----------- | :------------------------ | :-------------------------: | | W310×39 | sum(b×t^3/3) for elements | 370 | | W410×60 | sum(b×t^3/3) for elements | 650 | | W530×82 | sum(b×t^3/3) for elements | 1,100 | | W610×125 | sum(b×t^3/3) for elements | 982 | | HSS 152×152×9.5 | ~t×A_m^2 | 375,000 | | HSS 203×203×9.5 | ~t×A_m^2 | 943,000 |

HSS sections have 300-1,000× higher J than equivalent W-shapes, making them dramatically better for torsional resistance.

Warping Constant Cw

Bi-Moment and Warping Stress

Warping torsion produces normal stresses (bi-moment B) and shear stresses in the flanges:

Warping normal stress: sigma_w = B × w_n / Cw

Where:

• B = E × Cw × phi'' (bi-moment)
• w_n = normalized warping function (mm^2)

For W-shapes: sigma_w at the flange tip = B × d × b_f / (4 × I_y)

Total normal stress at a point: sigma_total = sigma_bending + sigma_w

Per CSA S16 Clause 13.9.3, when warping stresses exceed 10% of bending stresses:

Interaction: (sigma_bending / (phi × Fy)) + (sigma_w / (phi × Fy)) ≤ 1.0

Torsional Resistance — W-Shapes

Note: T_r is the elastic torsional resistance at 1°/m rate of twist. For HSS, the capacity is much higher.

Torsional Resistance — HSS Sections

HSS 203×203×9.5 has approximately 20× the torsional capacity of a W610×125 at 1°/m twist.

Combined Torsion + Bending + Shear

Per CSA S16 Clause 13.9.2, the interaction for combined torsion, bending, and shear:

T_f / Tr + V_f / Vr + M_f / Mr ≤ 1.0 (linear interaction)

For severe torsion (T_f / Tr > 0.30), more detailed interaction is required per Clause 13.9.3:

sigma_combined / (phi × Fy) ≤ 1.0

Where sigma_combined combines:

• Normal stress from bending (M/S)
• Normal stress from warping (B × w_n / Cw)
• Shear stress from shear (V/A_w)
• Shear stress from torsion (T × r / J for closed, or T_f × Q / (J × t) for open sections)

Worked Example — HSS Cantilever Beam with Eccentric Load

Given: HSS 203×203×9.5 cantilever, 2.0 m long, fixed at one end. A 20 kN factored load at the tip, applied 150 mm eccentric from the section centreline. 350W steel. No other bending load.

Step 1 — Torsional Moment: T_f = 20 × 0.15 = 3.0 kN·m

Step 2 — HSS Torsional Capacity: HSS 203×203×9.5: J = 943 × 10^3 mm^4 Wall thickness t = 9.5 mm, mean dimension = 203 - 9.5 = 193.5 mm Shear stress due to torsion (closed section): tau = T_f / (2 × A_m × t) A_m = (193.5)^2 = 37,442 mm^2 tau = 3.0 × 10^6 / (2 × 37,442 × 9.5) = 3.0 × 10^6 / 711,400 = 4.2 MPa

Step 3 — Shear from Vertical Load: V_f = 20 kN (direct shear) tau_shear = V_f / (0.9 × A_web_total) ≈ 20 × 1000 / (0.9 × 2 × 193.5 × 9.5) = 20,000 / 3,309 = 6.0 MPa

Step 4 — Bending: M_f = 20 × 2.0 = 40 kN·m S_x = 1311 × 10^3 mm^3 sigma = 40 × 10^6 / 1,311,000 = 30.5 MPa

Step 5 — Interaction Check (Clause 13.9.2): T_r = tau_yield × 2 × A_m × t / 10^6 = (0.90 × 0.577 × 350) × 2 × 37,442 × 9.5 / 10^6 = 181.8 × 711,400 / 10^6 = 129.3 kN·m V_r = 0.90 × 0.66 × 350 × (2 × 193.5 × 9.5) / 1000 = 0.90 × 231 × 3,677 / 1000 = 765 kN M_r = 0.90 × Z_x × Fy = 0.90 × 1808 × 350 / 10^6 = 569.5 kN·m

Interaction: T_f/T_r + V_f/V_r + M_f/M_r = 3.0/129.3 + 20/765 + 40/569.5 = 0.023 + 0.026 + 0.070 = 0.120 ≤ 1.0. OK.

Result: HSS 203×203×9.5 is adequate with substantial margin.

Design Recommendations for Torsion

Avoiding Torsion in Structural Design

1. Use HSS sections for members subject to significant torsion (spandrel beams, edge beams)
2. Reduce eccentricity by moving load application points closer to the shear centre
3. Provide torsional bracing at supports — full torsional restraint (warping fixed) reduces twist
4. Use closed sections (HSS, CHS, box sections) — they have 300-1000x the torsional stiffness of W-shapes
5. Avoid long unbraced cantilevers with eccentric loads — the rate of twist accumulates over length

Torsional Bracing

Frequently Asked Questions

What is the difference between St. Venant torsion and warping torsion? St. Venant (pure) torsion causes shear stresses that flow around the cross-section with no longitudinal deformation. It is the only torsion mechanism in closed sections (HSS, CHS). Warping torsion occurs in open sections (W-shapes, channels) where the flanges bend in opposite directions, creating longitudinal normal stresses and additional shear. For W-shapes, warping typically governs 80-95% of the torsional resistance.

What is the torsional constant J for a W310x39? J = 370 × 10^3 mm^4 for W310×39. For comparison, HSS 152×152×9.5 has J = 375,000 × 10^3 mm^4 (about 1000× higher). This is why HSS sections are strongly preferred for torsionally loaded members. The St. Venant constant J for open sections is calculated as sum(b × t^3 / 3) for each plate element.

How is torsion handled for W-shape beams in Canadian design? Per CSA S16 Clause 13.9, torsion in W-shapes is checked using combined stress interaction. W-shapes have very low St. Venant torsional stiffness (small J) and rely on warping resistance. For torsionally loaded W-shapes: (a) provide full warping restraint at supports, (b) keep torsional loads (T_f) below 10-15% of member capacity, and (c) check the interaction of bending + torsion + shear.

When is torsion significant enough to check in design? Torsion should be checked when: (a) the load application point is eccentric from the shear centre by more than 5% of the member depth, (b) the member supports spandrel panels or curtain walls on one flange, (c) the beam supports an eccentric crane runway, or (d) the member is a cantilever with an unbalanced load. A common rule: if the torsional moment T_f exceeds 5% of the bending moment M_f, a torsion check is warranted.

Related Pages

• Canadian HSS Section Properties
• CSA S16 Beam Design
• CSA S16 Combined Loading
• Canadian Beam Sizes — W-Shape Table
• CSA S16 Fatigue Design
• Beam Capacity Calculator
• All Canadian References

This page is for educational reference. Torsional design per CSA S16:24 Clause 13.9. Verify torsional section properties against CISC Handbook. For significant torsion, use closed sections (HSS, CHS). Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent PE/SE verification.

Design Resources

Calculator tools

• Torsion Analysis Calculator

Reference pages

• CA Torsion