**2. Novelty and Significance of This Study**

As reported in the introduction, the estimation of the buckling capacity of circular opening steel beams is important for the safety of structures under subjected loads. As instability is a complex (nonlinear) problem that is affected by various parameters, the determination of the critical buckling load remains challenge for researchers (engineers) in the fields of mechanics and civil engineering. Despite various experimental works having investigated this problem, it is not easy to derive a generalized expression that considers all the parameters that govern the instability of circular opening steel beams. To overcome this difficulty, the use of ML techniques, such as ANFIS optimized by the PSO algorithm proposed in this study, could be a good choice as a surrogate model. This soft computing method could help to explore the nonlinear relationships between the buckling capacity and the input variables, especially the geometrical parameters of the beams. In addition, the investigation of PSO parameters based on the Monte Carlo random sampling technique could contribute to better knowledge on selection of suitable parameters to achieve better performance with the PSO algorithm, which could be further recommended for other problems. Finally, the proposed ML-based model

of other numerical schemes (i.e., finite element models).

could be a potential tool for researchers or structural engineers in accurately estimating the buckling capacity of circular opening steel beams, which could (i) work within the ranges of values used in this study for the input variables and (ii) save time and costs in development of other numerical schemes (i.e., finite element models). The database in this study was obtained by analyzing 3645 different configurations of circular opening steel beams (Figure 1). It should be noted that the database was extracted from a validated finite element model, which was previously proposed in the literature by Abambres et al. [34]. It consisted of 8 input parameters, namely the length of the beam (denoted as L), the end opening

Materials 2020, 13, x FOR PEER REVIEW 4 of 27

ranges of values used in this study for the input variables and (ii) save time and costs in development

#### **3. Database Construction** distance (denoted as d0), circular opening diameter (denoted as D), the inter-opening distance (denoted as d), the height of the section (denoted as H), the thickness of the web (denoted as tweb), the

parameters.

3. Database Construction

The database in this study was obtained by analyzing 3645 different configurations of circular opening steel beams (Figure 1). It should be noted that the database was extracted from a validated finite element model, which was previously proposed in the literature by Abambres et al. [34]. It consisted of 8 input parameters, namely the length of the beam (denoted as L), the end opening distance (denoted as d0), circular opening diameter (denoted as D), the inter-opening distance (denoted as d), the height of the section (denoted as H), the thickness of the web (denoted as tweb), the width of the flange (denoted as wflange), the thickness of the flange (denoted as tflange), and the buckling capacity, which was considered as the target variable (denoted as Pu). It should be pointed out that the database was generated for one material type (with a typical Young's modulus of 210 GPa and Poisson's ratio of 0.3). The results of the statistical analysis of the Pu and the corresponding influential parameters are presented in Table 1. width of the flange (denoted as wflange), the thickness of the flange (denoted as tflange), and the buckling capacity, which was considered as the target variable (denoted as Pu). It should be pointed out that the database was generated for one material type (with a typical Young's modulus of 210 GPa and Poisson's ratio of 0.3). The results of the statistical analysis of the Pu and the corresponding influential parameters are presented in Table 1. The input and target variables in this work were scaled in the range of [0, 1] to minimize the numerical bias of the dataset. After performing the simulation part, a transformation into the normal range was conducted to better interpret the obtained results. Concerning the development phase, the dataset was split into two parts, namely the training part (70% of the total data) and the testing part (the remaining 30% of the data), which served as the learning and validation phases of the proposed ANFIS-PSO model, respectively.

Figure 1. Diagram of circular opening steel beam under uniform loading and its geometrical **Figure 1.** Diagram of circular opening steel beam under uniform loading and its geometrical parameters.

**Table 1.** Initial statistical analysis of the dataset.


Mean 6.0 265.4 383.6 112.51 560.00 12.0 216.0 20.0 225.7 <sup>a</sup> Median. <sup>b</sup> Standard deviation. <sup>c</sup> Coefficient of variation (%).

The input and target variables in this work were scaled in the range of [0, 1] to minimize the numerical bias of the dataset. After performing the simulation part, a transformation into the normal range was conducted to better interpret the obtained results. Concerning the development phase, the dataset was split into two parts, namely the training part (70% of the total data) and the testing part (the remaining 30% of the data), which served as the learning and validation phases of the proposed ANFIS-PSO model, respectively.
