The span cast in the afternoon

The strand in the sun beside the strand in the shade followed a stressed ribbon during its erection, with one span’s deck segments hung and the next span bare tendon, and found that the pier between them reads the difference between the two spans’ temperatures — and that a hung span drops a few millimetres for every degree its strand warms, so the hour at which its joints are cast fixes the level it is cast at. It ended on what happens after that hour. A span whose joints are cast in the afternoon becomes a continuous concrete band at a sag deeper than designed, and then cools through the night. As a band it is about eleven times stiffer than the bare strand. A uniform cooling of the finished bridge moves nothing at its interior piers, because every span cools alike. But a span cast warm and its neighbour cast cool are not alike: whether a few tens of degrees between two spans’ casting temperatures leave a permanent unbalance at the pier between them, of the size of a crowd or of a fraction of one, was the question.

A span keeps the state it was cast in #

The ribbon is two 100 m spans at a sag of 2 per cent under 35 kN/m, every segment hung, the strands continuous over a pier whose top resists a horizontal force with a stiffness of 200 MN/m. Before its joints are cast a span is bare strand with an axial stiffness of 3,900 MN, carrying the segments; once they are cast it is a concrete band of 43,000 MN.

The casting fixes one thing: the band’s stress-free length. At the moment the joints harden, the span has the length and the thrust the strand gives it at that day’s temperature, and the band’s stress-free length is that length less the stretch the thrust gives it. After that the band lengthens and shortens with the temperature from its casting temperature, exactly as the strand did from the day it was cut. Strand and concrete share a coefficient of expansion, so a span cast warm and cooled is not shortened by anything the strand was not already shortened by. What it keeps is subtler: the state it was in when it was cast, now held by a much stiffer member.

On the warm afternoon both spans are bare strand 20 °C above the reference, and both have lost 570 kN of thrust: the strand has lengthened, the spans have sagged into the extra length, and the thrust has fallen with the sag. The first span’s joints are cast now. That night the bridge cools back to the reference, the first span as a band and the second still as strand, and the two respond differently. The strand recovers the 570 kN it lost and a little more from the pier’s lean. The band recovers 1,380 — more than twice what warming took — and ends 800 kN above the design thrust. Casting the second span the next morning, at the reference temperature, changes nothing: it is cast in the state it is in. The difference between the two spans, 710 kN, is what the finished pier carries.

Why the band takes back more than the strand lost #

A cable’s thrust responds to a change in its length through two flexibilities in series. One is elastic, the span’s length over its axial stiffness, $L/EA$. The other is geometric: a sagging cable lengthens by taking up sag, and its length grows with the thrust’s fall as $w^2L^3/12H^3$. A temperature change $\alpha\Delta T\,L$ changes the thrust by that length over the two together:

$$\Delta H = \frac{\alpha\,\Delta T\,L}{L/EA + w^2L^3/12H^3}.$$

For the bare strand the elastic flexibility is $100/3.93 \times 10^6 = 2.5 \times 10^{-5}$ m/kN and the geometric one $9.8 \times 10^{-6}$; 20 °C on 100 m is 20 mm, and the thrust falls by $0.020/3.5 \times 10^{-5} = 570$ kN. For the band the elastic flexibility is eleven times smaller, $2.3 \times 10^{-6}$, the geometric one the same, and the same 20 mm of cooling raises the thrust by $0.020/1.29 \times 10^{-5} = 1{,}560$ kN on rigid piers — the geometric part a little larger now that the thrust has fallen — and by 1,380 with the pier leaning towards it.

The two halves of the round trip are taken by different structures. The warming is absorbed by a soft strand that sags into it; the cooling is resisted by a stiff band that cannot. The pier carries the difference between a span that made the round trip and a neighbour that did not.

A fraction of a crowd, proportional to the difference #

The unbalance grows almost in proportion to the casting difference: 353 kN for 10 °C, 713 for 20, 1,081 for 30. A crowd of 20 kN/m on one span, the load case a ribbon’s piers are designed against, puts 4,953 kN on the pier. So for this ribbon the answer to the question is a fraction: a seventh of a crowd for a casting difference of 20 °C, which is about the difference between a summer afternoon and the following dawn.

A seventh of the pier’s governing load is not nothing, and it is permanent rather than occasional — it is there under the dead load every day of the bridge’s life, so the crowd’s unbalance adds to it, on one side of the pier, rather than replacing it. But it does not decide the pier. What decides whether the casting temperature matters is the ribbon’s sag.

The flatter the ribbon, the more it matters #

The share climbs steeply as the ribbon flattens: 1 per cent of a crowd at a sag of 4 per cent, 3 at 3, 14 at 2, 82 at 1 — and at 0.8 per cent the casting difference leaves more on the pier than the crowd does. The formula above says why. The geometric flexibility goes as $w^2L^3/H^3$, and since $H = wL/8f$ it goes as the cube of the sag. At a deep sag it dominates both the strand and the band, they are almost equally flexible, and the round trip comes back almost where it started. At a shallow sag the elastic flexibility dominates, and that is exactly the part in which the strand and the band differ eleven-fold.

The crowd’s unbalance moves the other way over part of the range, falling from 4,953 kN at a sag of 2 per cent to 2,316 at 1, because a flatter ribbon is stiffer against a crowd’s sag too and leans its pier less. So the ratio climbs from both ends at once. A flatter ribbon, which is the one people prefer to walk on, is the one whose casting has to be timed.

A level set by the night, not the afternoon #

The thrust is the pier’s concern. The span’s level is everyone’s.

On the afternoon it is cast, the warm span hangs 53 mm below its designed sag, because the strand has lengthened and the thrust has fallen. The earlier essay found that drop, a few millimetres a degree, and concluded that the hour of casting fixes the finished level. It does — but the finished level is not the one the span had that afternoon. Cooling as a band, the span gains 1,380 kN of thrust, and its sag falls with it: it ends 72 mm above its designed level. The span cast low ends high, by more than it was low, because the band does not stretch back down as the strand would have.

Its neighbour, cast cool, ends 9 mm high, pulled a little by the pier leaning towards the warm span. And if both spans are cast on equally warm afternoons, nothing reaches the pier, but each span ends 99 mm high — the round trip’s overshoot with no difference to carry. That case points at the remedy. An overshoot that is the same in every span is a level error that a designer can predict and allow for, by setting the sag at casting deeper by what the cooling will take back. A difference between spans is a level error that varies from span to span and a force that stays on the piers, and the only thing that removes it is casting every span at the same temperature.

A cable that changes what it is made of #

Most cables keep their stiffness. A stay or a suspension cable warms and cools for its whole life with the same axial stiffness, and its round trips come back where they started — which is what lets the cable that is a spring treat a stay as one spring of fixed stiffness for the life of a bridge. A stressed ribbon is the exception, because it is built as one material and finished as another: the strand that hangs on the day of casting is soft, the band that hangs ever after is stiff, and any change made while it was soft is answered while it is stiff.

That makes the ribbon a member with a memory of exactly one day. It does not remember the weeks it hung as bare strand, because every one of their temperature swings came back; it does not remember anything after the joints set, because every later swing is shared with its neighbours. It remembers the temperature on the afternoon its joints hardened, against the temperature on the afternoon its neighbour’s did. The tendon that pulls before the deck arrives found the erection’s pier forces set by which span was hung and which was still bare; the finished bridge’s are set by when each stopped being strand.

Alternate castings, and a block of them #

A programme that casts one span’s joints by day and the next by night loads every pier, each with a force of 500 to 700 kN alternating in direction along the bridge: each pier stands between a warm span and a cool one. A programme that casts two warm spans and then two cool loads mainly the pier between the warm pair and the cool pair, with 774 kN, and the others lightly. The crowd on one span, for comparison, loads its own pier with 5,318 kN and the next one with 1,361, the effect decaying along the ribbon as the pier that carries only the difference found it does.

So the casting sequence decides which piers carry the unbalance, and how many. For a ribbon with several spans the cheapest programme is the one that keeps the casting temperature the same everywhere; the next cheapest groups the warm spans together, so that one pier carries the difference and is designed for it.

Stiffer piers carry more of both #

A flexible pier lets the spans settle their difference by leaning: on 50 MN/m it carries 337 kN of the casting’s unbalance and 2,011 of the crowd’s. A stiff pier refuses to lean and carries more of both: on 5,000 MN/m, 1,117 and 9,115. The ratio between them stays between a sixth and an eighth. The deck that is its own cable found that a ribbon’s piers are designed as a compromise between the stiffness that keeps the deck level under a crowd and the flexibility that keeps the forces down; the casting temperature does not change where that compromise lies, because it scales with the crowd at every stiffness.

What a programme can do about it #

Four remedies follow from the figures, and they are not equally cheap. The first is to cast every span’s joints at the same time of day — before dawn, when the strand is coolest and nearest the air temperature, and least disturbed by the sun. That removes the unbalance and leaves a uniform overshoot that the casting sag can be set to anticipate, and it costs only a night shift. The second is to accept the differences and compute them, setting each span’s casting sag deeper or shallower by what its own casting temperature will do to it; that keeps the level right and leaves the force on the piers, which the piers then have to be designed for. The third is to group the warm castings together, so that the unbalance falls on one pier rather than on all of them. The fourth is to use the second-stage stressing, made on one day for the whole bridge, to take back what the castings locked in — which works for the level and only partly for the force, because the stressing acts through the same stiff band.

For a ribbon at a 2 per cent sag the force is small enough that the first two are both reasonable. For a ribbon at 1 per cent, where a warm afternoon can leave most of a crowd’s force permanently on a pier, only the first is. The decision belongs to the programme, but it is a structural decision, and like the summer that is worse than the last it is one that a calculation made only for the finished bridge at its reference temperature will never see.

By hand, for the warm span #

The dead-load thrust is $wL/8f = 35 \times 100/(8 \times 0.02) = 21{,}875$ kN. Warming the strand 20 °C lengthens each span by $10^{-5} \times 20 \times 100 = 20$ mm; through the strand’s two flexibilities, $2.54 \times 10^{-5}$ and $9.75 \times 10^{-6}$ m/kN, that takes 570 kN off, to 21,310. The sag grows by the same proportion, $2.000 \times 21{,}875/21{,}310 = 2.053$ m: 53 mm low.

Cooling the band 20 °C shortens it by the same 20 mm, through flexibilities of $2.31 \times 10^{-6}$ and, at the now lower thrust, $1.05 \times 10^{-5}$: 1,560 kN on rigid piers, 1,380 with the pier leaning 3.6 mm towards the warm span. The thrust is 22,690 kN, the sag $2.000 \times 21{,}875/22{,}690 = 1.928$ m: 72 mm high. The neighbour, cooled as strand, recovers its 570 kN and gains another 100 from the lean, 21,980. The pier holds $22{,}690 - 21{,}980 = 710$ kN, leaning by $710/200{,}000 = 3.6$ mm.

What the stages leave out #

The joints are cast in an instant at one temperature. A span’s joints are cast over a morning or an afternoon, and the concrete sets over hours while the temperature moves; the casting temperature here is the temperature at which the band became stiff, which is a few hours after it was poured and somewhere between the day’s extremes.

The band’s stiffness is its uncracked one. A ribbon’s band carries the cable’s thrust as tension and is kept in compression only by its second-stage prestress; where that runs out — to creep over the years, or locally under a crowd — the band cracks, and a cracked band is softer, closer to the strand, so the round trip’s overshoot is smaller. The temperature through the band’s thickness is taken as uniform; a band warmer on its sunlit top than its shaded underside also curves.

The second-stage tendons are left out. Most ribbons are stressed a second time after their joints are cast, to put the band into compression, and that stressing — if it is done span by span at the casting temperature — locks in the same state again, while a stressing done on a single day for the whole bridge can be used to correct it. Neither creep nor shrinkage, which shorten the band for years, is in the figures; both shorten every span alike and so, like a uniform cooling, reach the piers only through differences — in age, since a span cast earlier has shrunk more by the time its neighbour is cast. The movement nobody applied found the same rule for any imposed strain: a uniform one costs a structure nothing it cannot accommodate, and a differential one is a force.

Still open: the ribbon whose spans are cast a month apart #

Every casting here differs from its neighbour only in temperature. A real programme casts its spans weeks apart, so the first span’s band has already crept and shrunk for a month when the next is cast — shortening by a few hundred microstrain, the equivalent of a cooling of 20 or 30 °C, and continuing to shorten faster than its younger neighbour for years. That is a casting difference that never stops growing, and it is held by a band whose own creep relaxes it. Whether the shrinkage of a span cast a month early leaves a larger unbalance on the pier than a warm afternoon does — and whether the band’s creep, which relaxes every imposed strain as the strain that was imposed found, takes back most of it before the bridge carries its first crowd — is the question the calendar puts to the thermometer.