$1\mathrm{.}9-1.$ The bar shown has unifo

rm cross$-$sectional area A and is fixed at both ends. An idealized

stress$-$strain relation is also shown. Assume that the relation is valid in compression as

well as in tension.

$($a$)$ Determine the value of load $P$ that initiates yielding.

$($b$)$ Determine the fully plastic load $P_{fp}.$

$($c$)$ Determine the state of residual stress after load $P_{fp}$ is removed.

$($a$)$

$P=\frac{{}_{Y}AL}{b}$

$($b$)$

$P_{f_P}=2A{}_Y$

$($c$)$

$({}_a{)}_{\text{r}\text{e}\text{s}}={}_Y(1-\frac{2b}{L})$

$({}_b{)}_{\text{r}\text{e}\text{s}}={}_Y(\frac{2a}{L}-1)$