Consider the finite element analysis illustrated in Figure $2.$

Young's modulus $E,$ Poisson's ratio $v,$ Density $,$ Gravity $g,$ Thickness $t,$ Plane stress condition: $\left[\begin{array}{c}{}_{}\\ {}_{yy}\\ {}_{xy}\end{array}\right]=\frac{E}{1-v^2}\left[\begin{array}{ccc}1& v& 0\\ v& 1& 0\\ 0& 0& \frac{1-v}{2}\end{array}\right]\left[\begin{array}{c}{}_{}\\ {}_{yy}\\ {}_{xy}\end{array}\right]$ with $v=0.($a$)(5$pt$.)$ Referring to Figure $3,$ establish the $H$ and $B$ matrices of an element, in which the nodal DOF vector for the element is defined by hat$(u)={\left[\begin{array}{ccccccc}{u}_1& u_2& u_3& u_4& v_1& v_2& v_3{v}_4\end{array}\right]}^T.$

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$($b$)(10$pt$.)$ Calculate the components of the $K$ matrix, $K_{U_2{U}_2},K_{U_6{U}_7},K_{U_7{U}_6}$ and $K_{U_5{U}_{12}}$ of the structural assemblage.Young's modulus $E,$ Poisson's ratio $v,$ Density $,$ Gravity $g,$ Thickness $t,$

Plane stress condition: $\left[\begin{array}{c}{r}_n\\ r_x\\ r_n\end{array}\right]=\frac{E}{1-v^2}\left[\begin{array}{ccc}1& v& 0\\ v& 1& 0\\ 0& 0& \frac{1-v}{2}\end{array}\right]\left[\begin{array}{c}{c}_n\\ c_n\\ {}_n\end{array}\right]$ with $v=0.$

$($a$)(5$pt$.)$ Referring to Figure $3,$ establish the $H$ and $B$ matrices of an element, in which the nodal

DOF vector for the element is defined by hat$(u)=[u_1{u}_2{u}_3{u}_4{v}_1{v}_2{v}_3{v}_4{]}^7.$

$($b$)(10$pt$.)$ Calculate the components of the $K$ matrix, $K_{U_2{U}_2},K_{U_6{U}_7},K_{U7U6}$ and $K_{U5U12}$

of the structural asscmblage.