Chapter 7.2, Problem 7.30P

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 x < L the moment M = Px creates a region of plastic yielding with an associated elastic core having a height 2y'. This situation has been described by Eq.6–30 and the moment M is distributed over the cross section as shown in Fig.63–48e. Prove that the maximum shear stress in the beam is given by ${\tau }_{\max }=\frac{3}{2}(P/A^{\prime })$, where A′ = 2y ′b, the cross-sectional area of the elastic core.

Prob. 7–30