Understanding Stress in Beams: Bending vs. Shear

When analyzing the stress in a beam subjected to external loads, it is crucial to differentiate between the types of stress that come into play. Specifically, bending stress and shear stress have distinct origins and behaviors. This article explores the primary causes and calculation methods for both types of stress, along with the reasons why shear stress is not obtained using a simple tau F/A.

Bending stress is a critical aspect of beam analysis, primarily due to the bending moments that create normal stresses in the material. The formula for calculating bending stress is given by

sigma M/W

Bending Stress

Bending stress sigma is caused by the moment M applied to the beam. It varies linearly across the height of the beam and reaches its maximum at the outermost fibers, specifically on the top and bottom surfaces.

Shear Stress in Beams

Shear Stress

Shear stress tau arises due to transverse loads F acting on the beam. It is calculated using the formula

tau V/A

where V is the internal shear force at the section, and A is the area over which the shear force is acting. In beams, shear stress is typically not as significant as bending stress, particularly when bending moments dominate.

Why Not Use tau F/A for Maximum Stress

Bending and shear stresses are fundamentally different. Bending stress arises from moments, whereas shear stress arises from forces acting parallel to the cross-section. When evaluating maximum stress for design or failure analysis, bending stress is often more critical due to the nature of most loading scenarios.

Moreover, in a beam, the shear force can be distributed across the cross-section, and the maximum shear stress does not necessarily occur in the same location as the maximum bending stress. This highlights the importance of considering both types of stress together.

Location of Shear Stress

In a beam, the shear force can be distributed, and the maximum shear stress typically occurs at the neutral axis of the beam. This is different from the location of the maximum bending stress, which is at the extreme fibers.

Combined Stress Analysis

In practical applications, both bending and shear stresses should be considered together when analyzing the overall stress state in a beam. This is often done using a combined stress approach, where both stresses are calculated and appropriate design criteria are applied. For example, the von Mises criterion is commonly used for ductile materials to ensure structural integrity.

Conclusion

In summary, bending stress is calculated using sigma M/W due to the significance of bending moments, while shear stress is calculated using tau V/A to account for transverse forces acting on the beam. Each stress type plays a crucial role in the overall structural analysis and design of beams but addresses different loading conditions and failure modes.

Understanding the difference between these two types of stress and their unique characteristics is essential for any engineer or student in the field of structural analysis.