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.
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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.
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