$1\mathrm{.}4-1.$ A prismatic bar is loaded by an axial force $P.$ Show that $P$ must be directed through cen$-1\mathrm{.}9-1.$ The bar shown has uniform 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.

troids of cross sections if axial stress $$ is not to vary over a cross section.