1 Introduction

In the area of structural performance, one of the key issues is how to use the available materials in an optimized way. In current construction processes, the two most widely used materials are steel and concrete. They can be used together in such a way that each one of them can improve the other’s performance, which finally results in a better overall behavior of the structure under various loads. As a result, when concrete and steel are combined appropriately, their performance will be more improved compared to the cases where they are utilized separately. Recently, composite material has been widely applied to different construction projects [1, 2] as well as to retrofitting and rehabilitation purposes [3, 4] across the world. Composite columns offer many benefits; they can be easily produced, they enjoy some improved features compared to other columns such as steel structures, and they reduce the construction expenses [1].

Accordingly, several researchers have attempted to test how the concrete-filled hollow steel columns behave in different conditions [5,6,7,8]. Based on the findings of the study conducted by He et al. [9], among different types of composite columns, the concrete-filled steel tube (CFST) can outperform the other types of columns. The concrete in CFST is employed inside, while the steel’s hollow sections are in the surrounding periphery. It helps the steel column not to be suddenly buckled, improves the way it performs, and delays the settlement because of the external loads. When these columns are being designed, a key issue is to examine the interactive impacts that occur between steel and concrete. This issue has not been investigated adequately in literature. Although there are some studies carried out into composite columns on the basis of experiments and theories, in most of the cases (e.g., [8]), the authors have not sufficiently described the columns’ behaviors under a variety of loading conditions.

In recent years, the civil engineering field has witnessed the advance of numerous methods. One of the most significant methods is soft computing and intelligent technique [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. Such techniques are presented in a variety of models in various sections of engineering [31]. For example, artificial intelligence is used in civil engineering for prediction and optimization purposes, e.g., the prediction of the concrete’s compressive strength [20, 29,30,31,32,33,34,35,36,37,38,39,40], the identification of the risk areas of regional transportation corridors [41], the prediction of final strength of reinforce concrete beams with fiber-reinforced plastic (FRP) strengthened in shear [8], the determination of damage in skeletal structure [42], and the optimization of the arch dam forms [43].

To improve the precision level in the computation of the parameters, the artificial neural network (ANN) can be effectively used [12,13,14,15, 27, 41]. Even though ANNs have dynamic research realms in science and engineering, directly map the input to output patterns, and are able to use all effective prediction characteristics, they have limitations such as slow learning rate and entrapment in local minima [44]. The use of optimization techniques such as particle swarm optimization (PSO) in different engineering optimized affairs can address the shortcomings of ANN. In fact, PSO is able to optimize weights and biases of ANN to get better performance prediction.

In the engineering field, a recently introduced technique called gene expression programming (GEP) has been found successful in enhancing the accuracy level. It is actually a combined form of genetic algorithm (GA) and genetic programming (GP). It has shown a high capability of introducing mathematical equations to do the predictions with a higher quality and solving problems of a high complexity [45, 46]. Lots of studies have reported the success of GEP in different fields of civil engineering, including the environmental issues of blasting [47, 48], piling [49], tunneling and rock mechanics [50, 51], concrete technology [52, 53], highway construction [54], and river engineering [55, 56].

The present study is aimed to propose efficient models based on these artificial intelligence approaches in order to predict the axial load bearing capacity of the composite columns. To this end, experimental data are gathered and required tests are carried out on parameters that affect the bearing capacity of columns. Next, the gathered data are applied to the formation and development of various models of GEP. Then, a comparison is made between the results of the proposed model and those of hybrid PSO–ANN (or neuro-swarm) and ANN networks. At the final step, to achieve the optimum cross-sections, a whale optimization algorithm (WOA), which is considered as a novel optimization algorithm, is proposed. In the following, the theoretical underpinning of the models used in this study is explained first, the datasets are described, and the prediction phase of the study is then discussed. To achieve the optimum values of the variables, the best predictive model will be selected and then it will be utilized as an input model in the WOA optimization technique.

2 Predictive model background

2.1 Artificial neural network (ANN)

The concept of neural networks was first introduced in 1950s by Donald Hebb [57] with the introduction of a simple learning mechanism. He developed this method by investigating human brain neurons and the effect of learning on them. In each neuron of ANN, dendrites receive information from the previous neuron, and axons transfer the results to the next section (i.e., next neuron) after an initial processing. Chemical signaling is done through synopses between the cells. The performance of a computational neuron, which is used in neural networks, is similar (assuming sigmoid activation function) to that of a biological neuron with inputs and outputs. An ANN contains two or more layers, and each layer has a series of neurons. The strength of the links between the layers is associated with the weights constituting a network. The weights associated with each neuron linearly transform the input vectors, which become the arguments of each neuron’s non-linear activation function (transfer function). Two main algorithms, i.e., feed-forward multilayer and back-propagation (BP), are used in neural networks. BP is more common and recommended by different researchers [58,59,60]. This algorithm updates the weights in order for the loss functions to reach the minimum error (loss) in the system. This training process is repeated for a few times so that it can reach the termination criterion. The BP phrase is associated with conditions in which gradient is calculated for non-linear multilayer networks (the networks that are used to solve most of the engineering problems). The sigmoid transfer function receives the input values and presents them as an interval of 0–1, regardless of the initial input interval [61,62,63,64].

2.2 Particle swarm optimization (PSO)

Kennedy and Eberhart [65] introduced a solution for optimal continuous problems called particle swarm optimization (PSO). PSO is a non-linear procedure inspired by social systems such as fish shoals. Indeed, PSO has been formed based on the number of particles, which is established randomly. Seeking an optimal value goal is another PSO stage as an iterative process. In this stage, the particles are adjusted to their positions based on their own experience and that of other particles. To gain the best position, each particle follows its own best position (PBEST) and global best position (GBEST) among other particles. Moreover, each particle tends to move toward its own PBEST and GBEST during the training process based on a new velocity term and distance of its best positions in the learning stage. Respectively, the new position of each particle depends on the new velocity value in next iteration [66, 67]. In PSO, Eqs. (1) and (2) are used to gain the velocity updated and movement. Equation (1) calculates the particle’s real movement through its velocity vector and Eq. (2) adjusts the vector to the PBEST and GBEST.

$$\overrightarrow {{v_{\text{new}}}} = \vec{v} + C_{1} \times \left({\overrightarrow {\text{pbest}} - \vec{p}} \right) + C_{2} \times \left({\overrightarrow {\text{gbest}} - \vec{p}} \right)$$
(1)
$$\overrightarrow {{p_{\text{new}}}} = \vec{p} + \overrightarrow {{v_{\text{new}}}}$$
(2)

where \(\overrightarrow {{v_{\text{new}} }}\) is the new particle velocity, \(\vec{v}\) the current particle velocity, \(\overrightarrow {{p_{\text{new}} }}\) the new particle position, \(\vec{p}\) the current particle position, \(C_{1}\) and \(C_{2}\) are the coefficients, and \(\overrightarrow {\text{pbest}}\) and \(\overrightarrow {\text{gbest}}\) are the personal and global positions of particles, respectively.

The flowchart of PSO is shown in Fig. 1. More details about PSO and its structure are available in the literature [68].

Fig. 1
figure 1

The flowchart of PSO algorithm [65]

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2.3 Gene expression programming (GEP)

To explain more, GEP is known as one of the innovative methods developed in the field of artificial intelligence. It is actually a developed version of GA and GP. GEP, comprising various parts, suggests appropriate solutions to a variety of problems [69]. It uses two main chromosomes, and the expression tree used in this algorithm shows capacity for removal of the limitations of the two antecedents (GA and GP). In GEP, codifications are normally presented in the shape of a string attained from the Karva programming language; it is able to behave similar to ETs. Interestingly, GEP is capable of presenting its own models by means of mathematical equations that, in turn, form relationships between dependent and independent parameters. In the context of the engineering field, it is of a high importance and practicality to create models with the capacity of providing equations. Such methods are efficient substitutes for the ANN models in solving the problems. These issues have led the scholars in this field to further develop such methods.

In GP, a variety of mathematical functions, including \(-\), +, \(\times\), sin, etc., are noted and applied to the variables; thus, a mathematical set is obtainable through combining them for the purpose of problem examination. In chromosomes with more than one gene, each gene denotes a sub-ET comprising a head and a tail. Such symbolic chromosomes need to be defined as trees of various forms and sizes (expression trees). According to Fig. 2, the GEP modeling process begins with the random creation of chromosomes for determined numbers, which follows Karva language (Karva is a symbolic language to introduce chromosomes). These points are tested considering the functions that control the models and their adaptability level. Such functions are of various types each of which can be defined using a variety of criteria. The functions include root relative squared error (RRSE), mean absolute error (MAE), and root mean square error (RMSE). In the next step, in case of not satisfying the termination criterion (that is the achievement of the maximum iteration or the proper fitness value), the best chromosomes chosen by means of the Roulette Wheel method for the first process will enter the next structure. Then, the most important genetic operators, namely mutation, transfer (RIS, IS, and gene transfer), and reconstruction (one point, two points, and gene reconstruction), will be applied to existing chromosomes in accordance with their proportions, which can be defined using the codes and the experts’ opinions of the GP method. This way, new chromosomes replace the remains, and the process continues until the termination criteria are fully satisfied [70,71,72]. Literature consists of more detailed information regarding GEP and the way it can be implemented initially [73].

Fig. 2
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A view of GEP system

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2.4 Whale optimization algorithm (WOA)

The whale optimization algorithm (WOA) was proposed in 2016, mimicking the hunting mechanism of humpback whales in nature [74]. The most interesting thing about the humpback whales is their special hunting method. This foraging behavior is called bubble-net feeding method [75]. Humpback whales prefer to hunt school of krill or small fish close to the surface. It has been observed that this foraging is done by creating distinctive bubbles along a circle or a ‘9’-shaped path as shown in Fig. 3. The three-dimensional hunting behavior was studied in 2011; before that, the hunting behavior was studied based on observation from the sea surface. Nevertheless, Goldbogen et al. [76] conducted a different investigation based on the use of tag sensors. In this new method of study, they captured 300 tag-derived bubble-net feeding events of 9 individual humpback whales and introduced two new bubble movement plans, i.e., upward spirals and double loops. In the previous movement plan, humpback whales create bubble from depth of around 12 m in a spiral shape around the prey and swim up toward the surface. However, the new movement plan found by Goldbogen et al. [76] consisted of three different stages: coral loop, lobtail, and capture loop [76]. It is important to note that bubble-net feeding is a unique behavior that can only be observed in humpback whales. More details regarding WOA can be found in the original reference [74].

Fig. 3
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The behavior of humpback whales in WOA designing process [74]

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3 Established database

The CFST columns/structures, as noted earlier, have been implemented in the main lateral resistance systems of both braced and unbraced building constructions. Moreover, this type of columns has been widely recommended for the retrofitting projects to improve the columns bearing capacity, particularly when situated in dynamic conditions. As a result, it is of a high importance to appropriately predict the CFST bearing capacity. To this end, a database containing totally 303 test results was taken into consideration for the purpose of this study. The data, which were on the basis of wide-ranging laboratory experiments, were gathered from the literature [9, 77,78,79,80,81,82,83]. In the following, the testing procedure used in this study is simply explained.

During the concrete casting process carried out in laboratory using formworks, a support system was created, which included steel bars, wooden plates, European pallet, and bottom steel blocks. It was used for the purpose of ensuring the steel tube stability in the course of casting the concrete. The top surface of the outer steel tube was approximately 25 mm higher than the top surface of the inner concrete core. Following the casting operation, all of the composite columns and cylinder specimens for reference tests were cured at room temperature within the laboratory conditions. Prior to the start of testing process, the steel tube (L) and overall length of the concrete core (Lc) were precisely determined. At both ends of the concrete core, two steel blocks were positioned in a way to apply the axial load upon the concrete core. After that, we made use of the servo-controlled hydraulic testing system to compress concentrically the prepared specimens. A load cell was employed to record the applied force. Then, with the rate of 0.01 mm/s, the axial load was amplified upon the columns in a gradual manner, and simultaneously, the axial displacement was measured using the linear varying displacement transducers (LVDTs). This process went on until the measured axial displacements were roughly 20 mm. Next, the maximum axial compressive strength of the columns, i.e., their bearing capacity, was obtained as output of the study.

In case of all gathered datasets, to predict the bearing capacity of CFST columns (Pexp), five parameters were taken into account as input parameters; they were the concrete compressive strength (fc), the column length (L), outer diameter (D), tensile yield stress of the steel column (fy), and steel cover thickness (t). This is worth noting that ranges of (23.2–188.1 MPa), (60–450 mm), (180–4000 mm), (0.86–10.37 mm), (185.7–853 MPa), and (215–13,776 kN) were considered for concrete compressive strength, outer diameter, length of column, steel cover thickness, tensile yield stress of the steel column, and bearing capacity of CFST columns, respectively. Table 1 shows 100 items of data (as examples) out of 303 data used in the modeling of this study. In addition, distributions of the data are displayed in Figs. 4, 5, 6, 7, 8, and 9. In addition, the flowchart of this paper with details is shown in Fig. 10.

Table 1 100 data out of 303 data used in this research
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Fig. 4
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Statistical distribution of concrete compressive strength

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Fig. 5
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Statistical distribution of outer diameter

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Fig. 6
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Statistical distribution of length of column

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Fig. 7
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Statistical distribution of steel cover thickness

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Fig. 8
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Statistical distribution of tensile yield stress of the steel column

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Fig. 9
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Statistical distribution of bearing capacity of CFST columns

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Fig. 10
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Flowchart of this study

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4 Prediction modeling

4.1 ANN modeling

As the previous sections mentioned, ANN is capable of suggesting proper solutions to both linear and non-linear engineering problems. This section presents the neural network models in such a way that their obtained results could be compared with those of the PSO–ANN and the new GEP models discussed in the following subsections. For the purpose of designing the required networks, 80% of all data (i.e., 242 cases) were allocated to the training section in order to be applied to the development process of the models and the remaining 20% were assigned to the testing section to make required evaluations on the model developments [84,85,86,87,88]. Such categorization helps to effectively assess the performance of artificial models regarding the prediction of the CFST columns bearing capacity.

Generally, a key criterion applied to designing of ANN is RMSE; it is considered as the initial termination criterion of the network training process. The RMSE value is obtainable using the values coming from the system (network) and the measured values. Remember that when RMSE = 0, the most appropriate model is achieved.

$$\Delta = t - E_{\text{st}} ,$$
(3)
$${\text{RMSE}} = \sqrt {{\text{Average}}\left[ {\mathop \sum \limits_{k = 1}^{NT} \Delta_{k}^{2} } \right]} .$$
(4)

The parameters Est, t, Δ, and k stand for the measured values, predicted values, error, and number of network outputs, respectively. Furthermore, the determination coefficient (R2) value was also employed; it was responsible for determining the correlation between the measured value and the predicted one. When R2 = 1, it is in its best condition. The two criteria of R2 and RMSE were applied to the evaluation of the prediction models proposed in the present study. We took into consideration different explanations provided in the related literature; then, various ANN models were designed and configured in a way to be effectively used for the prediction of the CFST columns bearing capacity. Figures 11 and 12 present the models of this method for training and testing sections, respectively. As it is clearly observable, the most proper performance of the model was achieved when the iteration value was fixed at 250 and the number of neuron was set to 8. More required details in regard to the most appropriate ANN model for the prediction of bearing capacity of the CFST columns will be presented later.

Fig. 11
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Performance of ANN model for training datasets

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Fig. 12
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Performance of ANN model for testing datasets

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4.2 PSO–ANN modeling

ANNs have been developed through applying optimization algorithms like GA and PSO for solving engineering problems [26, 61, 89,90,91,92]. Regarding BP as a local seeking learning algorithm, the optimal seeking procedure of ANN might be failed with unfavorable solution. Therefore, PSO could be used to adjust the biases and weights of ANN in its performance developing. Considering the local minimal of ANN, there would be occasionally high possibility of convergence; however, PSO has the capability of finding a global minimum. Thus, PSO–ANN would enjoy the search characteristics of their technique. Regarding the search space, PSO searches the global minimum and the ANN will use them to find the best performance with lowest system error.

As previously mentioned, there would be few effective parameters on performance of PSO–ANN like swarm size, inertia weight, and coefficient velocity. According to Armaghani et al. [93], in this study, inertia weight, which is equal to 1 (among the other proposed values of 0.25, 0.5, and 0.75), was selected and applied in PSO–ANN model. Through a parametric research, different combinations of C1 and C2 were regarded to make PSO–ANN models. While, the best model based on the lowest system error was assigned to the combination of C1 = C2 = 2. Therefore, the variables are set as the best C1 and C2 in the PSO system. To determine the swarm size (SS) and the maximum number of iteration (IMax), different SS values ranging from 50 to 400 with incremental step of 50 were taken into account with a total number of iteration = 500. Thus, eight PSO–ANN models were built to predict bearing capacity of CFST columns. Figure 13 displays the effects of SS and IMax on performance prediction of PSO–ANN models. According to this figure, the lowest system error was gained by blue line or SS = 300, showing that a PSO–ANN model with SS = 300 is able to provide the highest performance capacity in predicting bearing capacity of CFST columns. On the other hand, RMSE values of SS were gradually reduced from iteration number 1 until iteration number 400. After iteration number 400, the RMSE results are constant with no alteration. As a result, in this paper, 400 was selected as IMax to predict bearing capacity of CFST columns.

Fig. 13
figure 13

The effect of IMax and SS on performance of PSO–ANN model

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To sum up, inertia weight of 1, C1 = C2 = 2, IMax = 400, SS = 300, and the number of hidden nodes = 8 were optimized as PSO and ANN parameters. It is important to mention that in modeling of PSO–ANN, the same architecture obtained from ANN section was used. More discussion regarding the best PSO–ANN model in estimating bearing capacity of CFST columns will be given later.

4.2.1 GEP-based tree modeling

After obtaining the results of ANN and PSO–ANN networks, GEP predictive models are implemented in this stage, aiming at developing equation for predicting bearing capacity of CFST columns. The values and way of implementing them until obtaining the results of GEP models as well as presenting the relations will be all presented in the mathematical way. The process used in this research for implementing GEP is as follows:

  1. 1.

    In the first step, the fitness function was selected as a criterion for each chromosome’s merit occurrence. RMSE is the common fitness function that is used in modeling process of GEP. However, based on the problem’s conditions, different modes can be used for investigating the models’ performance more accurately. Therefore, each chromosomes’ fitness was determined as follows:

    $${\text{RMSE}}^{\prime} = \frac{1}{{ 1 {\text{ + RMSE}}}} \times 1000.$$
    (5)
  2. 2.

    The second step was to allocate two important sections called the set of terminals (T) and functions (F) to the chromosomes’ structure, which created a mixture of them. The independent variables are considered as the terminal set, and the function set is usually defined according to the main core of the problem. In the current study, trigonometry and mathematical functions were used as follows:

    $$F = \left\{ { + ,\; - ,\; \times ,\; \cdot ,\;{\text{Sin,}}\;{\text{Cos}},Arc{\text{Tan}},\tanh ,\;{\text{sqrt}}} \right\}$$
    (6)
  3. 3.

    In the third step, structural parameters of GEP were introduced and applied to the system. The number of genes parameter was introduced for ET subsections specified for each chromosome. According to Ferreira’s investigation [72, 73, 94] and some other researchers [79,80,81,82,83], the best way to obtain proper values for structural parameters of GEP is the method of trial and error. The analysis process started with the increasing values of the above-mentioned parameters of GEP, and then the predictive performance of the GEP model was checked. Several GEP models are designed and implemented with different parameters for predicting compressive strength of composite columns. Finally, after executing these processes for several times, values of the number of chromosomes, head size, and number of genes were found to be 40, 5, and 3, respectively, for this section.

  4. 4.

    Step 4 was to select the rates of genetic operators. In this step, assuming the proposed values by previous researchers [64, 65, 84], some other GEP models were created using the trial and error method. The obtained values of GEP parameters are tabulated in Table 2.

    Table 2 GEP model parameters
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  5. 5.

    In the final step, the linking function for connecting the created genes was defined. There are various linking functions like addition (\(+\)), subtraction (\(-\)), division (\(\div\)), and multiplication (\(\times\)). In this research, addition of different sections was used to connect sub-ETs because it provides a better connection in comparison with other functions.

To evaluate the predictive performance of the GEP models, R2 and RMSE were used as performance indices. Several parameters of the GEP model were examined in this section to determine its impact on the performance of models. Figure 14 displays the effects of the most influential parameters on GEP results including the number of generations, the number of genes, and size of head. Therefore, based on Fig. 14, generation, gene, and size of head were considered as 3000, 3, and 5, respectively. Eventually, after the aforementioned implementation, the results of some different models (as equations) are presented in Table 3. This way, different mathematical equations can be compared and their performance can be evaluated for prediction of uniaxial compressive strength in composite columns. In the end, the model No. 4 was chosen as the selected model based on the R2 results.

Fig. 14
figure 14

Effects of the most influential parameters on GEP results: a number of generations, b number of genes/size of head

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Table 3 The equations to predict the bearing capacity of CFST columns with GEP
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According to Fig. 15, the expression tree of each gene of model/Eq. (4) was presented. All functions and terminal sets were illustrated in the circles. To extract mathematical equations, reading the circles from left to right and top to bottom is recommended (models of 1–5). After extracting the equation of each gene, the final predictive model of GEP was obtained by subtracting of Eq. (4). All statistical data of model No. 4 are presented in Table 4 for both training and testing sections. More discussions regarding the best GEP model will be presented later.

Fig. 15
figure 15

The tree expression of Eq. (4)

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Table 4 Statistical data of Eq. (4)
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5 Results

Three intelligent systems, i.e., ANN, PSO–ANN, and GEP were developed in this study to predict the bearing capacity of CFST columns. As discussed earlier, many parametric studies were constructed and the best one for each method (ANN, PSO–ANN, and GEP-based tree) was selected. The selection of the best predictive models was based on the results of R2. The mentioned results for training and testing datasets of the best predictive models are tabulated in Table 5. According to the obtained results, 0.895 and 0.881 as R2 of training and testing sections, respectively, are considered as acceptable finding to estimate bearing capacity of CFST columns. However, if more accurate predictive result is of interest, PSO–ANN can be applied herein. When PSO–ANN model is developed, R2 results of 0.910 and 0.904 for training and testing datasets can be achieved to estimate bearing capacity of CFST columns. However, if a perfect predictive model (among the applied ones) is of interest, GEP-based tree equation with R2 values of 0.928 and 0.939 for training and testing datasets has shown its capability. Actually, this technique can provide higher performance capacity compared to other predictive models in estimating bearing capacity of CFST columns. The GEP-based tree can be introduced as a new model that is able to propose new equation for solving problems in structural engineering fields. In the next section, GEP equation will be used as a cost function for optimizing purposes.

Table 5 Coefficient of determination results in predicting bearing capacity of CFST columns
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6 Optimization modeling

In this section, the whale optimization algorithm (WOA) is developed to optimize the results of bearing capacity of CFST columns. To examine the WOA algorithm, the selected functions (Eqs. 7, 8) were employed. The minimum values of these functions in the mentioned intervals are 3 and − 1.9133, respectively. Figures 16 and 17 illustrate the three-dimensional graph of these two functions in the specific interval. Figures 18 and 19 demonstrate the results obtained by WOA, which are based on these two equations. As it can be seen, the written code of this algorithm can identify the minimums well. That is why this code can be run for the research conditions obtained in the previous section.

Fig. 16
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The three-dimensional graph of Eq. (7)

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Fig. 17
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The three-dimensional graph of Eq. (8)

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Fig. 18
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The result of WOA algorithm for Eq. (7)

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Fig. 19
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The result of WOA algorithm for Eq. (8)

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$$\begin{aligned} F_{1} (x) = & \left[ {1 + (x_{1} + x_{2} + 1)^{2} (19 - 14x_{1} + 3x_{1}^{2} - 14x_{2} + 6x_{1} x_{2} + 3x_{2}^{2} )} \right] \\ & \times \left[ {30 + (2x_{1} - 3x_{2} )^{2} (18 - 32x_{1} + 12x_{1}^{2} + 48x_{2} - 36x_{1} x_{2} + 27x_{2}^{2} )} \right], \\ \end{aligned}$$
(7)
$$F_{2} \left( x \right) = \sin (x_{1} + x_{2} ) + \left( {x_{1} + x_{2} } \right)^{2} - 1.5x_{1} + 2.5x_{2} + 1.$$
(8)

To optimize bearing capacity of CFST columns, the selected predictive model (GEP-based tree) was used. In fact, the GEP equation is considered as a cost function in WOA technique. Different models of WOA algorithm (using various parameters) were designed, each of which was executed by adjusting the parameters of the optimization algorithm. After a set of conducted analyses, the most appropriate parameters of WOA algorithm were obtained. The best parameters that can deliver well the performance of WOA algorithm for optimizing this problem are presented in Table 6.

Table 6 The effective parameters of WOA for optimization of the problem
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With the use of the best model results, the optimum parameters that can provide bearing capacity of CFST columns were determined. The best cost function is presented in Fig. 20 for this problem. The proposed parameters are given in Table 7. It should be noted that the changes in these parameters are assumed to be the values considered for modeling (Table 1). As it can be seen, in cases where optimization was done, appropriate enhancements were gained in performance of bearing capacity of CFST columns. For example, optimum values of 186.31 MPa, 416.09 mm, 364.31 mm, 6.79 mm, and 823.88 MPa increased the bearing capacity by 23,436.63 kN in comparison with the initial sample. Therefore, different patterns of designing can be applied under various conditions and the best performance can be reached. As a result, WOA is able to optimize the model inputs in a way that maximum amount can be obtained for bearing capacity of CFST columns. As a result, WOA can be introduced as a powerful optimization algorithm in maximizing bearing capacity of CFST columns.

Fig. 20
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The best cost of WOA for optimization of the problem

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Table 7 The proposed values for optimization of the problem
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7 Conclusions

The specification of the bearing capacity of CFST columns made up of hollow steel cylinders filled with concrete is among important issues in constructing different structures. To this end, this paper presented a novel intelligent method to predict and optimize the bearing capacity of such structures. The data used in this research were collected from conducted experimental frameworks. The data comprise parameters of the outer diameter, concrete compressive strength, tensile yield stress of the steel column, steel cover thickness, and length of the used samples. The ANN and neuro-swarm methods were used for prediction purposes. The GEP-based tree model was implemented and developed under different conditions to predict the bearing capacity of CFST columns and finally, an equation was proposed to estimate bearing capacity of CFST columns. Based on the obtained results, the GEP-based tree equation with R2 values of 0.928 and 0.939 for training and testing datasets, respectively, indicates higher performance capacity compared to ANN and neuro-swarm predictive models. On the other hand, neuro-swarm provided higher performance capacity compared to ANN. Therefore, the GEP-based tree technique was selected as the best predictive model for estimating the bearing capacity of CFST columns. A GEP-based tree equation was used as a cost function in the WOA optimization algorithm. Different WOA models were constructed to obtain higher values of the bearing capacity. According to the obtained results, it was found that WOA (with increasing bearing capacity to 23436.63 kN) was able to maximize significantly the bearing capacity of the CFST columns. The optimum values obtained in this study allow engineers and researchers to reach the best performance in designing and constructing CFST columns.