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Torsional buckling of generally laminated conical shell

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Abstract

Buckling of generally laminated conical shells under uniform torsion with simply-supported boundary conditions is investigated. The Donnel type strain displacement relations are used to obtain potential strain energy of the shell and membrane stability equation is applied to acquire the external work done by torsion. The Ritz method is used to solve the governing equations and critical buckling loads are obtained. The accuracy of the results is validated in comparison of with other investigations and finite element method. The effects of lamination sequence, semi-vertex angle and length to radius ratio of the cone are evaluated and mode shapes are presented for two types of lamination sequences. To find a design criterion, effects of lamination angles and semi-vertex angle for two types of lamination sequence on torsional buckling of conical shells are investigated.

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Correspondence to M. A. Kouchakzadeh.

Appendix

Appendix

Stretching coupling

$$\begin{array}{*{20}l} {{\text{U}}_{\text{s}} = \frac{1}{{R(x)^{2} }}\left( {2A_{12} R\left( x \right)\frac{{\partial U\left( {x,\theta } \right)}}{\partial x}\left( {\sin \left( \alpha \right)U\left( {x,\theta } \right) + \frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta } - \cos \left( \alpha \right)W\left( {x,\theta } \right)} \right)} \right.} \hfill \\ {\quad \quad - \;2A_{26} \left( { - R\left( x \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial x} - \frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta } + \sin \left( \alpha \right)V\left( {x,\theta } \right)} \right)\left( {\sin \left( \alpha \right)U\left( {x,\theta } \right) + \frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta } - \cos \left( \alpha \right)W\left( {x,\theta } \right)} \right)} \hfill \\ {\quad \quad + \;2A_{16} R\left( x \right)\frac{{\partial U\left( {x,\theta } \right)}}{\partial x}\left( {R\left( x \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta } - \sin \left( \alpha \right)V\left( {x,\theta } \right)} \right) + A_{66} \left( {R\left( x \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta } - \sin \left( \alpha \right)V\left( {x,\theta } \right)} \right)^{2} } \hfill \\ {\quad \quad + \;A_{11} R\left( x \right)^{2} \left( {\frac{{\partial U\left( {x,\theta } \right)}}{\partial x}} \right)^{2} + \left. {A_{22} \left( {\sin \left( \alpha \right)U\left( {x,\theta } \right) + \frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta } - \cos \left( \alpha \right)W\left( {x,\theta } \right)} \right)^{2} } \right)} \hfill \\ \end{array}$$
(17)

Bending coupling

$$\begin{array}{*{20}l} {{\text{U}}_{\text{b}} = \frac{1}{{R(x)^{4} }}\left( {4D_{16} R\left( x \right)^{2} \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial x^{2} }}\left( {R\left( x \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{\partial x\,\partial \theta } - \sin \left( \alpha \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta }} \right)} \right.} \hfill \\ {\quad \quad + \;4D_{26} \left( {\sin \left( \alpha \right)R\left( x \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \right)\left( {R\left( x \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{\partial x\,\partial \theta } - \sin \left( \alpha \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta }} \right)} \hfill \\ {\quad \quad + \;4D_{66} \left( {\sin \left( \alpha \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta } - R\left( x \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{\partial x\,\partial \theta }} \right)^{2} + 2D_{12} R\left( x \right)^{2} \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial x^{2} }}\left( {\sin \left( \alpha \right)R\left( x \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \right)} \hfill \\ {\quad \quad + \;D_{11} R\left( x \right)^{4} \left( {\frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial x^{2} }}} \right)^{2} + \left. {D_{22} \left( {\sin \left( \alpha \right)R\left( x \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \right)^{2} } \right)} \hfill \\ \end{array}$$
(18)

Stretching-bending coupling

$$\begin{array}{*{20}l} {U_{bs} = \frac{1}{{R(x)^{4} }}\left( {B_{16} \left( {4R\left( x \right)^{2} \frac{{\partial U\left( {x,\theta } \right)}}{\partial x}} \right.\left( {\sin \left( \alpha \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta } - R\left( x \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{\partial x\,\partial \theta }} \right)} \right.} \hfill \\ {\quad \quad - \;2R\left( x \right)^{3} \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial x^{2} }}\left. {\left( {R\left( x \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta } - \sin \left( \alpha \right)V\left( {x,\theta } \right)} \right)} \right)} \hfill \\ {\quad \quad + \;2B_{26} R\left( x \right)\left( { - 2R\left( x \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{\partial x\,\partial \theta }} \right.\left( {\sin \left( \alpha \right)U\left( {x,\theta } \right) + \frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta } - \cos \left( \alpha \right)W\left( {x,\theta } \right)} \right)} \hfill \\ {\quad \quad - \;\sin \left( \alpha \right)R\left( x \right)\frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta }\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} + \sin^{2} \left( \alpha \right)R\left( x \right)V\left( {x,\theta } \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x}} \hfill \\ {\quad \quad + \;\sin \left( \alpha \right)R\left( x \right)^{2} \left( { - \frac{{\partial V\left( {x,\theta } \right)}}{\partial x}} \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} - R\left( x \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial x}\frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \hfill \\ {\quad \quad + \;2\sin^{2} \left( \alpha \right)U\left( {x,\theta } \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta } - \frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta }\frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }} + \sin \left( \alpha \right)V\left( {x,\theta } \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \hfill \\ {\quad \quad + \;2\sin \left( \alpha \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta }\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta } - \left. {2\sin \left( \alpha \right)\cos \left( \alpha \right)W\left( {x,\theta } \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta }} \right)} \hfill \\ {\quad \quad - \;4B_{66} R\left( x \right)\left( {R\left( x \right)\frac{{\partial^{2} W\left( {x,\theta } \right)}}{\partial x\,\partial \theta } - \sin \left( \alpha \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial \theta }} \right)\left( {R\left( x \right)\frac{{\partial V\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial U\left( {x,\theta } \right)}}{\partial \theta } - \sin \left( \alpha \right)V\left( {x,\theta } \right)} \right)} \hfill \\ {\quad \quad + \;B_{12} \left( { - 2R\left( x \right)^{3} \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial x^{2} }}} \right.\left( {\sin \left( \alpha \right)U\left( {x,\theta } \right) + \frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta } - \cos \left( \alpha \right)W\left( {x,\theta } \right)} \right)} \hfill \\ {\quad \quad - \;2R\left( x \right)^{2} \frac{{\partial U\left( {x,\theta } \right)}}{\partial x}\left. {\left( {\sin \left( \alpha \right)R\left( x \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \right)} \right)} \hfill \\ {\quad \quad - \;2B_{22} R\left( x \right)\left( {\sin \left( \alpha \right)R\left( x \right)\frac{{\partial W\left( {x,\theta } \right)}}{\partial x} + \frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial \theta^{2} }}} \right)\left( {\sin \left( \alpha \right)U\left( {x,\theta } \right) + \frac{{\partial V\left( {x,\theta } \right)}}{\partial \theta } - \cos \left( \alpha \right)W\left( {x,\theta } \right)} \right)} \hfill \\ {\left. {\quad \quad - \;2B_{11} R\left( x \right)^{4} \frac{{\partial U\left( {x,\theta } \right)}}{\partial x}\frac{{\partial^{2} W\left( {x,\theta } \right)}}{{\partial x^{2} }}} \right)} \hfill \\ \end{array}$$
(19)

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Shakouri, M., Sharghi, H. & Kouchakzadeh, M.A. Torsional buckling of generally laminated conical shell. Meccanica 52, 1051–1061 (2017). https://doi.org/10.1007/s11012-016-0429-8

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