The angle that doubles the force

Almost every structure in this collection is analysed in the state it will spend its life in. The exception is the day it is put up, and on that day the governing calculation is often not about the structure at all — it is about the things holding it in the air.

A lift is a statics problem of the plainest kind. A hook takes the weight; a set of legs runs down from it to the object; each leg is straight, pinned at both ends and loaded only there, so each is a two-force member carrying force along its own line. There is nothing else in it. And yet it produces one of the few numbers in this subject that gets people killed when it is got wrong.

Which free body produced the number #

Take the hook. Three forces act on it: the crane’s pull upwards, and the two leg forces along the two legs. Three concurrent forces in equilibrium, which is the oldest construction in graphic statics.

Resolve vertically. If each leg makes an angle $\beta$ with the horizontal and the load is shared equally,

$$T = \frac{W}{n \sin\beta}$$

Resolve horizontally and the two legs’ horizontal components cancel at the hook — but they do not disappear. Each leg exerts $T\cos\beta$ on the object, pulling its two pick points towards one another, so the object carries

$$H = T\cos\beta = \frac{W \cot\beta}{n}$$

as an internal compression between them. That force is applied by the rigging and carried by the thing being lifted, and it is invisible in the lift plan.

The numbers are worth having by heart, because they are the whole subject. At sixty degrees each of two legs carries 0.58 of the load; at forty-five, 0.71; at thirty, exactly 1.00; at fifteen, 1.93. The thirty-degree limit that appears in every lifting standard is where the curve crosses one, and the reason it is a hard limit rather than a preference is what the curve does below it: the force rises faster than the angle falls, and no rigger can judge fifteen degrees from twenty by eye at the top of a lift.

The compression that is not in the lift plan #

The horizontal component is the part that turns a rigging problem into a structural one, and it is the reason spreader beams exist.

A long slender object — a column, a truss, a precast beam, a length of pipe — picked up on two legs at thirty degrees carries 0.87 of its own weight as axial compression between the pick points. For a twenty-metre steel girder weighing 200 kN that is 173 kN of compression in a member with no lateral restraint at all, no load applied along its length to hold it straight, and its weak axis free. Some objects are buckled by being lifted, and the check is an Euler calculation on the weak axis with an effective length nobody has designed.

The bow is not a defect anybody has to invent. A girder is fabricated with a camber, it sags under its own weight the moment it leaves the bed, and it deflects further as soon as the rigging squeezes it — so the offset in that drawing is a lower bound rather than a tolerance. The useful form of the result is the ratio rather than the moment.

A spreader turns that compression into a designed member. Run the crane’s legs down to the ends of a horizontal beam, and vertical legs from the beam’s ends down to the object; the beam takes the whole horizontal thrust as a strut sized for it, and the object below sees only vertical force. A lifting frame does the same in two directions. Both are heavier and more trouble than a pair of slings, which is why they are used only when the arithmetic says they must — and the arithmetic that says so is exactly the $W\cot\beta$ above.

The distinction between a spreader and a lifting beam is worth keeping straight, because they fail differently. A spreader is in compression and is checked for buckling; a lifting beam is picked up in the middle and is in bending, and it is checked for the moment. Confusing them means checking a strut for bending.

Why the angle is shallow, which is a question about headroom #

If the leg force falls monotonically as the angle steepens, the obvious question is why anybody rigs at thirty degrees rather than sixty. The answer is not ignorance; it is the ceiling.

For pick points a distance $2a$ apart, the hook must sit $a\tan\beta$ above them. At thirty degrees that is $0.577a$; at forty-five, $a$; at sixty, $1.732a$. Steepening the legs from thirty to sixty degrees triples the height the hook needs, and it needs that height above the highest point of the load, under whatever the crane hook can reach. For a six-metre pick width the hook goes from 1.7 m above the load to 5.2 m, and the crane’s own hook block, the sling hardware and the hoist rope sit above that again.

That is why the trade is real rather than a matter of care. Inside a building, under an existing roof, in a lift shaft, beneath a bridge soffit — the available headroom is fixed and it is often what sets the angle. So the sequence in practice runs the other way from the formula: the headroom fixes $\beta$, $\beta$ fixes the leg force and the horizontal thrust, and the question becomes whether the object can carry the thrust or needs a spreader. And a spreader is itself a member requiring headroom above it, which is why the alternative under a tight ceiling is not a spreader but a different pick arrangement altogether.

The same arithmetic explains the shape of the standard’s rule. Thirty degrees is not where the force becomes unacceptable — it is where the force reaches one times the load, which is a number a rigger can hold in mind, and where the headroom is still modest enough that the rule can be obeyed. Set the limit at forty-five and it would be broken routinely.

The angle in the drawing is not the angle in the sling #

One further geometric trap belongs here, because it is produced by the drawings rather than by the rigging.

A four-point pick on a rectangular object is set out on a plan and an elevation, and each drawing shows a leg leaning by some angle. Neither is the leg’s angle. A leg running from a hook above the centre to a corner of a rectangle leans in two directions at once, and its true inclination to the horizontal satisfies $\cot^2\beta = \cot^2\beta_1 + \cot^2\beta_2$, where the two are the apparent angles on the two orthogonal views.

The error is always in the same direction. Two apparent angles of forty-five degrees give a true angle of $\arctan(1/\sqrt2) = 35.3°$, so a rigger reading either drawing believes the legs are at forty-five and carrying $0.71W$ each while they are at thirty-five and carrying $0.87W$ — twenty-three per cent more. Two apparent thirty-degree angles give a true angle of $22.2°$, which is below the limit the drawings appear to satisfy.

The check is one line and it is the three-dimensional habit this collection keeps recommending: take the leg’s own vector between its two ends and read its angle from that, rather than from any view of it. A projection of a length is shorter than the length, and every apparent angle on a drawing is steeper than the real one.

The centre of gravity, which decides the tilt #

A rigid object hanging from a single hook comes to rest with its centre of gravity directly beneath the hook. That is not a design decision, it is the only position in which the moments balance — so the object rotates until it is true.

Two consequences follow, and both are practical rather than theoretical.

The first is that an object whose centre of gravity is not midway between its pick points hangs out of level, and the legs then carry unequal shares — the near leg taking more, in proportion to how much of the weight is on its side. Riggers deal with that by moving a pick point or by shortening one leg, and both of those are ways of putting the hook back over the centre of gravity.

The second is that if the centre of gravity is above the pick points, the hanging position is unstable. The object will roll over until the centre of gravity is below them, which for a tall, narrow, top-heavy lift is a violent event that nobody was planning. It is the same stability criterion a floating body obeys, with the hook standing in for the metacentre, and it is why a tall panel is lifted from points near its top rather than near its middle.

Where the equal-share assumption comes from #

The formula at the top of this essay divided the load by $n$, and it is worth asking when that division is allowed.

For two legs it follows from symmetry, provided the centre of gravity is midway between the pick points and both legs are the same length. Neither is guaranteed and both are checkable, which is what a rigger is doing when a load is trial-lifted a few centimetres and looked at before it goes up: an out-of-level load is a direct reading of an unequal share.

For three legs it follows from equilibrium rather than from symmetry, and that is worth separating. Three legs from one hook to three points on a rigid body give three unknowns against three force equations, so the tensions are determinate whatever the geometry — unequal, in general, and calculable. Three is the largest number of legs for which the sharing can be known rather than assumed, and it is the reason a three-point pick is preferred wherever the object allows one.

Past three, the division by $n$ stops being a derivation and becomes a hope. The next section is what happens to it.

Four legs, three equations #

Three legs to a rigid load are statically determinate. Four are not, and the reason is worth being precise about because it is easy to count the equations wrong.

All four leg forces pass through the hook, so they exert no moment about it. The moment equations therefore say nothing about the distribution of force between the legs — what they say is that the load’s weight must also pass through the hook, which is the two conditions that fix the hook’s position in plan. That leaves the three force-equilibrium equations and four unknown tensions: one degree of static indeterminacy.

What settles an indeterminate problem is stiffness, and a wire rope leg’s stiffness is $EA/L$ — for a twenty-millimetre rope four metres long, about 3.9 kN per millimetre. So a one-millimetre difference in leg length transfers about two kilonewtons from one diagonal pair to the other. On a hundred-kilonewton lift with the legs at sixty degrees, each nominally carrying 29 kN, that is a seven per cent redistribution from a tolerance no fabricator would consider worth measuring — and the redistribution is proportional, so five millimetres takes it to a third.

The honest design assumption, and the one every lifting standard uses, is therefore that two diagonal legs carry everything. That is not conservatism for its own sake: it is the correct answer for a leg mismatch of about fifteen millimetres, and nothing about a four-leg sling guarantees better than that. Where the object being lifted is flexible rather than rigid the picture improves, because the object deflects and takes up the difference — but a stiff object on four legs is genuinely a two-leg lift and is rated as one.

This is the clearest small example on the site of a general property: an indeterminate structure’s force distribution depends on things statics cannot see. Here the invisible thing is a manufacturing tolerance, and the structure is four pieces of rope.

The tandem lift, and why it is a different problem #

Two cranes on one load looks like a bigger version of the same arithmetic and is not, because the two hooks are not one hook.

With a single hook the load hangs where moment equilibrium puts it and the geometry is a result. With two hooks the positions are imposed — each crane holds its hook where the operator puts it — and the load’s attitude is fixed by the two of them together. The force in each is then decided by moments about the other, which is a determinate calculation only as long as both hooks stay where they are.

They do not. Cranes deflect, their booms shorten under load, their outriggers settle, and the two machines are not built to the same tolerance. A tandem lift is an indeterminate problem in which the redundancy is resolved by the two cranes’ load–deflection curves, and the outcome is that one of them takes more than its share and neither operator knows which. It is the four-legged sling again with the legs replaced by machines, and the industry’s answer is the same: derate both cranes substantially, and assign one person to watch the geometry rather than the load.

There is a second effect that has no counterpart in the single-crane case. If the load tilts — because one crane lifts slightly faster, or one boom deflects more — the geometry changes and the sharing changes with it, and it can change in the direction that makes the tilt worse. That is a stability question rather than an equilibrium one, and it is why tandem lifts are made with both hooks well outboard of the centre of gravity rather than close in.

Where the model stops #

Nothing here is dynamic. Every force above is the static one, and a real lift accelerates. Hoisting a load off the ground applies it suddenly, which doubles the force in the limit; snatching a load that has gone slack does worse. Lifting standards handle this with a dynamic factor of 1.1 to 1.6 depending on the crane class and how the lift is made, and the factor multiplies everything in this essay.

The legs are not two-force members at their ends. A sling round a hook bends over it, a shackle carries the force through a pin with friction, and a chain sling’s links transmit force through a contact patch. None of that changes the line of action enough to matter for the arithmetic, and all of it decides the sling’s own capacity, which falls sharply with the bend radius it is taken round.

And the object is being lifted from points it may not have been designed for. A precast unit has cast-in lifting sockets, which are designed; a steel member is often picked up wherever a sling can be put round it. The local check at the pick point — bearing, tearing, block shear on a lifting lug — is a connection design in its own right, and it is done on a drawing that says “lift here” rather than on one that says how hard.

The day the structure is weakest #

The reason this arithmetic belongs in a collection about statics rather than in a rigging manual is that it is the case where a structure’s design condition and its worst condition are different structures.

A frame is designed as a braced, connected, restrained assembly. It is erected as a sequence of pieces, each of which is at some point supported by two ropes and a crane, with no bracing, no continuity and no lateral restraint. The most dangerous day is before it is finished, and it is dangerous because on that day the structure is a mechanism plus friction rather than a structure.

The rigging arithmetic is the bridge between those two conditions. It converts a weight and a geometry into a set of forces on the piece, and the piece then has to be checked for them as though they were a load case — because they are one. The check is nearly always about buckling and about local bearing, and it is nearly never in the calculation package for the finished structure.

The other place this arithmetic decides a design outright is a precast concrete unit, and it decides it in a direction nobody expects. A wall panel is designed for wind and for the load above it, both of which act in its own plane or across its thickness. It is lifted flat off a casting bed, which puts it in bending about its weak axis under its own weight, with a suction force from the mould on top of it — and for a thin panel that is the largest bending moment it will ever see. Panels are reinforced for the lift and not for the building, which is the clearest case in this collection of a structure whose governing load case is a manufacturing operation.

The generalisation #

The sentence worth carrying away is one line long and it is not about cranes.

A force resolved into a wanted component and an unwanted one still contains both, and the unwanted one has to be carried by something. The sling wants a vertical component and produces a horizontal one, which the lifted object carries. A stay wants a vertical component and produces a horizontal one, which the deck carries as compression. A guy wants to restrain a mast and produces a vertical one, which the mast carries. An inclined brace does the same to its beam.

In every case the unwanted component is proportional to $\cot\beta$ and therefore unbounded as the member goes flat, which is why every one of those structures has a minimum angle in its rules and why the number is always somewhere between fifteen and thirty degrees. It is the same curve each time.