A rectangular plate has the aspect ratio $3$ ; the length along the x direction is $\backslash (3$ L$[\backslash$mathrm$\{$~m$\}]\backslash )$ and the length along the y direction is $\backslash ($ L$[\backslash$mathrm$\{$~m$\}]\backslash ).$ The plate is isotropic with Young's modulus of $\backslash ($ E$[\backslash$mathrm$\{$~Pa$\}]\backslash )$ and Poisson's ratio of $\backslash ($ v $\backslash ).$ The plate is supported by a line series of springs in the middle, as sketched below. The spring constant per area is set as $\backslash ($ k$\backslash$left$[\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash$right$]=\backslash$mathrm$\{$k$\}\_\{0\}\backslash$frac$\{$E t$^\{3\}\}\{12\backslash$left$(1-$v$^\{2\}\backslash$right$)\}\backslash ),$ and $\backslash (\backslash$mathrm$\{$k$\}\_\{0\}\backslash )$ is a constant. The plate is simply supported at all the edges, and applied with the compressive stress resultant $\backslash ($ N x $\backslash )$ only in the $\backslash ($ x $\backslash )$ direction.

a$)$ Think of two single$-$mode virtual displacements to describe the plate deformation due to buckling.

b$)$ Solve for the critical buckling loads associated with each of the two virtual displacement.

c$)$ Based on the above calculation $($or based on your engineering knowledge, in case you have trouble working on part b$),$ provide your observation about the critical buckling loads difference$($s$)$ between the two virtual displacements.