Interfacial bond capacity prediction of concrete-filled steel tubes utilizing artificial neural network

Abstract

Concrete-filled steel tubes (CFSTs) have demonstrated superior performance compared to other types of composite columns. The evaluation of interactions between concrete and steel composites is crucial due to their significant impact on the overall structural behavior in various loadings. This study develops an artificial neural network (ANN) that predicts the ultimate interfacial bond strength ($P_u$) of circular and square CFSTs. The length of the interfacial bond between the tube and the concrete; the thickness, shape, and inner perimeter of the tube; and the cubic compression strength and age of the concrete are considered as model inputs. The modeling process uses 397 experimental datasets from 18 studies of push-out tests, more specifically, 143 square and 254 circular CFSTs. ANN with a hidden layer of error-propagation, feed-forward, and sigmoidal activation function is trained, tested, optimized, and validated to achieve a good estimate. The resulting model can predict the $P_u$ with a satisfactory coefficient of determination (R²) of 0.87. The ability of the developed model to predict the $P_u$ is compared to the proposed formulas in the literature. It is found that the ANN model provides the most accurate predictions among all suggested formulas, in terms of R² and Taylor diagram analyses. Furthermore, the inclusion of the shape factor enabled the ANN model to predict the $P_u$ of both squared and circular shapes of CFSTs. The validated ANN model is then used to examine the sensitivity of the parameters to the $P_u.$ The correlations agreed well with the expected trend of published studies.

Keywords:

• Concrete-filled steel tubes
• artificial neural networks
• interfacial bond capacity
• load capacity
• shear stress
• predictive model

Reviewing Editor:

• Sanjay Kumar Shukla
• School of Engineering
• Edith Cowan University
• AUSTRALIA

Subjects:

• Stress Analysis
• Machine Learning
• Concrete & Cement

1. Introduction

Steel and concrete are the two most used materials in modern construction. They can be combined in such a way that each improves the performance of the other, resulting in an improved overall response of the structure to various loads. Concrete-filled steel tubes (CFSTs) are an excellent example of using composite sections of steel and concrete, which possess more advantages than their pure counterparts. They have been extensively employed in the construction of underground infrastructure, high-rises, flyovers, and other engineering assemblies since the 1950s, as this system possesses better ductility and weighs less than reinforced concrete structures (Huang et al., 2012). Consequently, when concrete is being poured, steel tube provides it with a form and confines the core of the concrete to improve its ductility and strength. The concrete core, meanwhile, increases the compression strength and stiffness of the steel tube and decreases the likelihood of localized buckling occurring.

Several researchers (Ahmadi et al., 2017; Beheshti-Aval, 2012; Gupta et al., 2007; Han et al., 2014; Perea et al., 2014; Yu et al., 2007; Zheng et al., 2000) have used experiments and theories to investigate the structural behavior of CFST columns under various conditions. Frequently, the authors of these studies fail to adequately define the behavior of the columns under these various loading conditions (Ahmadi et al., 2017). When designing these columns, it is essential to analyze the interactive effects that result from the steel-concrete interface. Numerous researchers have chosen the push-out test due to its ability to evaluate the ultimate bond strength between concrete and steel, as well as the performance throughout the entire loading process. Consequently, it has become the method of choice for most researchers (Chen et al., 2015; Yan et al., 2014).

Roeder et al. (1999) experimentally examined the behavior of the bonds in 20 circular CFSTs (cCFSTs) under push-out loading, specifically those with large diameter-to-thickness $(D/t)$ ratios compared with extant previous studies (Shakir-Khalil, 1993a; Tomii et al., 1980; Virdi & Dowling, 1980). In addition, the authors proposed a formula to predict the ultimate bond strength of such columns, which is only valid for cCFST with $(D/t)$ ratio as a variable and its value is limited to 80. Parsley and Yura (2000) similarly developed an equation to calculate the strength of the bond upon testing a total of eight square CFSTs (sCFSTs) in push-out tests. The equation was given as a function of the steel tube wall thickness and the square root of the tube width. Chen et al. (2009) further proposed a formula that considers the impact of altering the recycled aggregate content (δ), the diameter-to-length ratio $(L_e/d$), and the $f_{\mathit{\text{cu}}}$ on the strength of the bonds in cCFSTs that comprise aggregates that have been recycled. In addition, experimental push-out experiments were conducted on nine CFST short columns. In the test, different slenderness ratios and interface treatments (with or without butter) were considered. Furthermore, Xue and Cai (1996) tested 32 CFST specimens under push-out loading to examine bond strength and deduced that the strength of interfacial bonding primarily depends on the strength of concrete, the type of interfacial treatment used, and the conditions under which the concrete was cured, whereas the length of the interface did not significantly impact the strength of the bond. The authors also suggested a formula to anticipate the bond strength of CFSTs, based only on $f_{\mathit{\text{cu}}}.$ Lyu and Han (2019) developed two equations with which the ultimate strength of the bonds in cCFSTs and sCFSTs is predicted. The equations were developed using the results of 56 push-out tests conducted on CFST specimens with square and round cross-sections of variable dimensions, interface treatment methods, and coarse recycled aggregate content. The concrete’s compressive strength, $f_{\mathit{\text{cu}}},$ ranged between 20 and 60 MPa to yield accurate predictions, while the width or diameter of the specimens should be between 120 and 600 mm, and the $\mathrm{ }D/t$ ratio should be <60. Martinelli (2021) investigated the influence of creep and shrinking on composite structures using finite element (FE) analysis and integrated time to reflect the strains that accumulate due to these long-term impacts.

Parameters, such as width-to-length ratio, section size, and wall thickness (t) have been extensively examined for CFSTs using computational and experimental methods under different loadings, leading to the formulation and refinement of multiple design-related formulas for CFSTs. It was found that the behavior of CFST columns is primarily influenced by several factors, including the cubic compression strength of the concrete ($f_{\mathit{\text{cu}}}),$ as well as the geometric properties, yield strength, and wall thickness of the steel tube (Liu et al., 2018; Sakino et al., 2004; Tao et al., 2013). Additionally, research indicated that the steel-to-concrete bonding behavior of CFSTs is affected by shrinkage, compaction, cross-sectional shape, interfacial roughness, concrete age (A), and the lubricant used (Qu et al., 2013; Roeder et al., 1999; Shakir-Khalil, 1993a; Virdi & Dowling, 1975). Reducing the width-to-thickness (w/t) ratio may increase the degree of a concrete’s confinement, while the inverse may encourage more localized buckling and crushing to occur in the concrete (Giakoumelis & Lam, 2004). Moreover, studies indicate that the confinement that concrete provides in sCFST columns is inefficient, as they lose their rigidity and only provide effective restraint around the center and edges of such columns (De Oliveira et al., 2009). Nevertheless, different types of infill concrete may exhibit behavior that is distinct from normal concrete. It is crucial to acknowledge that the properties of a material can significantly affect the strength of the bonds between concrete and steel (Hunaiti, 1996).

Various innovative machine learning approaches, such as multi-expression programming (MEP), artificial neural networks (ANNs), and gene expression programming (GEP), have been employed to analyze the bond behavior of CFSTs (Farooq et al., 2021; Sarir, Chen, et al., 2021; Sarir, Shen, et al., 2021). Machine learning (ML) approaches have become essential tools to project and optimize various civil engineering domains (Faradonbeh et al., 2018; Jahed Armaghani et al., 2018). As such, they have been used to predict and model the nonlinear behavior of various geo-engineering elements; such as concrete, soil, asphalt, and clay; leading to their ground-breaking use in the domains of optimization and material science (El Tabach et al., 2007; Javed et al., 2020). More specifically, ANNs have demonstrated an ability to predict the mechanical attributes of CFSTs. This includes the extent of their axial compression, ultimate pure bending, and fire performance (Basarir et al., 2019; Moradi et al., 2021; Tran et al., 2020). Moreover, ANNs can use a nonlinear activation function to identify and comprehend intricate patterns that are nonlinear amidst numerous variables in datasets that are very complex (Manasrah et al., 2022). They also do not require a predetermined model form, which distinguishes them from regression techniques. As such, it has been widely employed to predict the mechanical behavior of concrete as well as its structural elements (Ilyas et al., 2021). Allouzi et al. (2022) employed ANN modeling to create a complete stress-slip behavior curve, which was then employed to simulate CFST bonds in FE models. The reasonable ability of ANN to predict the steel-to-concrete interfacial bonds based on the steel tube’s geometry and the concrete’s attributes was demonstrated.

Recently, the interfacial bonding capacity ($P_u$) of CFST structures has received considerable attention. However, the majority of studies have independently modeled cCFSTs or sCFSTs. But none of the existing equations can simultaneously estimate the interfacial $P_u$ of cCFSTs or sCFSTs; consequently, there is an urgent need for a model that can accurately predict the interfacial $P_u$ of CFSTs, considering the correlations between them and their geometry, as well as concrete characteristics, such as $f_{\mathit{\text{cu}}}$ and A. The objective of this study is to propose an efficient model based on an artificial intelligence approach for predicting the interfacial $P_u$ of both cCFSTs and sCFSTs using ANN modeling. The generated ANN model predicts and models interfacial ($P_u$) using previously tested specimens with various parameters, including the age and characteristics of the concrete specimens, as well as the dimensions and shapes of the CFSTs.

2. Artificial neural network (ANN)

The Artificial Neural Network (ANN) is a popular soft computing tool that draws inspiration from the information processing capabilities of the human brain. It is widely recognized for its ability to provide more accurate outcomes for intricate issues compared to traditional numerical analytic methods while requiring considerably fewer computational resources (Ly et al., 2021; Psyllaki et al., 2018). The two most common types of ANN are the Feed-Forward network and the Recurrent network, with the former being the most used. In this study, an error backpropagation, Feed-Forward neural network with a hidden layer, and a sigmoidal activation function was used.

2.1. Data

The data consisted of 397 test result datasets from 18 existing studies (Abendeh et al., 2022; Aly et al., 2010; Fu et al., 2018; Lyu & Han, 2019; Parsley & Yura, 2000; Roeder et al., 1999; Shakir-Khalil, 1993a, 1993b; Tao et al., 2011; Virdi & Dowling, 1975). The interfacial length, between the steel tube and concrete core, of the specimens in these datasets were untreated and relied solely on the naturally-occurring bond between the surfaces. Studs and shear connectors had not been utilized either. The parameter of $f_{\mathit{\text{cu}}}$ was also used to account for the additives used to substitute aggregates in the concrete mix of some of the specimens in these datasets.

The specimen variables that were considered included the square or round shape (S), tube thickness (t), and inner perimeter (C) of the steel tube; the $f_{\mathit{\text{cu}}}$ and age of the concrete mix A; and the length of the steel tube-to-concrete interface (L_i). Figure 1 shows a schematic presentation of the circular and square geometries of the CFSTs.

Figure 1. A schematic representation of circular and square CFSTs.

Table 1 provides an overview of the dataset characteristics used to formulate the proposed ANN model.

Table 1. Characteristics of the utilized dataset.

Source	Shape	# Of samples	fcu (MPa)		Age (days)		Li (mm)		t (mm)		B (mm)		D (mm)		Pu (kN)
			Max	Min	Max	Min	Max	Min	Max	Min	Max	Min	Max	Min	Max	Min
Roeder et al. (1999)	Circular	2	53.54	53.54	29	29	1775	1775	7.11	7.11	–	–	341.4	341.4	356.15	333.29
Shakir-Khalil (1993a)	Circular/square	12	52.2	52.2	28	28	600	204	5	5	150	150	168.3	168.3	228.5	70.99
Parsley and Yura (2000)	Square	4	45.16	40.4	28	28	1498.6	1193.8	6.35	6.35	254	203.2	–	–	283.44	280.95
Lyu and Han (2019)	Circular/square	15	48.83	39.75	70	70	550	310	6.6	4	200	120	200	120	239.23	40.77
Virdi and Dowling (1975)	Circular	49	46.33	21.99	47	7	463.55	149.35	9.73	5.74	–	–	158.9	148.49	609.66	138.71
Shakir-Khalil (1993b)	Circular/square	8	45.4	42.7	28	28	400	397	6.3	5	200	150	–	–	253.76	80
Aly et al. (2010)	Circular	7	96.43	51.25	98	28	400	400	3.2	3.2	–	–	114.3	114.3	176.74	87.65
Tao et al. (2011)	Circular/square	11	97.38	51.85	111	31	1200	600	8	3.6	200	120	400	120	905.14	81.6
Fu et al. (2018)	Circular	27	37.38	28.71	28	28	780	350	4	2.5	–	–	114	114	430.37	156.75
Ahmad et al. (2023)	Circular/square	100	24.31	9.11	365	28	190	190	5.38	4.02	141.58	90.85	154.54	107.7	225.09	32.99
Abendeh et al. (2016)	Circular/square	87	24.31	14.58	365	28	190	190	5.38	4.02	141.58	90.85	154.54	107.7	235.1	37.00
Ke et al. (2016)	Circular	4	79.44	70.52	28	28	400	300	3	3	–	–	110	110	254.44	194.98
Lu et al. (2018)	Circular	16	54.4	53.9	28	28	480	480	4.25	2.5	–	–	165	165	522.72	156.82
Lv et al. (2020)	Circular	9	67.1	44.7	28	28	1533	657	6	6	–	–	351	351	1287.27	293.93
Tao et al. (2016)	Circular	2	52.71	51.85	28	28	1200	1200	8	8	–	–	400	400	1538.74	890.06
Zou et al. (2019)	Square	3	35.5	31.1	28	28	260	260	3	3	120	120	–	–	67.39	58.66
Sangeetha et al. (2018)	Circular	5	30	30	28	28	460	460	6	6	–	–	113	113	248.81	156.67
Abendeh et al. (2022)	Circular/square	36	55.3	37	365	30	190	190	5.38	4.48	141.58	141.58	154.54	154.54	344.21	64.56
All datasets	Circular/square	397	97. 83	9.11	365	7	1775	149.35	9.73	2.5	254	90.85	400	107.7	1538.74	32.99

2.2. Performance assessment

Several performance statistical indices have been proposed in the literature to evaluate the prediction capacity and accuracy of various models. This study employed average squared error (ASE), mean absolute relative error (MARE), and coefficient of determination (R²) as evaluation metrics to analyze and determine the most suitable neural network architecture. Lower values of ASE and MARE imply higher levels of accuracy in predicting results (zero denotes a perfect fit). Whereas R² measures the variation attributed by the model. R² values range from 0 to 1, a number close to 1 suggests a strong predictive ability, while a value close to 0 indicates inadequacy. These performance metrics are a reliable indicator of the comprehensive predictive precision of ANN.

It should be noted that the R² value is mostly employed to judge the accuracy of performance in comparison to other methodologies. Nevertheless, in cases where two predictive models yield distinct R² values together with various slopes, it becomes impossible to make a meaningful comparison between the two models. In this context, an engineering indicator called the a20-index has been recently proposed to enhance the dependability of model assessments (Apostolopoulou et al., 2020; Asteris et al., 2021; Asteris & Mokos, 2020; Zhou et al., 2020). (1) $$a20-\mathit{\text{index}}=\frac{m20}{M}$$(1)

Where M represents the total number of samples in the dataset, and m20 is the number of samples for which the ratio of the experimental value to the anticipated value falls within the range of 0.80–1.20. It should be noted that in the case of an ideal predictive model, the predicted values for the a20-index equals one. The suggested a20-index possesses the benefit of tangible engineering significance, as it quantifies the number of samples that meet the expected values within a deviation range of ±20% when compared to the experimental values.

2.3. Model structure and training

The datasets used for the ANN model were categorized into three sets; (1) training, (2) testing, and (3) validation. The training datasets were used for optimizing the weights of the connecting links of the network across various networks with different numbers of hidden nodes that ranged from 1 to 15. The statistical accuracy of all the trained networks; namely, MARE, ASE, R², and a20-index; were computed. The testing datasets were then employed to assess the precision of the trained networks. The network with the lowest ASE was deemed to have the optimal quantity of nodes that were hidden. The validation datasets, that had not been utilized during testing or training, were then employed to substantiate and verify the model by comparing the outputs. It is noteworthy, that to efficiently train an ANN, the inputs as well as the outputs should be scaled and normalized so that the entire data ranges from 0.1 to 0.9 before modeling (Allouzi et al., 2023). As such, all the datasets were normalized before modeling. After the inputs of the ANN model had been scaled, its outputs were scaled as well. Table 2 presents the inputs and output utilized in the formulated ANN model while Figure 2 depicts a schematic of the 6 inputs, n hidden nodes, and 1 output (6-n-1) ANN model. The mathematical formulas for the 6-n-1 ANN can be expressed as seen in Equations (2)(2) $${\mathit{\text{Output}}}_m=\frac{1}{1+e^{-{\sum }_{i=1}^n{\mathit{\text{HN}}}_i\mathrm{*}\left({\mathit{\text{HN}}}_i-{\mathit{\text{Output}}}_m\mathit{\text{Connection}}\right)+{\mathit{\text{Bias}}}_{2m}}}$$(2) and (3)(3) $${\mathit{\text{HN}}}_i=\frac{1}{1+e^{-{\sum }_{j=1}^5{\mathit{\text{Input}}}_j\mathrm{*}\left({\mathit{\text{Input}}}_j-{\mathit{\text{HN}}}_i\mathit{\text{Connection}}\right)+{\mathit{\text{Bias}}}_{1i}}}$$(3) . (2) $${\mathit{\text{Output}}}_m=\frac{1}{1+e^{-{\sum }_{i=1}^n{\mathit{\text{HN}}}_i\mathrm{*}\left({\mathit{\text{HN}}}_i-{\mathit{\text{Output}}}_m\mathit{\text{Connection}}\right)+{\mathit{\text{Bias}}}_{2m}}}$$(2) (3) $${\mathit{\text{HN}}}_i=\frac{1}{1+e^{-{\sum }_{j=1}^5{\mathit{\text{Input}}}_j\mathrm{*}\left({\mathit{\text{Input}}}_j-{\mathit{\text{HN}}}_i\mathit{\text{Connection}}\right)+{\mathit{\text{Bias}}}_{1i}}}$$(3)

Figure 2. A schematic of the six inputs, the n hidden nodes, and the one output of the ANN.

Table 2. The parameters used to model the ANN.

	Parameter		Unit	Notes
Input1	C	Inner perimeter of the tube	mm	–
Input2	t	Tube thickness	mm	–
Input3	Li	Interfacial length	mm	–
Input4	fcu	Concrete compressive strength	MPa	–
Input5	A	Age	Days	–
Input6	S	Shape factor	–	1 for circular and 0 for squared
Output1	Pu	Bond capacity	kN	–

The datasets were split into sets containing 60, 297, and 40 datasets to test, train, and validate the formulated ANN model, respectively. All three sets were selected randomly, and the maximum and minimum values for each parameter were further incorporated into the training set. Figure 3 depicts the ASE_training, ASE_testing, MARE_training, MARE_testing, R²_training, and R²_testing of the ANN model during the optimization stage. The optimal quantity of hidden nodes was deemed to be six, as it yielded the lowest ASE_testing. Therefore, the architecture of the model was 6-6-1.

Figure 3. The statistical accuracy of the networks with varying quantities of hidden nodes. (a) ASE. (b) MARE. (c) R².

Table 3 provides the statistical accuracy of the training and testing of the 6-6-1 ANN model, while Figure 4 compares the prediction accuracy of the experimental dataset of the developed ANN with that of its testing and training datasets. Meanwhile, Table 4 lists the weights of the connection links of the optimal network for Equations (2)(2) $${\mathit{\text{Output}}}_m=\frac{1}{1+e^{-{\sum }_{i=1}^n{\mathit{\text{HN}}}_i\mathrm{*}\left({\mathit{\text{HN}}}_i-{\mathit{\text{Output}}}_m\mathit{\text{Connection}}\right)+{\mathit{\text{Bias}}}_{2m}}}$$(2) and (3)(3) $${\mathit{\text{HN}}}_i=\frac{1}{1+e^{-{\sum }_{j=1}^5{\mathit{\text{Input}}}_j\mathrm{*}\left({\mathit{\text{Input}}}_j-{\mathit{\text{HN}}}_i\mathit{\text{Connection}}\right)+{\mathit{\text{Bias}}}_{1i}}}$$(3) .

Figure 4. The prediction results of the training and testing vs. the experimental datasets of the 6-6-1 ANN.

Table 3. The statistical accuracy of the training and testing datasets of the proposed 6-6-1 ANN.

Model	6-6-1 ANN
ASEtraining	0.000995
ASEtesting	0.002656
MAREtraining	24.584
MAREtesting	31.379
R^2training	0.871
R^2testing	0.634
a20-indextraining	0.734
a20-indextesting	0.883

Table 4. The weights of the connection links for the 6-6-1 ANN.

Inputj	Connection links weights between inputs and hidden nodes (Inputj–HNi connection)
	HN1		HN2		HN3		HN4		HN5		HN6
Input1 C	2.507500		−7.230200		3.062600		5.977200		−1.684200		−0.044326
Input2 t	−5.363200		1.552700		0.159110		−1.775900		0.324500		−0.323030
Input3 Li	−9.213300		−9.468900		0.305460		6.861300		1.408200		0.193230
Input4 fcu	1.331600		10.747000		−1.652800		4.521500		−0.508530		−0.058638
Input5 A	−6.502100		−4.779500		−11.720000		−2.187100		−1.903400		−1.198100
Input5 S	−1.967400		−3.215600		3.782300		−2.334000		0.590650		0.243100
Bias1	9.936400		10.618000		−3.544800		0.200600		−0.682390		−0.972860
	Connection links weights between hidden nodes and outputs (HNi–Outputm connection)
Outputm	HN1	HN2		HN3		HN4		HN5		HN6		Bias2
Output1 Pu	−2.881200	7.431100		4.785500		1.609400		1.362600		0.031348		−8.078700

The importance factor (IF) of every input parameter was computed using the approach proposed by Olden and Jackson (2002), which computes the total weight of the connections of each input by multiplying the weights of the hidden connection links for all the inputs and outputs. The products of all the hidden nodes were then summed to determine the IFs. Figure 5 provides the calculated IFs of the inputs utilized in the 6-6-1 ANN. As shown, the age was the most significant predictor of $P_u$ followed by the length of the interfacial bond, which ascertains the concrete-to-steel tube bond area and the tube inner perimeter. Conversely, the compressive strength and the tube shape carried nearly equal importance in the model while tube thickness contributed the least to the estimation of $P_u$ in the ANN model.

Figure 5. The IF of the inputs of the 6-6-1 ANN.

2.4. Model validation

Forty validation datasets were used to compare the predictions of $P_u$ to the experimental findings to validate the 6-6-1 ANN. The validated predictions vs. the findings of the experimental data (Figure 6) and validated statistical accuracy (Table 5) demonstrate that the predictions of the ANN were highly acceptable and plausible. It should be noted that the data used for the validation was not used to develop the model.

Figure 6. The validation results of the 6-6-1 ANN vs. the experimental data.

Table 5. The validated statistical accuracy of the 6-6-1 ANN.

Model	6-6-1 ANN
ASEvalidation	0.001173
MAREvalidation	27.127
R^2validation	0.848
a20-indexvalidation	0.675

Figure 7 depicts the error frequency, which is the difference between the experimental and predicted $P_u,$ of the 397 datasets plotted against the normal distribution using the same mean (μ = −3.23 kN) and standard deviation (σ = 67.58 kN). The prediction error was almost normally distributed around the mean, as seen in the comparison of the normal distribution and frequency of the errors.

Figure 7. The prediction error frequency vs. the error normal distribution of the 397 datasets used in the 6-6-1 ANN.

3. Results and discussion

3.1. Comparisons with published models

The $P_u$ of the developed 6-6-1 ANN were compared with equations obtained from extant studies on cCFSTs; namely, Lyu and Han (2019), Roeder et al. (1999), and Xue and Cai (1996); while the equations for sCFSTs were obtained from Lyu and Han (2019), Parsley and Yura (2000), and Xue and Cai (1996). Table 6 shows the σ and μ of the ratio between the predicted and experimental findings as well as the R² of the $P_u$ estimated by the ANN model and extant models. As seen in Table 6, the proposed ANN more accurately described the majority of the specimens than extant equations for cCFSTs and sCFSTs.

Table 6. A comparison of the σ, μ, and R² of the $P_u.$

References	Prediction model	Mean value, μ	Standard deviation, σ	R^2
Circular CFST
Roeder et al. (1999)	${\tau }_u=2.109-0.026(D/t)$	0.976	0.534	0.478
Xue and Cai (1996)	${\tau }_u=0.1{(f_{\mathit{\text{cu}}})}^{0.4}$	0.314	0.294	0.400
Lyu and Han (2019)	${\tau }_u=0.071+4900(t/D^2)$	0.972	0.485	0.469
6-6-1 Proposed ANN		1.080	0.314	0.801
Square CFST
Parsley and Yura (2000)	${\tau }_u=0.013+1751(t/B^2)$	0.825	0.447	0.313
Lyu and Han (2019)	${\tau }_u=0.043+1100(t/B^2)$	0.574	0.305	0.337
Xue and Cai (1996)	${\tau }_u=0.1{(f_{\mathit{\text{cu}}})}^{0.4}$	0.607	0.566	0.391
6-6-1 Proposed ANN		1.123	0.387	0.563

Where: D: circular tube diameter; B: square tube width; t: tube thickness; ${\tau }_u:$ bond strength.

The Taylor diagram analysis was used to compare the developed 6-6-1 ANN to the extant models listed in Table 6. Figures 8(a,b) show the differences between extant models for sCFSTs and cCFSTs, respectively. As seen in Figure 8(a), the correlation and σ of the 6-6-1 ANN's predictions for cCFSTs were closer to those of the experimental findings (dotted red line) than the other models. The correlation and σ of the 6-6-1 ANN's predictions for sCFSTs were, similarly, closer to those of the experimental findings than the other models (Figure 8(b)).

Figure 8. The Taylor diagram comparison of the proposed 6-6-1 ANN and extant models. (a,b) Normalized standard deviation.

3.2. Trends analysis

Overfitting is a frequently reported issue that arises in the mathematical simulation of datasets (Armaghani & Asteris, 2021; Asteris et al., 2021; Mansour et al., 2004). This implies that a model has the potential to perform well in anticipating data utilized for its development and training, yet it may also exhibit very unusual behavior when applied to different datasets. Consequently, it is important to compare the overall behavior of the developed 6-6-1 ANN model in anticipating the ultimate bond strength to the expected behavior based on the parameters considered in this study. This also allows the model to investigate the trend and sensitivity of the variables influencing the steel-to-concrete bonds in CFSTs. To test this, two parameters, namely the thickness of the steel tube and the compressive strength of the concrete, were chosen and plotted against the $P_u.$

Figure 9 depicts the correlations between tube thickness and $P_u$ for the proposed ANN model and several models available in the literature. These correlations were examined for both circular and squared CFSTs, while maintaining a constant value for the remaining parameters (C = 400 mm, L_i = 300 mm, A = 28 days, and $f_{\mathit{\text{cu}}}$ = 30 MPa). As shown, the curves are smooth and follow the expected trend, which provides a good indication that overfitting is not occurring in the model such that a higher tube stiffness causes the $P_u$ to increase as tube thickness increases (Shaker et al., 2022). The reason for this phenomenon is that the thicker steel tube offers increased confinement to the concrete core and demonstrates enhanced stripping resistance to the lateral strains induced in the concrete (Lu et al., 2018). It was also observed that the use of a thicker steel tube resulted in an extended chemical adhesion stage of the load vs. slip curves, consequently leading to a delay in the bond failure of CFST columns (Lu et al., 2018). Moreover, it is worth mentioning that the influence of tube thickness on $P_u$ was found to be more pronounced in CFSTs with circular cross-sections compared to those with square cross-sections, as indicated by Figure 9. This can be attributed to the circular section that possesses uniformly distributed frictional points along its entire surface, leading to more effective confinement compared to specimens with square geometry, which only exhibit friction marks at the corners (Abendeh et al., 2022; Shakir-Khalil, 1993a). Furthermore, research has indicated that the subscription percentages of micro-interlocking to bond strength in cCFSTs vary between 32 and 75. In contrast, for sCFSTs, this range is between 10 and 20 (Chen et al., 2009; Virdi & Dowling, 1975).

Figure 9. Bond capacity vs. tube thickness.

Furthermore, the correlations between the concrete compressive strength and $P_u$ for the proposed ANN model and several models available in the literature are presented in Figure 10. These correlations were examined for both circular and squared CFSTs while maintaining a constant value for the remaining parameters (C = 400 mm, L_i = 300 mm, A = 28 days, and t = 5 mm). As seen in Figure 10, the curves are smooth and follow the expected trend, which provides a good indication that overfitting is not also occurring in the model such that raising the $f_{\mathit{\text{cu}}}$ from 10 to 60 MPa augments the $P_u$ of cCFSTs and sCFSTs, but with small effect, as observed in several studies (Allouzi et al., 2022; Roeder et al., 1999). Notably, once the $f_{\mathit{\text{cu}}}$ exceeded 60 MPa, the $P_u$ decreased. Due to considerable shrinkage, the $P_u$ of CFSTs loaded with superior $f_{\mathit{\text{cu}}}$ decreases (Martinelli, 2021; Qu et al., 2015; Szadkowska & Szmigiera, 2021).

Figure 10. Bond capacity vs. concrete compressive strength.

It should be noted that the relationships of other parameters influencing the ultimate bond strength, such as age, cannot be applied to such comparisons, i.e. overfitting issues or trend analysis. This is due to the variety of concrete mixtures used in the literature as infill for steel tubes, which influence the ultimate bond strength differentially with age. Normal concrete, for example, exhibits a decrease in ultimate strength with age due to shrinkage problems that reduce the contact area between the concrete core and steel tube, thereby decreasing the bond strength (Aly et al., 2010; Roeder et al., 1999; Tao et al., 2016), whereas self-stressing concrete exhibits self-expansion behavior over time (Parsley & Yura, 2000; Xu et al., 2009). This expansion behavior compresses air pockets that form during pouring and enlarges the contact surface. Furthermore, the expansion enhances the bonding behavior of CFST columns by increasing the normal force at the interface. For the study of the sensitivity analysis to be reliable in regard to how age affects the ultimate bond strength, as well as for the use of such a trend to ensure no overfitting issues, mixture types must be studied separately. It is also essential to note that this paper investigates the influence of CFST parameters on the ultimate bond load, and parameters, such as interfacial length and interior perimeter of the steel tube cannot be correlated with the ultimate load in such research.

4. Limitations and future work

It is important to emphasize that the neural network models can be effectively utilized within the parameter range specified in Table 4. Any predictions made outside of this range may not be considered accurate. Notably, additional comparisons and investigations with accurate laboratory data are required, so future work in this field can concentrate on conducting more experiments. For instance, investigate the high-performance or high-strength infill concrete for CFSTs. This could help in making predictions about $P_u$ more general. So that other data points can occupy the gap between 600 and 1600 kN for the ultimate bond capacity range, allowing for more accurate estimations, as shown in Figure 6. Moreover, the data in this study was collected from multiple studies involving various types of concrete mixtures. Future research can also be conducted by studying the effect of several concrete mixtures separately; this will enable the sensitivity analysis to be conducted on all parameters influencing the bond. Mixtures of normal concrete, for instance, exhibit a decrease in ultimate strength with age due to shrinkage issues. In contrast, other types of concrete, such as self-stressing concrete, exhibit self-expansion over time, which enlarges the contact surface and improves bonding.

5. Conclusions

It is important to note that obtaining a precise and dependable determination of $P_u$ in CFSTs can result in time savings during the production of such systems, as well as cost and performance optimization. In this study, an extensive database of 143 squared CFSTs specimens and 254 circular CFSTs specimens from multiple extant studies was utilized to generate one ANN predictive model to include both shapes with six model inputs, i.e. L_i, t, S, C, f_cu, and A. The findings of this study can be succinctly described as follows:

• The statistical evaluations conducted on the validation datasets revealed that the proposed ANN model exhibited adequate predictive accuracy for $P_u.$ For both cCFSTs and sCFSTs, the assessment indices ASE, MARE, R², and a20-index yielded values of 0.001173, 27.127, 0.848, and 0.675, respectively. Furthermore, the R² value of the training datasets was 0.871. It is important to mention that the test results were derived from a range of experiments conducted under diverse conditions. The discrepancies in the outcomes can be attributed to variances in specific characteristics, manufacturing processes, and variations in internal forces. Hence, it is reasonable to observe differences between the outcomes obtained by the constructed ANN and the experimental data.

• The developed ANN model was also compared to other available regression models. Mean values, standard deviations, R², and Taylor diagram analysis results also indicate that the ANN model was the most suitable model for predicting $P_u$ of all the examined extant equations.

• Using the generated ANN model, the correlations between $P_u$ and the model’s input parameters, namely the thickness of the steel tube and the compressive strength of the concrete mixture, were described. This enables the model to examine the sensitivity of variables influencing the steel-to-concrete interactions in sCFSTs and cCFSTs following design, optimization, and validation. The generated correlations were found to correspond well with the findings and expected trend of published research studies, moreover, they substantiate the absence of overfitting.

Acknowledgments

Does not apply.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

No funding received for this work.

Notes on contributors

Hatem H. Almasaeid

Dr. Hatem Almasaeid holds the position of assistant professor in civil engineering at Al al-Bayt University in Mafraq, Jordan. He earned his BS and MS degrees from Jordan University of Science and Technology in 2013 and 2015, respectively, before completing his PhD at the University of Mississippi in 2018. His research pursuits encompass a range of interests, prominently featuring applications of Artificial Neural Networks (ANN), alongside a focus on monitoring, assessing, and retrofitting concrete structures.

Donia G. Salman

Donia Salman is a Ph.D. Student in the department of civil Engineering at the University of Mississippi, Oxford, MS. She received her BS from Hashemite University, Zarqa, Jordan, in 2014 and her MS from the University of Jordan, Amman, Jordan, in 2018. Her research interests include construction and building materials, materials modeling, and finite element analysis.

Raed M. Abendeh

Raed Abendeh is Associate Professor in the department of civil and infrastructure engineering at Al-Zaytoonah University of Jordan, Amman, Jordan. He received his BS and MS from Jordan University of Science and Technology, Irbid, Jordan, in 1998 and 2001, and his Ph.D. from Technical University of Hamburg, Hamburg, Germany. His research interests include construction and building materials and finite element analysis.

Rabab A. Allouzi

Rabab Allouzi is a professor of civil engineering at the University of Jordan in Amman, Jordan. She received her BS and MS from the University of Jordan in 2008 and 2010 and PhD from Purdue University in 2015. Her research interests include seismic response of reinforced concrete structures, design of innovative structural systems, nanotechnology in structures and finite element analysis of structural systems.

Hesham S. Rabayah

Hesham Rabayah is Associate professor in the department of civil and infrastructure engineering at Al-Zaytoonah University of Jordan, Amman, Jordan. He received his MS from Jordan University of Science and Technology, Irbid, Jordan, in 1997, his MS from Amman Arab University for Graduate Studies, Amman, Jordan, in 2005, and his Ph.D. from University of Birmingham, Birmingham, West Midland, England, UK, in 2010. His research interests include sustainability, construction materials, and management. He was EPPM association president from 2020-2022.