Torsional Buckling — Warping & Twist in Thin-Walled Steel Members

Torsional buckling is a column instability mode distinct from flexural (Euler) buckling. Instead of bending about its weak axis, the cross-section twists about its longitudinal axis. The member rotates without significant lateral translation, like wringing a towel.

This mode governs when the torsional stiffness is low relative to flexural stiffness — a common situation for thin-walled open sections (cruciform, tee, angle, channel) where the St. Venant torsional constant J and warping constant Cw are small. Torsional buckling often controls the design of single-angle compression members, star-battened cruciform columns, and unrestrained tee-section chords in trusses.

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.

Two Types of Torsion

Understanding torsional buckling requires understanding the two distinct mechanisms of torsion in thin-walled sections:

St. Venant (Uniform) Torsion

St. Venant torsion is resisted by shear stresses that circulate around the cross-section. The torque-twist relationship is:

T = G × J × dθ/dz

Where:

• G = shear modulus (≈ 11,200 ksi for structural steel, G = E / [2(1+ν)])
• J = St. Venant torsional constant (in⁴)
• dθ/dz = rate of twist (radians per unit length)

For a solid circular section: J = πd⁴/32 (relatively large). For a thin-walled open section: J ≈ (1/3) × Σ(bt³) (very small — thin elements contribute little to J because of the t³ term).

St. Venant torsion alone is the torsional resistance of circular and thick-walled sections. For thin-walled open sections, St. Venant torsion is typically negligible.

Warping Torsion

When a thin-walled open section twists, cross-section points move longitudinally — the section warps. If warping is restrained (e.g., at a fixed support or where torsional moment changes), warping normal stresses σw develop. These stresses form a self-equilibrating couple that resists twist:

T_warping = −E × Cw × d³θ/dz³

Where Cw is the warping constant (in⁶). Cw is typically LARGE for I-sections (flanges act as independent beams in opposite bending) and ZERO for angle sections (all elements intersect at a common point — no warping stiffness).

The total torque resistance is the sum:

T_total = G × J × dθ/dz − E × Cw × d³θ/dz³

Warping Constant Cw — Physical Interpretation

For a W-shape, warping torsion works as follows:

1. The section twists about the shear center
2. Flanges bend in opposite directions (one upward, one downward) — this is warping
3. The flange bending creates a couple that resists twist
4. Cw = Iy × h²/4 for a doubly-symmetric I-section (approximately)

Where Iy = moment of inertia of one flange about the y-axis (≈ tf × bf³/12) and h = distance between flange centroids. Because Cw depends on bf³ and h², wide-flange sections have very high warping stiffness — W-shapes rarely suffer torsional buckling.

For an angle: Cw = 0 because the two legs intersect at a point — there's no flange pair to create a warping couple. Angles resist torsion through St. Venant torsion alone (and J for an angle with leg lengths b, b and thickness t is J ≈ 2 × b × t³/3, which is very small). Angles are highly susceptible to torsional and flexural-torsional buckling.

Torsional Buckling Stress — AISC 360 E4

Doubly-Symmetric Sections (Flexural-Torsional)

For doubly-symmetric sections (W-shapes, HSS, cruciform), the elastic torsional buckling stress is:

Fe = [π² × E × Cw / (Kz × L)² + G × J] / (Ix + Iy) × (1/A)

Or more intuitively:

Fe = (warping term + St. Venant term) / (polar moment of inertia / area)

Key parameters:

• Kz = effective length factor for twisting (typical Kz = 1.0 unless ends are specially detailed to prevent warping)
• Ix + Iy = polar moment of inertia about the shear center

For W-shapes: Cw is large, so Fe is typically higher than the flexural buckling stress (Fey = π²E/(KLy/ry)²). Flexural buckling governs; torsional buckling rarely controls for standard W-shapes.

Singly-Symmetric Sections (Channels, Tees)

For singly-symmetric sections (symmetry about one axis only — channels about web, tees about stem), flexural buckling about the axis of symmetry couples with torsion, producing flexural-torsional buckling. The buckling stress Fe is the smaller root of:

(Fe − Fex)(Fe − Fey)(Fe − Fez) − Fe²(Fe − Fey)(xo/ro)² − Fe²(Fe − Fex)(yo/ro)² = 0

Where:

• Fex = π²E/(KL/r)x² — flexural buckling about x-axis
• Fey = π²E/(KL/r)y² — flexural buckling about y-axis
• Fez = torsional buckling stress = [π²ECw/(KzL)² + GJ] / (A × ro²)
• xo, yo = shear center coordinates relative to centroid
• ro² = xo² + yo² + (Ix + Iy)/A — polar radius of gyration about shear center

For a channel: Fey (flexural about weak axis) and Fez (torsional) couple — the actual buckling stress is lower than either individually because the modes interact.

Unsymmetric Sections (Angles, Z-Shapes)

For angles and Z-sections (no axis of symmetry), all three modes (flexural-x, flexural-y, torsional) couple. The solution for Fe is the smallest root of a cubic equation — always lower than any individual flexural or torsional buckling stress. This is why single-angle compression members require special analysis and are typically very conservative when designed by simplified methods.

Which Sections Are Most Vulnerable?

Flexural-Torsional Buckling of Angles

The single-angle compression member is the poster child for flexural-torsional buckling. Consider an L4×4×1/4 used as a web member in a truss:

• rx = ry = 1.25 in (equal-leg angle)
• rz = 0.795 in (minimum — governs for single-angle buckling per AISC 360 E5)
• J ≈ 2 × 4 × (0.25)³/3 = 0.0417 in⁴ (very small)
• Cw = 0 (no warping stiffness)
• ro = distance from centroid to shear center ≈ 1.1 in

For L = 5 ft: (KL/r)z = 60/0.795 = 75.5 — flexural buckling stress Fey = π² × 29,000/75.5² = 50.3 ksi. But torsional-flexural buckling stress Fe (solving the cubic) is considerably lower because the weak torsional stiffness couples with flexure.

AISC 360 E5 provides simplified provisions for single angles, using an effective slenderness ratio (KL/r)eff that accounts for flexural-torsional interaction without solving the cubic.

Warping Restraint — How Supports Affect Torsional Buckling

The boundary conditions for torsion differ from flexure:

Most beam-to-column connections provide some warping restraint, but it's difficult to quantify. Conservative practice assumes Kz = 1.0 unless warping fixity is specifically designed and detailed (e.g., end plates with full-penetration welds all around).

Worked Example — Tee Section

WT6×13 (A992, Fy = 50 ksi), used as a truss chord, L = 8 ft:

Properties: A = 3.81 in², Ix = 11.4 in⁴, Iy = 7.28 in⁴, J = 0.088 in⁴, Cw = 0.215 in⁶, xo = 0 (symmetric about y-axis — tee stem), yo = 0.944 in (shear center is outside the section, at the flange-web junction)

rx = √(11.4/3.81) = 1.73 in → (KL/r)x = 96/1.73 = 55.5
ry = √(7.28/3.81) = 1.38 in → (KL/r)y = 96/1.38 = 69.6
ro² = 0 + 0.944² + (11.4+7.28)/3.81 = 0.891 + 4.90 = 5.79 in²

Fey = π² × 29000 / 69.6² = 59.1 ksi
Fez = [π² × 29000 × 0.215 / (1.0×96)² + 11200 × 0.088] / (3.81 × 5.79)
    = [61500/9216 + 986] / 22.06 = (6.67 + 986) / 22.06 = 45.0 ksi

Solving cubic: Fe (flexural-torsional) ≈ 38.4 ksi < Fey = 59.1 ksi
→ Flexural-torsional buckling governs (Fe = 38.4 ksi vs Fey = 59.1 ksi)

Fcr = 0.658^(50/38.4) × 50 = 30.1 ksi
Pn = 30.1 × 3.81 = 115 kips
φPn = 0.90 × 115 = 103 kips (LRFD)

Frequently Asked Questions

Why don't W-shapes typically experience torsional buckling?

W-shapes have very high warping stiffness Cw because the two flanges act as a couple — when the section twists, flanges bend in opposite directions, creating a strong restoring moment. Cw ≈ Iy × h²/4, and for a W14×48: Iy_flange ≈ 2.0 in⁴, h² ≈ 13.8² = 190 in², giving Cw ≈ 95 in⁶ — this is enormous compared to a tee or angle. The warping term π²ECw/L² dominates over GJ, making the torsional buckling stress much higher than the flexural buckling stress.

How does torsional buckling differ from lateral-torsional buckling (LTB)?

Torsional buckling applies to columns under axial compression — the member twists without significant bending. Lateral-torsional buckling applies to beams under flexure — the compression flange buckles laterally while the section simultaneously twists. Both involve torsion, but the triggering mechanism is different (axial force vs. bending moment), and they're checked using different code provisions (AISC 360 E4 vs. F2).

Can an HSS experience torsional buckling?

Closed sections (HSS round and rectangular) have very high St. Venant torsional stiffness because J is large (thin-walled tube J ≈ 4A²t/P where A is enclosed area and P is perimeter). The shear flow around the closed perimeter resists torsion efficiently. Torsional buckling of HSS columns is theoretically possible but almost never governs for practical geometries — flexural buckling controls.

Related Terms and Pages

• Buckling — Definition, Types & Euler Load
• Elastic Buckling — Euler Load Pe Formula
• Lateral Torsional Buckling — LTB Explained
• Slenderness Ratio (KL/r) — Column Classification & Limits
• Moment of Inertia (I) — Definition & Formula
• Effective Length Factor (K) — Definition & Values
• Column Buckling Equations — Reference Guide
• Steel Buckling — Reference Guide

Educational reference only. Torsional and flexural-torsional buckling must be evaluated per AISC 360 Section E4, EN 1993-1-1 Section 6.3.1.4, or AS 4100 Section 6.3 by a licensed Professional Engineer for all construction applications.

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.