The joint cast over the pier

The angle made in the casting yard followed the end of a pretensioned precast beam through thirty years and found that the largest term in the rotation its bearing must take is none of the ones a bearing schedule lists. It is the upward turn the prestress gives the beam’s ends at release, and the creep that goes on turning them before the beam ever reaches its bearing. Every load the beam later carries turns the ends back down a little, and none of them turns them back as far as the yard turned them up.

A bridge deck is often built from such beams made continuous over its piers. The beams are set on their bearings as simple spans, the slab is cast over them, and at the piers a joint — an in-situ diaphragm, with the slab’s top steel running across it — is cast at the same time, so that the superimposed load and the traffic are carried by a continuous beam. At the abutments the beams’ ends go on turning freely. At the pier, the two ends meet in a joint that will not let them turn apart. This essay asks what the joint is then asked to carry, and whether the date it is cast decides it as the bearing essay found the date the beam was set decides the bearing.

Two beams, and a joint between them #

The beams are the bearing essay’s: pretensioned, 300 mm wide and 700 mm deep, with a 160 mm in-situ slab 1.2 m wide over them, 12 m long, in air at 60 per cent humidity, released at three days. The slab is cast on them at an age that is the variable here, and the pier joint with it. The superimposed finishes and the traffic go on afterwards, onto a continuous beam.

Each beam’s section is followed fibre by fibre through its life, with two concretes of different ages bonded at the slab’s underside and a bonded tendon whose force falls as the concrete beside it shortens. That is the calculation a section made of two materials set up, and the bearing essay drew the end rotation from it. Here the end rotation a beam would make if nothing held it is computed for its whole life, and from the day the joint is cast every change in it is a rotation the joint must undo.

For two equal spans simply supported at their abutments and joined at the pier, a moment $M$ at the joint turns each beam’s pier end by $ML/3EI$. A joint that must undo a free rotation $\Delta\theta$ therefore carries $M = 3EI\,\Delta\theta/L$ — if the beams were elastic. They are not; the restraint arrives slowly, over years, and the beams creep under it while it arrives. The moment is computed step by step: each increment of rotation the joint holds is resisted by the composite section, and each increment of moment it produces creeps away by the beam concrete’s own creep coefficient from the day it arrived. The superimposed load goes on after the joint and is carried continuous from the start, so it drives no restraint and is added separately.

The date decides the sign #

The hero figure is the whole answer drawn three times. A joint cast when the beams are 28 days old is asked, for its first few weeks, for a small hog — the slab’s early shrinkage — and then for a sagging moment that grows for thirty years to 211 kN·m. The joint’s cracking moment in sagging, its bottom fibre at the in-situ concrete’s tensile strength, is 140 kN·m; the restraint passes it after seven months.

A joint cast at 90 days is hogged by about 50 kN·m in its first months and then drifts to a sag of 40. A joint cast at a year is hogged hard and fast, to 249 kN·m within three years, and relaxes slowly to 214. The three lines start from the same zero, are driven by the same two mechanisms and end on opposite sides of the axis.

The reason is in the end rotation each joint inherits.

The beam’s end is turned upward by its prestress at release and goes on turning upward as the prestress creeps, faster in its first weeks and slower after. The slab, when it is cast, turns it part of the way back down by its weight. After the slab, two things continue: the beam’s creep under its prestress, which is still turning the end upward, and the slab’s shrinkage, which shortens the top of a composite section and turns the end downward. Which wins depends on how much of the beam’s creep is left.

A slab cast at 28 days lands on beams with most of their creep still to come. After the joint, the free end would turn another 1.74 milliradians upward, and the joint, holding it, is sagged. A slab cast at a year lands on beams whose creep has nearly finished; the slab’s own shrinkage is then the larger movement and turns the end 1.64 milliradians the other way, and the joint is hogged. The joint is asked to undo whatever the casting yard has not finished, plus whatever the slab is about to do, and the date it is cast decides which is the larger.

Where the two pulls balance #

Between the two there is a casting age at which the joint ends its life holding nothing.

The moment at thirty years falls steadily with the casting age, from a sag of 211 kN·m at 28 days through zero at about 113 days to a hog of 214 at a year. A joint cast at sixteen weeks is asked to hold nothing at the end of its life. It is not asked to hold nothing on the way: the slab’s shrinkage arrives faster than the beam’s remaining creep, so the joint is hogged first, by about 50 kN·m, and sagged back to zero over the following decades. The dashed line, the largest moment of either sign, never falls below about 50 kN·m. There is no casting age at which the joint is asked for nothing at every moment of its life.

The band in which neither the final nor the largest moment reaches the joint’s cracking moment runs from about seven weeks to about six months. That is the window for a joint with no special detailing either way. Cast earlier, and the joint needs bottom reinforcement across it — a positive-moment connection — to hold the beams’ ends together at the soffit. Cast later, and its top steel must carry the restraint’s hog on top of the traffic’s.

Two pulls in opposite directions #

The restraint is the sum of two mechanisms of opposite sign, and the next figure separates them.

The first bar is the beams: their prestress creeping and their self-weight creeping, with the slab given the beams’ own shrinkage, so that the composite section shrinks evenly and nothing curves it. That part always sags the joint, because the prestress’s creep always lifts the ends more than the self-weight’s creep lowers them, and it falls with the casting age from 317 to 114 kN·m as the yard finishes more of the creep before the joint is there to hold it.

The second bar is the slab’s extra shrinkage: a young slab on older beams shrinks faster than they do, and the difference curves the composite section downward. That part always hogs the joint, and it grows with the casting age, from −105 to −327 kN·m. A slab cast on young beams shares its shrinkage with beams that are still shrinking too; a slab cast on year-old beams shrinks against beams that have nearly stopped, which is what the slab that shrinks onto a finished beam found for the beam’s soffit. The two bars move in opposite directions as the casting age rises, and that is why the total crosses zero rather than merely shrinking.

The support that had no moment when it was cast made two ordinary reinforced beams continuous a month after they took their own weight, and found a restraint that only ever hogged, because nothing but the self-weight was creeping and the self-weight only ever turns a beam’s ends down. A pretensioned beam carries a load that turns its ends up, and a composite one carries a second concrete that turns them down, and the restraint is the difference between them.

Half of it creeps away #

How the restraint is calculated matters as much as when the joint is cast.

The free rotation the joint must undo arrives over years, and the beams that resist it creep while it arrives. Each increment of restraint moment, from the day it appears, relaxes by the creep coefficient of concrete loaded at that age. The result is about half of what the composite section’s elastic stiffness would give for the same rotation: 211 kN·m against 425 for a joint cast at 28 days, and −214 against −402 for one cast at a year.

This is the effect that the age-adjusted effective modulus approximates with a single ageing coefficient, and the stress that leaks away is the case of a held strain where it is exact. A designer who skips it doubles the moment and, for an early joint, specifies a positive-moment connection twice as strong as it need be; for a late one, top steel over the pier for twice the restraint’s hog. The elastic estimate is safe, but it is not close.

What the load sees #

The restraint does not act alone; the pier carries the superimposed load and the traffic as well, and the last figure puts them together.

The restraint moment at the pier is carried by a moment that varies linearly from nothing at each abutment to its full value at the joint, so it changes the moment everywhere along both spans. The superimposed load and the traffic on both spans hog the pier by 259 kN·m and sag the spans by 146. A joint cast at 28 days adds its 211 kN·m of sag: the pier is hogged by only 48, and the spans sag to 236, close to the 259 the same load would give a simple span. The continuity the joint was built to provide has been nearly cancelled by the restraint, and the beams carry their traffic almost as simple spans. A joint cast at a year does the opposite: the pier is hogged by 473, nearly twice the load’s, and the spans carry a sag of only 77.

Neither is unsafe if it is designed for, and each is unsafe if it is not. An early joint makes the beams’ midspan the critical section and needs a bottom connection that may crack every summer; a late joint makes the pier’s top steel critical. A deck whose beams arrive on site at an age set by the precaster’s schedule rather than the designer’s has a restraint that nobody chose.

What a specification can fix #

The American bridge specification has, since a study of exactly this joint in the early 2000s, allowed a designer to neglect the sagging restraint at a continuity joint when the girders are at least 90 days old when it is made. On this deck the restraint at thirty years crosses zero at 113 days, three weeks after the rule’s date, and the rule’s logic is visible in the age figure: before 90 days the sag grows quickly as the age falls, and by 90 days it is down to 40 kN·m, well inside the joint’s cracking moment, with the slab’s hog taking over a few weeks later. The rule is a statement about the shape of the beams’ creep curve, written as a date.

It does not say what the joint is asked for after 90 days, and the figures say it is not nothing: a hog that grows with the casting age, about 70 kN·m on the way even for a joint cast at the crossing, and 214 at a year, to be carried by the same top steel that carries the traffic. A precaster who stores the beams for a year because the site is late has turned the problem from the soffit to the deck. The rule moves the risk from one face of the joint to the other; it does not remove it.

The two other choices are older. A designer can detail every joint for both signs — a bottom connection good for the early sag and top steel for the late hog — and stop caring when the beams arrive. Or a designer can control the age itself, which needs the precaster’s schedule in the contract, and is what the angle the casting yard made found the bearing would also have liked. The yard’s upward turn, which the deflection that arrives three years late follows in a reinforced beam the other way round, is the one movement every one of these choices is negotiating with, and it is the one nobody on site can see.

A deck’s bearings, finally, see the joint’s choice too. A continuous beam’s abutment ends turn by less than a simple span’s under the traffic, and by an amount that depends on the restraint. An early joint, sagging, turns the abutment ends further down than the simple spans would have; a late joint, hogging, turns them up. The prestress the member takes back is part of what moves both: every loss of prestress after the joint is a loss of the very upward turn the joint was holding.

By hand, at a year #

The elastic restraint for a joint cast at a year can be checked from the free rotation alone. The composite section’s second moment, with the slab transformed at the ratio of the two concretes’ moduli, is $2.63 \times 10^{10}$ mm⁴, and the beam concrete’s modulus 37,300 N/mm². The joint’s stiffness against a rotation of both ends is $3EI/L$ = 3 × 37,300 × 2.63 × 10¹⁰/12,000, or 245,000 kN·m a radian, and a free rotation of 1.64 milliradians held at the pier gives 245,000 × 0.00164 = 402 kN·m, the elastic figure.

The relaxed figure can be estimated with the ageing coefficient’s shortcut. The beam concrete’s creep coefficient from a year to thirty years is 0.86, and with an ageing coefficient of 0.8 the restraint is the elastic value divided by 1 + 0.8 × 0.86 = 1.69, or 59 per cent of it, against the step-by-step 53. The shortcut overestimates a little here because most of the rotation arrives in the joint’s first two years, while the beams can still creep under it for nearly thirty.

Where the two-concrete model stops #

The joint is rigid. A diaphragm cast between two beam ends, with the slab’s steel running over it and perhaps bent bars or strands from the beams’ bottoms projecting into it, is stiff, but once it cracks at the soffit it is not rigid: a cracked positive-moment connection rotates, and the restraint it carries falls. The 211 kN·m for an early joint is an upper bound on what an uncracked joint would be asked for, and the real joint cracks and carries less.

The beams are identical and the spans equal. Two beams of different ages — cast weeks apart in the yard — have different creep left, and a joint between them is asked for an asymmetric restraint the model does not show.

The temperature is constant. A deck’s daily and seasonal temperature differences produce restraint moments of their own, sagging in summer and hogging in winter, of the same order as these, and they come and go rather than accumulating.

The crack at the soffit and the precaster’s schedule #

They cannot show the bearings, which see the abutment ends turn with the continuous beam’s rotation rather than the simple span’s; nor the crack at the joint’s soffit, which on a real deck is often visible within a year and is the evidence of an early joint. And they cannot show the precaster’s schedule, which in practice is what sets the casting age — a decision about storage space and lorries that turns out to decide the sign of a moment in the finished structure.

Still open: the joint cast in two stages #

Some decks are made continuous in two steps: the diaphragm is cast first, with the beams’ bottom connection, and the slab over it weeks later. The bottom connection then holds the beams’ creep from one date and the slab’s shrinkage acts only on the continuous beam from another, and a joint cast early at its bottom and late at its top could see each mechanism start where it does least. Whether splitting the pour lets a deck take the early joint’s lack of hog and the late joint’s lack of sag at once — or whether the slab’s shrinkage on a beam already continuous simply hogs the pier from a different starting point — is the question the single joint leaves, and it turns again on the angle nobody limits at the moment each pour is made.