Abstract
Bond–slip is an important characteristic that determines the stiffness, displacement, and loadbearing capacity of a reinforced concrete (RC) beam. It is essential for performing a precise numerical analysis of the beam. In most cases, bond–slip models can define the bond–slip curve only when there are experimental data. However, many bond test data have been obtained from pullout tests, and the dominant view is that the bond–slip behavior observed in the pullout test is quite different from that in an actual RC beam. Therefore, a mapping function that makes it possible to estimate the bond–slip behaviors of beam specimens using those of pullout specimens was developed in this study. A total of 255 pullout specimen data and 75 beam specimen data were collected from previous studies, and the importance and influence of each feature of the two groups were analyzed using random forest and Kmeans clustering. The mapping function was derived using genetic programming, and its accuracy was verified through a comparison with existing models. The proposed model exhibits a high degree of accuracy in estimating bond–slip and bond strength in beam specimens and can provide useful information for understanding the difference in bond–slip behaviors between the two groups.
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1 Introduction
Reinforced concrete (RC) is a composite material composed of concrete and rebar, and the bond between the two materials is a major factor in determining the stiffness, deformation, and loadcarrying capacity of RC members (Alharbi et al., 2021; Darwin et al., 1992; Eligehausen et al., 1982; Tepfers, 1979). Therefore, various empirical models to estimate bond strength (\(\tau_{\max }\)) have been proposed (Esfahani & Kianoush, 2005; Harajli et al., 1995; Orangun et al., 1977; Wu & Zhao, 2013; Xu, 1990). On the other hand, estimation models for bond–slip (\(\tau  s\)) were very rare (Wu & Zhao, 2013), and bond–slip (\(\tau  s\)) models presented by most previous research and design codes require experimental data, which makes it difficult to apply them to the numerical analysis of RC members without testing. (Comité EuroInternational du Béton, 1993; Eligehausen et al., 1982; Martin, 1973; Mirza & Houde, 1979; Nilson, 1968; Rehm, 1961).
However, the test data available for the derivation of \(\tau  s\) are quite limited, and the reason is explained in Fig. 1. The pullout test is a simple bond testing method by directly pulling out a reinforcing bar embedded in concrete. In this case, tensile stress occurs in the rebar, and compressive stress arises in the surrounding concrete (Alharbi et al., 2021). The other bond test method is the beam test in which a reinforcing bar in an RC beam is pulled out by an external moment. In this case, both the rebar and the surrounding concrete are subjected to tensile stress (Alharbi et al., 2021). Owing to this difference in the stress state, the \(\tau  s\) developed in the RC beam may be quite different from the \(\tau  s\) obtained from the pullout specimen (Alharbi et al., 2021). Nevertheless, because of the simplicity of the pullout test, most of the \(\tau  s\) data are pullout test results, and the beam test data are very limited.
If a mapping function that can convert the \(\tau  s\) of pullout specimens to those of beam specimens is developed, the \(\tau  s\) curve of RC beams can be estimated using the large amount of \(\tau  s\) data obtained from the pullout test. Therefore, a mapping function between the \(\tau  s\) of pullout and beam specimens was developed in this study. The overall research process complies with the data science pipeline consisting of data acquisition, analysis, modeling, and verification steps. In the next section, the background and existing models are described. The data and machinelearning methods used in this study are introduced in Sect. 3 and the data analysis results and proposed models are presented in Sect. 4. The verification of the accuracy of the proposed model is described in Sect. 5.
2 Backgrounds
2.1 Bond and stress components
Fig. 2 shows the bond and stress components of the rebar embedded in concrete. As shown in Fig. 2a, the bond components are chemical adhesion, friction, and mechanical interlock (Lutz & Gergely, 1967; Tepfers, 1979). Chemical adhesion is a bond component arising from physical and chemical bonding between concrete and rebar and makes little contribution to bond strength, because it disappears when a slip occurs (Alharbi et al., 2021). Friction is caused by the roughness of the surface and normal stress between the concrete and rebar. Lutz and Gergely reported that normal stress is caused by the shrinkage of concrete (Lutz & Gergely, 1967). Mechanical interlocking is a bond component arising from the shear resistance of the rib and adjacent concrete, which has the greatest effect on the bond strength of the deformed bar (Alharbi et al., 2021; Lutz & Gergely, 1967).
Fig. 2b shows the stress component caused by the bond. The stress that occurs in the longitudinal direction of the rebar is shear stress (\(\tau_{l}\)). The average shear stress (\(\tau\)) caused by the bond is calculated as
where \(d_{b}\) is the rebar diameter, and \({\Delta }T\) is the increment of the tensile force generated in a unit length (\({\Delta }x\)) of rebar. The stress generated in the horizontal direction of the rebar section is divided into radial stress (\(\sigma_{{\text{r}}}\)) and tangential stress (\(\sigma_{\theta }\)), and the relationship between \(\sigma_{r}\) and \(\tau\) is (Tepfers, 1979)
where \(\alpha\) is the angle between the principal compressive bond stress and the axis of the reinforcing bar, which is approximately 45° (Tepfers, 1979). Based on the thickwall cylinder theory (Timoshenko, 2002), the tangential stress (\(\sigma_{\theta }\)) can be represented using the following equation:
where \(c\) is the cover depth, and \(r\) is the radial distance from the center of the rebar. As shown in Eq. (3) and Fig. 2b, \(\sigma_{\theta }\) is maximum at the surface of the reinforcing bar, and it decreases as the \(r\) increases.
2.2 Failure Mode, Bond Strength, and Bond–Slip Behavior
Fig. 3 shows the bond–slip curve according to the pullout failure or splitting failure (Mazumder & Gilbert, 2019). Pullout failure is a failure mode in which concrete around the rib is crushed. This mainly occurs when the cover depth (\(c\)) is great, the compressive strength of concrete (\(f^{\prime}_{{\text{c}}}\)) is low, or the stress of the concrete near the rib is large, because the embedded length \(\left( {L_{{\text{e}}} } \right)\) is small. Splitting failure is a failure mode in which splitting cracks occur in the longitudinal direction when \(\sigma_{\theta }\) exceeds the tensile strength of concrete (\(f_{{\text{t}}}\)). This mainly occurs when \(c\) is shallow or \(f_{{\text{t}}}\) is small. As shown in Fig. 3, splitting failure shows a rapid decrease in bond strength (\(\tau_{{{\text{max}}}}\)) compared with pullout failure. There is almost no residual strength in the unconfined specimen, whereas the residual strength is partially maintained in the confined specimen. In addition, the bond strength (\(\tau_{{{\text{max}}}}\)) of the confined specimen is larger than that of the unconfined specimen. Therefore, the bond characteristic of rebar varies depending on various influencing parameters and ultimately has a great impact on the behavior of the flexural member.
2.3 Influencing Parameters
Various tests have been conducted on the influencing factors of bonds (Darwin et al., 1992; Eligehausen et al., 1982; Esfahani & Kianoush, 2005; Harajli et al., 1995; Orangun et al., 1977; Soroushian & Choi, 1989; Walker et al., 1997), including \(f^{\prime}_{{\text{c}}}\), \(c\), \(d_{{\text{b}}}\), the crosssectional area of one leg of transverse reinforcement (\(A_{{{\text{st1}}}}\)), and the spacing of transverse reinforcement (\(S_{{{\text{st}}}}\)). Here, \(f^{\prime}_{{\text{c}}}\) is the most important influencing factor that increases \(\tau_{{{\text{max}}}}\)(Wu & Zhao, 2013), and the effect on \(\tau_{{{\text{max}}}}\) is mainly proportional to \(\sqrt {f^{\prime}_{{\text{c}}} }\) (Esfahani & Kianoush, 2005; Harajli et al., 1995; Orangun et al., 1977). Furthermore, \(c\) is an influencing factor that increases \(\tau_{{{\text{max}}}}\), but the effect gradually decreases as \(c\) increases (Walker et al., 1997). \(d_{{\text{b}}}\) serves as a factor that decreases \(\tau_{{{\text{max}}}}\) and is often reflected in the form of \(c/d_{{\text{b}}}\) (Ichinose et al., 2004). \(A_{{{\text{st1}}}}\) and \(S_{{{\text{st}}}}\) are influencing factors confining longitudinal reinforcing bars, and \(\tau_{{{\text{max}}}}\) tends to increase as \(A_{{{\text{st1}}}}\) increases or \(S_{{{\text{st}}}}\) decreases.
2.4 Existing Models
The empirical models to estimate \(\tau_{\max }\) are presented in Table 1. Orangun et al. (1977) proposed an empirical model based on nonlinear regression analyses of bond test data. In this model, the influences of \(c\) and \(f_{{{\text{yt}}}}\) (the yield strength of rebar) on the bond strength are reflected linearly. Xu (1990) and Harajli et al. (1995) proposed simpler models, compared to Orangun's model, where the influence of \(c\) is reflected nonlinearly and \(f_{{{\text{yt}}}}\) is excluded. Esfahani and Kianoush (2005) proposed an empirical model with high complexity, where the median of side cover, bottom cover, and rebar spacing (\(c_{{{\text{med}}}}\)) is implemented. Wu and Zhao (2013) also proposed a complex bond strength model that provides a unified result with their bond–slip model.
The bond–slip (\(\tau  s\)) models are shown in Table 2. The top six models require \(\tau  s\) test data for completing the curve. On the other hand, Wu and Zhao’s model (2013) gives a complete \(\tau  s\) curve without bond test data once the influencing factors, such as compressive strength of concrete (\(f^{\prime}_{{\text{c}}}\)), cover depth (\(c\)), rebar diameter (\(d_{{\text{b}}}\)), crosssectional area (\(A_{{{\text{st}}}}\)), and spacing of transverse reinforcement (\(S_{{{\text{st}}}}\)), etc., are provided.
3 Materials and Methods
3.1 Materials
As shown in Table 3, 255 and 31 \(\tau  s\) data were collected from the previous pullout and beam tests, respectively. Seven data points were extracted from each \(\tau  s\) curve. The data included the points at which slip (\(s\)) is the maximum and the minimum, the points at which it is \(\tau = \tau_{{{\text{max}}}}\), and the points which are the closest points to each of 4 points splitting the maximum and the minimum of slip with equal distance. As a result, 1785 and 217 data points were collected from the pullout and beam test data. The \(f^{\prime}_{{\text{c}}}\) of pullout specimens ranged from 5 to 150 MPa, the \(c\) from 15 to 96 mm, the \(d_{{\text{b}}}\) from 6 to 32 mm, the \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) from 0 to 3.6, the s from 0 to 25 mm, and the \(\tau\) from 0 to 48 MPa. Similarly, the \(f^{\prime}_{{\text{c}}}\) of the beam specimens ranged from 24 to 93 MPa, the \(c\) from 15 to 59 mm, the \(d_{{\text{b}}}\) from 8 to 40 mm, the \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) from 0 to 2, the s from 0 to 4.2 mm, and the \(\tau\) from 0 to 28 MPa. In addition, as shown at the bottom of Table 3, 44 beam test data with only \(\tau_{{{\text{max}}}}\) reported were collected to compensate for the lack of data in the beam test. The \(f^{\prime}_{{\text{c}}}\) of the data ranged from 29 to 93 MPa, the \(c\) from 15 to 76 mm, the \(d_{b}\) from 12 to 26 mm, the \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) from 0 to 1.6, and the \(\tau\) from 4 to 18 MPa. Fig. 4 shows the histograms of \(\tau_{{{\text{max}}}}\) data. The features and counts are presented in x and yaxes, respectively. The highest frequency occurred at \(f_{{{\text{ck}}}}\) = 30 MPa, \(c\) = 67 mm, \(d_{{\text{b}}}\) = 16 mm, \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) = 0 mm, and \(\tau_{\max }\) = 13 MPa.
3.2 Methods
3.2.1 Random Forest
A regression tree (RT) (Breiman et al., 1984) can be used advantageously in modeling and data analysis, because it can provide a regression model with moderate accuracy and feature importance. Fig. 5a shows the regression model of data consisting of two features (\(x_{1,i} ,x_{2,i} \in X\)) and a target (\(y_{i} \in Y\)) to explain the learning process of RT. As shown in Fig. 5b, the learning process of an RT is to build up splits (\(s_{p},\) i.e., ifelse function) that can minimize the impurity (\(I\)) of data from the root node.
The first step in the learning process is to define the impurity (\(I\)), and the \(I\) that a node \(t\) has in a regression problem can be defined as variance (\({\text{Var}}\left( t \right)\)), as follows:
where \(N\left( t \right)\) is the number of data in node \(t\), and \(\overline{y}\left( t \right)\) is the average of \(y\) in node \(t\). The next step is to define a split set (\(S_{p}\), \(s_{p} \in S_{p}\)), which is created by sorting the values of each feature in order and adopting the midpoint. For example, when the \(x_{1}\) of the data included in \(t\) are 0.5, 1.5, and 2.5 and the \(x_{2}\) are 20, 40, 60, and 80, \(S_{p} = \left\{ {\left( {x_{1} :1} \right),\left( {x_{1} :2} \right),\,\left( {x_{2} :30} \right),\left( {x_{2} :50} \right),\left( {x_{2} :70} \right)} \right\}\). The next step is to derive the \(s_{{{\text{p,max}}}} \in S_{p}\) that can minimize the impurity change (\({\Delta }I\)), and \({\Delta }I\) can be calculated as
where \(t_{{\text{L}}}\) and \(t_{{\text{R}}}\) are the left and right child nodes generated from \(s_{p}\), respectively. For each \(s_{p}\), \({\Delta }I\) is calculated, and \(s_{{{\text{p,max}}}}\) is derived based on the \(\mathop {\arg \max }\limits_{s_{p}\, \in \,S_{p}} {\Delta }I\) value. Through the repetition of the above process, an RT is built, and the feature importance is derived based on the \({\Delta }I\).
The RT is prone to overfitting, because it can build a regression model that fits the training set perfectly. In addition, because the RT selects \(s_{p}\) based on variance, it is greatly affected by the data distribution. One method devised to compensate for these problems is the random forest (RF) (Breiman, 2001), and the main concept of this method is the bagging and random selection of features. Bagging is a method of creating multiple trees using various data sets generated through sampling with replacements and averaging the predicted values of all trees to determine the final predicted value. The random selection of features is a method of limiting the number of features that can be used to generate \(S_{p}\), and it serves to decorrelate the trees that make up the RF. The regression model derived from this method exhibits superior generalization performance to that of the RT (Breiman, 2001).
3.2.2 KMeans Clustering
Clustering is the task of grouping similar data and is useful in data mining, because it can summarize data (Xu & Wunsch, 2008). Kmeans clustering, one of the most common methods used in clustering, is an algorithm that groups a given data set into a set of K clusters (Ding & He, 2004). Fig. 6 shows the process of Kmeans clustering, where \(\times\) is each datum, and \({\text{o}}\) is the center of the cluster. The dotted line represents the distance (\(D\)) between a cluster center and a datum, and the Euclidean distance (\(L_{2}\)) can be used to measure \(D\), as shown below:
where \(x\) is the datum, and \(\mu\) is the center of the cluster. The objective of Kmeans clustering is to minimize the variance between \(\mu\) and \(x\) of each cluster, which can be represented as
To do this, the algorithm repeats the following process.

1.
Determine which cluster the data belong to: Identify the \(D\) between \(x\) and \(\mu\) and then determine the cluster to which each \(x\) belongs based on Eq. (7).

2.
Move the centroid of the cluster: Move \(\mu\) to the centroid of each cluster.
The process continues until the centroid of the cluster no longer changes.
3.2.3 Genetic Programming
Genetic programming (GP) is an optimization algorithm that simulates evolution and can be used for symbolic regression, classification, and feature selection (Koza, 1994). Fig. 7 shows the symbolic regression using GP. The algorithm creates several expression trees (a formula expressed as a tree type) to generate a population. Subsequently, it evaluates the fitness (\(f\), i.e., estimation error) of each tree, where the mean squared error (MSE) or mean absolute error (MAE) can be used to calculate \(f\) based on true (y_{i}) and prediction values (y_{pi}):
Based on the previously evaluated \(f\), two excellent parent trees are selected to perform a crossover that exchanges a part of each tree. Subsequently, a mutation is carried out to change a part of the tree to a random tree. The generated offspring tree is merged into the nextgeneration population, and this process continues until a userdefined termination condition is met.
4 Results and Discussion
4.1 Parametric Study
4.1.1 Feature Importance
There are some machine learningbased methods to analyze the feature importance of the data set (Kuhn & Johnson, 2019). For example, recursive feature elimination (RFE), which observes the change of the model performance by recursively eliminating a variable in the training data set, can be used for deriving feature importance. However, RFE can underestimate the importance of the features with colinearity (Granitto et al., 2006). Meanwhile, although impuritybased feature importance, such as RF tends to underestimate the importance of categorical features, it is less affected by collinearity than RFE (Deng et al., 2011). In this paper, there was no categorical feature in the data sets, but there was collinearity between some features. Thus, the feature importance of \(\tau_{{{\text{max}}}}\) was analyzed using the RF introduced in Sect. 3.2.1. To confirm the difference in the feature importance of the pullout and beam specimens, two data sets were trained separately, and the ratio of training data to test data was 2:1. Hyperparameter optimization was performed to derive the feature importance from the optimized RF model. For the pullout data, the lowest test error occurred when the number of trees was 100, the depth of the tree was 20, and the maximum number of features was 3. The optimal result was obtained when they were 1000, 10, and 4, respectively, for the beam data set.
Fig. 8a, b shows the feature importance results of the pullout and beam specimens derived by RF, respectively. As shown in Fig. 8a, \(f^{\prime}_{{\text{c}}}\) had a significant influence on the bond strength of the pullout specimens (\(\tau_{{\text{p,max}}} )\). The level of importance was the highest in \(f^{\prime}_{{\text{c}}}\), followed by \(c\), \(d_{{\text{b}}}\), and \(\frac{{A_{{{\text{st1}}}} }}{{S_{{{\text{st}}}} }}\). However, the degree of influence of \(f^{\prime}_{{\text{c}}}\) on the bond strength of the beam specimens (\(\tau_{{\text{b,max}}}\)) was reduced compared with that of \(\tau_{{\text{p,max}}}\), and the influence of \(\frac{{A_{{{\text{st1}}}} }}{{S_{{{\text{st}}}} }}\) increased significantly. As a result, the level of importance was the highest in \(f^{\prime}_{{\text{c}}}\), followed by \(\frac{{A_{{{\text{st1}}}} }}{{S_{{{\text{st}}}} }}\), \(c\), and \(d_{{\text{b}}}\), for the bond strength of the beam specimens (\(\tau_{{\text{b,max}}}\)).
4.1.2 Correlation Analysis
The correlation between the features and \(\tau_{{{\text{max}}}}\) was analyzed in this section. To identify the correlation between a target and a selected feature more clearly, the influence of other features should be minimized. Thus, Kmean clustering, the distancebased clustering method introduced in Sect. 3.2.2, was used to cluster the data that are close to each other. The number of clusters was selected based on the silhouette coefficient (Rousseeuw, 1987), and the highest silhouette coefficient was obtained from four and three clusters, respectively, for the pullout and beam specimens.
Fig. 9a, b shows the results of the correlation analysis between the target and the features constituting the pullout and beam specimens. The xaxis of each graph represents the features shown at the bottom, and the yaxis represents \(\tau_{{{\text{max}}}}\). The scatter plot and linear regression model of the unclustered data and clustered data are shown at the top and bottom of the figure, respectively. The graphs are numbered in the upperleft corner of each graph.
As shown in Fig. 9a Graph 1, the variable with the greatest slope in the pullout specimens was \(f^{\prime}_{{\text{c}}}\), and the effect of \(f^{\prime}_{{\text{c}}}\) decreased slightly with increasing \(f^{\prime}_{{\text{c}}}\), as shown in Graph 5. As \(c\), which showed the secondhighest feature importance, increased, \(\tau_{{\text{p,max}}}\) increased (Graph 2), and the increase in \(\tau_{{\text{p,max}}}\) gradually decreased as \(c\) increased (Graph 6). In addition, \(d_{b}\) showed different tendencies for each cluster (Graph 7), but \(\tau_{{\text{p,max}}}\) decreased slightly as \(d_{b}\) increased for unclustered data (Graph 3). Moreover, \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) slightly increased \(\tau_{{\text{p,max}}}\) (Graph 4), and a consistent tendency was found in all clusters (Graph 8).
In the beam specimens, \(f^{\prime}_{{\text{c}}}\) increased \(\tau_{{\text{b,max}}}\) as in pullout specimens, but the increase was smaller than that of the pullout specimens (Fig. 9b Graph 1). The influence of \(c\), which showed the thirdhighest degree of influence, was obscure (Graph 2 and 6). Unlike the pullout specimens in which \(d_{{\text{b}}}\) had the lowest degree of influence, \(d_{{\text{b}}}\) showed a distinct influence to decrease \(\tau_{{\text{b,max}}}\) (Graph 7). Finally, \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\), which showed the secondlargest feature importance, was found to increase \(\tau_{{\text{b,max}}}\) (Graph 4).
Tables 4 and 5 show the average feature values of each cluster obtained from the pullout and beam specimens, respectively. As shown in Table 4, clusters 2 and 3 with underlines showed the lowest and highest τ_{p,max}, respectively, and there were significant differences in f'_{c} and c/d_{b}. It can be confirmed that the effect of f'_{c} was much larger than that of c/d_{b}, when comparing the differences in clusters 2, 3, and 4. In the beam specimens, clusters 2 and 3 with underlines in Table 5 showed the lowest and highest τ_{max,b}, respectively, and major differences occurred in f'_{c} and A_{st1}/S_{st}. Although cluster 1 had the highest f'_{c}, the cluster with the highest τ_{b,max} was cluster 3 which has the secondbest f'_{c} and firstbest A_{st1}/S_{st}. This is because the influence of f'_{c} decreased and because the influence of A_{st1}/S_{st} increased in the beam specimens, as shown in Figs. 8b and 9b.
4.2 Derivation of Mapping Function
4.2.1 Evolutionary Parameter
In this section, the mapping function (\(\tau  s\) relationship between pullout and beam specimens) was modeled based on data, and the GP introduced in Sect. 3.2.3 was used to do this. Before deriving the mapping function, a generalized \(\tau  s\) model (\(\tau_{{\text{p}}} \left( s \right)\)) of the pullout specimen was derived. The evolutionary parameters used in the GP are summarized in Table 6. The instructions used to model \(\tau_{{\text{p}}} \left( s \right)\) were \(f^{\prime}_{{\text{c}}}\), \(c\), \(d_{{\text{b}}}\), \(\frac{{A_{{{\text{st1}}}} }}{{S_{{{\text{st}}}} }}\), \(s\), and arithmetic operators (i.e., \(+\), \(\), \(\div\), \(\times\)). Since the mapping function between the bond–slip model of the beam specimens (\(\tau_{{\text{b}}}\)) and \(\tau_{{\text{p}}}\) is
\(\tau_{{\text{p}}}\), \(f^{\prime}_{{\text{c}}}\), \(c\), \(d_{{\text{b}}}\), \(\frac{{A_{{{\text{st1}}}} }}{{S_{{{\text{st}}}} }}\), and the arithmetic operators were used as instructions. The ramped halfandhalf (Koza, 1994) method was used to generate the initial population, and the size of the tree was set at 1 to 4. Because a larger training data set requires a higher computational cost for evaluation, the population size was set to 1000 in the \(\tau_{{\text{p}}}\) modeling, while it was set to 100,000 in the modeling of \(m\left( {\tau_{{\text{p}}} } \right)\). Tournament selection (Miller & Goldberg, 1995) was used as a selection method, and the tournament size was set to 7. The crossover and mutation probabilities were set to 90% and 10%, respectively, and the maximum size of the tree was set to 40 and 30, respectively, for \(\tau_{{\text{p}}} \left( s \right)\) and \(m\left( {\tau_{{\text{p}}} } \right)\) modeling to derive a concise equation. In \(\tau_{{\text{p}}} \left( s \right)\) modeling, the MAE of the GP did not decrease significantly after approximately 250 generations, and the algorithm was thus terminated in the 250th generation. Even in \(m\left( {\tau_{{\text{p}}} } \right)\) modeling, the algorithm was terminated in the 50th generation for the same reason. As with the RF, the ratio of the training and test data was 2:1.
4.2.2 Bond–Slip and Bond Strength Model for PullOut Specimens
The \(\tau_{{\text{p}}} \left( s \right)\) obtained using GP is shown below.:
To simplify Eq. (11), \(\left( {\frac{155}{{f^{\prime}_{{\text{c}}} }} + \frac{310}{c}} \right)s\) is removed from the denominator of Eq. (11), and constant optimization is performed using the Levenberg–Marquardt (Levenberg, 1944; Marquardt, 1963) algorithm. The modified \(\tau_{{\text{p}}} \left( s \right)\) is expressed as
In Eq. (12), \(f^{\prime}_{{\text{c}}}\) is modeled in the form of \(f^{\prime}_{{\text{c}}}  \frac{\alpha }{{f^{\prime}_{{\text{c}}} }}\)(where \(\alpha\) is a constant), which is a combination of a rational function and a linear function. This is the result of reflecting the effect that the increase in bond strength decreased with increasing \(f^{\prime}_{{\text{c}}}\), as shown in Fig. 9a Graph 5. In addition, \(c\) appears in the form of \( \frac{\alpha }{c}\) (where \(\alpha\) is a constant), which is a result of reflecting the decrease of influence on \(\tau_{{\text{p}}}\) as \(c\) increase. However, \(d_{{\text{b}}} ,\) and \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) are not reflected in Eqs. (11) and (12). This is because their feature importance is low, as shown in Fig. 8a, and their influence on bond strength is relatively low, as shown in Fig. 9a. \(\tau_{{\text{p}}}\) is differentiated with respect to \(s\) to derive the bond strength of the pullout specimens (\(\tau_{{\text{p,max}}}\)) from Eq. (12). The \(d\tau_{{\text{p}}} /ds\) is
Equation 13 confirms that a maximum point is obtained at \(s = 1\). \(s = 1\) is substituted into Eq. (12) to obtain the bond strength of the pullout specimens (\(\tau_{{\text{p,max}}}\)). The \(\tau_{{\text{p,max}}}\) is
4.2.3 Mapping Function Between PullOut and Beam Bond–Slip Data
The \(m\left( {\tau_{{\text{p}}} } \right)\) derived using GP is
Unlike in Eq. (11), the effects of \(d_{{\text{b}}}\) and \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) are reflected in Eq. (15). This is because, as shown in Fig. 9b, their influence increase in the beam tests. Meanwhile, in Eq. (15), \(\tau_{b}\) is not equal zero at \(s = 0\) because of the \(\frac{c}{{d_{{\text{b}}} + 7\tau_{{\text{p}}} }}\) term. Accordingly, the \(\frac{c}{{d_{{\text{b}}} + 7\tau_{{\text{p}}} }}\) term is removed from Eq. (15), and the influence of \(c\) is then reflected in the denominator of \(d_{{\text{b}}}\) to introduce \(c/d_{{\text{b}}}\), a term used widely in the \(\tau_{{{\text{max}}}}\) prediction model. In addition, a constant term is added to the denominator to build a model with a higher level of accuracy, and constant optimization is performed. The modified \(m\left( {\tau_{{\text{p}}} } \right)\) is
In Eq. (16), the influence of \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) is reflected linearly, and \(d_{{\text{b}}}\) is reflected in the form of a rational function that gradually reduce \(\tau_{{\text{b}}}\). \(\tau_{{\text{b}}}\) is differentiated with respect to \(s\) to derive the bond strength of the beam specimens (\(\tau_{{\text{b,max}}}\)) from Eq. (16) as below:
As in Eq. (17), \(\tau_{{\text{b}}}\) has a maximum point at the same position (\(s = 1\)) as \(\tau_{{\text{p}}}\) according to the chain rule, and the bond strength of the beam specimens (\(\tau_{{\text{b,max}}}\)) is
A safety factor (\({\upgamma }\)) is introduced to use Eq. (18) as a design equation as below:
5 Verification
5.1 Accuracy of Proposed Models
Fig. 10a shows the prediction accuracy of the bond–slip models for the pullout and beam specimens in Eqs. (12) and (16), respectively. Onethird of 1785 data points from the pullout tests and onethird of 217 data points from the beam tests were used for the validation of the proposed models. (i.e., 590 data points from the pullout tests and 72 data points from the beam tests). In the figure, the black dots indicate the \(\tau\) of the pullout specimens predicted using Eq. (12), while the red dots indicate the \(\tau\) of the beam specimens predicted using Eq. (16). As shown in the figure, both models predicted \(\tau\) with high levels of accuracy, and Eqs. (12) and (16) showed MAEs of 2.04 and 3.11 MPa, respectively.
Fig. 10b shows the \(\tau_{{{\text{max}}}}\) prediction accuracy of Eqs. (14) and (18) for 101 data points from the pullout test and 56 data points from the beam test (including additional 44 data points of \(\tau_{{{\text{max}}}}\)) except for the \(\tau_{{{\text{max}}}}\) used for training. The black dots indicate the \(\tau_{{{\text{max}}}}\) of the pullout specimens predicted using Eq. (14), while the red dots indicate the \(\tau_{{{\text{max}}}}\) of the beam specimens predicted using Eq. (18). As shown in the figure, Eqs. (14) and (18) provided prediction results close to the ideal curve and showed MAEs of 2.80 and 1.53 MPa, respectively.
A comparison between the accuracy of Eqs. (12) and (14) and that of Eqs. (16) and (18) for beam specimens reveals the accuracy improvement effect owing to the mapping function. The black and red dots in Fig. 11a indicate the \(\tau\) of the beam specimens predicted using Eqs. (12) and (16), respectively. When the \(\tau\) of the beam specimens was predicted using Eq. (12), the MAE was 4.43 MPa. When the mapping function was used for prediction, the MAE was 3.11 MPa, reducing the error by approximately 30%.
The black and red dots in Fig. 11b indicate the \(\tau_{{{\text{max}}}}\) of the beam specimens predicted using Eqs. (14) and (18), respectively. In general, the measured \(\tau_{{{\text{max}}}}\) values of the pullout specimens are larger than those of the beam specimens. Therefore, Eq. (14), trained using pullout data, evaluated the \(\tau_{{{\text{max}}}}\) of the beam specimens as very unsafe. As a result, the MAE of Eq. (14) was 8.26 MPa, and the MAE of Eq. (18) was 1.53 MPa, reducing the error by 81%.
In Fig. 12, the \(\tau\) prediction accuracy of the model proposed by Wu and Zhao (2013) and that of Eq. (16) are compared. The mean, standard deviation (SD), and coefficient of variation (COV) are statistical values based on the values which are divided the predicted value into the measured value. The model by Wu and Zhao showed an MAE of 3.6 MPa, while Eq. (16) had an MAE of 2.9 MPa, exhibiting a predicted behavior closer to the ideal curve. The proposed model provides conservative prediction results at the interval with a lower \(\tau\) than that of Wu and Zhao’s model. Consequently, the mean of those were 0.9 and 0.8, and the SD and COV of the two models were 0.5 and 0.6, respectively.
Fig. 13 compares the \(\tau  s\) curve obtained by the model proposed by Wu and Zhao (2013) and Eq. (16) with the beam test data. Both models exhibited excellent accuracy. However, in the case of Fig. 13a, c with a high \(\tau_{{{\text{max}}}}\), the proposed model simulated the \(\tau  s\) behavior more closely, whereas, in the case of Fig. 13b, d with low \(\tau_{{{\text{max}}}}\), the model proposed by Wu and Zhao (2013) showed results that were slightly closer to the experimental results. The low accuracy in the high \(\tau\) region of Wu and Zhao's model was due to the lack of highstrength specimen data used for modeling. On the other hand, the reason for the low accuracy in the low \(\tau\) region of the proposed model seemed to be due to the limited model complexity during evolution.
Fig. 14 shows comparison results of \(\tau_{{{\text{max}}}}\) obtained from the existing models, Eqs. (18), and (19). In the model proposed by Xu (Xu, 1990), \(f_{{\text{t}}}\) was calculated (ACI committee318, 2019) as
In the model used by Orangun et al. (1977), \(f_{{{\text{yt}}}} =\) 400 MPa was applied to specimens for which \(f_{yt}\) was not reported. As shown in Table 8, 0.7, which is a value at 0.08 fractile, was applied as a safety factor (\({\upgamma }\)) in Eq. (19).
As shown in Fig. 14h, Eq. (18) showed the lowest MAE (1.8 MPa) and a mean of 1.0, and Eq. (19) provided conservative \(\tau_{{{\text{max}}}}\) values. Equations (18) and (19) showed the lowest SD and COV, and the models of Esfahani and Kianoush (2005) and Orangun et al. (1977) also showed a low SD and COV. These results indicate that Eq. (18) can predict bond–slip and bond strength close to the measured values and also provide bond strength on the safe side by introducing a safety factor.
5.2 Bond–Slip Behavior of Proposed Models
In this section, Eq. (12) (i.e., the \(\tau  s\) model for pullout specimens) and Eq. (16) (i.e., the \(\tau  s\) model for beam specimens), the accuracies of which were verified in Sect. 5.1, were used to analyze the change in \(\tau  s\) resulting from the influence of each variable and to compare the difference in the \(\tau  s\) behavior between the pullout and beam specimens. In Fig. 15, the black and red lines indicate the \(\tau  s\) behavior of Eqs. (12) and (16), and the values of each feature increase in the order of solid, dashed, and dotted lines. Fig. 15a shows the effect of \(f^{\prime}_{{\text{c}}}\). As \(f^{\prime}_{{\text{c}}}\) increased, the \(\tau\) values of Eqs. (12) and (16) were found to increase significantly. In addition, the \(\tau_{{{\text{max}}}}\) of Eq. (16) was approximately 6–7 MPa lower than that of Eq. (12), and Eq. (16) showed a less sharp decrease in \(\tau\) at the postpeak stage compared with that of Eq. (12). Fig. 15b shows the effect of \(c\). In both models, \(\tau\) increased with increasing \(c\). However, the rate of increase in \(\tau\) gradually decreased as \(c\) increased. The effects of \(d_{{\text{b}}}\) and \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) are shown in Fig. 15c, d, respectively. Because the effects of \(d_{{\text{b}}}\) and \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) were not reflected in the pullout bond–slip curve of Eq. (12), the \(\tau  s\) behavior was the same, even when \(d_{{\text{b}}}\) and \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) changed—that is, it is represented by a single curve in the graphs in Fig. 15c, d. However, the effects of \(d_{{\text{b}}}\) and \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\) were reflected in Eq. (16). As a result, the \(\tau\) of Eq. (16) decreased as \(d_{{\text{b}}}\) increased, and the rate of decrease gradually decreased. In addition, the \(\tau\) of Eq. (16) increased almost linearly with increasing \(A_{{{\text{st1}}}} /S_{{{\text{st}}}}\).
6 Conclusions
In this study, a mapping function was developed to simplify estimating \(\tau  s\) in RC beams. A total of 255 pullout specimen data and 75 beam specimen data were collected, and the feature importance and correlation of the two groups were analyzed based on the data. In addition, the \(\tau  s\) model for pullout specimens (base model) and the \(\tau  s\) mapping function between the pullout and beam specimens were derived using GP. The rationality of the proposed model was verified by comparing its accuracy with that of existing models. The mapping function proposed can be used to estimate the \(\tau  s\) of the beam through a relatively simple pullout test. In addition, if the mapping function is used together with the bondslip model of pullout specimens, the \(\tau  s\) curve can be derived without experimental data. The following conclusions were drawn from the results of this study.

1.
It was found that \(f^{\prime}_{{\text{c}}}\) had the greatest influence on the \(\tau\) of the pullout specimens, and the degree of influence was the highest in \(f^{\prime}_{{\text{c}}}\), followed by \(c\), \(d_{{\text{b}}}\), and \(\frac{{A_{{{\text{st1}}}} }}{{S_{{{\text{st}}}} }}\). When compared with the pullout specimens, the influence of \(f^{\prime}_{{\text{c}}}\) was slightly decreased in the beam specimens. In addition, the influence of \(A_{st1} /S_{st}\) increased significantly, and \(f^{\prime}_{{\text{c}}}\) had the highest degree of influence, followed by \(\frac{{A_{{{\text{st1}}}} }}{{S_{{{\text{st}}}} }}\), \(c\), and \(d_{{\text{b}}}\).

2.
While the MAEs for the \(\tau  s\) and \(\tau_{max}\) of the proposed model were 2.04 and 2.80 MPa, respectively, for the pullout specimens, the MAEs for the \(\tau  s\) and \(\tau_{{{\text{max}}}}\) of the proposed model were 3.11 and 1.53 MPa, respectively, for the beam specimens. The proposed model exhibited the lowest error when compared with the existing models.

3.
The proposed model was used to compare the \(\tau  s\) behaviors of the two groups. The comparison revealed that the \(\tau_{{{\text{max}}}}\) of the beam specimens was lower than that of the pullout specimens, and the beam specimens exhibited a more gradual decrease in \(\tau\) at the postpeak stage when compared with the pullout specimens.
Availability of data and materials
All data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT). (No. 2019R1A2C2086388).
Funding
National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT). (No. 2019R1A2C2086388).
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HJ Methodology, data curation, writing—original draft, conceptualization. SJ Data curation, investigation. JHK Validation, investigation. SHC Validation, visualization. IH Data curation, visualization. KSK Writing—review and editing, funding acquisition. All authors read and approved the final manuscript.
Authors’ Information
Hoseong Jeong is a Ph.D. candidate in the department of architectural engineering of University of Seoul. He received B.S. degrees in material science & engineering and architectural engineering and an M.S. degree in architectural engineering from University of Seoul. His research interest focus on machine learning in structural engineering.
Seongwoo Ji is a graduate student in the department of architectural engineering of University of Seoul. His research interest is the bond–slip behavior in reinforced concrete.
Jae Hyun Kim is a Ph.D. candidate in the department of architectural engineering of University of Seoul. He received a B.S. and M.S. in architectural engineering from University of Seoul. His research interest is the analysis of beam–column joints.
SeungHo Choi is a postdoctoral researcher in the department of architectural engineering of University of Seoul. He received a B.S., M.S., and Ph.D. in architectural engineering from University of Seoul. His research interest is the analysis of beam–column joints.
Inwook Heo is a Ph.D. candidate in the department of architectural engineering of University of Seoul. He received a B.S. and M.S. in architectural engineering from University of Seoul. His research interest is the fire safety.
Kang Su Kim is a professor in the department of architectural engineering of University of Seoul. He received a B.S., M.S. in architectural engineering from Inha University and a Ph.D. in civil engineering from University of Illinois UrbanaChampaign. His research interest is the structural analysis of prestressed concrete members.
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Jeong, H., Ji, S., Kim, J.H. et al. Development of Mapping Function to Estimate Bond–Slip and Bond Strength of RC Beams Using Genetic Programming. Int J Concr Struct Mater 16, 49 (2022). https://doi.org/10.1186/s40069022005366
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DOI: https://doi.org/10.1186/s40069022005366