The beam has a rectangular cross section and is made of wood...

The beam has a rectangular cross section and is made of wood having an allowable shear stress of $\tau_{\text {allow }}=200$ psi. Determine the maximum shear force $V$ that can be developed in the cross section of the beam. Also, plot the shear-stress variation over the cross section.

Key Concepts

Cross-sectional Geometry

The geometry of a beam’s cross section, including its shape and dimensions, plays a significant role in its ability to resist loads. Properties derived from the geometry, such as the area and moment of inertia, directly influence the distribution and magnitude of internal stresses, like shear stress. Understanding cross-sectional geometry is crucial in analyzing and designing structural elements for strength and stability.

Shear Stress Distribution

Shear stress distribution refers to the way shear stress varies over the cross-section of a beam. In many beam cross-section shapes, such as rectangular sections, the shear stress is not uniform across the area. The stress typically reaches a maximum value in a specific region of the cross-section and varies according to the geometry, methods of force distribution, and loading conditions.

Allowable Shear Stress

The allowable shear stress is the maximum shear stress that a material can safely endure without failure. This concept is critical in design, as it incorporates factors of safety and material strength into determining load limits. It ensures that during operation, the material's internal stresses remain within predictable and safe limits.

Shear Force

Shear force is an internal force within a beam that arises due to external loading. It is the force that attempts to slide one part of the material over an adjacent part. Accurately determining the magnitude of the shear force is essential for ensuring structural integrity and avoiding failure.

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