popovic s 1973

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CEMENT and CONCRETE RESEARCH. Vol. 3, pp. 583-599, 1973. Pergamon Press, Inc

Printed in the United States.

A NUMERICAL APPROACH TO THE COMPLETE

STRESS-STRAIN CURVEOF CONCRETE

Sandor Popovics

College of Engineering

Northern Arizona University

Flagstaff, Arizona 86001

Communicated by R. E. Philleo)

ABSTRACT

This paper presents the experimental jus tification of two

previously published formulas, Eqs. 2) and 6), for the

estimation of the complete stress-strain diagram of con-

crete. Eq. 2) combined with Eq. 3) differs from the other

formulas offered in the literature for similar purpose in

that provides more relative curvature in the diagram for

concretes of lower strengths. Also, with Eq. 6), i t can

take the fact into consideration that the value of ¢0 in-

creases with increasing concrete strength. The result of

these refinements is that the stress-strain diagrams cal-

culated by these formulas f i t better the experimentally

obtained diagrams and within wider limits than the similar

formulas available in the li terature. Figs. 5a through 5d,

8a through 8d, and 9.)

Diese Arbeit veranschaulicht die experimentelle Rechtfertigung

zweier frUher verBffentlichter Formeln, Gleichungen 2) und 6)

zur Bestimmung des vollst~ndigen Spannungs-Dehnungs-Diagramms

des Betons. Die mit Gleichung 2 verbundene Gleichung 3 weicht

insofern yon den anderen, bisher in der Literatur bekannten

Formeln der gelichen Richtung ab, als sie mehr relative

KrUmmungen in dem Diagramm fur Beton niederer Festigkeit auf-

zeit. Auch kann man mit Gleichung 6 annehmen, dab der Wert

¢o mit zunehmender Beton festigkeit zunimmt. Das Resultat

d]eser Verbesserungen zeigt, dab die mit diesen Formeln berech-

neten Spannungs-Dehnungs-Diagramme sich besser und im weiteren

Rahmen den experimentell erhaltenen angleichen, als ~hnliche

Formeln, die man der Literatur entnehmen kann.

58

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584 Vol. 3, No. 5

STRESS STRAIN CURVES, CONCRETE, THEORY, DATA

Introduction

This paper is the continuation of the previous work of the wr it er

on the stress-strain diagram. I) - 3) The purpose of th is study is

a) to present formulas for the estimation of the complete st ress-s train

diagram of normal-weight concrete, made with a given aggregate and tested

with a given procedure, under short-term loading either sole ly from the

f compressive strength or from the combination of the compressive strength

o

and the measured C unit st ra in in concrete at the f ult imate stress Fig.

o

l ) ; and b) to show the applicabi l i ty of these formulas by demonstrating

thei r goodness of f i t to pertinent experimental results.

The discussion of the stress-strain curve of concrete is timely from

a theoretical point of view because deformations may provide ind irec t

information concerning the internal structure as well as the fai lur e mech-

anism of concrete. From a practical standpoint, the ultimate-strength

design of reinforced concrete elements brought the stress-st ra in relat ion-

ship into focus. Also, a knowledge of the deformabil ity of concrete is

necessary to compute def lections of st ructures, to compute stresses from

observed st ra ins, to design sections of highway slabs, to compute loss pre-

stress in prestressed elements, etc. 4) In the study of models of concrete

structures, i t is also necessary to know the st ress -st rain characteristics

of the model material so that dimensional simi lar ity may be obtained.

Stress-Strain Diagram for Compression

The diagram is influenced considerably by the test ing conditions type

of the test ing machine, rate and duration of loading, size and shape of the

specimen, size and location of the st ra in gages, number of load repet it ions,

etc. ) as well as by the age and composition of concrete, especial ly by

the type and quanti ty of aggregate and by the porosity, in a similar but

not ident ical way as the concrete strength is influenced by most of these

factors. 2)

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Vol. 3, No. 5 585


/ / ~ C o n s t a n t r a t e o f s t r e s s

/ - : -

/ / ~ C o n s t a n t

,

~ / / r a t e o f s t r a i n ~ .

/ / i - - . . .

t I

E U n i t S t r a i n

F i g . 1 . Two t y p i c a l s t r e s s s t r a i n c u rv e s f o r c o n c r e t e u nd er u n i a x i a l l o a d .

The top curve i s c h a ra c t e r i s t i c o f a load ing p rocess where the

r a t e o f s tr e s s i n c r e a s e is k e p t c o n s t a n t d u r i n g t he t e s t i n g . The

bo ttom curve i s o b ta ined by keep ing the ra te o f s t r a in inc rease

c o n s t a n t .

Another interest ing fact is that the stress-strain diagrams for stones

and hardened cement pastes under uni-axial loading are practica ll y straight

lines almost up to the ultimate stress. Yet the same diagram is curved

for mortars consisting essential ly of the same two comoonents, and even more

curved for concretes Figs. 2 and 3), as has been pointed out by Gilkey and

Murphy 5). This paradox can be explained in qualitative terms by the

internal cracking 7) and creep of the hardened paste in concrete under

load that are produced by the stress, especially by the stress concentrations,

result ing from the embedded aggregate particles 8). However, the exact

nature of this problem is so complex that only empirical formulas are avai l-

able in the literature for numerical approximation of the stress-strain

diagram. These formulas have been discussed elsewhere. 1)

In most of these formulas the E/Eo rat io is a fixed number regardless

of the concrete strength. Thi s res tr icts the limits of val id it y of these

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586 Vol. 3, No. 5

STRESS STRAIN CURVES CONCRETE THEORY DATA

Fig. 2.

m

o -

or)

Q

t i

Q

I Z :

. . o

I .O O i

0 8 0

0 6 0

0 . 4 0 .

O ZO

O K

0

P a s t e s

n ~ 1 2

/7 f o , p s i

. . . . . . . . 5 8 2 5

- - - - - - 5 2 2 0

- - 4 3 5 O

8 3 3 5

0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0

I

I

I

I

I

I

I

I

I

I

m

1.00

1.00 '

w 0 8 0

Ik..

11-

0.60

_u 0 4 0

n,,

~ 0 .20 ,

0

M o r t a r s

f o , p s i n

. 17 45 1.9

2 . 5 4 5 0 2 .0

3 . 5 8 0 0 Z I

4 . 4 5 6 0 2 .4

5 . 4 7 0 5 2 . 5

6 . 7 5 6 0 2 ,8

7 . 1 0 9 7 0 3 . 0

8 . 1 5 4 7 0 3 . 7

0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0

~ ' / ~ o R e l a t i v e S t r a i n

1.00

Relative stress-strain diagrams for hardened pastes and mortars of

various strengths along with the best f i t values of n. The curves

were published by Gilkey and Murphy. 5)

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Vol. 3, No. 5 587


Fig. 3.

1.0

e e 0 . 8

m

Q

t .

0 -

0.6

_ >

¢}

O 0 . 4 -

0 .2 -

oncrete

f o p s i

i . 1200

2 . 1600

3 . 2 2 5 0

4 . 3 0 0 0

5 . 4 5 0 0

6 . 6 0 0 0

n

1.4

1.7

2.0

2.5

3.0

4 . 5

0 0 . 2 0 . 4 0 . 6 0 6 1 .0

• ~ R e l a t i v e S t r a i n

Re la t i ve s t re ss -s t ra in d iagrams fo r concretes o f var ious compressive

stren gths a lon~ w i th the best f i t va lues of n. The curves were

pub l ished by Rusch. (6)

formulas because experimental data show that the E/Eo ratio varies from

near 4 for normal concretes of l,O00 psi to about 1.3 for concretes of

lO,O00 psi . Consequently, when such a formula f i t s a concrete of medium

strength, i t wi l l over estimate the stress fo r a given strain in the ascend-

ing branch of the st ress-s train diagram for the high-strength concretes and

under estimate i t for low-strength concretes. (9) Formulas with variable

E/Eo rat ios are more fl ex ible because they can take the composition of the

concrete into consideration di rect ly or indi rect ly , such as through the

on rete

strength. Such a formula is the fol lowing (3):

n - l

f = EE l)

n - l + (E/Eo)n

or, considering that at = EO~E = (fo/Eo)n/(n - I ) ,

f - o

n

2

n - l + (~/~_)n

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588 Vol. 3 No. 5


where the n power can be expressed as an approximate function of the

compressive strength of normal-weight concrete as follows:

nconcrete = 0 4xi0-3 fo + l.O 3)

Similar formulas for cement mortars and pastes are:

nmortar = O'15xlO-3 fo + 1.5 4)

and

npast e = 12. 5)

The long fractions in Eqs. I) and 2) represent the deviation from

the linear el as ti ci ty . This might be ut il ized in the future for the analysis

of crack propagation in concrete. Note also that these equations, combined

with Eq. 3), di ff er from the other formulas offered in the li te rature for

simi lar purpose in that they provide more re lat ive curvature in the diagram

for concretes of lower strengths.

Figure 4 il lus tra tes Eq. 2) in relative terms for normal-weight con-

cretes and pastes. I t is impossible to produce a single formula similar

to Eq. 3) for lightweight concretes in general because the various types

of such concretes have greatly dif fering deformabi lit ies. For instance,

gas concretes show nearly linear elasticity while concretes made with ex-

panded blast-furnace slag as aggregate have highly curved stress-strain

diagrams. (lO) I t should also be pointed out that the formulas above

are valid only for standard concrete specimens with a height-width ratio

not less than two, and when the uniaxial compressive load is a short-term

load which is applied at a rate that produces constant rate of strain in

the specimen. (Fig. l)

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Vo l . 3 No . 5 589


1.0 - - f,,= 2 0 0 0 ps i

0 8 0

/ / / / / A / f _ _ . ~.L~. n _ \ ~ 6 0 0 0

,0 00

- ~ H i l l / . I \

= /

, .= 0 .2 / ~

~ . ~ : . . . . .

0 0 .2 0 . 4 0 .6 0 .8 1 .0 1 .2 1 .4 1 .6 1 .8 2 .0

6 /£0 Relat ive Stra in

Fig. 4. Calculated relative stress-strain diagrams for normal-weight

concretes of various compressive strengths and for cement pastes.

Compa.rison to Experimental Data

It has been shown in qualitat ive terms 3) that Eqs. 2) through 5)

are more suitable for the description of the stress-strain diagram of a

concrete than the formulas with fixed E/Eo rat io. However, i t has not

been examined yet that to what extent experimental data support these for-

mulas. This examination is presented below for two di fferent situations:

a) when both the fo and theG o values are available; and b) the more

practical case, when only the value of fo is available. Incidental ly,

Eq. l ) is not recommended for practical purposes, although i t is an inter-

esting formula because, for a given n value, i t expresses the stress values

purely in terms of deformations.

In the fi rs t situation, that is, when both fo and Lo are measured

for a concrete, mortar, or paste, the corresponding value of n can be cal-

culated from one of Eqs. 3) through 5), and used with Eq. 2). Since Eqs.

3) through S) were obtained by f i t t ing the curves of Figure 4 to those in

Figures 2 and 3, the stress-strain diagrams calculated in this way obviously

f i t the pertinent experimental data by Gilkey 5), and by R~sch 6). Other

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590 Vol. 3, No. 5


8

7

6 -

. S-

9

~ 4 -

,23

I -

0

8

7 -

6

~ , . ~ w / ¢ = 3 3

.50 ~

~ a

.

o

E . S t r a i n , i n / In X I O 3 )

W/C 3340

/ / i i f . . . . o o

/S

2 3 4 5

L~. S t r a i n , i n / i n X I O s )

7

6

5 ~

X 4 -

?.

3 -

I

o

o

i i i i g

8 -

~ - ' w / ~ , ~ . 33

7

/ /

x 4

l b o y s l ~

.67 ~

t o o I

o

o

w / c , 3 3

K

// I// ~ i 90 days

LO

I 2 3 4 5

E S t r a i n , i n /i n X I O s )

L~: F . S t r a i n , i n / i n X I 0 3 )

Figs. 5a through 5d. Comparison of experimental stress-strain diagrams

of various concretes (continuous lines) to those plotted from

Eq. 2) with Eq. 3) (dash lines). The experimental curves were

taken from Reference (ll).

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Vol. 3, No.


L e g e n d

591

T h e c o m p l e t e c u rv e w a s o b ta in e d

e x p e r i m e n t a l l y .

f : f

- I + ( F . / ' & . )

n - o . 4 x lO S ~ o l

f . a n d . w e r e o b t a in e d e x p e r i m e n t a l l y .

f : f . ~ n - , . r } , ~ . .

n . o . , x , o f . I

( ; 2 . 7 x I 0 4

w a s o b t a in e d e x p e r i m e n t a l l ~

curves calculated with Eqs. 2) and 3) are compared in Figures 5a through

5d with experimental curves published by Hognestad et. al . l l ) The com-

parison is done with 20 pairs of diagrams representing five water-cement

ratios and four ages. I t can be seen that despite the wide ranges in water-

cement rat io from 0.33 to l.O by weight) and age at testing from 7 to

90 days), the calculated diagrams f i t the experimental curves quite wel l.

In the second si tuat ion, when only fo is given, the value of Eo can

be estimated from one of the available formulas in the li te ra tu re , and

then this value used with Eq. 2) again, as mentioned above. A convenient

form of such a formula was presented earl ier - 3) from certain considerations,

but without experimental just i fi ca ti on , as follows:

4 4

kx1o

6

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592 Vol. 3, No. 5


4

0

3

X

_=

r

°

. 2

7

6

o =

5

=

o

7 d a y s

x 1 4 d a y s

o 2 8 d a y s

A 9 0 d a y s

£ , = 2 . 7 X 1 0 4 X

m

A

I X

0

I J w i I I l,og ~ool~

1 .3 2 3 4 5 6 7 8

f~ p s i x I 0 ~

Fig. 6. Example for the relationship between~o and fo for normal-weight

concretes of various ages. The experi6ental data were taken

from a paper by Hognestad et al. l l ) .

where k is a function of the type of mineral aggregate used and the applied

test method. As can be seen from Figure 6, Eq. 6) is supported quite well

by the test results on normal-weight concretes that were published by

Hognestad et al. l l ) , although the f i t could be improved by including

the age as an extra variable). Figure 7 shows that test results published

by Watanabe 12) again support Eq. 6). The difference between the defor-

mabilities of lightweight and normal-weight concretes is also illustrated

in this figure.

Returning to the original question, Figures 8a through 8d demonstrate

the goodness of f i t of the curves calculated from the pertinent combination

of Eqs. 2), 3) and 6) the latter with k = 2.7 in~/Ib~) to the same exper-

imental curves that were discussed in connection with Figures 5a through

5d. Figure 9 shows another comparison, with k = 2.2 in~/Ib~, to a set of

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Vol. 3, No. 5 593

STRESS STPJ~IN CURVES, CONCRETE, THEORY, DATA

7

6

4

u

X

¢

o l i g h t w e i g h t c o n c r e t e

'--~ o e + n o r m a l - w e i g h t c o n c r e t e

C ° = 2 . 8 5 x 1 0 4 ,~ '~

( l o g s c a l e ]

I I I I I

2 3 5 6 7 8

f o , p s i x I 0 s

Fig. 7. Examples for the relationship between sand fo for l ight-weight

as well as normal-weight concretes. The experimental data were

taken from a paper by Watanabe (12).

experimentally obtained stress-strain diagrams by Smith and Young (9).

I t can be seen again that the presented formulas provide a good estimate

of the complete stress-strain diagram of a concrete. As a matter of fact,

the approximation of the proper combination of these formulas is better

within the given wide ranges of water-cement ra tio, compressive strength,

and age than the approximations of other formulas recommended in the l i t -

erature for the same purpose. (9) (13) - (16)

The stress-strain diagram for a concrete under short-term uniaxial

tension is similar to the diagram produced by compression, except that the

same curve is applicable for a ll the relative tensile stress-st rain diagrams

regardless of the strength. (17) I t is probably more than coincidental

that this single tension curve is very close to the compression curve

presented in Figure 4 for pastes.

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594 Vol. 3 No. 5


8 - 8

7 7

_ _ W 1 C , . 3 3

f ' - ' - z t , . -

I /

/j

o ....

~ ¢

. -

1 7 a a y ] i C 4 a . s q

- / . 6 7

I , / / / / I Loo

1 0

0 , , ~

~ ~

0 I 2 3 4 - 5 0 I 2 :3 4 5

£ S t r a i n , i n / i n X l O 3 ) ~ E S t r a i n , i n / i n X I O 3 )

8

8 - - . . - - - - - / ~ / / ~ / c . = . ] 3

/ F. ' ° ' ' • /,./I- -...........

/ /

/ /' ......50

/ . . .. .- - 7 / ~ i ~ ° ~ ~

~

I I

- ~ / , 7 / ~ / / / /

l I I z e d o y l i ,,/,~ . ~ . go ~ o y s

/ /

u 2 l / / / / / J o o

~ _ /~/ . . . . ~ . , ~ , ~ - - - - ~ - - , e -

l

~ o o

0

o o i 3 : 4 ~

o i ~ ~ ~

~ . £ S t r a i n , i n / i n ( X I 0 ) ~ . £ S t r a i n , i n / i n ( X l O )

Figs. 8a through 8d. Comparison of experimental stress-strain diagrams

of various concretes continuous lines) to those plotted from

Eq. 2) with Eqs. 3) and 6) dot lines). The experimental curves

were taken from Reference l l).

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Vo l . 3 No. S 595


Fig. 9.

t

0

8

6

x

w

= 4

( D

2

f = 7 1 8 0 p s i

4 4 4 0

3 5

1 2 7 5

I I i t l I ~ ,

0 I 0 0 0 2 0 0 0 3 0 0 0

S t r a i n m i c r o i n / i n

Comparison of stress-strain diagrams of concrete cylinders (con-

tinuous lines) to those plotted from Eq. 2) with Eqs. 3) and 6)

(dash lines). The experimental curves were taken from Reference

(9).

Supplementary Remarks

(a) The area under any of the stress-strain diagrams calculated from

the presented equations can be determined by numerical integration for

which the computer programming is quite simple.

(b) Di fferentiat ion of Eq. 2) provides the E tangent modulus of

elasticity of the concrete as follows:

n - 1 + E / ~ o ) n - n E / E o ) n

E = df/d = nfo/6 o =

[n -

1

+ ~ l ~ o ) n ] z

1 - , E / ~ o ) n

[ n - 1 + ~ / 8 o ) n ] 2

n n - 1 ) f o / £ o

7

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596 Vol. 3,

N o ,

5


This becomes E at ~= O, that is

n 8

E / E o = n - 1

which, with Eq. 3), provides the following relationship for normal-weight

concretes:

EIEo : l +

2,5oo 9

fo

This means that E/Eo is 3.5 when fo = l,O00 psi, and is 1.25 when fo =

lO,O00 psi. These figures are in accordance with experimental data.

(c) Since Eo = fo/ o , the combination of Eq. 9) with Eqs. 3) and 6)

provides the following relationship between the compressive strength and

ini t ial modulus of elasti ci ty of concrete:

4

lO fo + 2,500

E - lO)

This equation f i ts reasonably well, within 2,000 and I0,000 psi compres-

sive strength limi ts, the tradit ional empirical formula for normal-weight

concretes:

E = K~ o ll )

(d) I t follows from Eq. 8) that

nf o

12)

to ='(n - I)E

Therefore, with the consideration of Eq. 3):

£o = ~ fo + 2,500) 13)

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Vol. 3, No. 5 597


If E is given, or i t can be estimated (18), Eq. 13) provides ~o in

terms of the concrete strength. For instance, by substi tut ing Eq. ll ) for

E

2 s o o > 1 4 >

The peculiarity of Eq. 14) is that the value of Eo decreases with the

decrease of fo up to a point (which is in our case 2,500 psi) , then i t

starts increasing so that as fo approaches zero, £o approaches inf in i ty .

Since experimental data, such as shown in Figure 6, do not seem to show

such an increase in ~o at low strengths, one can conclude that Eq. 14) may

not hold for low strength normal-weight concretes because Eq. l l ) may not

hold for such concretes.

Conclusions

The proper combination of Eqs. 2), 3), and 6) seems suitable for the

estimation of the complete stress-strain diagram of normal-weight concretes,

made with a given aggregate and tested with a given procedure, either from

the fo and ~o values, or solely from the values of fo As Figures 5a through

9 demonstrate, the approximation of these formulas is bet ter within the given

wide ranges of water-cement ratio, strength and age than the approximation of

other formulas recommended in the l iterature for the same purpose. Math-

ematical analysis of Eqs. l) through 6) provides addit ional relationships

between the strength and deformation of a concrete.

REFEREN ES

( I ) Popovics, S., ACl Journal, Proc. 67, March 1970, pp. 243 - 248.

(2) Popovics, S., Symposium on Concrete Deformation, Highway Research

Record Number 324, Highway Research Board, 1970, pp. l - 14.

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598 Vol. 3, No. 5


(3)

Popovics, S., Mechanical Behavior of Materials, Proceedings of the

International Conference on Mechanical Behavior of Materials, Vol.

IV., The Society of Materials Science, Japan, 1972, pp. 172 - 183.

(4) Phil leo, R. E., Significance of Tests and Properties of Concrete

and Concrete Making Materials, ASTM STP No. 169-A, Philadelphia,

1966, pp. 160 - 175.

(5) Gilkey, H. J. , and Murphy, G., Proceedings ASTM, 38, Part I, 1938,

pp. 318 - 326.

6 )

R~sch, H., Versuche zur Festigkeit der Biegedruckzone (Experiments

Concerning the Strength of the Compression Zone in Bending), Deutscher

Ausschuss f~r Stahlbeton, Heft 120, Wilhelm Ernst Sohn, Berl in, 1955.

(7) Popovics, S., Journal of the Engineering Mechanics Division, Proc.

ASCE, EM 3, June 1969, pp. 531 - 544.

(8) Brown, C. B., and Mostaghel, N., Journal of Materials, 2, No. l ,

March, 1967, pp. 120 - 130.

(9)

(lO)

Smith, G. M., and Young, L. E., ACI Journal, Proc. 53, December 1956,

pp. 597 - 609.

Rbsch, H., and Sell, R., Deutscher Ausschuss f~r Stahlbeton, Heft 143,

Wilhelm Ernst Sohn, Berl in, 1961.

( l l ) Hognestad, E., Hanson, N. W., and McHenry, D., ACI Journal, Proc. 52,

December 1955, pp. 455 - 479.

(12) Watanabe, F., Mechanical Behavior of Materials, Proceedings of the

International Conference on Mechanical Behavior of Materials, Vol.

IV., The Society of Materials Science, Japan, 1972, pp. 153 - 161.

(13) Liebenberg, A. C., Magazine of Concrete Research, L4, No. 41, London,

July 1962, pp. 85 - 90.

(14) Alexander, S., Indian Concrete Journal, 39, No. 7, July 1965, pp. 274 -

277.

(15) Desayi, P., and Krishnan, S., ACI Journal Proc. 61, March 1964, pp.

345 - 350.

(16) Desayi, P., Publication No. 30, Annual Report of the Department of

Civil and Hydraulic Engineering, Indian Ins ti tu te of Science, Bangalor,

pp. 79 - 82.

(17) Johnston, C. D., Symposium on Concrete Deformation, Highway Research

Record Number 324, Highway Research Board, 1970, pp. 66 - 76.

(18) Popovics, S., American Ceramic Society Bulletin, 48, No. I I , November

1969, pp. I060 - I064.

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Vol. 3 No. 5 599


NOTATION

The letter symbols used in this paper and thei r defini tions are

as follows:

E

E

f

f

k K

= i n i t i a l m o d u l u s o f e l a s t i c i t y o f t h e c o n c r e t e ,

= s e c a n t m o d u l u s o f e l a s t i c i t y a t t h e f u l t i m a t e

o

s t r e s s , t h a t i s , E o = f o / ~ o '

= t a n g e n t m o d u l u s o f e l a s t i c i t y a t t h e ~ s t r a i n ,

t h a t i s , E = d f / d ,

= a x i a l s t r e s s i n t h e c o n c r e t e s p e cim e n

= u l t i m a t e s t r e s s ; i n c o m p r e ss io n i t m e a ns t h e

c y l i n d e r s t r e n g t h

= e x p e r i m e n t a l p a r a m e t e r s

= u n i t s t r a i n i n c o n c r e t e ca u se d b y t h e f s t r e s s

and

= u n i t s t r a i n i n c o n c r e te a t t h e f o u l t i m a t e s t r e s s .

popovic s 1973

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