A Machine Learning Approach to Predict Relative Residual Strengths of Recycled Aggregate Concrete after Exposure to High Temperatures

Abstract

In recent years, there has been a heightened focus among researchers and policymakers on assessing the environmental impact and sustainability of human activities. In this context, the reutilization of construction materials, particularly recycled aggregate concrete, has emerged as an environmentally friendly choice in construction projects, gaining significant traction. This study addresses the critical need to investigate the mechanical properties of recycled aggregate concrete under diverse extreme scenarios. Conducting an extensive literature review, key findings were synthesized on the relative residual strength of recycled aggregate concrete following exposure to high temperatures. Leveraging these insights, innovative hybrid machine learning models were developed, offering practical equations and model trees for predicting the relative residual compressive strength, flexural strength, elasticity modulus, and splitting tensile strength of recycled aggregate concrete post high temperature exposure. Uncertainty analysis was performed on each model to assess the reliability, while sensitivity analysis was performed to find out the significance of each input variable for each predictive model. This paper presents interpretable models achieving high levels of performance, with $R^2$ values of 0.91, 0.94, 0.9, and 0.96 for predicting the relative residual compressive strength, flexural strength, modulus of elasticity, and splitting tensile strength of RCA concrete exposed to high temperatures, respectively. The unique contribution of the paper lies in the provision of easily applicable equations and model trees, enhancing accessibility for practitioners seeking to estimate mechanical properties of recycled aggregate concrete. Notably, our hybrid machine learning models stand out for their user-friendly nature compared with conventional ML algorithms, without compromising on accuracy. This paper not only advances our understanding of sustainable construction practices but also equips industry professionals with efficient tools for practical implementation.

1. Introduction

What really happens when concrete is exposed to high temperatures? This question has had the attention of concrete specialists since the 1920s as to how abundant physicochemical changes occur and cause thermomechanical changes in concrete as the temperature increases [1,2,3]. Indeed, the relative residual mechanical properties deteriorate in terms of aggregate type, cement matrix, and their interaction. Based on intensive investigations of research that has studied concrete performance at high temperatures, Figure 1 summarizes the processes that concrete experiences at high temperature exposure [4,5,6,7,8,9]. It is a process started by a minor loss of strength at low temperatures. This loss becomes more significant at higher temperatures up to 600 °C, where the concrete undergoes intense microcracking and aggregate melting starts to occur, depending on mineralogical composition [10]. Thus, the resistance of concrete to high temperatures can be defined as “the ability of concrete’s elements to perform their intended load-bearing functions at high temperatures” [11]. Truly, the impact grade is influenced by several factors that also may cause explosive spalling, among which are heating rate, maximum temperature, moisture content, calcium silicate hydrates, amount of Ca(OH)₂, cement matrix, and aggregate type [12]. Increasing the maximum temperatures leads to a thermal mismatch between the aggregate and cement matrix: the aggregate expands while the cement matrix shrinks, leading to the bond breaking at the interfacial transition zone [2].

Given the influence of aggregate, which occupies 60–80% of the volume of concrete, on concrete behavior at high temperatures, it should receive more attention when designing concrete that resists high temperatures [13,14]. Recycled aggregate concrete seems to behave differently than conventional concrete, as recycled concrete aggregate (RCA) has higher water absorption capacity, higher porosity, and more interfacial transition zones [15,16]. Recycled aggregate concrete performance has been extensively studied in the literature [17,18] and is recognized as structural concrete by the European Standard (EN 12620) [19]. It is known as an acceptable sustainable alternative to be used in the construction industry. Enterprise Park in the United States of America, Middlehaven in the United Kingdom, Okanagan in Canada, Council House in Australia, and J-Cube in Singapore are examples of commercial and residential structures around the world that use recycled aggregate concrete [20].

However, recycled aggregate concrete performance under special circumstances has not been studied enough, including its behavior after exposure to high temperatures. Studies concerning the behavior of recycled aggregate concrete after high temperature exposure such as relative residual compressive strength, splitting tensile strength, flexural strength, elasticity modulus, and ultrasonic pulse velocity are summarized in Table 1. Notably, the scope of this study concerns recycled aggregate concrete produced using different proportions of recycled aggregate and no other cementitious materials other than cement. Herein is a discussion that describes the relative residual strengths of recycled aggregate concrete produced with different recycled aggregate proportions based on the collected data shown in Table 1.

Prediction of concrete properties based on the proportions of its materials has always been a focus of researchers. Initial models to achieve this goal were based on statistical techniques; however, in recent years, with advancements in the field of machine learning, many researchers have shown interest in the prediction of the mechanical properties of concrete [21,22,23] and other materials [24]. Yuan et al. [25] explored the adaptive network-based fuzzy inference system and genetic algorithm as a basis for a hybrid model for the prediction of the compressive strength of concrete. To optimize the weights and thresholds of the back-propagation artificial neural network, the genetic algorithm was used. Two construction techniques were looked at for the adaptive network-based fuzzy inference system model. The pricey and time-consuming laboratory procedures might be avoided by using these forecasting techniques for concrete strength. The results demonstrated that this hybrid model performed well in terms of acceptable accuracy and applicability in practical production, with a high potential to replace the standard regression models in actual engineering practice. They concluded that fuzzy logic may be a beneficial modeling tool for engineers and research scientists working in the cement and concrete industries, according to the model’s successful predictions of the observed cement strength. Mozumder et al. [26] examined the uniaxial compressive strength of fiber-reinforced polymer-constrained concrete using the support vector machine regression approach. A glass fiber-reinforced polymer-confined concrete vector machine regression model and a carbon fiber-reinforced polymer-confined concrete support vector regression model were both constructed. When compared with artificial neural networks and current empirical fiber-reinforced concrete strength prediction models, the prediction efficacy of generated vector machine regression models performed much better. Furthermore, a sensitivity analysis was provided to evaluate the variable contributions to the strength prediction process. The study showed that vector machine regression-based techniques may be a potent alternative for the prediction of concrete performance since they have less variability in their prediction error distributions than other models.

Feng et al. [27] suggested an adaptive boosting technique to combine several weak learners to create a strong learner for estimating the compressive strength of concrete. A total of 1030 sets of concrete compressive strengths, coarse/fine aggregates, cement, water, additives, and curing times were gathered, To show the suggested mode’s generalizability, a fresh dataset of 103 samples for concrete compressive strengths was also employed. The suggested methodology outperformed existing techniques including artificial neural networks and support vector machines in every way. It was demonstrated that the decision tree was the best option for the weak learner in the boosting framework when utilizing 80% of the total data for training. In their study, Bingol et al. [28] employed the artificial neural network approach to develop a model for assessing the compressive strength of lightweight concrete that included pumice aggregate and was exposed to high temperatures. This model integrated three input factors: the desired temperature, the pumice-to-aggregate ratio, and the heating duration.

The increasing preference for recycled aggregate concrete highlights the necessity for user-friendly methods to predict its mechanical properties under diverse conditions, especially when exposed to elevated temperatures. This study addresses this need by aiming to provide engineers and researchers with a versatile and easily applicable tool. This tool aims to facilitate precise assessment of various mechanical characteristics of recycled aggregate concrete when subjected to high-temperature conditions.

One practical application of these predictive models is in assessing the functionality and structural integrity of buildings following fire incidents. This assessment relies on predicting the compromised mechanical attributes resulting from exposure to fire. Importantly, the existing literature lacks models for such predictions, and the utilization of machine learning techniques in predicting properties of recycled aggregate concrete under high temperatures has been significantly overlooked.

This investigation places emphasis on providing an optimal solution by initially exploring linearity within the dataset. If linear models prove insufficient, a proposed algorithm leverages the M5p decision tree technique and various linear regressions to provide a interpretable and accurate model for mechanical properties of RCA concretes. This combination aims to predict the residual compressive strength, flexural strength, elasticity modulus, and splitting tensile strength of recycled aggregate concrete. The models are evaluated rigorously using evaluation metrics and cross-validation. An uncertainty analysis is performed to determine the reliability of the models and compare them to each other. A sensitivity analysis is performed to determine the importance of the variables in the prediction of the mechanical properties”.

Table 1. Summary of research on the relative residual properties of concrete made with different proportions of RCA after exposure to high temperatures.

Reference	Maximum Temperature, °C	Exposure Time, h	RCA Proportion, %	Experiments	Test Type
[12]	800	1	0%, 20%, 50%, 100%	CS; STS; EM	Residual
[29]	500	1, 4	0%, 75%	CS; EM; UPV	Residual
[30]	750	1	0%, 25%, 50%, 100%	CS; STS; EM	Residual
[31]	800	2	0%, 30%, 50%, 70%, 100%	CS; FS	Residual
[32]	650	1	0%, 50%, 100%	CS; EM; STS	Residual
[33]	800	2	0%, 25%, 50%	CS; FS, UPV	Residual
[34]	800	2	100%	CS; FS	Residual
[35]	800	2	100%	CS	Residual
[36]	600	1	0%, 30%, 60%, 100%	CS; UPV	Residual
[37]	800	3	0%, 25%, 50%, 0.75%, 100%	CS; STS	Residual
[38]	800	1	0%, 30%, 50%, 70%, 100%	CS; EM	Residual
[39]	500	1	0%, 100%	CS; UPV	Residual
[40]	800	2	0%, 25%, 50%	CS; FS	Residual
[41]	800	3	0%, 25%, 50%, 0.75%, 100%	CS; EM	Residual
[42]	500	1	0%, 75%	CS; EM; UPV	Residual
[43]	800	4	0%, 50%, 100%	CS; STS	Residual
[44]	400	6	0%, 30%, 50%, 70%, 100%	CS	Residual
[45]	500	1	0%, 30%, 70%, 100%	CS	Residual
[46]	800	1	0%, 50%, 100%	CS; EM	Residual
[47]	800	2	0%, 25%, 50%	CS; FS	Residual
[48]	800	3	0%, 50%, 100%	STS	Residual
[49]	600	1	0%, 100%	EM	Residual
[50]	450	2	0%, 30%	EM	Residual

CS: compressive strength, FS: flexural strength, STS: splitting tensile strength, ME: modulus of elasticity, UPV: ultrasonic pulse velocity.

Table 1. Summary of research on the relative residual properties of concrete made with different proportions of RCA after exposure to high temperatures.

Reference	Maximum Temperature, °C	Exposure Time, h	RCA Proportion, %	Experiments	Test Type
[12]	800	1	0%, 20%, 50%, 100%	CS; STS; EM	Residual
[29]	500	1, 4	0%, 75%	CS; EM; UPV	Residual
[30]	750	1	0%, 25%, 50%, 100%	CS; STS; EM	Residual
[31]	800	2	0%, 30%, 50%, 70%, 100%	CS; FS	Residual
[32]	650	1	0%, 50%, 100%	CS; EM; STS	Residual
[33]	800	2	0%, 25%, 50%	CS; FS, UPV	Residual
[34]	800	2	100%	CS; FS	Residual
[35]	800	2	100%	CS	Residual
[36]	600	1	0%, 30%, 60%, 100%	CS; UPV	Residual
[37]	800	3	0%, 25%, 50%, 0.75%, 100%	CS; STS	Residual
[38]	800	1	0%, 30%, 50%, 70%, 100%	CS; EM	Residual
[39]	500	1	0%, 100%	CS; UPV	Residual
[40]	800	2	0%, 25%, 50%	CS; FS	Residual
[41]	800	3	0%, 25%, 50%, 0.75%, 100%	CS; EM	Residual
[42]	500	1	0%, 75%	CS; EM; UPV	Residual
[43]	800	4	0%, 50%, 100%	CS; STS	Residual
[44]	400	6	0%, 30%, 50%, 70%, 100%	CS	Residual
[45]	500	1	0%, 30%, 70%, 100%	CS	Residual
[46]	800	1	0%, 50%, 100%	CS; EM	Residual
[47]	800	2	0%, 25%, 50%	CS; FS	Residual
[48]	800	3	0%, 50%, 100%	STS	Residual
[49]	600	1	0%, 100%	EM	Residual
[50]	450	2	0%, 30%	EM	Residual

CS: compressive strength, FS: flexural strength, STS: splitting tensile strength, ME: modulus of elasticity, UPV: ultrasonic pulse velocity.

2. Methodology

2.1. Handling Imbalanced Data

One of the most common engineering problems is that the data are imbalanced. Dealing with this issue in a categorical variable is much easier in comparison with continuous numeric variables since this means at least one category undergoes substantially more observation than other categories. The most common approaches for these problems are adding more observations to the minority groups (over-sampling) or decreasing the number of observations in the majority groups (under-sampling) [51].

In regression problems, and due to the fact that the target variable is continuous, defining a minority group is much more complicated than classification problems. Furthermore, defining the target value for newly synthesized points can add to the complications of the problem [52]. The SMOTER process can be divided into the following three steps:

• Using a relevance function for identifying the minority section of data.
• Synthesizing the new examples.
• Calculate the target variable for newly synthesized examples.

In this research, a Sigmoid function is used in order to calculate the relevance of the existing data points and find the minority portions. New examples are created by interpolating between randomly chosen data points belonging to the minority portion of the data. This interpolation is performed using a weight variable between 0 and 1. Even though this is a weighted interpolation, and the weight is random, still the resulting values are between the two given values. Therefore, all the synthesized data follow the range of the given data, which we know are applicable and logical. In the end, the target value is calculated by averaging the corresponding target value for the two source data points based on their distance to a synthesized point. Figure 2 illustrates the minority selection and synthesizing process of new data points in a 2D problem in SMOTER.

2.2. Predictive Models

2.2.1. Model Tree

One of the main disadvantages of regression trees is the fact that, at each leaf, only a constant prediction value is assigned. This can lead to either the creation of very big tree structures or the oversimplification of the problem. In order to generalize this approach, the model tree develops a model on each leaf of the tree [53]. This simple change can make this technique very robust while keeping it more understandable than other techniques such as artificial neural networks [54,55]. One of the most robust algorithms used for model trees is called M5p [56]. Model tree algorithms can be divided into three main steps including tree construction, pruning, and smoothing. The M5p model uses the divide and conquer technique [57] to build the tree. In this technique, standard deviation is considered a measure of the error in subspaces [58]; therefore, the goal would be to create subspaces with the least standard deviation. This standard deviation reduction (SDR) is calculated as shown in Equation (1):

$$SDR=sd\left(T\right)-\sum _i^n\frac{T_i}{\left|T\right|}\times sd\left(T_i\right)$$

(1)

where T is all the data points at the node, $T_i$ is the ith examining subspace, and $sd$ is the standard deviation. The variable and splitting condition that leads to the most standard deviation reduction is chosen for the split. This process is repeated until the standard deviation becomes less than a threshold or the number of data points at the node is less than a predefined amount.

One of the issues with this technique can be overfitting. Merging lower-level nodes can be a solution, which is called pruning. In this technique, by defining a minimum error threshold for each leaf, the leaves with errors less than the threshold are merged into upper-level nodes. In other words, a certain amount of error is accepted just to assure the generalization of the predictions and prevent overfitting. The pruning can lead to sharp discontinuities between two adjacent nodes [54]. This is handled with a smoothing process. Quinlan observed that smoothing can lead to higher accuracy [56]. Figure 3 shows a schematic view of the M5p model tree for a two-dimensional problem.

2.2.2. Linear Models

In this research, three different linear regression models are used, including ordinary least squares (OLS), least absolute shrinkage and selection operator (LASSO), and ridge regression. In this section, each of these methods is briefly explained.

2.2.3. OLS

The OLS assumes that the model can be represented in the following vector form (Equation (2) [59]):

$$\sum _{j=1}^{n_v}{X}_{ij}{\alpha }_j=y_i$$

(2)

where $X_{ij}$ is the ith observation of the jth variable, ${\alpha }_j$ is the coefficient of the jth variable, and $y_i$ is ith observation. The goal of this method is to minimize the following objective function (Equation (3) [59]):

$$e\left(\alpha \right)=\sum _{i=1}^{n_o}{\left|y_i-\sum _{j=1}^{n_v}{X}_{ij}{\alpha }_j\right|}^2$$

(3)

where $n_v$ is the number of variables and $n_o$ is the number of observations. Since there is no exact solution for the above formulation, the goal would be to find ${\alpha }_j$ in a way to minimize the above objective function, which can also be interpreted as an error of the model.

2.2.4. Lasso Regression

LASSO regression is a type of regression that uses L1 regularization in order to penalize a high value of the coefficient. The LASSO objective function can be written as shown in Equation (4) [60]:

$$e\left(\alpha \right)=\sum _{i=1}^{n_o}{\left(y_i-\sum _{j=1}^{n_v}{X}_{ij}{\alpha }_j\right)}^2+\lambda \sum _{j=1}^{n_V}\left|{\alpha }_j\right|$$

(4)

where $\lambda$ is a tuning parameter and can be interpreted as the intensity of the penalization. Again, the goal would be to find the set of coefficients that minimizes the objective function.

2.2.5. Ridge Regression

Like LASSO, ridge regression puts more constraints on the coefficients of linear regression. Ridge regression adds an L2 regularization term to the objective function of the linear regression (see Equation (5) [60]):

$$e\left(\alpha \right)=\sum _{i=1}^{n_o}{\left(y_i-\sum _{j=1}^{n_v}{X}_{ij}{\alpha }_j\right)}^2+\lambda \sum _{j=1}^{n_V}{\alpha }_j^2$$

(5)

This penalty ensures that no coefficient is ignored, which can happen in the case of multicollinearity.

2.3. Performance Metrics

To compare the performance of the models in predicting the mechanical properties of high-temperature exposed concrete, four different metrics are considered. These performance metrics include the Pearson correlation coefficient ($R^2$) and the root mean squared error (RMSE), which are calculated using Equations (6) and (7):

$$R^2={\left(\frac{{\sum }_{i=1}^N\left(A_i-\overline{A}\right)\left(P_i-\overline{P}\right)}{\sqrt{{\sum }_{i=1}^N{\left(A_i-\overline{A}\right)}^2{\sum }_{i=1}^N{\left(P_i-\overline{P}\right)}^2}}\right)}^2$$

(6)

$$RMSE={\left(\frac{{\sum }_{i=1}^N{\left(P_i-A_i\right)}^2}{N}\right)}^{0.5}$$

(7)

2.4. Random Search Hyper-Parameter Tuning

Each machine learning technique includes multiple predefined hyper-parameters that control their algorithm. Finding the best combination of these parameters can be a daunting task. The random search technique tries a predefined number of the random combination (Figure 4) of values to find the combination that develops the most accurate model.

2.5. Cross-Validation

In order to make sure the train-test splitting does not lead to any specification on the dataset, cross-validation techniques are essential for any prediction model. K-fold cross-validation divides the training data into K folds, where each fold contains randomly selected, not repeated, data points. Then, it removes one fold and trains the model using the remaining (K − 1) folds. After that, it validates the developed model in the previous step with the left-out fold. This process is repeated until all folds have been used for validation once. The K-fold cross-validation can be considered a computationally expensive technique among validation techniques. However, the fact that all data points have been used once as validation can assure higher generalization of the model. A schematic view of the five-fold cross-validation process is shown in Figure 5.

2.6. Sensitivity Analysis

Most predicting models incorporate many known and unknown factors. These degrees of freedom can increase the complexity and nonlinearity of the problem. Sensitivity analysis examines how the uncertainty of input variables can proportionally affect the uncertainty in the output of a prediction model. Therefore, it should be a vital part of corroboration, quality assurance, and the defensibility of any machine learning model [61]. Sensitivity analysis gives us a better understanding of the relation between the dependent and independent variables and their importance, and can lead to simplification of the problem.

To perform sensitivity analysis, one input feature is changed to its average value at all data points, and the rest of the features are kept unchanged. Then, a new model is trained, and performance metrics are subsequently calculated. A lower value of the correlation coefficient and a higher value for errors are indicators that the excluded parameter is more relevant to the problem and contributes more toward the accuracy of the model.

2.7. Uncertainty Analysis

Since many sources of error, including human and machine error, can play a role in the measurement of TBMs’ ROP and other related parameters, uncertainty analysis must be a vital part of any prediction model. In this section, an uncertainty analysis based on the probability distribution of errors is conducted on all points of the dataset. The difference between prediction and observation is defined as the error of that data point. Since our dataset is large enough, one can assume a normal distribution, and hence the mean and standard deviation can be measured using the mean and standard deviation of the dataset using the following equations (Equations (8) and (9)), respectively:

$$\overline{e}=\frac{1}{N}\sum _i^N{e}_i$$

(8)

$$S_e=\sqrt{\frac{{\sum }_i^N{\left(e_j-\overline{e}\right)}^2}{N-1}}$$

(9)

where $\overline{e}$ is the mean value, e is the variable value, $S_e$ is the standard deviation, and N is the size of the dataset.

The prediction is overestimating when the mean error is positive, and it is underestimating otherwise. Using these expressions, Ref. [62] states that a margin of error (m) with a 95% level of significance for the error distribution can be calculated by Equation (10):

$$m=1.96S_e$$

(10)

Using the mean and margin of error, one can calculate the confidence interval (CI), which can be calculated by using Equation (11):

$$CI=\overline{e}\pm m$$

(11)

2.8. Data Collection

A dataset for the relative residual mechanical properties of recycled aggregate concrete after being exposed to high temperatures was collected from the literature, as presented in Table 1. The features in the dataset are exposure time, cement content, max temperature, fine-to-coarse aggregate (F/C) ratio, water-to-cement (W/C) ratio, and RCA proportion. The target values are relative residual mechanical properties of concrete. However, the heating rate was hard to collect as many researchers have not provided it in their research; however, the majority of the research considers the ISO 834 [63] curve or 0.5–12 °C/min. Figure 6 graphically presents the relative residual strengths of the main mechanical properties, which are compressive strength, splitting tensile strength, flexural strength, and elasticity modulus.

The relative residual strengths are presented as a function of high temperatures, which is basically the ratio between the relative residual strength at a certain maximum temperature and the strength of the same concrete at ambient temperature.

Table 2. Data characteristics of the datasets.

Statistical Properties	Exposure Time, h	Cement Content, Kg/m³	Max Temp, °C	F/C, %	W/C, %	RCA Proportion, %	Residual Property, %
Residual Compressive Strength
mean	2.12	434.28	377.43	62.99	44.71	48.24	74.1
std	1.29	83.21	259.78	18.79	9.19	37.13	26.21
min	1	240	20	42.81	30.98	0	9.76
max	6	581	800	117.86	77.14	100	133.91
variation coef.	0.61	0.19	0.69	0.3	0.21	0.77	0.35
skewness coef.	1.33	−0.28	0.06	0.99	1.42	0.06	−0.64
Residual Flexural Strength
mean	2	469.7	399.09	68.14	39.02	40.15	51.61
std	0	33.83	273.66	19.9	4.62	33.42	34.87
min	2	430	20	42.82	35	0	0
max	2	500	800	84.19	48	100	100
variation coef.	0	0.07	0.68	0.29	0.12	0.83	0.67
skewness coef.	0	−0.24	0.06	−0.5	0.45	0.48	0.15
Residual Elasticity Modulus
mean	2.12	434.28	377.43	62.99	44.71	48.24	73.9
std	1.29	83.21	259.78	18.79	9.19	37.13	26.39
min	1	240	20	42.81	30.98	0	7.33
max	6	581	800	117.86	77.14	100	133.91
variation coef.	0.61	0.19	0.69	0.3	0.21	0.77	0.36
skewness coef.	1.33	−0.28	0.06	0.99	1.42	0.06	−0.65
Residual Splitting Tensile Strength
mean	1.95	416.01	386.67	66.67	47.31	48.97	67.81
std	1.09	89.5	277.04	11.29	11.81	39.77	28.66
min	1	268.2	20	56.21	30.98	0	11.11
max	4	581	800	88.05	77.14	100	106.87
variation coef.	0.56	0.21	0.71	0.17	0.25	0.81	0.42
skewness coef.	0.58	0.5	0.05	0.82	0.86	0.06	−0.41

provides statistical characteristics of each dataset where std is a standard coefficient, and min and max are the minimum and maximum values of that parameter in the acquired dataset.

3. Overview of Adopted Algorithm

This paper introduces a series of models developed to predict diverse mechanical properties of RCA concrete, including compressive strength, flexural strength, modulus of elasticity, and splitting tensile strength. The primary objective is to create models that are easily applicable for practitioners, engineers, and scientists to estimate these mechanical properties in real-world scenarios. The sample sizes for each property are as follows: 403 for compressive strength, 65 for flexural strength, 323 for modulus of elasticity, and 85 for splitting tensile strength. To ensure the robustness of our models, we employ the synthetic minority over-sampling technique with edited nearest neighbors (SMOTER) in the initial step to assess data balance. In the event of imbalances, over-sampling, as detailed in preceding sections, is utilized to rectify the data distribution.

Subsequently, the proposed methodology employs linear regression models to characterize the mechanical properties. Should the linear regression models prove sufficiently accurate, further development of more complex models is deemed unnecessary and redundant. Conversely, if the accuracy of the linear models falls short, a hybrid model combining a model tree and linear regression is suggested. This hybrid approach, distinct from other machine learning techniques like support vector machines (SVM), exploits the rule-based structure of decision trees, yielding regressive equations for easy future use. To assess the performance of the proposed models in comparison to sophisticated machine learning methods, neural network (NN) and SVM models are trained. The NN model adopts a fully connected architecture with three hidden layers having neuron counts of 128, 64, and 32, utilizing the rectified linear unit as the activation function. All the models provide an $R^2$ and MSE as the performance metrics.

Figure 7 provides the schematic overview of the procedural steps employed in this study to predict the relative residual mechanical properties of recycled aggregate concrete through hybrid models. Following this, a five-fold cross-validation is executed to ensure the model’s generalization across the dataset. Subsequently, an uncertainty analysis is conducted to establish a 95% confidence interval for the prediction error, assessing the reliability of the model’s predictions. Finally, a sensitivity analysis is undertaken to identify the features with greater importance in the proposed models.

4. Results and Discussion

4.1. Performance of the Models

4.1.1. Relative Residual Compressive Strength

Table 3. Performance metrics of the developed models for the prediction of relative residual compressive strength.

Model	${\mathit{R}}^2$	MSE
M5p-Lasso	0.84	96.19
M5p-linear	0.91	58.10
M5p-ridge	0.90	58.60
Lasso	0.75	148.16
Linear	0.76	146.92
Ridge	0.76	146.93
NN	0.87	87.49
SVM	0.77	142.48

shows the performance of models developed for predicting the relative residual compressive strength of recycled aggregate concrete after exposure to high temperatures. To deal with imbalanced data, 175 synthesized samples were added to 403 collected data samples. Based on the results, the Lasso, linear, and ridge models are incapable of capturing the complexity of the problem at an acceptable rate, with $R^2$ values of 0.75, 0.76, and 0.76, respectively. However, the proposed hybrid models illustrate a performance improvement. The M5p-Lasso model has the lowest $R^2$ increase (12%) to 0.84. The M5p-linear and M5p-ridge models show superior performance, with an $R^2$ equal to 0.9 for the latter model, and a slightly better performance for M5p-linear, with an $R^2$ equal to 0.91. It is demonstrated that the hybrid models outperform both NN and SVM models in terms of accuracy. The SVM model exhibits slightly superior performance compared with linear models, while the NN model surpasses the performance metrics of M5p-Lasso. However, it falls short in comparison with the performance achieved by the M5p-ridge and M5p-linear models.

Figure 8 and Figure 9 show a strong correlation between the predicted values of M5p-linear and the observed values of M5p-Lasso, see Figure 10 The tree consists of three levels, seven rule nodes, and eight leaves.

Table 4. Sub-models of the model tree of relative residual compressive strength of recycled aggregate concrete.

NO.	Equation
$M_1$	$-0.34ET+0.02CC-0.096MT+0.05F/C+0.19W/C+0.019RCA+85$
$M_2$	$-0.36ET-0.02CC-0.04MT+0.35F/C-0.47W/C+0.11RCA+104$
$M_3$	$0.21ET+0.011CC-0.06MT-0.11F/C+0.47W/C-0.03RCA+80$
$M_4$	$1.48ET-0.02CC-0.20MT+0.13F/C+0.69W/C-0.03RCA+119$
$M_5$	$-6.87ET+0.023CC-0.17MT-4.72F/C-1.03W/C-0.08RCA+414$
$M_6$	$-4.36ET+0.41CC-0.13MT-1.24F/C4.87W/C-0.03RCA-174$
$M_7$	$-3.8ET-0.04CC-0.16MT-0.08F/C-0.38W/C+0.17RCA+214$
$M_8$	$-16.73ET-0.09CC-0.1MT-1.82F/C-2.44W/C+11RCA+423$

provides the details for sub-models of the M5p-linear model tree.

Table 5. Cross-validation results of hybrid models for the prediction of relative residual compressive strength.

Fold	${\mathit{R}}^2$	MSE
M5p-Lasso
Fold 1	0.76	143
Fold 2	0.84	101
Fold 3	0.84	129
Fold 4	0.76	143
Fold 5	0.85	105
M5p-linear
Fold 1	0.87	82
Fold 2	0.84	108
Fold 3	0.84	119
Fold 4	0.79	131
Fold 5	0.85	106
M5p-ridge
Fold 1	0.70	182
Fold 2	0.86	94
Fold 3	0.85	123
Fold 4	0.87	77
Fold 5	0.87	89

lists the cross-validation results of the hybrid models. The cross-validation results show the same pattern of testing results. M5p-Lasso illustrates the least accurate performance between hybrid models, with an average $R^2$ of 0.81.

Furthermore, the M5p-linear model demonstrates a slightly better performance than the M5p-ridge model, with an average $R^2$ of 0.84 and 0.83, respectively. Importantly, the results reveal a significant increase in testing accuracy from cross-validation accuracy for M5p-linear and M5p-ridge in comparison with M5p-Lasso. This could be due to a better generalization of these two models.

4.1.2. Relative Residual Flexural Strength

The performance of the developed models for the prediction of the relative residual flexural strength is provided in

Table 6. Performance metrics of the model developed for the prediction of relative residual flexural strength.

Model	${\mathit{R}}^2$	MSE
Lasso	0.91	109.74
Linear	0.93	86.86
Ridge	0.94	86.66

. To deal with imbalanced data, 34 synthesized samples were added to 65 collected data samples. Based on these results, non-hybrid models are capable of predicting the target with an accuracy high enough for there to be no need for the utilization of hybrid models. All three models show high accuracy, with $R^2$ values of 0.91, 0.93, and 0.94, corresponding to the Lasso, linear, and ridge models, respectively. The most significant observation emerging from these results can be the linearity of the problem of predicting the relative residual flexural strengths of heat-exposed recycled aggregate concrete. From Figure 11 and Figure 12, a high correlation between predicted and observed relative residual flexural strength for all three models can be noted. The proposed equation by the ridge model is as follows

$$-1.31CC-0.11MT+1.54\frac{F}{C}-6.24\frac{W}{C}+0.06RCA+84.8$$

(12)

The five-fold cross-validation results are provided in

Table 7. Cross-validation results of models for the prediction of relative residual flexural strength.

Model	${\mathit{R}}^2$	MSE
Lasso
Fold 1	0.92	110
Fold 2	0.89	122
Fold 3	0.91	117
Fold 4	0.85	173
Fold 5	0.86	112
Linear
Fold 1	0.93	95
Fold 2	0.89	121
Fold 3	0.91	117
Fold 4	0.86	166
Fold 5	0.87	105
Ridge
Fold 1	0.92	110
Fold 2	0.89	121
Fold 3	0.92	116
Fold 4	0.85	172
Fold 5	0.85	114

. The average validation $R^2$ values for M5p-Lasso, M5p-linear, and M5p-ridge are 0.88, 0.89, and 0.88. Therefore, all three models show excellent results in cross-validation and the differences between them are negligible.

4.1.3. Relative Residual Elasticity Modulus

Table 8. Performance metrics of the developed models for the prediction of the relative residual elasticity modulus.

Model	${\mathit{R}}^2$	MSE
M5p-Lasso	0.84	96.35
M5p-linear	0.90	58.60
M5p-ridge	0.85	92.61
Lasso	0.74	152.16
Linear	0.76	146.16
Ridge	0.75	147.03
NN	0.80	114.05
SVM	0.76	145.14

provides the performance results of the developed models in the prediction of the relative residual elasticity modulus. To deal with imbalanced data, 137 synthesized samples were added to 327 collected data samples. As it is seen, the non-hybrid models provide a sub-optimal accuracy, with $R^2$ values of 0.74, 0.76, and 0.75 for M5p-Lasso, M5p-linear, and M5p-ridge, respectively. Therefore, hybrid models were developed. M5p-linear shows the highest accuracy, with an $R^2$ of 0.9. Both M5p-Lasso and M5p-ridge provide a sub-optimal performance, with an $R^2$ of around 0.85. The MSE also shows the same pattern, with M5p-linear showing the best performance.

Regarding the performance of the NN and SVM models, there is a subtle shift in the pattern when compared with the results for relative residual compressive strength models. Specifically, the SVM model yields an $R^2$ value identical to that of the linear model, but notably lower than the hybrid models, much like the case with residual compressive strength models. On the other hand, the NN model exhibits a lower performance here compared with all three hybrid models. This is a departure from the scenario with residual compressive strength models, where the NN’s performance outpaces that of the M5p-Lasso model. Furthermore, Figure 13 and Figure 14 illustrate a significant correlation between the prediction of the M5p-linear model and the observed relative residual elasticity modulus. The details of the M5p-linear model are provided in Figure 15, which includes four sub-models. Details of these sub-models are provided in

Table 9. Sub-models of the model tree of the relative residual elasticity modulus of recycled aggregate concrete.

NO.	Equation
$M_1$	$-0.04ET+0.02CC-0.06MT+0.01F/C+0.022W/C+0.03RCA+211$
$M_2$	$1.71ET+0.02CC+0.28MT+0.66F/C+0.55W/C+0.02RCA-74$
$M_3$	$-1.87ET-0.03CC+0.13MT-1.01F/C-0.99W/C0.021RCA+240$
$M_4$	$-7.21ET-0.11CC-0.11MT-0.56W/C+211$

.

Table 10. Cross-validation results of hybrid models for the prediction of the relative residual elasticity modulus.

Model	${\mathit{R}}^2$	MSE
M5p-Lasso
Fold 1	0.69	203
Fold 2	0.86	92
Fold 3	0.84	134
Fold 4	0.79	128
Fold 5	0.85	103
M5p-linear
Fold 1	0.81	124
Fold 2	0.78	165
Fold 3	0.82	133
Fold 4	0.77	141
Fold 5	0.80	141
M5p-ridge
Fold 1	0.70	201
Fold 2	0.86	92
Fold 3	0.83	138
Fold 4	0.79	129
Fold 5	0.85	106

specifies the cross-validation results of the hybrid models for each fold. Again, the M5p-linear model with an average $R^2$ of 0.83 provides the best performance and generalization for the problem. Both M5p-Lasso and M5p-ridge scored 0.80 for an average $R^2$.

4.1.4. Relative Residual Splitting Tensile Strength

Prediction of the model for the last mechanical properties under study in this paper is provided in

Table 11. Performance metrics of the model developed for the prediction of the relative residual splitting tensile strength.

Model	${\mathit{R}}^2$	MSE
Lasso	0.953	43.54
Linear	0.959	40.57
Ridge	0.958	40.61

. To deal with imbalanced data, 42 synthesized samples were added to 116 collected data samples. Based on the results, all three models show excellent performance in the prediction of the relative residual splitting tensile strength of recycled aggregate concrete, with all models capturing more than 95% of the problem complexity based on the $R^2$ value. This is an indicator of extreme linearity of the problem. Figure 16 and Figure 17 demonstrate predicted relative residual strength versus observed values for the developed models, which are an indicator of the high accuracy of the models too. The proposed equation of the linear model with slightly higher accuracy than other models is as follows

$$-2.77ET-0.04CC-0.09MT+0.29\frac{F}{C}-0.27\frac{W}{C}-0.02RCA+120$$

(13)

The cross-validation results (

Table 12. Cross-validation results of hybrid models for the prediction of the relative residual splitting tensile strength.

Model	${\mathit{R}}^2$	MSE
Lasso
Fold 1	86	0.89
Fold 2	85	0.92
Fold 3	128	0.80
Fold 4	181	0.52
Fold 5	143	0.84
Linear
Fold 1	0.89	83
Fold 2	0.91	96
Fold 3	0.81	125
Fold 4	0.56	169
Fold 5	0.83	149
Ridge
Fold 1	83	0.89
Fold 2	96	0.91
Fold 3	125	0.81
Fold 4	169	0.56
Fold 5	149	0.84

) are also an indicator of the high accuracy of the model, with an average $R^2$ of almost 0.81 for all the models. The reason for the difference between the average accuracy of cross-validation and testing split could be explained by the limited number of data available. This leads to insufficient training of cross-validation models and subsequently non-generalization of the results.

4.2. Uncertainty Analysis

A comprehensive uncertainty analysis was performed on the developed models. Corresponding results are provided in

Table 13. Results of uncertainty analysis.

Model	Mean Error	95% Uncertainty Bandwidth	95% Uncertainty Band Lower Limit	95% Uncertainty Band Upper Limit
Relative residual compressive strength
M5p-Lasso	−0.22	15.30	−15.53	15.08
M5p-linear	−0.31	14.92	−15.24	14.60
M5p-ridge	−0.33	14.98	−15.32	14.65
Lasso	0.01	23.85	−23.84	23.86
Linear	0.005	23.75	−23.75	23.76
Ridge	0.005	23.75	−23.75	23.76
Relative residual flexural strength
Lasso	2.12	20.10	−17.97	22.23
Linear	1.54	17.99	−16.44	19.54
Ridge	2.06	19.84	−17.78	21.91
Relative residual elasticity modulus
M5p-Lasso	0.11	15.43	−15.32	15.54
M5p-linear	0.60	14.31	−13.71	14.92
M5p-ridge	0.57	14.47	−13.88	15.03
Lasso	0.19	23.93	−23.73	24.12
Linear	0.18	23.89	−23.70	24.07
Ridge	0.18	23.89	−23.70	24.07
Relative residual splitting tensile strength
Lasso	−1.39	12.64	−14.03	11.25
Linear	−1.56	12.10	−13.66	10.53
Ridge	−1.55	12.11	−13.67	10.55

. For relative residual compressive strength, as was expected, the M5p-linear model shows the best result, with a 95% uncertainty range of −15.24 to 14.60. In the case of the relative residual flexural strength, a linear model with a 95% uncertainty bandwidth of 17.99 has the lowest uncertainty. Also, for the relative residual elasticity modulus, the M5p-linear model indicates the lowest level of uncertainty, with a 95% uncertainty bandwidth of 14.31 and a 95% range of −13.71 to 14.92. The ridge model for the relative residual splitting tensile strength provides the best uncertainty result in comparison with the linear and Lasso models. Overall, the proposed models provide a high level of certainty in predicting the parameters under study.

4.3. Sensitivity Analysis

Table 14. Results of sensitivity analysis for each proposed model.

Variable	${\mathit{R}}^2$	MSE
M5p-linear model for residual compressive strength
None	0.91	58.10
Exposure time, h	0.87	79.22
Cement content, Kg/m³	0.91	61.04
Max temp, °C	0.15	700.27
F/C, %	0.50	301.99
W/C, %	0.47	323.98
RCA proportion, %	0.61	259.78
Linear model for residual flexural strength
None	0.94	86
Exposure time, h	0.94	87
Cement content, Kg/m³	0.05	143,895
Max temp, °C	0.02	1300
F/C, %	0.01	52,248
W/C, %	0.02	35,849
RCA proportion, %	0.91	101
M5p-linear model for residual elasticity modulus
None	0.90	58
Exposure time, h	0.89	62
Cement content, Kg/m³	0.60	902
Max temp, °C	0.14	628
F/C, %	0.01	17,924
W/C, %	0.02	97,733
RCA proportion, %	0.81	157
Ridge model for residual splitting tensile strength
None	0.96	41
Exposure time, h	0.95	49
Cement content, Kg/m³	0.95	55
Max temp, °C	0.19	874
F/C, %	0.96	43
W/C, %	0.95	55
RCA proportion, %	0.96	42

shows the results of sensitivity analysis for the proposed models for each parameter under study. For the case of the compressive strength prediction model, the most important parameter is maximum temperature, with the highest reduction in the $R^2$ value. In contrast, the contribution of exposure time and cement content to the accuracy of the model is infinitesimal, with a 5% and 0% reduction in the F/C and W/C and the RCA proportions are somewhat important to the model since all of them caused a reduction in the accuracy of the model in the 50% to 40% range.

For flexural strength, the importance order of the variables is slightly different. Non-contributing factors for the flexural strength predictive model are exposure time, the same as compressive strength, and RCA proportion. All other parameters are crucial for the model and ignoring any of them leads to more than a 90% reduction in the accuracy of the model. For the elasticity modulus predictive model, exposure time is the only non-important variable. The variables in order of importance are F/C, W/C, max temperature, cement content, and RCA proportion.

The sensitivity analysis for splitting tensile strength led to interesting observations. The only variable that with replacement by its average led to a reduction in accuracy was max temperature. The change in accuracy for the rest of the variables was almost nothing. This can be explained by the fact that these variables capture the same variance of the target variable. Therefore, changing only one of them does not reduce the accuracy of the model since another variable can capture the same variance of the target variable.

5. Applicability and Limitations

The proposed models can be valuable in various scenarios where RCA concrete structures are subjected to fire. For instance, following a fire incident in a building, engineers can evaluate the mechanical properties of the exposed concrete by understanding the mixture design and determining the maximum temperature reached during the fire. This approach is equally applicable to tunnel liners post-accidents. By considering these parameters, structural integrity and stability can be thoroughly examined through various structural analyses.

The proposed models exhibit certain limitations. While they leverage linear regression to offer user-friendly and interpretable results, their reliance on this approach restricts their ability to capture nonlinearity. Consequently, the models face constraints in terms of accuracy and generalization. A potential avenue for future research involves exploring the applicability of nonlinear regression techniques within the proposed framework.

Additionally, the use of SMOTER to address data imbalance raises concerns about potential biases. The synthesized database generated by SMOTER may deviate from the distribution of the actual database, impacting the reliability of the models. It is crucial to acknowledge that the training data are confined to those collected for the study, and real-world applications may pose challenges for materials with characteristics divergent from those present in the used database.

Importantly, uncertainty analysis is inherently tied to the assumptions made during both the modeling process and subsequent analysis. For instance, assuming a normal distribution for the error of the models influences the interpretation of confidence intervals. Caution is advised to prevent overestimation or underestimation of errors. Additionally, sensitivity analysis is contingent on the chosen evaluation metrics and does not account for codependency among features, potentially leading to higher or lower sensitivity toward specific variables.

Note that the data were gathered through a comprehensive literature review, introducing the possibility of certain biases. For instance, the tendency to publish positive results may skew the findings in favor of better performance for RCA concrete exposed to high temperatures. Additionally, the selected data are sourced from public databases of published papers, potentially introducing bias, as private information is not considered. Consequently, caution is advised when utilizing the results of this research, taking into account the aforementioned biases.

Nevertheless, the proposed technique, when applied to a larger dataset, has the potential to yield models with a similar level of performance. In cases where more data are available, it is recommended to adopt the algorithm proposed in this study and thoroughly evaluate the models before implementation.

6. Conclusions

In this study, predictive models were developed to predict the relative residual mechanical properties of recycled aggregate concrete after exposure to elevated temperatures. The mechanical properties include compressive strength, flexural strength, elasticity modulus, and tensile splitting strength. The proposed algorithm initially tried linear models and, in the case of non-desirable performance, multiple hybrid models based on M5p and linear models were developed. This process ensured the creation of an easily navigable and interpretable predictive model tailored for engineers and researchers. A notable advantage of this approach, in contrast to other machine learning techniques, lies in its versatility for addressing real-world challenges, such as evaluating the functionality and structural integrity of buildings following fire incidents. Unlike models like NN and SVM, which lack the ability to transfer knowledge through interpretable rules and equations, the proposed hybrid method offers a transparent and comprehensible framework. The linear models were highly capable of modeling flexural strength and splitting tensile strength of recycled aggregate concrete. However, their performance for compressive strength and elasticity modulus was insufficient, and therefore hybrid models were proposed. The hybrid models provided higher accuracy compared with the NN and SVM models. Uncertainty analysis was performed for each of the developed models to acquire a 95% confidence interval for each model. It was shown that not only were the proposed models the most accurate ones, but also they provided the highest confidence level among all developed models. Sensitivity analysis was performed to investigate the contribution of each input feature in the prediction of each of the proposed models. Overall, the max temperature was the most important feature, and the exposure time was the least important one. The RCA proportion was a contributor to the prediction of compressive strength and modulus of elasticity; however, it played a small role in predicting flexural strength and splitting tensile strength. The proposed models are beneficial for real-world applications following fire incidents in any type of RCA structures. Using the proposed models, engineers can assess the mechanical properties of exposed concrete by analyzing mixture design and max temperature in the aftermath of a fire. The model enables a thorough examination of structural integrity and stability using diverse structural analyses.