The deflection that arrives three years late

Every deflection calculation on this site has one variable in it that has been treated as a constant: the elastic modulus. For steel that is nearly exact — the number is the same today and in fifty years. For concrete it is not remotely true, and the size of the error is not a percentage. It is a factor of three.

A concrete beam loaded on the day the props come out deflects by some amount, computed correctly from $5wL^4/384EI$ with the right modulus. A year later it has deflected by three times as much. Nothing has been added to it, nothing has cracked that was not already cracked, and the strength calculation was never in doubt at any point.

What creep is #

Under a sustained stress, concrete goes on straining. The mechanism is water: cement paste holds water in pores of a range of sizes, and a sustained compressive stress squeezes it out of the finer ones and into the coarser ones and eventually out of the member altogether. The solid skeleton takes up the slack, and the member gets shorter.

The strain that arrives instantly is elastic and reversible. The strain that arrives afterwards is creep and is mostly not. Both are proportional to the applied stress over the working range, which is the one thing that makes the subject tractable: creep is linear in stress up to about 40% of the strength, so the whole of it can be carried in a single dimensionless number.

That number is the creep coefficient $\phi$, defined as the creep strain divided by the elastic strain. A $\phi$ of 2 means the creep strain is twice the elastic one, so the total is three times the elastic — and since deflection is proportional to strain, the deflection multiplier is $1 + \phi$.

The whole of creep, for design purposes, is therefore the statement that concrete’s modulus should be replaced by an effective modulus

$$E_{\text{eff}} = \frac{E}{1+\phi}$$

and every elastic calculation re-run. That is the trick, and it works because creep is linear.

The size of it, and what it depends on #

$\phi$ for ordinary structural concrete runs between about 1.5 and 3. It depends on the humidity — a member drying out creeps far more than a sealed one — on the size of the member, since drying is a surface phenomenon and a thick member has less surface per unit volume, on the mix, and on the age at which the load was applied.

That last dependency is the one that separates creep from any other material property on this site. A load applied to concrete at 7 days produces substantially more creep than the same load applied at 28 days, and a load applied at 10 years produces about a third as much as one applied at 28 days. The material is not the same material at different ages, because hydration is still going on.

That is a real design consequence and it is not intuitive. Propping a slab for an extra fortnight has almost no effect on its strength, which was going to be adequate at 28 days regardless, and a measurable effect on where it will be sitting in ten years. The decision looks like a programme question and is a serviceability one.

What it does to a cracked section #

A concrete beam is almost always cracked in service, and the cracked section’s analysis runs on the modular ratio $n = E_s/E_c$ — the factor that converts an area of steel into an equivalent area of concrete so that a two-material section can be analysed as one.

If the concrete’s effective modulus falls by a factor of $1+\phi$, then $n$ rises by the same factor. And $n$ is not a bystander in that analysis: it decides where the neutral axis sits.

Two things in that comparison are worth stating separately, because they point in opposite directions.

The stresses barely move, and the concrete’s actually falls. That is creep doing what creep does: relieving stress in the part of the structure that is creeping, and shedding it to the part that is not. The concrete lets go and the steel picks up.

The stiffness moves a great deal. The cracked second moment of area rises by a factor of 2.34, which sounds like an improvement and is not: the quantity that governs deflection is $E_c I_{cr}$, and $E_c$ has fallen by 3.38 while $I_{cr}$ has risen by 2.34. The product has fallen by 1.44, and that is on top of the direct $1+\phi$ multiplication. Long-term deflection is not a simple multiple of the short-term one, and the two effects have to be carried through together.

What is actually being deflected #

A further complication is worth drawing rather than describing, because it is the reason long-term deflection calculations are as awkward as they are: the beam is not cracked everywhere.

So the beam has a stiffness that varies along its length, and the variation moves with time — because creep raises $I_{cr}$ and because shrinkage lowers the moment at which the section cracks. Design practice handles this with an interpolation between the cracked and uncracked stiffnesses weighted by how far the moment has passed the cracking moment, which is an approximation of an approximation and is the single largest source of scatter in predicted deflections.

Which free body produced the number #

The creep coefficient is not derived here — it is an empirical function fitted to measurements of specimens under sustained load, and this site is not in a position to derive it from anything. What is computed is what follows from it, and the free body for that is the same cracked section as before: cut, with the stress block on the face, the concrete carrying a triangular compression above the axis and nothing below it, and the steel carrying tension at its own depth.

Two conditions fix the answer. The axial force on the face is zero, which puts the neutral axis wherever the first moment of the effective area vanishes — $bx^2/2 = nA_s(d-x)$, a quadratic with one positive root. And the couple formed by the two resultants equals the applied moment, which fixes the stresses.

The check is that the couple closes. At the short-term ratio the compression resultant is 371 kN on a lever arm of 404 mm, which multiplies back to 150 kNm — the moment applied. At the long-term ratio the resultants and the arm have both changed and the product is still 150 kNm, because the applied moment is an input rather than an output. That closure is what makes the neutral-axis position trustworthy: if the axis were in the wrong place the two resultants would not be equal, and if the stresses were wrong the couple would not come back.

Where the model stops #

The linearity fails at high stress. Above about 40% of the compressive strength, creep becomes nonlinear in stress and accelerates; above about 75% it can run away entirely, and a specimen held there long enough fails at a load it carried indefinitely at first. That is tertiary creep, it is a genuine failure mode, and design keeps sustained stresses well below it precisely so that the linear treatment holds.

The effective-modulus method is the crudest of the available treatments and is exactly right only for a stress that has been constant since it was applied. Where the stress varies — because the structure is redundant and creep is redistributing its moments, or because more load arrived later — the correct treatment is a superposition integral over the whole stress history, and the shortcut has to be adjusted. That adjustment gets its own essay, and the coefficient it uses turns out not to be the constant it is quoted as.

Steel creeps too, and at ordinary temperatures it does not matter. Prestressing tendons are the exception: held at 70% of their strength for fifty years, they relax measurably, and the loss is a design quantity. Structural steel at working stress and ambient temperature does not creep usefully at all — but at 500°C it does, which is why a steel member in a fire has a time-dependent capacity as well as a temperature-dependent one.

And the number is not knowable in advance to better than about ±30%. $\phi$ depends on the humidity the member will actually experience, the mix that will actually be delivered, and the age at which the load will actually arrive. A calculated long-term deflection carries that uncertainty, which is worth remembering when it is compared against a span/250 limit quoted to the millimetre.

What the picture cannot show #

None of these plots has a real building in it. The curve is a single member under a load applied on one day and never changed. A real floor is loaded in stages — its own weight when the props come out, the floor above a fortnight later, the finishes at six months, the occupants at a year — and each of those increments starts its own creep curve from its own age. The total is a superposition of a dozen curves offset in time, and its shape is nothing like any one of them.

And the construction sequence is invisible. A slab cast on props that bear on the slab below is carrying load into a member that is itself creeping, so the load-sharing between them changes over months. The moment a member carries is not fixed and the calculation on this page assumes it is.

The curve also cannot show what it costs. The multiplier is dimensionless and a serviceability limit is not: what matters is millimetres against a partition, a door frame or a drainage fall. A flat roof whose deflection admits water is the case where those millimetres feed back into the load, and creep and ponding together are the mechanism behind a category of roof collapse that has no overload in it anywhere.

The generalisation #

The pattern is that a serviceability limit can be exceeded by a structure whose strength was never in question, and time is one of the ways it happens.

This site has met the shape twice before. A beam that is strong enough and too flexible fails a limit that contains no strength. A flat roof that deflects, admits water, and deflects further fails by a mechanism whose equation is a buckling equation with rain in it. Creep is the third member of the family and the slowest: the load is constant, the geometry is constant, the material’s strength is constant, and the deflection triples.

What all three share is that the failure has no event in it. There is no moment at which something happens, no sound, no crack, no visit from an engineer. There is a door that will not close, a floor that ponds, a partition that has been pushed out of plumb — and by the time any of those is noticed the deflection has been accumulating for years and cannot be taken back out.

A surprising place this turns up #

Creep is the reason concrete structures survive things they should not.

An imposed deformation in a redundant structure — a settled support, a restrained shrinkage, a thermal movement — generates forces proportional to the stiffness restraining it. In steel that stiffness is constant and the forces stay. In concrete the stiffness falls by a factor of three over a few years, and the forces fall with it.

So a concrete frame subjected to a settlement quietly disposes of two-thirds of the moments the settlement caused, without anything yielding, cracking or being repaired. The same is true of the forces from restrained shrinkage, of the locked-in effects of construction sequence, and of differential temperature. Concrete’s worst property as a material for deflection is its best property as a material for imposed deformation, and it is the same property.

That is the sense in which creep is not merely a nuisance to be multiplied around. It is a mechanism by which a structure relaxes its own self-inflicted forces, and it is the reason a material with almost no tensile strength and a large shrinkage can be used to build continuous frames hundreds of metres long.

Where the ladder goes next #

Later rungs on this anchor: the superposition integral in full, and the age-adjusted effective modulus that approximates it. The separation of creep into basic and drying components, and why member size enters through the second. Shrinkage as the other time-dependent strain, which is not a response to stress at all and adds to the same deflections. Creep redistribution of moments in a redundant frame, and the theorem that the redistribution tends towards the solution for a structure built monolithically. Prestress losses, where creep, shrinkage and steel relaxation all consume the same tendon extension. Tertiary creep and sustained-load strength. And the composite steel-concrete beam, where one material creeps and the other does not and the section redistributes internally for decades.

Historically creep was discovered by builders rather than by researchers. Nineteenth-century reinforced concrete was built for two decades before anybody had a theory of it at all, and long-term deflections were observed, recorded and worked around empirically well before Hatt gave them a name in 1907. The systematic study belongs to the 1930s, and the modern prediction models — with their coefficients for humidity, notional size and age at loading — are the descendants of test programmes that necessarily had to run for years, which is the other thing that makes this property unlike every other one in this field.