QUESTION $1(25$ MARKS$)$

$($a$)$ Consider a single $1-$dimensional bar element with two nodes as shown in Figure Q$1($a$).$ Assuming that the displacement field, $u(x)$ varies linearly from node $1$ to node $2$ and can be described as $u(x)=a_1+a_{2}x..$ By using the information given:

i$.$ Establish the general equation of $u(x)$ in term of nodal displacement

ii$.$ Derive the strain and stress equations in the matrix form

$[8$ marks$]$

$($b$)$ Consider two linear elastic bars that having similar length and modulus of elasticity but different cross section as shown in Figure Q$1($b$).$ The Bar $1$ is rigidly supported at the left end and Bar $2$ is free on the right end. Two concentrated forces of $2P$ and $P$ are applied at the location as shown. Let solve the problem numerically by making use of the finite element method.

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

ii$.$ Write a complete column vector for the external force, $\{f\}.$

iii. Determine the stiffness matrix for each element in term of $A,E$ and $L$

iv$.$ Write the system of linear equation in the form of $\{F\}=[K]\{Q\}$ for a complete structural problem as shown in Figure Q$1($b$)$

v$.$ Write the reduced system of linear equation in the form of $\{F\}=[K]\{u\}$ by considering all the boundary conditions.

vi$.$ Determine whether Bar $2$ has touched and compressed on the right wall or not?

$|$~$17|$ marks$$

Given:

$[K^{(e)}]=\frac{A_e{E}_e}{L_e}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$