Problem $3$ A bar reinforced plane stress square panel is subjected to loads as shown in the figure. The material and geometric properties are given as $E=1{10}^6,v=0\mathrm{.}25,a=30$ in $.,$ thickness of panel $t=0\mathrm{.}02in.,$ cross sectional area of bar $A=0\mathrm{.}2i{n}^2.$ The spring constant $k=2000l\frac{b}{i}n.$ The panel is subjected to one uniform distributed $($normal$)$ load $p_1=5000$ psi and a uniformly distributed tangential $($shear$)$ load $p_2=4000,$ as shown in the figure. A bar is used to reinforce the panel. A concentrated $P=2000lb$ is also applied at node $2$ as shown.

a$)$ Express the reduced element stiffness matrix $[k_{\text{b}\text{a}\text{r}}],[k_{Q4S}],$ and $[K_{S\text{t}\text{r}\text{u}\text{c}\text{t}\text{u}\text{r}\text{e}}].$

b$)$ Express the reduced $FE$ system equation $\{F\}=[K]\{d\}$ in numerical form $(\{d\}$ is unknown$).$

c$)$ Find the nodal displacements for the system $($spring$,$ bar, and $Q_4$ Square elem.