The beam in Fig. 6-48 f is subjected to a fully plastic moment 𝐌p . Prove that the longitudinal

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The beam in Fig. 6-48 f is subjected to a fully plastic...

The beam in Fig. $6-48 f$ is subjected to a fully plastic moment $\mathbf{M}_{p} .$ Prove that the longitudinal and transverse shear stresses in the beam are zero. Hint: Consider an
element of the beam as shown in Fig. $7-4 c$.

Key Concepts

Pure Bending

Pure bending refers to a condition in which a beam is subjected solely to bending moments and no transverse shear force. In such a scenario, the internal stresses are produced entirely by bending, and the curvature of the beam is uniform. The analysis of an infinitesimal beam element under pure bending shows that the derivative of the moment with respect to the beam's length is zero, ensuring there is no local imbalance that could give rise to shear stresses.

Plastic Moment Capacity

The plastic moment capacity is the maximum bending moment that a cross-section can sustain when the entire section has yielded. At this point, the beam is said to be fully plastic, and its stress distribution reaches a constant yield value in the tension and compression zones. This uniform stress state contributes to the absence of longitudinal stress gradients, which, under equilibrium conditions, results in zero induced shear stress.

Differential Equilibrium of Beam Elements

Differential equilibrium involves analyzing a very small segment of the beam to assess the balance of internal forces and moments. In a beam under a fully plastic moment, the moment remains constant over small elements, producing no moment gradient. Since the shear force is the spatial derivative of the bending moment, a zero gradient implies that the shear force, and hence both the longitudinal and transverse shear stress components, are zero.

Stress Distribution in Fully Plastic Beams

In the fully plastic state of a beam, the entire cross-section yields uniformly, leading to a constant stress level across the section. This absence of stress variation means there is no differential force driving shear stress within the beam. Therefore, under the conditions of pure bending and fully developed plastic behavior, the typical shear stress components associated with stress gradients vanish.

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