Development of a multiple linear regression model for predicting the 28-day compressive strength of Portland pozzolan cement

Transcription

1 Materials and Structures/Matériaux et Constructions, Vol. 32, March 1999, pp SCIENTIFIC REPORTS Development of a multiple linear regression model for predicting the 28-day compressive strength of Portland pozzolan cement G. CH. Kostogloudis 1, J. Anagnostou 1, CH. Ftikos 1, J. Marinos 2 (1) Lab. of Inorganic Materials Technology, Department of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Str., Zografou Campus, GR Athens, Greece (2) HERACLES General Cement Company, P. O. B. 3500, GR Athens, Greece Paper received: December 24, 1997; Paper accepted: April 21, 1998 A B S T R A C T A multiple linear regression model was developed for the prediction of the 28-day compressive strength of Portland Pozzolan Cement (PPC) containing Santorin Earth as an admixture. The independent variables of the model were: (1) the compressive strength at the age of 2 days, (2) the compressive strength after autoclave hydrothermal treatment at 2.07 N/mm 2 (300 psi) and 214 C for 3 h, (3) the C 3 S/C 2 S ratio of the clinker, (4) the insoluble residue of the cement, (5) the pozzolanic activity factor and (6) the C 3 A content of the clinker. The evaluation of the proposed model was performed by various statistical tests, all of which were successful. These tests included: multiple correlation, test of the significance of coefficients (t-test), estimation of confidence intervals for coefficients, test for outliers and unusual residuals, test for influential points, conditional sums of squares, R-squared and analysis of variance. There was very good agreement between the strength predicted by the multiple regression model and experimental results. R É S U M É Un modèle de régression multiple linéaire a été developpé pour la prédiction de la résistance en compression du ciment pozzolanique Portland, contenant de la Terre de Santorin comme mélange. Les variables indépendantes du modèle étaient : (1) la résistance en compression pour l âge de 2 jours, (2) la résistance en compression après un traitement de 3 heures dans un autoclave à 2,07 N/mm 2 (300 psi) et 214 C, (3) le rapport C 3 S/C 2 S du clinker, (4) la quantité d insolubles du ciment, (5) le facteur d activité pozzolanique et (6) la quantité de C 3 A du clinker. L évaluation du modèle proposé à été effectuée par une série d essais statistiques : une corrélation multiple, un essai de signification des coefficients (t-test), estimation des intervalles de confiance pour les coefficients, essai pour les substances étrangères et les résidus inhabituels, essai pour les points d influence, somme des moindres carrés conditionnels, carré et une analyse de variance. Un très bon accord entre la résistance prévue par le modèle de régression et les résultats expérimentaux, a été constaté. 1. INTRODUCTION Many efforts have been made to identify the relationship between various physicochemical and mineralogical parameters of cement clinker and the strength of cement [1]. Although such relationships have not been fully justified [2], many models have been proposed for the prediction of the compressive strength of Portland cement [1-11]. A carefully designed model can serve as a valuable tool during the production of cement and the control of its quality. However, the use of such models is restricted to the specific applications for which they are designed. In the case of blended cements, which have drawn much attention over the past two decades mainly due to energy saving reasons, little information is available concerning strength prediction [12]. In the present work, a multiple linear regression model was developed for the prediction of the 28-day compressive strength of Portland Pozzolan Cement (PPC) containing Santorin Earth as an admixture. The evaluation of the proposed model was performed by various statistical tests. 2. EXPERIMENTAL Seventeen industrial PPC samples were used for the development of the model. The natural pozzolan material used as an admixture for the production of PPC was Santorin Earth. The sampling was random, and it was performed during a period of three months at the place of production. The chemical analysis, potential mineral composition and ratios of the Portland clinker used for the production of PPC, are shown in Table 1. The fineness (Blaine), R90µm, R32µm, SO 3 content, insoluble residue (IR) and loss on ignition (LOI) of all PPC samples are also shown in Table 1. It should be noted that the IR of the PPC cement refers to the pozzolan content of the cement, and should not exceed 20%, according to Greek specifications /99 RILEM 98

2 Kostogloudis, Anagnostou, Ftikos, Marinos Table 1 Analysis of Portland clinker and Portland pozzolan cement (PPC) Sample number * The CaO-CaO f content was used for the calculation. Portland clinker chemical analysis (%) SiO Al 2 O Fe 2 O CaO MgO K 2 O Na 2 O SO CaO f Portland clinker potential mineral composition (%) C 3 S* C 2 S C 3 A C 4 AF Portland clinker ratios LSF* SR AR C 3 S/C 2 S PPC analysis Blaine (cm 2 /g) R90µ m (%) R32µ m (%) SO IR LOI Table 2 Compressive strength development, strength after autoclave hydrothermal treatment, and pozzolanic activity factor (PAF) of PPC Sample number Compressive strength of PPC (2, 7 and 28 days) S2 (N/mm 2 ) S7 (N/mm 2 ) S28 (N/mm 2 ) Compressive strength of PPC after autoclave hydrothermal treatment Sa (N/mm 2 ) day compressive strength of standard Portland cement/pozzolan mixture (S28sp), and PAF of PPC S28sp (N/mm 2 ) PAF The compressive strength of the PPC samples was measured at the ages of 2, 7 and 28 days, according to EN specifications, and the results are shown in Table 2. The compressive strength of PPC mortar specimens after hydrothermal treatment at 2.07 N/mm 2 (300 psi) and 214 C for 3h was measured. The results of this test are also illustrated in Table 2. The activity of the pozzolan used for the production of each cement was tested. The test involved the measurement of the 28-day compressive strength of a mixture of a standard high strength Portland cement of known strength (80%), and the material of which the pozzolanic activity was to be determined (20%). The pozzolanic activity factor (PAF) was then calculated by: 99

3 Materials and Structures/Matériaux et Constructions, Vol. 32, March 1999 S28sp S28ss (1) PAF = S28s S28ss where S28sp is the 28-day strength of the standard Portland cement (80%)/pozzolan (20%) mixture, S28ss is the 28-day strength of the standard Portland cement (80%)/AFNOR sand (20%) mixture (Blaine: 3810 cm 2 /g, R90µm = 6.8%, S28s s= 36.1 N/mm2), and S28s in the 28-day strength of the standard Portland cement (Blaine: 3820 cm2/g, R90µm=2%, S28s = 45.9 N/mm 2 ). The preparation of the specimens was performed according to EN specifications. The results on the measured S28sp and calculated PAF values are presented in Table DEVELOPMENT OF THE MULTIPLE REGRESSION MODEL 3.1 Formulation of the model The multiple linear regression model for the prediction of the 28-day compressive strength (S28) of PPC had 6 independent variables, and it is described by the following equation: S28 = b 0 + b 1 S2 + b 2 Sa + b 3 C 3 S/C 2 S (2) + b 4 IR + b 5 PAF + b 6 C 3 A where b i are the coefficients to be determined. The independent variables of the model were: (1) the compressive strength at the age of 2 days (S2), (2) the compressive strength after autoclave hydrothermal treatment at 2.07 N/mm 2 (300 psi) and 214 C for 3 h (Sa), (3) the C 3 S/C 2 S ratio of the clinker, (4) the insoluble residue of the cement (IR), (5) the pozzolanic activity factor (PAF), and (6) the C 3 A content of the clinker. The determination of the coefficients of the model and all statistical tests were performed using the EXE- CUSTAT statistical computer package [13]. 3.2 Multiple correlation Before the determination of the coefficients of the model it is necessary to ensure that the independent variables do not correlate with each other. In practice the question is whether there is a significant correlation among the independent variables. A table was constructed (Table 3) with the correlation coefficients between pairs of variables. The correlation coefficient measures the strength of the linear relationship between two variables on a scale of -1 to +1. The P values, which are included in Table 3, are used to test whether the coefficients are significantly different from zero. If the P value is less than 0.05, there is a significant correlation between the pair of variables with 95% confidence. As can be seen, P > 0.05 for all pairs, indicating that all variables are independent from one another. 100 Table 3 Correlation coefficients. Values in parentheses are P values for the hypothesis of zero correlation S2 Sa C 3 S/C 2 S IR PAF C 3 A S (0.6826) (0.4525) (0.7083) (0.0900) (0.9714) Sa (0.6826) (0.1661) (0.1644) (0.0875) (0.1880) C 3 S/C 2 S (0.4525) (0.1661) (0.2947) (0.3292) (0.1122) IR (0.7083) (0.1644) (0.2947) (0.4183) (0.3460) PAF (0.0900) (0.0875) (0.3292) (0.4183) (0.9700) C 3 A (0.9714) (0.1880) (0.1122) (0.3460) (0.9700) Table 4 Estimated coefficients for the multiple regression model and their confidence intervals Standard 95% confidence limits Estimate error t value P value Lower Upper Constant S Sa C 3 S/C 2 S IR PAF C 3 A R-squared = 97.76% Standard error of estimation = Mean absolute error = Estimation of coefficients, test of significance and confidence intervals The estimated coefficients for the multiple regression model are shown in Table 4. The P values correspond to tests of the hypotheses that the coefficients are equal to zero. Values of P less than 0.05 indicate statistically significant nonzero coefficients at a 95% confidence level. In Table 4, all coefficients have P < The standard error of estimation provides an estimate of the standard deviation of the residuals around the regression model, equal to Table 4 also shows the 95% confidence intervals for each of the coefficients in the fitted model. These intervals quantify the sampling variability in the estimation of the coefficients and are useful for determining how well each coefficient has been determined. Since none of the intervals includes 0, there is an actual linear relationship between the dependent variable and each of the independent variables, and so, all variables should be included in the model. 3.4 Test for outliers and unusual residuals Fig. 1 shows the Studentized residuals as a function of the predicted S28 values. The residuals show the difference between the actual values and the predictions,

4 Kostogloudis, Anagnostou, Ftikos, Marinos the residuals, shown in Fig. 2, can be used to judge whether the residuals could reasonably be considered to follow a normal distribution, and may also be helpful in detecting outliers. The residuals fall fairly well along a straight line, while no outliers can be observed. In Fig. 1, it can be seen that none of the residuals have Studentized values greater than 2 or less than -2. Consequently, no unusual residuals appear for this model. 3.5 Test for influential points Fig. 1 Plot of Studentized residuals versus predicted S28 values. A point may be considered as influential if it has an unusually large impact on the fitted model. This is measured through a statistic called leverage, which measures each point s influence on the coefficients of the fitted equation. The average leverage was calculated to be As none of the points has a leverage more than 2.5 times the average leverage, there are no influential points. 3.6 Conditional sums of squares Fig. 2 Normal probability plot of residuals. The conditional sums of squares, shown in Table 5, test the significance of each variable when added sequentially to the fit. These sums of squares are useful for quantifying the additional contribution of each variable after accounting for the contribution of those entered earlier. Since all variables have small P values (less than 0.05), they are all added significantly to the fit when they enter the model. S2 is the most significant variable, while the significance of Sa is also high. The contribution of the other four variables is far lower than that of the previous two. and the Studentized residuals, express each deviation in terms of how many deviations it is away from the fitted line. The Studentized residuals that were calculated, are based on the estimated residual standard deviation if the fitting were performed without that data value. This kind of residuals is particularly useful for detecting outliers (i.e. points that do not follow the same pattern as the others). As can be seen in Fig. 1, the plot appears reasonably random, and none of the residuals is noticeably distinct from the others. The normal probability plot of Table 5 Conditional sums of squares Source Sum of squares D.F. Mean square F-ratio P value S Sa C 3 S/C 2 S IR PAF C 3 A Model Analysis of variance The analysis of variance (ANOVA) table (Table 6) decomposes the total variability in S28 into two components: one due to the regression, and the second due to deviations around the fitted model. The R-squared statistic, based upon the ratio of the model sum of squares divided by the total (corrected) sum of squares, indicates that the model accounts for 97.76% of the variation of S28. The mean squared error estimates the variance of the deviations around the model to be equal to Since the P value corresponding to the F-ratio is less than 0.05, the model as a whole is statistically significant. Table 6 Analysis of variance Source Sum of squares D.F. Mean square F-ratio P value Model Error Total (corr.)

5 Materials and Structures/Matériaux et Constructions, Vol. 32, March DISCUSSION The multiple regression model that was developed in the present work, was shown to satisfy all the statistical tests used for its evaluation. Fig. 3 shows a plot of the observed (experimental) values of S28 versus the values predicted from the fitted model. A line with slope equal to 1 is also included. All points of the plot fell very close to this line, while they show approximately the same scatter around the line everywhere. Therefore, the proposed model has a very good predicting ability. The choice of the independent variables of the model was based on statistical as well as physical considerations. A large number of variables with physical importance was considered, and those satisfying certain statistical criteria were actually selected. The variable S2 is the most significant among those used, and its presence in the model is essential. The lack of a satisfactory mechanism for the strength development of blended cements [14], does not yet permit the substitution of S2 with other physicochemical parameters. The variable Sa is also of high significance. This was expected, since the strength development under conditions of accelerated hardening provides a good measure of the actual hydraulic behavior of the cement. The C 3 S/C 2 S ratio is an important variable, which has also been used in other works [3]. It incorporates trends arising from the variation of other important parameters that affect the C 3 S and C 2 S contents. These parameters include the CaO f content, the alkalis occurring in solid solution with the above calcium silicate phases, and the degree of burning of the clinker. The IR is also an important variable, since it is directly related to the concentration of pozzolan in the blended cement. The negative dependence of S28 on IR can be explained by the fact that an increase of the pozzolan content is known to reduce the compressive strength of blended cements. The PAF is a measure of the contribution of the pozzolan in the strength development of the cement, and it is also included as a variable in the model. However, it should be pointed out that the need of knowing this coefficient may induce some delay, as compared to the rest of the input parameters. The C 3 A content is the less significant variable among those used. It reflects the composition of the liquid phase of the clinker, and incorporates factors related to the burning process and the cooling rate. 5. CONCLUSION The developed multiple regression model was shown to be able to predict with adequate accuracy the 28-day compressive strength of Portland pozzolan cement (PPC) containing Santorin Earth as an admixture. The model was evaluated by various statistical tests, which were all successful. The usefulness of the proposed model to the cement industry, is based on its ability to make predictions on the 28-day strength of PPC, only 2 days after its production. This gives the opportunity to cement engineers to make the necessary adjustments to improve the quality of the product in due time. Fig. 3 Plot of observed versus predicted values of S28. ACKNOWLEDGMENTS The authors wish to thank HERACLES General Cement Company for providing the industrial cement samples used in this work. REFERENCES [1] Bruggemann, H. and Brentup, L., Relationship between cement strength and the chemico-mineralogical parameters of the clinker, Translation ZKG, 1/90, [2] Aldridge, L. P., Estimating strength from cement composition, Proceedings of the 7th International Congress on the Chemistry of Cement, Vol VI, Paris, 1980, [3] Alexander, K., Taplin, J. and Wardlaw, J., Correlation of strength and hydration with composition of Portland cement, Proceedings of the 5th International Congress on the Chemistry of Cement, Vol III, Tokyo, 1968, [4] Alexander, K., The relationship between strength and the composition and fineness of cement, Cem. Concr. Res. 2 (1972) [5] Popovics, S., Calculations of strength development from the compound composition of Portland cement, Proceedings of the 7th International Congress on the Chemistry of Cement, Vol VI, Paris, 1980, [6] Dragicevic, L. J., Prognosis of the physical-mechanical characteristics of Portland cement, Proceedings of the 8th International Congress on the Chemistry of Cement, Vol II, Rio de Janeiro, 1986, [7] Hargrave, R., Venkateswaran, D., Deshmukh, V. and Chatterjee, A., Quantification of OPC clinker microstructure - An approach for prediction of cement strength, Proceedings of the 8th International Congress on the Chemistry of Cement, Vol II, Rio de Janeiro, 1986, [8] Popovics, S., Model for the quantitative description of the kinetics of hardening of Portland cement, Cem. Concr. Res. 17 (1987) [9] Relis, M., Ledbetter, W. and Harris, P., Prediction of mortarcude strength from cement characteristics, Cem. Concr. Res. 18 (1988) [10] Philippou, Th., Marinos, J. and Kostakis, G., On the strength behavior and phase composition of low lime saturation clinkers, Ibid. 18 (1988) [11] Petrasinovic, L. and Duric, M., Predicting the compressive strength of Portland cement and optimizing its raw mixture composition, Ibid. 20 (1990) [12] Ftikos, Ch., Study of the action of Santorin Earth during the hydration of cements, PhD Thesis (Athens, 1977). [13] EXECUSTAT (PWS-Kent, Boston, 1991). [14] Baragano Coronas, J., Vasquez de la Torre, P. Y., Difficulties in the manufacture of blended cements, Translation ZKG, 11/86,