Finite Element Analysis of Orthotropic Thin Plates Using Analytical Solution

Abstract

Despite the high importance of orthotropic bending plate, up to now, only a few studies have been performed based on the analytical solution for analysis of these structures. In this paper, an efficient element is proposed by utilizing the hybrid-Trefftz method to analyze structures. The proposed quadrilateral element (QHT-OR) has four nodes and twelve degrees of freedom. Moreover, two independent displacement fields are defined for inner and boundaries of element. The internal field is achieved by solving the governing equation of orthotropic plates. Furthermore, the boundary field is related to the nodal degrees of freedom by using interpolation functions. It should be mentioned that the interpolation functions of two-node beam elements are applied for boundary field. The responses obtained by using numerical tests show the high accuracy of the QHT-OR element even for coarse meshes.

Appendix: The Interpolation Matrix of the Boundary Displacement Field

The entries of the interpolation matrix of the boundary displacement field (45) have the coming appearance:

$$\left\{ \begin{aligned} {\tilde{w}} \hfill \\ \tilde{\theta }_{x} \hfill \\ \tilde{\theta }_{y} \hfill \\ \end{aligned} \right\} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {N_{11} } & {\begin{array}{*{20}c} {N_{12} } & {N_{13} } & {N_{14} } \\ \end{array} } & {\begin{array}{*{20}c} {N_{15} } & {N_{16} } \\ \end{array} } \\ \end{array} } \\ {\begin{array}{*{20}c} {N_{21} } & {\begin{array}{*{20}c} {N_{22} } & {N_{23} } & {N_{24} } \\ \end{array} } & {\begin{array}{*{20}c} {N_{25} } & {N_{26} } \\ \end{array} } \\ \end{array} } \\ {\begin{array}{*{20}c} {N_{31} } & {\begin{array}{*{20}c} {N_{32} } & {N_{33} } & {N_{34} } \\ \end{array} } & {\begin{array}{*{20}c} {N_{35} } & {N_{36} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]\left\{ \begin{aligned} \tilde{w}_{i} \hfill \\ \tilde{\theta }_{xi} \hfill \\ \tilde{\theta }_{yi} \hfill \\ \tilde{w}_{j} \hfill \\ \tilde{\theta }_{xj} \hfill \\ \tilde{\theta }_{yj} \hfill \\ \end{aligned} \right\}$$

$$\begin{aligned} N_{11} & = \frac{1}{4}\left( {2 - 3s + s^{3} } \right) \\ N_{12} & = \frac{{y_{ji} }}{8}\left( {1 - s - s^{2} + s^{3} } \right) \\ N_{13} & = \frac{{ - x_{ji} }}{8}\left( {1 - s - s^{2} + s^{3} } \right) \\ N_{14} & = \frac{1}{4}\left( {2 + 3s - s^{3} } \right) \\ N_{15} & = \frac{{y_{ji} }}{8}\left( { - 1 - s + s^{2} + s^{3} } \right) \\ N_{16} & = \frac{{ - x_{ji} }}{8}\left( { - 1 - s + s^{2} + s^{3} } \right) \\ N_{21} & = \frac{{3y_{ji} }}{{2l_{ji}^{2} }}\left( { - 1 + s^{2} } \right) \\ N_{22} & = \frac{1}{{4l_{ji}^{2} }}\left( { - 1 + s} \right)\left( { - 2x_{ji}^{2} + \left( {1 + 3s} \right)y_{ji}^{2} } \right) \\ N_{23} & = \frac{{ - 3x_{ji} y_{ji} }}{{4l_{ji}^{2} }}\left( { - 1 + s^{2} } \right) \\ N_{24} & = \frac{{ - 3y_{ji} }}{{2l_{ji}^{2} }}\left( { - 1 + s^{2} } \right) \\ N_{25} & = \frac{1}{{4l_{ji}^{2} }}\left( {1 + s} \right)\left( {2x_{ji}^{2} + \left( { - 1 + 3s} \right)y_{ji}^{2} } \right) \\ N_{26} & = \frac{{ - 3x_{ji} y_{ji} }}{{4l_{ji}^{2} }}\left( { - 1 + s^{2} } \right) \\ N_{31} & = \frac{{ - 3x_{ji} }}{{2l_{ji}^{2} }}\left( { - 1 + s^{2} } \right) \\ N_{32} & = \frac{{ - 3x_{ji} y_{ji} }}{{4l_{ji}^{2} }}\left( { - 1 + s^{2} } \right) \\ N_{33} & = \frac{1}{{4l_{ji}^{2} }}\left( { - 1 + s} \right)\left( { - 2y_{ji}^{2} + \left( {1 + 3s} \right)x_{ji}^{2} } \right) \\ N_{34} & = \frac{{3x_{ji} }}{{2l_{ji}^{2} }}\left( { - 1 + s^{2} } \right) \\ N_{35} & = \frac{{ - 3x_{ji} y_{ji} }}{{4l_{ji}^{2} }}\left( { - 1 + s^{2} } \right) \\ N_{36} & = \frac{1}{{4l_{ji}^{2} }}\left( {1 + s} \right)\left( {2y_{ji}^{2} + \left( { - 1 + 3s} \right)x_{ji}^{2} } \right) \\ \end{aligned}$$