Steel Torsion Design — Saint-Venant, Warping, and AISC Design Guide 9
Torsion in steel members arises when applied loads do not pass through the shear center. For open sections (W-shapes, channels), torsion produces both shear stresses (Saint-Venant torsion) and normal stresses (warping torsion). For closed sections (HSS, pipes), Saint-Venant torsion dominates and warping is negligible. AISC 360-22 Section H3 covers combined torsion with other forces, while AISC Design Guide 9 provides comprehensive torsion design procedures.
Two types of torsion resistance
Saint-Venant (pure) torsion: Resisted by shear stresses circulating around the cross-section. The resistance depends on the torsional constant J. For closed sections, J is large and shear stresses are low. For open sections, J is small and shear stresses are high.
Warping torsion: Resisted by flange bending in opposite directions (bi-moment). The resistance depends on the warping constant Cw. Warping occurs only in open sections where the flanges are free to bend independently. Closed sections have negligible warping because shear flow around the closed cell prevents differential flange movement.
Key section properties
| Property | Symbol | Units | Open sections | Closed sections |
|---|---|---|---|---|
| Torsional constant | J | in^4 | Small (0.5-20) | Large (50-500) |
| Warping constant | Cw | in^6 | Large (500-50000) | ~0 |
| Shear center | e | in | May not coincide with centroid | At centroid |
For W-shapes: J = (2bftf^3 + (d-2tf)tw^3)/3 (approximate). For rectangular HSS: J = 2t(b-t)^2*(d-t)^2/(b+d-2*t).
AISC 360-22 Section H3 -- combined forces with torsion
For HSS members under torsion:
Tn = 0.6*Fy*C [Section H3.1]
phiTn = 0.90*Tn
Where C = torsional shear constant. For rectangular HSS: C = 2*(B-t)*(H-t)*t - this represents the area enclosed by the mid-thickness line times the wall thickness.
For round HSS: C = pi*(D-t)^2*t/2.
Interaction with shear and axial force
When torsion acts simultaneously with other forces, AISC Section H3.2 requires:
(Pr/Pc + Mr/Mc)^2 + (Vr/Vc + Tr/Tc)^2 <= 1.0
This elliptical interaction recognizes that torsional shear adds directly to transverse shear (both produce shear stress on the same cross-section planes), while axial and flexural normal stresses interact separately.
Open section torsion -- the differential equation
For open sections (W-shapes, channels), torsion is governed by:
GJ*(d_theta/dz) - ECw*(d^3_theta/dz^3) = t(z)
Where theta = angle of twist, z = length along member, t(z) = distributed torque. The first term is Saint-Venant resistance; the second is warping resistance. The relative contribution of each depends on the parameter:
a = sqrt(ECw/(GJ)) [characteristic length]
When the member length L >> a, Saint-Venant torsion dominates. When L << a, warping dominates. For typical W-shapes, a = 30-100 inches, so warping is significant for short members and Saint-Venant governs for long members.
Worked example -- W18x50 spandrel beam with eccentric cladding
Given: W18x50, Fy = 50 ksi, L = 25 ft, simply supported. Eccentric cladding load = 0.5 kip/ft at 8 inches from the shear center.
Properties: J = 1.24 in^4, Cw = 3040 in^6, Sx = 88.9 in^3.
Torque: t = 0.5*8 = 4.0 kip-in/ft = 0.333 kip-in/in.
Characteristic length: a = sqrt(290003040/(112001.24)) = sqrt(748,200) = 865 in. L/a = 300/865 = 0.347 -- warping is very significant at this length.
Maximum normal stress from warping (at midspan): sigma_w = (tL^2Wno)/(8*ECw) -- from DG9 tables. For simply supported with uniform torque: sigma_w is approximately 8.5 ksi.
Combined check: The warping normal stress adds to flexural bending stress. If the beam is at 60% utilization for gravity bending (sigma_b = 30 ksi), the combined stress = 30 + 8.5 = 38.5 ksi, which is 77% of Fy. The torsion increased stress by 28%.
Practical tip: avoiding torsion in design
The best torsion design is no torsion at all. Load the member through or near its shear center whenever possible:
- Use stiffened seats or direct bearing rather than eccentric clip angles
- Frame beams into the web of columns (load passes near shear center) rather than to the flange face
- Brace spandrel beams at cladding attachment points to prevent twist
- Use closed sections (HSS) when torsion cannot be avoided -- a HSS 8x8x3/8 has J = 159 in^4, compared to J = 1.24 in^4 for a W18x50 at similar weight
The torsional stiffness ratio (closed/open) can exceed 100:1 for similar weight members.
Common mistakes
- Ignoring torsion on spandrel beams. Eccentric cladding, canopy, or curtain wall loads on edge beams always produce torsion. Even small eccentricities produce significant warping stresses in W-shapes.
- Using open sections when closed sections are better. For members subject to significant torsion (canopies, sign structures, crane bridges), HSS or pipe sections are far more efficient.
- Neglecting warping stresses at supports. For fixed-end conditions, the maximum warping stress occurs at the supports, not midspan. The boundary conditions fundamentally change the stress distribution.
- Forgetting the shear center location for channels. Channel shear centers are outside the web, so gravity loads on channels always produce torsion unless the load is applied through the shear center.
- Not combining torsional and flexural stresses. Warping normal stresses add directly to flexural bending stresses at the flange tips. Both must be combined for a complete strength check.
Run this calculation
Related references
- Combined Loading — Chapter H Interaction
- Lateral-Torsional Buckling
- Stress-Strain Relationship
- Steel Box Girder
- How to Verify Calculations
Disclaimer
This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against AISC 360-22 Section H3 and AISC Design Guide 9. The site operator disclaims liability for any loss arising from the use of this information.