The internal force with no diagram

Cut a member and four quantities appear on the face: an axial force, a shear, a bending moment, and a torque. This collection has drawn diagrams for the first three about forty times, and has not once drawn the fourth.

That is not an oversight in the writing. It is a property of the drawings: a plane diagram has one axis to turn about, and a torque turns about the axis the drawing is looking along. So the internal force that this field is named for has, until now, been invisible — and it obeys precisely the same rules as the others.

Read that figure with the axis label covered and it is a shear diagram: constant between load points, stepping at each applied action, closing at the end. Everything the relationship between load, shear and moment established carries across whole, because the derivation used nothing but a free body and a sum.

Where a torque comes from #

A member is twisted when the line of action of its load misses a particular point in its cross-section. Not the centroid — the shear centre, which for an asymmetric section is somewhere else entirely and is frequently outside the material.

The everyday sources are all of that form: a beam supporting a slab on one side only, a crane runway with the wheel load offset from the web, an edge beam carrying a facade hung outside its face, a curved beam in plan where the load’s line of action cannot pass through any straight axis. In each case the torque is small compared with the bending, and in each case it is carried by a mechanism far weaker than bending — which is the whole difficulty.

Two mechanisms, and one of them is hundreds of times better #

A closed section resists a torque by a shear flow that circulates round the walls: constant round the perimeter, equal to $T/2A_m$ where $A_m$ is the area the walls enclose. An open section cannot do that — a shear flow going round would have to cross the slit — so it resists instead by a shear flow that runs up one face of each plate and back down the other, which is a far smaller lever arm.

That figure is the most extreme geometry-beats-material result in the collection, and it is worth comparing with the others. Rearranging the same steel changes a section’s bending stiffness by perhaps a factor of ten; a saw cut two millimetres wide, removing about a tenth of a per cent of the material — 24 mm² of 19,200 — changes its torsional stiffness by a factor of several hundred. Nothing else on this site is that sensitive to anything.

The arithmetic behind it is short. For a thin closed tube, Bredt’s formula gives

$$J = \frac{4A_m^2}{\oint ds/t}$$

which contains the enclosed area squared — so it grows as the fourth power of the size. For an open section made of thin plates,

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

which contains the thickness cubed and does not know how far apart the plates are at all. One formula is about the shape’s extent; the other is about how thick its pieces are. A closed section resists torsion with its outline and an open one resists it with its wall thickness, and no amount of making an open section deeper helps.

It is the same shear flow in both cases, and the shear nobody draws is the argument for why the flow is the right thing to think in rather than the stress. A closed tube lets that flow circulate round a loop enclosing a large area, so every millimetre of wall is working at the same lever arm from the centre of twist. An open section leaves it nowhere to go but back along the plate it came from, at a lever arm of a third of the plate’s own thickness. The two mechanisms are the same physics arranged round different topology, and topology is not a quantity that any amount of steel changes.

What holds the ends decides more than the section does #

The formula $J = \frac{1}{3}\sum bt^3$ describes uniform torsion, in which every cross-section is free to warp — to move out of its own plane, so that the flanges of an I-section slide along the member relative to each other. Prevent that warping at a support and a second mechanism appears: the flanges bend in their own planes, in opposite directions, and the pair of flange shears is itself a torque.

The decay length is $1/\lambda$ where $\lambda = \sqrt{GJ/EI_w}$, and it does the same job in this field that Saint-Venant’s principle does in the connections field: it says how far a disturbance at a boundary reaches into a member. For open sections it is a fraction of a metre, so a long member is almost entirely in uniform torsion and the restraint at its ends buys nothing.

The interesting case is a short one, where $\lambda L$ is a small number and the whole member is inside its own boundary layer:

A 1.5 m member is 57% stiffer in torsion than its torsion constant says, and a 12 m one is 5% stiffer. That is a large effect appearing and disappearing over ordinary lengths, and it is the reason torsional stiffness quoted as a section property is a partial answer: in torsion, the length is a property of the section too.

For closed sections the same calculation gives $\lambda L = 109$ for the box above, and the stiffening is 1.009. Warping restraint is a phenomenon of open sections, and the reason is the one already established — their uniform torsional stiffness is so small that anything else is comparable with it.

A torque that has to exist, and a torque that does not #

The distinction this field turns on has no analogue in bending, and it is the reason a competent designer can make a torsion problem disappear rather than solve it.

Equilibrium torsion is a torque required by statics. A cantilevered canopy hung off one side of a beam delivers a torque that nothing else can carry; take the beam’s torsional stiffness to zero and the canopy falls off. There is no design decision here — the torque is as real as any bending moment and the member must be sized for it.

Compatibility torsion exists only because two members are joined and must rotate together. A floor beam framing into the side of an edge beam wants to rotate at its end; the edge beam resists, and the torque that arises is whatever it takes to make the two rotations equal. Nothing about equilibrium requires it.

That curve is the practical content of this whole essay. Faced with a compatibility torque, three responses are available: make the member strong enough to carry it, make it stiff enough that it attracts even more of it, or make it soft and let the torque leave. The third is usually the right answer, and it is the opposite of the instinct that adding stiffness is adding safety.

The one condition is that the load must have somewhere else to go. Softening the edge beam sends the moment back into the floor beam, which now behaves as a simply supported member and needs the midspan capacity to match. That is a redistribution, and it is legitimate exactly when the receiving member has the ductility to accept it — the same argument, in a different field, with the same condition attached.

The check that governs is usually the rotation #

A member in torsion is rarely limited by its strength. The shear stresses a working torque produces in a closed section are modest — Bredt’s flow spread over the whole perimeter is an efficient way to carry anything — and what runs out first is the amount of twist the things attached to the member will tolerate.

The last line of that figure is the one to keep: no code gives a limit for the sideways movement of a flange tip. It is the one deformation in this collection that is computed only after somebody has complained about it, and the complaint usually arrives as a cladding joint that will not close.

The numbers make the case. An edge beam supporting a facade twists by $TL/GJ$, and a rotation of a hundredth of a radian at the top of a 3 m cladding panel is 30 mm of movement at its head — a serviceability failure with no stress anywhere near a limit. The same beam as a closed box twists by 1/683 of that, which is nothing at all.

This is why the answer to a torsion problem is so often a change of section rather than a change of size. Deepening an open member increases its bending stiffness and leaves its torsional stiffness essentially unchanged; welding a plate across the open face of a channel to make a box multiplies the torsional stiffness by hundreds and the bending stiffness by a few per cent. The two stiffnesses respond to completely different changes, which is unusual — for most of this collection, making a member better at one thing makes it better at most things.

The concrete member has to be hollow, whether it is or not #

Reinforced concrete answers this field in a way that makes Bredt’s formula physical rather than analytical, and the answer is worth having because it explains three detailing rules that otherwise look like folklore.

A concrete member cracks in torsion at a shear stress of a few newtons per square millimetre, and the cracks spiral round it at roughly 45 degrees — the direction of the principal tension, which for pure shear is at 45 degrees to the axis. Past that point the member is no longer a solid section carrying a circulating flow. It is a space truss: the concrete between the cracks acts as a helix of diagonal compression struts, the closed stirrups act as transverse ties holding the struts from pushing the faces apart, and longitudinal bars run the corners as chords.

The first consequence is the one that matters most on site. Only a closed stirrup works. The struts push outward all the way round, so the tie has to complete the circuit; a stirrup with an open top, or one lapped at a face rather than in a corner with a proper anchorage, unzips at its weakest point and the loop is gone. That is the same statement Bredt’s formula makes about a slit, arriving as a bar-bending schedule.

The second is that longitudinal steel is required all round the perimeter, not only at the tension face where bending would put it. The diagonal struts are inclined, so their axial component pushes along the member as well as around it, and something has to carry that push in every face. A torsion cage therefore has bars at mid-depth on the side faces that a bending design would never place, and a member designed for bending and then found to carry torsion needs steel where nothing was drawn.

The third is the striking one. Because the truss forms in the outer shell, a solid section and a hollow one of the same outline have nearly the same torsional strength, provided the wall of the hollow one is thick enough to contain the struts. The core of a solid member contributes almost nothing after cracking — it is inside the loop, where the enclosed area already counted it. That is exactly the elastic statement that $A_m$ includes the hole, surviving into a completely different material model, and it is why hollow-core bridge piers and box girders lose nothing by being empty.

Where the largest stress sits, and why it is not the corner #

The shear stress in an open section has a distribution that Coulomb’s circular-shaft solution predicts wrongly, and the error is instructive because the correct answer follows from a boundary condition rather than from any analysis.

Coulomb’s result — stress proportional to the distance from the centre — says the maximum in a rectangular bar is at the corners, which are the furthest points from the axis. The true stress at a corner is zero.

The reason is complementary shear. A shear stress on one plane is always accompanied by an equal one on the perpendicular plane, and a free surface can carry no stress across itself. At a corner the material has two free surfaces at right angles, so both components of shear must vanish there, and no distribution of anything can put a stress at a point where two free faces meet.

The maximum sits instead at the middle of the longer side, where the wall is furthest from the ends of the plate and the flow doubling back has its longest run. For a thin rectangle $b \times t$,

$$\tau_{\max} = \frac{3T}{b t^2}$$

which is the flat plate’s version of $\tau = Tt/J$ with $J = bt^3/3$. For a section built of several plates it is the thickest plate that carries the peak, because the stress goes with the local thickness — so a section with one heavy flange and a thin web is worst at the flange, which is the opposite of where a transverse shear would put it.

That last point catches people. Under a transverse shear the web carries almost everything and the flanges very little; under a torque the flanges carry the peak stress and the web very little. Two shear stresses in the same section, from two different actions, with their maxima in different plates — which is why a combined check has to be made plate by plate rather than on a single reported maximum.

The one place plane analysis is completely blind #

A plane frame analysis cannot report a torque, because in its coordinate system there is nowhere for one to be. Every commercial frame program will happily analyse a grillage of beams at right angles and report torsion in the members — but only if the model was built as a grillage, which is a decision made before any analysis ran.

This is the missing three equations at their most concrete. The moment at the end of a floor beam, in the plane analysis of that beam, is a number resisted by “the support”. In the real building it is resisted by the edge beam twisting, and the amount depends on a torsional stiffness that appears in neither model. Both analyses can be correct and the pair of them can still miss a member’s governing action, because the action lives in the joint between two models.

The history is a case of the tail wagging the dog #

Torsion was solved for a circular shaft by Coulomb in 1784, and for anything else in 1855 by Saint-Venant, whose treatment introduced warping as the thing a non-circular section does that a circular one does not. The engineering pressure came from machinery: a drive shaft’s whole purpose is to carry a torque, so the theory was developed by people for whom torsion was the design case rather than a nuisance.

Structural engineering inherited it and needed the awkward half. Machine shafts are circular and closed; building members are open and thin-walled, which is exactly where the simple theory stops working and where the warping term Vlasov added in the 1930s is required. The order of events is worth noticing: the case that is easy to derive is the one structures never use, and the theory that structures need was the correction.

The practical consequence in the standards is a marked reluctance. Design codes treat torsion with more conservatism than anything else, and the recommended approach for compatibility torsion is usually to detail for it rather than to compute it — a piece of engineering judgement that the curve above justifies exactly.

What the picture cannot show #

The whole treatment is elastic and thin-walled. Bredt’s formula assumes the shear flow is uniform through the wall thickness, which is a good approximation for a wall thinner than a tenth of the section and a poor one for a solid rectangle.

Combined actions are absent. A member carrying bending and torsion together has shear stresses from both adding on one face of a flange and subtracting on the other, and the check that matters is the combined one. The figures above draw each mechanism alone, which is what makes them legible and what makes them incomplete.

The torsional restraint at the ends is drawn as perfect or absent. Real end conditions are neither, and the torque a member attracts is decided by the same stiffness comparison as everything else here — so an assumption about the support decides the answer, and the assumption is rarely stated.

Where the ladder goes #

The first rung is the one that has appeared twice already and has its own essay waiting: what happens to a member whose load misses its shear centre by an amount nobody intended, which is a question about tolerance rather than about analysis.

The second is the failure this field produces. A beam in torsion does not usually fail by twisting; it fails because the torsion has reduced its capacity in bending, or because the twist has moved its compression flange sideways and it has buckled laterally. Torsion is more often the mechanism of some other failure than a failure of its own.

The third is the one this essay opened with. A cut reveals four internal forces and this collection has now drawn all four — which means the next question is what happens when a section carries several of them at once, and how an interaction surface is drawn in more than two dimensions.