Bond behaviour of rebar in concrete at elevated temperatures: A soft computing approach

This paper assesses the capability of using a new data‐driven approach to predict the bond strength between steel rebar and concrete subjected to high temperatures. The analysis has been conducted using a novel evolutionary polynomial regression analysis (EPR‐MOGA) that employs soft computing techniques, and new correlations have been proposed. The proposed correlations provide better predictions and enhanced accuracy than existing approaches, such as classical regression analysis. Based on this novel approach, the resulting correlations have achieved a lower mean absolute error ( MAE ), and root mean square error ( RMSE ), a mean ( μ ) close to the optimum value (1.0) and a higher coefficient of determination (R2) compared to available correlations, which use classical regression analysis. Based on their enhanced performance, the proposed correlations can be used to obtain better optimised and more robust design calculations.


| INTRODUCTION
International standards, such as the CEB/FIB Model Code 2010 1 and Eurocode 2, 2 provide means of evaluating concrete behaviour for structural fire design. Both simplified and more advanced methods based on tables or curves describing concrete compressive strength evolution and other properties at high temperatures are presented in these standards. Nonetheless, the existing guidelines do not address the loss of bond between the steel reinforcement and the concrete during or after exposure to high temperatures. The bond between steel rebar and concrete is critical to the structural strength of a reinforced concrete (RC) element. After an RC component is exposed to high temperatures, the bond between reinforcing steel and concrete is weakened. The main reason for this bond degradation is the reduction in concrete strength and possible plastic deformation of the embedded steel rebar. 3 The reduced concrete-steel bond can significantly impact the structural behaviour of the RC members as this affects the transfer of tensile stress. 4 One other factor that plays a significant role in bond strength degradation is the corrosion of the steel, as it can impact the bond between the steel and concrete, causing deterioration of the structural performance. 5 The effect of the bond subjected to elevated temperatures has been explored to some extent in the existing literature in an effort to determine the main variables that impact performance and hence develop analytical correlations to predict the bond strength. [6][7][8][9] Nevertheless, the bond between steel and concrete at high temperatures remains one of the least studied phenomena in concrete research. 10 The failure mechanism of a RC member, including the bond between the concrete and steel, is affected by heat application; this is in part due to the concrete's temperature gradient. 10,11 Since there is a difference in heat conductivity between concrete and steel, this property can be influential during a fire because significant temperature gradients within the structural element can emerge. The resulting temperature change directly affects the compressive strength of the concrete due to the dehydration of the C-S-H gel and can induce thermal spalling. The following factors have been identified as causes of thermal spalling: the heating rate that the concrete experiences, 12 the incompatibility of thermal strains of components (cement paste, aggregates, steel) or layers with various temperatures and coefficients of thermal expansion, 11,[13][14][15][16][17] and the pore pressure build-up induced by steam ejected during the dehydration of the C-S-H gel and portlandite 12 as well as CO 2 expelled during the calcination of limestone aggregates (in cases where limestone aggregates predominate). 18 When steel fibres are added to the concrete mix, the steel fibre content influences the temperature gradient as steel fibres, being distributed throughout the concrete section, distribute the heat much faster within the concrete. 7 A variety of parameters must be considered when evaluating the performance of concrete in a structural member subjected to high temperatures. Key parameters include the exposure period, temperature degree, peak temperature, member dimensions, concrete humidity, concrete age, aggregate type, cement chemical composition, water-cement ratio (w/c) and the structural member's loading circumstances. 3 The main variables that control bond strength, according to the literature, [6][7][8][9] are the compressive strength after high-temperature exposure (f c ), the concrete age at testing (A), where fibres are used, the total volume of fibres within the concrete (V), the surface temperature at failure (T), the ratio of the duration of thermal saturation at the maximum target temperature to the minimum size of the pull-out specimen squared (Δ), the length-to-diameter ratio ( l = d ) (i.e., the bond length of the embedded ribbed bar to the diameter of the bar), and finally, the cover-to-diameter ratio of the embedded ribbed bar to bar Where β can be taken as 3.5 for T = 20 C to 400 C and as 2.5 for T = 600 C to 800 C.
For normal strength concrete;

| Data collection
The data used in the study is based on the database developed by  Table 1 presents the statistics of the data obtained from Varona et al. 9 used in this study. In addition, Figure 1 illustrates the histograms of the frequency of input and output variables. The visual representation of the data shows a good variation between the input and output. The need for further testing in the future is also recommended.

| Multi-objective evolutionary polynomial regression analysis (EPR-MOGA)
EPR-MOGA can be defined as an intelligent computational method that uses the input data to create an innovative novel solution for practical problems. 22,33,34 This approach is based on regression analysis and uses a genetic algorithm (GA) to produce a mathematical correlation that can describe the relationship between the physical input variables. 30,31 The EPR-MOGA uses regression analysis and implements a GA to search for the best correlation. This GA is also enhanced by adding more than one objective to control the correlation complexity and ensure the accuracy and fitness of the new correlations. 35 Thus, the advantages of this regression technique over the classical regression approach are as follows: 1. The best mathematical correlation is found automatically by a search algorithm. Thus, the user needs to specify the correlation structure, the number of terms of the correlation, and the exponents' range and step, unlike classical regression analysis, where the user needs to try every possible correlation manually.
2. The EPR-MOGA overcomes the overfitting problem correlated with other regression analyses that do not include AI approaches.
Overfitting usually occurs when the developed correlation learns the details of the data, including the noise; this adversely affects the performance of the developed new correlation when used to predict the results of newly generated data. This implies that the developed correlation picks the noise and the random variations in the data used in the correlation development.
To conduct the analysis, the user only needs to identify the structure of the required correlation, the range of the exponents and the number of terms. A full background description of the EPR-MOGA can be found in. 30,31 The performance of both the new and existing analytical correlations has been examined using statistical indicators. These indicators include the mean absolute error (MAE), root mean square error (RMSE), mean (μ), and coefficient of determination (R 2 ) as shown in Equations 4-7. The same approach of accuracy examination has been used in many previous studies, albeit in a different application. [36][37][38][39][40][41] The statistical meaning for the MAE and the RMSE values describe the best fit as the lower means. Meanwhile, the μ value is correlated with an optimum value of 1.0, any higher value can be considered as an overall overprediction of the correlation and vice versa.
In Equations 4-7, n is the number of data points used in the assessment, V p ð Þ is the predicted bond strength, and V m ð Þ is the measured bond strength.

| DEVELOPMENT OF CORRELATIONS
The correlations proposed in the present work are developed using a multi-objective GA evolutionary polynomial regression analysis (EPR-MOGA). As previously described, a database collected from existing studies has been used in the EPR-MOGA analysis. The database has been divided into two sets: training data and testing data. The training data comprises 80% of the total data (316 data points), while the testing data takes the remaining (81 data points, i.e., [20%]) in line with. 23 The first set is used to train the correlation to develop the mathematical correlation, and the second set is used to validate the correlation.
This means that only the training data has been used in the correlation development, and the testing data has been used only to assess the ability of the new correlation to predict the bond strength using a database that has not guided the correlation training. Division of the database into training and testing is carried out by shuffling the data using a random-sort function, and so forth, as available in Microsoft Excel, and then splitting the data based on the required percentages. However, significant efforts have been made to ensure that the testing data is within the statistical range of the training data to avoid correlation extrapolation in the testing stage. 23,42 Tables 2 and 3 present the training and testing statistics of the data.
The selection of the input variables assumed for the modelling is based on the relevant literature; then, a trial-and-error process is applied using the EPR-MOGA technique to fit a mathematical correlation between the input and output data to formulate the correlations. It is worth stating that the authors decided to keep these small coefficients for the developed correlations (which will be discussed in the following subsections) as the work aimed to improve the accuracy of the existing correlations.
The effect of all the main variables presented in the literature [5][6][7] has been included in the developed correlation. In the case of the existing correlations, the crucial variables were often only implicitly included. In contrast, this paper explicitly proposes three correlations to include all the relevant variables and obtain more accurate correlations to predict bond strength. The suitability of the correlation to be used in practice was also considered in its development.   .

| Proposed EPR correlation 1
The results from Equation 7 have been used in the predictions of for both training and testing sets is shown in

| EPR Proposed Correlation 2
For the second proposed correlation (EPR Correlation 2), all the input variables were included in the prediction of the bond strength (T b ) except for Δ(the ratio of the duration of thermal saturation, where the water in the pores reaches boiling point, at maximum target temperature to a minimum size of pull-out specimen squared). This approach was taken to produce a practical and convenient predictive correlation as the variable Δ cannot be obtained easily for practical applications.
After conducting the EPR-MOGA analysis, the best correlation obtained from the intelligence computing analysis is shown in Equation 8.
The predictions produced by this correlation are compared with the corresponding measured values in Figure 3.   However, this correlation seems to have less accuracy than EPR Correlation 1 and EPR Correlation 2 (see Figure 4). This shows that the concrete (A) age impacts the accuracy of the prediction of the data.

| EPR proposed Correlation 3
When the statistical performance parameters (MAE, RMSE, μ, and R 2 ) shown in Table 6 were compared to the first two correlations (Tables 4 and 5

| COMPARISONS OF THE PERFORMANCE OF THE NEW CORRELATIONS WITH EXISTING EMPIRICAL CORRELATIONS
The statistical performance of the three correlations developed in this study was compared to the existing correlations. [5][6][7]9 It should be mentioned that there is no correlation developed in the available literature that has included all the crucial variables except for the correlation developed by Varona et al. 6 shown in Equation 10. However, that correlation has not included the effect of the ratio of the duration of the thermal parameter (Δ), which was included in EPR Correlation

Another shortcoming with the Varona et al. correlation is that it
produces negative values in specific conditions [6], that is, for temperatures above 550 C, a 28-day plain concrete with c/d = 5 and l/d greater than 10, the correlation would predict a negative value of coefficient "kb" due to a limitation in the CEB/FIB Model Code Where the k b is the coefficient for plain concrete And for the concrete with fibres   practical reasons, EPR Correlation 2 excludes only one of the variables, that is, the ratio of the duration of thermal saturation at maximum target temperature to a minimum size of pull-out specimen squared (Δ), this correlation could be used in practice to give reliable predictions for the bond strength at elevated temperatures as the statistical performance of the correlation R 2 is 0.86 for training and 0.80 for testing, and the μ is 1.06 for training and 1.13 for testing. Nevertheless, with the third correlation from EPR Correlation 3, even if it was shown to be a less accurate prediction, the first two were still better than the other correlations proposed in the literature, 5-7 with R 2 being 0.81 for training and 0.78 for testing; meanwhile, the R 2 for the existing correlations is below 0.50.

| CONCLUSIONS AND FUTURE WORK
This study applied the novel EPR-MOGA technique to predict the bond strength between concrete and steel rebar at elevated temperatures using data obtained from the literature. The results showed that much higher accuracy was achieved using the new approach. Three new correlations were developed. Consequently, the outcomes of this work provide a potentially powerful and straightforward approach for designers and have shown improved accuracy compared to existing correlations. Taking the limitations of the presented work into consideration, the following conclusions can be drawn: