The beam is to be cut longitudinally along both sides as...

The beam is to be cut longitudinally along both sides as shown. If it is made from a material having an allowable shear stress of $\tau_{\text {allow }}=75 \mathrm{MPa}$, determine the maximum allowable internal shear force $\mathbf{V}$ that can be applied before and after the cut is made.

Key Concepts

Effect of Cross-Sectional Modification

When a beam’s cross-section is altered—such as by cutting it longitudinally along both sides—the effective area that resists shear is reduced. This modification alters the shear stress distribution and consequently decreases the maximum allowable internal shear force that the beam can tolerate. Analyzing these changes is critical in ensuring that the remaining section maintains adequate strength under load.

Allowable Shear Stress

This concept refers to the maximum shear stress that a material can sustain without failure. In design problems, such as the one described, knowing the allowable shear stress helps determine the maximum load or internal shear force that can be applied safely to the beam. It is a material property used to ensure that the induced stresses remain within safe limits under service conditions.

Internal Shear Force

Internal shear force is the component of internal forces that acts parallel to the cross-sectional area of a beam. It arises due to external loads and is critical in assessing the beam’s capacity to resist sliding failure along the plane of the cut section. Understanding and calculating the internal shear force is essential in the analysis and design of structural members.

Shear Stress Distribution in Beams

This key concept involves understanding how shear stress is distributed across the cross-section of a beam. In many beams, such as rectangular sections, the shear stress is not uniform; it typically reaches a maximum at or near the center of the cross-section and tapers towards the edges. The distribution influences the design and analysis since different portions of the beam will experience varying levels of stress.

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