The deck that is its own cable

A suspension bridge has a cable and a deck, and the two are different objects doing different jobs: the cable carries and the deck distributes. A cable-stayed bridge has stays and a deck, and again they are different. A stressed ribbon has neither arrangement. The deck is the cable — a prestressed concrete slab, three hundred millimetres thick, hanging in a catenary between two abutments with nothing above it and nothing below it — and people walk on the curve.

Everything about the structure follows from one decision, and the decision is not structural.

The sag is set by a ramp gradient #

A parabola of sag $f$ over a span $L$ leaves each end at a slope of $4f/L$. That is not a small number at the sags cables like: at $f/L = 1/10$ the deck arrives at the abutment at 40 per cent — a 22° slope — which is a roof, not a footpath.

Accessibility limits for a pedestrian route sit around 8 per cent, or 1 in 12.5. Setting $4f/L \le 0.08$ gives

$$\frac{f}{L} \le 0.02,$$

exactly. The sag of a stressed ribbon is a wheelchair gradient rearranged, and it comes out at a fiftieth of the span for reasons that have nothing to do with any force in the structure. The 100 m span drawn therefore sags 2 m, arrives at 4.57°, and hands over the rest of the design as a consequence.

What the abutment is asked for #

$$H = \frac{wL^2}{8f} = \frac{35 \times 100^2}{8 \times 2} = 21{,}875\ \text{kN}$$

against a total deck weight of $wL = 3{,}500$ kN. The horizontal pull at each end is 6.25 times the weight of the entire bridge, and it is permanent, and it does not go away when the bridge is empty.

That number is what makes stressed ribbons rare rather than common, because it is a foundation problem rather than a structural one. It is also a number that gets no smaller when the bridge gets lighter: $H/wL = L/8f$ contains no load at all, so the abutment force in deck-weights is a pure function of the sag ratio. Making the deck out of carbon fibre would halve the force and leave the ratio exactly where it was, and the anchorage would still be sized at 6.25 times whatever the deck weighs. Twenty-two meganewtons of horizontal thrust has to be resisted by something: rock anchors into a competent stratum, a very large gravity block, or a tie under the ground connecting the two abutments — which is the same three answers an arch’s thrust gets, because it is the same problem with the sign of the curvature reversed.

Two things soften it slightly and neither by much. The full live load raises $H$ from 21,875 to 34,375 kN, a factor of 1.57, so the anchorage is sized for the loaded case and lives most of its life at two thirds of it. And a ribbon can be built with its two abutments tied to one another below ground when the geometry permits, which converts the whole structure into a self-anchored system and the thrust into an internal force — at the cost of a tie 100 m long carrying 22 MN.

The end that cannot be clamped #

The ribbon has bending stiffness. It is a concrete slab, not a chain. But that stiffness is almost entirely irrelevant, and where it is not irrelevant it is catastrophic.

Away from the ends, the deck’s own $EI$ competes with the geometric stiffness the tension provides, and it loses badly: the length over which bending matters at all is

$$\ell = \sqrt{EI/H} = \sqrt{324{,}000/21{,}875} = 3.85\ \text{m},$$

which is 3.85 per cent of the span. Over the other 96 per cent the ribbon is a cable and its flexural stiffness contributes nothing.

At the abutment the deck has to be turned from its 4.57° end slope to whatever the approach requires. If it were clamped, the moment would be

$$M = \theta\sqrt{EI\,H} = 0.08\sqrt{324{,}000 \times 21{,}875} = 6{,}735\ \text{kNm},$$

which is 4.21 times the deck’s own capacity. The clamped detail is not conservative or expensive; it is impossible.

The alternative, used where a saddle will not fit, is to haunch the deck locally — two or three times the depth over the last few metres — which raises $M_{cap}$ faster than it raises $\ell$, since the first goes as $d^2$ and the second as $d^{3/2}$.

Which free body produced the number #

Every quantity above comes from one of two cuts, and it is worth naming them because the second is the one people skip.

The whole ribbon, cut at mid-span. The half-bridge is in equilibrium under half the weight, the reaction at one abutment, and the horizontal force $H$ acting through the mid-span point. Taking moments about the abutment gives $H f = wL^2/8$ directly, which is the entire derivation of the headline number. Nothing in it is about concrete, prestress, or the deck’s depth; it is the same moment equation a funicular polygon draws with a ruler.

A short length at the abutment. This one has the deck’s bending in it. Cut a metre of ribbon just inside the support and the forces on the cut face are the tension along the deck, a shear, and a moment — and the moment exists only because something is stopping the deck being the curve it wants to be. Away from the ends nothing does, so the moment is zero and stays zero. That is why the whole of the flexural design of a stressed ribbon happens in the last five metres at each end.

The contrast between the two cuts is the structure’s whole character. One free body decides everything about the forces and knows nothing about the deck; the other decides everything about the deck and is 4 per cent of the span long.

Temperature, which is the reason it is prestressed #

A cable’s arc length is longer than its chord by

$$s - L \approx \frac{8f^2}{3L},$$

107 mm for the ribbon drawn. That is a small number and it depends on $f^2$, which is what makes the ribbon so sensitive: differentiating, a change of arc length $ds$ produces

$$df = \frac{3L}{16f}\,ds,$$

and at $f = L/50$ the multiplier $3L/16f$ is 9.375. A 30 °C rise lengthens 100 m of concrete by 30 mm, and 30 mm of extra arc becomes 281 mm of extra sag — 14 per cent of the sag itself, and enough to change the horizontal force from 21,875 kN to 19,178.

This is why the deck is prestressed rather than merely tensioned by its own weight. Prestress raises $H$ above what equilibrium requires, which does two things at once: it stiffens the geometry against sag change, and it keeps the whole section in compression so that the curvature reversals at the ends — and the tension a live load on half the span produces — never crack it.

Why it is lively #

The last consequence of hanging a thin slab on a very large tension is the frequency, and it is the property that has closed more of these bridges than any structural check.

The ribbon behaves as a taut string, so its modes are

$$f_n = \frac{n}{2L}\sqrt{\frac{H}{\mu}} = n \times 0.427\ \text{Hz}.$$

Evenly spaced. Not merely low — a beam’s modes go as $n^2$ and spread out, so a beam has a first mode and then a gap; a string has modes at 0.427, 0.854, 1.281, 1.708, 2.135 Hz and onward, one every 0.427 Hz forever.

The damping of a prestressed concrete ribbon is around 1 per cent, and there is very little mass per unit length to hide a crowd in. Every stressed ribbon of any span built since about 1990 has a tuned mass damper on it, and what a damper does is buy the only thing that stops a resonance.

What it is for #

It is worth being clear about what this arrangement buys, because the list above is nearly all cost.

It is the lightest way to cross a gap. There is no separate cable, no tower, no stay, no hanger, no truss and no pier. The structure is a slab three hundred millimetres thick and nothing else, and for a 100 m footbridge that is a smaller quantity of material than any alternative by a wide margin. Ranking materials and forms is a question about the load case, and for a long light pedestrian span with good rock at both ends this one wins outright.

It needs almost no maintenance. A prestressed concrete slab in permanent compression has no bearings, no expansion joints, no cables to inspect and no coating to renew. The two details that do need attention — the saddles and the anchorages — are at the ends, on land, and reachable.

It is very difficult to make ugly. The shape is the funicular of its own weight, which is to say the structure is showing exactly what it is doing, and there is nothing else in the picture. That is not a structural argument and it is the reason most of these bridges got built.

Where the model stops #

The cable is a parabola. Under uniform load along the span it is; under uniform load along the arc, which is what self-weight actually is, it is a catenary. At a sag ratio of 1 in 50 the two differ by less than a millimetre and the parabola is the honest choice, but nothing in the arithmetic here would survive a deeper sag.

The deck is one member with one $EI$. A real ribbon is precast segments post-tensioned together, so it has a joint every few metres, and its bending stiffness depends on whether those joints are open — which depends on the prestress, which is the quantity being designed. The stiffness is a function of the load case.

Half-span loading is quoted as a force and not as a shape. A cable under an unsymmetric load changes shape rather than merely stretching, and the deflected form is not a scaled version of the dead-load one. Getting it right needs the cable equation solved with the geometry updated, which is a nonlinear problem the numbers here approach only through $H$.

The abutment is a number. The 21,875 kN has to be resisted by ground, and ground moves: a few millimetres of abutment spread lengthens the chord, which flattens the sag, which raises $H$, which spreads the abutment further. Whether that converges is a question about the foundation’s stiffness rather than the bridge’s, and it is the check a ribbon actually lives or dies by.

The live load is uniform. It is not — a crowd is patchy, and half the span loaded is the case that moves a cable most, because it is the case that asks the shape to change rather than merely to stretch. The 1.57 ratio quoted for $H$ under full load is the easy case.

And no drawing here shows a person on it. The dynamic behaviour that decides whether the bridge is usable is a crowd, correlated in time, on a structure with 3,000 kg per metre and 1 per cent damping — and the picture of that is a lock-in rather than a load.

The ladder from here #

Later rungs on this anchor: the nonlinear cable analysis proper, with the shape updated as the load goes on and the tangent stiffness recomputed at each step. The multi-span ribbon over intermediate piers, where each span’s thrust nearly cancels its neighbour’s and the pier carries only the difference — which is how the long ones are built. Erection, which is the hardest part: the ribbon is a catenary of bare tendons before it is a deck, its sag at that stage is very different, and the segments are hung on it in an order that determines the finished geometry. The saddle detail resolved as a beam on a curved support with slip. Ribbon-and-arch hybrids, where an arch below the deck takes the thrust and the ground takes almost nothing. And the anchorage itself, which is a rock-mechanics problem rather than a structural one and is the reason these bridges appear in gorges and almost nowhere else.