The restraint that beats the gradient

The shape of the moment diagram is worth a factor of up to 2.7 on a beam’s lateral-torsional capacity, and it is tabulated in every code. There is a second factor of similar size that is not tabulated anywhere, is decided by a detail rather than by a load case, and is usually thrown away without anybody noticing it was available.

Two mechanisms, and one of them needs an end #

An open section resists twist two ways and they are not alternatives.

Saint-Venant torsion is a shear flow circulating through the thickness of each plate. It needs no boundary condition at all: a section free at both ends resists twist this way, and the resistance per unit length is $GJ$ times the rate of twist.

Warping torsion is the two flanges bending in opposite directions in their own planes. It requires the section to be prevented from warping somewhere, because if every section is free to warp the flanges do not bend at all.

So the second mechanism is not a property of the member; it is a property of the member plus its ends. A beam held against warping at one end has it and the same beam free at both ends does not.

Which free body produced the number #

The free body is a slice of the member, cut on two sections a small distance apart.

Crossing each cut is a torque, and the torque is the sum of two things: the moment of the circulating shear flow about the shear centre, and the moment of the two flange shears about the same point. Equilibrium of the slice says those two add to the applied torque at every section, which is what the first figure asserts and checks.

Their split between the two is not decided by equilibrium at all. It is decided by compatibility — the flanges’ bending and the section’s twist are the same displacement seen two ways — and the resulting differential equation has $\kappa = \sqrt{GJ/EI_w}$ in it, whose reciprocal is a decay length.

Warping torsion is a boundary-layer effect. It is everything at a held section and nothing several decay lengths away, and the length over which it fades is a property of the section rather than of the member.

How much it is worth #

The whole benefit is one bracket, and it is a function of $\kappa L$.

Short members get everything and long ones get nothing, and the boundary between them is $\kappa L$ of order one. For a rolled I-section the decay length is a metre or two, so a segment between restraints of 3 m is transformed and one of 12 m is barely touched.

That is the same shape as a moment gradient factor and it acts on the same quantity, and the two are multiplicative rather than alternatives.

Why the tee and the angle have none of it #

The sweep also carries a comparison that decides which sections this argument applies to.

The same three plates arranged as an I-section, as a tee and as an angle give warping constants in the ratio 1 : 1/288 : 1/236. A tee and an angle have essentially no warping resistance to offer, and the reason is geometric: their plates meet at a point, so there is no pair of flanges at a distance apart to bend against each other.

Their $\kappa L$ is therefore very large — 37 and 34 against the I-section’s 2.76 — and their stiffening is about 3 per cent. An angle in torsion has $GJ$ and nothing else, which is why an angle is such a poor thing to twist and why the earlier essay’s factor of 432 between an open and a closed section is a lower bound for these shapes rather than an upper one.

What “restrained against warping” means on a drawing #

Since the whole benefit rests on a boundary condition, it is worth being concrete about which details deliver it and which do not.

A full-depth end plate welded to both flanges does. The plate is stiff in its own plane, the two flanges cannot rotate in opposite directions in plan without bending it about its strong axis, and the restraint is very nearly complete.

A pair of full-depth stiffeners on either side of a support does, for the same reason: they tie the two flanges together in a plane that resists the differential rotation.

A web-cleat or a fin plate does not. Both connect the web only. The flanges are free to rotate in plan independently, which is precisely the deformation warping is, and the restraint is essentially zero.

And a beam continuous over a support restrains itself, provided the adjacent segment is long enough to develop the boundary layer — which for a decay length of 2 m means about 4 m of beam beyond the support. A short back-span does not deliver it.

That list explains the near-universal $k_w = 1.0$. Two of the four commonest beam connections supply no warping restraint at all, and the two that do are moment connections designed for something else — so a designer claiming the benefit is usually claiming it from a detail chosen for a different reason, and would have to check that it is stiff enough for this one.

The stress nobody’s diagram carries #

Warping torsion produces a longitudinal stress, and it is the reason the mechanism cannot be treated as a free improvement.

The bimoment is a genuinely odd object. It is four equal flange-tip forces, two positive and two negative, arranged so that their resultant force is zero, their resultant moment about every axis is zero and their resultant torque is zero. It is invisible to every equilibrium check that has ever been written, and it puts 53.8 N/mm² into the flange corner where the bending stress is already highest.

So a warping-restrained end is not free. It buys stiffness against twist and pays in a longitudinal stress at exactly the fibre that governs bending — and the payment lands at the support, where the moment on a continuous beam is also largest.

How the two mechanisms divide a torque #

The first figure shows the split changing along the member, and the arithmetic behind it is worth having because it says which mechanism a designer is actually relying on.

The governing equation is $EI_w\phi''' - GJ\phi' = -T$, whose solution for a member built in at one end and free at the other contains $\cosh$ and $\sinh$ of $\kappa x$. The Saint-Venant share at a distance $x$ from the held end is $1 - \cosh\kappa(L-x)/\cosh\kappa L$, which is zero at $x = 0$ and approaches one for large $\kappa x$.

Three readings of that expression are useful.

At the held end the split is exactly 0 : 100. Not approximately — the boundary condition is $\phi' = 0$, and the Saint-Venant torque is $GJ\phi'$, so it vanishes identically. Everything is carried by the flanges.

One decay length along, the split is about 37 : 63, which is $1/e$ of the way. That is the number the first figure prints, and it is the same $1/e$ that appears in every boundary-layer problem.

And several decay lengths along, the split is 100 : 0. The flanges have stopped bending, the member has settled into uniform torsion, and the end might as well not be there.

So the mechanism a member is using depends on where along it the question is asked, which is unlike almost every other resistance in this collection. A section does not have a torsional resistance; a section at a position in a member with particular ends has one.

Two ends, and the shape changes #

Holding both ends rather than one changes the distribution as well as the amount.

That is worth having because it says where the bimoment lands. A member held at both ends has two smaller bimoments rather than one large one, so the flange-tip stress at each end is roughly halved — which makes the double-restrained case better on both counts and is the arrangement worth detailing for.

What it is worth on the buckling curve #

The benefit reaches lateral-torsional buckling because the critical moment contains both torsional stiffnesses.

$M_{cr}$ contains $\sqrt{EI_w \cdot \pi^2/L^2 + GJ}$, so the warping term is divided by the square of the segment length and the Saint-Venant term is not. The short segments where the warping benefit is largest are the ones where warping already dominates the critical moment, so preventing warping at the segment ends helps most in exactly the range codes design in.

The design device is an effective length factor for warping, $k_w$, alongside the one for lateral bending. It is conventionally taken as 1.0 — that is, warping is assumed free — and taking it as 0.7 is worth roughly the same as a moment-gradient factor of 1.5.

Why the benefit is usually thrown away #

If it is worth so much and costs a detail, the obvious question is why $k_w = 1.0$ is the near-universal assumption.

Because the detail has to be real. Preventing warping means preventing the two flanges rotating in opposite directions in plan, which needs something attached to both flanges that is stiff in its own plane — a full-depth end plate, a pair of stiffeners, a torsion box. A simple web-cleat connection prevents none of it.

Because it is hard to verify. Lateral restraint is visible on a drawing; warping restraint is a statement about a connection’s in-plane stiffness that nobody computes.

And because the penalty for being wrong is unbounded. A beam designed with $k_w = 0.7$ and built with a connection that does not deliver it has a critical moment well below what the check assumed, in a failure mode with no warning in it.

So $k_w = 1.0$ is a conservatism with a known cost, which is the right decision when the alternative is a benefit with an unknown one — and it is worth knowing what is being left on the table, because on a short heavily loaded segment it is a great deal.

Where the same argument reappears #

The decay length is the useful export from this page, and it turns up wherever a boundary condition has a reach.

A beam on an elastic foundation forgets its own length past three or four characteristic lengths, and the mechanism is identical: a fourth-order equation with a length in it, an effect that is everything at a disturbance and nothing far from it. A shell’s edge disturbance is the same equation again.

What warping adds to that family is a case where the disturbance is wanted. In the foundation and the shell the boundary layer is a nuisance to be checked; here it is a stiffness worth having, and the design question is how to make the boundary layer as long as possible relative to the member — which means a small $\kappa$, which means a large $I_w$ relative to $J$.

That is a section-shape argument rather than a length one. A deep I-section has a large warping constant and a small torsion constant, so its $\kappa$ is small and its boundary layer long; a hollow section has the opposite and needs none of this. Which is a fair summary of why the whole subject belongs to open sections: a closed section carries torque by circulating shear and never needs the flanges to bend at all.

What to carry away #

Warping resistance requires an end and dies within a decay length of it. $\kappa = \sqrt{GJ/EI_w}$, its reciprocal is the reach, and $\kappa L$ decides everything.

It is worth more than a moment-gradient factor on a short segment and nothing on a long one. The two multiply.

It costs a longitudinal stress at the flange tip, from a stress resultant that no equilibrium diagram contains and that lands where the bending stress is already largest.

And it does not exist for a tee or an angle, whose plates meet at a point and have no pair of flanges to bend against each other.

Where the model stops #

The restraint is treated as complete or absent. A real connection is somewhere between, and the intermediate case needs a warping spring rather than a boundary condition.

The section is doubly symmetric. For a monosymmetric section the shear centre is not at the centroid, the two mechanisms couple with bending through the Wagner effect, and the split above is not the whole of the torsional resistance.

The analysis is linear and elastic. A beam near its plastic moment has yielded in the flanges, which is where the warping stiffness lives, so the benefit falls away exactly as the beam approaches the capacity it is being claimed for.

Nothing here is a check on the restraining member. A connection preventing warping is carrying two flange forces in opposite directions — a bimoment applied to it — and it has to be designed for that.

The restraint at a support is assumed to be at the support. The section that is actually held against warping is the one at the end plate, and there is often a length of beam beyond it — an overhang, a haunch, a cleat extension — over which the boundary layer has already started.

And the decay length is computed from nominal properties. $J$ for a rolled section is sensitive to the root fillets and $I_w$ to the flange width, and the ratio of the two is what $\kappa$ is.

The asymmetry worth remembering #

There is one more contrast between the two factors, and it is about who controls them.

The moment gradient is given. It comes from the loading and the support conditions, a designer does not choose it, and the tabulated factor is a way of recognising a benefit the structure already has. It costs nothing and can be claimed with confidence, because the load case is what it is.

The warping restraint is bought. It comes from a detail, a designer does choose it, and claiming it means committing to a connection that has to be built. It costs a stiffener or an end plate and it can be lost on site.

That asymmetry is why one is in every table and the other is in none. A code can tabulate a property of a load case and cannot tabulate a property of a detail — and the profession’s response has been to leave the second at its conservative value permanently rather than to specify what would justify a better one.

It is a defensible response and it is not free. On a short, heavily loaded segment — a crane girder between restraints, a transfer beam, a portal haunch — the factor left unclaimed is comparable with the one carefully looked up, and it is sitting in a connection that was probably detailed to deliver it anyway.

The three quantities that decide a beam’s lateral-torsional capacity are all geometric and none of them is in the section table. The mode itself is the starting point; which flange the brace is on decides whether the restraint does anything; and the height the load is applied at can be worth more than either. A design that gets the moment gradient right and any of those three wrong has optimised the smallest of the four terms.

The ladder from here #

Later rungs on this anchor: the quarter-point formula for the gradient factor and what it is a fit to. Load height and its interaction with the gradient. Monosymmetric sections and the Wagner effect, where the two torsional mechanisms couple with bending. Restraint placement as an optimisation over segment factors. Cantilevers, where the root condition dominates and the tables disagree with each other. Moment gradient in the inelastic range, where the benefit is capped because the flanges have yielded. And the warping spring, which is what a real connection supplies and which turns a boundary condition into a stiffness.

Vlasov’s theory of thin-walled beams, in which the bimoment is a stress resultant on the same footing as a moment or a torque, was published in Russian in 1940 and reached the West in translation twenty years later. It is the reason a quantity with units of kN·m² has a name at all — and it remains the only stress resultant in structural engineering that a designer can meet without ever having been told it exists.