Section D: Long Problems

You must attempt all questions $(3$ questions$)$ from this section

$1-(35pts$ The below figure shows a stepped bar structure that consists of three elements. The structure is fixed in the left$-$hand side.

A $20(kN)$ load is applied at the position of $100(mm)$ from left and a $10(kN)$ load is applied at the end right$-$hand side.

Develop a finite element model to find the below:

a$)$ Using direct stiffness method, determine the nodal displacement of stepped bar. Elastic Modulus of the material $($E$)$ is $200(GPa).$

b$)$ Determine reaction at the support and

c$)$ Determine the stresses in each element.

Please note that you must solve the problem by using the Partition Method.

$2-(30$ pts$)$ The circular rod shown in the below figure has an outside diameter of $60(mm),$ length of $1(m),$ and is perfectly insulated on its circumference. The left half of the cylinder is aluminium, for which thermal conductivity $k_{x1}=200(\frac{W}{mK})$ and the right half is copper having thermal conductivity $k_x=389(\frac{W}{mK}).$ The extreme right end of the cylinder is maintained at a temperature of $80(C),$ while the left end is subjected to a heat input rate $4000(\frac{W}{m^2}).$ Using four equal$-$length elements, determine the steady$-$state temperature distribution in the cylinder.

Please note that you must solve the problem by using the Deletion Method.