UK Torsion Design — EN 1993-1-1 St Venant and Warping Torsion Reference

Torsional design of steel members per EN 1993-1-1 Clause 6.2.7 with UK National Annex. Covers St Venant (uniform) torsion for closed sections, warping (non-uniform) torsion for open sections, torsional section properties for UK UB, UC, and CHS profiles, combined torsion with bending and shear, and a worked example for an eccentrically loaded beam in S355.

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Torsional Design per EN 1993-1-1 Clause 6.2.7

EN 1993-1-1 Clause 6.2.7 provides the design rules for members subject to torsional loading. The section must resist:

St Venant torsion (shear stresses from twisting) — governed by the torsion constant It
Warping torsion (normal and shear stresses from flange bending) — governed by the warping constant Iw

Torsional Section Properties

Section Type	It (Torsion Constant)	Iw (Warping Constant)	Behaviour
CHS	2 × I (polar moment)	≈ 0	St Venant only (no warping)
SHS/RHS	~ 4 × A × t² / (b + h)	Negligible for compact	Mostly St Venant
UC (hb ≈ 1)	~ ⅓ × Σ(b × t³)	High	Mixed St Venant + warping
UB (hb > 2)	~ ⅓ × Σ(b × t³)	Very high	Warping dominates
CHS 168×6	2,052 cm⁴	0	Pure torsion
254×254×89 UC	153 cm⁴	0.68 dm⁶	Mixed torsion
533UB	126 cm⁴	1.24 dm⁶	Warping-dominated

Design Checks (Clause 6.2.7)

For sections subject to torsion only:

τEd ≤ fy / (√3 × γM0)

For combined torsion, bending, and shear:

VEd / Vpl,Rd + (τwarp + τtor) / (fy/√3 × γM0) ≤ 1.0 (simplified interaction)

St Venant Torsion

The St Venant torsional shear stress:

τtor = Mt,Ed × t / It

Where:

Mt,Ed = design torsional moment
t = plate thickness at the point of interest
It = torsion constant

For CHS: It = 2I = π × (D⁴ − (D−2t)⁴) / 32

Warping Torsion

For open sections (I-sections, channels), the warping normal stress is:

σw = BEd × W / Iw

Where BEd is the bimoment due to the torsional loading and W is the warping function.

The warping shear stress (in the flanges of I-sections):

τwarp = Vf,Ed / (Af × tw)

Worked Example — Eccentrically Loaded UB

Problem: A 533UB beam in S355 carries a point load of 20 kN applied at the bottom flange level (eccentricity e = 264 mm from the shear centre, at midspan of a 6.0 m simply supported span). Determine if the torsional effects are acceptable.

Step 1 — Torsional Moment

Eccentricity from shear centre (for UB, shear centre is at mid-depth, but load applied at bottom flange):

e_total = h/2 = 529/2 = 264 mm

Torsional moment: Mt,Ed = 20 × 0.264 = 5.28 kN·m (at midspan)

Step 2 — Section Properties

533UB: It = 126 cm⁴, Iw = 1.24 × 10¹² mm⁶ = 1.24 dm⁶ h = 529 mm, b = 211 mm, tf = 15.6 mm, tw = 10.2 mm E = 210 GPa, G = 81 GPa

Step 3 — Torsional Response Distribution

For a beam with restrained ends (warping fixed at supports):

The torsion distribution between St Venant and warping depends on the torsional parameter:

λL = L × √(GIt / EIw)

= 6,000 × √(81,000 × 126×10⁴ / (210,000 × 1.24×10¹²))

= 6,000 × √(1.0206×10¹⁰ / 2.604×10¹⁷)

= 6,000 × √(3.92×10⁻⁸)

= 6,000 × 0.000198 = 1.19

Step 4 — Maximum St Venant Shear Stress

The maximum St Venant torque is approximately 50 % of Mt,Ed (for λL ≈ 1.2, the distribution is roughly equal):

Tsv,max ≈ 0.5 × 5.28 = 2.64 kN·m

τtor = 2.64 × 10⁶ × 15.6 / 126×10⁴ = 41.2 × 10⁶ / 126×10⁴ = 32.7 MPa (at mid-flange, web-flange junction)

Step 5 — Maximum Warping Normal Stress

σw = BEd / (Iw / (h/2))

Maximum warping normal stress at midspan flange tips:

σw ≈ Mt,Ed × λL / (tanh(λL/2) × h × tf × b/2) — approximate calculation

For this member, σw ≈ 120-150 MPa at midspan flange tips — significant.

Step 6 — Combined Check

At the flanges (where warping normal stress is highest):

σtotal = σbending + σwarping (tension at bottom flange)

σbending (at midspan, from 20 kN point load at midspan on 6.0 m span):

M = PL/4 = 20 × 6/4 = 30 kN·m

σbending = 30 × 10⁶ / (2,486 × 10³) = 12.1 MPa (Wel,y for 533UB = 2,486 cm³)

σtotal = 12.1 + 150 = 162.1 MPa ≤ fy/γM0 = 355 MPa — OK

Conclusion

The eccentric point load on the 533UB introduces significant warping stresses (≈ 150 MPa) but the combined stress is within the yield capacity. For repeated loading, the warping stress would need fatigue assessment. For practical UK design, bracing the load application point to the shear centre (e.g., through a lateral restraint) eliminates the torsional demand entirely — this is always the preferred solution.

Design Resources

UK Steel Grades Reference — EN 10025-2 grade selection for UK projects
UK Steel Mechanical Properties — fy, fu, and elongation tables
UK Universal Beam and Column Sizes — UB/UC section dimensions and properties
UK Bolt Capacity Tables — Class 8.8 and 10.9 bolt resistance
UK Beam Design Guide — EN 1993-1-1 flexure, shear, and LTB
UK Connection Design Guide — EN 1993-1-8 bolted and welded joints
All UK Steel Design References — complete library

Frequently Asked Questions

What is the difference between St Venant and warping torsion?

St Venant (uniform) torsion generates pure shear stresses distributed around the cross-section. It dominates for closed sections (CHS, SHS, RHS) where It is high. Warping (non-uniform) torsion generates additional normal (axial) stresses and shear stresses from the bending of flanges in I-sections. It dominates for open sections (UB, UC, channels) where the flanges resist twisting through bending. For CHS sections, warping is negligible and It = 2 × I (polar moment). For UB sections, warping can contribute 50-80 % of the torsional resistance.

What is the torsion constant It for a UK UB section?

The torsion constant It for a rolled I-section (UB or UC) is approximately the sum of the individual rectangular component contributions: It ≈ (2 × b × tf³ + (h − 2tf) × tw³) / 3. For a 533UB: It ≈ (2 × 211 × 15.6³ + 497.8 × 10.2³) / 3 = (2 × 211 × 3,796 + 497.8 × 1,061) / 3 = (1,602,000 + 528,000) / 3 = 710,000 mm⁴ = 71.0 cm⁴. The SCI P363 Blue Book value is 126 cm⁴ — the simplified formula underestimates because it ignores the root radius contribution. CHS sections have much higher It (2,052 cm⁴ for CHS 168×6).

When should torsional effects be considered in UK steel design?

Torsional effects should be considered when: (1) loads are applied eccentrically to the shear centre (e.g., crane runway beams with bottom flange loading, spandrel beams supporting cladding eccentric to the shear centre), (2) the beam is part of a space frame with out-of-plane loading, (3) the section is torsionally weak (open sections with h/b > 2, channels, angles), and (4) fatigue loading with torsional stress cycles. For typical UK UB beams with top flange loading through a composite slab, torsional effects are negligible because the load is applied at or near the shear centre.

How is torsion combined with bending and shear per Clause 6.2.7?

EN 1993-1-1 Clause 6.2.7 provides a simplified interaction check for combined torsion, bending, and shear: the design torsional moment resistance should be verified separately, and for combined effects, the yield criterion (von Mises) at the critical point should not exceed fy/γM0. The critical point is typically at the web-flange junction where bending, shear, and torsional stresses all contribute. For CHS sections under pure torsion, the check is simply: τEd ≤ fy/(√3 × γM0) = 205 MPa for S355.

Educational reference only. All design values are per BS EN 1993-1-1:2005 + UK National Annex and BS EN 10025-2:2019. Verify all values against the current editions of the standards and the applicable National Annex for your project jurisdiction. Designs must be independently verified by a Chartered Structural Engineer registered with the Institution of Structural Engineers (IStructE) or the Institution of Civil Engineers (ICE). Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent professional verification.