The L $-$shaped frame is subjected to a force P at point C and a uniformly distributed load of intensity $500\frac{N}{m}$ along AB $.$ The members AB and BC lie in the $x-$z plane and are mutually perpendicular to each other. Both $AB$ and $BC$ have a circular cross$-$section with $d=60mm,AB=2m,BC=1\mathrm{.}2m,G=60$GPa and $v=0\mathrm{.}3.$

a$)$ Use Castigliano's theorem to determine the deflection at C $($as a function of P$)$ and the equivalent stiffness $(K_{\text{e}\text{f}\text{f}})$ of the structure. Consider bending and torsion only. $(25$ pts $)$

b$)P$ is changed into a time varying force $P(t)=P_{0}sin(t)$ at $C.P_0=500N.$ The equivalent mass of the structure is $Mef{l}_{ef}=4kg$ and the damping ratio $=0\mathrm{.}1.$ Find the value of $$ which maximizes the deflection at C and the corresponding maximum deflection at C $.(8$ pts $)$

c$)$ For the $$ found in part $($b$),$ find the maximum bending stress at A $.(7$ pts$)$