Sobre e Metodologia

Abrams' Law

Duff Abrams demonstrated in 1918 that the compressive strength of fully compacted concrete is governed primarily by the water-to-cement ratio. His empirical law takes the form:

where:

• f_c is the compressive strength (MPa)
• A is a constant representing the maximum achievable strength for a given cement and aggregate combination — typically 80–100 MPa for OPC with good aggregates
• B is a constant that controls the rate of strength decrease with increasing w/c ratio — typically 8–25, with higher values producing steeper curves
• w/c is the water-to-cement ratio by mass

Abrams' law works because the w/c ratio determines the capillary porosity of the hardened cement paste. Lower w/c ratios produce denser paste with fewer and smaller pores, resulting in higher compressive strength. The relationship holds well for w/c ratios between approximately 0.30 and 0.80.

Bolomey Correction

Bolomey (1935) extended Abrams' approach by accounting for how different cement types affect the constants A and B. Blended cements such as PPC (Portland Pozzolana Cement) and PSC (Portland Slag Cement) hydrate differently from OPC:

• OPC — standard Portland cement. Rapid early strength gain, highest 28-day strength for a given w/c ratio. Uses baseline A and B constants.
• PPC — blends OPC with fly ash (15–35%). The pozzolanic reaction is slower, so early strength is lower, but long-term strength can match or exceed OPC. A and B are adjusted downward to reflect the slower 28-day development.
• PSC — blends OPC with ground granulated blast furnace slag (25–70%). Like PPC, early strength is reduced, but PSC offers excellent chemical resistance. Constants are further adjusted.

In MixRight, the Bolomey correction modifies the Abrams constants based on the selected cement type, so the strength prediction accounts for the different hydration kinetics of blended cements.

Nurse-Saul Maturity Method

Nurse (1949) and Saul (1951) established that concrete strength development depends on both temperature and time. The maturity function combines these into a single index:

where:

• M is the maturity index (°C·hours or °C·days)
• T is the average curing temperature during interval Δt
• T₀ is the datum temperature below which hydration effectively ceases — conventionally −10 °C
• Δt is the time interval

The key insight is that concrete cured at 40 °C for 7 days can reach the same maturity — and approximately the same strength — as concrete cured at 20 °C for 14 days. MixRight uses reference conditions of 20 °C and 28 days as the baseline, then scales the predicted strength using the maturity ratio.

This allows the Strength Predictor to account for real-world curing conditions: hot climates, cold weather concreting, or accelerated steam curing.

Monte Carlo Simulation

Deterministic models give a single point estimate: "this mix will reach 32 MPa." In reality, every input — w/c ratio, cement strength class, aggregate quality, curing temperature — carries uncertainty. Monte Carlo simulation addresses this by:

1. Modelling each uncertain input as a probability distribution (normal, log-normal, or uniform as appropriate)
2. Drawing thousands of random samples from these distributions
3. Running the combined Abrams–Bolomey–Nurse-Saul model for each sample
4. Collecting the resulting strength values into a histogram that shows the full range of likely outcomes

The output is not a single number but a distribution. From this, MixRight reports the mean, median, 5th and 95th percentiles, and the characteristic strength — giving engineers the information they need to design with appropriate safety margins.

References

1. Abrams, D.A. (1918). “Design of Concrete Mixtures.” Bulletin 1, Structural Materials Research Laboratory, Lewis Institute, Chicago.
2. Bolomey, J. (1935). “Granulation et prévision de la résistance probable des bétons.” Travaux, 19(30), 228–232.
3. Nurse, R.W. (1949). “Steam Curing of Concrete.” Magazine of Concrete Research, 1(2), 79–88.
4. EN 206:2013+A2:2021. “Concrete — Specification, performance, production and conformity.”
5. BRE (1997). “Design of Normal Concrete Mixes.” 2nd Edition, Building Research Establishment.