Mathematical model for the prediction of cement compressive strength at the ages of 7 and 28 days within 24 hours
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1 Materials and Structures / Matériaux et Constructions, Vol. 36, December 2003, pp Mathematical model for the prediction of cement compressive strength at the ages of 7 and 28 days within 24 hours G. F. Kheder 1, A. M. Al Gabban 2 and S. M. Abid 3 (1) University of Mustansiriyah, Baghdad P.O. BOX 14150, Iraq (2) National Center for Constructional Laboratories, Iraq (3) University of Diyala, Iraq ABSTRACT In this study 450 cement mortar cubes were cast from 50 different cement samples taken from 9 different cement factories, to develop a mathematical model that can predict Portland cement compressive strength at ages 7 and 28 days within 24 hours only. This is in order to save time and expense, that is lost in waiting for such a long period, and for quality control assurance for both produced cement (in cement factories), and concrete mixes in constructions. In addition, attention has been made on the right choice of variables involved in this model, especially the characteristics of the cement itself (phase composition and fineness). In addition, an attempt has been made to use other variables that are believed to affect compressive strength of Portland cement as the minor oxides MgO, SO 3 and soundness. Other variables obtained from chemical analysis of the cement as LOI, IR, and LSF were also included in the model. The most important thing in this study is to get use of the concept of using early age strength to predict Portland cement strength at later ages for the first time. An attempt was made to combine both accelerated strength testing (as an early strength and UPV of cement mortar specimens), with the characteristics of the cement mentioned above, in predicting the compressive strength of cement. It was found that the accelerated strength yields good and high correlation with the compressive strength of cement, especially at the age of 28 days. In this work too, the importance of the ultrasonic pulse velocity (UPV) and mortar density were evident and the usefulness of using these variables in predicting the compressive strength of the cement was proved (because of fixing most of the factors affecting this property). Thus, it is possible to have good results that can be used in the prediction of compressive strength of cement. It was found that using each of the accelerated compressive strength f acc, UPV and density of the mortar cubes yielded high correlation with the compressive strength than any of the other variables. Different combinations of variables were introduced into the model, in order to choose the variables that can significantly predict the cement compressive strength. In this work, it was possible to obtain a model that can predict the cement strength with standard errors of only and MPa and coefficients of correlation of and 0.928, for cement strengths at 7 and 28 days respectively. RÉSUMÉ Dans cette étude, 450 cubes de mortier de ciment ont été coulés à partir de 50 échantillons différents de ciment provenant de 9 usines cimentaires différentes, dans le but de développer un modèle mathématique capable de prévoir la résistance à la compression du ciment Portland à des âges de 7 et 28 jours, seulement en l espace de 24 heures. Le but est de gagner du temps et de l argent, à cause de la très longue période d attente, et d assurer un meilleur contrôle de qualité à la fois pour le ciment produit (dans les usines cimentaires), et pour les mélanges de bétons dans les constructions. En outre, il faut faire attention à bien choisir les variables impliquées dans ce modèle, notamment les caractéristiques du ciment lui-même (composition de phase et finesse). On a également essayé d utiliser d autres variables qui sont censées affecter la résistance à la compression du ciment Portland comme les oxydes mineurs MgO, SO 3 et la solidité. D autres variables obtenues de l analyse chimique du ciment comme LOI, IR et LSF ont aussi été inclues dans le modèle. La chose la plus importante dans cette recherche est de faire usage du concept de résistance au jeune âge pour prévoir la résistance du ciment Portland à des âges plus avancés pour la première fois. On a essayé de combiner les essais de caractérisation de la résistance accélérée (comme une résistance au jeune âge et la vitesse de l impulsion ultrasonore des échantillons de mortiers de ciment), avec les caractéristiques du ciment mentionnées ci-dessus, prévoir la résistance à la compression du ciment. Il s est révélé que cette résistance accélérée offrait une forte corrélation avec la résistance à la compression du ciment, surtout à l âge de 28 jours. Dans ce travail également, l importance de la vitesse de l impulsion ultrasonore et la densité du mortier étaient évidentes et l utilité d employer ces variables pour prévoir la résistance à la compression du ciment a été prouvée (pour déterminer la plupart des facteurs affectant cette propriété). Ainsi, il est possible d obtenir de bons résultats qui peuvent être utilisés pour prévoir la résistance à la compression du ciment. Il a été trouvé que la résistance à la compression accélérée f acc, la vitesse de l impulsion ultrasonore et la densité cubes de mortier offraient une plus forte corrélation avec la résistance à la compression que toute autre variable. Différentes combinaisons de variables ont été introduites dans le modèle, afin de choisir les variables pouvant prévoir de façon significative la résistance à la compression du ciment. Il a été possible dans cette étude d obtenir un modèle capable de prévoir la résistance à la compression du ciment avec des erreurs normales de seulement 1,887 et 1,904 MPa et des coefficients de corrélation de 0,903 et 0,928, pour des résistances à 7 et 28 jours respectivement /03 RILEM 693
2 Kheder, Al Gabban, Abid 1. INTRODUCTION Cement, like any other construction materials has a number of specifications that have been placed all over the world. The object of these specifications has been to fix values of certain readily determined properties of cement, which are found to be satisfactory in practice, in order that inferior materials could be detected by their deviation from such standards [1]. The compliance of any produced cement with these specifications is the aim of both the producer and purchaser. The specifications, generally, include a statement of physical and chemical requirements. Among all, strength tests are prescribed by all specifications for cement, because mechanical strength of cement in the set and hardened condition is very important, and perhaps it is the most obviously required for structural use [2]. Specifications usually specify test method as well as age of test. Strength of cement, as specified by all the standards, is very important (from 1 to 28 days), because different cements are used for different purposes, and these cements differ in their early development of strength (early gain in strength). But, as early strength of Portland cement is important, strength at later ages is more important, because after all, it is this property which is relied upon in structural design of concrete as a construction material. The traditional 28 days standard test has been found to give general index of the overall quality (used in quality control process) and acceptance of concrete, and has served well for so many years. This age has already been introduced in most of the international standards, and some of which have adopted this age of test lately, for its importance, as a part of their specification (compulsory) after it was only optional, like the British and ASTM standards. 2. RESEARCH SIGNIFICANCE The presence of such model would possibly obtain the hard balance and equality between controlling the quality (quality control process) and economics (saving time and expense, i.e., this model could be used in cement factories in quality control process to provide a chance for the producer to make the necessary corrections during the process of cement production. Moreover, in construction to make the necessary adjustments on mix proportion used, to avoid situations where concrete does not reach the required design strength or by avoiding concrete that is unnecessarily strong. 3. MODELING OF THE COMPRESSIVE STRENGTH OF CEMENT Under the currently quicker pace of construction, there was a great need for more production of cement with persisting on the conformability of the quality of the produced cement with the standards and specifications. Neither waiting 28 days for such a test would serve the rapidity of construction, nor neglecting it would serve the quality control process of the produced cement (in cement factories for example), or concrete in large construction sites. Moreover, for the advantage of both producer and purchaser or user, rapid and reliable prediction of the results of 28 days strength test as early as possible, would be of satisfaction for all parties instead of waiting for the traditional 28 days results [3]. Rapid determination or prediction of the strength of cement could be attained by: - Suitable mathematical model with variables affecting the hydration and strength development of cement. - Accelerated strength testing results [4]. - Velocity of ultrasonic pulse and density of mortar or concrete. The attempts that have been made in the past used either one of the first or second approaches mentioned above. In this study, an attempt is made out, for the first time, to combine both approaches together with a non destructive testing, namely the ultrasonic pulse velocity (which is expected to be very effective because the materials and mix proportions are specified when testing the strength of the cement) in a suitable proposed model, capable of predicting strength of Portland cement at the age of 7 days as well as the age of 28 days. The variables used in the mathematical model in this work are: 1. Phase composition (the four main compounds of Portland cement C 3 S, C 2 S, C 3 A and C 4 AF). 2. Fineness of cement 3. Chemical analysis parameters (MgO, SO 3, LOI, LSF, IR) and soundness day accelerated strength of cement. 5. Density of accelerated strength mortar cubes. 6. The Ultra-sonic pulse velocity of accelerated strength mortar cubes. The basic concept of this model is that, it produces a reliable relationship between strength of cement and its own characteristics (depending on the above mentioned variables). The problem with such mathematical cement models is that they cannot be formed without internal contradictions, and, thus, with an adequate fit to experimental results [5]. In an attempt to eliminate these contradictions, the current proposed cement model has been introduced, with the fact that using strength at early ages to forecast strength at later ages increases the accuracy of the used equation. 4. EXPERIMENTAL WORK In this study, 50 different cements were tested. These cements were Ordinary Portland produced in nine different cement factories and taken randomly from monthly production of these factories; Table 1 shows the number of samples taken from each factory. Tables 2 and 3 respectively; show the limits of chemical Table 1 - Number of cement specimens taken from each factory No. of specimens Factory 8 F1 7 F2 7 F3 7 F4 6 F5 7 F6 4 F7 2 F8 2 F9 694
3 Materials and Structures / Matériaux et Constructions, Vol. 36, December 2003 and physical properties of the tested cements. The compliance of the cement was carried out according to the British Standard B.S 12:1989. The four main compounds in Portland cement C 3 S, C 2 S, C 3 A and C 4 AF were calculated for each cement specimen using Bogues equations. The results were also listed in Table 2 with the chemical properties. 4.1 Curing Normal curing Normal curing in water, for the cement mortar cubes, was made (after demolding) by immersing these cubes in water bath, with a curing temperature of 20 C for 7 and 28 days Accelerated curing In this study, the accelerated curing method that has been adopted was that of the British Standard B.S.1881 part 112:1983 method B [6]. The Cement mortar cubes were placed in a water bath with their molds for 20 hrs curing (water temperature 55 C). The reasons for choosing this method were: - Suitability with working hours (i.e., it dose not need overtimes). - Producing normal hydration products compared with high temperature methods (Boiling at 100 C). - It has no hazard effects like burning with boiling water or steam. Table 2 - Limits of chemical properties for the tested cements Factory C 3 S % C 2 S % C 3 A % C 4 AF % SO 3 % MgO % LOI % IR % F F F F F F F F F
4 Kheder, Al Gabban, Abid 4.2 Density The density of the accelerated strength mortar cubes (in g/cm 3 ) was found by weighting these cubes and dividing the values (mass in grams) by the volume of these cubes ( ) mm. 4.3 Ultra-sonic pulse velocity This test was carried out according to the British Standard BS 1881: part 203: 1986 [7], using the portable ultra-sonic non-destructive indicating tester (PUNDIT). All mortar cubes of accelerated curing; were tested after removal from the cooling water at the age of 24 hrs. The readings of the PUNDIT were recorded as the transit time in microseconds. The transducers used to read transit time were 150 khz arranged to give direct transmission. The choice of 150 khz transducers was based on what have been stated in the BS 1881: part 203: DISCUSSION 5.1 The effect of the accelerated compressive strength It has been mentioned earlier that accelerated testing of compressive strength of cement is based on accelerating the gain and development in strength by accelerating the reactions involved in the hydration process, thus producing strength that has reasonable correlation with strengths at later ages and could be used to predict these strengths. In this study, this concept has been used successfully and it was found that the accelerated 1 day compressive strength yields good correlation with compressive strength of cement at the age of 7 and 28 days (especially with 28 days compressive strength). The introduction of this variable was found to be important, because the elevated curing temperature (55 o C) aggregates the deleterious effects of free CaO, MgO and SO 3, thus improves the accuracy of the strength predicted values. The relationships between the accelerated strength and normal 7 and 28 days strengths are plotted in Fig. 1. Furthermore, it is very important to mention here that the correlation of the accelerated compressive strength with both strengths (7 and 28 days) was higher than the correlation between any of the variables used in this study. In fact the correlation coefficients were 0.79 and 0.90 for 7 and 28 days strengths respectively. This assures the possibility of using the accelerated strength as a variable in a mathematical model to predict the compressive strength of cement, as an effective variable and can serve for this purpose successfully. Fig. 1 - Relationship between accelerated compressive strength with 7 and 28 days compressive strength. hydration, as the chemical reactions proceed, the hydration will progress and there would be certainly a continuous formation of hydration products and thus, development in strength with reduction in porosity and consequently the density of the mass will be increased. It is said that the denser the concrete, the stronger it is and more durable (less permeable), thus less voids and formation of cracks. This could be more understandable from Fig. 2, it is obvious that the density of cement mortars increases with the increase in its compressive strength and with age. It is also important to mention here, that the introduction of density as a variable in the mathematical model to predict the cement strength is justified if we kept in mind that the efficiency of compaction of the cube specimen would largely affect the compressive strength of cement. Thus, accuracy or reproducibility of the compressive strength tests can be checked and controlled here, as this have been one of the main problems that may lead to scattered results of the compression test of cement in practice. 5.2 Effect of density The expected relationship between density of the mortar cubes and the compressive strength is a positive one, i.e. increasing density increases compressive strength. The most possible explanation is the one concerned with 5.3 The ultrasonic pulse velocity This is a long established, non-destructive test method, which determines the velocity of longitudinal (compressional) waves. This determination consists of measurement of the time taken by a pulse to travel a measured distance. The tests of 696
5 Materials and Structures / Matériaux et Constructions, Vol. 36, December 2003 Therefore, by fixing all the previously mentioned factors, the only variable remaining will be the cement, and particularly its intrinsic hydration and binding capabilities, which obviously means that the reliability of this method will be improved, and it can detect the internal structure of the cast mortar cubes, thus giving good results that can evaluate the cement compressive strength, This can be seen clearly from Fig. 3, in which the UPV acc was plotted with the cement strength at 1day accelerated, 7 and 28 days. From this figure, it can be seen that the UPV acc correlated excellently with the cement compressive strength. 6. THE MATHEMATICAL REGRESSION MODEL Fig. 2 - Relationship between density of cement mortar and compressive strength at (a) 1d accelerated (b) 7 d and (c) 28 d. ultrasonic pulse velocity have been used to evaluate concrete structures [8]. Several attempts have been made to correlate the compressive strength with the pulse velocity but this is subject to strict limitations. These limitations are [9, 10]: 1. Modulus of elasticity of the aggregate. 2. Aggregate content in the mix. 3. The type of the aggregate. 4. Mix proportions. 5. Moisture condition of the mortar or concrete. In this study, it was believed that the test of the ultrasonic pulse velocity could give excellent correlation with the compressive strength of cement. The reasons for this are: 1. The samples taken through this study have constant water/cement ratio and constant mix proportion. 2. Constant aggregate type (standard sand). 3. Curing under the same method, at same degree of temperature. All the regression methods are designed to fit functions that minimize the sum of the squares of the residuals between the data and the function. Such methods are termed least- squares regressions. Linear least-squares regression is used for cases where a dependent and independent variables are related to each other in a linear fashion [11]. The general form of such relationship is: Y= a o + a 1.X 1 (1) And when a dependent variable is a linear function of two or more independent variables, multiple linear regressions are utilized. Y= a o + a 1.X 1 + a 2.X 2 +a 3.X a m X m (2) This form of relationship was widely used by most of the researchers to relate strength of cement to its phase composition, i.e f t = a 0 +a 1 C 3 S+a 2 C 2 S+a 3 C 3 A+a 4 C 4 AF (3) Many attempts have been made to introduce additional variables other than main compounds [12-15]. For situations where the multiple dependencies are nonlinear logarithmic transformation can also be applied to this type of regression [10]: Log(y) = log(a o )+a 1.log(X 1 )+ a 2.log (X 2 ) +a 3.log(X 3 )+... a m.log (X m ) (4) This equation can be transformed back to a form that predicts the dependent variable (Y) by taking its antilogarithm to yield an equation of the type: Y= a o.x a1 1.X a2 2.X a3 am 3... X m (5) This equation is also called multivariable power equation. In engineering, material properties are often related to several independent variables, this functional dependency is best characterized by the equation above and is said to give more realistic results too. In this study, the multivariable power equation was found to be very suitable for predicting strength of Portland cement (as a dependent variable) from factors affecting this strength like; f acc, UPV acc, density, phase composition (C 3 S, C 2 S, C 3 A, C 4 AF), fineness (S s ), minor oxides (MgO, SO 3 ), soundness, LOI, LSF and IR. 697
6 Kheder, Al Gabban, Abid Fig. 3 - Relationship between ultrasonic pulse velocity at 1 day accelerated and (a) 1 day accelerated, (b) 7 days and (c) 28 days. Compressive strengths. Table 4 shows the relationship between the compressive strength at the age of 7 and 28 days with the selected variables that are going to be used in the proposed model. This relationship is represented by the correlation coefficient between each variable and each strength. From this table too, it can be seen that some the variables have significant correlation with the predicted strength at one age and not at the other. The highest significant correlations were with the accelerated compressive strength followed by the ultrasonic pulse velocity; these significant correlations were for both strengths at 7 and 28 days. Table 4 - Correlation between 7- and 28- day compressive strength with the variables used in the proposed model f 28 f 7 Variable -0.41* MgO -0.31* -0.33* SO LOI IR * LSF 0.28* 0.32* S s -0.31* Soundness * C 3 S * C 2 S * C 3 A -0.29* C 4 AF 0.90* 0.79* f acc 0.73* 0.67* 0.76* 0.70* UPV acc * Marked correlation is significant. These factors where considered to be independent variables in the regression. The power equation has been used to relate all these variables to 7 and 28 days strength, until getting the final and best form of the mathematical model. Combinations of variables were also used in such an alternative choices between the mathematical models, in the power equation to determine which of the variables to be used (or group of them) could give significant improvement to the fit of the equation. These variables or combinations of variables are. 1. C 3 S/C 2 S, C 3 A/C 4 AF,C 3 S+ C 3 A,C 3 S+C 2 S,C 3 A + C 4 AF 2. Initial setting time (IST) and final setting time (FST) When analyzing the results using the model that comprises combined variables, it was found that there is no significant improvement in correlation coefficient of the proposed model. Furthermore, the existing variables in the model (without the new variables) yielded good and reasonable results. Also, it is not proffered to load the prediction model with large number of variables, because it is preferred to use a model with lesser number of variables with higher possible accuracy to assure the rapid and easy use of the model. Considering the main compounds, it was found that using the main compounds of cement individually (i.e. C 3 S, C 2 S, C 3 A, C 4 AF) yielded better results than using the combinations mentioned above (when combining every two compounds in a certain form, for example, C 3 S/C 2 S etc), thus the model with the variables used was: f t = a o. f a1 acc. UPV a2 acc. a3. C 3 S a4. C 2 S a5. C 3 A a6. C 4 AF a7.s s a8. MgO a9. SO a10 3. Unsoundness a11. LSF a12. IR a13. LOI a14 (6) Tables 5 and 6 give the regression coefficients of the prediction model above for the prediction of 7 and 28 days 698
7 Materials and Structures / Matériaux et Constructions, Vol. 36, December 2003 Table 5 - Regression coefficients for the 7-day compressive strength prediction models Coefficient Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 a C 3 S C 2 S C 3 A C 4 AF S s MgO SO Soundness LOI IR LSF F acc UPV acc Density S.E C.C Table 6 - Regression coefficients for the 28 days compressive strength prediction models Coefficient Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 a C 3 S C 2 S C 3 A C 4 AF S s MgO SO Soundness LOI IR LSF F acc UPV acc Density S.E C.C compressive strength respectively, as well as the value of coefficient of correlation (C.C) and standard error of estimate (S.E) corresponding to each set of variables used in each model. In these tables several notes have to be pointed out: 1. f acc, UPV acc and density were taken together. 2. The components of the phase composition and fineness were used all together as they have been found to be interrelated and to affect strength of cement, although at different levels. 3. MgO, SO 3 and unsoundness were all taken together as they are factors causing deleterious expansion i.e. have a negative effect on the compressive strength. 4. LOI, IR and LSF were taken as one group. The most important thing to concentrate on, is the value of correlation coefficient and the standard error to determine which variable have significant improvement on the equation, and will result in accurately predicted strength. For example in the two tables below, the following points have been recorded: a) The effect of the four main compounds on the compressive strength of cement at the age of 7days was more than their effect on the 28 days compressive strength. This could be clearer from the value of the standard error for the equation used in the prediction. This looks very reasonable, as the effect of these compounds at early ages is more affected by their rate of hydration, more than their effect at later ages (especially the silicates). b) Effect of fineness (as expected) on the strength at 7days is more pronounced than that at the age of 28days, also, it can be seen that the introduction of fineness in model (3) reduces (slightly) the effect of the silicates. This could be because fineness is a strong effective factor that contributes to strength, especially at early age as 7 days. c) Using the value of each of 1day accelerated, the ultrasonic pulse velocity and the density was found to give excellent fit for the data. These three variables had a contribution on the correlation that exceeded that of the main cement compounds and fineness, especially at the age of 28 days. This can be seen clearly from the values of the coefficients of correlations of model (1) and model (3). The 699
8 Kheder, Al Gabban, Abid coefficients of correlations for these equations were versus at 7 days, and versus at 28 days respectively. This clearly shows the importance of the introduction of these factors in the regressions. d) Adding models (1) and (3) yielded significant improvement in the value of correlation coefficient especially at the age of 7 days (the value of the standard error also reduced to about half of its value). But it is important to note that introducing model (1) into model (3) i.e. combining these two models together decreases the effect of the four main compounds (the power of each of the four compounds and fineness in the model was decreased). The reduction in the effect of the four main compounds and fineness at the age of 28 days was more than that at the age of 7 days (the power of the fineness was turned from positive to negative value), this is attributed to fact that the effect of these factors (f acc, density, UPV acc ) on the 28 days compressive strength was higher than their effect on the 7 days compressive strength, while the effect of the four main compounds and fineness was on the reverse. e) Introducing other variables such as MgO, SO 3 and unsoundness was found to increase the value of the correlation coefficient and reduces the value of the standard error of the prediction equation. The same thing was observed when introducing LOI, IR and LSF. In the proposed model, a fact was proved, this was the role of the variables affecting strength of cement is not additive (especially the four main compounds), but there is an interaction between them, i.e., the introduction of one variable may enhance the effect of some variables and weaken the effect of other variables. Another important thing to note is the increase of the coefficient of correlation with the decrease in the value of the standard error from model (1) until model (6). Fig. 4 show the predicted versus observed compressive strengths of 7 and 28 days respectively, using Model (6) proposed by this study. In Fig. 4a, boundary limits of prediction for ± 10% from line of equality were drawn. It can be seen that 90% of the data used or obtained in this study were within ± 10% limits of the prediction, while only 10% of the data were out of these limits. In case of the 28 days compressive strength, it can be seen from Figure 4b that only 6% of the data were out of ± 10% limits, while 94% of the data were within ± 10%. 7. CONCLUSIONS The following conclusions were drawn from this study: 1. Two groups of regressions were proposed, these groups predict the compressive strength of cement at the ages of 7 and 28 days. In these regressions multivariable relationships were used. This type of regressions proved to yield better predicted results than the multivariable linear regressions used by other researches. Each group contained six models of regressions, each containing different Fig. 4 - Predicted versus observed compressive strength of cement: a) Observed versus predicted 7-day compressive strength of cement; b) Predicted versus observed 28-day compressive strength of cement. variables. These variables were divided into combinations, which included the following variables: * Accelerated compressive strength (f acc ), ultrasonic pulse velocity (UPV acc ) and density. * Chemical composition of main cement compounds and cement fineness. * MgO, SO 3, and soundness. * Insoluble residue (IR), loss on ignition (LOI), lime saturation factor (LSF). The total number of independent variables was (13). The best correlation coefficients obtained were and for 7 and 28 days respectively, these regressions included all the 13 variables. 2. In this work new variables were introduced in the prediction regressions for the first time, these variables were the 1day accelerated compressive strength of the cement and its corresponding ultrasonic pulse velocities and densities. These factors proved to be of equal importance as the chemical composition of the cement and fineness, as they were able alone to yield prediction regressions with accuracies equal or even better in predicting the compressive strength of Portland cement and especially at the age of 28 days. These three variables yielded excellent information on the cement binding capacity, internal structure of the mortar mix and accuracy of workmanship in 700
9 Materials and Structures / Matériaux et Constructions, Vol. 36, December 2003 preparation of the test specimens. This is because, the mix proportions were all fixed, and the only variable is the cement properties itself. 3. From the comparison between the regressions obtained, it can be said that including the insoluble residue IR, loss on ignition LOI and lime saturation factor LSF did not result in significant improvement in the correlation coefficient. Therefore these variables may not be included in the regression. The correlation coefficients obtained for this were and for 7- and 28-day compressive strength respectively, compared with and for regressions in which these variables were included with the other ones. NOTATIONS IR: Insoluble residue LOI: Loss on ignition LSF: Line saturation factor S.E: Standard error C.C: Coefficient of correlation REFERENCES [1] Lea, F.M., Chemistry of Cement and Concrete, third edition, 1970 (Edward Arnold Ltd, London). [2] Neville, A.M., Properties of Concrete, 4th edition (Pitman publishing, London, 1995). [3] Lapinas, R.A., Accelerated concrete strength testing by modified boiling method: concrete producer s view, SP 56, Accelerated Strength Testing, American Concrete Institute, ACI (1978) [4] Tsivilis, S. and Parrisakis, G., Mathematical model for the prediction of cement strength, Cement and Concrete Research 25 (1) (1995) [5] Popovics, S., Model for the quantitative description of the kinetics of hardening of Portland cement, Cement and Concrete Research 17 (5) (1987) [6] British standards, BS 1881: part 112, Methods of Accelerated curing of test cubes, [7] British Standards BS 1881: Part 203: Recommendations of measurement of velocity of ultrasonic pulse in concrete, [8] Tomsett, H.N., the practical use of ultrasonic pulse velocity measurements in the assessment of concrete quality, Magazine of Concrete Research 32 (110) (1980) [9] Malhotra, V.M., Non-destructive methods for testing concrete, Mines Branch Monograph 875, (Department of Energy, Mines and Resources, Ottawa, Canada), 1971, 55 p. [10] Jones, R., Non-Destructive Testing of Concrete, (Cambridge University, London, 1962). [11] Steven, C., Chapra, R. and Canale, P., Numerical Methods for Engineers with Personal Computer Applications, [12] Blain, R.L., Arni, H.T. and Defore, M.R., Compressive strength test of mortars, Section 7, Interrelations between cement and concrete properties, part 3 (Building Science Series 8, U.S. Department of Commerce, National Bureau of Standards, Washington, D.C.) 1968, [13] Alexander, K.M., The relationship between strength and the composition and fineness of cement, Cement and Concrete Research (1972) [14] Odler, I., Strength of cement (Final report), Mater. Struct. 24 (1991) [15] Aldridge, L.P., Estimating strength from cement composition, Proceedings of 7 th ICCC, Paris, 1980, Vol. 3, VI-83 to VI-86. Paper received: January 30, 2002; Paper accepted: August 27,
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