$(22)9.$ a$)$ For the frame shown, derive the equilibrium equation for translation $$ at joint $(3).$ Neglect the effects of axial strain. EI has the same value for all members.

b$)$ Derive the equilibrium equations for moment equilibrium at joints $(2)$ and $(3).$

c$)$ Present your results in matrix form by giving the terms of the stiffness matrix $[$K$]$ and the load vector $\{P\}$ in the following equation:

$[K]\{[],[{}_2],[{}_3]\}=\{P\}$

DO NOT SOLVE THE EQUATIONS.

The unknowns of the problem shall be:

$=$ translation at joint $(3)($positive in the direction shown$)$

${}_2=$ rotation $of$ joint $(2)$

${}_3=$ rotation $of$ joint $(3)$