Short Answer:
The shear stress in an I-beam is the internal resisting stress developed when the beam is subjected to transverse shear force. It varies across the depth of the section, being maximum in the web and very small in the flanges. The web of the I-beam carries almost the entire shear force because of its smaller thickness and large height.
In simple words, when a shear force acts on an I-beam, the stress is not uniform throughout the section. The web resists most of the shear stress, while the flanges mainly resist bending moments, making the I-beam strong and economical.
Shear Stress in an I-Beam
Detailed Explanation :
An I-beam (or H-beam) is one of the most commonly used structural sections in mechanical and civil engineering. It is designed to carry loads primarily by resisting bending and shear. The beam’s geometry — having wide flanges and a thin web — provides high strength with minimum material use.
When an I-beam is subjected to a transverse shear force (V), shear stress develops over its cross-section. However, the distribution of this shear stress is not uniform; it is concentrated mainly in the web, while the flanges carry very little shear.
Concept of Shear Stress in an I-Beam
When a beam carries transverse loading, the internal shear stress at any point in the cross-section is given by the general shear formula:
where,
- = shear stress at a given point (N/mm²),
- = total shear force acting on the beam (N),
- = first moment of area about the neutral axis (mm³),
- = second moment of area of the entire section about the neutral axis (mm⁴),
- = width of the section at the point considered (mm).
This equation helps determine the shear stress at different points in the I-beam — in the flanges and in the web.
- Shear Stress in the Web
The web is the vertical portion connecting the two flanges. Since the web is relatively thin and the shear stress is inversely proportional to width , the stress becomes significant in the web region.
Let the web thickness be and the overall depth of the beam be . The first moment of area for a point within the web can be computed by considering the area above (or below) that point and its distance from the neutral axis.
At the neutral axis, , the shear stress in the web is maximum and is given by:
In most standard I-sections, this maximum shear stress occurs approximately at the mid-depth of the web and carries almost all the shear force acting on the beam.
Because the web has a smaller width (), even a moderate shear force produces significant stress, hence the web is designed to resist this load safely.
- Shear Stress in the Flanges
The flanges are the horizontal top and bottom parts of the I-beam. Their primary purpose is to resist bending stresses because they are located far from the neutral axis (where bending stress is highest).
In the flange region, the shear stress is very small because:
- The flange width is large.
- The distance from the neutral axis is large, making the value of comparatively smaller.
Thus, most of the shear force passes through the web, while the flanges contribute very little.
Near the junction of the web and flange, the shear stress is slightly higher due to the continuity of stress across the section, but it quickly decreases toward the outer edge of the flange, where it becomes nearly zero.
- Distribution of Shear Stress in an I-Beam
The variation of shear stress across the I-beam can be summarized as follows:
- In Flanges:
Shear stress starts from zero at the top and bottom surfaces, increases slightly near the junction of the flange and web, but remains relatively small. - In Web:
The shear stress is maximum at the neutral axis (center of the web) and decreases parabolically to zero at the top and bottom edges of the web.
This means that almost 90% of the total shear force is carried by the web, and only 10% by both flanges together.
Hence, the web plays the major role in resisting shear forces in an I-beam.
- Expression for Maximum Shear Stress in I-Beam
For practical calculations, it is often sufficient to approximate the maximum shear stress in the web using:
where is the area of the web, and is the depth of the web between the flanges.
However, since the stress varies parabolically, the average stress over the web is smaller than the maximum value.
For an accurate result, the maximum shear stress in the web is obtained by using:
This formula is similar in form to that of a rectangular section, since the web behaves like a rectangle in shear.
- Importance of Web in Shear Resistance
- The web of the I-beam provides resistance to transverse shear, while the flanges carry bending moments.
- The design of the web thickness () ensures that the maximum shear stress remains below the allowable shear stress for the material.
- If the web is too thin, it may buckle under high shear load; hence, stiffeners (vertical plates) are sometimes welded to the web to increase its strength.
- Simplified Shear Stress Formula for I-Beam
The shear stress at any point within the web of an I-beam is approximately given by:
and the maximum shear stress in the web is:
This formula shows that:
- varies linearly within the web thickness.
- The stress is maximum at the center and decreases to zero at the top and bottom edges.
- Key Observations
- Shear stress is parabolic across the web height.
- Maximum shear stress occurs at the neutral axis.
- Flanges carry negligible shear; their main function is bending resistance.
- Web resists nearly all the shear load.
- The web is often treated as a rectangular section for calculating shear stress.
Conclusion
In conclusion, the shear stress in an I-beam is mainly concentrated in the web, while the flanges carry very little of it. The maximum shear stress occurs at the neutral axis of the web and is given approximately by
The distribution of shear stress across the web is parabolic, being zero at the top and bottom surfaces. Understanding this behavior helps engineers design I-beams efficiently — ensuring that the web thickness and flange dimensions are optimized for both bending and shear strength.