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Question $3(15$ points$)$: A square plate with simply supported edges is subject to an areal load $q$ as in the figure below

The deflection in the $z$ dimension can be determined by solving the elliptic PDE:

$\frac{de{l}^{4}z}{del{x}^4}+2\frac{de{l}^{4}z}{del{x}^{2}del{y}^2}+\frac{de{l}^{4}z}{del{y}^4}=\frac{q}{D}$

Subject to the boundary conditions that, at the edges, the deflection and slope normal to the boundary are zero. The parameter $D$ is the flexural rigidity

$D=\frac{E{z}^3}{12(1-{}^2)}$

where $E=$ the modulus of elasticity, $z=$ the plate's thickness, and $=$ Poisson's ratio. If a new variable is defined as

$u=\frac{de{l}^{2}z}{del{x}^2}+\frac{de{l}^{2}z}{del{y}^2}$

Which reformulate the deflection equation into

$\frac{de{l}^{2}u}{del{x}^2}+\frac{de{l}^{2}u}{del{y}^2}=\frac{q}{D}$

Determine the deflections using Liebmann method for a square plate subject to a constant areal load. Given the plate has $2\mathrm{.}7-m-$long edges, $q=33\mathrm{.}6k\frac{N}{m^2},=0\mathrm{.}3,z={10}^{-2}m,$ and $E=2{10}^{11}Pa.$ Employ $x=y=0\mathrm{.}3m.$