Understanding Shape Factor for Rolled I-Sections

The concept of shape factor is fundamental in structural engineering, particularly when dealing with the plastic analysis and design of steel structures. It provides insight into the reserve strength of a cross-section beyond its yield point, up to its full plastic moment capacity.

Shape Factor Definition and Calculation

The shape factor (often denoted as 'f' or 'SF') for a cross-section is defined as the ratio of its plastic moment capacity ($M_p$) to its yield moment capacity ($M_y$).

Mathematically, it can be expressed as:

$$ \text{Shape Factor} (f) = \frac{M_p}{M_y} $$

Where:

• $M_p$ is the plastic moment, which is the maximum moment a section can resist when the entire cross-section has yielded and reached its plastic stress distribution. It is calculated as $M_p = Z_p \cdot f_y$.
• $M_y$ is the yield moment, which is the moment at which the outermost fiber of the cross-section first reaches the yield stress. It is calculated as $M_y = Z_e \cdot f_y$.

Here, $Z_p$ is the plastic section modulus, $Z_e$ is the elastic section modulus, and $f_y$ is the yield stress of the material. Since $f_y$ cancels out, the shape factor can also be defined as the ratio of the plastic section modulus to the elastic section modulus:

$$ f = \frac{Z_p}{Z_e} $$

Rolled I-Sections and Flexure about Strong Z-axis

Rolled I-sections are widely used structural steel members, often found in beams and columns. They are characterized by their "I" shape, consisting of two flanges (top and bottom) and a web connecting them.

Flexure about the strong z-axis refers to bending of the I-section about its major principal axis, which is typically the horizontal axis passing through the centroid (also known as the x-axis in some conventions, or the z-axis as per the question's notation). This axis offers the greatest moment of inertia, making the section most resistant to bending in this direction.

Shape Factor Range for Rolled I-Sections

For most standard rolled I-sections when subjected to flexure about the strong z-axis, the shape factor typically falls within a specific range. This range reflects the efficiency of the I-section's geometry in developing plastic resistance.

The generally accepted range for the shape factor of common rolled I-sections in bending about their strong axis is 1.09 to 1.18. This value is slightly higher than 1.0 (which would be for a perfectly rectangular section, where the shape factor is 1.5) because the I-section's material is concentrated at the flanges, further away from the neutral axis, contributing more effectively to both elastic and plastic resistance.

A shape factor greater than 1 indicates that a cross-section has a reserve of strength beyond the point where the outermost fiber yields, allowing for plastic redistribution of stresses before the full plastic moment is reached. This is a crucial concept in plastic design, where the full plastic capacity of the member is utilized.

Comparing this with the given options:

• 1.09 to 1.18
• 1.42 to 1.61
• 1.20 to 1.42
• 1.18 to 1.20

The range 1.09 to 1.18 accurately represents the shape factor for most rolled I-sections in flexure about the strong z-axis.