Concrete formula solvers

Categories — 68 of them, 1611 solvers in all

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A C E G M N S W

Abrams' Water-Cement Ratio Law

f_c = \dfrac{A}{B^{\,w/c}}

ConcreteDuff Abrams' 1918 finding, and still the single most useful sentence in concrete technology: for a workable mix made with sound materials, compressive strength depends on the ratio of water to cement and on very little else. Strength falls exponentially as that ratio rises, which is why every extra litre of water added at the truck to make placing easier is paid for in strength.

Absolute Volume of a Mix Component

V_i = \dfrac{m_i}{SG_i \, \rho_w}

ConcreteThe volume one ingredient of a mix actually occupies as solid matter: its mass divided by its specific gravity times the density of water. This is the arithmetic the whole absolute-volume method of mix design rests on — every component's solid volume, plus the air, must add up to the volume of concrete you meant to make.

Aggregate Moisture Correction (Free Water)

W_{free} = m_{st} \, \dfrac{MC - Ab}{1 + MC}

ConcreteHow much water a stockpiled aggregate gives to the mix — or takes from it. Moisture content and absorption are both quoted against the OVEN-DRY mass, while the aggregate is batched at its wet stock mass, so the correction is not simply the difference of two percentages. This is the arithmetic that keeps a mix design and the concrete in the truck describing the same material.

Arrhenius Equivalent Age (Freiesleben Hansen and Pedersen)

t_e = t \, \exp\!\left[ -\dfrac{E}{R} \left( \dfrac{1}{T} - \dfrac{1}{T_r} \right) \right]

ConcreteThe maturity function Freiesleben Hansen and Pedersen put forward in 1977, and the one most maturity systems now use: hydration is treated as a thermally activated reaction, so the rate follows an Arrhenius exponential in absolute temperature rather than a straight line above a datum. It holds over a far wider temperature range than the linear form, at the price of a second parameter that has to be measured.

Concrete Batch Yield

Y = \dfrac{\sum m}{\rho_{fresh}}

ConcreteThe volume of concrete a batch actually produced: everything that went into the mixer, weighed, divided by the density of the fresh concrete that came out. Mass is conserved and volume is not, which is why yield is checked by weighing rather than by adding up what the mix design said.

Elastic Modulus from Compressive Strength

E_c = k \sqrt{f'_c}

ConcreteThe square-root correlation every concrete code carries: stiffness estimated from compressive strength. The coefficient k is a USER INPUT here and always will be, because ACI, CSA and Eurocode fit different lines through different test populations and their coefficients are not interchangeable — which is exactly the thing a built-in constant would hide.

Gravimetric Air Content

A = \dfrac{T - D}{T}

ConcreteAir content worked out from two densities: what the concrete WOULD weigh per unit volume with no air in it at all, and what it actually weighs. The shortfall is air. It needs no pressure meter and no special apparatus beyond a scale and a measure of known volume, and it is the method that catches a pressure meter that has drifted.

Modulus of Rupture Predicted from Compressive Strength

f_r = k \sqrt{f'_c}

ConcreteThe flexural tensile stress at which plain concrete cracks, PREDICTED from its compressive strength. It is the stress a deflection calculation uses to decide when a section cracks, and it is the basis of rigid pavement and slab-on-grade thickness design. The coefficient k is a USER INPUT, because the codes do not agree on it and one of them does not even use a square root.

Nurse-Saul Equivalent Age

t_e = t \, \dfrac{T_a - T_0}{T_r - T_0}

ConcreteSaul's 1951 maturity function, written as an equivalent age: a stretch of curing at some concrete temperature, converted into the number of hours at a reference temperature that would have done the same amount of hydration. It assumes the rate of strength gain is a straight line in temperature above a datum below which nothing happens, which is the simplest assumption that works.

Split Cylinder Tensile Strength

f_{ct} = \dfrac{2P}{\pi L D}

ConcreteA cylinder laid on its side and squeezed across a diameter splits along that diameter, because the diametral compression sets up a nearly uniform TENSILE stress across the loaded plane. This is the elasticity solution for that stress, it is pure test geometry, and it is how the tensile strength of concrete, rock and asphalt is actually measured.

Water-Cement Ratio

\dfrac{w}{c} = \dfrac{m_w}{m_c}

ConcreteThe mass of free water in a batch divided by the mass of cementitious material in it. Two weights and a division, and it is the single number that governs strength, permeability, durability and very nearly everything else about hardened concrete.