# Practical Hybrid Machine Learning Approach for Estimation of Ultimate Load of Elliptical Concrete-Filled Steel Tubular Columns under Axial Loading

Authors: Tien-Thinh Le
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

In this study, a hybrid machine learning (ML) technique was proposed to predict the bearing capacity of elliptical CFST columns under axial load. The proposed model was Adaptive Neurofuzzy Inference System (ANFIS) combined with Real Coded Genetic Algorithm (RCGA), denoted as RCGA-ANFIS. The evaluation of the model was performed using the coefficient of determination (R2) and root mean square error (RMSE). The results showed that the RCGA-ANFIS (R2 = 0.974) was more reliable and effective than conventional gradient descent (GD) technique (R2 = 0.952). The accuracy of the present work was found superior to the results published in the literature (R2 = 0.776 or 0.768) when predicting the load capacity of elliptical CFST columns. Finally, sensitivity analysis showed that the thickness of the steel tube and the minor axis length of the elliptical cross section were the most influential parameters. For practical application, a Graphical User Interface (GUI) was developed in MATLAB for researchers and engineers and to support the teaching and interpretation of the axial behavior of CFST columns.


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## Body

## 1. Introduction

In recent decades, composite concrete-filled steel tubular (CFST) columns are considerably employed in the construction of infrastructures thanks to their excellent structural behavior [1]. These structural members exhibit many benefits than single material columns (i.e., concrete columns or hollow steel columns). These advantages could be listed as fire, axial capacity, and earthquake resistance [2, 3]. In practical engineering, various cross section geometries of CFST columns have been considered, such as circular [4], square [5], or rectangular cross sections [6]. Recently, the elliptical cross section was adopted in several works [3, 7, 8]. Indeed, the use of elliptical CFST columns has gained attention from the scientific community and applied engineering as it provides specific advantages compared to other cross sections of CFST, including a better strength and rigidity as well as fire resistance [9]. Due to its reasonable distribution of the major-minor axis, elliptical CFST column exhibits a better architectural aesthetic appearance and a small fluid resistance coefficient [10, 11]. Moreover, the prevention of local buckling in the elliptical CFST columns could be well-established thanks to the concrete core [12, 13]. The elliptical section possesses aesthetic qualities along with more effective bending resistance when compared to circular section due to having different second moments of area around its principal axes [14]. Therefore, analyzing the structural behavior, especially the ultimate load of elliptical CFST columns, is essential to facilitate the use in civil engineering structures.However, there are currently no standards or codes in any countries for assessing the load-carrying capacity of elliptical CFST columns [15]. Besides, there were several empirical formulations in the available literature such as Liu and Zha 2011 [16] and Shen et al. [17] for predicting the ultimate load of elliptical CFST members. However, these equations were derived using assumptions and experimental observations, which led to a simplification of the prediction model. Consequently, the application of these models could not be extended to other results. All these limit the application of elliptical CFST columns in engineering practice. Although previous studies provided significant contributions to the progress in modeling and prediction of axial behavior of CFST members, a more robust and efficient model should be developed to reduce the cost and time consumed in experiments and field works.Recently, machine learning (ML) approaches have been employed in various mechanical and civil engineering applications [18, 19], particularly for structural members under compression [20, 21]. As an example, Sarir et al. [22] proposed a tree-based and whale optimization model for predicting the load-bearing capacity of circular CFST members. Besides, Ahmadi et al. [23, 24] applied an artificial neural network for predicting the axial capacity of circular CFST short columns. In another work, Tran et al. [25] developed a neural network-based model for predicting the load-bearing capacity of square CFST columns. The obtained results in the literature demonstrated that ML methods have a very promising potential for predicting the mechanical behavior of structural elements. Despite the importance of elliptical CFST columns, most ML-based studies focused on circular and square cross sections [22, 26, 27]. Therefore, more investigations should be carried out to assess the potential applications of ML-based models for studying the axial behavior of elliptical CFST columns.Therefore, the primary objective of the present work was to develop an ML-based model to predict the ultimate load of elliptical CFST columns under axial loading. For this purpose, a hybrid ML model, namely Adaptive Neurofuzzy Inference System (ANFIS) combined with Real Coded Genetic Algorithm (RCGA), was developed. The RCGA was chosen because of its higher optimization capability than the conventional gradient descent (GD) technique, as highlighted in this study. As the present work mainly focused on elliptical CFST columns, the input data included the length of the column, the major and minor axis lengths of the elliptical cross section, the thickness of the steel tube, and the mechanical properties of steel and concrete (i.e., yield strength and compressive strength, respectively). In order to train and validate the developed hybrid ML model, statistical quality assessments such as coefficient of determination (R2) and root mean squared error (RMSE) were employed. Monte Carlo simulations were also carried out in order to estimate the robustness of the proposed ML model. A sensitivity analysis was conducted to investigate the influence of input variables on the prediction results. The prediction capacity of the RCGA-ANFIS model was also compared with existing equations in the literature for estimating the ultimate load of elliptical CFST columns. Finally, a Graphical User Interface (GUI) based on the developed ML model was provided, aiming at quick and efficient estimation of the ultimate load of elliptical CFST columns.


## 2. Materials and Methods

### 2.1. Database

In this work, a database was constructed by extracting available datasets from experimental research of Uenaka [28], Yang et al. [29], Liu et al. [30], Ren et al. [12], Dai et al. [31], Jamaluddin et al. [32], Yang et al. [33], McCann et al. [34], and Zhao and Packer [35]. From these investigations, a total number of 94 configurations were collected and summarized (Table 1), including the number of data points and proportion (in %). As revealed in the literature, the experimental procedure was conducted following the steps below:(i)
Design of specimens(ii)
Manufacturing of steel tube(iii)
Manufacturing of concrete core(iv)
Assembly of composite columns(v)
Loading and measurement (see Figure1 for a schematic description of the test as well as geometrical parameters of the members)Table 1
Organization of database.
Source of dataNumber of data pointsProportion of data (%)Uenaka [28]1920.2Yang et al. [29]22.1Liu et al. [30]1819.1Ren et al. [12]66.4Dai et al. [31]1313.8Jamaluddin et al. [32]1718.1Yang et al. [33]99.6McCann et al. [34]22.1Zhao and Packer [35]88.5Total94100Figure 1
Schematization for (a) the CFST columns under axial loading, (b) the elliptical cross section, and (c) the load-axial shortening curve (a drawing based on experimental curves of Uenaka [28]).
(a)(b)(c)In terms of the experimental studies, various geometrical parameters, as well as mechanical properties of the constituent materials, were considered in order to test the failure of elliptical CFST columns under axial compression. For that reason, the input parameters of the problem regarding the geometry were the length of the column (denoted byL), the major axis length of the elliptical cross section (denoted by D), the minor axis length of the elliptical cross section (denoted by d), and the thickness of the steel tube (denoted by δ). Regarding the mechanical properties of constituent materials, the yield strength of the steel tube (denoted by fy) and the compressive strength of the filled concrete (denoted by fc′) were considered. The ultimate load of the column under axial compression was the output of the problem, denoted by Qn. A primarily statistical analysis of the database is indicated in Table 2, including the min, average, max, standard deviation (StD), and coefficient of variation (CV) values of all variables. It should be noticed that several statistical correlation techniques such as Principal Component Analysis [36] were applied, and no significant correlations were found in the input space. This confirmed that, for the prediction problem, all input parameters in this study were independent, and the selection of inputs was relevant. Finally, all data were scaled into the range of [−1, 1] in order to minimize numerical bias in the training phase.Table 2
Initial statistical analysis of the database.
ParameterUnitNotationMinAverageMaxStDCV (%)Length of columnmmL160991.863600923.90893.1Major axis length of cross sectionmmD136.5177.281318.535.98620.3Minor axis length of cross sectionmmd63.193.69315521.46622.9Thickness of steel tubemmδm13.8549.721.67943.6Yield strength of steel tubeMPafy201360.657439.359.37816.5Compressive strength of concreteMPafc′13.1848.638102.2620.84342.9Ultimate loadkNQn413.31130.4622607484.16442.8


### 2.2. Methods Used

#### 2.2.1. Adaptive Neurofuzzy Inference System (ANFIS)

The Adaptive Neurofuzzy Inference System, referred to as ANFIS, is an ML model constructed from the combination between a set of fuzzy if-then rules and the fuzzy inference systems through an adaptive network [37, 38]. The main idea of ANFIS is to construct a set of fuzzy if-then rules, including suitable membership functions to create the stipulated output and input variables [39, 40]. Supposing that the ANFIS model has two input variables such as X and Y and one output variable such as Z, we apply the following Takagi and Sugeno’s if-then rules [41, 42]:(1)IfXisA1andYisB1,thenZ1=a1X+b1Y+c1rule1;IfXisA2andYisB2,thenZ2=a2X+b2Y+c2rule2.Here, A and B are linguistic labels characterized by appropriate membership functions, and a,b, and c are the linear output parameters.Consider the above ANFIS model with two input variablesX and Y. Its structure can be divided into five main layers as follows [43]:Layer 1: each node in this layer corresponds to a node function, which can be chosen to be bell-shaped with a minimum value equal to 0 and a maximum value equal to 1, for example, the Gaussian function, such that(2)μAix=exp−x−aibi2,wherex is problem input and ai,bi are input parameters.In fact, any continuous and differentiable functions can be chosen for the nodes in this layer.Layer 2: each node in this layer is a node function that multiplies the incoming inputs and sends the results to the next layer:(3)wi=μCi1x1×μCi2x2×⋯×μCinxn.Layer 3: each node in this layer computes the ratio between theith rule’s firing strength and the sum of all rules’ firing strength:(4)w¯i=wi∑k=1nwk.Layer 4: each node in this layer is a node function chosen such that(5)fi=w¯ic0+∑k=1nckXk.Layer 5: the circle node in this layer calculates the sum of all incoming results and exports as the overall output(6)Overall ouput=∑iw¯ifi.The training algorithm uses a combination of the least-squares and backpropagation gradient descent methods to model the training dataset [44].


#### 2.2.2. Real Coded Genetic Optimization Algorithm

Real Coded Genetic Algorithm, referred to as RCGA, is a metaheuristic optimization technique which is inspired by the principles of biological evolution. The basic idea of RCGA is to move a population of chromosomes, which are composed of strings of ones and zeros (or genes), to a new one that performs better than the old one [45]. There are two primary operations in RCGA, which are crossover and mutation [46, 47]. Crossover is a phase where the chromosomes in the population randomly share their features. This is the most significant operation in the RCGA, as more powerful offspring are created taking useful features from their parent’s genes. Mutation is a process that is operated within each offspring, meaning that some of the bits in the bit string can be flipped. The main objective of the mutation process is to maintain the diversity of the population after new offspring are created from crossover [48].The RCGA can be divided into five main steps as follows [48, 49]:(i)
Initial population. In this step, a set of chromosomes called population is defined. Each individual of the population corresponds to a solution of the considered problem. Each chromosome is formed by joining genes into a string. Typically, chromosomes are composed of strings of ones and zeros.(ii)
Fitness function. In this step, the fitness score of each individual in the population is calculated. It defines how to fit the chromosome or the ability of that chromosome to compete with others. A higher fitness score means that the individual is more likely to be reproduced.(iii)
Selection. In this step, the chromosomes with the highest values of fitness score will be selected in order to share their features in the next step.(iv)
Crossover. In this step, the crossover process will be operated on the most fitting chromosomes. Their genes are randomly exchanged to create new offspring.(v)
Mutation. In this step, the mutation process is done within each individual offspring to maintain the diversity of the population.The algorithm is terminated when the model has converged, meaning that the newly created offspring are not different from the previous ones. In the literature, RCGA was used mainly in hybrid ML approaches [49]. For instance, Kim and Shin [50] used a hybrid approach based on neural networks and genetic algorithms for detecting temporal patterns, Le et al. [51] in steel structures applications, or Yan et al. [52] for engineering design problems. Finally, a complete review of the RCGA technique could be found in Lee [53].


#### 2.2.3. Random Sampling Technique: Monte Carlo Method

The main idea of the Monte Carlo method is that the output is computed by repeating the sampling of variables randomly from the input space [54–56]. That way, (i) the Monte Carlo method is widely applied in order to propagate the variability of inputs on the output response; (ii) based on statistical analysis of output, several posttreatments such as robustness and/or sensitivity analyses could be thoroughly achieved [57] (see Figure 2 for a typical statistical problem using the Monte Carlo method). As shown in Figure 2, each input exhibits a probability distribution describing its variability. Due to the variabilities of input variables, the response also exhibits its statistical behaviors, which are necessary to be characterized [58]. The robustness of the model and/or sensitivity of input variables could then be deduced based on statistical analysis of output response [59–61].Figure 2
Monte Carlo simulation taking into account variability in the input space for robustness and sensitivity analysis.Using Monte Carlo simulation, the bigger the number of realizations, the higher the reliability of the response archived. In this work, in order to optimize the number of Monte Carlo runs, a statistical estimator of convergence was applied, such as [62–65](7)NMC↦INMC=1W¯1NMC∑i=1NMCWi,where W¯ is the mean of the considered variable W and NMC is the number of Monte Carlo runs.


### 2.3. Quality Assessment Criteria

In the present work, statistical criteria, namely, the coefficient of determination (R2) and Root Mean Squared Error (RMSE), have been used in order to validate and test the developed ML model. The R2 allows us to identify the statistical relationship between two data points. This measurement of the linear correlation yields a value between 0 and 1 inclusively, where 0 is no correlation and 1 is total correlation. R2 could be calculated using the following equation [66, 67]:(8)R2=∑k=1Npk−p¯wk−w¯2∑k=1Npk−p¯2∑k=1Nwk−w¯2,where N is the number of the observations, pk and p¯ are predicted and mean predicted values, and wk and w¯ are measured and mean measured values of ultimate load, respectively (k=1:N). The formulation of RMSE is described by the following equation [68–70]:(9)RMSE=1N∑k=1Npk−wk2.Finally, the slope criterion is defined, such as the slope of the linear regression fit between predicted and observed vectors.


## 2.1. Database

In this work, a database was constructed by extracting available datasets from experimental research of Uenaka [28], Yang et al. [29], Liu et al. [30], Ren et al. [12], Dai et al. [31], Jamaluddin et al. [32], Yang et al. [33], McCann et al. [34], and Zhao and Packer [35]. From these investigations, a total number of 94 configurations were collected and summarized (Table 1), including the number of data points and proportion (in %). As revealed in the literature, the experimental procedure was conducted following the steps below:(i)
Design of specimens(ii)
Manufacturing of steel tube(iii)
Manufacturing of concrete core(iv)
Assembly of composite columns(v)
Loading and measurement (see Figure1 for a schematic description of the test as well as geometrical parameters of the members)Table 1
Organization of database.
Source of dataNumber of data pointsProportion of data (%)Uenaka [28]1920.2Yang et al. [29]22.1Liu et al. [30]1819.1Ren et al. [12]66.4Dai et al. [31]1313.8Jamaluddin et al. [32]1718.1Yang et al. [33]99.6McCann et al. [34]22.1Zhao and Packer [35]88.5Total94100Figure 1
Schematization for (a) the CFST columns under axial loading, (b) the elliptical cross section, and (c) the load-axial shortening curve (a drawing based on experimental curves of Uenaka [28]).
(a)(b)(c)In terms of the experimental studies, various geometrical parameters, as well as mechanical properties of the constituent materials, were considered in order to test the failure of elliptical CFST columns under axial compression. For that reason, the input parameters of the problem regarding the geometry were the length of the column (denoted byL), the major axis length of the elliptical cross section (denoted by D), the minor axis length of the elliptical cross section (denoted by d), and the thickness of the steel tube (denoted by δ). Regarding the mechanical properties of constituent materials, the yield strength of the steel tube (denoted by fy) and the compressive strength of the filled concrete (denoted by fc′) were considered. The ultimate load of the column under axial compression was the output of the problem, denoted by Qn. A primarily statistical analysis of the database is indicated in Table 2, including the min, average, max, standard deviation (StD), and coefficient of variation (CV) values of all variables. It should be noticed that several statistical correlation techniques such as Principal Component Analysis [36] were applied, and no significant correlations were found in the input space. This confirmed that, for the prediction problem, all input parameters in this study were independent, and the selection of inputs was relevant. Finally, all data were scaled into the range of [−1, 1] in order to minimize numerical bias in the training phase.Table 2
Initial statistical analysis of the database.
ParameterUnitNotationMinAverageMaxStDCV (%)Length of columnmmL160991.863600923.90893.1Major axis length of cross sectionmmD136.5177.281318.535.98620.3Minor axis length of cross sectionmmd63.193.69315521.46622.9Thickness of steel tubemmδm13.8549.721.67943.6Yield strength of steel tubeMPafy201360.657439.359.37816.5Compressive strength of concreteMPafc′13.1848.638102.2620.84342.9Ultimate loadkNQn413.31130.4622607484.16442.8


## 2.2. Methods Used

### 2.2.1. Adaptive Neurofuzzy Inference System (ANFIS)

The Adaptive Neurofuzzy Inference System, referred to as ANFIS, is an ML model constructed from the combination between a set of fuzzy if-then rules and the fuzzy inference systems through an adaptive network [37, 38]. The main idea of ANFIS is to construct a set of fuzzy if-then rules, including suitable membership functions to create the stipulated output and input variables [39, 40]. Supposing that the ANFIS model has two input variables such as X and Y and one output variable such as Z, we apply the following Takagi and Sugeno’s if-then rules [41, 42]:(1)IfXisA1andYisB1,thenZ1=a1X+b1Y+c1rule1;IfXisA2andYisB2,thenZ2=a2X+b2Y+c2rule2.Here, A and B are linguistic labels characterized by appropriate membership functions, and a,b, and c are the linear output parameters.Consider the above ANFIS model with two input variablesX and Y. Its structure can be divided into five main layers as follows [43]:Layer 1: each node in this layer corresponds to a node function, which can be chosen to be bell-shaped with a minimum value equal to 0 and a maximum value equal to 1, for example, the Gaussian function, such that(2)μAix=exp−x−aibi2,wherex is problem input and ai,bi are input parameters.In fact, any continuous and differentiable functions can be chosen for the nodes in this layer.Layer 2: each node in this layer is a node function that multiplies the incoming inputs and sends the results to the next layer:(3)wi=μCi1x1×μCi2x2×⋯×μCinxn.Layer 3: each node in this layer computes the ratio between theith rule’s firing strength and the sum of all rules’ firing strength:(4)w¯i=wi∑k=1nwk.Layer 4: each node in this layer is a node function chosen such that(5)fi=w¯ic0+∑k=1nckXk.Layer 5: the circle node in this layer calculates the sum of all incoming results and exports as the overall output(6)Overall ouput=∑iw¯ifi.The training algorithm uses a combination of the least-squares and backpropagation gradient descent methods to model the training dataset [44].


### 2.2.2. Real Coded Genetic Optimization Algorithm

Real Coded Genetic Algorithm, referred to as RCGA, is a metaheuristic optimization technique which is inspired by the principles of biological evolution. The basic idea of RCGA is to move a population of chromosomes, which are composed of strings of ones and zeros (or genes), to a new one that performs better than the old one [45]. There are two primary operations in RCGA, which are crossover and mutation [46, 47]. Crossover is a phase where the chromosomes in the population randomly share their features. This is the most significant operation in the RCGA, as more powerful offspring are created taking useful features from their parent’s genes. Mutation is a process that is operated within each offspring, meaning that some of the bits in the bit string can be flipped. The main objective of the mutation process is to maintain the diversity of the population after new offspring are created from crossover [48].The RCGA can be divided into five main steps as follows [48, 49]:(i)
Initial population. In this step, a set of chromosomes called population is defined. Each individual of the population corresponds to a solution of the considered problem. Each chromosome is formed by joining genes into a string. Typically, chromosomes are composed of strings of ones and zeros.(ii)
Fitness function. In this step, the fitness score of each individual in the population is calculated. It defines how to fit the chromosome or the ability of that chromosome to compete with others. A higher fitness score means that the individual is more likely to be reproduced.(iii)
Selection. In this step, the chromosomes with the highest values of fitness score will be selected in order to share their features in the next step.(iv)
Crossover. In this step, the crossover process will be operated on the most fitting chromosomes. Their genes are randomly exchanged to create new offspring.(v)
Mutation. In this step, the mutation process is done within each individual offspring to maintain the diversity of the population.The algorithm is terminated when the model has converged, meaning that the newly created offspring are not different from the previous ones. In the literature, RCGA was used mainly in hybrid ML approaches [49]. For instance, Kim and Shin [50] used a hybrid approach based on neural networks and genetic algorithms for detecting temporal patterns, Le et al. [51] in steel structures applications, or Yan et al. [52] for engineering design problems. Finally, a complete review of the RCGA technique could be found in Lee [53].


### 2.2.3. Random Sampling Technique: Monte Carlo Method

The main idea of the Monte Carlo method is that the output is computed by repeating the sampling of variables randomly from the input space [54–56]. That way, (i) the Monte Carlo method is widely applied in order to propagate the variability of inputs on the output response; (ii) based on statistical analysis of output, several posttreatments such as robustness and/or sensitivity analyses could be thoroughly achieved [57] (see Figure 2 for a typical statistical problem using the Monte Carlo method). As shown in Figure 2, each input exhibits a probability distribution describing its variability. Due to the variabilities of input variables, the response also exhibits its statistical behaviors, which are necessary to be characterized [58]. The robustness of the model and/or sensitivity of input variables could then be deduced based on statistical analysis of output response [59–61].Figure 2
Monte Carlo simulation taking into account variability in the input space for robustness and sensitivity analysis.Using Monte Carlo simulation, the bigger the number of realizations, the higher the reliability of the response archived. In this work, in order to optimize the number of Monte Carlo runs, a statistical estimator of convergence was applied, such as [62–65](7)NMC↦INMC=1W¯1NMC∑i=1NMCWi,where W¯ is the mean of the considered variable W and NMC is the number of Monte Carlo runs.


## 2.2.1. Adaptive Neurofuzzy Inference System (ANFIS)

The Adaptive Neurofuzzy Inference System, referred to as ANFIS, is an ML model constructed from the combination between a set of fuzzy if-then rules and the fuzzy inference systems through an adaptive network [37, 38]. The main idea of ANFIS is to construct a set of fuzzy if-then rules, including suitable membership functions to create the stipulated output and input variables [39, 40]. Supposing that the ANFIS model has two input variables such as X and Y and one output variable such as Z, we apply the following Takagi and Sugeno’s if-then rules [41, 42]:(1)IfXisA1andYisB1,thenZ1=a1X+b1Y+c1rule1;IfXisA2andYisB2,thenZ2=a2X+b2Y+c2rule2.Here, A and B are linguistic labels characterized by appropriate membership functions, and a,b, and c are the linear output parameters.Consider the above ANFIS model with two input variablesX and Y. Its structure can be divided into five main layers as follows [43]:Layer 1: each node in this layer corresponds to a node function, which can be chosen to be bell-shaped with a minimum value equal to 0 and a maximum value equal to 1, for example, the Gaussian function, such that(2)μAix=exp−x−aibi2,wherex is problem input and ai,bi are input parameters.In fact, any continuous and differentiable functions can be chosen for the nodes in this layer.Layer 2: each node in this layer is a node function that multiplies the incoming inputs and sends the results to the next layer:(3)wi=μCi1x1×μCi2x2×⋯×μCinxn.Layer 3: each node in this layer computes the ratio between theith rule’s firing strength and the sum of all rules’ firing strength:(4)w¯i=wi∑k=1nwk.Layer 4: each node in this layer is a node function chosen such that(5)fi=w¯ic0+∑k=1nckXk.Layer 5: the circle node in this layer calculates the sum of all incoming results and exports as the overall output(6)Overall ouput=∑iw¯ifi.The training algorithm uses a combination of the least-squares and backpropagation gradient descent methods to model the training dataset [44].


## 2.2.2. Real Coded Genetic Optimization Algorithm

Real Coded Genetic Algorithm, referred to as RCGA, is a metaheuristic optimization technique which is inspired by the principles of biological evolution. The basic idea of RCGA is to move a population of chromosomes, which are composed of strings of ones and zeros (or genes), to a new one that performs better than the old one [45]. There are two primary operations in RCGA, which are crossover and mutation [46, 47]. Crossover is a phase where the chromosomes in the population randomly share their features. This is the most significant operation in the RCGA, as more powerful offspring are created taking useful features from their parent’s genes. Mutation is a process that is operated within each offspring, meaning that some of the bits in the bit string can be flipped. The main objective of the mutation process is to maintain the diversity of the population after new offspring are created from crossover [48].The RCGA can be divided into five main steps as follows [48, 49]:(i)
Initial population. In this step, a set of chromosomes called population is defined. Each individual of the population corresponds to a solution of the considered problem. Each chromosome is formed by joining genes into a string. Typically, chromosomes are composed of strings of ones and zeros.(ii)
Fitness function. In this step, the fitness score of each individual in the population is calculated. It defines how to fit the chromosome or the ability of that chromosome to compete with others. A higher fitness score means that the individual is more likely to be reproduced.(iii)
Selection. In this step, the chromosomes with the highest values of fitness score will be selected in order to share their features in the next step.(iv)
Crossover. In this step, the crossover process will be operated on the most fitting chromosomes. Their genes are randomly exchanged to create new offspring.(v)
Mutation. In this step, the mutation process is done within each individual offspring to maintain the diversity of the population.The algorithm is terminated when the model has converged, meaning that the newly created offspring are not different from the previous ones. In the literature, RCGA was used mainly in hybrid ML approaches [49]. For instance, Kim and Shin [50] used a hybrid approach based on neural networks and genetic algorithms for detecting temporal patterns, Le et al. [51] in steel structures applications, or Yan et al. [52] for engineering design problems. Finally, a complete review of the RCGA technique could be found in Lee [53].


## 2.2.3. Random Sampling Technique: Monte Carlo Method

The main idea of the Monte Carlo method is that the output is computed by repeating the sampling of variables randomly from the input space [54–56]. That way, (i) the Monte Carlo method is widely applied in order to propagate the variability of inputs on the output response; (ii) based on statistical analysis of output, several posttreatments such as robustness and/or sensitivity analyses could be thoroughly achieved [57] (see Figure 2 for a typical statistical problem using the Monte Carlo method). As shown in Figure 2, each input exhibits a probability distribution describing its variability. Due to the variabilities of input variables, the response also exhibits its statistical behaviors, which are necessary to be characterized [58]. The robustness of the model and/or sensitivity of input variables could then be deduced based on statistical analysis of output response [59–61].Figure 2
Monte Carlo simulation taking into account variability in the input space for robustness and sensitivity analysis.Using Monte Carlo simulation, the bigger the number of realizations, the higher the reliability of the response archived. In this work, in order to optimize the number of Monte Carlo runs, a statistical estimator of convergence was applied, such as [62–65](7)NMC↦INMC=1W¯1NMC∑i=1NMCWi,where W¯ is the mean of the considered variable W and NMC is the number of Monte Carlo runs.


## 2.3. Quality Assessment Criteria

In the present work, statistical criteria, namely, the coefficient of determination (R2) and Root Mean Squared Error (RMSE), have been used in order to validate and test the developed ML model. The R2 allows us to identify the statistical relationship between two data points. This measurement of the linear correlation yields a value between 0 and 1 inclusively, where 0 is no correlation and 1 is total correlation. R2 could be calculated using the following equation [66, 67]:(8)R2=∑k=1Npk−p¯wk−w¯2∑k=1Npk−p¯2∑k=1Nwk−w¯2,where N is the number of the observations, pk and p¯ are predicted and mean predicted values, and wk and w¯ are measured and mean measured values of ultimate load, respectively (k=1:N). The formulation of RMSE is described by the following equation [68–70]:(9)RMSE=1N∑k=1Npk−wk2.Finally, the slope criterion is defined, such as the slope of the linear regression fit between predicted and observed vectors.


## 3. Results and Discussion

### 3.1. Optimization of ANFIS’s Weight Parameters

In this section, the optimization of ANFIS’s weight parameters is presented. Such optimization procedure was done using both conventional GD and advanced RCGA techniques, respectively, to identify the best training algorithm. Table3 indicates the characteristics of ANFIS, including the type of membership function, the number of weights per membership function, and the number of membership functions per input as well as the number of nodes. It is seen that there were 190 consequent and antecedent ANFIS parameters to be optimized as ANFIS was generated using the c-means clustering algorithm for the considered six-dimensional input space [71, 72]. In this study, a maximum number of iterations of 1000 was employed as the stopping condition when optimizing. The cost function was selected as RMSE. The parameters of RCGA during the training phase are also indicated in Table 3. Figures 3(a) and 3(c) present the evolution of RMSE during the optimization process, using GD and RCGA, respectively. The same illustration is presented in Figures 3(b) and 3(d), but for the evolution of R2. It should be noticed that, in these figures, the value of RMSE (i.e., R2) for training and testing data was also highlighted during the learning phase. It is seen that at least 600 iterations were needed for obtaining a convergence with respect to both RMSE and R2. At the same time, the evolution of RMSE and R2 is plotted using the testing data, which were totally new when applied. Such evolution exhibits efficiency during the training process; i.e., no overfitting or underfitting was observed.Table 3
Parameters of ANFIS and RCGA used in this study.
Parameter of ANFISValueParameter of RCGAValueNumber of inputs6Population size100Number of outputs1Length of chromosome190Membership functionGaussianFitness functionLinear rankingNumber of parameters per membership function2Crossover typeRandom pairNumber of membership functions per input (rules)10Crossover probability0.4Number of nodes149Number of offsprings12Number of nonlinear parameters of the antecedent membership function120Mutation typeRandomNumber of linear parameters of the consequent membership function70Mutation probability0.7Total number of parameters190Number of mutants21Cost functionRMSEMutation rate0.15Selection functionFitness proportionate selectionFigure 3
Evolution during optimization process for RMSE using (a) GD and (c) RCGA; forR2 using (b) GD and (d) RCGA.
(a)(b)(c)(d)The values of all quality assessment criteria at the end of the training process are indicated in Table4, whereas the results in terms of regression plots and error distribution are shown in Figures 4(a)–4(c), respectively. As indicated in Table 4, using the training data, RCGA-ANFIS provided the highest value of R2, which is 0.971, while the R2 value of GD-ANFIS is 0.933. In terms of RMSE, RCGA-ANFIS yielded the smallest value, which is 70.379 kN, whereas the RMSE value of GD-ANFIS is 105.428 kN. In terms of linear fit, the RCGA-ANFIS model produced the highest value of slope (0.98) corresponding to a slope angle of 44.425°, while the slope value of GD-ANFIS was 0.937 corresponding to slope angle of 43.125°. Regarding error analyses, using the training data, the mean values are 1.409 and 0.972%, while the corresponding standard deviation values are 11.082 and 8.497% for GD-ANFIS and RCGA-ANFIS, respectively. It can be seen that the RCGA-ANFIS model yielded an error mean, which is the closest to zero and the smallest standard deviation value (see also Figure 4(c)). The application of the two ML models to the validating data is presented in the next section.Table 4
Summary of prediction capability.
Data usedModelR2RMSEMeanerror (%)StDerror (%)SlopeSlope angle (°)TrainingGD-ANFIS0.933105.4281.40911.0820.93743.125RCGA-ANFIS0.97170.3970.9728.4970.98044.425TestingGD-ANFIS0.952130.065−0.4568.9670.92042.600RCGA-ANFIS0.974100.3402.5418.0421.01945.536Figure 4
Results after training process for (a) using GD, (b) using RCGA, and (c) distribution of errors.
(a)(b)(c)


### 3.2. Validation of Model

The previously developed GD-ANFIS and RCGA-ANFIS models were applied to the validating data for validation. As a result, Figures5(a) and 5(b) present regression graphs between actual and predicted ultimate load, whereas Figure 5(c)shows error distribution, respectively. All quantitative values of quality assessment criteria are indicated in Table 4. As indicated in Table 4, using the validating data, GD-ANFIS provided R2 = 0.952, RMSE = 130.065 kN, Meanerror = −0.456 kN, StDerror = 8.967 kN, and slope = 0.920, whereas RCGA-ANFIS provided R2 = 0.974, RMSE = 100.340 kN, Meanerror = 2.541 kN, StDerror = 8.042 kN, and slope = 1.019, respectively. The same remarks were obtained for the training data, RCGA-ANFIS yielded the best prediction performance. It could be stated that the RCGA-ANFIS model is validated because it performs well the prediction of ultimate load using the validating data. Thus, RCGA-ANFIS model was selected as the final prediction model for estimating the ultimate load of elliptical CFST columns.Figure 5
Results after validating process for (a) using GD, (b) using RCGA, and (c) distribution of errors.
(a)(b)(c)


### 3.3. Sensitivity Analysis

In this section, the influence of input variables on the prediction of column load-carrying capacity is presented. For this purpose, the probability distribution of each input was characterized by 11 levels of quantiles such asQ0, Q10, Q20, Q30, Q40, Q50, Q60, Q70, Q80, Q90, and Q100. For a given input, a local influence index, denoted by θ (in %), was computed by the following equation:(10)θqk=Qqk−QmedianallQmedianall×100,where Qmedianall is the output, the ultimate load when all inputs are equal to their Q50 values. Qqk is the output of the ML model when applying kth input at its qth levels (quantiles from 0 to 100 every 10, respectively) (k = 1, …, 6 and q = 1, …, 11). That way, the global influence index of the kth input, denoted by Mk, is calculated as follows:(11)Mk=∑q=111θqk.Figures6(a) and 6(b) present the global influence index of all inputs parameters using GD-ANFIS and RCGA-ANFIS, respectively (see the appendix for statistical convergence of Monte Carlo simulations). It could also be noticed that the bar graphs are reorganized in decreasing order of the mean value for all six input variables. All values are indicated in Table 5. It is clearly observed that all input variables affect the axial capacity of structural members considerably under axial compression from a minimum of 6.1% to a maximum of 22.5% on average. It is also seen that the axial capacity is in function of inputs under a nonlinear form (i.e., a linear equation could not join all mean values of sensitivity index). It is seen that there are at least four levels of influence ranking. Indeed, the two most important variables are d and δ, which exhibit more than 20% of influence each. Next, L and D could be classified in the second group, which exhibit about 18% of influence each. The third group contains the compressive strength of concrete, whereas the yield strength of steel has about 6% of influence and is in the last group. Last but not least, it is seen that the fluctuation of the influence index obtained by GD-ANFIS is higher than the ones obtained by RCGA-ANFIS. This points out that RCGA-ANFIS is more robust and efficient than GD-ANFIS, which confirms the higher performance of RCGA than GD, as identified in Section 3.2.Figure 6
Sensitivity analysis of input variables using (a) GD-ANFIS and (b) RCGA-ANFIS.
(a)(b)Table 5
Statistical analysis of global influence index (in %).
ParameterModelLDdδfyfc′MeanGD-ANFIS18.89818.83222.50520.8826.42211.378RCGA-ANFIS17.69217.78821.34422.2646.15113.210StDGD-ANFIS4.1986.8955.0454.7954.6833.671RCGA-ANFIS4.0854.3003.8073.8393.6824.113


### 3.4. Comparison with Existing Models

In this section, the best prediction model, namely RCGA-ANFIS, is compared with existing models in the literature for the axial capacity of elliptical CFST columns. Liu and Zha [16] have proposed the following equation:(12)QnLiu−2011=1+1.5d/D0.31+As/Ac×AsfyAc,where As and Ac are the cross-sectional area of the steel tubular and the concrete core, respectively. Another formula for predicting the axial capacity of elliptical CFST columns was developed by Shen et al. [17], such as(13)QnShen−2015=fc′As+Ac0.0075×AsfyAcfc′3+0.0624×AsfyAcfc′2+0.7080×AsfyAcfc′+1.3625.Figures7(a)–7(c) present the regression graph between actual and predicted ultimate load, using Liu et al. 2011, Shen et al. 2015, and RCGA-ANFIS model, respectively. All performance indicators are also highlighted in Table 6. It is seen in Figure 7 and Table 6 that the RCGA-ANFIS model provided better performance than the literature, with respect to all error measurement criteria. In Table 6, the percentage of gain is also indicated. The percentage of gain is calculated based on the following equation:(14)%Gain=Ωthis−study−1−Ωliterature−1×100,in case of:R2and Slope;Ωliterature−Ωthis−studyΩliterature×100,in case of:RMSE and ErrorStD.Figure 7
Regression graphs between predicted and actualQn (all data) using (a) Liu and Zha [16], (b) Shen et al. [17], and (c) RCGA-ANFIS model.
(a)(b)(c)Table 6
Comparison between RCGA-ANFIS model and literature.
ParameterModel usedR2RMSEMeanerrorStDerrorSlopeSlope angle (°)Performance indicatorLiu et al. 20110.776348.43515.52333.8171.12148.277Shen et al. 20150.768515.33329.88739.7761.31052.636This work0.97480.4891.4398.3521.00345.092% of gainLiu and Zha [16]+19.8+76.9+90.7+75.3+11.8+7.1Shen et al. [17]+20.6+84.4+95.2+79.0+30.6+16.8Figure8 shows the comparison regarding the performance indicators between RCGA-ANFIS and existing models. Obviously, the RCGA-ANFIS model showed an excellent performance in predicting the ultimate load of the elliptical CFST columns.Figure 8
Comparison between RCGA-ANFIS model and literature: (a) in terms ofR2 and slope, (b) in terms of RMSE, and (c) in terms of StDerror.
(a)(b)(c)


### 3.5. Practical Application

For further application of RCGA-ANFIS model, a Graphical User Interface (GUI) was developed in MATLAB 2018a [73]. Figure 9 presents the main GUI, which is simple and easy to use. Users can enter the values of input variables; the ultimate load of elliptical CFST columns is then displayed directly by clicking the Start Predict button. The GUI is provided freely at https://github.com/Tien-ThinhLe/EllipticalCFST_AxialCapacityPrediction.Figure 9
MATLAB’s GUI for the prediction of the ultimate load of elliptical CFST columns based on RCGA-ANFIS model.


### 3.6. Proposed Empirical Formula

It is not convenient for researchers/engineers to employ machine learning models in practice, because such a model contains weights, bias parameters, and transfer functions. Thus, an empirical formula based on the developed machine learning model should be derived to be employed in the engineering field. Based on the results obtained from the machine learning model, a mathematical method was used to derive a practical equation for the prediction of ultimate load of elliptical CFST columns. Such a procedure was inspired by a recent development of Nikbin et al. [74] in deriving an empirical formula for prediction of fracture energy of concrete based on machine learning models. Figure 10 presents the diagram of the procedure. More details could be found in Nikbin et al. [74].Figure 10
Methodology flowchart for the development of empirical formula.Based on the procedure presented in Figure10, the ultimate load of elliptical CFST columns can be predicted using(15)QnProposed formula=CL×CD×Cd×Cδ×Cfy×Cfc′,where(16)CL=−0.0075912×L10002−0.15675×L1000+1.2827,(17)CD=−0.25383×D1802+1.313×D180−0.016222,(18)Cd=0.06122×d2−2.7245×d+501.50,(19)Cδ=0.11857×δ42+0.1051×δ4+0.83073,(20)Cfy=0.51644×fy3502−0.70249×fy350+1.2644,(21)Cfc′=0.015364×fc′502+0.25698×fc′50+0.80208.The coefficients presented in (16)–(21) were deduced based on a least square optimization process (see also Nikbin et al. [74]). In order to evaluate the performance of the proposed equation, 94 experimental data points have been employed for a comparison purpose. Details of the experimental dataset, including input variables (geometric variables and strength of constituent materials), output variable (measured ultimate load), and three ratios QnLiu−2011/Qn, QnShen−2015/Qn, QnProposed formula/Qn, are indicated in Table 7. At the end of Table 7, statistics of the three ratios are also indicated, including the min, average, max, standard deviation, and coefficient of variation values. Finally, Figure 11 shows the probability density distribution of the three ratios.Table 7
Comparison of performance between the proposed formula and existing equations.
LDdδfyfc′QnQnLiu−2011/QnQnShen−2015/QnQnProposed formula/QnmmmmmmmmMPaMPakN———300150.475.64.18376.526.938391.021.100.96300150.5775.524.19376.547.39741.041.130.93300150.3975.674.18376.584.5712651.021.200.86300150.1275.655.1236926.939810.991.140.88300150.2375.745.0836947.310841.031.150.89300150.2875.675.0936984.5712961.071.270.90300148.7875.456.32400.526.9311931.011.350.83300148.9275.566.43400.547.312801.071.270.88300149.5375.356.25400.584.5714831.071.290.90500150.1875.214.5139569.210751.161.320.96500150.4975.265.4135869.211631.111.290.92500150.0575.426.5636969.213101.111.330.92600200.21100.125.239769.215981.301.471.11600200100.356.141169.220681.101.250.95600200.6100.028.1738369.221331.191.411.08600200.19100.419.7236769.222901.191.461.15698220.7110.76.1642148.221091.121.211.01300150.1754.1431.435.89001.111.180.99299150.175.24.2431.492.1412391.161.360.97398197.8100.15.1347.936.8712321.191.271.16398197.5100.25.1347.953.5417370.971.080.90398197.4100.15.1347.9102.2621161.101.350.941497150.975.44431.417.9650.81.321.541.031498150.475.24.1431.451.29742.81.511.631.101496150.375.24.1431.477923.21.421.621.011499197.5100.25.2347.920.33938.41.351.451.181498197.7100.15.1347.97714801.351.591.011785150.775.24.2431.451.67663.21.721.871.181786150.775.44.1431.486.08871.21.591.841.061785197.6100.25.1347.931.32967.51.441.521.151786197.7100.15.1347.950.2712371.331.461.001786197.31005.2347.983.871411.21.491.771.052499197.8100.15.1347.946.29471.681.831.102498197.7100.15.1347.987.281072.31.992.391.20160160107.8120727.3768.70.620.741.20160159.4106.51.629627.38440.720.771.12160159.7107.42.334127.3921.30.840.841.12250159.9105.5120727.3681.30.680.821.31250160.1105.51.629627.3783.30.770.831.19250160.81072.334127.3850.70.910.911.20160159.480.51.627925699.70.630.680.99160158.880.72.320125761.50.580.640.95250160.874.9121125468.40.680.811.35250158.3822.320125630.10.710.781.15160159.263.2120727.34960.580.701.16160159.663.31.629627.3500.60.770.861.20160159.564.22.334127.3665.30.760.810.98250158.564.5120727.3413.30.710.861.39250159.363.11.629627.3499.30.770.861.19250158.863.22.334127.3620.60.800.861.02200158.679.7120727.3484.60.730.881.3820015880.61.629627.3613.30.770.841.14200159.282.12.334127.37240.850.871.061678150.475.24.141045.647431.401.511.001679197.51005.235020.339381.361.461.151678197.71005.135077.214801.361.590.981965150.575.44.141013.184841.652.201.211965150.775.24.241052.136631.671.821.111966150.775.44.141086.188711.551.811.001965197.61005.135031.329681.451.521.111966197.71005.135050.2712371.331.470.971966197.31005.235084.1714111.501.781.012681150.1754.141086.185472.452.861.352678197.51005.235020.338391.521.631.042679197.81005.135046.569471.691.841.062678197.71005.135087.1810721.992.391.1536001921243.82439.348.4111211.681.730.8936001921243.82439.348.4111571.631.670.8627001921243.82439.348.4113891.361.390.9327001921243.82439.348.4113221.431.460.9818001921243.82439.348.4118960.991.020.8318001921243.82439.348.4118291.031.060.862154148.4575.786.3369.132886.61.311.610.821154148.3775.636.3369.1331059.31.101.350.82271136.5136.52.75376.450.361296.31.031.051.162711371372.75376.450.361325.31.011.031.14271137.8137.82.75376.450.3613431.011.031.143381701122.75376.450.361310.60.981.061.03338169.61112.75376.450.361299.20.981.061.03338168112.52.75376.450.361294.40.991.071.04407202992.75376.450.361298.71.011.131.01407199.8100.82.75376.450.3613251.001.121.01407201.5100.42.75376.450.361381.10.961.070.9747523695.82.75376.450.361309.21.091.261.07475237.5962.75376.450.361364.61.061.221.0347523696.52.75376.450.361354.21.061.231.046363181552.75376.450.3626071.061.241.21636318.5151.52.75376.450.362497.31.091.281.22636317153.52.75376.450.362521.51.081.271.23279139682.75376.450.36687.21.061.161.0427913868.22.75376.450.36688.11.051.151.04279137.5682.75376.450.36699.21.031.131.022670.4199.7105.72.6376.44511401.111.220.791910.4204.3103.12.6376.4459661.301.441.09Min0.580.640.79Average1.161.301.05Max2.452.861.39StD∗0.340.400.13CV∗∗29.2730.6212.55StD: standard deviation, CV: coefficient of variation (%).Figure 11
Distribution of ratio predictedQn/actual Qn using different equations.It is seen in Table7 (statistics of the three ratios) and Figure 11 that the prediction based on the proposed formula exhibits the highest agreement with the experimental data points or, in other words, the lowest error measurements (an average value of 1.05 compared to 1.16, 1.30 using Liu and Shen equations; a standard deviation value of 0.13 compared to 0.34, 0.40 using Liu and Shen equations; and a coefficient of variation of 12.55% compared to 29.27, 30.62 using Liu and Shen equations, respectively). It can be concluded that the prediction performance based on the proposed formula is superior to those available in the literature. Thus, with a simple form, the proposed formula can be used in practice. Moreover, if more experimental data are available in the future, the model will be improved (i.e., for a wider range of data).


## 3.1. Optimization of ANFIS’s Weight Parameters

In this section, the optimization of ANFIS’s weight parameters is presented. Such optimization procedure was done using both conventional GD and advanced RCGA techniques, respectively, to identify the best training algorithm. Table3 indicates the characteristics of ANFIS, including the type of membership function, the number of weights per membership function, and the number of membership functions per input as well as the number of nodes. It is seen that there were 190 consequent and antecedent ANFIS parameters to be optimized as ANFIS was generated using the c-means clustering algorithm for the considered six-dimensional input space [71, 72]. In this study, a maximum number of iterations of 1000 was employed as the stopping condition when optimizing. The cost function was selected as RMSE. The parameters of RCGA during the training phase are also indicated in Table 3. Figures 3(a) and 3(c) present the evolution of RMSE during the optimization process, using GD and RCGA, respectively. The same illustration is presented in Figures 3(b) and 3(d), but for the evolution of R2. It should be noticed that, in these figures, the value of RMSE (i.e., R2) for training and testing data was also highlighted during the learning phase. It is seen that at least 600 iterations were needed for obtaining a convergence with respect to both RMSE and R2. At the same time, the evolution of RMSE and R2 is plotted using the testing data, which were totally new when applied. Such evolution exhibits efficiency during the training process; i.e., no overfitting or underfitting was observed.Table 3
Parameters of ANFIS and RCGA used in this study.
Parameter of ANFISValueParameter of RCGAValueNumber of inputs6Population size100Number of outputs1Length of chromosome190Membership functionGaussianFitness functionLinear rankingNumber of parameters per membership function2Crossover typeRandom pairNumber of membership functions per input (rules)10Crossover probability0.4Number of nodes149Number of offsprings12Number of nonlinear parameters of the antecedent membership function120Mutation typeRandomNumber of linear parameters of the consequent membership function70Mutation probability0.7Total number of parameters190Number of mutants21Cost functionRMSEMutation rate0.15Selection functionFitness proportionate selectionFigure 3
Evolution during optimization process for RMSE using (a) GD and (c) RCGA; forR2 using (b) GD and (d) RCGA.
(a)(b)(c)(d)The values of all quality assessment criteria at the end of the training process are indicated in Table4, whereas the results in terms of regression plots and error distribution are shown in Figures 4(a)–4(c), respectively. As indicated in Table 4, using the training data, RCGA-ANFIS provided the highest value of R2, which is 0.971, while the R2 value of GD-ANFIS is 0.933. In terms of RMSE, RCGA-ANFIS yielded the smallest value, which is 70.379 kN, whereas the RMSE value of GD-ANFIS is 105.428 kN. In terms of linear fit, the RCGA-ANFIS model produced the highest value of slope (0.98) corresponding to a slope angle of 44.425°, while the slope value of GD-ANFIS was 0.937 corresponding to slope angle of 43.125°. Regarding error analyses, using the training data, the mean values are 1.409 and 0.972%, while the corresponding standard deviation values are 11.082 and 8.497% for GD-ANFIS and RCGA-ANFIS, respectively. It can be seen that the RCGA-ANFIS model yielded an error mean, which is the closest to zero and the smallest standard deviation value (see also Figure 4(c)). The application of the two ML models to the validating data is presented in the next section.Table 4
Summary of prediction capability.
Data usedModelR2RMSEMeanerror (%)StDerror (%)SlopeSlope angle (°)TrainingGD-ANFIS0.933105.4281.40911.0820.93743.125RCGA-ANFIS0.97170.3970.9728.4970.98044.425TestingGD-ANFIS0.952130.065−0.4568.9670.92042.600RCGA-ANFIS0.974100.3402.5418.0421.01945.536Figure 4
Results after training process for (a) using GD, (b) using RCGA, and (c) distribution of errors.
(a)(b)(c)


## 3.2. Validation of Model

The previously developed GD-ANFIS and RCGA-ANFIS models were applied to the validating data for validation. As a result, Figures5(a) and 5(b) present regression graphs between actual and predicted ultimate load, whereas Figure 5(c)shows error distribution, respectively. All quantitative values of quality assessment criteria are indicated in Table 4. As indicated in Table 4, using the validating data, GD-ANFIS provided R2 = 0.952, RMSE = 130.065 kN, Meanerror = −0.456 kN, StDerror = 8.967 kN, and slope = 0.920, whereas RCGA-ANFIS provided R2 = 0.974, RMSE = 100.340 kN, Meanerror = 2.541 kN, StDerror = 8.042 kN, and slope = 1.019, respectively. The same remarks were obtained for the training data, RCGA-ANFIS yielded the best prediction performance. It could be stated that the RCGA-ANFIS model is validated because it performs well the prediction of ultimate load using the validating data. Thus, RCGA-ANFIS model was selected as the final prediction model for estimating the ultimate load of elliptical CFST columns.Figure 5
Results after validating process for (a) using GD, (b) using RCGA, and (c) distribution of errors.
(a)(b)(c)


## 3.3. Sensitivity Analysis

In this section, the influence of input variables on the prediction of column load-carrying capacity is presented. For this purpose, the probability distribution of each input was characterized by 11 levels of quantiles such asQ0, Q10, Q20, Q30, Q40, Q50, Q60, Q70, Q80, Q90, and Q100. For a given input, a local influence index, denoted by θ (in %), was computed by the following equation:(10)θqk=Qqk−QmedianallQmedianall×100,where Qmedianall is the output, the ultimate load when all inputs are equal to their Q50 values. Qqk is the output of the ML model when applying kth input at its qth levels (quantiles from 0 to 100 every 10, respectively) (k = 1, …, 6 and q = 1, …, 11). That way, the global influence index of the kth input, denoted by Mk, is calculated as follows:(11)Mk=∑q=111θqk.Figures6(a) and 6(b) present the global influence index of all inputs parameters using GD-ANFIS and RCGA-ANFIS, respectively (see the appendix for statistical convergence of Monte Carlo simulations). It could also be noticed that the bar graphs are reorganized in decreasing order of the mean value for all six input variables. All values are indicated in Table 5. It is clearly observed that all input variables affect the axial capacity of structural members considerably under axial compression from a minimum of 6.1% to a maximum of 22.5% on average. It is also seen that the axial capacity is in function of inputs under a nonlinear form (i.e., a linear equation could not join all mean values of sensitivity index). It is seen that there are at least four levels of influence ranking. Indeed, the two most important variables are d and δ, which exhibit more than 20% of influence each. Next, L and D could be classified in the second group, which exhibit about 18% of influence each. The third group contains the compressive strength of concrete, whereas the yield strength of steel has about 6% of influence and is in the last group. Last but not least, it is seen that the fluctuation of the influence index obtained by GD-ANFIS is higher than the ones obtained by RCGA-ANFIS. This points out that RCGA-ANFIS is more robust and efficient than GD-ANFIS, which confirms the higher performance of RCGA than GD, as identified in Section 3.2.Figure 6
Sensitivity analysis of input variables using (a) GD-ANFIS and (b) RCGA-ANFIS.
(a)(b)Table 5
Statistical analysis of global influence index (in %).
ParameterModelLDdδfyfc′MeanGD-ANFIS18.89818.83222.50520.8826.42211.378RCGA-ANFIS17.69217.78821.34422.2646.15113.210StDGD-ANFIS4.1986.8955.0454.7954.6833.671RCGA-ANFIS4.0854.3003.8073.8393.6824.113


## 3.4. Comparison with Existing Models

In this section, the best prediction model, namely RCGA-ANFIS, is compared with existing models in the literature for the axial capacity of elliptical CFST columns. Liu and Zha [16] have proposed the following equation:(12)QnLiu−2011=1+1.5d/D0.31+As/Ac×AsfyAc,where As and Ac are the cross-sectional area of the steel tubular and the concrete core, respectively. Another formula for predicting the axial capacity of elliptical CFST columns was developed by Shen et al. [17], such as(13)QnShen−2015=fc′As+Ac0.0075×AsfyAcfc′3+0.0624×AsfyAcfc′2+0.7080×AsfyAcfc′+1.3625.Figures7(a)–7(c) present the regression graph between actual and predicted ultimate load, using Liu et al. 2011, Shen et al. 2015, and RCGA-ANFIS model, respectively. All performance indicators are also highlighted in Table 6. It is seen in Figure 7 and Table 6 that the RCGA-ANFIS model provided better performance than the literature, with respect to all error measurement criteria. In Table 6, the percentage of gain is also indicated. The percentage of gain is calculated based on the following equation:(14)%Gain=Ωthis−study−1−Ωliterature−1×100,in case of:R2and Slope;Ωliterature−Ωthis−studyΩliterature×100,in case of:RMSE and ErrorStD.Figure 7
Regression graphs between predicted and actualQn (all data) using (a) Liu and Zha [16], (b) Shen et al. [17], and (c) RCGA-ANFIS model.
(a)(b)(c)Table 6
Comparison between RCGA-ANFIS model and literature.
ParameterModel usedR2RMSEMeanerrorStDerrorSlopeSlope angle (°)Performance indicatorLiu et al. 20110.776348.43515.52333.8171.12148.277Shen et al. 20150.768515.33329.88739.7761.31052.636This work0.97480.4891.4398.3521.00345.092% of gainLiu and Zha [16]+19.8+76.9+90.7+75.3+11.8+7.1Shen et al. [17]+20.6+84.4+95.2+79.0+30.6+16.8Figure8 shows the comparison regarding the performance indicators between RCGA-ANFIS and existing models. Obviously, the RCGA-ANFIS model showed an excellent performance in predicting the ultimate load of the elliptical CFST columns.Figure 8
Comparison between RCGA-ANFIS model and literature: (a) in terms ofR2 and slope, (b) in terms of RMSE, and (c) in terms of StDerror.
(a)(b)(c)


## 3.5. Practical Application

For further application of RCGA-ANFIS model, a Graphical User Interface (GUI) was developed in MATLAB 2018a [73]. Figure 9 presents the main GUI, which is simple and easy to use. Users can enter the values of input variables; the ultimate load of elliptical CFST columns is then displayed directly by clicking the Start Predict button. The GUI is provided freely at https://github.com/Tien-ThinhLe/EllipticalCFST_AxialCapacityPrediction.Figure 9
MATLAB’s GUI for the prediction of the ultimate load of elliptical CFST columns based on RCGA-ANFIS model.


## 3.6. Proposed Empirical Formula

It is not convenient for researchers/engineers to employ machine learning models in practice, because such a model contains weights, bias parameters, and transfer functions. Thus, an empirical formula based on the developed machine learning model should be derived to be employed in the engineering field. Based on the results obtained from the machine learning model, a mathematical method was used to derive a practical equation for the prediction of ultimate load of elliptical CFST columns. Such a procedure was inspired by a recent development of Nikbin et al. [74] in deriving an empirical formula for prediction of fracture energy of concrete based on machine learning models. Figure 10 presents the diagram of the procedure. More details could be found in Nikbin et al. [74].Figure 10
Methodology flowchart for the development of empirical formula.Based on the procedure presented in Figure10, the ultimate load of elliptical CFST columns can be predicted using(15)QnProposed formula=CL×CD×Cd×Cδ×Cfy×Cfc′,where(16)CL=−0.0075912×L10002−0.15675×L1000+1.2827,(17)CD=−0.25383×D1802+1.313×D180−0.016222,(18)Cd=0.06122×d2−2.7245×d+501.50,(19)Cδ=0.11857×δ42+0.1051×δ4+0.83073,(20)Cfy=0.51644×fy3502−0.70249×fy350+1.2644,(21)Cfc′=0.015364×fc′502+0.25698×fc′50+0.80208.The coefficients presented in (16)–(21) were deduced based on a least square optimization process (see also Nikbin et al. [74]). In order to evaluate the performance of the proposed equation, 94 experimental data points have been employed for a comparison purpose. Details of the experimental dataset, including input variables (geometric variables and strength of constituent materials), output variable (measured ultimate load), and three ratios QnLiu−2011/Qn, QnShen−2015/Qn, QnProposed formula/Qn, are indicated in Table 7. At the end of Table 7, statistics of the three ratios are also indicated, including the min, average, max, standard deviation, and coefficient of variation values. Finally, Figure 11 shows the probability density distribution of the three ratios.Table 7
Comparison of performance between the proposed formula and existing equations.
LDdδfyfc′QnQnLiu−2011/QnQnShen−2015/QnQnProposed formula/QnmmmmmmmmMPaMPakN———300150.475.64.18376.526.938391.021.100.96300150.5775.524.19376.547.39741.041.130.93300150.3975.674.18376.584.5712651.021.200.86300150.1275.655.1236926.939810.991.140.88300150.2375.745.0836947.310841.031.150.89300150.2875.675.0936984.5712961.071.270.90300148.7875.456.32400.526.9311931.011.350.83300148.9275.566.43400.547.312801.071.270.88300149.5375.356.25400.584.5714831.071.290.90500150.1875.214.5139569.210751.161.320.96500150.4975.265.4135869.211631.111.290.92500150.0575.426.5636969.213101.111.330.92600200.21100.125.239769.215981.301.471.11600200100.356.141169.220681.101.250.95600200.6100.028.1738369.221331.191.411.08600200.19100.419.7236769.222901.191.461.15698220.7110.76.1642148.221091.121.211.01300150.1754.1431.435.89001.111.180.99299150.175.24.2431.492.1412391.161.360.97398197.8100.15.1347.936.8712321.191.271.16398197.5100.25.1347.953.5417370.971.080.90398197.4100.15.1347.9102.2621161.101.350.941497150.975.44431.417.9650.81.321.541.031498150.475.24.1431.451.29742.81.511.631.101496150.375.24.1431.477923.21.421.621.011499197.5100.25.2347.920.33938.41.351.451.181498197.7100.15.1347.97714801.351.591.011785150.775.24.2431.451.67663.21.721.871.181786150.775.44.1431.486.08871.21.591.841.061785197.6100.25.1347.931.32967.51.441.521.151786197.7100.15.1347.950.2712371.331.461.001786197.31005.2347.983.871411.21.491.771.052499197.8100.15.1347.946.29471.681.831.102498197.7100.15.1347.987.281072.31.992.391.20160160107.8120727.3768.70.620.741.20160159.4106.51.629627.38440.720.771.12160159.7107.42.334127.3921.30.840.841.12250159.9105.5120727.3681.30.680.821.31250160.1105.51.629627.3783.30.770.831.19250160.81072.334127.3850.70.910.911.20160159.480.51.627925699.70.630.680.99160158.880.72.320125761.50.580.640.95250160.874.9121125468.40.680.811.35250158.3822.320125630.10.710.781.15160159.263.2120727.34960.580.701.16160159.663.31.629627.3500.60.770.861.20160159.564.22.334127.3665.30.760.810.98250158.564.5120727.3413.30.710.861.39250159.363.11.629627.3499.30.770.861.19250158.863.22.334127.3620.60.800.861.02200158.679.7120727.3484.60.730.881.3820015880.61.629627.3613.30.770.841.14200159.282.12.334127.37240.850.871.061678150.475.24.141045.647431.401.511.001679197.51005.235020.339381.361.461.151678197.71005.135077.214801.361.590.981965150.575.44.141013.184841.652.201.211965150.775.24.241052.136631.671.821.111966150.775.44.141086.188711.551.811.001965197.61005.135031.329681.451.521.111966197.71005.135050.2712371.331.470.971966197.31005.235084.1714111.501.781.012681150.1754.141086.185472.452.861.352678197.51005.235020.338391.521.631.042679197.81005.135046.569471.691.841.062678197.71005.135087.1810721.992.391.1536001921243.82439.348.4111211.681.730.8936001921243.82439.348.4111571.631.670.8627001921243.82439.348.4113891.361.390.9327001921243.82439.348.4113221.431.460.9818001921243.82439.348.4118960.991.020.8318001921243.82439.348.4118291.031.060.862154148.4575.786.3369.132886.61.311.610.821154148.3775.636.3369.1331059.31.101.350.82271136.5136.52.75376.450.361296.31.031.051.162711371372.75376.450.361325.31.011.031.14271137.8137.82.75376.450.3613431.011.031.143381701122.75376.450.361310.60.981.061.03338169.61112.75376.450.361299.20.981.061.03338168112.52.75376.450.361294.40.991.071.04407202992.75376.450.361298.71.011.131.01407199.8100.82.75376.450.3613251.001.121.01407201.5100.42.75376.450.361381.10.961.070.9747523695.82.75376.450.361309.21.091.261.07475237.5962.75376.450.361364.61.061.221.0347523696.52.75376.450.361354.21.061.231.046363181552.75376.450.3626071.061.241.21636318.5151.52.75376.450.362497.31.091.281.22636317153.52.75376.450.362521.51.081.271.23279139682.75376.450.36687.21.061.161.0427913868.22.75376.450.36688.11.051.151.04279137.5682.75376.450.36699.21.031.131.022670.4199.7105.72.6376.44511401.111.220.791910.4204.3103.12.6376.4459661.301.441.09Min0.580.640.79Average1.161.301.05Max2.452.861.39StD∗0.340.400.13CV∗∗29.2730.6212.55StD: standard deviation, CV: coefficient of variation (%).Figure 11
Distribution of ratio predictedQn/actual Qn using different equations.It is seen in Table7 (statistics of the three ratios) and Figure 11 that the prediction based on the proposed formula exhibits the highest agreement with the experimental data points or, in other words, the lowest error measurements (an average value of 1.05 compared to 1.16, 1.30 using Liu and Shen equations; a standard deviation value of 0.13 compared to 0.34, 0.40 using Liu and Shen equations; and a coefficient of variation of 12.55% compared to 29.27, 30.62 using Liu and Shen equations, respectively). It can be concluded that the prediction performance based on the proposed formula is superior to those available in the literature. Thus, with a simple form, the proposed formula can be used in practice. Moreover, if more experimental data are available in the future, the model will be improved (i.e., for a wider range of data).


## 4. Conclusions

The research presented in this article proposed a robust surrogate tool for the estimation of the ultimate load of elliptical CFST members under axial compression. Based on the developments and analyses, the following conclusions may be made:(i)
An experimental dataset was collected from the available literature for the development of the models including two groups of variables: geometric dimensions of cross section and mechanical properties of constituent materials (concrete and steel).(ii)
Two hybrid ML models, namely, the conventional GD-ANFIS and metaheuristic-based RCGA-ANFIS, were proposed to predict the ultimate load of the columns. The results showed that the RCGA-ANFIS model outperformed GD-ANFIS. In addition, the performance of the RCGA-ANFIS model was superior to two empirical equations in the literature.(iii)
The robustness of the proposed models was assessed by conducting Monte Carlo simulations taking into account the variability in the input space.(iv)
Sensitivity analysis showed that the steel pipe wall thickness and the short side length of the cross section were the most critical parameters affecting the bearing capacity of elliptical CFST columns (i.e., 22.264% and 21.344%, respectively).(v)
A Graphical User Interface was developed and provided freely for researchers/engineers/interested users. The results of the present work could simplify the design of elliptical CFST columns. The optimum values obtained in this study could allow quick and accurate determining of the bearing capacity of elliptical CFST columns for practical purposes.However, it is worth noticing that, in this research, only elliptical CFST columns were considered. It is well-known that the cross section of columns has other forms; thus, the extension of the GUI to other cross sections would be the main perspective of the next study. In further research, a generic model should be developed for different types of cross section (i.e., circular, rectangular, square, hexagonal, etc.). Such a model can be highly beneficial for the research and practical purposes. Finally, in terms of practical application, a GUI based on Excel should be developed for wider applicability.



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*Source: 8832522-2020-10-28.xml*