## 2 Materials and Methods
### Input and Output Variables  
Based on results from previous studies [3; 47], the following three input variables were used to characterize each RC beam and predict the torsional capacity \(T_{R\_Exp}\): the cross-section area \(A_{c}\) (area enclosed within the outer perimeter of the cross-section), the average concrete compressive strength \(f_{c}\), and the reinforcement factor \(A_{1}f_{ly}\frac{A_{tfy}}{s}\) (factor accounting for the ratio between the yielding forces of both the longitudinal and transverse reinforcements). To compute the last input variable, the units of each parameter are the same as the ones given in Table 1 in Appendix A. The reinforcement factor combines both the ratio of the longitudinal to the transverse torsional reinforcement, as well as the ratio of their yielding stresses. Hence, it can be used to characterize beams with balanced (which is a usual design criterion for pure torsion) or unbalanced (for which the reinforcement factor is different from unity) yielding forces of the longitudinal and transverse reinforcements. For this reason, the authors considered that there is no need to consider separately the effects of longitudinal and transverse reinforcement in developing the ML models in this study, as it was carried out in previous studies using ANN ML models (for instance, [40; 41; 43; 48]) which usually require more input variables. The reinforcement factor was firstly proposed by [PERSON] [1] as an input variable and further used successfully in recent studies from the  
Figure 2: Histograms of key parameters for the RC beams: (**a**) total reinforcement ratio, (**b**) concrete compressive strength, (**c**) yielding stress of longitudinal reinforcement, (**d**) yielding stress of transverse reinforcement. authors [3; 47] using nonlinear regression and MP5 models to predict the torsional capacity of RC beams under pure torsion. Figure 3 presents the histograms of the input variables (the concrete compressive strength was added again), as well as the output variable (torsional capacity), for the 202 reference RC beams from the database. Figure 3 shows that for most of the beams, the reinforcement factor ranges between \(1.1\times 10^{6}\) and \(2\times 10^{8}\), and the torsional capacity ranges between 9.0 kNm and 200 kNm. As it can be can noticed from Figures 2 and 3, some of the predictors and outcome variables do not obey the normal distribution curve. However, DT and ET methods do not require feature scaling to be performed as they are not sensitive to the variance in the data [55], as previously mentioned in Section 1. For this reason, data normalization was not performed in this study. Following a previous study by [PERSON] et al. [32], the correlation between each pair of parameters is also important to be analyzed. Figure 4 and Table 1 show the scatter plot and the Pearson correlation coefficient between pairs of the inputs and the output variables, respectively. Both Figure 4 and Table 1 show that there is a strong correlation between some of the input variables, namely the cross-section area and the reinforcement factor, and the output, whereas there is a moderate correlation between the concrete compressive strength and the output. In general, low degrees of correlation exist between pairs of input variables, indicating that these will not cause multicollinearity problems on the models [69]. A high absolute value of the Pearson correlation coefficient between pairs of inputs could  
Figure 3: Histograms of input and output variables: (**a**) cross-section area, (**b**) concrete compressive strength, (**c**) reinforcement factor, and (**d**) torsional capacity. affect the accuracy of the model and the interpretation of the effects of the inputs on the output [33].