Short Answer:

Shear stress in beams is the internal resistance developed in the material of a beam when it is subjected to transverse loads. It acts parallel to the cross-section of the beam and helps balance the applied shear force. Shear stress arises because different layers of the beam tend to slide over each other due to external loading.

In simple words, shear stress in beams is produced when the beam is loaded in such a way that one part of it tries to move or slide past the other. It is maximum at the neutral axis and zero at the top and bottom surfaces of the beam cross-section.

Detailed Explanation :

Shear Stress in Beams

When a beam is subjected to external transverse loads, two types of internal effects are produced — bending moment and shear force. While the bending moment causes bending stresses (tension and compression), the shear force produces shear stress in the beam section. Shear stress plays an important role in maintaining the equilibrium and structural integrity of beams, especially near the supports where shear force is maximum.

1. Meaning of Shear Stress in Beams

Shear stress in beams is the intensity of the internal tangential force developed per unit area at any cross-section of the beam. It occurs because of the variation in bending moment along the length of the beam.

If the bending moment were constant, there would be no shear stress. However, in practical beams, the bending moment changes from point to point, leading to internal shear forces that create shear stress.

Shear stress acts in the longitudinal plane of the beam (parallel to its length) and is directed along the cross-section. It helps resist the sliding tendency of one layer of the beam material over another.

Mathematically, average shear stress can be expressed as:

where,
= average shear stress,
= shear force acting on the section,
= area of the cross-section.

However, this equation gives only the average shear stress. The actual shear stress distribution across the section varies and depends on the beam’s shape.

2. Distribution of Shear Stress in Beams

Shear stress is not uniform throughout the cross-section of the beam. It varies according to the distance from the neutral axis and the shape of the cross-section. The general formula for shear stress at a distance  from the neutral axis is:

where,
= shear stress at a given point,
= shear force at the section,
= first moment of area about the neutral axis of the portion above or below the layer considered,
= moment of inertia of the entire cross-section about the neutral axis,
= width of the section at the layer where shear stress is considered.

This formula helps find the shear stress distribution for different shapes like rectangular, circular, or I-sections.

3. Shear Stress Distribution in Common Sections

(a) Rectangular Section:
For a rectangular beam of width  and depth , the shear stress distribution is parabolic. The maximum shear stress occurs at the neutral axis and is given by:

At the top and bottom surfaces, the shear stress is zero.

(b) Circular Section:
For a circular cross-section, shear stress is also parabolic but with a different constant:

(c) I-section or T-section:
For I-shaped beams, the shear stress is mostly concentrated in the web, as it has less width compared to flanges. Hence, in structural design, the web thickness is chosen to safely resist shear stress.

4. Importance of Shear Stress in Beams

• Structural Safety: Shear stress determines how much shear force a beam can safely carry without failure.
• Design Consideration: Engineers design the beam’s web and cross-section based on the maximum expected shear stress.
• Failure Prevention: If shear stress exceeds the permissible limit, it may lead to shear failure, especially near the supports.
• Complementary Role: Shear stress works along with bending stress to maintain equilibrium and stiffness in beams.

5. Relation Between Shear Force and Shear Stress

The shear force acting on a beam section causes the development of shear stress across that section. The magnitude of the shear stress depends on the intensity of the shear force and the geometry of the section.

If  (no shear force), then  — meaning no shear stress. Thus, the presence of shear stress is directly linked with the variation in bending moment along the beam.

6. Shear Stress Characteristics

• Shear stress varies from zero at the extreme top and bottom surfaces to maximum at the neutral axis.
• The variation of shear stress across the section is parabolic for rectangular and circular beams.
• The average shear stress is always less than the maximum shear stress.
• In an I-section beam, most of the shear stress acts through the web due to its narrow width.
• Shear stress helps prevent the sliding of one fiber of the beam over another, maintaining overall integrity.

7. Example for Better Understanding

Consider a simply supported rectangular beam carrying a uniformly distributed load. The shear force is maximum at the supports and zero at the center. Therefore, shear stress will be maximum near the supports and gradually decrease toward the middle. This practical example shows how shear stress depends on both load distribution and beam geometry.

Conclusion :

Shear stress in beams is the internal tangential stress that resists the sliding of layers within a beam under transverse loading. It arises due to the variation of bending moment and is maximum at the neutral axis. The distribution of shear stress depends on the shape and dimensions of the beam cross-section. Understanding shear stress is essential for safe and efficient beam design, ensuring that the structure can safely carry the applied loads without shear failure.