The counterweight that follows the jib

A counterweight is a moment, not a weight. A bascule leaf of 900 kN with its centroid 9 m from the pivot is balanced by 2,700 kN at 3 m, and the trunnion then carries both, four times the leaf. The leaf and its counterweight turn together, so the balance holds at every angle. A jib crane is harder, because its load moves along the jib on a trolley while the counterweight stays where it is: a counterweight sized for the middle of the trolley’s travel leaves the mast 1,110 kN·m one way with the load out, 1,110 the other with it in, and 1,290 backwards with the jib empty. A load that moves cannot be balanced by a weight that does not.

The obvious answer is a weight that does. The commonest one in practice is not on the jib crane at all but on the luffing crane, the crane whose jib is raised and lowered to change its radius. There the counterweight is linked to the luffing motion — by a lever, a rope or a second jib — so that it moves outward as the jib comes down. The interesting question is what such a counterweight is actually following, because on a luffing crane the payload is not the largest thing that moves.

Two moments about the mast #

The crane drawn has a 50 m jib weighing 300 kN, luffing between 15° and 85° above horizontal, lifting 60 kN at its tip. The moment about the mast has two parts, and they have different sizes and different speeds.

The jib’s weight acts at the jib’s centroid, half-way along it, so its moment is $W_j (L_j/2)\cos\theta$: 7,244 kN·m with the jib nearly flat and 654 with it nearly vertical. That change, 6,591 kN·m, is more than twice the largest moment the payload ever adds, which is 60 kN at the tip’s full radius of 48.3 m. On a luffing crane the jib is the larger moving load, and it moves every time the crane changes radius, loaded or not.

The two parts also differ in speed. The jib luffs over a minute or two, smoothly, under the crane’s own control, and its weight is known to the kilogram. The payload arrives when it leaves the ground — over a second or so, as the rope takes up — and leaves the same way when it is set down, and its weight is whatever is on the hook.

A counterweight that stays put #

With one fixed counterweight, sized so that the mast’s largest moment is the same either way, the mast takes the full range of both parts.

The worst case towards the jib is the obvious one, the jib flat with the payload at its tip. The worst case backwards is the crane at rest with the jib raised and nothing on the hook — the case the trolley crane also had, the one that does not feel like a load case at all — and here it is as large as the forward one, because the counterweight that balances a flat jib is far too much for a raised one. Of the 4,744 kN·m each way, 3,295 is half the jib’s change of moment and 1,449 half the payload’s. Most of what the mast is designed for is the crane’s own geometry.

A counterweight that follows the jib #

Link the counterweight to the luff so that, at every angle, it balances the jib’s moment and half the payload’s, and leave the rest to fixed ballast.

The jib’s whole change of moment has gone. What is left is the payload’s moment less the half the linkage was set for, which is plus or minus half the payload’s moment at whatever radius the jib is at: ±1,449 kN·m at the longest radius, ±131 at the shortest. The mast’s governing moment has fallen to less than a third.

The linkage has to supply the jib’s change plus half the payload’s — 7,909 kN·m of moment between the jib flat and the jib raised — which is a large counterweight moving a long way, or a heavy one on a short lever. Real level-luffing cranes use a counterweight hung from a lever or a second jib that swings as the main jib luffs; the geometry is chosen so that the counterweight’s moment tracks $\cos\theta$, which is exactly what a weight on an arm turning with the jib does.

The floor #

Half the payload’s moment is not a property of this linkage. It is a floor under every linkage.

Each share of the moment the linkage takes is a share off the mast, in a straight line, until about 85 per cent of it is linked. From there on nothing changes: the mast’s largest moment is 1,449 kN·m and stays there. The jib’s contribution has been removed; what remains is the payload’s, and the payload cannot be followed.

The reason is the instant of pick-up. However the counterweight is linked, it moves with the jib, and the jib does not move when the payload leaves the ground. At that instant the moment in the mast changes by the payload’s whole moment, $Qx$, in about a second, and the counterweight is wherever it was. The best place for it to be is halfway, so that the mast goes from $-Qx/2$ to $+Qx/2$ rather than from 0 to $Qx$; any other position makes one side of the step larger.

The two cranes take exactly the same step at every radius, because a step is the payload’s moment and nothing else. What the linked counterweight changes is where the step starts. The fixed crane’s step sits at a different height at every radius, because it rides on the jib’s own unbalanced moment, which runs from strongly backwards with the jib up to strongly forwards with it down. The linked crane’s step is centred on zero at every radius. A counterweight that moves can choose where the step sits; it cannot make the step smaller.

The trolley crane, where the floor is everything #

The same floor answers the question the trolley crane left open. On a crane whose jib is horizontal and whose payload runs out along it on a trolley, the jib does not move: its own moment is constant, a fixed counterweight balances it exactly, and the only moving load is the payload. A counterweight that followed the trolley would be following the payload — and the payload is the one thing whose step no counterweight can soften.

For the trolley crane of the first counterweight essay, 60 kN running from 3 m to 40 m, the floor is half the payload’s moment at the tip, $60 \times 40/2 = 1{,}200$ kN·m. The fixed counterweight already leaves 1,290, with the jib empty. A counterweight on the counter-jib, motorised to track the trolley, could bring the mast from 1,290 kN·m down to 1,200: seven per cent, for a second machine and the moving mass that goes with it.

The luffing crane gains seventy per cent from the same idea, and the difference is entirely in what is moving. A moving counterweight is worth what the structure’s own weight changes by, and nothing for the payload, because the structure’s weight moves slowly and predictably and the payload neither. Read the other way, the counterweight is the bascule’s counterweight again — a weight that turns with the thing it balances — and a bascule leaf, which has no payload at all, can be balanced perfectly.

The bascule inside the luffing crane #

The resemblance is closer than an analogy. A weight $C_m$ on an arm $a$ behind the jib’s pivot, fixed to the jib so that it turns with it, has a moment $C_m a\cos\theta$ about the pivot, which follows the jib’s own $W_j(L_j/2)\cos\theta$ at every angle if $C_m a = W_j L_j/2$. That is a tail on the jib, and the crane is then a bascule with a hook on its leaf. To balance half the payload too, $C_m a = W_j L_j/2 + Q L_j/2 = 7{,}500 + 1{,}500 = 9{,}000$ kN·m — a counterweight of 1,500 kN six metres behind the pivot, which is heavier than the jib it balances.

Real level-luffing cranes rarely carry the counterweight on the jib itself, because a heavy tail swinging below the jib’s pivot at low angles needs clearance a city site does not have. They hang it from a lever linked to the jib by a rope or a strut, which gives the same $\cos\theta$ moment with a different path through space. The arithmetic of four times as heavy comes with it: the mast now carries the jib, the payload and a counterweight heavier than the jib, all as vertical load, in exchange for carrying almost none of their moment.

The mast carries the payload, and only the payload #

With the jib balanced, what is left in the mast is a simple thing.

The linked crane’s mast moment is the payload’s alone, a straight line through the origin: no payload, no moment. The fixed crane’s starts at 3,295 kN·m with an empty hook, because the jib’s change of moment is there whether the crane is working or not, and climbs from there. The two lines are parallel: the fixed crane’s is the linked crane’s moved up by half the jib’s change of moment, 3,295 kN·m, at every payload. Its mast carries more than its payload ever puts on it in every case drawn, and the excess does not shrink as the payload grows.

That changes what limits the crane’s capacity, in the way an influence line changes what limits a bridge: the governing position of the load is found from the shape of the response, not from intuition. With the jib balanced, the mast’s design moment is set by the payload at the longest radius, and a crane’s load chart — the payload it may lift at each radius — can be drawn as nearly a constant moment, $Q \cdot x$ fixed. With a fixed counterweight, the jib’s own imbalance uses part of the mast’s capacity at every radius, more at the extremes of the luff than in the middle, and the load chart has to bend round it.

The crane left for the night #

One case on the fixed crane’s figure is not a working case at all, and it is often the one that governs. A luffing crane out of service is parked with its jib raised, empty, free to turn with the wind — and with a fixed counterweight, a raised empty jib is exactly where the mast’s backward moment is largest, 4,744 kN·m here, with a storm’s wind on the crane added to it. The design case for many fixed-counterweight cranes is the night, not the lift.

The linked counterweight removes it. With the jib parked at 85° the counterweight has swung in with it, the mast carries 131 kN·m from the half-payload allowance and nothing else, and the storm’s wind is the only large load left. A crane whose counterweight follows its jib is designed for working and for weather separately, and neither has to carry the other’s worst case on top of its own.

The same arithmetic at the mast’s base #

The mast’s moment is only the first place the balance matters. It passes into the slewing ring, the tower, and finally the foundation, where weight is the only thing resisting overturning. A base designed for ±4,744 kN·m has to carry a resultant that swings from one side of its footprint to the other as the jib luffs, empty, through its range; one designed for ±1,449 sees a resultant that stays near the middle until a payload is on the hook.

That is the larger saving. A crane base’s size is set by keeping its resultant within the footprint, whether it tips or slides, and a moment three times smaller needs a base whose width grows with the moment over the weight — or, for a crane standing on a fixed base, ballast blocks to the same ratio. The counterweight that moves with the jib is paid for by the ballast that does not have to be cast.

The jib, by hand #

Every number above comes from two moments. The jib’s is $W_j(L_j/2)\cos\theta = 300 \times 25 \times \cos\theta$: 7,244 kN·m at 15° and 654 at 85°, a change of 6,591. The payload’s is $Q L_j\cos\theta = 60 \times 50 \times \cos\theta$: 2,898 kN·m at 15° and 261 at 85°.

A fixed counterweight centres the extremes. The mast moment runs from the jib down and loaded, $7{,}244 + 2{,}898 = 10{,}142$, to the jib up and empty, 654; centred, it is $\pm(10{,}142 - 654)/2 = \pm 4{,}744$. A linked counterweight takes away the jib’s moment and half the payload’s at every angle, leaving $\pm Q L_j\cos\theta/2$, largest at 15°: $\pm 2{,}898/2 = \pm 1{,}449$. The ratio between them is the jib’s change of moment against the payload’s, which is the crane’s own proportions: a heavier jib, or a lighter payload, makes the linked counterweight worth more.

A rigid crane and a static lift #

The calculation rests on choices that limit it.

The crane is rigid. A real jib and mast deflect, the jib’s radius grows under load, and the payload’s moment is a little larger than its nominal radius gives.

The pick-up is static. The payload’s step is treated as a moment applied at once. In fact a payload lifted off the ground is a load that arrives with a velocity and swings: the dynamic factor on a crane’s hoist load is typically 1.1 to 1.5, and the step in the mast is that much larger for an instant. The floor of half the payload’s moment is a static floor; the dynamic one is higher, and no counterweight lowers it either.

The linkage is exact. A real level-luffing linkage tracks $\cos\theta$ to within a few per cent, and the residual is a small moment that varies with the angle, added to the payload’s.

Slewing is ignored. A crane that turns while luffing carries its moment round the mast, and the mast’s base sees the moment in every direction; the balance is the same in each, but the foundation sees it as a rotating resultant.

Thousands of lifts, and a counterweight with mass of its own #

The figures cannot show fatigue. A crane lifts thousands of times, and each lift cycles the mast through the payload’s step and the luff’s change of moment. With a fixed counterweight that cycle can span the whole of ±4,744 kN·m when the crane picks with the jib down and sets down with it up; with the linked counterweight it spans at most the payload’s own step. Small cycles in large numbers are what a crane’s welded mast details are designed against, and a counterweight that follows the jib removes the largest repeated range the mast would otherwise see.

They also cannot show the counterweight’s own inertia. A counterweight on a lever that swings with the jib is a large mass moving whenever the jib does, and its inertia adds to the luffing drive’s work and to the forces when the luff is stopped quickly — a counterweight is a mass with a spring when something makes it ring.

Still open: the counterweight that is positioned for the next lift #

The linked counterweight waits at the half-load position because it does not know the next payload. A crane with a load cell does know, the moment the rope takes up — too late to move a counterweight before the pick, but early enough to move it before the jib luffs out with the load. A counterweight driven by its own motor, positioned from the measured payload during the luff rather than linked to the luff alone, could balance the payload as well as the jib for everything except the instant of pick-up and set-down. Whether the mast is then designed by the step at pick-up alone — half of $Qx$ at the pick radius rather than the longest — and whether a crane that always picks close in and luffs out can carry much more at the tip as a result, is a question about when the crane knows what it is lifting.