Structural Engineering & Statics
where the load goes
Concept

Critical moment — where it appears

The bending moment at which a beam buckles sideways and twists, set by the minor-axis, torsional and warping stiffnesses together. It combines three stiffnesses that come from three different section properties, and for an open section the warping term is a large part of it.

Named by 5 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 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 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.

Lateral-torsional bucklingCompression flangeWarpingEffective lengthBracingLateral restraintBraceBuckling modeCantileverContinuityDistortionEigenvalue

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