**Parametric Investigation of Particle Swarm Optimization to Improve the Performance of the Adaptive Neuro-Fuzzy Inference System in Determining the Buckling Capacity of Circular Opening Steel Beams**

**Quang Hung Nguyen 1,\* , Hai-Bang Ly 2,\* , Tien-Thinh Le 3,\* , Thuy-Anh Nguyen <sup>2</sup> , Viet-Hung Phan <sup>4</sup> , Van Quan Tran <sup>2</sup> and Binh Thai Pham <sup>2</sup>**


Received: 20 April 2020; Accepted: 6 May 2020; Published: 12 May 2020

**Abstract:** In this paper, the main objectives are to investigate and select the most suitable parameters used in particle swarm optimization (PSO), namely the number of rules (nrule), population size (npop), initial weight (wini), personal learning coefficient (c1), global learning coefficient (c2), and velocity limits (fv), in order to improve the performance of the adaptive neuro-fuzzy inference system in determining the buckling capacity of circular opening steel beams. This is an important mechanical property in terms of the safety of structures under subjected loads. An available database of 3645 data samples was used for generation of training (70%) and testing (30%) datasets. Monte Carlo simulations, which are natural variability generators, were used in the training phase of the algorithm. Various statistical measurements, such as root mean square error (RMSE), mean absolute error (MAE), Willmott's index of agreement (IA), and Pearson's coefficient of correlation (R), were used to evaluate the performance of the models. The results of the study show that the performance of ANFIS optimized by PSO (ANFIS-PSO) is suitable for determining the buckling capacity of circular opening steel beams, but is very sensitive under different PSO investigation and selection parameters. The findings of this study show that nrule = 10, npop = 50, wini = 0.1 to 0.4, c<sup>1</sup> = [1, 1.4], c<sup>2</sup> = [1.8, 2], f<sup>v</sup> = 0.1, which are the most suitable selection values to ensure the best performance for ANFIS-PSO. In short, this study might help in selection of suitable PSO parameters for optimization of the ANFIS model.

**Keywords:** particle swarm parameters; adaptive neuro-fuzzy inference system; circular opening steel beams; buckling capacity

### **1. Introduction**

Circular opening steel beams have been increasingly acknowledged in structural engineering because of their many remarkable advantages [1], including their ability to bridge the span of a large aperture or their lighter weight compared with conventional steel beams. In general, the industrial approach to producing such a structural member is the rolled method, involving a single steel piece. This is then cut so that the two halves can be assembled, making an I-section, which is also called an H-section steel beam. Hoffman et al. [2] showed that the flexural stiffness and specific gravity per unit

length was improved significantly in circular opening steel beams structures. In addition, economic and aesthetics factors are also beneficial points that deserve significant attention [3,4]. A typical structural member has a regular circular openings along its length [1–8], and is about 40–60% deeper and 40–60% stronger than a regular I-section [5,6]. Because of these advantages, circular beams are not only used in lightweight or large-span structures, but are also used for other complex civil engineering structures, such as bridges [9]. Due to the possibility of using circular opening steel beams in various engineering applications, investigation of the failure behavior is crucial to ensure the safety of structures. Several previously published studies on the failure modes of circular beams, for instance the work by Sonck et al. [3], have shown that the web openings are the leading causes of the complex failure behavior of cellular beams, including web post-buckling (WPB), the Vierendeel mechanism (VM), rupture of the web post-weld [1], local web buckling (LWB), and web distortional buckling (WDB) [5,6].

Miscellaneous analysis-related research studies have been conducted to study the behavior of circular opening steel beams [10–12], which have mainly focused on the web openings using various numerical approaches [7,9]. As an example, Chung et al. [11] used finite element models with material and geometrical nonlinearity to calculate the behavior of circular beams, resulting in approximately 15.8% of error. Numerical methods help create various case studies in order to gain more knowledge about the working principles of the structures. Taking the work of Panedpojaman and Thepchatri [4] as an example, the authors created a total of 408 nonlinear finite element models using ANSYS software to investigate the behavior of circular steel beams. The results indicated that there is always a small difference between the finite element model and the theoretical formulation. In another study, Sonck et al. [3] generated 597 numerical models, which were calibrated with laboratory tests for 14 geometrically different full-scale steel cellular beams and verified with 1948 numerical analyzes. The results showed that the experimental and numerical curves were identical, with a maximum load gap range of 5.1% to 6.5%. Typically, the numerical models are useful for evaluating the behavior of circular beams [1,3,6,9,13]. However, these model require much effort and the use of modern software and equipment.

Machine learning (ML) algorithms, a branch of artificial intelligence (AI) techniques, have been constantly developed during the past few decades due to the significant increase in computer science [14–21]. Various ML models have been effectively implemented to solve countless specific engineering problems, including in material sciences [22–24], geotechnical engineering [25–29], and especially structural engineering [18,30–32]. As an example, Vahid et al. [33] selected an artificial neural network (ANN) algorithm, the most popular ML model, to predict the shear capacity of a web opening steel I-beam. The proposed ANN model had better accuracy compared with other existing formulas or theoretical predictions derived from the ACI 318-08 standard. Abambres et al. [34] also used the ANN method to investigate the buckling load capacity of cellular beams under uniformly distributed vertical loads, using eight geometrical parameters. Good results were achieved by the ANN, giving 3.7% for the total error and 0.4% for the average relative error. Blachowski and Pnevmatikos [35] proposed an ANN model for the design and control of the vibration of structural elements under earthquake loading. In the same context of seismic excitation, Pnevmatikos and Thomos [36] employed a stochastic control approach to determine the influence of random characters on the dynamic behavior of engineering structures. The neuro-fuzzy system is another efficient ML algorithm, which has been employed in many structural and material engineering applications, including for steel structures. Seitllari and Naser [37] investigated the performance of an adaptive neuro-fuzzy inference system (ANFIS) in predicting a fire-induced spalling phenomenon in steel-reinforced concrete structures. Naser [38] derived a material model for steel structures, taking into account the dependency of temperature based on machine learning techniques. Basarir et al. [39] compared the performance between conventional regression techniques and ANFIS in predicting the ultimate pure bending of concrete-filled steel tubular members. Naderpour and Mirrashid [40] used ANFIS to predict the shear strength of beams that had been reinforced with steel stirrups. Mermerda¸s et al. [41] applied ANFIS

to evaluate the flexural behavior of steel circular hollow section (CHS) beams. It was stated that the ANFIS was a promising tool for quick and accurate evaluation of the mechanical behavior of steel-based engineering structures.

In general, the ML algorithms are excellent and effective for evaluating the behavior of structural members, including circular beams. However, their performance depends significantly on the selection of parameters used to learn the models [42]. Therefore, the process of determining such parameters is crucial to obtain highly reliable and accurate prediction results. Concerning the ANN, many parameters could be involved, such as the initial weights, biases to start the training phase, the learning rate, the stopping criterion, the choice of features in the training phase, the choice of the splitting dataset ratio, the number of hidden layers and the corresponding activation functions, the training algorithm, and the number of neurons in each hidden layer [43–45]. Considering the ANFIS, two groups of parameters can be considered, namely the nonlinear parameters of the antecedent membership function (MF) and linear parameters of the consequent MF, which depends on the partitioning of the fuzzy space, as well as the type of Sugeno model [46,47]. Besides, many optimization techniques, such as particle swarm optimization (PSO), differential evolution (DE), evolutionary algorithm (EA), genetic algorithm (GA), artificial bee colony (ABC). or cuckoo search (CS) techniques, have been proposed to optimize the parameters of the ML models [48,49]. Each optimization technique also possesses many different parameters that need to be tuned to obtain good prediction performances, inducing the time required to adjust the combination of these parameters [48,49]. Among the well-known optimization techniques, PSO is considered as one of the most popular and effective techniques [50]. Many hybrid ML algorithms have used PSO for the parameter tuning process, including ANN, ANFIS, and Support Vector Machine (SVM) algorithms [51–53]. In the literature, limited studies have used ANFIS optimized by PSO (ANFIS-PSO) to predict the mechanical properties of structural members. Moreover, a systematic investigation of ANFIS-PSO parameters under random sampling has not been performed, as the sampling method has been proven to greatly affect the accuracy of the ML algorithms [54].

In this study, the main purpose was to carry out a parametric investigation of PSO parameters to improve the performance of ANFIS in predicting the buckling capacity of circular opening steel beams, which is an important mechanical property that is crucial for the safety of structures under subjected loads. The database used in this work consisted of 3645 data samples, which were derived from numerical results using ANSYS and available in the literature. The parametric studies were carried out with the help of Monte Carlo simulations, which are natural variability generators, in the training phase of the algorithm. Various statistical measurements, such as the root mean square error (RMSE), mean absolute error (MAE), Willmott's index of agreement (IA), and Pearson's coefficient of correlation (R), were used to evaluate the performance of the model.
