# Shearing Stresses in a Beam: Problem Solving

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Mechanical Engineering
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JoVE Core Mechanical Engineering
Shearing Stresses in a Beam: Problem Solving

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A cantilever beam with a rectangular cross-section is subjected to distributed and point loads.

To calculate shearing stress in the beam, the loads acting upon it are first identified. Reactions at the fixed end are then calculated using equilibrium equations.

Vertical reaction is the sum of distributed and point loads, and the moment reaction is the sum of their moments.

The shear force distribution along the beam due to the loads is determined by drawing a shear force diagram starting from reaction forces at the fixed end, including the distributed load and the point load.

Shearing stress is calculated using a formula considering the shear force at that point, the first moment of the cross-section area about the neutral axis, and the cross-section's moment of inertia.

The shearing stress is calculated at various critical points along the beam, typically near the supports and at the point load location.

The maximum shearing stress should be compared with the material's allowable shear stress to determine if the beam is safe.

## Shearing Stresses in a Beam: Problem Solving

A cantilever beam with a rectangular cross-section under distributed and point loads experiences shearing stresses. The analysis begins by identifying the loads acting on the beam. Then, the reactions at the beam's fixed end are calculated using equilibrium equations. The vertical reaction is a combination of the distributed and point loads, while the moment reaction is the sum of their moments. The shear force distribution along the beam, resulting from these loads, is established by creating a shear force diagram, starting from the fixed end and incorporating the effects of both distributed and point loads.

Shearing stress is determined using a specific formula that considers the shear force at the point of interest, the cross-section area's first moment about the neutral axis Q, the cross-section's moment of inertia I, and the width of the beam b:.

This calculation is performed at various critical points along the beam, especially near supports and where point loads are applied. The highest shearing stress found is then compared to the material's allowable shear stress, to assess the beam's safety under the given loads. This ensures the beam's structural integrity by confirming that the stress levels do not exceed the material's limits.