The tendon that pulls before the deck arrives

The pier that carries only the difference found that a pier between two finished spans of stressed ribbon carries nothing under the dead load and a great deal under a crowd: 20 kN/m on one of two 100 m spans asks a rigid pier for 9,440 kN, and a pier of ordinary stiffness carries half of that and lets the ribbon turn the rest into movement. It ended on the state the ribbon is in before it is finished — one span hung and its neighbour not — where the balance that protects the pier under the dead load does not yet exist.

The fear is easy to state. A finished span pulls on its pier with $wL^2/8f = 21{,}875$ kN, more than twice what the crowd asks for, and during erection there is a moment when nothing pulls back. If that moment is real, the pier is designed by a stage that lasts a few weeks rather than by a bridge that lasts a century, and the temporary works are a structure in their own right.

This essay computes the stage, and finds that the fear is right about one way of building a ribbon and wrong about the usual one. What decides it is not the hung span but what is already across the empty one.

How a ribbon is put up #

A stressed ribbon carries its deck on tendons, and the tendons have to be there before the deck is. The usual sequence is the one the ribbon’s form invites. First the bearing tendons are pulled across every span, over a saddle on each pier, anchored in the abutments at each end and tensioned. Then the precast deck segments are hung on them, slid or lowered into place one after another until a span is full. Then the joints between segments are cast, and a second set of tendons is stressed through the whole band, which turns a chain of segments hanging on cables into the continuous prestressed ribbon the finished-bridge calculation assumes.

During the hanging the tendons carry everything. The segments are not yet joined, so the ribbon’s axial stiffness is the steel’s alone — a few thousand meganewtons for the strands of a 100 m span, rather than the 43,200 MN of the finished concrete band — and each span’s shape is set by its weight and by the length of strand in it. That length is fixed when the tendons are strung, because it is the length the finished span needs: cut it differently and the finished ribbon has the wrong sag. A tendon cut to a length is a member built to a length, and in a structure with more supports than it needs, a member’s length is a force waiting for somewhere to go.

The question of this essay is what that fixed length does to the pier between a span with its segments on and a span still waiting for them.

A span with its deck, beside one without #

The ribbon is the one the earlier essays used: spans of 100 m, laid at a sag of 2 m, carrying 35 kN/m when finished. The bearing tendons are sized to carry the finished thrust at 1,100 N/mm², a little over half their tensile strength, which needs about 19,900 mm² of strand — some 130 strands of 15.7 mm — and gives the span an axial stiffness of 3,880 MN. Bare, the strands weigh about 3 kN/m. The pier is the one before: a concrete column 10 m tall with a horizontal stiffness of 200,000 kN/m.

Hang the segments on the left span. It sags to its finished shape — 2.04 m, slightly more than 2 because the pier has leaned toward it — and pulls on the pier with 21,442 kN. The right span carries only its strands, and sags 0.20 m. So far this is what the fear predicted.

What it did not predict is the right span’s pull. The bare tendon pulls on the pier with 18,363 kN, nearly as much as the span with its deck. The pier carries only the difference, 3,080 kN, and leans 15 mm. A crowd on the finished bridge asks the same pier for 4,953 kN.

The stretch the empty span already holds #

The bare tendon pulls because it is short. A tendon that will carry $H_0$ in the finished span stretches by $H_0 L/EA$ under that force — here $\varepsilon L$, with $\varepsilon = 1{,}100/195{,}000 = 0.0056$ the working strain, which is 564 mm over 100 m — so it is cut 564 mm shorter than the length it will finally hang at.

The sag sets that length. A cable hanging with sag $f$ over chord $L$ is longer than its chord by $8f^2/3L$, which for the finished span is 107 mm. So the finished tendon hangs 107 mm longer than the span and was cut 564 mm shorter than that: it is cut about 457 mm shorter than the span it crosses. Strung across a span with no deck on it, it still has to reach from support to support, and a light cable under a large tension hangs nearly straight, so its own sag buys back almost nothing. It must stretch nearly the whole 457 mm — and stretching 457 mm of the 564 that $H_0$ produces takes four fifths of $H_0$.

Written as a balance, the hung span’s thrust and the bare span’s differ by what the hung span’s slack lets go of:

$$(H_0 - H_\text{bare})\,\frac{L}{EA} = \frac{8f^2}{3L}, \qquad H_\text{bare} = H_0\left(1 - \frac{1}{\rho}\right), \qquad \rho = \frac{H_0 L/EA}{8f^2/3L} = \frac{3\varepsilon}{8s^2},$$

with $s = f/L$ the sag ratio. The ratio $\rho$ is the tendon’s stretch against the slack its sag needs — 564 mm against 107, and it depends on nothing but the working strain and the sag ratio. For these spans it is 5.3. The bare tendon then holds $1 - 1/5.3 = 81$ per cent of the finished thrust before a single segment is hung, and a rigid pier carries the remaining fifth:

$$\Delta H = \frac{H_0}{\rho} = \frac{wL\,s}{3\varepsilon}.$$

That last form is worth reading slowly, because it reverses the intuition the deck that is its own cable built, where flattening a ribbon made everything harder. The stage asks the pier for the deck’s weight times $s/3\varepsilon$ — a fiftieth over three times the strain, a little over one deck-weight’s worth — and the thrust has dropped out of it. Halve the sag and the thrust doubles, but the stage force halves, because the flatter span needs a quarter of the slack and its neighbour holds more of the thrust in stretch. For the ribbon above it gives 4,137 kN on a rigid pier, against 4,109 from the full solution, which keeps the bare tendon’s own sag.

Nothing beyond the pier #

The other way to build the ribbon is the one the fear had in mind: tendons that end at each pier, so that the first span’s strands are anchored on the pier top and nothing beyond it pulls back until the next span’s strands are strung. Then there is no stretch held across the empty span, because there is no tendon across it.

The three curves are the whole comparison. With nothing beyond the pier, a rigid pier carries the hung span’s entire thrust, 21,875 kN, and the 200,000 kN/m pier carries 19,290 kN while leaning 96 mm — four times the crowd’s force, and about what the fear predicted. Beside bare tendon the same pier carries 3,080 kN, two thirds of the crowd’s.

The curves also differ in shape. The pier reaches half of its rigid-pier force at a stiffness of about 67,000 kN/m for the stage beside bare tendon and about 180,000 kN/m for the crowd, because the spans either side of a pier during erection — a hung span on bare strand, which stretches far more easily than finished concrete, and a bare tendon beside it — give way to its lean more readily than two finished spans do. In the language of the earlier essay, where the stiffest path takes the load and the ribbon is one of the paths, the ribbon’s own stiffness against the pier is lower during erection, so a pier takes a smaller share of the stage than of the crowd, on top of the stage being smaller in the first place.

So the answer to whether the stage designs the pier is a statement about the erection scheme. Tendons strung continuously over the saddles, at their finished length, make the stage a fraction of the service unbalance. Tendons anchored span by span make it four times the service unbalance, on a pier that is otherwise designed for a difference.

Worst at a sag nobody builds a ribbon at #

The closed form has a limit built into it. $H_\text{bare} = H_0(1 - 1/\rho)$ needs $\rho$ greater than one: the tendon must be stretched by more than the sag’s slack, or the bare tendon has nothing left to pull with and simply hangs slack across the empty span. That happens when the sag ratio exceeds $\sqrt{3\varepsilon/8}$, which for a working strain of 0.0056 is 4.6 per cent.

Below the turn the stage grows with the sag, as $wL\,s/3\varepsilon$; above it the bare tendon is slack and the pier takes the whole of $H_0 = wL/8s$, which falls with the sag. The two branches meet at the turn, so the stage is worst at the sag where the tendon’s stretch exactly pays for the sag’s slack, and the bare tendon’s own weight rounds the corner off: the full solution peaks at 7,739 kN at 4.2 per cent. Below 3.5 per cent the stage is smaller than the crowd’s unbalance on a rigid pier; above it, larger.

No stressed ribbon is laid at 4 per cent. The walkable gradient that sets a ribbon’s sag holds it near 2 per cent, and the ribbon’s whole case against a suspension bridge is that it is flat. So for the ribbons that are built, the stage sits on the rising branch, well short of its peak, and the flatness that makes the finished ribbon pull hard on its abutments is exactly what makes its erection gentle on its piers. A shallow cable is a cable that is mostly stretch, and stretch is what the empty span holds in advance — the same fact, seen from the side, as a flat cable being stiff because of its shape rather than its steel.

The same figure says what happens to a deeper cable built the same way. A hung roof or a pipeline bridge at a sag of 8 or 10 per cent — nearer the proportions of a suspension bridge’s main cable than of a ribbon — has a bare tendon that goes slack under its own weight before it can pull, and its intermediate supports see nearly the whole thrust at every stage: at 8 per cent, nine tenths of it. For those the stage, not the finished structure, is the case to check first.

More strand, more trouble #

The working strain is the other half of $\rho$, and it is a design choice. A designer who wants a stiffer ribbon, a longer fatigue life or a margin against corrosion specifies more strand, and more strand carrying the same thrust works at a lower stress.

The figure turns a virtue into a hazard. Halving the tendon stress roughly doubles the stage force, because $\Delta H = wLs/3\varepsilon$ is inversely proportional to the strain. At 1,100 N/mm² the stage asks a rigid pier for 4,109 kN, under half the crowd’s 9,440. At 400 N/mm² — strand sized for stiffness rather than strength — it asks for 11,045 kN, and the stage has become the pier’s governing horizontal load. The crossing is at about 475 N/mm² for a rigid pier and about 575 for the 200,000 kN/m one.

The mechanism is the one in the bolt that was already stretched, turned round. A preloaded bolt hides an external load in its clamp, and the more it is stretched the more load it can hide. A bearing tendon hides the hung span’s thrust in its own stretch across the empty span, and a tendon stretched less has less room to hide it in. Generous strand is a stiffer bridge and a harsher erection, and the two decisions are usually made by different people at different times.

One pier at a time #

A ribbon of more than two spans meets the stage at every pier, and the order of hanging decides when.

Hanging from one end walks the unbalance along the ribbon. With the first span full the first pier carries 3.1 MN and the second and third a few hundred kilonewtons between them, passed along by the first pier’s lean. Fill the second span and the first pier is balanced again; the second now sits between a hung span and a bare one and carries 3.2 MN. Fill the third and the third pier takes its turn. Each pier meets the stage once, for as long as it takes to hang one span, and is then released.

Hanging the middle spans first loads the two outer piers together and leaves the middle one balanced from the start. It changes which piers are loaded at once, not how hard: the force on each loaded pier is the same 3.1 MN, because it is set by the two spans either side of it, and those are a hung one and a bare one whichever way the sequence goes. Where the order does matter is in the temporary works: a scheme that loads two piers at once needs two sets of whatever restrains them.

This is the shape of every prop’s worst day in a braced excavation, at a gentler scale. There the force locked into a prop at its installation stage decided its design, and the finished arrangement was nobody’s worst case. Here no pier’s worst case is the finished ribbon under dead load, which loads none of them, but neither is it necessarily the erection stage. It is whichever of the stage and the crowd is larger, and that comparison is a property of the tendon’s working strain and the ribbon’s sag rather than of the piers.

A tie to the ground #

Where the stage does govern — tendons anchored at each pier, a deep sag, or generous strand — the standard response is temporary: a tie from the pier top back to a ground anchor or to the abutment, removed when the next span is hung.

For tendons at 500 N/mm² the 200,000 kN/m pier carries 5,417 kN at the stage, a little over the crowd’s 4,953. A tie of about 40,000 kN/m — a fifth of the pier’s own stiffness — brings the pier back to the crowd’s force while carrying 964 kN itself.

The figure has a feature that a simple share would not. The tie and the pier divide the unbalance in proportion to their stiffnesses, but the unbalance grows as the pair stiffens: the tie relieves the pier by 464 kN while carrying 964, because a pier top that moves less lets the two spans relieve less of their difference by moving. Held rigidly the pier top would carry 8,923 kN, two thirds more than the free pier’s 5,417. A tie is therefore sized against the ribbon’s response, not against the free pier’s force, and a tie much stiffer than it needs to be is a tie carrying load that a flexible pier would simply have let the ribbon absorb.

The numbers at the first pier, by hand #

For the default ribbon: $H_0 = 35 \times 100^2/(8 \times 2) = 21{,}875$ kN; strand at 1,100 N/mm² is $21{,}875{,}000/1{,}100 = 19{,}890$ mm², so $EA = 3{,}880$ MN and the working strain is 0.0056.

The tendon is cut $0.0056 \times 100 = 0.564$ m short. The finished sag uses $8 \times 2^2/(3 \times 100) = 0.107$ m of slack, so $\rho = 0.564/0.107 = 5.3$, and the bare tendon holds $21{,}875 \times (1 - 1/5.3) = 17{,}740$ kN — against 17,766 kN from the solution on a rigid pier. A rigid pier carries the rest: $21{,}875/5.3 = 4{,}140$ kN, or equivalently $35 \times 100 \times 0.02/(3 \times 0.0056) = 4{,}140$ kN.

On the 200,000 kN/m pier, the spans resist the lean too, and the pier carries 3,080 kN with a lean of $3{,}080/200{,}000 = 15$ mm. Its base moment is $3{,}080 \times 10 = 30{,}800$ kN·m, against the 49,500 kN·m the crowd produces on the finished ribbon. On this ribbon the stage is not the pier’s design case; the finished bridge is.

Shallow cables, free saddles and segments hung all at once #

Each span is a shallow elastic cable, as before: its length is its chord plus $w^2c^3/24H^2$, and it stretches as $H/EA$, with $EA$ the strands’ alone because the segments are not yet joined. The segments’ bending stiffness is ignored, which is exact while their joints are open.

The tendons slide freely over each saddle, so the pier receives the difference of the two thrusts. A saddle that grips during erection would hold some of the difference in the tendons themselves and pass less to the pier; one that slips in steps would pass it in steps.

A span is hung all at once. In practice its segments are placed one after another, and while a span is part-hung its own shape is asymmetric and its thrust is somewhere between the bare value and the finished one. The stage force rises through the hanging to the full value computed here, so the figures give the largest it reaches, not its history.

The tendons are cut once and never adjusted. Like the stress that was there before the load in a rolled section, the stage force is a consequence of how the parts were made rather than of anything applied, and a real erection usually re-tensions the bearing tendons between stages, with jacks at the abutments, to keep the sag on its target as the weight arrives — which is exactly the freedom that would let a contractor choose the bare tendon’s force, and with it the pier’s stage force, rather than inherit it from the finished length.

The strand in the sun #

None of the figures carries a temperature, and the bare tendon’s force is all stretch, which moves with temperature. A strand across an empty span holding 18,363 kN by being stretched nearly half a metre loses some of that force on a warm afternoon, when it grows by $\alpha\,\Delta T\,L$: 36 mm for a 30° rise, against a stretch of a few hundred. The hung span’s thrust barely changes, because its slack takes the growth up as sag, so the whole change appears at the pier. None of the figures carry a temperature.

Still open: the strand that warms before the deck is on #

A finished ribbon of equal spans is balanced at every temperature: both spans warm, both sag a little, and the pier sees nothing — the movement nobody applied reaches the pier only where the spans differ. During erection the two spans either side of a pier are not alike — one is a sagging cable that absorbs a change of length as sag, the other a taut strand that absorbs it as force — and a uniform change of temperature unbalances them. With 3,880 MN of strand the bare tendon loses about $EA\,\alpha\,\Delta T = 1{,}400$ kN on a 30° rise, a third of the stage force itself. Whether a day’s swing of temperature, acting on a ribbon whose spans are half hung, adds to the stage force enough to make the timing of the hanging — morning or afternoon, summer or winter — part of the pier’s design, and whether a bare tendon in the sun can lose enough of its stretch to go slack, is the question the stage leaves open.