Torsion and Warping of Steel Sections (It, Iw): Where the Numbers Come From

A twisted steel member resists torque through two distinct mechanisms. St. Venant torsion (uniform torsion) develops pure shear stresses that circulate within the cross-section; its stiffness is the product G·It, where It is the torsion constant and G the shear modulus, obtained from E and Poisson's ratio in the usual way. Warping torsion (non-uniform torsion) appears when cross-sections cannot warp freely out of their plane, a phenomenon called warping or deplanation. Restrained warping generates longitudinal normal stresses, primarily in the flanges, and the associated stiffness is E·Iw, with Iw the warping constant.

Every real member carries torque through a mix of both. The split is governed by the characteristic length √(E·Iw/(G·It)): short members with warping-restrained ends push the load into warping torsion, while long members shed it into St. Venant torsion. For open sections this length can be comparable to the span itself, which is exactly why they behave so differently from hollow sections.

For a thin-walled open section, the classical result is It ≈ Σ b·t³/3 summed over the plates: the width enters linearly, but the thickness enters cubed. Prandtl's membrane analogy explains why: shear flow in an open wall must go out along one face and return along the other, so the internal lever arm is only a fraction of the wall thickness. Doubling plate width barely helps; only thickness matters, and thin walls are catastrophically inefficient.

A thin-walled closed section obeys Bredt's formula It = 4·A0²/∮(ds/t), where A0 is the area enclosed by the wall midline. Here the shear flow travels around the whole cell, and its lever arm is the dimension of the section, not the wall thickness. The enclosed area enters squared. Cut a slit in a hollow section and you switch from Bredt's formula to the strip formula: It collapses by orders of magnitude while area, Iy and Iz stay essentially unchanged. This single distinction (whether the shear flow can close a loop) dominates every torsion design decision. Compare an open I-section with an RHS of similar mass and the gap is dramatic.

For a doubly symmetric I-section, warping is easy to visualise: twisting the member makes the two flanges bend laterally in opposite directions. That is why Iw ≈ Iz·hs²/4 for such sections, with hs the distance between flange centroids: the warping constant is essentially the minor-axis flange stiffness working on the lever arm of the section depth.

This matters most in lateral-torsional buckling (LTB). The elastic critical moment Mcr of an I-beam contains both torsional contributions: a St. Venant term with G·It and a warping term with E·Iw divided by the buckling length squared. For short spans and deep slender sections the warping term dominates: the beam is stabilised mainly by its flanges resisting opposite lateral bending. For long spans the warping term fades and the modest G·It of the open section is all that remains, which is why long unrestrained IPE beams lose so much moment capacity to LTB. The table below shows how It, Iw and Iz grow through the IPE range.

Designing an open section for sustained torsion is usually a mistake. The St. Venant stiffness is tiny, so rotations become large; restrained warping then adds a bimoment whose normal stresses superimpose on ordinary bending stresses in the flanges. EN 1993-1-1 permits this (clause 6.2.7 points to elastic interaction of St. Venant shear, warping shear and warping normal stresses), but the calculation is laborious and the resulting utilisation is dominated by a load effect the section is fundamentally bad at carrying.

The professional reflex is to detail the torsion away: bring loads in through the shear centre, use pairs of beams so torque becomes a push-pull couple of bending forces, add diaphragms or end plates that convert twist into flange bending over a short length, or hang the eccentric load from a bracket that spans between two members. If a genuine, unavoidable torque remains (crane runway surge arms, façade outriggers, spiral stair stringers), switch to a hollow section and the problem largely dissolves.

The circular hollow section is the optimum torsion shape, and not by a small margin. Every part of the wall sits at the same radius, so the uniform shear flow of Bredt's model is not an approximation but the exact solution: the whole cross-section works at full efficiency, and for a circular section the torsion constant coincides with the polar moment of inertia. Just as important, a CHS does not warp (its Iw is essentially zero), so there is no warping torsion, no bimoment, and no interaction between torsion and longitudinal stresses to check. Torque produces pure, uniformly distributed shear, full stop.

SHS and RHS follow close behind: still closed cells with excellent It, but the flat walls and corners are slightly less efficient than a circle enclosing the same area, and a small warping effect exists for rectangular cells. The comparison below groups hollow sections of similar mass so the geometry, not the steel tonnage, does the talking.