Structural Engineering & Statics
where the load goes
Concept

Load reversal — where it appears

A change in the sign of a member's force between one load case and another, turning a tie into a strut or a sagging moment into a hogging one. It changes which variable decides the member — an area for tension, a radius of gyration for compression — so the previous design contains no information about the new requirement.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Balanced, and four times as heavy on the bearing. A bascule leaf of 900 kN whose centroid is 9 m from the trunnion, balanced by 2700 kN at 3 m on the other side. What balancing achieves is exactly one thing: the moment about the pivot is zero at every opening angle, because both terms carry the same cosine. What it costs is two things that are not zero. The reaction on the trunnion becomes 4.0 times the leaf's own weight, since both weights are still there. And the rotational inertia rises by 33%, so the balanced leaf is the hardest one to start and to stop — which is why the counterweight is put as close to the pivot as it will fit, at the price of being heavy: the same balance at twice the radius weighs 1350 kN and carries 1.25 times the inertia.

Balanced, and four times as heavy

A counterweight cancels a moment about a pivot, and that is the only thing it cancels. The bearing beneath carries both weights, the inertia rises as the square of the radius, and a load that moves cannot be balanced at more than one position at all.

equilibrium · Counterweight A Pratt truss of 8 panels. A Pratt truss under equal panel-point loads. The joint equilibrium equations were assembled and solved; 14 members came out in tension, 13 in compression and 2 carrying nothing.

The tie that spends an afternoon as a strut

A tension member is chosen by its area and nothing else. A compression member is chosen by how that area is arranged. So a member whose force reverses under some load case is not merely being asked for the same number with the other sign — it is being designed against a different variable, and the same steel can carry seventy times more or less depending on a shape nobody chose for that purpose.

stability · Load reversal The two braces balance until one of them buckles. An inverted-V brace after the compression member has gone. While both braces are elastic they carry equal and opposite forces and their vertical components cancel on the beam above, which is why the beam in a chevron bay is usually sized for gravity alone. The compression brace buckles at 445 kN and then sheds most of what it was carrying — 30% is left here — while the tension brace goes on to yield at 1065. The difference between the two vertical components is 659 kN, applied at the middle of the span with no help from either brace, and it asks the beam for 1317 kNm against the 200 kNm the gravity load asks for — 6.6 times as much. The beam drawn does not: 1517 kNm against a capacity of 731. The force is not a load case anybody applies; it is what the frame leaves behind on its way to the state it will actually be in.

The force the brace leaves behind

Two braces meeting under a beam carry the storey shear as a tension and a compression whose vertical components cancel, so the beam above sees nothing. They cancel only while both braces are elastic. Once the compression brace buckles it sheds most of its force, the tension brace goes on to yield, and the difference is a point load at midspan that nobody applied.

structures · Chevron brace Influence line for the shear force at x = 10.5. The shear force at one fixed station, plotted against the position of a unit load walking across the span. The beam was re-solved at 301 load positions. The worst position is x = 10.56, giving 0.560.

Two diagonals, one of which is absent

A truss diagonal is sized for the shear in its panel, and near mid-span that shear changes sign depending on where the load stands. A member that can only pull cannot carry the reversed case, so the panel gets a second diagonal — and at any instant one of the pair is not there.

structures · Truss A counterweight that follows the jib leaves only the payload, halved. The moment in the mast of a crane whose 50 m jib weighs 300 kN and luffs between 15° and 85°, lifting 60 kN at its tip, against the luffing angle, with a counterweight linked to the luff so that it balances the jib and half the payload at every angle: 1,449 kN·m loaded and −1,449 empty at 15°, 131 and −131 at 85°. What is left is the payload's moment less the half the counterweight was set for, plus or minus 1,449 kN·m at most, against 4,744 with a fixed counterweight.

The counterweight that follows the jib

A counterweight that stays put balances one position and leaves the mast a moment at every other. Make it move and it can balance anything that moves slowly enough to be followed. On a luffing crane the thing that moves most is not the payload but the jib itself, whose own moment about the mast changes more than twice as much through its luff as the payload's ever is — and a counterweight linked to the luff cancels that change exactly. What it cannot cancel is the payload, which leaves the ground in a second. The best any counterweight can do with that is wait halfway, and the mast is left with half the payload's moment at the longest radius, whatever else the counterweight does.

equilibrium · Counterweight Balancing the payload is a hedge between lifting it and losing it. The mast's design moment over the payload's own moment, against the share of the payload a counterweight driven from the load cell balances, for a crane whose 50 m jib weighs 300 kN, lifting 60 kN at 48.3 m, with a hoisting factor of 1.20 and a rebound factor of 0.50 for a payload let go at once. Lifting off, the mast carries 1.20 less the share balanced; let go, the share balanced plus 0.50, backwards. The larger of the two is least at a share of 0.35, where it is 0.85. The linked counterweight's half gives 1.00; balancing the whole payload gives 1.50, the worst of the three, because a crane balanced for its load is thrown backwards by all of it when the load goes.

The counterweight that knows the load

A counterweight driven from the hoist's load cell can follow the payload as the rope takes it up, so the step at pick-up that defeated every linked counterweight becomes a ramp a motor can keep pace with. Balancing the whole payload is then the worst thing to do with it. A crane balanced for its load is thrown backwards by all of it when the load is let go — 1.5 times the payload's moment with a rebound factor of 0.5, against 1.0 for the linked crane's half. The best share is a hedge between lifting the payload and losing it: 0.35 of it, for 0.85, which buys 18% more at the tip from the same mast.

equilibrium · Counterweight A slow release moves the hedge up, and never past half the surge. The best share of the payload for the counterweight to balance, and the mast's design moment over the payload's at that share, against the release time the design can rely on, for the crane lifting 60 kN at 48.3 m, its mast and jib rocking backwards with a period of 2.0 s and a damping ratio of 0.02, with a hoisting factor of 1.20. Let go at once, 0.13 and 1.07; over 1.2 s, 0.36 and 0.84; over 6 s, 0.56 and 0.64. However slow the release, the share cannot pass 0.60, half the hoisting factor, because the payload's own moment is still let go even when its rebound is not.

The release the rope cannot slow

A crane whose counterweight knows the payload balances only a third of it, because a payload can be let go and the crane is then thrown backwards by whatever the counterweight was holding, and half as much again in the rebound. Slow the release and the rebound goes, so the hope was that a device on the rope could slow it. Nothing on the rope can: the rope holds only what the hook holds. How fast a load goes belongs to the payload, and what the crane owns is how fast it answers — its period and its damping. And with no rebound at all, the counterweight still should not balance more than 0.6 of the payload, because the payload's own moment is still let go.

equilibrium · Counterweight

Named alongside it

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

BalanceCounterweightEquilibriumOverturningBucklingLever armMoving loadBracingDynamic amplificationFree bodyLoad arrangementLoad path

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