The deck the stays squeeze

The bridge that pays for its own anchorage took the suspension cable’s pull off the ground and put it into the deck, and found that a self-anchored deck loses the cable’s stiffening and pays for it in bending, and that its compression softens it further. It ended on the observation that every cable-stayed bridge is self-anchored. A stay pulls the deck towards the tower; the deck carries the horizontal component as compression; and there is no cable tension along the deck to offset it, because a stay is straight and pulls at one point.

That essay asked whether the compression is what limits how far a stayed deck can span. This one answers it with a model a designer could check by hand: a deck as a strut, held up by the stays as a row of springs, under the compression the same stays put into it.

A strut held up by its own load #

The bridge is a fan. Stays every 15 m along the deck all rise to the top of a tower whose height above the deck is two fifths of the half-span; each stay carries its own 15 m of a 150 kN/m dead load vertically, and is sized for that tension at 600 N/mm². The deck is a steel box with a bending stiffness of $0.5 \times 10^{12}\ \text{N·m}^2$, about what a box three metres deep provides. Half the main span is modelled, from the tower to mid-span, with the deck held vertically at the tower and either its slope or its deflection held at mid-span, for the symmetric and antisymmetric buckled shapes.

Each stay does two things to the deck. Its horizontal component is a pull towards the tower, and every pull from beyond a point has to pass through the deck at that point, so the deck’s compression at a section is the sum of the horizontal components of all the stays further out. And its vertical stiffness — the stay’s axial stiffness $EA/\ell$ resolved vertically, $EA\sin^2\theta/\ell$ — is a spring under the deck. The deck is a strut on springs, and both the strut’s load and the springs come from the same stays.

The compression is nothing at mid-span and climbs towards the tower in steps, one per stay, each step larger than the last because the outer stays are flatter and pull harder sideways for the same vertical share. At the tower it reaches 95 MN. The stays’ stiffness runs the other way: the short, steep stays near the tower are stiff, the long, flat ones near mid-span are an eighth as stiff. The compression is largest exactly where the support is stiffest, and the question is whether that is enough.

Waves between the stays #

A strut on closely spaced springs does not buckle as a whole. Held everywhere, and it forgets its length found that a member restrained along its length chooses its own wavelength, set by the ratio of its stiffness to the restraint’s, and buckles in as many half-waves as fit. A stayed deck does the same.

At 6.91 times its dead load the deck buckles in half-waves about 125 m long, eight stays to each, crossing its own axis four times between the tower and mid-span. The largest wave is not at the tower, where the compression is largest but the stays are stiffest, and not at mid-span, where the stays are softest but the compression has gone; it sits about 180 m out, where the two have traded off. The buckle is a property of the whole arm, but it is decided in the inner half of it.

The model leaves out the tower’s own flexibility and the cable stays’ coupling to the back span, both of which make a real bridge somewhat softer, and it leaves out the deck’s live load, which adds compression. It is a lower bound on how near the deck is to buckling — the factor a real bridge has is less than this, not more.

The span, to the power one and a half #

The factor is a comfortable 6.9 at 1,000 m. The question the span limit asks is how it changes as the bridge grows.

On logarithmic axes the factor falls on a straight line with a slope of about one and a half. Double the span and the deck is nearly three times nearer buckling. At 300 m the question does not arise, at 41 times the dead load. At 1,000 m the factor is below ten, which in the European rules is the threshold below which a structure’s second-order effects must be included in its analysis: the deck’s deflections under load now change its own bending moments enough to matter, and the design has to follow them. At 2,000 m the deck buckles at 2.3 times its own weight, which is within reach of the traffic, temperature and wind a bridge carries.

The span the record has reached sits where this line says it should. The longest cable-stayed spans built — the Russky Island bridge at 1,104 m, Sutong at 1,088 m, Stonecutters at 1,018 m — cluster just past 1,000 m, all with steel box decks, while suspension bridges have passed 2,000 m. Nothing in the model is calibrated to those bridges, and many things besides the deck’s compression decide where a record stops: the towers, the stays’ weight and wind load, the cost of a second form against the first. But the compression is the one limit that is built into the stayed form itself, and it falls steeply in exactly the range where the records stop.

The slope has a simple origin. The compression at the tower is the sum of the stays’ horizontal pulls, each its share of the dead load times its reach over the tower’s height, so it grows as $g a^2 / 2H$ — as the square of the half-span $a$ when the tower grows in proportion. The stays near the tower, which decide the buckle, have a stiffness per metre of deck of about $E g /\sigma H$, because a stay’s area is sized to its share of $g$ and its length is about $H$: it falls as the span grows. A strut on a uniform foundation of stiffness $k$ buckles at $2\sqrt{k\,EI}$, so the factor is

$$\lambda \approx \frac{2\sqrt{k\,EI}}{N_0} = \frac{4}{a^2}\sqrt{\frac{E}{\sigma}\,\frac{EI}{g}\,H},$$

and with $H$ in proportion to $a$ it goes as $a^{-1.5}$. The compression pays the square of the span; the springs pay the root of its inverse.

The deck buys the square root #

The obvious remedy is a stiffer deck. The formula says how much it buys.

Four times the stiffness — a box half as deep again, with heavier plates — buys twice the factor, at every span. That is the square root the strut on springs predicts, and it is what makes the compression a real limit rather than a detail. The span the deck can reach at a given factor goes as the cube root of its stiffness: to carry the same factor from 1,000 m to 2,000 m the deck would need $2^3 = 8$ times the stiffness, and to carry it to 1,600 m it would need four. The deck’s self-weight rises with its stiffness, and that raises the compression it is trying to resist.

The other terms in the formula are no more generous. The ratio $E/\sigma$ is the stays’ stiffness per unit of load: a stay worked harder is a smaller stay and a softer spring, so a 25 per cent higher dead-load stress costs about 10 per cent of the factor. And the dead load $g$ appears below the root, as the deck’s own stiffness over its own weight — the one ratio a lighter deck improves, which is why the longest stayed spans have steel decks rather than concrete ones.

A taller tower helps twice #

The remaining term is the tower’s height, and it enters twice: once in the compression, which a taller tower reduces because its stays stand steeper and pull less sideways, and once in the stays’ stiffness, which steeper stays increase.

From a tower a fifth of the half-span tall to one two fifths of it, the compression at the tower halves and the factor nearly doubles. The cable that is a spring found that the cheapest tower for the stays’ steel alone is about 0.4 of the half-span tall, and that real bridges use 0.2 to 0.3, because a tower has to be built and stood up in wind. The deck’s compression is another reason to go higher. At 0.25 of the half-span — a common proportion — the 1,000 m deck would buckle at about four times its weight rather than seven.

The sag that does not matter #

A long stay sags under its own weight, and a sagging stay is softer than a straight one, because part of any extension it is given goes into pulling out the sag rather than stretching the steel. Ernst’s equivalent modulus, $E/(1 + (\gamma \ell_h)^2 E / 12\sigma^3)$, is how the softening is usually taken into account, and it is often named as the thing that limits stayed spans. For the deck’s stability it is not.

The sag costs the long, flat stays near mid-span a tenth of their stiffness, and the buckle hardly uses them: it is decided where the compression is large, and the compression is large near the tower, where the stays are short and steep and their sag is negligible. Leaving Ernst’s correction out changes the critical factor by less than one per cent. The sag matters to how a long stay shares traffic load and to its vibration, which held by something that goes soft traced for the guys of a mast. It is the compression, not the sag, that brings a long stayed deck near to buckling.

The day before the two arms meet #

A stayed deck is usually built outwards from each tower as a balanced cantilever, a segment and a stay at a time, until the two arms meet at mid-span and are joined. On the day before the closure the arm is free at its tip: nothing holds its slope or its deflection at mid-span, and the most dangerous day for a structure is often one on which a restraint the finished structure relies on is missing.

Here it hardly matters. With its tip free, the 1,000 m arm buckles at 6.72 times its dead load, against 6.91 once it is joined — three per cent less. At 500 m the difference is seven per cent and at 2,000 m it is under one. The reason is the same one that made the long stays’ sag irrelevant: the buckle is decided in the inner half of the arm, where the compression is, and the tip carries almost none. A free tip is a free end on a part of the deck that was not being asked to do anything. The erection stage has real stability problems — a tower standing free before its back stays are installed, a deck caught by wind on a long cantilever — but the deck’s own compression is not among the worst of them.

Why the stays are not simply made stiffer #

The formula offers one more lever, the stays’ stiffness, through $E/\sigma$: a stay designed to a lower stress is a larger stay and a stiffer spring. It is not usually pulled, for three reasons that have nothing to do with the deck.

The first is that the stay’s stress under dead load is set by fatigue rather than by strength. The cable that is a spring found that a stay carries a large, steady dead-load tension and a small, frequently repeated traffic one, and the strand’s endurance limits the range rather than the peak; designers choose a dead-load stress that leaves room for the traffic range, around 0.45 of the strand’s strength in total. The second is that a stay twice as large weighs twice as much and catches twice the wind, and its anchorages at deck and tower grow with it. The third is the arithmetic: halving the stress buys only $\sqrt 2$ in the factor, because the stays’ stiffness sits under the same square root as the deck’s.

So the levers a designer has all pay in square roots — of the deck’s stiffness, of the stays’ area, of the tower’s height — and the compression they are working against grows as the square of the span. That mismatch is the limit the self-anchored suspension bridge found in its deck, arriving in the stayed bridge by a different route and with no cable tension to postpone it.

The 1,000 m deck by hand #

The figures can be checked with the strut-on-springs formula. The compression at the tower is the sum over the 33 stays of $g s x_i / H$, with $s = 15$ m, $H = 200$ m and $x_i = 15, 30, \dots, 495$ m: $150 \times 15 \times 8{,}415/200 = 94{,}669$ kN, the figure’s 95 MN.

The stiffness of the first stay per metre of deck is $E g \sin\theta / (\sigma \ell)$, with $\sin\theta = 0.997$ and $\ell = 200.6$ m: $1.95 \times 10^{11} \times 1.5 \times 10^5 \times 0.997 / (6 \times 10^8 \times 200.6) = 2.42 \times 10^5$ N/m², the figure’s 242 kN/m per metre. A strut on that foundation buckles at

$$2\sqrt{k\,EI} = 2\sqrt{2.42 \times 10^5 \times 5 \times 10^{11}} = 6.96 \times 10^8 \text{ N} = 696 \text{ MN},$$

which is 7.3 times the 95 MN. The full solution gives 6.91, a little less, because the foundation softens away from the tower faster than the compression falls. The formula is a good first estimate and on the unsafe side by about 5 per cent. Scaled to 2,000 m it predicts $7.3 / 2^{1.5} = 2.6$, against the full solution’s 2.3: the estimate keeps its shape and loses a little more of its accuracy, because a longer arm’s foundation is softer further out.

What the springs leave out #

The deck here is elastic, its towers rigid, its stays linear springs fixed at their tops, and its compression the dead load’s alone. A real tower leans when the deck buckles, because the stays’ horizontal components no longer balance, and that couples the deck’s buckle to the tower’s sway and lowers the factor. The back span and its backstays, which hold the tower top, are themselves springs. Traffic, which a bridge of this size carries at perhaps a tenth of its dead load, adds compression where it stands. The deck’s own plate panels buckle locally at a stress the box’s section design decides, and a deck with a square metre of steel in its cross-section is at 95 N/mm² from the compression alone, before any bending — well inside a steel deck’s capacity, but enough that its stiffeners are sized for it. Each of these takes the factor down. The model’s 6.9 at 1,000 m is an upper limit on what the deck has, and the slope of one and a half is what each of them leaves standing.

The deck is not there to carry the load began this line of argument with a suspension cable that needs a deck to keep it in shape. A stayed deck is the inversion: the deck is a strut the stays have to keep straight, and it is the strut that runs out first.

Still open: the deck stayed from both ends #

Every stay here runs from one tower to the deck, so the compression accumulates from mid-span towards each tower and is nothing at mid-span. A multi-span stayed bridge with intermediate towers, or a single span stayed from both towers with stays that cross at mid-span, loads its middle differently: the stays from the far tower pull the mid-span deck away from the near one, and the deck’s compression no longer vanishes in the middle, where its stays are softest. Whether crossing stays at mid-span — which some long-span designs adopt for stiffness under traffic — move the buckle towards the middle and lower the factor, or whether the extra stiffness of the crossing stays wins, is the question the single fan leaves for the tension that was left out to answer in stays rather than in a cable.