Radius of Gyration (r) — Definition, Formula & Column Buckling

The radius of gyration (r) is a geometric property defined as the square root of the ratio of moment of inertia to cross-sectional area:

r = sqrt(I / A)

Physically, r represents the distance from the centroidal axis at which the entire area A could be concentrated (as a thin ring) and still produce the same moment of inertia I. It is a measure of how efficiently a cross-section distributes its material relative to its centroid — the larger r is for a given area, the more buckling-resistant the section.

The radius of gyration is the single most important parameter governing column compressive strength, as it appears in the slenderness ratio KL/r that controls elastic and inelastic buckling behavior across all major design codes.

Physical Interpretation

PRELIMINARY — NOT FOR CONSTRUCTION. All content is for educational and reference use only. Must be independently verified by a licensed Professional Engineer (PE) or Structural Engineer (SE) before use in any project.

Consider two sections with the same cross-sectional area A = 10 in^2:

Both sections have the same area and weight. But the W-shape has a radius of gyration 4.5x larger, resulting in a slenderness ratio 4.5x smaller and dramatically higher column capacity. This illustrates why structural steel shapes (I-beams, HSS tubes) are far more efficient for columns than solid sections of equivalent weight.

Formulas for Common Shapes

Rectangle

I = b * d^3 / 12
A = b * d
r = sqrt(I/A) = sqrt(b * d^3 / (12 * b * d)) = d / sqrt(12) = 0.289 * d

Circle

I = pi * D^4 / 64
A = pi * D^2 / 4
r = sqrt(I/A) = sqrt((pi*D^4/64) / (pi*D^2/4)) = D / 4

Thin-Walled Tube (HSS Round)

I = pi * R^3 * t      (approximate, R = mean radius)
A = 2 * pi * R * t
r = sqrt(I/A) = sqrt(pi * R^3 * t / (2 * pi * R * t)) = R / sqrt(2) ≈ 0.707 * R

I-Shape (W, UB, UC)

rx = sqrt(Ix / A)    (strong-axis radius of gyration)
ry = sqrt(Iy / A)    (weak-axis radius of gyration)

For standard rolled shapes, rx and ry are tabulated in the AISC Manual. Weak-axis (ry) typically governs column design because it produces the larger slenderness ratio.

Radius of Gyration — Common W-Shapes (AISC)

Key observation: rx/ry ratios of 1.7 to 6.8 show how dramatically W-shapes differ between strong and weak axes. For unbraced columns, the weak axis (ry) nearly always controls. A column 20 ft tall with KL = 20 ft has:

• KL/rx = 240/11.7 = 20.5 (W30x99, strong axis)
• KL/ry = 240/2.10 = 114 (W30x99, weak axis)

The weak-axis slenderness ratio is 5.6x larger and will produce a substantially lower buckling capacity.

Role in Column Buckling — Slenderness Ratio

The slenderness ratio KL/r is the universal parameter governing column compressive strength. All major design codes base their column strength curves on KL/r (or the equivalent non-dimensional slenderness).

AISC 360 Chapter E — Column Strength

Elastic buckling stress (Euler):

Fe = pi^2 * E / (KL/r)^2

Critical stress Fcr:

When KL/r <= 4.71*sqrt(E/Fy) = 113 (Fy = 50 ksi):
    Fcr = (0.658^(Fy/Fe)) * Fy     (inelastic buckling)

When KL/r > 113:
    Fcr = 0.877 * Fe                (elastic buckling)

AS 4100 Section 6 — Column Strength

The modified slenderness lambda_n is:

lambda_n = (KL/r) * sqrt(kf) * sqrt(Fy/250)

The member slenderness reduction factor alpha_c depends on lambda_n and the section constant alpha_b.

EN 1993-1-1 Clause 6.3.1

Non-dimensional slenderness:

lambda_bar = sqrt(A*fy / Ncr)
Ncr = pi^2 * E * I / (KL)^2
lambda_bar = (KL/r) / (pi * sqrt(E/fy))

The reduction factor chi is determined from the appropriate buckling curve (a0, a, b, c, d).

Effect of r on Column Capacity — W12x50 Example

A 15-ft tall W12x50 column (Fy = 50 ksi):

If the column is unbraced in the weak direction, capacity drops 34% to 399 kips. Providing weak-axis bracing at mid-height (Ky*Ly = 7.5 ft, KL/ry = 45.9) restores capacity to 610 kips.

Design Implications

1. Weak-Axis Governs for Unbraced Columns

Unless bracing is provided in both directions, ry (weak axis) will produce the controlling slenderness ratio. A column that is equally unbraced in both directions will buckle about the weak axis.

2. rx/ry Ratio Guides Bracing Requirements

Sections with high rx/ry ratios (like W21x44 at 6.40) require weak-axis bracing at much closer intervals than strong-axis bracing. If the column height is H:

• Strong-axis unbraced length: Kx * Lx = H (braced at top and bottom)
• Weak-axis unbraced length: Ky _ Ly <= H _ (ry/rx) to make slenderness ratios equal

3. HSS Sections Are Balanced

Square HSS sections have rx = ry, making them equally efficient in both directions. This simplifies bracing requirements and makes them attractive for free-standing columns.

4. Built-Up Sections Can Optimize r

Lacing, batten plates, or perforated cover plates can increase the radius of gyration of built-up columns while adding minimal weight.

Frequently Asked Questions

What is radius of gyration? Radius of gyration r = sqrt(I/A) measures how a cross-section's area is distributed relative to its centroid. A larger r means material is farther from the center, providing greater buckling resistance for the same cross-sectional area.

How does radius of gyration affect column design? r appears in the slenderness ratio KL/r, which controls buckling capacity. For a column of given length, doubling r halves the slenderness ratio and dramatically increases compressive strength. The weak-axis r (ry) typically governs design for unbraced W-shape columns.

What is a good radius of gyration for a column? For a 12-ft column, a minimum ry of 1.5-2.0 in (KL/ry ~ 72-96) is typical for W-shapes, yielding reasonable efficiency. For taller columns, select sections with larger ry or provide intermediate bracing. W14 sections offer the best rx/ry balance; W8 and W10 sections are more square-shaped with better weak-axis properties.

How do I calculate radius of gyration? For standard rolled sections, use tabulated r values from the AISC Manual Table 1-1 (rx and ry columns). For custom shapes, compute I (moment of inertia) and A (area), then r = sqrt(I/A). The calculation accounts for the entire cross-section including fillets.

Related Terms and Pages

• Effective Length Factor — K Factor Guide
• Compact Section — Definition & Limits
• Plastic Modulus — Definition & Formula
• Lateral Torsional Buckling — LTB Explained
• Column Capacity Calculator — Free Online Tool
• Column Design Guide — AISC 360
• Column Buckling Equations
• Section Properties Database

Educational reference only. Radius of gyration values should be taken from the governing design standard (AISC Manual Table 1-1) for final design. All column designs must be independently verified by a licensed Professional Engineer.

Disclaimer: This content is for educational purposes only. Results must be verified by a licensed professional engineer. Steel Calculator provides preliminary design tools — NOT a substitute for professional engineering judgment.