The slit that costs a factor of six hundred

Nothing else in this collection behaves like torsional stiffness.

Bending stiffness rewards putting material far from the middle, and the reward is large but bounded: the same steel in an I-section instead of a flat bar is worth a factor of forty, and past that the section runs into local buckling and stops improving. Axial stiffness does not care where the material is at all. Shear stiffness cares a little.

Torsional stiffness cares about exactly one thing, and it is a topological question rather than a geometric one: does the material form a closed loop? The penalty for not doing so is not a factor of two or forty. It is $3(r/t)^2$, which for an ordinary wall is several hundred, and it is decided by a slit a millimetre wide.

Which free body produced the number, in the closed case #

Cut the tube on a plane perpendicular to its axis and take the piece beyond. The torque has to be carried across the cut face by shear stresses lying in it, and the question is what distribution of them can be in equilibrium.

Take a second free body: a small element of the wall, cut by two planes perpendicular to the axis and two cuts across the wall. Axial equilibrium of that element says that the shear flow $q = \tau t$ entering it must equal the shear flow leaving it — so $q$ is constant all the way round the loop, whatever the wall thickness does. That is the whole of Bredt’s theory and it is a statement about equilibrium alone, with no compatibility in it at all.

The torque is then the moment of that shear flow about any axis. For an element $ds$ of the wall at perpendicular distance $h$ from the chosen axis, the moment is $q \, ds \, h$, and $ds \cdot h$ is twice the area of the triangle the element subtends. Integrating right round,

$$T = 2 q A_m \qquad\Longrightarrow\qquad q = \frac{T}{2 A_m}$$

with $A_m$ the area enclosed by the wall’s midline. The corresponding torsion constant follows from the strain energy:

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

The one thing worth extracting from that is the $A_m^2$. The stiffness goes as the square of the enclosed area, and the area is what a closed section has and an open one does not.

And in the open case, where there is no loop #

An open section cannot carry a shear flow round a loop, because there is no loop for it to go round. What it does instead is much less efficient and much easier to picture.

Each plate of the section behaves as a narrow rectangle in torsion on its own. The shear stress runs one way along one face of the plate and the other way along the other face, and the pair of opposed flows forms a couple with a lever arm of about two thirds of the plate’s thickness. The resulting constant for a plate of length $b$ and thickness $t$ is $bt^3/3$, and the section’s total is the sum:

$$J_{open} = \sum \frac{b t^3}{3}$$

The arrangement has vanished. The formula contains each plate’s length and its thickness, cubed, and it does not contain where the plate is, which way it points, or what it is connected to. Roll the same plate into a channel, an angle, a tee or leave it flat, and the torsion constant is identical.

That is a genuinely strange property, and it is worth measuring against the alternatives. The four open sections in the figure have second moments of area that differ by more than a hundred to one, radii of gyration that differ by an order of magnitude, and shear centres in four completely different places — one of them outside the section altogether. Their torsion constants agree to within a factor of about one.

The two ratios, and which one bites #

For a thin circular tube the comparison is exact and reduces to nothing but the slenderness of the wall. With mean radius $r$ and thickness $t$:

$$J_{closed} = 2\pi r^3 t, \qquad J_{open} = \frac{2\pi r t^3}{3}, \qquad \frac{J_{closed}}{J_{open}} = 3\left(\frac{r}{t}\right)^2$$

and the shear stresses under the same torque are in the ratio

$$\frac{\tau_{open}}{\tau_{closed}} = 3\left(\frac{r}{t}\right)$$

At $r/t = 15$ — a perfectly ordinary wall — those are 675 and 45. The stiffness penalty is the square of the stress penalty, and that ordering decides which limit state arrives first.

An open section carrying a torque it was not designed for will nearly always twist unusably far before it shears through. A 200 mm channel carrying enough torque to raise 100 N/mm² of shear will twist through several degrees per metre; the same box section carrying the same torque twists through a few hundredths. The failure is a serviceability failure, and it looks like a door that will not close, a cladding rail that has rotated off its fixing, or a crane rail out of line — which is the same category of failure that decides most floor beams and for the same reason. The strength check is not what stops the section; the deflection is.

The half of it that is not St Venant #

Everything above is St Venant torsion: the mode in which the section is free to warp out of its own plane and does. It is the only mode a closed section really needs, because a closed section is so good at it.

An open section is so bad at it that the second mode is usually the larger half. Restrain the warping — build the end into a column, weld a plate across it, or simply have the torque vary along the length — and the section resists through a completely different mechanism: the flanges bend in their own planes, in opposite directions, and the pair of flange moments forms a bimoment. The two mechanisms share the torque, and how they share it is decided by a length:

$$\lambda = \sqrt{\frac{E I_w}{G J}}$$

For a rolled I-section of ordinary proportions that is on the order of one to two metres — several times the section depth, not a fraction of it. So a member shorter than a few metres is carrying its torque mostly by flange bending, and the $J$ this essay is about barely enters.

That is a striking exception to a rule this site has made elsewhere. Saint-Venant’s principle says that a self-equilibrating load dies away within about a depth. A bimoment is self-equilibrating — no force, no moment, nothing a resultant can see — and it travels metres. The principle is a statement about a length scale, and for an open section in torsion the length scale is not the section’s depth.

Which is why a closed section is asked for by name #

The design consequence is unusually blunt. Wherever a member has torque it cannot shed, the specification says hollow section, and the reason is the ratio at the top of this essay rather than any subtlety.

The clearest cases are the ones where the torque cannot be designed away. A curved beam turns part of its bending into twist by geometry alone, and no amount of restraint at the ends removes it. An eccentrically loaded spandrel beam carries the floor’s reaction at an offset and there is nowhere else for the moment to go. A crane runway girder takes the surge load at rail level and the section below it has to twist, and a beam braced on the wrong flange discovers the same thing at the point where its restraint was supposed to act. Each of them ends up with a torque that no diagram on this site draws. Every one of those is drawn as a hollow section or as an I-section with a closing plate welded across the bottom flanges, which is the same thing done on site.

The opposite decision is the interesting one. Where the torque can be shed, an open section is fine and the twist is the mechanism that sheds it. Compatibility torsion is exactly that: the torque exists only because the member is stiff enough to attract it, and a member with a torsion constant six hundred times smaller attracts six hundred times less. The bad property becomes the design. A slab edge beam that cracks, softens in torsion and hands its twisting moment back to the slab is doing on purpose what the ratio in this essay does by accident.

The soap film that made it visible #

The reason torsion was the last of the four internal forces to be understood is that its stress field cannot be got at by the method that settles the other three.

Bending, shear and axial force all yield to a cut and a sum. Torsion does not, because Saint-Venant’s own discovery in 1855 was that a non-circular section warps — points on the cut face move out of the plane — and the whole difficulty of the problem is finding the warping function that lets the section do so while the surface stays free of traction. For a circle it is zero and the answer is elementary; for anything else it is a boundary-value problem in two dimensions and there is no elementary answer at all.

Ludwig Prandtl noticed in 1903 that the governing equation is the same one a soap film obeys. Stretch a membrane over a hole cut in the shape of the section, blow gently on it, and the height of the film satisfies the same Poisson equation as the stress function: the slope of the film at any point is the shear stress there, and twice the volume under it is the torsion constant.

That is why every result in this essay is intuitive once it is said out loud. A film over a long thin slot is a shallow ridge with very little volume under it, so $J$ is small — and its slope across the narrow direction is steep, so the stresses are large. Doubling the plate’s length doubles the volume, but doubling its thickness increases the ridge’s height as well as its width and gives eight times the volume, which is the cube. Cut a slit in a closed shape and the film that spanned the enclosed area collapses into a pair of narrow ridges along the walls, which is the factor of hundreds.

The analogy was a laboratory instrument for decades — sections whose torsion constants were literally measured by photographing a soap film — and it survives as the reason engineers reach for the right answer about torsion before doing any arithmetic. Nothing else in this collection has an experiment that gives the answer to a problem in elasticity by blowing on something.

Where the model stops #

Three things are missing from the arithmetic, and the third is the one that catches people out on site.

The junctions. $\sum bt^3/3$ treats the section as a set of disconnected plates, and at the corners where they meet the shear flow has to turn — which stiffens the section slightly. Rolled sections carry a fillet at the junction that adds more. The usual correction is a factor $\eta$ of about 1.2 to 1.3 on the sum, tabulated per shape, and it is why a published $J$ for a rolled section is larger than the plate sum. The correction is a percentage; the ratio this essay is about is a factor of hundreds, and the two do not compete.

Thick walls. Bredt’s $q = T/2A_m$ is a thin-wall result, exact only in the limit. For a wall a tenth of the section dimension the enclosed area is not well defined to better than a few per cent, and for a solid section it is meaningless — a solid circular shaft has $J = \pi r^4/2$ from the exact elasticity solution and a solid square has $0.141 a^4$ from Saint-Venant’s own membrane analogy, neither of which is a limit of Bredt’s formula.

And a single opening destroys it entirely. This is the practical trap. A hollow section with a hole cut in one wall — for a service penetration, an access hatch, a bolt group — has, over the length of that hole, an open section. The torque arriving at the hole cannot cross it as a shear flow, so it has to convert into a bimoment, travel past the hole as flange bending, and convert back. The local stresses and the local twist are both governed by the open constant over that length, which is to say by the number six hundred times smaller. Cutting a 100 mm hole in a box girder can be a bigger structural decision than removing a metre of flange, and nothing about the arithmetic of bending would suggest it.

The generalisation, and what it says about diaphragms #

The last item is worth pulling out, because it is the assumption the whole of Bredt’s theory rests on and it is never stated with the formula.

$q = T/2A_m$ assumes the cross-section retains its shape. If the walls can distort — if the box turns into a parallelogram as the torque is applied — then the enclosed area changes, the shear flow is no longer constant, and the section has a second, much softer mode available to it. That is why a box girder has diaphragms at intervals along it, and why the spacing of those diaphragms is a real calculation rather than a detailing convention.

Read that way, the closed–open distinction is less absolute than the ratio suggests. A box with widely spaced diaphragms is partly open, in the sense that some of its torque is being carried by a mechanism that is not a shear flow round the loop. The topology holds only as long as the geometry does, and the geometry is held by something the torsion constant does not contain.

The habit worth carrying is a question rather than a number: before quoting a torsion constant, ask whether the loop is continuous over the length being checked. A section that is closed at midspan and open at a penetration is two sections, and the twist is the sum of what each of them does over its own length. That sum is dominated, always, by the open part — however short it is.