For the two$-$bar pin joint system shown in Figure Q$1,$ the section areas are both $A=200m{m}^2$ and the Young's modulus E $=50$ GPa. A load F$=10$ kN is applied at Point $3.$

a$)$ Assemble the stiffness matrices of the rod elements in a global stiffness matrix. $(8$ Marks$)$

b$)$ Apply the boundary conditions, solve the linear systems of equations and obtain the vertical displacement at joint $3.$

$(7$ Marks$)$

c$)$ Apply the boundary conditions, solve the linear systems of equations and obtain the horizontal displacement at joint $3.$

$(7$ Marks$)$

d$)$ Calculate the axial force in bar $1-3$

$(3$ Marks$)$