The least radius of gyration represents a critical geometric property of a cross-section, particularly important in structural engineering for assessing resistance to buckling. It is fundamentally derived from the section's moment of inertia and its cross-sectional area.
The radius of gyration ($r$) for any axis is defined by the formula:
$r = \sqrt{\frac{I}{A}}$
Where:
- $I$ is the moment of inertia about the axis.
- $A$ is the cross-sectional area.
Understanding the Least Radius of Gyration ($r_{min}$)
The least radius of gyration, denoted as $r{min}$, is the minimum value of the radius of gyration that a cross-section possesses. This minimum value is obtained by using the minimum moment of inertia ($I{min}$) of the cross-section in the formula:
$r{min} = \sqrt{\frac{I{min}}{A}}$
Why is the least radius of gyration important? Columns and other compression members tend to buckle about the axis with the least stiffness, which corresponds to the axis with the minimum moment of inertia and, consequently, the least radius of gyration. A smaller radius of gyration indicates a greater slenderness and a higher propensity for buckling.
Calculating the Least Radius of Gyration
To find the least radius of gyration for a given cross-section, follow these steps:
- Calculate the Cross-Sectional Area (A): Determine the total area of the material in the cross-section.
- Determine the Principal Axes: Identify the centroidal axes about which the moments of inertia are either maximum or minimum. For standard shapes like rectangles or circles, these align with the geometric axes of symmetry.
- Calculate Moments of Inertia about Principal Axes: Compute the moment of inertia ($I$) about each principal axis.
- Find the Minimum Moment of Inertia ($I_{min}$): Compare the calculated moments of inertia about the principal axes and select the smallest value. This $I_{min}$ occurs about the axis where the cross-section is "thinnest" or least spread out from the axis.
- Apply the Formula: Use the minimum moment of inertia ($I{min}$) and the cross-sectional area ($A$) in the formula $r{min} = \sqrt{I_{min}/A}$ to calculate the least radius of gyration.
Example: Rectangular Cross Section
Consider a rectangular cross section with height H and base (width) B.
- Area (A): $A = BH$
- Centroidal Moments of Inertia:
- About the x-axis (parallel to B): $Ix = \frac{BH^3}{12}$
- About the y-axis (parallel to H): $Iy = \frac{HB^3}{12}$
To find the least radius of gyration for this rectangle, we need to determine the minimum moment of inertia ($I_{min}$) between $Ix$ and $Iy$. This depends on whether H or B is smaller. The minimum moment of inertia will be about the axis parallel to the smaller dimension.
Using the Reference:
According to the provided information from Equation 1.8, the moment of inertia about the y-axis used to compute the minimum radius of gyration for a rectangular cross section is $Iy = HB^3/12$. This indicates that, in the context of the reference, $Iy$ is considered the minimum moment of inertia for this calculation (implying B is the smaller dimension or the buckling axis aligns with the y-axis).
Based on this reference, we would compute the minimum radius of gyration as:
$r_{min} = \sqrt{\frac{Iy}{A}} = \sqrt{\frac{HB^3/12}{BH}} = \sqrt{\frac{B^2}{12}} = \frac{B}{\sqrt{12}}$
In general, for a rectangle, $I{min} = \min(Ix, Iy)$. If H < B, $I{min} = Ix = BH^3/12$, and $r{min} = H/\sqrt{12}$. If B < H, $I{min} = Iy = HB^3/12$, and $r_{min} = B/\sqrt{12}$. The reference specifically uses $Iy$ for the minimum radius computation for a rectangle, suggesting the context where the base (B) governs the minimum dimension or the relevant buckling axis.
Understanding $I_{min}$ and how to calculate it for various cross-sections is key to determining the least radius of gyration, a vital parameter in structural design against buckling failure.
