Structural Engineering & Statics
where the load goes
Concept

Centre of gravity — where it appears

The point through which a body's whole weight can be taken to act. A hanging load rotates until it lies directly beneath the hook, because that is the only attitude in which the moments balance, so a lift whose pick points do not straddle it arrives out of level.

Named by 4 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 At thirty degrees each leg carries the whole load. A 100 kN lift on two legs at 60 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) = 57.7 kN, which is 0.58 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 57.7 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.

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.

equilibrium · Rigging A fourth power, and then a cliff. The factor of safety against rolling, against beam length, for one section hung from a roll axis 0.9 m above its centre of gravity. Nothing about the section changes along this axis. z̄ goes as the fourth power of the length — 0.236 m at 30 m becomes 0.747 m at 40 — and the factor of safety is proportional to (y_r − z̄), so it does not decline gently: it falls away and then stops existing. The working factor of 1.5 is lost at about 41 m, and past 42 m there is no hook height at all at which this beam hangs stably. Which is why long girders are lifted with the picks moved inboard, or with the beam braced, or not in one piece.

Hung from above and still unstable

A rigid body hanging from a point above its centre of gravity is a pendulum and cannot fall over. A beam is not rigid, and tilting it puts a component of its own weight sideways — which bows it, which moves its centre of gravity further out. Past a length there is no hook height at which it hangs stably at all, and the length arrives as a fourth power.

stability · Lift stability A cambered girder is an arch on its slings. A 40 m precast girder of 11 kN/m, lateral stiffness 2.55 × 10⁵ kN·m², hung 0.90 m below its roll axis, picked 2 per cent of its length in from each end, with 100 mm of camber at mid-length, drawn in elevation with the camber exaggerated. The girder's centre of gravity is two-thirds of the camber above the line through its ends, 67 mm; its lifting points, on the arc 2 per cent in from each end, are 8 mm above it. It rolls about the line through the picks (dashed), so the camber has lifted its centre of gravity 59 mm towards the roll axis: the girder hangs as though its hook were that much lower.

The camber that lowers the hook

A camber is built into a girder so that it ends up level in the finished structure, and nothing about it matters until then — except on the day it is lifted. A cambered girder hanging from its ends is a shallow arch, its centre of gravity two-thirds of the camber above the line through its lifting points, and it rolls about that line as though its hook were that much lower. On a long precast girder already close to its lifting limit, a hundred millimetres of camber takes the factor of safety against roll from 2.3 to 1.4. The camber grows while the girder waits in the yard, so the same girder becomes harder to lift every week it is stored. Pick it a fifth of its length in from each end and the camber drops out of the problem altogether.

deflection · Camber

Named alongside it

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

EquilibriumFree bodyLift stabilityStabilityBalanceBearingBucklingCamberConstruction sequenceCounterweightCreepDead load

All concepts