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.

The shape itself is not in question. A ribbon takes up the funicular of the load it carries the moment it is built, which under uniform load is a parabola, and it has no bending in it anywhere except where something stops it being that curve. What is in question is only how deep that parabola is allowed to be.

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.

The duality is worth stating plainly, because it is where the intuition comes from: a cable in tension and an arch in compression under the same load take the same curve and hand the same horizontal force to the same abutments. A stressed ribbon is a very flat inverted arch that people walk on, and the shallower an arch is the harder it pushes. It is the identical reciprocal, arrived at from the other side, and it has the identical consequence — the curve has no flat part to settle on.

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.

What has to happen instead is that the end curves over a saddle rather than meeting the abutment at a point. The curvature the deck can take is set by its own capacity, $1/R = M_{cap}/EI$, so the radius is $EI/M_{cap} = 324{,}000/1{,}600 = 203$ m; turning through the 0.08 rad end slope at that radius takes an arc of $0.08 \times 203 = 16.2$ m. The saddle is not a refinement of the detail. It is the detail, and it is a sixth of the span long.

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.

The general statement behind that arithmetic is that an axial strain is a movement proportional to length, and it is applied to every structure of any material. What is peculiar to a ribbon is the geometry that stands between the movement and the shape: a flat cable multiplies a small change of arc length into a large change of sag, by the factor $3L/16f$, and that factor is 9.375 only because the sag is a fiftieth of the span. It is the reason a stressed ribbon’s shape is a function of the weather.

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.

The abutment has a stiffness it has to exceed #

The feedback between the anchorage and the thrust is not merely a nuisance to be noted; it has a criterion, and the criterion is a number the foundation either beats or does not.

Spread the abutments by $\delta$ with the cable’s own length fixed. A longer chord means less sag, and less sag means more thrust — so the spread makes the force that caused it larger. For a shallow parabola the arc length is $S \approx L(1 + 8f^2/3L^2)$, and holding $S$ fixed while $L$ grows gives $\Delta f = -3L\delta/16f$. Substituting into $H = wL^2/8f$:

$$\frac{dH}{d\delta} = \frac{3wL^3}{128\,f^3}$$

A stiffness, in kilonewtons per millimetre of spread, and the abutment must be stiffer than it or the loop runs away.

Put numbers to a 100 m ribbon of 1.5 m sag carrying 30 kN/m: $H = 25{,}000$ kN and

$$\frac{dH}{d\delta} = \frac{3 \times 30 \times 10^6}{128 \times 3.375} = 208{,}000\ \text{kN/m}$$

— 208 kN for every millimetre. An abutment in sound rock beats that comfortably; one on a compressible fill does not, and the bridge would flatten progressively over years rather than failing at a moment.

Two features of the expression are the design content. It goes as the cube of the span over the cube of the sag, so a flatter ribbon is dramatically more sensitive — halving the sag multiplies the required abutment stiffness by eight, on top of doubling the thrust itself. And it contains no property of the deck at all: stiffening the ribbon does nothing whatever for it, because the mechanism is a geometry chasing a foundation.

Which is the same parallel-stiffness criterion a snapping frame obeys — a negative stiffness cancelled by a positive one in parallel — arriving here with the ground supplying the positive term. And it says plainly why these bridges are built where they are built. The structure is a cable with a walkway on it; the design is an anchorage.

A point load stays where it is put #

A suspension bridge separates the cable from the deck and inserts a stiffening girder between them, whose whole job is to spread a concentrated load along the span. A stressed ribbon has merged the two, so it has the cable’s geometric stiffness and the deck’s own bending stiffness and nothing in between — and the deck’s bending stiffness is very small.

The length over which a point load is felt follows from the two acting together. A member with bending stiffness $EI$ under a tension $H$ has a characteristic length

$$\ell = \sqrt{\frac{EI}{H}}$$

and for a 300 mm deep concrete ribbon 3 m wide — $EI \approx 230{,}000$ kNm² — under 25,000 kN of thrust, that is

$$\ell = \sqrt{\frac{230{,}000}{25{,}000}} = 3.0\ \text{m}$$

Three metres. A person standing on a stressed ribbon produces a dip about three metres either side of them and essentially nothing beyond it, and the dip travels with them.

That is the whole of why a ribbon feels the way it does underfoot, and it is a stiffness statement rather than a strength one. It is also why the deck’s own bending stiffness matters so little to the global behaviour and so much to the local: it appears under a square root divided by an enormous tension, so quadrupling $EI$ only doubles the reach — and the reach is what a pedestrian is actually experiencing.

The same length decides the segment joints. A precast ribbon has a joint every few metres, comparable with $\ell$, so a load can sit within one characteristic length of a joint and open it — which is why the prestress is sized against a local condition rather than against the global thrust the rest of this page has been computing.

It also explains the one thing a ribbon is unambiguously good at. Because a load is not spread, it is not shared either — so a load applied anywhere is carried by the tension that was already there, and adding load does not change the structure’s shape appreciably beyond three metres. A cable-supported deck of any other kind redistributes; a ribbon absorbs. That is a poor property for a bridge carrying vehicles, whose loads are large and moving, and an excellent one for a footbridge, whose loads are small and whose users are the only thing that has to be comfortable.

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.