The bridge that pays for its own anchorage

The tension that was left out compared the two theories suspension bridges have been designed by. Rankine’s elastic theory treats the deck as a beam carrying whatever the cable does not; Melan’s deflection theory keeps one more term, the cable’s whole horizontal tension $H$ acting on the deck’s deflected shape, and that term is a stiffness. On a long span it is most of what carries the live load, the deflection-theory moments are a fraction of the elastic ones, and the slender decks of the twentieth century’s great suspension bridges are its consequence.

Every bridge in that essay delivered $H$ to the ground, through anchorage blocks at the ends of the backstays. A self-anchored suspension bridge ties the cable to the ends of the deck instead. No anchorage blocks are needed, which is the point where the ground is poor, and the pull the ground would have taken is carried by the deck as a compression. That looks like a change of anchorage. It is a change of theory, and it takes away exactly the term the long-span bridge was built on.

One term, and its sign #

Write the deck’s equation with both effects in it. The cable’s tension $H$ acting on the deflected shape contributes $-H v''$, a stiffness. A compression $N$ in the deck acting on the same deflected shape contributes $+N v''$ — a softening, the term by which a compressed member makes its own load worse. In an earth-anchored bridge $N = 0$. In a self-anchored one the deck carries the cable’s horizontal pull between its ends, so $N = H$ at every section, and

$$EI\,v'''' - H\,v'' + H\,v'' = p - H_p\,y'' \quad\Longrightarrow\quad EI\,v'''' = p - H_p\,y''.$$

The geometric term vanishes identically. What is left is Rankine’s elastic theory, term for term: the deck is a beam on its supports, relieved only by the extra cable tension $H_p$ the live load draws, and the whole of the stiffening the cable’s tension lent the deck is gone — cancelled by the compression that the tension put into the deck on the way to its anchorage.

The bridge drawn is the earlier essay’s: a 500-metre span, a 50-metre sag, a deck of flexural stiffness $1.03 \times 10^{12}\,\mathrm{N\,m^2}$, 100 kN/m of dead load and 20 kN/m of live load on half the span, the case a suspension bridge is designed by. Anchored to the ground, the deck’s largest moment is 56.3 MN·m. Anchored to itself, 83.0 — 1.47 times as much, and slightly more even than the elastic theory’s 80.1.

Why self-anchored is worse than the elastic theory #

The last detail needs a second look, because a cancellation should have landed the self-anchored bridge exactly on the elastic theory, not above it. The difference is in how the live load’s extra cable tension is found.

The extra tension $H_p$ comes from compatibility: the deck deflects, the cable must lengthen to follow it, and a lengthening cable is a cable carrying more tension. In an earth-anchored bridge the cable’s ends are fixed. In a self-anchored one they are fixed to the deck’s ends, and the deck, carrying $H_p$ as extra compression, shortens: its ends move toward each other by $H_p L/EA_{\text{deck}}$ and give the cable slack. So the same deflection draws less extra tension from the cable, the cable relieves the deck less, and the deck carries more.

For this bridge the effect is small, 4 per cent over the elastic theory, because the deck — a steel box of about 0.3 m² — is stiff in compression compared with the cable’s extension. But it is present at every span, and on short spans, where the geometric term was small to begin with, it is the whole of the penalty.

The number, by hand #

The self-anchored figure can be checked with the oldest formula in the subject. In Rankine’s theory, with the cable’s extension neglected, a suspension bridge under a uniform load on half its span behaves as a simply supported beam relieved by a cable force that takes exactly the uniform half of the load; what is left for the deck is an antisymmetric load, $+w/2$ on one half and $-w/2$ on the other, and its largest moment, at the quarter points, is

$$M = \frac{wL^2}{64} = \frac{20 \times 500^2}{64} = 78 \text{ MN·m}.$$

The computed elastic theory gives 80.1 — a little more, because a cable that stretches relieves the deck a little less — and the self-anchored bridge 83.0, because its deck’s shortening slackens the cable further. The earth-anchored deflection theory’s 56.3 is what the geometric term takes off that. It depends on one number, $\mu = L\sqrt{H/EI}$, the number of the deck’s characteristic lengths in the span; here $H$ is 62.5 MN, so $\mu$ is 3.9, and a deck several characteristic lengths long is a deck that the cable’s tension holds straighter than its own stiffness can.

A self-anchored deck has $\mu = 0$ whatever its span, because the net tension on its deflected shape is nothing. Every long-span economy the earlier essay found lived in $\mu$.

Every share the deck takes costs #

Some bridges are anchored partly to the deck and partly to the ground — the deck taking a share of the pull through a restrained expansion joint, or an anchorage that yields. The geometric stiffness the deck receives is then the cable’s tension less the deck’s compression, $(1 - s)H$ for a share $s$, and the moment climbs steadily with the share: 56.3 MN·m with none, 61.6 with a quarter, 67.6 with half, 74.6 with three-quarters, 83.0 with all of it. There is no threshold and no free portion. Every newton of the pull the deck carries is a newton of stiffening the deck loses.

What self-anchoring costs as the span grows #

The two theories diverge as the span grows, and so do the two anchorages. The earth-anchored deck’s live-load moment rises with the span and then stops rising — at 1,100 m it is lower than at 800 — because the cable’s tension, which stiffens the deck, grows as the square of the span too. The self-anchored deck gets none of that: its moment is the elastic theory’s, which grows as the square of the span without limit. With the same deck and a sag of a tenth of the span, the self-anchored bridge carries 1.21 times the earth-anchored moment at 200 m, 1.47 at 500, 2.83 at 800 and 5.85 at 1,100 m.

That is the whole explanation of why self-anchored suspension bridges are short. The longest built, across the eastern part of San Francisco Bay, spans 385 metres; earth-anchored bridges span five times that. A long span is a structure in which the deck has been made light by trusting the cable’s tension to stiffen it, and a self-anchored cable does not stiffen anything.

A deck compressed past its own buckling load, and standing #

The compression itself looks alarming. The deck of a self-anchored bridge carries the cable’s whole dead-load pull, $wL^2/8f$, and for this deck that pull passes the deck’s own Euler load over the span, $\pi^2EI/L^2$, at about 450 metres; at 1,100 metres it is sixteen times it. A deck on its own would buckle long before it was strong enough to matter.

It does not buckle, and the reason is the same cancellation. A buckle is a deflected shape the compression can hold by itself: the softening term $+Nv''$ outweighing the bending stiffness. In the self-anchored bridge every such shape is also a shape the cable is pulled into, and the cable’s tension on it supplies $-Hv''$ exactly equal and opposite. The system has no geometric softening to buckle with — and no geometric stiffening either. The compression is not a stability problem for the deck as a whole. It is a strength problem, a large axial force added to the bending in every section, and a local one, since a panel of the deck’s plating does not know that the cable is there.

So the refutation runs both ways. A designer who treats the self-anchored deck as a strut and sizes it against Euler buckling wastes steel on a failure that cannot happen; a designer who treats it as a deflection-theory deck saves steel on a stiffening that is not there.

The deck also rings lower #

The same term sets the deck’s natural frequencies, and there the loss is less visible and matters as much. A deck vibrating in a shape with wavenumber $k$ resists it with its bending stiffness, $EIk^4$, and — if its cable is anchored to the ground — with the tension on the curved shape, $Hk^2$; the square of the frequency is their sum over the deck’s mass per metre. For the lowest antisymmetric mode of the 500-metre bridge, one full wave along the span, which does not stretch the cable, the bending term is 25.7 kN/m² per metre of amplitude and the tension term 9.9: with the deck’s ten tonnes a metre that is 0.30 Hz anchored to the ground and 0.25 Hz anchored to itself, where the tension term is cancelled.

A sixth off a frequency is not a strength problem. It is a problem for everything that is decided by frequency — the wind’s vortices, which lock on at a wind speed proportional to it, so that a lower frequency is met by a commoner wind — and it grows with the span as the tension term does. The self-anchored deck is not only more heavily bent; it is also the less stiff structure in every sense that the cable’s tension had been supplying.

How much stiffer the deck must be #

Turn the question round: how much stiffer must a self-anchored deck be to deflect no more under the same live load? The answer has a minimum, and the minimum is informative. At 800 metres the deck must be 3.3 times as stiff, and the factor climbs steeply beyond — the lost geometric stiffening. At 200 metres it must also be 3.4 times as stiff, for the other reason: on a short span the geometric term was small anyway, but the deck’s shortening under its extra compression slackens a cable that is short and stiff, and the cable’s contribution to carrying the live load is then what is lost. Between the two the factor falls to 1.7 near 400 metres.

That is roughly where self-anchored suspension bridges have been built. It is the span at which neither of the two things a self-anchored deck gives up is yet large.

One more thing the deck must do first #

The cancellation has a consequence for how the bridge is built, and it is the other reason self-anchored bridges are rare. An earth-anchored cable is hung first, between its towers and anchorages, and the deck is lifted and hung from it piece by piece: the cable carries the deck’s weight as it arrives. A self-anchored cable has nothing to pull against until the deck exists, because the deck is its anchorage. The deck must be built first, complete, on temporary supports across the whole span, and the cable is then tensioned against it until the deck lifts off. The falsework that the suspension form normally avoids entirely is, for a self-anchored bridge, the first thing built.

That order also decides the deck’s permanent state. On its temporary supports the deck carries its own weight as a continuous beam over many short spans, with moments nobody would design a suspension deck for; as the cable is tensioned, hangers lift the deck off one support after another, and the dead-load moments migrate toward the state the parabola was chosen to give — a deck carrying almost no dead-load bending. Whether they arrive there depends on the sequence and amount of jacking, which in an earth-anchored bridge is a matter of geometry and in a self-anchored one is a matter of design: the compression the deck will carry for its whole life is being put into it, a hanger at a time, by the operation that removes its falsework.

The tied arch, which is the same trade the other way round #

A tied arch is the self-anchored suspension bridge turned upside down. Its rib is in compression, its thrust is carried not by abutments but by a tie at deck level — usually the deck itself — and the tie is in tension. The signs are all reversed, and so is the conclusion. The arch rib, compressed, is softened by its own thrust acting on its deflected shape, which is why an arch’s buckling is a design case. But the deck-tie, tensioned by the same thrust, is stiffened by it, exactly as an earth-anchored suspension deck is stiffened by its cable: a tension acting on a deflected deck pulls it straight.

So the two self-anchored forms trade the same term in opposite directions. The suspension bridge’s deck carries the compression and loses the stiffening; the tied arch’s deck carries the tension and gains it, and the compression goes into the rib, which is where an arch expected it. This is one reason tied arches reach spans of five hundred metres and self-anchored suspension bridges have stopped at under four hundred, although the two look like mirror images: the member that ends up compressed is, in the arch, the one designed for it.

Why the form is chosen anyway #

None of this makes the self-anchored bridge a mistake. It removes the most expensive single element of a suspension bridge on poor ground — the anchorage blocks, which must resist the cable’s whole pull by weight and friction, as any block does that holds a horizontal force — and it keeps the cable’s form, which for a crossing that is also a landmark is the reason the form was wanted. At 300 to 400 metres the penalty in the deck is a factor of 1.2 to 1.5 on moment and less than two on stiffness, which a deep box girder carries without difficulty; the deck of a bridge that short would have been substantial in any case.

What the calculation does is put a price on the choice, and show that the price is not in the anchorage but in the deck. Anchoring to the deck saves the blocks and spends the stiffening, and the stiffening is the one thing that made a long suspension deck light.

The deck, as one series #

The deck is solved as a simply supported beam on a tensioned string, each term of a sine series independent: for the $n$-th term the deck’s resistance is its bending stiffness times the fourth power of the wavenumber plus the net geometric force — tension less compression — times its square. The live load’s extra cable tension follows from one compatibility equation, the cable’s extension plus any shortening of the deck equalling the cable’s geometric lengthening, and the cable’s total tension is iterated because it appears in the geometric term. With no share of the pull on the deck this reproduces the deflection-theory bridge exactly; with all of it, the geometric term is zero and the only difference from Rankine’s theory is the deck’s shortening.

A single span, a parabola, and a deck that stays elastic #

A single span with pinned ends. Real self-anchored bridges usually have side spans, whose decks also carry the compression and whose cables also pull on the main span; the cancellation holds span by span wherever the deck carries the cable’s pull.

The dead-load shape is a parabola. The deck’s dead load is taken as uniform, so the cable’s shape is exactly parabolic and the deck carries no dead-load moment. A self-anchored deck’s dead-load state also includes whatever the erection sequence locked into it, since the deck existed before the cable pulled on it.

The deck stays elastic and straight in plan. The large axial compression reduces the deck’s bending capacity in every section, and its interaction with bending, and the torsional and lateral stability of a compressed deck, are checks this calculation does not make.

Still open: the cable-stayed deck, which is always self-anchored #

Every cable-stayed bridge is self-anchored: its stays pull the deck toward the towers, and the deck carries their horizontal components as a compression that grows toward the towers. Unlike the suspension cable, a stay is straight and pulls the deck at a point, so its tension does not act on the deck’s deflected shape as a distributed string does — there was no geometric stiffening to lose in the first place — but the deck’s compression still softens it, and near the tower it is largest. Whether that compression, which the cable-stayed bridge carries without any compensating tension, is what finally limits how far a stayed deck can span, is a question about the self-anchored form in which nothing cancels.