Machine Learning the Concrete Compressive Strength From Mixture Proportions

Concrete mixture proportioning, or concrete mixture design, is the process of selecting the type and quantity of individual constituents to yield the concrete that meets requirements of specific practical applications [1]. Among all the characteristics, the concrete’s compressive strength after 28 days of aging is the most commonly used parameter of its engineering properties and performance, which is also known to be proportional to other mechanical properties, such as the flexural and tensile strength [2]. The high-performance concrete is usually made with good quality aggregates, high cement content, and supplementary materials, such as water, fly ash (FA), ground granulated blast furnace slag (GBFS), and superplasticizer. Each constituent has its own characteristics and affects the concrete mixture and performance differently.

Empirical results have shown that the concrete compressive strength (CCS) is strongly influenced by the water-to-cement ratio, but the amount of other individual constituents also plays a significant role in the concrete workability and final mechanical strength [3,4]. FAs are one of the residues generated by coal combustion, which can be siliceous or calcareous, depending on the coal type. FAs consist of glassy spheres, some crystalline matter, and unburnt carbon [5,6]. Concretes mixed with added FAs show better workability due to the increased content of fine fractions in the concrete composition. The replacement of cement with the same fly ash mass also provides a larger paste volume [7]. GBFS is formed by melting the waste rock from iron ore in temperature of 1300–1600 °C, which contains calcium and magnesium compounds from the composition of limestone and dolomite, as well as residues from the combustion of coke [8]. Both FA and GBFS are the most known and widely used supplements of Portland clinker. The introduction of these two components as substitutions for Portland cement can lead to the reduction of the kinetics of the heat production rate, which in some structural elements allows for avoidance of cracks caused by thermal stresses [7,9]. In addition, these two components can reduce the CCS in early hardening periods, but over time, concrete with additions of FA and GBFS can achieve strength similar to or even higher than those without these two components [7,10–12]. Johari et al. [13] observed a 20% increment on the static elastic modulus of high strength concretes at 28 days of age, with the GBFS substitution up to 60% in the concrete mixture. Additionally, aggregates, coarse or fine, are commonly used as one of concrete proportions, which account for 60–80% of the volume and 70–85% of the weight [14]. Aggregates are inert fillers in the concrete mixture, but they have significant effects on concrete’s thermal and elastic properties. The coarse aggregate is usually greater than 4.75 mm, while the fine aggregate is less than that. The aggregate size and amount affect workability of fresh concrete and CCS of hardened concrete [15,16]. Besides, superplasticizer is used as water reducers in the concrete mixture after the introduction of FAs to cement so that the cement paste can retain constant workability without having to increase the water requirement [17]. The amount of superplasticizer depends highly on the substitution proportion of Portland cement with FAs. New requirements on concrete mechanical performance and other characteristics demand fast and reliable mixture design. A new study has asked for adding superconducting materials in the cement to obtain superconducting concrete [18,19]. There are other reports about new innovations in the concrete construction technology, such as the electronically conductive concrete, photovoltaic concrete, and green concrete [20–23], which expand concrete applications beyond traditional areas. In summary, relationships between each component and those with the final mechanical performance are complicated. A feasible and optimal concrete mixture that can achieve a desired mechanical performance, i.e., a desired CCS value, requires significant amounts of efforts in proportioning design and experimental work that includes mixing and testing. Therefore, it is of great importance to develop a robust model that can lead to accurate estimations of CCS values based on mixture proportions. This will enable rapid mixture design and decrease labor-intensive approaches.

Despite experimental approaches, mathematical modeling and simulation methods are valuable tools for studying various processes in science and engineering disciplines [24–30]. Recently, artificial intelligence (AI) and machine learning (ML) have been used to predict many properties in chemical, petroleum, and energy systems [31–35]. For example, the permeability of heterogeneous oil reservoirs was modeled using the least-square support vector machine (LSSVM) optimized with coupled simulated annealing. The obtained results indicated increased robustness, efficiency, and reliability compared with the previous multilayer perceptron artificial neural network (ANN) model [31]. Another example is to use the LSSVM methodology to predict the unloading gradient pressure in continuous gas-lifting systems during petroleum production operations. The implementation of AI/ML models on process optimizations and property predictions has saved a large amount of time and manpower in testing and measurement, and provided mathematical correlations among process parameters, input parameters, and final performance. Similarly, many models have been developed to predict several mechanical properties of the concrete [36–38]. As mentioned above, concretes are heterogeneous materials made up of several ingredients. The mixture proportions, sources of the ingredients, initial properties of each component, and mixing techniques are all influencing factors on the compressive strength. Experimentally, the compressive strength of the concrete is typically evaluated through laboratory tests by crushing the cylinders or cubes of concrete samples in standard dimensions at a certain timepoint after the concrete is casted [39–41]. If tests at multiple timepoints are required, increased amounts of test samples will be made, tested, and eventually destroyed [42,43]. While it is a standard method for concrete property evaluations, it is very costly and time-consuming, which also generates a lot of laboratory waste. To achieve cleaner production, constructing AI/ML models using experimental data from a limited number of tests can provide guidance on future design of concrete mixture proportions. Common methods include the ANN, decision tree (DT), random forest, support vector machine, deep learning, and gene expression programming (GEP). Back in 1998, Yeh demonstrated the possibility of using the ANN to predict CCS of high-strength concrete [44]. The experimental dataset has since been used by different AI/ML model development researches to compare model performance [45,46]. Sobhani et al. used adaptive network-based fuzzy inference systems to predict CCS and obtained better results than traditional regression models [46]. Several optimization algorithms have been proposed to enhance the capability of the ANN, such as the harmony search algorithm, simulated annealing method, gray wolf optimizer, metaheuristic algorithm, genetic algorithm, and ensemble method [47–52]. ANN methods have also been applied to a wide range of problems in cement and concrete research and predicting other concrete properties [53]. The shear strength of steel-fibre-reinforced concrete beams, mechanical properties of silica fume concretes, compressive strength and tensile strength of waste concretes, compressive strength of lightweight foamed concretes, and durability of reinforced concrete structures have been predicted through aforementioned models in the literature [54–62]. In Song’s study on the prediction of CCS via mixture proportions, four approaches, the bagging regressor (BR), GEP, ANN, and DT, were developed and compared [36]. The models were validated through the k-fold cross validation approach using the correlation coefficient (CC) (R²) and root mean square error (RMSE). It was found that the BR showed high accuracy as indicated by the R² value of 0.95, while R² values for GEP, ANN, and DT were all below 0.90. However, there still exists room for improvements in model performance so that a higher R² and a lower RMSE can be achieved.

Soft computing techniques have been found to be useful in modeling different mechanical properties. For example, Asteris et al. developed an ANN model to predict the CCS incorporating metakaolin from six different parameters, i.e., the age at testing, metakaolin percentage in relation to the total binder, water-to-binder ratio, percentage of superplasticizer, binder to sand ratio, and coarse to fine aggregate ratio, and achieved remarkable accuracy [63]. Mahmood et al. constituted both linear and nonlinear models for assessing the compressive strength of the cement grout with different grain sizes, water to cement ratios, percentages of polymers, and curing ages [64]. Mahmood et al. built a nonlinear Vipulanandan p-q equation for predicting the stress–strain relationship of the modified cement with polymers, which achieved better accuracy than the β model [65]. Liao et al. proposed two novel hybrid fuzzy systems for creating a new framework to estimate the axial compression capacity of circular concrete-filled steel tubular columns and determined that the hybrid fuzzy systems could improve accuracy based on base models and existing design code methodologies [66]. Nguyen et al. put forward general semi-empirical formulas that involve nondimensionalization and optimization techniques for revealing explicit relations between the compressive strength and mixture proportions and reached high accuracy with universal capacities [67]. Asteris et al. adopted an ANN model for predictions of the compressive strength of masonry, which achieved better estimates than formulas existing in codes or the literature [68].

Here, we develop the Gaussian process regression (GPR) model to shed light on the relationship between mixture proportions, in terms of the cement, blast furnace slag, fly ash, water, superplasticizer, coarse aggregates, and fine aggregates, and the CCS aged for 28 days. The model is well generalized with only a few descriptors, where machine learning algorithms are capable of learning and recognizing patterns. This modeling approach is highly stable and accurate that contributes to fast and low-cost 28-day CCS estimations and understandings of which from mixture proportions. The developed model is also applied on an out-of-sample dataset to show its robustness.

The reminder of the paper is organized as follows. Section 3 discusses the methodology used. Section 4 presents the data analyzed. Section 5 reports the result and Sec. 6 provides the conclusion. Figure 1 shows the structure of this work.

It is a big challenge to design mixture proportions of the high-performance concrete due to complicated relationships between each component and those with the final mechanical performance. The high-performance concrete is usually made with good quality aggregates, high cement content, and various supplementary materials. The concrete mixture and performance are affected by each constituent, which has its own individual characteristics. Significant amounts of efforts, including experimental mixing and testing, are usually required to obtain an optimal concrete mixture with desired mechanical properties, among which, the CCS after 28 days of aging is the most commonly used parameter. To save experimental cost and enable rapid mixture design, it is of great significance to develop a robust model that can lead to accurate estimations of CCS values based on mixture proportions.

The present paper focuses on predictions of CCS values after 28-day aging through the GPR model using the contents of cement, blast furnace slag, fly ash, water, superplasticizer, coarse aggregates, and fine aggregates as descriptors. Despite various models developed to predict mechanical properties of concretes, the GPR model has been rarely used to predict the CCS aged for 28 days in the literature. This work is to introduce the GPR method and provide a more accurate and relatively easier machine learning approach in the field of concrete mixture proportion design. As one of machine learning methods, the GPR has been used in various material systems to predict important physical parameters in diverse application fields. This model could serve as a guideline for concrete mixture design and could be utilized to aid understandings of relationships between the mixture design variables and strength. Therefore, this work aims at coming up with a practical solution to the aforementioned problem.

GPRs are nonparametric probabilistic models. They could model complex relations while handling uncertainties in principled manners [69–79]. Similar to other machine learning approaches, GPRs learn from the data and serve as an approximation tool for predictions of the concrete compressive strength considered in this work. They are not deterministic calculations and there should exist room for improvements in prediction accuracy through explorations of different machine learning models, datasets, predictors, and algorithms. Let {(x_i, y_i); i = 1, 2, …, n}, where $xi∈Rd$ and $yi∈R$⁠, be the training dataset from a distribution that is unknown. Provided x^new—the input matrix, constructed GPRs make predictions of y^new—the response variable. In this study, x’s include the cement, blast furnace slag, fly ash, water, superplasticizer, coarse aggregate, and fine aggregate, all measured with kg/m³, and y is the CCS, measured with MPa.

Let y = x^Tβ + ɛ, where ɛ ∼ N(0, σ²), be the linear regression model. The GPR tries to explain y through incorporating l(x_i)—the latent variable, where i = 1, 2, …, n, from a Gaussian process such that l(x_i)’s are jointly Gaussian-distributed, and b basis functions that project x into the p-dimensional feature space. Note that the smoothness of y is captured by l(x_i)’s covariance function.

The covariance and mean characterize the GPR. Let $k(x,x′)=Cov[l(x),l(x′)]$ represent the covariance, m(x) = E(l(x)) represent the mean, and y = b(x)^Tβ + l(x), where l(x) ∼ GP(0, k(x, x′)) and $b(x)∈Rp$⁠, represent the GPR. The parameterization of k(x, x′) usually is through θ—the hyperparameter—and thus one could have k(x, x′|θ). Different algorithms generally make estimations of β, σ², and θ during the process of model training and allow one to specify b and k, as well as parameters’ initial values.

For parameter estimations, we use cross validation and Bayesian optimizations. For the former, we adopt 20 randomized folds (see Tables 5, 1, and 2), and for the latter, we adopt both the lower confidence bound (LCB) and expected improvement per second plus (EIPSP) algorithms. With a GPR of f(x), the Bayesian algorithm evaluates y_i = f(x_i) for N_s points x_i taken at random within the variable bounds, where N_s points stand for the number of initial evaluation points and 4 is used. If there are evaluation errors, it takes more random points until N_s successful evaluations are arrived at. The algorithm then repeats the following two steps: (1) updating the Gaussian process model of f(x) to obtain a posterior distribution over functions Q(f | x_i, y_i for i = 1, …, n) and (2) finding the new point x that maximizes the acquisition function a(x). It stops after reaching 100 iterations. The acquisition function, a(x), evaluates the goodness of a point, x, based on the posterior distribution function, Q. The LCB acquisition function examines the curve G two σ_Q—standard deviations below μ_Q—the posterior mean at each point: G(x) = μ_Q(x) − 2σ_Q(x). Thus, G(x) is the objective function model’s 2σ_Q lower confidence envelope. The LCB algorithm then tries to maximize the negative of G: LCB = 2σ_Q(x) − μ_Q(x). The expected improvement family of acquisition functions evaluates the expected amount of improvement in the objective function, ignoring values that cause an increase in the objective. Let x_best be the location of the lowest posterior mean and μ_Q(x_best) be the lowest value of the posterior mean. The expected improvement is EI(x, Q) = E_Q[max(0, μ_Q(x_best) − f(x))]. Sometimes, the time to evaluate the objective function can depend on the region. If so, the Bayesian algorithm could obtain a better improvement per second by using time-weighting in its acquisition function. Specifically, during objective function evaluations, the optimization process maintains another Bayesian model of objective function evaluation time as a function of position x. The expected improvement per second (EIPS) that the acquisition function uses is $EIPS(x)=$$EIQ(x)/μS(x)$⁠, where μ_S(x) is the posterior mean of the timing Gaussian process model. To escape a local objective function minimum, behavior of acquisition functions can be modified when they estimate that they are overexploiting an area. Let σ_F(x) be the standard deviation of the posterior objective function at x and σ_PN be the posterior standard deviation of the additive noise so that $σQ2(x)=σF2(x)+σPN2$⁠. Let $tσPN>0$ be the exploration ratio. The EIPSP acquisition function, after each iteration, further evaluates whether the next point x satisfies $σF(x)<tσPNσPN$⁠. If this is the case, the EIPSP algorithm declares that x is overexploiting and the acquisition function modifies its kernel function by multiplying θ by the number of iterations [80]. This modification, as compared to EIPS, raises the variance σ_Q for points in between observations. It then generates a new point based on the new fitted kernel function. If the new point x is again overexploiting, the EIPSP acquisition function multiplies θ by an additional factor of 10 and tries again. It continues in this way up to five times, trying to generate a point x that is not overexploiting. The EIPSP algorithm accepts the new x as the next exploration ratio and therefore controls a tradeoff between exploring new points for a better global solution and concentrating near points that have already been examined.

The data used in Table 5 (columns 1–9) are from Ref. [44]. The dataset covers a wide range of concrete mixtures. A total of 399 concrete mixtures with 28-day CCS ranging from 8.54 MPa to 62.94 MPa are examined. The cement, blast furnace slag, fly ash, water, superplasticizer, coarse aggregate, and fine aggregate, all measured with kg/m³, are used as descriptors. The ranges are [102,540], [0,359.4], [0,200.1], [121.8,247], [0,28.2], [801,1145], [594,992.6], and [8.54,62.94] for the cement, blast furnace slag, fly ash, water, superplasticizer, coarse aggregate, and fine aggregate, respectively. Figure 4 visualizes the data that reveal nonlinear patterns modeled via the GPR.

While our dataset might not be considered large as compared to some studies utilizing thousands of, tens of thousands of, or hundreds of thousands of observations in building machine learning models, it represents sufficient coverage of samples for our purpose and has enough degrees-of-freedom for parameter estimates with the use of seven predictors. This is an important consideration as the dataset used for modeling could be as important as the modeling approach itself when constructing machine learning models for predictions of different properties [81–85].

The relationship between performance of a model and the size of the training data is first examined in Fig. 5, which reveals the benefit to train the GPR with all observations. As an attempt to investigate the potential boundary of model performance, model 1 in Table 1 is constructed by utilizing all observations for training, whose predicted results are detailed in Table 5 (column “Prediction”) and visualized in Fig. 6 (legend “Prediction”). It results in CC, MAE, and RMSE of 99.89%, 0.2840 (0.82% of the average experimental CCS), and 0.5821 (1.69% of the average experimental CCS), respectively. It leads to the mean percentage error of 0.14% and the maximum percentage error of 13.94%, where the maximum percentage error happens for the 320th sample shown in Table 5. Section 5 discusses how our final model is arrived at and its performance.

The final GPR model is reported in Table 1 as model 2 that reveals good alignments between predictions and experimental values, as shown in Table 5 (columns “Prediction (Average)” and “Concrete Compressive Strength”) and visualized in Fig. 6. The CC, MAE, and RMSE are 99.85%, 0.3769 (1.09% of the average experimental CCS), and 0.6755 (1.96% of the average experimental CCS), respectively, reflecting good performance and approaching performance of model 1 in Table 1 discussed in Sec. 4.2. The final GPR model leads to the mean percentage error of 0.18% and the maximum percentage error of 13.97%, where the maximum percentage error happens for the 320th sample shown in Table 5.

From Table 1, it is observed that model parameter estimates are, in general, stable. Two observations are worth noting. First, β₅ associated with the predictor “superplasticizer” shows flipping coefficient signs for several times among model 2.CV Fi (i = 1, 2, …, 20), which might lead to certain prediction instabilities. Second, σ₂ associated with the predictor “blast furnace slag” shows rather small estimates for several times among model 2.CV Fi (i = 1, 2, …, 20), which should be caused by a relatively high ratio of zero values and the cross validation segmenting. To get an idea about implications of these parameter estimates on the prediction stability, Table 2 lists model performance measures across the 20 folds. It is found that all folds maintain high and rather stable CCs from the training to validation sub-sample. On average, the RMSE and MAE are 10.66% and 6.54% of the experimental mean of the validation sample, meaning that prediction errors beyond training samples are generally in a controllable range. The average mean percentage error across the validation folds is 0.80% and the average maximum percentage error across the validation folds is 28.50%. Figure 7 further visualizes the cross validation results, where one could observe that model accuracy is generally maintained from training to validation sub-samples across the 20 folds, except for several sporadic validation points. To cancel out each fold’s idiosyncratic irregularities and arrive at final stable predictions, model 2 is built as a combined prediction approach. Specifically, model 2’s predicted results are obtained by applying model 2.CV Fi (i = 1, 2, …, 20) to all observations in Table 5 and taking the average of the 20 predictions for each observation. Performance of model 2 is discussed in Sec. 5.1, where we find that it is highly accurate.

To analyze the importance of different predictors under the GPR framework, we remove one predictor each time and rebuild model 1 and model 2. Model performance of the rebuilt models is compared with that of model 1 and model 2 that incorporate all predictors. The comparisons are reported in Table 3, where we could see that model 1 and model 2 incorporating all predictors lead to the best performance. Removing the superplasticizer has the largest impact on model performance, followed by the cement, fine aggregate, and coarse aggregate in order. This indicates that the superplasticizer, cement, fine aggregate, and coarse aggregate are the four most important predictors in order under the GPR framework for the concrete compressive strength.

To analyze the model’s capability of handling fluctuations in input parameters and extrapolations, we conduct out-of-sample tests using the 26 samples [44] shown in Table 4, where predictions from model 2 are included in the last column. These samples’ predictors and experimental CCS values have different ranges from the samples shown in Table 5. Particularly, the ranges for the cement, blast furnace slag, fly ash, water, superplasticizer, coarse aggregate, fine aggregate, and CCS are [250,540], [0,282.8], [0,132], [137.8,195], [0,32.2], [822,1130], [613,896], and [63.14,81.75], respectively, for the samples in Table 4, while the ranges are [102,540], [0,359.4], [0,200.1], [121.8,247], [0,28.2], [801,1145], [594,992.6], and [8.54,62.94] for the samples in Table 5. With fluctuations in input, model 2 achieves the CC, RMSE, MAE, mean percentage error, and maximum percentage error of 99.47%, 0.9132, 0.6747, −0.88%, and 0.41%, respectively, showing high accuracy. Figure 8 further visualizes the predictions for the out-of-sample tests, where we could observe that the predicted CCS values align well with the experimental ones.

We use the contents of cement, blast furnace slag, fly ash, water, superplasticizer, coarse aggregates, and fine aggregates as descriptors to develop the GPR model to predict the CCS at 28 days. This model requires only a few parameters and is simple and straightforward. It is also highly stable and accurate, which suggests the GPR’s usefulness to model and understand the relationship between the predictors and the CCS. The model applies to a wide range of concrete mixtures and it could be used to help design and the understanding of high-performance concrete. Future work of interest might include extending GPR models for predictions of different properties of the concrete. Additionally, models predicting CCS at various timepoints after casting can be developed and a correlation between aging time and CCS might be revealed mathematically.

This article does not include research in which human participants were involved. Informed consent not applicable. This article does not include any research in which animal participants were involved.

The authors attest that all data for this study are included in the paper.