The stress that leaks away

A concrete member’s deflection arrives three years late, multiplied by a factor that is nearly all creep.

That is creep acting on a force. The same property acting on a displacement does the opposite, and the reversal is what keeps most restrained concrete uncracked.

The same coefficient, running backwards #

The mechanism is one sentence. A member given a load holds the stress and gains strain; a member given a strain holds the strain and loses stress. Creep is a growth of strain at constant stress, so under a fixed strain the only way to accommodate it is for the stress to fall.

The elastic calculation is not slightly wrong here; it is wrong by a factor of six, and it predicts a phenomenon — universal cracking of every restrained pour — that does not happen.

That is worth stating because it is the reason the steel in a restrained wall decides how many cracks rather than how much: the stress that has to be carried at cracking is a fraction of the elastic one, and the question becomes crack width rather than crack existence.

Which free body produced the number #

The free body is a length of the member with the restraint replaced by whatever force it delivers.

Under a load-controlled action the force crossing the cut is known and the strain is the unknown. Creep multiplies the strain by $(1+\varphi)$ and the force is unchanged, so the deflection grows and the stress does not.

Under a displacement-controlled action the strain crossing the cut is known and the force is the unknown. The strain is fixed at whatever the restraint imposes, so as the creep strain accumulates the elastic strain must fall to keep the total constant — and with it the stress.

The distinction is not about the material. It is about which of the two quantities the boundary condition fixes, and every action in a structure is one or the other: gravity fixes a force, shrinkage fixes a strain, temperature fixes a strain, prestress fixes a force that then relaxes.

That gives a rule worth carrying. Ask what the restraint is holding constant, and the sign of creep’s effect follows immediately.

The two figures are the same material property, at the same age, doing opposite things — and which one a member gets is decided by whether anything is holding it.

The shortcut, and where its constant came from #

Doing the superposition integral properly is work, and the standard alternative is one line.

The age-adjusted effective modulus replaces $E$ by $E/(1 + \chi\varphi)$, with $\chi$ the ageing coefficient, and turns a hereditary integral into an elastic calculation at a modified modulus. It is used everywhere and $\chi$ is quoted as 0.8 almost universally.

The figure above says the shortcut leaves 3.05 N/mm² where the integral leaves 1.72 — a factor of 1.8, and this creep function implies an ageing coefficient of 1.67 rather than 0.8.

The disagreement is not an error in either. $\chi$ is a property of the creep function and the loading history, fitted so that the one-line formula reproduces the integral, and the 0.8 in circulation was fitted for a slowly varying stress under a particular creep model. A number fitted to one function is being used with another, which is the ordinary fate of a calibrated constant.

For this problem the difference does not change the conclusion — both answers are below the tensile strength — and the lesson is about which conclusions the shortcut can be trusted for. It is reliable for the sign and the order of magnitude and not for a value near a limit.

How much is left, and when #

The curve’s shape matters as much as its endpoint, because a member has to survive the early part of it.

Relaxation is fast at first and slow later, for the same reason creep is: the creep coefficient’s own development is logarithmic in time, so most of the relief arrives in the first weeks and the remainder takes decades. On the 14-day case above, the stress has fallen from 10.9 to about 4 within a month and takes fifty years to reach 1.72.

That is the wrong shape for a member drying out. Shrinkage is also fastest early, so the strain being imposed and the relief being granted are racing each other, and whether the member cracks is decided by which of the two curves is steeper over the first few weeks rather than by either endpoint.

The practical consequence is one every concrete detailer knows and few calculations contain. A member cracks early or not at all. Once the first weeks have passed the relief has arrived, the strain has largely been imposed, and a member that survived them is not going to crack from shrinkage afterwards — which is why the remedies are all about the early days: curing, insulation, pour sequence, and the age at which restraint is applied.

It also says why a pour cast against something already hard is the difficult case. The restraint is complete from the first hour, when the concrete has almost no strength and its shrinkage is fastest, and it is the one arrangement in which the race is lost before it starts.

Strength and stiffness do not arrive together #

Creep is a property of a material that is also changing, and the two rates are different.

A young structure is much nearer its final deflection than its final capacity, which has two consequences of opposite sign.

Good, for striking formwork: the member is stiff enough to hold its shape long before it is strong enough to carry its design load, so an early strike is a strength question rather than a deflection one.

Bad, for early loading: a member loaded at seven days carries the load at 93 per cent of its final stiffness — so its elastic deflection is nearly the final one — and creeps from there with a coefficient that is larger because the concrete was younger when loaded. The strength it had on the day is the quantity a striking calculation asks for, and the stiffness is the one the deflection depends on. Two different properties, two different development curves, and one age — which is why an early-striking decision that is safe on strength can be expensive on deflection, and why the two are argued about separately on every site.

There is a second reading of that figure which decides how a striking calculation is written. Because the modulus is at 93 per cent while the strength is at 78, the ratio between them is not constant with age — a young concrete is relatively stiffer than it is strong. So a member struck early is stiff enough not to sag and not yet strong enough to be safe, and the two checks that a striking decision needs point in opposite directions rather than being one check with a factor on it.

The curing that crosses over #

The rate at which the material matures depends on its temperature, and the dependence has a sting in it.

That crossing is the single most useful thing on this page for anybody on a site. A cube that looks good at a day is not evidence of a good concrete, and a hot pour reads well early and finishes badly.

It also compounds with creep. A member cured hot and loaded early is loaded onto a material whose final strength will be lower and whose creep coefficient is higher for having been loaded young, so both of the effects on this page run the wrong way at once.

The two coefficients are not the same number #

A subtlety worth separating, because it is where a hand calculation usually goes wrong.

The creep coefficient $\varphi$ that multiplies a deflection is the one for a stress applied at the age of loading and held. The one that governs relaxation is the same function evaluated over a stress history that is falling, and the two are numerically different even for the same member.

That is what the ageing coefficient exists to reconcile. In the age-adjusted formulation the effective modulus for a constant stress is $E/(1+\varphi)$ and for a varying one is $E/(1+\chi\varphi)$, with $\chi$ between zero and one for the ordinary case — the varying stress is applied progressively, at ages when the concrete creeps less, so it produces less creep than the same final stress applied at once.

A structure with both actions therefore needs two effective moduli, and applying one of them to both is the commonest simplification in long-term concrete analysis. It is conservative for one action and unconservative for the other, and which is which depends on the sign of the effect being computed.

The clean statement is that creep is a function of a stress history rather than of a stress, and every one-line rule is a way of avoiding an integral over that history. The rules work because the histories in practice are a small family — constant, or monotonically falling from a start — and stop working for anything else.

Where the reversal decides a design #

Four cases show the rule doing real work, and they divide cleanly.

Restrained shrinkage in a wall is displacement-controlled, so creep relieves it and the elastic stress is a gross over-estimate. Nothing needs to be designed for 10.9 N/mm².

A settled support imposes a displacement on a redundant beam, so the moment it produces relaxes — which is why a long-term settlement is far less damaging than a sudden one of the same magnitude.

A prestress force is force-controlled at the jack and displacement-controlled thereafter: the tendon holds a fixed extension, the concrete creeps and shortens, and the tendon force falls. Prestress loss is creep relaxation, and the same integral applies.

And a column carrying a sustained load is force-controlled, so its deflection grows — and because the second-order amplification depends on the effective modulus, the buckling load falls with it. That case is the one where creep can produce a failure rather than a deformation.

The limit of the linear model #

The relaxation integral is a linear-viscoelastic one, and pushing it hard enough produces an answer that is arithmetically fine and physically impossible.

At a creep coefficient of 3.5 with the strain imposed at three days, the integral relaxes the stress past zero — it returns a small compression in a member that has only ever been stretched. That cannot happen: a strain applied once and held cannot end by putting the material into the opposite state.

What has gone wrong is not the arithmetic but the range. Superposition assumes the response to a stress history is the sum of the responses to its increments, and that assumption is at its weakest for a falling stress — which is exactly what relaxation is. A young concrete with a large creep coefficient is the corner of the parameter space where the accumulated increments overshoot.

The generator refuses that combination rather than drawing it, which is the right answer: a caption reading “the stress that leaks away” over a negative number, with an implied ageing coefficient of −33, is a figure asserting something false about its own subject.

The design reading is the same one. Relaxation is reliable for the sign and the order of magnitude and not at the extremes, and the extremes are exactly where a hand rule is most tempting to apply.

What to carry away #

Creep grows a strain and relaxes a stress, and which one happens is decided by whether the boundary condition fixes a force or a displacement.

The elastic calculation of a restrained strain is a gross over-estimate. 10.9 N/mm² becomes 1.72 over the life of the member.

The one-line shortcut carries a fitted constant, and the constant belongs to a different creep function than the one being used. It is right about the sign and unreliable near a limit.

And the material is changing while all of this happens. Strength and stiffness develop at different rates, temperature changes both, and a hot cure is strong at a day and weaker for ever.

Where the model stops #

Creep is treated as linear in stress. Above about 40 per cent of the strength it is not, and a heavily loaded member creeps faster than proportionally — which is the regime a column near its long-term critical load is in.

Superposition is assumed. The hereditary integral assumes the response to a stress history is the sum of responses to its increments, which is a linear-viscoelastic assumption and is approximate for concrete under decreasing stress — exactly the case relaxation is.

Shrinkage and creep are separated. They are measured together and separated by convention, and the split between them depends on the specimen size, which enters through drying.

Nothing here is a crack-width calculation. Relaxation decides whether a member cracks; if it does, the width depends on the reinforcement and the bond and not on any of this.

The restraint is treated as complete. A wall on a base is restrained at its foot and free at its top, so the strain imposed varies over its height and the relief varies with it — which is why real shrinkage cracks are tall near the base and stop partway up.

And the coefficients are predictions. A creep coefficient from a code is a fitted function of humidity, notional size, age at loading and strength class, with a scatter of ±30 per cent — which is larger than most of the differences argued about on this page.

What a designer does with it #

Three habits follow, and none of them requires the integral.

Classify the action before computing it. Force or displacement — gravity, prestress after transfer, a jack, against shrinkage, temperature, settlement, a lack of fit. The first family grows and the second decays, and knowing which is which is more useful than any coefficient.

Never design a member for an unrelaxed imposed strain. The elastic value is an over-estimate by a factor between three and six, and using it produces reinforcement quantities that are large, expensive and aimed at a stress that will not arrive.

And do the relaxation properly when it lands near a limit. The shortcut’s constant belongs to a different function, and the case where the answer sits close to the tensile strength is exactly the case where a factor of 1.8 between the two methods decides the answer.

The fourth habit is not about creep at all. Ask when the restraint was applied, because the relief available depends on the age at which the strain started being imposed — and that is a construction-sequence question rather than a material one, decided by a pour order on a drawing.

The same slow deformation arrives under three other names in this collection. A deflection that arrives three years late is this effect measured downwards; a strain that was imposed and then relaxed is it measured as a force that goes away; and a strength that depends on how long the load was left on is it measured as a capacity. What they share is that the material’s answer depends on a duration, which is not a quantity any section property contains — and which is also why a structure that settles down does so over a time nobody specified.

The ladder from here #

Later rungs on this anchor: the superposition integral written out, and the age-adjusted effective modulus derived from it rather than quoted. The separation of creep into basic and drying components, and why member size enters through the second. Creep redistribution of moments in a redundant frame, and the theorem that the redistribution tends toward 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, which is the non-linear regime. And the composite steel–concrete beam, where one material creeps and the other does not and the section redistributes internally for decades.

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, long-term deflections were observed and worked around empirically, and Hatt gave the phenomenon a name in 1907. The relaxation half — that the same property removes a stress it would otherwise have produced — took longer to be believed, because it is a mechanism by which a structure gets better with age, and there are not many of those.