## 6 Results
### Problem 2  
The results achieved for the second case study are summarized in Figures 8 and 9. Figure 8: Problem 2. Experimental dispersion versus analytical dispersion and iteration versus accuracy for \(m\) in the range 1 to 4 and for: \(\delta\) = 1 and \(\beta\) = 0.2, 0.5, 1, 2, 7, 50. Figure 7: Problem 1. Shear strength _Vopt_ by Equation (15) versus \(f_{c}\) and \(a\)/\(d\), respectively. Differently to the previous case which was characterized by a homogeneous database, herein there is neither a progressive decrease in the experimental dispersion nor a significant increase in the analytical dispersion as \(\beta\) rises. This is due to the fact that the value of the function \(f\) (\(\sigma_{Ylim}\)) will not tend towards zero while the parameter \(\beta\) increases, since the inhomogeneity of the database implies that the ratio \(\frac{\sigma_{Ylim}}{\mu}\) results in a value greater than 1. So, for this dataset, the best trade-off between experimental and analytical dispersions is observed for values of \(\beta\) close to 1. Furthermore, the maximum accuracy, equal to about 0.8, is already achieved for values of \(\beta\) close to 1. Therefore, for inhomogeneous datasets, it is convenient to select the model with a value of \(\beta\) close to 1 and a complexity as low as possible. In the models constructed, the unit used for the \(CS_{opt}\) and the \(CG\) is [MPa] while all the other input parameters are dimensionless. The highest accuracy together with acceptable experimental and analytical dispersions are achieved by the following model, corresponding to the parameters \(\delta=1\), \(\beta=50\), \(m=4\):  
\[CS_{opt} =0.129\cdot\left\{AS^{0.1}\cdot CG^{1.5}\cdot\left(\frac{W}{B} \right)^{-0.7}\cdot\left(\frac{B}{S}\right)^{0.6}\right\}+4.9187\left\{AS^{0.8 }\cdot CG^{-1.}\cdot\left(\frac{W}{B}\right)^{-1.3}\cdot\left(\frac{B}{S} \right)^{1.6}\right\}\] \[-0.9903\cdot\left\{AS^{-1.2}\cdot CG^{0.8}\cdot\left(\frac{W}{B} \right)^{-0.8}\cdot\left(\frac{B}{S}\right)^{0.7}\right\}+22.8565\left\{AS^{-0.1}\cdot CG^{-0.3}\cdot\left(\frac{W}{B}\right)^{1.5}\cdot\left(\frac{B}{S} \right)^{0.7}\right\} \tag{16}\] \[R_{N}{}^{2} =0.823\]  
Nevertheless, the best compromise between accuracy, complexity and analytical/experimental dispersions is obtained by the following models, characterized by the parameters \(\delta=1\), \(\beta=7\), \(m=1\), 2, respectively:  
\[CS_{opt} =0.33786\cdot\left\{AS^{0.2}\cdot CG^{0.9}\cdot\left(\frac{W}{B} \right)^{-1.4}\cdot\left(\frac{B}{S}\right)^{0.2}\right\}\qquad{R_{N}}^{2}=0. 704 \tag{17}\] \[CS_{opt} =-3.0173\cdot\left\{AS^{-1.4}\cdot CG^{0.6}\cdot\left(\frac{W}{B} \right)^{-0.3}\cdot\left(\frac{B}{S}\right)^{0.7}\right\}+0.35213\cdot\left\{ AS^{0.1}\cdot CG^{1.2}\cdot\left(\frac{W}{B}\right)^{-0.8}\right.\] \[\left.\cdot\left(\frac{B}{S}\right)^{0.5}\right\}\] (18) \[R_{N}{}^{2} =0.814\]  
Conversely, by choosing \(\delta=0\), the best solution is given by Equation (18), with \(m=2\):  
Figure 9: Problem 2. (**a**) Experimental dispersion versus analytical dispersion and iteration versus accuracy for \(m\) in the range 1 to 4 and for \(\delta=0\); (**b**) experimental versus predicted concrete strength for Equations (16)–(19) (results for the whole database). \[ \begin{array}{l}CS_{opt}=-0.0011\cdot\left\{AS^{-0.7}.CG^{1.7}.\left(\frac{W}{B} \right)^{-1.2}.\left(\frac{B}{S}\right)^{-0.9}\right\}+0.18981\cdot\left\{AS^{0.2}.CG^{1.2}\right.\\ \left.\left(\frac{W}{B}\right)^{-0.9}.\left(\frac{B}{S}\right)^{0.3}\right\} \hskip 113.811024 ptR_{N}{}^{2}=0.806\end{array}  \tag{19}\]  
Also in this case, the indicated accuracies refer to the whole database; for each model, Figure 9 depicts the relationship between experimental and estimated concrete strengths. It is important to underline that the proposed formulations allow some physical interpretations of the studied phenomenon.