You are tasked to solve this problem so that you can achieve mastery of The Force Method and Castigliano's Theorem for statistically indeterminate structures. Observe that $A$ is a fixed support and $B$ is a roller support.

$2a[10$ points$]$ Verify that the degree of indeterminacy is to the $1^{st}$ degree.

$2$b $[10$ points$]$ In order to calculate the reactionary forces at A and $B,$ remove the reactionary moment at A and replace it with a variable moment. Draw the free$-$body diagram and write the reactionary forces at A and $B$ as functions of the variable moment and the applied load, $P.$

$2$c $[10$ points$]$ Determine the internal shear and bending forces attributable to the loading.

$2$d $[10$ points$]$ Solve for the variable moment by enforcing the compatibility condition at A using Castigliano's Theorem $($accounting for bending forces only$).$

$2$e $[10$ points$]$ Recalculate the reactionary forces at A and $B$ using the variable moment. Draw updated internal shear and bending moment diagrams attributable to the loading.

$2$f $[10$ points$]$ Sketch the deformed shape based on the updated $M(x)$ diagram and other constraints.

$2g[10$ points$]$ Compare your effort here with the "fixed end moment" for this beam found in typical engineering tables $($shown below$).$