Problem $8\mathrm{.}15$ from Plates and Shells: Analysis and Design, by Ugural. Derive an expression for the deflection of an orthotropic clamped rectangular plate under a uniform load $p_0($Figure $5\mathrm{.}19).$ Assume, by setting $m\frac{}{a}={}_m$ and $2n\frac{}{b}={}_n$

$w={\displaystyle \underset{m}{\overset{}{}}}{\displaystyle \underset{n}{\overset{}{}}}{a}_{mn}(1-cos{}_{m}x)(1-cos{}_{n}y)$

Use the Ritz method by retaining the first term of the series solution. Find the maximum deflection if $a=b.$