The bridge that does not feel the boat

The Pontcysyllte aqueduct in north Wales carries the Llangollen Canal across the valley of the Dee, thirty-eight metres up, in an iron trough barely wider than a narrowboat. Boats cross it in single file, and visitors on the towpath look down over the railing at the river far below. Thomas Telford designed it in the 1790s, and one of the things he did not have to design for was the weight of the boats. A barge crossing an aqueduct adds nothing to the load the structure carries.

This is Archimedes’ principle applied to the structure rather than to the boat. The weight of the water that is not there set it out: a floating body sinks until it has pushed aside its own weight of water. Where does that water go? In an aqueduct joined to a canal at each end, it flows out into the canal. The trough holds a little less water and a boat, and the boat weighs exactly what the missing water did.

A load that leaves as it arrives #

The clearest case is a boat lift, a structure that carries a tank of water with boats floating in it from one level of a canal to another.

The two lines are the whole idea. A caisson whose gates are open to the canal while the boat floats in keeps its water level, because the canal holds the level, so the boat’s arrival pushes water out of the caisson and into the canal. When the end gates close, the caisson holds the same depth of water it always holds, part of it replaced by a boat that weighs the same. If instead the end gates were closed first and the boat lowered in by a crane, the water would have nowhere to go, the level would rise, and the caisson would gain the boat’s whole weight. What matters is not whether there is a boat but whether the level is fixed while the boat enters.

That makes a boat lift fundamentally different from a goods lift. A goods lift carrying a ten-tonne lorry has to lift ten more tonnes. A boat lift carrying a ten-tonne barge lifts the same weight it lifts when empty, the weight of a full caisson. Pairs of caissons can therefore be balanced against each other permanently, and the boats on either side never upset the balance.

A wheel balanced by a water level #

The most spectacular example is the Falkirk Wheel in Scotland, opened in 2002, which lifts boats twenty-four metres between two canals by rotating two caissons on opposite ends of a pair of arms, like a giant fairground wheel with two gondolas. The two caissons always balance, whatever boats they hold, and the wheel is turned by a modest electric drive that has only friction and a small residual imbalance to overcome.

The residual imbalance comes from the water level, not from the boats. If one caisson holds a centimetre more water than the other, it carries 1.6 tonnes more, and on an arm of 17.5 metres that is a quarter of a meganewton-metre of turning moment. So the operation of such a lift depends on getting the levels right when the caissons are joined to the canals at the top and bottom. There is a second level that has to be kept: the caissons themselves must stay horizontal as the arms sweep round, or the water would run to one end, shift the centre of mass and slosh the boats against the end gates. At Falkirk each caisson is held level by a train of gears linking it to a fixed central gear, so that it turns against the arm exactly as fast as the arm turns, the way the gondolas of a Ferris wheel hang upright. The water then stays level and still throughout, and the balance holds at every angle of the wheel. The figures here are of Falkirk’s general proportions rather than its exact specification, but the principle is the one its engineers used: the lift needs to measure water depth to a few millimetres and need never weigh anything.

Driven by a foot of water #

The same fact was used in reverse, a century earlier, to power lifts with nothing but water. The hydraulic lifts of the late nineteenth century — at Anderton in Cheshire, opened in 1875, on Belgium’s Canal du Centre, and at Peterborough on the Trent–Severn Waterway in Canada — stood their two caissons on hydraulic rams whose cylinders were joined by a pipe. Each caisson pushed on its ram, the rams pushed on the water in the cylinders, and the pipe let the water flow from one cylinder to the other, so that one caisson went down as the other went up.

To make the lift move, the operators let the upper caisson take on a little more water than the lower one, about a foot at Peterborough. The extra water, a few tens of tonnes, is the only imbalance in the system, and it drives the upper caisson down and the lower one up. When they arrive, the extra water is let out into the lower canal. The lift is powered entirely by water falling from the upper canal to the lower, a few tonnes of it per operation, and it works the same way whether the caissons carry pleasure boats or heavy barges. Force multiplied and nothing gained described the hydraulic principle that let a small pressure difference move a large weight; here the large weight is moved by a small difference in water depth, and the boats do not appear in the balance at all.

The aqueduct and the reach #

An aqueduct is a boat lift that does not move: a trough of water joined permanently to canals at both ends.

When a boat enters a canal reach, it raises the level of the whole reach by its displaced volume divided by the reach’s surface area. For an eighty-tonne boat on ten kilometres of canal six and a half metres wide, that is about a millimetre. The aqueduct, part of the reach, carries water to that level whether the boat is on it or not, and its load changes by the weight of a millimetre of water over its own length, which is negligible. A road bridge carrying an eighty-tonne lorry carries the lorry. An aqueduct carrying an eighty-tonne barge carries water, as it always does.

The engineering consequence is that an aqueduct’s design load is its water, which is large and constant, rather than its traffic, which is small and variable. Telford’s iron trough at Pontcysyllte had to hold back the pressure of 1.6 metres of water against its sides, which the pressure that only knows depth described, and support the weight of that water across each span. It never had to support a boat.

When the boat moves #

The argument so far is static, and boats move. A moving boat in a narrow channel changes the water around it in a way that can matter.

As a boat moves along a channel, the water it displaces has to flow backwards past it, through the gap between hull and channel walls. The narrower the gap, the faster that return flow, and by Bernoulli’s principle faster water stands lower. The water level drops alongside the boat, the boat sinks with it — the effect called squat, which makes a boat in a narrow canal sit deeper and steer badly — and the water piles up ahead of it in a bow wave. Above a certain speed, which depends on how much of the channel the boat fills, there is no steady flow at all: the boat cannot push water past itself fast enough and starts to drive a wave ahead of it. The speed below which nothing on water leaves a wake followed the waves a moving body makes in open water; in a channel the limit comes much sooner, because the water has nowhere to go but back past the hull.

For the aqueduct, this means the load is redistributed as a boat moves, lower water beside the boat, higher ahead, but not increased. The total weight of water on the aqueduct still adjusts so that it carries no more than its own water. Canal speed limits, typically a few kilometres an hour, are set by the damage the drawdown and wash do to the banks, and on narrow aqueducts by the boat’s tendency to ground in its own drawdown.

The same principle, everywhere a thing floats #

The fact behind all of this is that a floating object’s weight is carried by the fluid, which passes it on to whatever holds the fluid. That has consequences in less engineered places. The body that displaces two things worked through the puzzles that arise when something floats in a container on a scale, and found the scale’s reading always determined by what the container holds, floating or not. The floating ice that still raises the sea found the subtle exception: floating ice displaces its own weight of salty water, and when it melts into fresh water it takes up a little more volume than it displaced, so even floating ice raises the sea slightly. In a canal the corresponding effect would be a boat that changed its density, which a boat does not.

There is also a pleasing inversion in ship design. Why a ship comes back upright explained how a hull’s shape makes the buoyant force swing to the low side when it heels, creating a righting moment. In a boat lift, the water does the opposite job: it makes the caisson indifferent to what floats in it. The same principle that makes a ship stable makes a lift balanced.

What a lock spends and a lift saves #

The comparison with a lock makes the economy of a balanced lift concrete. A lock raises a boat by filling a chamber with water from the upper canal. For a chamber of the same plan as the hydraulic lift’s caissons, 22.9 metres by 4.6 metres, and a rise of fifteen metres, about the rise at Anderton, each filling takes about 1,580 tonnes of water from the upper level and each emptying sends it down to the lower one. A flight of locks to climb that height would spend that much water for every boat, or for every group of boats sharing a passage.

The hydraulic lift spends the extra foot of water that drives it, about thirty tonnes, and the water needed to top up levels as the caissons join the canals: a fiftieth of the lock’s consumption. That difference is the reason lifts were built at all. A lock is simple, robust and needs no moving parts beyond its gates and paddles; a lift is complicated and expensive; but a canal summit fed by a small reservoir cannot afford to send a lockful of water downhill for every boat, and a lift lets it pass boats almost for free. The lock’s water does real work, raising boats against gravity with water’s own weight, and that work is paid for by the water falling through the full height. The lift’s boats are raised by a falling caisson, and the water that falls is only the small excess that tips the balance.

There is an elegant intermediate that shows the principle in another form. Some locks have side ponds, basins beside the chamber at intermediate levels, into which the upper part of the chamber’s water is drained when it empties and from which it is refilled next time, saving a large share of each lockful. The gates and sluices that do this are the ones the floodgate that opens at a chosen depth analysed, where the turning moment of water on a gate decides when it swings. A side pond and a balanced lift are two ways of not letting water fall further than it has to.

A floating dock and a boat in a box #

A floating dry dock lifts a ship out of the water by the same arithmetic, run deliberately. The dock is a hollow structure that can be flooded to sink beneath a ship, then pumped out to rise, lifting the ship clear. To lift a ship of ten thousand tonnes, the dock must pump out ten thousand tonnes of ballast water beyond what it needs to float itself. The ship’s weight is carried by the dock’s buoyancy, and the dock’s buoyancy is bought by pumping, tonne for tonne. There is no saving in a floating dock as there is in a lift: the ship really is lifted, against gravity, and the work done is the ship’s weight times the height.

A lift avoids that work only because it carries water as well as boats, and lowers as much water as it raises. A caisson rising with a barge is balanced by a caisson falling with a barge or without, since both weigh the same. The energy cost of a balanced lift is the friction in its bearings and the water used to make up levels, which at Falkirk is reported to be small enough that the wheel uses electricity comparable to boiling a few kettles per turn. Locks, by contrast, use water: each passage through a lock sends a lockful of water from the upper canal to the lower, which is why canal engineers on summit levels with little water built lifts.

What the picture cannot show #

The calculations assume water at rest, rigid structures, and boats that float freely. The caisson figures assume the level is held exactly while a boat enters; in practice the boat pushes a small wave ahead of it, and a caisson closed before the wave has settled may carry a few centimetres more or less water than intended, which is why lifts allow time for levels to equalise and measure them before moving. The proportions of the caissons are representative, not the specifications of particular lifts.

The drawdown calculation is the simplest one-dimensional model, with the return flow uniform across the gap and no friction. Real channels have sloping banks, irregular boats and turbulent return flows, and the speeds at which the curves end are approximate; detailed predictions of squat use corrections measured in towing tanks. The model also ignores the boat’s own change in trim, which in practice puts the stern down more than the bow.

And the aqueduct’s load is not entirely independent of its traffic. A boat that grounds in a shallow aqueduct, resting on the bottom, puts part of its weight directly onto the structure, because it is no longer floating. That is the one way a boat can load an aqueduct, and narrow aqueducts maintain their depth carefully for that reason among others.

Still open: how a crowded channel shares its water #

The simple squat model treats one boat in an otherwise still channel. Busy canals and ports have many boats passing and overtaking in confined waterways, and the interactions between their drawdowns and wave systems can draw two boats together or push them into a bank, sometimes violently. These interaction forces are studied with towing-tank experiments and increasingly with computation, because ships have grown much larger relative to the channels they use, and the margins of depth and width that kept older vessels safe no longer apply. How to set speed limits and passing rules in narrow, crowded waterways from first principles, rather than from accident records, is still an active engineering problem.

A floating boat pushes out its own weight of water, so a caisson or aqueduct whose water level is held carries the same weight with a boat or without: 788 tonnes for the lift caisson, 5,265 for the aqueduct, raised by a 1.2 mm rise of the whole reach; a lift is balanced by its water levels, 1.6 tonnes per centimetre, and hydraulic lifts were driven by leaving a foot more water in the upper caisson; a moving boat only redistributes the water, lowering the level beside it. The habit worth carrying is to ask where displaced fluid goes, because the answer decides who carries the weight.