Novel Fuzzy-Based Optimization Approaches for the Prediction of Ultimate Axial Load of Circular Concrete-Filled Steel Tubes

Abstract

An accurate estimation of the axial compression capacity of the concrete-filled steel tubular (CFST) column is crucial for ensuring the safety of structures containing them and preventing related failures. In this article, two novel hybrid fuzzy systems (FS) were used to create a new framework for estimating the axial compression capacity of circular CCFST columns. In the hybrid models, differential evolution (DE) and firefly algorithm (FFA) techniques are employed in order to obtain the optimal membership functions of the base FS model. To train the models with the new hybrid techniques, i.e., FS-DE and FS-FFA, a substantial library of 410 experimental tests was compiled from openly available literature sources. The new model’s robustness and accuracy was assessed using a variety of statistical criteria both for model development and for model validation. The novel FS-FFA and FS-DE models were able to improve the prediction capacity of the base model by 9.68% and 6.58%, respectively. Furthermore, the proposed models exhibited considerably improved performance compared to existing design code methodologies. These models can be utilized for solving similar problems in structural engineering and concrete technology with an enhanced level of accuracy.

1. Introduction

Concrete-filled steel tube (CFST) members make better utilization of steel and concrete than traditional bare steel or reinforced concrete structures. The steel tube gives confinement to the concrete infill, while the concrete infill prevents the inward buckling of the steel tube. CFST members have a long history of being used in a broad range of construction projects due to their efficiency as structural components. As an example, CFSTs have been utilized as (1) mega columns in super high-rise buildings, (2) chord members in long-span arch bridges, (3) bridge piers, (4) floodwall piling, and (5) underwater pipeline structures, as described by researchers like Wang et al. [1]. For the most part, the CFST components in these situations are utilized to support compressive forces.

When it comes to improving the compressive strength of CFST components, there are primarily two approaches that are used. Using bigger cross sections is one approach. However, it may increase structural weight (and as a result the seismic impact) and decrease useable space, making it a less feasible or cost-effective solution. Alternatively, high strength steel (i.e., with a yield stress higher than 525 MPa) and high strength concrete (i.e., with a compressive strength greater than 70 MPa) are two additional viable methods [AISC 360 [2]].

According to experts like Nishiyama et al. [3], Kim [4] and Han [5] and others, many studies have been carried out to examine the behavior of conventional-strength members of the CFST. Several researchers have experimented with the behavior of high-strength CFST columns facilitating their adoption in practice. For example, Cederwall et al. [6], Varma [7], Uy [8], Liu et al. [9], Mursi and Uy [10], Sakino et al. [11], Lue et al. [12], Aslani et al. [13], and Xiong et al. [14] have performed experimental testing on high-strength rectangular CFST short columns. Lai and Varma [15] reviewed these experiments and provided design equations for calculating the cross-sectional strength of high-strength rectangular CFST columns and also effective stress-strain relationships for the steel tube and concrete infill of such high-strength components.

Additional experimental tests on CFST columns were conducted by Gardner and Jacobson [16], Bergmann [17], O’Shea and Bridge [18], Schneider [19], O’Shea and Bridge [20], Giakoumelis and Lam [21], Sakino et al. [11], Zeghiche and Chaoui [22], Yu et al. [23], de Oliveira et al. [24], Liew and Xiong [25], Chen et al. [26]. The experimental results from previous studies have been employed in this study in order to build an experimental database. It is noted, however, that experiments featuring columns with fibers in the concrete, stainless or aluminum steel tubing, grease on the inner surface of the tubing, or concrete infill alone were excluded from this database.

Machine learning (ML) methods have been widely used in many civil engineering applications [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55], particularly in compressive structures [56,57,58,59]. ML uses databases to develop models that can solve various linear and nonlinear problems with varying degrees of complexity. These methods, using computer processing, help considerably in solving problems more efficiently and quickly, and is introduced as a powerful alternative method for older, experimental and statistical models. An optimization and tree-based approach has been developed by Sarir et al. [60] to find out the maximum capacity of circular CFST members. Short CFST members’ load-bearing capacity was predicted by Ahmadi et al. using an artificial neural network [61,62]. A gene expression model for predicting circular CFST capability was established by Güneyisi et al. [63] and Ipek and Güneyisi [64]. In the study of Moon et al. [65], the load-bearing behavior of circular CFST was also examined using a fuzzy logic model. According to Al-Khaleefi et al. [66], the fire resistance of CFST columns has also been studied using a machine learning method that considers material characteristics and loading circumstances. For sections other than circular, Ren et al. [67] recently published a study on the prediction of square CFST members, using support vector machines and particle optimization methods. While for the same section Tran et al. [68] used a neural network model to predict the ultimate load. Also, Lee et al. [69] used a categorical gradient boosting algorithm to predict the strength of both circular and rectangular CFSTs under concentric or eccentric loading. Zarringol et al. [70] used ANN for the same problem. It can be concluded from these studies that ML methods prove quite promising in investigating the mechanical behavior of structures made up of CFST members.

In this research, the major goal is to develop a regression machine learning model for compressive circular CFST, particularly in contemporary buildings. This was achieved using a hybridizing fuzzy system (FS) with two optimization algorithms known as the firefly algorithm (FFA) and differential evolution (DE). The input data consist of column length, cross-section diameter and steel tube thickness in addition to concrete compressive and steel yield stress. Precise quality metrics such as root mean-squared-error (RMSE), and coefficient of determination (R²) were utilized throughout the model’s testing/validation phase. The FS-FFA and FS-DE models were evaluated and compared with existing design code methodologies to highlight the best predictive model for the examined problem.

2. Research Significance

CFST design can be done using different methods and codes around the world. Accurate, faster, and less costly design is one of the priorities of any structural project. Due to the fact that an accurate CFST design has important effects on the stability of structures, examining different techniques could give a better understanding of their effects and behaviour. Therefore, this research, using a new generation of computational methods developed by learning machines, is aimed at coming up with a practical solution to the aforementioned problem. Using a combination of FS and optimization algorithms (i.e., FFA and DE), new predictive models can be developed to more accurately and quickly evaluate CFST design. Optimal solutions of hybrid models consisting of these conditions can be used for new conditions and provide acceptable results considering practical applications in industrial fields.

3. Short Literature Review on Design Codes

The design of circular CFST columns is already supported by several steel and composite codes, available worldwide. Such codes include EN1994 [71] in Europe, AISC 360 [72] in the USA, AIJ [73] in Japan. Besides providing the squash load that is relevant for short columns, design codes also provide methodologies to predict the resistance against flexural buckling, which becomes the critical failure mode for long, slender columns. Local buckling of the steel tube is also a failure mode relevant for thin-walled steel sections. It is typically covered by placing section slenderness limits and depending on them, either accounting for a reduced effective steel sectional area (i.e., EN1994 [71]), or limiting the ultimate stress the composite section may reach (i.e., AISC 360 [72]). Regarding squash load, which involves the plastic strength of the steel and concrete parts of the CFTS section, the influence of the increased concrete confinement provided by the circular tube is typically expressed through an increase of the concrete strength contribution. The following formulas describe the squash loads, for the EN1994 [71], AISC 360 [72], AIJ [73] design codes (ignoring any safety factors):

$$N_p^{EN1994}=\{\begin{array}{ll}{\eta }_a{f}_y{A}_s+\left(1+{\eta }_c\frac{t{f}_y}{d{f}_c^{\prime }}\right)f_c^{\prime }{A}_c\hfill & ,\overline{\lambda }<0.5\hfill \\ f_y{A}_s+f_c^{\prime }{A}_c\hfill & ,\overline{\lambda }\ge 0.5\hfill \end{array}$$

(1)

$$N_p^{AISC360}=f_y{A}_s+0.95f_c^{\prime }{A}_c$$

(2)

$$N_p^{AIJ}=1.27f_y{A}_s+0.85f_c^{\prime }{A}_c$$

(3)

Factors ${\eta }_a$ and ${\eta }_c$ account for the member slenderness $\overline{\lambda }$. For slender columns, the squash loads given above fail to represent the ultimate compressive load. In such cases, buckling phenomena emerge that cause an earlier failure, depending on the global column slenderness. The methodologies provided by the aforementioned design codes are differentiated in this context. Due to space limitations the relevant expressions are not reproduced herein however.

All design codes place specific limits on their field of application. These are related to material strength limits, steel tube slenderness, global slenderness or steel to concrete ratio.

Table 1. Design codes application limits, related to circular CFSTs.

Code	${\mathit{f}}_{\mathit{y}}$ (MPa)	${\mathit{f}}_{\mathit{c}}^{\prime }$ (MPa)	Section Slenderness	Other
EN1994 [71]	$235\le f_y\le 460$	$25\le f_c^{\prime }\le 50$	$\frac{d}{t}\le 90\frac{235 \mathrm{MPa}}{f_y}$	$0.2\le \frac{A_s{f}_y}{N_p}\le 0.9$
AISC 360 [72]	$f_y\le 525$	$21\le f_c^{\prime }\le 69$	$\frac{d}{t}\le \frac{0.31E_s}{f_y}$	$A_s\ge 0.01A_{sc}$
AIJ [73]	$235\le f_y\le 355$	$18\le f_c^{\prime }\le 60$	$\frac{d}{t}\le 1.5\frac{23500 \mathrm{MPa}}{\sqrt{\min \left\{f_y;0.7f_u\right\}}}$	$\frac{L_e}{B}\le 50$

A_sc, A_s, and A_c are the areas of the total cross section, the steel tube and the concrete core, respectively L_e is the column effective length.

presents the relevant application limits for the codes examined.

4. Modeling Approaches

4.1. Fuzzy System (FS)

A chapter titled “fuzzy sets” by Professor Lotfizadeh in 1965 presented the fuzzy theory [74]. Initially, his primary objective was to create a more accurate model of how natural language processing works. Fuzzy sets, fuzzy events, fuzzy numbers, and phases are only a few of the innovations he made to mathematics and engineering thanks to these ideas. A rule base, which includes If–Then rules created by application specialists, constitutes FS’s core component [75]. Membership functions are used to deploy the fuzzy sets. For the FS process, the most popular fuzzifiers are Gaussian, Singleton, and Triangular. In addition, the most often used defuzzifiers in the literature are center of gravity, center average, and maximum. Fuzzy logic principles govern the firing of If–Then rules while the inference engine is operating. A fuzzy rule has the following syntax [76]:

$$If x_1 is A_{1 }and\dots x_n is A_n then Y is B$$

(4)

Fuzzy sets in U R (U is the input space) are Ai and V R (V is the output space) are B, and X is equal to the product of the variables in the input space and the variables in the output space, respectively. There are two types of FS controller: closed-loop and open-loop. The product inference engine and Gaussian fuzzifier were used in the following ways:

$$f_{1\left(x\right)}=\{\begin{array}{c}\exp [-\frac{1}{2}\left(\frac{x-m_1}{\sigma }\right)] x\le m_1\\ 1 otherwise\end{array}$$

(5)

$$f_{2\left(x\right)}=\{\begin{array}{c}1 x\le m_2\\ \exp [-\frac{1}{2}\left(\frac{x-m_1}{\sigma }\right)] otherwise\end{array}$$

(6)

4.2. Firefly Algorithm (FFA)

Yang was the first to propose FFA as a nature-inspired, meta-heuristic algorithm [77]. Engineers have used this method to address a variety of issues. The most critical aspects of the FFA process are the formulation of attraction and the change in light intensity. Fireflies will operate virtually independently in FFA modelling, which is advantageous for parallel implementation in particular. Fireflies in this algorithm tend to congregate closer to the optimum, making it superior to the particle swarm optimization (PSO) and the genetic algorithm (GA) [76,78]. Figure 1 depicts FFA’s foundation for a better understanding. Several studies, including Yang [77], Zhang and Wu [79], and Apostolopoulos and Vlachos [80], go into great depth regarding FFA. Reviewing past research shows that the FFA may be utilized as a powerful tool for engineering optimization in almost all fields [81,82]. Gholizadeh and Barati [83], for example, used the PSO, FFA, and harmony search (HS) to explore the size and form optimization of truss systems. In terms of optimizing the size and geometry of truss structures, FFA outperformed PSO and HS.

4.3. Differential Evolution (DE)

Storn and Price (1997) first proposed the concept of differential evolution (DE) as a stochastic population-based search technique [84]. NP members are randomly selected from the original population (parents) before the search may begin. Using crossover, mutation, and selection operators, the DE technique then produces a new population (i.e., offspring). This iteration’s members are chosen by comparing how similar they are to the previous iteration’s members. This cycle is repeated until the desired outcome is achieved. The following sections describe the major stages of this algorithm [84].

4.3.1. Generating the Initial Population

If the problem’s decision variables are indicated by D, the initial population vector is produced with a random size N*D inside the decision variables allowed range, according to the following equation:

$$x_{io}=x_{imin}+round\left({\phi }_i\times x_{imax}-x_{imin}\right), i=1,\dots ,NP$$

(7)

There are lower and higher limits on the choice variables x_imin and x_imax, respectively, while index i is a random number between 0 and 1.

4.3.2. Mutation

To carry out the mutation procedure, the following equation is used:

$$v_{i,G+1}=x_{r1,G}+F\times \left(x_{r2,G}-x_{r3,G}\right)$$

(8)

There are three randomly selected members of the population in this case, and the scaling factor F ranges from zero to two, giving us a mutant vector $v_{i,G+1 }$ and three randomly picked members of the population.

4.3.3. Crossover

This operator combines the modified particles with the members of the target group that were chosen in the first stage as follows:

$$u_{ij,G+1}=\{\begin{array}{c}{v}_{ji,G+1 r<Cr or j=r{n}_i}\\ x_{ji,G} otherwise\end{array}$$

(9)

where j = 1, 2, …, D; ∈ r_j [0, 1] is the random number; C_r stands for the crossover constant ∈ [0, 1]; and ∈ rn_i (1, 2, …, D) is the randomly chosen index.

4.3.4. Selection

Once all operators have been initialised to their respective goal functions, a new measurement vector and target member are created. If the measurement vector’s value exceeds the target member’s, the member is promoted to the next generation. If this does not happen, the target member will be added to the population of the following generation. Figure 2 depicts the DE’s pseudo-code.

4.4. Hybridization of FS

The CCFST is predicted using two hybrid FS-FFA and FS-DE in this research. Five characteristics were utilized as inputs in the hybrid FS modelling procedure, with CCFST ultimate load being the output. The proposed FS-FFA and FS-DE models were trained and tested using 328 datasets of data in the training phase and 82 datasets of data in the testing phase. Fuzzy-based modifications to FFA and DE are suggested in this research to remove or minimize model drawbacks. In this structure, the member (population) of optimization algorithms in each step may affect each other’s movements. For determining progress in the program, we used two metrics to indicate how close the algorithms are getting to the ideal answer. We call this loop counter (iteration) Count and its value is decided by expertise or via trial and error [85]. The fuzzy controller will have to deal with this last problem. The following is an introduction to the delta parameter:

$$Delt{a}^i=F\left(Bes{t}^i\right)-F\left(TBes{t}^{i-1}\right)$$

(10)

Iteration i yields Bestⁱ, which is the best solution, whereas iteration i−1 yields TBestⁱ⁻¹, which is the best solution. One of the benefits of the hybrid FS is that it regulates the fundamental database, i.e., physics of the examined problem. As a result, convergence speed may be improved by making the appropriate initial adjustments. A MATLAB programme was used to implement the hybrid FS model’s code. The following equation is used to standardize datasets before beginning hybrid FS modelling:

$$Xnorm=\frac{X-Xmin}{Xmax-Xmin}$$

(11)

where X, Xmin, and Xmax represent the parameters’ real values, minimum and maximum values, respectively, while Xnorm represents the parameter’s normalized value. FS-FFA modeling’s most critical parameters are Npop (swarm size), Alpha (mutation coefficient), Gamma (light absorption coefficient), Beta (attraction coefficient base value) and Maxiter (maximum number of iterations), according to prior research [81,82,85]. Parameters’ number of iteration, crossover constant and population (Npop) are also effective for the DE algorithm [84,86]. Following the trial and error technique, Npop of FFA was set to 50, Npop of DE was set to 80, Alpha was set to 0.25, Gamma was set to 1, crossover constant was set to 0.8, Beta was set to 2, and the number of iterations for both algorithms was set to 500. Figure 3 depicts the steps involved in putting hybrid FS into practice.

5. Data Setup

A variety of sources were used to compile the database for this study such as Wang et al. [87], Geng [88], Dong et al. [89], Wang et al. [90], Chen et al. [91], Yang et al. [92], Wang et al. [93], Wei et al. [94] Hoang and Fehling [95], He et al. [96]. These sources include axial compression tests on circular CFST columns that make up 410 samples in total (whole used datasets for modeling are presented in Appendix A). Several geometrical factors and mechanical characteristics were utilized in these tests to investigate the failure of CFST columns under axial stress. These are column length (L), diameter (D), and thickness (t) as geometrical input variables. Additionally, the steel tube yield stress (fy) and the compressive strength (fc) of the filling concrete are the material specific variables representing their mechanical properties. The only output of the problem is the CFST column ultimate experimental axial compressive load (Pexp).

Table 2. General information of dataset.

Parameter	Unit	Min	Average	Max	SDT
L	mm	180	720.73	4000	594.56
D	mm	60	169.41	550	74.04
t	mm	0.86	4.47	16.72	2.59
fy	MPa	184.8	388.38	1153	170.29
fc	MPa	23.2	74.98	188.1	44.01
Pexp	KN	215	2992.71	29590	3213.2

shows a statistical examination of the dataset.

6. Development of the Hybrid Models

The performance of FS-FFA and FS-DE models is discussed in this part, presenting how well it can predict the circular CFST ultimate compressive load. In order to do this, three quantitative standard statistical performance measures, namely R², the a20-index, and RMSE, have been used, as described by Equations (12)–(14) [52,58,97,98,99,100,101,102]:

$$RMSE=\sqrt{\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(y_{fr,i}-{\widehat{y}}_{fr,i}\right)}^2}$$

(12)

$$R^2=1-\frac{{{\displaystyle \sum }}_{i=1}^n{\left(y_{fr,i}-{\widehat{y}}_{fr,i}\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(y_{fr,i}-{\overline{y}}_{fr,i}\right)}^2}$$

(13)

$$a20-index=\frac{m20}{n}$$

(14)

where, the predicted and measured values for n data are indicated by ${\widehat{y}}_{fr,i}$ and $y_{fr,i}$, respectively, and m20 is the number of samples with a value of (experimental value)/(predicted value) ratio, between 0.80 and 1.20. The best performance of the models is achieved when the errors (RMSE) are zero and the R² is close to one. The performance of the developed FS-FFA and FS-DE is presented in

Table 3. The final result of hybrid fuzzy system (FS) models.

Model	Training			Testing
	a20-Index	R2	RMSE	a20-Index	R2	RMSE
FS-FFA	0.9604	0.9854	482.0362	0.8659	0.9880	415.4471
FS-DE	0.9634	0.9571	655.4708	0.8659	0.9876	419.4502

in terms of R², RMSE, and a20-index. The optimal model values have been optimally picked having as objective to achieve the best possible performance metrics.

In Table 3, it can be shown that the proposed FS-FFA and FS-DE have a good performance for predicting CFST values. In the training section, the performance of the FS-FFA model seems better in terms of RMSE and a20-index, compared to the FS-DE model whereas the latter scores a higher R² value. However, in the test section, the performance of the two models proves more closely matched, with the FS-FFA achieving slightly better metrics. Given that the FS-FFA model has been able to provide better predictions in both sections, it becomes the preferred one for the estimation of circular CFST ultimate compressive load. Figure 4 and Figure 5 show separately for the training and the testing datasets graphs of predicted vs. experimental loads for both hybrid models. It can be seen that both models exhibit a consistent performance throughout the range of available compressive loads.

7. Discussion

7.1. Comparison against Alternative Hybrid Models

In this section, a comparison is made between the developed hybrid models with the base model and the other two traditional hybrid models. Given that optimization algorithms have different performances for each problem, it is possible to identify their differences and compare their performances by examining several algorithms together. Therefore, two hybrid models, FS-genetic algorithm (GA) and FS-particle swarm optimization (PSO), were developed to compare with the two developed models in this study. The findings showed that the FS-FFA, FS-DE, FS-GA, and FS-PSO models outperformed FS in terms of prediction accuracy for training data by 9.68%, 6.58%, 5.68%, and 1.56% respectively. Among the hybrid models, the FS-FFA model provided the best performance according to the various criteria for predicting circular CFST ultimate load values.

7.2. Comparison against Design Codes

In this section a comparison between the two developed models against the predictions of the three design codes, mentioned earlier in the text (EN1994, AISC360 and AIJ) is presented. The design code calculations were performed ignoring any safety factors. Also, the calculations were not focused on the squash load only but took into account the relevant for more slender columns buckling failure methodologies available in each code.

Table 4. Performance indices on the testing datasets.

Ranking	Model	a20-Index	R2	RMSE
1	FS-FFA	0.8659	0.9880	415.4471
2	FS-DE	0.8659	0.9876	419.4502
3	AIJ [73]	0.6341	0.9842	786.3858
4	EN1994 [71]	0.5732	0.9681	1119.6477
5	AISC 360 [72]	0.3659	0.9814	1330.6249

presents the performance indices for the two developed models and the respective ones for the design codes. The results correspond to the designated testing datasets among the specimens in the experimental database, amounting to 82 specimens. The various models in the Table are ranked according to their RMSE index. It appears that the two developed models achieve a considerable improvement in almost all indices, compared to the design codes. In particular the improvement from the best performing code which proves the Japanese AIJ [73], is remarkable for both the RMSE and a20-index. A marginal improvement is also found in terms of R² index. Among design codes, the AIJ [73] achieves improved performance compared to the other codes.

Figure 6 illustrates for the two hybrid models and for the examined design codes, the individual experimental vs. predicted load values for all specimens in the testing datasets. It can be visually inspected that the FS-FFA hybrid model achieves a better fit to the experimental values, with less outliers, over the entire range of specimens.

7.3. Limitations and Future Works

This research developed several hybrid models using artificial intelligence to predict the ultimate compressive load of CCFST columns. These models are based on data collected from laboratory works of previous research. Given that the structure of models is highly dependent on the number of parameters, their types must be taken into account in measuring and using such data. Laboratory outline data reduce the accuracy of prediction. On the other hand, the purpose of this study is to develop non-linear models to more accurately evaluate the target parameter. Therefore, a balance between the input data and their statistical characteristics must be elaborated. By doing this, a wider range of data can be analyzed and a model with higher power can be developed. Since these models have used two optimization algorithms to improve the performance of the base model, some other optimization techniques such as the whale optimization algorithm can be examined to increase the base model performance. The base model in this research was made using the FS model, which has special features, while by changing the basic model to other predictive models such as neuro-fuzzy, new results can be achieved. The last future direction of this research can be related to increasing the number of data samples with circular cross sections to develop a new model with a level of more generalization.

8. Conclusions

In this study, a novel artificial intelligence-based prediction model is used to correctly evaluate CCFST columns’ axial compression capacity. FS and two recent nature meta-heuristic optimization methods known as FFA and DE were combined to create two hybrid FS-FFA and FS-DE models. It was also combined with two common optimization techniques, the GA and PSO. An extensive database of 410 experimental tests for the CCFST columns was gathered from openly available papers for this research project. A statistical and visual analysis was undertaken to determine the effectiveness and correctness of the findings, and the following conclusions can be reached.

According to the research findings, the suggested hybridization models outperformed the basic FS model when it came to resolving the axial compression capacity problem. The findings showed that the FS-FFA, FS-DE, FS-GA, and FS-PSO models outperformed FS in terms of prediction accuracy for training data by 9.68%, 6.58%, 5.68%, and 1.56%, respectively. According to all performance assessments, the new suggested FS-FFA model is optimal for the prediction of the axial compression capacity of CCFTS columns, with improved RMSE and a20-index compared with FS-DE. Additionally, the proposed model achieved a significantly improved prediction of the ultimate compressive load compared to available design code predictions. In particular, RMSE of the FS-FFA model was reduced by 47% from AIJ [73] and more from the EN1994 [71] and AISC360 [72], whereas a20-index was also considerably increased.

The base model in this research was made using the FS model, which has special features, while by changing the basic model to other predictive models such as neuro-fuzzy, new results can be achieved. The last future direction of this research can be related to increasing the number of data samples with circular cross sections to develop a new model with a greater generalization level.

This model’s performance and machine learning methods are largely reliant on the database used. However, a more sophisticated and bigger database may have significant effects on the hybrid FS model’s final outcomes. Other optimization techniques such as the whale optimization algorithm can be examined to increase the base model performance. Furthermore, other sections’ geometries of CFTS, such as squares and round-ended squares, can also be investigated by the proposed hybrid models in this research. Close form equations of this issue using machine learning models will be very beneficial to engineering in the future.