Structural Engineering & Statics
where the load goes
Concept

Moment gradient — where it appears

The variation of a beam's bending moment between its restraints, expressed as a factor raising the buckling resistance above the uniform-moment case. A moment that changes along the span puts only part of the length at its peak, so the member is harder to buckle.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

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 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 Three braces do more where the moment is. The 24 m plate girder, its minor-axis second moment 2.67 × 10⁸ mm⁴ and its warping constant 1.00 × 10¹⁴ mm⁶, under a uniform load, held by three torsional braces asked to bring its peak moment to 5,627 kN·m: above, the moment diagram; below it, the twist of the buckled girder with the braces equally spaced, 6.0 m apart, and clustered 4.0 m either side of midspan, each at the stiffness that just reaches the moment. Equally spaced the row needs 8,537 kN·m/rad in all; clustered, 6,325, 26 per cent less. The clustered braces sit where the girder twists most, and the bays they leave at the ends carry little moment.

Three braces where the moment is

A row of torsional braces is spaced equally because that is how it is drawn, and under a uniform moment equal spacing is very nearly the best there is. Under a moment that peaks at midspan it is not. Moving the outer two of three braces toward the middle cuts the total stiffness a 24 m girder needs by a quarter under a uniform load and a third under a point load, and the bays it leaves at the ends cost nothing, because they carry so little of the moment.

stability · Beam bracing Braces that cannot move can still be made stiffer where the moment is. The 24 m plate girder, its minor-axis second moment 2.67 × 10⁸ mm⁴ and its warping constant 1.00 × 10¹⁴ mm⁶, under a uniform load, held by braces at midspan and 6.0 m either side and asked to bring its peak moment to 5,627 kN·m: above, the moment diagram; below it, the twist of the buckled girder with the three braces equally stiff and with the middle one given 61 per cent of the row's stiffness, each row at the total that just reaches the moment; the ticks are drawn thicker for a stiffer brace. Equal, the row needs 8,537 kN·m/rad, 2,846 in each brace; graded, 7,067, 4,280 at midspan and 1,394 at each side, 17 per cent less.

The braces that stay where the framing puts them

Three cross-frames on a girder usually go where the floor beams or the bridge's bays put them, and cannot be moved toward the moment. They can still be made unequal. Giving the middle one three-fifths of the row's stiffness saves a sixth of the total under a uniform load and more than a quarter under a point load — most of what moving them would save. Moved and graded together, they save no more than moved alone, because both are ways of making every brace twist by the same amount.

stability · Beam bracing

Named alongside it

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

Lateral-torsional bucklingWarpingCritical momentBracingRestraintTorsionBimomentCritical loadDecay lengthEffective lengthEigenvalueEnergy method

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