Torsional Analysis of Structural Steel Members — Part 1: Theory & Section Properties

1. Why torsion is the check everyone avoids

In steel design, the torsion check has long been a grey area most engineers avoid. There are three concrete reasons.

Common analysis software does not support it

SAP2000, ETABS and STAAD.Pro model members as beam elements with 6 degrees of freedom per node.
The 7th degree of freedom — warping — is ignored entirely.
Consequence: the software returns only the total torque T, and never the detailed torsional stresses.

The engineer must calculate by hand

The process is involved, requiring several tables and charts.
The formulas use the 1st, 2nd and 3rd derivatives of the twist angle — hard to approach in practice.
No step-by-step guide exists that truly serves the practising engineer.

This article systematises torsion theory, gives a step-by-step design procedure, covers both AISC (Design Guide 9) and Eurocode 3 (SCI P385), and includes fully worked examples in Part 2.

2. Section classification — the most critical premise

This classification determines the entire analysis approach. Get it wrong and everything downstream is wrong.

Figure 1. Torsional behaviour compared — open versus closed sections. — **Figure 1.** Torsional behaviour compared — open versus closed sections.

Open sections

Examples: W-shapes (I-beams), Channels (C/MC), Angles (L), Tees (WT/MT/ST).
Torsional stiffness is very low — small torsional constant J.
When twisted the section warps: planar elements displace out of their original plane.
Warping torsion dominates, generating large normal stresses at the flange tips.

Closed sections

Examples: rectangular HSS, round HSS (pipe), box sections.
Torsional stiffness is very high — 100 to 1,000 times that of a comparable open section.
A closed shear flow runs around the perimeter; St. Venant torsion dominates completely.
Warping torsion is generally negligible.

Property	Open sections	Closed sections
Torsional constant J	Small	Very large (100–1,000×)
Warping constant Cw	Large, significant	Small, usually neglected
Primary mechanism	Warping torsion	St. Venant
Normal stress from torsion	Present, often large	Very small, neglected
Computational complexity	High	Low
Under high torsion	Limit use	Preferred

3. The shear center — the core concept

The shear center is the point through which a transverse load must act to cause bending without twisting. A load offset from it induces a torque T = P × e, where e is the eccentricity.

Figure 2. Shear center locations for common cross-section types. — **Figure 2.** Shear center locations for common cross-section types.

Section type	Shear center location
W-shape — doubly symmetric	Coincides with the centroid
Channel — singly symmetric	Outside the web, on the flange side, at distance e₀
HSS rectangular / round	Coincides with the centroid
Angle (L)	At the corner where the legs meet
Tee (WT/MT/ST)	On the axis of symmetry, near the flange–web junction

Shear center formula for Channels

e₀ = 3·b_f²·t_f / ( h·t_w + 6·b_f·t_f )

b_f = flange width        t_f = flange thickness
h   = clear web depth     t_w = web thickness

4. Decomposing an eccentric load

When a load misses the shear center, decompose it into two independent components and superpose.

Figure 3. Eccentric load = pure bending through the shear center + pure torsion. — **Figure 3.** Eccentric load = pure bending through the shear center + pure torsion.

Concentrated torque:   T = P × e
Distributed torque:    t = w × e     (per unit length)

P = concentrated load (kN)   w = distributed load (kN/m)
e = eccentricity to the shear center

Four scenarios you will actually meet

Figure 4. Common sources of torsion in steel structures. — **Figure 4.** Common sources of torsion in steel structures.

Spandrel beam: wall and cladding loads applied eccentrically on the flange.
Crane runway beam: wheel loads on the rail offset from the shear center. The usual fix is a channel cap welded to the top flange.
Cantilever with eccentric load: the worst case — the free end has no restraint at all.
Asymmetrically loaded beam: slab on one side only, or eccentric secondary-beam connections.

5. The two torsional resistance mechanisms

The applied torque is resisted by two mechanisms that are physically quite different.

T = T_sv + T_w

T_sv = pure (St. Venant) torsion     T_w = warping torsion

5.1. Pure torsion — St. Venant

The section rotates about its longitudinal axis while staying planar.
Resisted by shear stress varying linearly through the wall thickness.
Peak stress in the thickest element, directed parallel to its edge.
Dominates in closed sections. In open sections J is small, so this is usually not the governing component.

T_sv = G·J·θ'
τ_t  = G·t·θ'

G = shear modulus = 11,200 ksi = 77,200 MPa
J = St. Venant torsional constant
θ' = 1st derivative of the twist angle along z

5.2. Warping torsion

As the section twists, flanges and web tend to displace out of plane — that is warping.
If warping is restrained (fixed connections, continuity), extra stresses develop.
It produces two stress types: normal stress σ_w and shear stress τ_w.

T_w = −E·C_w·θ'''
σ_w = E·W_no·θ''
τ_w = E·S_w·θ''' / t

E   = modulus of elasticity = 29,000 ksi = 200,000 MPa
C_w = warping constant        W_no = normalized warping function
S_w = warping statical moment  t = element thickness

Figure 5. Distribution of the three torsional stress types on an I-section. — **Figure 5.** Distribution of the three torsional stress types on an I-section.

Stress type	Symbol	Location of maximum
St. Venant shear	τ_t	Thickest element, usually the flange
Warping normal	σ_w	Flange tip
Warping shear	τ_w	Flange–web junction

6. Section properties you must look up

Symbol	Name	Meaning
J	St. Venant torsional constant	Resistance to pure torsion
C_w	Warping constant	Resistance to warping torsion
W_no	Normalized warping function	Gives σ_w at a point
S_w	Warping statical moment	Gives τ_w at a point
a	Torsional property	a = √(E·Cw / G·J), in length units
e₀	Shear center eccentricity	Shear center to centroid distance

Open section, approximate:  J ≈ (1/3)·Σ b_i·t_i³
Doubly symmetric W-shape:   C_w = I_y·h₀² / 4        (h₀ = d − t_f)
Combined property:          a  = √( E·C_w / (G·J) )

What the L/a ratio tells you

Small L/a → warping torsion dominates.
Large L/a → St. Venant torsion dominates.
Transition zone is roughly L/a ≈ 1 to 5.

Source	Content
AISC Steel Construction Manual — Dimensions and Properties	J, Cw for W, M, S, HP, C, MC, WT, MT, ST, L
AISC Design Guide 9 — Appendix A	W_no, S_w, Q_f, Q_w for common sections
SCI P385 — Appendix	Torsional properties for Eurocode sections (UB, UC, PFC)

7. Boundary conditions — what really drives the answer

Torsional behaviour is extremely sensitive to end conditions. Two distinct restraints matter: torsional restraint prevents rotation about the axis (affects θ), and warping restraint prevents the flanges moving out of plane (affects θ').

Figure 6. The three typical torsional boundary conditions. — **Figure 6.** The three typical torsional boundary conditions.

Practical connection	Torsion (θ)	Warping (θ')	Model
Clip angle / fin plate	θ = 0	θ'' = 0	Pinned
Thin end plate (flush)	θ = 0	θ'' = 0	Pinned
Thick / extended end plate	θ = 0	θ' = 0	Fixed
Flange welded to column	θ = 0	θ' = 0	Fixed
Composite concrete slab	θ = 0	≈ θ' = 0	≈ Fixed (top flange)
Free cantilever tip	θ ≠ 0	θ' ≠ 0	Free

Fixed–Fixed gives the smallest stresses and twist — the most material-efficient.
Pinned–Pinned gives significantly larger values.
Fixed–Free (cantilever) gives the largest — the most unfavourable.

8. The governing equation and the 12 loading cases

Internal–external equilibrium:  T = G·J·θ' − E·C_w·θ'''

Standard form:                  θ''' − (1/a²)·θ' = −T / (E·C_w)
where                           a² = E·C_w / (G·J)

This is a 3rd-order differential equation in θ' (4th-order in θ). AISC Design Guide 9 gives closed-form solutions for 12 representative loading cases, referred to as Cases 1–12.

Case	Boundary condition	Load type	Note
1	Pinned – Pinned	Concentrated T at midspan	Most common
2	Pinned – Pinned	Concentrated T anywhere	More general than Case 1
3	Pinned – Pinned	Uniform t over full span	Spandrel with wall load
4	Pinned – Pinned	Uniform t over half span	Asymmetric loading
5	Fixed – Fixed	Concentrated T at midspan	Lower stress than Case 1
6–8	Fixed – Fixed	Other load patterns	Case 7 is best for spandrels
9	Fixed – Free	Concentrated T at free end	Typical cantilever
10–12	Fixed – Free	Other load patterns	—

How to use the solution tables

Identify the Case from the boundary conditions and load type.
Compute a = √(E·Cw / G·J) and the ratio L/a.
Read the charts or evaluate the equations for θ, θ', θ'', θ''' at the location of interest.
Compute the stresses from those derivatives.

9. When may you simplify?

When may warping torsion be neglected?

Closed sections (HSS, box): almost always — St. Venant dominates.
Open sections with very large L/a: St. Venant starts to dominate, but still worth checking.
Angles and Tees: warping stiffness is inherently small.

The reverse — neglecting St. Venant — only holds for open sections with very small L/a, and is rare in practice.

Figure 7. Decision flowchart for a torsion analysis. — **Figure 7.** Decision flowchart for a torsion analysis.

10. Part 1 summary

Section classification comes first — open or closed decides the whole method.
The shear center is the core concept — any load eccentric to it induces torsion.
Two resisting mechanisms: St. Venant (shear) and warping (normal + shear).
Warping normal stress is usually the governing component in open sections.
Boundary conditions matter enormously — determine them, do not guess.
The 12 loading cases in Design Guide 9 cover most real situations.
a = √(E·Cw / G·J) is the key parameter linking the two mechanisms.

Part of the “Industrial structural design guide” series by Roberto Structural. The content is technical guidance only; the engineer remains responsible for verifying and adapting it to the specific conditions of each project and the governing code.