**2. Conventional concrete methods**

To obtain a concrete having the desired properties according to climatic and other requirements, and to use local materials for economic reasons and in order to know the proper method of concrete formulation, we have seen that it is necessary to mention some of the conventional methods widely used in the formulation of concrete.

Methods were adopted for the formulation of concrete whose first principles of physical relations emerged at the end of the 19th century, and these methods of formulation have varied depending on the materials available and our need for the required concrete quality.

#### **2.1 Strength formulas**

#### *2.1.1 Formulas of Feret*

René Féret [11] in 1892 was one of the first to research the law governing the prediction of the compressive strength of concrete *f <sup>c</sup>* (1).

Its formula based on the strength of the cement (the true class), the nature of the aggregate, the cement/water dosage ratio and taking into account the volume of voids. But does not take into account neither the shape of the aggregate nor the granular distribution, nor the resistance to fragmentation of the aggregate. The latter is formulated using the following expression:

*Application of a Granular Model to Identify the Particle Size of the Granular Mixtures… DOI: http://dx.doi.org/10.5772/intechopen.99969*

In 1892, Féret [11] to whom the first researches are attributed, worked on a principle of the mechanical resistance of concrete *f <sup>c</sup>* (1). *Kf*́e*ret* coefficient of the resistance of the cement and the type of the aggregate. *c*, *e* are dosages of cement and water. *v* is the volume of the area. But it does not directly take into account neither the shape nor the type of gravel used, as well as the granular distribution of the granular concrete mixture.

$$f\_c = K\_f f\_{mc} \left(\frac{\mathbf{V}\_c}{\mathbf{V}\_c + \mathbf{V}\_w + \mathbf{V}\_a}\right)^2 \tag{1}$$


#### *2.1.2 Methods of fuller*

Fuller and Thomson [12] in 1907 established their method based on the maximum compactness of the continuous granular mixture, and it depends mainly on the porosity of the granular mixture (2) and the granular expansion. However, it does not directly take into account the shape of the grains, nor the resistance to friability of the aggregate used, and its relation is written as soot:

$$P\_{F\ T} = \mathbf{100} \sqrt[s]{\left(\frac{d}{\mathbf{D}}\right)}\tag{2}$$


#### *2.1.3 Methods of Abrams*

Abrams [13] in 1918, Regardless of the European school, he empirically proposed an exponential equation to predict the compressive strength of concrete, still used in North America, which has two adjustable parameters [Popovics, 1995].

The cement/water ratio, and involves through a coefficient (improved K\_Féret) which indirectly presents nature and shape of the aggregates.

We note the absence of a direct representation of the resistance to fragmentation of the aggregate and of the granular distribution, as is the case in the rest of the previous methods, its formula (3) is written:

$$f\_c = K\_{Abrams} \left(\frac{1}{7.5^{(1.5/W/C)}}\right) \tag{3}$$


### *2.1.4 Methods of Bolomey*

Bolomey [14] in 1925 is based on a formula (4), (improved iron) to determine the dosages of cement and water. This formula for predicting the mechanical compressive strength of concrete, which depends on the shape of the aggregates as well as the consistency of the concrete, and the dosages of cement and water, and the volume of voids. But does not take the representation of resistance to aggregate fragmentation.

This formula, like that of Féret, is the product of three terms which share, in order of factors, the influence of aggregates, cement and concrete formulation. The difference, compared to the relation of Féret, relates exclusively to the third term, parabolic in Féret, linear in Bolomey. It has been shown that the Bolomey relation is a good approximation of that of Féret for the values of the E/C ratio between 0.40 and 0.70; within this range, the error is less than or equal to 3%.

$$f\_c = K\_{Bolemy} \left(\frac{\mathbf{C}}{\mathbf{W} + \mathbf{V}} - \mathbf{0.5}\right) \tag{4}$$


#### *2.1.5 Methods of Caquot*

The scientist Caquot [15] circulated his research during the year 1937, through which he sought to find the optimal aggregate distribution in which the porosity of the aggregate mixture is minimal, according to the basic hypothesis of compatibility between two aggregates classes without influence due to the presence of another aggregates class.

This basic idea was taken up by F. de Larrard [3], who had previously embarked on a vast process of developing other concrete formulation programs.

The relation is determined empirically by assuming that the volume of the voids depends on the dimensions of the small grains, then on the addition of grains, then on a constant determined empirically according to the relation of Caquot (5).

$$\mathbf{V} = \mathbf{V\_0} \sqrt[s]{\left(\frac{d}{D}\right)}\tag{5}$$


*Application of a Granular Model to Identify the Particle Size of the Granular Mixtures… DOI: http://dx.doi.org/10.5772/intechopen.99969*

#### *2.1.6 Methods of Faury*

We find in the work of Faury [16] and Joisel [17] that they made modifications to the work of Caquot in 1942 and 1952, and Faury extends to the granular range up to 6.5 μm, incorporating the cement as a granular material and taking into account the effect of the wall. And Joisel gave a reference straight line (at a complex scale) taking into account the cement, water, voids, granulometry and the compactness of the granular classes. Here, we note an indirect representation of the granular distribution with the mechanical resistance of the aggregates [18].

The optimum grain size of a concrete is a mixture (in a certain proportion) of two kinds of grains of the aggregate.

The reference curve to be followed consists of two straight sections.

The first AB gives the granulometry of fine grains. The second straight line is that of coarse grains. The y coordinate of, called the break point, indicates the percentage by volume of the grains. Its value is given by the experimental formula (6).

$$\mathbf{Y} = \mathbf{A} + \mathbf{1}\mathbf{7}. \sqrt[3]{\left(\frac{\mathbf{d}}{\mathbf{D}}\right)} + \frac{\mathbf{B}}{\frac{\mathbf{R}}{\mathbf{D}} - \mathbf{0}.\mathbf{7}\mathbf{5}} \tag{6}$$


#### *2.1.7 Methods of Dreux-Gorisse*

He method of Dreux and Gorisse [19] is based on the optimal granularity which is still current for the design of the concrete formulation. This is an empirical approach according to an OAB granular reference curve (segments of two lines in a semi-logarithmic plot). Contrary to the moment, the cement is not part of the reference curve of the mixture, since its mass is determined separately. It is a method which takes into account a large number of parameters [18]. But it does not take into consideration the direct representation of the granular distribution of the aggregate, and indicates what the true class of cement represents, and the dosage of cement and water, type, shape, quality and dimensions. of aggregates, the smoothness, consistency and strength of concrete.

This method is fundamentally empirical in nature, unlike the Faury method which predates it [Faury, 1942] and which is based on Caquot's theory of the granular optimum [Caquot, 1937]. Dreux carried out a large survey to collect data on satisfactory concretes [de Larrard, 2000]. On the basis of a statistical analysis of this large number of concretes and by combining the granular curves obtained, they were able to base an empirical approach to determine a reference granular curve.

It is also very easy to use since it only requires knowing the grain size curves of the aggregates used.

A test batch is necessary to be carried out in the laboratory in order to make any usage corrections.

"B" (on the ordinate 100%) corresponds to the dimension D of the largest aggregate.

"O" (at ordinate 0%) corresponds to the dimension d of the smallest aggregate. The break "A" has the following coordinates:


Si: D ≤ 20 mm; the abscissa is D / 2.

If: D ≥ 20 mm; the abscissa is located in the middle of the "gravel segment" limited by the modulus 38 (5 mm) and the modulus corresponding to D.

in ordinates (7)

$$\mathbf{Y} = \mathbf{S}\mathbf{0} - \sqrt{\mathbf{D}} + \mathbf{K} + \mathbf{K}\_s + \mathbf{K}\_p \tag{7}$$


### *2.1.8 Baron and Lesage*

The method of Baron and Lesage [20] is based on a technique proposed in 1976 to improve the granular skeleton according to the principle of relating the minimum flow time specified by the LCL Maniabilimeter according to Standard 18–452 [21] with the quantity optimal granularity for constant cement and water ratios.

The principle is to measure the time taken for a concrete sample to flow under vibration to a certain mark. The optimum proportions of aggregate are assumed to give the minimum flow time, for a given amount of cement and water. Once the granular proportions have been identified, the water and cement dosages are adjusted experimentally, so that the mixture has the desired workability and resistance. It is assumed, in this method, that the optimum proportions of aggregates do not depend on the quantity of cement.
