The water the valley floor takes

The angle that saves the most concrete cut an arch dam into horizontal rings and gave each ring the water at its own depth. A ring of radius $R$ under a pressure $p$ carries a hoop thrust $pR$ — every pressure on a curved surface points at the centre, so the ring sees only the chord’s projection, and the force is only a radius — and with an allowable stress the thickness follows, and the least concrete comes at a central angle of 133.6°.

Every metre of water went to a ring. But a dam is one shell, not a stack of rings, and it is fixed along its base to the valley floor as well as along its sides to the abutments. At its crown it is also a vertical cantilever, a strip of concrete standing on the valley floor with the reservoir behind it, and that cantilever can carry water too. This essay asks how the water divides between the two, and what the division does to the ring formula.

Rings and a cantilever that must agree #

The method is the oldest in arch-dam design, the trial load: at each depth the water is split between the horizontal ring there and the vertical cantilever at the crown, and the split is adjusted until the two deflect by the same amount at every depth. A ring deflects in proportion to its share and the square of its radius over its thickness, as a thin ring does; the cantilever, fixed at the base and free at the crest, deflects at each depth by the effect of its share of the water at every depth.

The dam is 100 m high, in a canyon 200 m wide at the crest, with its rings at 133.6°. It thickens from 5 m at the crest to 25 m at the base, as arch dams do: the rings at depth carry more water, and the cantilever’s base carries the most moment.

In a V-shaped canyon the rings carry nearly all of it. The canyon narrows with depth, so the lower rings are short — at 90 m below the crest the canyon is 20 m wide and the ring’s radius 11 m — and a short ring is very stiff, since its deflection goes as the square of its radius. Against those rings the cantilever, 100 m tall, is soft, and it takes 6% to 13% of the water down most of its height and 10% of the reservoir in all.

Two places break the pattern. At the very foot, the cantilever is fixed into the rock and cannot deflect, so neither can the ring there, and a ring that does not deflect carries nothing: the bottom metres of water go to the cantilever whatever the canyon. And at the top the shares go the other way.

The top rings carry more than their water #

Near the crest the cantilever’s share is negative. The cantilever is loaded from below — by the large pressures at depth that it carries a small share of — and it is free at the top, so it leans downstream there, further than the stiff upper rings would deflect under their own water. The rings hold it back. They carry their own water and a reaction from the cantilever besides.

4 m below the crest the ring carries 1.95 times the water at that depth. The crest of an arch dam is therefore not lightly loaded, as the ring formula’s small pressure would suggest: it is the arch that stops the whole cantilever from leaning. It is one reason arch dams are given a crest thicker than the ring formula would ask for, and why the crest arch is treated as a member in its own right.

The canyon decides the division #

The V is the canyon that favours the rings most. A canyon whose sides are steeper — narrowing slowly at first and closing near the floor — leaves wide rings at depth, and wide rings are soft.

In the U-shaped canyon, whose width falls slowly until close to the floor, the canyon is still 100 m across at 90 m depth, and the cantilever carries 87% of the water there. Over the whole reservoir it carries 51%: the same dam carries half its water in arch action in a U and nine-tenths in a V. The curved canyon is between, at 43%.

This is the reason arch dams are built in narrow, V-shaped gorges and gravity dams in wide valleys. A gravity dam is all cantilever, and in a U-shaped valley the cantilever is what an arch dam would become anyway. Where the canyon allows rings to be short, arch action is cheap; where it does not, the dam is a cantilever whether it was drawn as one or not.

The same dam in a U #

The U-shaped canyon is worth drawing whole, because there the division is not a correction.

The top rings carry two and a half times their water, because the cantilever, which now takes much more of the deep water, leans harder on them. By mid-depth the cantilever takes a quarter; below about 70 m it takes most of the water, and the rings there — still a hundred metres across — are almost idle. The lower half of this dam is a cantilever wall with arches resting along its top, and the shape that was chosen as an arch’s, curved in plan at 133.6°, is doing little for it there.

The division also explains a habit of arch-dam design that the ring formula cannot: the dams in wide valleys are double-curved. A shell curved in the vertical plane as well as in plan carries its cantilever’s load partly by arching vertically, and the cantilever stops being a plain beam. In a V the plan curvature does the work and the section can be nearly straight; in a U it cannot, and the section is curved to make the cantilever an arch too.

What the division does to the rings #

The ring formula is right where the rings carry their own water, and wrong where they do not.

In the middle of the V the two curves are close: the rings carry 87% to 90% of the water, and their stress is that fraction of the ring formula’s — 1.77 N/mm² at the peak against 2.04. Near the crest they part the other way. The ring formula gives the top rings almost nothing, because the water is shallow; divided, the ring 4 m down carries 1.48 N/mm², twice the formula’s figure. Near the foot, where the cantilever takes over, the rings carry almost nothing and the formula overstates them.

So the ring formula sizes the middle of a V-shaped dam about right, the top too thin and the bottom too thick. In a U-shaped canyon it is further out: there the rings below mid-depth carry well under half of what it gives them.

The cantilever’s bending, which the ring formula cannot see #

The water the cantilever takes is carried by bending, and the bending is largest at its foot.

Near the crest the cantilever bends backwards — upstream — because the rings are holding it back there, and its downstream face is in tension. Lower down it bends downstream, carrying its share of the deep water, and at the foot the upstream face, the heel, is in tension: 1.09 N/mm² in the V, 3.90 in the U.

That tension is offset by the dam’s own weight, which is how a gravity dam answers the same question: it is all cantilever, and it keeps its heel in compression by being heavy enough that the resultant stays in the middle third of its base. The concrete above the base presses down with about $24 \times 15 \times 100 / 25 \approx 1{,}440$ kPa averaged over the 25 m base, about 1.4 N/mm². In the V that is more than the heel’s bending tension, and the heel stays in compression. In the U it is not: the net is about 2.5 N/mm² of tension on the upstream face at the foot, which is enough to crack unreinforced concrete, open the joint between the dam and the rock, and let reservoir water in under pressure. The ring formula, which gives every metre of water to a ring, has no cantilever in it and no heel; it cannot see any of this.

Wider canyons hand the water down #

The canyon’s width, at a fixed height, moves the division steadily.

In a canyon 80 m wide at the crest, 0.8 heights, the cantilever carries 4%; at two heights, 10%; at six heights, 43%. A ring’s flexibility goes as the square of its radius, and the radius goes with the width; the cantilever’s flexibility depends on its height and thickness and not on the canyon at all. So every metre of width softens every ring and leaves the cantilever as it was, and the water moves down to the valley floor.

The same comparison runs through the stiffest path takes the load, with the difference that here the two paths are in two directions at once and meet at every depth. The ring’s stiffness falls with the square of the width; the dam’s proportions decide which path is stiffer at each level, and the water goes there.

Thickness moves both paths at once #

The canyon is given; the dam’s thickness is the designer’s, and it changes the division in a way that is not obvious, because it stiffens both paths.

Thickening the base from 25 m to 40 m, with the crest left at 5 m, doubles the cantilever’s share in the V-shaped canyon, from 10% to 19%, and raises it from 51% to 62% in the U. A ring’s stiffness goes as its thickness and a cantilever’s as the cube of it, so extra concrete at depth favours the cantilever. In the V the heel’s bending stress hardly moves, 1.09 to 0.96 N/mm²: the cantilever carries twice the water on a section far deeper. In the U it falls from 3.90 to 2.35 N/mm², because the base section’s modulus grows with the square of its thickness while the water it takes grows more slowly. The rings’ peak stress falls in both, 1.77 to 1.22 N/mm² in the V.

Thinning the dam to 3 m at the crest and 15 m at the base goes the other way, and fast. The cantilever’s share in the V falls to 5% — the rings carry almost everything, at a peak of 3.15 N/mm² — and in the U the heel’s bending stress reaches 6.54 N/mm², well past anything the dam’s weight could offset. A thin dam in a wide valley is the worst of both: its rings are too soft to carry the water at depth and its cantilever is too thin to carry it without cracking at the heel.

Two directions, one rule #

The division is the same arithmetic as a slab that spans both ways, which sends its load in two directions in proportion to their stiffness — a fourth power of the spans, for a slab, because a strip’s stiffness goes as the inverse fourth power of its length. Here the horizontal direction is a ring whose stiffness goes as its thickness over the square of its radius, and the vertical direction is a cantilever whose stiffness goes as the cube of its thickness over the fourth power of its height. Every depth has its own pair, and they are coupled because the cantilever is one member carrying the water of every depth at once.

The method that makes the division is older than the computers that now do it. The trial-load method was developed by the United States Bureau of Reclamation in the 1920s and 1930s for the great dams of the American West, and it was literally a trial: engineers guessed a division, computed the deflections of a set of rings and a set of cantilevers by hand, adjusted the guess where they disagreed, and repeated, for months. The single crown cantilever here is the first of those adjustments, and it already contains most of the answer.

The division, checked at one depth #

At 49.4 m below the crest in the V-shaped canyon the canyon is 101 m wide, the ring’s radius $101/(2\sin 66.8°) = 55.1$ m and the dam 14.9 m thick. The water presses with $9.81 \times 49.4 = 484$ kPa. The ring carries 89% of it, 430 kPa, and its hoop stress is $430 \times 55.1 / 14.9 = 1{,}590$ kPa, 1.59 N/mm². Its deflection, for a modulus $E$, is $430 \times 55.1^2 / (14.9 E)$, and the cantilever’s at the same depth, under its 11% share of the water at every depth above and below, has to equal it. That equality is the whole method: a ring and a cantilever that have to be in the same place.

A crown cantilever is not a dam #

The calculation is the trial load’s crown adjustment with one cantilever, and the full method does more.

One cantilever. A real trial-load analysis uses several cantilevers across the dam, each fixed into the sloping canyon side at its own height, and the rings meet each of them. The crown cantilever is the tallest and the softest; the shorter ones near the abutments are stiffer and take more of the water there.

Thin rings, abutments not modelled. A ring’s deflection here is a thin ring’s under uniform pressure. A ring fixed into its abutments bends near them, and the rock behind the abutments gives — and a deflection that belongs to the support adds to the ring’s own, softening it and sending more of the water to the cantilever.

Radial deflection only. The full method also matches the twist and the tangential movement of rings and cantilevers, which changes the shares by a few per cent and adds the torsion that couples them.

A full reservoir. Every figure has the water at the crest. With the reservoir drawn down, the cantilever is loaded only below the water line and still leans downstream above it, so the dry rings at the top carry nothing but the cantilever’s reaction — a load the ring formula, with no water there, would call zero. The case is checked separately for exactly that reason: the crest arch is loaded hardest, relative to its own water, when it has none.

Elastic concrete. The division assumes the rings and the cantilever stay elastic and uncracked. A cracked heel, or a ring joint that opens in winter when the arches shorten, changes the stiffnesses the division was made from, and the water moves to whatever is still whole.

No temperature, no self-weight in the shell. An arch dam’s rings shorten in winter and their thrust falls; the cantilevers carry the weight; both change the division, and the weight changes the heel’s net stress as above.

Still open: the joint at the heel #

The cantilever’s heel goes into tension in a U-shaped canyon, and the tension can open the joint between the dam and the rock. An opened joint is a cantilever no longer fixed at its foot: it rotates, it deflects more, and it hands its water back to the rings — which are wide and soft in a U, and were not sized for it. The water that enters the opened joint presses up under the dam as well, as water that comes back under a basement does, on an area the crack itself decides. Whether the rings can take back what a cracked heel gives up, before the uplift in the crack lifts the cantilever further, is the question of whether an arch dam in a wide valley is stable once its foot has let go, and it is the question that some arch dams in wide valleys have had to answer after they were built.