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Dynamic analysis of laminated composite double cylindrical and conical shells with bulkheads using meshfree method

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Abstract

In this paper, the free vibration and dynamic response of laminated composite double cylindrical and conical shells with bulkheads are investigated by means of the meshfree method. The laminated composite double conical shell with bulkheads (DCOSB) is divided into several conical shells and annular plates to ensure the numerical stability. The theoretical formulation of the individual segment is established by using the energy principle in framework of first order shear deformation theory (FSDT). The displacement components of each segment are approximated by a meshfree shape function along the meridional direction and Fourier series for the circumferential direction. The formulation of entire system are derived by using coupling conditions obtained through the geometrical relations between the segments. The double cylindrical shell with bulkheads (DCYSB) is considered as a DCOSB with semi-vertex angle of α = 0. The harmonic load and stationary stochastic excitation are considered as the external force. Convergence and verification studies are performed to confirm the reliability and accuracy of the present formulation. Satisfactory agreements are achieved between the numerical results by the proposed method and those from the published literature. Finally, the effect of some parameters on the free vibration and dynamic response of laminated composite DCYSB and DCOSB is presented.

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Acknowledgements

I would like to take the opportunity to express my hearted gratitude to all those who make a contribution to the completion of my article.

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Authors and Affiliations

  1. School of Economics and Management, University of Science and Technology Beijing, Beijing, 100083, China

    Jangsu Kim & Kwonryong Hong

  2. Institute of Information Technology, High-Tech Research & Development Centre, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea

    Jangsu Kim, Cholnam Om & Dokgil Kang

  3. Faculty of Energy Science, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea

    Kwonryong Hong

  4. Department of Transport Mechanical Engineering, Pyongyang University of Mechanical Engineering, Pyongyang, 99903, Democratic People’s Republic of Korea

    Tong Ho Choe

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  1. Jangsu Kim
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Appendices

Appendix A: The components of the matrices M and F

$$ C_{c} = \left\{ {\begin{array}{*{20}c} \pi & {n \ne 0} \\ {2\pi } & {n = 0} \\ \end{array} } \right.\quad ,\quad S_{s} = \pi $$

-Matrix M

$$ {\varvec{M}} = \int\limits_{x} {\left[ {\begin{array}{*{20}c} {M_{11} } & {M_{12} } & {M_{13} } & {\quad M_{14} } & \quad \cdots & { \quad M_{110} } \\ {} & {M_{22} } & {M_{23} } & { \quad M_{24} } &\quad \cdots & {\quad M_{210} } \\ {} & {} & {M_{33} } & {\quad M_{34} } & \quad \cdots & { \quad M_{310} } \\ {} & {Symmetric} & {} & { \quad M_{44} } & \quad \cdots & { \quad M_{410} } \\ {} & {} & {} & {} & \ddots & \vdots \\ {} & {} & {} & {} & {} & {M_{1010} } \\ \end{array} } \right]Rdx} $$
$$M_{11} = M_{33} = I_{0} \omega^{2}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c},$$
$$M_{14} = I_{1} \omega^{2}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c},$$
$$M_{22} = I_{0} \omega^{2}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s},$$
$$M_{25} = I_{1} \omega^{2}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s},$$
$$M_{44} = I_{2} \omega^{2}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c},$$
$$M_{55} = I_{2} \omega^{2}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s},$$
$$M_{66} = M_{88} = I_{0} \omega^{2}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s},$$
$$M_{69} = I_{1} \omega^{2}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s},$$
$$M_{710} = I_{1} \omega^{2}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c},$$
$$M_{77} = I_{0} \omega^{2}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c},$$
$$M_{99} = I_{2} \omega^{2}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s},$$
$$M_{1010} = I_{2} \omega^{2}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c},$$

Other elements of the matrix M are zero.

-Matrix F

$$ \begin{gathered} {\varvec{F}} = \int\limits_{x} {\int\limits_{\theta } {\left[ {q_{u}{\varvec{\varPhi}}\cos n\theta \quad q_{v}{\varvec{\varPhi}}\sin n\theta \quad q_{w}{\varvec{\varPhi}}\cos n\theta \quad m_{x}{\varvec{\varPhi}}\cos n\theta \quad m_{\theta }{\varvec{\varPhi}}\sin n\theta } \right.} } \hfill \\ \quad \quad \quad \left. {q_{u}{\varvec{\varPhi}}\sin n\theta \quad q_{v}{\varvec{\varPhi}}\cos n\theta \quad q_{w}{\varvec{\varPhi}}\sin n\theta \quad m_{x}{\varvec{\varPhi}}\sin n\theta \quad m_{\theta }{\varvec{\varPhi}}\cos n\theta } \right]^{T} Rdxd\theta \hfill \\ \end{gathered} $$

Appendix B: The components of the matrix K

$$ {\varvec{K}} = {\varvec{K}}_{c} + {\varvec{K}}_{bl} + {\varvec{K}}_{br} $$
$$ \begin{gathered} {\varvec{K}}_{bl} = diag\left[ {k_{u0}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} R\quad k_{v0}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} R\quad k_{w0}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} R\quad k_{x0}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} R\quad k_{\theta 0}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} R} \right. \hfill \\ \left. {\quad \quad \quad \quad k_{u0}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} R\quad k_{v0}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} R\quad k_{w0}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} R\quad k_{x0}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} R\quad k_{\theta 0}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} R} \right]_{x = 0} \hfill \\ \end{gathered} $$
$$ \begin{gathered} {\varvec{K}}_{br} = diag\left[ {k_{u1}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} R\quad k_{v1}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} R\quad k_{w1}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} R\quad k_{x1}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} R\quad k_{\theta 1}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} R} \right. \hfill \\ \quad \quad \quad \quad \left. {k_{u1}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} R\quad k_{v1}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} R\quad k_{w1}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} R\quad k_{x1}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} R\quad k_{\theta 1}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} R} \right]_{x = L} \hfill \\ \end{gathered} $$
$$ {\varvec{K}}_{c} = \int\limits_{x} {\left[ {\begin{array}{*{20}c} {k_{{11}} } & {\quad k_{{12}} } & {\quad k_{{13}} } & {\quad k_{{14}} } & {\quad \cdots } & {\quad k_{{110}} } \\ {k_{{12}}^{T} } & {\quad k_{{22}} } & {\quad k_{{23}} } & {\quad k_{{24}} } & {\quad \cdots } & {\quad k_{{210}} } \\ {k_{{13}}^{T} } & {\quad k_{{23}}^{T} } & {\quad k_{{33}} } & {\quad k_{{34}} } & {\quad \cdots } & {\quad k_{{310}} } \\ {k_{{14}}^{T} } & {\quad k_{{24}}^{T} } & {\quad k_{{34}}^{T} } & {\quad k_{{44}} } & {\quad \cdots } & {\quad k_{{410}} } \\ \vdots & {\quad \vdots } & {\quad \vdots } & {\quad \vdots } & {\quad \ddots } & {\quad \vdots } \\ {k_{{110}}^{T} } & {\quad k_{{210}}^{T} } & {\quad k_{{310}}^{T} } & {\quad k_{{410}}^{T} } & {\quad \cdots } & {\quad k_{{1010}} } \\ \end{array} } \right]Rdx} $$
$$ \begin{gathered} k_{11} = A_{11} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{n^{2} A_{66} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{A_{22} \sin^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} + \frac{{A_{12} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} \hfill \\ \qquad\quad + \frac{{A_{12} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} \hfill \\ \end{gathered} $$
$$ k_{12} = \frac{{nA_{12} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nA_{66} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{nA_{22} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} + \frac{{nA_{66} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ k_{13} = \frac{{A_{12} \cos \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} + \frac{{A_{22} \sin \alpha \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{14} = B_{11} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{n^{2} B_{66} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{B_{12} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} + \frac{{B_{12} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{B_{22} \sin^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{15} = \frac{{nB_{12} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nB_{66} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{nB_{22} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} + \frac{{nB_{66} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ k_{16} = \frac{{nA_{16} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nA_{16} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} - \frac{{nA_{26} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nA_{26} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ \begin{gathered} k_{17} = A_{16} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{n^{2} A_{26} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{A_{26} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} - \frac{{A_{16} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} \hfill \\ \qquad\quad - \frac{{A_{26} \sin^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} \hfill \\ \end{gathered} $$
$$ k_{18} = - \frac{{nA_{26} \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ k_{19} = \frac{{nB_{16} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nB_{16} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} - \frac{{nB_{26} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nB_{26} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{110} = B_{16} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{n^{2} B_{26} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} - \frac{{B_{16} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} + \frac{{B_{26} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} - \frac{{B_{26} \sin^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ \begin{gathered} k_{22} = A_{66} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} - \frac{{A_{66} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} - \frac{{A_{66} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} + \frac{{n^{2} A_{22} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} \hfill \\ \quad\qquad + \frac{{A_{66} \sin^{2} \alpha + A_{44} \cos^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} \hfill \\ \end{gathered} $$
$$ k_{23} = \frac{{nA_{44} \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nA_{22} \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{24} = \frac{{nB_{12} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} - \frac{{nB_{66} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nB_{66} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nB_{22} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ \begin{gathered} k_{25} = B_{66} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{n^{2} B_{22} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} - \frac{{B_{66} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} - \frac{{B_{66} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{B_{66} \sin^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} \hfill \\ \qquad\quad - \frac{{A_{44} \cos \alpha }}{R}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} \hfill \\ \end{gathered} $$
$$ \begin{gathered} k_{26} = A_{16} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{n^{2} A_{26} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} + \frac{{A_{26} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} - \frac{{A_{16} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} \hfill \\ \qquad\quad- \frac{{A_{26} \sin^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} \hfill \\ \end{gathered} $$
$$ k_{27} = - \frac{{nA_{26} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nA_{26} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{nA_{26} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} - \frac{{nA_{26} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{28} = \frac{{A_{26} \cos \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} - \frac{{A_{45} \cos \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} - \frac{{A_{26} \sin \alpha \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ \begin{gathered} k_{29} = B_{16} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{n^{2} B_{26} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} + \frac{{B_{26} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} - \frac{{B_{16} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} - \frac{{B_{26} \sin^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} \hfill \\ \quad\qquad - \frac{{A_{45} \cos \alpha }}{R}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} \hfill \\ \end{gathered} $$
$$ k_{210} = - \frac{{nB_{26} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nB_{26} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{nB_{26} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} - \frac{{nB_{26} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{33} = A_{55} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{n^{2} A_{44} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{A_{22} \cos^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{34} = A_{55} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} + \frac{{B_{12} \cos \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{B_{22} \sin \alpha \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{35} = - \frac{{nA_{44} }}{R}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nB_{22} \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{36} = \frac{{nA_{26} \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{37} = \frac{{A_{26} \cos \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} - \frac{{A_{45} \cos \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} - \frac{{A_{26} \sin \alpha \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{38} = \frac{{nA_{45} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nA_{45} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} $$
$$ k_{39} = - \frac{{nA_{45} }}{R}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nB_{26} \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{310} = A_{45} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} + \frac{{B_{26} \cos \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} - \frac{{B_{26} \sin \alpha \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ \begin{gathered} k_{44} = D_{11} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{n^{2} D_{66} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{D_{22} \sin^{{2}} \alpha }}{{R^{{2}} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} + A_{55}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} + \frac{{D_{12} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} \hfill \\ \qquad\quad + \frac{{D_{12} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} \hfill \\ \end{gathered} $$
$$ k_{45} = \frac{{nD_{12} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nD_{66} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{nD_{66} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nD_{22} \sin \alpha }}{{R^{{2}} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{46} = - \frac{{nB_{16} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{nB_{16} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} + \frac{{nB_{26} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nB_{26} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ \begin{gathered} k_{47} = B_{16} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{n^{2} B_{26} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{B_{26} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} - \frac{{B_{16} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} - \frac{{B_{26} \sin^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} \hfill \\ \quad\qquad- \frac{{A_{45} \cos \alpha }}{R}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} \hfill \\ \end{gathered} $$
$$ k_{48} = \frac{{nA_{45} }}{R}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nB_{26} \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ k_{49} = \frac{{nD_{16} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nD_{16} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{nD_{26} \sin \alpha }}{{R^{{2}} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nD_{26} \sin \alpha }}{{R^{{2}} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ \begin{gathered} k_{410} = D_{16} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{n^{2} D_{26} }}{{R^{{2}} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} - \frac{{D_{16} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} + \frac{{D_{26} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} - \frac{{D_{26} \sin^{{2}} \alpha }}{{R^{{2}} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} \hfill \\ \quad\qquad + A_{45}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} \hfill \\ \end{gathered} $$
$$ \begin{gathered} k_{55} = D_{66} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{n^{2} D_{22} }}{{R^{{2}} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} + \frac{{D_{66} \sin^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + A_{44}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} - \frac{{D_{66} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} \hfill \\ \quad\qquad - \frac{{D_{66} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} \hfill \\ \end{gathered} $$
$$ k_{56} = B_{16} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{n^{2} B_{26} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} - \frac{{B_{16} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{B_{26} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} - \frac{{B_{26} \sin^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ k_{57} = \frac{{nB_{26} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} - \frac{{nB_{26} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} - \frac{{nB_{26} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} + \frac{{nB_{26} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ k_{58} = A_{45}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{B_{26} \cos \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} - \frac{{B_{26} \sin \alpha \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ \begin{gathered} k_{59} = D_{16} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{n^{2} D_{26} }}{{R^{{2}} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} - \frac{{D_{16} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{D_{26} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} - \frac{{D_{26} \sin^{{2}} \alpha }}{{R^{{2}} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} \hfill \\ \quad\qquad + A_{45}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} \hfill \\ \end{gathered} $$
$$ k_{510} = - \frac{{nD_{26} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nD_{26} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{nD_{26} \sin \alpha }}{{R^{{2}} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} - \frac{{nD_{26} \sin \alpha }}{{R^{{2}} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ \begin{gathered} k_{66} = A_{11} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{n^{2} A_{66} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} + \frac{{A_{22} \sin^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{A_{12} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} \hfill \\ \quad\qquad + \frac{{A_{12} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} \hfill \\ \end{gathered} $$
$$ k_{67} = - \frac{{nA_{12} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nA_{66} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} - \frac{{nA_{22} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} - \frac{{nA_{66} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ k_{68} = \frac{{A_{12} \cos \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} + \frac{{A_{22} \sin \alpha \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ k_{69} = B_{11} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{n^{2} B_{66} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} + \frac{{B_{12} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} + \frac{{B_{12} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{B_{22} \sin^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ k_{610} = - \frac{{nB_{12} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nB_{66} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} - \frac{{nB_{22} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} - \frac{{nB_{66} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} $$
$$ \begin{gathered} k_{77} = A_{66} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} - \frac{{A_{66} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} - \frac{{A_{66} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} + \frac{{n^{2} A_{22} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} \hfill \\ \quad\quad+ \frac{{A_{66} \sin^{2} \alpha + A_{44} \cos^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} \hfill \\ \end{gathered} $$
$$ k_{78} = - \frac{{nA_{44} \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nA_{22} \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ k_{79} = - \frac{{nB_{12} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{nB_{66} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nB_{66} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nB_{22} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ \begin{gathered} k_{710} = B_{66} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} + \frac{{n^{2} B_{22} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} - \frac{{B_{66} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}C_{c} - \frac{{B_{66} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} \hfill \\ \qquad\quad+ \frac{{B_{66} \sin^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} - \frac{{A_{44} \cos \alpha }}{R}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} \hfill \\ \end{gathered} $$
$$ k_{88} = A_{55} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{n^{2} A_{44} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} + \frac{{A_{22} \cos^{2} \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ k_{89} = A_{55} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} + \frac{{B_{12} \cos \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{B_{22} \sin \alpha \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ k_{810} = \frac{{nA_{44} }}{R}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nB_{22} \cos \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ \begin{gathered} k_{99} = D_{11} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} + \frac{{n^{2} D_{66} }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} + \frac{{D_{22} \sin^{{2}} \alpha }}{{R^{{2}} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + A_{55}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} + \frac{{D_{12} \sin \alpha }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}S_{s} \hfill \\ \quad\qquad+ \frac{{D_{12} \sin \alpha }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} \hfill \\ \end{gathered} $$
$$ k_{910} = - \frac{{nD_{12} }}{R}\frac{{\partial{\varvec{\varPhi}}}}{\partial x}^{T}{\varvec{\varPhi}}S_{s} + \frac{{nD_{66} }}{R}{\varvec{\varPhi}}^{T} \frac{{\partial{\varvec{\varPhi}}}}{\partial x}C_{c} - \frac{{nD_{66} \sin \alpha }}{{R^{2} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}C_{c} - \frac{{nD_{22} \sin \alpha }}{{R^{{2}} }}{\varvec{\varPhi}}^{T}{\varvec{\varPhi}}S_{s} $$
$$ \begin{aligned} k_{{1010}} & = D_{{66}} \frac{{\partial \user2{\Phi }}}{{\partial x}}^{T} \frac{{\partial \user2{\Phi }}}{{\partial x}}C_{c} + \frac{{n^{2} D_{{22}} }}{{R^{{\text{2}}} }}\user2{\Phi }^{T} \user2{\Phi }S_{s} + \frac{{D_{{66}} \sin ^{2} \alpha }}{{R^{2} }}\user2{\Phi }^{T} \user2{\Phi }C_{c} + A_{{44}} \user2{\Phi }^{T} \user2{\Phi }C_{c} - \frac{{D_{{66}} \sin \alpha }}{R}\user2{\Phi }^{T} \frac{{\partial \user2{\Phi }}}{{\partial x}}C_{c} \\ & \quad - \frac{{D_{{66}} \sin \alpha }}{R}\frac{{\partial \user2{\Phi }}}{{\partial x}}^{T} \user2{\Phi }C_{c} \\ \end{aligned} $$

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Kim, J., Om, C., Kang, D. et al. Dynamic analysis of laminated composite double cylindrical and conical shells with bulkheads using meshfree method. Acta Mech 234, 4775–4800 (2023). https://doi.org/10.1007/s00707-023-03628-w

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