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Nurse-Saul Equivalent Age

Also known as Nurse Saul · maturity method · temperature time factor · equivalent age maturity · datum temperature concrete · degree hours concrete · Saul maturity function · concrete maturity equivalent age · TTF

te=t Ta−T0Tr−T0t_e = t \, \dfrac{T_a - T_0}{T_r - T_0}

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See also
Arrhenius Equivalent Age (Freiesleben Hansen and Pedersen) — The same conversion with an exponential temperature law instead of a linear one — better over a wide temperature range, and the reason the linear form has a limited window.Abrams' Water-Cement Ratio Law — The other half of what sets strength: the water-cement ratio, which maturity cannot see at all.Elastic Modulus from Compressive Strength — What an early strength buys in stiffness, once a strength-maturity curve has given you a strength to use.

Learning zone

Concrete does not gain strength with time. It gains strength with hydration, and hydration is a chemical reaction that runs faster when it is warm. A slab poured in July is stronger at eighteen hours than the same slab poured in October is at three days, and neither the calendar nor a set of cylinders sitting in a curing tank at twenty degrees can tell you that. The maturity method is the arithmetic that converts a temperature history into a measure of how far the reaction has actually got.

A.G.A. Saul set it out in 1951, working on steam-cured precast, and the assumption is the simplest one that could possibly work: the rate of strength gain is a straight line in temperature above some datum below which nothing happens at all. Multiply the temperature above the datum by the time spent there, sum over the whole history, and the total — the maturity index — is a measure of hydration. Two specimens of the same mix with the same maturity have, in Saul's formulation, the same strength however they got there.

This site ships equivalent age rather than the maturity index, and it is worth saying why. The index has the dimensions of temperature multiplied by time — degree-hours — and this site's units engine has no such type among its ninety-six. Rather than fake it with a bare number and a printed label, the page divides through by the reference temperature above the datum and returns an equivalent age: the number of hours at a reference temperature that would have produced the same hydration. Time in, time out, and no unit problem anywhere. It is also the more useful of the two quantities, because equivalent age is what a contractor compares against a strength curve plotted on an age axis, and because it is directly comparable between the linear and the Arrhenius formulations. The index and the equivalent age carry exactly the same information and differ by a single division.

The datum temperature is the parameter everybody borrows and almost nobody measures. T0T_0 is the temperature below which the model says hydration has stopped. Every textbook prints a value near freezing, and it circulates as though it were a physical constant. It is not. It is a fitted parameter of a particular mixture, it moves with the cement chemistry, with fly ash or slag replacement, and with accelerating or retarding admixtures, and a five-degree error in it moves the answer by more than most people would accept in a strength estimate. ASTM C1074 is the standard practice for determining it for your mix, it is the authority, and it is copyrighted — so no procedure or default from it appears anywhere on this site. Reaching for a textbook datum and applying it to a mixture you have never tested is the commonest maturity error in the field, and the arithmetic gives you no warning at all.

Two structural limitations sit on the linear form. It works well over ordinary curing temperatures and drifts at the ends, which is why the Arrhenius form exists and why most modern maturity systems use that instead. And no maturity function of any kind can see the crossover: concrete cured warm gains early strength quickly and ends up lower in ultimate strength than the same mix cured cool, because the hydration products form a coarser and less uniform structure. Maturity predicts the early gain honestly and then over-predicts the long-term strength of a hot-cured element. It is the known blind spot of the whole method, it is not a small effect, and no amount of care with the datum will fix it.

Finally, the thing that matters more than everything above put together. An equivalent age is not a strength. It becomes a strength only against a strength-maturity curve previously established for that specific mixture, by casting specimens, curing them, and breaking them at intervals against their own measured maturity. Change the mix and the curve is void. A maturity meter reading quoted as megapascals with no curve behind it is not an estimate — it is a unit conversion of nothing. And even with the curve in hand: formwork stripping, shore removal and post-tensioning are specification and code decisions, made by the person the specification names, on evidence the specification defines. This site computes an equivalent age. It does not authorise a strip.

Nurse-Saul Equivalent Age
te=t Ta−T0Tr−T0t_e = t \, \dfrac{T_a - T_0}{T_r - T_0}
TaT0t
Where
  • tet_e= Equivalent age at the reference temperature (h)
  • tt= Elapsed time at this temperature (h)
  • TaT_a= Average concrete temperature (°C)
  • T0T_0= Datum temperature (°C)
  • TrT_r= Reference temperature (°C)
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