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Robust Machine Learning Framework for Modeling the Compressive Strength of SFRC–Database Compilation, Predictive Analysis, and Empirical Verification

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24 October 2023

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25 October 2023

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Abstract
Abstract: In recent years, the construction engineering sector has undergone a transformative shift towards the integration of machine learning (ML) techniques, particularly in predicting the properties of steel fiber-reinforced concrete (SFRC). Despite the theoretical sophistication of existing models, a persistent challenge remains – their opacity, lacking transparency and real-world relevance for practi-tioners. To address this gap and advance our current understanding, this study employs the extra gradient (XG) Boosting algorithm, crafting a comprehensive ap-proach. Grounded in a meticulously curated database drawn from 43 seminal pub-lications, encompassing 420 distinct records, this research focuses predominantly on three primary fiber types: crimped, hooked, and mil-cut. Complemented by hands-on experimentation involving 20 diverse SFRC mixtures, this empirical campaign is further illuminated through the strategic use of Partial Dependence Plots (PDPs), revealing intricate relationships between input parameters and con-sequent compressive strength. A pivotal revelation of this research lies in the identification of optimal SFRC formulations, offering tangible insights for real-world applications. The developed ML model stands out not only for its sophistication but also its tangible accuracy, evidenced by exemplary performance against inde-pendent datasets, boasting a commendable mean target-prediction ratio of 99%. To bridge the theory-practice gap, we introduce a user-friendly digital interface, thri-ughly designed to guide professionals in optimizing and accurately predicting the compressive strength of SFRC. This research thus contributes to the construction and civil engineering sectors by enhancing predictive capabilities and refining mix designs, fostering innovation, and addressing the evolving needs of the industry.
Keywords: 
Subject: Engineering  -   Civil Engineering

1. Introduction

Today, the world consumes 14 billion cubic meters of concrete annually, equating to approximately 4.4 tons per individual [1]. Despite its popularity, concrete has some inherent difficulties that limit its use under harsh weather conditions and in modern architectural designs. A major drawback of concrete is its relatively low tensile strength. Reinforced concrete (RC) structures are traditionally designed without considering this property, constituting approximately 6 to 12% of the normal concrete’ compressive strength (CS) [2,3,4]. Using discrete, randomly distributed, and discontinuous steel fibers in concrete reinforces the mix to mitigate problems [5,6,7]. This type of composite material is known as steel fiber-reinforced concrete (SFRC). Consequently, RC members can be enhanced in crack resistance by using this easy-to-manufacture and highly effective technology [8].
Concrete stands out as the foremost building material employed extensively across the global construction industry, thanks to its unparalleled attributes of durability, strength, and sustainability [9,10,11]. Within this context, Steel Fiber Reinforced Concrete (SFRC), fortified with short discrete fibers as mass reinforcement, emerges as a highly efficient cement-based composite capable of mitigating, to a considerable extent, the inherent brittleness of plain concrete [12]. The incorporation of fibers into concrete represents a pivotal enhancement, notably augmenting its strength, toughness, crack resistance, and tension performance. This transformative effect occurs because the haphazardly distributed fibers adeptly curtail the unstable propagation of cracks, operating both at the micro and macro levels [13]. Remarkably, SFRC exhibits a phenomenon known as strain-softening behavior, persisting even after the emergence of macro-cracks—a testament to the significant crack control offered by these fibers [14]. The introduction of steel fiber reinforcement into concrete goes a step further, markedly elevating its ductility and toughness. This outcome is primarily attributed to the introduction of supplementary fracture mechanisms and the energy expended in overcoming the interlocking and adhesion forces between the fibers and the cementitious matrix [15]. Assessing the enhancement of the ultimate properties necessitates mechanical tests and can only be determined upon reaching the conclusion of the loading phase. Recent developments have underscored the utility of acoustic emission as a means to evaluate the performance of SFRC beams [16]. Additionally, it's worth noting that the formation of macro-cracks can potentially lead to the corrosion of steel reinforcement. Therefore, the reduction in crack width, achieved through the presence of fibers, assumes paramount importance in enhancing the durability of Reinforced Concrete (RC) structural elements [17]. Furthermore, compelling evidence suggests that the inclusion of steel fibers may reduce the need for conventional steel shear reinforcement. This conclusion is drawn from a comprehensive array of tests and analyses conducted by the authors and other esteemed researchers in the field [18,19,20].
The SFRC was patented by Bernard in 1874 for strengthening concrete in tension, using steel splinters to achieve this purpose [21]. In the ensuing years, this exploration has led to many studies being conducted on its microstructure [22,23], flowability and SFRC’s behavior under tension [24,25], durability [26,27,28], strength under extreme and cyclic loadings [28,29,30]. In recent years, several new types of fibers have been proposed as reinforcements for the SFRC, along with the use of magnetic fields to align the steel fibers during casting [30,31], and microscale numerical analyses have been published to illustrate the fundamental failure mechanism of SFRC under varying external loads [32,33]. According to these studies, SFRC has substantially different strength and elasticity properties than traditional concrete. For the post-cracking response, the SFRC is positively influenced by the high tensile resistance and elasticity that results in a crack-bridging mechanism. Hence, this composite material has excellent behavior under tension and shear loadings. A large extent of mechanical property variability is, however, a result of material heterogeneity. A steel fiber reinforcement can prevent macrocrack propagation in concrete, but the resulting SFRC may be less flowable than its conventional equivalent. The quality reduction is likely caused by the interference between aggregates and fibers [33].
A design-oriented formulation for the key mechanical properties of SFRCs is necessary for the calculation of their key performance characteristics to facilitate their successful implementation in practical applications. Further, the existing models, however, tend to be either insufficiently accurate or lack physical relevance. In the literature, there have been several studies [34,35,36] proposing linear empirical formulas for predicting SFRC mechanical properties. There is a general consensus that fiber dosage and water-binder ratio play the greatest role in determining SFRC behavior. A composite material mixing theory indicates that SFRC properties strongly correlate with matrix and steel fiber elastic properties. This theoretical background has been used as a foundation for the development of various empirical models [37]. In spite of the fact that some of the currently available empirical models are capable of accurately predicting the results of a small number of tests [7,38], the development and implementation of comprehensive and robust models are still in the early stages of development.
Machine learning (ML) and ML interpretability algorithms find extensive application in various aspects of structural engineering. These applications span a wide spectrum of areas within structural engineering, encompassing: structural analysis and design, structural health monitoring and damage detection, fire resistance of structures, resistance of structural members under various loads, andthe characterization of mechanical properties and mix design of concrete [39]. As an illustration of the capabilities of these techniques, Zheng et al. [40] successfully employed a YOLO-v5 model to detect surface cracks on wind turbines. Similarly, Cardellicchio et al. [41] leveraged ML to facilitate the recognition and interpretation of defects for the purpose of risk management in heritage bridge preservation.
The use of machine learning (ML) to model the SFRC's mechanical properties has become increasingly popular over the last few years. In this context, the back-propagation Artificial Neural Network (ANN) approach has been utilized to calculate the SFRC's CS by Açikgenç et al. [38]. Their study used the aggregate size at maximum, fiber dosage, length, and size, as well as fiber length and size, as input variables. In a similar manner, Awolusi et al. [42] used an ANN method to predict the flowability, CS, and splitting strength of SFRC. In addition, Karahan [43] used multiple nonlinear regression techniques in conjunction with ANN to predict the long-term strength of SFRC containing varying levels of fly ash. While these closed-source databases may be able to provide accurate estimates of SFRC mechanical properties, their limitations remain a concern. There have been criticisms of these models for their deficiencies, which include a lack of a physical mechanism illustration and a lack of an application tool. Recently, Pakzad et al. [7] have employed various data-driven machine-learning algorithms to predict the CS of SFRC. Sensitivity and parametric analyses were performed to demonstrate the capabilities of ML algorithms in their study.
In this study, we present solutions to the above-mentioned by focusing on the CS of SFRC-containing mono-fibrous (crimped, hooked, and mil-cut) systems. The research has been completed in several phases to achieve the goal. First, a comprehensive database of 420 records collected from 43 published studies was compiled and refined to establish a representative sample population. The second component of this research involves constructing and evaluating a numerical model that uses a state-of-the-art ML technique (XG Boost). A third aspect of the study involved narrowly focused experiments designed to verify the developed model using 30 SFRC mixes. Fourth, the model parameters were ranked according to their importance, and partial dependence plots were constructed to visualize their relationships. An intuitive graphical user interface was developed to enhance the model's applicability.
ML has seen increased application in modeling SFRC properties. However, many current ML models for this purpose are "black boxes" that lack transparency and physical relevance, and also present implementation challenges. Moreover, globally, there is significant consumption of concrete, which, despite its popularity, has inherent limitations, including its relatively low tensile strength. The necessity to improve the transparency, relevance, and implement-ability of ML models for SFRC properties, combined with the global demand for more durable and versatile concrete solutions, drives this research. Existing models for predicting and optimizing SFRC properties using ML lack transparency, do not have a clear physical basis, and are challenging to implement in practical scenarios. This study introduces an innovative approach that utilizes the extreme gradient (XG) Boosting algorithm for predicting and optimizing the CS of SFRC. The approach is grounded in a comprehensive database, which is a compilation from 43 publications, and is validated through extensive experimental studies. The ML model developed in this research not only promises superior predictive capabilities against independent experimental data but also introduces a user-friendly interface, paving the way for more accessible and efficient predictions and optimizations in the realm of SFRC. This study represents a transformative effort within the construction industry landscape. Although ML techniques have gained prominence in predicting material properties, a significant challenge endures—the lack of transparency and practical relevance in existing models. To address this challenge and enhance real-world applicability, this research adopts a meticulous approach, leveraging the XG Boosting algorithm. Furthermore, the investigation gains clarity through the application of Partial Dependence Plots (PDPs), which reveal intricate connections between input parameters and the CS of SFRC. Of utmost importance, the current study uncovers optimal SFRC formulations, offering practical insights into the most effective combinations of fibers for real-world applications.

2. Methodological approach

2.1. Background

The Extreme Gradient Booting (XG Boost) algorithm is a collective ML method that integrates a gradient boosting algorithm with decision trees to convert training data into a regression model suitable to classify new data [44]. Due to Friedman et al.'s [45] introduction of the gradient-boosting methodology, Chen et al. [46] developed an algorithm designed to enhance the performance of the gradient-boosting methodology. In comparison to gradient boosting, the XG Boost algorithm can be distinguished from gradient boosting by several advantages, such as efficient tree partitioning, shorter nodes, randomized with Newton-Raphson boosting, and multi-objective optimization [47]. In recent years, it has become a very popular programming language because of its inclusion in Python and its use in several Kaggle [48] competitions. The model has demonstrated excellent predictive capabilities in identifying a person's medical condition [49], forecasting the effects of the COVID-19 pandemic [50], and predicting a company's probability of bankruptcy [51].

2.2. XG boost algorithm development

In Figure 1, a conceptual diagram illustrates the steps involved in the preparation of an XG Boost algorithm, while the following equations provide a detailed formulation of the algorithm. It is noteworthy that Python [52] for coding was used.
In this study, the database of the SFRC involves N variables with x ∈ R M × N and the model’s output, y ∈ R M × 1 , where M and N equal 420 and 12, respectively. Here, the first step involves the creation of a model’s output constant (initial) value [ f ^ 0 ] using Eq. (1). In the following step, evaluate the scores ( g ^ m ) and canvases ( h ^ m ) by employing Eqs. (2)–(3), respectively. This step involves all nodes ( m = 1 – M ) having weak responses. As a result, the multi-objective condition [Eq. (4)] can be solved using the training set { x i , − ( g ^ m ( x i ) ) / ( h ^ m ( x i ) ) } , as a basis for fitting the base learner (tree). The model is then iteratively enhanced in accordance with Eq. (5), following a process of optimization. The final results are then evaluated using Eq. (6) following the calibration of the model. In Eq. (1)–(6), L y ,   f ( x ) is a loss function that behaves differently depending on differentiability, and α is the learners' rate of progress. It is important to note that, as part of the XG Boost single tree analysis, L y , f ( x ) is continually evaluated during the modeling process of each node to determine the node that will result in the highest gain over time. When features in f ^ m x are split into subsets, it is possible to create an additional regression tree. The accuracy of the model is calculated after adding up each predictor's score to determine the total score for the model.
f ^ 0 x = arg θ ⁡ m i n ∑ i = 1 N L y i , θ
g ^ m x i = ∂ L y i , f ( x i )     ∂ f x i f x = f ^ m − 1 x
h ^ m x i = ∂ L y i , f ( x i )     ∂ f x i 2 f x = f ^ m − 1 x
ϕ ^ m = ar g ϕ ∈ Φ ⁡ m i n ∑ i = 1 N 1 2 h ^ m x i − g ^ m x i h ^ m x i − ϕ x i 2 f ^ m   ( x ) = α ϕ ^ m ( x )
f ^ m   x = f ^ m − 1 x + f ^ m   ( x )
f ^ x = f ^ M x = ∑ m = 0 M f ^ m   x

2.3. Indicators of prediction performance

As a means of testing the accuracy of the developed models about the test observations of the characteristic CS of SFRC, four performance metrics [Eqs. (7)–(10)] were calculated in this study. In these formulas, a i , a i ^ , and a ¯ i are the tested, calculated, and mean of tested SFRC’s CS, respectively.
R 2 = 1 − ∑ i = 1 n a i − a i ^ 2 ∑ i = 1 n a i ^ 2
R M S E = ∑ i = 1 n a i − a i ^ 2 n
M A P E = 100 n ∑ i = 1 n a i − a i ^ a i ^
M S E = 1 n ∑ i = 1 n a i − a i ^ 2  

3. Data collection, characteristics, and handling

3.1. Data compilation

In this study, the primary focus was on the 28d CS of SFRC with three types of mono-fibrous systems (crimped, hooked, and mil-cut) as a response to the SFRC ingredients and their characteristics. In total, the unprocessed population of the study consists of 422 datasets collected from 43 independent reports that were published between the period of 1994–2021. The careful selection of these reports was guided by a comprehensive evaluation of accessibility, data richness, and alignment with research objectives in order to ensure the robustness and relevance of the datasets. As listed in Table 1 and Table 2, the variables in the study are coded according to the database's inputs (X) and output (y). Moreover, Table 3 provides a summary of the collected datasets that were used in the study.

3.2. Data wrangling and statistical analysis

3.2.1. Treatment of outliers

Statistically significant outliers are data points that deviate from the norm, which indicates there could be an anomaly in the data [95]. It is common practice in regression analysis to deal with outliers as the first factor, which can significantly impact the outcome [96]. Several factors can contribute to detecting an outlier in a given data set; these include errors in measurement, mistakes made in capturing data, and signals detected in newly acquired data. In statistical models and analyses, outliers pose a challenge, particularly when the data involved in the analysis is excessive [97,98]. However, outliers present an exciting opportunity for exploring new possibilities. It is possible to identify outliers using various methods based on the type of data being analyzed and the type of outlier being sought. Furthermore, these methods can detect emerging phenomena or anomalous behavior. Some methods, such as Chauvenet's criteria and Grubb's test, are available for identifying outliers that use averages and standard deviations and assume a normal data distribution [48].
The variables included in the study were analyzed using descriptive statistics according to the method described in [99] to identify any outliers. Here, Grubb's test was used during preprocessing to detect outliers, errors, and even distributions in the data by checking them for outliers, errors, and odd distributions. For this purpose, we used the p-test of the hypothesis (with a 5% significance level), where the null hypothesis is that all data values come from the same normal population and the alternative hypothesis is that the largest or smallest data value is an outlier. It is noteworthy that further critical analysis of the data is an essential measure for determining the weaknesses of the approach to enhance its effectiveness. This analysis has been carried out via bivariate boxplots, confirming the datasets’ regularity. An evaluation of the rationality of the datasets has been conducted using bivariate boxplots (Figure 2), which indicated the rationality of the datasets for further regression analysis. This figure provides an informative summary of the distribution characteristics (e.g., median, interquartile range, outlier, and skewness) of model variables, making them valuable tools for data exploration

3.2.2. Data descriptive statistics and visualization

After cleaning up the database, two outliers were removed [(i) y = 88   MPa, and (ii) x 7 = 40   mm], thus leaving 420 observations available for developing the AI model. A summary of the statistical information obtained from the refined database after removing outliers is presented in Table 5, while Figure 3 displays their graphical visualization. The figure demonstrates that the distribution of the majority of these variables is well-suited for applications in machine learning.

4. Results and discussion

4.1. Features and label relations

In the current study, a Pearson correlation constant ( r x y , Eq. 11) was calculated during data preprocessing to assess the linear correlation between the model variables. A constant with a value between -1 and 1 is always present in this equation [100]. In this equation, n is the number of records ( x i , y i ) is the number i feature–label set having an average value of x ¯ ,   y ¯ . In a linear relationship between two random variables, the constant represents the average degree of variability of the linear relationship. The resulting correlation constant coefficients for the features–label relations are presented in Figure 4. As shown in the figure, cement, HRWR contents, and fiber tensile strength (i.e., X 4 , X 8 , and X 9 ) had the largest positive impact on SFRC CS. The results obtained here are consistent with those of Ayan et al. [101], in which it was shown that the type and amount of binder and the volume fraction of steel fiber had the most significant impact on SFRC compression strength. In contrast, increasing the proportions of water-binder ratio, water and coarse aggregates (i.e., X 5 , X 3 , and X 6 ) would likely reduce CS. The finding can be explained by poor microstructural properties and packing densities resulting from increased water-binder ratio and coarse aggregate contents [102]. Furthermore, a comprehensive examination of the data depicted in Figure 4 explains that variables X6 (fine aggregate content) and X11 (diameter of fiber) will likely exert minimal influence on the CS of SFRC. Given their negligible impact, these specific variables have been thoughtfully excluded from subsequent analyses and modeling.
r x y = ∑ i = 1 n x i − x ¯ y i − y ¯ ∑ i = 1 n x i − x ¯ ∑ i = 1 n y i − y ¯

4.2. Development and performance of the initial model

The default hyperparameters (Table 6) were used as a starting point for the development of the model (Model-0). It is worth noting that only the fine-tuned hyperparameters are included in this table. The prediction performance measures for this benchmark model are listed in Table 6. The initial model's performance indicators for the test data were lower than those for the training data. Based on this finding, it appears that the initial model was prone to overfitting. Therefore, a multi-objective optimization process was used to fine-tune the default hyperparameters to maximize the model's performance.

4.3. Fine-tuned model

In the present study, hyperparameters (Table 6) most affecting the model's performance, were optimized by trial and error to achieve the best accuracy. In the pursuit of refining the model's default hyperparameters, we adopted a multi-objective optimization strategy that combines aspects of both random search and grid search. At each iteration of the approach, we methodically documented the model's performance, facilitating the identification of an optimal configuration that effectively balances diverse performance objectives, encompassing the R 2 scores for both training and testing data. The favorable result was achieved using a multitarget optimization technique based on Pareto's [103] frontier approach. Figure 5 depicts the results of this multi-objective optimization process. The optimized model exhibited adequate prediction performance with scores of 0.966 and 0.879 (Table 7) for training and testing data. As presented in Figure 6, the model-target results were close to the ±95%, and ±85% accuracy ranges for both training and testing results, respectively. Additionaly, the error of the predictions by the constructed model are rarely exceeds ±30%. The model seems to be able to make accurate predictions for the database used during the modeling process. The next stage of the study involved a narrowly focused experimental campaign to verify the accuracy of the proposed ML model. The following section provides details of the experimental programs.

4.4. Experimental verification

4.4.1. Materials

Ordinary Portland cement (OPC) type I (according to ASTM C 150) was used to prepare the SF-HSC, which was sourced from a local plant. This cement is estimated to have a median particle size of 13 microns. An analysis of the physicochemical properties of the OPC used is provided in Table 8, and the distribution of its grain size is shown in Figure 7. In addition, this figure illustrates a photograph obtained by scanning electron microscopy (SEM) of the OPC. This figure illustrates that OPC particles have several features, including polyangular shapes, asymmetrical distribution, and sizes ranging from 1 to 20 µm.
The desired workability of the concrete mixes was successfully achieved through the application of a modified polycarboxylic ether polymer high-range water-reducing (HRWR, commercially known as MasterGlenium 51). This HRWR contains 36% dry powder and has a relative density of 1.1, according to the manufacturer. To determine the amount of HRWR to be added to the concrete mix, we divided the dry extract (D.E.) by the cement weight. Optimum concrete workability for this mix has been achieved by optimizing this ratio.
Additionally, all concrete mixes were prepared with coarse aggregates (Ag) whose maximum aggregate size is 10 mm. Figure 8 displays the particle size curves of the aggregates used in this study..
Three different hook-ended steel fibers were used in the production of SFRCs. A steel fiber of three different lengths and diameters was used. A summary of the physicomechanical properties of steel fibers used is given in Table 9.
The present study involved the preparation and testing of 20 concrete mixes using three different water-cement ratios (0.25, 0.35, and 0.45), three types of fiber (Table 9), and four levels of fiber dosage (0.0, 0.5, 1.0, and 1.5). Table 10 summarizes the proportions of these mixes, including the amount of steel fibers included in each one. Here, the letters U, H, and N denote concrete that contains a water-cement ratio of 0.25, 0.35, and 0.45, respectively. Thus, the “U-F1-0.5” mix means that it was prepared with water–cement ratio of 0.25, steel fiber type "F1" at a dosage of 0.5 percent (vol.). A noteworthy feature of the control mixes is that the slump (as a workability measure) was set to 100 ±25 mm and measured according to ASTM C143.

4.4.2. Methods

4.4.2.1. Preparation of the test specimens

This study was carried out by stirring a variety of aggregates in a typical concrete mixer for a few minutes, with the associated absorption water being introduced simultaneously. In the following, cement was mixed in a dry state for a few minutes. The HRWR was mixed with water for two minutes, then re-mixed with the aggregates for three minutes, followed by another three minutes without mixing before the final mixing of two minutes was conducted. A concrete mixture was eventually poured into different molds in accordance with the specimen size and the mixer was turned off. In the case of the SFRC mixes the fibers were introduced into the concrete mixture after the concrete mixture had been prepared and mixed well for five minutes so that the fibers were dispersed well within the concrete mixture.
A series of 100 (dia.) × 200 (ht.) mm concrete cylinder specimens were cast in rigid plastic molds to evaluate the CS of the concrete. To maintain moist conditions, the specimens were covered with plastic sheets after excess material was removed from the mold's surface. In this study, the specimens were demolded after 24 h and cured at 22 ± 2 °C and a relative humidity of 100%. Test specimens were kept in this condition until the time of testing. A total of three specimens were cast for each type of test as well as for each mix. For the CS specimens, tests were conducted after seven and 28 days. As a result of the study, we calculated and presented average strength results for the three specimens.

4.4.2.2. Detail of testing

The cylindrical specimens are coated with sulfur mortar before conducting the uniaxial compression test to ensure the load is evenly distributed across the top and bottom surfaces. An assessment of the 7d and 28d CSs of cement-based materials was performed in this study according to ASTM C39 specifications. A ToniTech universal testing machine equipped with a 3000 kN load capacity (Figure 9) was utilized to carry out this test. The samples were attached to two linear variable displacement transducers (LVDTs) and a compressometer ring at approximately 100 mm, the height of the center of the samples, to measure in-plane and transverse strains. The test was conducted under displacement-controlled conditions with a rate of 2.5×10-3 mm/s. The compressed test was performed on two or three duplicate samples, and the mean result was reported to ensure its reliability.

4.4.3. Test and model results

The observed and calculated CS of the studied SFRC mixes is listed in Table 11. The developed ML model yielded reasonable predictions. The mean and COV of the tested-to-predicted results were about 0.99 and 9%, respectively. A demonstration of this superior predictive capability is shown in Figure 10. In this figure, the predicted and tested data points show a low error rate of less than 10% in most cases.

5. Model implementation

5.1. Feature ranking

In this research, two [(Gini index [104] and SHapley Additive exPlanations (SHAP)] techniques have been used to assess the significance of the model's features. The practice of analyzing features based on Gini coefficients has proven to be more effective in detecting the significance of features with unique values [105]. The results of this analysis are shown in Figure 11.
The SHAP approach unveiled that the model's most influential variables are X9, X4, X5, X7, and X8, while the Gini index indicates that the importance factors include X7, X5, X3, X4, X9, and X13. Consequently, both approaches concur that the key parameters influencing the CS of SFRC are X9 (high-range water reducer content), X4 (cement content), X5 (water-binder ratio), and X7 (gravel content). It can be seen from Figure 11 (a) that a high value of the cement increases the strength and vice versa at low content. The two methods also were consistent about the rank for other variables [i.e., coarse aggregate content ( x 6 ) and fiber type ( x 1 and x 2 )]. Interestingly, these results are consistent to some extent with those reported in 4.1. This conclusion is made because the Pearson correlation constant analysis also showed that the most significant features are X5, X4, and X9.

5.2. Partial dependence plots

This study conducted a partial dependence analysis for each independent variable employed in the ML model. Figure 9 shows the PDPs of the CS of SFRC in response to different predictors, except the dummy ones. The figure suggests the optimum water content ( X 3 ) and HRWR ( X 8 ) are in the range of 100–150 and more than 10–20 kg/m3 to maximize the CS and strength notably decreases as the content increases. Further, the strength will likely increase as the cement content ( X 4 ) increase. Also, the ideal content for coarse aggregate ( X 7 ) content is perhaps 900–1100 kg/m3. Moreover, the results in the figure suggested that the best fibrous combination that has a tensile strength ( X 10 ) of about 1000 MPa, dosage ( X 12 ) of around 1.0%, and length ( X 13 ) of 40–50 mm. As expected, Figure 12 also illustrated that the CS decreased as the water–binder ratio ( X 5 ) increases.

5.3. Graphical user interface development

In this study, we provide an intuitive graphical user interface (GUI) for interacting with the developed XG Boost model. Python and Gradio [106] have been used to implement sliding control systems that allow input values to be limited to minimums and maximums (Table 5). Figure 13 shows three main components: input features with slider controls, output results, and SHAP-based explanations. The model produces SFRC's strength and the concrete class ("normal strength" if it has a strength lower than 60 MPa, otherwise "high strength concrete").

6. Conclusions, implications, and future research

This study involved the compilation and refinement of an extensive database, comprising 420 entries sourced from 43 scholarly publications. We conducted experimental analyses on 20 different SFRC mixtures to assess the predictive accuracy of the constructed model. Furthermore, we employed PDPs to elucidate the relationships between the model's input variables and its outcomes. The significance of these input variables within the model was also explored. To enhance the model's usability, we developed a user-friendly graphical interface. It is worth noting that the research specifically focused on three distinct fiber types: crimped, hooked, and mil-cut. Therefore, the findings may not be directly applicable to SFRCs with alternative fiber types or mixtures. Additionally, while the model consistently performed well with experimental data, its effectiveness may vary under different conditions or when using different raw materials. Regarding the research findings:
1.The analysis, including Pearson correlations, Gini indices, and SHAP analyses, highlighted that the most significant factors influencing the CS of SFRC were the cement and HRWR contents, as well as the fiber tensile strength-water-binder ratio. Notably, increasing the proportion of water and coarse aggregates is likely to reduce the compressive strength of the concrete.
2. We utilized the Pareto frontier multi-criterion method to develop an optimized version of the standard XG Boost model. Based on training and testing datasets, the optimized model demonstrated satisfactory predictive performance, achieving scores of 0.97 and 0.88, respectively.
3. The developed ML model consistently exhibited superior predictive capability when tested against independent experimental data conducted by the authors, with average and COV values of the tested-predicted results at 0.99 and 6%, respectively.
4. Through the application of PDPs, we determined that the optimal water and HRWR contents for achieving maximum CS are in the range of 100–150 kg/m³ and 10–20 kg/m³, respectively. Similarly, for coarse aggregates, ideal contents fall in the ranges of 900–1100 kg/m³. Additionally, the most effective fibrous combination exhibited a tensile strength of 1000 MPa, a diameter length of 40–50 mm, and a dosage of about 1.0%.
The adoption of ML techniques, particularly the XG Boosting methodology, offers the construction and civil engineering sectors an enhanced predictive toolset for determining the CS of SFRC. This research not only elucidates optimal SFRC formulations, pinpointing effective fiber combinations but also facilitates the development of concrete with superior strength and durability. In future investigations, it would be valuable to compare the predictive competence of the current numerical model against existing empirical and analytical frameworks. Additionally, the inclusion of data on ultra-high-performance concrete could enhance the model's universality. Addressing the size effect might benefit from the incorporation of a conversion factor as an input variable for various types of test samples, effectively contributing to handling this aspect of the study. While our present research offers a robust database, there is a compelling case for expanding this repository by incorporating newer studies and a wider array of SFRC mix variations, ensuring it remains at the forefront of technological progress. Despite the current study's reliance on the XG Boosting technique, exploring alternative ML schemes (e.g., neural networks or stacked ensemble alogrithms) may reveal novel perspectives and enhance predictive accuracy. Beyond immediate CS predictions, there is a growing need to investigate the enduring resilience of SFRC in diverse scenarios. In an era emphasizing ecological responsibility, future research should critically assess the environmental implications of various SFRC formulations, including aspects such as lifecycle assessments, carbon emissions, and potential for recycling.

Acknowledgments

The authors extend their appreciation to Researcher Supporting Project number (RSPD2023R692), King Saud University, Riyadh, Kingdom of Saudi Arabia.

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Figure 1. XG Boost methodology.
Figure 1. XG Boost methodology.
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Figure 2. Boxplots of the model variables.
Figure 2. Boxplots of the model variables.
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Figure 3. A graphic depiction of a model's variables.
Figure 3. A graphic depiction of a model's variables.
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Figure 4. (a) Pearson’s encoded matrix, and (b) linear correlations between features and label.
Figure 4. (a) Pearson’s encoded matrix, and (b) linear correlations between features and label.
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Figure 5. Pareto frontier-based model’s multi-objective optimization.
Figure 5. Pareto frontier-based model’s multi-objective optimization.
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Figure 6. Predicted vs. target results for the: (a) training, and (b) testing data.
Figure 6. Predicted vs. target results for the: (a) training, and (b) testing data.
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Figure 7. SEM analysis and particle size curves for the used OPC.
Figure 7. SEM analysis and particle size curves for the used OPC.
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Figure 8. Particle size curves for various aggregates.
Figure 8. Particle size curves for various aggregates.
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Figure 9. The uniaxial compression test: (a) representatives’ diagram and (b) detailed setup.
Figure 9. The uniaxial compression test: (a) representatives’ diagram and (b) detailed setup.
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Figure 10. Predicted vs. target results for experimental verification data.
Figure 10. Predicted vs. target results for experimental verification data.
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Figure 11. Variable significance by: (a SHAP value, and (b) Gini index methods.
Figure 11. Variable significance by: (a SHAP value, and (b) Gini index methods.
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Figure 12. Compressive strength of SFRC partial dependence plots.
Figure 12. Compressive strength of SFRC partial dependence plots.
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Figure 13. A GUI for predicting the compressive strength of SFRC using the XG Boost model.
Figure 13. A GUI for predicting the compressive strength of SFRC using the XG Boost model.
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Table 1. Coding system used for the dummy variables.
Table 1. Coding system used for the dummy variables.
Fiber type Hooked Mil-cut Crimped
X 1 1 0 0
X 2 0 1 0
Table 2. Coding system used for other variables.
Table 2. Coding system used for other variables.
variable Unit Description variable Unit Description
X 3 kg/m3 Water content X 9 kg/m3 HRWR content
X 4 Cement content X 10 MPa Tensile strength of the fiber
X 5 –– water-binder ratio X 11 mm Diameter of the fiber
X 6 kg/m3 Fine aggregate content X 12 %, vol. Dosage of the fiber
X 7 Coarse aggregate content X 13 mm Length of the fiber
X 8 mm Maximum aggregate size y MPa 28d CS of the SFRC
Note. HRWR: high-range water reduction superplasticizer.
Table 3. Summary of the collected CS database of SFRC.
Table 3. Summary of the collected CS database of SFRC.
Ref. Data-sets X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10 X 11 X 12 X 13 y
[53] 17 1 0 180–192.5 280–400 0.5|0.6|0.7 713.8–762.6 1027.2–1097.4 31.5 3.4–6.6 –– 0.19–0.64 0.55–0.9 30–60 29.7–46.7
[54] 12 1 0 162–177 354–450 0.4|0.5 893–920 858–887 19 2.3–6.5 1050 0–1 0.75 60 31.5–53.3
[55] 4 1 0 181.2 453 0.4 624 1242 22 1 1100 0–0.8 0.75 60 21.7–25.3
[56] 5 1 0 230.2 338 0.7 1049.2 760 10 0.7 –– 0–1.25 0.75 60 25.6–28.1
[57] 40 1 0 175 350 0.5 952.2 1321.2–1398.8 20 4.5 1100–1250 0–1.5 0–1.5 50–60 17.9–40.8
[58] 14 1 0 178–179 325–396 0.4|0.6 842–891 913–965 19 0.9–1.4 1050–2000 0–1 0.71–0.75 60 47.2–61.1
[59] 2 1 0 175.6 351 0.5 914.8 994.7 22 2.7 –– 0.26–0.45 1 50 33.3–33.4
[60] 5 1 0 205–398 311–458 0.6|0.7 408–665 1156–1407 20 3.1–7.3 1700 0–1 0.4 25 31.2–88.0
[61] 2 1 0 169.8–177.6 283–296 0.6 705–838 968–1071 10 2.8–3.3 –– 1–1.5 0.62 30 23.3–30.0
[62] 7 1 0 220 440 0.5 1193–1225 356–366 10 3–4 –– 0–1.5 0.75 25–31 35.1–43.8
[63] 8 1 0 191.2–191.6 324 0.6 1052.8–1064.2 750.0–758.3 22 3.2 1200 0.19–0.76 0.5–0.75 30–60 34.0–48.0
[64] 2 1 0 185 410 0.5 915 866 20 2–4 1100 0.5–1.0 1 50 50.0–52.0
[65] 10 0|1 0 150–168 341–429 0.3|0.4|0.5 800 975 14 2.8–11 –– 0.51 1 54–60 40.9–67.6
[66] 18 1 0 123–167 325–439 0.6 741–885 846–998 14 0.5–11.4 –– 0.48–0.75 1 54–60 38.2–69.6
[36] 6 1 0 195 342 0.3|0.5|0.6 701 1143 20 10.3 –– 0–1.5 0.75 30 24.6–30.6
[67] 23 1 0 164–185 308.3–529.1 0.3|0.6 639.3–763.9 928.6–1232.7 20 0–5.8 600–1000 0-2 0.55–0.75 35 26.7–65.4
[68] 32 1 0 172–195 336–521 0.6 652–750 1080–1145 10–40 0–5.2 1100 0-1 0.75 30–60 25.8–67.4
[69] 21 0|1 0 166 286 0.4 739 1170–1259 20 0–0.6 600 0–2 0.75 30–40 28.0–33.0
[35] 5 0 0 165 367 0.3|0.4|0.5 702–765 1053–1146 16 2.2 380 0–2 0.5 30 26.8–31.0
[70] 12 1 0 195–200 415–651 0.4|0.7 527–610 1022–1182 20 0 1000 0–1.5 0.75 32 30.9–58.5
[71] 6 1 0 185 268–524 0.4|0.5|0.6 488–725 1056–1185 31.5 0 1000 0–0.63 1 50 16.0–34.4
[72] 22 0|1 0|1 172–196 321–475 0.4|0.5 622.8–886.0 960–1171 20 0–4.5 –– 0–2 0.75–0.94 32.3–62 26.6–43.6
[73] 6 1 0 156–165 312–381 0.6 817–1200 700–1214 20 2.2–5.3 1000 0.38–0.77 1 50 63.9–67.5
[74] 13 0|1 0 264 480 0.4 716.5–769.1 895.1–989.5 20 0 380–500 0–2 0.9–1.2 30.2–32.3 27.7–34.3
[75] 4 1 0 175 461 0.4 512 1252 10 0 600 0–1.5 1.0 35 28.5–31.4
[76] 2 1 0 160 400 0.4|0.6 750 1140 20 8 1700 0–0.5 0.5 30 42.7–45.1
[77] 2 1 0 230 535 0.5 556 1079 16 0 1345 0–1.0 0.6 30 36.5–41.1
[37] 2 0 0 165 300–471 0.4 690–760 1038–1140 20 1.8–2.8 380 1 0.5 30 23.8–39.0
[78] 18 0 1 195–228 361–475 0.4 630–875 715–1180 20 0 380 0–2 0.94 32.2 28.1–34.1
[79] 2 1 0 166 415 0.4 838.6 1024.9 20 0 1345 0–1 0.55 35 40.8–42.6
[80] 5 0 1 161 460 0.4 1150 1048.8 20 18.4 808.6 0–1 0.8 32 30.0–40.6
[81] 4 0 1 215.5 480 0.4 825 880 25 8 –– 0–0.51 0.5 38 40.6–49.3
[82] 16 0|1 0|1 160–200 258–513 0.4|0.5|0.6 540–868 1012–1283 20 0 500–1325 0–0.7 0.55–1.15 32–50 27.0–47.0
[83] 5 1 0 172 400 0.4 730 1046–1100 15 0 –– 0 0.8 32 27.7–37.0
[84] 10 0 0 161.0–167.7 453.3–460.0 0.4 699.2 1594.2 20 18.1–18.4 808.6 0–1.6 0.8 32–40 43.5–56.0
[85] 28 1 0 161.7 437 0.4 756 1210 20 1.3–6.1 –– 0–1.5 0.7 35–50 31.8–50.3
[86] 3 1 0 168.1 410 0.4 1073 645 16 4.1–5.7 1100 0–1 0.8 50 32.1–38.7
[87] 3 1 0 167.7 390 0.4 1075 758 16 2.3–3.1 1100 0–1.45 0.8 50 36.9–40.5
[88] 9 1 0 198–205 440–460 0.4|0.5 924–985 721–846 12 11.1–12.8 1225 0.76 0.62 40 52.4–70.0
[89] 5 1 0 165 300 0.6 1128.8 806.3 16 2.5 1050–1100 0–0.5 0.55–0.75 35–60 17.6–29.2
[90] 6 1 0 190 380 0.6 1082.0 742.0 12 0.0–2.4 2200 0–0.63 0.35 30 21.6–43.92
[91] 3 1 0 169.1 412.4 0.4 927.8 890.7 12 0.0 1100 0–0.63 1 50 40.1–41.6
[92] 3 1 0 188.0 400.0 0.5 610.0 1132.0 10 3.3 1270 0–1 0.62 30 36.4–40.5
Tot. 422 0|1 0|1 123–398 258–651 0.3–0.7 408–1225 356.0–1594.2 10–40 0.0–18.4 380–2200 0–2 0.1–1.2 25–62 16–88
Note. In this table, X 5 is rounded to one decimal place.The datasets for the current study were derived from the comprehensive database compiled by Wang et al. [8], which served as a basis for the present study. Their study focused on the mechanical characteristics of normal and high-strength SFRC mixtures that only contained Portland cement (type I), natural aggregate, and a single type of steel fiber. A further aspect of ensuring the integrity of the compressive data has been achieved by considering the test specimen cylindrical with a diameter of 150 and a height of 300 mm. As a result, the conversion factors in Table 4 were applied to ensure consistency in the test results.
Table 4. Data consistency conversion factors.
Table 4. Data consistency conversion factors.
Specimen type Conversion factor Source Note
Cubic 0.80 ACI 318 [93] Below 60 MPa strength
Cubic 0.90 ACI 318 [93] Above 60 MPa strength
Prismatic 0.96 Wu et al. [94] ––
Table 5. Descriptive statistics of the processed datasets.
Table 5. Descriptive statistics of the processed datasets.
Variable Mean StDev Minimum Q1 Median Q3 Maximum
X 1 0.8012 0.3997 0.0000 1.0000 1.0000 1.0000 1.0000
X 2 0.0865 0.2814 0.0000 0.0000 0.0000 0.0000 1.0000
X 3 181.19 23.62 123.00 165.00 175.00 192.50 264.00
X 4 394.97 74.01 258.00 342.00 392.00 450.00 651.00
X 5 0.472 9.23 0.285 0.390 0.4800 0.550 0.690
X 6 804.05 152.58 488.00 713.00 756.00 911.00 1225.00
X 7 1067.6 210.6 356.0 945.0 1088.6 1206.3 1594.2
X 8 19.542 5.218 10.000 19.000 20.000 20.000 40.000
X 9 3.451 4.131 0.000 0.000 2.750 4.963 18.400
X 10 989.4 378.5 380.0 600.0 1050.0 1200.0 2200.0
X 11 0.7246 0.5377 0.0000 0.3840 0.5780 1.0000 2.0000
X 12 0.7553 0.1970 0.1000 0.7000 0.7500 0.9000 1.2150
X 13 44.015 12.178 25.000 32.300 40.000 60.000 62.000
y 39.677 11.034 16.000 31.800 37.418 45.358 70.000
Table 6. XG Boost hyperparameters for the initial and fine-tuned models.
Table 6. XG Boost hyperparameters for the initial and fine-tuned models.
Hyperparameter Role Range Default value (Model-0) Optimized value (Model-1)
max_depth Tree Maximum Depth: Modifying this parameter to higher values results in a more intricate model, increasing the risk of overfitting. 0 – ∞ 6 52
n_estimators This hyperparameter dictates the quantity of boosting iterations or trees incorporated within the ensemble. 1 – ∞ 100 325
learning_rate step size shrinkage employed during updates is intended to mitigate overfitting 0–1 0 0.2
colsample_bytree The subsample ratio of columns determines the proportion of features used when constructing each tree. 0–1 1 0.1
subsample Subsample ratio of training instance (e.g., 0.5 indicates 50% of data used prior to growing trees). >0–1 1 0.5
reg_alpha L1 regularization (increasing its value makes the model more conservative). –– 0 0.01
reg_lambda L2 regularization (increasing its value makes the model more conservative). –– 1 10
gamma Regularization parameter for tree pruning that specifies the minimum loss reduction required to make a split. 0 – ∞ 0 0.1
Table 7. Performance metrics of the initial and fine-tuned models.
Table 7. Performance metrics of the initial and fine-tuned models.
Performance indicator Model-0 Model-1
Training set Testing set Training set Testing set
M A P E 0.742 3.541 1.239 2.797
N M B E 2.576 28.413 4.008 16.997
R M S E 1.605 5.330 2.002 3.933
R 2 0.978 0.776 0.966 0.879
Table 8. The physicochemical properties of the used OPC.
Table 8. The physicochemical properties of the used OPC.
Oxide composition (%) L.O.I. (%) Specific gravity Fineness (m2/kg)
SiO2 Al2O3 Fe2O3 CaO MgO Na2Oeq SO3
20.20 5.49 4.12 65.43 0.71 0.26 2.61 1.38 3.14 373
Table 9. Properties of the used steel fibers.
Table 9. Properties of the used steel fibers.
Fiber Length
(mm)		 Diameter
(mm)		 Aspect ratio Young’s Modulus
(GPa)		 Tensile strength
(MPa)
shape ID
Preprints 88584 i001 F1 40 0.62 65 210 1250
F2 50 0.62 80
F3 60 0.75 80
Table 10. Test features and response.
Table 10. Test features and response.
Model’s input variables Strength   ( MPa ) ,   y y 1 / y 2
No. ID X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10 X 11 X 12 X 13 Test ,   y 1 Model ,   y 2
1 H-F1-0.5 1 0 157.5 450 0.35 716 1053 10 0.68 1250 0.62 0.5 65 69.4 74.1 0.937
2 N-F1-0.5 1 0 157.5 350 0.45 798 1078 10 0.42 1250 0.62 0.5 65 60.7 56.3 1.078
3 H-F1-1.0 1 0 157.5 450 0.35 716 1053 10 0.68 1250 0.62 1.0 65 72.3 65.8 1.099
4 N-F1-1.0 1 0 157.5 350 0.45 798 1078 10 0.42 1250 0.62 1.0 65 64.1 61.8 1.037
5 H-F1-1.5 1 0 157.5 450 0.35 716 1053 10 0.68 1250 0.62 1.5 65 75.9 79.7 0.952
6 N-F1-1.5 1 0 157.5 350 0.45 798 1078 10 0.42 1250 0.62 1.5 65 62.3 59.1 1.054
7 H-F2-0.5 1 0 157.5 450 0.35 716 1053 10 0.68 1250 0.62 0.5 80 73.8 71.7 1.029
8 N-F2-0.5 1 0 157.5 350 0.45 798 1078 10 0.42 1250 0.62 0.5 80 55.8 60.4 0.924
9 H-F2-1.0 1 0 157.5 450 0.35 716 1053 10 0.68 1250 0.62 1.0 80 76.4 81.2 0.941
10 N-F2-1.0 1 0 157.5 350 0.45 798 1078 10 0.42 1250 0.62 1.0 80 62.3 61.1 1.020
11 H-F2-1.5 1 0 157.5 450 0.35 716 1053 10 0.68 1250 0.62 1.5 80 77.3 80.3 0.963
12 N-F2-1.5 1 0 157.5 350 0.45 798 1078 10 0.42 1250 0.62 1.5 80 64.2 69.6 0.922
13 H-F3-0.5 1 0 157.5 450 0.35 716 1053 10 0.68 1250 0.75 0.5 80 68.0 63.2 1.076
14 N-F3-0.5 1 0 157.5 350 0.45 798 1078 10 0.42 1250 0.75 0.5 80 58.9 60.7 0.970
15 H-F3-1.0 1 0 157.5 450 0.35 716 1053 10 0.68 1250 0.75 1.0 80 70.6 68.9 1.025
16 N-F3-1.0 1 0 157.5 350 0.45 798 1078 10 0.42 1250 0.75 1.0 80 59.7 61.8 0.966
17 H-F3-1.5 1 0 157.5 450 0.35 716 1053 10 0.68 1250 0.75 1.5 80 78.6 83.4 0.942
18 N-F3-1.5 1 0 157.5 350 0.45 798 1078 10 0.42 1250 0.75 1.5 80 66.2 68.6 0.965
19 H-CTRL 1 0 157.5 450 0.35 716 1053 10 0.68 –– –– –– –– 61.7 59.8 1.032
20 N-CTRL 1 0 157.5 350 0.45 798 1078 10 0.42 –– –– –– –– 52.4 57.2 0.916
μ 0.992
σ 0.058
CV 0.059
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