The pier that carries only the difference

The deck that is its own cable found what laying a footbridge’s walking surface as a catenary at a fiftieth of its span does: it hands the abutments six and a quarter times the bridge’s weight as a horizontal pull, and it makes the abutment’s stiffness a criterion rather than a detail, because a spreading abutment lets the ribbon flatten, and a flatter ribbon pulls harder. That essay’s single span ended between two abutments. It named how the long ribbons are actually built — as several spans, the tendons running continuously over a saddle on each intermediate pier, where “each span’s thrust nearly cancels its neighbour’s and the pier carries only the difference”.

The phrase is accurate under dead load and misleading under everything else. This essay computes the difference, and finds that it is neither small nor fixed: it depends on the pier’s own stiffness, against a stiffness the ribbon supplies itself, and whatever the pier declines to carry, the ribbon carries as movement.

Two spans and a pier #

The ribbon throughout is two spans of 100 m, each laid at a sag of 2 m — the fiftieth of the span a wheelchair gradient allows — carrying 35 kN/m of its own weight and with an axial stiffness of 43,200 MN, the numbers of the single span before it. Under the dead load each span pulls on the pier with $H = wL^2/8f = 21{,}875$ kN, and the two pulls cancel. The pier is a column whose top can lean; its horizontal stiffness $k$ is what the rest of the argument turns on, and a concrete pier 10 m tall and 3 by 2 m in plan, fixed at its foundation, has $k = 3EI/h^3 \approx 200{,}000$ kN/m.

Now add 20 kN/m of crowd to the left-hand span. That span’s thrust rises. The right-hand span’s does not, at first. The pier top is pulled toward the loaded span, leans, and in leaning shortens the loaded span’s chord — which lets that span sag more and pull less — and lengthens the other’s, which flattens it and makes it pull more. The lean stops where the difference between the two pulls equals the pier’s resistance.

With a rigid pier the loaded span behaves as if it were alone: its thrust climbs to 31,300 kN, it sags a further 195 mm, and the pier carries the whole difference, 9,440 kN — more than 40 per cent of a span’s dead thrust, applied at the top of a 10 m column. With the 200,000 kN/m pier the top leans 25 mm, and that small movement changes the picture: the pier carries 4,953 kN, about half; the loaded span sags 358 mm, nearly twice as much; and the unloaded span rises 191 mm, lifted by a load that is nowhere near it.

The stiffness the ribbon supplies #

The single-span essay found the number that decides this. Spread a span’s ends apart by $\delta$ with its length fixed, and its sag falls and its thrust rises, at a rate

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

which for these spans is 102,500 kN/m. In a single span that is the stiffness the abutment must beat. Between two spans it is the stiffness with which the ribbon itself resists the pier’s lean: leaning toward the loaded span by $\delta$ reduces that span’s thrust by $g\delta$ and increases the other’s by $g\delta$, so the ribbon pushes the pier back with $2g\delta$ — in parallel with the pier’s own $k\delta$. The unbalanced thrust is shared between two springs, and the pier’s share is

$$\frac{k}{k + 2g}.$$

The full calculation follows the formula closely, and the formula says what the figure shows. A pier ten times stiffer than $2g$ carries 93 per cent of the difference — the same parallel-stiffness arithmetic that decides whether a roof snaps through, with the ribbon’s geometric stiffness in place of the roof’s; a pier equal to it, half; a pier a tenth of it, an eighth. The pier described above, at about 200,000 kN/m, sits almost exactly at $2g$ — which is not a coincidence of these numbers but a consequence of proportions: a pier stocky enough to stand up to a ribbon’s construction loads, beside a span flat enough to be walkable, lands near the middle of the curve. The pier and the ribbon are the same stiffness to within a factor of a few, so neither dominates, and the question of which carries the unbalance has no default answer.

This is the stiffest path takes the load in a form where one of the paths is geometry. The pier is a column with an elastic stiffness; the ribbon is a cable whose stiffness comes from its shape, and which is larger the flatter it is. Flattening the ribbon to ease its gradient — the whole reason ribbons are flat — stiffens it against the pier by the cube of the flattening and pushes the load off the pier and into the ribbon’s movement.

Where the difference goes when the pier does not take it #

What the pier does not carry is not lost; it is carried by the ribbon rearranging itself until the two thrusts nearly balance. The loaded span’s ends come closer together, so it sags more and pulls less. The unloaded span is stretched and flattens, so it pulls more. With the softest pier drawn the thrusts are within a few kilonewtons of each other, and the price is movement: 520 mm of extra sag in the loaded span, 2.7 times the rigid-pier value, and 393 mm of rise in the other.

For a footbridge those numbers are the design. A rise of four tenths of a metre in a hundred-metre span changes the gradient at its ends by more than one and a half per cent, a fifth of the eight per cent the ribbon was laid at to be walkable, and a crowd moving from one span to the other drives the ribbon through that change twice. A soft pier trades a horizontal force at the top of a column for a vertical movement of the walking surface, and the second is the one the users notice.

Four spans, and a load that travels #

A longer ribbon repeats the exchange at every pier. With piers of 200,000 kN/m, a load on the first span leans the first pier, which unloads the second span a little, which leans the second pier less, and so on: the pier forces fall from 5,318 to 1,361 to 314 kN along the ribbon, and the rises from 152 to 40 to 12 mm. The disturbance decays by a factor of about four per span.

With soft piers it hardly decays at all. The loaded span sags 691 mm, and every other span rises by around 200 mm, because each pier passes nearly the whole unbalance to the next. The ribbon then behaves as one long cable over rollers, and a crowd on one span moves the entire walkway — which is the right description of the behaviour, and the wrong one for a structure whose users are walking on it.

Why the dead load hides it #

A pier between two ribbon spans is designed first for the state it spends its life in, and that state is balanced. Under the dead load the two pulls cancel exactly, the pier stands straight, and its horizontal load is nothing. The loads that unbalance it are live loads, which are occasional, and temperature, which is cyclic — so the pier’s horizontal force is entirely made of the load cases a designer is most tempted to treat as secondary. A pier sized for the dead load and checked for the live load “as a small difference” is checked for the difference a rigid pier would see, which is the right number for strength and the wrong one for everything else: a pier that is actually flexible sees half of it, and the ribbon’s walking surface sees the other half as movement that nobody computed.

The two failures are different in kind. A pier too weak for the rigid-pier difference is a strength problem, found by a routine check. A pier flexible enough to shed most of it is a serviceability problem: a walking surface that rises and falls by tenths of a metre as a crowd moves, gradients that exceed the walkable limit near the piers for as long as the crowd stands still, and joints in the precast segments opened by the change of curvature. None of it is visible in the dead-load design, because under the dead load it does not exist.

The same choice at a suspension bridge’s middle tower #

The ribbon’s pier is the smallest example of a problem that suspension bridges meet at their largest. An ordinary suspension bridge has two towers, each with a main span on one side and a side span on the other, and its towers are made deliberately slender: the main cable runs over a saddle at the top, the tower leans a little when the main span is loaded, and the cable rebalances itself across the saddle. The tower carries the vertical load and very little of the horizontal difference, which is why suspension bridge towers can be tall and thin.

A suspension bridge with three or more towers cannot do that at its middle tower. Load one main span and the middle tower is asked for the difference between two main spans’ pulls, and the choice is exactly the one above: make the tower stiff and it carries the difference, as a very large bending moment and as a friction demand on the saddle the cable might slip through; make it flexible and the loaded span sags and the neighbouring span rises, by amounts that govern the deck. The stiffness that decides between the two is again the cable’s own geometric stiffness, the stiffness that a stiffening girder was never there to supply. Multi-span suspension bridges have been built only since that trade was settled by making the middle tower an intermediate stiffness, neither the slender tower of a two-tower bridge nor a rigid one, with its saddle designed for the friction the resulting difference demands.

Temperature, which a balanced ribbon was not balanced for #

Two equal spans are balanced at every temperature: warm them and both sag a little more and pull a little less, by the same amount. Unequal spans are balanced under dead load only if their sags are in proportion to the square of their spans, so that $wL^2/8f$ is the same in each. They are then not equally flat — the shorter span is flatter — and a flatter span’s thrust changes more for the same change of length.

So a ribbon with spans of 80 and 120 m, balanced at its installation temperature, loads its pier every time the temperature changes. A rigid pier between them carries 2,155 kN on a 30° cooling and 1,314 kN on a 30° warming — the asymmetry because the cable’s thrust is not linear in its length — and the 200,000 kN/m pier carries 923 and 754 kN. The force reverses with the season and cycles with the day. The movement nobody applied reaches a structure through its restraints; here the restraint is the neighbouring span, and the only way to avoid the force is to make the spans equal.

The cycle matters more than the peak. A pier foundation pushed one way by day and the other by night, year after year, is loaded like the abutment of an integral bridge, whose backfill ratchets a little further each summer because granular soil pushed and released does not return to where it was. A pier on a shallow footing in granular ground, carrying a reversing horizontal force of a meganewton or two, is a candidate for the same slow walk — and a pier that has leaned permanently by a few millimetres has shifted the ribbon’s dead-load balance and started to carry a permanent difference of its own. The equal-span ribbon avoids all of it; the unequal one needs a pier foundation designed for a cyclic load that the dead-load balance says is not there.

Balanced thrusts or walkable gradients #

The temperature finding is the milder form of a choice every ribbon of unequal spans has to make.

Lay every span at the gradient a wheelchair allows and the sags are in proportion to the spans, so the thrusts are in proportion to the spans too: the longer span pulls harder, and the pier between them carries the difference under the dead load, permanently — 5,969 kN for the 200,000 kN/m pier beside a span half as long again as its neighbour, with the pier leaning 30 mm for ever. Balance the thrusts instead and the pier is free, but the sags must go as the square of the spans, the longer span sets the gradient, and every span pulls as hard as the longest: half as hard again as two equal spans would at a ratio of 1.5, twice as hard at 2, with the shorter span laid at half the gradient it could have had and its abutment taking the long span’s thrust.

A ribbon cannot have balanced thrusts, walkable gradients and unequal spans at once. It can have any two. The long multi-span ribbons are built with nearly equal spans for this reason, and where the ground forces unequal ones, the pier that takes the permanent difference is designed as a foundation rather than as a column.

The numbers at the first pier, by hand #

For each 100 m span at 2 m sag and 35 kN/m, the geometric stiffness is $g = 3 \times 35 \times 100^3/(128 \times 2^3) = 102{,}500$ kN/m. The pier at 200,000 kN/m against the ribbon’s $2g = 205{,}000$ gives a share of $200{,}000/405{,}000 = 0.49$, so the pier should carry about half of the rigid-pier difference of 9,440 kN: 4,660 kN, against 4,953 from the full solution, which follows the thrusts as they change and the stiffness with them.

The pier’s lean is its force over its stiffness, $4{,}953/200{,}000 = 25$ mm, and the base moment of a 10 m pier carrying it is $4{,}953 \times 10 = 49{,}500$ kN·m — the moment a rigid pier would see halved, and still enough to govern the pier’s design.

Shallow cables, fixed abutments and a sliding saddle #

Each span is a shallow elastic cable. Its length is its chord plus $w^2c^3/24H^2$, which is the parabola’s arc length to first order, and it stretches as $H/EA$. The deck’s bending stiffness, which spreads a point load over a few metres, does nothing to these global numbers.

The tendons run continuously over each pier, free to slide on the saddle, so the pier receives the difference of the two thrusts and nothing else; a saddle that grips transfers a vertical load and a moment the model leaves out. The abutments are fixed — the single-span essay’s stiffness criterion is assumed met — and the piers lean without settling.

The live load is uniform over a whole span. A crowd over half a span produces the same thrust difference with an asymmetric sag within the span, which is the single-span essay’s own case.

What the pictures cannot show #

That the ribbon moves in time. Every figure is a static equilibrium, and a crowd walking from one span to the next drives the ribbon from one of these shapes to its mirror image over the minutes it takes to cross. The ribbon’s lowest modes involve exactly this exchange — one span down as the other comes up — and with soft piers they are low in frequency and lightly damped, which is where the load that will not hold still finds its purchase.

Still open: the ribbon that is built span by span #

A multi-span ribbon is erected one span after another, each span’s tendons pulled across and anchored before the next, and the piers see the unbalanced thrust of a finished span against an empty one before the ribbon is complete. Whether the pier stiffness that is right for the finished bridge is also enough for the stage at which one span is hung and its neighbour is not — or whether temporary ties are what actually decide the pier’s size — is the construction question the same arithmetic asks, with the unbalance at its largest and the ribbon’s own balancing stiffness absent from one side.