QUESTION $4(25$ MARKS$)$

A two$-$bar planar truss structure has geometry as shown in Figure Q$4.$ A concentrated force $P=500kN$ is applied to the truss at joint $C,$ in the direction shown. Member $AC$ is made of steel $=200$GPa $)$ and has a cross$-$sectional area of $600m{m}^2$ while member $BC$ is made of aluminum $(E=70$GPa $)$ with cross$-$sectional area of $1000m{m}^2.$

$($i$)$ Sketch a finite element model of the whole structure, labelling all nodes and elements, and indicate the number of degrees of freedom.

$($ii$)$ Obtain the relevant reduced stiffness matrix for each element, considering only the active degrees$-$of$-$freedom.

$($iii$)$ Set up the related equilibrium equation for the whole truss structure, putting in all the known values of the relevant parameters

$($iv$)$ Obtain the unknown reactions at point A $.$

$($v$)$ Determine the stress in bar aluminum.

Given:

$[K^{(e)}]=\frac{A_e{E}_e}{L_e}\left[\begin{array}{cccc}{c}^2& cs& -c^2& -cs\\ cs& s^2& -cs& -s^2\\ -c^2& -cs& c^2& cs\\ -cs& -s^2& cs& s^2\end{array}\right]$

Where: $,c=cos,s=sin$