Problem $1(20$ points$)$

The structure shown in the figure has length $3L$ and a square crosssection with side $a.$ The structure is made of a material with a modulus of elasticity $E.$ The structure is simply supported at its ends. Using an appropriate finite$-$element formulation, show how you will determine the deflection and slope at the left end and at distance of $L$ from the left end.

a$.$ Write down the condensed equations.

b$.$ Show all boundary conditions.

c$.$ Solve for nodal unknowns $($in terms of $E,$ a and $L)$

d$.$ Assume that there is a spring of stiffness $k$ is attached at the middle of the structure, as shown i: the second image. How will your analysis and results change? Discuss and show the nodal unknowns

Problem $2$a $(10$ points$)$

The figure below shows a finite$-$element mesh. In the figure, all numbers within circles denote element numbers and all numbers without circles around them denote global node numbers. Assume that each node has two degrees of freedom. Elements $1$ and $5$ are plane elements, while the remaining elements are truss elements.

a$)$ determine the size of each element matrix and global asse$$nuy mau

b$)$ determine the half bandwidth of the assembly matrix

c$)$ rearrange the node numbers to minimize half bandwidth

d$)$ determine the new half bandwidth

Problem $2$b $(10$ points$)$

The figure below shows a truss structure with two degrees of freedom per node

a$)$ Determine the assembly stiffness matrix size

b$)$ Show which of the cells in the stiffness matrix will be zero and which will be non$-$zero $($ show entire matrix$)$

c$)$ Determine the half bandwidth of the matrix

d$)$ Rearrange the node numbers so that the half bandwidth is minimum. Determine the the half bandwidth

need handwriting solution for all the questions

please do urgently