The slab that shrinks onto a finished beam

The essays on creep so far have made the creep coefficient a property of successively smaller things. The deflection that arrives three years late treated it as a property of the concrete. The creep that belongs to the member showed that half of it is the member drying out, so it belongs to the member’s size and the air around it. The slab that dries from one face made it a property of a depth within the member, and the support that had no moment when it was cast and the prestress the member takes back made it a property of a construction schedule and of the tendon inside it.

The case left is the commonest arrangement in precast construction and the one none of those could represent: a member made of two concretes. A pretensioned beam is cast in a factory, released after three days, stored, delivered and set on its bearings; weeks or months later a slab is cast on top of it, wet, and bonded to it along the whole of their shared face. From then on the section is one section and has to stay plane. But its two halves were born at different times, from different mixes, and each goes on creeping and shrinking by its own clock.

The member, and the two ways of calculating it #

The beam is 300 mm wide and 700 mm deep, of C50 concrete, pretensioned with 1,150 mm² of strand stressed to 1,300 N/mm² and placed 200 mm below the beam’s centroid. It spans 12 m. The strand is released on day 3, when the beam’s own weight is already on it; the slab — 1,200 mm wide, 160 mm thick, C30 — is cast unpropped on day 42, so its wet weight is carried by the beam alone; a superimposed dead load arrives on the composite section four weeks later. The air is at 60 per cent humidity throughout.

The calculation steps through thirty years fibre by fibre. Every fibre of concrete carries the history of every stress increment it has ever received, each creeping from the age at which it arrived, according to that concrete’s own creep function; every fibre also shrinks by its own concrete’s law, from the day that concrete was cast. The slab’s strains count from the moment it joined the section, because it was cast onto a beam that had already moved. The strand is a bonded steel fibre, so every shortening of the concrete at its level comes straight off its force. At every step, the strain at the top of the beam and the curvature are fixed by the one thing that is known — the section carries the moment of its loads and no axial force it was not given.

That calculation is run twice. Once with two concretes: the slab has its own mix, its own size and its own age. And once with one: the slab is given the beam’s concrete and the beam’s age, so it creeps and shrinks exactly as the beam would have. The second is what a composite section with a single creep coefficient amounts to, and the difference between the two is what the second concrete does.

The slab shrinks the same whenever it is cast #

Start with what each half would do on its own. The slab, whenever it is poured, is young concrete of modest size drying through its top face, and over thirty years it shrinks about 350 microstrain. That figure barely changes whether the slab goes on at six weeks or at a year, because the slab’s clock starts when the slab is cast.

The beam’s clock started on day 0. By six weeks it has done a third of its drying; by a year, most of it. What it has left to shrink when the slab arrives is 236 microstrain if the slab comes at six weeks and 70 if it comes at a year.

A one-concrete calculation gives the slab the beam’s age, so it assumes the slab shrinks exactly what the beam has left — 236 or 70. The slab actually shrinks 351 or 349. The difference is the strain the section has to resist rather than share: 115 microstrain for the early slab, 279 for the late one. The second concrete’s whole effect comes from that gap, and the gap is set almost entirely by the beam’s age.

Where the difference goes #

The slab wants to shorten more than the beam will let it. Bonded along its underside, it cannot, so it is held in tension relative to where it would have been, and that tension has to be balanced by an equal compression somewhere else in the section — which can only be the beam. The compression arrives at the top of the beam, next to the interface, and because it acts well above the beam’s centroid it is also a sagging moment on the beam.

A sagging moment is exactly what the prestress was put there to oppose. The strand’s eccentricity puts the beam’s soffit into compression, so that when the service load arrives the bottom fibre stays compressed or barely goes into tension and the beam does not crack. The slab’s shrinkage takes that precompression away, and it does it silently over years, with no load changing.

The figure measures how much. With one concrete the soffit ends at −8.37 N/mm² whichever day the slab was cast. With two it ends at −7.43 when the slab went on at six weeks — 0.94 N/mm² less, a ninth of the precompression — and at −6.12 when it went on at a year: 2.15 N/mm² less, a quarter. On a beam designed to keep its soffit in compression under a live-load stress of a few N/mm², that is the difference between a beam that stays uncracked and one that cracks at every loading.

The size of it, by hand #

The order of magnitude can be had without stepping through thirty years, and doing it once shows why the answer is a fraction of what the slab could exert.

If the beam were rigid, the slab would be held at its original length and the whole of the 115 microstrain gap would become stress in the slab. A tension that builds slowly is relaxed by creep as it builds, so the slab’s stiffness against it is not its elastic modulus but an age-adjusted one, $E/(1 + \chi\varphi)$ — about 33,000 N/mm² over $1 + 0.8 \times 2$, or roughly 12,700. On the slab’s 192,000 mm² that is a force of $115 \times 10^{-6} \times 12{,}700 \times 192{,}000 \approx 280$ kN: the most the slab’s differential shrinkage could ever deliver to the section.

The beam is not rigid. It is shortened and bent by the force the slab puts into it, and every millimetre it gives is a millimetre the slab does not have to be held back by. The stepped calculation finds the slab’s force changed by 87 kN relative to the one-concrete case, a little under a third of the rigid-beam bound, and that third is what reaches the soffit as lost precompression. For the one-year slab the gap is 279 microstrain, the bound about 680 kN, and the transfer is 192 kN — the same third again, because the ratio is set by the relative stiffness of the two parts of the section, which does not change with the casting day.

That gives a rule a designer can carry: the slab’s force change is roughly a third of $\Delta\varepsilon_{sh} E_{\text{eff}} A_{\text{slab}}$ for sections of these proportions, where $\Delta\varepsilon_{sh}$ is the slab’s shrinkage minus what the beam has left. Every term in it is known on the day the programme fixes the casting date.

Shrinkage, not creep #

The young slab differs from the beam in two ways, not one. It shrinks more over the period that matters, and it creeps more, because concrete loaded young creeps more than concrete loaded old. A natural guess is that creep, which is what a design calculation has a coefficient for, carries most of the effect.

It carries almost none of it. Give the slab its own creep but the beam’s shrinkage and the soffit loses 0.16 N/mm² at six weeks. Give it its own shrinkage but the beam’s creep and it loses 0.83. Both together, 0.94. At a year the split is starker: 0.22 from creep, 2.14 from shrinkage, 2.15 together.

This is why no choice of creep coefficient can repair the one-concrete calculation. A designer who knows the slab is younger can raise the coefficient, or use the slab’s own; either adjusts the part of the effect that barely matters. The part that matters is a strain the slab imposes on itself whether or not anything loads it — the same kind of effect the stress that leaks away found for a restrained member, where creep is not the cause but the relief. Here too creep is on the relief side: the young slab’s high creep lets it relax some of the tension its shrinkage builds, which is why the “creep only” bar is small and why, in the first months, the slab’s own tension barely appears.

The slab gives up its compression, then goes into tension #

Seen from the slab’s side the same exchange is a transfer of force. With one concrete the slab settles as a compression flange carrying 130 kN — its share of the superimposed load, plus what it picks up as the beam beneath it goes on creeping under the prestress and shortening its top. With two concretes, the six-week slab ends at 43 kN of compression: two thirds of its job as a compression flange has been handed back to the beam. The one-year slab ends in tension, at 70 kN.

That force has to get into and out of the slab through the interface. At midspan the slab carries it; at the beam’s ends, where the slab stops, it carries nothing. So the whole difference is passed across the interface in the end regions of the member, as a shear that no load produced. It is the time-dependent counterpart of the effect that makes the connection busiest where the beam is not, and it arrives after the connection has been designed for everything else.

The day the slab is cast #

Sweep the casting day and the shape is the argument. The precompression lost rises steadily with the beam’s age at casting: 0.59 N/mm² at two weeks, 0.94 at six weeks, 2.15 at a year, 2.61 at a thousand days. The slab’s own stress follows the same curve from the other side: in compression throughout if it is cast within about 120 days, and increasingly in tension after that.

The intuition here runs backwards. A slab cast late meets a beam that has done most of its creeping and shrinking, and it is natural to think of that as the easy case — the beam has stopped moving, so the slab has less to cope with. The opposite is true. A beam that is still shortening shares the slab’s shrinkage by shortening with it; a beam that has finished resists it, and resisting it is what puts the stresses into the section. The early slab is carried along by the beam’s own contraction. The late one is held back by a beam that no longer contracts.

The casting day is not normally a design decision. It is set by the construction programme — how long the beams sat in the yard, when the deck was reached, whether the slab was held up by weather or by the trade before it. A number that decides a quarter of the precompression at the soffit is being chosen by whoever sequences the site, and nothing on the drawings records it.

The camber the second concrete takes away #

The same sagging moment shows in the shape. A pretensioned beam is cambered upward by its eccentric prestress, and creep grows that camber over time; that growth is one of the serviceability quantities a precast designer predicts, because the finished deck level depends on it. The slab’s differential shrinkage bends the section the other way. With one concrete the midspan curvature ends at −1.16 per km for the six-week slab; with two, at −0.96, a sixth of the hogging gone. For the one-year slab, −1.26 against −0.83: a third of it gone.

A deck whose levels were set from the one-concrete prediction therefore finishes lower than predicted, by an amount that depends on the same casting day. This is the same problem stated backwards that staged construction poses at the level of loads: the section’s history, not its final shape, decides where it ends up.

The late slab’s full profile is the most striking picture of what has happened. With two concretes the beam’s top, next to the slab, has gone from −2.34 to −6.41 N/mm²; its soffit from −8.27 to −6.12. The prestress was designed to make the bottom of the beam the more compressed face. Thirty years after a slab cast at a year, the top is the more compressed face, and the slab above it is in tension. Nothing that was applied to the beam changed; the second concrete rearranged the stresses the first one was carrying.

The section at mid-span, and everything that crosses it #

The free body is the composite section at midspan, cut through both concretes and the strand. Crossing the cut are the concrete stresses in every fibre and the strand’s force, and they must together make the section’s moment at that time — self-weight from day 3, the slab’s weight on the beam alone from day 42, the superimposed load on the composite section from day 70 — and no axial force at all. Each fibre’s stress is whatever its creep history and its shrinkage allow at the strain the plane section imposes on it.

The creep functions are those of EN 1992-1-1’s Annex B, evaluated for each concrete at its own strength, its own notional size — 210 mm for the beam, drying on all sides, and 320 mm for the slab, drying only through its top — and the age at which each increment of stress arrived. Shrinkage is the same Annex’s drying shrinkage plus autogenous shrinkage, from each concrete’s own casting. The strand’s force falls from 1,495 kN before release to 1,411 at release and about 1,245 at thirty years, within half a per cent in both calculations, because its loss is governed by the beam’s concrete at the strand’s level, which both calculations share.

The ends, the cracks, the bars and the continuity #

The ends of the member. Every stress here is at midspan, where the section is fully composite. Toward the ends, the slab’s force has to be transferred into the beam across the interface, and the stresses there are those of a disturbed region, not a plane section.

Cracking. The slab reaches 0.65 N/mm² of tension when cast at a year and a thousand-day slab more than 1 — well below the tensile strength of C30 concrete, but tension at an interface in a slab that also has restraint from its supports and from temperature. If the slab cracks, it sheds the tension and the beam keeps more of its precompression; the calculation stops describing it.

Reinforcement in the slab. Bars in the slab restrain its shrinkage themselves and change how much of it reaches the beam. They are left out, which overstates the slab’s free shrinkage slightly.

Continuity. The beam here is simply supported. A slab made continuous over the supports turns the differential shrinkage into restraint moments at the supports as well, which is where the redistribution of the support that had no moment when it was cast and this one meet.

Two shrinkage laws, right in their ratio #

That the shrinkage laws are right in the ratio between them. Every conclusion here is driven by the gap between two shrinkage curves — the slab’s from its casting, the beam’s from its own — and a code’s shrinkage prediction has a scatter of thirty per cent or more for any one mix. The shape of the conclusion survives that scatter: the gap still grows with the beam’s age, the soffit still pays for it, and creep still barely enters. How large the gap is, for a particular pair of mixes in a particular climate, is a measurement the calculation cannot supply, and on a project where the casting day may slip by months it is worth making.

Still open: the slab that is continuous over the supports #

Every beam here is simply supported, so the slab’s differential shrinkage is a self-equilibrating stress within each span and bends each beam freely. Most precast decks are not left that way. The slab is cast continuous over the piers, or the beams are stitched together there, and then the curvature the shrinkage wants to put into each span is not free to happen: the continuity resists it, and a restraint moment grows at the supports as the slab shrinks. It is the same redistribution the support that had no moment when it was cast computed for creep, driven now by a differential shrinkage acting in the opposite sense — sagging where the prestress creep was hogging — so the two partly cancel at the support, by amounts that depend on two different casting days. Which of them wins at a real pier, and whether the support can end up with the positive moment it was never reinforced for, is the question after this one.