Problem D $(40$ points$)$

A wire of length $L=4ft$ and diameter $d=0\mathrm{.}125in.$ is stretched by tensile forces $P=600lb.$ The wire is made of a copper alloy having a stress$-$strain relationship that may be described mathematically by the following equation:

$=\frac{18,000lon}{1+300lon},0lon0\mathrm{.}03$

in which $lon$ is nondimensional and $$ has units of kips per square inch $($ksi$).$ A plot of the stressstrain curve is provided below. Determine:

a$)$ the elongation of the wire due to the forces $P=600lb,$

b$)$ the modulus of elasticity $E$ of the copper alloy by calculating the slope of the stress$-$strain curve at the origin $(lon,)=(0,0),$

c$)$ the yield stress of the copper alloy using the $0\mathrm{.}2\%$ offset method $($where the yield stress is determined by the intersection of the stress$-$strain curve with a straight line of slope $E$ passing through the point $(lon,)=(0\mathrm{.}002,0),$

d$)$ the permanent set of the wire if the tensile forces $P=600lb$ are removed, and

e$)$ the proportional limit of the copper alloy if the forces are applied again.