C$2.(40$ pts$).$ Determine the displacements and stresses in the $4-$inch long bar subjected to the traction

$T=1000*(5x-x^2)$ pounds per inch using the following models:

$($a$)$ one linear element $(2-$node element$)$ assuming linearly varying traction force;

$($b$)$ one quadratic element $(3-$node element$)$ assuming constant traction force;

$($c$)$ two linear elements assuming constant traction force on each element.

The traction is distributed along the length of the beam as shown in the figure $($Note: x is measured in

inches while traction $T(x)$ is measured in $l\frac{b}{i}n$ and the coefficients are such that the resulting traction force

is in lb$/$in$).$ Make $($and explain$)$ your decision on the approximation of the traction force.

Hint: For linearly varying traction, assume the straight line connecting the traction values at two ends of

the element. For constant traction, approximate the traction force over the element by its average value. To

compute average of a polynomial use $T_{\text{a}\text{v}\text{g}}=\frac{1}{b-a}{\displaystyle \underset{a}{\overset{b}{}}}T(x)dx.$