Solved: The beam has a rectangular cross section and is

Chapter 7, Problem 7-30

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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 \(M_{p}=P L\) 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 \(2^{\prime}\). This situation has been described by Eq. 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 \(\tau_{\max }=\frac{3}{2}\left(P / A^{\prime}\right)\), where \(A^{\prime}=2 y^{\prime} b\), the cross-sectional area of the elastic core.