Structural Engineering & Statics
where the load goes
Equilibrium

The angle that doubles the force

A crane hook takes the weight and nothing else. What holds the load is a set of legs running down to it at an angle, and each leg is a two-force member — so at thirty degrees from the horizontal each of two legs carries the entire load, and the difference goes into the thing being lifted as compression.

Assumes The member with only one direction, Six equations, and the drawing shows three and The free body is a choice, and choosing it well is the whole skill.

17 min read 9 figures The statics of things that do not moveThe load must go somewhereOne support too many

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.

At thirty degrees each leg carries the whole loadA 100 kN lift on two legs at 30 degrees to the horizontal. Each leg is a two-force member, so its force is the vertical share it carries divided by the sine of its angle: T = W/(2 sin β) = 100.0 kN, which is 1.00 times the whole load in each. The curve on the right is that division, and it is the reason the rule about sling angles exists rather than a convention: at sixty degrees a leg carries 0.58 W, at forty-five 0.71, at thirty exactly 1.00, and at fifteen 1.93. And the horizontal components do not disappear — they run through the thing being lifted, which here carries 173.2 kN of compression between the two pick points. That is what a spreader beam is for: it takes the compression as a designed strut so the legs above it can stand up.hook100 kN100.0 kN100.0 kN30°173.2 kN of compression, in the load2040608000.511.52sling angle (degrees)leg force ÷ load30°
Fig. 1 A hundred-kilonewton lift on two legs at thirty degrees to the horizontal. Each leg carries the whole load, and the object carries 173 kN of compression between the pick points.

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 β\beta with the horizontal and the load is shared equally,

T=Wnsin⁡βT = \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 Tcos⁡βT\cos\beta on the object, pulling its two pick points towards one another, so the object carries

H=Tcos⁡β=Wcot⁡βnH = 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 polygon closes at every angle, and gets bigger at all of themThe force polygon for a 100 kN lift at four sling angles, drawn tip to tail. Every one of them closes — equilibrium is satisfied at any angle, which is exactly why the angle is not decided by equilibrium — and every one of them is larger than the last. The vertical side is the same in all four, because it is the weight; the two inclined sides grow as 1/sin β, and the polygon flattens towards a straight line as the legs go flat. At 20 degrees the legs carry 1.46 times the load each. The drawing makes the reason visible in a way the formula does not: a flat polygon needs long sides to close a short height.60°58 kN a leg0.58 × the load45°71 kN a leg0.71 × the load30°100 kN a leg1.00 × the load20°146 kN a leg1.46 × the loadthe weight is the same in all four; the legs are not
Fig. 2 The same lift at four angles, drawn tip to tail. Every polygon closes — equilibrium is satisfied at any angle, which is exactly why the angle is not decided by equilibrium — and every one is larger than the last.

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 column curveFailure load against slenderness, as a fraction of the squash load. A stocky column crushes; a slender one buckles at the Euler load; the crossover is where the two curves meet, and real columns fall below both near it.5010015020000.20.40.60.811.2slenderness (effective length ÷ radius of gyration)they cross at λ = 76squashingEuler bucklingreal columns, which are neither
Fig. 3 The curve the lifted object has to be checked against, on its weak axis, over a length nobody chose. A slender member picked up flat is a strut whose load was put there by the rigging.

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 Wcot⁡βW\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.

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.

A beam, its loads and its reactionsA free body: the beam cut away from everything it touches, with the forces that were doing the touching drawn on it. The reactions are computed from the loads, so their arrows are to scale relative to each other.1289.510.5ΣM about one support gives the other reaction; ΣF then gives the first
Fig. 4 The free body that fixes the attitude of a hanging load. Moment equilibrium about the hook has exactly one solution and the object finds it whether or not anybody planned for it.

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.

One counterweight, two governing cases, and neither is balancedA jib crane whose trolley runs between 3 m and 40 m. A counterweight is a moment, so it can balance the load at one radius only: sized for the mean radius it is 3130 kN, and the structure then carries 1110 kNm one way with the load out and 1290 kNm the other way with the jib empty. The empty case is the one people forget, and it is the larger of the two here: a crane with nothing on the hook is a crane leaning backwards.3130 kN60 kNor here40 mbalanced at 21.5 mempty jib -1290 kNm · load in -1110 · load out 1110the governing case is "empty jib", at 1290 kNm
Fig. 5 Stability as a question about where the resultant is, rather than about how strong anything is. A hanging load answers the same question about the hook that a wall answers about its toe.

Where the equal-share assumption comes from

The formula at the top of this essay divided the load by nn, 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.

Three legs, three equations, one answerA rigid top on three legs carrying 100 kN at (0.4, 0.25) m. The three equilibrium equations available — one vertical and two moments — leave three unknowns, so the system is exactly determinate and the reactions are 60.4, 3.1, 36.5 kN. Move the load anywhere and the answer moves with it; nothing about the legs' stiffness enters.60.4 kN3.1 kN36.5 kN100 kNthree legs · rank 3 · determinateΣV, ΣMx and ΣMy all close to 1e-14 kN
Fig. 6 Three legs to one hook, which is the largest arrangement whose forces come out of equilibrium alone. Every leg force here is a result; on four legs it would be an assumption.

Past three, the division by nn 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.

Four legs, three equations, and a millimetre decides the restA 100 kN load on a four-legged sling at 60 degrees. All four leg forces pass through the hook, so they contribute no moment about it and the only equations available are the three of force equilibrium — four unknowns against three equations, which is one degree of static indeterminacy. What settles it is stiffness, and a leg's stiffness is EA/L: a mismatch of 1 mm in a 2.40 m leg is 3.3 kN transferred from one diagonal to the other, on a nominal 28.9 kN a leg. The two stiff legs go to 32.1 kN and the slack pair to 25.6, and at a mismatch of 8.8 mm the slack pair carries nothing at all. Which is why the honest design assumption is that two diagonal legs carry everything — 57.7 kN each, drawn as the line — and why a four-leg sling is rated as though it had two.hook32.125.632.125.6100 kN1234an equal quarterdesigned on twomismatch 1 mm
Fig. 7 Four legs, three equations, and a millimetre of manufacturing tolerance deciding the rest. The two stiff legs go up and the slack pair goes down, and at fifteen millimetres the slack pair carries nothing at all.

What settles an indeterminate problem is stiffness, and a wire rope leg’s stiffness is EA/LEA/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.

Counting unknowns against equationsThree frames differing by one member. Two equilibrium equations per joint, one unknown per member and one per restraint: fewer unknowns than equations is a mechanism, equal is solvable by statics, more needs stiffness.m 4 + r 3 − 2j 8 = -1a mechanismm 5 + r 3 − 2j 8 = 0statically determinatem 6 + r 3 − 2j 8 = +1one member too manystatics can answer only the middle case
Fig. 8 Counting restraints against equations. A structure part-way through erection has a different count from the finished one, and the difference is often the whole of the risk.

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⁡β\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.

The moment is the force times the distanceOne force applied at five distances from a pivot, with the moment it produces drawn as a bar. The force never changes; only the arm does, and the moment follows it exactly.pivotthe same force of 100, moved along the levermoment about the pivot1100 × 1 = 1002100 × 2 = 2003100 × 3 = 3004100 × 4 = 400
Fig. 9 The other half of every rigging calculation: a force times an offset. Where the sling angle decides the magnitude, the geometry of the pick points decides how it is shared.

Where this goes next

200 kN200 kNthe chord — and the line of action1.20 mbending moment = P × offsetpeak 240 kNmbending stress is 24.0 times the axial stress at the worst section The member with only one direction If a body is in equilibrium under forces applied at exactly two points, those forces are equal, opposite and along the line joining the points. It is three conclusions from two equations, it is the shortest real theorem in statics, and nearly everything on this site depends on it without saying so. the loadroller: vertical onlypin: any directionall three lines meet here Three forces must meet at a point, and a drawing can find it A body held by exactly three forces has their lines of action concurrent. That is a theorem, it is enough to solve for direction and magnitude, and for a century it was done with a straightedge. 02040608000.20.40.60.811.21.41.6brace stiffness (units of EI/L³)effective length factor k23.2 EI/L³ = 1.41 N꜀ᵣ/L0.77 — held1.32 — freepast the threshold the frame buckles in a mode the brace does not hold The most dangerous day is before it is finished A structure is analysed once, complete, with every restraint present. It spends weeks in states nobody drew — a beam landed with no deck on it holds 17% of the moment its section is worth, a frame not yet braced buckles at a third of the load it will, and a bolt not yet tightened is a pin where the analysis assumed a fixity. 56.8 kN13.6 kN49.6 kN120 kNthree legs · rank 3 · determinateΣV, ΣMx and ΣMy all close to 1e-14 kN Six equations, and the drawing shows three Ninety essays have taken place on a piece of paper, where equilibrium is three equations and a structure is a diagram. The real object has six, the extra three are the ones nobody writes down, and the difference between three legs and four is not a matter of degree.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

BucklingCentre of gravityEquilibriumErectionForce polygonFree bodyGraphic staticsIndeterminacyLoad pathSling angleSpreader beamStiffnessThree dimensional equilibriumToleranceTwo force member