QUESTION $3(25$ MARKS$)$

A three$-$bar planar truss structure has geometry as shown in Figure Q$3.$ Concentrated forces of $1kN$ and $2kN$ are applied to the truss structure at joint A and C respectively in the direction shown. Each member of the structure is made of Aluminium Alloy $(E=70$GPa $)$ and has a crosssectional area of $300m{m}^2.$

$($a$)$ Sketch a finite element model of the whole bar structure, labelling all nodes and elements, and indicate the number of degrees of freedom. $[3$ marks$]$

$($b$)$ Determine the stiffness matrix for each element. $[9$ marks$]$

$($c$)$ By applying the boundary conditions, write the reduced global system of linear equations.

$[8$ marks$]$

$($d$)$ Determine the unknown degrees of freedom. $[3$ marks$]$

$($e$)$ Calculate the force reactions at joint A$.[2$ marks$]$

rigure ys

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: $,c=cos,s=sin$