$(21)8.$ a$)$ For the frame shown, derive the equilibrium equation for the translation shown at

joint$(3)$ Neglect the effects of axial strain. EI has the same value for both

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$(3)(positiveto$ the right$)$

${}_2=$ rotation $of$ joint$(2)(counter$ clockwise positive$)$

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

$($counter clockwise positive$)$