Torsion — where it appears
Named by 31 essays across 6 fields — each of them below, with the objects they name alongside it.
What a cut reveals, and why it was there all along
Cut a beam anywhere and two quantities appear on the face — a shear force and a bending moment. Nothing was applied there. They are what the material was already doing.
The moment that will not lie flat
A plane cut exposes three actions. A real cut exposes six, and the fourth of them behaves unlike the others — torsion is resisted by a loop of shear, and one slit down the length of a tube destroys it.
The same steel in a different shape, and a factor of forty
Four sections of identical area, identical weight and identical cost. The stiffest is dozens of times the stiffest of the flattest, and the only thing that changed was the arrangement.
The point that is not in the section
A channel loaded down its web twists. To stop it, the load must be applied through a point outside the steel entirely — in the air beside the section, where nothing can be attached.
The beam that fails sideways
A deep narrow beam bending in its strong plane can, at a moment well below its capacity, swing out of that plane and twist. The failure has nothing to do with how much it can carry and everything to do with what is holding it.
The bolt that carries more than its share
Six bolts, one hundred kilonewtons, and a worst bolt carrying fifty. The load is shared equally and the torque is not, and the second one is invisible on any drawing where the connection is a point.
The corner that is not the worst point
Check the point furthest from the centroid. It is the standard rule for a weld group under an eccentric load, it is exactly right for some shapes, and for others it misses the peak by sixteen per cent — or picks one of four points it cannot tell apart whose stresses differ by two thirds.
The internal force with no diagram
A cut through a member reveals four things, and this collection has drawn diagrams for three of them. The fourth is a torque, it obeys exactly the same rules, and whether it exists at all can depend on a decision the designer is free to make.
The section that cannot stay flat
Twist an I-section and its cross-section dishes out of its own plane. Stop that happening at one end and the member finds a second way to resist — the flanges bend in opposite directions — and the stress resultant that describes it has units nothing else in statics has.
The column that twists instead of bending
Euler's column has one mode. A real column has three, and which of them governs is settled by where the shear centre sits. A cruciform strut buckles by rotating about its own length at a load that does not change no matter how short it is made.
Bending that arrives as twist
A straight beam under a vertical load carries no torsion unless something applies one. A beam whose axis curves on plan carries torsion everywhere, from the same load, with nothing applied off the axis — and it cannot be simply supported at all.
The torsion that goes away if you let it
A spandrel beam attracts a torque in proportion to its own torsional stiffness. Crack it and the stiffness falls by a factor of four, the torque falls with it, and nothing has failed — because the floor beam it was competing with picks up exactly what was shed. A canopy hung off the same spandrel is a different animal entirely.
The section that will not keep its shape
A box girder is closed, so torsion costs it almost nothing. What an eccentric load actually does to it is something a torsion calculation contains no term for — the rectangle becomes a parallelogram, in its own plane, along the whole length of the span.
Two moments and a neutral axis that obeys neither
Tilt the load on a rolled beam by five degrees and the neutral axis swings by seventy. The section is doubly symmetric, its product of inertia is exactly zero, and none of that helps — because what decides the axis is the moment ratio multiplied by a stiffness ratio of thirty.
The slit that costs a factor of six hundred
Bending stiffness cares where the material is, and changes by a factor of two or three between sensible sections of the same area. Torsional stiffness cares whether the material forms a closed loop, and the penalty for not doing so is an order of magnitude squared.
A third of the load crosses sideways
A slab spanning both ways is usually explained as two beams sharing a load by a fourth power, and the explanation is not merely approximate — it is missing a mechanism. A real plate carries load three ways, and the third one has no beam strip in it: it is twisting, it accounts for a third of the load on a square panel, and it is why the corners lift.
Two volumes, and both of them are torques
Torsion of a solid section that is not a circle has no elementary answer, and for thirty years after Saint-Venant posed it the only way to get one was to blow a soap film over a hole cut in a plate and measure it. The film is not an illustration of the solution. It is the solution, and so is a heap of sand poured on the same hole.
Two cells, one equation, and a web with nothing in it
Bredt's formula answers a single closed cell because a single closed cell has one unknown and one equation. Put a web down the middle and there are two unknowns and still one equation — and the answer, when the missing statement is supplied, is that the new web carries exactly nothing.
The buckling load with no compression in it
Twist a straight bar hard enough and it snaps into a helix, with no axial load on it anywhere. The load at which that happens is 2πEI/L — first power of the length, and no shear modulus in it at all, so how stiff the bar is in torsion has nothing to do with the torque that buckles it in torsion.
The movement with no limit against it
Every code in the world gives a deflection limit. None gives a twist limit — and a beam loaded off its shear centre twists. On an open section a modest eccentricity moves the flange tip further sideways than four fifths of the sag that does get checked, and nothing anywhere says whether that is acceptable.
The deck that spans square
A slab bridge crossing a road at an angle is loaded uniformly and does not carry uniformly. Load takes the shortest route between the abutments, which is not the direction the carriageway runs, and the reaction piles up in two corners.
The shape of the diagram, and not its peak
A beam's lateral-torsional capacity is quoted against uniform moment, which is the one case a beam carrying a load never has. Change the shape of the moment diagram without changing its peak and the buckling moment moves by a factor of nearly three.
The torque that has nowhere to go
A curved beam on two supports splits its torsion between them, and the two halves cancel at mid-span. A curved cantilever has one end, so every increment of torque accumulates toward it — and the largest action at the root of a curved balcony is one that a straight beam does not have at all.
The eccentricity a purlin cannot avoid
A channel's shear centre is outside the material, so a load applied anywhere on the section misses it. The distance is fixed by the proportions rather than by the detailing, it is 38 mm on an ordinary purlin, and the torque it produces is not an error anybody made.
The restraint that beats the gradient
A moment-gradient factor is worth up to 2.7 on a beam's critical moment and is tabulated everywhere. Holding the ends against warping is worth more, is achieved by a detail rather than by a load case, and appears in no table at all.
The radius rule, and where it fails
A weld group under an eccentric load is checked at the point furthest from its centroid, on the reasoning that the stress from the twist grows with the radius. That reasoning ignores the direction the two stresses point in, and for one common shape it misses the peak by nine per cent.
The torque that should not be shed
A compatibility torque can be let go, because the load has somewhere else to go. What the rule does not say is what it costs the somewhere else — a twenty per cent rise in a floor beam's midspan moment, a crack width nobody limits, and a rotation the spandrel has to actually deliver. There is a size of torque past which shedding is the wrong answer, and no code states it.
One diaphragm is nearly none
A box girder's distortion decays over a length the section decides, and on a sixty-metre span that length is twenty-four metres. So a single diaphragm at midspan sits further from each end than the distortion can reach and removes a third of the problem; three diaphragms remove nine tenths. The spacing rule is not span over five — it is a property of the plates.
Three actions on one web
A load applied off the centreline of a box girder is three actions at once, and every textbook decomposes it into them. What the decomposition does not say is that all three land on the same piece of plate — added on one web and subtracted on the other, so the two walls of one section differ by a factor of ten.
The twist the combination rule invents
Two modes close together respond together, and the square root of the sum of squares assumes they do not. The error has the sign of the two modal contributions: where they agree, as they do in base shear, the rule comes up short, and where they oppose, as they always do in torque, it comes up long — by a factor of five for a floor whose stiffness sits twenty centimetres off its mass, a torque the building does not have.
The bracket pushed from the wrong side
A six-bolt bracket checked for a load straight down carries 198.5 kN. Push the same load through the same point at 138 degrees and it carries 135.8 — the direction changes the torque as well as the shear, and the worst direction is one no drawing shows. The capacity of a group whose load can turn is a closed curve, the bolt that governs it changes as the load turns, and what the curve rewards is not a larger polar moment but a smaller distance to the furthest bolt.
Named alongside it
The objects these essays reach for when they reach for this one.
WarpingShear flowEccentricityShear centreFree bodyLoad pathServiceabilityStiffnessTorsional constantOpen sectionDecay lengthDiaphragm