The torque that has nowhere to go

A beam curved in plan turns bending into twist for a reason that has nothing to do with how it is loaded: the moment vector at one section is not parallel to the moment vector at the next, because the axis has turned underneath it. Every curved member has that property. What decides how much trouble it causes is where the ends are.

Two ends, and the torque splits #

The difference is not a matter of degree.

The plan says where the torque comes from and the diagrams say how large it is along the member, which is the pair of readings a design actually needs.

A supported arc has two places for a torque to leave. By symmetry each end takes half, and the two halves are opposite in sign, so the torsion diagram crosses zero at the middle. The section at mid-span carries the largest bending moment and no torsion at all, and the section carrying the largest torsion carries a smaller moment.

That is a considerable convenience and it is the reason curved beams on two supports are so often built without much fuss. The two worst actions are in different places, so the combination that governs is milder than either peak suggests.

The cantilever has one place for the torque to leave, so nothing cancels. Each element of load contributes a torque about the root proportional to its perpendicular offset from the tangent there, and the offsets all have the same sign — so the torque accumulates monotonically and arrives at the support alongside the largest bending moment.

Which free body produced the number #

Take the free body of the arc from the free end round to some angle $\phi$, cut on a radial plane, and resolve the moment of everything on it about the two axes of the cut section.

The load on an element at angle $\theta$ is $wR\,d\theta$. Its moment about the cut’s bending axis is that load times the perpendicular distance $R\sin(\phi - \theta)$, and about the cut’s torsion axis it is the load times $R(1 - \cos(\phi - \theta))$. Integrating from the free end,

$$M(\phi) = wR^2 \sin\phi \quad\text{and}\quad T(\phi) = wR^2(\phi - \sin\phi)$$

Two features of that pair are the whole essay. The torsion term is zero to first order in $\phi$ — it starts as $\phi^3/6$ — which is why a gently curved member has almost no torsion and why the effect sneaks up on people. And neither expression contains $EI$ or $GJ$: a cantilever is statically determinate, so the two actions are decided by the geometry and the load and by nothing else at all.

The second point is worth pressing. The supported arc is redundant and its torsion depends on the ratio of its two stiffnesses; the cantilever’s does not. So the one case where a designer might hope to reduce the torsion by choosing a stiffer section in bending is the case where it cannot be done.

How fast it arrives #

The ratio of the two actions is a function of the sweep angle alone.

Between a straight cantilever and a quarter-circle one, the torsion goes from nothing to 57 per cent of the bending. That is fast enough that the intermediate cases matter: a 30° curve gives 0.16, a 45° gives 0.26. A member curved by an amount an architect would describe as “slightly” is already carrying a fifth of its bending as twist, and there is no straight-beam calculation anywhere in which that quantity appears.

There is a second reading of the same pair of expressions that is worth having, because it explains why the effect is so easy to underestimate on a drawing.

Write the two as $M = wR^2\sin\phi$ and $T = wR^2(\phi - \sin\phi)$ and expand for small $\phi$: the bending starts as $wR^2\phi$ and the torsion as $wR^2\phi^3/6$. The bending grows linearly with the arc and the torsion cubically. So the ratio between them is $\phi^2/6$, which is 0.005 at 10°, 0.02 at 20°, 0.16 at 60° and 0.57 at 90°.

A cubic that starts at zero is invisible until it is not. Everything a designer’s experience contains about slightly curved members is about the flat part of that curve, and the extrapolation from “a gentle curve caused no trouble” to “a sharper one will not either” is exactly the wrong shape of reasoning to apply to it.

The support is the difficulty #

The largest torsion is at the root, and a root that can hold a torque is a specific and often expensive detail.

A straight cantilever needs a support that resists a vertical force and a moment about one axis. A curved cantilever needs the same plus a moment about the axis along the member, and the three are resisted by different things:

Two bearings a distance apart convert the torque into a couple of vertical forces, at the price of a wider support and an uplift on one of them. This is the usual answer and the uplift is the surprise — the far bearing can be in tension under nothing but gravity.

A built-in end into a wall or a core resists the torque as a moment in the wall, which is a wall bending out of plane about a horizontal axis and is usually the weakest thing available.

A ring beam or a spine takes the torque away as bending in a member running the other way, which is the arrangement every curved balcony on a building actually uses and which turns a torsion problem into somebody else’s bending problem.

None of those is free, and all three are decisions about the support rather than about the member — which is why a curved cantilever’s difficulty is frequently discovered by the person detailing the connection rather than by the person who sized the beam.

What the section has to be #

The torsion is fixed by the geometry, so the only remaining freedom is what carries it.

The ratio is 1,449 to one, and the slit section’s numbers are not a design case — they are an arithmetic demonstration that the member does not exist. One slit costs a factor of hundreds, and a curved cantilever is the clearest place in this collection where the choice between an open and a closed section is not an optimisation.

That has a consequence for what gets built. A curved steel balcony is a box or a tube; a curved concrete one is a solid or hollow section whose torsional stiffness comes for free. An I-section curved in plan is possible only when the sweep is small or when something else — a slab, a diaphragm, a second beam — is taking the torque, which is the same substitution a beam whose load misses its shear centre relies on.

Where the two cases meet #

Between the cantilever and the two-support beam there is a family, and knowing where a real member sits in it is more useful than either limit.

A curved beam on two supports with one end free to twist is a hybrid: the torque cannot leave at the free-twisting end, so all of it goes to the other one, and the member behaves like a cantilever in torsion while behaving like a simply supported beam in bending. That is the commonest real arrangement and it is the one nobody draws — a curved beam sitting on two pads, one of which is a single bearing.

A curved beam on three supports is redundant in torsion as well as in bending, and the middle support’s torsional restraint decides how much of the torque each end takes. Since a redundant structure shares load by stiffness, a torsionally soft middle support does nothing and a stiff one takes most of the torque — which means the detail that looks least structural on the drawing is deciding the largest action in the member.

The general rule is that torsion goes wherever it is restrained, and only there. Bending is distributed along a member and torsion is delivered to points, so a curved member’s torsion diagram is shaped almost entirely by which of its supports can hold a torque and which cannot. Getting that wrong on a drawing does not produce a slightly wrong answer; it produces the other structure.

The deflection nobody has a limit for #

The twist is not only a strength question, and the serviceability side of it is worse served.

A cantilever that twists 0.378° over its arc has rotated its free end by that much, which lifts one edge of whatever it carries relative to the other. On a 2 m wide balcony 0.378° is 13 mm of differential across the width — visible, and enough to pond water or to crack a screed.

There is no code limit on that rotation. Deflection limits are written for displacement and there is nothing equivalent for twist, so the check that governs the appearance of a curved cantilever is one the designer has to invent. The usual invention is to limit the differential displacement across the member as though it were a deflection, which is a reasonable thing to do and is not what any document says.

And the twist is much more sensitive to the section than the strength is. Doubling the wall thickness of the box above doubles its torsion constant and halves the twist, while the peak shear stress falls by only half as much — so a curved cantilever detailed for appearance is a thicker member than one detailed for strength, and by a factor that depends on a limit nobody wrote down.

Where the curvature stops being the point #

Two limiting cases are worth carrying, because they say when to stop worrying.

A very gentle curve has torsion in the ratio $\phi^2/6$ to its bending, so a 10° sweep gives 0.005 and a 20° gives 0.02. Below about 15° the torsion is smaller than the uncertainty in the load and the member can be designed as straight — which is the justification, usually left unstated, for treating a slightly cambered or slightly skewed beam as a straight one.

A full circle has no free end at all and is a different structure again: a ring, whose bending and torsion are both self-equilibrating and whose whole behaviour is governed by how many supports it stands on. The arc’s formulae do not approach the ring’s as the sweep tends to 360°, because the boundary conditions change discontinuously when the two ends meet.

Between them, the interesting range is 30° to 180°, which is where balconies, curved canopies, spiral stairs and highway ramps all live — and in that range the torsion is between a fifth and one and a third of the bending, always present, and never produced by anything eccentric.

What it costs, against the straight member it replaced #

It is worth putting a number on what curvature costs, because the comparison is usually made against nothing.

The quarter-circle cantilever above has an arc length of 9.42 m and a plan projection of 6 m. A straight cantilever of 6 m under the same 16 kN/m carries 288 kNm at its root and no torsion. The curved one carries 576 kNm of bending and 329 of torsion — twice the moment and a torque as well, for the same load intensity reaching the same distance from the wall.

The doubling is not curvature’s fault; it is arc length’s. The curved member is 57 per cent longer, and a cantilever’s root moment goes as the square of its length, so 1.57² = 2.47 accounts for most of it. What curvature adds on top of that is the torsion, and the torsion is the part with no straight-member equivalent.

That split is the useful way to present the cost to somebody choosing the shape. A curve buys plan geometry and pays for it twice: once in length, which is arithmetic anybody expects, and once in torsion, which is a new action needing a closed section, a wider support and a serviceability limit that does not exist. The second payment is the one that turns up late.

Where the model stops #

The torsion is taken as uniform torsion. For an open section a large part of the resistance is warping rather than St Venant, the warping is restrained at the root, and the actual twist is much smaller than the $GJ$ calculation says while the stresses are larger and in different places.

The section is assumed to be free to warp at the support. A built-in curved member is warping-restrained there, which stiffens it and puts longitudinal stresses into the flanges that no torque diagram contains.

The load is vertical and uniform. A point load, a line load along one edge, or a load applied off the shear centre each add their own torque, and they add to a torsion that is already the largest thing at the root.

Nothing here is a stability check. A curved member in bending is also a member whose compression flange is curved in plan, and its lateral-torsional behaviour is not the straight-beam case with a correction.

The arc is circular and of constant radius. A ramp on a transition spiral has a radius that changes along its length, so the torsion accumulates at a rate that varies too, and the closed-form integrals above are not available.

And the deflections are small. A member that twists appreciably has moved its own load’s line of action relative to its shear centre, so the torque grows with the twist — a second-order effect that is negligible for a box and is not obviously negligible for anything else.

What to carry away #

Count the ends before anything else. One end means the torque accumulates and arrives with the largest moment; two ends mean it splits, changes sign and vanishes where the moment is largest. Those are two structures and the second is very much the easier of them.

Torsion goes as the cube of the sweep and bending as the first power. Below about 15° the effect is inside the noise; by a quarter circle it is more than half the bending; past a half circle it is the larger of the two. Nothing in between behaves like either limit.

And the section is not a free choice. A closed section is not preferable here, it is required, and the arithmetic that says so is a factor of 1,449 rather than a factor of two. Where a section cannot stay flat the warping term rescues an open section in some places; on a member with 329 kNm of uniform torsion to carry over nine metres, it does not.

Two other essays follow the same torque to the places it eventually arrives. A torque that should not be shed is the case where a member could give it away and had better not, and a deck that spans square is the plan geometry that produces one without any curvature at all.

The ladder from here #

Later rungs on this anchor: the continuous curved beam over several supports, where the torsion at an interior support has two adjacent spans arguing about it. The curved beam with radial restraint — a ring beam held by a shell or a slab — where the argument inverts and hoop force takes over from bending. The full ring on discrete supports, which the arc’s formulae do not approach. Curved bridge decks, where the whole deck is one wide box and the analysis is a grillage rather than a member. The distortion of a box under an eccentric load, which is a fourth mode after bending, shear and torsion. Beams curved on a transition spiral, where the constant-radius result quietly stops being true. And warping restraint at the root worked properly, which is the largest single omission in everything above.

The curved cantilever is a modern problem in the sense that it is a modern shape. The mechanics were settled by the middle of the nineteenth century and the members were rare, because a curved balcony in masonry is a vault and carries its load a different way entirely. What made it common was reinforced concrete, in which a curved cantilever is no harder to make than a straight one — and the torsion, which the shape had always implied, arrived with it.