# Predicting the Ultimate Axial Capacity of Uniaxially Loaded CFST Columns Using Multiphysics Artificial Intelligence

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## Abstract

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_{st}by GEP, ANN and ANFIS for training are 0.0416, 0.1423, and 0.1016, respectively, and for N

_{lg}these values are 0.1169, 0.2990 and 0.1542, respectively. Corresponding OF values are 0.2300, 0.1200, and 0.090 for N

_{st}, and 0.1000, 0.2700, and 0.1500 for N

_{lg}. The superiority of the GEP method to the other techniques can be seen from the fact that the GEP technique provides suitable connections based on practical experimental work and does not rely on prior solutions. It is concluded that the GEP model can be used to predict the bearing capacity of circular CFST columns to avoid any laborious and time-consuming experimental work. It is also recommended that further research should be performed on the data to develop a prediction equation using other techniques such as Random Forest Regression and Multi Expression Program.

## 1. Introduction

#### 1.1. Concrete Filled Steel Tube Artificial Modelling

_{c}′) of 67 MPa and 90 MPa and steel tube yield stress (f

_{y}) of 420 MPa and 440 MPa, respectively. The recently released AISC 360-16 [9] and AS/NZS 2327 [10] is applicable to f

_{c}′ of 69 MPa and 100 MPa and f

_{y}is limited to 525 MPa and 690 MPa, respectively. The Eurocode [11] permits f

_{c}′ and f

_{y}up to 50 MPa and 460 MPa, respectively. In comparison, Liew and Xiong [12] extended these limits to 90 MPa and 460 MPa, respectively. The use of high strength material is valuable for reduction of the size of CFST columns which eventually leads to the savage of floor space and lesser construction cost. The high strength steel tube enhances the elastic behavior and thus improves the confinement effect towards the concrete core. Use of concrete in CFST helps in the functional optimization of both materials. Advancements in the construction industry permit high strength materials to be practically utilized. In addition, the equations available in the mentioned standard codes do not agree with each other. Moreover, these codes are based on the pre-assumed stress–strain curve of CFST, which makes the validity of the presented equations suspicious. To tackle this issue, many researchers have conducted experimental studies on the utilization of high strength materials in CFST columns. Khan et al. [13] use f

_{c}′ and f

_{y}up to 113 MPa and 762 MPa, respectively, in CFST columns. Mursi and Uy [14] and Sakino et al. use normal strength concrete in CFST columns with f

_{y}up to 761 MPa and 853 MPa, respectively.

_{t}: eccentricity at the top face or loading face, e

_{b}: eccentricity at the bottom face, f

_{y}: yield strength of the tube and f

_{c}: compressive strength of the infilled concrete. The schematic layout showing the input variables used to predict the capacity of CFST columns has been provided in Figure 1. The GEP algorithm delivers a simplistic empirical hand-based expression that can be used for future unseen data. The ANN and ANFIS algorithms are also used as a predictor to confirm the validity of the equation. A detailed and comprehensive database has been developed from peer-reviewed internationally published articles. This widespread database ensures the applicability of the model for new data. Statistical error checks are employed to verify the performance of the established models. In the end, a permutation feature analysis and a parametric study were also conducted to arrive at an accurate, reliable model.

#### 1.2. Detailed Description of Machine Learning Algorithms (ANN, ANFIS, GEP)

_{st}/N

_{lg}), eight inputs were selected, while in the hidden layer the input parameters (D, t, L, L/D, e

_{t}, e

_{b}, f

_{y}and f

_{c}) were multiplied by a suitable weight factor for connection. A threshold value at every node (θ

_{j}) is added to the weighted input values after their summation. The resultant input (Ij) is passed through the linear transfer function, which is called the transfer phase. The various activation transfer function (AFs) usually used in ANNs are the linear sigmoid, the stepped hyperbolic tangent and the logistic, among others [45]. An activation function is the key feature of a neural network which plays an important role in the artificial neural network model. It can be observed that these activated functions assist in appointing nonlinearity to the neural networks, due to which the selection of the appropriate activated function becomes very important [46]. Activated functions which have been used in the past include tangent hyperbolic and logistic sigmoid activated functions [47], the transcendental type parametric algebraic activated function [48], swish activated functions [49], and Multistate AF’s to improve the DNN models [50] etc.

_{st j}or N

_{lg j}) is obtained as the resulting output, and the input of a PE is basically the output of the previous PE. For the hidden and output layer, every neuron utilizes the Logistic function (Equation (1)) as an activated function [54]. Moreover, the complete process can be observed from Equations (1)–(3).

- (a)
- data collection,
- (b)
- ANFIS growth,
- (c)
- variables selection,
- (d)
- training and testing,
- (e)
- results

_{t}, e

_{b}, f

_{y}and f

_{c}) is shown in Figure 3. The circle denotes the set nodes, while the square denotes the adaptive nodes. The two statements used for the presentation of the architecture of the ANFIS are IF-THEN statements are as follows.

_{1}) and (t is B

_{1}) THEN,

_{2}) and (t is B

_{2}) THEN,

_{n}denotes the fuzzy outputs (N

_{st}, N

_{lg}) for the fuzzy inputs (D, t, L, L/D, e

_{t}, e

_{b}, f

_{y}and f

_{c}), according to the fuzzy statement, A

_{i}and B

_{i}denote the fuzzy sets, and p

_{i}, q

_{i}, and r

_{i}denote the arrangement elements determined in the training cycle.

#### 1.2.1. Layer 1

_{Ak}(D) and µ

_{Bk-2}(t) differentiate the method of applying any fuzzy membership function. Equation (8) gives the µ

_{Ak}(D) for a bell-shaped membership function

_{k}, b

_{k}and c

_{k}are the factors affecting this membership function.

#### 1.2.2. Layer 2

#### 1.2.3. Layer 3

#### 1.2.4. Layer 4

#### 1.2.5. Layer 5

_{k}, b

_{k}, and c

_{k}, also known as premise parameters, are linked to input membership functions. Similarly, the three adaptable parameters p

_{k}, q

_{k}, and r

_{k}, also known as consequent parameters, are analogous to first-order polynomials and are found in the fourth sheet [59].

- (a)
- Head consisting of function or terminal symbols
- (b)
- Tail containing only the terminal symbols.

#### 1.3. The Aim of the Research

## 2. Methods

#### 2.1. Description and Division of Collected Data

_{t}, e

_{b}). The material properties of steel and concrete include the yield stress (f

_{y}) and compressive strength (f

_{c}) of concrete. The concrete compressive strength obtained from the experimental tests collected from the literature was based on both available cylinder and cube specimens. Cube strength was converted to cylinder strength through related conversion factors. Furthermore, cylinder strength was used in the design equations [70] to avoid errors. Other material properties, such as steel and concrete moduli and steel ultimate stress, were considered of minor significance. In the case of concrete, the compressive strength for all the specimens is given, and the modulus is directly affected by this compressive strength, so there is no need to establish any relation between compressive strength and modulus of concrete, and only strength was incorporated in the model as a significant factor. A similar strategy was used to eliminate the need for steel’s ultimate stress, while in the case of steel modulus, for example, 200 Gpa are probable, and this value is normal for all steel grades used in all columns.

#### 2.2. Structure of ANN, ANFIS and GEP Models

_{st}and N

_{lg}were found to be dependent on the following factors (Equation (14)):

_{t}and e

_{b}is eccentricity at top and bottom face, f

_{y}is the yield strength of tube, and f

_{c}is the compressive strength of the tube.

_{t}, e

_{b}, f

_{y}, f

_{c}), and the output layer had N

_{st}and N

_{lg}for ANN. After using the Levenberg-Marquardt algorithm and choosing random data division, the number of hidden neurons was set to ten. In addition, the network form was chosen as feed-forward back-propagation. To achieve a better output at the required number of hidden layers, trial and error methods should be used [54]. Table 2 lists the statistical parameters of modelling for ANNs in this research.

**.**

#### 2.3. Evaluation of Models through Statistical Measures

_{st}and N

_{lg}prediction (ANN, ANFIS and GEP models) were measured using five standard statistical metrics, including correlation coefficient (R), determination coefficient (R

^{2}), root mean square error (RMSE), mean absolute error (MAE), relative squared error (RSE), mean absolute percent error (MAPE) and relative root mean square error (RRMSE) in the training testing sets [77,78,79]. In addition, for all the proposed models, a performance index (PI) has been calculated as another metric, ruled primarily by RRMSE and R [36]. Equation (15) to Equation (21) define these performance measures:

^{2}was used because of its impartial assessment and comparatively better performance. R

^{2}values equal to 1 and closer to each other demonstrate that much of the model’s variation between input parameters was used [81]. In addition, RMSE is a common metric since significant errors in comparison to smaller errors are resolved very effectively. RMSE closer to 0 indicates that the prediction error is negligible [58]. It does not, however, ensure optimum efficiency in any conditions. MAE was also calculated and is enormously advantageous in the presence of smooth and continuous data [82]. To sum up, a greater value of R and smaller values for RMSE, MAE, RSE, and RRMSE provides a better standard calibration for model performance. In addition, Gandomi et al. [83] proposed that PI ranges from 0 to infinity and closer to zero indicate a good model performance.

## 3. Results and Discussion

#### 3.1. Regression Analysis of ANN, ANFIS and GEP Model

^{2}) for all the proposed models is greater than 90% in both training and testing stages following the trend: R

^{2}(ANN) > R

^{2}(GEP) > R

^{2}(ANFIS), reflecting the shortcoming of neural and fuzzy arrangement in the projected ANFIS model. The mean correlation coefficient (R) in the projected models for N

_{st}is also maximum for ANN (0.9986) tracked by GEP (0.9922) and ANFIS (0.9874). For a stronger correlation, the R-value will be higher (i.e., R > 0.8) for an acceptable model [77,87,88]. In the case of N

_{st}, the R

^{2}for ANN model is highest, i.e., 99.73% and 99.72% for training and testing data, respectively, and 98.20% and 98.74% for the GEP model, respectively.

#### 3.2. GEP Based Formulation of Bearing Capacity of CFST Columns

_{st}and N

_{lg}use four basic mathematical operations, i.e., +, −, × and ÷, as presented in Figure 6. The ETs in Figure 6a,b corresponding to three different numbers of genes with addition used as a linking function, are decoded to derive a respective mathematical equation for N

_{st}and N

_{lg}as shown in Equations (23) and (24). Based on the total number of records, the projected formulae are in close agreement with standard limits for an ideal model and can be confidently and reliably used for the prediction of bearing capacity of short and long circular CFST columns [64,89,90].

#### 3.3. Performance Evaluation of Proposed Models Using Statistical Indicators

_{st}and 84.5 and 36.13, respectively, for N

_{lg},

^{2}, enumerates the linear dependency of response and explanatory variables. An acceptable value of R greater than 0.8 shows a strong correlation between actual and predicted values [92]. Thus, the evaluation of the proposed models based on the slope of the regression line and regression or correlation coefficient is insufficient [92]. Therefore, the developed models are also assessed using different statistical metrics for evaluating their robustness.

#### 3.3.1. ANN Model

_{st}and N

_{lg}, respectively, have an error less than 10% As reflected in Figure 7a,b, the error values are scattered near zero, showing the outburst performance of the developed multi-physics-based ANN model. Furthermore, both the MAE and RMSE in Table 3, enumerate the magnitude of average error values and have their own importance. The RMSE squared the error before average and gave more weightage to larger error values [93]. At the same time, MAE gives low weightage to larger error values and is always lower than RMSE [80,93]. For the ANN model for N

_{st},

_{testing}MAE

_{testing}(145.96) is lower than RMSE

_{testing}(193.20), satisfying the stated condition. Similarly, these values for N

_{lg}are (196.22) and (325.12), respectively. The RSE

_{testing}of ANN models for N

_{st}and N

_{lg}is also minimal and nearly equals zero, i.e., (0.00274) and (0.0086), respectively. The details of the statistical analysis of all the proposed models for N

_{st}and N

_{lg}in both stages (training and testing), are provided in Table 3.

#### 3.3.2. ANFIS

_{st}and N

_{lg}, respectively, have percent error less than 10%.

_{testing}and RMSE

_{testing}of ANFIS models is (55.6)% and (71.7)% greater than that of ANN models in the case of N

_{st}and (60.8)% and (63.2)% greater in the case of N

_{lg}, respectively. Like the ANN model, the RSE

_{testing}for ANFIS models is also near to 0. Thus, based on the above facts, the prediction of bearing capacity of short and long CFST columns can also be obtained through these reliable and accurate models.

#### 3.3.3. GEP Model

^{2}and the magnitude of error statistics. However, as shown in Figure 9, around 42% and 27% (nearly equal to the results deduced for the ANFIS model) of the GEP predicted values for N

_{st}and N

_{lg}have percent error below 10%, which are less than the ANFIS model. In the GEP model, the RMSE and RSE values for the testing set are lower than the ANFIS model by 22.6% and 36% for N

_{st}, respectively, and 44.6% and 69.3% lower for N

_{lg}, respectively.

#### 3.4. Comparison of Models Using External Testing Criteria

_{st}and N

_{lg}fall within the prescribed limit and show an outstanding performance. However, the PI in the training and testing stage of the ANFIS model for N

_{lg}are equal to 0.2990 and 0.2652. Its OF value is also equal to 0.2700. In the ANN model for N

_{st}, the PI in the testing stage is 0.361, which is considerably higher than the prescribed limit. Consequently, the OF for the model is 0.2300. Thus, the performance of the ANN and ANFIS model based on PI and OF is ambiguous and marked as satisfactory, although the ANFIS model for short and ANN model for long CFST columns perform well. Lewis [94] suggested that the model can be categorized as either excellent prediction $\left(\mathrm{MAPE}\le 10\%\right)$, good prediction $(10\%<\mathrm{MAPE}\le 20\%)$, acceptable prediction ($20\%<\mathrm{MAPE}\le 50\%)$ and inaccurate prediction ($50\%<\mathrm{MAPE})$ [79,95]. In accordance with the mentioned criteria of model categorization based on MAPE, the ANN provides excellent forecasting results for N

_{st}, while all other models including N

_{lg}-ANN falls in the “acceptable prediction” category.

_{st}and N

_{lg}tests utilizing the corresponding GEP models is significantly quicker considering the time needed by the traditional test technique [41]. Thus, the suggested mathematical formulas provide a feasible fast method for finding N

_{st}and N

_{lg}.

_{0}

^{2}), and amongst experimental and actual data (R

_{0}′

^{2}) must be nearer to 1 [76]. It can be observed in Table 4 that the suggested GEP models satisfy all the necessary criteria, thus indicating the better level of accurateness of both the models.

#### 3.5. Sensitivity and Parametric Study of GEP Models

_{y}), and eccentricity at bottom (e

_{b}) parameters is negligible for both models with relative contribution of 0.09%, 0.03%, 0.3% for short and 0.03%, 0.3%, 0.5% for long CFST columns, respectively. The impact of length (L) in long columns and eccentricity at the top (e

_{t}) in short columns is also less, with a relative contribution of 0.44% and 0.41%, respectively. According to [99], the bearing capacity of CFST columns is mainly governed by the diameter and thickness of the steel tube.

_{st}and N

_{lg}with varying input variables, i.e., D, t, L, L/D, e

_{t}, e

_{b}, f

_{y}, and f

_{c}. It is well understood that D and t are important factors controlling the bearing capacity of CFST columns. The capacity of both short and long columns follows a degree polynomial curve with variation diameter and thickness of the tube. Similar trends were also noticed for the compressive strength and yield strength of the steel tube. Increasing the compressive strength of concrete will divert the failure control mode to the yield strength of the steel tube and vice versa. The strength of the concrete inside the CFST is responsible for the stiffness of the CFST columns [100]. Stiffness rises along with concrete compressive strength, yet columns fracture owing to concrete crushing and brittle behavior when filled with high strength concrete. However, regardless of the length to diameter ratio, an increase in concrete core strength enhances the strength of filled columns to a greater extent. Linear decreasing pattern was observed for variation in the length or length to diameter ratio of the steel tubes with the prescribed limits. The effect of the eccentricity at the top or bottom face of the column also adversely affecting the capacity of CFST columns. Compared to zero eccentricity, in the case of eccentric loading the contact stresses will not be distributed non-uniformly, causing outward buckling [101]. Moreover, changes in the diameter and thickness of the steel tube greatly influence the bearing capacity of CFST columns for N

_{lg}and N

_{st}, as observed and stated by many researchers in the past [6].

_{lg}and N

_{st}.

## 4. Conclusions

_{st}) and long columns (N

_{lg}) were developed through ANN and ANFIS and GEP. Two databases were extracted from the literature by collecting 702 datasets of short and 965 datapoints of long circular CFST columns. The conclusion is drawn below.

- The GEP model can efficiently predict N
_{st}and N_{lg}with high accuracy and best performance. Moreover, the bearing capacity prediction model from GEP is better than the ANFIS and ANN models. The diversity of the GEP technique can be seen from the simplified formulation, with higher accuracy and correlation among the experimental and predicted data with the consideration of linear and non-linear data. - The statistical indicators used to evaluate the performance of the model were mean absolute error (MAE), root square error (RSE), root means square error (RMSE), correlation coefficient (R), relative root mean square error (RRMSE), performance index (PI) and objective function (OF). The PI of the predicted N
_{st}by GEP, ANN and ANFIS for training are 0.0416, 0.1423, and 0.1016, respectively, and for N_{lg}these values are 0.1169, 0.2990 and 0.1542, respectively. Corresponding OF values are 0.2300, 0.1200, and 0.090 for N_{st}, and 0.1000, 0.2700, and 0.1500 for N_{lg}. The superiority of the GEP method to the other techniques can be seen from the fact that the GEP technique provides suitable connections based on the practical experimental work and does not undertake prior solutions. In reference to MAPE indicator, the ANN provides excellent forecasting results for N_{st}, while all other models including N_{lg}-ANN fall in the “acceptable prediction” category. - Sensitivity analysis was performed and the following input importance with increasing pattern was observed for N
_{st}: D (55.45) > T (20.45) > L (17.109) > f_{c}(5.526) > e_{t}(0.41) > L/D (0.34) > e_{b}(0.32) > f_{y}(0.096); whereas, in the case of N_{lg}, it followed the order: D (59.83) > T (33.73) > e_{t}(3.844) > f_{c}(1.302) > L/D (0.541) > L (0.443) > e_{b}(0.282) > f_{y}(0.033). Parametric analysis showed a trend similar to the findings in previous literature. The effect of input parameters on the bearing capacity of circular short (N_{st}) and long (N_{lg}) CFST columns was studied. Thus, it can be concluded from this research that artificial intelligence techniques can be effectively employed to solve various complex engineering problems, especially in structural and material engineering. A simple, reliable, and accurate model can be developed which can perform better on unseen data. - The overall comparison shows that the most reliable and accurate technique for developing prediction models is GEP. The prediction models developed through the GEP technique are simpler than ANN and ANFIS models. It is, therefore, suggested that the developed GEP equations (Equations (22) and (23)) are used in routine design for circular short (N
_{st}) and long (N_{lg}) CFST columns with eccentric loading using simple geometric and material properties. These models can replace tedious, time consuming and costly experimental work for finding the bearing capacity of CFST columns.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Romero, M.L.; Espinós, A.; Lapuebla-Ferri, A.; Albero, V.; Hospitaler, A. Recent developments and fire design provisions for CFST columns and slim-floor beams. J. Constr. Steel Res.
**2020**, 172, 106159. [Google Scholar] [CrossRef] - Suizi, J.; Wanlin, C.; Zibin, L.; Wei, D.; Yingnan, S. Experimental study on a prefabricated lightweight concrete-filled steel tubular framework composite slab structure subjected to reversed cyclic loading. Appl. Sci.
**2019**, 9, 1264. [Google Scholar] [CrossRef] [Green Version] - Ayough, P.; Ibrahim, Z.; Sulong, N.R.; Hsiao, P.-C. The effects of cross-sectional shapes on the axial performance of concrete-filled steel tube columns. J. Constr. Steel Res.
**2021**, 176, 106424. [Google Scholar] [CrossRef] - Ibañez, C.; Hernández-Figueirido, D.; Piquer, A. Shape effect on axially loaded high strength CFST stub columns. J. Constr. Steel Res.
**2018**, 147, 247–256. [Google Scholar] [CrossRef] - Phan, D.H.H.; Patel, V.I.; Al Abadi, H.; Thai, H.-T. Analysis and design of eccentrically compressed ultra-high-strength slender CFST circular columns. Structures
**2020**, 27, 2481–2499. [Google Scholar] [CrossRef] - Thai, S.; Thai, H.-T.; Uy, B.; Ngo, T. Concrete-filled steel tubular columns: Test database, design and calibration. J. Constr. Steel Res.
**2019**, 157, 161–181. [Google Scholar] [CrossRef] - Chinese, S. Technical Code for Concrete Filled Steel Tubular Structures; GB 50936; Ministry of Housing and Urban Rural Construction of the People’s Republic of China: Beijing, China, 2014.
- AIJ. Recommendations for Design and Construction of Concrete Filled Steel Tubular Structures; AIJ: Tokyo, Japan, 1997. [Google Scholar]
- ANSI/AISC 360-05. Specification for Structural Steel Buildings; American Institute of Steel Construction: Chicago, IL, USA, 2016; p. 586. [Google Scholar]
- Uy, B.; Hicks, S.J.; Kang, W.-H.; Thai, H.-T.; Aslani, F. The New Australia/New Zealand Standard on Composite Steel-Concrete Buildings. In Proceedings of the 8th International Conference on Composite Construction in Steel and Concrete, Jackson, WY, USA, 30 July–2 August 2017. ASNZS2327. [Google Scholar]
- European Committee for Standardization. 1-1; Eurocode 4: Design of Composite Steel and Concrete Structures—Part 1-1: General Rules and Rules for Buildings; Europian Committee for Standardization: Brussels, Belgium, 2004. [Google Scholar]
- Liew, J.R. Design Guide for Concrete Filled Tubular Members with High Strength Materials to Eurocode 4; Research Publishing: Singapore, 2015; ISBN 978-981-09-3267-1. [Google Scholar]
- Khan, M.; Uy, B.; Tao, Z.; Mashiri, F. Behaviour and design of short high-strength steel welded box and concrete-filled tube (CFT) sections. Eng. Struct.
**2017**, 147, 458–472. [Google Scholar] [CrossRef] - Mursi, M.; Uy, B. Strength of slender concrete filled high strength steel box columns. J. Constr. Steel Res.
**2004**, 60, 1825–1848. [Google Scholar] [CrossRef] - Vatulia, G.; Orel, Y.; Rezunenko, M.; Panchenko, N. Using statistical methods to determine the load-bearing capacity of rectangular CFST columns. MATEC Web Conf.
**2018**, 234, 04002. [Google Scholar] [CrossRef] - Le, T.-T. Practical machine learning-based prediction model for axial capacity of square CFST columns. Mech. Adv. Mater. Struct.
**2020**, 1–16. [Google Scholar] [CrossRef] - Mai, S.H.; Seghier, M.; Nguyen, P.L.; Jafari-Asl, J.; Thai, D.-K. A hybrid model for predicting the axial compression capacity of square concrete-filled steel tubular columns. Eng. Comput.
**2020**, 1–18. [Google Scholar] [CrossRef] - Xu, J.; Wang, Y.; Ren, R.; Wu, Z.; Ozbakkaloglu, T. Performance evaluation of recycled aggregate concrete-filled steel tubes under different loading conditions: Database analysis and modelling. J. Build. Eng.
**2020**, 30, 101308. [Google Scholar] [CrossRef] - Nguyen, H.Q.; Ly, H.-B.; Tran, V.Q.; Nguyen, T.-A.; Le, T.-T.; Pham, B.T. Optimization of Artificial Intelligence System by Evolutionary Algorithm for Prediction of Axial Capacity of Rectangular Concrete Filled Steel Tubes under Compression. Materials
**2020**, 13, 1205. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ly, H.-B.; Pham, B.T.; Le, L.M.; Le, T.; Le, V.M.; Asteris, P.G. Estimation of axial load-carrying capacity of concrete-filled steel tubes using surrogate models. Neural Comput. Appl.
**2021**, 33, 3437–3458. [Google Scholar] [CrossRef] - Luat, N.-V.; Shin, J.; Lee, K. Hybrid BART-based models optimized by nature-inspired metaheuristics to predict ulti-mate axial capacity of CCFST columns. Eng. Comput.
**2020**, 1–30. [Google Scholar] - Dao, D.V.; Ly, H.-B.; Vu, H.-L.T.; Le, T.-T.; Pham, B.T. Investigation and optimization of the C-ANN structure in predicting the compressive strength of foamed concrete. Materials
**2020**, 13, 1072. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chahnasir, E.S.; Zandi, Y.; Shariati, M.; Dehghani, E.; Toghroli, A.; Mohamad, E.T.; Shariati, A.; Safa, M.; Wakil, K.; Khorami, M. Application of support vector machine with firefly algorithm for investigation of the factors affecting the shear strength of angle shear connectors. Smart Struct. Syst.
**2018**, 22, 413–424. [Google Scholar] - Han, Q.; Gui, C.; Xu, J.; Lacidogna, G.J.C.; Materials, B. A generalized method to predict the compressive strength of high-performance concrete by improved random forest algorithm. Constr. Build. Mater.
**2019**, 226, 734–742. [Google Scholar] [CrossRef] - Khan, M.A.; Memon, S.A.; Farooq, F.; Javed, M.F.; Aslam, F.; Alyousef, R. Compressive strength of fly-ash-based geopolymer concrete by gene expression programming and random forest. Adv. Civ. Eng.
**2021**, 2021, 6618407. [Google Scholar] [CrossRef] - Shariati, M.; Mafipour, M.S.; Haido, J.H.; Yousif, S.T.; Toghroli, A.; Trung, N.T.; Shariati, A. Identification of the most influencing parameters on the properties of corroded concrete beams using an Adaptive Neuro-Fuzzy Inference System (ANFIS). Smart Struct. Syst.
**2020**, 34, 155. [Google Scholar] - Singh, V.; Bano, S.; Yadav, A.K.; Ahmad, S. Feasibility of artificial neural network in civil engineering. IJTSRD
**2019**, 3, 724–728. [Google Scholar] [CrossRef] - Hajihassani, M.; Armaghani, D.J.; Kalatehjari, R.J.G.; Engineering, G. Applications of particle swarm optimization in geotechnical engineering: A comprehensive review. Geotech. Geol. Eng.
**2018**, 36, 705–722. [Google Scholar] [CrossRef] - Abdollahzadeh, G.; Jahani, E.; Kashir, Z. Genetic programming based formulation to predict compressive strength of high strength concrete. Civ. Eng. Infrastruct. J.
**2017**, 50, 207–219. [Google Scholar] - Ali Khan, M.; Zafar, A.; Akbar, A.; Javed, M.F.; Mosavi, A.J.M. Application of Gene Expression Programming (GEP) for the prediction of compressive strength of geopolymer concrete. Materials
**2021**, 14, 1106. [Google Scholar] [CrossRef] [PubMed] - Nguyen, Q.H.; Ly, H.-B.; Tran, V.Q.; Nguyen, T.-A.; Phan, V.-H.; Le, T.-T.; Pham, B. A novel hybrid model based on a feedforward neural network and one step secant algorithm for prediction of load-bearing capacity of rectangular concrete-filled steel tube columns. Molecules
**2020**, 25, 3486. [Google Scholar] [CrossRef] - Al-Khaleefi, A.M.; Terro, M.J.; Alex, A.P.; Wang, Y. Prediction of fire resistance of concrete filled tubular steel columns using neural networks. Fire Saf. J.
**2002**, 37, 339–352. [Google Scholar] [CrossRef] - Zarringol, M.; Thai, H.-T.; Thai, S.; Patel, V. Application of ANN to the design of CFST columns. Structures
**2020**, 28, 2203–2220. [Google Scholar] [CrossRef] - Balasubramanian, S.; Jegan, J.; Sundarraja, M. ANFIS-Based Accurate Estimation of the Confinement Effect for Concrete-Filled Steel Tubular (CFST). Int. J. Fuzzy Syst.
**2020**, 22, 1760–1771. [Google Scholar] [CrossRef] - Basarir, H.; Elchalakani, M.; Karrech, A. The prediction of ultimate pure bending moment of concrete-filled steel tubes by adaptive neuro-fuzzy inference system (ANFIS). Neural Comput. Appl.
**2019**, 31, 1239–1252. [Google Scholar] [CrossRef] - Gandomi, A.H.; Roke, D.A. Assessment of artificial neural network and genetic programming as predictive tools. Adv. Eng. Softw.
**2015**, 88, 63–72. [Google Scholar] [CrossRef] - Javed, M.F.; Farooq, F.; Memon, S.A.; Akbar, A.; Khan, M.A.; Aslam, F.; Alyousef, R.; Alabduljabbar, H.; Rehman, S.K.U. New prediction model for the ultimate axial capacity of concrete-filled steel tubes: An evolutionary approach. Crystals
**2020**, 10, 741. [Google Scholar] [CrossRef] - Ipek, S.; Güneyisi, E. Ultimate axial strength of concrete-filled double skin steel tubular column sections. Adv. Civ. Eng.
**2019**, 2019, 6493037. [Google Scholar] [CrossRef] [Green Version] - Güneyisi, E.M.; Gültekin, A.; Mermerdaş, K. Ultimate capacity prediction of axially loaded CFST short columns. Int. J. Steel Struct.
**2016**, 16, 99–114. [Google Scholar] [CrossRef] - Zhang, Q.; Barri, K.; Jiao, P.; Salehi, H.; Alavi, A.H. Genetic programming in civil engineering: Advent, applications and future trends. Artif. Intell. Rev.
**2021**, 54, 1863–1885. [Google Scholar] [CrossRef] - Jalal, F.E.; Xu, Y.; Iqbal, M.; Javed, M.F.; Jamhiri, B. Predictive modeling of swell-strength of expansive soils using artificial intelligence approaches: ANN, ANFIS and GEP. J. Environ. Manag.
**2021**, 289, 112420. [Google Scholar] [CrossRef] [PubMed] - Venkatesh, K.; Bind, Y.K. ANN and Neuro-Fuzzy Modeling for Shear Strength Characterization of Soils. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci.
**2020**, 1–7. [Google Scholar] [CrossRef] - Sada, S.; Ikpeseni, S. Evaluation of ANN and ANFIS modeling ability in the prediction of AISI 1050 steel machining performance. Heliyon
**2021**, 7, e06136. [Google Scholar] [CrossRef] - Kourgialas, N.N.; Dokou, Z.; Karatzas, G.P. Statistical analysis and ANN modeling for predicting hydrological extremes under climate change scenarios: The example of a small Mediterranean agro-watershed. J. Environ. Manag.
**2015**, 154, 86–101. [Google Scholar] [CrossRef] - Das, S.K. 10 Artificial Neural Networks in Geotechnical Engineering: Modeling and Application Issues. Metaheuristics Water Geotech. Transp. Eng.
**2013**, 45, 231–267. [Google Scholar] - Koçak, Y.; Şiray, G.Ü. New activation functions for single layer feedforward neural network. Expert Syst. Appl.
**2021**, 164, 113977. [Google Scholar] [CrossRef] - Xu, B.; Huang, R.; Li, M. Revise saturated activation functions. arXiv
**2016**, arXiv:1602.05980. preprint. [Google Scholar] - Naresh Babu, K.; Edla, D.R. New algebraic activation function for multi-layered feed forward neural networks. IETE J. Res.
**2017**, 63, 71–79. [Google Scholar] [CrossRef] - Ramachandran, P.; Zoph, B.; Le, Q.V. Searching for activation functions. arXiv
**2017**, arXiv:1710.05941. preprint. [Google Scholar] - Cai, C.; Xu, Y.; Ke, D.; Su, K. Deep neural networks with multistate activation functions. Comput. Intell. Neurosci.
**2015**, 2015, 721367. [Google Scholar] [CrossRef] [Green Version] - Tang, Y.-J.; Zhang, Q.-Y.; Lin, W. Artificial neural network based spectrum sensing method for cognitive radio. In Proceedings of the 2010 6th International Conference on Wireless Communications Networking and Mobile Computing (WiCOM), Chengdu, China, 23–25 September 2010; pp. 1–4. [Google Scholar]
- Tahani, M.; Vakili, M.; Khosrojerdi, S. Experimental evaluation and ANN modeling of thermal conductivity of graphene oxide nanoplatelets/deionized water nanofluid. Int. Commun. Heat Mass Transf.
**2016**, 76, 358–365. [Google Scholar] [CrossRef] - Dorofki, M.; Elshafie, A.H.; Jaafar, O.; Karim, O.A.; Mastura, S. Comparison of artificial neural network transfer functions abilities to simulate extreme runoff data. Int. Proc. Chem. Biol. Environ. Eng.
**2012**, 33, 39–44. [Google Scholar] - Hanandeh, S.; Ardah, A.; Abu-Farsakh, M. Using artificial neural network and genetics algorithm to estimate the resilient modulus for stabilized subgrade and propose new empirical formula. Transp. Geotech.
**2020**, 24, 100358. [Google Scholar] [CrossRef] - Alavi, A.H.; Gandomi, A.H. A robust data mining approach for formulation of geotechnical engineering systems. Eng. Comput. Int. J. Comput.-Aided Eng.
**2011**, 28, 242–274. [Google Scholar] - Nosratabadi, S.; Mosavi, A.; Duan, P.; Ghamisi, P.; Filip, F.; Band, S.S.; Reuter, U.; Gama, J.; Gandomi, A.H. Data science in economics: Comprehensive review of advanced machine learning and deep learning methods. Mathematics
**2020**, 8, 1799. [Google Scholar] [CrossRef] - Shahin, M.A. Artificial intelligence in geotechnical engineering: Applications, modeling aspects, and future directions. In Metaheuristics in Water, Geotechnical and Transport Engineering; Curtin University: Perth, Australia, 2013; Volume 169204. [Google Scholar]
- Golafshani, E.M.; Behnood, A.; Arashpour, M. Predicting the compressive strength of normal and High-Performance Concretes using ANN and ANFIS hybridized with Grey Wolf Optimizer. Constr. Build. Mater.
**2020**, 232, 117266. [Google Scholar] [CrossRef] - Islam, M.R.; Jaafar, W.Z.W.; Hin, L.S.; Osman, N.; Hossain, A.; Mohd, N.S. Development of an intelligent system based on ANFIS model for predicting soil erosion. Environ. Earth Sci.
**2018**, 77, 186. [Google Scholar] [CrossRef] - Gao, W. A comprehensive review on identification of the geomaterial constitutive model using the computational intelligence method. Adv. Eng. Inform.
**2018**, 38, 420–440. [Google Scholar] [CrossRef] - Shishegaran, A.; Boushehri, A.N.; Ismail, A.F. Gene expression programming for process parameter optimization during ultrafiltration of surfactant wastewater using hydrophilic polyethersulfone membrane. J. Environ. Manag.
**2020**, 264, 110444. [Google Scholar] [CrossRef] - Ferreira, C. Gene expression programming in problem solving. In Soft Computing and Industry; Springer: Berlin/Heidelberg, Germany, 2002; pp. 635–653. [Google Scholar]
- Wang, M.; Wan, W. A new empirical formula for evaluating uniaxial compressive strength using the Schmidt hammer test. Int. J. Rock Mech. Min. Sci.
**2019**, 123, 104094. [Google Scholar] [CrossRef] - Soleimani, S.; Rajaei, S.; Jiao, P.; Sabz, A.; Soheilinia, S. New prediction models for unconfined compressive strength of geopolymer stabilized soil using multi-gen genetic programming. Measurement
**2018**, 113, 99–107. [Google Scholar] [CrossRef] - Ferreira, C. Mutation, Transposition, and Recombination: An Analysis of the Evolutionary Dynamics. In Proceedings of the 6th Joint Conference on Information Sciences, Research Triangle Park, Raleigh, NC, USA, 8–13 March 2002; pp. 614–617. [Google Scholar]
- Armaghani, D.J.; Safari, V.; Fahimifar, A.; Monjezi, M.; Mohammadi, M.A. Uniaxial compressive strength prediction through a new technique based on gene expression programming. Neural Comput. Appl.
**2018**, 30, 3523–3532. [Google Scholar] [CrossRef] - Vyas, R.; Goel, P.; Tambe, S.S. Genetic programming applications in chemical sciences and engineering. In Handbook of Genetic Programming Applications; Springer: Berlin/Heidelberg, Germany, 2015; pp. 99–140. [Google Scholar]
- Mazari, M.; Rodriguez, D.D. Prediction of pavement roughness using a hybrid gene expression programming-neural network technique. J. Traffic Transp. Eng. (Engl. Ed.)
**2016**, 3, 448–455. [Google Scholar] [CrossRef] [Green Version] - Lam, D.; Goode, C. Concrete Filled Steel Tube Columns-Test compared with Eurocode4. In Proceedings of the International Conference on Composite Construction in Steel and Concrete 2008, Devil’s Thumb Ranch, CO, USA, 20–24 July 2008. [Google Scholar]
- Mansur, M.A.; Islam, M.M. Interpretation of concrete strength for nonstandard specimens. J. Mater. Civ. Eng.
**2002**, 14, 151–155. [Google Scholar] [CrossRef] - Ahmad, M.R.; Chen, B.; Dai, J.-G.; Kazmi, S.M.S.; Munir, M. Evolutionary artificial intelligence approach for performance prediction of bio-composites. Constr. Build. Mater.
**2021**, 290, 123254. [Google Scholar] [CrossRef] - Maeda, T. How to Rationally Compare the Performances of Different Machine Learning Models? PeerJ Preprints: London, UK, 2018; pp. 2167–9843. [Google Scholar]
- Jalal, M.; Grasley, Z.; Nassir, N.; Jalal, H. Strength and dynamic elasticity modulus of rubberized concrete designed with ANFIS modeling and ultrasonic technique. Constr. Build. Mater.
**2020**, 240, 117920. [Google Scholar] [CrossRef] - Ferreira, C. Gene Expression Programming: Mathematical Modeling by an Artificial Intelligence; Springer: Berlin/Heidelberg, Germany, 2006; Volume 21. [Google Scholar]
- Alavi, A.H.; Gandomi, A.H.; Nejad, H.C.; Mollahasani, A.; Rashed, A. Design equations for prediction of pressuremeter soil deformation moduli utilizing expression programming systems. Neural Comput. Appl.
**2013**, 23, 1771–1786. [Google Scholar] [CrossRef] - Ferreira, C. Genetic representation and genetic neutrality in gene expression programming. Adv. Complex Syst.
**2002**, 5, 389–408. [Google Scholar] [CrossRef] [Green Version] - Iqbal, M.F.; Liu, Q.-f.; Azim, I.; Zhu, X.; Yang, J.; Javed, M.F.; Rauf, M. Prediction of mechanical properties of green concrete incorporating waste foundry sand based on gene expression programming. J. Hazard. Mater.
**2020**, 384, 121322. [Google Scholar] [CrossRef] - Çanakcı, H.; Baykasoğlu, A.; Güllü, H. Prediction of compressive and tensile strength of Gaziantep basalts via neural networks and gene expression programming. Neural Comput. Appl.
**2009**, 18, 1031. [Google Scholar] [CrossRef] - Ağbulut, Ü.; Gürel, A.E.; Biçen, Y. Prediction of daily global solar radiation using different machine learning algorithms: Evaluation and comparison. Renew. Sustain. Energy Rev.
**2021**, 135, 110114. [Google Scholar] [CrossRef] - Alade, I.O.; Bagudu, A.; Oyehan, T.A.; Abd Rahman, M.A.; Saleh, T.A.; Olatunji, S.O. Estimating the refractive index of oxygenated and deoxygenated hemoglobin using genetic algorithm–support vector regression model. Comput. Methods Programs Biomed.
**2018**, 163, 135–142. [Google Scholar] [CrossRef] - Zhang, W.; Zhang, R.; Wu, C.; Goh, A.T.C.; Lacasse, S.; Liu, Z.; Liu, H. State-of-the-art review of soft computing applications in underground excavations. Geosci. Front.
**2020**, 11, 1095–1106. [Google Scholar] [CrossRef] - Shahin, M.A. Use of evolutionary computing for modelling some complex problems in geotechnical engineering. Geomech. Geoengin.
**2015**, 10, 109–125. [Google Scholar] [CrossRef] [Green Version] - Gandomi, A.H.; Alavi, A.H.; Mirzahosseini, M.R.; Nejad, F.M. Nonlinear genetic-based models for prediction of flow number of asphalt mixtures. J. Mater. Civ. Eng.
**2011**, 23, 248–263. [Google Scholar] [CrossRef] - Emamgholizadeh, S.; Bahman, K.; Bateni, S.M.; Ghorbani, H.; Marofpoor, I.; Nielson, J.R. Estimation of soil dispersivity using soft computing approaches. Neural Comput. Appl.
**2017**, 28, 207–216. [Google Scholar] [CrossRef] - Chu, H.-H.; Khan, M.A.; Javed, M.; Zafar, A.; Khan, M.I.; Alabduljabbar, H.; Qayyum, S. Sustainable use of fly-ash: Use of gene-expression programming (GEP) and multi-expression programming (MEP) for forecasting the compressive strength geopolymer concrete. Ain Shams Eng. J.
**2021**, 12, 3603–3617. [Google Scholar] [CrossRef] - Khan, M.A.; Shah, M.I.; Javed, M.F.; Khan, M.I.; Rasheed, S.; El-Shorbagy, M.; El-Zahar, E.R.; Malik, M. Application of random forest for modelling of surface water salinity. Ain Shams Eng. J.
**2021**, in press. [Google Scholar] - Erzin, Y. Artificial neural networks approach for swell pressure versus soil suction behaviour. Can. Geotech. J.
**2007**, 44, 1215–1223. [Google Scholar] [CrossRef] - Aslam, F.; Elkotb, M.A.; Iqtidar, A.; Khan, M.A.; Javed, M.F.; Usanova, K.I.; Khan, M.I.; Alamri, S.; Musarat, M.A. Compressive strength prediction of rice husk ash using multiphysics genetic expression programming. Ain Shams Eng. J.
**2021**, in press. [Google Scholar] [CrossRef] - Gholampour, A.; Gandomi, A.H.; Ozbakkaloglu, T. New formulations for mechanical properties of recycled aggregate concrete using gene expression programming. Constr. Build. Mater.
**2017**, 130, 122–145. [Google Scholar] [CrossRef] - Mohammadzadeh, S.; Kazemi, S.-F.; Mosavi, A.; Nasseralshariati, E.; Tah, J.H. Prediction of compression index of fine-grained soils using a gene expression programming model. Infrastructures
**2019**, 4, 26. [Google Scholar] [CrossRef] [Green Version] - Frank, I.E.; Todeschini, R. The Data Analysis Handbook; Elsevier: Amsterdam, The Netherlands, 1994. [Google Scholar]
- Nguyen, T.; Kashani, A.; Ngo, T.; Bordas, S. Deep neural network with high-order neuron for the prediction of foamed concrete strength. Comput. Civ. Infrastruct. Eng.
**2019**, 34, 316–332. [Google Scholar] [CrossRef] - Iqbal, M.F.; Javed, M.F.; Rauf, M.; Azim, I.; Ashraf, M.; Yang, J.; Liu, Q.-F. Sustainable utilization of foundry waste: Forecasting mechanical properties of foundry sand based concrete using multi-expression programming. Sci. Total Environ.
**2021**, 780, 146524. [Google Scholar] [CrossRef] - Lewis, C.D. Industrial and Business Forecasting Methods: A Practical Guide to Exponential Smoothing and Curve Fitting; Butterworth-Heinemann: Oxford, UK, 1982. [Google Scholar]
- Chen, R.J.C.; Bloomfield, P.; Cubbage, F.W. Comparing Forecasting Models in Tourism. J. Hosp. Tour. Res.
**2008**, 32, 3–21. [Google Scholar] [CrossRef] - Mollahasani, A.; Alavi, A.H.; Gandomi, A.H. Empirical modeling of plate load test moduli of soil via gene expression pro-gramming. Comput. Geotech.
**2011**, 38, 281–286. [Google Scholar] [CrossRef] - Roy, P.P.; Roy, K. On some aspects of variable selection for partial least squares regression models. QSAR Comb. Sci.
**2008**, 27, 302–313. [Google Scholar] [CrossRef] - Trucchia, A.; Frunzo, L. Surrogate based Global Sensitivity Analysis of ADM1-based Anaerobic Digestion Model. J. Environ. Manag.
**2021**, 282, 111456. [Google Scholar] [CrossRef] [PubMed] - Wang, W.; Ma, H.; Li, Z.; Tang, Z. Size effect in circular concrete-filled steel tubes with different diameter-to-thickness ratios under axial compression. Eng. Struct.
**2017**, 151, 554–567. [Google Scholar] [CrossRef] - Yadav, R.; Chen, B. Parametric study on the axial behaviour of concrete filled steel tube (CFST) columns. Am. J. Appl. Sci. Res.
**2017**, 3, 37–41. [Google Scholar] [CrossRef] [Green Version] - Cai, J.; Pan, J.; Lu, C.; Li, X. Nonlinear analysis of circular concrete-filled steel tube columns under eccentric loading. Mag. Concr. Res.
**2020**, 72, 292–303. [Google Scholar] [CrossRef]

**Figure 5.**Regression plot between actual and predicted bearing capacity of (

**a**) ANN model for short CFST columns (

**b**) ANN model for long CFST columns (

**c**) ANFIS model for short CFST columns (

**d**) ANFIS model for long CFST columns (

**e**) GEP model for short CFST columns and (

**f**) GEP model for long CFST columns.

**Figure 7.**Variation of mean absolute error and error histograms of bearing capacity established using ANN algorithm (

**a**,

**c**) short CFST (

**b**,

**d**) long CFST columns.

**Figure 8.**Variation of mean absolute error and error histograms of bearing capacity established using ANFIS algorithm (

**a**,

**c**) short CFST (

**b**,

**d**) long CFST columns.

**Figure 9.**Variation of mean absolute error and error histograms of bearing capacity established using GEP algorithm (

**a**,

**c**) short CFST (

**b**,

**d**) long CFST columns.

**Figure 10.**Relative importance of input variables on the bearing capacity of short and long circular CFST columns.

**Figure 11.**Summarized parametric study for formulation of bearing capacity of short and long circular CFST columns using Gene expression programming (GEP) in reference to the input variables (D: diameter of tube, t: thickness of tube, L: length of tube, L/D: length to diameter ratio, e

_{t}and e

_{b}: eccentricity at top and bottom surface, f

_{y}: yield strength of tube, f

_{c}: compressive strength of concrete).

Category | Parameters | Mean | Median | Max | Min | S.D. | Kurtosis | Skewness |
---|---|---|---|---|---|---|---|---|

Long | Inputs | |||||||

D (mm) | 147.2 | 121.0 | 1020.0 | 44.5 | 89.9 | 31.78 | 4.38 | |

t (mm) | 4.4 | 4.0 | 16.5 | 0.5 | 2.4 | 5.12 | 1.79 | |

L (mm) | 1438.3 | 1040.0 | 5560.0 | 152.3 | 1094.5 | 1.12 | 1.23 | |

L/D | 11.2 | 8.6 | 51.5 | 0.8 | 8.9 | 1.88 | 1.35 | |

e_{t} (mm)
| 13.0 | 0.0 | 300.0 | 0.0 | 28.1 | 25.87 | 4.08 | |

e_{b} (mm)
| 11.2 | 0.0 | 300.0 | 0.0 | 27.5 | 29.50 | 4.38 | |

f_{y} (MPa)
| 332.1 | 322.0 | 853.0 | 178.3 | 81.7 | 7.83 | 1.98 | |

f_{c} (MPa)
| 46.6 | 40.1 | 193.3 | 7.7 | 26.8 | 7.75 | 2.36 | |

Output | ||||||||

Nexp (kN) | 1616 | 848.5 | 46,000 | 45.2 | 3181.1 | 73.86 | 7.53 | |

Short | Inputs | |||||||

D (mm) | 169.2 | 133.1 | 1020.0 | 48.0 | 112.5 | 23.19 | 4.17 | |

t (mm) | 4.2 | 4.0 | 13.3 | 0.5 | 2.3 | 1.56 | 1.15 | |

L (mm) | 498.7 | 399.5 | 3060.0 | 152.3 | 334.0 | 24.32 | 4.16 | |

L/D | 3.0 | 3.0 | 4.0 | 0.8 | 0.6 | 0.19 | −0.55 | |

e_{t} (mm)
| 2.8 | 0.0 | 105.0 | 0.0 | 10.9 | 30.38 | 5.03 | |

e_{b} (mm)
| 2.8 | 0.0 | 105.0 | 0.0 | 10.9 | 30.38 | 5.03 | |

f_{y} (MPa)
| 336.8 | 322.7 | 853.0 | 185.7 | 97.5 | 10.53 | 2.52 | |

f_{c} (MPa)
| 58.8 | 46.6 | 193.3 | 7.7 | 35.9 | 2.60 | 1.54 | |

Output | ||||||||

Nexp (kN) | 2782.5 | 1678.1 | 46,000 | 199.9 | 4304.5 | 39.20 | 5.39 |

Parameters | Class and Value | |
---|---|---|

N_{lg} | N_{st} | |

Training dataset (70%) | 676 | 495 |

Testing dataset (30%) | 289 | 207 |

ANN | ||

Network type | Feed-forward back-propagation | |

Data division | Random (un-biased) | |

No. of hidden layer | 8 | |

No. of hidden neurons | 10 | |

Training algorithm | Levenberg-Marquardt | |

Hidden layer’s Transfer function | TANSIG | |

Output layer’s Transfer function | PURELIN | |

No. of non-linear parameters | 16 | |

No. of epochs | 40 | |

Learning rate | 0.01 | |

ANFIS | ||

No. of linear parameters | 72 | 65 |

No. of nonlinear parameters | 140 | 120 |

Total No. of parameters | 176 | 154 |

No. of fuzzy rules | 5 | 8 |

No. of MFs | 5 | 8 |

No. of nodes | 20 | 45 |

No. of Training epoch | 30 | 30 |

Training error goal | 0 | 0 |

Membership Function type | Trimf | |

Fuzzy structure | Sugeno | |

Type of FIS | Sub clustering | |

Method of Optimization | Back propagation and least square | |

Output function | Linear | |

GEP | ||

Parameters | ||

General | ||

Number of chromosomes | 100 | |

Number of Genes | 3 | |

Head size | 8 | |

Linking function | Addition | |

Function set | +, −, ×, ÷ | |

Numerical constants | ||

Constant per gene | 10 | |

Type of data | Floating number | |

Maximum complexity | 8 | |

Ephemeral random constant | [−10,10] | |

Genetic operators | ||

Rate of mutation | 0.00138 | |

Inversion rate | 0.00546 | |

IS transposition rate | ||

RIS transposition rate | ||

One-point recombination rate | 0.00277 | |

Two-point recombination rate | ||

Gene recombination rate | ||

Gene transposition rate |

**Table 3.**Statistical indicators for ANN, ANFIS and GEP models developed for bearing capacity of short and long circular CFST columns.

Model | Statistical Metrics | ANN | ANFIS | GEP | |||
---|---|---|---|---|---|---|---|

Training | Testing | Training | Testing | Training | Testing | ||

Long | MAE | 214.98 | 196.22 | 556.86 | 500.30 | 306.34 | 290.36 |

MAPE | 27.59 | 25.28 | 36.29 | 42.91 | 40.50 | 39.74 | |

RSE | 0.0148 | 0.0086 | 0.0931 | 0.0635 | 0.0256 | 0.0195 | |

RMSE | 369.15 | 325.12 | 925.33 | 882.60 | 485.46 | 489.06 | |

R | 0.9929 | 0.9959 | 0.9534 | 0.9678 | 0.9871 | 0.9906 | |

RRMSE | 0.2330 | 0.1922 | 0.5841 | 0.5219 | 0.3064 | 0.2892 | |

PI | 0.1169 | 0.0963 | 0.2990 | 0.2652 | 0.1542 | 0.1452 | |

OF | 0.1000 | 0.2700 | 0.1500 | ||||

Short | MAE | 155.29 | 145.96 | 360.05 | 328.73 | 350.56 | 387.93 |

MAPE | 8.79 | 9.17 | 24.55 | 27.20 | 22.39 | 20.15 | |

RSE | 0.00270 | 0.00274 | 0.0345 | 0.0197 | 0.0179 | 0.0126 | |

RMSE | 235.77 | 193.20 | 750.68 | 681.67 | 552.17 | 527.47 | |

R | 0.9986 | 0.9986 | 0.9842 | 0.9907 | 0.9909 | 0.9936 | |

RRMSE | 0.0833 | 0.0273 | 0.2824 | 0.2213 | 0.2023 | 0.1812 | |

PI | 0.0416 | 0.361 | 0.1423 | 0.1111 | 0.1016 | 0.0908 | |

OF | 0.2300 | 0.1200 | 0.090 |

Equation | Condition | GEP Model | |
---|---|---|---|

Long | Short | ||

$k={{\displaystyle \sum}}_{i=1}^{n}\frac{\left({q}_{i}\times {p}_{i}\right)}{{q}_{i}{}^{2}}$ | 0.85 < $k$ < 1.15 | 0.989 | 1.00 |

${k}^{\prime}={{\displaystyle \sum}}_{i=1}^{n}\frac{\left({q}_{i}\times {p}_{i}\right)}{{p}_{i}{}^{2}}$ | 0.85 < ${k}^{\prime}$ < 1.15 | 0.995 | 0.974 |

${R}_{m}={R}^{2}(1-\sqrt{\left|{R}^{2}-{R}_{0}{}^{2}\right|}$ | 0.5 < ${R}_{m}$ | 0.847 | 0.877 |

${R}_{0}{}^{2}=1-\frac{{\sum}_{i=1}^{n}{\left({p}_{i}-{q}_{i}{}^{0}\right)}^{2}}{{\sum}_{i=1}^{n}{\left({p}_{i}-{p}_{i}{}^{0}\right)}^{2}}$ | ${R}_{0}{}^{2}\cong 1$ | 0.999 | 0.999 |

${{R}^{\prime}}_{0}{}^{2}=1-\frac{{\sum}_{i=1}^{n}{\left({q}_{i}-{p}_{i}{}^{0}\right)}^{2}}{{\sum}_{i=1}^{n}{\left({q}_{i}-{q}_{i}{}^{0}\right)}^{2}}$ | ${{R}^{\prime}}_{0}{}^{2}\cong 1$ | 0.979 | 0.958 |

${q}_{i}{}^{0}=k\times {p}_{i}$ | |||

${p}_{i}{}^{0}=k\times {q}_{i}$ |

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**MDPI and ACS Style**

Khan, S.; Ali Khan, M.; Zafar, A.; Javed, M.F.; Aslam, F.; Musarat, M.A.; Vatin, N.I.
Predicting the Ultimate Axial Capacity of Uniaxially Loaded CFST Columns Using Multiphysics Artificial Intelligence. *Materials* **2022**, *15*, 39.
https://doi.org/10.3390/ma15010039

**AMA Style**

Khan S, Ali Khan M, Zafar A, Javed MF, Aslam F, Musarat MA, Vatin NI.
Predicting the Ultimate Axial Capacity of Uniaxially Loaded CFST Columns Using Multiphysics Artificial Intelligence. *Materials*. 2022; 15(1):39.
https://doi.org/10.3390/ma15010039

**Chicago/Turabian Style**

Khan, Sangeen, Mohsin Ali Khan, Adeel Zafar, Muhammad Faisal Javed, Fahid Aslam, Muhammad Ali Musarat, and Nikolai Ivanovich Vatin.
2022. "Predicting the Ultimate Axial Capacity of Uniaxially Loaded CFST Columns Using Multiphysics Artificial Intelligence" *Materials* 15, no. 1: 39.
https://doi.org/10.3390/ma15010039