A plate of thickness t is bent as shown and then used as a...

A plate of thickness $t$ is bent as shown and then used as a beam. For a vertical shear of $600 \mathrm{lb}$, determine (a) the thickness $t$ for which the maximum shearing stress is 300 psi, $(b)$ the corresponding shearing stress at point $E$. Also sketch the shear flow in the cross section.

Key Concepts

Shear Flow

Shear flow is the measure of shear force per unit length distributed along the cross section of a beam, often denoted by q. It is defined by the relationship q = ?t, where ? is the local shear stress and t is the thickness at that point. The concept of shear flow is especially useful in the analysis of thin-walled and built-up sections, where it helps in visualizing how the shear force 'flows' along the structure and in designing connections that resist shear forces effectively.

Shear Stress in Beams

This concept refers to the internal stress that develops within a beam due to the application of an external shear force. The distribution of shear stress across the beam's cross section is not uniform and is generally calculated using the formula ? = VQ/It, where V is the applied shear force, Q is the first moment of area about the point in the cross section under consideration, I is the moment of inertia of the entire cross section, and t is the thickness at that point. Understanding this distribution is critical in ensuring that the maximum shear stress does not exceed the material's design limits.

Moment of Inertia and First Moment of Area

These two concepts are fundamental in determining how stresses are distributed within a beam's cross section. The moment of inertia (I) is a geometric property that indicates the resistance of a cross section to bending, while the first moment of area (Q) is used in shear stress calculations to account for the distribution of area relative to the neutral axis. Together, they are essential for quantitatively analyzing the bending and shear behavior of beams.

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