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 shown in Fig. $7-4 d$

Key Concepts

Pure Bending

In pure bending, the beam is subjected to a constant bending moment along its length, meaning that there is no moment gradient to generate shear forces. This condition implies that the deformation is characterized solely by bending, and the curvature is uniform across any cross?section of the beam. With no additional net forces acting on the section aside from bending, the shear stresses in both the longitudinal and transverse directions are zero.

Plastic Moment

The plastic moment is the load at which the entire cross section of a beam yields, resulting in a fully plastic behavior across the section. When a beam reaches its plastic moment, it forms a plastic hinge and can rotate without any further increase in moment. This state of full yielding ensures that the load is carried entirely by plastic resistance and confirms that the beam experiences pure bending without accompanying shear, leading to zero longitudinal and transverse shear stresses.

Equilibrium of Forces and Moments

Analyzing a free?body element of a beam, like the one indicated in the provided hint, involves applying the equilibrium of forces and moments. In the case of a beam under a fully plastic bending moment, the constancy of the moment across the element means that there are no net shear forces required to balance any changes in moment. This equilibrium analysis is central to proving that the shear stresses, both longitudinal and transverse, must vanish since there is no shear force acting along the free-body element.

Stress Distribution in Pure Bending

Under the assumption of pure bending, the stress distribution within a beam follows a linear variation across the cross section, according to the bending theory. In such a state, all contributions to the beam's response are due to bending, and any shear component, which would normally arise from a non-uniform moment distribution, is absent. This results in zero longitudinal and transverse shear stresses, as the material only carries bending normal stresses.

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