$2\mathrm{.}3-13$ A wood pile, driven into the earth, supports a load $P$ entirely by friction along its sides isee figure part a$).$ The friction force $f$ per unit length of the pile is assamed to be uniformly distributed over the surface of the pile. The pile has a length $L,$ cross$-$sectional area $A,$ and modulus of elasticity $E$

$($a$)$ Derive a formula for the shortening $$ of the pile in terms of $P,L,E,$ and $A.$

$($b$)$ Draw a diagram showing how the compressive stress ${}_C$ varies throughout the length of the pile.

$($c$)$ Repeat parts $($a$)$ and $($b$)$ if skin friction $f$ varies linearly with depth $($see figure part b$)$