Nowadays, engineers should have a deeper understanding, than just specifying say 1:2:4 concrete mix ratio for their construction works. Under normal and well controlled conditions, 1:2:4 mix ratio should yield a concrete with 28 days compressive strength greater than 20 N/mm^{2}. However, there are variations in results when batching by weight and when batching by volume. This situation demands that to produce concrete of a specified compressive strength, you have to accurately carry out a mix design, which incorporates the tendency of failure during tests.

Concrete mix design is therefore the process of specifying the quantity of concrete ingredients required to produce a concrete with a desired fresh and hardened properties. A little consideration from mix design results will show that the mix ratio for batching by weight, and for batching by volume are not basically the same for obtaining a specified grade of concrete.

Compressive strength of concrete is a very variable quantity. Therefore, when we are carrying out mix design, we target a higher average strength so that we can expect every part of the structure to meet the specified strength. We normally employ the statistical standard deviation in taking care of this, which is a measure of scatter or dispersion of strength about the mean.

Therefore, the target strength of concrete (f_{m})is given by;

f_{m} = f_{min} + KSD ————– (1)

Where;

- f
_{m}is the mean compressive strength - f
_{min}is the minimum compressive strength of the concrete. In the Eurocodes, this is called the characteristic strength of the concrete (f_{ck}), while in the US, it is called the design strength (f_{c}‘). - K is the probability factor which is usually taken as 1.64 or 2.33 to express the probability of 1 in 20 and 1 in 100 respectively, for the strength to fall below the minimum strength.
- SD is the standard deviation which is best obtained by considering the previous test results obtained using the same materials, the same procedure, and under the same supervision.

The term KSD is normally referred to as the margin.

The probability of strength values in the range fck ± KSD and below fck – KSD (risk) for normal distribution is shown in Table 1.0 below;

In the Eurocodes and in many other codes, the range of the risk of 1 in 20 is recommended for concrete tests. Which means that in 20 concrete cubes, there is a probability of only one cube not meeting the required strength.

When statistical data is not available for obtaining the standard deviation, the following table below according to ACI code could be used.

Let us consider the trial mix design for a concrete of minimum specified strength of 25 N/mm^{2}, to be employed in the construction of the floor beams and slab of building.

**Materials Analysis**

*Coarse aggregate*: Crushed granite of nominal maximum size of 20mm

Oven dry relative density = 2.68

Fineness modulus = 2.60

Absorption = 0.4% (saturated surface dry)

Bulk density = 1650 kg/m^{3}

Specific gravity of cement = 3.15

*Fine aggregate*: Sharp sand from river

Relative density = 2.64

Absorption = 0.60 %

Bulk density = 1600 kg/m^{3}

The target strength can be obtained from the relation below;

f_{m} = f_{ck} + KSD

f_{m} = 25 + 8.5 = 33.5 N/mm^{2}

**WATER-CEMENT RATIO**

The relationship between water-cement ratio and the 28 day compressive strength of concrete based on ACI code is as shown in the table above.

Therefore, Water – Cement ratio for non-air entrained 33.4 N/mm^{2} concrete = 0.491 (interpolating from the Table above)

**CEMENT AND WATER CONTENT**

Water content is normally estimated from workability requirements, which is guided by slump. The range of slump required for different types of construction is given in the Table below;

For maximum size of aggregate of 19mm, and a slump of 75mm, the table below gives a water demand of 205 kg/m^{3}

^{
}

Therefore cement content;

205/C= 0.491; Therefore the cement content (C) = 205 / 0.491 = 417.515 kg/m^{3}

**MASS OF COARSE AGGREGATE**

**
** For 19mm aggregate with fineness modulus of 2.60, the bulk volume of dry rodded coarse aggregate per m

^{3}of concrete is 0.64 (see Table below).

Therefore, mass of coarse aggregate (Mc) per m^{3} of concrete = 0.64 × 1650 = 1056 kg/m^{3}

^{
} Approximate air content = 2%

**MASS OF FINE AGGREGATE**

The mass of coarse aggregate can be estimated using the relationship below;

Mass of fine aggregate M^{f} = γ_{f} [1000 – (W – C/γ + Mc/γ_{c} + 10A)] ————- (2)

Where;

γ_{f} = Specific gravity of fine aggregate (saturated surface dry)

W = Mixing water requirement

C = Cement Content

γ = Specific gravity of cement (take value as 3.15 unless otherwise specified)

Mc = Coarse Aggregate content

γ_{c} = Specific gravity of coarse aggregate (saturated surface dry)

A = Air content (%)

Therefore:

M_{f} = 2.64 [1000 – (205 + (417.515/3.15) + (1056/2.68) + (10 × 2))] = 655.843 kg/m^{3}

**Final Volume computations**

Water = 205 / (1 × 1000) = 0.205 m^{3}

Cement = 417.515 / (3.15 × 1000) = 0.1325 m^{3}

Air = 2/100 = 0.02

Coarse aggregate = 1056 / (2.68 × 1000) = 0.394 m^{3}

Fine aggregate = 655.843 / (2.64 × 1000) = 0.248 m^{3}

**SUMMARY OF TRIAL MIX DESIGN**

*By weight*

Water = 205 kg/m^{3}

Cement = 417.515 kg/m^{3}

Coarse Aggregate = 1056 kg/m^{3}

Fine Aggregate = 655.843 kg/m^{3}

^{
} Yield of concrete = 2334.36 kg/m^{3}

^{
} Mix ratio by weight (Cement : Fine Aggregate : Coarse Aggregate) = (1:1.57:2.53)

*By volume*

Water = 0.205 m^{3}

Cement = 0.1325 m^{3}

Coarse Aggregate = 0.394 m^{3}

Fine Aggregate = 0.248 m^{3}

Mix ratio by volume (Cement : Fine Aggregate : Coarse Aggregate) = (1:1.87:2.97)

**MIX DESIGN WITHOUT CONSIDERING TEST MARGIN**

However, carrying out mix design for 25 N/mm^{2} grade of concrete without considering the margin, we can obtain the following result using the steps described above;

**Water-cement ratio**

Water – Cement ratio for non-air entrained 25 N/mm^{2} concrete = 0.61 (see Table 9-3 above)

**Water and Cement Demand**

For maximum size of aggregate of 19mm, and a slump of 75mm, gives a water demand of 205 kg/m^{3}.

Therefore the cement content C = 205/0.61= 336.06 kg/m^{3}

**Mass of Coarse Aggregate**

For 19mm aggregate with fineness modulus of 2.60, the bulk volume of dry rodded coarse aggregate per m^{3} of concrete is 0.64. Therefore;

Weight per m^{3} = 0.64 × 1650 = 1056 kg/m^{3}

Approximate air content = 2%

**Mass of Fine Aggregate**

Mass of fine aggregate M_{f} = γ_{f} [1000 – (W – C/γ + Mc/γ_{c} + 10A)]

M_{f} = 2.64 [1000- (205 + (336.06/3.15) + (1056/2.68) + 10(2))] = 724.11 kg/m^{3}

**Volume computations**

Water = 205 / (1 × 1000) = 0.205 m^{3}

Cement = 336.06 / (3.15 × 1000) = 0.1066 m^{3}

Air = 2/100 = 0.02%

Coarse aggregate = 1056 / (2.68 × 1000) = 0.394 m^{3}

Fine aggregate = 724.11 / (2.64 × 1000) = 0.274 m^{3}

**Summary of trial mix design without considering the margin**

*By weight*

Water = 205 kg/m^{3}

Cement = 336.06 kg/m^{3}

Coarse Aggregate = 1056 kg/m^{3}

Fine Aggregate = 724.11 kg/m^{3}

Yield of concrete = 2321.17 kg/m^{3}

Mix ratio by weight (Cement:Fine Aggregate:Coarse Aggregate) = (1: 2.15: 3.142)

*By volume*

Water = 0.205 m^{3}

Cement = 0.1066 m^{3}

Coarse Aggregate = 0.394 m^{3}

Fine Aggregate = 0.274 m^{3}

Mix ratio by volume (Cement:Fine Aggregate:Coarse Aggregate) = (1:2.57:3.696)

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