Structural Engineering & Statics
where the load goes
Materials

The slab that dries from one face

A creep coefficient is one number for a member, and a member drying through one face does not have one. Give every depth its own humidity and the section a strain profile it cannot deliver, and two things follow that no single coefficient contains — a six-metre slab lifts eleven millimetres at its edges with nothing on it, and its top surface is past cracking before it has been loaded.

Assumes The deflection that arrives three years late and Plane sections stay plane, and what the assumption costs.

16 min read 7 figures Scale changes everythingThe load that will not hold stillWhich failure arrives first

The notional size is a member’s area over the part of its perimeter that can dry, and it is one number for the whole section. That is a statement that a member dries evenly, and it is the modelling step the essay below ended by naming: a slab on a steel deck, a ground-bearing slab, a roof with a vapour barrier beneath it, all dry through their top face and none of them dries evenly.

The notional size handles that by doubling — one face open instead of two — and the doubling is a correction to a mean. What it cannot describe is the thing a reader can see on a warehouse floor, which is that the slab has lifted at its edges.

The humidity through a slab's depth, and the strain it asks for. A slab 200 mm thick drying through its top face only into air at 50 per cent, with its underside sealed by a steel deck. Left, the pore humidity against depth at 28 days — 50 per cent at the surface and 100 at the base; 90 days — 50 per cent at the surface and 100 at the base; 1 year — 50 per cent at the surface and 99 at the base; 5 years — 50 per cent at the surface and 74 at the base; 20 years — 50 per cent at the surface and 51 at the base. Right, the free strain each of those profiles asks for, taken as shrinkage at the local humidity. At 28 days the top wants to be 461 microstrain shorter than it was and the bottom 0; at 20 years the two are 461 and 456, and the gradient that produced the curl has gone. Not one of the profiles is straight, and a section that stays plane cannot deliver any of them.
Fig. 1 A 200 mm slab drying through its top face only into air at 50 per cent, its underside sealed. Left, the pore humidity against depth: at 28 days the drying is confined to the top 50 mm and the base is still saturated; at a year the profile runs the whole depth; at twenty years the slab has equalised at the ambient humidity. Right, the free shrinkage each of those profiles asks for. Not one of them is straight.

Moisture has a clock and the clock is a square

The diffusion arithmetic is the whole reason the profile is a profile rather than a step.

Pore humidity in concrete obeys a diffusion equation with a diffusivity of order 10−10 m2/s10^{-10}\,\mathrm{m^2/s}, and a diffusion front reaches a depth LL in a time of order L2/DL^2/D. For a slab drying through one face, LL is the whole thickness rather than half of it, so the 200 mm slab in the figure has 200 mm to empty rather than 100 — and 0.22/10−100.2^2/10^{-10} is about 4×1084\times10^8 seconds, thirteen years.

That is why the profiles in the figure look the way they do. At 28 days the drying has not reached a quarter of the way down; at a year it is most of the way through; at five years the base is at 74 per cent and still falling; at twenty the slab is uniform. The gradient exists for the whole of that time and is largest somewhere in the middle of it.

The square law is also the difference from the essay below. The notional size enters the creep coefficient through a cube root and enters the drying time through a square, and the second of those is the one that decides when a floor is flat enough to put a finish on.

What plane sections do to a strain that is not straight

Every depth now wants a different length, and the section is not going to let them have it.

What plane sections do to a strain that is not straight. The same slab at 1 year. Left, the free strain through the depth and the straight line the section is obliged to take instead: a mean of −185 microstrain and a curvature of 2.43 per kilometre, which lifts the edges of a 6 m slab 10.9 mm above its middle. Right, the stress the difference costs, tension positive: 0.42 N/mm² at the drying face, −0.67 at its most compressed inside, and 1.35 at the base — tension at both surfaces and compression between them, which is the shape a free strain curving one way always produces. That stress integrates to zero force and zero moment — nothing is holding the slab, and it is stressed anyway.
Fig. 2 The same slab at a year. Left, the free strain through the depth against the straight line the section imposes instead: a mean of −185 microstrain and a curvature of 2.43 per kilometre. Right, the stress the difference costs — 0.42 N/mm² of tension at the drying face, 0.67 of compression inside and 1.35 of tension at the base — integrating to zero force and zero moment.

The decomposition is the one a plane section always performs on a free strain that is not linear. Whatever profile the material asks for, the section delivers a+bza + bz, with aa and bb chosen so that the difference carries no net force and no net moment. Three things come out of it and they are separate quantities:

The mean, a+bh/2a + bh/2, which is the shortening. At a year it is 185 microstrain and by twenty years it is 461 — and this is the number a single shrinkage strain is about.

The curvature, bb. At a year it is 2.43 per kilometre, and a slab of length LL curling freely lifts its edges bL2/8bL^2/8 above its middle: 10.9 mm on a six-metre panel. Nothing has been applied to the slab. It is not resting on anything that holds it down and it is not carrying anything.

The residual, which is the stress. It is what is left when the mean and the curvature have been taken out of the free strain, and it is self-equilibrating by construction: it exists on a member nobody is restraining, and no equilibrium check made on the member can detect it.

That last quantity is the one a bridge deck develops when the sun comes out, and the mechanism is identical. What is not identical is the clock, and the clock changes the answer.

Creep takes the stress and leaves the curl

Creep takes the stress away and leaves the curl. The tension at the drying face of the same slab, computed twice: once at the short-term modulus, as a thermal gradient would be, and once with every depth at its own effective modulus E/(1+φ), which is what a gradient taking years to develop actually meets. At 28 days the two are 9.10 and 5.42 N/mm², a relief of 40 per cent; at a year, 0.90 against 0.43, a relief of 53; at twenty years, 68 per cent. The curvature meanwhile moves by almost nothing — 2.47 per kilometre against 2.43 at a year — because a change of stiffness that is nearly uniform cancels out of the ratio that sets it and does not cancel out of a stiffness times a strain. Both curves stay above the tensile strength of 2.90 N/mm² for the first months, so the relief decides how badly the surface crazes and not whether it does.
Fig. 3 The tension at the drying face, computed twice: once at the short-term modulus, as a thermal gradient would be, and once with every depth at its own effective modulus E/(1+φ). At 28 days the two are 9.10 and 5.42 N/mm², a relief of 40 per cent; at a year, 0.90 against 0.43; at twenty years, 68 per cent. The curvature moves by two per cent over the same range.

A temperature gradient arrives in hours. Concrete does not creep measurably in hours, so the stress a thermal profile leaves is an elastic stress and the short-term modulus is the right one to compute it with.

A moisture gradient takes years, and the concrete carrying it is creeping the whole time. So the right modulus for each depth is that depth’s own effective one, E/(1+φ)E/(1+\varphi) — which is the quantity the second of these essays is about, arriving here through the depth of a section rather than along the length of a member.

The relief is between 40 and 68 per cent and the curvature barely moves, and the reason those two behave differently is worth stating because it is not obvious.

The curvature comes from a ratio — a first moment of the free strain weighted by stiffness, divided by a second moment of the stiffness. Scale every depth’s stiffness by roughly the same factor and both integrals scale with it and the ratio does not move. The stress is a stiffness times a strain difference, with nothing to cancel against, so it moves in full.

So the two halves of a self-equilibrating profile respond to creep in opposite ways, and a design that treats the moisture gradient as a slow temperature gradient gets the deflection right and overestimates the cracking by a factor of two. The curvature is a deformation and survives; the stress is a force and leaks away.

The surface is past cracking before the slab is loaded

The relief is a factor of two and the early stress is a factor of three above what concrete can take, so the relief does not save the surface.

What plane sections do to a strain that is not straight. The same slab at 28 days. Left, the free strain through the depth and the straight line the section is obliged to take instead: a mean of −48 microstrain and a curvature of 1.21 per kilometre, which lifts the edges of a 6 m slab 5.5 mm above its middle. Right, the stress the difference costs, tension positive: 5.28 N/mm² at the drying face, −2.12 at its most compressed inside, and 1.67 at the base — tension at both surfaces and compression between them, which is the shape a free strain curving one way always produces. That stress integrates to zero force and zero moment — nothing is holding the slab, and it is stressed anyway.
Fig. 4 The same slab at 28 days, when the drying has reached only the top fifth of it. The free strain profile has a sharp knee, so the residual is large: 5.28 N/mm² of tension at the drying face against a mean tensile strength of 2.90, 2.12 of compression inside, and 1.67 of tension at the base. The mean shrinkage is 48 microstrain and the edge lift 5.5 mm.

At 28 days the drying face is at 5.28 N/mm² against a tensile strength of 2.90. That is not a marginal exceedance, and it is the origin of a defect every concrete floor has and nobody records as one: the fine map cracking on the surface of a slab that has never been loaded. It is not a structural crack, it does not go anywhere, and it is produced entirely by the section refusing the strain profile its top surface asked for.

The number to be careful with is the earliest one. The model here puts the ambient humidity onto the surface the instant drying begins, with no surface film resistance and no curing membrane, so what it reports in the first days is an upper bound rather than a reading. What survives that caveat is that the tension exceeds the strength for the first three months whichever way the early boundary is treated, which is a statement about the shape of the curve and not about its first point.

The base is in tension too — 1.67 N/mm² at 28 days — and that is the half of the picture nobody looks for. A free strain profile that curves one way produces tension at both surfaces and compression between them, so a slab on a steel deck can craze its soffit as well as its top, in the one place a finish will never be applied to hide it.

Whether the curl goes away depends on the underside

The curl arrives, and whether it leaves depends on the base. The lift of the edges of a 6 m slab 200 mm thick above its own middle, against age on a logarithmic axis, for a base sealed by a deck and a base the ground keeps wet. Both reach about 11.0 mm, at 307 days and 1 year. The sealed slab then comes flat again — 0.0 mm at a hundred years — because a sealed member eventually reaches the ambient humidity throughout and has no gradient left. The slab on the ground settles at 10.4 mm and stays there, because its base is a source that never runs out.
Fig. 5 The lift of the edges of a 6 m slab above its own middle against age, for a base sealed by a deck and a base kept wet by the ground. Both reach about 11 mm, at 307 days and at a year. The sealed slab then comes flat again — 0.0 mm at a hundred years — and the slab on the ground settles at 10.4 mm and stays there.

Two boundary conditions, and the difference between them is not a coefficient.

A sealed base passes no moisture. The only sink is the top face, so the whole slab eventually reaches the ambient humidity, the gradient disappears, and the curl with it. That takes a few decades and it does happen: the curl is a transient.

A base kept wet by the ground is a source that never runs out. The profile settles to a steady gradient — dry at the top, saturated at the bottom, for ever — and the curl settles with it. A ground-bearing slab’s curl is permanent, and no amount of waiting will take it out.

Both slabs reach about the same peak at about the same age, so an inspection at a year cannot tell them apart. The distinction is made entirely by what is under the slab, it is decided before any concrete is placed, and it is invisible in every calculation that uses one creep coefficient and one shrinkage strain.

That has a practical edge on a warehouse floor, where a curled edge at a joint is what a forklift wheel hits. On a suspended slab the curl relaxes and the joint closes up; on a ground slab it does not, and the joint arris keeps being hit until it spalls.

Thinner is worse, and sooner

A thin slab curls more, and gets there first. The largest lift a 6 m slab ever reaches, and the age at which it reaches it, against the slab's thickness, on a base sealed by a deck. 100 mm lifts 21.8 mm at 71 days; 150 mm lifts 14.6 mm at 161 days; 200 mm lifts 11.0 mm at 277 days; 300 mm lifts 7.3 mm at 1.7 years; 450 mm lifts 4.9 mm at 3.9 years. The lift falls and the date recedes together, and the second is the square law: drying time goes as the square of the distance to a free surface, so doubling the thickness pushes the peak out by about four. A notional size has a cube root in it and no time-to-peak at all.
Fig. 6 The largest lift a 6 m slab ever reaches, and the age at which it reaches it, against thickness: 100 mm lifts 21.8 mm at 71 days, 150 mm lifts 14.6 at 161 days, 200 mm lifts 11.0 at 277, 300 mm lifts 7.3 at 1.7 years and 450 mm lifts 4.9 at 3.9 years.

Both curves run the way a designer would want and their rates are the interesting part.

The lift falls roughly as the inverse of the thickness, because the same free strain difference across a smaller depth is a larger curvature. The date recedes roughly as the square of it, because that is the diffusion clock. So halving a slab’s thickness roughly doubles its curl and brings it forward by a factor of four — and a thin slab is therefore both worse and finished with sooner, which is the opposite trade from the one thickening usually buys.

The humidity through a slab's depth, and the strain it asks for. A slab 450 mm thick drying through its top face only into air at 50 per cent, with its underside sealed by a steel deck. Left, the pore humidity against depth at 90 days — 50 per cent at the surface and 100 at the base; 1 year — 50 per cent at the surface and 100 at the base; 5 years — 50 per cent at the surface and 99 at the base; 20 years — 50 per cent at the surface and 79 at the base; 100 years — 50 per cent at the surface and 51 at the base. Right, the free strain each of those profiles asks for, taken as shrinkage at the local humidity. At 90 days the top wants to be 461 microstrain shorter than it was and the bottom 0; at 100 years the two are 461 and 456, and the gradient that produced the curl has gone. Not one of the profiles is straight, and a section that stays plane cannot deliver any of them.
Fig. 7 A 450 mm slab, on the same axes as the hero. At 90 days the drying has reached a tenth of the depth and at five years the base is still at 99 per cent; the profile does not run the whole section until twenty years and the slab is not uniform until a hundred. At ten years its curvature is 0.885 per kilometre, a fifth of the 200 mm slab’s peak, and its edges are 4.0 mm up.

A 450 mm slab spends twenty years with a moisture gradient in it, and the thing to take from the pair of figures is that the gradient is not smaller in a thick member, it is shallower and it lasts longer. The top 50 mm of both slabs reaches the ambient humidity in about the same time, because that depth’s own clock does not know how much concrete is below it. What changes is how much of the section is still wet while that happens.

What a designer does with a curvature nobody applied

Three habits follow, and none of them is a calculation the model above has to be run for.

Put the drying face where the curl helps. A slab that curls edges-up is a slab whose top surface has shrunk more than its bottom, and that is decided by which face is open. On a suspended slab on a metal deck the choice has been made by the deck; on a ground-bearing slab a polythene membrane under it makes the base wet rather than sealed, which is the boundary condition that makes the curl permanent. That membrane is specified for damp-proofing and it decides a serviceability problem nobody attributes to it.

Compute the two ends of any difference. The creep of a member already varies from one member to another by a quarter; within one section it varies from face to face by considerably more than that, and the quantity that matters is always a difference. A slab spanning onto a deep beam, a column beside a core wall, a topping cast on a precast unit — each is two objects with different profiles bonded together, and using one coefficient for both sets their difference to zero, which is the one value it certainly does not have.

And treat the surface cracking as expected rather than as a defect. Map cracking on a fresh slab is the section refusing the strain its top face asked for, it is self-limiting because the stress that caused it is self-equilibrating, and it is not the restraint cracking that runs the full depth and is a durability problem. The two look similar on a surface and are produced by opposite mechanisms: one is a gradient with no restraint, the other is restraint with no gradient. The steel that controls a crack is designed for the second and does very little about the first, because there is no force in the first for it to take over.

The same decomposition, three times in this collection

It is worth naming the family, because the arithmetic above has now been done three times in this collection with three different causes.

A nonlinear temperature profile in a bridge deck, arriving in hours and relieved by nothing. A drying profile in a slab, arriving over years and relieved by half. And the strain a restrained member is given, which is the same relief computed along a member’s length rather than through its depth.

In every one the free strain is not linear, the section insists on a straight line, and the difference is a stress with no external cause. What separates them is entirely the time the strain takes to arrive, measured against the material’s own creep. Hours: full elastic stress. Years: about half. Decades: almost none, which is why the deflection that arrives three years late is a deflection problem and not a cracking one.

So the useful question about any imposed strain is not how large it is but how fast it came, and that question is answered by a diffusion clock, a weather pattern or a construction programme rather than by anything in a materials table.

Which free body produced the number

The free body is a slice of the slab of unit width, cut on two vertical planes, and nothing crosses its boundary except moisture.

That is the unusual part. The two vertical cuts carry the self-equilibrating stress, which is why it is in the free body and invisible to it: the stress on each face integrates to zero force and zero moment, so the free body’s equilibrium equations are satisfied identically whatever the profile is. A check made on this free body cannot detect the stress it contains, and the only way to see it is to compute the strain profile the material wanted and subtract.

Three modelling steps go into that profile and each is a choice rather than a measurement:

The diffusion constant. Taken as 10−10 m2/s10^{-10}\,\mathrm{m^2/s}, which is an order of magnitude for ordinary concrete and varies by a factor of three between mixes. Every date in this essay scales inversely with it and no curvature does.

Shrinkage at the local humidity. The free shrinkage is taken as the humidity function a code applies to the ambient humidity, evaluated at the local pore humidity instead. That substitution is the whole of what turns a one-number rule into a profile, and it is a model rather than a code provision.

Creep at the local humidity too, with a notional size of twice the thickness because one face is open. So each depth gets its own creep coefficient, which is what makes the effective-modulus column of the decomposition vary through the section at all.

What the picture cannot show

Nothing here is reinforced. A real slab has steel in it, usually nearer one face than the other, and that steel restrains the shrinkage it is nearest to — which produces a curvature of its own with the opposite sign when the steel is in the bottom. The two curvatures are computed the same way and they add, and a slab with heavy bottom steel and a drying top face has them in opposition. Nothing in these figures knows about it.

The slab is free to curl. A real one is not: it is sitting on the ground, or it is continuous over supports, or its corners are held down by the weight of whatever is on them. A restrained curl becomes a moment instead of a movement, and the moment is what lifts and cracks a floor at its joints rather than lifting it smoothly. Which of the two arrives is decided by the slab’s own weight against bL2/8bL^2/8, and that comparison is not in any figure here.

And the time axis is a model of the weather. The ambient humidity is one number for a hundred years, where a real slab meets a winter, a summer, a building being heated for the first time and a roof going on. Each of those reverses the gradient near the surface for months, and the surface stress follows them while the curvature, which depends on the whole depth, does not.

Still open: two concretes in one member

The essay below made the creep coefficient a property of the member rather than of the mix. This one has made it a property of a depth. The obvious next question is a member made of two concretes of different ages, which is the commonest arrangement in precast construction: a beam cast in a factory, dried for six weeks, and an in-situ slab cast on top of it while it is still wet.

The two halves then have different creep coefficients, different shrinkage remaining, and different notional sizes, and they are bonded along a plane so that the section must still stay plane. The same decomposition applies with a discontinuity in the middle of it, and the stress redistribution between them runs for years. After that, creep redistribution of moments in a frame built in stages, and the theorem that the redistribution moves toward the answer for a frame built all at once — and prestress losses, where a tendon’s force is consumed by exactly the drying creep of the member it is in, so a thin-flanged box girder loses more of it than a solid slab of the same concrete.

Where this goes next

Half of the creep is the member drying out. The creep coefficient of a concrete of mean strength 38 N/mm² loaded at 28 days, in a member of notional size 150 mm, against time since loading: exposed to air at 50 per cent relative humidity, and the same member sealed so that it cannot dry. After a year the exposed member's coefficient is 1.93 and the sealed member's 0.81; after 50 years they are 2.45 and 1.28. The shaded difference, 48 per cent of the exposed member's final creep, is what drying adds. The sealed member's coefficient has no size in it; the drying part is where the member comes in. The creep that belongs to the member A creep coefficient is quoted for a concrete, and half of it is not a property of the concrete. It is the member drying out, and drying goes through the surface — so one mix creeps a quarter more in a thin slab than in a deep beam, gets there years sooner, and in saturated air forgets its size altogether. The strain it wants, the strain it is allowed, and the difference. A bridge deck 1.40 m deep with 18 °C at the top face falling away over 10% of the depth. The left curve is the free thermal strain αT(y); the straight line beside it is what a plane section will actually take, ε₀ + κy with ε₀ = 32.0 microstrain and κ = 0.063 per km. The right-hand block is E times the difference, and it reaches -3.98 N/mm² of compression at the surface and 1.83 of tension 140 mm below it. Its resultant force is 8.3e-14 kN and its resultant moment 3.0e-12 kNm, which is what self-equilibrating means: the field is invisible to every equilibrium check that could be made on the member. The stress nobody restrained A bridge deck lying loose on its bearings, with nothing holding it anywhere, develops four newtons per square millimetre when the sun comes out. The stress is not caused by restraint. It is caused by plane sections, and it is invisible to every equilibrium check that could be made on the member. The least reliable number in the material decides the answer, briefly. What a twenty per cent error in the concrete's tensile strength does to a computed deflection, against how far past cracking the beam is. Well past the cracking moment it does almost nothing — at 1.9 times M_cr the spread is 56 per cent — because the section is nearly fully cracked and the interpolation has run out. Just above cracking it does everything: at 1.19 times M_cr the same twenty per cent moves the deflection by 3658 per cent. Tensile strength is the property with the widest scatter and the least direct test, and a beam designed to sit near its cracking moment has put the answer on it. The curvature nobody applied Concrete shrinks as it dries, by about half a millimetre in every metre. In a symmetrically reinforced member that is a shortening and nothing else. In a member with more steel in one face than the other — which is every beam and every slab — the steel holds one side back and the section bends, with no load on it at all. Where plane sections stop staying plane. Strain across a cut face at four span-to-depth ratios, with the straight line the theory assumes drawn faintly behind. For a slender beam the two coincide; for a beam as deep as its span the real distribution is nothing like a straight line, and beam theory has no claim on it. Plane sections stay plane, and what the assumption costs Beam theory rests on one sentence about geometry. It is very nearly true for a slender member, wrong for a deep one, and everything in the subject that fails does so where it stops holding.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

AgeingCreepCurvatureEffective modulusImposed deformationSelf equilibrating stressServiceabilityShrinkage