The joint poured twice over the pier

The joint cast over the pier made two pretensioned beams continuous over their pier with a joint cast together with the slab, and found that the date of that one pour decided the sign of everything the joint would carry. A joint cast when the beams are four weeks old is sagged for thirty years by the creep their prestress has still to do, to 211 kN·m against a cracking moment of 140. A joint cast at a year is hogged by the slab’s shrinkage on beams that have nearly stopped, to 214 the other way. Between them, at about 113 days, a joint ends its life holding nothing.

That essay ended on a question that bridge builders ask in practice, because many decks are not made continuous in one pour. The diaphragm over the pier is often cast first, with the bars or strands projecting from the beams’ ends lapped inside it so that the beams are joined at their soffits, and the slab and its top steel follow weeks or months later. The hope behind the split is easy to state. The first pour, made early, catches the beams while their creep is still lifting their ends; the second, made late, lands on beams whose creep has nearly finished. If each mechanism began where it does the least, a deck might have the young joint’s freedom from hog and the old joint’s freedom from sag at once. This essay asks whether it does.

Two pours, and the section each one finds #

The beams, the slab and the climate are the single-pour essay’s, so that every difference below is the split and nothing else. Two pretensioned beams, 300 mm wide and 700 mm deep and 12 m long, are released at three days and stand as simple spans; a 160 mm in-situ slab 1.2 m wide is cast over them; the air is at 60 per cent humidity. What changes is that the joint is now made on two days. On the first, when the beams are 28 days old unless stated otherwise, the diaphragm is cast around their ends and the beams become continuous. On the second the slab is cast.

The calculation follows the end rotation each beam would make if nothing held it, from release to thirty years, with the section followed fibre by fibre in two concretes of different ages and a bonded tendon whose force falls as the concrete around it shortens — the model that a section made of two materials set up and the angle made in the casting yard read the bearing’s rotation from. Every change in that free rotation after the first pour is a rotation the joint has to undo, and it undoes it with a moment.

What the split changes is the section that does the undoing. A joint between two spans is one support too many: equilibrium cannot say what it carries, and the answer comes from stiffness — here, from whichever stiffness the joint has on the day each rotation arrives. Between the two pours the joint holds bare beams, 700 mm of precast concrete with nothing over it; for two equal spans the stiffness that turns a rotation into a moment is $3EI/L$, and with the bare beam’s second moment of area that is $8.0 \times 10^{10}$ N·mm per radian. From the slab’s day on the joint holds the composite section, whose second moment of area is three times larger, at $2.45 \times 10^{11}$. Each increment of free rotation is therefore restrained by the section that exists when it happens, and each increment of moment it produces relaxes from that day by the beam concrete’s creep, as in the single-pour essay — the step-by-step creep integral that turns a slowly arriving restraint into something near half its elastic value.

One increment is different in kind. When the slab is cast, its weight — 4.8 kN/m on each beam — lands on beams that are already continuous. Poured once, the wet slab is carried by two simple spans and the joint is born after it; poured twice, the first stage is there to receive it, and a uniform load on two continuous spans hogs the support by $wL^2/8$. On the bare beams that is 86 kN·m of hog, put into the first stage on the slab’s day, before the slab has hardened enough to share it.

Why the slab’s section holds the single pour’s moment #

The hero figure shows the result before it is explained. Each solid line starts on the first pour’s day, climbs as the bare beams’ creep sags the joint, drops sharply into hog on the slab’s day, and then runs beside the dashed line of the single pour on the same slab day for the rest of the deck’s life. With the slab at 90 days the two lines end one kilonewton-metre apart, 41 kN·m against 40. With the slab at six weeks the split ends at 139 against 160, and with the slab at a year at −165 against −214. The split moves each ending by somewhere between nothing and a quarter, and it never changes a sign.

The reason is the decomposition that the calculation is built on, and the next figure draws it.

Restraint is linear in the rotation it holds, and creep is linear in the moment that causes it, so the joint’s moment is the sum of what it holds before the slab and what it holds after — the same superposition that the beam built towards its own middle used to add up a continuous beam stressed span by span, where each closure locked in what the pieces had done before it. The second part is the moment that the composite section develops from undoing every change in free rotation after the slab’s day, relaxing by the beams’ creep from the day each change happens. That is, term by term, the calculation of a joint poured once on the slab’s day. The free rotation after the slab is the same whether or not the beams were already joined, because the free rotation is what the beams would do with nothing holding them. The stiffness is the same composite stiffness. The creep that relaxes it is the same. So the composite section’s share, the pale violet bar, is the single pour’s moment — 160 kN·m with the slab at six weeks, −213 with the slab at a year — to within the step of the time grid.

The first stage adds its own account, and only that. It is the moment the bare beams develop between the two pours, from the beams’ creep lifting their ends, which sags the joint, and from the wet slab’s weight on the slab’s day, which hogs it; and both parts go on relaxing for thirty years by the same creep. With the slab at six weeks the first stage has two weeks of creep to hold and then the whole slab’s weight, so its account ends in hog, at −21 kN·m. With the slab at 90 days the two nearly cancel. With the slab at a year the first stage has eleven months of creep and the same slab, and its account ends at 48 kN·m of sag.

This is why the split cannot do what it was hoped to do. The young joint’s freedom from hog and the old joint’s freedom from sag are not two separate mechanisms that the two pours can share out. The slab’s day fixes the composite section’s share, completely, and whatever the first stage was doing before then is a smaller second term laid over it. A first stage that held all of the beams’ remaining creep could only ever add sag; it could not remove the slab’s shrinkage on old beams, because the composite section feels that shrinkage whatever was holding the beams before.

The balance moves by days #

The single-pour essay’s most useful result was that a joint cast at about 113 days ends its life holding nothing. The next figure asks where the split puts that age.

The two curves lie almost on top of each other. Where the slab is young, the split sits a little below the single pour, because the slab’s weight on continuous bare beams subtracts more hog than two weeks of creep added sag. Where the slab is old, the split sits above it, because the first stage gathered months of creep sag and adds it to a composite share that is hogging. The two curves cross the axis six days apart, at 113 and 119 days, and over the whole range from six weeks to a year the split never moves the ending by more than 49 kN·m.

A first stage cast earlier moves more. With the first pour at 14 days rather than 28, a slab at a year ends at −149 kN·m instead of −165, and a slab at six weeks at 155 instead of 139. The sense of each change is the same as before — an earlier first stage has more creep to catch, and catching it adds sag — but even the earliest practical diaphragm, a fortnight after the beams are stressed, removes less than a third of a year-old slab’s hog.

A precaster who is told that the slab cannot be poured before a year is therefore not rescued by an early diaphragm. The joint still needs top steel for a restraint hog of about 165 kN·m on top of the traffic’s 259, which is 424 kN·m over the pier against the 473 of a single pour. The saving is real, about a tenth of the top steel, and it is a trim, not a cure.

The joint that holds the beams on half its strength #

The decomposition makes the first stage look like a small correction, and for the finished joint it is. For the first stage itself it is not small at all, because what that stage holds is held by a section with half the strength.

The bare beam’s cracking moment is computed with the joint concrete’s tensile strength on its extreme fibre, top or bottom, and for a 300 by 700 mm rectangle it is 71 kN·m either way, half the composite joint’s 140. That is the moment the first stage can hold before its soffit or its top opens.

The beams’ creep starts sagging the first stage the day it is cast, and the sag grows for as long as the slab waits. With the slab at six weeks the first stage has only reached 22 kN·m before the slab arrives. With the slab at 90 days it reaches 58. With the slab at about 124 days it reaches the cracking moment, and with the slab at a year it is sagged to 112, more than half as much again as it can carry uncracked. A first stage cast early and left for months is cracked at its soffit, by the creep it was cast to catch, before the slab that would have made it strong enough is there.

The dashed line is the other half of the first stage’s life. On the slab’s day the wet slab hogs it by up to 86 kN·m, elastically, and if the stage had gathered less sag than that, it swings into hog. With the slab at six weeks it reaches 61 kN·m of hog, close to its cracking moment the other way, and with the slab at 90 days about 24. With the slab later than about five months the sag gathered before the slab exceeds what the slab can take away, and the first stage never hogs at all.

Two details follow, and they pull opposite ways. A first stage that will see the wet slab within about five months must carry hog on the bare beams, so the diaphragm needs tie bars near the beams’ tops as well as the lapped strands at their soffits. Built with soffit steel only, it opens at its top under the slab’s weight, lets the beams turn as simple spans, and the joint is no longer the one this calculation describes. A first stage that waits longer than about four months for its slab must carry sag beyond its cracking moment, which needs bottom reinforcement sized for the bare section, not the composite one. Between about four and five months, with the first pour at 28 days, neither limit binds and the first stage is the simple locating pour it is usually drawn as. The window is narrow, and it sits just where the single pour’s balance age already was.

The age of the first pour #

The next figure fixes the slab at a year — the case for which a split would be most wanted — and moves the first pour.

The earlier the first stage, the more of the beams’ creep it catches and the less hog the finished joint ends with: −135 kN·m with the first pour a week after release, −165 at four weeks, −201 at three months. The largest hog on the way falls in step, from the single pour’s 249 to 162. But the dotted line rises as fast: the first stage’s sag before the slab is 151 kN·m with the first pour at a week, more than twice its cracking moment, and 112 at four weeks. Every kilonewton-metre of hog the split takes off the finished joint is paid for with sag held by the half-strength first stage in the months before the slab.

The right-hand end of the figure holds a result that runs the other way. A first stage cast a few weeks before a year-old slab catches almost no creep — the beams have nearly stopped — but it is there on the slab’s day, so the wet slab’s weight is carried continuous and hogs the joint by 86 kN·m that a single pour would never have seen. That hog relaxes, but not to nothing, and it adds to the composite share’s hog instead of offsetting it. A first stage cast 37 days before the slab ends at −251 kN·m, against −214 for the single pour on the slab’s day. A late first stage is worse than no first stage. The split helps only when the first pour is early enough that the creep it catches outweighs the slab weight it receives, which for these beams, with the slab at a year, means within about the first four months; past about 125 days the first stage makes the joint worse.

What an elastic estimate promises #

The single-pour essay found that holding the free rotation elastically roughly doubles the restraint at every casting age. The last figure asks what that does to the case for splitting.

Elastically, the split looks far more useful than it is. With the slab at a year, the elastic calculation puts the single pour at −402 kN·m and the split at −279, a saving of 123; relaxed, the same pair is −214 and −165, a saving of 49. With the slab at six weeks the elastic saving is 52 and the relaxed one 21. In every case the elastic estimate promises about two and a half times the trim the split delivers.

The reason is the same relaxation that halves the moment itself, acting harder on the first stage than on the rest. The first stage’s account is made early, while the beams are young and their creep coefficient for a load applied at that age is large, and it then has thirty years to relax. The slab’s weight on the continuous bare beams, the 86 kN·m of elastic hog that offsets the creep sag, relaxes most of all, because it arrives in one step and then sits on young concrete for the rest of the deck’s life. The parts that depend on early events are the parts creep erodes most, and the split’s whole contribution is made of early events.

The stress that leaks away is the general form of this: a strain held from a given day relaxes by the creep of concrete loaded on that day, and the younger the day, the more it loses. A designer who sizes the first stage’s reinforcement, or decides on a split at all, from an elastic restraint calculation is being misled about both — the first stage’s moments are smaller than elastic, and the benefit to the finished joint is much smaller still.

By hand, with the slab at a year #

The decomposition gives a check that needs no time grid. The composite share is the single pour’s moment on the slab’s day, −214 kN·m with the slab at a year. The first stage’s share has two parts. The beams’ creep between 28 days and a year turns their free ends upward by a further 2.5 milliradians, which the bare beams’ $8.0 \times 10^{10}$ N·mm per radian would hold elastically with 201 kN·m of sag. It does not arrive at once, and the young beams creep under it as it builds, so the first stage reaches only 112 kN·m by the eve of the slab — a relaxation to 56 per cent, close to the half the single-pour essay found for its whole restraint. Then the wet slab turns the free ends down by 1.0 milliradian, which the bare beams hold with 80 kN·m of hog, a little under the $4.8 \times 12^2/8 = 86$ of a continuous beam because the restraint is figured on the 28-day modulus of concrete that is now a year old. On the slab’s day the first stage holds $112 - 80 = 32$ kN·m, which is the calculation’s value.

Over the following thirty years both parts go on relaxing, but not equally. The slab’s hog was applied in one step to concrete a year old, with a creep coefficient of 0.86 still ahead of it; the sag was built up from four weeks and much of its relaxation is already spent. The hog therefore loses a larger share of what remains, and the first stage’s account drifts upward from 32 to 48 kN·m. Added to the composite share, −214 + 48 = −166, against the calculation’s −165. The hand check is not independent of the model — it uses the model’s free rotation — but it confirms that the split’s effect is the first stage’s account and nothing else, and that the account is made of two terms of opposite sign with the creep deciding which survives.

An uncracked first stage and a slab that acts at once #

The calculation assumes that the first stage holds whatever it is asked to, in sag and in hog, with the bare beams’ uncracked stiffness. The previous sections showed that it is asked for more than it can hold uncracked over much of the practical range. A cracked first stage is softer and so holds less, which would move the split further towards the single pour: the essay’s conclusion that the split is a trim would be strengthened, not overturned, but the numbers in the cracked range are upper bounds, not values.

It assumes the composite section’s stiffness from the slab’s day on, although a slab a few days old is soft and does not yet act with the beam. The slab’s first days carry little of the restraint in either case, and the creep that belongs to the member set out why a thin slab, drying through both faces, creeps and shrinks faster than its age alone suggests. It treats the diaphragm as a joint of no length, and it ignores the pier’s own flexibility and the bearings’ — anything that lets the two beam ends turn together softens the restraint, for both stages alike. And it keeps the beams on their bearings throughout: the support that had no moment when it was cast found that a restraint which only ever hogs can still be large, and a first stage receiving the wet slab is that case for a few weeks.

And it keeps the beams’ prestress and creep coefficients deterministic. The single-pour essay’s balance age was itself a point estimate on a creep model with a scatter of twenty per cent or more, so that 113 days is better read as somewhere between three and five months. On that reading the six days by which the split moves it are invisible, which is the plainest statement of the result.

Still open: setting the joint instead of timing it #

Both essays have treated the joint’s moment as something the programme decides and the designer then accommodates. The arm connected late found the other approach in an outrigger: a connection whose force can be set by jacking at the moment it is made, so that the structure starts from the state the designer chose. A pier joint can be treated the same way — the beams’ pier ends lifted or lowered on jacks before the first stage is cast, so that the joint is born holding a chosen moment and the restraint that follows is laid over it. Whether a jacked first stage can be set so that the restraint’s sag and the slab’s hog cancel at thirty years, for any slab date the programme happens to produce, or whether the jacked moment relaxes away like everything else the first stage holds early, is the question the timing leaves, and it turns again on the angle nobody limits at the moment the jacks are released.