{h}_{{N_{\theta } n}} } \overline{u}_{\theta jn} } \right);\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$\begin{gathered} \overline{a}_{64}^{cccc} = \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{2\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{rjn} - \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} + \frac{h}{{R_{m} \theta_{m} }}\frac{{2\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} + \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} ;\,\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered}$,

$$ \overline{a}_{65}^{cccc} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{66}^{cccc} = - \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{66} }}\overline{\tau }_{r\theta jn} - \frac{{2\overline{\tau }_{r\theta jn} }}{{\overline{r}}};\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right). $$

1.3 Clamped–simply–clamped–simply

$$ {\overline{\mathbf{G}}}_{CSCS} = \left[ {\begin{array}{*{20}c} {\overline{a}_{11}^{\csc s} } & {\overline{a}_{12}^{\csc s} } & {\overline{a}_{13}^{\csc s} } & {\overline{a}_{14}^{\csc s} } & {\overline{a}_{15}^{\csc s} } & {\overline{a}_{16}^{\csc s} } \\ {\overline{a}_{21}^{\csc s} } & {\overline{a}_{22}^{\csc s} } & {\overline{a}_{23}^{\csc s} } & {\overline{a}_{24}^{\csc s} } & {\overline{a}_{26}^{\csc s} } & {\overline{a}_{26}^{\csc s} } \\ {\overline{a}_{31}^{\csc s} } & {\overline{a}_{32}^{\csc s} } & {\overline{a}_{33}^{\csc s} } & {\overline{a}_{34}^{\csc s} } & {\overline{a}_{35}^{\csc s} } & {\overline{a}_{36}^{\csc s} } \\ {\overline{a}_{41}^{\csc s} } & {\overline{a}_{42}^{\csc s} } & {\overline{a}_{43}^{\csc s} } & {\overline{a}_{44}^{\csc s} } & {\overline{a}_{45}^{\csc s} } & {\overline{a}_{46}^{\csc s} } \\ {\overline{a}_{51}^{\csc s} } & {\overline{a}_{52}^{\csc s} } & {\overline{a}_{53}^{\csc s} } & {\overline{a}_{54}^{\csc s} } & {\overline{a}_{55}^{\csc s} } & {\overline{a}_{56}^{\csc s} } \\ {\overline{a}_{61}^{\csc s} } & {\overline{a}_{62}^{\csc s} } & {\overline{a}_{63}^{\csc s} } & {\overline{a}_{64}^{\csc s} } & {\overline{a}_{65}^{\csc s} } & {\overline{a}_{66}^{\csc s} } \\ \end{array} } \right] $$

$$ \overline{a}_{11}^{\csc s} = \frac{1}{{\overline{r}}}\left( {\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }} - 1} \right)\overline{\sigma }_{rjn} ;\,\,\,\,\,\left( {\,j,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right) $$

$$ \overline{a}_{12}^{\csc s} = \frac{h}{L}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{zjn} + \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\left( {\frac{{\overline{\mathbb{Q}}_{12} - \overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{zjn} ;\,\,\,\,\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right) $$

$$ \begin{gathered} \overline{a}_{13}^{\csc s} = \frac{h}{{R_{m} \theta_{m} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} - \frac{h}{{R_{m} \theta_{m} }}\frac{{2\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\left( {\frac{{\overline{\mathbb{Q}}_{22} - \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} - \frac{h}{{R_{m} \theta_{m} }}\frac{2}{{\overline{r}^{2} \overline{\mathbb{Q}}_{33} }}\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} ;\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{14}^{\csc s} = \frac{h}{{R_{m} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{rjn} + \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{z}^{0} \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{rjn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\left( {\frac{1}{{\theta_{m}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{rjn} - \overline{u}_{rjn} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \overline{u}_{rjn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\left( {\frac{{\overline{\mathbb{Q}}_{22} - \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{rjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{rjn} + \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{33} }}\left( {\frac{1}{{\theta_{m}^{2} }}\overline{P}_{m}^{2} \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{rjn} - \overline{u}_{rjn} } \right) + \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{rjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\mathbb{Q}}_{13} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{rjn} - \frac{{R_{m} h}}{{L^{2} }}\overline{\mathbb{Q}}_{55} \left( {\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{i1} \overline{g}_{1j} \overline{u}_{rjn} + \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{{iN_{z} }} \overline{g}_{{N_{z} j}} \overline{u}_{rjn} } } } \right);\,\,\,\,\,\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \overline{a}_{15}^{\csc s} = - \frac{{R_{m} }}{L}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\tau }_{rzjn} - \frac{{R_{m} }}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\tau }_{rzjn} - \frac{{R_{m} }}{L}\overline{\sigma }_{r}^{0} \frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} \overline{\mathbb{Q}}_{55} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\tau }_{rzjn} ;\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) $$

$$ \overline{a}_{16}^{\csc s} = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\tau }_{r\theta n} - \frac{{R_{m} }}{{\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\tau }_{r\theta n} - \frac{1}{{\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{66} \overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\tau }_{r\theta n} ;\left( \begin{gathered} \,i = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right) $$

$$ \overline{a}_{21}^{\csc s} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{22}^{\csc s} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{23}^{\csc s} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right)} \right)}} ,\;\overline{a}_{24}^{\csc s} = - \frac{{R_{m} }}{L}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{25}^{\csc s} = \frac{{R_{m} }}{h}\frac{{\overline{\tau }_{rzjn} }}{{\overline{\mathbb{Q}}_{55} }}\,;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right),\;\overline{a}_{26}^{\csc s} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right)} \right)}} , $$

$$ \overline{a}_{31}^{\csc s} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{32}^{\csc s} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{33}^{\csc s} = \frac{{\overline{u}_{\theta j} }}{{\overline{r}}}\,\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right), \hfill \\ \,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right),\;\overline{a}_{34}^{\csc s} = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) $$

$$ \overline{a}_{35}^{\csc s} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{36}^{\csc s} = \frac{{R_{m} }}{h}\frac{{\overline{\tau }_{r\theta jn} }}{{\overline{\mathbb{Q}}_{66} }};\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{41}^{\csc s} = \frac{{R_{m} }}{h}\frac{1}{{\overline{\mathbb{Q}}_{33} }}\overline{\sigma }_{rjn} ;\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{42}^{\csc s} = - \frac{{R_{m} }}{L}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{zjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{43}^{\csc s} = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} ;\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{44}^{\csc s} = - \frac{1}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{rjn} ;\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right),\;\overline{a}_{45}^{\csc s} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{46}^{\csc s} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right)} \right)}} , $$

$$ \overline{a}_{51}^{\csc s} = - \frac{{R_{m} }}{L}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\sigma }_{rjn} - \frac{{R_{m} }}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\left( {1 + \frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{55} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\sigma }_{rjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \begin{gathered} \overline{a}_{52}^{\csc s} = \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{z}^{0} \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} + \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {\frac{h}{{R_{m} }}} \right)^{3} \frac{{\overline{\omega }^{2} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\overline{u}_{zjn} - \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{66} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{55} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} - \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{11} - \frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{55} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} - \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \overline{u}_{zjn} - \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\mathbb{Q}}_{44} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} - \frac{{R_{m} h}}{{L^{2} }}\left( {\overline{\mathbb{Q}}_{11} - \frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}\left( {\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{i1} \overline{g}_{1j} \overline{u}_{zjn} + \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{{iN_{z} }} \overline{g}_{{N_{z} j}} \overline{u}_{zjn} } } } \right);\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{53}^{\csc s} = \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{23} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{\theta jn} - \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{L\theta_{m} }}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{\theta jn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{54}^{\csc s} = \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{23} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} - \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} - \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{L}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \overline{a}_{55}^{\csc s} = - \frac{{\overline{\tau }_{rzjn} }}{{\overline{r}}};\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right),\;\overline{a}_{56}^{\csc s} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right)} \right)}} $$

$$ \overline{a}_{61}^{\csc s} = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\sigma }_{rjn} - \frac{1}{{\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\left( {1 + \frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{66} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\sigma }_{rjn} ;\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \begin{gathered} \overline{a}_{62}^{\csc s} = \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{zjn} - \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{66} }}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{L\theta_{m} }}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{zjn} ;\,\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{63\,}^{\csc s} = \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{z}^{0} \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{\theta jn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\left( {\frac{1}{{\theta_{m}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} - \overline{u}_{\theta jn} } \right) + \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{66} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{\theta jn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\left( {\frac{1}{{\theta_{m}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} - \overline{u}_{\theta jn} } \right) - \left( {\frac{h}{{R_{m} }}} \right)^{3} \frac{{\overline{\omega }^{2} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{66} }}\overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} - \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{r}^{0} \frac{{\overline{\mathbb{Q}}_{44} }}{{\overline{\mathbb{Q}}_{66} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{\theta jn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\left( {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{m1} \overline{h}_{1n} } \overline{u}_{\theta jn} + \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{{mN_{\theta } }} \overline{h}_{{N_{\theta } n}} } \overline{u}_{\theta jn} } \right);\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{64}^{\csc s} = \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{2\sigma_{\theta }^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{rjn} - \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} + \frac{h}{{R_{m} \theta_{m} }}\frac{{2\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} + \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} ;\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \overline{a}_{65}^{\csc s} = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right),n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{66}^{\csc s} = - \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{66} }}\overline{\tau }_{r\theta jn} - \frac{{2\overline{\tau }_{r\theta jn} }}{{\overline{r}}};\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right). $$

1.4 Simply–clamped–simply–clamped

$$ {\overline{\mathbf{G}}}_{SCSC} = \left[ {\begin{array}{*{20}c} {\overline{a}_{11}^{s\csc } } & {\overline{a}_{12}^{s\csc } } & {\overline{a}_{13}^{s\csc } } & {\overline{a}_{14}^{s\csc } } & {\overline{a}_{15}^{s\csc } } & {\overline{a}_{16}^{s\csc } } \\ {\overline{a}_{21}^{s\csc } } & {\overline{a}_{22}^{s\csc } } & {\overline{a}_{23}^{s\csc } } & {\overline{a}_{24}^{s\csc } } & {\overline{a}_{26}^{s\csc } } & {\overline{a}_{26}^{s\csc } } \\ {\overline{a}_{31}^{s\csc } } & {\overline{a}_{32}^{s\csc } } & {\overline{a}_{33}^{s\csc } } & {\overline{a}_{34}^{s\csc } } & {\overline{a}_{35}^{s\csc } } & {\overline{a}_{36}^{s\csc } } \\ {\overline{a}_{41}^{s\csc } } & {\overline{a}_{42}^{s\csc } } & {\overline{a}_{43}^{s\csc } } & {\overline{a}_{44}^{s\csc } } & {\overline{a}_{45}^{s\csc } } & {\overline{a}_{46}^{s\csc } } \\ {\overline{a}_{51}^{s\csc } } & {\overline{a}_{52}^{s\csc } } & {\overline{a}_{53}^{s\csc } } & {\overline{a}_{54}^{s\csc } } & {\overline{a}_{55}^{s\csc } } & {\overline{a}_{56}^{s\csc } } \\ {\overline{a}_{61}^{s\csc } } & {\overline{a}_{62}^{s\csc } } & {\overline{a}_{63}^{s\csc } } & {\overline{a}_{64}^{s\csc } } & {\overline{a}_{65}^{s\csc } } & {\overline{a}_{66}^{s\csc } } \\ \end{array} } \right] $$

$$ \overline{a}_{11}^{s\csc } = \frac{1}{{\overline{r}}}\left( {\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }} - 1} \right)\overline{\sigma }_{rjn} ;\,\,\,\,\,\left( {\,j,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right), $$

$$ \overline{a}_{12}^{s\csc } = \frac{h}{L}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{zjn} + \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\left( {\frac{{\overline{\mathbb{Q}}_{12} - \overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{zjn} ;\,\,\,\,\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) $$

$$ \begin{gathered} \overline{a}_{13}^{s\csc } = \frac{h}{{R_{m} \theta_{m} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} - \frac{h}{{R_{m} \theta_{m} }}\frac{{2\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} \theta_{m} }}\frac{{\sigma_{r}^{0} }}{{\overline{r}^{2} }}\left( {\frac{{\overline{\mathbb{Q}}_{22} - \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} - \frac{h}{{R_{m} \theta_{m} }}\frac{2}{{\overline{r}^{2} \overline{\mathbb{Q}}_{33} }}\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} ;\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{14}^{s\csc } = \frac{h}{{R_{m} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\overline{u}_{rjn} + \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{z}^{0} \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{rjn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\left( {\frac{1}{{\theta_{m}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{rjn} - \overline{u}_{rjn} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{33} }}\left( {\frac{1}{{\theta_{m}^{2} }}\overline{P}_{m}^{2} \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{rjn} - \overline{u}_{rjn} } \right) + \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{rjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\mathbb{Q}}_{13} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{rjn} - \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{\overline{\mathbb{Q}}_{66} }}{{\overline{r}^{2} }}\left( {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{m1} \overline{h}_{1n} } \overline{u}_{\theta jn} + \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{{mN_{\theta } }} \overline{h}_{{N_{\theta } n}} } \overline{u}_{\theta jn} } \right);\,\,\,\,\,\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \overline{a}_{15}^{s\csc } = - \frac{{R_{m} }}{L}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\tau }_{rzjn} - \frac{{R_{m} }}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\tau }_{rzjn} - \frac{{R_{m} }}{L}\overline{\sigma }_{r}^{0} \frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} \overline{\mathbb{Q}}_{55} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\tau }_{rzjn} ;\left( \begin{gathered} \,j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) $$

$$ \overline{a}_{16}^{s\csc } = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\tau }_{r\theta n} - \frac{{R_{m} }}{{\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\tau }_{r\theta n} - \frac{1}{{\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{66} \overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\tau }_{r\theta n} ;\left( \begin{gathered} \,i = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) $$

$$ \overline{a}_{21}^{s\csc } = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{22}^{s\csc } = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{23}^{s\csc } = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{24}^{s\csc } = - \frac{{R_{m} }}{L}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{25}^{s\csc } = \frac{{R_{m} }}{h}\frac{{\overline{\tau }_{rzjn} }}{{\overline{\mathbb{Q}}_{55} }}\,;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right),\;\overline{a}_{26}^{s\csc } = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{31}^{s\csc } = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{32}^{s\csc } = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{33}^{s\csc } = \frac{{\overline{u}_{\theta j} }}{{\overline{r}}}\,\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ \,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{34}^{s\csc } = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{35}^{s\csc } = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right)} \right)}} ,\;\overline{a}_{36}^{s\csc } = \frac{{R_{m} }}{h}\frac{{\overline{\tau }_{r\theta jn} }}{{\overline{\mathbb{Q}}_{66} }};\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{41}^{s\csc } = \frac{{R_{m} }}{h}\frac{1}{{\overline{\mathbb{Q}}_{33} }}\overline{\sigma }_{rjn} ;\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{42}^{s\csc } = - \frac{{R_{m} }}{L}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{zjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{43}^{s\csc } = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{\theta jn} ;\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \overline{a}_{44}^{s\csc } = - \frac{1}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\overline{u}_{rjn} ;\,\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\, \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right),\;\overline{a}_{45}^{s\csc } = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{46}^{s\csc } = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),\,\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{51}^{s\csc } = - \frac{{R_{m} }}{L}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\sigma }_{rjn} - \frac{{R_{m} }}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{33} }}\left( {1 + \frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{55} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{\sigma }_{rjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \begin{gathered} \overline{a}_{52}^{s\csc } = \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{z}^{0} \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} + \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {\frac{h}{{R_{m} }}} \right)^{3} \frac{{\overline{\omega }^{2} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\overline{u}_{zjn} - \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{66} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{55} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} - \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{11} - \frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{55} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} - \left( {\frac{h}{{R_{m} }}} \right)^{3} \overline{\omega }^{2} \overline{u}_{zjn} - \frac{h}{{\theta_{m}^{2} R_{m} }}\frac{{\overline{\mathbb{Q}}_{44} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{zjn} - \frac{{R_{m} h}}{{L^{2} }}\left( {\overline{\mathbb{Q}}_{11} - \frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{{R_{m} h}}{{L^{2} }}\left( {\overline{\mathbb{Q}}_{11} - \frac{{\overline{\mathbb{Q}}_{13}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\left( {\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{i1} \overline{g}_{1j} \overline{u}_{zjn} + \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{{iN_{z} }} \overline{g}_{{N_{z} j}} \overline{u}_{zjn} } } } \right);\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{53}^{s\csc } = \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{23} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{\theta jn} - \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{L\theta_{m} }}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{\theta jn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } } \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{54}^{s\csc } = \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} \overline{\mathbb{Q}}_{23} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} - \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{55} }}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} - \frac{h}{L}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{L}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij} } \overline{u}_{rjn} ;\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \overline{a}_{55}^{s\csc } = - \frac{{\overline{\tau }_{rzjn} }}{{\overline{r}}};\,\,\left( \begin{gathered} j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right),\;\overline{a}_{56}^{s\csc } = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right),\,n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right)} \right)}} $$

$$ \overline{a}_{61}^{s\csc } = - \frac{1}{{\theta_{m} }}\frac{1}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\sigma }_{rjn} - \frac{1}{{\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{33} }}\left( {1 + \frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{66} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{\sigma }_{rjn} ;\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), $$

$$ \begin{gathered} \overline{a}_{62}^{s\csc } = \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}}}\frac{{\overline{\mathbb{Q}}_{13} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{zjn} - \frac{h}{{L\theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{66} }}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{zjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{L\theta_{m} }}\frac{1}{{\overline{r}}}\left( {\overline{\mathbb{Q}}_{12} + \overline{\mathbb{Q}}_{44} - \frac{{\overline{\mathbb{Q}}_{13} \overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{j = 1}^{{N_{z} }} {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{g}_{ij} \overline{h}_{mn} } } \overline{u}_{zjn} ;\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right), \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{63\,}^{s\csc } = \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{z}^{0} \sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{\theta jn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\left( {\frac{1}{{\theta_{m}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} - \overline{u}_{\theta jn} } \right) + \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{{R_{m} h}}{{L^{2} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{\mathbb{Q}}_{66} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{\theta jn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\left( {\frac{1}{{\theta_{m}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} - \overline{u}_{\theta jn} } \right) - \left( {\frac{h}{{R_{m} }}} \right)^{3} \frac{{\overline{\omega }^{2} \overline{\sigma }_{r}^{0} }}{{\overline{\mathbb{Q}}_{66} }}\overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{\theta jn} - \frac{{R_{m} h}}{{L^{2} }}\overline{\sigma }_{r}^{0} \frac{{\overline{\mathbb{Q}}_{44} }}{{\overline{\mathbb{Q}}_{66} }}\sum\limits_{j = 1}^{{N_{z} }} {\overline{g}_{ij}^{\left( 2 \right)} } \overline{u}_{\theta jn} + \frac{h}{{R_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\overline{u}_{\theta jn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{1}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}\left( {\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{m1} \overline{h}_{1n} } \overline{u}_{\theta jn} + \sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{{mN_{\theta } }} \overline{h}_{{N_{\theta } n}} } \overline{u}_{\theta jn} } \right);\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \begin{gathered} \overline{a}_{64}^{s\csc } = \frac{h}{{R_{m} \theta_{m}^{2} }}\frac{{2\overline{\sigma }_{\theta }^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn}^{\left( 2 \right)} } \overline{u}_{rjn} - \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} + \frac{h}{{R_{m} \theta_{m} }}\frac{{2\overline{\sigma }_{r}^{0} \overline{\sigma }_{z}^{0} }}{{\overline{r}^{2} \overline{\mathbb{Q}}_{66} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} + \frac{h}{{R_{m} \theta_{m} }}\frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}^{2} }}\frac{{\overline{\mathbb{Q}}_{23} }}{{\overline{\mathbb{Q}}_{33} }}\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - \frac{h}{{R_{m} \theta_{m} }}\frac{1}{{\overline{r}^{2} }}\left( {\overline{\mathbb{Q}}_{22} - \frac{{\overline{\mathbb{Q}}_{23}^{2} }}{{\overline{\mathbb{Q}}_{33} }}} \right)\sum\limits_{n = 1}^{{N_{\theta } }} {\overline{h}_{mn} } \overline{u}_{rjn} ;\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \overline{a}_{65}^{s\csc } = \left[ 0 \right]_{{\left( {j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right),n = 1,...,\left( {N_{z} } \right) \times \left( {N_{\theta } - 2} \right)} \right)}} , $$

$$ \overline{a}_{66}^{s\csc } = - \frac{{\overline{\sigma }_{r}^{0} }}{{\overline{r}\overline{\mathbb{Q}}_{66} }}\overline{\tau }_{r\theta jn} - \frac{{2\overline{\tau }_{r\theta jn} }}{{\overline{r}}};\left( \begin{gathered} j = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right), \hfill \\ n = 1,...,\left( {N_{z} - 2} \right) \times \left( {N_{\theta } - 2} \right) \hfill \\ \end{gathered} \right). $$