Understanding Shear Area in Steel I-Sections

The question asks about the shear area of a rolled steel I-section when it is subjected to bending about its minor axis. Understanding how different parts of an I-section resist shear force during bending is crucial in structural design.

Minor Axis Bending Explained

A rolled steel I-section has two principal axes of bending: the major axis (usually horizontal, parallel to the flanges) and the minor axis (usually vertical, perpendicular to the flanges, passing through the web). When bending occurs about the minor axis, the bending stresses are distributed across the flanges and web based on their distance from this axis. The shear force associated with minor axis bending is applied parallel to the major axis.

Consider an I-section under shear force parallel to the major axis. This shear force is primarily resisted by the material parallel to its direction. In the case of minor axis bending, the shear force acts horizontally (parallel to the flanges). Therefore, the parts of the I-section that effectively resist this horizontal shear are the flanges.

Calculating Shear Area for Minor Axis Bending

For minor axis bending, the shear stress distribution shows that the flanges carry the significant majority of the shear force. The web contributes very little to resisting shear when bending is about the minor axis.

Let's look at the geometry:

• Breadth of flange = $b$
• Thickness of flange = $t_f$

There are two flanges in an I-section. Each flange is a rectangle with dimensions $b \times t_f$. The area of one flange is $b \times t_f$.

The total area of the two flanges is the area that resists the shear force during minor axis bending. This area is the sum of the areas of both flanges:

Shear Area for Minor Axis Bending = Area of top flange + Area of bottom flange

Shear Area for Minor Axis Bending = $(b \times t_f) + (b \times t_f) = 2bt_f$

Comparing with Given Options

Let's evaluate the provided options:

1. $ht_w$: This represents the approximate shear area of the web ($h$ is overall depth, $t_w$ is web thickness). This area is primarily responsible for resisting shear during major axis bending.
2. $2bt_f$: As derived above, this represents the total area of the two flanges, which is the effective shear area for minor axis bending.
3. $bt_f$: This is the area of only one flange.
4. $ht_f$: This combination does not represent a standard shear area calculation for an I-section under either major or minor axis bending.

Based on our analysis, the shear area for minor axis bending of a rolled steel I-section is $2bt_f$.

Conclusion

The shear area of a rolled steel I-section for minor axis bending is $2bt_f$, which is the combined area of the two flanges. This is because when bending occurs about the minor axis, the shear force is primarily resisted by the flanges.

Revision Table: Steel Section Properties

Understanding these basic dimensions is fundamental to calculating various section properties like area, moment of inertia, and shear area for structural design.

Additional Information: Shear Distribution in I-Sections

The distribution of shear stress within an I-section is not uniform. For major axis bending, the shear stress is maximum at the neutral axis (middle of the web) and significantly lower in the flanges. The web carries the bulk of the shear force. This is why the shear area for major axis bending is approximated by the web area ($ht_w$).

Conversely, for minor axis bending, the shear stress is highest in the flanges and very low in the web. The flanges are oriented parallel to the shear force direction in this case, making them effective in resisting it. Thus, the shear area for minor axis bending is based on the combined area of the flanges ($2bt_f$). These simplified shear areas ($ht_w$ and $2bt_f$) are often used in design formulas for calculating shear strength.