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Benchmark solution for free vibration of thick open cylindrical shells on Pasternak foundation with general boundary conditions

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Abstract

In the present article, a new three-dimensional exact solution for free vibration of thick open cylindrical shells on Pasternak foundation with general boundary conditions is presented. The three-dimensional elasticity theory is employed to formulate the theoretical model. The admissible functions of the thick shells are described as a combination of a three-dimensional (3-D) Fourier cosine series and auxiliary functions. Compared with the traditional Fourier series, the improved Fourier series can eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions. The excellent accuracy and reliability of the current solutions are demonstrated by numerical examples and comparison of the present results with those available in the literature and obtained by using ABAQUS which is based on the finite element method. Numerous new results for thick open cylindrical shells on Pasternak foundation with elastic boundary conditions are presented. In addition, comprehensive studies on the effects of the elastic restraint parameters, geometric parameters and elastic foundation coefficients are also reported.

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Acknowledgments

The authors would like to thank the anonymous reviewers for their very valuable comments. The authors also gratefully acknowledge the financial support from the National Natural Science Foundation of China (No. 51209052).

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Authors and Affiliations

  1. College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin, 150001, People’s Republic of China

    Qingshan Wang, Dongyan Shi & Fazl e Ahad

  2. College of Shipbuilding Engineering, Harbin Engineering University, Harbin, 150001, People’s Republic of China

    Fuzhen Pang

Authors
  1. Qingshan Wang
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  2. Dongyan Shi
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  3. Fuzhen Pang
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  4. Fazl e Ahad
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Corresponding author

Correspondence to Qingshan Wang.

Appendix 1: Detailed expressions of the matrices M, K and G

Appendix 1: Detailed expressions of the matrices M, K and G

$${\mathbf{K}} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{uu} } & {{\mathbf{K}}_{uv} } & {{\mathbf{K}}_{uw} } \\ {{\mathbf{K}}_{uv}^{T} } & {{\mathbf{K}}_{vv} } & {{\mathbf{K}}_{vw} } \\ {{\mathbf{K}}_{uw}^{T} } & {{\mathbf{K}}_{vw}^{T} } & {{\mathbf{K}}_{ww} } \\ \end{array} } \right],{\mathbf{M}} = \left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{uu} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{M}}_{vv} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{M}}_{ww} } \\ \end{array} } \right],{\mathbf{G}} = \left[ {\begin{array}{*{20}c} {{\mathbf{G}}_{u} } \\ {{\mathbf{G}}_{v} } \\ {{\mathbf{G}}_{w} } \\ \end{array} } \right]$$
(18)
$$\begin{aligned} {\mathbf{K}}_{uu} = \iiint {\left\{ {\left( {\chi + 2\varUpsilon } \right)r\frac{{\partial {\mathbf{U}}^{T} }}{\partial x}\frac{{\partial {\mathbf{U}}}}{\partial x} + \varUpsilon r\frac{{\partial {\mathbf{U}}^{T} }}{\partial r}\frac{{\partial {\mathbf{U}}}}{\partial r} + \frac{\varUpsilon }{r}\frac{{\partial {\mathbf{U}}^{T} }}{\partial \theta }\frac{{\partial {\mathbf{U}}}}{\partial \theta }} \right\}dV} \hfill \\ + \iint {\left\{ {k_{x0}^{u} {\mathbf{U}}^{T} {\mathbf{U}}} \right\}_{|x = 0} dS_{0} } + \iint {\left\{ {k_{xL}^{u} {\mathbf{U}}^{T} {\mathbf{U}}} \right\}_{|x = L} dS_{1} } \hfill \\ + \iint {\left\{ {k_{\theta 0}^{u} {\mathbf{U}}^{T} {\mathbf{U}}} \right\}_{|\theta = 0} dS_{2} } + \iint {\left\{ {k_{\theta 1}^{u} {\mathbf{U}}^{T} {\mathbf{U}}} \right\}_{|\theta = \phi } dS_{3} } \hfill \\ \iint {\left\{ {K_{w} {\mathbf{W}}^{T} {\mathbf{W}} + K_{s} \frac{{\partial {\mathbf{W}}^{T} }}{\partial r}\frac{{\partial {\mathbf{W}}}}{\partial r} + \frac{{K_{s} }}{{r^{2} }}\frac{{\partial {\mathbf{W}}^{T} }}{\partial \theta }\frac{{\partial {\mathbf{W}}}}{\partial \theta }} \right\}_{|z = 0} dS_{4} } \hfill \\ \end{aligned}$$
(19)
$${\mathbf{K}}_{uv} = \iiint {\left\{ {\chi \left( {\frac{{\partial {\mathbf{V}}^{T} }}{\partial \theta }\frac{{\partial {\mathbf{U}}}}{\partial x} + \frac{{\partial {\mathbf{V}}}}{\partial \theta }\frac{{\partial {\mathbf{U}}^{T} }}{\partial x}} \right) + \varUpsilon \left( {\frac{{\partial {\mathbf{V}}^{T} }}{\partial x}\frac{{\partial {\mathbf{U}}}}{\partial \theta } + \frac{{\partial {\mathbf{V}}}}{\partial x}\frac{{\partial {\mathbf{U}}^{T} }}{\partial \theta }} \right)} \right\}dV}$$
(20)
$${\mathbf{K}}_{uw} = \iiint {\left\{ \begin{aligned} &\chi r\left( {\frac{{\partial {\mathbf{W}}^{T} }}{\partial r}\frac{{\partial {\mathbf{U}}}}{\partial x} + \frac{{\partial {\mathbf{W}}}}{\partial r}\frac{{\partial {\mathbf{U}}^{T} }}{\partial x}} \right) + \chi \left( {{\mathbf{W}}^{T} \frac{{\partial {\mathbf{U}}}}{\partial x} + {\mathbf{W}}\frac{{\partial {\mathbf{U}}^{T} }}{\partial x}} \right) \hfill \\ &+ \varUpsilon \left( {\frac{{\partial {\mathbf{W}}^{T} }}{\partial x}\frac{{\partial {\mathbf{U}}}}{\partial r} + \frac{{\partial {\mathbf{W}}}}{\partial x}\frac{{\partial {\mathbf{U}}^{T} }}{\partial r}} \right) \end{aligned} \right\}dV}$$
(21)
$$\begin{aligned} {\mathbf{K}}_{vv} & = \iiint {\left\{ {\frac{{\left( {\chi + 2\varUpsilon } \right)}}{r}\frac{{\partial {\mathbf{V}}^{T} }}{\partial \theta }\frac{{\partial {\mathbf{V}}}}{\partial \theta } + \varUpsilon r\left( {\frac{{\partial {\mathbf{V}}}}{\partial r} - \frac{{\mathbf{V}}}{r}} \right)\left( {\frac{{\partial {\mathbf{V}}^{T} }}{\partial r} - \frac{{{\mathbf{V}}^{T} }}{r}} \right) + \varUpsilon \frac{{\partial {\mathbf{V}}^{T} }}{\partial x}\frac{{\partial {\mathbf{V}}}}{\partial x}} \right\}dV} \\ & \quad + \iint {\left\{ {k_{x0}^{v} {\mathbf{V}}^{T} {\mathbf{V}}} \right\}_{|x = 0} dS_{0} } + \iint {\left\{ {k_{xL}^{v} {\mathbf{V}}^{T} {\mathbf{V}}} \right\}_{|x = L} dS_{1} } \\ & \quad + \iint {\left\{ {k_{\theta 0}^{v} {\mathbf{V}}^{T} {\mathbf{V}}} \right\}_{|\theta = 0} dS_{2} } + \iint {\left\{ {k_{\theta 1}^{v} {\mathbf{V}}^{T} {\mathbf{V}}} \right\}_{|\theta = \phi } dS_{3} } \\ \end{aligned}$$
(22)
$${\mathbf{K}}_{vw} = \iiint {\left\{ \begin{aligned} &\left( {\chi + 2\varUpsilon } \right)\frac{1}{r}\left( {{\mathbf{W}}^{T} \frac{{\partial {\mathbf{V}}}}{\partial \theta } + {\mathbf{W}}\frac{{\partial {\mathbf{V}}^{T} }}{\partial \theta }} \right) + \chi \left( {\frac{{\partial {\mathbf{W}}^{T} }}{\partial r}\frac{{\partial {\mathbf{V}}}}{\partial \theta } + \frac{{\partial {\mathbf{W}}}}{\partial r}\frac{{\partial {\mathbf{V}}^{T} }}{\partial \theta }} \right) \hfill \\ &+ \varUpsilon \left\{ {\frac{{\partial {\mathbf{W}}^{T} }}{\partial \theta }\left( {\frac{{\partial {\mathbf{V}}}}{\partial r} - \frac{{\mathbf{V}}}{r}} \right) + \frac{{\partial {\mathbf{W}}}}{\partial \theta }\left( {\frac{{\partial {\mathbf{V}}^{T} }}{\partial r} - \frac{{{\mathbf{V}}^{T} }}{r}} \right)} \right\} \hfill \\ \end{aligned} \right\}dV}$$
(23)
$$\begin{aligned} {\mathbf{K}}_{ww} & = \iiint {\left\{ \begin{aligned} \left( {\chi + 2\varUpsilon } \right)r\frac{{\partial {\mathbf{W}}^{T} }}{\partial r}\frac{{\partial {\mathbf{W}}}}{\partial r} + \left( {\chi + 2\varUpsilon } \right)\frac{1}{r}{\mathbf{W}}^{T} {\mathbf{W}} + \varUpsilon \frac{1}{r}\frac{{\partial {\mathbf{W}}^{T} }}{\partial \theta }\frac{{\partial {\mathbf{W}}}}{\partial \theta } + \varUpsilon r\frac{{\partial {\mathbf{W}}^{T} }}{\partial x}\frac{{\partial {\mathbf{W}}}}{\partial x} + \chi \left( {\frac{{\partial {\mathbf{W}}^{T} }}{\partial r}{\mathbf{W}} + \frac{{\partial {\mathbf{W}}}}{\partial r}{\mathbf{W}}^{T} } \right) \hfill \\ \end{aligned} \right\}dV} \\ & \quad + \iint {\left\{ {k_{x0}^{w} {\mathbf{W}}^{T} {\mathbf{W}}} \right\}_{|x = 0} dS_{0} } + \iint {\left\{ {k_{xL}^{w} {\mathbf{W}}^{T} {\mathbf{W}}} \right\}_{|x = L} dS_{1} } + \iint {\left\{ {k_{\theta 0}^{w} {\mathbf{W}}^{T} {\mathbf{W}}} \right\}_{|\theta = 0} dS_{2} } \\ & \quad + \iint {\left\{ {k_{\theta 1}^{w} {\mathbf{W}}^{T} {\mathbf{W}}} \right\}_{|\theta = \phi } dS_{3} } + \iint {\left\{ {K_{w} {\mathbf{W}}^{T} {\mathbf{W}} + K_{s} \frac{{\partial {\mathbf{W}}^{T} }}{\partial x}\frac{{\partial {\mathbf{W}}}}{\partial x} + K_{s} \frac{1}{{r^{2} }}\frac{{\partial {\mathbf{W}}^{T} }}{\partial \theta }\frac{{\partial {\mathbf{W}}}}{\partial \theta }} \right\}_{{|r = R_{1} }} R_{1} dS_{4} } \\ \end{aligned}$$
(24)
$${\mathbf{M}}_{uu} = \iiint {\left\{ {\rho r{\mathbf{U}}^{T} {\mathbf{U}}} \right\}dV};{\mathbf{M}}_{vv} = \iiint {\left\{ {\rho r{\mathbf{V}}^{T} {\mathbf{V}}} \right\}dV};{\mathbf{M}}_{ww} = \iiint {\left\{ {\rho r{\mathbf{W}}^{T} {\mathbf{W}}} \right\}dV}$$
(25)
$${\mathbf{G}}_{u} = \left\{ \begin{aligned} A_{000} , \ldots ,A_{mnq} , \ldots ,A_{MNQ} , \ldots ,a_{00}^{1} , \ldots ,a_{MN}^{1} ,a_{00}^{2} , \ldots ,a_{MN}^{2} \hfill \\ ,a_{00}^{3} , \ldots ,a_{MQ}^{3} ,a_{00}^{4} , \ldots ,a_{MQ}^{4} ,a_{00}^{5} , \ldots ,a_{NQ}^{5} ,a_{00}^{6} , \ldots ,a_{NQ}^{6} \hfill \\ \end{aligned} \right\}e^{j\omega t}$$
(26)
$${\mathbf{G}}_{v} = \left\{ \begin{aligned} B_{000} , \ldots ,B_{mnq} , \ldots ,B_{MNQ} , \ldots ,b_{00}^{1} , \ldots ,b_{MN}^{1} ,b_{00}^{2} , \ldots ,b_{MN}^{2} \hfill \\ ,b_{00}^{3} , \ldots ,b_{MQ}^{3} ,b_{00}^{4} , \ldots ,b_{MQ}^{4} ,b_{00}^{5} , \ldots ,b_{NQ}^{5} ,b_{00}^{6} , \ldots ,b_{NQ}^{6} \hfill \\ \end{aligned} \right\}e^{j\omega t}$$
(27)
$${\mathbf{G}}_{w} = \left\{ \begin{aligned} C_{000} , \ldots ,C_{mnq} , \ldots ,C_{MNQ} , \ldots ,c_{00}^{1} , \ldots ,c_{MN}^{1} ,c_{00}^{2} , \ldots ,c_{MN}^{2} \hfill \\ ,c_{00}^{3} , \ldots ,c_{MQ}^{3} ,c_{00}^{4} , \ldots ,c_{MQ}^{4} ,c_{00}^{5} , \ldots ,c_{NQ}^{5} ,c_{00}^{6} , \ldots ,c_{NQ}^{6} \hfill \\ \end{aligned} \right\}e^{j\omega t}$$
(28)

where

$${\mathbf{U}} = \left\{ \begin{aligned} &\cos \lambda_{L0} x, \ldots ,\cos \lambda_{L0} x\cos \lambda_{\phi n} \theta \cos \lambda_{hq} r, \ldots ,\cos \lambda_{LM} x\cos \lambda_{\phi N} \theta \cos \lambda_{hQ} r, \hfill \\ &\zeta_{r}^{1} \left( r \right), \ldots ,\zeta_{r}^{1} \left( r \right)\cos \lambda_{L0} x\cos \lambda_{\phi n} \theta , \ldots ,\zeta_{z}^{1} \left( z \right)\cos \lambda_{LM} x\cos \lambda_{\phi N} \theta , \hfill \\ &\zeta_{\theta }^{2} \left( \theta \right), \ldots ,\zeta_{\theta }^{2} \left( \theta \right)\cos \lambda_{L0} x\cos \lambda_{hq} r, \ldots ,\zeta_{\theta }^{2} \left( \theta \right)\cos \lambda_{LM} x\cos \lambda_{hQ} r, \hfill \\ &\zeta_{x}^{2} \left( x \right), \ldots ,\zeta_{x}^{2} \left( z \right)os\lambda_{\phi n} \theta \cos \lambda_{hq} r, \ldots ,\zeta_{z}^{2} \left( z \right)\cos \lambda_{\phi N} \theta \cos \lambda_{hQ} r \hfill \\ \end{aligned} \right\}$$
(29)
$${\mathbf{V}} = {\mathbf{W = U}}$$
(30)

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Wang, Q., Shi, D., Pang, F. et al. Benchmark solution for free vibration of thick open cylindrical shells on Pasternak foundation with general boundary conditions. Meccanica 52, 457–482 (2017). https://doi.org/10.1007/s11012-016-0406-2

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