The beam has a rectangular cross section and is

subjected to a load $P$ that is just large enough to develop a

fully plastic moment $M_p=PL$ at the fixed support. If the

material is elastic perfectly plastic, then at a distance $M=Px2y^{\text{'}}M{}_{\text{m}\text{a}\text{x}}=\frac{3}{2}(\frac{P}{A^{\text{'}}})A^{\text{'}}=2y^{\text{'}}bx$

the moment $M=Px$ creates a region $of$ plastic yielding

with $an$ associated elastic core having a height $2y^{\text{'}}.$ This

situation has been described $byEq.6-30$ and the moment $M$

$is$ distributed over the cross section $as$ shown $in$ Fig. $6-48e.$

Prove that the maximum shear stress $in$ the beam $is$ given

$by{}_{\text{m}\text{a}\text{x}}=\frac{3}{2}(\frac{P}{A^{\text{'}}}),$ where $A^{\text{'}}=2y^{\text{'}}b,$ the cross$-$sectional

area $of$ the elastic core.

Prob. $7-30$