Problem $3(10$pts$)$

The structure in the diagram contains three rods joined by a pin at A $.$ The rods are supported by pins at $B,C$ and $D.$ Only the rod $AB$ is subjected to a temperature change of $T.$ All the rods have the same length $L,$ same Young's modulus $E,$ same cross$-$sectional area $A,$ and same thermal expansion coefficient $.$

$($a$).$ Prove that the relation between the changes in length ${}_{AB}$ and ${}_{AC}$ is

$2{}_{AB}={}_{AC}$

Assume the angle between rod AC and AD negligibly changes. Hint: you can directly assume symmetry and the relation can be derived purely geometrically.

$($b$).$ Find the magnitude of internal axial force in the member AB with respect to $,T,E,$A and $L.$

$($c$).$ Find ${}_{AB}$ with respect to $,T$ and $L.$

$($d$).$ Given the yield stress ${}_y$ of AB $,$ determine the temperature at which AB will yield.