$(35$ points$)$

Given: a circular rod with a diameter of $10$ mm

has an original length of $20$ m $.$ The material

has a Poisson's ratio of $v=0\mathrm{.}25$ and has a

stress strain curve below. A force of $P=45kN$

is applied to the beam.

a$)$ Calculate the new length of the rod $($in meters$)$

while the force $P$ is applied.

b$)$ Calculate the decrease in diameter while the

force $P$ is applied.

c$)$ Calculate the permanent strain after the force

$P$ is removed.

d$)$ Calculate the new length of the rod $($in meters$)$

after the force $P$ is removed.