Bending that arrives as twist

Nothing has to be applied eccentrically for a beam to twist. The axis only has to turn.

Take a beam whose centreline follows a circle in plan — a curved balcony, an approach ramp, a bridge deck round a bend, a ring beam under a dome — and load it with its own weight, straight down, applied on the axis. At one section the internal moment vector points one way; at the next section, a little further round, the axis has rotated underneath it and the same moment vector is no longer aligned with the section. Its component along the axis is a torque, and it was not applied by anything. A moment has to be told which point and which axis it is about before it means anything, and this is the case where the axis moves out from under it.

The straight beam, for comparison #

A straight simply supported beam under a uniform load of 20 kN/m over 12.566 m has a mid-span moment of $wL^2/8 = 394.8$ kNm and no torsion at all. Bend the same beam into a 60° arc of 12 m radius, keep every other number, and the mid-span moment becomes 445.5 kNm — 12.9% larger — while a torsion of 154.8 kNm appears at each support, which is 35% of the peak bending.

Both numbers grew out of the geometry. The load did not move; the developed length did not change; nothing was applied off the axis. The moment is larger because the load is now further from the chord between the supports, and the torsion is there because a moment vector that stays horizontal cannot stay perpendicular to a section whose normal is turning.

It cannot be simply supported #

This is the fact that decides how such a beam is built, and it comes before any calculation.

Consider the free body of the whole arc, and take moments about the chord joining the two supports. Two vertical reactions sit on that chord, so neither has a lever arm about it. The load does: the arc bulges away from the chord, and every element of load has an offset. So the sum of moments about the chord is not zero and cannot be made zero.

A straight beam escapes this because its axis is the chord and every load offset is zero. A curved one does not. This is the three-dimensional equilibrium the drawing only shows three of doing its work: the two equations a plan view hides are precisely the two that matter here.

A beam curved on plan on two simple supports is a mechanism — it rolls off them, turning about the line joining them — and the drawing of it looks entirely reasonable.

So a curved beam’s supports must hold a torque. That gives it four restraint components — a vertical reaction and a torque at each end — against the three equations of out-of-plane equilibrium, which makes the simplest possible curved beam statically indeterminate. There is no determinate curved-beam-on-two-supports to start from, and that is unusual: nearly every other structure in this collection has a determinate ancestor.

The one arrangement that is determinate #

The exception is a cantilever, and it is worth having because it has closed forms.

For a circular cantilever of radius $R$ under a uniform load $w$ per unit length, at an angle $\theta$ from the free end,

$$M = wR^2(1 - \cos\theta), \qquad T = wR^2(\theta - \sin\theta)$$

Both are worth reading rather than merely quoting. The bending expression starts at zero and grows like $\theta^2/2$ for small angles, which is $wL^2/2$ with $L = R\theta$ — the straight cantilever’s fixing moment, recovered exactly. The torsion starts like $\theta^3/6$, so it is third order in the angle and vanishes much faster than the bending as the beam straightens. That is the analytic form of the observation that a nearly straight beam has nearly no torsion.

At 60° and the numbers above, the fixing moment is 1,440 kNm and the fixing torque 521.8 kNm, which is 36% of it.

The generator behind these figures does not use those closed forms. It integrates the moment of every element of load about every section and resolves the result onto the section’s own tangent and radius, which is the free-body argument this collection is built on applied section by section, and would work for any curve at all. The closed forms are then the check: over the whole arc the largest disagreement is 2.5 parts in ten million of $wR^2$, which is the numerical integration and not the physics.

Which free body produced the number, for the supported case #

Return to the redundant one. The primary structure is made by releasing the torque at one support, leaving a vertical reaction at each end and a torque at one — three restraints, three equations, determinate.

And its two vertical reactions are not equal. This is the trap in the problem and it is worth stating plainly, because symmetry is exactly the assumption an experienced reader reaches for. The retained torque acts about the tangent at its own support, which is not parallel to the chord, so it has a component across the chord; the two vertical reactions must differ to balance that component. Assuming $wL/2$ at each end produces a left-hand free body that does not agree with the right-hand one, and every internal force after that is a plausible answer to a structure not in equilibrium.

With the reactions solved properly, the released torque $T_0$ follows from compatibility — the rotation about the tangent at the released support has to be zero:

$$\int \frac{M\,m_1}{EI}\,ds + \int \frac{T\,t_1}{GJ}\,ds = 0$$

which is the unit-load method with two flexibilities in it instead of one, because a curved beam stores strain energy in twisting as well as in bending.

The self-stress is pure torsion, and everything follows #

Solve that equation and something unexpected falls out. The unit-load state — a unit torque applied at the released support and carried by the primary structure — has internal forces of exactly zero bending and constant torsion, at every station, for any radius and any subtended angle.

It is not a coincidence and it can be seen directly. Apply $T_0 = -1$ at one support and a vertical reaction pair $\pm 1/R$ at the two ends. The moment vector at angle $\theta$ is then

$$\mathbf{M}(\theta) = -\mathbf{t}(0) + \tfrac{1}{R}\big(R(1-\cos\theta),\; R\sin\theta\big) = -\mathbf{t}(\theta)$$

— a vector of unit length that rotates with the section, keeping its whole magnitude in the tangential direction and none in the radial one. The vertical reaction pair supplies exactly the moment needed to turn the constant applied torque round the arc.

Three consequences, and each is worth more than the algebra that produced it.

The stiffnesses cancel. With $m_1 = 0$ everywhere, the compatibility equation loses its bending term entirely and becomes $\int T t_1 \,ds / GJ = 0$, from which $GJ$ divides out. The redundant is $T_0 = -\overline{T_p}$, the negative of the mean of the released structure’s torsion — a statement with no material property in it at all.

The bending moments are statically determinate. Since the redundant contributes no bending, $M = M_p$ exactly: the moments in this redundant structure are the released structure’s, unchanged. The same curved beam in a thin-walled open section and in a closed box carries identical bending moments, which is not true of any other redundant structure in this collection.

The torsion diagram has zero mean. $T = T_p - \overline{T_p}$ by construction, so the area under the torque diagram round the arc is exactly zero — and since the arrangement is symmetric, the torsion is antisymmetric and passes through zero at mid-span.

How fast the torsion arrives #

The curve rises steeply and never flattens. A beam that turns through 15° — a shallow bend on a road, the sort of curvature nobody would call curved — already has a torsion nine per cent of its bending. At 90° it is 52%, and the mid-span moment is 34% above the straight beam’s. At 150° the moment is 3.3 times the straight beam’s and the torsion 85% of it.

A half circle is where it ends, and not gradually. For a semicircular beam the chord is a diameter, both supports sit on it, and both end tangents are perpendicular to it — so neither torque has any component about the chord, neither reaction has a lever arm about it, and the load’s moment about it is balanced by nothing. The structure is a mechanism again, this time in the arrangement that was supposed to have cured the first one. A ring beam under a dome escapes only because it is supported continuously rather than at two points.

The section has to be chosen for both #

Bending and torsion peak in different places, which sounds convenient and is not: the section is continuous, and it has to carry the largest of each somewhere along its length.

This is the practical content of the whole subject. Closing a section multiplies its torsional stiffness by a factor of hundreds, and a curved beam is the one member type where that factor is not a refinement. It is why curved bridge girders are boxes, why a curved balcony edge beam is usually a rectangular section rather than a rolled I, and why a curved steel beam made from a plate girder needs its own torsional design rather than a bending one with a note attached.

An open section curved on plan does not fail immediately, because the torque is small in absolute terms and an open section has a second mechanism to fall back on — the flanges bending in opposite directions.

Where the deflection is not determinate at all #

The forces contain neither stiffness. The deflection contains both, and the split between them is the thing a section choice actually buys.

A curved beam of open section is therefore a member whose stresses are respectable and whose deflection is not. Every strength check passes with the same numbers a closed section would give, and the beam drops four times as far. The failure mode this produces on site is a floor that is level on paper and visibly sagging along a curve, and the calculation that missed it is a bending calculation that never asked about $GJ$. It belongs with the other deflections that are not bending — real, unbounded by any strength check, and invisible to the one that was made.

Where the model stops #

Everything above needs a constant radius. The self-stress being pure torsion is a property of the circle: it is the vertical reaction pair’s moment turning at exactly the same rate as the section. A beam curved on a transition spiral, or on a compound curve, has a self-stress state with bending in it, and then the bending moments do depend on $GJ/EI$ like any other redundant structure.

Nothing here is a curved beam in its own plane. A beam bent in elevation is an arch and belongs to a different argument entirely — its curvature converts bending into axial force, which is the opposite trade. Curvature on plan converts bending into torsion, which buys nothing.

The section is assumed to twist about its shear centre. For an open section curved on plan the load is applied on the axis, the shear centre is somewhere else, and the eccentricity between them adds a torque that this calculation has not included. The correction is small compared with the curvature term and it is not zero.

The supports are assumed rigid in torsion. A real bearing that holds a torque does so by being a pair of bearings some distance apart, and the pair has a rotational flexibility. Softening it lets the beam relax its end torque — and since the end torque is exactly the mean of the released torsion, a support that yields in torsion changes the whole diagram rather than a corner of it.

And the whole treatment is first-order and elastic. A curved beam that yields does so under a combination of bending and torsion, and the interaction between them is not the one this page has drawn.

What the pictures cannot show #

The plan figures draw bending and torsion as offsets from the axis, in two colours, on one scale. That is a convenient lie: they are components of one vector quantity resolved onto two axes that turn, and there is no direction in the drawing that either offset really points along.

The deflection figure plots a ratio against a stiffness ratio, and both of its ends are unphysical. A section with $GJ/EI = 0.02$ exists — a thin open channel is close to it — but such a beam would not be built curved, so the left of that curve is a limit rather than a design.

And the figures of the cantilever draw a member fixed at one end by a hatched wall. Nothing supplies a perfect torsional fixity, and the difference between a good one and a poor one moves the free end’s deflection much more than it moves any of the forces.

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. 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 and is what actually governs a thin-walled curved deck. Beams curved on a transition spiral, where the constant-radius result above quietly stops being true. And the reverse problem: a straight beam supported on a curved line of bearings, which has the same coupling from the other direction.