Short Answer:

Shear stress in an I-section is the internal tangential force per unit area developed when the section is subjected to a transverse shear force. In an I-beam, the shear stress is not uniform across the depth. It is maximum at the neutral axis in the web and nearly zero at the top and bottom flanges. Most of the shear force is carried by the web portion of the I-section, while the flanges mainly resist bending stresses.

The distribution of shear stress in an I-section is parabolic in the web region and very small in the flange areas. Therefore, while designing I-beams, the web thickness is selected to safely carry this shear stress without failure.

Detailed Explanation :

Shear Stress in an I-Section

Overview
An I-section, commonly called an I-beam or girder, is widely used in mechanical and civil engineering structures such as bridges, frames, and machine components. Its cross-section consists of two horizontal flanges connected by a vertical web. When the beam is subjected to a transverse shear force , internal shear stresses are produced. The magnitude and distribution of these shear stresses depend on the geometry of the cross-section.

The general formula for shear stress at any point in a beam is:

where

•  = shear force at the section,
•  = first moment of area above or below the layer considered,
•  = second moment of area (moment of inertia) of the entire section,
•  = width of the section at that layer,
•  = shear stress at that layer.

Distribution of Shear Stress in an I-Section

1. Structure of the I-section
   An I-beam has three main parts:

• Top flange: Upper horizontal plate that resists compression due to bending.
• Web: Vertical central portion that connects flanges and carries most of the shear force.
• Bottom flange: Lower horizontal plate that resists tension due to bending.

The overall height of the section is much greater than the web thickness, which means the bending strength is increased, but the web primarily resists the shear stress.

2. Shear stress in the web
   In the web region, the width is small (equal to the web thickness ), and the first moment of area changes rapidly with distance  from the neutral axis.
   Using the shear formula,

the shear stress is maximum at the neutral axis (center of the web) because  is maximum there.
As we move towards the flange–web junction, the value of  decreases, and so does the shear stress.
Thus, the shear stress in the web varies parabolically, with the highest value at the neutral axis and zero at the top and bottom surfaces.

Maximum shear stress in the web:

Since  (the web thickness), most of the shear load acts through this thin central web area.

3. Shear stress in the flanges
   In the flange region, the width is much larger (equal to the flange width ), but the first moment changes very little across the flange thickness. Hence, the shear stress is quite small in the flanges compared to the web.

At the top and bottom surfaces of the beam, the shear stress becomes zero.
This shows that flanges mainly resist bending, while the web resists shear.

4. Shape of shear stress distribution
   If we draw a vertical line across the height of the I-section, the shear stress distribution appears as:

• Zero at the top flange surface.
• Small but nonzero in the flange region.
• Suddenly increases at the junction of flange and web.
• Maximum value at the neutral axis (center of the web).
• Symmetrical decrease towards the bottom, becoming zero again at the bottom flange surface.

This results in a parabolic distribution within the web and very low constant stresses in the flanges.

5. Practical significance

• Because most of the shear stress is carried by the web, designers often provide sufficient web thickness to resist the maximum shear force safely.
• Shear buckling of the web is an important consideration in design, especially for thin-walled I-beams.
• The flanges are primarily designed for bending moment resistance and are made wider to increase the section modulus.
• Stiffeners may be provided in the web to prevent excessive shear deformation in long beams.

6. Simplified analysis
   In practical design, it is common to assume that the entire shear force is resisted by the web area alone, ignoring the flanges.
   Therefore, the average shear stress in the web is:

where  (web area).
This gives an approximate but safe estimation for design purposes.

7. Comparison with rectangular beam
   In a rectangular section, the shear stress distribution is also parabolic but covers the full width of the section.
   In an I-section, since the web is narrow and most of the shear is carried by it, the stress concentration in the web is much higher than in a rectangular section of the same height.

Conclusion

Shear stress in an I-section is highly non-uniform. It is almost zero at the flanges and maximum at the neutral axis in the web. The web carries most of the shear load, while the flanges mainly resist bending. The distribution of shear stress is parabolic across the web, and its maximum value can be calculated using . This understanding is essential in beam design to ensure safety against shear failure and efficient material usage.