Comprehensive multiscale techniques to estimate the compressive strength of concrete Incorporated with carbon nanotubes at various curing times and mix proportions

Concrete as a building material is classified as either normal or high strength based on its compressive strength. The compressive strength of conventional concrete ranges somewhere around 20 to 40 MPa. The incorporation of high-performance nanomaterials, such as carbon nanotubes (CNT), into the concrete mix, is gaining popularity to produce multifunctional composite materials with improved mechanical, physical, and electrical properties. However, the compressive strength of normal concrete (NC) increased with the addition of CNT to the mix design. Therefore, a reliable mathematical model is required to estimate the amount of CNT to gain the necessary compressive strength. In this research, five different models were proposed to forecast the compressive strength of conventional concrete modified with carbon nanotubes, including the artificial neural network model (ANN), M5P tree model, nonlinear regression model (NLR), multilinear regression model (MLR), and linear regression model (LR). For this purpose, 282 data were collected from the literature review to examine and develop the models. During the model development, the most powerful parameters influencing concrete's compressive strength were found, i.e., curing time ranged from 1-180 days, cement varied between 250-475 kg/m, water to binder ratio ranged from0.4-0.87, coarse aggregate 498-1466.8 kg/m, fine aggregate 175.51285 kg/m, and carbon nanotube varied between (0-10%). Based on statistical assessment parameters such as coefficient of determination R2, mean absolute error (MAE), root mean square error (RMSE), scatter index (SI), and objective (OBJ), the ANN model execute better performance in predicting the compressive strength of NC modified with CNT. Jo urn al Pr e-p roo f


Methodology
The overall number of 282 data was analyzed statistically and classified into two categories. The greater group contained 188 data points used to construct models, although the smaller group contained 94 data points used to test models [50,51]. In Table 1, representative samples of the database are given, including the compressive strength of normal concrete changed with carbon nanotubes at various mix proportions. Fig. 1 illustrates the flow chart process followed in this study.
The input dataset includes the cement content (C, kg/m 3 ), water to binder ratio (w/b), coarse aggregate (CA, kg/m 3 ), fine aggregate (FA, kg/m 3 ), curing time (t, days), carbon nanotube (CNT, %). The above data were then used to evaluate the compressive strength of normal concrete using various models, and the predicted value was compared to the measured (actual) compressive strength (MPa). The following sections provide further information about the data set, modeling, and results.

Statistical evaluation of normal strength concrete properties modified with CNT
The standard error of Skewness and Kurtosis is calculated in this section to determine if the considered data (i) curing time, (ii) CA content, (iii) FA content, (iv) CNT content, (v) cement content, and (vi) compressive strength are normally distributed. A strong negative value (SNV) for the Kurtosis indicates that the distribution's tails are shorter than the normal distribution. For positive values, the reverse is true (longer tails). SNV denotes a long-left tail in terms of skewness, while for a positive value, the converse is valid (right tail). Research [52] provides more information on each of these methods of statistical analysis.

i. Curing Time (t)
To aid in the hydration process, the curing time should be extended to provide suitable early and late age compressive strength. Thus, based on published data, the curing period for NC modified with CNT ranged from 1 day to 180 days, with a median of 28 days. The variance, standard deviation, kurtosis, and skewness are 1064.69, 32.63, 3.24, and 1.51, respectively, based on statistical study. The link between compressive strength and curing time of NC mixes enhanced with carbon nanotubes using a histogram Fig. 2.

ii. Cement Content
Ordinary Portland cement OPC type 1 conforms to ASTM C 150 was used. The cement content had a specific gravity of 3050-3200. Based on data gathered from literature, the cement content J o u r n a l P r e -p r o o f varied between (250 -475 kg/m3) with a standard deviation of 45.32 kg/m3, a median of 400 kg/m3, and a variance of 2053.51 kg/m3. The statistical factors for the cement quality of normal concrete mixtures such as Kurtosis and Skewness are -0.298 and -0.11, respectively. The relationship between compressive strength and curing time of NC mixes enhanced with carbon nanotubes using a histogram Fig. 3.

iii. w/b
The w/b for regular concrete varied from 0.4 to 0.87, with a median of 0.49, a standard deviation of 0.08, and a variation of 0.01. Skewness, a function of the possibility of the meaning being asymmetric, is 1.89 for real-valued distributed variables. Additionally, Fig. 4 illustrates the relationship between compressive intensity and w/b and the histogram of NC updated with CNT.
The findings indicate that compressive strength and w/b are inversely proportional.

iv. Coarse aggregate (CA)
In the literature, crushed stone or gravel with a particle size of between 10 and 20 mm was used as coarse aggregate in the production of NC. The minimum and maximum coarse aggregate

v. Fine aggregate (FA)
In previous experiments, river sand with a maximum aggregate size of 4.75 mm and a specific gravity of 2.60-2.8 was employed as the fine aggregate. Additionally, its gradation met the requirements of ASTM C 33. The highest and minimum fine aggregate concentrations in the NC mixes were 175.5 and 1285 kg/m3, respectively, with a median of 608.38 kg/m3, a variance of 26617.49 kg/m3, and a standard deviation of 163.15 kg/m3. Kurtosis and skewness are additional functional factors for the fine aggregate dose in NC mixes. They are 3.18 and 1.14, respectively.

CNT
The variation in compressive strength and the CNT material is shown in Fig. 7a, and there is little association between them. According to data obtained 282 from the literature review, the CNT utilized in the mix proportions had a particle size diameter of 20-100 nm, a surface area of 50-260 m2/g, and 94-98 percent purity. The lowest and highest percentage of CNT between 0 and 10% by weight of cement was utilized in mixture design (Fig. 7a). Furthermore, the standard deviation, variance, skewness, and kurtosis are correspondingly 1.89, 3.56, 4.36, and 18.47. (Fig. 7b).

vii. Compressive strength
Based on total data gathered from literature

Modeling
In accordance with a coefficient of correlation (R) and root mean square error (RMSE), no direct association between the composition of conventional concrete and the compressive strength, for instance, cement, sand, gravel, CNT content, and w/b up to 180 days of curing, was seen. As a result, the following models (sections 4.1-4.5) were utilized to investigate the influence of the Where CNT represents the carbon nanotube content (percent), C represents the cement content (kg/m 3 ), w/b represents the water to binder ratio (percent), t represents the curing period (days),

4.2.Multilinear regression model (MLR)
MLR is a regression technique utilized when the criterion variable has a more considerable value than two phases. In other words, the MLR is similar to multiple linear regression. It may be used to indicate the connection between nominal dependent variables and two or more independent variables (Eq. 3).

= * ⁄ (3)
However, Eq. 3 has a restriction in that it cannot be utilized to forecast the compressive strength of NC that is devoid of CNT. As a result, the CNT content should be bigger than zero in this model (the constraint of Equation 3 is a CNT content more significant than 0%). The least-square approach was also used to determine the model's parameters (a, b, c, d, e, f, and g) and model variables.  Where:, t is the curing time in days, C is the cement content (kg /m 3 ), w/b is the water to binder ratio, CA is the coarse aggregate content (kg /m 3 ), FA is the fine aggregate, and CNT is the carbon nanotube content (percent), and the model parameters are a, b, c, d, e, f, g, h, i, j, k, l, and m were calculated using the least square method.

4.4.Artificial neural network (ANN)
The ANN feed is the inverse of the forward neural network feed [53, 58-60]. It consists of three sorts of layers: the input layer, the output layer, and the hidden layer, as seen in Fig. 8.
The input layer is where the signal to be examined will be received. The output layer performs the necessary tasks, such as predicting and categorizing. The true computational ANN engine is composed of an infinite number of hidden layers placed among the input and output layers. The data travels forward from the input to the output layer, similar to the feed-forward network in the ANN. The -hidden layer output performance improved during trial iterations to select the optimal number of hidden layers for a model to reduce error and enhance R 2 . However, because of the complexity of the Equation for many hidden layers, a single hidden layer with four neural networks was chosen in this study by error and trial in order to obtain the lowest RMSE and MAE in Fig. 9 and a higher R 2 . Eq. 5 illustrates an ANN with a single hidden layer.  maximizes the predicted error reduction from assessing each attribute at that node is chosen for node division. As a result of the branching strategy, the data for child nodes (subtree or smaller nodes) has a lower StD value. Nodes that serve as parents (greater nodes). After examining all viable structures, choose the one that has the most potential for mistake reduction. Additionally, this division results in the formation of a vast tree-like structure, which promotes overfitting. The massive tree is pruned in the second stage, and the cut subtrees are substituted with linear regression functions.

Assessment criteria for models
Various output parameters, including the R 2 , RMSE, MAE, SI, and OBJ which are specified, have been used to test and evaluate the efficiency of the suggested models. J o u r n a l P r e -p r o o f

5.2.Multilinear regression model (MLR)
The MLR model parameter indicates that the cement content and w/b ratios have a more significant influence on the compressive strength of typical concrete mixes than the other mix components Eq. 12. This is matches with previous studies in the literature [65,66]. Figure  respectively, for the training dataset.

5.3.Nonlinear regression model (NLR)
The predicted compressive strength against the actual compressive strength derived from training and testing datasets of NC mixes modified with CNT is shown in Fig. 12a  J o u r n a l P r e -p r o o f

5.4.Artificial neural network (ANN)
To forecast compressive strength values for the appropriate input parameters, the network was fed both training and test data Fig. 8. The process of developing an ANN model is iterative (such as the number of hidden layer neurons, learning rate, momentum, and iteration). The number of hidden layers utilized in this study is one with nineteen neural networks. The learning rate is 0.1, the momentum is 0.1, and the training duration is 50,000. Additionally, the number of epochs is a hyperparameter that determines how many times the learning algorithm may process the training dataset. As the error is reduced, the greater the number of epochs, the higher the R 2 , the lower the RMSE, and the lower the MAE. The projected compressive strength vs. the actual value is displayed in Fig. 13, illustrating the primary concept of generating data using an ANN model.  MPa and 0.03.  Table 3. The study dataset has a 25% error line, suggesting that almost all measured values fall inside the 25% error line (Fig. 14).

5.6.Comparative analysis of several models
As mentioned before, five distinct statistical metrics, RMSE, MAE, SI, R 2 , and OBJ, were used to evaluate the suggested models' effectiveness. Fig. 15 compares models for the training and testing datasets of NC modified with CNT using RMSE, MAE, and R 2 . Compared to the LR, MLR, M5P, and NLR models, the ANN model has a higher R 2 value and a lower RMSE and MAE value.
Additionally, Fig. 16 shows the residual error for all models created during the dataset preparation, training, and testing processes. Both graphs demonstrate that the real and estimated values of CS are closer to the ANN model, indicating the ANN's better efficiency over other models.
The OBJ rates for each of the proposed models are shown in Fig. 17 Figure 18 illustrates the SI assessment parameter values for the proposed models during the training and testing stages. As seen in Fig. 18, the SI values for all models and phases (training J o u r n a l P r e -p r o o f and testing) ranged between 0.03 and 0.18, suggesting that all models performed admirably.
However, like with the other performance characteristics, the ANN model has a lower SI value of 0.03 for the training dataset than other models. In the training phase, the ANN model has a SI value 75% lower than the LR model, 72.73 percent lower than the MLR model, 63% lower than the NLR model, and 62.5 percent lower than the M5P model.
Among all other models, the ANN model has the lowest SI value. Additionally, this demonstrated that the ANN model is more efficient and outperformed the LR, MLR, NLR, and M5P models when predicting the CS of NC mixes changed with CNT. Additionally, Fig. 19 compared the actual and forecasted CS of NC modified with CNT across all various models trained on the same data set.

5.7.Sensitivity
Sensitivity Analysis examines the relationship between the variation in an output of a numerical  Table 2 summarizes the findings of the sensitivity analysis.
As a consequence of the results, it is clear that curing time is the most critical and influential variable in predicting the CS of NC mixes modified with CNT. The curing duration spanned from 1 to 180 days in this study, indicating that extending the curing time significantly improved the CS of NC mixes including CNT. Almost all of the experimental data in Table 1 corroborate this.

Conclusions
The compressive strength of NC may be increased by precisely adding nanomaterials such as Additionally, the proportion of carbon nanotubes in the cement varied between 0 and 10%.
The cure period for data acquired from diverse experimental programs varied between one and one hundred and eighty days. J o u r n a l P r e -p r o o f