Short Answer:

The shear stress distribution in a beam refers to how the internal shear stress varies across the cross-section of the beam when it is subjected to transverse shear force. It is not uniform; instead, it depends on the shape of the beam’s cross-section.

In simple words, shear stress is zero at the top and bottom surfaces of the beam and maximum at the neutral axis. The distribution of shear stress helps in determining the safe design and performance of structural members under transverse loading conditions.

Detailed Explanation :

Shear Stress Distribution in a Beam

When a beam is subjected to transverse loading, internal forces develop to resist the external loads. These internal forces include bending moment and shear force. The shear force produces shear stress within the beam’s cross-section. The variation of this shear stress across the section is known as the shear stress distribution.

The study of shear stress distribution is very important in beam design because it determines the strength and safety of the beam under loads. The distribution depends on the shape of the cross-section (rectangular, circular, I-section, etc.) and the magnitude of the shear force acting on the beam.

Definition

Shear stress distribution in a beam is defined as:

“The variation of shear stress across the cross-section of a beam subjected to transverse shear force.”

Mathematically, the shear stress at any point in the cross-section is given by the formula:

Where,

= shear stress at a given point (N/mm²),
= total shear force acting on the section (N),
= first moment of area about the neutral axis (mm³),
= moment of inertia of the entire section about the neutral axis (mm⁴),
= width of the section at the point where shear stress is calculated (mm).

This equation is derived from the principles of equilibrium and material elasticity and shows that shear stress depends on geometry and loading.

Derivation of Shear Stress Formula

Consider a beam subjected to a transverse shear force . The internal shear stress developed resists this force and maintains equilibrium.

For a small beam element, the bending moment changes along its length due to the applied shear force. If is the bending moment at one end and at the other, then:

This changing bending moment produces different normal stresses across the section, and the difference between these stresses in the longitudinal direction gives rise to shear stress.

The horizontal shear stress at a point is found using:

This relation helps in calculating the shear stress at any point in the beam’s cross-section.

Explanation of Terms

Shear Force (V):
It is the total internal force acting parallel to the beam cross-section to resist external transverse loads.
Moment of Inertia (I):
It represents the beam’s resistance to bending or deflection and depends on the cross-sectional shape.
First Moment of Area (Q):
It is the product of the area above (or below) the point where stress is to be found and the distance of its centroid from the neutral axis.

where = area considered, and = distance of centroid from the neutral axis.

Width (b):
It is the thickness of the beam cross-section at the level where shear stress is being calculated.

Shear Stress Distribution in Different Cross-Sections

Rectangular Section:
- The shear stress is zero at the top and bottom surfaces and maximum at the neutral axis.
- The maximum shear stress occurs at the neutral axis and is given by:

where = cross-sectional area.

- The distribution is parabolic in shape.
Circular Section:
- The shear stress is also maximum at the neutral axis and zero at the outer surface.
- The maximum shear stress is given by:

- The distribution is elliptical in shape.
Triangular Section:
- The shear stress is zero at the vertices and maximum at the neutral axis.
- The distribution is non-symmetrical, depending on the triangle’s orientation.
I-Section or T-Section:
- In these sections, shear stress distribution is non-uniform.
- Most shear stress is concentrated in the web, while the flanges carry very little shear stress because they are wide and thin.
- Hence, in practical design, the web is made strong enough to resist the majority of the shear force.

Important Observations about Shear Stress Distribution

Shear stress is zero at the extreme fibers (top and bottom surfaces) of the beam.
Shear stress is maximum at the neutral axis.
The variation of shear stress between these two points is parabolic for rectangular sections.
The average shear stress across the section is less than the maximum shear stress.
The shape of the beam cross-section strongly influences the shear stress distribution.

Significance of Shear Stress Distribution

Beam Design:
It helps engineers determine the required strength and thickness of beams to resist shear stresses safely.
Material Utilization:
Understanding the stress distribution ensures efficient use of materials, especially in I-beams and T-beams.
Failure Prevention:
Shear failure can occur if the material exceeds its shear strength, so accurate analysis prevents structural damage.
Determination of Safe Load:
By knowing the maximum shear stress, the allowable load on the beam can be calculated safely.
Optimization of Cross-Section:
It assists in selecting the most efficient cross-sectional shape for given loading conditions.

Example (Rectangular Section):

For a beam with width and depth , subjected to a shear force :

Hence, the maximum shear stress at the neutral axis is , decreasing parabolically to zero at the outer fibers.

Conclusion

In conclusion, shear stress distribution in a beam represents the variation of shear stress across the cross-section when a beam resists transverse loads. The stress is zero at the top and bottom fibers and maximum at the neutral axis. The exact pattern of distribution depends on the cross-sectional shape. For rectangular beams, the variation is parabolic, and for I-sections, it is concentrated mainly in the web. Understanding this distribution is essential for safe and economical beam design, preventing shear failure in mechanical and structural systems.