The concrete maturity formula converts a concrete element’s recorded temperature history into a single index value, from which in-place strength can be estimated using a strength-maturity relationship developed for that mix. The most widely used version in North America is the Nurse-Saul temperature-time factor, standardised in ASTM C1074:
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M(t) = Σ (Ta – T0) · Δt
where M(t) is the maturity index in degree-hours or degree-days, Ta is the average concrete temperature during the time interval, T0 is the datum temperature (the temperature below which strength gain is taken to stop, commonly 0°C or 32°F), and Δt is the length of the time interval. Two other functions are in use: the equivalent age method, based on the Arrhenius equation, which handles curing temperatures outside the 0°C to 40°C range more accurately, and weighted maturity, which is standardised in the Netherlands rather than by ASTM. All three convert temperature and time into an index. None of them measures strength directly.
| Method | Formula | Relationship to temperature | Where it is standardised | Use it when |
|---|---|---|---|---|
| Temperature-time factor (Nurse-Saul) | M(t) = Σ (Ta – T0) · Δt | Linear | ASTM C1074, standardised 1987. Most common method in North America | Curing temperatures stay roughly within 0°C to 40°C (32°F to 104°F) and simplicity matters |
| Equivalent age (Arrhenius) | te = Σ e−Q(1/Ta − 1/Ts) · Δt | Exponential | ASTM C1074 and European standards | Curing temperatures fall outside 0°C to 40°C, or higher accuracy is needed and the activation energy is known |
| Weighted maturity | Area under the temperature curve with a cement-specific correction factor applied | Non-linear, cement-specific | NEN 5790 (Netherlands), not standardised by ASTM | Working to Dutch or European practice with a cement C-value available from the producer |
Temperatures in the equivalent age equation are in kelvin. T0 is the datum temperature, Ts is the specified reference temperature (usually 23°C in North America and 20°C in Europe), and Q is the activation energy divided by the gas constant, which ASTM C1074 allows to be taken as 5000 K for Type I cement without admixtures.
Temperature-Time Factor (Nurse-Saul Method)
The temperature-time factor (TTF) method, also known as the Nurse-Saul maturity function, was the first maturity method developed in the early 1950s, thanks to the work of Nurse, McIntosh and Saul. The goal of their work was to understand the effect of accelerated curing and the impact of different curing temperatures on the strength development of concrete. This method was standardized in 1987 by ASTM C1074. Today, the Nurse-Saul function is the most commonly used maturity method in North America because of its simplicity.
This approach takes into consideration that the maturity value is linearly dependent on temperature and can simply be represented by the area below the temperature curve, as graphically shown in Figure 1. In this approach, the area under the temperature curve is taken as the difference between the average recorded temperature and the datum temperature (T). The datum temperature is defined as the temperature at which the hydration of the cement stops, in other words, the temperature at which concrete stops developing strength. The Nurse- Saul equation is mathematically represented as follow:
M(t) = Σ (Ta – T0) · Δt
M(t) is the maturity index at age t, expressed in degree-hours or degree-days. Ta is the average concrete temperature over the interval. T0 is the datum temperature. Δt is the length of the interval, typically 30 minutes or 1 hour depending on how often the sensor logs.
Most of the variables in Eq. 4-1 can easily be obtained without a complex analysis. “T” is simply obtained by the maturity monitoring system at a given time. “Δt” is the default value given by the frequency of measurements taken by the maturity meter and is usually defined as 1 hour, 30 min, or less. The only variable that is unknown and needs to be calculated or estimated is the datum temperature. For better accuracy, T0 can be determined through laboratory testing as specified in ASTM C1074, but, in most cases, it can be defined as 0°C (32°F), -5°C (23°F) or -10°C (14°F).
ASTM C1074 states that: “for type I cement without admixture and a curing range between 0 to 40°C, the recommended datum temperature is 0°C”. Originally, the datum temperature was defined as -10°C. However, studies have shown that the datum temperature for any given type of cement is within 0 to -10°C. Assuming a 0°C datum temperature in most cases is often considered a conservative approach, as it assumes that there is no strength gain if the temperature of the concrete falls below freezing point. Since concrete cannot lose strength as it hydrates, any given condition that would cause (Ta – T0) is less than or equal to 0 would result in no strength gain, M(t) = 0.
During the summer, when the temperature of the concrete is higher, assuming any datum temperature from 0 to -10°C for calibration will not necessarily have a significant impact on the results. However, in winter when the temperature can easily fall below freezing in certain regions, closer attention needs to be given when determining the actual datum temperature. In general, project specifications require that the temperature of the concrete remain above a certain temperature (>5°C) for a certain period of time during curing.
Example
Figure 2 and Figure 3 below show an example of maturity and strength calculations using different datum temperature (0°C, -5°C, -10°C). Each maturity calibration curve was developed by taking into account the different datum temperature to calculate maturity for the same strength data. The difference of those three mix calibrations are shown in Figure 2, where it is possible to observe that any difference is simply a shift to the right because the maturity corresponding to each strength gets larger as the datum temperature decreases (more area under the temperature curve). Given a temperature curve that falls below freezing point, it is possible to observe how the maturity and strength value can vary for different datum temperatures (Figure 3).
Figure 3: Effect of datum temperature on the maturity and strength calculations
How to Determine the Datum Temperature
As noted above, T0 can be taken as 0°C for simplicity (other temperatures also used in the industry are -5°C and -10°C). Nevertheless, it is possible to determine the datum temperature by following the steps provided in ASTM C1074 A1, which are summarized below.
The datum temperature procedure consists of making a minimum of 54 mortar cubes (ASTM C109) representing the concrete mix. The cubes should be divided into 3 different sets; each containing 18 cubes. Each of these three sets will be cured at a different temperature. The temperature chosen for curing should be based on the maximum and minimum expected curing temperature as well as an average temperature of the jobsite. Three cubes are then broken at six different times to obtain the strength of the concrete (ASTM C109).
The Rate Constant (k-value)
To calculate the datum temperature, the k-value must be determined for each curing condition. The k-value is the reaction rate constant which is dependent on time and temperature. It represents the rate of the chemical reaction, in this case, the cement hydration reaction which represents the strength development in concrete.
There were originally 3 approaches suggested by ASTM C1074 to determine the k-value, in the last revision of ASTM C1074 only one approach was kept as the standard procedure to determine the reaction rate k.
The k-values can be solved using a computer program for Eq. 4-2. In this equation, in addition to the rate constant, “to” and “Su” are also unknowns that need to be solved, ASTM C1074- 17 provides an example of calculation and spreadsheet set up to solve for those parameters.
The rate constant k is solved from the strength gain data for each curing temperature, together with the limiting strength Su and the age at which strength development begins, t0. ASTM C1074 sets out the calculation and provides a worked example and spreadsheet layout.
Once the k-value has been calculated for each curing temperature, the reciprocal of k vs. curing temperature can be plotted. By fitting a linear interpolation, the intersection with the x-axis represents the datum temperature (Figure 4).
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Equivalent Age (Arrhenius Method)
Even though the temperature-time factor equation is widely used and accepted worldwide, the linear approach to determine maturity is not accurate for all curing conditions, especially at temperatures outside the range of 0-40°C. To mitigate this effect, in 1977 Freiesleben-Hansen and Pedersen proposed a more accurate equation in which the maturity is exponentially dependent on temperature. This maturity method is known as the equivalent age method and is based on the Arrhenius equation. This approach is standardized in most codes including ASTM C1074 and European standards (European countries typically allow the Arrhenius method instead of the Nurse-Saul). Despite the fact that this approach is a bit more complex than the temperature-time factor, if the assumptions are done properly, it can lead to a more accurate prediction of the in-place strength. The equivalent age can be calculated using Eq. 4-3.
te = Σ e−Q(1/Ta − 1/Ts) · Δt
te is the equivalent age at the specified temperature. Ta is the average absolute concrete temperature over the interval, in kelvin. Ts is the specified reference temperature, in kelvin. Q is the activation energy divided by the gas constant, in kelvin. Δt is the length of the interval.
Similar to the Nurse-Saul equation, “Ts” and “Δt” can be obtained from the maturity meter. “Ts” represents a specified temperature and is usually defined as 23°C in North America and 20°C in Europe. The activation energy divided by gas constant must be determined experimentally using very similar steps as shown in the calculation of datum temperature. ASTM C1074 also proposes a standard value where Q can be defined as 5000 K for type I cement without admixtures.
To determine “Q”, the k-value must be obtained following the same steps presented in the Rate Constant (k-value) section above. By plotting the natural logarithm of k vs. the reciprocal of the curing temperature (in Kelvin), the negative of the linear slope represents “Q”.
Weighted Maturity
The third method proposed to calculate maturity is the weighted maturity, which was developed in the 1970s by Papadakis and Bresson and later modified by de Vree in 1979. This method is not typically used in North America as it is not standardized by ASTM C1074. However, it is currently standardized in the Netherlands (NEN5790) and accepted in Europe.
The general approach of the weighted maturity method is described in Eq. 4-4. This equation is very similar to the Nurse-Saul equation as “tkTk” represents the area under the temperature curve while the “Cnk” represents a correction factor.
Using this equation is, however, not practical as the “nk” factor is temperature-dependent. A discontinuous function to simplify the calculation of the proposed linear equation (Eq. 4-5), can be used in determining the n parameter. Therefore, the weighted maturity equation can be rewritten in a simpler form by taking the integral of “Cnk” from datum temperature (-10°C) to average temperature. The weighted maturity equation can now be defined as Eq. 4-6 with a continuous n function. It is graphically represented in Figure 5.
The weighted maturity function follows the same shape as Nurse-Saul, the area under the temperature curve above a datum temperature, but multiplies each interval by a correction factor derived from the cement’s C-value. The C-value expresses how sensitive that cement is to temperature and is obtained from the cement producer or determined under NEN 5790. It typically falls between 1.25 and 1.75.
The C-value is a cement-specific value which indicates the influence of the sensitivity of the cement to temperature. The C-value can be obtained directly from the cement producer or through the standardized procedure in NEN 5790. The value typically varies from 1.25 to 1.75.
Concrete Maturity Formula FAQ
What is the formula for concrete maturity?
The standard formula in North America is the Nurse-Saul temperature-time factor: M(t) = Σ (Ta – T0) · Δt. It sums, over every measurement interval, the amount by which the concrete temperature exceeded the datum temperature, multiplied by the length of that interval. The result is a maturity index in degree-hours or degree-days, which is converted to an estimated strength using a strength-maturity relationship developed in the lab for that specific mix.
What is the datum temperature in the maturity formula?
The datum temperature is the temperature below which concrete is assumed to stop gaining strength. ASTM C1074 recommends 0°C (32°F) for Type I cement without admixtures over a curing range of 0°C to 40°C, and it can be determined experimentally for other cases. Values of -5°C (23°F) and -10°C (14°F) are also used in industry. A lower datum temperature produces a larger maturity value for the same temperature history, so the datum temperature used to build the calibration curve must be the same one used on site.
What is the difference between Nurse-Saul and the Arrhenius method?
Nurse-Saul treats the rate of strength gain as increasing linearly with temperature, which is a good approximation between about 0°C and 40°C and is simple to compute. The Arrhenius based equivalent age method treats it as increasing exponentially, which is more accurate outside that range, particularly at high curing temperatures. The cost of the extra accuracy is that the activation energy has to be determined or assumed, whereas Nurse-Saul only needs a datum temperature.
Does the maturity formula measure concrete strength?
No. The formula converts a temperature history into an index number. Strength is then estimated from that index using a strength-maturity relationship built in the lab by breaking cylinders of the same mix at several ages, per ASTM C1074. The formula is one half of the method, the calibration curve is the other, and the estimate is only as good as the calibration behind it.
How accurate is the maturity method?
Accuracy depends almost entirely on the calibration, not on the formula. Standard practice is to validate the relationship by comparing estimated strength against broken cylinders on a later pour, with a difference of up to about 10 percent generally treated as acceptable (ACI 306R-16, section 8.4). Differences beyond that consistently indicate the mix has changed and a new calibration is needed. In-place elements also retain heat longer than cylinders, so in-place maturity estimates typically run ahead of field-cured cylinder results.
**Editor’s Note: This post was originally published On July 19, 2019 and has been updated for accuracy and comprehensiveness.
