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Tensile behavior evaluation of two-stage concrete using an innovative model optimization approach

  • Muhammad Nasir Amin EMAIL logo , Faizullah Jan , Kaffayatullah Khan EMAIL logo , Suleman Ayub Khan EMAIL logo , Muhammad Tahir Qadir and Marcin Kujawa
Published/Copyright: March 13, 2025
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REVIEWS ON ADVANCED MATERIALS SCIENCE
From the journal REVIEWS ON ADVANCED MATERIALS SCIENCE Volume 64 Issue 1

Abstract

Two-stage concrete (TSC) is a sustainable material produced by incorporating coarse aggregates into formwork and filling the voids with a specially formulated grout mix. The significance of this study is to improve the predictive accuracy of TSC’s tensile strength, which is essential for optimizing its use in construction applications. To achieve this objective, novel and reliable predictive models were developed using advanced machine learning algorithms, including random forest (RF) and gene expression programming (GEP). The performance of these models was evaluated using important evaluation metrics, including the coefficient of determination (R 2), mean absolute error (MAE), mean squared error, and root mean square error (RMSE), after they were trained on a comprehensive dataset. The results suggest that the RF model outperforms the GEP model, as evidenced by a higher R 2 value of 0.94 relative to 0.91 for GEP and reduced MAE and RMSE error values. This suggests that the RF model has a superior predictive capability. Additionally, sensitivity analyses and SHapley Additive ExPlanation analysis revealed that the water-to-binder (W/B) ratio was the most influential input parameter, accounting for 51.01% of the predictive outcomes presented in the model. This research emphasizes optimizing TSC design, enhancing material performance, and promoting sustainable, cost-effective construction.

Graphical abstract

Keywords: two-stage concrete; tensile strength; machine learning

1 Introduction

Two-stage concrete (TSC) is a specialized type of concrete that is produced using a process that is distinct from conventional concrete. The initial step in TSC entails the stuffing of the formwork with coarse aggregate particles. Subsequently, a highly fluid grout mixture is employed to occupy the cavities between the particles of aggregate. The fundamental structural component of TSC is the coarse aggregate, which comprises approximately 60% of the material [1]. By directing stresses onto the aggregate particles’ points of contact, TSC achieves an exceptionally accurate stress distribution [2]. These forces may cause aggregate particles to shatter or separate from grout [3]. The grout used in TSC is composed of water, well-graded sand, chemical admixtures, and ordinary Portland cement (OPC). In certain TSC formulations, blended binders are now included as a result of the presence of supplementary cementitious materials (SCMs). The flowability of grout, water requirements, and heat generated during hydration can all be enhanced by replacing approximately 33% of OPC with fly ash [4,5]. Despite its unique characteristics and potential advantages in construction, the existing body of research on TSC remains limited. Many studies focus on traditional concrete, with comparatively fewer investigations exploring TSC’s mechanical behavior and durability. As the demand for sustainable and high-performance concrete solutions grows, understanding TSC’s properties becomes increasingly important. However, the time-consuming and costly nature of experimental evaluations has hindered extensive research in this area.

It is anticipated that the utilization of TSC will experience a substantial increase in the years ahead due to the ongoing expansion of concrete in construction projects. However, despite the multiple benefits that TSCs provide, only a small number of study examinations have been conducted on them. It was determined that the highest quality mortar is produced by a cement–sand ratio of 1 and a water-to-cement ratio of 0.47 when developing TSC [6]. Furthermore, the incorporation of silica fume (SF) as a partial replacement for OPC exhibits two opposing impacts on the properties of the TSC: it simultaneously improves the strength of the TSC while diminishing the workability of the grout [2]. The workability of TSC grouts is decreased as a result of the incorporation of SF as a micro-filler that is designed to fill the crevices between the particles of sand and cement during the preparation process [2]. This is because SF has a very large surface area, which increases the water demand. Simultaneously, SF undergoes a vigorous reaction with CH, resulting in the production of additional calcium silicate hydrate (C-S-H), the primary hydration product that contributes to the enhancement of mechanical strength [7]. The mechanical characteristics of TSC have been the subject of extensive research [1,2,4]. A number of factors considerably affect the mechanical characteristics of TSC, including the w/c and s/c ratios of the cement and sand, as well as the parameters of the coarse aggregate. SCMs were suggested as a potential material for enhancing the performance and durability of TSC [7]. Evaluating the mechanical properties of TSC necessitates considerable time, financial resources, and extensive experimentation [8,9]. The computational discipline of machine learning (ML) has emerged in recent decades to predict a variety of properties, thereby avoiding the costly and time-consuming process of conducting experiments [10,11,12,13]. Given these complexities, there is a growing need for advanced computational techniques to facilitate the prediction and optimization of TSC properties. ML offers a powerful approach to predicting concrete properties with high accuracy while reducing reliance on extensive experimental work. ML-based models can analyze vast datasets, recognize patterns, and provide reliable estimations of mechanical properties such as tensile strength (T-S). However, despite the success of ML in conventional concrete studies, its application in TSC research remains underexplored.

The properties of concrete have been predicted using a variety of ML algorithms, including artificial neural networks (ANNs) [14,15,16], gradient boosting [17,18,19], extreme learning machines [20,21], support vector regression [22,23], random forest (RF) [24,25], decision trees (DT) [26,27], gene expression programming (GEP) [28,29], XGBoost [30,31], and adaptive boosting [16,32]. The recent developments in ML approaches have generated new opportunities for the precise prediction of T-S. T-S is important because it ensures structural integrity and durability in applications subject to stretching or pulling forces [11,33]. For instance, an optimized support vector machine (SVM) model has been successfully implemented to estimate the splitting T-S at the bonding interface, achieving an error margin of less than 5% between the predicted and actual values [34]. Gradient boosting machine models were used to illustrate additional advancements in predictive modeling. These models demonstrated superior performance in estimating concrete T-S when compared to SVM [35]. ANNs have also demonstrated substantial potential as they circumvent conventional mathematical equations and rapidly adjust to new data, thereby expanding their applicability [36]. The accuracy of predictions and the number of errors have been significantly enhanced and reduced by ensemble learning techniques, such as bagging and boosting when applied to 1,030 datasets [16]. The optimization of predictive models remains a persistent challenge, particularly in the determination of the initial parameter values, despite these advancements [37]. Optimization techniques are employed to overcome these obstacles and improve the efficacy of the model [38,39]. These techniques enhance the model’s performance by fine-tuning the relevant parameters, ensuring better accuracy and generalization [40,41,42,43]. These methods are essential for improving the accuracy of models and circumventing the constraints of traditional methods. Table 1 provides the details of the ML techniques employed in previous research studies.

Table 1

Previous researchers employed ML methodologies [44]

S. no Applied ML methods Predicted properties Materials used Total data points Publication year Ref.
1 MV CS Crumb rubber with SF 21 2020 [45]
2 RKSA Slump, CS FA 40 2018 [46]
3 GEP CS NZ (natural zeolite) 54 2019 [47]
4 ANFIS CS — 55 2018 [48]
5 GEP, MLR, and MNLR CS Bagasse ash 65 2020 [49]
6 ANN CS FA 69 2017 [50]
7 RSM, GEP CS Steel fibers 108 2020 [51]
8 IREMSVM-FR with RSM CS FA 114 2019 [52]
9 ANN CS FA 114 2017 [53]
10 M5MARS CS, Slump FA 114 2018 [26]
11 DEA CS, Slump FA 114 2021 [54]
12 SVM Slump, CS FA 115 2020 [55]
13 RF CS FA, GGBS 131 2019 [56]
14 ANN, DT, RF T-S Recycled aggregates 166 2022 [57]

Experimental testing is a resource-intensive and time-consuming procedure that necessitates specialized laboratory apparatus, an experienced workforce, and competent labor to guarantee the quality of TSC. Furthermore, the ecosystems in which they are introduced are adversely affected by sand, additives, and cement. Another environmental concern is the persistence of TSC specimens following the completion of testing. Consequently, the utilization of TSC in structures may prove to be affordable for the endeavor as a whole and more straightforward for the engineers involved if the necessary number of experimental experiments can be reduced. The objective of this investigation is to create a surrogate prediction model that employs ML algorithms to predict the T-S of TSC. This will result in a decrease in the number of environmental issues that must be addressed, as well as time and financial savings. Modeling material behavior is essential because it provides detailed insights into how materials will perform under various conditions, enabling better design and application strategies. GEP and RF are two powerful ML algorithms known for their effectiveness in predictive modeling [58,59]. Consequently, GEP was employed to develop a T-S equation, and RF was incorporated as a result of its precise predictions and modeling capabilities. Ferreira proposed the concept of GEP in 2001 as an enhanced variant of genetic programming (GP) [60]. It employs a parse tree with branches of varying lengths and a string that is formatted in a straight line. The RF approach, also known as RF, was introduced in 2001 and is now considered an enhanced classification regression technique [61]. Consequently, the objective of this study is to establish a precise approach for predicting the T-S of TSC through the application of ML techniques. Two distinct methodologies were implemented to optimize the forecast’s efficacy. This study is crucial as it addresses the challenge of accurately predicting TSC’s T-S, a key property influencing its performance in construction applications.

Although ML-based predictions for TSC properties have been conducted in the past [62,63,64], no equation has been developed for predicting T-S. The experimental evaluation of TSC’s mechanical properties remains time-consuming and costly, limiting large-scale research and practical implementation. Additionally, conventional methods for predicting TSC performance often lack accuracy and adaptability, while the influence of diverse input parameters on TSC properties remains underexplored in ML applications. To address these gaps, this study aims to develop a predictive equation for T-S using GEP and enhance modeling accuracy through RF. By leveraging advanced ML techniques, this research seeks to improve the precision of TSC predictions, optimize material performance through data-driven insights, and reduce reliance on extensive experimental work. Sensitivity analysis and SHapley Additive ExPlanation (SHAP) analysis were conducted to elucidate the link between the inputs and the outcome. The analysis sought to determine the most significant input factors that affect the target variable. The findings are anticipated to advance sustainable construction practices and facilitate better-informed decisions by contributing valuable knowledge to materials science and construction engineering.

2 Research methodology

The T-S of TSC is predicted in this investigation using GEP and RF. The research employs a structured methodology that commences with data acquisition and is subsequently followed by preprocessing to guarantee the quality of the data. Subsequently, the T-S is predicted using GEP and RF models in accordance with the input variables. Standard metrics are implemented to assess the models’ functionality. Figure 1 illustrates the flowchart of the research methodology, which delineates the primary stages in the model development and validation process.

Figure 1 Flowchart of the study.
Figure 1

Flowchart of the study.

2.1 GEP

GEP has the ability to generate explicit mathematical equations, making it highly interpretable compared to black-box ML models. It effectively captures nonlinear relationships in complex datasets while offering robust generalization. Additionally, GEP’s evolutionary approach optimizes model performance without overfitting, making it well-suited for predicting TSC properties. Ferreira was the pioneer in introducing GEP, a novel computational technique that serves as a subset of GP [60]. Experiments have demonstrated that GEP converges more quickly than traditional genetic algorithms [60]. Also, because the genetic manipulations are performed at the chromosomal level, there is more openness about the process [65]. Figure 2 is a simplified diagram of the primary GEP procedures. The procedure kicks off with the chromosomes of the founding population being generated at random. Then, the performance of each individual is measured against a collection of fitness instances based on the expression of these chromosomes. The individuals are then filtered based on their fitness (how well they do in that environment) to reproduce with alteration, producing offspring with novel characteristics. The same developmental process – genome expression, environmental conflict, selection, and modified reproduction – is then applied to these newly formed individuals. The procedure is continued until a satisfactory answer is discovered [66], which may take several generations.

Figure 2 The stepwise procedure of the GEP technique [71].
Figure 2

The stepwise procedure of the GEP technique [71].

In GEP, individuals are initially represented as linear strings of fixed-length (genomes). These genomes are then translated into complex structures of varying sizes and forms. Expression trees (ETs) identify these entities [67,68]. The typical human being consists of a single chromosome, each copy of which might contain anywhere from one to many genes. To produce offspring, a chromosome’s expression, or ET, is subjected to a selection mechanism (often fitness proportional) based on the ET’s fitness. The genetic operators do not alter the respective ET but rather the chromosomes during reproduction [47]. Open reading frames (ORFs) provide a better framework for understanding the structural organization of GEP genes. A gene’s ORF consists of a “start” codon, the amino acid codons themselves, and a termination codon [60]. The genes speak one language, and ETs speak another in GEP.

In GEP, the phenotype can be directly derived from the genetic code, owing to the fundamental principles that govern the formation and functioning of ETs. Karva language is the name given to this mutually understandable bilingual notation [60,66]. Genetic operators make changes to the chromosomes during reproduction. Genetic operators in the GEP system include various processes such as duplication, alteration, rearrangement of genetic material, root and insertion sequence transpositions, recombination through single or double crossover, and gene transposition [60]. Each genetic operator in the GEP system has extensive documentation outlining its capabilities, which can be found in previous literature [65,69,70].

2.2 RF

The performance of individual trees within a forest can be predicted using a method called RF. This approach relies on values from a randomly generated vector, collected without bias, and applies a uniform distribution applied uniformly to all trees in the forest. As the quantity of trees in a forest significantly grows, the generalization error will eventually approach a limit. A forest of tree classifiers will have a generalization error that is proportional to the effectiveness of each tree in the forest and the interactions between the trees [56]. When a subset of features is chosen randomly to divide each node, the error rates are equal to those of AdaBoost; however, they are more resilient in terms of noise. Improvements in error, robustness, and connection measurement are achieved by internal assessments in reaction to increasing the amount of characteristics utilized for partitioning. This is done as a result of the increased number of features. Estimations made internally can also be utilized in the process of determining whether or not a variable is significant. These concepts are also relevant when thinking about regression. RFs are an effective method for making predictions [72]. They are able to avoid becoming too snug because of the law of huge numbers. They become reliable classifiers and regressors when the appropriate form of randomness is introduced into the process. Additionally, the framework gives insight into the forecasting capabilities of the RF with regard to the interactions between individual predictors and their efficacy. Instead, theoretical parameters like strength and correlation are estimated using out-of-bag data, and the results are more tangible [61]. Figure 3 shows the diagrammatic representation of the RF method.

Figure 3 A diagrammatic representation of the RF method [73].
Figure 3

A diagrammatic representation of the RF method [73].

2.3 Data collection

The information used to compile the TSC database could be found in a variety of published sources [3,6,8,74,75,76,77,78,79,80,81,82,83,84]. These sources were consulted in order to collect the necessary information. The frequency distribution of the database, which contains a total of 226 data points for tensile testing, is illustrated in Figure 4. The data was divided into two parts: 30% for assessment and 70% for training. Data preprocessing is the crucial stage of ML techniques. Preprocessing of data involves scanning it for any error or invalid entity. The current study was preprocessed using various statistical checks. Table 2 and Figure 5 provide information on the various statistical measures, including the highest and lowest possible values for each parameter and correlation coefficients. Table 2 also describes the standard deviation, median, mean, and sum of all the target and input regressors. These parameters provide a comprehensive overview of the data distribution, including the total, range, and central tendency. The minimum and maximum values indicate the spread of the data, while the mean and median offer insights into its average behavior and central position. The standard deviation highlights the variability, and the sum represents the cumulative value across all observations. It has been previously stated that the minimum ratio between the number of data points and input variables should be three. For a robust model, it is recommended that this ratio exceeds 5 [85]. In this case, the ratio is approximately 20.5, using 226 data points and 11 input factors, ensuring the adequacy of the dataset used for the ML modeling. Prior to the building of a model, the most important step that has an impact on the TSC’s characteristics is input selection, and it takes place in this context. The running time of the developed models varies depending on their complexity and dataset size [86,87]. In order to produce a function that can be used in a more broad sense, the components of concrete which have the most significant influence on the qualities of the concrete are extracted from the mixture. As part of this investigation, the TSC properties were looked at in relation to the following equation:

(1) TS = f ( C, W, SF, SL, S, S/B, G, W/B, EA, FA, SP ) .

Figure 4 Frequency distribution: (a) cement, (b) fly ash, (c) SF, (d) slag, (e) sand, (f) water, (g) gravel, (h) W/B, (i) S/B, (j) superplasticizer, (k) expanding admixture, and (l) tensile strength.
Figure 4 Frequency distribution: (a) cement, (b) fly ash, (c) SF, (d) slag, (e) sand, (f) water, (g) gravel, (h) W/B, (i) S/B, (j) superplasticizer, (k) expanding admixture, and (l) tensile strength.
Figure 4

Frequency distribution: (a) cement, (b) fly ash, (c) SF, (d) slag, (e) sand, (f) water, (g) gravel, (h) W/B, (i) S/B, (j) superplasticizer, (k) expanding admixture, and (l) tensile strength.

Table 2

Description of the parameters involved for TSC (inputs similar to [64])

Parameters Units Symbol Total Minimum Maximum Mean Median Standard deviation Sum
Input
Cement (kg·m−3) C 226 176.00 873.00 379.44 376.74 105.00 85754.02
Water (kg·m−3) W 226 100.00 431.00 203.54 191.85 63.14 46000.78
Slag (kg·m−3) SL 226 0.00 228.00 11.69 0.00 43.08 2643.00
W/B — W/B 226 0.30 0.85 0.52 0.47 0.17 117.44
SF (kg·m−3) SF 226 0.00 57.00 5.23 0.00 12.57 1183.02
S/B — S/B 226 0.00 2.00 0.98 1.00 0.58 220.91
Sand (kg·m−3) S 226 0.00 873.00 363.40 396.00 179.81 82129.26
Expanding admixture (kg·m−3) EA 226 0.00 10.50 1.27 0.00 2.95 287.86
Gravel (kg·m−3) G 226 1.56 2001.00 1375.06 1474.00 349.12 310764.27
Fly ash (kg·m−3) FA 226 0.00 262.00 13.51 0.00 45.43 3052.40
Superplasticizer (kg·m−3) SP 226 0.000 10.500 2.977 2.115 3.336 672.770
Output
Tensile strength MPa T-S 226 1.200 5.500 2.756 2.600 0.939 622.900
Figure 5 Pairwise correlation coefficient of T-S data.
Figure 5

Pairwise correlation coefficient of T-S data.

A mix of domain knowledge and statistical methodologies were used to assess the TS of TSC. A detailed literature study and expert consultation resulted in the identification of a number of possible indicators, as shown in Eq. (1). These factors were chosen because they have been shown to influence the TS of TSC. A thorough feature importance analysis was conducted utilizing SHAP to evaluate the relative significance of the chosen indicators. This method facilitated a thorough evaluation of each feature’s contribution to the model’s predictions [30]. In addition to highlighting the most impactful input parameter, the SHAP analysis also helped identify potential multicollinearity issues, ensuring that the model remained robust and interpretable. As a result of this method, a more comprehensive knowledge of the underlying linkages among the attributes and the output parameter was made possible, which further enhanced the model’s transparency and dependability. Moreover, as demonstrated by low pairwise correlations and consistent feature priority rankings across many model iterations, multicollinearity was not deemed to be a serious problem among the chosen features.

Data autoscaling and standardization are crucial for ML model performance because they mitigate the impact of features with greater values on model accuracy. After data normalization, the model is more accurate and stable since each variable is considered to have contributed equally. This process is essential for achieving results that are reliable and interpretable, particularly in the case of complex datasets. It was essential to normalize the original data before the ML algorithms began their analysis. The normalizing procedure not only improves computational stability but also reduces undesired feature scale effects [88]. Using Eq. (2), the range of values for each parameter was set to be between 0 and 0.9

(2) y ′ = 0.9 × y − y min y max − y min ,

where y is the initial value, y min is the lowest value of the function, y max is the highest value of the function, and y′ is the normalized value.

2.4 Hyperparameter tuning of the models

Defining appropriate hyperparameters is essential for developing generalized and robust GEP and RF models. Hyperparameter optimization is achieved through systematic tuning techniques that combine trial and error with guidance from prior studies [89], ensuring the models align effectively with the data. In the GEP model, hyperparameters such as population size, crossover rate, and mutation rate are adjusted to balance exploration and convergence, which enhances the model’s ability to find optimal solutions while avoiding overfitting. As the population size increases, the model becomes more complex and accurate, although it may take longer to align optimally with the data. Similarly, in the RF model, hyperparameters like the number of trees, maximum depth, and minimum samples split to control the model’s complexity and generalization ability. Proper tuning of these parameters is crucial for optimizing model performance, ensuring it captures relevant patterns while maintaining a balance between accuracy and generalization. Table 3 outlines the hyperparameter ranges optimized for predicting TSC T-S.

Table 3

Configuration of hyperparameter values

Hyperparameter GEP (range) GEP (value) RF (range) RF (value)
Generations 50–400 200 — —
Crossover rate 0.5–1.0 0.7 — —
Population size 50–600 400 — —
Tail length 5–50 20 — —
Mutation rate 0.01–0.1 0.07 — —
Head length 5–50 40 — —
Maximum depth — — 5–50 30
Bootstrap samples — — True/false True
Number of trees — — 100–1,000 700
Minimum samples leaf — — 1–10 4
Random state — — Integer or none None
Minimum samples split — — 2–20 10

2.5 Validation of models

The correctness of the model was determined for the dataset by examining its performance based on three metrics: the mean absolute error (MAE), mean squared error (MSE), the root mean squared error (RMSE), and the coefficient of determination (R 2). When estimating regression models for this kind of issue, these measures are utilized extensively [15,90,91]. Based on MAE and RMSE, the error criteria are employed to terminate the development of the models involved. To prevent overfitting, the models were suspended when these metrics reached predefined acceptable thresholds or demonstrated minimal improvement across iterations.

The MAE is a measurement that determines the mean size of the errors, which is the size of the disparity between the values that were seen and those that were anticipated [92]. Large mistakes brought about by outliers are almost never significant because the value being discussed is absolute and not quadratic [93]. Eq. (3) is used to calculate the MAE, which can range from zero to positive infinity, where P i represents the actual value obtained from the experiment, M i denotes the anticipated value, and n represents the overall count of occurrences. The model is considered to be of higher quality when its MAE is lower

(3) MAE = 1 n ∑ i = 1 n ∣ P i − M i ∣ .

MSE is a widely utilized metric for evaluating the accuracy of regression models. It computes the average of the squared deviations between expected and observed values, as shown in Eq. (4). MSE is sensitive to outliers since it squares the data, giving greater mistakes a higher weight. A lower MSE suggests that the model is better fitted to the data. MSE is commonly used for model improvement and performance evaluation [43]

(4) MSE = 1 n ∑ i = 1 n ∣ P i − M i ∣ 2 .

The RMSE is a statistic that is frequently utilized for the purpose of measuring the extent of mistakes [92]. However, in contrast to the MAE, the RMSE tends to increase quite a bit when the error associated with each estimate grows due to the fact that it possesses a quadratic characteristic [93]. Eq. (5) is used to calculate RMSE. In the same vein as the MAE, having a low RMSE number is preferable

(5) RMSE = ∑ i = 1 n ( P i − M i ) 2 N .

A statistic known as R 2 has values that can vary all the way from negative infinity to one [94]. As a result, R 2 will equal 1 if the model provides a perfect fit to the data, and the model will understand all of the data’s variability [93]. Because the performance of the evaluated model is differentiated from that of a flat line, which is a reference model in which every forecast will correspond to the average of the outputs, the R 2 will be negative if the evaluated model offers a fit that is less satisfactory than that of the flat line of mean values. It is possible to calculate the R 2 by using Eq. (6), in which P i represents the actual value received from the experiment, M i represents the value anticipated by the model, and n represents the overall count of occurrences

(6) R 2 = ∑ i = 1 n ( M i − M ¯ i ) ( P i − P ¯ i ) ∑ i = 1 n ( M i − M ¯ i ) 2 ∑ i = 1 n ( P i − P ¯ i ) 2 .

3 Results and discussion

3.1 Development of the GEP equation

In order to successfully evolve computer programs, genexpro tool software was used for GEP; the GEP method makes use of a number of different factors. The length of the chromosome, which determines the size of the program representation, and the gene set, which includes all of the accessible functions and genes that are responsible for the construction of the programs, are the essential parameters. The length of the main program that is contained within the chromosome is characterized by the head length parameter, whereas the length of the remaining section is characterized by the tail length parameter. GEP is able to optimize its algorithmic behavior in order to provide the required outcomes in tasks such as symbolic regression, classification, and feature selection if these parameters are adjusted with care and attention. The configuration of the variables is specified in Table 4.

Table 4

Variables used to configure the GEP algorithm for implementation

Variables Settings
Linking function Addition
Chromosome 100
Function set Ln, +, ×, ÷, −, ∛. -
Head size 11
Genes 3
General T-S

After configuring the GEP algorithm’s settings in accordance with Table 4, as shown above, ETs, also known as ETs, are constructed by merging genes and functions from the gene set using the aforementioned criteria. The ETs are a representation of possible equations or solutions. The ETs are shown in Figure 6. They are assessed with fitness measurements that are unique to the issue and then optimized, utilizing those results. Techniques like genetic crossover, mutation, and selection are used during a series of iterative generations to bring about the evolution of the ETs. Eq. (7) displays the resulting equation that was obtained from the ET.

(7) T − S = A + B + C + D ,

where

(7a) A = ( S ) 3 × C + Ln ( W / B ) 9.61 2 – W ,

(7b) B = Ln ( ( 12.28 × EA ) + ( SL + S ) + EA 2 − ( G × Ln ( W / B ) ) ) ,

(7c) C = 1 ( 8.92 2 + e SP ) + ( 2 × SF ) × ( FA − 0.32 ) − ( FA − W / B ) ,

(7d) D = − 4.53 − 1 ( C + S / B ) − ( − 2.47 − C ) − ( S + SL ) + EA ,

where T-S represents the tensile strength, S represents the sand, C represents the cement, W/B represents the water-to-binder ratio, W represents the water, EA represents the expanding admixture, SL represents the slag, G represents the gravel, SP represents the superplasticizer, FA represents the fly ash, S/B represents the superplasticizer-to-binder ratio, and SL represents the slag.

Figure 6 ET of TSC for T-S: (a) Sub-ET 1; (b) Sub-ET 2; (c) Sub-ET 3; and (d) Sub-ET 4.
Figure 6

ET of TSC for T-S: (a) Sub-ET 1; (b) Sub-ET 2; (c) Sub-ET 3; and (d) Sub-ET 4.

3.2 The outcome of the GEP model

The GEP method was employed to predict the T-S of TSC. The analysis revealed a coefficient of determination (R 2) of 0.91, indicating an excellent agreement between the predicted values and the experimentally observed results, as shown in Figure 7. The maximum prediction error recorded was 0.73 MPa, while the minimum error observed was 0.01 MPa, with an MAE of 0.19 MPa. An analysis of the error distribution demonstrated that 31.3% of the predictions had an error of 0.1 MPa or less, reflecting a high level of accuracy. Additionally, 62.6% of the predictions fell within an error range of 0.1–0.5 MPa, while only 6.1% exhibited errors exceeding 0.5 MPa. Figure 8 presents the distribution of errors for the experimental and predicted data. The close alignment between experimental and predicted values highlights the robustness of the GEP model in capturing the underlying relationship between input parameters and T-S. Furthermore, the low error margin reinforces the capability of GEP to approximate experimental outcomes with a high degree of precision. Collectively, these findings confirm that the GEP method is effective in predicting the T-S of TSC, supported by a high R 2 value and a low average prediction error.

Figure 7 Performance evaluation of the experimental and predicted values for the GEP model.
Figure 7

Performance evaluation of the experimental and predicted values for the GEP model.

Figure 8 GEP model error distribution.
Figure 8

GEP model error distribution.

3.3 The outcome of the RF model

The RF technique was also employed to predict the T-S of TSC. The analysis was conducted using Python programming, implemented via the Spyder (Anaconda software). The R 2 for the predictions was determined to be 0.94, indicating a strong correlation between the predicted values and the experimentally observed results, as shown in Figure 9. The maximum prediction error recorded was 0.55 MPa, while the minimum error observed was 0.02 MPa, with an MAE of 0.18 MPa. An analysis of the error distribution revealed that 28.3% of the predictions had an error of 0.1 MPa or less, demonstrating a high degree of accuracy. Additionally, 70.1% of the predictions fell within an error range of 0.1–0.5 MPa, while only 1.6% exhibited errors exceeding 0.5 MPa. The alignment between the experimental and predicted values reinforces the reliability of the RF model in capturing the underlying relationships between input parameters and tensile strength. Figure 10 illustrates the distribution of errors for the experimental and predicted data.

Figure 9 Performance evaluation of the experimental and predicted values for the RF model.
Figure 9

Performance evaluation of the experimental and predicted values for the RF model.

Figure 10 RF model error distribution.
Figure 10

RF model error distribution.

The high R 2 value and the relatively low average prediction error affirm the effectiveness of the RF method in predicting the T-S of TSC. The RF model outperformed the GEP model in terms of predictive performance, as evidenced by a higher R 2 value. Furthermore, the concentration of predictions within acceptable error margins underscores the reliability of RF as a practical tool for forecasting the T-S of concrete. These findings imply that the RF model is better suited for practical applications where accurate and reliable predictions of T-S are crucial, particularly in TSC design and material optimization. This is primarily because the RF model can capture complex relationships between input parameters and T-S more effectively, making it a more reliable and accurate tool for predicting T-S in TSC.

3.4 Modeling based on statistical data evaluation

Presently, there are no models capable of accurately forecasting the T-S of TSC. Given the same range of input variables, Figure 11 demonstrates that RF models perform better than GEP models in predicting the T-S of TSC. The RF model, in contrast to other ML models, has the capability of effectively capturing the relationship between complex input factors and the output variables. When evaluated with the outcomes achieved by the GEP model, the MAE was determined to be 0.197 MPa, MSE was found to be 0.064 MPa, RMSE was found to be 0.254 MPa, and R 2 was found to be 0.91. Similarly, the RF model yielded MAE values of 0.181 MPa, MSE of 0.046 MPa, RMSE of 0.215 MPa, and R 2 of 0.94, respectively.

Figure 11 Statistical checks for the GEP and RF models.
Figure 11

Statistical checks for the GEP and RF models.

Following the completion of the investigation, the GEP model was developed. This model was subsequently used to generate predictions regarding the T-S of TSC and to assess the accuracy of those forecasts in comparison to those generated by RF models. Given that the observed and anticipated values are more closely aligned, there is statistical evidence to imply that RF models are successful in forecasting the T-S of TSC. This proof comes in the form of statistics. The RF model demonstrates a minimum rate of error and higher R 2 values in comparison to the GEP ML approaches.

3.5 Sensitivity analysis

A model’s expected output can be affected by changes to its input variables; therefore, conducting a sensitivity analysis is helpful. This method is crucial for assessing the stability and predictability of the model [95,96]. First, the problem has to be narrowed down by identifying the input factors that have an effect on the model’s predicted value. After the variables were recognized, their respective intervals of variation were calculated. For the parameters under discussion, this range should contain values that are both acceptable and meaningful. By testing different values within the specified ranges, the relative importance and influence of each input variable on the model’s output were determined using sensitivity analysis. This technique helps determine which factors have the most impact on forecasts and makes model-based decisions easier. A sensitivity assessment was performed to evaluate the effect of different factors on the T-S performance of TSC. The sensitivity analysis was conducted using the following equations:

(8) N i = f max ( x i ) − f min ( x i ) ,

(9) SA = N i ∑ n j = 1 N j .

The results illustrated the relative significance of the factors by displaying the percentage of the impact each input had on the output. It was revealed that the W/B had the greatest impact of around 51.01% among the components that were studied, which demonstrates the huge influence that it has on the T-S performance of the TSC. It was also observed in the previous investigation that the critical parameter for concrete strength is W/B [97]. Following that, gravel, water, and cement had the biggest influence on the T-S, contributing 23.52, 5.33, and 5.03%, respectively. The contributions of the remaining seven parameters on the T-S were less than 5%, as shown in Figure 12.

Figure 12 Sensitivity analysis of the TSC for the T-S.
Figure 12

Sensitivity analysis of the TSC for the T-S.

3.6 SHAP analysis results

The SHAP summary graphic illustrates the influence of various input characteristics on an ML model that forecasts T-S in TSC, as shown in Figure 13. The features, including the W/B, gravel, water, superplasticizer, cement, and other components, are ranked in order of their significance. The predicted T-S is considerably influenced by the most impactful feature, W/B, which is depicted at the top. Each dot on the figure represents the SHAP value for a single observation, indicating the degree to which each characteristic contributes to the T-S prediction. Positive SHAP values signify that a characteristic augments the T-S prediction, while negative ones attenuate it. The hue of the dots signifies the true value of the characteristic, with red dots representing high values and blue dots representing low values. The predicted T-S is increased by a high W/B ratio (represented by red dots), whereas lesser values (blue dots) have a reducing effect. The impacts of other features, such as water, superplasticizer, and gravel, are also variable. The horizontal spread of the dots for each feature indicates the extent of influence. Consistent effects are indicated by tightly concentrated dots, while a broader range of impacts is implied by more dispersed dots. These features importance provides essential insights for TSC design, as they highlight which material properties significantly influence T-S. For instance, the high impact of the W/B ratio suggests that adjusting this parameter can lead to better control over T-S, which is crucial for optimizing the mix design in TSC production. This plot provides valuable insights for optimizing T-S predictions by facilitating the interpretation of the contributions of individual material properties in the TSC mixture.

Figure 13 The influence of input parameters is illustrated in the SHAP plot.
Figure 13

The influence of input parameters is illustrated in the SHAP plot.

4 Discussion

TSC refers to the process of depositing coarse aggregate particles in an established formwork and subsequently occupying the gaps with a specialized cementitious mix. There are numerous differences between TSC and regular concrete. Initially, the components of traditional concrete are combined and thereafter placed in the formwork [63]. When making TSC, however, the grout components are mixed individually and then injected into the pre-arranged aggregate mass in the same way as previously explained. Not only that, but TSC has a greater percentage of coarse aggregate, which accounts for around 60% of the overall volume, whereas regular concrete normally accounts for approximately 40%. Through the placement of the aggregates in the formwork in advance, the TSC technique is able to assist in the resolution of the issue of coarse aggregate segregation, which is particularly prevalent in high-density aggregate concretes. In addition, TSC does not need to be consolidated, vibrated, or compressed in order to obtain a compact structure, which results in a reduction in the expenses associated with its manufacture [98]. As a result of these advantages, researchers are consistently investigating the mechanical characteristics of TSC.

The current trend in investigating the mechanical characteristics of TSC requires lab trials, which conversely influence the time and cost. However, in recent years, ML applications have been widely employed in the field of civil and material engineering to study the performance and characteristics of materials [99,100]. These ML techniques include SVM, DT, ANN, GEP, RF, bagging, and boosting [101,102,103,104]. Most of the studies in the past utilized ANN to predict the mechanical characteristics of concrete successfully. The current study is compared to the work of previous researchers who have implemented ML algorithms, as shown in Table 5. In addition, prior researchers have employed metrics such as R 2, MAE, and RMSE to assess the accuracy and performance of their models in predicting outputs [105]. In analytical studies, these statistical measures are frequently employed to evaluate the model’s model fit and to quantify the discrepancies between predicted and actual values, thereby enabling a thorough comprehension of the model’s predictive capabilities [105,106]. However, to study the T-S of TSC, this study utilized GEP and RF techniques. GEP is an advanced technique utilized for symbolic regression and the creation of new features. It utilizes mathematical formulas to adapt computer programs to match the provided data. GEP provides the mathematical expression to compute the T-S of TSC, which can easily be computed and cross-checked by users. Similarly, RF is an effective ensemble learning method that builds many DTs and then combines their predictions. When training, each tree uses a different subset of the available data and parameters. By integrating the predictions of numerous trees, RF enhances the model’s generalization performance and reduces overfitting. The results of this experiment show that the RF model performed better than the GEP model, as indicated by the RF model’s greater values for R 2 and reduced MAE and RMSE results.

Table 5

ML evaluation of the present study’s findings in relation to pertinent previous studies

Ref. Material studied Property studied Applied ML algorithms Best-performing model
Present study TSC T-S GEP and RF RF
[64] TSC Compressive strength GEP and RF RF
[107] Fly-ash based concrete Compressive strength GEP and RF RF
[108] High-strength concrete Compressive strength GEP and RF RF
[109] Concrete-filled steel tube columns Bearing capacity of concrete GEP, PSO, and ANN GEP
[110] Fly-ash based concrete Compressive and T-S GEP, MNLR, and RSM GEP

The RF and GEP models devised in this study offer advantages by operating within a predefined set of 11 independent variables (i.e., cement, water, SF, slag, sand, S/B, gravel, W/B, expanding admixture, fly ash, and superplasticizer). This characteristic guarantees that the forecasts generated are tailored to the application of these materials in the context of predicting T-S. The model's adherence to a standardized testing procedure and the use of identical unit measurements establish the dependability of the T-S predictions they generate. The prescribed arithmetic equations of the ML models are of the utmost significance in understanding the ratio of the design of the mix and the impact on each variable that is independent. Nevertheless, the relevance of the ML forecasting algorithms may be compromised if additional variables are incorporated into the model equations in addition to the 11 variables previously mentioned [111]. It is possible that additional input variables will cause the produced models to fail because they were tailored to handle the particular collection of 11 input variables. The forecast models may also generate incorrect results if the input variable units are altered or if there are discrepancies. It is imperative that the units of the input parameter models match those employed in this study in order to guarantee that the models are deemed effective.

5 Conclusions and recommendations

The T-S characteristics of TSC are examined in this investigation using both GEP and RF ML algorithms. In order to enhance the predictive capabilities, both the GEP and RF strategies were implemented. Each input variable was meticulously analyzed to uncover its relative frequency distribution. In order to conduct RF analysis, the Spyder (Anaconda program) was modified using Python code, and the Genexpro tool was employed for GEP to conduct the requisite model simulations for further research. Multiple statistical tests were examined and evaluated to ensure the accuracy of the models being employed, including MAE, RMSE, and R 2. Furthermore, a sensitivity analysis was performed to investigate the potential impacts of all of the input variables. The subsequent findings were derived from the investigation’s results.

  • When it comes to predicting the T-S of TSC, the RF model provides a more accurate answer with less degree of variation.

  • The value of the coefficient of correlation (R 2) for T-S according to the RF model is equal to 0.94, but the value of R 2 according to the GEP model is equal to 0.91.

  • The excellent accuracy of the model is indicated by the relatively high values of R 2 for the RF regressor’s contribution to the prediction of T-S.

  • According to the statistical tests, the fact that the RF method has a lower value for errors (MAE and RMSE) shows that it has a higher level of performance when compared to the GEP algorithm.

  • According to the results of the sensitivity analysis, the contribution of W/B to the prediction of the T-S of TSC was much higher than that of any other parameter, amounting to 51.01%.

  • SHAP analysis also revealed that the W/B had the highest impact on T-S compared to all other components, followed by gravel. This insight can guide targeted adjustments to improve the T-S of TSC.

The study highlights that while the RF model outperforms GEP in predicting T-S of TSC, its effectiveness is limited by the dataset’s scope and the use of only 11 input variables. Future research should focus on expanding the dataset and incorporating advanced ML techniques like gradient boosting, convolutional neural networks, and DTs. Exhaustive sensitivity analyses and exploring additional variables will improve predictive accuracy and understanding. The study emphasizes the importance of advanced ML in enhancing material performance for sustainable construction.

Acknowledgments

The authors acknowledge the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. KFU250709). The authors extend their appreciation for the financial support that made this study possible.

  1. Funding information: This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. KFU250709).

  2. Author contributions: M.N.A.: funding acquisition, supervision, project administration, investigation, writing, reviewing, and editing. F.J.: conceptualization, methodology, software, formal analysis, writing, reviewing, and editing. K.K.: visualization, resources, funding acquisition, formal analysis, writing, reviewing, and editing. S.A.K.: data acquisition, software, methodology, validation, writing-original draft. M.T.Q.: conceptualization, project administration, investigation, funding acquisition, writing, reviewing, and editing. M.K. investigation, resources, visualization, writing, reviewing, and editing. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The datasets generated and/or analyzed during the current study are available as supplementary materials to this article.

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Received: 2024-06-28
Revised: 2025-02-01
Accepted: 2025-02-17
Published Online: 2025-03-13

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/rams-2025-0097
Keywords for this article
two-stage concrete; tensile strength; machine learning
Creative Commons
BY 4.0

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  57. Establishing strength prediction models for low-carbon rubberized cementitious mortar using advanced AI tools
  58. Investigating the strength performance of 3D printed fiber-reinforced concrete using applicable predictive models
  59. An eco-friendly synthesis of ZnO nanoparticles with jamun seed extract and their multi-applications
  60. The application of convolutional neural networks, LF-NMR, and texture for microparticle analysis in assessing the quality of fruit powders: Case study – blackcurrant powders
  61. Study of feasibility of using copper mining tailings in mortar production
  62. Shear and flexural performance of reinforced concrete beams with recycled concrete aggregates
  63. Advancing GGBS geopolymer concrete with nano-alumina: A study on strength and durability in aggressive environments
  64. Leveraging waste-based additives and machine learning for sustainable mortar development in construction
  65. Study on the modification effects and mechanisms of organic–inorganic composite anti-aging agents on asphalt across multiple scales
  66. Morphological and microstructural analysis of sustainable concrete with crumb rubber and SCMs
  67. Structural, physical, and luminescence properties of sodium–aluminum–zinc borophosphate glass embedded with Nd3+ ions for optical applications
  68. Eco-friendly waste plastic-based mortar incorporating industrial waste powders: Interpretable models for flexural strength
  69. Bioactive potential of marine Aspergillus niger AMG31: Metabolite profiling and green synthesis of copper/zinc oxide nanocomposites – An insight into biomedical applications
  70. Preparation of geopolymer cementitious materials by combining industrial waste and municipal dewatering sludge: Stabilization, microscopic analysis and water seepage
  71. Seismic behavior and shear capacity calculation of a new type of self-centering steel-concrete composite joint
  72. Sustainable utilization of aluminum waste in geopolymer concrete: Influence of alkaline activation on microstructure and mechanical properties
  73. Optimization of oil palm boiler ash waste and zinc oxide as antibacterial fabric coating
  74. Tailoring ZX30 alloy’s microstructural evolution, electrochemical and mechanical behavior via ECAP processing parameters
  75. Comparative study on the effect of natural and synthetic fibers on the production of sustainable concrete
  76. Microemulsion synthesis of zinc-containing mesoporous bioactive silicate glass nanoparticles: In vitro bioactivity and drug release studies
  77. On the interaction of shear bands with nanoparticles in ZrCu-based metallic glass: In situ TEM investigation
  78. Developing low carbon molybdenum tailing self-consolidating concrete: Workability, shrinkage, strength, and pore structure
  79. Experimental and computational analyses of eco-friendly concrete using recycled crushed brick
  80. High-performance WC–Co coatings via HVOF: Mechanical properties of steel surfaces
  81. Mechanical properties and fatigue analysis of rubber concrete under uniaxial compression modified by a combination of mineral admixture
  82. Experimental study of flexural performance of solid wood beams strengthened with CFRP fibers
  83. Eco-friendly green synthesis of silver nanoparticles with Syzygium aromaticum extract: characterization and evaluation against Schistosoma haematobium
  84. Predictive modeling assessment of advanced concrete materials incorporating plastic waste as sand replacement
  85. Self-compacting mortar overlays using expanded polystyrene beads for thermal performance and energy efficiency in buildings
  86. Enhancing frost resistance of alkali-activated slag concrete using surfactants: sodium dodecyl sulfate, sodium abietate, and triterpenoid saponins
  87. Equation-driven strength prediction of GGBS concrete: a symbolic machine learning approach for sustainable development
  88. Empowering 3D printed concrete: discovering the impact of steel fiber reinforcement on mechanical performance
  89. Advanced hybrid machine learning models for estimating chloride penetration resistance of concrete structures for durability assessment: optimization and hyperparameter tuning
  90. Influence of diamine structure on the properties of colorless and transparent polyimides
  91. Post-heating strength prediction in concrete with Wadi Gyada Alkharj fine aggregate using thermal conductivity and ultrasonic pulse velocity
  92. Special Issue on Recent Advancement in Low-carbon Cement-based Materials - Part II
  93. Investigating the effect of locally available volcanic ash on mechanical and microstructure properties of concrete
  94. Flexural performance evaluation using computational tools for plastic-derived mortar modified with blends of industrial waste powders
  95. Foamed geopolymers as low carbon materials for fire-resistant and lightweight applications in construction: A review
  96. Autogenous shrinkage of cementitious composites incorporating red mud
  97. Mechanical, durability, and microstructure analysis of concrete made with metakaolin and copper slag for sustainable construction
  98. Special Issue on AI-Driven Advances for Nano-Enhanced Sustainable Construction Materials
  99. Advanced explainable models for strength evaluation of self-compacting concrete modified with supplementary glass and marble powders
  100. Analyzing the viability of agro-waste for sustainable concrete: Expression-based formulation and validation of predictive models for strength performance
  101. Special Issue on Advanced Materials for Energy Storage and Conversion
  102. Innovative optimization of seashell ash-based lightweight foamed concrete: Enhancing physicomechanical properties through ANN-GA hybrid approach
  103. Production of novel reinforcing rods of waste polyester, polypropylene, and cotton as alternatives to reinforcement steel rods
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