Static solutions for functionally graded simply supported plates

Abstract

In this article mixed semi-analytical and analytical solutions are presented for a rectangular plate made of functionally graded (FG) material. All edges of a plate are under simply supported (diaphragm) end conditions and general stress boundary conditions can be applied on both top and bottom surface of a plate during solution. A mixed semi-analytical model consists in defining a two-point boundary value problem governed by a set of first-order ordinary differential equations in the plate thickness direction. Analytical solutions based on shear-normal deformation theories are also established to show the accuracy, simplicity and effectiveness of mixed semi-analytical model. The FG material is assumed to be exponential in the thickness direction and Poisson’s ratio is assumed to be constant.

Appendix

The coefficients of matrix [A] are,

$$ \begin{aligned} &A_{1,1} = \frac{{\bar{E}_{h} - \bar{E}_{o} }}{\lambda }\,\quad A_{1,2} = \frac{{h\bar{E}_{h} - A_{1,1} }}{\lambda }\quad A_{1,3} = \frac{{h^{2} \bar{E}_{h} - 2A_{1,2} }}{\lambda }\quad A_{1,4} = \frac{{h^{3} \bar{E}_{h} - 3A_{1,3} }}{\lambda } \hfill \\ &A_{1,5} = \upsilon A_{1,1} \quad A_{1,6} = \upsilon A_{1,2} \quad A_{1,7} = \upsilon A_{1,3} \quad A_{1,8} = \upsilon A_{1,4} \quad A_{1,9} = A_{1,5} \hfill \\ &A_{1,10} = A_{1,6} \quad A_{1,11} = A_{1,7} \hfill \\ \end{aligned} $$

$$ \begin{aligned} &A_{2,2} = A_{1,3} \quad A_{2,3} = A_{1,4} \quad A_{2,4} = \frac{{h^{4} \bar{E}_{h} - 4A_{2,3} }}{\lambda }\quad A_{2,5} = \upsilon A_{1,2} \quad A_{2,6} = \upsilon A_{2,2} \hfill \\ &A_{2,7} = \upsilon A_{2,3} \quad A_{2,8} = \upsilon A_{2,4} \quad A_{2,9} = A_{2,5} \quad A_{2,10} = A_{2,6} \quad A_{2,11} = A_{2,7} \hfill \\ \end{aligned} $$

$$ \begin{aligned} &A_{3,3} = A_{2,4} \quad A_{3,4} = \frac{{h^{5} \bar{E}_{h} - 5A_{3,3} }}{\lambda }\quad A_{3,5} = \upsilon A_{1,3} \quad A_{3,6} = \upsilon A_{2,3} \quad A_{3,7} = \upsilon A_{3,3} \hfill \\ &A_{3,8} = \upsilon A_{3,4} \quad A_{3,9} = A_{3,5} \quad A_{3,10} = A_{3,6} \quad A_{3,11} = A_{3,7} \hfill \\ \end{aligned} $$

$$ \begin{aligned} &A_{4,4} = \frac{{h^{6} \bar{E}_{h} - 6A_{4,3} }}{\lambda }\quad A_{4,5} = \upsilon A_{1,4} \quad A_{4,6} = \upsilon A_{2,4} \quad A_{4,7} = \upsilon A_{3,4} \quad A_{4,8} = \upsilon A_{4,4} \hfill \\ &A_{4,9} = A_{4,5} \quad A_{4,10} = A_{4,6} \quad A_{4,11} = A_{4,7} \hfill \\ \end{aligned} $$

$$ A_{5,5} = A_{1,1} \quad A_{5,6} = A_{1,2} \quad A_{5,7} = A_{1,3} \quad A_{5,8} = A_{1,4} \quad A{}_{5,9} = A_{1,5} \quad A_{5,10} = A_{1,6} \quad A_{5,11} = A_{1,7} $$

$$ A_{6,6} = A_{2,2} \quad A_{6,7} = A_{2,3} \quad A_{6,8} = A_{2,4} \quad A_{6,9} = A_{2,5} \quad A_{6,10} = A_{2,6} \quad A_{6,11} = A_{2,7} $$

$$ A_{7,7} = A_{3,3} \,\,\,\,\,\,A_{7,8} = A_{3,4} \,\,\,\,\,\,A_{7,9} = A_{3,5} \,\,\,\,\,\,A_{7,10} = A_{3,6} \,\,\,\,\,\,A_{7,11} = A_{3,7} $$

$$ A_{8,8} = A_{4,4} \quad A_{8,9} = A_{4,5} \quad A_{8,10} = A_{4,6} \quad A_{8,11} = A_{4,7} $$

$$ A_{9,9} = A_{1,1} \quad A_{9,10} = A_{1,2} \quad A_{9,11} = A_{1,3} $$

$$ A_{10,10} = A_{2,2} \quad A_{10,11} = A_{2,3} $$

$$ A_{11,11} = A_{3,3} $$

where, $\bar{E}_{h} = \left( {\frac{{E_{o} }}{{1 - \upsilon^{2} }}} \right)e^{\lambda } \quad {\text{and}}\quad \bar{E}_{o} = \frac{{E_{o} }}{{(1 - \upsilon^{2} )}}$ and A _i,j = A _j,i, (i, j = 1 to 11)

The coefficients of matrix [B] are,

$$ \begin{aligned} &B_{1,1} = \frac{1 - \upsilon }{2}A_{1,1} \quad B_{1,2} = \frac{1 - \upsilon }{2}A_{1,2} \quad B_{1,3} = \frac{1 - \upsilon }{2}A_{1,3} \quad B_{1,4} = \frac{1 - \upsilon }{2}A_{1,4} \hfill \\ &B_{2,2} = \frac{1 - \upsilon }{2}A_{2,2} \quad B_{2,3} = \frac{1 - \upsilon }{2}A_{2,3} \quad B_{2,4} = \frac{1 - \upsilon }{2}A_{2,4} \hfill \\ &B_{3,3} = \frac{1 - \upsilon }{2}A_{3,3} \quad B_{3,4} = \frac{1 - \upsilon }{2}A_{3,4} \hfill \\ &B_{4,4} = \frac{1 - \upsilon }{2}A_{4,4} \hfill \\ \end{aligned} $$

and B _i,j = B _j,i (i, j = 1 to 4)

[D] and [E] matrix are same as [B] matrix.

The coefficients of vector {I} are

$$ \begin{aligned} &I_{1} = \frac{{\rho_{h} - \rho_{o} }}{{\lambda_{1} }}\,\quad I_{2} = \frac{{h\,\rho_{h} - I_{1} }}{{\lambda_{1} }}\quad I_{3} = \frac{{h^{2} \rho_{h} - 2\,I_{2} }}{{\lambda_{1} }} \hfill \\ &I_{4} = \frac{{h^{3} \rho_{h} - 3\,I_{3} }}{{\lambda_{1} }}\quad I_{5} = \frac{{h^{4} \,\rho_{h} - 4\,I_{4} }}{{\lambda_{1} }}\quad I_{6} = \frac{{h^{5} \rho_{h} - 5\,I_{5} }}{{\lambda_{1} }} \hfill \\ &I_{7} = \frac{{h^{6} \rho_{h} - 6\,I_{6} }}{{\lambda_{1} }} \hfill \\ \end{aligned} $$

The coefficients of matrix [X] are

$$ \begin{aligned} &X_{1,1} = A_{1,1} \alpha^{2} + B_{1,1} \beta^{2} \quad \,X_{1,2} = A_{1,5} \alpha \beta + B_{1,1} \alpha \beta \quad X_{1,3} = 0\quad X_{1,4} = A_{1,2} \alpha^{2} + B_{1,2} \beta^{2} \quad X_{1,5} = A_{1,6} \alpha \beta + B_{1,2} \alpha \beta \quad X_{1,6} = - A_{1,9} \alpha \,\quad X_{1,7} = A_{1,3} \alpha^{2} + B_{1,3} \beta^{2} \quad X_{1,8} = A_{1,7} \alpha \beta + B_{1,3} \alpha \beta \quad X_{1,9} = - 2A_{1,10} \alpha \quad X_{1,10} = A_{1,4} \alpha^{2} + B_{1,4} \beta^{2} \quad X_{1,11} = A_{1,8} \alpha \beta + B_{1,4} \alpha \beta \quad X_{1,12} = - 3A_{1,11} \alpha \hfill \\ \end{aligned} $$

$$ \begin{aligned} &X_{2,2} = A_{1,5} \beta^{2} + B_{1,1} \alpha^{2} \quad X_{2,3} = 0\quad X_{2,4} = A_{5,2} \alpha \beta + B_{1,2} \alpha \beta \quad X_{2,5} = A_{5,6} \beta^{2} + B_{1,2} \alpha^{2} \quad X_{2,6} = - A_{5,9} \beta \quad X_{2,7} = A_{5,3} \alpha \beta + B_{1,3} \alpha \beta \quad X_{2,8} = A_{5,7} \beta^{2} + B_{1,3} \alpha^{2} \quad X_{2,9} = - 2A_{5,10} \beta \quad X_{2,10} = A_{5,4} \alpha \beta + B_{1,4} \alpha \beta \quad X_{2,11} = A_{5,8} \beta^{2} + B_{1,4} \alpha^{2} \quad X_{2,12} = - 3A_{5,11} \beta \hfill \\ \end{aligned} $$

$$ \begin{aligned} &X_{3,3} = D_{1,1} \alpha^{2} + E_{1,1} \beta^{2} \quad X_{3,4} = D_{1,1} \alpha \quad X_{3,5} = E_{1,1} \beta \quad X_{3,6} = D_{1,2} \alpha^{2} + E_{1,2} \beta^{2} \quad X_{3,7} = 2D_{1,2} \alpha \quad X_{3,8} = 2E_{1,2} \beta \quad X_{3,9} = D_{1,3} \alpha^{2} + E_{1,3} \beta^{2} \quad X_{3,10} = 3D_{1,3} \alpha \quad X_{3,11} = 3E_{1,3} \beta \quad X_{3,12} = D_{1,4} \alpha^{2} + E_{1,4} \beta^{2} \hfill \\ \end{aligned} $$

$$ \begin{aligned} &X_{4,4} = A_{2,2} \alpha^{2} + B_{2,2} \beta^{2} + D_{1,1} \quad X_{4,5} = A_{2,6} \alpha \beta + B_{2,2} \alpha \beta \quad X_{4,6} = - A_{2,9} \alpha + D_{1,2} \alpha \quad X_{4,7} = A_{2,3} \alpha^{2} + B_{2,3} \beta^{2} + 2D_{1,2} \quad X_{4,8} = A_{2,7} \alpha \beta + B_{2,3} \alpha \beta \quad X_{4,9} = - 2A_{2,10} \alpha + D_{1,3} \alpha \quad X_{4,10} = A_{2,4} \alpha^{2} + B_{2,4} \beta^{2} + 3D_{1,3} \quad X_{4,11} = A_{2,8} \alpha \beta + B_{2,4} \alpha \beta \quad X_{4,12} = - 3A_{2,11} \alpha + D_{1,4} \alpha \hfill \\ \end{aligned} $$

$$ \begin{aligned} &X_{5,5} = A_{6,6} \beta^{2} + B_{2,2} \alpha^{2} + E_{1,1} \quad X_{5,6} = - A_{6,9} \beta + E_{1,2} \beta \quad X_{5,7} = A_{6,3} \alpha \beta + B_{2,3} \alpha \beta \quad X_{5,8} = A_{6,7} \beta^{2} + B_{2,3} \alpha^{2} + 2E_{1,2} \quad X_{5,9} = - 2A_{6,10} \beta + E_{1,3} \beta \quad X_{5,10} = A_{6,4} \alpha \beta + B_{2,4} \alpha \beta \quad X_{5,11} = A_{6,8} \beta^{2} + B_{2,4} \alpha^{2} + 3E_{1,3} \quad X_{5,12} = - 3A_{6,11} \beta + E_{1,4} \beta \hfill \\ \end{aligned} $$

$$ \begin{aligned} &X_{6,6} = D_{2,2} \alpha^{2} + E_{2,2} \beta^{2} + A_{9,9} \quad X_{6,7} = - A_{9,3} \alpha + 2D_{2,2} \beta \quad X_{6,8} = - A_{9,7} \beta + 2E_{2,2} \beta \quad X_{6,9} = D_{2,3} \alpha^{2} + E_{2,3} \beta^{2} + 2A_{9,10} \quad X_{6,10} = - A_{9,4} \alpha + 3D_{2,3} \alpha \quad X_{6,11} = - A_{9,8} \beta + 3E_{2,3} \beta \quad X_{6,12} = D_{2,4} \alpha^{2} + E_{2,4} \beta^{2} + 3A_{9,11} \hfill \\ \end{aligned} $$

$$ \begin{aligned} &X_{7,7} = A_{3,3} \alpha^{2} + B_{3,3} \beta^{2} + 4D_{2,2} \quad X_{7,8} = A_{3,7} \alpha \beta + B_{3,3} \alpha \beta \quad X_{7,9} = - 2A_{3,10} \alpha + 2D_{2,3} \alpha \quad X_{7,10} = A_{3,4} \alpha^{2} + B_{3,4} \beta^{2} + 6D_{2,4} \quad X_{7,11} = A_{3,8} \alpha \beta + B_{3,4} \alpha \beta \hfill \\ \end{aligned} $$

$$ \begin{aligned} &X_{8,8} = A_{7,7} \beta^{2} + B_{3,3} \alpha^{2} + 4E_{2,2} \quad X_{8,9} = - 2A_{7,10} \beta + 2E_{2,3} \beta \quad X_{8,10} = A_{7,4} \alpha \beta + B_{3,4} \alpha \beta \quad X_{8,11} = A_{7,8} \beta^{2} + B_{3,4} \alpha^{2} + 6E_{2,3} \quad X_{8,12} = - 3A_{7,11} \beta + 2E_{2,4} \beta \hfill \\ \end{aligned} $$

$$ \begin{aligned} &X_{9,9} = D_{3,3} \alpha^{2} + E_{3,3} \beta^{2} + 4A_{10,10} \quad X{}_{9,10} = - 2A_{10,4} \alpha + 3D_{3,3} \alpha \\ & X_{9,11} = - 2A_{10,8} \beta + 3E_{3,3} \beta \\ &X_{9,12} = D_{3,4} \alpha^{2} + E_{3,4} \beta^{2} + 6A_{10,11} \hfill \\ \end{aligned} $$

$$\begin{aligned} &X_{10,10} = A_{4,4} \alpha^{2} + B_{4,4}\beta^{2} + 9D_{3,3} \\ &X_{10,11} = A_{4,8} \alpha \beta +B_{4,4} \alpha \beta \quad A_{10,12} = - 3A_{4,11} \alpha +3D_{3,4} \alpha \end{aligned} $$

$$\begin{aligned} &X_{11,11} = A_{8,8} \beta^{2} + B_{4,4}\alpha^{2} + 9E_{3,3} \\ &X_{11,12} = - 3A_{8,11} \beta + 3E_{3,4}\beta \end{aligned} $$

$$ X_{12,12} = D_{4,4} \alpha^{2} + E_{4,4} \beta^{2} + 9A_{11,11} $$

in which, $\alpha = \frac{m\pi }{a}$ and $\beta = \frac{n\pi }{b}$ and X _i,j = X _j,i, (i, j = 1 to 12)