The moment that will not lie flat

A cut through a beam exposes three quantities: an axial force, a shear force and a bending moment. That is the plane problem, and it is the whole of most of this subject.

A cut through a member in three dimensions exposes six — three forces and three moments, one of each along and about each axis. Two of the extra three are simply the plane case turned on its side: a second shear and a second bending moment, obeying every rule the first pair obeys. The sixth is different in kind.

Torsion is the moment about the member’s own axis, and nothing else on the list resists the way it does. A bending moment is carried by direct stress varying linearly across the section, which makes the second moment of area the governing property. A torsional moment is carried by shear stress circulating around the section — and whether it can circulate at all depends on something no other action cares about: whether the section is a closed loop.

That factor is not a detail. It is the reason a member expected to twist is specified as a hollow section, and the reason an eccentric load on an I-beam is normally resolved by adding a member to take the eccentricity out rather than by calculating the twist. Torsion is the one action in this subject that is more often designed away than designed for, and the ratio above is why: the cheapest torsional design is almost always a load path that does not produce any.

Two mechanisms, and only one of them is a loop #

A closed section carries torque by shear flow: a constant flow $q$ running all the way round the wall, at every point tangential to it. Bredt’s formula follows from equilibrium alone. Take moments of the flow about any point inside the section, and the total is $2A_m q$ where $A_m$ is the area the wall’s centreline encloses — twice the area, because a flow round a closed loop sweeps the enclosed area twice in the moment integral. So

$$q = \frac{T}{2A_m}, \qquad J = \frac{4A_m^2}{\oint \mathrm{d}s/t}$$

Both of those are properties of the hole, not of the metal. The enclosed area appears squared in the torsion constant, which is why a hollow section’s torsional stiffness is so brutally sensitive to its outside dimensions and so mildly sensitive to its wall.

Slit the tube and the loop is gone. There is no path for a circulating flow, and all that remains is each wall plate twisting through its own thickness — a shear flow that runs up one face of the plate and back down the other, cancelling at the mid-thickness. Its lever arm is the plate’s thickness rather than the section’s size, and the resulting constant is

$$J = \frac{1}{3}\sum b t^3$$

with the thickness cubed. For a wall a fortieth of the section’s width, the ratio between the two mechanisms is on the order of a thousand.

Which free body produced the number #

Take the closed 200 × 200 × 8 box. The wall centreline dimensions are 192 by 192, so the enclosed area is $A_m = 36{,}864\ \text{mm}^2$ and the centreline perimeter is 768 mm.

Bredt: $J = 4 A_m^2 / (\oint \mathrm{d}s/t) = 4 \times 36{,}864^2 / (768/8) = 5.44 \times 10^9 / 96$, which the figure prints as $5.66 \times 10^7\ \text{mm}^4$.

The open case: $J = \frac{1}{3} \times 768 \times 8^3 = 1.31 \times 10^5\ \text{mm}^4$. The ratio is 432, and the closed form for a square tube — $J_{\text{closed}}/J_{\text{open}} = 3b^2/4t^2$ — reproduces it exactly — $3 \times 192^2 / (4 \times 8^2) = 432$ — which is the independent check the generator is held to. Halve the wall and the same two formulas give 1801, because one of them contains the thickness once and the other contains it three times.

The free body in each case is a slice of the member one unit long, cut on two planes perpendicular to the axis. The torque applied to that slice must be balanced by the moment of the shear stresses on the cut face about the axis. For the closed section those stresses run round the loop and their moments add. For the open one they run up and back within each plate, and their moments nearly cancel — what survives is the small residue from the two halves of the plate being at slightly different radii, and “slightly” is the whole factor of hundreds.

The assumption the figures rest on is that the section is free to warp: that nothing at the ends prevents the cross-section from distorting out of its own plane. That assumption is what makes $J$ the only property needed, and the next section is about what happens when it fails.

Warping, which no other stress resultant does #

Under torsion, an open section does something unique among the six stress resultants: the cross-section does not stay plane. The flanges of an I-section displace along the beam’s length, one forward and one back, so a plane cut becomes a warped surface. Plane sections staying plane — the assumption the whole of bending theory rests on — is simply false here, and not as a small correction.

If the warping is free, nothing follows from it; the section warps and that is all. If it is restrained — at a fixed end, at a stiffener, at a moment connection — then the fibres that wanted to move along the beam cannot, and direct stresses appear to stop them. Those stresses are not in any of the six stress resultants. They are a seventh action, the bimoment, and warping torsion is a separate calculation added to the St Venant one.

The two mechanisms share the torque in a proportion that depends on length:

$$T = \underbrace{GJ\,\theta'}_{\text{St Venant}} + \underbrace{-EI_w\,\theta'''}_{\text{warping}}$$

The first term does not care about length; the second does, and dominates for short members. So a short cantilevered I-section in torsion is held mostly by warping restraint, and a long one mostly by the feeble St Venant mechanism — the opposite of the intuition that a short member is stiffer in every respect.

This is the same $I_w$ that appears in lateral-torsional buckling, and for the same reason: a beam buckling sideways twists as it goes, and the twist is resisted by both mechanisms at once. The two-regime shape of the buckling curve is this equation’s shape.

The flow that also does the shearing #

The shear flow in torsion is the same object as the shear flow that carries transverse shear — the same $q$, in the same wall, measured in force per unit length. The two superpose.

The interaction has a consequence that catches people out. In a closed section under combined shear and torsion, the two flows add on one wall and cancel on the opposite one, so the critical wall is not the one carrying the most shear and not the one carrying the most torsion, but the one where they happen to agree in sign.

The shear centre is where the two subjects meet. It is defined by torsion — it is the point about which the transverse shear flow has no net moment — and it is the reason ordinary beams end up twisting: the load is applied at the web, the shear centre is somewhere else, and the difference is a torque.

Where the model stops #

Thin walls. Bredt’s formula assumes the shear flow is uniform across the wall thickness, which is accurate when the wall is thin compared with the section’s size and progressively wrong as it thickens. For a solid section neither formula applies, and the classical solutions — Saint-Venant’s, obtained by solving a Poisson equation over the cross-section — are needed instead. A solid circular shaft is the one case where the elementary treatment is exact.

Multiple cells. A box girder with internal webs has more than one loop, and the flows in them are not independently determined by statics: the section is torsionally redundant, and compatibility of twist between cells has to be enforced. A three-cell box is a small simultaneous system before it is anything else.

Distortion. A thin-walled box under eccentric load does not only twist; the cross-section also distorts, the rectangle becoming a parallelogram in its own plane. Distortion is neither torsion nor bending, it has its own stresses, and it is the reason box girders carry internal diaphragms at intervals.

Uniform torque. The two-mechanism split assumes the torque is applied at the ends and constant along the member. A torque applied gradually along the length — as it is on a curved girder, or on a spandrel beam picking up a floor along one edge — makes the twist a function with its own differential equation, and the proportions carried by each mechanism vary from point to point.

Plasticity. Everything above is elastic. The plastic torsional capacity of a closed section is reached when the shear flow yields all the way round, which is a clean sand-heap analogy and a much larger capacity than first yield suggests.

The figures cannot show the deformation they are about. Twist is a rotation per unit length along an axis pointing out of the page, and the drawing is of the cross-section, so what the reader sees is a shape with arrows on it and a number in words beneath. The one honest way to convey the difference is the number itself: at 5 kNm over 3 metres the closed section here twists by 0.19 degrees, which nobody would see, and the slit one by 81 degrees, which is not a structural member any more. The peak shear stress tells the same story from the other side — 8.5 N/mm² against 305, from the identical torque on the identical steel.

The section that is closed and the section that is nearly closed #

Between the box and the slit box there is a case that arises constantly and behaves like neither: the section closed by something that is not the parent material — a cover plate on bolts, a concrete slab on shear studs, a lipped channel whose lips nearly touch.

The rule is unforgiving. A loop is a loop only if it can carry the flow all the way round, so the connection closing it must carry the full $q = T/2A_m$ along its entire length. Get that right and the section has the closed constant; get it wrong anywhere and it has the open one. There is no intermediate: the flow either completes the circuit or it does not.

This is the same requirement that turns two plates into one built-up section, and it is the more severe version of it, because a bending connection that is locally weak sheds a little load to its neighbours while a torsional loop that is locally broken is simply not a loop.

That last point is worth dwelling on, because it explains why torsion is under-taught rather than merely difficult. Every drawing convention in statics is two-dimensional. Shear and bending have natural pictures; torsion has a curly arrow that means “into the page and round”, which is a symbol rather than a depiction. A subject drawn on paper will tend to neglect the action that paper cannot show.

The generalisation #

The pattern that makes torsion behave unlike its five siblings is worth stating on its own, because it recurs whenever a structural action is resisted by a circulation rather than by a distribution.

Bending, shear and axial force are all resisted by stresses that integrate across the section; their capacity is an integral of material times lever arm, and adding material anywhere helps a little. Torsion in a closed section is resisted by a loop, and a loop is a topological object. Its capacity depends on the area the loop encloses, which is not a property of any material at all — a fact made vivid by the observation that the enclosed area in Bredt’s formula includes the hole, where there is nothing.

The consequence is that torsional capacity can be destroyed without removing material. A slit of zero width, taking away no steel, divides the constant by hundreds. Nothing else in structural engineering has that property: there is no cut that halves a section’s bending capacity without removing anything.

Coulomb solved the circular shaft in 1784, and the result is so clean — plane sections stay plane, stress proportional to radius — that it was assumed to generalise. It does not: applied to a rectangular bar it predicts the maximum stress at the corners, where the true stress is zero, since a free surface cannot carry shear across itself. Saint-Venant sorted it out in 1855 with the warping function, and Prandtl gave the memorable version in 1903 with the membrane analogy — stretch a soap film over a hole shaped like the section, blow gently, and the volume under the film is proportional to the torsion constant while the slope is the shear stress. The analogy makes the whole of this essay obvious in one image: a film over a long thin slot bulges hardly at all, and a film over a closed annulus is a different problem entirely.

The ladder from here #

Later rungs on this anchor: warping torsion in full, with the bimoment as a stress resultant in its own right. The multi-cell box and its torsional redundancy. Distortion and the diaphragm. Combined bending, shear and torsion, and the interaction surface. Plastic torsion and the sand-heap analogy. And the curved girder, in which bending and torsion are inseparable because the geometry couples them — the case where torsion stops being an accident to be avoided and becomes part of the primary load path.