The strain that was imposed, and the stress that leaked away

Concrete shrinks. A member drying from the inside out loses perhaps 300 microstrain over a few years — three tenths of a millimetre per metre — and if something is holding it, it cannot.

The arithmetic for what that costs takes one line. A strain of $300 \times 10^{-6}$ prevented in a material of modulus 32,000 N/mm² produces a tensile stress of 9.6 N/mm². The tensile strength of the same concrete is about 3.5. The stress is nearly three times the strength, so the member cracks, and since almost every concrete member is restrained by something — a wall on a foundation, a slab on columns, a deck on abutments — the conclusion is that everything cracks from shrinkage alone before any load arrives.

Some do. Most do not. The line of arithmetic is wrong, and what is wrong with it is that it computes an elastic response to a strain that is imposed over years by a material that does not stay elastic for minutes.

Relaxation is creep’s other question #

Creep is what happens when the stress is held and the strain is watched. Relaxation is what happens when the strain is held and the stress is watched. They are the same material behaviour, and the second is the one that matters whenever the thing being applied to a structure is a deformation rather than a force.

That distinction has run through this whole field and it is worth restating in its sharpest form. A force has to be carried: something outside the structure insists on it, equilibrium has to be found with it, and no amount of deforming makes it go away. A deformation does not: it produces stress only where it is restrained, and anything the material does to accommodate it — yielding, cracking, creeping — reduces the stress it causes.

A settled support is the elastic version of this and its moments are proportional to $EI$. What this page adds is that for concrete the $E$ in that expression is a function of time, so the moments a settlement causes are largest on the day it happens and decay from there.

Which free body produced the number, and how it was computed #

The free body is a bar of concrete held between two immovable points — nothing simpler exists — and the number wanted is the force in it.

The governing statement is that the total strain is zero throughout: whatever shrinkage the material wants, an equal and opposite mechanical strain is being forced on it. Writing that with creep in it needs a superposition integral, because the mechanical strain at time $t$ is the response to the whole history of stress and not to its current value:

$$\varepsilon = \sigma(t_0)\,J(t,t_0) + \int_{t_0}^{t} J(t,\tau)\,d\sigma(\tau)$$

where $J(t,\tau)$ is the creep compliance — the strain at time $t$ produced by a unit stress first applied at time $\tau$, which for concrete is $(1+\phi(t,\tau))/E$.

That is a Volterra integral equation and it is solved here by discretisation: divide the time axis into steps, treat the stress as a sequence of increments applied at each, and require the strain condition at each step. Everything but the current increment is known, so the system is triangular and forward substitution solves it in one pass with no iteration.

The routine that does that was checked against a case with a closed-form answer before being trusted here. Fed a Maxwell material, whose compliance is $1/E + (t-\tau)/\eta$ and whose relaxation is $\sigma(t) = \varepsilon E e^{-E(t-t_0)/\eta}$ exactly, it reproduces the exponential to within 0.6% at 800 steps and closer as the grid refines — and it lands nowhere near the effective-modulus answer for the same kernel, which is what a broken implementation would have produced.

The shortcut, and where it disagrees #

Solving an integral equation is not what a designer does. The standard shortcut replaces the whole thing with

$$\sigma(t) = \frac{\varepsilon E}{1 + \chi\phi}$$

where $\chi$ is the ageing coefficient, usually quoted as about 0.8. It is the effective-modulus method with one correction: a strain imposed gradually is not the same as one imposed at the start, and $\chi$ discounts the creep accordingly.

Run both on this creep function and they do not agree. The step-by-step integral gives 2.32 N/mm². The shortcut at $\chi = 0.8$ gives 3.31. Solving for the $\chi$ that would reproduce the integral gives 1.32, not 0.8 — a 40% discrepancy in the coefficient and a 43% one in the answer.

That disagreement is recorded here rather than reconciled, and it is worth being explicit about what has been checked and what has not. The integration is not the suspect: it is validated against a closed form above, it is stable to four significant figures under a sixty-four-fold refinement of the time grid, and it is unchanged by moving each stress increment to its interval midpoint. What the disagreement says is that $\chi$ is a property of the creep function it was fitted to and not a constant, and the creep function here is not the one the tabulated values were derived against.

The honest conclusion is narrower than the discrepancy looks. The two methods disagree about the number and agree about the answer: 2.32 and 3.31 are both below 3.5, so the member does not crack either way. What decides the outcome is whether creep is counted at all, and not which of the two ways it is counted. A quantity that varies by 40% between methods and changes no decision is a quantity worth knowing the uncertainty of and not worth arguing about.

When it does crack #

Nothing above says restrained shrinkage is harmless. It says the elastic arithmetic overstates it by a factor of four, which turns a certainty into a question, and the question has a real answer that depends on several things the figure holds fixed.

The variables that decide it are the shrinkage strain, the creep coefficient, the tensile strength and the degree of restraint — and the awkward feature of the list is that the first two work in opposite directions and are both driven by drying. A member that dries quickly shrinks more and creeps more, so the outcome depends on which effect wins, which depends on the size of the member and the humidity around it. That is the same shape of trade as the one between a section’s shape factor and its curvature demand — one physical cause driving two quantities that push in opposite directions — and it has the same consequence, which is that neither quantity means anything quoted on its own.

The same argument at the scale of a structure #

A bar between two immovable points is the simplest restraint there is. A real structure restrains itself, and the quantity that decays is not a stress in a bar but a whole set of bending moments.

This is why settlement is a far more serious matter in a steel frame than in a concrete one of comparable stiffness, and why the standard advice to compute settlement effects with a long-term modulus is not a fudge but the same calculation as everything on this page. The two-thirds of the moment that creep disposes of was never a load; it was a stiffness times a displacement, and one of the two factors went away.

Where the model stops #

Full restraint is a limiting case and nearly nothing is fully restrained. A wall cast on a foundation is restrained at its base and free at its top, so the stress varies over the height and is highest where the restraint is. A slab between columns is partially restrained, and the degree depends on the columns’ bending stiffness. Everything on this page is the upper bound.

Concrete’s tensile strength is a badly behaved quantity, in the way any strength read off a specimen is and rather more so. It scatters far more than its compressive strength, it is lower under sustained load than under a short-term test, and it depends on the size of the member being tested. Comparing a computed stress with it to two significant figures is not a comparison that carries that much precision.

And cracking is not the end of the calculation. A cracked member has relieved its restraint stress at the crack, so the crack is a mechanism for accommodating shrinkage rather than a failure — much as a plastic hinge relieves a moment rather than ending a beam. The design question is not usually whether it cracks but how wide the cracks are and how far apart, which depends on bond and bar spacing and not on anything in this analysis — and which is why minimum reinforcement is specified for members that carry no load at all.

Steel relaxes too, at high sustained stress. A prestressing tendon held at 70% of its strength loses a few per cent of its force over its life, and this is the same phenomenon in a different material. The interesting complication in a prestressed member is that all three time-dependent effects — the concrete’s creep, the concrete’s shrinkage and the steel’s relaxation — consume the same tendon extension, so they cannot be added independently.

What the picture cannot show #

The curve is a stress and cracking is not decided by a stress alone. Concrete does not fail at a stress; it fails when a flaw large enough to run is loaded hard enough to run it, which is an argument about the largest defect present and not about a nominal value. The tensile strength drawn as a horizontal line is a statistical summary of that, taken from specimens far smaller than a wall.

The horizontal axis is logarithmic, which flatters the shape. Nine tenths of the plot’s width covers the period from a month to twenty-seven years, and the first month — where the stress rises fastest and the concrete is weakest — is compressed into the left-hand edge. Early-age behaviour is where a large fraction of real restraint cracking happens, and this plot is the wrong instrument for it.

And there is only one member on it. Real restraint is mutual: a wall restrains its base and the base restrains the wall, both are creeping, and both are shrinking on different timescales because they were cast on different days and have different surface-to-volume ratios. The single relaxing bar is the free body that makes the mechanism visible, not the geometry anybody builds.

The generalisation #

The pattern is that the right question about an imposed deformation is how fast it arrives.

A strain imposed instantly is resisted by the instantaneous stiffness. A strain imposed over decades is resisted by whatever stiffness the material has left by the time it is imposed, which for a creeping material is a fraction of the first. The same imposed deformation therefore costs a factor of four more if it happens in an afternoon than if it happens over ten years, and the material is identical in both cases.

That has a design consequence with no calculation attached: arrange for imposed deformations to be slow. Pouring a long slab in bays with a gap left to be filled later, delaying the fixing of a restraint, sequencing a deck’s connection to its abutments — each is a way of letting a substantial fraction of the shrinkage happen before anything is holding it. All of them are programme decisions rather than design ones, and none of them appears in the analysis of the finished structure.

A surprising place this turns up #

The same reasoning explains something about redundant structures that is otherwise puzzling: a concrete frame is far less sensitive to construction inaccuracy than an equivalent steel one, and the difference is not workmanship.

A member fabricated a few millimetres too long and forced into place is an imposed deformation. In steel it generates a force proportional to the axial stiffness $EA/L$, and that force stays for the life of the structure. In concrete it generates the same force initially and then loses two-thirds of it over a few years, because the member creeps into the position it was forced towards.

So concrete frames quietly forgive their own construction, and steel frames do not. It is the same mechanism as the shrinkage relief on this page, applied to a strain imposed by a tape measure rather than by drying, and it is one of the reasons that erection tolerances in steelwork are a much larger part of the specification than their concrete equivalents.

Where the ladder goes next #

Later rungs on this anchor: the ageing coefficient derived rather than tabulated, and the family of creep functions it is a property of. Restraint factors for real geometries — wall on base, slab on columns, deck on abutments — and how the stress varies over a member rather than being uniform. Crack width and spacing, which is where the design actually lands, and the bond argument behind it. Early-age thermal cracking, where the imposed strain is the heat of hydration leaving rather than water, and which arrives in days rather than years. Prestress losses in full, with all three time-dependent effects sharing one extension. Creep redistribution of moments in a redundant frame. And the equivalent argument for a composite steel-concrete beam, where one material creeps against another that does not and the section redistributes internally for decades.

Historically the recognition that relaxation and creep are the same phenomenon came late enough to have caused damage. Early reinforced concrete design treated shrinkage by the elastic arithmetic at the top of this page, concluded that restrained members would crack, and either accepted the cracking or over-reinforced against a force that was never going to be there. Dischinger’s rate-of-creep method in the 1930s was the first systematic attempt at the integral, Trost’s ageing coefficient followed in 1967 and Bažant put it on a proper footing in 1972 — and the tabulated $\chi \approx 0.8$ that appears everywhere is a fit to particular creep functions, which is exactly what the disagreement on this page is a reminder of.