22.3: 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.
