jl59-12

Title No. 59- l 2

Modulus of Elasticity of Concrete Affected by Elastic Moduli

of Cement Paste Matrix and Aggregate

By TEDDY J. HIRSCH

A general equation is derived which expresses the modulus of elasticity of concrete or mortar in terms of an empirical constant, the elastic moduli of the cement matrix and aggregate constituents, and the mix proportions. Laboratory tests showed the equation to produce good results over a wide range of variables. The average deviation was found to be written ±I 0 percent, and the maximum deviation was within +35 percent.

The variety of aggregate materials used included steel punchings, crushed glass, lead drops, Ottawa sand, crushed limestone, and a calcareous-siliceous river gravel.

• IN THE DESIGN oF reinforced and prestressed concrete, a know ledge of the elastic strain properties of the material is important. This term, elastic strain, is usually defined as the unit deformation in a material parallel to the direction of stress which is recoverable after the stress is removed.

The stress-strain curve for cement paste and concrete is seldom a straight line, and, furthermore, the strains for the first loading cycle are seldom entirely recoverable. However, if a specimen of cement paste or concrete is loaded and unloaded once to eliminate the plastic strains encountered in the first loading cycle, a stress-strain curve is usually found to be relatively straight for compressive stresses less than about one-half the ultimate.

Because the stress-strain curve of concrete is not entirely straight, it is preferred to define the modulus of elasticity as the ratio of an increment of stress to a corresponding increment of instantaneous strain. This definition is applicable to all materials, and only elastic materials which obey Hooke's law have a constant modulus of elasticity independent of stress magnitude and time.

427

428 jOURNAL OF THE AMERICAN CONCRETE INSTITUTE March 1962

NOTATION

A = length of aggregate particle Aa = cross-sectional area of aggre

gate Ac cross-sectional area of concrete A.. cro.;;s-sectional area of cement

paste matrix dA infinitesimal cross-sectional

area c = ratio of bulk volume of aggre-

gate to volume of concrete, Va/Vc

D. deformation of aggregate par-ticles

D. deformation of concrete D., deformation of matrix E. modulus of elasticity of aggre

gate Em modulus of elasticity of cement

paste matrix E,.. modulus of elasticity of mortar

matrix F c total stress on concrete F. total stress on aggregate par

ticles F.. = total stress on cement paste

matrix j( ) K1

dL L. Lc

.3ymbol for a function of average stress factor for aggregate average stress factor for matrix infinitesimal length effective length of aggregate length of concrete

L.. effective length of matrix A L. = small finite length of aggre-

gate .l L,~ small finite length of matrix M length of matrix particle d P infinitesimal load on concrete S. unit stress in aggregate s. (avg) = average unit stress in ag

gregate S, (max) = maximum unit stress in

aggregate s. = unit stress on concrete s.. = unit stress in matrix s ... (avg) = average unit stress in ma

trix S.., (min) = minimum unit stress in

matrix V. = bulk volume of aggregate V. = volume of concrete V... volume of cement paste matrix V ... volume of mortar d V, infinitesimal volume of aggre

gate d V. = infinitesimal volume of con

crete d V m = infinitesimal volume of cement

paste matrix v unit shear stress Y1 = vertical coordinate y, = vertical coordinate dy1 = infinitesimal vertical length dy. = infinitesimal vertical length Z = a suitable empirical constant fa unit strain in aggregate E., = unit strain in matrix

CURRENT METHODS OF PREDICTING THE MODULUS OF ELASTICITY OF CONCRETE

The modulus of elasticity of concrete has been a difficult value to predict since the properties and quantities of the individual constituents composing concrete vary considerably. To date, several empirical formulas have been proposed to serve as a guide for use when the modulus cannot be determined by tests. Formulas proposed by ACI Committee 318, Standard Building Code, ACI-ASCE Committee 323, Prestressed Concrete, and Jensen and Hognestad all relate the modulus of elasticity to the compressive strength. Pauw10 has recently proposed a formula which relates the modulus to the compressive strength and the unit weight.

MODULUS OF ELASTICITY 429

ACI member Teddy J. Hirsch is an assistant professor of civil engineering and an assistant research engineer with the Texas Agricultural and Mechanical College System, College Station, Tex. Dr. Hirsch first joined the A & M College staff in 1956 as an instructor in civil engineering. He is the coauthor of over seven publications in the field of structures and materials. This paper is a condensation of his PhD thesis.

All of these equations have obvious limitations. The first four equations contain no provision for evaluating the effect of different aggregate properties, except as they may affect strength. The modulus of concrete changes with its age, and the empirical constants of the first four equations were based only on concrete specimens tested at 28 days. None of the equations can evaluate the effect of different mix proportions. Most important of all, more recent investigators8 ·n have found no definite relationship between the modulus of elasticity and the compressive strength when different aggregate types are used. Concretes with the same compressive strength but made from different aggregates were observed to have considerably different moduli.

Assuming a relationship between the modulus of elasticity and the compressive strength is primarily a method of convenience since the compressive strength test is relatively simple to perform and is commonly used to control concrete quality. In addition, when a given aggregate is used in a concrete, it has been observed that factors which affect the compressive strength also affect the modulus in a similar manner.

OTHER PROPOSED FORMULAS

LaRue,H Dantu,~ and Kaplan''· 11 have proposed formulas which relate the modulus of elasticity of concrete to the elastic properties of the individual constituents. LaRue's equation was developed empirically and contained three empirical constants which are difficult to evaluate. Dantu and Kaplan separately proposed the same two equations. When the compressive stresses within the individual constituents of the heterogeneous concrete mass were assumed to be equal, the equation they developed was

where c = v.;v. V .. = volume of aggregate V c = volume of concrete

1 E,

__c:__ + (1- c) E. Emo ·········· (1)

When the strains in the individual constituents of the heterogeneous body were assumed equal, the equation derived was

E, = cE, + (1- c)Em .... . ......... (2)

In both equations the Poisson's ratio of the constituents was assumed equal to zero.

430 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE March 1962

The major shortcoming of both of these equations lies in the basic <~ssumption of either equal stresses or strains in the individual constituents. In addition, these equations provide no means of compensating for the effect of size, shape, or arrangement of the aggregate particles or other variables. The following hypothesis is a refinement of the approaches of Kaplan and Dantu and overcomes the major shortcomings of their two equations.

HYPOTHESIS

Modulus of elasticity of concrete

The modulus of elasticity of concrete must be determined from the average stress-strain ratio of a relatively large portion of a concrete mass. This condition must be placed on the term since concrete is a macroscopic heterogeneous mass composed of rather large aggregate particles embedded in a cement paste matrix. If based on the stress-strain relationship at only a single point in the mass, the modulus of concrete would reflect the elastic properties of only one of the constituents. The constant for the combined mass would not be represented since the two constituents usually have greatly different elastic properties.

In laboratory test procedure the modulus is determined from the average stress-strain relationship of a specimen which is relatively large compared to the maximum size aggregate particle as illustrated in Fig. 1. The deformation of this specimen under stress must be the result of strains in both the cement paste matrix and in the embedded aggregate. The strains in the constituents depend on the stresses in them and on their elastic properties. Thus

Ea s. E.

and

f,,/1 Sm Em

where Ea and Em = the unit strain in the aggregate and cement paste matrix, re-

spectively

S, and Sm = the unit stress in the aggregate and cement paste matrix, re-spectively

E, and E, = the modulus of elasticity of the aggregate and cement paste matrix, respectively

If a uniform stress S,, is applied to the top and bottom of the concrete, · it can be shown that the stresses and strains within the constituents of the specimen are variable if their elastic moduli are different. To


Fig. I -Typical cross section of concrete specimen (left); assumed unit compressive stress variation throughout the infinitesimal volume of concrete when E, > Em. dVc = Lr = dA

0 S (min) Sc Sa (max.)

X

(right)

illustrate this point, reference is made to Fig. 2, which is an enlargement of a small part of Fig. 1. At points where the aggregate and matrix have a common horizontal face bonded together, it is reasonable to assume that the compressive stresses in the aggregate and in the matrix are equal. On the other hand, at points where the aggregate and matrix have a common vertical face bonded together, the compressive stresses in the aggregate and in the matrix cannot be equal and hence are changing. These normal stresses cannot be equal because if they were. then the normal strains in the two constituents would be different due to their different elastic properties. Consequently, shear stresses would be produced at the common vertical face where they are bonded together. and these shear stresses would cause the normal stress to vary as indicated in Ffg. 2. The exact nature of the stress variation is complex and is discussed in more detail in the Appendix. However, the following discussion will illustrate how it may be possible to predict the resultant strain or deformation of a concrete specimen under load and. consequently, its modulus of elasticity.

Consider the idealized infinitesimal volume of concrete defined by the area dA and length L 0 in Fig. 1 (left). The deformation of this volume of concrete is

De = D. + D.,, .................... .. (3)

in which D,., Da, and Drn is the deformation of concrete, aggregate. and cement matrix, respectively.

Now the deformations of the aggregate and cement paste are La

D. = l Ea Ll L. 0

and [,..,

D .. :s '"' .> L ... "


Sa (max.)

h-:--=-:-=--.-1 ~*t=t==!dL I ~~~~~~~~~dL2 ==~~~==~~~dL3

y, M

Fig. 2-Enlarged cross-sectional area of idealized concrete specimen (left); assumed unit compressive stress variation along the infinitesimal volume of

concrete when En > E,11 , dV c = Lc dA (right)

where L,, L., and L .. = the effective lengths of concrete, aggregate, and cement

matrix, respectively and

Ll L. and Ll Lm = very small finite lengths of aggregate, and cement matrix, respectively

Substituting these values of Da and D.,, into Eq. (3) yields

La L11l

De = ::S Ea Ll L., + ::S Em Ll L,., .................................. (3a) 0 0

From this equation it follows that

L La S 1-::;z Sm S, ··- '- = ::S -" Ll L. + ..,. - Ll L,, ...

E, II E, () E., .... (3b)

where E, is the modulus of elasticity of the concrete

In Eq. (3b) Poisson's effect is neglected. This simplifying assumption was made because the Poisson's values of the constituents are frequently nearly equal. In addition, little lateral restraint of the individual constituents is present when the concrete specimen or member is relatively small and has definite geometric boundaries.

As pointed out previously, the unit compressive stresses in the aggregate S, and in the cement matrix Sm are variable. This stress variation in the idealized infinitesimal concrete volume of Fig. 1 (left) can be assumed to be as pictured in Fig. 1 (right). This variation will be discussed in more detail later. However, the average unit stress through-


out the total length of aggregate may be expressed in terms of the unit stress on the concrete as

s. (avg) = K, S.

Similarly the average unit stress on the cement matrix is

Sm (avg) = K, S,

in which K, and K, are the average stress factors for the aggregate and cement matrix, respectively.

Using these average stress values, Eq. (3b) may be written as

Eq. (3c)

If the infinitesimal volume of concrete is representative of the total volume, then Lc = Vc/Ac, L,, = V,,/A,, and L,11 = V",/Ac, where Ac is the cross-sectional area of the concrete specimen and V "' Va, and Vm is the volume of concrete, aggregate, and cement matrix, respectively.

Substituting these values of L,, Ln, and L 11 , into Eq. (3c) yields

.... ( 4)

where E. and E,, > 0.

Eq. ( 4) expresses the modulus of elasticity of the infinitesimal concrete volume and, consequently, the modulus of the total concrete specimen.

Eq. ( 4) may be rewritten as follows

1 E. Ern K, V. E., + K, ( 1 _ V. ) .....

V.E. V. where E, and E.., > 0.

Eq. (4a) is composed of the dimensionless parameters EJE,,, E,JE,11 ,

and V"/V, .. These ratios will be useful in analyzing the hypothesis with laboratory data. This leaves the stress factors K 1 and K~ to be determined, which relate the average stress in the aggregate and cement matrix respectively, to that in the concrete.

These factors, which are dimensionless, are dependent on each other and could be evaluated empirically with only one empirical constant. The Appendix, however, presents a rational determination of these factors which are modified with one empirical constant Z. These average factors derived in the Appendix are

K, = 1 - 2Z [ 1 ;t

t See Eq. (A7) in Appendix.

E.., ( 1 E,

1~) + ~J·· ................ (5)t

v, v,.

434 jOURNAL OF THE AMERICAN CONCRETE INSTITUTE March I 962

and

Ko=l- 2: [1-(1~~;+ ~:-::J· (6)t

The value of the empirical constant Z will be shown to be approximately 0.785.

DISCUSSION

It will be noted that the foregoing hypothesis makes no specific allowances for the effect of the size, shape or arrangement of the aggregate particles. These factors are believed to have only a minor effect on the over-all modulus of a concrete specimen provided the specimen is large compared to the maximum aggregate size and provided the aggregate particles have a random shape and arrangement within the concrete mass. In any case this hypothesis may be modified to compensate to some extent for these and other unknown factors by an empirical evaluation of the constant Z which appears in the stress factors K 1 and K 2 .

It can be seen that Eq. ( 4) expresses the modulus of elasticity of concrete in terms of the elastic moduli and quantities of the aggregate and cement matrix constituents as follows:

E, = f (Em, E .. , V m, V,)

This equation can be applied to any heterogeneous mass of two phases. It can also be used to determine the elastic modulus of a concrete of three or more phases. Consider, for example, a concrete composed of a coarse and fine aggregate each with a different elastic modulus. The modulus of the concrete can be considered a function of the properties of the coarse aggregate and mortar phase. In turn, the modulus of the mortar may be considered a function of the fine aggregate and cement paste phase.

CONCRETE TESTS

In the laboratory investigation several materials, each with a different elastic modulus, were selected for use as concrete aggregates. These aggregates were mixed in various proportions with a selected cement paste in order that the effects of their elastic moduli and of batch proportions on the modulus of elasticity of concrete could be observed.

Selected aggregates and their properties Six materials, each with a different elastic modulus, were selected for

use as aggregates. Their moduli ranged from 2.18 X 106 to 30 X 106 psi. These materials were steel punchings, Ottawa sand. crushed glass. a

t See Eq. (A6) in Appendix.

' TABLE I- SELECTED AGGREGATE PROPERTIES

Crushed Aggregate Steel Ottawa glass

type punchings sand (common (silica) plate)

Designation ST DS GL

Elastic properties Dynamic E X 10-• psi 30.0t u.o; 10.6 Secant E x 10-• psi 30.0t 11M 10.5§ Dynamic G X I0-6 psi 11.5t 4.26 Dynamic 1L 0.30t 0.22

Specific gravity 7.85+ 2.58 2.58 (SSD)

Absorption, percent - 0.25 -Unit weight, lb per cu ft 275 105 86

(dry loose)

Sieve analysis Percent passing I

% in. 100 %in. 0 lh in. 100 % in. 77 No.4 26 No. 8 0 No. 16 100 No. 30 0.6

-- --

t-Typical handbook values for ordinary structural steel ASTM A 7. * These values were estimated and are based on typical modulus of elasticity values of siliceous quartz and flint.•· 6.12 •

§ Secant modulus of elasticity of common plate glass as given by Birch.• Secant moduli of other aggregate materials were determined from stress-strain curves of

representative specimens. Dynamic moduli of other aggregate materials were determined by ASTM C 215-551T) on

representative specimens.

Gravel (calcareous-siliceous)

GR

8.64 8.94

2.64

1.2

96

100 97 61 10 1.4

Crushed limestone

LI

4.45 4.62

2.32

7.0

80

100

39 12 0

----

Lead drops

PB

2.41 2.18

0.46

11.37

-

395

100 97.2 0.2

s:: 0 Cl c r c V1

0 -n

m r )> v: -1 n ~ -<

~ w U1


calcareous-siliceous gravel, a crushed limestone, and lead drops. The elastic properties, specific gravity, absorption, unit weight, and gradation of these materials are given in Table 1. The steel punchings, crushed glass and lead drops were selected because they had a desirable range of elastic properties, were relatively homogeneous and isotropic and were available within the financial limitations of this project. The Ottawa sand, calcareous-siliceous gravel and crushed limestone were selected because they had a desirable range of elastic properties, were representative of materials commonly used as concrete (or mortar) aggregates and were available within the financial limitations of the project. No attempt was made in this investigation to evaluate the effects of size, shape, texture, and gradation of the aggregates on the elastic moduli of concrete.

The elastic properties of the steel punchings and Ottawa sand were taken from handbooks and typical test data as indicated in Table 1. The secant modulus of the glass is also based on handbook data.~ The dynamic moduli of the other aggregate materials were determined by ASTM

TABLE 2 - CONCRETE BATCH DATA

Ratio of Batch absolute desig- Quantities per cubic foot volume of Water-nation of concrete Concrete aggregate cement

unit to volume ratio Aggregate Cement, Aggregate Water, weight, of concrete by

type lb (SSD), lb lb lb per cu ft V,JV. weight ~~

CM - 87.1 0 34.1 121.8 0 0.39t

ST-5 Steel 43.5 245 17.4 305.9 0.50 0.40 ST-4 Steel 52.2 196 20.9 269.1 0.40 0.40 ST-3 Steel 61.0 147 24.4 232.4 0.30 0.40

SD-4 Sand 51.8 64.8 20.8 137.4 0.40 0.40 SD-2 Sand 69.4 32.5 27.8 129.7 0.20 0.40

GL-6 Glass 37.4 91.7 15.0 144.1 0.57 0.40 GL-5 Glass 43.5 80.5 17.4 141.4 0.50 0.40 GL-4 Glass 52.2 64.4 20.9 137.5 0.40 0.40 GL-3 Glass 61.0 48.3 24.4 133.7 0.30 0.40 GL-2 Glass 69.7 32.2 27.9 129.8 0.20 0.40

GR-6 Gravel 37.4 94.1 15.0 146.5 0.57 0.40 GR-5 Gravel 43.5 82.6 17.4 143.5 0.50 0.40 GR-4 Gravel 52.2 65.9 20.9 139.0 0.40 0.40 GR-3 Gravel 61.0 49.6 24.4 135.0 0.30 0.40 GR-2 Gravel 69.7 33.0 27.9 130.6 0.20 0.40

LI-6 Limestone 37.4 82.5 15.0 134.9 0.57 0.40 LI-5 Limestone 43.5 72.3 17.4 133.2 0.50 0.40 LI-4 Limestone 52.2 57.8 20.9 130.9 0.40 0.40 LI-3 Limestone 61.0 43.4 24.4 128.8 0.30 0.40 LI-2 Limestone 69.7 28.9 27.9 126.5 0.20 0.40

PB-5 Lead 43.5 355 17.4 415.9 0.50 0.40 PB-4 Lead 52.2 284 20.9 357.1 0.40 0.40 PB-2 Lead 69.7 142 27.9 239.6 0.20 0.40

t Water-cement ratio corrected for bleeding.


c 215-55 (T) using the fundamental flexural frequency of vibration. Various size representative aggregate specimens were selected for the test. The secant moduli were determined from stress-strain curves on the same prismatic specimens of aggregate. Two electric strain gages were placed on opposing sides of each specimen to measure the compressive strain. The secant modulus was determined from the ratio of stress to strain as given by the slope of a straight line drawn from the point of zero stress to a stress of 1000 psi (300 psi for lead).

Concrete batch proportions Twenty-four batches of concrete were made with the six different

aggregates and a cement paste with a water-cement ratio of 0.4 by weight. This water-cement ratio was selected because its consistency was thick enough to keep segregation of the concrete solid constituents to a minimum, and because it bled little and was convenient to place and finish. In general, the amount of aggregate used in the concrete varied from a maximum of 57 percent to a minimum of 20 percent by absolute volume. The batch designated as "CM" contained no aggregate, and it provided the elastic properties of the cement paste matrix. Table 2 gives the batch proportions of the twenty-four concrete mixes from which the test specimens were cast.

Test schedule

One 3 x 4 x 16 in. prism specimen was prepared from each batch of concrete, and tests to determine its secant modulus of elasticity, and dynamic modulus of elasticity were conducted at various ages. In general, tests were conducted at 3, 7, 14, 28, and 60 days to observe the effect of the changing properties of the cement paste on the concrete.

Results of concrete tests The secant modulus of elasticity and dynamic modulus of elasticity of

the concrete specimens are presented in Table 3. The initial water-cement ratios of all mixes were 0.4; however, since the mix proportion "CM," which contained only cement and water, bled slightly, its actual ratio was reduced to 0.39. The elastic properties of the cement matrix with water-cement ratio of 0.4 were estimated from these values and experience gained from a cement paste test series (see Fig.6).

From data in Table 3 it can be seen that the modulus of elasticity of concrete (secant or dynamic) increases with age. This increase, of course, is due to the increasing stiffness or elastic modulus of the cement paste constituent.

A comparison of the dynamic modulus of elasticity values with the secant modulus of elasticity results indicates that the secant moduli are, in general, approximately 10 percent less than the dynamic moduli.

Typical stress-strain curves for concrete are shown in Fig. 3.

TABLE 3- MODULUS OF ELASTICITY OF CONCRETE SPECIMENS (f X I0- 6 PSI, 3 x 4 x 16 IN. PRISMS WATER CURED, E DYNAMIC AND

E SECANT) Age of concrete, days

Batch 3 7 14 designation

Dynamic Dynamic I Secant Dynamic Secant

CM W /C = 0.39t 2.17 2.64 2.26 2.92 2.66 W /C = 0.40 estimated 2.08 2.56 2.19 2.84 I 2.59

ST-5 6.05 7.50 6.97 8.09 7.04 ST-4 5.09 5.93 5.29 6.44 6.25 ST-3 4.38 5.12 4.41 5.59 4.67

SD-4 4.15 4.83 4.08 5.21 4.81 SD-2 3.01 3.56 3.31 3.91 3.55

GL-6 5.09 5.68 5.34 5.89 5.85 GL-5 4.86 5.38 4.72 5.65 5.35 GL-4 4.17 4.73 4.18 5.00 4.52 GL-3 3.59 4.11 3.70 4.40 4.03 GL-2 3.03 3.53 3.31 3.80 3.45

GR-6 4.67 5.20 4.51 5.50 5.02 GR-5 4.14 4.65 4.31 4.95 I 4.59 GR-4 3.77 4.28 3.88 4.59 ! 4.17 GR-3 3.14 3.63 3.47 3.94 3.74 GR-2 2.78 3.26 2.92 3.57 3.29

LI-6 3.33 3.66 3.17 3.74 3.27 LI-5 3.18 3.54 3.13 3.64 3.20 LI-4 2.99 3.36 2.90 3.52 3.04 LI-3 2.66 3.07 2.72 3.26 2.91 LI-2 2.55 2.99 2.58 3.20 2.86

PB-5 2.69 2.03 PB-4 2.81 2.21 PB-2 3.09 2.46

t Water-cement ratio corrected for bleeding. Secant moduli were determined by the slope of a straight line drawn from the point of

zero stress on the stress-strain curve of a specimen to a stress of 1000 psi ( 300 psi for concrete with lead aggregate).

f\'1~--- • • •• ~-~~---~~--..:1 1..-· A....,rn"l\11'",.., 01P"" ...... ff"r"l\

28

Dynamic

3.14 3.06

8.56 I 6.85

5.99

5.60 4.25

6.17 5.90 5.27 4.66 4.05

5.74 5.21 4.86 4.20 3.83

3.80 3.74 3.63 3.38 3.37

2.87 2.97 3.29

Secant

2.85 2.78

7.75 6.45 5.41

4.92 3.72

6.21 5.43 5.12 4.46 3.79

5.24 4.72 4.31 3.98 3.33

3.40 3.37 3.24 2.98 2.93

2.35 2.42 2.50

60

Dynamic I Secant

3.32 I 2.90 3.24 2.83

8.94 i

8.55 7.19 7.46 6.29 5.60

5.78 ! 5.11 4.41 4.05

6.42 ' 6.02 6.11 5.72 5.46 5.21 4.83 I

4.54 4.21 ' 3.85

I

5.97 I 5.68 5.39 I

4.98 5.11 4.58 4.40 4.18 4.01 4.00

3.96 3.47 3.90 3.47 3.78 3.31 3.58 3.19 3.58

I 3.09

2.96 ; 2.33 3.10 ' 2.46 3.45 I 2.61

.,.. w 00

0 c "' z )> r

0 "T1 _, I m

)>

~ m ;c

fi )> z n 0 z n ::0 m _, m

z Vl _, -1 c _, rn

~ ., ;:; ::r

->() 0.

""'


1400~--------_,-------------r--------

1200 .

CJ) a.. z 1000

CJ) CJ) u.1 0::: ..... 800 CJ)

UJ > CJ)

600 CJ) UJ

ValVe 0::: a.. • GR-6 .57 ~ 0 GR-5 .50 0 400 0 GR-4 .40 (.)

t:::. GR-3 .30 + GR-2 .20 • CM .0

200

0 0 100 200 300 400 500

STRAIN IN MICROINCHES PER INCH

Fig. 3 -Stress-strain curves of concrete and cement paste


EVALUATION OF EMPIRICAL CONSTANT AND COMPARISON OF HYPOTHESIS WITH LABORATORY OAT A

To determine if the elastic moduli and proportions of the aggregate and cement matrix constituents are significant variables which affect the modulus of elasticity of concrete, Fig. 4 and 5 were plotted. The dimensionless parameter Ec/Ern was plotted as the ordinate and Ea/E,,, as the abscissa. Variations in the concrete batch proportions were indicated by the dimensionless parameter V a!V c· As can be seen in Fig. 4 and 5, a rather distinct family of curves resulted.

These data indicate conclusively that the modulus of elasticity of concrete is a function of the elastic moduli of the constituents. An increase or decrease in the modulus of either the aggregate or matrix constituent will produce a corresponding effect on the concrete. Furthermore, the degree to which the elastic modulus of one constituent will affect th€ modulus of the concrete is a function of the mix proportions.

To evaluate the empirical constant and to make a quantitative comparison of the hypothesis with the laboratory data, Eq. (4a) was used. This equation was written as

1

+ K, ( 1 - V, ) .. v. (4a)

where

K1 = 1 _ !2[ 1 _ E .. ,--( 1 _ ~~) + ~J···.

E. V, V,

. ..... (5)

and

.... (6)

where Z is an empirical constant.

The expressions K 1 and Ke, which are the average stress factors for the aggregate and cement matrix, respectively, have not been substituted in Eq. ( 4a) so that they may keep their identity. If all of the assumptions and idealizations on which the hypothesis was based were completely accurate, the constant Z would be one and, of course, would be unnecessary. This empirical constant, however, allows the modification of these average stresses so that the hypothesis can be made to agree more closely with the laboratory data. It may be considered that the empirical constant Z is compensating to some extent for such uncontrollable variables as size. shape, and arrangement of the aggregate particles.

E w 2.0t----....... 0

w

l,5t----

1.0

0.5

2

MODULUS OF ELASTICITY

4

HYPOTHESIS REPRESENTED

BY CURVES

0 CONCRETE DATA

V0 /V0

• .57 0 • 50 + .40 0 .30 c. .20

6 8

Ea!Em

441

10 12

Fig. 4- Comparison of the hypothesis with dynamic modulus of elasticity data on concrete specimens

442 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE

E UJ 2.0 I-----

' (,) UJ

1.51-----

1.0

0,5

2 4

HYPOTHESIS REPRESENTED

BY CURVES

0 CONCRETE DATA

V0 /Vc

• .57 D • 50 + .40 0 ,30 6 .20

6 8

Ea/Em

March 1962

10 12

Fig. 5- Comparison of the hypothesis with secant modulus of elasticity data on concrete specimens


For a given value of V a!V c, the constant Z can only be evaluated so that Eq. ( 4a) will produce a curve that will pass through only one point as defined by specific values of Ec/E"' and Ea/Em. Since the data presented here consisted of numerous points defined by the variables E0/E 111 , Ea!Em, and also ValVe, the empirical constant was evaluated by trial and inspection. The constant Z was first assumed to have a particular value, and then the hypothesis was compared with the test data as in Fig. 4 and 5. The solid curves in these figures were developed by the computation of Eo/Em with Eq. ( 4a) for various values of Ea/Em and Va/V, .. Curves similar to those on Fig. 4 and 5 were developed with the use of several other assumed values of Z until the best over-all agreement of

(/) Q.

"' I

Q X

>-

--- -~ --- -, ---r~-• 3 DAYS 1 ~ .. 1 " t. 0 14 • 28 • c 60 " 1

s 4 1----c-t- TYPE I CEMENT

f;; <t ...J UJ 3

u.. 0

~ 2 1-----+----':,"-.;~4,-,;:::~,---+---i ...J ::> c 0 :::E

t-z <t (.) UJ 0 ~._ _ ___.__ __ ..,L..__~:------7-----: (/) .2 .3 .4 .s ,6 ,7

WATER I CEMENT RATIO BY WEIGHT

Fig. 6- Effect of water-cement ratio and age of the secant modulus of

elasticity of cement pastet

tWater-cement ratios were corrected for bleeding and all specimens were water cured.

the curves with the plotted data was determined by visual inspection. By this method the value of Z has tentatively been set at 0.785, and

the curves in Fig. 4 and 5 were developed with the use of this value. It can be seen that the hypothesis gives values of Ec/E, in good agreement with the laboratory data taken in this study. Table 4 presents a numerical comparison of the 28 day elastic moduli of the concrete specimens with values computed with the use of the hypothesis proposed in this report. The maximum deviation of the hypothesis from all these data is within ± 13 percent.

COMPARISON WITH OTHER DATA

In the review of the literature, considerable data were found concerning the effect of the elastic moduli of the mortar and aggregate constituents on the modulus of elasticity of concrete. These data on the elastic moduli of concrete and mortar taken by Willis and De Reus,2 LaRue,8

Kaplan, 11 and Dantu3 were also compared with values computed with the use of the hypothesis proposed in this report. When the uncontrollable variables such as size, shape, and arrangement of the aggregate particles are considered, the values of the moduli obtained from the hypothesis were found to be in good agreement with the data. The

TABLE 4- COMPARISON OF HYPOTHESIS WITH MODULUS OF ELASTICITY DATA

-------- --~ ------- -- -----

Secant modulus of elasticity X lQ-•• psi r----

Aggregate v. type v.- Cement paste

Concrete Comparison Aggregate, (28 day), E,

Ee,mputed (28 day), E. I Ecompuled E .. Etut E~e.t

Steel punchings 0.50 2.78 30.0 7.75 7.70 0.99 Steel punchings 0.40 2.78 30.0 6.45 6.56 1.02 Steel punchings 0.30 2.78 30.0 5.41 5.61 1.04 Ottawa sand 0.40 2.78 11.0 4.92 4.76 0.97 Ottawa sand 0.20 2.78 11.0 3.72 3.70 0.99 Crushed glass 0.57 2.78 10.5 6.21 5.70 0.92 Crushed glass 0.50 2.78 10.5 5.43 5.23 0.96 Crushed glass 0.40 2.78 10.5 5.12 4.67 0.91 Crushed glass 0.30 2.78 10.5 4.46 4.14 0.93 Crushed glass 0.20 2.78 10.5 3.79 3.67 0.97 Gravel 0.57 2.78 8.98 5.24 5.31 1.01 Gravel 0.50 2.78 8.98 4.72 4.89 1.04 Gravel 0.40 2.78 8.98 4.31 4.42 1.03 Gravel 0.30 2.78 8.98 3.98 3.95 0.99 Gravel 0.20 2.78 8.98 3.33 3.53 1.06 Crushed limestone 0.57 2.78 4.62 3.40 3.64 1.07 Crushed limestone 0.50 2.78 4.62 3.37 3.53 1.05 Crushed limestone 0.40 2.78 4.62 3.24 3.37 1.04 Crushed limestone 0.30 2.78 4.62 2.98 3.20 1.07 Crushed limestone 0.20 2.78 4.62 2.93 3.03 1.03 Lead drops 0.50 2.78 2.18 2.35 2.28 0.97 Lead drops 0.40 2.78 2.18 2.42 2.45 1.01 Lead drops 0.20 2.78 2.18 2.50 2.59 1.04

- ------------ -----Dynamic moduli of elasticity X w-•• psi

--

Steel punchings 0.50 3.06 30.0 8.56 8.26 0.96 Steel punchings 0.40 3.06 30.0 6.85 7.07 1.03 Steel punchings 0.30 3.06 30.0 5.99 6.06 1.01 Ottawa sand 0.40 3.06 11.0 5.60 5.05 0.90 Ottawa sand 0.20 3.06 11.0 4.25 3.98 ' 0.94 Crushed glass 0.57 3.06 10.6 6.17 6.06 0.98 Crushed glass 0.50 3.06 10.6 5.90 5.57 0.94 Crushed glass 0.40 3.06 10.6 5.27 5.02 0.95 Crushed glass 0.30 3.06 10.6 4.66 4.47 0.96 Crushed glass 0.20 3.06 10.6 4.05 3.95 0.98 Gravel 0.57 3.06 8.64 5.74 5.45 0.95 Gravel 0.50 3.06 8.64 5.21 5.08 0.98 Gravel 0.40 3.06 8.64 4.86 4.62 0.95 Gravel 0.30 3.06 8.64 4.20 4.16 0.99 Gravel 0.20 3.06 8.64 3.83 3.76 0.98 Crushed limestone 0.57 3.06 4.45 3.80 3.73 0.98 Crushed limestone 0.50 3.06 4.45 3.74 3.64 0.97 Crushed limestone 0.40 3.06 4.45 3.63 3.52 0.97 Crushed limestone 0.30 3.06 4.45 3.38 3.40 1.01 Crushed limestone 0.20 3.06 4.45 3.37 3.27 0.97 Lead drops 0.50 3.06 2.41 2.87 2.51 0.87 Lead drops 0.40 3.06 2.41 2.97 2.69 0.91 Lead drops 0.20 3.06 2.41 3.29 2.85 0.87 ----------'------------------------------------


average deviation of the hypothesis from these data was within ±11.5 percent, and the maximum deviation was within ±35 percent.

When all the data noted in this report are considered, the average deviation of the hypothesis is less than ±10 percent and the maximum deviation is still only ±35 percent. These results are considerably more reliable than values obtained from any other equation currently in use.

MODULUS OF ELASTICITY OF CEMENT PASTE

Limited tests were conducted to determine the effects of the watercement ratio and of the age of cement paste on its modulus of elasticity. The results of these tests are presented in Fig. 6. It is pointed out that these tests were performed on water-cured specimens, since dry specimens free from cracks are extremely difficult if not impossible. to obtain.

GENERAL DISCUSSION

This report of this study is being presented for its academic and scientific value only. At present there is not enough data on the moduli of elasticity of various cement pastes and on the moduli of various aggregate types to support the general use of the proposed equation in design work. More extensive and more refined research needs to be done to determine the stress-strain properties of different aggregates, cement pastes, and even concretes.

There is also a need for laboratory data concerning the elastic and creep properties of the various new manufactured lightweight aggregates. The major difficulty of such a study is that of obtaining from the material representative test specimens of the required size.

Additional work is needed concerning the modulus of elasticity of concrete with three or more phases. One example is lightweight aggregate blended with siliceous sand used in air-entrained concrete.

This investigation of the modulus of elasticity of concrete has been limited to a consideration of the relationship between stress and the resulting immediate strain. The creep strain, or delayed strain which is a function of time and sustained stress level, is often considered highly important by structural engineers. Most of the research concerning creep strain in concrete has been limited to attempts to relate this property to the compressive strength or some other familiar property of concrete. It is felt that a study concerning the effects of the creep strain of the cement paste matrix and aggregate on the creep strain of concrete would yield fruitful results. Since creep strain. like elastic strain, is believed to be proportional to stress, the hypothesis presented in this paper may prove to be useful in such an investigation.

446 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE March I 962

CONCLUSIONS

The main points brought out by this investigation of the effects of the elastic moduli of the cement paste matrix and aggregate on the modulus of elasticity of concrete are as follows:

1. The tests have shown that the modulus of elasticity of concrete is a function of the elastic moduli of its cement paste matrix and of its aggregate constituents.

2. The degree to which the elastic properties of one of the ingredients affects the modulus is a function of the quantity present in the batch.

3. The general form of an equation has been derived which expresses the modulus of elasticity of concrete or mortar in terms of one empirical constant, the elastic moduli of the matrix and aggregate constituents and the quantities of each present in the mix. If Ec, E,, and Em are the moduli of elasticity of the concrete, aggregate. and matrix, respectively, and Vc, V,, and V, are the volume of concrete aggregate, and matrix, respectively, then for E, and E, > 0

(4)

where

K, 1- ~ [ 1 (5)

and

2Z r 1 J K" = 1 - ~ 1 - ( 1 - V,) + V. ~ . V, V, Em

(6)

where Z is an empirical constant.

The empirical constant has been tentatively evaluated as 0.785, and modulus values computed with this equation were shown to be in good agreement with laboratory data which covered an unusually wide range of variables. The average deviation is less than ±10 percent, and the maximum deviation was ±35 percent. These results are considerably more reliable than values obtained from any other equation currently in use.

4. The modulus of elasticity of cement paste is a function of its water-cement ratio and of its age. Typical values range from I X 1 or. to 4 X 106 psi.


ACKNOWLEDGMENTS

Recognition is due to Henson K. Stephenson, Truman R. Jones, and Robert M. Holcomb for their encouragement, suggestions and constructive criticisms which were invaluable in carrying out this project. Acknowledgment is also due the Texas Transportation Institute and the Civil Engineering Department of the Texas A & M College System in whose laboratories this research was conducted.

REFERENCES

1. Bingham, E. C., and Reiner, "Rheological Properties of Cement-MortarStone," Physics, American Institute of Physics, V. 4, No. 3, New York, N. Y .. 1933, pp. 88-96.

2. Birch, Francis, "Handbook of Physical Constants," Special Papers, G.S.A .. No. 36, 1942.

3. Dantu, P., "Etude des Contraintes dans les Milieux Heterogenes Application au Beton," Publication No. 57-6, Laboratoire Central des Fonts et Chaussees (Paris), Mar. 5, 1957.

4. Washa, George W., and Fluck, Paul G., "Effect of Sustained Loading on Modulus of Elasticity," ACI JouRNAL, Proceedings V. 46. No. 9, May 1950, pp. 693-700.

5. Kaplan, M. F., "Flexural and Compressive Strength of Concrete as Affected by the Properties of Coarse Aggregate," ACI JouRNAL, Proceedings V. 55, No. 11, May 1959, pp. 1193-1208.

6. Kaplan, M. F., "Ultrasonic Pulse Velocity, Dynamic Modulus of Elasticity, Poisson's Ratio and the Strength of Concrete Made with Thirteen Different Coarse Aggregates," International Association of Testing and Research Laboratories for Materials and Structures (RILEM), Paris, June, 1959.

7. Koenitzer, L. H.. "Elastic and Thermal Expansion Properties of Concrete as Affected by Similar Properties of the Aggregate,'' Proceedings, ASTM, Part II, v. 36, 1936, pp. 393-410.

8. LaRue, H. A., "Modulus of Elasticity of Aggregates and Its Effect on Concrete," Proceedings, ASTM, V. 46, 1946.

9. Noble, P.M., "Effect of Aggregate and Other Variables on Elastic Properties of Concrete,'' Bulletin No. 29, Kansas State College, Manhattan, Kan., 1932, p. 27.

10. Pauw, Adrian, "Static Modulus of Elasticity of Concrete as Affected by Density," ACI JouRNAL, Proceedings V. 57, No. 6, Dec. 1960, pp. 679-687.

11. Walker, Stanton. "Modulus of Elasticity of Concrete," Structural Materials Research Laboratory, Bulletin 5, Lewis Institute, Chicago, 1923.

12. Willis. T. F., and De Reus, M. E., "Thermal Volume Change and Elasticity of Aggregates and their Effect on Concrete," Proceedings, ASTM, V. 39, 1939, p. 919.

Received by the Institute May 16, 1961. Title No. 59-12 is a part of copyrighted Journal of the American Concrete Institute, Proceedings V. 59, No. 3, Mar 1962. Separate printe

are available at 60 cents each

AmP.rican Concrete Institute, P. 0 Box 4754, Redford Station, Detro.,t 19, Mich

Discussion of this paper should reach ACI headquarters in triplicate by June 1, 1962, for publication in September 1962 JOURNAL.


APPENDIX

STRESS VARIATION AND AVERAGE STRESSES

To aid in this development of the average stress factors K1 and K" reference is made to Fig. 2 which is an enlargement of a small part of Fig. 1. The aggregate is assumed regular in shape, and its modulus of elasticity will be considered larger than that of the cement matrix, E. > E,., for convenience only. The infinitesimal length dL, intersects common horizontal faces between the aggregate particles and the cement matrix. At these points it is reasonable to assume that the unit compressive stress in the aggregate S, is equal to the compressive stress in the cement matrix S,. Therefore, at dL"

Sa = S, =Sa ... . ...... (Al)

On the other hand, the infinitesimal lengths dL1 and dLa intersect the aggregate particles at mid-depth. Here the cement matrix and aggregate particles have a common vertical face bonded together. As pointed out previously, the stresses in the aggregate and matrix are variable up and down the depth of the aggregate particle and are equal at the top and bottom. Therefore at these points, at middepth, it seems reasonable to assume that the stresses will reach their maximum or minimum value. At these points the change of normal stress with respect to depth would be zero, and hence the shear stress here would be zero. Consequently, it is assumed that the unit strains in the aggregate Ea and adjacent cement matrix e, are equal.

Thus

Ea == Em

~dL E.

and

s.

at dL1 and dLa. Since the normal unit stresses in the matrix at dL, and dL, are different, shear

stresses along the vertical faces of the aggregate particles and matrix must account for this change. These vertical shear stresses are indicated in Fig. 2 (left), and the assumed variation of the normal compressive stress is indicated accordingly in Fig. 2 (right).

At these points corresponding to dL, and dL, the hypothetical maximum and minimum values of compressive stress occur. For example if it is assumed that E.> E,, then

S. (max) = S, (min)~ ..... Em

. ..... (A2)

Now if it is assumed that the infinitesimal length dL, slices across a representative plane of the concrete specimen, then the total stress on the aggregate F. and matrix F,, in this plane must equal the applied external stress F,.

Thus

Fa (A3)


and

S, A, = S. (max) A. + S.., (min) Am . (A3a)

Substitution of Eq. (A2) into Eq. (A3a) yields

S, A, = Sm (min) ;: A. + Sm (min) A .... . ......... (A3b)

If the small slice of concrete cut by dL" is representative of the total volume, then A, = VjL,, A, = V./L,, and A,, = V,,/L, .. Substituting these values in Eq. (A3b) produces

Solving for Sm (min) yields

S,. (min)

s,. (min) [~ v. + vmJ ....... E"t

s, V. E. VcEm

. (A3c)

...... (A4)

Eq. (A4) gives the hypothetical minimum unit compressive stress in the matrix when E. > E,,, On the other hand if E, < Em, it would express the hypothetical maximum compressive stress in the matrix of the hypothetical concrete mass illustrated in Fig. 2.

Eq. (A4) can now be substituted in Eq. (A2) to produce an expression for the maximum compressive stress in the aggregate. This stress is also in terms of the concrete stress.

Thus

S. (max) s, Em ( 1 ·- V., ) + E, V,

v. v,

..... (A2a)

It can be seen that the values of the minimum and maximum unit compressive stresses in the cement matrix are expressed by Eq. (A4) and (Al), respectively. Likewise, the minimum and maximum unit compressive stresses in the aggregate are given by Eq. (Al) and (A2a), respectively.

With these hypothetical values of the maximum and minimum compressive stresses, values of K, and Kz in Eq. ( 4) could be mathematically determined if the manner of variation of the stress were known. One possible assumption of the variation is that it is sinusoidal. Therefore, the compressive stress in the cement matrix in Fig. 2 (right) may be expressed as

S .. = S, - [S, - S,. (min)] Z sin :rr:ij' .. . ....... (A5)

in which Z is an empirical constant which will allow some empirical modification of the assumed variation. This sine curve variation has been chosen because it satisifes the boundary conditions, it is a continuous function and because past experience has shown that stress and strain variations can frequently be approximated by such a function. In addition, this stress variation can easily be modified at some future time to fit experimental results by the addition of additional sine terms to form a Fourier series.


The average stress in the matrix over the length M is therefore

S,. (avg) ir .!M1 S, - [S, - S, (min)] Z sin ~, ~ dy,

0

Sm (avg) s .. 11 - ?;~[ 1 -( ~ 1-----=-v.--,-----)1 +----=-v.=--E, ]t t v,. -V, E, )

and

where

K 2 = 1 - 2: [ 1 - ( 1 _ ~) + ~ -t.:l· .......... (A6) t

This K2 is the average stress factor required in Eq. ( 4) for the modulus of elasticity of the concrete. Similarly the variable compressive stress in the aggregate is

s .. S, + [Sa (max) - S,] Z sin Jt}{" . . ......... A

The average stress is

S, (avg) ! /1 S, + [S, (max) - S,] Z sin Jt}{." ~ dy2,

0

S, (avg) s,) 1-~ 1

l Jtl Em ( 1 E ..

1 ]t Va) + ~ f v, v.

and

S, (avg) S, K,

where

. (A7):j:

This expression for K, is the other stress factor required in Eq. ( 4) for the modulus of concrete.

t See Eq. (6). t See Eq. (5).

MODULUS OF ELASTICITY

Modulo de Elasticidad del Concreto Afectado por los M6dulos Ell:isticos de la Matriz de Pasta y Agregado

451

Se deriva una ecuaci6n general para expresar el modulo de elasticidad del concreto o mortero en terminos de una constante empirica, los m6dulos elasticos de la matriz de cimento y los agregados constituyentes, y las proporciones de la mezcla. Los ensayes de laboratorio han demonstrado que la ecuaci6n da buenos resultados para un gran numero de variables. La desviaci6n promedio se encontra, estuvo de ± 10 porciento y que la desvaci6n maxima dentro de ±35 porciento.

La variedad de materiales usados en el agregado incloyeron escoria de acero, vidrio triturado, gotas de plomo, arena de Ottawa, piedra caliza triturada y grava de rio de tipo calcarea siliceo.

L'Effet des Modules d'Elasticite de la Matrice de PaJe de Ciment et de l'Agregat sur le Module d'Elasticite du Beton

Une equation generale est developpee qui exprime le module d'elasticite du beton ou du mortier en termes de constante empirique, les modules d'elasticite de la matrice de ciment et des agregats constituants, et sur la proportion du melange. Des essais en laboratoire ont prouve que l'equation produit de bons resultats pour un champ tres large de variables. L'ecart moyen se trouva entre plus ou moins 10 pourcent, et l'ecart maximum entre plus ou moins 35 pourcent.

La variete des materiaux utilises comme agregats incluaient des rebuts d'etampage d'acier, du verre, des egouttures de plomb, sable standard Ottawa, de la pierre calcaire concassee, et un gravier pluvial de silice calcaire.

Einfluss der Elastiziti:itsmodule des Zementmi:irtels und der Zuschlage auf den Elastizitatsmodul des eBtons

Es wird eine allgemeine Gleichung abgeleitet, die den Elastizitatsmodul des Betons oder Mi:irtels in Form einer empirischen Konstanten, der Elastizitatsmodule der Zementpaste und der Zuschlagstoffe und auf die Mischungsverhaltnisse ausdriickt. Laboratoriumversuche zeigten, dass die Gleichung tiber einen weiten Bereich der Variablen gute Ergebnisse liefert. Es stellte sich heraus, dass die Durchschnittsabweichung bis zu ± 10 Prozent betragt, bei einer maximalen Abweichung bis zu ± 35 Prozent.

Als Zuschlagstoffe wurden u.a. verwendet: Stahlstanzschrott, zerstossenes Glas, Bleiabflille, Ottawa Sand, gemahlener Kalkstein und ein kalk- und siliziumhaltiger Flusskies.

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