Structural Engineering & Statics
where the load goes
Equilibrium

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.

Assumes Weight is the only thing resisting it, Moving a force, and what it costs and The free body is a choice, and choosing it well is the whole skill.

16 min read 8 figures The statics of things that do not moveThe load must go somewhereThe load that will not hold still

A bascule bridge is a beam that has to lift. Its leaf weighs 900 kN, its centre of gravity is nine metres from the trunnion it turns about, and the moment that has to be overcome to open it is 8,100 kNm — which, delivered through machinery, would need a motor and a gear train of a size nobody wants to buy or maintain.

So a weight is hung on the other side of the pivot. Three metres out, 2,700 kN of it, and the moment about the trunnion is zero at every angle of opening. The machinery now has to overcome friction and inertia and nothing else.

That is the whole of what balancing achieves, and it is worth being exact about it because two other quantities went in the wrong direction while the moment went to zero.

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.
Fig. 1 The balanced leaf, with the two quantities the balance did not improve marked on it. The moment about the trunnion is zero at every angle. The force on the trunnion is 3,600 kN, four times the leaf’s own weight. And the rotational inertia — the thing the machinery actually has to accelerate — is a third larger than the leaf’s alone.

Which free body produced the number

Draw the whole moving assembly — leaf, counterweight and the structure connecting them — and take moments about the trunnion.

The leaf’s weight acts at aa on one side and the counterweight’s at rr on the other, and both lever arms shorten by the same cosine as the assembly rotates:

M(θ)=(Wa−Cr)cos⁡θM(\theta) = (Wa - Cr)\cos\theta

Balance is Wa=CrWa = Cr: a product, not a weight. There is no counterweight that balances a leaf; there is a family of them, and the family is a hyperbola. 2,700 kN at three metres, 1,350 at six, 810 at ten — all of them balance the same leaf exactly, at every angle.

The cosine is worth noticing rather than cancelling. It is there because both weights are vertical and both lever arms are horizontal projections of arms that rotate together, and it means balance achieved at one angle is balance at all of them. That is a property of this geometry and not of balancing in general: a counterweight on a crank, or one whose arm is not collinear with the leaf’s, balances at two angles and nowhere else.

Choosing where on that hyperbola to sit is the counterweight designer’s only real freedom, and nothing in the moment equation helps make the choice, because a moment is a product and the product is the only thing it constrains. The two quantities that do decide it are both invisible in it, and the way to see them is to draw the same balanced leaf at a different point on the same hyperbola.

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 810 kN at 10 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 1.9 times the leaf's own weight, since both weights are still there. And the rotational inertia rises by 111%, 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 405 kN and carries 1.53 times the inertia.
Fig. 2 The same leaf balanced by 810 kN at 10 m instead. The moment about the trunnion is still exactly zero at every opening angle, because the product is unchanged. The reaction on the bearing has fallen from 4.0 times the leaf’s own weight to 1.9 — and the rotational inertia, which was 33% above the leaf’s alone, is now 111% above it. One choice along a curve of constant moment, and the two consequences move in opposite directions.

The reaction goes up, not down

Sum vertical forces on the same free body and the counterweight does the opposite of what it does to the moment.

R=W+C=900+2,700=3,600 kNR = W + C = 900 + 2{,}700 = 3{,}600\ \text{kN}

Four times the leaf’s own weight, at a bearing that had to be provided anyway. The trunnion of a balanced bascule is one of the most heavily loaded bearings in civil engineering, and the reason is entirely visible in one line of arithmetic: the moment cancelled because the two weights are on opposite sides, and the force added because they are on the same side of the only equation that counts them.

This is the general and slightly counter-intuitive statement about all self-balancing structures. Balance removes a moment by adding mass, and every other consequence of that mass remains. The foundation is bigger, the bearing is bigger, the pier is bigger, and the seismic mass — which nobody was thinking about — has quadrupled.

The bearing is therefore the one quantity that wants the counterweight light, which by the hyperbola means far out. Halving it is available and the price is stated on the same drawing.

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 1350 kN at 6 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 2.5 times the leaf's own weight, since both weights are still there. And the rotational inertia rises by 67%, 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 675 kN and carries 1.40 times the inertia.
Fig. 3 The middle of the family: 1,350 kN at 6 m, the same 8,100 kNm of balancing moment as every other member of it. The trunnion reaction is 2.5 times the leaf’s weight rather than 4.0 — a bearing carrying 2,250 kN instead of 3,600 — and the inertia is 67% above the leaf’s alone rather than 33%. Sum the moments and the two weights subtract; sum the forces and they add, which is why a structure can be moment-free and heavily loaded at once.

Balance makes it harder to move, not easier

The second thing that went the wrong way is the one the machinery cares about most.

An unbalanced leaf’s machinery has to supply a moment against gravity, which is large and known. A balanced leaf’s machinery has to supply a moment against inertia, which is what accelerates and decelerates the assembly, and the counterweight contributes to it as Cr2/gCr^2/g.

For the leaf drawn: the leaf alone has Wa2/g=7,430Wa^2/g = 7{,}430 tonne·metres squared. Add the counterweight at three metres and the total is 9,900 — 33 per cent more inertia to accelerate, from the object provided to make it easier to move.

And that is the good case, because the radius was chosen small. Balance the same leaf with 1,350 kN at six metres and the counterweight’s own inertia quadruples: the total goes to 12,400, two thirds more than the leaf alone. The lightest counterweight is the one furthest out, and it is the worst one to have to start and stop.

That trade — a heavy weight close in against a light one far out, with the same balance and different dynamics — is the design decision the whole subject reduces to, and neither the moment equation nor the force equation contains 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 4050 kN at 2 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 5.5 times the leaf's own weight, since both weights are still there. And the rotational inertia rises by 22%, 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 2025 kN and carries 1.18 times the inertia.
Fig. 4 The end of the family the machinery wants: 4,050 kN at 2 m, balancing the same leaf. The inertia penalty has fallen to 22%, the smallest of any arrangement drawn here, and the trunnion reaction has risen to 5.5 times the leaf’s own weight — 4,950 kN through one bearing. The motor gets its easiest leaf and the bearing gets its hardest, from the same decision.

Set the four drawings beside each other and the pattern is a line rather than a set of cases. At two metres the reaction is 5.5 times the leaf and the inertia penalty 22 per cent; at three, 4.0 and 33; at six, 2.5 and 67; at ten, 1.9 and 111. Nothing in between is unavailable and nothing on the curve is wrong. What decides it is which of the two numbers is being bought.

20 kN applied at once and held, on a structure of 0.500 s period. Displacement against time for a single-degree-of-freedom structure of natural period 0.500 s and 2.0% damping, under 20 kN applied at once and held. The static deflection under the same peak force is 12.67 mm and the peak response is 24.56 mm — a factor of 1.94.
Fig. 5 What an inertia does to a structure that is asked to move. Everything above is statics; the moment being zero says nothing about the forces that appear while the assembly is accelerating, and those are proportional to the inertia the balance has just increased.

The structure between the two weights is not balanced at all

The moment about the trunnion is zero, and it is worth asking exactly where that statement is true, because it is true at one point and nowhere else.

Cut the assembly on the counterweight side of the trunnion — a vertical plane between the bearing and the counterweight arm — and take the counterweight alone as the free body. It weighs 2,700 kN acting three metres from the cut, so the section carries 8,100 kNm of bending and 2,700 kN of shear. Cut on the leaf side instead and the leaf alone gives 900 kN at nine metres: the same 8,100 kNm, the other way round.

The balancing has not reduced a single internal force. It has arranged for two equal and opposite demands to meet at the bearing so that the bearing does not have to supply a moment — which is precisely what a bearing is bad at and precisely why the arrangement exists — and the structure on either side of it carries the full couple exactly as it would have without any counterweight at all.

That is the free-body argument this collection keeps returning to, arriving with an unusually sharp edge. A quantity is internal or external according to where the boundary was drawn, and here two different boundaries give two answers that are both correct and read as opposites. Take the whole assembly and the moment is zero; take either half and it is 8,100 kNm. Nothing about the steel changed between the two sentences.

Two practical consequences follow, and both are routinely missed by people who have just been told the leaf is balanced.

The connecting structure is sized by the unbalanced moment. The girders running from the trunnion to the counterweight carry 8,100 kNm and are as heavy as a cantilever holding the counterweight up — because that is what they are. The saving from balancing appears in the machinery and the bearing’s moment capacity, and nowhere in the steelwork tonnage.

And a heavy counterweight close in is worse for that structure than a light one far out. The bending is CrCr either way — the same 8,100 kNm — but the shear on the counterweight arm is CC, and the arm is shorter, so the heavy option puts more shear over less length and concentrates a very large force into a short cantilever. Which adds a fourth quantity to the three tabulated below: the moment is indifferent to the choice, the reaction and the inertia want opposite things, and the connecting structure wants the counterweight light and far out for shear while wanting it close in for its own self-weight moment. There is no arrangement that is best for everything, which is why the pit usually decides it.

A load that moves cannot be balanced

The bascule is the easy case, because the two weights are fixed relative to one another. A crane is not.

A jib crane has a fixed jib and a trolley that runs along it, and the moment the counterweight has to oppose depends on where the trolley is. Choose the counterweight for the mean radius — jib plus half the load’s travel — and the results are symmetric and unhelpful:

  • Load at the smallest radius: 1,110 kNm of moment one way.
  • Load at the largest radius: 1,110 kNm the other way.
  • Empty jib: 1,290 kNm, backwards, and the largest of the three.

The last is the case that gets forgotten, because it does not feel like a load case at all. A crane with nothing on its hook is a crane whose counterweight is unopposed, and it is leaning backwards harder than it ever leans forwards.

Every structure that carries a moving load has this shape of problem, and the general form is an influence line: for every quantity being designed there is a position of the load that maximises it, and the positions are different for different quantities. What the counterweight adds is a permanent action of the opposite sign, which turns a set of maxima into a set of maxima and minima, with the zero-load case sitting at one extreme.

One counterweight, two governing cases, and neither is balanced. A 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.
Fig. 6 The three cases, and the one that is easy to forget. The counterweight is a permanent action and the load is a variable one, so the governing case for one direction has the load on and the governing case for the other has it off. A tower crane is ballasted for both, and the empty case is often the one that sizes the ballast.

What the counterweight is really for

There are three different reasons to add a counterweight, and they want different answers.

To reduce the machinery. This is the bascule’s reason, and it wants exact balance — or slightly less, so that the leaf closes under gravity if the drive fails.

To reduce the moment on a foundation. This is a crane’s reason and a retaining structure’s, and it wants the worst moment reduced rather than the mean one cancelled. That is an optimisation over load cases, and its answer is not the balance point.

To hold something down. This is uplift’s answer and a cantilever’s during construction: the weight is there to keep a resultant inside a base, not to cancel anything. It is sized by a factor of safety on a ratio of two weights.

The three get called the same word and the arithmetic is different in each. The first has an exact answer, the second has an optimum, and the third has a factor.

Weight is the only thing holding it down. A body 4 m wide and 12 m tall weighing 900 kN, under a wind pressure of 1 kN/m². The wind delivers 36 kN and an overturning moment of 216 kNm about the leeward toe; the weight restores 1800 kNm, a factor of 8.33. The resultant lands 0.24 m from the centre against a middle third of ±0.67 m, so the base is still wholly in bearing.
Fig. 7 The overturning check the third reason belongs to. A ratio of a restoring moment to a disturbing one, with no material strength in it anywhere. A counterweight raises the numerator; whether that is what is wanted depends on which of the three problems is being solved.

The counterweight during construction

There is a fourth case, and it is the one that has caused the most trouble in practice: the balance that is correct in the finished structure and wrong on the way there.

A balanced cantilever bridge is built outward from a pier in segments, and it is balanced only if the two arms grow together. One segment ahead on one side is a moment of the segment’s weight times its lever arm, at a pier designed for a balanced condition — which is why balanced-cantilever construction uses temporary props, temporary ties, or a counterweight that is moved as the arms grow.

A bascule under repair with its deck removed is unbalanced by the weight of the deck, in the direction the counterweight pulls, at a bearing carrying the counterweight’s full moment.

The general statement is one this collection keeps arriving at: a structure is a sequence rather than a state, and a balance is a relationship between two parts that both have to exist.

The props decide where the stress ends up. Bottom-fibre stress in the steel of a 12 m composite beam carrying 12 kN/m of wet concrete and 18 kN/m afterwards. Unpropped, the bare steel takes the first stage alone and reaches 292 MPa; propped, the finished composite section takes everything and reaches 186 MPa — a ratio of 1.57. 62% of the unpropped beam's final stress was locked in before the slab was structural at all. The deflections differ by 1.73 times for the same reason, and no drawing of the finished beam distinguishes the two.
Fig. 8 The same idea in its usual clothes. Load applied to a structure that is not yet the finished one goes where the incomplete structure sends it, and no analysis of the completed geometry can see it. A counterweight is simply a very large piece of that argument.

The three numbers a balance produces, and their different units

It is worth setting the whole thing out as a table of consequences, because the three quantities respond to the same decision in three different ways and the units make it obvious why they cannot all be improved at once.

The moment goes as CrCr — a force times a length. Setting it to zero fixes the product and leaves both factors free.

The reaction goes as CC alone. It is minimised by making the counterweight as light as possible, which means putting it as far out as possible.

The inertia goes as Cr2Cr^2, which with CrCr fixed is proportional to rr. It is minimised by making the counterweight as close in as possible, which means making it as heavy as possible.

So the second and third want opposite things, and the first is indifferent. There is no optimum without a statement of what is being paid for — a bearing, or a motor — and the answer for a bascule bridge (heavy, close in, because the trunnion is a one-off and the drive runs every day) is the opposite of the answer for a machine part on a rotating shaft (light, far out, because the bearing is a catalogue item and the acceleration is continuous).

That is a small instance of a rule that runs through this whole collection: a design decision is only well posed once the objective is stated, and a great many arguments about the right answer are arguments about which quantity was being minimised.

Where the model stops

The assembly is rigid. A real bascule leaf deflects, and its centre of gravity moves as it does — by a few millimetres on a nine-metre arm, which is a few thousandths of the balancing moment and is buried in the tolerance.

The pivot is frictionless. It is not, and the friction moment at a trunnion carrying 3,600 kN is not small. In practice it is the reason a bascule is deliberately balanced slightly out — a small permanent moment ensures the leaf seats itself rather than floating on the drive.

And the loads are dead. Wind on an open leaf is a large moment about the trunnion, applied to a structure whose gravity moment is zero, and it is the governing case for the machinery brake rather than for the drive.

Why the empty case is forgotten

The empty-jib case deserves a paragraph on its own, because the reason it gets missed is structural rather than careless.

A load case is normally assembled by adding things. Dead load, then imposed load, then wind, each with a factor, each making the answer larger. The habit of the whole exercise is additive, and the arrangement that maximises an effect is the arrangement with the most load on it.

A counterweight breaks that habit, because it is a permanent action whose effect has the opposite sign to the variable one. The worst case for the reversed direction is therefore the case with the minimum variable load, combined with the maximum permanent one — which is the arrangement a designer working additively will never construct.

Codes handle this by requiring two partial factors on permanent actions, a large one and a small one, and by requiring the unfavourable one to be applied where the permanent action helps. It is a rule that reads as bureaucratic and is the entire content of the paragraph above.

The same trap appears wherever a permanent action opposes a variable one: an uplift check where the building’s own weight is the resistance, a retaining wall whose surcharge helps, a cantilever whose back span holds it down. In every one of them the dangerous case is the light one.

What the pictures cannot show

The counterweight is drawn as a disc on an arm. In a real bascule it is a block of concrete inside the pit below the road, and it swings up into the space the leaf leaves. Its size is constrained by that pit rather than by the arithmetic, and the pit is usually the reason the radius is short and the weight is large.

Nor can any figure here show the tolerance. A leaf is balanced to within a few per cent by calculation, and to something better than that on site by adding or removing ballast until it behaves. Balance is one of the few structural quantities that is adjusted after construction by measurement.

One more thing no figure here contains: the counterweight is usually not one object. A bascule’s is cast in a pit and adjusted with removable blocks; a tower crane’s is a stack of individual slabs whose number is set on site from a chart; a mobile crane’s is bolted on in sections and taken off again for transport. The single hatched disc on the drawing is a family of configurations, and the structure has to be safe in each of them — including the one where somebody has fitted the wrong number of slabs.

The assumption the figure rests on

The leaf’s centre of gravity is at nine metres and treated as known. It is a weighted mean over a deck, a set of stringers, a surfacing, a kerb, a railing and whatever services were run along it — every one of which is estimated at design and different when built. An error of 200 mm in that arm is 180 kNm of residual moment, which is 2 per cent of the unbalanced value and a permanent load on machinery designed for none.

The ladder from here

Later rungs on this anchor: the moving counterweight, which balances a load at every position and is how a level-luffing crane works. Balance about two axes at once, which is what a slewing crane needs. The counterweight as a dynamic mass, where its inertia is not a nuisance but the point — a tuned mass damper is a counterweight with a spring. Balanced cantilever construction and the out-of-balance moment that governs the pier. And the aircraft and lift versions of the same arithmetic, where balance is about a centre of gravity within a permitted envelope rather than about a moment being zero.

Where this goes next

Weight is the only thing holding it down. A body 2.5 m wide and 6 m tall weighing 120 kN, under a wind pressure of 1 kN/m². The wind delivers 48 kN and an overturning moment of 144 kNm about the leeward toe; the weight restores 150 kNm, a factor of 1.04. The resultant lands 1.20 m from the centre against a middle third of ±0.42 m, so the base is lifting over 2.35 m of its width. Weight is the only thing resisting it A structure that is strong enough everywhere can still be blown over, and nothing in its material properties has any part in whether it is. The whole answer is a weight and a width — and the failure begins long before anything tips, at the moment one edge stops pressing down. A general force system is a screw, not a force. Two forces that are neither parallel, nor concurrent, nor coplanar. Their resultant is (30, 0, -60) and no point in space reduces them to it alone: the moment left on the central axis is (14.40, 0.00, -28.80), which is parallel to the resultant and has a pitch of 0.4800. The simplest object equivalent to the system is therefore a force and a couple about the same line — a wrench — and a planar drawing can never produce one, because a coplanar system has its moment perpendicular to its resultant by construction. Moving a force, and what it costs Every free body begins by putting a force somewhere convenient. That move is free along the force's own line, costs a couple across it, and in three dimensions leaves behind something no choice of point can remove. The same beam, cut at x = 5. A beam separated at one station. On the exposed face a shear force and a bending moment appear, equal and opposite on the two pieces, with values obtained by summing the forces on whichever piece is easier. The free body is a choice, and choosing it well is the whole skill Cutting a structure open is not a step in the method. It is the method — and where the cut is made decides whether the answer takes one line or twenty. Influence line for the bending moment at x = 3. The bending moment 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 = 3.00, giving 2.100. The worst place to stand A bridge is not designed for a load. It is designed for a load that moves, and for every station along it there is a different position of that load that does the most damage.

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BalanceBearingCentre of gravityCounterweightDead loadEquilibriumFree bodyLever armLoad reversalMomentMoving loadOverturningReactionRotational inertiaStability