$2-$ The two uniform linearly elastic rods shown below are welded together at $B,$ and the resulting two$-$segment rod is attached to rigid supports at A and $C.$ Rod $(1)$ has a modulus $E_1=180$GPa, cross$-$sectional area $A_1=550m{m}^2,$ length $L_1=150mm,$ and coefficient of thermal expansion ${}_1=8\frac{{10}^{-6}}{}C.$ Rod $(2)$ has a modulus $E_2=90$GPa, cross$-$sectional area $A_2=900m{m}^2,$ length $L_2=100mm,$ and coefficient of thermal expansion ${}_2=17\frac{{10}^{-6}}{}C.$

a$)$ Determine the axial stresses in the rods if the temperature of both is raised by $50C.$

b$)$ Determine whether joint $B$ moves to the right or left and by how much?

Rigid support