# Modelling of Modulus of Elasticity of Low-Calcium-Based Geopolymer Concrete Using Regression Analysis

Authors: Ali A. Khalaf; Katalin Kopecskó
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

Despite the unremitting efforts to model the modulus of elasticity of low-calcium-based geopolymer concrete, the state-of-the-art models need much improvement to reduce the error signals and increase the reliability. This study represents a comprehensive regression analysis to model the modulus of elasticity of low-calcium-based geopolymer concrete in terms of its compressive strength. The proposed model’s assumptions are based on taking into account the chemical composition and compressive strength class and considering the normal density of concrete. The modelling is based on 67 different mix-design samples collected from peer-reviewed literature, which are divided into two groups. The first group consists of 59 samples that are used to construct the proposed model, while the second group consists of 8 samples that are used to test the validity of the proposed model. The analysis showed that the proposed model gives the root mean squared error value (RMSE) of 3.122 GPa and the mean absolute percentage error value (MAPE) of 15.0%. Therefore, the proposed model gives 41% and 52.2% reductions in RMSE and MAPE, respectively, from the state-of-the-art model in the literature. Furthermore, other statistical parameters to evaluate the goodness of fitness of the proposed model have been considered, such as the relative root mean squared error (RRMSE) (15.5%), the coefficient of determination (R-squared) (0.773), and the coefficient of correlation (R) (0.88); all of which indicated that the goodness of fitness is good and the proposed model has a high correlation to the actual values. Applying the proposed model in future applications will help reduce the time and cost of geopolymer production, as the proposed model has significantly reduced the error signals.


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## Body

## 1. Introduction

Geopolymers are three-dimensional networks of silico-aluminate structures produced by reactivating amorphous aluminosilicate (precursor or raw material) in the presence of alkali metal cation (alkali activation solution) through a process that is so-called geopolymerization [1–4]. Geopolymer concrete is considered an environmentally friendly construction material because its manufacturing emits much less carbon dioxide (CO2) gas than the manufacturing of Portland cement concrete [5, 6]. Studies stated that 1-ton production of cement produces approximately 1 ton of carbon dioxide (CO2). Moreover, many industrial wastes and by-product materials such as red mud, slag, mine tailings, and volcanic pumice ash have been successfully utilised in the process of geopolymer production, which help to produce sustainable construction materials [7–13]. Therefore, geopolymer binders can be an efficient alternative to Portland cement in the concrete industry [1, 2, 5].Machine learning techniques have been proven to be valuable and beneficial in civil engineering applications for the prediction of the mechanical properties of geopolymer concrete due to the fact that these techniques, such as neural networks, genetic programming, linear regression, and nonlinear regression, have the ability to capture the complex behaviour [14–17]. In practical applications of concrete structures, the most important feature to evaluate concrete strength is compressive strength. All other mechanical properties, such as the modulus of elasticity, are expressed as functions of compressive strength [18]. Machine learning techniques help establish a functional approximation solution with acceptable error from the unknown relationship based on experimental observations to eventually express an approximate mathematical model between the dependent and independent variables [17, 19]. The nonlinear regression analysis has the advantages of expressing more complex relationships and produces a visualised discrete relationship that is easy to comprehend and apply [19].Young’s modulus of elasticity plays a pivotal role in evaluating the load-deformation behaviour of concrete, which describes the reversible part of the stress-strain curve [20]. In engineering practices such as ACI 318 [21] and the CEB FIP model [22], prediction models are developed to calculate the modulus of elasticity of concrete instead of testing each time; these models are established based on empirical equations depending on historical tests [21, 22]. Since Portland cement concrete practice manuals and design codes have been instituted over decades, most cement alternatives in the primary stage are compared to cement properties to examine their validation [23]. Despite that, these prediction models of the modulus of elasticity of Portland cement concrete overestimate the results when used for geopolymer concrete [24]. Moreover, the results of the modulus of elasticity of these models for geopolymer concrete are smaller than the results for Portland cement concrete, even when the compressive strength results of geopolymer concrete are higher [25]. Consequently, it has been found that high-strength concrete prediction models are in agreement margins when applied to high-strength geopolymer concrete [24, 26]. However, this finding did not solve the problem because the predicted modulus of elasticity values will be far from the experimental values when the compressive strength decreases [27].The other approach to research work is to build a new model to predict the modulus of elasticity of geopolymer concrete. Most researchers have constructed their models only based on their experiment results [24, 28–33]. In return, these models seem to be fit only for a limited range of compressive strength. Some researchers considered the previous results to build the prediction models [27, 34–37]. However, these models also have flaws. The prediction model proposed by Prachasaree et al. and Xie et al. had neither been tested against experimental data nor provided statistics to assure their reliability. In contrast, the prediction model proposed by Prachasaree et al. [27] improved their previous model [35]; however, the new model provided only a small difference in statistics compared to the prediction model proposed by Nath and Sarker [34]; the error percentage is still considerably high. In the construction of prediction models proposed by Nath and Sarker and Cui et al. [34, 36], only a few data were selected without clarifying why those few data particularly have been chosen and leaving the rest of the previous data, which made these models subjected to bias. Therefore, it is necessary to build a prediction model based on firm and explicit assumptions by incorporating all the data of interest, giving engineers a clear idea when applying this model in practical applications.


## 2. Research Significance

Many researchers have introduced prediction models for the modulus of elasticity of low-calcium-based geopolymer concrete in terms of its compressive strength [27, 30, 31, 34, 37, 38]. However, the error percentage is still relatively high; for example, ACI 318 [21] determined the MAPE in the range of ±20.0%, while the best performance model in the literature, Prachasaree et al. [27], has an MAPE value of 31.4%. Thus, this research aimed to establish a reliable regression model for the modulus of elasticity, Ec, of low-calcium-based geopolymer concrete in terms of its compressive strength, fc′, by reducing the error signals and increasing the correlation between the proposed model and the actual values. The modelling process was carried out on all applicable data collected from peer-reviewed literature, and the proposed model was built with a 95% confidence interval based on explicit and firm assumptions. It is reported that increasing the compressive strength increases the quality of other properties of concrete and, more importantly, the modulus of elasticity [18]. Unfortunately, with an increase in compressive strength, there is a corresponding increase in the cost of geopolymer production [39]. Therefore, by reducing the error signals of the proposed model of modulus of elasticity, its application increases the safety margins and controls the cost management.


## 3. The Proposed Model Assumptions

As a first step prior to analysis, the margins and characteristics of the population and samples should be clearly determined and described to ensure that the prediction model will be representative. Therefore, the following assumptions are constructed:(1)
Geopolymers can behave differently because of the chemistry of the binding material. Basically, geopolymers can be divided into two main groups, namely, low-calcium-based geopolymers, CaO <8% [37], and high-calcium-based geopolymers, CaO ≥8% [37]. This division is due to the difference in the reaction mechanism. In low-calcium-based geopolymers, the reaction mechanism is classified as pozzolanic, while high-calcium-based geopolymers undergo hydraulic reactions as well [37]. The gel type produced in the high-calcium geopolymers, such as ground granulated blast furnace slag based geopolymers, is C-A-S-H. On the other hand, low-calcium geopolymers such as class F fly ash based geopolymers, the gel type of N-A-S-H [40, 41], are produced. Therefore, it should be noted that only low-calcium-based geopolymers will be dealt with in the formulation of this model.(2)
The type of concrete should be normal unit weight and normal compressive strength concrete. High-strength concretes are very rigid materials with high modulus of elasticity values and with very low elastic deformation. When the specified design compressive strength is 55 MPa or more, concrete is considered high-strength and exhibits different material properties [42]. Therefore, the samples with a specified design compressive strength in the range of 55 MPa or more will be excluded from the proposed model’s formulation and the future applications of this model.(3)
Compressive strength and modulus of elasticity values are the mean values of specimens [21]. In cases where the authors give the values of specimens for compressive strength and modulus of elasticity, the sample means are taken.(4)
The cylindrical compressive strength,fc′, will be related to the modulus of elasticity, Ec, in the formulation of the proposed model. In cases where the authors gave the cubic compressive strength and performed the modulus of elasticity on cylindrical specimens, a correlation factor for the normal compressive strength should be considered [43]. All samples should be at 28 days of age, or they are reported to have an equivalent hardening rate to 28 days of age [44].(5)
The confidence interval that can be seen in the literature is from 99.9% to 90.0%. However, it is traditionally constructed as 95.0% [45]. This value, 95.0%, will be set as a confidence interval in the construction of the proposed model.(6)
The software SPSS version 23 will be employed to perform the statistical work of this research.


## 4. Data Collection, Classification, and Testing

The data collected from peer-reviewed literature consist of 67 different mix-design samples, in accordance with the aforementioned assumptions, which are displayed in Table1. Most of the data (59 samples, analysing data) are used to construct the proposed model. In comparison, some data (8 samples, testing data) are selected to be testing samples to examine the goodness of fitness and compare the proposed model to the common and most recent models.Table 1
Data collected to generate the regression analysis and test the proposed model.
Sample IDReferencefc′ (MPa)Ec (GPa)RemarksM26[24]44.023.00SolutionN[25]31.711.90The sample means have been takenSolutionW41.517.104[30]40.428.60547.629.48646.729.36746.828.52846.126.461047.425.641112.27.041212.86.811320.77.961410.37.461546.628.74165.54.621943.425.612620.211.272752.628.092832.522.812923.413.183022.712.313236.824.73U1[46]52.229.13U25432.10U353.830.62GPC0[47]45.324.00GPC1–60[48]50.825.10GPC1–9054.425.40GPC2–6052.0421.90GPC2–9053.625.50GPC3–6046.3422.80GPC3–9049.7322.60GPC3–12052.4328.00A40S00[34]25.617.40A40S1038.322.60A40S1546.624.60A35S0032.519.80A35S1033.319.20A40P0643.223.20A40P0834.420.60A35P0635.321.40A40C024222.40A40C0341.521.60A35C0236.822.20M-0[49]26.811.84A correlation factor has been used toconvert cubic strength to cylindricalM-129.414.79M-230.815.44M-332.816.931[36]35.424.29The sample means have been taken22921.25346.221.19436.118.47540.518.95641.418.87733.516.74839.118.293516.971037.117.781137.716.991237.217.9M1[33]12.413.36A correlation factor has been used toconvert cubic strength to cylindricalM213.714.3GP-1[50]15.0247A correlation factor has been used toconvert cubic strength to cylindricalP1[51]43.7625.8P227.9221.5P340.0824.6P438.1623


### 4.1. Testing the Analysing Data for Normality

The data of the samples are considered representative of the population when they follow the same distribution. According to the central limit theorem [52], most natural and physical phenomena follow a normal distribution. As a result, the data should be checked for normal distribution. The best technique to perform the normality test is the quantile-quantile plot [53]. The outcome of the normality check is shown in Figure 1 for both the compressive strength, fc′, in MPa and the modulus of elasticity, Ec, in GPa. Since the data adhered to the straight line, the data are normally distributed for the independent variable, the compressive strength, and the dependent variable, the modulus of elasticity. Therefore, these samples are considered representative of the populations. The mean values of compressive strength and modulus of elasticity are 36.79 MPa and 20.284 GPa, respectively, while the standard deviation values are 12.15 MPa and 6.55 GPa, respectively.Figure 1
Normality distribution check: (i) compressive strength,fc′ (MPa), and (ii) modulus of elasticity, Ec (GPa). (a) Normal Q-Q plot of fc′(MPa). (b) Normal Q-Q plot of Ec (GPa).
(a)(b)


## 4.1. Testing the Analysing Data for Normality

The data of the samples are considered representative of the population when they follow the same distribution. According to the central limit theorem [52], most natural and physical phenomena follow a normal distribution. As a result, the data should be checked for normal distribution. The best technique to perform the normality test is the quantile-quantile plot [53]. The outcome of the normality check is shown in Figure 1 for both the compressive strength, fc′, in MPa and the modulus of elasticity, Ec, in GPa. Since the data adhered to the straight line, the data are normally distributed for the independent variable, the compressive strength, and the dependent variable, the modulus of elasticity. Therefore, these samples are considered representative of the populations. The mean values of compressive strength and modulus of elasticity are 36.79 MPa and 20.284 GPa, respectively, while the standard deviation values are 12.15 MPa and 6.55 GPa, respectively.Figure 1
Normality distribution check: (i) compressive strength,fc′ (MPa), and (ii) modulus of elasticity, Ec (GPa). (a) Normal Q-Q plot of fc′(MPa). (b) Normal Q-Q plot of Ec (GPa).
(a)(b)


## 5. Results and Discussion

### 5.1. Initial Modelling

The first step of the modelling is to investigate the proper relationship that governs and fits the modulus of elasticity,Ec, in terms of the corresponding compressive strength, fc′. This step can be accomplished by running the analysing data in the SPSS software. The analysis is performed as shown in the flowchart in Figure 2. The results of the analysis are shown in Table 2. The results show that the highest value of the coefficient of determination, R-squared, corresponds to the power relationship, equal to 0.809, with initial values of constants equal to 1.094 and 0.809. The p values are ≤ 0.001; thus, the variables are statistically significant and related. Therefore, the governing equation of the modulus of elasticity is mathematically expressed as (1)Ec=a∗fc′b,where Ec is the modulus of elasticity in GPa, fc′ is the cylindrical compressive strength in MPa, and a and b are constants. This relationship complies with the form of ACI 318-14 and most of the other previous models in the literature [21].Figure 2
The initial modelling process in SPSS.Table 2
Model summary and parameter estimates (the dependent variable isEc_GPa, and the independent variable is Fc_MPa).
EquationR-squaredp valueConstant,ab1b2b3Linear0.772≤0.0012.9770.471Logarithmic0.723≤0.001−21.51311.869Inverse0.531≤0.00126.480−183.139Quadratic0.773≤0.0011.9970.544−0.001Cubic0.775≤0.0014.7540.1770.0120.000Compound0.769≤0.0016.2041.031Power0.808≤0.0011.0940.809S0.687≤0.0013.394-13.425Growth0.769≤0.0011.8250.030Exponential0.769≤0.0016.2040.030Logistic0.769≤0.0010.1610.970


### 5.2. Final Modelling

After selecting the proper governing equation and its initial values, the final model can be obtained by running these results in SPSS software by following the flowchart shown in Figure3. The results of the analysis are shown in Table 3. The final values of model parameters, a and b, are 1.088 and 0.815, respectively. The determination coefficient, R-squared, of the model is 0.773.Figure 3
The process of the final modelling in SPSS.Table 3
Results of final modelling for the dependent variableEc_GPa.
Iteration historybIteration no.aResidual sum of squaresParametersab1.0581.1081.0940.8091.1575.0861.0880.8152.0575.0861.0880.8152.1575.0861.0880.8153.0575.0861.0880.8153.1575.0861.0880.815Parameter estimatesParameterEstimateStd. error95% confidence intervalLower boundUpper bounda1.0880.3080.4701.705B0.8150.0760.6620.967ANOVAcSourceSum of squaresdfMean squaresRegression23632.371212946.106Residual523.8775710.089Uncorrected total24156.24959Corrected total2294.66158Derivatives are calculated numerically.aMajor iteration number is displayed to the left of the decimal, and the minor iteration number is to the right of the decimal. bRun stopped after 6 model evaluations and 3 derivative evaluations because the relative reduction between successive residual sums of squares is at most SSCON = 1.000E − 8. cR-squared = 1 − (residual sum of squares)/(corrected sum of squares) = 0.773.The final modelling formula is expressed as(2)Ec=1.088∗fc′0.815,where Ec is the modulus of elasticity in GPa and fc′ is the cylindrical compressive strength in MPa.Both confidence bands and prediction bands are constructed by applying equations (3) and (4), respectively, with a 95% probability [54, 55].(3)YPred.±t0.05∑Y−YPred2n−2⋅1n+X−X¯2∑X−X¯2,(4)YPred.±t0.051+∑Y−YPred2n−2⋅1+1n+X−X¯2∑X−X¯2,where YPred is the predicted value of the dependent variable, t is the critical value for 95% of probability, Y is the true value of the dependent variable, n is the number of samples, and X¯ is the mean value of the independent variable of X.The relationship between the compressive strength,fc′, and modulus of elasticity of low-calcium-based geopolymer concrete, Ec, is described in Figure 4 with the corresponding 95% confidence bands and 95% prediction bands. It can be seen that the 95% confidence bands are tight and narrow, confining the proposed model through a long distance with only a slight expansion at the edges. This means that the true regression line is very close to the proposed model line with a probability of 95% [55]. The proposed model also performs well in terms of prediction with a 95% probability. The 95% prediction bands confine all the experimental data with much inside space between the outer experimental data and the prediction bands.Figure 4
The relationship between compressive strength,fc′, and modulus of elasticity, Ec, of geopolymer concrete.The root mean squared error (RMSE) of the proposed equation is equal to 3.122. This value is much lower than the results of the most recent findings, Lee and Lee [31], Nath and Sarker [34], and Prachasaree et al. [27], which are equal to 8.900, 6.01, and 5.260, respectively. Therefore, the proposed model gives a 41% reduction of RMSE from the least error model [27]. The mean absolute percentage error (MAPE) of the proposed equation is equal to 15.00%, which is much lower than the results in the most recent findings, Lee and Lee [31], Nath and Sarker [34], and Prachasaree et al.[27], which are equal to 40.2%, 35.0%, and 31.4%, respectively. Therefore, the proposed model gives a 52.2% reduction in MAPE from the least error model, Prachasaree et al. [27].Moreover, the data have been run by neural networks in the SPSS software. The optimised neural network architecture was one hidden layer with two neurons for the multilayer perceptron neural network type. The activation function for the hidden layer was the hyperbolic tangent. The results of the neural network analysis have yielded anR-squared value of 0.752 and an RMSE value of 4.764 MPa. As a result, the neural network results are comparable to the proposed model but with less accuracy since the RMSE and R-squared values are 3.122 MPa and 0.773 for the proposed model in this study.


### 5.3. Error Evaluation

In addition to the RMSE,R-squared, and MAPE criteria, other error evaluation criteria such as the correlation coefficient (R), mean absolute error (MAE), relative squared error (RSE), and relative root mean squared error (RRMSE) have been calculated as recommended by Khan et al. [56] and Li et al. [57], and the results are shown in Table 4. The R and R-squared are indications of the correlation between the proposed model and the actual values of the modulus of elasticity of low-calcium geopolymer concrete. The closer the values are to 1.0 of the R-squared for a model, the higher the variation by the independent variable is reflected [56]. When the R of a model is greater than 0.8, then the model is highly correlated [56]. The proposed model showed R and R-squared values equal to 0.88 and 0.773, respectively. As a result, the proposed model has a high correlation and represents a high percentage of variation. On the other hand, RMSE, MAPE, RSE, and RRMSE are evaluation criteria for the error of a model, and they reflect the deviation of the predicted value from the actual values [58]. The proposed model is very close to the actual values, as it has only 2.632 MPa and deviates only 3.122 MPa on average in terms of MAE and RMSE, respectively. The model accuracy is also expressed by the RRMSE criterion. When RRMSE <10%, a model is classified as excellent, 10 < RRMSE < 20 is good, 20 < RRMSE < 30 is fair, and RRMSE >30 is poor [58]. Interestingly, the proposed model demonstrated 15.5% of RRMSE, which is classified as good.Table 4
Statistical parameters to evaluate the goodness of fitness of the proposed model.
Statistical parameterValueR0.880R-squared0.773RMSE3.122MAE2.632RSE0.290MAPE0.150RRMSE0.155Moreover, the error of the proposed model can be evaluated by plotting the residuals against the fitted values of the dependent variable. The residuals are the differences between the experimental values and the predicted values obtained by the proposed model. The fitted values are the values predicted by the proposed model. It can be seen from Figure5 that the residuals are scattered in a symmetrical and balanced way about the horizontal axis. Moreover, the outer points of the residuals perform almost a circle shape. This distribution confirms that the residuals are independent of each other and randomly distributed around the centreline [54]. The determination coefficient, R-squared, is calculated to ensure this situation. It is found to be almost equal to zero (0.0002). That means that the residuals are independent of each other. Therefore, the goodness of fitness of the proposed model is very well.Figure 5
Distribution of the residuals against fitted values.


### 5.4. Testing the Proposed Model against Overfitting and Comparison with the Previous Models

The testing data, 8 samples separated earlier from the collected data, are employed to evaluate the proposed model against overfitting and its advantage over the previous models described below:Hardjito et al. [38]:(5)Ec=2707fc′+5300,whereEc and fc′ are in MPa.Diaz and Eleazar [30]:(6)Ec=580fc′,whereEc and fc′ are in MPa.Lee and Lee [31]:(7)Ec=5300fc′3,whereEc and fc′ are in MPa.Nath and Sarker [34]:(8)Ec=3.51fc′,whereEc is in GPa and fc′ is in MPa.Xie et al. [37]:(9)Ec=3.65fc′,whereEc is in GPa and fc′ is in MPa.Prachasaree et al. [27]:(10)Ec=1.55fc′3/4,whereEc is in GPa and fc′ is in MPa.The predicted values of modulus of elasticity are calculated by the models from equations (2) and (5)–(10), and the results are presented in Table 5.Table 5
The results of the comparison between the previous models and the proposed model.
Sample IDfc′ (Mpa)Ec (Gpa) exp.Equation (5)/exp.Equation (6)/exp.Equation (7)/exp.Equation (8)/exp.Equation (9)/exp.Equation (10)/exp.Equation (2)/exp.1943.4025.610.900.980.730.900.941.020.922620.2011.271.551.041.281.401.461.311.12U254.0032.100.780.980.620.800.840.960.88GPC16050.8025.100.981.170.781.001.041.181.06A35P0635.3021.401.000.960.810.971.011.050.93M-230.8015.441.321.161.081.261.311.311.15346.221.191.121.260.901.131.171.301.17436.1018.471.171.130.951.141.191.241.10Max.1.551.261.281.401.461.311.17Min.0.780.960.620.800.840.960.88The results show that equation (5) overpredicts the modulus of elasticity. The predicted value of the modulus of elasticity reached 55% of overprediction. The authors did not take into account the compressive strength class in the development of their model [38]. Equation (6) gives only a 26% overprediction. This result is because the author developed equation (6) depending on relatively large data. However, the error percentage value did not reach the best possible value because the author did not take into account the chemical composition. The author treated the class F fly ash and class C fly ash the same, while the first mentioned is low-calcium fly ash and the latest is high-calcium fly ash [30]. Equation (7) shows 38% of underprediction. The authors only included their experimental data, which were relatively a few. Moreover, their experimental data were mixed with low-calcium and high-calcium precursors [31]. Equation (8) overpredicts the testing data by up to 40%. Although the authors included some data from previous studies, the error percentage was still relatively high. It was noticed that the authors did not consider the compressive strength class [34]. However, their data were not sufficient to generate a model with a low percentage of error. Although equation (9) was introduced after equation (8), it did not reduce the error percentage. It overpredicts the testing data by up to 46% [37]. Equation (10) shows a 31% overprediction. The authors took into account a wide range of data from the literature to generate their model, but they included low-calcium based geopolymer and high-compressive strength data [27]. Therefore, they did not come up with the best possible value of the error percentage.All previous models present a considerably high percentage of error. Moreover, they tend to either overpredict or underpredict the modulus of elasticity values. On the contrary, the proposed model shows a considerably low percentage of error. The maximum overprediction is +17%, which is very close to the MAPE value (15%). On the other hand, the minimum prediction is −12% of the error percentage, which is very close to the MAPE as well; therefore, the proposed model gives overprediction and underprediction of 15% on average. This means that the proposed model evidently shows consistency in terms of overprediction and underprediction. Therefore, the upper and lower bands of ±15% are constructed versus the testing data. As shown in Figure6, all testing data were confined between the upper and lower bands. Therefore, the proposed model is valid for the application of future data and it is not overfitted.Figure 6
The proposed model and ±15% bands against the testing data andEc_Exp (GPa).


## 5.1. Initial Modelling

The first step of the modelling is to investigate the proper relationship that governs and fits the modulus of elasticity,Ec, in terms of the corresponding compressive strength, fc′. This step can be accomplished by running the analysing data in the SPSS software. The analysis is performed as shown in the flowchart in Figure 2. The results of the analysis are shown in Table 2. The results show that the highest value of the coefficient of determination, R-squared, corresponds to the power relationship, equal to 0.809, with initial values of constants equal to 1.094 and 0.809. The p values are ≤ 0.001; thus, the variables are statistically significant and related. Therefore, the governing equation of the modulus of elasticity is mathematically expressed as (1)Ec=a∗fc′b,where Ec is the modulus of elasticity in GPa, fc′ is the cylindrical compressive strength in MPa, and a and b are constants. This relationship complies with the form of ACI 318-14 and most of the other previous models in the literature [21].Figure 2
The initial modelling process in SPSS.Table 2
Model summary and parameter estimates (the dependent variable isEc_GPa, and the independent variable is Fc_MPa).
EquationR-squaredp valueConstant,ab1b2b3Linear0.772≤0.0012.9770.471Logarithmic0.723≤0.001−21.51311.869Inverse0.531≤0.00126.480−183.139Quadratic0.773≤0.0011.9970.544−0.001Cubic0.775≤0.0014.7540.1770.0120.000Compound0.769≤0.0016.2041.031Power0.808≤0.0011.0940.809S0.687≤0.0013.394-13.425Growth0.769≤0.0011.8250.030Exponential0.769≤0.0016.2040.030Logistic0.769≤0.0010.1610.970


## 5.2. Final Modelling

After selecting the proper governing equation and its initial values, the final model can be obtained by running these results in SPSS software by following the flowchart shown in Figure3. The results of the analysis are shown in Table 3. The final values of model parameters, a and b, are 1.088 and 0.815, respectively. The determination coefficient, R-squared, of the model is 0.773.Figure 3
The process of the final modelling in SPSS.Table 3
Results of final modelling for the dependent variableEc_GPa.
Iteration historybIteration no.aResidual sum of squaresParametersab1.0581.1081.0940.8091.1575.0861.0880.8152.0575.0861.0880.8152.1575.0861.0880.8153.0575.0861.0880.8153.1575.0861.0880.815Parameter estimatesParameterEstimateStd. error95% confidence intervalLower boundUpper bounda1.0880.3080.4701.705B0.8150.0760.6620.967ANOVAcSourceSum of squaresdfMean squaresRegression23632.371212946.106Residual523.8775710.089Uncorrected total24156.24959Corrected total2294.66158Derivatives are calculated numerically.aMajor iteration number is displayed to the left of the decimal, and the minor iteration number is to the right of the decimal. bRun stopped after 6 model evaluations and 3 derivative evaluations because the relative reduction between successive residual sums of squares is at most SSCON = 1.000E − 8. cR-squared = 1 − (residual sum of squares)/(corrected sum of squares) = 0.773.The final modelling formula is expressed as(2)Ec=1.088∗fc′0.815,where Ec is the modulus of elasticity in GPa and fc′ is the cylindrical compressive strength in MPa.Both confidence bands and prediction bands are constructed by applying equations (3) and (4), respectively, with a 95% probability [54, 55].(3)YPred.±t0.05∑Y−YPred2n−2⋅1n+X−X¯2∑X−X¯2,(4)YPred.±t0.051+∑Y−YPred2n−2⋅1+1n+X−X¯2∑X−X¯2,where YPred is the predicted value of the dependent variable, t is the critical value for 95% of probability, Y is the true value of the dependent variable, n is the number of samples, and X¯ is the mean value of the independent variable of X.The relationship between the compressive strength,fc′, and modulus of elasticity of low-calcium-based geopolymer concrete, Ec, is described in Figure 4 with the corresponding 95% confidence bands and 95% prediction bands. It can be seen that the 95% confidence bands are tight and narrow, confining the proposed model through a long distance with only a slight expansion at the edges. This means that the true regression line is very close to the proposed model line with a probability of 95% [55]. The proposed model also performs well in terms of prediction with a 95% probability. The 95% prediction bands confine all the experimental data with much inside space between the outer experimental data and the prediction bands.Figure 4
The relationship between compressive strength,fc′, and modulus of elasticity, Ec, of geopolymer concrete.The root mean squared error (RMSE) of the proposed equation is equal to 3.122. This value is much lower than the results of the most recent findings, Lee and Lee [31], Nath and Sarker [34], and Prachasaree et al. [27], which are equal to 8.900, 6.01, and 5.260, respectively. Therefore, the proposed model gives a 41% reduction of RMSE from the least error model [27]. The mean absolute percentage error (MAPE) of the proposed equation is equal to 15.00%, which is much lower than the results in the most recent findings, Lee and Lee [31], Nath and Sarker [34], and Prachasaree et al.[27], which are equal to 40.2%, 35.0%, and 31.4%, respectively. Therefore, the proposed model gives a 52.2% reduction in MAPE from the least error model, Prachasaree et al. [27].Moreover, the data have been run by neural networks in the SPSS software. The optimised neural network architecture was one hidden layer with two neurons for the multilayer perceptron neural network type. The activation function for the hidden layer was the hyperbolic tangent. The results of the neural network analysis have yielded anR-squared value of 0.752 and an RMSE value of 4.764 MPa. As a result, the neural network results are comparable to the proposed model but with less accuracy since the RMSE and R-squared values are 3.122 MPa and 0.773 for the proposed model in this study.


## 5.3. Error Evaluation

In addition to the RMSE,R-squared, and MAPE criteria, other error evaluation criteria such as the correlation coefficient (R), mean absolute error (MAE), relative squared error (RSE), and relative root mean squared error (RRMSE) have been calculated as recommended by Khan et al. [56] and Li et al. [57], and the results are shown in Table 4. The R and R-squared are indications of the correlation between the proposed model and the actual values of the modulus of elasticity of low-calcium geopolymer concrete. The closer the values are to 1.0 of the R-squared for a model, the higher the variation by the independent variable is reflected [56]. When the R of a model is greater than 0.8, then the model is highly correlated [56]. The proposed model showed R and R-squared values equal to 0.88 and 0.773, respectively. As a result, the proposed model has a high correlation and represents a high percentage of variation. On the other hand, RMSE, MAPE, RSE, and RRMSE are evaluation criteria for the error of a model, and they reflect the deviation of the predicted value from the actual values [58]. The proposed model is very close to the actual values, as it has only 2.632 MPa and deviates only 3.122 MPa on average in terms of MAE and RMSE, respectively. The model accuracy is also expressed by the RRMSE criterion. When RRMSE <10%, a model is classified as excellent, 10 < RRMSE < 20 is good, 20 < RRMSE < 30 is fair, and RRMSE >30 is poor [58]. Interestingly, the proposed model demonstrated 15.5% of RRMSE, which is classified as good.Table 4
Statistical parameters to evaluate the goodness of fitness of the proposed model.
Statistical parameterValueR0.880R-squared0.773RMSE3.122MAE2.632RSE0.290MAPE0.150RRMSE0.155Moreover, the error of the proposed model can be evaluated by plotting the residuals against the fitted values of the dependent variable. The residuals are the differences between the experimental values and the predicted values obtained by the proposed model. The fitted values are the values predicted by the proposed model. It can be seen from Figure5 that the residuals are scattered in a symmetrical and balanced way about the horizontal axis. Moreover, the outer points of the residuals perform almost a circle shape. This distribution confirms that the residuals are independent of each other and randomly distributed around the centreline [54]. The determination coefficient, R-squared, is calculated to ensure this situation. It is found to be almost equal to zero (0.0002). That means that the residuals are independent of each other. Therefore, the goodness of fitness of the proposed model is very well.Figure 5
Distribution of the residuals against fitted values.


## 5.4. Testing the Proposed Model against Overfitting and Comparison with the Previous Models

The testing data, 8 samples separated earlier from the collected data, are employed to evaluate the proposed model against overfitting and its advantage over the previous models described below:Hardjito et al. [38]:(5)Ec=2707fc′+5300,whereEc and fc′ are in MPa.Diaz and Eleazar [30]:(6)Ec=580fc′,whereEc and fc′ are in MPa.Lee and Lee [31]:(7)Ec=5300fc′3,whereEc and fc′ are in MPa.Nath and Sarker [34]:(8)Ec=3.51fc′,whereEc is in GPa and fc′ is in MPa.Xie et al. [37]:(9)Ec=3.65fc′,whereEc is in GPa and fc′ is in MPa.Prachasaree et al. [27]:(10)Ec=1.55fc′3/4,whereEc is in GPa and fc′ is in MPa.The predicted values of modulus of elasticity are calculated by the models from equations (2) and (5)–(10), and the results are presented in Table 5.Table 5
The results of the comparison between the previous models and the proposed model.
Sample IDfc′ (Mpa)Ec (Gpa) exp.Equation (5)/exp.Equation (6)/exp.Equation (7)/exp.Equation (8)/exp.Equation (9)/exp.Equation (10)/exp.Equation (2)/exp.1943.4025.610.900.980.730.900.941.020.922620.2011.271.551.041.281.401.461.311.12U254.0032.100.780.980.620.800.840.960.88GPC16050.8025.100.981.170.781.001.041.181.06A35P0635.3021.401.000.960.810.971.011.050.93M-230.8015.441.321.161.081.261.311.311.15346.221.191.121.260.901.131.171.301.17436.1018.471.171.130.951.141.191.241.10Max.1.551.261.281.401.461.311.17Min.0.780.960.620.800.840.960.88The results show that equation (5) overpredicts the modulus of elasticity. The predicted value of the modulus of elasticity reached 55% of overprediction. The authors did not take into account the compressive strength class in the development of their model [38]. Equation (6) gives only a 26% overprediction. This result is because the author developed equation (6) depending on relatively large data. However, the error percentage value did not reach the best possible value because the author did not take into account the chemical composition. The author treated the class F fly ash and class C fly ash the same, while the first mentioned is low-calcium fly ash and the latest is high-calcium fly ash [30]. Equation (7) shows 38% of underprediction. The authors only included their experimental data, which were relatively a few. Moreover, their experimental data were mixed with low-calcium and high-calcium precursors [31]. Equation (8) overpredicts the testing data by up to 40%. Although the authors included some data from previous studies, the error percentage was still relatively high. It was noticed that the authors did not consider the compressive strength class [34]. However, their data were not sufficient to generate a model with a low percentage of error. Although equation (9) was introduced after equation (8), it did not reduce the error percentage. It overpredicts the testing data by up to 46% [37]. Equation (10) shows a 31% overprediction. The authors took into account a wide range of data from the literature to generate their model, but they included low-calcium based geopolymer and high-compressive strength data [27]. Therefore, they did not come up with the best possible value of the error percentage.All previous models present a considerably high percentage of error. Moreover, they tend to either overpredict or underpredict the modulus of elasticity values. On the contrary, the proposed model shows a considerably low percentage of error. The maximum overprediction is +17%, which is very close to the MAPE value (15%). On the other hand, the minimum prediction is −12% of the error percentage, which is very close to the MAPE as well; therefore, the proposed model gives overprediction and underprediction of 15% on average. This means that the proposed model evidently shows consistency in terms of overprediction and underprediction. Therefore, the upper and lower bands of ±15% are constructed versus the testing data. As shown in Figure6, all testing data were confined between the upper and lower bands. Therefore, the proposed model is valid for the application of future data and it is not overfitted.Figure 6
The proposed model and ±15% bands against the testing data andEc_Exp (GPa).


## 6. Conclusions

A comprehensive regression analysis of the modulus of elasticity of low-calcium-based geopolymer concrete has been conducted on 59 different mix-design samples and tested by other 8 different mix-design samples; all are collected from the peer-reviewed literature based on firm assumptions such as classifying the geopolymers as low-calcium and specifying the strength class and density of concrete. Therefore, the following conclusions are drawn:(i)
The proposed model performs very well in predicting the modulus of elasticity when taking into account the chemical composition in terms of low- and high-calcium contents and compressive strength class. It reduced RMSE from ±5.260 GPa to ±3.122 GPa and MAPE from 31.4% to 15.0% compared to the best performance model found in the literature described by Prachasaree et al. [27]. Therefore, the proposed model gives a 41% and 52.2% reduction of RMSE and MAPE, respectively.(ii)
The proposed model has a high correlation with the actual values since the value ofR (0.88) > 0.8.(iii)
The goodness of fitness of the proposed model is characterised as good since its RRMSE (15.5) < 20.0%.


## 7. Future Recommendations

This comprehensive regression analysis study has yielded a reliable proposed model to predict and visualise the relationship between the modulus of elasticity of low-calcium-based geopolymer concrete and its compressive strength. It is recommended to utilise the proposed model in the prediction of the modulus of elasticity if and only if the basic assumptions of deriving the proposed model equation, which are stated in Section3, are met.



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*Source: 4528264-2022-05-19.xml*