$(30pt)$ Beam $AB$ with a length of $8ft$ is under an eccentric

loading $P.$ The end $A$ is fixed on the bottom, and the end $B$ is

rolling tangentially but the slope remains unchanged. The

eccentricity is $e=1in.$ The cross$-$section of the beam is a square

shape with $4$ in by $4$ in$.$ The Young's modulus of the beam is $E=$

$29{10}^6.$

Hint: $B$ end boundary: slope remains zero

$($a$)$Use the shown coordinate system to construct the governing

equation to obtain the general solution for beam deflection. $(8$pt$)$

$($b$)$Apply the boundary conditions at ends A and B to determine

the expression of the deflection of B$.(8pt$

$($c$)$Determine the value of the critical load $P_{cr}.(6$pt$)$

$($d$)$When the applied load is $\frac{P_{cr}}{2},$ determine the value of the

maximum deflection and maximum normal stress within the beam

AB$.(8pt)$