## Materials and Methods  
### Fractal Dimension  
Fractal dimension (\(F_{D}\)) is a value that characterizes fractal objects [24] by assigning them a non-integer value between two topological dimensions (0 in the case of a point, 1 for a line, 2 for a plane and 3 for a volume). The so-called Hausdorff-Besicovitch dimension (\(D_{HB}\)) is applied for the analytical determination of an object's fractal dimension:  
\[D_{HB}=\frac{\ln[L(\delta)]}{\ln[\delta]} \tag{1}\]  
where \(L[\delta]\) represents the number of repetitions of the unit and \(\delta\) is the observation scale. Equation (1) is used when fractal objects have an infinite detail (i.e., they are theoretical objects such as mathematical fractals). By contrast, real objects are limited by a finite range of scales, and thus the iterations for generating them are finite. Therefore, estimation procedures are frequently used for computing the fractal dimension of real objects. The box-counting method is one of the most widely used procedures due to its simplicity and the reliability of the results obtained. Hence, box-counting is chosen in this work for calculating the fractal dimension of the beams.  
Box-counting is one of the so-called counting methods [25]. In these methods, magnitudes are quantified by using successive partitions of different sizes and by counting the amount of portions of the object that belong to the aforementioned partitions. In the case of box-counting, the object is successively subdivided into boxes of equal size, varying the size of these boxes so that they become smaller and smaller. Next, the number of boxes containing portions of the fractal object are counted. When the relationship between the box size (\(\delta\)) and the number of boxes containing the object \(L(\delta)\) follow a power law, fractal properties are evinced (Figure 1a). When a double logarithmic plot represents the aforementioned relationship, the power law is transformed into a linear function. According to Equation (1), the fractal dimension can be estimated as the quotient between \(\ln[L(\delta)]\) versus \(\ln[\delta]\)--that is, the slope of the linear fit (Figure 1b). As an example, the estimation of the fractal dimension of the Sierpinski carpet is shown in Figure 1. The fractal dimension of the Sierpinski carpet can be analytically calculated by means the quotient between \(\ln(8)\) versus \(\ln(3)\), resulting around 1.893 ([[https://en.wikipedia.org/wiki/Sierpi%C5%84](https://en.wikipedia.org/wiki/Sierpi%C5%84) ski_carpet]([https://en.wikipedia.org/wiki/Sierpi%C5%84](https://en.wikipedia.org/wiki/Sierpi%C5%84) ski_carpet)) [accessed on 3 September 2021]. This fractal dimension between 1 and 2 indicates that the Sierpinski carpet is a fractal object with a dimension greater than a line, but less than a plane.  
The third iteration of the Sierpinski carpet (Figure 2) is employed for this example, obtaining an estimated fractal dimension of around 1.751. The estimated fractal dimension by means of the box-counting method is close to the analytical result. The difference is due to the use of a finite fractal (third iteration). The estimated fractal dimension should become closer to the theoretical value as the iterations increase.