QUESTION $3$

$($a$)$ What is the shape factor of a beam, and briefly explain the use of the shape factor in designing structures with beam members?

$($b$)$ A cantilevered beam of length $1\mathrm{.}2m$ is subjected to a load $P(kN)$ at its free end. The cross section of the beam is illustrated in Figure Q$3.$ The beam material behaves as elastic$-$perfectly plastic with yield strength of $200$MPa.

$($i$)$ Determine the magnitude of the load, $P$ at initial yield of the beam.

$($ii$)$ Determine the ultimate load at plastic collapse of the beam.

$($iii$)$ Calculate the shape factor of the beam section