The tension that was left out

The deck is not there to carry the load sets out the system: a beam on a tensioned string, $EI v'''' - H v'' = p$, solved for a stated $H$. The girder’s job is to distribute the live load so that what reaches the cable is close enough to uniform for the cable’s own shape to be right.

Every number in that essay depends on $H$, and $H$ was handed to it. Where it comes from is the whole of the difference between the two theories a suspension bridge has been designed by, and the difference is not a refinement.

The two theories, in one term #

Both theories agree on everything except one term, and setting them side by side makes the term visible.

Elastic theory. The deck is a beam. The cable picks up an extra horizontal force $H_p$ when the live load goes on, that force acts on the cable’s existing curvature, and the product is an upward load on the deck. So

$$EI\,v'''' = p - H_p\,\kappa$$

with $\kappa = 8f/L^2$ the parabola’s curvature. The deck is a simply supported beam under the live load and an upward pressure, and $H_p$ comes from a compatibility integral between the cable’s extension and the deck’s deflection.

Deflection theory. Everything above, plus the observation that the deck is not a beam. It is a beam with a large axial tension in it, delivered by the hangers from a cable carrying $H_w + H_p$, and a tension member is stiffened by its own tension in exactly the way a compression member is softened by its own compression:

$$EI\,v'''' - (H_w + H_p)\,v'' = p - H_p\,\kappa$$

The extra term is a geometric stiffness with the opposite sign to the one that causes buckling. It is not a correction to the load; it is a second stiffness in parallel with $EI$, and on a long span it is very much the larger of the two.

Which free body produced the number #

The free body is a slice of the deck, and the difference between the theories is a question about what crosses its faces.

Both theories agree that a shear and a moment cross, and that the hangers deliver an upward force per unit length.

Deflection theory says something else crosses: an axial tension, inclined by the slope of the deflected deck. The deck is being pulled along by the cable system, and once it has bent, that pull has a transverse component proportional to the curvature. Integrating it along the slice gives $-H v''$, and that is the term.

Whether the term belongs is a question about geometry rather than about materials, and it is the same question a load that makes itself worse asks with the opposite sign. Equilibrium taken on the undeformed shape does not contain it; equilibrium on the deformed shape does. Elastic theory is a first-order analysis of a structure whose whole behaviour is second order.

What the number is, and what it is not #

The most surprising line in the first figure’s caption is the one about the cable: the extra tension is 6.20 MN by one theory and 6.11 by the other, a difference of one and a half per cent, while the moments differ by 30.

That is worth dwelling on because it says exactly where the effect lives. Nothing about the cable is in the difference. The compatibility integral that fixes $H_p$ is nearly the same integral in both, because it depends on the area under the deflection curve and both theories give similar areas. What differs is the deck’s own share of the work, and the geometric stiffness takes most of it.

So a designer using elastic theory does not get the cable wrong. The tower loads, the anchorage forces and the cable size are all very nearly right. What is wrong is the girder, and it is wrong by a factor.

What it does to the deflection #

The moments are what a designer sizes a girder by, and the deflections are what the public notices, so it is worth reading both.

On the 500 m span the elastic theory gives a mid-span deflection of 541 mm under the half-span live load and the deflection theory 394 — a 27 per cent reduction, almost exactly the same fraction as the moment. That agreement is not automatic: a geometric stiffness added in parallel with a flexural one changes the shape of the deflection as well as its size, and the two would only track each other if the shape were unchanged.

They very nearly are, and the reason is in the series. Every term of the solution has a denominator $EI k^4 + H k^2$, and adding $H$ raises every denominator by a different proportion — the low modes most, since $k^2$ is small for them and the $H$ term therefore dominates. So the geometric stiffness suppresses the long waves more than the short ones, and the deflected shape under deflection theory is slightly more localised around the loaded half than the elastic one.

That is a small effect at μ = 4 and a large one at μ = 20, where the deflection under a half-span load has become a local dip near the load rather than a global sagging of the whole span. A very long suspension bridge does not deflect as a span; it deflects where the load is, and that is one of the two reasons the deck’s own moments have almost disappeared.

The one dimensionless group #

The whole family collapses onto one number, and knowing it is what makes the argument portable.

$$\mu = L\sqrt{\frac{H}{EI}}$$

is the span divided by the characteristic length of a beam under tension $H$ — the same $\sqrt{EI/H}$ that governs a beam on an elastic foundation with the sign of the axial term reversed, and the same group that appears in every tension-stiffened problem — including the cable whose stiffness is its own shape, read from the deck’s side rather than the cable’s.

Below about $\mu = 2$ the girder’s own bending is most of the stiffness, the two theories agree, and a suspension bridge is a stiffened beam. That describes a short span or a very deep truss.

Above about $\mu = 10$ the geometric term is nearly all of it, the deck’s moments are small, and the structure is a cable with a road on it. The girder is there for aerodynamics, for local distribution between hangers, and to stop the deck folding under a concentrated load — not to carry the span.

The bridge in the figures has $\mu = 4.1$ at 500 m and 21 at 1,500. The transition happens across the range of spans that were built between 1880 and 1940, which is why the argument was live during exactly that period and settled afterwards.

Two bridges across one river #

The clearest evidence is in New York, three-quarters of a mile apart.

The Brooklyn Bridge (1883, 486 m main span) was designed by Roebling on elastic theory, and its stiffening truss is 5.2 m deep — deep enough to be the dominant visual element of the deck, with the diagonal stays added on top of that as a further stiffening system Roebling did not fully trust the arithmetic without.

The Manhattan Bridge (1909, 448 m — a slightly shorter span) was the first designed by Moisseiff on deflection theory, and its stiffening truss is 7.3 m deep but very much lighter, carrying four rail tracks and seven road lanes on a structure whose steel weight per metre is a fraction of what elastic theory would have demanded for the same job.

The trajectory from there runs one way. The George Washington (1931, 1,067 m) opened with no stiffening truss at all — a bare deck 3 m deep on a span of over a kilometre, which is $\mu$ well past 20 and a structure the older theory could not have permitted anybody to build.

Why the older engineers were not being careless #

It is tempting to read the factor of fourteen as an error, and it is worth resisting because the reasoning that produced elastic theory was sound at the time and the shape of it recurs.

A second-order analysis is a nonlinear one: the stiffness depends on the force, so superposition fails and every load case has to be solved on its own. Before 1950 that meant solving a differential equation by hand for each combination, and Melan’s equation has no closed form for a general load. Moisseiff’s practice used series solutions and tabulated coefficients, and it took a specialist.

Elastic theory, by contrast, is linear. Influence lines can be drawn, load cases superposed, and the whole design done with a slide rule by an ordinary drawing office. It was chosen because it was tractable, and it was known to be conservative, which is the correct pair of reasons to choose a method.

What changed was not the discovery of an error but the arrival of a span at which the conservatism stopped being affordable. At 300 m the older theory costs a few per cent of deck; at 1,000 m it costs a bridge that cannot be built. That is the general form worth carrying: a conservative method is a record of what could be computed, and the size of its conservatism is a function of the structure rather than of anybody’s caution.

Where it went wrong afterwards #

The end of that trajectory is the one every account of it reaches, and it belongs here because it is the same theory read past its limits.

Moisseiff’s later designs took the argument to its conclusion: if the geometric stiffness carries the live load, the girder can be shallow, and a shallow girder is cheaper and better-looking. The Tacoma Narrows bridge of 1940 had a plate girder 2.4 m deep on an 853 m span — a depth-to-span ratio of 1:350, against the George Washington’s 1:350 with a far wider deck.

Its failure was not a failure of deflection theory. Every static calculation was right, and the bridge carried its loads. What the theory contains no term for is aerodynamics: a shallow solid girder is a bluff body that sheds vortices and, at a certain wind speed, extracts energy from the flow through a coupling between its own torsion and heave. That is a load that does not arrive until the structure invites it, and no amount of static stiffness is a defence against it.

The lesson usually drawn is about humility, and the better one is about which quantity a design is being optimised against. Deflection theory removed bending as the constraint on a suspended deck. It did not supply a new constraint, and the profession took twelve years and a collapse to find out that torsional stiffness and aerodynamic shape were what had been left holding the answer.

Where the model stops #

One load case and a parabolic cable. The half-span live load is the governing case for the deck’s moment and it is not the only one; a cable whose dead load is not uniform is not a parabola, and the curvature $\kappa$ in the load term then varies along the span.

The girder is prismatic and continuous over the whole span. Real stiffening trusses are hinged at mid-span on many older bridges — the Brooklyn is — which removes a moment the calculation here carries and changes the series entirely, and they are often deeper at the towers than at mid-span.

The towers are rigid and the cable is anchored. A tower that deflects, a self-anchored bridge where $H$ goes into the deck as a compression, and a bridge with side spans all change the compatibility integral that fixes $H_p$ — and the last of them changes it a great deal, since the side spans’ cables extend too.

The deflection is small. Both theories here are linear once $H$ is fixed. A genuinely large-displacement analysis updates the geometry as the load goes on, which for the deck matters little and for the cable’s own shape matters more.

The live load is a pressure. A real live load is a train of axles or a queue of vehicles, and on a bridge whose deflection has localised around the load, the difference between a smeared pressure and a real distribution is largest exactly where the deck’s own moment is.

And it is entirely static. Everything decided by the wind — the aerodynamics that ended the argument above — is outside both theories, as is the deck’s torsional behaviour, which is the mode that actually governs a modern long span.

What the pictures cannot show #

The erection sequence, which is when a suspension bridge is at its most awkward and when $H$ is least like the value used here. The cable is spun first and carries only itself; the deck is placed segment by segment, and at every stage the structure has a different $H$, a different shape and a different stiffness. The structure that was never complete is the general form of the difficulty and a suspension bridge is its extreme case.

They also cannot show the hangers, which are drawn as a continuous connection and are not. A hanger every 10 to 20 m means the deck spans locally between them, with a local moment that has nothing to do with either theory — and it is that local moment, rather than the global one, that sizes the deck of a very long span where the global moment has nearly vanished.

The assumption the figure rests on #

That the deck’s axial force is the cable’s horizontal component.

It is not the deck’s axial force at all — the deck of an anchored suspension bridge carries very little axial load, and the $H$ in the equation is the cable’s. What the term really represents is the transverse component of the hanger forces as the cable and deck deflect together, and writing it as $Hv''$ is a compact way of saying that the cable’s tension resists any change in the shape it and the deck share.

The distinction matters in one case, and it is a common one now: a self-anchored suspension bridge takes $H$ into the deck as a genuine compression, so the deck has a real axial force with the opposite sign — a geometric softening rather than a stiffening, and the same equation with the sign of $H$ reversed. That is a beam-column, it can buckle, and the whole comfortable conclusion of this page runs backwards for it.

What to carry away #

Three things, in the order a reader is most likely to need them.

The stiffness of a suspended deck is mostly not its own. It is the cable’s tension acting on the deck’s curvature, it is a geometric stiffness, and it is in the equation only if equilibrium was taken on the deflected shape.

The two theories agree about the cable and disagree about the girder, so the error in the older one is confined to one member — which is why bridges designed by it stand up perfectly well and are simply heavier than they need to be.

And the group that decides how much it matters is μ = L√(H/EI), which contains the span, the sag, the dead load and the girder. The same bridge is two different structures at two different spans, and the theory it needs is decided by arithmetic rather than by preference.

The ladder from here #

Later rungs on this anchor: Melan’s equation solved rather than summarised, with the cable’s extensibility and the tower flexibility both in the compatibility. The self-anchored bridge, where $H$ compresses the deck and the geometric term changes sign. Torsional stiffness of the deck and the flutter speed, which is the criterion that actually governs a long span. Cable-stayed bridges, where the stays are inclined and the deck is a beam on elastic springs rather than a beam on a string. The erection stages, where every quantity here is a function of time. And the anchorage, which is where $H$ finally has to be delivered to something — a block of concrete sized by a horizontal force nobody has yet asked where it goes.