(Solved): A rectangular plate, simply supported at all edges, is loaded by a uniformly distributed loading q ...

A rectangular plate, simply supported at all edges, is loaded by a uniformly distributed loading $q$ and an in-plane compressive load of $N_x$ per unit length on the edge $x=0$ and $x=\mathrm{a}$, as shown in Figure 2 . Find an approximate solution for the deflection of the plate using energy method. The one term deflection function for the plate can be assumed as follows: $w(x,y)=W_{mn}\sin \frac{n?x}{a}\sin \frac{n?y}{b},$ where $w(x,y)$ is the lateral deflection of the plate, $W_{mn}$ is the unknown deflection coefficient, ' $a$ ' and ' $b$ ' are the length and width of the plate, respectively. The following equation may be used. Strain energy: $U=\frac{D}{2}{?}_{\text{area }}\left\{{\left(\frac{{?}^{2}w}{?x^2}+\frac{{?}^{2}w}{?y^2}\right)}^2?2(1?v)\left[\frac{{?}^{2}w}{?x^2}\frac{{?}^{2}w}{?y^2}?{\left(\frac{{?}^{2}w}{?x?y}\right)}^2\right]\right\}dxdy$ A rectangular plate, simply supported at all edges, is loaded by a uniformly distributed loading $q$ and an in-plane compressive load of $N_x$ per unit length on the edge $x=0$ and $x=\mathrm{a}$, as shown in Figure 2 . Find an approximate solution for the deflection of the plate using energy method. The one term deflection function for the plate can be assumed as follows: $w(x,y)=W_{mn}\sin \frac{n?x}{a}\sin \frac{n?y}{b},$ where $w(x,y)$ is the lateral deflection of the plate, $W_{mn}$ is the unknown deflection coefficient, ' $a$ ' and ' $b$ ' are the length and width of the plate, respectively. The following equation may be used. Strain energy: $U=\frac{D}{2}{?}_{\text{area }}\left\{{\left(\frac{{?}^{2}w}{?x^2}+\frac{{?}^{2}w}{?y^2}\right)}^2?2(1?v)\left[\frac{{?}^{2}w}{?x^2}\frac{{?}^{2}w}{?y^2}?{\left(\frac{{?}^{2}w}{?x?y}\right)}^2\right]\right\}dxdy$ Figure 2