Structural Engineering & Statics
where the load goes
Concept

Lateral-torsional buckling — where it appears

A beam strong enough in bending failing by moving sideways and twisting, because its compression flange is a column with nothing holding it. It is prevented by restraint rather than by strength, and a restraint on the tension flange does not prevent it at any stiffness whatever.

Named by 10 essays across one field — each of them below, with the objects they name alongside it.

The length at which a beam stops being a beam. Elastic critical moment against the distance between lateral restraints, with the section's plastic capacity drawn across it. The two cross at 3803 — beyond that length the beam buckles sideways before it reaches the strength its cross-section has, and the capacity is set by the restraints rather than by the steel.

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.

stability · Lateral-torsional How stiff a brace has to be before the frame stops swaying. The effective length factor of a swaying portal against the stiffness of a horizontal spring at its head. The curve starts at k = 1.317, the unbraced value, and falls to 0.774 — the factor for the same frame with its head held — at a brace stiffness of 23.2 EI/L³. Past that point the frame buckles in the non-sway mode, which the brace does not restrain, and further stiffness buys nothing at all. The threshold is worth stating as 1.41 N꜀ᵣ/L, which is the form the number is memorable in: for a storey carrying a thousand kilonewtons over four metres it is about 0.35 kN per millimetre of sway. Against the frame's own lateral stiffness of 12.0 EI/L³ it is a factor of 1.93.

The most dangerous day is before it is finished

A structure is analysed once, complete, with every restraint present. It spends weeks in states nobody drew — a beam landed with no deck on it holds 17% of the moment its section is worth, a frame not yet braced buckles at a third of the load it will, and a bolt not yet tightened is a pin where the analysis assumed a fixity.

stability · Erection stability A brace on the wrong flange never gets there, however stiff it is. The critical moment of an 8 m beam against the stiffness of a single midspan brace, drawn three times for the three heights the brace could sit at. On the compression flange it climbs from 143 kNm to the two-half-wave plateau of 447 — the beam braced into two 4.0 m beams — and reaches 99% of it at 447 kN/m. At the shear centre it needs 2252 kN/m, 5.0 times as much. On the tension flange it never arrives at all: at the stiffness that would have done the job on the other flange it has bought a factor of 1.068, and a stiffer brace in the same place buys the same nothing. Past the plateau the beam stops using the brace, which is where the idea of an ideal stiffness comes from.

The brace on the wrong flange

A brace on a column has one property that matters, and it is stiffness. A brace on a beam has two, and the second decides whether the first is worth anything: put the identical restraint on the tension flange and it does not reach the answer at any stiffness whatever.

stability · Beam bracing The restraint chooses the buckling length, and it is not the member's. A compression flange 12 m long held sideways not at points but everywhere, by a restraint of 0.35 N/mm per mm of length. Unrestrained it would buckle at 173 kN in a single half-wave, drawn faintly. Restrained it buckles at 1968 kN — 11.4 times as much — in two half-waves, because the sum n²π²EI/L² + kL²/n²π² has its minimum there and every other n is worse. The effective length that answer implies is 3555 mm, which is 0.30 of the member and is a property of the restraint rather than of the span.

Held everywhere, and it forgets its length

A brace at a point divides a member's buckling length. A restraint spread along the whole member does something else — the member chooses its own number of half-waves, and past a few of them the critical load stops depending on the length at all.

stability · Continuous restraint The length at which a beam stops being a beam. Elastic critical moment against the distance between lateral restraints, with the section's plastic capacity drawn across it. The two cross at 4086 — beyond that length the beam buckles sideways before it reaches the strength its cross-section has, and the capacity is set by the restraints rather than by the steel.

The load that moves with the twist

A beam about to buckle sideways is beginning to rotate, and everything attached to it rotates with it. A load hung from the top flange swings out over the side and drives the rotation on; the same load hung underneath swings back and stops it. Two identical beams, two different capacities, and the only difference is a height.

stability · Load height It is the square of the diagram that destabilises. Four moment diagrams normalised to the same peak, and the buckling factor each one earns. Eliminating the lateral displacement from the coupled buckling equations leaves one functional in the twist, and its destabilising side is ∫M(z)²φ²/EI_z — the SQUARE of the moment, weighted by where the beam wants to twist. A diagram with a peak over a short length has a much smaller weighted square than a flat one of the same maximum, so it buckles at a higher peak: uniform 1.00, uniformly distributed load 1.13, central point load 1.36, cantilever 1.71. The root-mean-square of each diagram, printed beside it, very nearly predicts the order — which is as close to an intuition for C₁ as the subject has.

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.

stability · Moment gradient Warping stiffens a short member and nothing at all a long one. The stiffening 1/[1 − tanh(κ)/κ] against kL, both axes logarithmic, over kL from 0.05 to 200. At the low end the curve is a straight line of slope −2, because for small kL the bracket is κ²/3 and the stiffening is 3/kL²: it reaches 1201 at kL = 0.05, falls to 1.005 at the top, and every open section ever rolled sits somewhere on it. The same three plates arranged three ways are marked: the 533 by 190 mm I-section at kL 2.76 and ×1.562, the tee at kL 37 and ×1.027, the angle at kL 34 and ×1.030. A tee's warping constant is 288 times smaller than the I-section's and an angle's 236 times, because their plates meet at a point and there is no pair of flanges to bend against each other — so they have no warping resistance to offer at all, and that is the reason an angle is a poor thing to twist.

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.

stability · Moment gradient What a torsional brace buys, and the stiffness it takes. The critical moment of an 8 m beam against the stiffness of a single rotational restraint at midspan — a cross-frame or a stiffened connection to a secondary beam, resisting the twist rather than the sideways movement. It climbs from an unbraced 143 kNm to the same two-half-wave plateau of 447 a lateral brace reaches, and gets to 99% of it at 253 kNm/rad. Nothing in the calculation refers to a height, which is the difference that matters: a torsional brace cannot be put on the wrong flange because it is not attached to a flange in the sense the lateral one is.

Twice the moment, four times the brace

A lateral brace has to be on the right flange and its demand is very nearly linear in the load. A torsional brace has no flange to be wrong about, and its demand is exactly quadratic — so the restraint that is indifferent to where it is attached is the one that gets expensive fastest.

stability · Beam bracing Four numbers read from a moment diagram. The moment diagram of the fixed-ended beam under a central point load, scaled to a largest value of one, with the four values the quarter-point formula reads: the peak, and the magnitudes at a quarter, a half and three quarters of the span — 1.00, 0.00, 1.00 and 0.00. The formula turns them into a gradient factor of 1.923. Solving the buckling problem for the whole diagram, on a beam 8 m between lateral restraints, fork-supported at its ends, gives 1.723: the formula is 12 per cent too high, on the unsafe side.

The formula that reads four numbers

The factor that credits a beam for the shape of its moment diagram is usually taken from a formula that reads the diagram at four places — its peak and its quarter points — and nowhere else. On the straight-line diagrams it was built around it is safe. On a fixed-ended beam under a central load, whose moment is zero exactly where the formula looks, it is twelve per cent unsafe, and it would give the same answer for a diagram that deserves twenty-two per cent less.

stability · Moment gradient At the tip it is the tension flange that swings. The buckled shape of a 3.00 m cantilever of 457 mm universal beam carrying a load at its tip at the shear centre, root built in: slope and warping held, seen from above: the sideways movement of the top flange and of the bottom flange along the length, scaled so the larger is one. The cantilever hogs, so its bottom flange is in compression, and that flange bows out furthest 0.58 of the way from the root, where it moves 0.15 — and has come back to 0.00 by the tip. The top flange, in tension, swings out to the full 1.00 at the tip. The section twists as it goes, so the flange that is doing the buckling is not the flange that moves most at the end. Critical moment at the root: 1266 kN·m.

The flange that swings is in tension

A cantilever hogs, so its bottom flange is the one in compression and the one that buckles. At its tip the bottom flange barely moves: the section turns about it, and the flange that swings out is the top one, in tension. So the restraint that matters at a cantilever's tip is on the flange every rule for spans calls the wrong one — and the root, which a drawing shows as a single line, moves the critical moment by a factor of thirty-four.

stability · Moment gradient

Named alongside it

The objects these essays reach for when they reach for this one.

WarpingCompression flangeEffective lengthBracingCritical momentEigenvalueCritical loadImperfectionLateral restraintMoment gradientRestraintTorsion

All concepts