The younger span catches up

The span cast in the afternoon found that a stressed ribbon is a member with a memory of exactly one day. A span hangs as soft strand, its joints are cast, and it becomes a stiff concrete band whose stress-free length is whatever the strand gave it at that hour. Two neighbouring spans cast 20 °C apart keep the difference for good, and the pier between them carries 713 kN of it, about a seventh of what a crowd on one span would put there.

That essay kept every other property of the two spans the same, and a programme will not. Spans of a ribbon are closed in turn, weeks apart. The first span’s band has by then been shrinking for a month and creeping under its second-stage prestress for a month, and when its neighbour closes it is shorter than it was. The essay ended on a worry about this: a month’s shortening is the same kind of thing as a casting temperature, it never stops, and the older span surely goes on shortening ahead of the younger, so that the difference on the pier goes on growing for years. It asked whether that difference could outgrow a warm afternoon, and whether the band’s creep, which relaxes every imposed strain, would save it. Both parts have clear answers. One of them is the reverse of what was feared, and the other turns on something a concrete designer does not usually have to think about.

Two bands, followed for a century #

The ribbon is the one from before: two 100 m spans at a sag of 2 per cent under 35 kN/m, strands continuous over a pier of 200 MN/m. Before closure each span is bare strand with an axial stiffness of 3,900 MN. At closure it becomes a band with 1.19 m² of concrete around that strand, and second-stage tendons put 4 N/mm² of compression into the concrete, balanced by an equal extra pull in the steel, which changes the thrust not at all on the day it is applied.

From then on the concrete is followed by its own stress history. Each step’s increment of stress goes on producing strain through a compliance that depends on the age at which the increment arrived, and shrinkage shortens the concrete whatever the stress is doing. The creep and shrinkage functions are the CEB-FIP Model Code’s, for a band with a notional size of 280 mm in air of about 60 per cent humidity, with a final creep coefficient near 2.2 × 0.63 at 7 days and a final shrinkage of 300 microstrain. What makes the ribbon different is what closes each step. A beam’s strain is decided by plane sections; a band’s is decided by the cable. The band’s length must equal the length its chord and thrust require, $S = c + w^2c^3/24H^2$, and the two spans’ thrusts must differ by whatever the pier’s lean provides.

It is largest on the day it starts #

The picture is the opposite of the worry. The force on the pier does not grow from nothing. It appears all at once, at closure, because the older band has already shortened, and from that moment on it falls. Spans closed 30 days apart leave 350 kN on the pier at closure, 264 a day later, 206 after a week, 142 after a month, 75 after a year, and 54 for the rest of the bridge’s life. The difference is not being added to. It is being closed.

The reason is that both shrinkage and creep are spent fastest when they are young. A band 37 days old has already used up the steepest part of both curves, and the band that closes beside it has not. Between closure and a year later, the younger band shortens by more than the older one does, and the pier’s force falls as it does. The two bands’ curves are the same curve with one shifted along the time axis, and a concave curve shifted sideways approaches itself. That was available as a fact about shrinkage curves all along. What misled the question was the phrase continuing to shorten faster than its younger neighbour, which reads as a natural thing to say about the span with a head start and is exactly backwards.

Nothing about this says the force is harmless. For the pier’s design it is a load that arrives at a known moment, and it arrives during erection, when the pier is at its least supported and the rest of the bridge is not yet there to share it. But it is a temporary construction load, and a designer who has been carrying the first day’s force as a permanent one has been overcautious by a factor of six.

Most of it is creep #

Separating the causes changes what the remedy has to be. The original question was framed around shrinkage, a few hundred microstrain by its account. Over a month a band this size shrinks much less than that, because the time scale of shrinkage in a 280 mm section is about seven and a half years: 300 microstrain × ($\sqrt{37/2781} - \sqrt{7/2751}$) is about 20 microstrain, or 2 mm on a 100 m span. That is worth 94 kN at the pier.

Creep is worth nearly three times as much, because the prestress is pushing on young concrete. Four N/mm² on 33,000 N/mm² is 121 microstrain of elastic shortening, and the concrete, loaded at 7 days, has a creep coefficient of 0.48 after 30 more: 58 microstrain, about 6 mm on the span. That is worth 260 kN. The two together are 350, about their sum.

So the question about the month is mostly a question about the second-stage prestress, and that suggests a remedy nobody would have reached by thinking about shrinkage. Hold the first band’s prestress back until the second band closes, and stress both on the same day. The first span then has only its shrinkage to bring to the closure, and the force on the pier falls from 350 to 88 kN. It is the cheapest large reduction in the figure.

It is also not free, and here the worry about growth comes true in a form nobody asked about. Stressed on one day, the two bands are at different ages: one 37 days old, one 7. The younger creeps more under the same stress, for the rest of the bridge’s life, and instead of a difference that closes there is a difference that opens. The force on the pier rises steadily from 88 kN to 154 at the end of a century, in the opposite direction, three times the 54 kN the original order leaves. A difference in when two bands were loaded never closes; only a difference in how long they have been shortening does. Choosing between them is choosing between a large construction load that goes away and a smaller permanent one that does not.

What a month is worth against an afternoon #

A month is worth half a warm afternoon, about 10 °C of casting difference on the day it arrives. Six months is about the same as the afternoon. A gap of a year is more than the afternoon, 898 kN against 713, and a gap of four years, in which the first span has done most of what it will ever do, leaves about 1,370. Long gaps are not unknown: an interrupted contract, a span left standing over a winter closure.

The comparison reverses over time. The afternoon’s 713 kN is permanent, since nothing later changes the temperature at which each span was cast. A month’s 350 kN is at a fifth of its size within a year. A century later the 30-day gap has left 54 kN and the afternoon 713 (or somewhat less: the creep the next section measures applies to a casting difference as well). So for the pier’s permanent load, the hour of casting is more important than the month. For the pier during erection, the month is about half the afternoon, and for the year-long gap the erection load exceeds it.

The temperature is also the one a programme can control for nothing. Casting before dawn removes the difference altogether, as the span cast in the afternoon said, and a programme cannot remove a month without building both spans at once. A month’s force has to be designed for, and it is carried by the temporary state of the pier, so whether it governs depends on what else is present at that moment. The tendon that pulls before the deck arrives showed how lightly loaded a pier can be during erection, and how much force a small asymmetry between spans can then put on it.

A sag does not creep #

The other half of the question was whether creep rescues the pier, by relaxing whatever difference there is. The usual answer for imposed strain in concrete is yes, by a large factor. The slab that shrinks onto a finished beam and the curvature nobody applied both lean on it: an imposed deformation held between stiff restraints loses half or more of its force as the concrete creeps. This band with its strand, held between fixed ends, keeps 0.48.

To test it, impose a fixed difference of 200 microstrain between two bands and compare a century of creep with none. At a sag of 2 per cent the ribbon keeps 0.82 of the force. Creep takes away about a third of what it takes from the same band between fixed ends.

The reason is what is holding the force. Two flexibilities act in series in a cable, as the earlier essays set out. At this ribbon’s thrust the band’s elastic flexibility is $L/EA = 2.3 \times 10^{-6}$ m/kN and the geometric one, $w^2L^3/12H^3$, is $9.8 \times 10^{-6}$: about four fifths of the flexibility is the sag. Creep softens only the concrete in the first of those. Even with the concrete’s effective modulus cut in half, the band’s flexibility about doubles, to $4.4 \times 10^{-6}$, the series total grows from 12.1 to $14.2 \times 10^{-6}$, and the force falls by only 15 per cent. Between fixed ends there is no second spring, and the whole of the softening reaches the force.

The cable that is a spring made this geometric stiffness a matter of a stay’s design. Here it does something less expected: it is a spring whose stiffness comes from gravity and the shape of the curve, so nothing about it ages. As the figure shows, the deeper the sag, the larger the geometric share and the less creep can touch: 0.89 at 4 per cent, and on towards one. A flat ribbon keeps a little less, 0.75 at half a per cent, and even there it does not approach the 0.48, because the strand in the band does not creep either.

Both spans rise together #

The force the pier feels is a small difference between two large and similar things. Both bands shorten by far more than they differ, and a shortening band on a fixed chord can only rise. After a century both spans are about 260 mm above their designed level, 13 per cent of the 2 m sag, and they differ by 4 mm. That dwarfs the 72 mm overshoot of a warm afternoon, and it is the largest number in this essay that a person standing on the bridge would notice. It would be set out for at casting, as the earlier essay said of a uniform overshoot, but the camber allowance has to carry the whole lifetime’s shortening rather than the first year’s. Less than half of it has happened by the end of the first year.

The force behind that rise has to come from somewhere, and it comes out of the prestress. A band’s thrust is shared between concrete and strand, and as the span rises its thrust grows, from 21.9 MN to 25.2 MN over the century. The strand is simultaneously losing tension as the band shortens. So the extra thrust appears in the concrete as tension working against its compression, and by the end of the century the 4 N/mm² placed by the second-stage tendons has fallen to about 0.4. The model does not include relaxation of the tendons or any cracking under a crowd, and the number should not be taken as a prediction, but the direction is plain, and it is the reverse of how a prestress loss is usually drawn. It is a ribbon’s shortening, rather than the tendon’s slackening alone, that takes the compression away.

Old segments help, but not as much as their age #

A real ribbon is mostly precast. Its segments may be months old before they are hung, and only the joints between them are young, so a reasonable expectation is that nearly all of this disappears. It does not. With a band entirely of segments three months old, the force at closure falls from 350 to 203 kN; with year-old segments, to 147.

The shrinkage does largely go: a segment a year old shrinks little in the following month. What remains is the creep, which depends on the age at loading only weakly. A creep coefficient’s dependence on age goes as $1/(0.1 + t_0^{0.2})$, which falls from 0.63 at 7 days to 0.30 at a year, so that even year-old concrete has about half the creep of week-old concrete under the same stress. A precast ribbon therefore has less of a difference, in the proportion of that weak function, rather than none. Stressing both bands together remains the larger lever.

Flatter ribbons feel it more #

Sag sets how much of this reaches the pier. On a ribbon at 4 per cent neither the month nor the afternoon is worth noticing. At 2 per cent the month is a fourteenth of a crowd. At 1 per cent it is 0.29 of a crowd, at closure, though the afternoon is at 0.82 and for good. The geometric flexibility goes as the cube of the sag, as the earlier essay found, so a flat ribbon offers little sag to absorb a shortening. The ribbon people prefer to walk on is the one where the programme is a structural decision, for the month as for the afternoon, and the comparison between the two stays about the same at every sag: the month sits under the afternoon by a factor of two or so, until the sag is deep enough that neither matters.

The pier’s stiffness moves it in the way the pier that carries only the difference would predict. A pier of 50 MN/m carries 170 kN of the month’s force at closure, one of 200 MN/m 350, and one of 5,000 MN/m 530, because a pier that cannot lean makes the spans settle their difference among themselves.

The hand check, for the pier #

A span’s flexibility is the band’s and the sag’s in series: $2.31 \times 10^{-6} + 9.75 \times 10^{-6} = 1.21 \times 10^{-5}$ m/kN per span, or $k = 82{,}600$ kN/m. Let the first span have shortened by $\delta$ more than the second. The pier leans by $d$ towards the shorter span, relaxing its thrust by $kd$ and raising the other’s by $kd$, so the force the pier carries is $k(\delta - 2d) = k_p d$, and

$$F = \frac{k\,k_p\,\delta}{k_p + 2k}.$$

With $k_p = 200{,}000$ kN/m that is 45 kN per millimetre of difference. Shrinkage’s 1.95 mm gives 88 kN, against 94 from the integration, which also lets the concrete’s stiffness grow. Creep’s 5.8 mm gives 262, against 260. The two figures for the same cause come out within a few per cent of each other, and that is the full content of the essay’s largest number.

What the bands leave out #

Mean ambient conditions. A real band’s shrinkage and creep depend on humidity, section and mix, and the Model Code’s functions carry a scatter of a fifth or more, enough to swamp a difference of a few per cent. The force at closure is proportional to the difference between two parts of one curve, which is usually better known than the curve.

Only the thrust. The band’s bending near the saddles, the tendons’ own relaxation, and cracking across joints under a crowd are outside the model. So is the second-stage prestress’s own force on the pier during stressing, with one span stressed and the other not. A band that has cracked is softer, and softer bands relax a little more.

Two spans. A long ribbon closed span by span has a series of ages along it, and each pier reads the difference between its two neighbours. Since that difference is largest at each span’s closure and fades within a year, a sequential programme presents each pier with a travelling transient rather than a permanent pattern. Whether a pier ever sees two of them at once depends on how fast the closures are made, and the column that creeps under the arch is a warning that the assumption of independence between such slow processes needs to be checked rather than granted.

A rigid view of the deck’s shape. The designed sag is taken as parabolic, and a span’s level is read at midspan. Differential age does not change the deck’s shape along a span in this model, though a band’s joints are younger than its segments, and a segmental band has a stiffness and shrinkage that vary along it.

Still open: the joint younger than the segments either side of it #

Each band here is one material with one history, a weighted mean of old segment and young joint. A real one is a chain: a metre of three-month-old segment, a short length of concrete a few days old, another metre of segment. The strand runs through all of them. The young joint is cast against old concrete on both faces, shrinks against a neighbour that no longer shrinks, and is restrained along the strand while it does. Whether that joint cracks in its first week, at a tension the band’s mean stress never shows — and whether the second-stage prestress, stressed early to keep it closed, is the same prestress whose timing decides the force on the pier — is the question that the mean hides, and the deck that is its own cable leaves it wherever its joints are.