Torsion on T, L and I sections

What this page computes

Torsion on T, L and I sections to EN 1992-1-1 §6.3.1(3)–(5): decomposition into rectangles, distribution of the moment among them, and the §6.3.2 check on each.

The code says three things, in three sentences:

• (3) complex sections may be divided into elementary sections, each modelled as an equivalent thin-walled tube, and the resistance of the whole is the sum of the elementary resistances;
• (4) the moment is distributed in proportion to the torsional stiffness of the uncracked section;
• (5) each elementary section is calculated separately.

So it is not a new calculation. It is a distribution, followed by the usual calculation on each rectangle.

Why the web takes far more than it appears to

The distribution is by stiffness, not by area. And the torsion constant of a rectangle grows with the cube of the short side:

\(J = \beta\, b\, h^3\)

The consequence is harsher than intuition suggests. On an ordinary T — web 300×700, flange 900×150 — the web takes 91% of the moment, though by area it would take only 70%. The two outstands, despite together being as wide as the web, take 5% each.

From this follows something useful in design: thickening the flange shifts the moment towards it appreciably. Not widening — thickening. Ten extra centimetres of thickness change the distribution far more than half a metre of extra width.

Two things the code does not give

The stiffness of a rectangle

EC2 requires "the torsional stiffness of the uncracked section" but gives no formula. The page uses the St. Venant constant:

\(\beta = \frac{1}{3} - 0.21\,\frac{h}{b}\left(1 - \frac{h^4}{12b^4}\right), \qquad b \ge h\)

This is classical mechanics, not Eurocode — and it is flagged as such on every run. But it is checkable at both known ends: for a square it gives 0.1408 against 0.1406 tabulated, and for a thin strip it tends to 1/3. Both have tests, and the asymptotic one checks the law of approach — the deviation must be exactly \(0.21\,h/b\) — not merely a threshold.

Where the section is cut

The code says complex sections "may be divided", without prescribing where. Two conventions are common:

• the full web over the whole depth, with only the side outstands left from the flanges;
• the full flange over the whole width, with only the portion below it left as web.

The choice is not neutral: whichever puts more material into the wide rectangle gains stiffness. The web's share differs by more than five percentage points between the two, for the same section.

The page offers both. If you have no reason to prefer one, run both and take the more unfavourable case.

Reinforcement goes on each rectangle

§6.3.1(5) is easy to read past: each elementary section is calculated separately. That also means closed stirrups around each rectangle, not a single stirrup wrapping the outer contour of the T.

It is a detailing consequence that often gets missed, because the reinforcement drawing looks "natural" with one large stirrup on the contour — but that one does not close the shear flow of each equivalent tube.

What it does not cover

• Warping torsion (§6.3.3). The code states that for closed thin-walled and for solid sections it may usually be neglected, but for open thin-walled sections it may be necessary — and for very slender members the calculation moves to a grillage model. Not here.
• Interaction with bending in the same expression — the page gives the torsion–shear interaction, (6.29), on each sub-section.
• Detailing rules of §9.2.3, beyond the remark above.
• Hollow box sections, where §6.3.1(4) requires the fictitious wall thickness to be capped at the actual one.

The texts of §6.3.1(3)–(5) are transcribed from EN 1992-1-1:2004, page 87.

Related calculations

• Torsion — the rectangular section, with the full §6.3.2 calculation this page builds on.
• Shear between web and flange — the other check specific to T sections.
• RC section — \(M_{Rd}\) and \(V_{Rd}\) for the same beam.