Q$1.(20$ Marks$)$

A composite shaft ABCD, rigidly attached at $A,$ is made of three different materials and varies in

diameter as shown. The steel section $AB$ is hollow and has an external diameter of $30mm$ and an

internal diameter of $20mm.$ Section $BC,$ attached to $AB$ by a rigid plate, is a $15mm$ diameter

solid bronze bar, and section $CD$ is a solid copper bar with a diameter of $10mm.$ A force of

$P_1=100kN$ is applied on the rigid plate at $B,$ a force $P_2=150kN$ is applied at point $C$ and a

force $P_3=120kN$ is applied at point $D.$ Ignore the solid plate in your calculations.

a$.$ Determine the internal axial forces in members $AB,BC,$ and $CD($specify direction and

whether they are in tension or in compression$)$ and draw an axial force diagram.

b$.$ Calculate the lateral deformation, Elat, of each section, specify if the deformation is a

contraction or an expansion.

c$.$ If the temperature decreases from $30C$ initially to $10C,$ what will be the displacement

$($including displacement induced by the applied forces$)$ of point $C$ with regards to $A.$

d$.$ Draw an axial displacement diagram for the structure after the temperature reduction.