The strand in the sun beside the strand in the shade

A stressed ribbon is put up by stringing its bearing tendons over every span first, each cut to the length the finished span will need, and then hanging the deck segments on them one span at a time. Halfway through, a pier stands between a span full of segments and a span of bare strand. The hung span pulls with nearly its finished thrust; the bare one, which was cut short by the whole of its working stretch, is already pulling with four fifths of it. The pier carries the difference — 3,080 kN for two 100 m spans laid at a 2 per cent sag, two thirds of what a crowd on one span of the finished bridge asks of it.

That figure was computed for one temperature, the one at which the tendons were cut. A finished ribbon of equal spans is indifferent to temperature: both spans warm, both lengthen alike, and the movement nobody applied reaches the pier only where the spans differ. During erection the spans do differ — one is a sagging cable that can absorb a change of length as a change of sag, the other a nearly straight strand that can only absorb it as a change of force. A day in the sun, on a bare tendon of 3,880 MN axial stiffness, is worth EAαΔT = 1,400 kN for 30 °C, a third of the stage force itself. The question is whether that makes the hour and the season of hanging part of the pier’s design.

Both spans lose tension #

Warm both spans’ tendons by the same amount and follow each span’s thrust.

The bare tendon loses close to what a strand between fixed points would: 1,334 kN against 1,396, the difference being the pier’s give. That was expected. The surprise is the hung span, which loses 1,021 kN — three quarters as much as the bare tendon. The intuition behind the question was that a sagging span absorbs a change of length as sag, cheaply, and so keeps its tension. At the sags ribbons are laid at it does not.

The reason is the property the deck that is its own cable was built on. A ribbon is laid at a fiftieth of its span because a footbridge has to be walkable, and a cable that flat is mostly stretch. Its axial stiffness, softened by its sag, is Ernst’s: the steel’s EA divided by 1 + (wL)²EA/12H³. For the hung span that factor is 1.40, so the hung span is 71 per cent as stiff against a change of length as the bare strand — far closer to the bare strand than to a slack rope. A warming that lengthens both by the same amount takes tension off both, and the pier sees only the difference between their stiffnesses, about 29 per cent of the bare tendon’s loss: 312 kN on this pier for 30 degrees.

So a hot day shared by the whole bridge barely touches the stage. The pier carries 10 per cent more at 30 °C above the cutting temperature, and 6 per cent less at 20 °C below. Hanging in summer rather than winter, with the tendons and the segments at the same temperature, is not a design decision for the pier.

The difference is the shade #

The tendons of a half-hung ribbon are not at the same temperature, and the reason is visible from the ground. The strands of the hung span sit inside or under the precast segments, shaded by concrete with a large thermal mass; the strands of the bare span hang in the open, in full sun on a clear day. Steel strand in direct summer sun runs well above the air around it, while strand under a concrete segment follows the slower temperature of the concrete.

Warm the bare tendon alone and the pier feels nearly the whole of its loss: 3,080 kN becomes 4,126 at 30 °C, a third more, three times what the shared warming did. The weather moves the pier little; the shade moves it a lot. On this pier every degree by which the bare strand runs warmer than the hung strand adds about 35 kN.

The two lines do not reach the crowd’s 4,953 kN in any plausible temperature range, and that is the first answer to the question. Even a bare tendon 40 °C warmer than the shaded one — more than strand in sun usually manages over concrete in shade — leaves the stage below the finished bridge’s own service unbalance. The pier designed for the finished bridge holds the stage in the sun, on a ribbon whose tendons run continuously over its saddles. On a ribbon whose tendons are anchored span by span, with nothing beyond the pier, the bare tendon does not exist and nor does this effect; the pier there already carries four times the crowd’s force.

Two temperatures, one difference #

The two cases are slices of one picture.

The contours are straight, parallel lines crossing the dashed diagonal at a shallow angle. Moving along the diagonal — the whole bridge warming — crosses them slowly. Moving up — the bare strand warming alone — crosses them fast, and moving right — the hung span warming alone — crosses them the other way, because a warmer hung span sags further and pulls less. The pier’s force is, very nearly, a function of one combination: thirty-five times the bare tendon’s warming less twenty-five times the hung span’s, in kilonewtons.

That combination is what a site would need to measure, and it is not the air temperature. It is the difference between two strand temperatures, one in sun and one in shade, and the sensible check during erection is a thermocouple on each set of strands rather than a thermometer on the pier.

The map also says what a cold night does. The bare strand cools below the shaded one after dark, by radiation to a clear sky, and the pier force falls: 10 °C of difference the other way takes it to 2,730 kN. A day on which the bare strand runs 10 °C cooler than the shaded one at night and 30 °C warmer in the afternoon swings the stage force between 2,730 and 4,126 kN, about fourteen hundred kilonewtons, centred near the force the uniform calculation gives. For a concrete pier that swing is not a fatigue question; for a slender steel trestle used as a temporary pier, it might be.

The tendon that cannot go slack #

The other half of the question was whether a bare tendon in the sun could lose so much of its stretch that it hung slack — which would put the hung span’s whole thrust on the pier at once, the way the span-by-span scheme does.

The arithmetic is short. The bare tendon pulls because it was cut short by its working stretch, and at a 2 per cent sag it holds four fifths of its finished thrust as stretch — the ratio ρ of its stretch to the slack its sag needs is 5.3. To lose that by warming, it would have to lengthen by the stretch it holds, which at a working strain of 0.0056 and α of 12 × 10⁻⁶ per degree is 381 °C. No sun slackens the tendon of a flat ribbon.

The figure also says where that stops being true. A deeper sag needs more slack, so the bare tendon holds less stretch, and less warming takes it out: 272 °C at 3 per cent, 109 at 4 per cent, and nothing at all at 4.6 per cent, the sag at which the stage is already at its worst because the bare tendon holds no stretch even cold. A cable structure built the same way at a sag of 4 per cent — a hung roof, a pipe bridge — could have a bare tendon go slack on a hot afternoon, and its intermediate supports would then see the whole thrust of the hung span without warning.

The dashed line runs the other way. At flat sags the bare tendon is nearly straight and very stiff, so a warming costs it much tension and the pier much force; at deep sags the bare tendon is softer and holds less, and the same warming moves the pier less. A ribbon flattened for walkability is the case where the sun matters most to the pier, and still not enough to design it.

A deeper ribbon minds the sun less #

The flatness that makes the hung span mostly stretch is also what makes the bare tendon so stiff, and the two effects do not scale together. A ribbon laid deeper shows how they separate.

At a 3.5 per cent sag the stage force is twice what it was at 2 per cent — the bare tendon holds only about two fifths of its finished thrust as stretch, and the pier makes up the rest — but the sun’s share of it falls. Thirty degrees on the bare tendon adds 639 kN, 11 per cent of the stage, against 34 per cent on the flat ribbon. The bare tendon is still nearly straight and still loses most of EAαΔT, but there is less strand in it — the tendon was sized for a smaller finished thrust — so EA itself is smaller. And the hung span, deeper, is softer, so a shared warming now costs it half of what the bare tendon loses rather than three quarters, and the shared case moves the pier by about half of the bare-alone case rather than a third.

Read together, the two ribbons say that the sun’s importance is tied to the property that makes a ribbon a ribbon. A flat cable is stiff because of its shape, so it answers a change of length with force rather than sag, and the flatter it is the more of its tendon’s thermal stretch reaches the pier as force. That is the same reason a flat ribbon’s bare tendon pulls so hard in the first place: both are consequences of a tendon whose working stretch is large against the slack its sag needs. The deeper the ribbon, the more of the stage is ordinary unbalance and the less of it is temperature.

It also says which way the risk runs when a design changes during development. A ribbon flattened late in design for a gentler gradient gains thrust at its abutments, as every essay on stressed ribbons has found, and gains sensitivity to the sun at its piers during erection — modestly, and never past the finished bridge’s own unbalance on these numbers, but in the same direction as everything else flattening does. A tendon built to its length in a structure with more supports than it needs turns every change of that length into force, and a degree of sun is a change of length nobody drew.

What the hour does change #

None of this makes the hour matter for the pier. The hour matters for something else, which the stage arithmetic does not ask about: the level.

A sagging cable lengthened by ΔS drops at mid-span by about 3L·ΔS/16f — for a 100 m span with a 2 m sag, nearly ten times the change of length. The tendons’ thermal lengthening alone would drop the hung span 11 mm a degree; the tension the warming takes off shortens the strand elastically and claws two thirds of that back, leaving 3.4 mm a degree. Over a 30 °C range from a cool morning to a hot afternoon, the hung span’s mid-span moves 100 mm.

That matters because of what happens next. When the span is full, the joints between segments are cast and a second set of tendons is stressed through the band, and from then on the span is a continuous concrete ribbon whose geometry is fixed by where the segments were when the joints set. A span whose joints are cast in the afternoon sun is cast 100 mm lower than one cast at dawn, and it keeps most of that shape: the finished ribbon cools as a concrete band with an axial stiffness about eleven times the bare strands’, and a cable that stiff takes back much more of its thermal shortening as tension and much less as sag. The hour of casting sets the finished level, and the pier does not care. That is a reason to cast the joints at a known and uniform strand temperature — at night or at dawn — rather than whenever the programme reaches them, and it is a reason about geometry, not about the pier.

The numbers at the pier, by hand #

The pier’s sensitivity can be checked with nothing but Ernst’s factor. Between rigid supports the bare tendon, nearly straight, loses its full EAαΔT: 3,880 MN × 12 × 10⁻⁶ × 30 = 1,396 kN, divided by an Ernst factor of 1.004 for its own slight sag. The hung span loses the same EAαΔT divided by its Ernst factor, 1 + (35 × 100)² × 3.88 × 10⁶/(12 × 21,442³) = 1.40: 997 kN. The difference on a rigid pier is 397 kN for a shared warming, and 1,393 for the bare tendon alone. A pier of 200,000 kN/m, leaning about 15 mm under the stage, gives way a little and takes three quarters to four fifths of the rigid figures — 312 and 1,046 kN, as the solve gives — the share a pier that carries only the difference takes of any unbalance between its spans.

What the warm ribbon leaves out #

One temperature per span. Each span’s strands are at one temperature along their length. A bare tendon partly shaded by a pier or by the first segments of the next span to be hung is warmer in some places than others; only its mean matters for its length, and that mean is what the calculation needs.

Temperatures as inputs. Every warming here is a number put in, not one computed from the weather. How much hotter than the air a bundle of bare strand runs in sun, how far a shaded strand lags the concrete around it, and how quickly either follows a cloud depend on the strand’s surface, the bundle’s packing, the wind and the time of year, and they are measured on site rather than derived. The calculation’s job is the other half: given the two strand temperatures, what the pier and the level do. It turns a site’s two thermocouple readings into a force and a level.

Segments that do not restrain. The precast segments hang on the strands and are not joined, so they add weight and shade and nothing else. Once the joints are cast the concrete takes over, and the finished ribbon’s temperature response is a different calculation with a different stiffness.

A pier at the cutting temperature. A concrete pier warms and cools too, more slowly, and lengthens vertically, which moves the saddle up and changes both spans’ chords. For a 10 m pier that is a few millimetres, against a hung span that moves 3.4 mm per degree of its own.

Two spans. With four spans, the half-hung stage has spans at several temperatures and piers between several pairs; each pier reads the difference of its own two neighbours, to the extent the others’ movements reach it through the ribbon.

Still open: the joint cast warm and cooled under the band #

A span whose joints are cast in the afternoon becomes a continuous concrete ribbon at a sag 100 mm deeper than designed, and then cools through the night with its second-stage tendons stressed. As a concrete band its axial stiffness is about 43,000 MN, eleven times the bare strand’s, and a uniform cooling of the whole finished bridge moves nothing at its interior piers. But the span cast warm and its neighbour cast cool have different unstressed lengths, and the difference is now held by a stiff band rather than by soft strand. Whether a few tens of degrees between the casting temperatures of two adjacent spans leave a permanent unbalance at the pier between them — of the size of a crowd, or of a fraction of one — and whether the casting sequence should therefore fix the time of day for every span rather than only for each, is the question the hour leaves for the finished ribbon.