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A simplified four-unknown shear and normal deformations theory for bidirectional laminated plates

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Abstract

This paper presents a simplified 4-unknown shear and normal deformations theory for the bending analysis of cross-ply laminated plates. The present theory accounts for an adequate distribution of transverse shear strains through the plate thickness and tangential stress-free on the plate surfaces. The effect of normal strain is also included. The governing, equilibrium equations and boundary conditions are derived by employing the virtual work principle. Numerical results for stresses and displacements are compared well with those obtained using 3-D elasticity solution.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    A M ZENKOUR

  2. Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh, 33516, Egypt

    A M ZENKOUR

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  1. A M ZENKOUR
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Correspondence to A M ZENKOUR.

Appendices

Appendix A

The elements J ij = J ji of the matrix elements [J] appeared in Eq. (11) are given by

$$\begin{array}{@{}rcl@{}} J_{11} &=&A_{11} d_{11} +A_{66} d_{22} , \quad J_{12} =\left( {A_{12} +A_{66} } \right)d_{12} ,\\ J_{13} &=&-B_{11} d_{111} -\left( {B_{12} +2B_{66} } \right)d_{122}, \\ J_{14} &=&\alpha_{3} G_{13}^{a} d_{1} +\alpha_{1} B_{11}^{a} d_{111} +\left[ \alpha_{2} B_{12}^{a} +\left( \alpha_{1} +\alpha_{2} \right)B_{66}^{a} \right]d_{122} ,\\ J_{22} &=&A_{66} d_{11} +A_{22} d_{22} ,\quad J_{23} =-\left( {B_{12} +2B_{66} } \right)d_{122} -B_{22} d_{222} ,\\ J_{24} &=&\alpha_{3} G_{23}^{a} d_{2} +\alpha_{2} B_{22}^{a} d_{222} +\left[ \alpha_{1} B_{12}^{a} +\left( \alpha_{1} +\alpha_{2} \right)B_{66}^{a} \right]d_{122} ,\\ J_{33} &=&-D_{11} d_{1111} -2\left( {D_{12} +2D_{66} } \right)d_{1122} -D_{22} d_{2222} ,\\ J_{34} &=&\alpha_{3} \left( {H_{23}^{a} d_{11} +H_{23}^{a} d_{22} } \right)+\alpha_{1} D_{11}^{a} d_{1111} +\left( {\alpha_{1} +\alpha_{2} } \right)\left[ {D_{12}^{a} +2D_{66}^{a} } \right]d_{1122}\\&&+\alpha_{2} D_{22}^{a} d_{2222} ,\\ J_{44} &=&{\alpha_{3}^{2}} P_{33}^{a} +\left[ {2\alpha_{1} \alpha_{3} L_{13}^{a} -\left( {\alpha_{1} +\alpha_{3} } \right)^{2}A_{55}^{a} } \right]d_{11} +\left[ {2\alpha_{2} \alpha_{3} L_{23}^{a} -\left( {\alpha_{2} +\alpha_{3} } \right)^{2}A_{44}^{a} } \right]d_{22} \\ &&+{\alpha_{1}^{2}} F_{11}^{2} d_{1111} +\left[ {2\alpha_{1} \alpha_{2} F_{12}^{a} +\left( {\alpha_{1} +\alpha_{2} } \right)^{2}F_{66}^{a} } \right]d_{1122} +{\alpha_{2}^{2}} F_{22}^{a} d_{2222} , \end{array} $$

where d i , d ij , d ijl , d ijlm are the following differential operators:

$$d_{i} =\frac{\partial }{\partial_{x_{i}}}, \quad d_{ij} =\frac{\partial^{2}}{{\partial_{x_{i}}} \partial_{x_{j}}},\quad d_{ijl} =\frac{\partial^{3}}{{{{\partial_x}_{i} {\partial_{x_{j}}}}}{{\partial_{x_{l}}}}}, \quad d_{ijlm} =\frac{\partial^{4}}{\partial_{x_{i}} \partial_{x_{j}} \partial_{x_{l}} \partial_{x_{m}}}.$$

Appendix B

The elements \(\bar {J}_{ij} =\bar {J}_{ij} \) of the matrix elements \(\left [ \bar {J} \right ]\) appeared in Eq. (17) are given by

$$\begin{array}{@{}rcl@{}} \bar{J}_{11} &=&A_{11} \lambda^{2}+A_{66} \mu \,^{2}, \quad \bar{J}_{12} =\left( {A_{12} +A_{66} } \right)\lambda\mu,\\ {\bar{J}_{13} } &=&-\lambda \left[ {B_{11} \lambda^{2}+\left( {B_{12} +2B_{66} } \right)\mu^{2}} \right],\\ \bar{J}_{14}&=&\lambda \left\{ {\alpha_{1} B_{11}^{a} \lambda^{2}+\left[ {\alpha_{2} B_{12}^{a} +\left( {\alpha_{1} +\alpha_{2} } \right)B_{66}^{a} } \right]\mu^{2}-\alpha_{3} G_{13}^{a} } \right\}, \\ \bar{J}_{22} &=&A_{66} \lambda^{2}+A_{22} \mu^{2}, \quad \bar{J}_{23} =-\mu \left[ {\left( {B_{12} +2B_{66} } \right)\lambda^{2}+B_{22} \mu^{2}} \right],\\ \bar{J}_{24} &=&\mu \left\{ {\alpha_{2} B_{22}^{a} \mu^{2}+\left[ {\alpha_{1} B_{12}^{a} +\left( {\alpha_{1} +\alpha_{2} } \right)B_{66}^{a} } \right]\lambda^{2}-\alpha_{3} G_{23}^{a} } \right\},\\ \bar{J}_{33} &=&D_{11} \lambda^{4}+2\left( {D_{12} +2D_{66} } \right)\lambda^{2}\mu^{2}+D_{22} \mu^{4},\\ \bar{J}_{34} &=&\alpha_{3} \left( {H_{13}^{a} \lambda^{2}+H_{23}^{a} \mu^{2}} \right)-\alpha_{1} D_{11}^{a} \lambda^{4}-\left( {\alpha_{1} +\alpha_{2} } \right)\left[ {D_{12}^{a} +2D_{66}^{a} } \right]\lambda^{2}\mu^{2}-\alpha_{2} D_{22}^{a} \mu^{4},\\ \bar{J}_{44} &=&{\alpha_{3}^{2}} P_{33}^{a} -\left[ {2\alpha_{1} \alpha_{3} L_{13}^{a} -\left( {\alpha_{1} +\alpha_{3} } \right)^{2}A_{55}^{a} } \right]\lambda^{2}-\left[ {2\alpha_{2} \alpha_{3} L_{23}^{a} -\left( {\alpha_{2} +\alpha_{3} } \right)^{2}A_{44}^{a} } \right]\mu^{2}\\ &&+{\alpha_{1}^{2}} F_{11}^{a} \lambda^{4}+\left[ {2\alpha_{1} \alpha_{2} F_{12}^{a} +\left( {\alpha_{1} +\alpha_{2} } \right)^{2}F_{66}^{a} } \right]\lambda^{2}\mu^{2}+{\alpha_{2}^{2}} F_{22}^{a} \mu^{4}. \end{array} $$

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ZENKOUR, A.M. A simplified four-unknown shear and normal deformations theory for bidirectional laminated plates. Sadhana 40, 215–234 (2015). https://doi.org/10.1007/s12046-014-0306-x

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