PROBLEM $1$ A rigid bar is supported by three rods with the following properties:

Yield stresses:

${}_{1,Y}={}_{3,Y}=36$ksi,${}_{2,Y}=50$ksi

Young's moduli:

$E_1=E_3=10{10}^6,E_2=30{10}^6$

Lengths:

$L_1=L_2=L_3=50in$

Areas:

$A_1=A_3=0\mathrm{.}2i{n}^2,A_2=0\mathrm{.}1i{n}^2$

The bar remains always horizontal. Assuming all three rods to be elastoplastic. If $P$ is gradually increased from zero to $14\mathrm{.}7$ kips then reduced back to zero, DETERMINE

$($a$)$ the maximum loads in each rod during loading,

$($b$)$ the maximum horizontal deflection of the rigid bar,

$($c$)$ the permanent set when the load is reduced back to zero,

$($d$)$ the residual stress in each link. Assume that the links are braced so that they can carry compressive forces without buckling.