*3.2. Aggregate Particles Size*

This section discussed the developed algorithm of diameter generation of the particles (Figure 2). The particle size distribution in concrete is expressed in terms of cumulative percentage passing through a series of sieves with openings with different sizes. The contribution of particle size fractions

is usually presented in the form of a grading curve. The most popular and commonly used grading curve, which lies within the limiting grading curves proposed by design recommendations [23], is the empirical curve proposed by Fuller [22,24]. Fuller's curve provides optimum density and strength of the concrete and is described by the following formula:

$$P(d) = \left(\frac{d}{d\_{\text{max}}}\right)^n \cdot 100\% \tag{9}$$

where *P*(*d*) is the cumulative percentage passing through the sieve with diameter opening *d*, *d*max is the maximum particle size, and *n* is the exponential factor with a typical value of 0.4–0.7. The weight percentage of the individual grading segment can be calculated as

$$P[d\_{\rm s}, d\_{\rm s+1}] = P(d\_{\rm s}) - P(d\_{\rm s+1}) \tag{10}$$

**Figure 2.** Generation of the aggregate particles.

1 1 , ( ) ( )

1

max min () ( )

( )()

 ሾ<sup>௦</sup> , ௦ାଵሿ

*ρ*

1

,

max ( ) 100% 

The aggregate volume fraction in the concrete mixture is ሾ<sup>௦</sup> , ௦ାଵሿ

$$v\_a = \frac{m\_a}{\rho\_a V} \tag{11}$$

where *m<sup>a</sup>* is the mass of aggregate particles, ρ*<sup>a</sup>* is aggregate material density, and *V* is the volume of the concrete specimen. The volume of the particles in the grading segment *Va*[*d<sup>s</sup>* , *ds*+<sup>1</sup> ] can then be calculated as

1

$$V\_{\mathfrak{a}}[d\_{\mathfrak{s}\prime}d\_{\mathfrak{s}+1}] = \frac{P(d\_{\mathfrak{s}}) - P(d\_{\mathfrak{s}+1})}{P(d\_{\text{max}}) - P(d\_{\text{min}})} \cdot v\_{\mathfrak{a}} \cdot V \tag{12}$$

To generate the particle belonging to the grading segment with randomly chosen diameter, one can use the formula:

$$d\_p = \eta (d\_{s+1} - d\_s) + d\_s \tag{13}$$

where η is a uniformly distributed random number in the interval (0,1). The number of particles generated in this way must ensure that the difference between the sum of their volumes and the volume *Va*[*d<sup>s</sup>* , *ds*+<sup>1</sup> ] is smaller than the volume of the smallest particle vs. belonging to the considered segment: , , 2

$$\left[\boldsymbol{V}\_{a}\left[\boldsymbol{d}\_{s},\boldsymbol{d}\_{s+1}\right]-\sum\_{i}^{n}\boldsymbol{V}\_{p}^{i}>\boldsymbol{V}\_{s}\right] \tag{14}$$

Based on the described relationships, the algorithm of diameter generation of the particles was performed. , 0.05

0.05
