Title no. 78-10 Generalization of the Abrams' Law - Prediction of Strength Development of Concrete from Cement Properties by Sandor Popovics The paper presents a generalized form of the well-known Abrams' formula f = AiJJw~< for the strength versus water-cement ratio relationship. This generalization permits the prediction of the strength development of concrete regardless of the fineness and composition (type) of the portland cement and the age of the concrete at testing within the limits of 1 day and 1 year. Formulas are offered for the estimation of the compressive and flexural strengths of concrete at any age within these time limits from the water-cement ratio as well as from the experimentally obtained 1-day strengths. A rationale for the new formulas is also offered pointing out that they represent a catalytic action by the C,A on the calcium silicates of the portland cement during the hardening. It is demonstrated that the average difference between the calculated and measured 28-day strengths is approximately 10 percent in most cases; that is, the new formulas are supported well by a wide range of experimental results. Keywords: age-strength relation; compressive strength; concretes; C,A; C,S; flexural strength; mix proportioning; models; portland cement compound composition; water-cement ratio. A relationship between concrete strength and watercement ratio was published by Abrams in 1918.' It was: f= A (1) where f wlc A and B strength of concrete water-cement ratio empirical parameters that are independent of the strength and water-cement ratio of the concrete but may be a function of the units, type of cement, aggregate and admixture used, methods of making, curing and testing the specimen, age at testing, and type of strength This simple empirical formula has been known as "Abrams' Law" and has become one of the pillars of concrete technology. However, the apparent simplicity of the formula is deceptive. The difficulty is that a ACI JOURNAL I March-April 1981 different pair of A and B values is needed for each case where any factor affecting the concrete strength changes. Such factors can be the age at testing, the type and fineness of the cement, the curing temperature, and many others. In other words, the knowledge of many A and B parameters is needed to cover the range of practice by Eq. (1). Although numerical values for these parameters have been worked out for special cases, such as for ASTM Types I and III cements and for ages from 3 days through 1 year/ it seems desirable from both theoretical and practical points of view to generalize Eq. (1) so that it would include explicitly at least some of the strength-affecting factors beside the water-cement ratio. This then would reduce the number of A and B parameters needed for the prediction of concrete strengths. The purpose of this paper is to introduce such a generalization. More specifically, the new formulas presented are valid for the prediction of concrete strengths from the water-cement ratio for portland cements of different compound compositions (types) and fineness within the age limit of 1 day through 1 year with the same A and B values. Otherwise, the generalized formulas are subject to the same restrictions as the Abrams' law; for instance, that the quality of the aggregate be satisfactory. THE NEW FORMULAS The following generalized form of Eq. (1) is offered for the estimation of the compressive strength of normal weight, non-air-entrained concretes: (2) Received June 9, 1980, and reviewed under Institute publication policies. Copyright © 1981, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion will be published in the JanuaryFebruary I982 ACt JOURNAL if received by Oct. 1, I981. 123 Sandor PopoYics, FACI, is a professor of civil engineering at Drexel University, Philadelphia, Pa. He received his PhD from Purdue University and has been engaged in teaching, research, and consulting work during the past 30 years. Dr. Popovics is the author of numerous papers and three books, including the recently published Concrete Making Materials. He is currently a member of ACI Committees 211, Proportioning Concrete Mixes, and 309, Consolidation of Concrete. where L. estimated compressive strength of concrete in psi at the age of t days I5,500 psi (I07 MPa), when determined A on 6 x I2 in. (15 x 30 em) cylinders that were made, cured, and tested according to ASTM C I92 and C 39 6.4 for the same conditions as above B water-cement ratio by weight wlc specific surface of the cement in quesS, and s. tion and that of a typical ASTM Type I cement, respectively, cm 2 I g C 3 S content of the portland cement, perp cent/IOO age of testing, day b, and b 2 = rate parameters which are independent of the CJS and C2S contents of the cement, and the strength as well as the age of the concrete, but they may be a function of the CJA content and fineness of the cement, curing temperature, minor constituents, water-cement ratio, admixtures, and any other factor that influences the strength development, type of strength, and test method, 1/day The values of I5,500 psi (I07 MPa) for A and 6.4 for Bin Eq. (2) were established earlier from the analysis of several independent test series. 2 They may change if there is a change in: the units; type of aggregate; methods of making, curing, and testing the specimen; type and/ or quantity of admixture; or the type of strength. For instance, for air-entrained concretes, with cement and air contents used for structural purposes, the value of A is approximately I4,000 psi (96. 7 MPa) in Eq. (2), but otherwise the formula remains unchanged. Several pertinent values for the b parameters are shown in Table 1. For the flexural strength of normal weight, non-airentrained concretes, the following formula is valid: (3) where t; is the estimated flexural strength of concrete in psi at the age of t days when determined on beams having 6 x 6 in. (15 x I5 em) cross section, and when made, cured, and tested with third-point loading according to ASTM C I92 and C 78. The other symbols are identical with the symbols of Eq. (2). Several pertinent values for the flexural b rate parameters are given again in Table I. These were obtained by the traditional method of kinetics of chemical reactions, except that the form of the equations made it necessary to use stepwise approximation for optimization instead of the usual linear regression. The A and B values for Eq. (3) were established again earlier from the analysis of several independent test series. 2 The major difference between Eq. (2) and (3) is that the latter does not contain the square-rooted form of specific surface. This reflects the experimentally demonstrated fact; namely, that the cement fineness has no sizable effects on the flexural strength of concrete at later ages. The limits of validity of Eq. (2) and (3) are those of Eq. (I), as well as the age from I day through 1 year, and compound composition and fineness as they occur in the five ASTM standard portland cement types as defined in ASTM C I50. EXPERIMENTAL JUSTIFICATION OF EQ. (2) AND (3) Fig. I shows the calculated as well as the experimental results of a series of compressive strength tests performed with five portland cements representing the ASTM five standard types. The tests were performed on 6 x I2 in. (15 x 30 em) cylinders made, cured, and tested according to ASTM C 192 and C 39. Data concerning the reported compositions of the concretes are: Nominal cement content: approximately 5 bag/ cu yd = 470 lb/ cu yd (280 kg/m 3 ). Water-cement ratio: w/c = 0.49 by weight. For the Type III cements: S, = 2730 cm 2 /g (Turbidimeter). For the other cements: Ss = S. = I710 cm 2/ g (Turbidimeter). Table 1 - b parameters of Eq. (2) and (3) for selected concretes 3 For specimens made, cured, and tested according to ASTM standards Approximate w/c by weight b, in Jlday b, in !/day Eq. No. Compressive strength, 6 bag/cu yd* (modified cube test) 0.43 0.40 S.ISo (0.002 C,A + 0.01) S.ISo (4) Flexural strength, 6 bag/ cu yd • (third-point load) 0.43 0.70 S.ISo (0.001 C,A ~ (5) Compressive strength, 4Y2 bag/cu yd* 0.54 0.30 S.ISo 0.005 C,A S.!So (6) Flexural strength, 4Y2 bag/cu yd• 0.54 0.50 S.ISo 0.005 C,A S.!So (7) Compressive strength, 3 bag/cu yd* Flexural strength, 3 bag/cu yd* 0.80 0.80 0.15 S.ISo 0.25 S.ISo 0.003 C,A S.!So 0.004 C,A S,/So (9) Type of test 'I bag/cu yd 124 = 94 lb/cu yd = 0.02) S.ISo (8) 56 kg/m' ACI JOURNAL I March-April 1981 IO.-,----,---,------.----.------.-~ 70 60 70 ~ ~ 0 ~ 8 a.. ~ .r:7 c;, c c;, c iii iii Q) "' 0 ·~ +--+ (Type I) u----"• ( Type nl o ····---o (Type Ill) x-x (Type IV) t,----.". (Type V) "'"'0. 20 E 0 (.) Age, ~ 5 - > ·~4 0 L ~ -+-+Cement No.11 (Type I) u----n " No.21 ( ;, I!) O:>·······<• No. 31 ( " Ill) _____j___..__ x-x " No.41 ( " IV) 3- (.) 2 10 Points in the figure represent experimental values reported by Gonnerman; 4 lines represent Eq. (2) with Eq. (4). * Fig. 2 shows the calculated as well as the experimental results of another series of compressive strength tests again with five portland cements representing the five standard types. The cements were selected from the 29 cements included in the long-time studies (L TS) of the Portland Cement Association (PCA). The tests performed were 6 in. (15 em) modified cube tests according to ASTM C 116. The specimens were made and cured also according to pertinent ASTM standards. The cement compositions were reported by Lerch, s the concrete compositions and strength results by Klieger. 6 Data concerning the reported compositions of the concrete are: Nominal cement content: 6 bag/cu yd = 565 lb/cu yd (335 kg/m 3 ). For the Type III cement: Ss = 2600 cm 2 /g (Turbidimeter). For the other cement: Ss = Sa = 1800 cm 2 /g (Turbidimeter). Water-cement ratio: as needed to obtain 2- to 3-in. (5- to 7.5-cm) slump. Points represent the experimental values while the lines represent values calculated from Eq. (2) with A = 17,000 psi (117 MPa) and with Eq. (4). The reason for increasing the value of A in Eq. (2) to 17,000 psi is that the compressive strength measured by the modified cube test is about 10 percent greater than the strength of the same concrete measured on standard cylindrical specimens. 7 Fig. 3 shows the calculated as well as the experimental results of a third series of compressive strength tests performed with five portland cements representing the five standard ASTM types. The tests were performed on 6 x 12 in. (15 x 30 em) cylinders made, ACI JOURNAL I March-April 1981 No.51 ( " t,---~'".. V) oL-~--~----L------L----~----~--~ 3 7 days Fig. I - Compressive strengths of concretes as a function of cement type and age at testing, I. Points represent experimental values reported by Gonnerman,4 and Jines represent values calculated from Eq. (2) with Eq. (4). "'> ·u; en Q) iL-~--~--~----~~~9~0----~~~~73~0° t .s::! ~6 t 28 90 365 730 ~ c. 20 E 0 (.) 10 o Age, days Fig. 2 - Compressive strengths of concretes as a function of cement type and age at testing, II. Points represent experimental values reported by Klieger, 6 and lines represent values calculated from Eq. (2) with A = 17,000 psi and Eq. (4). 70 10 9 ~ 8 .<::. c;, 7 +--+ Type I Cement D----n " II ···----······0 » III x-x » IV :-...----{'.., " 60 ~ 50 v 40 iii "' 4 ~ 20 c. E 0 0 (.) 2 10 c- I 0 ~ iii en Q) !5. 3 E (.) -;, "'> 30 ·;;; Q) > .<::. c c ~ ·u; 0 a.. 3 28 7 t Age, 90 365 0 730 days Fig. 3 - Compressive strengths of concretes as a function of cement type and age at testing, III. Points represent experimental values reported by the U.S. Bureau of Reclamation, 8 and lines represent values calculated from Eq. (2) with Eq. (4). cured, and tested according to ASTM C 192 and C 39. The test results were reported by the U.S. Bureau of Reclamation. 8 The reported cement content of the concrete was 6 bag/cu yd = 565 lb/ cu yd (335 kg/m 3 ). The specific surfaces of the cements and the watercement ratios of the concretes are not reported, but it is reasonable to assume that the same situation exists here that was reported for the L TS cements in Fig. 2; namely, that (a) the specific surface of the Type III cement was 60 percent more than that of the other cements, and (b) the concrete made with the Type III cement required about 15 percent higher water-cement *Eq. (4) through (9) are presented in Table I. 125 70 10 Type I Type III Type IV Type III • CaCI 2 _ 60 '! _________.__·--~---- _____ , ---~- 1000 ·u; & 900- -,-- ~ c. 2 50 .s::: ----~ 40 ~ ~ Vl 30 -~Vl Vl ~ 20 - t-t f (J---c 0 u 100 ~~-~-~7~--~~--790~-~3~65~~~: 0 3 Cement No.I I (Type I) No.21 ( ' ' II ) o- ... ·· ·<> NC131 ( " Ill) X-)( NoAI ( , IV) ·•---~• No.51 ( , V) 7 28 t Age, days 90 10 365 730 Age, days Fig. 4 - Compressive strengths of concretes as a function of cement type and age at testing, IV. Thin lines represent experimental values reported by Orchard,9 and thick lines represent values calculated by Eq. (2) with A = 17,000 psi and B = 7.9, and with Eq. (6). ratio than the one needed for the same consistency of the concrete made with the Type I cement. Points represent the experimental values, and the lines represent Eq. (2) with Eq. (4). Fig. 4 shows the calculated as well as the experimental results of another series of compressive strength tests with three British cements representing three cement types, namely, "ordinary," "rapid-hardening," and "low-heat" portland cements. These are considered here as equivalent to the ASTM Types I, III, and IV, and a fourth case where calcium chloride was combined with the Type III cement. The tests performed, as reported by Orchard, 9 were 6 in. (15 em) modified cube tests according to the British standard, which is very much the same as ASTM C 116 except that the curing temperature was 54 F (12.2 C). Data concerning the reported compositions of the concretes are: Mix proportion: 1:6 by weight. Water-cement ratio: wlc = 0.6 by weight. For the Type III cement: S, = 4000 cm 2 /g (air permeability). For the other cements: S, = So = 3200 cm 2 /g (air permeability). Thin lines represent the experimental values while the corresponding thick lines represent Eq. (2) with A = 17,000 psi (I 17 MPa) as in Fig. 2, with B = 7.9, and with Eq. (6). The reason for increasing the value of Bin Eq. (2) to 7.9 is that the curing temperature in this series was lower than the value of 73.4 ± 3 F (23 ± 1.7 C) specified in ASTM C 116. Due to the present lack of any more fundamental method, the value of B corresponding to this lower temperature was obtained by adjusting its value until Eq. (2) gave the best fit to the experimental data. In the consideration of the ef126 Fig. 5 - Flexural strengths of concretes as a function of cement type and age at testing. Points represent experimental values reported by Klieger, 6 and lines represent values calculated from Eq. (3) with Eq. (5). feet of calcium chloride, it was assumed that it acts as a catalyst on the hardening of the cement; that is, it increases the values of the b, and b2 rate parameters. A stepwise numerical approximation provided the value of 40 percent increase in both b values. Fig. 5 shows the calculated as well as the experimental results of a series of flexural strength tests with the same five portland cements and concretes, the compressive strengths of which are presented in Fig. 2. The tests were performed on beams having 6 x 6 in. (15 x 15 em) cross sections and made, cured, and tested with third-point loading according to ASTM C 192 and C 78. Points represent experimental values reported by Klieger, 6 and the lines represent values calculated from Eq. (3) with Eq. (5). It can be seen that for these widely differing concretes the new formulas provide good fits both to the compressive and flexural strengths of all five standard ASTM types of portland cement within the tested age limits ranging from 1 day through 1 year. For instance, the average error in the estimates for the compressive strengths of the L TS cements at the ages of 1, 7, 28, 90, and 365 days (Fig. 2) is 522 psi (3.60 MPa). (See Table 2.) This average error is defined as the average of the absolute values of differences between the measured compressive strengths and those calculated by the new formulas. The average errors in the estimates of the 7- and 28-day compressive strengths are 447 and 800 psi (3.09 and 5.52 MPa), respectively. The average errors calculated in the same way for the flexural strengths of the same concretes (Fig. 5) are also summarized in the upper portion of Table 2. It should be recognized that only a portion of these average errors may be charged to the new formulas. Another, unknown portion is due to testing errors in the strength determinations (5 to 10 percent) and to testing errors in the determination of the compound compositions of the cements (10 percent?). ACI JOURNAL I March-April 1981 Table 2 - I Illustration of the goodness of fit of Eq. (2), (3), and (10) with PCA's LTS cements Related figure (2) with (4) and A= 17,000 psi 2 522 447 1, 7, 28, 90, 365 (3) with (5) 5 47 82 I, II, III, IV, V 7, 28, 90, 365 (10) with (4) 6 1102* I, III, IV, 7, 28, 90, 365 (10) with (4) 6 737t 336t I, III, IV, 7, 28, 90, 365 (10) with (5) - lilt 42t Test ASTM cement types Testing ages, days Concrete compressive strength I, II, III, IV, V I, 7, 28, 90, 365 Concrete flexural strength I, II, III, IV, V Concrete compressive strength Concrete compressive strength v Concrete flexural strength Average of errors in five strength estimates for Average of total errors in 25 strength estimates, psi v Eq. 28 days 7 days psi percent psi percent 800 12.2 67 9.5 6.4t 414t 6.3t 7.1t Bit 11.3t 10 15.3 •20 estimates t4 estimates; the Type II cement was omitted for reasons explained in the text. t 16 estimates To convert from psi to MPa, multiply by 0.0069. STRENGTH PREDICTION FROM EARLY STRENGTH A modified form of Eq. (2) or (3) can be used with the appropriate b parameters for the estimation of concrete strength at any age from a strength determined experimentally at a single age. The practically important case, of course, is when the, say, 28-day strength is estimated from the result of an early strength determination. The method is illustrated by Fig. 6. This compares the measured values and corresponding values calculated from the measured 1-day strengths with Eq. (10) for the standard compressive strength of 6 bag/cu yd = 565 lb/ cu yd (335 kg/m 3 ) concretes. The same five LTS cements and concretes were selected for this comparison that were discussed earlier in Fig. 2. Eq. (10) is an appropriate form of Eq. (2) and is as follows: f = f, 1 - pe·b,,- (1 - p)e·b2' 1 - pe-b'- (1 - p)e·b2 (10) where f, is the experimentally obtained 1-day strength. The other symbols are identical with the symbols of Eq. (2). It can be seen that the goodness of fit of the equations used is quite good, except for cement No. 21. Details of this fit, the average errors of strength estimates above, etc., are summarized in the lower portion of Table 2. The reason for the low estimates for the strength of cement No. 21 may be that something was wrong with the experimentally obtained results of this cement. This is suggested by the fact that its 1-day strength is much lower than the 1-day strengths of any other Type II cement in the L TS series, and also it is lower than the 1-day strength of cement No. 41 of Type IV. Had any other Type II cement of the L TS series been used for the illustration of Eq. (10), the goodness of fit would have been about the same as that demonACI JOURNAL I March-April 1981 10r--r---r--r---,-----.-,----,---, 70 9f-t--- 60 & 2 ~- 7·~----~---~---~--~~--+-----4--~ 50 0, ~ 6~~-~--~~~~~----+----~~ ~ Sr-+---~--~~L_r-~--~+----4--~ "'> -~ l) -~ 4~~--~~sa~L_--~-----+----4--~ Vl ~ ..,.-('" ~ 3~~-~,L%_,. --~ +-+Cement No.11 (Type I ) 20 8 _,./'" n---D , No.21 ( " II) 8 2 o·········o , No.31 ( , lll) _ 10 ~,~c--+--+--- x-x ., No.41 ('" IV) ,".r----t" '" No. 51 ( " V) f ~ 28 Age, days 90 365 730 ° Fig. 6 - Prediction of compressive strengths of concrete from the 1-day measured strengths for various cement types. Points represent experimental results reported by Klieger, 6 and lines represent values calculated from Eq. (10) with Eq. (4). strated in Fig. 6 for the other four cements. This abnormal behavior of cement No. 21 is a warning concerning the use of Eq. (10); namely, that only highly reliable test results should be used for such strength estimation. Flexural strengths estimated from the measured 1day flexural strength show approximately the same goodness of fit to corresponding experimental results that can be seen in Fig. 6 for compressive strengths. Details of this fit are also given in the lower portion of Table 2. RATIONALE FOR THE NEW FORMULAS Both Eq. (2) and (3) are the products of two previously published formulas. One of these is Eq. (1) by Abrams 1 with the appropriate A and B parameters. 127 The other is an exponential formula, or exponential cement model, first published in I967: 3 f,el = 100 I - pe-•,,- (I - p)e-•2' 1 _ pe-28a 1 _ (1 _ p)e-28a 2 (II) where relative strength of portland cement paste, mortar, or concrete, percent of the 28-day strength a, and a2 = rate parameters of the two hardening components which are independent of the strength, age, and CJS as well as C2S contents, but may be a function of the temperature, C 3A content, and any other factor that influences the course of hydration (fineness, gypsum content, admixtures, water-cement ratio, curing, and testing methods, etc.), 1/day. In terms of Eq. (2) or (3), a, = (SJS,)b, and a2 = (SJS,) b2. The other symbols are identical with the symbols of Eq. (2). As explained in References 3 and I3-I5, Eq. (II) is the mathematical form of a cement model, the hardening of which is the sum of two hardening processes of first-order reaction. In other words, this exponential cement model consists of only two hardening components. The first component is the CJS; the second component is the mixture of the other cement ingredients, mostly C2S. Each of these components has its own specific rate of hardening, that is, (rate of hardening)/ (remaining strength development) at a given age for the two components, represented by the two a parameters. It is significant that both of these parameters may increase linearly with an increase in the C,A content of the cement indicating the catalytic effects on the calcium silicates in the model. The decelerations of the hardening of both the C3 S and the second component are also proportional at any given age to the remaining strength development at that time, and the two porportionality factors are the squares of the same a, and a2 parameters. 10 The model does not explicitly include the effect of diffusion on the strength development. Numerically, the a parameters (Table I) can be obtained from pertinent experimental data by applying a suitable method of optimization similarly to the traditional method of the kinetics of chemical reactions. It has been shown that experimental data support Eq. (II) within wide limits. 3 • 11 -' 6 Since Eq. (1) is also supported by a wealth of experimental data, the good fit of the new formulas as exhibited in Fig. I through 6 should not be surprising. [,. 1 ADVANTAGES AND DISADVANTAGES The advantages of the cement model represented by Eq. (2) through (II) are as follows: I. It works; that is, it is supported by experimental results within wide limits. 128 2. The model is simple. It has only two independent variables, namely, the CJS and CJA contents, and it needs only two parameters (b, and b2 ) for each test method instead of the many of the original formula by Abrams. 3. It has an interpretation, an unconventional one, though, from the standpoint of cement chemistry. Specifically, the model reproduces the strength development as if: (a) the CJA acted as a catalyst on the hardening of the calcium silicates in the portland cement; (b) the specific surface of the cement affected the b rates of hardening in a linear manner. 4. The model may also be used conveniently for quantitative investigations of other effects on the strength development, such as the effect of admixtures or curing temperature. 5. The model is applicable for various strengths of concretes. 6. It may also be extended to other characteristics of the hydration process, such as the amount of chemically bound water, specific surface of hydration products, heat of hydration, etc. 11 7. Eq. (IO) and its modifications have considerable practical importance in predicting a late strength from the result of an earlier strength determination. 8. The model calls the attention to new directions for research concerning the hydration of portland cement, such as the direct proof or disproof of the catalytic action of CJA in cement. There is also room to polish the model, primarily to improve its fit to the experimental results, by including additional variables that affect the strength development. Some of the possibilities are: I. Inclusion of the effect of the diffusion process. 2. Inclusion of the SOJ content as well as the minor constituents of portland cement. As a closing note, it may be mentioned that the exponential model concept can be extended to estimate the strengths of cements in standard mortars, such as those specified in ASTM C 109 and C I90. This, however, is the topic for another paper. SUMMARY AND CONCLUSIONS The formula offered in this paper for the prediction of strength development of concrete is the generalization of the strength versus water-cement ratio formula by Abrams (Abrams' Law). Also, it is the mathematical form of the exponential cement model published earlier. The most significant feature of this model is that, despite its simplicity, it reproduces the effects of compound composition and fineness of the cement on the strength development within wide limits of validity better than any other method offered for the same purpose. For instance, the average errors of estimating concrete strengths including all five standard ASTM cement types and ages spanning from I day through I year were found to be 522 psi (3.60 MPa) for compression and 47 psi (0.32 MPa) for flexure. ACI JOURNAL I March-April 1981 Not only does this good fit support the applicability of the offered formulas for the prediction of the strength development of concrete, but it also points at the possibility that there is a fundamental similarity between the hardening mechanism represented by the exponential cement model and the actual hardening mechanism in the portland cement paste. This recognition can help cement chemists to aim their research at new, promising goals, such as to establish directly the role of C 3 A in the cement hydration and hardening by focusing the investigation to its possible catalytic action. The exponential cement model is considered simple because it uses much fewer experimental parameters than similar earlier methods, such as the linear models. 17 The arithmetic forms of the exponential model may seem somewhat complex, but there is nothing in them that an inexpensive pocket calculator could not handle. The exponential model in its present form disregards the effects of diffusion on the hardening process and that of the minor constituents. It is likely that the inclusion of these factors as variables will further improve the goodness of fit of the model, especially that of Eq. (10). REFERENCES I. Abrams, Duff A., "Design of Concrete Mixtures," Bulletin No. I, Structural Materials Research Laboratory, Lewis Institute, Chicago, Dec. 1918, 20 pp. 2. Popovics, S., "Factors Affecting the Relationship Between Strength and Water-Cement Ratio," Materials, Research and Standarcls, V. 7, No. 12, Dec. 1967, pp. 527-534. 3. Popovics, Sandor, "A Model for the Kinetics of the Hardening of Portland Cement," Highway Research Record No. 192, Highway Research Board, 1967, pp. 14-35. 4. Gonnerman, H. F., and Lerch, William, Changes in Characteristics of Portland Cement as Exhibited by Laboratory Tests Over the Period 1904 to 1950, STP 127, American Society for Testing and Materials, Philadelphia, 1952, 56 pp. ACI JOURNAL I March-April 1981 5. Lerch, William, and Ford, C. L., "Long-Time Study of Cement Performance in Concrete: Chapter 3 - Chemical and Physical Tests of the Cements," ACI JouRNAL, Proceedings V. 44, No. 8, Apr. 1948, pp. 745-795. 6. Klieger, Paul, "Long-Time Study of Cement Performance in Concrete: Chapter 10 - Progress Report on Strength and Elastic Properties of Concrete," ACI JouRNAL, Proceedings V. 54, No. 6, Dec. !957, pp. 481-504. 7. Popovics, Sandor, "Relations Between Various Strengths of Concrete," Highway Research Record No. 210, Highway Research Board, 1967, pp. 67-89. 8. Concrete Manual, 8th Edition, U.S. Bureau of Reclamation, Denver, 1975, 627 pp. 9. Orchard, D. F., Concrete Technology: V. 1, Properties of Materials, 3rd Edition, John Wiley and Sons, New York, 1973, 428 pp. 10. Popovics, Sandor, "Examples for the Application of Mathematics in Concrete Technology," Reports, IV, International Congress on the Application of Mathematics in Engineering (Weimar, 1967), VEB Verlag fur Bauwesen, Berlin, 1968, V. I, pp. 375-382. I I. Popovics, Sandor, "Comparison of Various Measurements Concerning the Kinetics of Hydration of Portland Cements," Proceedings, Fifth International Symposium on the Chemistry of Cement (Tokyo, 1969), Part III, Properties of Cement Paste and Concrete, pp. 129-137. 12. Popovics, Sandor, "Calculation of the Strength Development of Mortars and Concretes from the Compound Composition of the Portland Cement Used," Betonstein-Zeitung (Wiesbaden), V. 34, No. I I, Nov. !968, pp. 587-590. (in German) 13. Popovics, Sandor, "Effect of Kinetics on the Ultimate Strength of Portland Cement Pastes," Beton i Zhelezobeton (Moscow), Mar. 1972, pp. 23-24. (in Russian) 14. Popovics, Sandor, "Strength Development of Portland Cement Paste," Supplementary Paper, Sixth International Congress on the Chemistry of Cement (Moscow, Sept. 1974), Section II. 15. Popovics, Sandor, "Phenomenological Approach to the Role of C,A in the Hardening of Portland Cement Pastes," Cement and Concrete Research, V. 6, No. 3, May 1976, pp. 343-350. 16. Popovics, Sandor, Concrete Making Materials, McGraw-Hill Book Company, New York/Hemisphere Publishing Corporation, Washington, D.C., 1979, 370 pp. 17. Gonnerman, H. F., "Study of Cement Composition in Relation to Strength, Length Changes, Resistance to Sulfate Waters and to Freezing and Thawing, of Mortars and Concrete," Proceedings, ASTM, V. 34, Part II, !934, pp. 244-295. 129 I