Appendix
The detailed expressions of k
ij
are given as follows:
$$ \begin{aligned} k_{11} & = 2\pi \left( { - \frac{31}{90}\rho h\omega^{2} L^{10} \sin \alpha + \frac{1}{8}\left( { - A_{22} + 16A_{11} + 6\rho h\omega^{2} L^{2} } \right)L^{8} \sin \alpha + \frac{1}{7}\left( { - 50A_{11} L + \frac{{4A_{66} n^{2} L}}{{\sin^{2} \alpha }} - 4\rho h\omega^{2} L^{3} } \right.} \right. \\ & \left. {\quad +\,4A_{22} L} \right)L^{7} \sin \alpha + \frac{1}{6}\left( { - \frac{{6A_{66} n^{2} L^{2} }}{{\sin^{2} \alpha }} + 56A_{11} L^{2} - 6A_{22} L^{2} + L^{4} \rho h\omega^{2} } \right)L^{6} \sin \alpha + \frac{1}{5}\left( { - 26L^{3} A_{11} + 4L^{3} A_{22} } \right. \\ & \quad \left. {\left. { +\, \frac{{4A_{66} n^{2} L^{3} }}{{\sin^{2} \alpha }}} \right)L^{5} \sin \alpha + \frac{1}{4}\left( { - L^{4} A_{22} + 4L^{4} A_{11} - \frac{{A_{66} n^{2} L^{4} }}{{\sin^{2} \alpha }}} \right)L^{4} \sin \alpha } \right), \\ \end{aligned} $$
$$ \begin{aligned} k_{12} & = 2\pi \left( {\frac{1}{8}\left( { + \frac{{4A_{12} n}}{\sin \alpha } - \frac{{A_{22} n}}{\sin \alpha } + \frac{{3A_{66} n}}{\sin \alpha }} \right)L^{8} \sin \alpha + \frac{1}{7}\left( {\frac{{4A_{22} nL}}{\sin \alpha } - \frac{{14A_{12} nL}}{\sin \alpha } - \frac{{10A_{66} nL}}{\sin \alpha }} \right)L^{7} \sin \alpha + \frac{1}{6}\left( { - \frac{{6A_{22} nL^{2} }}{\sin \alpha }} \right.} \right. \\ & \quad \left. { + \frac{{12A_{66} nL^{2} }}{\sin \alpha } + \frac{{18A_{12} nL^{2} }}{\sin \alpha }} \right)L^{6} \sin \alpha + \frac{1}{5}\left( { - \frac{{10A_{12} nL^{3} }}{\sin \alpha } + \frac{{4A_{22} nL^{3} }}{\sin \alpha } - \frac{{6A_{66} nL^{3} }}{\sin \alpha }} \right)L^{5} \sin \alpha + \frac{1}{4}\left( { + \frac{{2A_{12} nL^{4} }}{\sin \alpha } - \frac{{A_{22} nL^{4} }}{\sin \alpha }} \right. \\ & \quad \left. {\left. { + \frac{{A_{66} nL^{4} }}{\sin \alpha }} \right)L^{4} \sin \alpha } \right), \\ \end{aligned} $$
$$ \begin{aligned} k_{13} & = 2\pi \left( {\frac{1}{8}\left( {\frac{{4A_{12} }}{\tan \alpha } - \frac{{A_{22} }}{\tan \alpha }} \right)L^{8} \sin \alpha + \frac{1}{7}\left( { - \frac{{14A_{12} L}}{\tan \alpha } + \frac{{4A_{22} a_{3} L}}{\tan \alpha }} \right)L^{7} \sin \alpha + \frac{1}{6}\left( { - \frac{{6A_{22} L^{2} }}{\tan \alpha } + \frac{{18A_{12} L^{2} }}{\tan \alpha }} \right)L^{6} \sin \alpha } \right. \\ & \quad \left. { + \frac{1}{5}\left( {\frac{{4A_{22} L^{3} }}{\tan \alpha } - \frac{{10A_{12} L^{3} }}{\tan \alpha }} \right)L^{5} \sin \alpha + \frac{1}{4}\left( { + \frac{{2A_{12} L^{4} }}{\tan \alpha } - \frac{{A_{22} L^{4} }}{\tan \alpha }} \right)L^{4} \sin \alpha } \right), \\ k_{21} & = 2\pi \left( {\frac{1}{8}\left( { - \frac{{A_{22} n}}{\sin \alpha } - \frac{{4A_{12} n}}{\sin \alpha } - \frac{{5A_{66} n}}{\sin \alpha }} \right)L^{8} \sin \alpha + \frac{1}{7}\left( {\frac{{4A_{22} Ln}}{\sin \alpha } + \frac{{18A_{66} Ln}}{\sin \alpha } + \frac{{14A_{12} Ln}}{\sin \alpha }} \right)L^{7} \sin \alpha + \frac{1}{6}\left( { - \frac{{6A_{22} L^{2} n}}{\sin \alpha }} \right.} \right. \\ & \quad \left. { - \frac{{18A_{12} nL^{2} }}{\sin \alpha } - \frac{{24A_{66} nL^{2} }}{\sin \alpha }} \right)L^{6} \sin \alpha + \frac{1}{5}\left( { + \frac{{4L^{3} A_{22} n}}{\sin \alpha } + \frac{{10L^{3} A_{12} n}}{\sin \alpha } - + \frac{{14L^{3} A_{66} n}}{\sin \alpha }} \right)L^{5} \sin \alpha + \frac{1}{4}\left( {\frac{{2L^{4} A_{12} n}}{\sin \alpha }} \right. \\ & \quad \left. {\left. { - \frac{{L^{4} A_{22} n}}{\sin \alpha } - \frac{{3L^{4} A_{66} n}}{\sin \alpha }} \right)L^{3} \sin \alpha } \right), \\ k_{22} & = 2\pi \left( { - \frac{31}{90}\rho h\omega^{2} L^{10} \sin \alpha + \frac{1}{8}(6\rho h\omega^{2} L^{2} + 15A_{66} - \frac{{A_{22} n^{2} }}{{\sin^{2} \alpha }}} \right)L^{8} \sin \alpha + \frac{1}{7}\left( { - 4L^{3} \rho h\omega^{2} + \frac{{4A_{22} n^{2} L}}{{\sin^{2} \alpha }}} \right. \\ & \quad \left. { - 46A_{66} L} \right)L^{7} \sin \alpha + \frac{1}{6}\left( {50A_{66} L^{2} + L^{4} \rho h\omega^{2} - \frac{{6A_{22} n^{2} L^{2} }}{{\sin^{2} \alpha }} - \frac{{D_{22} n^{2} \cos \alpha }}{{\tan \alpha \sin^{3} \alpha }} + \frac{{8D_{66} \cos \alpha }}{\tan \alpha \sin \alpha }} \right) \\ & \quad \times L^{6} \sin \alpha + \frac{1}{5}\left( {\frac{{4L^{3} A_{22} n^{2} }}{{\sin^{2} \alpha }} + \frac{{4D_{22} Ln^{2} \cos \alpha }}{{\tan \alpha \sin^{3} \alpha }} - 22L^{3} A_{66} - \frac{{22D_{66} L\cos \alpha }}{\tan \alpha \sin \alpha }} \right)L^{5} \sin \alpha + \frac{1}{4}\left( {3L^{4} A_{66} } \right. \\ & \quad \left. { - \frac{{L^{4} A_{22} n^{2} }}{{\sin^{2} \alpha }} + \frac{{20L^{2} D_{66} \cos \alpha }}{\sin \alpha \tan \alpha } - \frac{{6D_{22} L^{2} n^{2} \cos \alpha }}{{\tan \alpha \sin^{3} \alpha }}} \right)L^{3} \sin \alpha + \frac{1}{3}\left( {\frac{{4L^{3} D_{22} n^{2} \cos \alpha }}{{\tan \alpha \sin^{3} \alpha }} - \frac{{6L^{3} D_{66} \cos \alpha }}{\tan \alpha \sin \alpha }} \right) \\ & \quad \times L^{3} \sin \alpha + \frac{1}{2}\left. {\left( { - \frac{{L^{4} D_{22} n^{2} \cos \alpha }}{\sin \alpha \tan \alpha }} \right)L^{2} \sin \alpha )} \right), \\ \end{aligned} $$
$$ \begin{aligned} k_{23} & = 2\pi \left( {\frac{1}{8}\left( { - \frac{{A_{22} n}}{\sin \alpha \tan \alpha }} \right)L^{8} \sin \alpha + \frac{1}{7}\left( { + \frac{{4A_{22} Ln}}{\sin \alpha \tan \alpha }} \right)L^{7} \sin \alpha + \frac{1}{6}\left( {\frac{{24D_{66} n}}{\tan \alpha \sin \alpha } + \frac{{4D_{22} n}}{\tan \alpha \sin \alpha } + \frac{{12D_{12} n}}{\tan \alpha \sin \alpha }} \right.} \right. \\ & \left. {\quad - \frac{{D_{22} n^{3} }}{{\tan \alpha \sin^{3} \alpha }} - \frac{{6A_{22} nL^{2} }}{\sin \alpha \tan \alpha }} \right)L^{6} \sin \alpha + \frac{1}{5}\left( { - \frac{{72D_{66} Ln}}{\sin \alpha \tan \alpha } + \frac{{4L^{3} A_{22} n}}{\sin \alpha \tan \alpha } + \frac{{4D_{22} n^{3} L}}{{\tan \alpha \sin^{3} \alpha }} - \frac{{14D_{22} Ln}}{\tan \alpha \sin \alpha }} \right. \\ & \left. {\quad - \frac{{36D_{12} Ln}}{\tan \alpha \sin \alpha }} \right)L^{5} \sin \alpha + \frac{1}{4}\left( {\frac{{18D_{22} nL^{2} }}{\tan \alpha \sin \alpha } - \frac{{6D_{22} L^{2} n^{3} }}{{\tan \alpha \sin^{3} \alpha }} + \frac{{38D_{12} L^{2} n}}{\tan \alpha \sin \alpha } - \frac{{L^{4} A_{22} n}}{\sin \alpha \tan \alpha } + \frac{{76D_{66} nL^{2} }}{\tan \alpha \sin \alpha }} \right) \\ & \quad \times\, L^{3} \sin \alpha + \frac{1}{3}\left( { - \frac{{10L^{3} D_{22} n}}{\tan \alpha \sin \alpha } - \frac{{16L^{3} D_{12} n}}{\sin \alpha \tan \alpha } - \frac{{32L^{3} D_{66} n}}{\tan \alpha \sin \alpha } + \frac{{4L^{3} D_{22} n^{3} }}{{\tan \alpha \sin^{3} \alpha }}} \right)L^{3} \sin \alpha + \frac{1}{2}\left( {\frac{{2L^{4} D_{12} n}}{\tan \alpha \sin \alpha }} \right. \\ & \left. {\left. {\quad + \frac{{2L^{4} D_{22} n}}{\tan \alpha \sin \alpha } + \frac{{4L^{4} D_{66} n}}{\tan \alpha \sin \alpha } - \frac{{L^{4} D_{22} n^{3} }}{{\tan \alpha \sin^{3} \alpha }}} \right)L^{2} \sin \alpha } \right), \\ k_{31} & = 2\pi \left( {\frac{1}{8}\left( { - \frac{{4A_{12} }}{\tan \alpha } - \frac{{A_{22} }}{\tan \alpha }} \right)L^{8} \sin \alpha + \frac{1}{7}\left( {\frac{{14A_{12} L}}{\tan \alpha } + \frac{{4A_{22} L}}{\tan \alpha }} \right)L^{7} \sin \alpha + \frac{1}{6}\left( { - \frac{{6A_{22} L^{2} }}{\tan \alpha } - \frac{{18A_{12} L^{2} }}{\tan \alpha }} \right)L^{6} \sin \alpha } \right. \\ & \quad \left. { + \frac{1}{5}\left( {\frac{{10L^{3} A_{12} }}{\tan \alpha } + \frac{{4A_{22} L^{3} }}{\tan \alpha }} \right)L^{5} \sin \alpha + \frac{1}{4}\left( { - \frac{{L^{4} A_{22} }}{\tan \alpha } - \frac{{2L^{4} A_{12} }}{\tan \alpha }} \right)L^{4} \sin \alpha } \right), \\ \end{aligned} $$
$$ \begin{aligned} k_{32} & = 2\pi \left( {\frac{1}{8}\left( { - \frac{{A_{22} n}}{\tan \alpha \sin \alpha }} \right)L^{8} \sin \alpha + \frac{1}{7}\left( {\frac{{4A_{22} Ln}}{\tan \alpha \sin \alpha }} \right)L^{7} \sin \alpha + \frac{1}{6}\left( {\frac{{12D_{66} n\cos \alpha }}{{\sin^{2} \alpha }} - \frac{{D_{22} n^{3} \cos \alpha }}{{\sin^{4} \alpha }} - \frac{{6A_{22} nL^{2} }}{\tan \alpha \sin \alpha }} \right.} \right. \\ & \left. {\quad - \frac{{2D_{22} n\cos \alpha }}{{\sin^{2} \alpha }} + \frac{{6D_{12} n\cos \alpha }}{{\sin^{2} \alpha }}} \right)\sin \alpha L^{6} + \frac{1}{5}\left( {\frac{{6D_{22} nL}}{{\sin^{2} \alpha }} - \frac{{32D_{66} nL\cos \alpha }}{{\sin^{2} \alpha }} + \frac{{4L^{3} A_{22} n}}{\tan \alpha \sin \alpha } + \frac{{4D_{22} Ln^{3} \cos \alpha }}{{\sin^{4} \alpha }}} \right. \\ & \quad \left. { - \frac{{16D_{12} nL\cos \alpha }}{{\sin^{2} \alpha }}} \right)\sin \alpha L^{5} + \frac{1}{4}\left( { - \frac{{6D_{22} n^{3} L^{2} \cos \alpha }}{{\sin^{4} \alpha }} - \frac{{6D_{22} L^{2} n\cos \alpha }}{{\sin^{2} \alpha }} - \frac{{L^{4} A_{22} n}}{\tan \alpha \sin \alpha } + \frac{{14D_{12} nL^{2} \cos \alpha }}{{\sin^{2} \alpha }}} \right. \\ & \quad \left. { + \frac{{28l^{2} D_{66} n\cos \alpha }}{{\sin^{2} \alpha }}} \right)\sin \alpha L^{4} + \frac{1}{3}\left( { - \frac{{4L^{3} D_{12} }}{{\sin^{2} \alpha }}n\cos \alpha - \frac{{8L^{3} D_{66} n\cos \alpha }}{{\sin^{2} \alpha }} + \frac{{2L^{3} D_{22} n\cos \alpha }}{{\sin^{2} \alpha }} + \frac{{4L^{3} D_{22} n^{3} \cos \alpha }}{{\sin^{4} \alpha }}} \right) \\ & \quad \left. {\sin (\alpha )L^{3} + \frac{1}{2}\left( { - \frac{{L^{4} D_{22} \cos \alpha n^{3} }}{{\sin^{4} \alpha }}} \right)\sin \alpha L^{2} } \right), \\ k_{33} & = 2\pi \left( { - \frac{31}{90}\rho h\omega^{2} \sin \alpha L^{10} + \frac{1}{8}\left( {6\rho h\omega^{2} L^{2} - \frac{{A_{22} }}{{\tan^{2} \alpha }}} \right)\sin \alpha L^{8} + \frac{1}{7}\left( { - 4L^{3} \rho h\omega^{2} + \frac{{4A_{22} L}}{{\tan^{2} \alpha }}} \right)L^{7} \sin \alpha } \right) \\ & \quad + \frac{1}{6}\left( {\frac{{2D_{22} n^{2} }}{{\sin^{2} \alpha }} - 72D_{11} + \frac{{36D_{66} n^{2} }}{{\sin^{2} \alpha }} + L^{4} \rho h\omega^{2} + \frac{{18D_{12} n^{2} }}{{\sin^{2} \alpha }} - \frac{{D_{22} n^{4} }}{{\sin^{4} \alpha }} + 8D_{22} a_{3} - \frac{{6A_{22} L^{2} }}{{\tan^{2} \alpha }}} \right)L^{6} \sin \alpha \\ & \quad + \frac{1}{5}\left( { + \frac{{4D_{22} n^{4} L}}{{\sin^{4} \alpha }} - 22D_{22} L + \frac{{4L^{3} A_{22} }}{{\tan^{2} \alpha }} - \frac{{52D_{12} n^{2} L}}{{\sin^{2} \alpha }} + 168D_{11} L - \frac{{104D_{66} Ln^{2} }}{{\sin^{2} \alpha }} - \frac{{8D_{22} Ln^{2} }}{{\sin^{2} \alpha }}} \right)L^{5} \sin \alpha \\ & \quad + \frac{1}{4}\left( {\frac{{12D_{22} L^{2} n^{2} }}{{\sin^{2} \alpha }} - \frac{{6D_{22} n^{4} L^{2} }}{{\sin^{4} \alpha }} + \frac{{104D_{66} }}{{\sin^{2} \alpha }}n^{2} L^{2} - 120D_{11} L^{2} - \frac{{L^{4} A_{22} }}{{\tan^{2} \alpha }} - + 20D_{22} L^{2} + \frac{{52D_{12} L^{2} n^{2} }}{{\sin^{2} \alpha }}} \right) \\ & \quad\times \sin \alpha L^{4} + \frac{1}{3}\left( { - 6L^{3} D_{22} + 24L^{3} D_{11} - \frac{{8L^{3} D_{22} n^{2} }}{{\sin^{2} \alpha }} + \frac{{4L^{3} D_{22} n^{4} }}{{\sin^{4} \alpha }} - \frac{{20L^{3} D_{12} n^{2} }}{{\sin^{2} \alpha }} - \frac{{40L^{3} D_{66} n^{2} }}{{\sin^{2} \alpha }}} \right)\sin \alpha L^{3} \\ & \quad \left. { + \frac{1}{2}\left( { - \frac{{L^{4} D_{22} n^{4} }}{{\sin^{4} \alpha }} + \frac{{2L^{4} D_{12} n^{2} }}{{\sin^{2} \alpha }} + \frac{{4L^{4} D_{66} n^{2} }}{{\sin^{2} \alpha }} + \frac{{2L^{4} D_{22} n^{2} }}{{\sin^{2} \alpha }}} \right)\sin \alpha L^{2} } \right) \\ \end{aligned} $$
(33)
The detailed expressions of S
ij
are given as follows:
$$ \begin{aligned} S_{11} & = \frac{1}{1260}\rho h\omega^{2} L^{10} \sin \alpha + \frac{1}{1680}L^{6} \left( { - 16A_{11} L^{2} - 6A_{22} L^{2} - 6A_{66} L^{2} n^{2} \cos^{2} \alpha + 12A_{22} L^{2} \cos^{2} \alpha + 32A_{11} L^{2} \cos^{2} \alpha } \right. \\ & \quad \left. { - 6A_{22} L^{2} \cos^{4} \alpha - 16A_{11} L^{2} \alpha \cos^{4} \alpha - 6A_{66} L^{2} n^{2} } \right)/\sin^{3} \alpha , \\ S_{12} & = \frac{1}{1680}L^{6} \left( {6A_{22} L^{2} n\sin \alpha \cos^{2} \alpha + 6A_{66} L^{2} n\sin \alpha \cos^{2} \alpha - 6A_{22} L^{2} n\sin \alpha - 6A_{66} L^{2} n\sin \alpha } \right)/\sin^{3} \alpha , \\ S_{13} & = + \frac{1}{1680}L^{6} \left( { - 6A_{22} L^{2} \sin \alpha \cos \alpha + 6A_{22} L^{2} \sin \alpha \cos^{3} \alpha } \right)/\sin^{3} \alpha , \\ S_{21} & = \frac{1}{1680}L^{6} \left( {6A_{22} L^{2} n\sin \alpha \cos^{2} \alpha + 6A_{66} L^{2} n\sin \alpha \cos^{2} \alpha - 6A_{22} L^{2} n\sin \alpha } \right)/\sin^{3} \alpha , \\ S_{22} & = \frac{1}{1260}\rho h\omega^{2} L^{10} \sin \alpha + \frac{1}{1680}L^{6} \left( { - 22A_{66} L^{2} - 112D_{66} \cos^{2} \alpha + 112D_{66} \cos^{4} \alpha + 6A_{22} L^{2} n^{2} \cos^{2} \alpha } \right. \\ & \quad \left. { - 56D_{22} L^{2} n^{2} \cos^{2} \alpha + 44A_{66} L^{2} \cos^{2} \alpha - 22A_{66} L^{2} \cos^{4} \alpha - 6A_{22} L^{2} n^{2} } \right)/\sin^{3} \alpha , \\ S_{23} & = + \frac{1}{1680}L^{6} \left( {56D_{22} n\cos \alpha - 112D_{66} n\cos \alpha - 56D_{12} n\cos \alpha - 56D_{22} n^{3} \cos \alpha - 56D_{22} n\sin^{3} \alpha } \right. \\ & \quad \left. { + 112D_{66} n\cos^{3} \alpha + 56D_{12} n\cos^{3} \alpha - 6A_{22} L^{2} n\cos \alpha + 6A_{22} L^{2} n\cos^{3} \alpha } \right)/\sin^{3} \alpha , \\ S_{31} & = + \frac{1}{1680}L^{6} \left( { - 6A_{22} L^{2} \sin \alpha \cos \alpha + 6A_{22} L^{2} \sin \alpha \cos^{3} \alpha } \right)/\sin^{3} \alpha , \\ \end{aligned} $$
$$ \begin{aligned} S_{32} & = + \frac{1}{1680}L^{6} \left( { - 56D_{12} n\cos \alpha + 56D_{22} n\cos \alpha - 112D_{66} n\cos \alpha - 56D_{22} n^{3} \cos \alpha + 112D_{66} n\cos^{3} \alpha } \right. \\ & \quad \left. { + 56D_{12} n\cos^{3} \alpha - 56D_{22} n\cos^{3} \alpha - 6A_{22} L^{2} n\cos \alpha + 6a_{2} A_{22} L^{2} n\cos^{3} \alpha } \right)/\sin^{3} \alpha , \\ S_{33} & = \frac{1}{1260}\rho h\omega^{2} \sin \alpha L^{10} + \frac{1}{1,680}L^{6} \left( { - 56D_{22} n^{4} - 1344D_{11} \cos^{2} \alpha - 224D_{22} \cos^{2} \alpha - 112D_{22} \cos^{4} \alpha } \right. \\ & \quad - 672D_{11} \cos \alpha^{4} + 112D_{22} n^{2} - 112D_{12} n^{2} + 224D_{66} n^{2} + 6A_{22} L^{2} \cos^{4} \alpha - 6A_{22} L^{2} \cos^{2} \alpha \\ & \quad \left. { + 224D_{66} n^{2} \cos \alpha^{2} - 112D_{22} \cos^{2} \alpha n^{2} + 112D_{12} \cos^{2} \alpha n^{2} - 112D_{22} - 672D_{11} } \right)/\sin^{3} \alpha . \\ \end{aligned} $$
(34)
The differential operators L
i
(i = 1, 2, 3) are given by
$$ \begin{aligned} L_{1} & = A_{12} \frac{{\partial^{2} u}}{{\partial x^{2} }} + A_{22} \left( { - \frac{1}{{x^{2} \sin \alpha }}\frac{\partial v}{\partial \theta } + \frac{1}{x\sin \alpha }\frac{{\partial^{2} v}}{\partial \theta \partial x} - \frac{u}{{x^{2} }} + \frac{\partial u}{\partial x}\frac{1}{x} - \frac{w}{{x^{2} \tan \alpha }} + \frac{\partial w}{\partial x}\frac{1}{x\tan \alpha }} \right) \\ & \quad + D_{12} \left( { - \frac{{\partial^{3} w}}{{\partial x^{3} }}} \right) + D_{22} \left( {\frac{1}{{x^{2} \sin^{2} \alpha }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }} - \frac{1}{{x\sin^{2} \alpha }}\frac{{\partial^{3} w}}{{\partial \theta^{2} \partial x}} - \frac{\cos \alpha }{{x^{3} \sin^{2} \alpha }}\frac{\partial v}{\partial \theta } - \frac{\cos \alpha }{{x^{2} \sin^{2} \alpha }}\frac{{\partial^{2} v}}{\partial \theta \partial x}} \right. \\ & \left. {\quad + \frac{1}{{x^{2} }}\frac{\partial w}{\partial x} - \frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{1}{x}} \right) + A_{11} \left( {\frac{1}{x}\frac{\partial u}{\partial x}} \right) + A_{12} \left( {\frac{1}{{x^{2} \sin \alpha }}\frac{\partial v}{\partial \theta } + \frac{u}{{x^{2} }} + \frac{w}{{x^{2} \tan \alpha }}} \right) + D_{11} \left( { - \frac{1}{x}\frac{{\partial^{2} w}}{{\partial x^{2} }}} \right) \\ & \quad + D_{12} \left( { - \frac{1}{{x^{3} \sin^{2} \alpha }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }} + \frac{\cos }{{x^{3} \sin^{2} \alpha }}\frac{\partial v}{\partial \theta } - \frac{1}{{x^{2} }}\frac{\partial w}{\partial x}} \right) - A_{12} \left( {\frac{1}{x}\frac{\partial u}{\partial x}} \right) - A_{22} \left( {\frac{1}{{x^{2} \sin \alpha }}\frac{\partial v}{\partial \theta } + \frac{u}{{x^{2} }}} \right. \\ & \quad \left. { + \frac{w}{{x^{2} \tan \alpha }}} \right) - D_{12} \left( { - \frac{1}{x}\frac{{\partial^{2} w}}{{\partial x^{2} }}} \right) - D_{22} \left( { - \frac{1}{{x^{3} \sin^{2} \alpha }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }} + \frac{\cos \alpha }{{x^{3} \sin^{2} \alpha }}\frac{\partial v}{\partial \theta } - \frac{1}{{x^{2} }}\frac{\partial w}{\partial x}} \right) + \frac{1}{x\sin \alpha } \\ & \quad \left( {A_{66} \left( {\frac{{\partial^{2} v}}{\partial \theta \partial x} + \frac{1}{x\sin \alpha }\frac{{\partial^{2} u}}{{\partial \theta^{2} }} - \frac{1}{x}\frac{\partial v}{\partial \theta }} \right) + D_{66} \left( {\frac{2}{x\sin \alpha }\frac{{\partial^{3} w}}{{\partial \theta^{2} \partial x}} + \frac{1}{x\tan \alpha }\frac{{\partial^{2} v}}{\partial \theta \partial x} + \frac{2}{{x^{2} \sin \alpha }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }}} \right.} \right. \\ & \quad \left. {\left. { - \frac{2}{{x^{2} \tan \alpha }}\frac{\partial v}{\partial \theta }} \right)} \right) \\ \end{aligned} $$
$$ \begin{gathered} L_{2} = \frac{1}{x\sin \alpha }(A_{12} \,\left( {\frac{{\partial^{2} u}}{\partial x\partial \theta }} \right) + A_{22} \left( {\frac{1}{x\sin \alpha }\frac{{\partial^{2} v}}{{\partial \theta^{2} }} + \frac{1}{x}\frac{\partial u}{\partial \theta } + \frac{1}{x\tan \alpha }\frac{\partial w}{\partial \theta }} \right) + D_{12} \left( { - \frac{{\partial^{3} w}}{{\partial \theta \partial x^{2} }}} \right) \\ \left. { + D_{22} \left( { - \frac{1}{{x^{2} \sin^{2} \alpha }}\,\,\frac{{\partial^{3} w}}{{\partial \theta^{3} }} + \,\,\frac{\cos \alpha }{{x^{2} \sin^{2} \alpha }}\frac{{\partial^{2} v}}{{\partial \theta^{2} }} - \,\,\frac{1}{x}\frac{{\partial^{2} w}}{\partial x\partial \theta }\,} \right)\,} \right) + A_{66} \,\,\left( {\,\frac{{\partial^{2} v}}{{\partial x^{2} }} - \frac{1}{{x^{2} \sin \alpha }}\frac{\partial u}{\partial \theta }} \right. \\ \left. { + \frac{1}{x\sin \alpha }\frac{{\partial^{2} u}}{\partial x\partial \theta } + \,\,\frac{v}{{x^{2} }}\,\,} \right) + D_{66} \left( {\,\,\frac{2}{{x^{2} \sin \alpha }}\frac{{\partial^{2} w}}{\partial x\partial \theta } - \,\,\frac{2}{x\sin \alpha }\frac{{\partial^{3} w}}{{\partial \theta \partial x^{2} }} - \,\,\frac{1}{{x^{2} \tan \alpha }}\frac{\partial v}{\partial x}} \right. \\ + \frac{1}{x\tan \alpha }\,\,\frac{{\partial^{2} v}}{{\partial x^{2} }}\,\, - \,\,\frac{4}{{x^{3} \sin \alpha }}\frac{\partial w}{\partial \theta }\,\, + \,\,\frac{2}{{x^{2} \sin \alpha }}\,\,\frac{{\partial^{2} w}}{\partial x\partial \theta }\,\, + \,\,\frac{4}{{x^{3} \tan \alpha }}v\,\, - \,\,\frac{1}{{x^{2} \tan \alpha }}\,\,\frac{\partial v}{\partial x} \\ \, + \frac{2}{x}\left( {A_{66} \left( {\frac{\partial v}{\partial x} + \frac{1}{x\sin \alpha }\frac{\partial u}{\partial \theta } - \frac{v}{x}} \right) + D_{66} \left( {\frac{2}{x\sin \alpha }\frac{{\partial^{2} w}}{\partial x\partial \theta } + \frac{1}{x\tan \alpha }\frac{\partial v}{\partial x} + \frac{2}{{x^{2} \sin \alpha }}\frac{\partial w}{\partial \theta }} \right.} \right. \\ \left. {\left. { - \frac{1}{{x^{2} \tan \alpha }}\,} \right)} \right) + \frac{1}{x\tan \alpha }\,\left( {\,D_{12} \left( {\frac{1}{x\sin \alpha }\frac{{\partial^{2} u}}{\partial x\partial \theta }} \right) + D_{22} \,\,\left( {\frac{1}{{x^{2} \sin^{2} \alpha }}\,\,\frac{{\partial^{2} v}}{{\partial \theta^{2} }}\,\, + \,\,\frac{1}{{x^{2} \sin \alpha }}\,\,\frac{\partial u}{\partial \theta }} \right.} \right. \\ \left. { + \frac{1}{{x^{2} \sin \alpha \tan \alpha }}\frac{\partial w}{\partial \theta }} \right) + B_{12} \left( {\frac{1}{x\sin \alpha }\frac{{\partial^{3} w}}{{\partial x^{2} \partial \theta }}} \right) + B_{22} \left( {\frac{1}{{x^{2} \sin^{3} \alpha }}\frac{{\partial^{3} w}}{{\partial \theta^{3} }} + \frac{\cos \alpha }{{x^{3} \sin^{3} \alpha }}\frac{{\partial^{2} v}}{{\partial \theta^{2} }}} \right. \\ \left. { - \frac{1}{{x^{2} \sin \alpha }}\frac{{\partial^{2} w}}{\partial x\partial \theta }} \right) + D_{66} \left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} - \frac{1}{{x^{2} \sin \alpha }}\frac{\partial u}{\partial \theta } + \frac{1}{x\sin \alpha }\frac{{\partial^{2} u}}{\partial x\partial \theta } + \frac{v}{{x^{2} }}} \right) + B_{66} \left( {\frac{2}{{x^{2} \sin \alpha }}} \right. \\ \frac{{\partial^{2} w}}{\partial x\partial \theta } - \frac{2}{x\sin \alpha }\,\,\frac{{\partial^{3} w}}{{\partial x^{2} \partial \theta }} - \frac{1}{{x^{2} \tan \alpha }}\,\,\frac{\partial v}{\partial x} + \frac{1}{x\tan \alpha }\frac{{\partial^{2} v}}{{\partial x^{2} }}\,\, - \,\,\frac{4}{{x^{3} \sin \alpha }}\frac{\partial w}{\partial \theta } + \,\,\frac{2}{{x^{2} \sin \alpha }}\frac{{\partial^{2} w}}{\partial x\partial \theta }\,\, \\ \left. { + \frac{4}{{x^{3} \tan \alpha }}\frac{\partial v}{\partial x}} \right) + D_{66} \left( {\frac{2}{x}\frac{\partial v}{\partial x} + \frac{2}{{x^{2} \sin \alpha }}\frac{\partial u}{\partial \theta } - \frac{2}{{x^{2} }}v} \right) + B_{66} \left( { - \frac{4}{{x^{2} \sin \alpha }}\frac{{\partial^{2} w}}{\partial x\partial \theta } + \frac{2}{{x^{2} \tan \alpha }}} \right. \\ \quad \left. {\left. {\frac{\partial v}{\partial x} + \frac{4}{{x^{3} \sin \alpha }}\frac{\partial w}{\partial \theta } - \frac{4v}{{x^{3} \tan \alpha }}} \right)} \right) \\ \end{gathered} $$
$$ \begin{gathered} L_{3} = - \frac{1}{x\tan \alpha }\left( {A_{12} \left( {\frac{\partial u}{\partial x}} \right) + A_{22} \left( {\frac{1}{x\sin \alpha }\frac{\partial v}{\partial \theta } + \frac{u}{x} + \frac{1}{x\tan \alpha }w} \right) + D_{12} \left( { - \frac{{\partial^{2} w}}{{\partial x^{2} }}} \right) + D_{22} \left( { - \frac{1}{{x^{2} \sin^{2} \alpha }}} \right.} \right. \\ \left. {\left. {\,\frac{{\partial^{2} w}}{{\partial \theta^{2} }} + \frac{\cos \alpha }{{x^{2} \sin^{2} \alpha }}\frac{\partial v}{\partial \theta } - \frac{1}{x}\frac{\partial w}{\partial x}} \right)} \right) + D_{11} \left( { - \frac{1}{{x^{2} }}\frac{\partial u}{\partial x} + \frac{1}{x}\frac{{\partial^{2} u}}{{\partial x^{2} }}} \right) + D_{12} \left( { - \frac{2}{{x^{3} \sin \alpha }}\frac{\partial v}{\partial x} + \frac{1}{{x^{2} \sin \alpha }}\frac{\partial v}{\partial x}} \right. \\ \left. {\, - \frac{2u}{{x^{3} }}\frac{1}{{x^{2} }}\frac{\partial u}{\partial x} - \frac{2w}{{x^{3} \tan \alpha }} + \frac{1}{{x^{2} \tan \alpha }}\frac{\partial w}{\partial x}} \right) + B_{11} \left( {\frac{1}{{x^{2} }}\frac{{\partial^{2} w}}{{\partial x^{2} }} - \frac{1}{x}\frac{{\partial^{3} w}}{{\partial x^{3} }}} \right) + B_{12} \left( {\frac{3}{{x^{4} \sin^{2} \alpha }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }}} \right. \\ \,\left. { - \frac{1}{{x^{3} \sin^{2} \alpha }}\,\,\frac{{\partial^{3} w}}{{\partial \theta^{2} \partial x}}\,\, - \,\,\frac{3\cos \alpha }{{x^{4} \sin^{2} \alpha }}\frac{\partial v}{\partial \theta }\,\, + \,\,\frac{\cos \alpha }{{x^{3} \sin^{2} \alpha }}\frac{{\partial^{2} v}}{\partial x\partial \theta }\,\, + \,\,\frac{2}{{x^{3} }}\frac{\partial w}{\partial x}\,\, - \,\,\frac{1}{{x^{2} }}\frac{{\partial^{2} w}}{{\partial x^{2} }}\,} \right) + D_{11} \left( {\frac{{\partial^{3} u}}{{\partial x^{3} }}} \right) \\ \, + D_{12} \left( {\,\frac{2}{{x^{3} \sin \alpha }}\frac{\partial v}{\partial \theta }\,\, - \,\,\frac{1}{{x^{2} \sin \alpha }}\frac{{\partial^{2} v}}{\partial x\partial \theta }\,\, - \,\,\frac{1}{{x^{2} \sin \alpha }}\frac{{\partial^{2} v}}{\partial x\partial \theta }\,\, + \,\,\frac{1}{x\sin \alpha }\frac{{\partial^{3} v}}{{\partial x^{2} \partial \theta }} + \,\,\frac{2}{{x^{3} }}u - \frac{2}{{x^{2} }}\frac{\partial u}{\partial x}} \right. \\ \left. {\, + \,\,\frac{1}{x}\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{2}{{x^{3} \tan \alpha }}w - \frac{2}{{x^{2} \tan \alpha }}\frac{\partial w}{\partial x} + \frac{1}{x\tan \alpha }\frac{{\partial^{2} w}}{{\partial x^{2} }}\,} \right)\,\, + B_{11} \left( {\, - \frac{{\partial^{4} w}}{{\partial x^{4} }}} \right) + B_{12} \left( {\frac{6}{{x^{4} \sin^{2} \alpha }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }}} \right.\, \\ \,\, + \,\,\frac{4}{{x^{3} \sin^{2} \alpha }}\,\,\frac{{\partial^{3} w}}{{\partial x\partial \theta^{2} }}\,\, - \,\,\frac{2}{{x^{2} \sin^{2} \alpha }}\,\,\frac{{\partial^{4} w}}{{\partial x^{2} \partial \theta^{2} }}\,\, + \,\,\frac{6\cos \alpha }{{x^{4} \sin^{2} \alpha }}\,\,\frac{\partial v}{\partial \theta }\,\, - \,\,\frac{4\cos \alpha }{{x^{3} \sin^{2} \alpha }}\,\,\frac{{\partial^{2} v}}{\partial x\partial \theta } + \frac{\cos \alpha }{{x^{2} \sin^{2} \alpha }} \\ \left. {\,\frac{{\partial^{3} v}}{{\partial x^{2} \partial \theta }} - \frac{2}{{x^{3} }}\frac{\partial w}{\partial x} + \frac{2}{{x^{2} }}\frac{{\partial^{2} w}}{{\partial x^{2} }} - \frac{1}{x\sin \alpha }\frac{{\partial^{4} w}}{{\partial \theta^{2} \partial x^{2} }}} \right) + B_{22} \left( { - \frac{1}{{x^{2} \sin^{3} \alpha }}\frac{{\partial^{4} w}}{{\partial \theta^{4} }} + \frac{\cos \alpha }{{x^{3} \sin^{3} \alpha }}\frac{{\partial^{3} v}}{{\partial \theta^{3} }}} \right. \\ \left. {\, - \frac{1}{{x^{2} \sin \alpha }}\frac{{\partial^{3} w}}{{\partial \theta^{2} \partial x}}} \right) + D_{66} \left( {\frac{{\partial^{3} v}}{{\partial \theta \partial x^{2} }} - \frac{1}{{x^{2} \sin \alpha }}\frac{{\partial^{2} u}}{{\partial \theta^{2} }} + \frac{1}{x\sin \alpha }\frac{{\partial^{3} u}}{{\partial \theta^{2} \partial x}} + \frac{1}{{x^{2} }}\frac{\partial v}{\partial \theta }} \right) + B_{66} \left( {\frac{2}{{x^{2} \sin \alpha }}} \right. \\ \,\frac{{\partial^{3} w}}{{\partial \theta^{2} \partial x}} - \frac{2}{\sin \alpha }\frac{{\partial^{4} w}}{{\partial \theta^{2} \partial x^{2} }} - \frac{1}{x\tan \alpha }\frac{{\partial^{2} v}}{\partial \theta \partial x} + \frac{1}{x\tan \alpha }\frac{{\partial^{3} v}}{{\partial \theta \partial x^{2} }} - \frac{4}{{x^{3} \sin \alpha }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }} + \frac{2}{{x^{2} \sin \alpha }}\frac{{\partial^{3} w}}{{\partial \theta^{2} \partial x}} \\ \left. {\, + \frac{4}{{x^{3} \tan \alpha }}\frac{\partial v}{\partial \theta }\frac{\partial v}{\partial x} + \frac{4v}{{x^{3} \tan \alpha }}\frac{{\partial^{2} v}}{\partial \theta \partial x}} \right) + D_{66} \left( {\frac{2}{x}\frac{{\partial^{2} v}}{\partial \theta \partial x} + \frac{2}{{x^{2} \sin \alpha }}\frac{{\partial^{2} u}}{{\partial \theta^{2} }} - \frac{2}{{x^{2} }}\frac{\partial v}{\partial \theta }} \right) + \\ \,B_{66} \left( { - \frac{4}{{x^{2} \sin \alpha }}\frac{{\partial^{3} w}}{{\partial \theta^{2} \partial x}} + \frac{2}{{x^{3} \tan \alpha }}\frac{{\partial^{2} v}}{\partial \theta \partial x} + \frac{4}{{x^{3} \sin \alpha }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }} - \frac{4}{{x^{3} \tan \alpha }}\frac{\partial v}{\partial \theta }} \right)\, \\ \end{gathered} $$
(35)
where
$$ \begin{aligned} A_{11} & = \int\limits_{ - h/2}^{h/2} {Q_{11} } dz,\quad A_{12} = \int\limits_{ - h/2}^{h/2} {Q_{12} } dz,\quad A_{22} = \int\limits_{ - h/2}^{h/2} {Q_{22} } dz,\quad A_{66} = \int\limits_{ - h/2}^{h/2} {Q_{66} } dz, \\ D_{11} & = \int\limits_{ - h/2}^{h/2} {zQ_{11} } dz,\quad D_{12} = \int\limits_{ - h/2}^{h/2} {zQ_{12} } dz,\quad D_{22} = \int\limits_{ - h/2}^{h/2} {zQ_{22} } dz,\quad D_{66} = \int\limits_{ - h/2}^{h/2} {zQ_{66} } dz, \\ B_{22} & = \int\limits_{ - h/2}^{h/2} {z^{2} Q_{22} } dz,\quad B_{12} = \int\limits_{ - h/2}^{h/2} {z^{2} Q_{12} } dz,\quad B_{11} = \int\limits_{ - h/2}^{h/2} {z^{2} Q_{11} } dz,\quad B_{66} = \int\limits_{ - h/2}^{h/2} {z^{2} Q_{66} } dz \\ \end{aligned} $$
