Baiant, l.P., and Panula, L. (1978-79). 'Practical prediction of time-dependent deformations of concrete.' Materials and Structures (RILEM, Paris); Part III, 'Drying creep' Vol. 11 (1978),415-424.

Practical prediction of time-dependent

deformations of concrete

Part III: Drying Creep

Z. P. BAZANT e), L. PANULA e)

The practical model for predicting creep and shrinkage developed in Parts I and II is extended to creep at drying environment and constant temperature. The increase of creep due to drying is related to shrinkage. Formulas for determining material parameters from concrete strength and mix composition are presented and verified by extensive comparisons with test data from the literature.

INTRODUCfION

The prediction models for shrinkage and basic creep, developed in Parts I and II, must now be extended to cover creep in a drying environment. The expressions describing this behavior were developed in a preceding work [4] e), and they were shown to allow good fits of test data. However, each data set was fitted indivi­dually, and no formulas that correlate the material parameters and predict them from the given concrete strength and mix composition were derived. This will be done in this part, in which data analysis of un­precedented scope (24 different mixtures) will be undertaken.

FORMULAS FOR DRYING CREEP

In [4], the creep function at simultaneous drying has been expressed as

J (t, t:)= L +Co (t, t') +Cd (t, t', to)-C p (t, t', to), (25)

in which Co (t, t') gives basic creep [double power law, equation (11)], Cd represents the increase of creep due to drying, C p represents the decrease of creep

(') Professor of Civil Engineering, Northwestern University, Evanston, Illinois 60201, U.S.A.; Active Member, RILEM.

CZ) Graduate Research Assistant, Northwestern University, Evanston, Illinois 60201, U.S.A.; presently Eng., Tippetts, Abbett McCarthy and Stratton, New York.

(') Reference numbers not listed at the end of this part are found in the preceding parts.

after drying,t' is the age at load application, and to is the age at the start of drying (both in days). The expressions for Cd and Cp , which represent only slight modifications of the expressions derived in [4], are as follows.

Creep during drying:

C (t t' t )- ({J~ ,-m/2 k' S ( ') ) d , , 0 - If: t h cshoo d t, t , o

, (t'-tO)-1 /2 \ ({Jd= 1+ lO<sh ({Jd' )

(26)

Creep after drying (predried):

C p (t, t', to) = c p k~ S p (t, to) Co (t, t'). (27)

Time shapes (similar to shrinkage):

( 10 < )-Cd

n

./ S (t t')= 1+~ d , t-t"

( 100 <sh )-n

Sp(t,to)= 1+-- . t-to

(28)

Humidity dependence:

k~= Ihb· 5 -h1. 5 1, (29)

Here h = relative humidity of environment (constant); ho = initial relative humidity at which the specimen was in moisture equilibrium before time to ~ t' (usually ho=0.98 to 1.00). For a detailed discussion of these formulas see reference [4].

415

VOL. 11 - N° 66 - MATERIAUX ET CONSTRUCTIONS

Equations (25)-(29) reflect various experimental pro­perties of drying creep: (a) Drying that is simultaneous with creep intensifies creep (function Cd), but some time after moisture equilibrium is reached the creep is less than that of sealed wet concrete and is lower for a lower equilibrium moisture content (function C pl. (b) The increase of creep due to drying depends on environmental humidity and size similarly as shrinkage, as reflected in k~ and 'sit, and for very thick specimens ('sit ~ 00) the drying effect must vanish. (c) The increase of creep due to drying is higher at a lower age (cf t,-ml 2) and is higher for a concrete that shrinks more. (d) The later the concrete is loaded after the start of drying, the smaller the increase of creep due to drying; this is more pronounced for a thinner specimen (cf <fJ~). (e) At the beginning of drying, and after drying as well, the creep curves have the shape of power functions. (f) The creep increase due to drying is delayed behind drying itself and shrinkage by approximately one log-decade, which is why equation (26) for <fJ~ contains 10 'sit instead of 'sIt.( g) The creep decrease after drying is delayed by approximately one more log-decade, i. e., it begins long after the end of drying (reaching moisture equilibrium), which is why equation (28) for Sp contains l00,sIt instead of 'sit. (h) The size­dependence conforms to the diffusion theory; indeed, all half-times ('sit, to'sIt, l00,sIt) are proportional to the size-square [see equation (4), Part I].

The shrinkage-like function Sd causes that after the drying reaches moisture equilibrium, the slope of the creep curve in log-time decreases. From this fact, the existence of a final value of creep has often been inferred. However, according to our model, only the increase of creep due to drying reaches a final value, while the basic creep for the new equilibrium humidity probably continues without approaching a bound.

Carbonation effects have not been included in the present model. In dense, sound, uncracked concretes, the penetration of carbonation is very shallow (2 to 10 rom) and has a negligible effect unless the thickness of concrete is very small. Effects of humidity cycling are not included either. They also affect only a rela­tively thin surface layer, but they are undoubtedly important for thin structures, e. g., thin shells.

For derivation and detailed discussion of equations (25)-(29), the previous work [4] may be consulted. The modifications with respect to that work consist solely in using exponents -m/2 and -1/2 instead of -m and - 1 in equation (26), which is purely empirical.

FREDICTION OF MATERIAL PARAMETERS

By analysis of numerous test data, the following empirical formulas were derived:

(30)

for r> 0:

<fJd=0.OO8+0.027 u, (31)

for r ~ 0: <fJd=0.OO8,

416

(5 )0.3 r= 56 000 af~

(g )1.3 (W/c )1.5 x - -- -0.85 s 1:.'",

(32)

Here n = exponent given in Part II (equation 17); f: = 28-day cylinder strength (ksi); w/ c = water­cement ratio; g/ s = gravel-sand ratio; s/ a = sand­aggregate ratio (all by weight): D,y=final shrinkage in to- b given in Part I (equation to a).

According to equation (32), an increase in strength, or in water-cement ratio, or in gravel-sand ratio, each causes an increase of the drying creep effect relative to the ultimate shrinkage, I:,h ,and to the elastic defor­mation. An increase of tlle final shrinkage has the opposite effect. In view of equation (30), a higher ratio of the long-time creep to the short-time creep, as indicated by n (and discussed in Part II), means also a higher ratio of later to early drying creep.

COMPARISON WITH TEST DATA

The method of optimizing creep test data was the same as described for shrinkage.

Fits of 24 different comprehensive data sets available in the literature ([16], [18], [19], [36], [25], [15], [23], [20], [56], [43], [22], [58]) are exhibited in figures 19-24. All tests represent creep during drying, except Wittmann's test which gives creep after drying, and McDonald's and Kennedy's tests which give creep after resealing. Three different types of fits are shown.

The solid lines in figures 19-21 represent the fits where Eo is determined from the reported measured static elastic modulus, using equation (12) or, ifunavail­able, determined by optimizing the basic creep data, <fJd is optimized, and all other parameters are calculated from the formulas [equations (30), (16-18) and (9-10)]. These fits give a picture of accuracy of the shape of the functions given in equation (26)-(29) but not of the accuracy of the magnitudes. The dashed lines in figures 19-21 represent the fits when Eo is determined similarly as for the solid lines but <fJd is obtained from equation (31). These dashed line fits give an idea of accuracy when there is no error in the elastic modulus. Finally, figures 22-24 show fits when all material parameters, including 1/ Eo, are calculated from the preceding formulas [equations (26-32) and 1/ Eo from equation (19)]. A picture of the accuracy of the formula for <fJd may be gained from figure 25.

With regard to Keeton's data (fig. 20), it must be mentioned that the last data point of each reported curve (marked as + in figure 20) was disregarded in fitting test data because it seems that the reported curves were smoothed by hand in the actual time scale, in which it is not possible to see the' data 'trend near the end of the curve. Regarding Lambotte's data it should be noted that the curing conditions were quite unusual (exposure to drying environment at age of 24 hours).

'iii c.

<it-'Q .= --~ ...,

1.0

0,8

0,6

0,4

1.5

1.0

• 0.5 0

10

0.5

0,4

U,3

0.2

0.8

0.6

04

L' HermHe, Mam111an, Lefevre, 1965, 1971 RH = 50%

1/Eo = 0.091·lQ'"lps1 ~ = 3,113 n = 0.132 m = 0.316 ~ = 0.051 Cd = 1,810

- 'f,= 0.030 - - 'f',= 0.0306

1.0

0.8

0,6

Z. P. BAZANT - L. PANULA

L' HermHe, Mam111an. 1965 1/Eo = 0,091'lQ'6/ps1 t~ 2 8 da~s ~ = 3,111 n = 0.132 m = 0.316 ox = 0,051 Cd = 1.811

-'1'0=0.038 -- '1'0= 0,0305

o ..._----~

---/' ...-/'/'

..-"­/'

---

o

o

•

------------0.4,..,-_-......-- _-. -­-------

o

0 0

100

Rostas)' et al, 1971, 1/Eo = 0,1'lQ'6/pS1

if, = 2,982 n = 0.127 m = 0.307 0( = 0,045 Cd = 1.845

- 'fd= 0,0330

Hansen, Mattock, 1966, 1/Eo = 0.080 'lQ'6/pS1

'f, = 4.295 n = 0.176 m = 0,308 '" = 0.035 Cd = 1.477

- ifd = 0,0138

RH = 65%

10

RH = 50%

100

C, = 1.899 --'1', = 0.0162

1000

------ 'f.,= 0.0329

100

t'-edQys

-- '1', = 0.0130

1000

D~ __ --~D----~D~-----------------

•

0.3

0,2

t- t' in days

0.6 ,------------------;c-;;r­/./.o

/.

0.5

0,4

0.3

Hunmel et al, 1962 l/Eo = 0.111' lQ'6/pS1

4', = 3,025 n = 0,136 m = 0,309 '" = 0,066 Cd = 1.779

-'i'd= 0,0191

10

RH = 65%

-- ifd= 0,0202

100

11 Mossloss~an and Gamble. 1972, I RH = 50% I

/ lIEo = 0.117 'lQ'6/psi I

4', = 3,814 / 10 n = 0.168 I

/ 0 m = 0.3 / 0< = 0.051 I

0.9 Cd = 1.537 ;/" 0

0

-if,=0,030 /1;)4.0Ci 0

AI" 0 /' 0

0.8 / / 0

/ 0 / / 0

/ 0.7 /.

/. /. 0

/. 0 /.

/. 0 --- - '1',= 0.0342 0,6 /.

0 0.5

10 100

York, Kennedy, Perry, 1970 RH = 60% 1/Eo= 0,089'10'6/ps1

'1', = 3459 n = 0.156 m = 0.303 0< = 0,059 c,= 1.635

--'1',= 0,020

-- 4',= 0,0257

o

1000

I I

0

0 0

10 100

Fig. 19, - Fits of Tests for Drying Creep by L'Hermite, Mamillan and Lefevre (1965, 1971) [16], L'Hermite and MamilJ'an (196'5) [16], Troxell, Raphael and Davis (1958) [18], Hummel et al., (1962) [19], Rostasy et al., (1971) [36], Mossiossian and Gamble (1972) [25], Hansen and Mattock (1966) [15], York, Kennedy and Perry (1970) [23]. lPd optimized - solid line, lPd by formula - dashed line, 1/ Eo calculated from experimental value E, if available, or optimized from basic creep data.

417

VOl. 11 - N° 66 - MATt:RIAUX ET CONSTRUCTIONS

CJ)

a.

t2 .S

--_. -,

1.1 r-----------------------------------------~

1.0

0.9

0.8

0.7

0.6

0.5

0.4

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.7

0.6

0.5

0.4

0.3

0.2

0.6

0.5

0.4

0.3

0.2

a ) Keeton, 1965 Hunu.dity effect, small 51..Z8 cyolinders 3" x 9" 11E,= 0.125·10""/psi

'R = 3.373 n = 0.150 m = 0.303 0< = 0.055 c. = 1.673

- If.= 0.0081 -- 'fl= 0.0081

+

100 1000

c ) Keeton, 1965 Humidity effect large Sl.ze c)'ll.nders 6 11

)( 1811

liE, = 0.125 '10""/psi If, = 3.373 n = 0.150 m = 0.303 '" = 0.055 c. = 1.673

10

LambOtte, _ns, 1976 Concrete P2 liE. = 0.118 ·10-";PS1

4', = 3.265 n = 0.121 c. = 1.892 m = 0.349

-- <&= 0.0319 0< = 0.041 --'1\$ 0.0335 Concre te P52 liE. = 0.093 ·10-";psi

'1', = 3.679 n = 0.163 /-c.= 1.577 m = 0302

-'i'.= 0.050 '" ;:.0 ~ --'(,= 0.0342 ....... ..-(j

.Jj .....

0 .......... ....0--

10

Lambotte, Manneno, 1976 Concre te P50, P51: liE. = 0.094 '101'psi

If, = 3.467 n = 0.153 m =0.304 0< = 0.051 c. = 1.654

_'1'.=0.0329

0 • 0 0

• • • • 10

•

- '1'.= 0.0081 -- 'f.o= 0.0081

100 1000

Concre te P54 lIE.= 0.090 ·10O<;/pS1

If, = 3.148 n = 0.131 Go = 1.816 m = 0.319

- <fl. = 0.0365 ... = 0.048 if. - 0.0336

100 1000

o

o

-- '1'.= 0.0339

100 1000

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.8

0.7

0.6

0.5

0.4

0.3

0.6

0.5

0.4

0.3

0.2

b) Keeton, 1965 hum~d1ty effect, medl.um size c1'll.nders 411 x 1211 1/E.= 0.125 '10~psi

<fI, = 3.373 n = 0.150 m = 0.303 '" = 0.055 c. = 1.673

10

Lambotte, Mannens, 1976 Concre te p13: 1/E.= 0.103 'lO-'1psi

If, = 3.193 n = 0.133 c. = 1.806 m = 0.323

-If,,= 0.0285 ex = 0.050 - -<4'.= 0.0335 Concrete P31: liE. = 0.090 '1O-6/psi

If: = 4.261 n = 0.184 / c. = 1.418 m = 0.294

-'f,= 0.0231 '" = 0.043 --'1\= 0.0342 /'

/'/'

///////'/

o

10

Laml>Otte, Monmens 1976 Concra te P23: 1/E.= 0.102 '101'psi

If, = 3.629 n = 0.163 m = 0.301 '" = 0.056 Co= 1.581

-".=0.0355 --'(,=0.0339

10

Laml>Otte, Monmens, 1976 Concrete P34, P35 lIE. = 0.101·10-<;psi

If, = 3.633 n = 0.160 m = 0.306 '" = 0.056 Go = 1.597

-'P.= 0.0344 - - If. = 0.0338

• • •

10

+

- "'.=0.0081 - - '1'.= 0.0081

100

100

1000

Concre te P32: liE, = 0.118 ·1O-6/pS1

4', = 3.338 n = 0.148 c. = 1.687 m = 0.311

-- '1'.= 0.0318 '" = 0.063 - - '(,= 0.0334

1000

Concra te p40, P41 liE. = 0.097 ·1O-6/pS1

'f. = 3.528 n = 0.157 Cd = 1.620 m = 0.305

- '/\= 0.0209 '" = 0.058 -- If.= 0.0338

100 1000

100

•

Concra te P49 liE. = 0.108 ·10-6/psi

If. = 3.194 c, $l.765 n = 0.138

- If. = 0.0340 m = 0.312 -- 't.= 0.0338 0< = 0.052

1000

I-I' in days

Fig. 20. - Fits of Tests for Drying Creep by Keeton (1965) [20) and Lambotte and Mommens (1976) [56). CPd optimized - solid line, CPd by formula - dashed line, 1/ Eo calculated from experimental value E, if available, or optimized from basic creep data,

418

The formulation has also been extended to cover the cases when the specimen is let to dry before loading and is then resealed at the time of loading, t=t'. In these cases <Pd and Cd should be multiplied by

[1- (~)0.lJ2

l00Tsh

and

I I

(33)

cP 1.0 Hunmel et al, 1962 RH-S5%/ 00

0.9

0.8

'8. 0.7 t

.!: 0.6 --

~ ...., 0.5

0.4

0.3

". 4 Q .S "CI

~ S2 CD :::>

"CI

.S 0 ... U;

0.1

1/Eo = 0.111·10"/ps1 / ~ = 3.721 I n = 0.160 / m =0.306 I 0 ex = 0.045 I 0 c, = 1.600 ~ II 0

- '1',= 0.0180 If/. 0 -- If,= 0.0193 :.F'/ 0

/, /,

/, 0 .h

10

Wit tmann, 1970

100

Sealed Predried Spec1m8ns 1/Eo 0.l· 10-<l/psi

If, 4.183 n 0.190 0( 0.063

00

1000

10

Z. P. BAZANT - L. PANULA

respectively, to being the start of drying. Fits of such tests (McDonald's and Kennedy's) are given in figures 19, 21, 22 and 24.

A number of other test data ([24], [28], [29], [50], [57]) were analyzed. Fits of these data are not shown, however, because various important information on the tests was missing and could not be obtained.

The basic information on the test data used is sum­marized in Appendix III.

0.8

0.1

0.6

0.5

0.4

0.3

0.3

0.2

// I

Meyers and Mai ty, 1970 RH = 50% /1 1IEo= 0.087·10-<l/psi I

." = 4.394 /1 n = 0.181 I m = 0.301 0 0( = 0.029 c,= 1.445

-Of,= 0.0310

o

-- '1',= 0.0332 o

10

McDonald, 1975 RH = 50 % 1 lEo = 0.086 ·10"/pS1

100

If, =3.272 _ n = 0.147 OQi' _--m = 0.305 ,''''IfJ _-'" = 0.059 _---c,= 1.719 _---

-'f',= 0602 __ ---------

0.6

'8. 0.5 'P. .......

Q .s

0.4

::5 ....,

--'fI,= 0.0230

10 100

LamtlOtte, Monmens, 1976 Concrete P26-P29 1 lEo = 0.075 'lO"/psi

«, = 4.176 n = 0.187 m = 0.303 c,= 1.398

-«,=0.0442

o

o

o

-- 'i',= 0.0342

100

o

t-t' in days

Fig. 21. - Fits of Tests for Drying Creep by Hummel et al., (1962) [19], Meyers and Maity (1970) [43]. Wittmann (1970) [58]. McDonald (1975) [22]. and Lambotte and Mommens (1976) [56]. CfJd optimized-solid line, CfJd by formula-dashed line. l/ Eo calculated fr~ experi­mental value, E. if available, or optimized from basic creep data.

419

VOL. 11 - N° 66 - MATI:RIAUX ET CONSTRUCTIONS

0

1.0 L'Herm~te. Maml.llan. LefE!Vre, 1965. 1971 1.0 L'Hermite, Mamillan. 1965

RH; 50% o ...A 1/Eo; 0.113·10-<;/psl. t'. 28 days

0 1/Eo; 0.113·1O-6/psl. 0." If, ; 3.111 0

n ; 0.132 ~; 3.113 n ; 0.132 • m ; 0.316 0.8 m ; 0.316 • 0°10

0< ; 0.051 '"

0.8 0< ; 0.051 "".<i c,; 1.810 '" Co; 1.811 ~ 0 -4\;0.0306 "'''' -'1',=0.0305

0.6 0.6

0.4 0.4 ~

0 0

0 0.2 0.2

10 100 1000

1000 0.6

0 0

1.5 Troxell. Raphael. OaVl.s. 1958 0 Hummel e tal. 1962 RH = 65% 0 0

1 lEo; 0.175 '1O-6/psl. t',28 days 0.5 1 lEo = 0.104 '1O-6/psl. 00

0

n = 0.120 'f: = 4.196 • If, = 3.025 'OoOi~ 0 m = 0.440 n = 0.136 ,,,'l.' 0

Cfo. = 0.042 m = 0.309 0

c, = 1.899 0.4 0< ; 0.066 c, = 1.779

1.0 -'1',= 0.0202 0

0 0.3

10 100 1000 'en • a. 0.5

CD ...... 'Q 10 100 1000 .!: ...:::. 1.1 0 -- MOS5~os'S~an and Gamble, 1972 - RH = 50% J 0.5 Rostasy et al. 1971 RH = 65% 1 lEo = 0.100 '1O-6/psl. 0

0 1/Eo= 0.103·1O-6/psl. 1.0 If, = 3.814

If ; 2.982 n = 0.168 0

0.4 n = 0.127 m = 0.300 0 m = 0307 C1- ; 0.051 0< = 0.045 0.9 c, = 1.537

0 0

c, = 1.845 -Cf,=0.0342 0 ., -'1',=0.0329 0 I><lS 0.3 0 ,9

0.8 0 ,

0

0.2 0

0.7 0.1 10 100 1000

Hansen, Mattock. 1966. 0.6 0

1 lEo = 0.080 '1O-6/psl. 'P, ; 4.295

0 n = 0.176 0.5

0.8 m = 0.308 10 100 0< = 0.035 c, = 1.477

• York. Kennedy. Perry • 1970. • 0.3 1/Eo= 0.loo·10-6/pSl. • <ii, = 3.459

0.6 n = 0.156 0 • 0 • m = 0.303 1fSl~ • 0< ; 0.059 ,.cj) 0

• • C," 1.635 - '1',=0.0257 0 0

t t'.8day, 0 0000

• -'f' .. =O.OI30 0.2

• 0 0

0 0 0

100 1000 0.1 10 100

t-t' in days

Fig. 22. - Fits of Tests for Drying Creep by L'Hermite, Mamillan and Lefevre (1965, 1971) [16], L'Hermite and Mamillan (1965) [16], Troxell, Raphael and Davis (1958) [18), Hummel et al., (1962) [19), Rostasy et al., (1971) [36], Mossiossian and Gamble (1972) [25), Hansen and Mattock (1966) [15), York, Kennedy and Perry (1970) [231. (f), and 1/ Eo calculated by proposed formulas.

420

.[

'f'Q 5

-<=: -....,

1.0

0.9

0.8

0.7

0.6

0.5 0

1.0

0.9

0.8

0.7

0.6

0.5

0.7

0.6

0.5

0) Keeton. 1965 Humidity affect. small size C11inders 3" J( 9" lIE.= 0.102·lo-"/psi

'I, = 3.373 n = 0.150 m = 0.303 do = 0.055 0

0 • 0 • •

0

•

10

C) Keeton. 1965 HLmidity af fact. large size cylinders 6 11 x 18 11

1/E. = 0.102 '10-l/psi 'P, = 3.373 n = O.lSO m = 0.303 .... = 0.055

{.eday,

0

• 0

•

LamDOtta. Monmens. 1976 Concr. te P2: . liE. = 0.126 ·10"/pll.

1/, = 3.265 n = 0.121 'iA = 1.892 m = 0.349

- '1;0= 0.0335 0( = 0.041 Cqncre ta PS2: lIE. = 0.097·10"/pSi

lj', : 3.679 n = 0.163

0

•

+ 1.0 0

0 + •

0 0.9

0.8

0.7

0.6

0.5 t'.8da~5

Cd = 1.673 - '1',= 0.0081 0.4

100 1000 0.9

+ 0

0 • + 0.8

• 0

0.7 •

0.6

0.5

0.4

100 1000

0.8

• • 0.7

0.6

0.5

0.4 6 Concre te P54

Co = 1.577 m = 0.302 0

;;7'= 0.0342 = 80053

o • o o

0.4

o 0.3

0.2 !

0.6

0.5

0.4

0.3

0.2 0

• 6 • 6 0.

10

LamIlOtte. Mannens. Coner. te P50. PSI lIE.= 0.101·10""lpsi

cp, = 3.467 n : 0.153 m =0.304 '" = 0.051 c,= 1.654

o 0

• • o

• 10

o •

1976

•

lIE. = 0.106 '10-</psi If, = 3.148 c,,= 1.816

n = 0.131 m =0.319 0.3

-If,= 0.0336 0( = 0.048

100 1000

0 0.6

0

• 0 0.5 •

0.4

• •

-'1',= 0.0339

100 1000

t-t' in days

0

I!I

Z. P. BAZANT - l. PANULA

0 b) Keeton. 1965

0 + Humidity affect. a small size c:flinders 4" x 12" 1/E.= 0.102·10"'/psi

'P. = 3.373 n = 0.150 m = 0.303 01 = 0 .. 055

0

0 0 a

• •

0

•

10

LamDOtte. _ns. 1976

10

LamDOtte. Mannens. 1976 Concre ta P23: llE.= 0.099·11T"/psi

If, = 3.629 n = 0.163 m = 0.301 '" = 0.056 c = 1.581

-",=0.0339

10

LambOt.te. Momnens I 1976 Concr. te P34. p35. lIE. = 0.101 '10"/psi

,/, = 3.633 n = 0.160 m =0.306 ",.= 0.056 c, = 1.597

-'/,=0.0338

• a

10

0

t',eday.

c,= 1.673 -'1',=0.0081

o o

o

Concra te P32 l/E.= 0.104·10-6/psi

'1', = 3.338 n = 0.148 c, = 1.687 m = 0.311

-<f',= 0.0334 '" = 0.063

100

IJ$I~

toP'! ~

D<'" a

o

o

1000

",/-'O . ~ ~oncre ta P40. P41

e liE.: 0.100·10-l /psi 'P. = 3.528 n = 0.157 c, = 1.620 m = 0.305

- '1',= 0.0338 '" = 0.058

100 1000

•

Concre te p49: lIE. = 0.108 '10-6/osi

't, = 3.194 n = 0.138 c,= 1.765 m = 0.312

- Of,= 0.0338 '" = 0.052

100 1000

Fig. 23. - Fits of Tests for Drying Creep by Keeton (1965) [20], and Lambotte and Mommens (1976) [56]. ({id and 1/ Eo calculated with proposed formulas.

421

VOL. 11 - N° 66 - MATERIAUX ET CONSTRUCTIONS

cf' 0.8 /

1.0 Meyers and Ma1 ty • 1970 RH = 50% Hunmel et aI, 1962 RH·65% 00

11E. = 0.100 ·10-<)<ps1 (J)0

11E. = 0.104 '1O"/ps1 00 'I, = 4.394 0

0.9 ~, = 3.721

0.7 n = 0.181 n = 0.160 m = 0.301 m = 0.306 0 oc. = 0.029 " = 0.045 0 c, = 1.445

0.8 c,= 1.600

",#' 0

0 •• 0.6 -'1,=0.0332 -'1,=0.0193 0

~, 00 0

0.7 0 0.5 0

0.6 0 0

0.4

0 0 0

.~ 0.5 0.3 0

0

~ 10 100

.5 <:;:: ..: -:>

Lambotte, Mommens, 1976 Concrete P26-P29

06 11E.= D.I02.1O"/pS1

10 100 1000 'P. = 4.176 n = 0.187 m = 0.303

" McDonald, 1975 RH-50% 0.5 0( = 0.071 o ~~ i O

0.3 1/E.= 0.101·10-6/pS1 c, = 1.398 '1', = 3.272 -'1,=0.0342 ",!-'O ~ n = 0.147 0 ' . 00

0

m = 0.305 0.4 ~ '" = 0.059

~ ~:l Ii - 'P.o= 0.0230 0.3 • " 0.2 ~ " 0

0

10 100 10

tot' in days

Fig. 24. - Fits of Tests for Drying Creep by Hummel et al., (1962) [19), Meyers and Maity (1970) [43), McDonald (1975) [22), and Lambotte and Mommens (1976) [56). <Pd and 1/ Eo calculated with proposed formulas.

0.04

~ 0.02

for r > 0.0 r ~ 0.0

'f. = 0.008 + O. 027u 'f.,= 0.008

022

0 111

ODO~ ____ ~~ ____ ~~ ____ ~~ ____ ~~ ____ ~~ ____ ~~ ____ ~~ ____ ~~ ____ ~ ______ ~ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

U

1 L I Herrn te et al., RH = 50% 9 Keeton 17 LamtlOtte, Mommens P32 2 L I HermJ. te et al., var num 10 MossJ.ossJ.an, Gamole 18 Lamootte, Mommens P34, P35

3 Troxell, Raphael, Davis 11 Meyers, MaJ.ty 19 Lamootte, Mommens P40, P41 4 Hummel et al., w/c = O. 38 12 Lamootte, Mommens P2 20 Lamootte, Mommens p49 5 Hummel et al., w/c=0.55 13 Lamootte, Mommens p13 21 Lamootte, Mommens P50, P51 6 York, Kennedy, Perry 14 Lamootte, Mommens P26-P29 22 Lamootte, Mommens P52 7 McDonald 15 Lamootte, Mommens p23 23 Lamootte, Mommens P54 8 Hansen, Mattock 16 Lamootte, Mommens P31 24 Rostasy

Fig. 25. - Coefficient ({Jd'

422

APPENDIX III

Basic Information on Test Data Used

L'Hermite, Mamillan and Lefevre's Tests of lJrying Creep (1965,1971) [9]. - For various ages at loading. Specimens prisms 7 x 7 x 28 cm cured in water; at to = 2 days exposed to drying at 50% relative humidity and 20°e. Portland cement 350 kg/m3. Water-cement-sand-gravel ratio 0.49 : 1 : 1.75 : 3.07. 28-day strength 370 kp/cm2 (36.3 N/ mm 2

). Aggregate Seine gravel (siliceous calcite). max. aggregate size 20 mm.

L'Hermite and Mamillan's Tests of Drying Creep (1965) [16]. - At various humidities. Specimens 7 x 7 x 28 cm cured in water 28 days, loaded at the age of 28 days and exposed to drying at relative humidities 50, 75 and 99%, temperature 20°C. Portland cement 350 kg/m', water­cement-sand-gravel ratio 0.49 : 1 : 1. 75 : 3.07. 28-day strength 370 kp/cm2 (36.3 N/mm2). Aggregate Seine gravel (siliceous calcite), max. size of aggregate 20 mm.

Troxell, Raphael and Davis' Tests of Drying Creep (1958) [18]. - At various humidities. Cylinders 4 x 14 inch (102 x 356 mm) exposed at age of 28 days to drying at relative humidities 50, 75 and 99%, temperature 70°F (21°C). Cement type I, water-cement-sand-gravel ratio 0.59 : 1 : 2 : 3.67. Granite aggregate, max. aggregate size 1.5 inch (38 mm), 28-day cylinder strength 2,500 psi (17.2 N/mm2). .

Rostasy et al. 's Tests of Drying Creep (1971) [36]. - After 7 days of curing cylinders 20 x 140 cm were exposed to drying at 65% relative humidity and 20°C temperature. Axial load applied at age of 28 days. 28-day cube strength 498 kp/cm2 (48.8 N/mm2). Cement content 275 kg/m3. Water-cement-sand-gravel ratio 0.56 : I : 3.08 : 4. Rhine sand and Rhine gravel, max. aggregate size 30 mm.

Hansen and Mattock's Tests of Drying Creep (1966) [15]. - For various sizes of specimens. Cylinders of dia­meters D = 10.2 to 61.0 cm, and lengths 45.7, 55.9, 66.0, 86.4, 106.7, 127.0, 147.3 cm); 2 days in mold, 6 days in fog at 70°F (21°C). At the age of 8 days specimens loaded and exposed to drying at 50% relative humidity, 28 days cylinder strength 6,000 psi (41.4 N/mm2). Elgin gravel (92% calcite, 8% quartz), max. aggregate size 0.75 inch (19 mm). ASTM type III cement (362 kg/m3). Water­cement-sand-gravel ratio 0.71: 1 :3.3 :2.7.

Hummel et al.'s Tests of Drying Creep (1962) [19]. -Mix A (fig. 19): After 7 days curing, cylinders 20 x 80 cm were exposed to drying at 65% relative humidity and 20°C. Axial load applied at age of 28 days. Portland cement, PZ225, 350 kg/m" 28-day cyl. strength 414 kp/cm2 (40.6 N/mm2). Water-cement-aggregate ratio 0.38 : 1 : 5.4. Mix B (fig. 21): Cement PZ425 (334 kg/ml). Water-cement­aggregate ratio 0.55: 1 : 5.4. 28-day cyl. strength 435 kp/ cm 2 (42.7 N/ mm2

). Specimens loaded at the ages of 3, 28 and 90 days. Rhine gravel, max. size 30 mm.

MoSsiossian and Gamble's Tests for Drying Creep (1972) [25]. - Cylinders 6 x 12 inch (152 x 305 mm) were loaded and exposed to drying after 4 days of curing at 50% relative humidity and 70°F (21°C). Cement type III (418 kg/ml). Water-cement-sand-gravel ratio 0.49: 1 : 1.35 : 2.98. Coarse aggregate: crushed limestone, max. size 1 inch (25.4 mm); 29-day cyl. strength 7,160 psi (49.4 N/mm2).

York, Kennedy and Perry's Tests of Drying Creep (1970) [23]. - After 7 days of curing cylinders 6 x 16 inch (152 x 406 mm) were exposed to drying at 60% relative humidity and' 75°F (24°C). At age of 83 days specimens were sealed in copper jackets. Load applied at age of90 days. Cement type II, 404 kg/ml. 28-day cyl. strength 6,650 psi (45.9 N/mm2). Water-cement-sand-grave1 ratio

Z. P. BAZANT - L. PANULA

0.425 : 1 : 2.03 : 2.62. Limestone aggregate, max. size 0.75 inch (19 mm).

Keeton's Tests of Drying Creep (1965) [20]. - At various humidities and various sizes of cylinders. At age of 24 hours specimens were demolded and placed in 100% relative humidity. Load was applied and specimens were exposed to drying at 75°F (24°C) at age of 8 days. Portland cement type III (451.2 kg/m3). Water-cement-sand-gravel ratio 0.46 : I : 1.66 : 2.07. Max. aggregate size 0.75 inch (19 mm). Fine aggregate: Saticoy River sand, coarse aggregate: Santa Clara River gravel. 28-day cyl. strength 6,550 psi (45.2 N/mm2).

Lambotte and Mommens' Tests of Drying Creep (1976) [56]. - High strength Portland cement, except concretes P40 and P41, where early strength cement was used. Speci­mens cured in mold for 24 hours, then exposed to drying environment of 20°C and relative humidity 60% for all concretes, except 95% for concretes P49-P54. Specimens P2 through P41 were of size 15 x IS x 60 cm, specimens P49 through P54 of size 10 x 10 x 40 cm.

For further data see table I below, in which t ' is in days, f: is cube strength at the time of loading, in N/mm>, c is cement content in kg/m" and w : c : s : g is water-cement­sand-gravel ratio.

Concrete

P2 ........ .. P13 ....... .. P26-P29 .... . P23 ....... .. P31 ........ . P32 ........ ~ P34, P35 .... . P40, P41 .... . P49 ....... .. P50, PSi .... . P52 ........ . P54 ........ .

TABLE I

t' fc'

28 32.6 28 40.5 28 53.3 (*) 18 52.7 7 47.6

28 46.8 28 I 51.2 28 152.0 48 49.8 49 56.6 35 57.3 50 I 45.2

c

270 350 400 360 400 400 400 450 350 350 362 350

(*) Average of four.

w : c : s: g

0.611 : I : 2.33: 4.74 0.5 : I : 1. 71 : 3.56

0.3511 : 1.06 : 3.49 0.445 : I : 1. 76 : 3.55 0.575: I : 1.71 : 3.04

0.4: I: 1.5 :3.17 o . 45 : I : I. 5 : 3. ) 7

0.433: I : 1.28 : 2.82 0.48: I: 1.85: 3.71 0.49 : I : I .85 : 3.59 o .47 : I : 1. 79 : 3. 98 0.52: 1: 1.85: 3.71

Wittmann's Tests of Predried Cement Paste (1970) [58]. - At various constant water content. Solid cement paste cylinders 18 x 60 mm, water-cement ratio 0.4; cured sealed for 28 days at 20°C; then dried in oven at 105°C for 2 days; then resaturated for 3 months at various constant relative humidities shown in the figure at 20°e. Then tested for creep in the same environment under stress 150 kp/cm2 (14.7 N/mm2) equal 0.2 offailure load; E = 210,000 kp/cm2 (20590 N/mm2) for 1 minute loading; strain at 20 minutes under load was subtracted to get the values shown.

Maity and Meyers' Tests of Drying Creep (1970) [43]. -Prisms 3.5 x 3.5 x 14 inch (89 x 89 x 356 mm), after 4 days of curing exposed to drying at 50% relative humidity and 70°F (21°C). Cement type III (253 kg/m3). Water-cement­sand-gravel ratio 0.85 : 1 : 3.81 : 3.81. Coarse aggregate: crushed limestone. 12-day cylinder strength 5,200 psi (35.9 N/mm2), cyl. 4 x ~ inch (102 x 203 mm).

McDonald's Tests of Drying Creep (1975) [22]. - After 7 days of wet curing, cylinders 6 x 16 inch (152 x 406 mm) were allowed to dry in air at 50% relative humidity and temperature 73°F (23°C). After exposing the specimens to this environment for 75 days, specimens w,ere resealed. Load was applied at the age of 90 days. Portland cement type II (404 kg/ml). 28-day average cyl. strength 6,300 psi (43.4 N/mm2). Water-cement-sand-gravel ratio 0.425 : 1 : 2.03 : 2.62. Limestone aggregate, max. size 0.75 inch (19 mm).

423

VOL. 11 - N° 66 - MATERIAUX ET CONSTRUCTIONS

REFERENCES

[56] LAMBOTTE H., MOMMENS A. - L'Evolution du fluage du heton en fonction de sa composition, du taux de contrainte et de l'age, groupe de travail GT 22. Centre national de Recherches scientifiques et techniques pour l'Industrie cimentiere, Bruxelles, July 1976.

[57] PIHLAJAVAARA S. E. - A review of some of the main results of a research on the aging phenomena of concrete. Effect of'moisture conditions on strength, shrinkage and creep of mature concrete. Cement and Concrete Research, Vol. 4, 1974, pp. 761-771.

[58] WITTMANN F. - Kriechverf'ormung des Betons unter statischer und unter dynamischer Belastung. Rheal. Acta., Vol. 10, 1971, pp. 422-428.

Part IV: Temperature effect on basic creep

Development of the model for basic creep in Part I I is followed here by a prediction model jor creep at various temperatures that are kept constant during creep. The model, which prese~ves the form of the double power law, reflects two opposing effects of temperature: the increase of creep rate due to heating, and the reduction of creep due to thermally accelerated hydration. Prediction of material parameters from mix composition is studied and extensive comparisons with test data indicate a good agreement.

INTRODUCfION

The double power law for basic creep, developed in Part II, will now be extended to model creep at various temperatures that are kept constant during creep. Unlike the preceding parts of this study, here we must not only model the composition influence but also decide on the proper form of the temperature effect because the model for variable temperature that we are going to investigate has not yet been proposed.

Realizing that the choice of reference temperature To is subjective and largely arbitrary, we must conclude that the creep formula for any temperature (within a certain range) should have the same basic form. In particular, the form of double power law should be preserved for heated sealed concrete.

FORMULAS PROPOSED FOR TEMPERATURE EFFECf ON BASIC CREEP

Preserving its basic form, we may generalize the double power law as

1 J (t, t')= E +Co (t, t'),

o (34)

Co (t, t')= ~T (t~-m+ 0::) (t_t')nT, o

where

t~= L fJT (t")dt", (35)

<PT=<Pl (1 +CT), fJT=exp (4 ~ - 4 ~). (36)

Here CT, nT and f3T are functions of temperature, and t~ represents the equivalent hydration period (or

424

maturity) [5] e), which is defined as the period at reference temperature To needed to achieve the same degree of hydration as period t' at temperature T. Equation (36) results from assuming that the tempe­rature effect on hydration is governed by an activation energy, Q. In equation (36), T and To must be absolute temperatures. The constant 40000K (representing Q divided by gas constant) has been derived empirically from the data fitted in the sequel.

Following a theoretical analysis by Wittmann [58], function CT was previously [4] suggested to also obey the activation energy concept. However, an in-depth analysis of test data revealed that this is true only for a limited range of temperatures, from about 35 to about 75°C. Beyond this range significant deviations occur, which may be due to phase changes and chemical changes, as well as simultaneous operation of several processes controlled by different activation energies. Therefore, function CT has been identified empirically, although the basic, product form of equation (36) for <PT' as indicated by activation energy effects, has been retained. Function CT' which is plotted in figure 26 a in comparison with the activation energy dependence, has the form

19.4 CT= 1+(I00/(T-253.2))3.5 -1,

1

I (37)

(38)

where Co is a composition parameter, t~ is the age of concrete when temperature T is applied and T is abso­lute temperature. Note that CT is defined not only as a function of temperature but also as a function of t~.

According to the activation energy model for power­type creep functions [58], exponent nT would be a

(') Reference numbers not listed at the end of this part are found in the preceding parts.

constant. Again, for a broader range of temperatures ( - 20 to 140°C) this is unacceptable. Nevertheless the form of equation (34), conforming to the activation energy model, may be retained and it suffices to take nT as temperature-dependent. By data fitting, the following empirical function has been found:

0.25 BT = 1+(74/(T-253.2))7 +1. (39)

Equation (37) is approximately valid from about - 200e to perhaps 120°e. Near the ends of the range the rise of CT with temperature is milder (fig. 26 a).

Function BT indicates that exponent nT increases with temperature, i. e., the ratio of long-time to short­time creep increases as temperature is raised. This may be explained by the larger effect of the accele­ration of aging during the early creep period.

Equations (35), (37), (38) reflect the fact that the tempe­rature effect on creep is twofold ([60], [5]): (a) an increase in temperature increases the creep rate, but (b) it also accelerates hydration, i. e., aging. These effects, modeled by coefficients CT' 'T and t;, respectively, oppose each other. When a young concrete is heated well before it is loaded, the equivalent hydration period t~ for the moment of load application may get sharply increased, causing a reduction of the creep increase due to heating. On the other hand, when an old concrete is heated, the change in t~ has little effect on subsequent creep, and so a strong increase of creep with temperature takes place. Modeling of both these opposing tendencies is essential for successful fitting of test data.

The elastic modulus E is known to decrease with temperature beyond 50oe, the drop reaching about 20% at 1000e ([61], [62 D. Like the double power law which gives proper age-dependence of elastic modulus, equation (34) seems to give approximately correct temperature dependence of the elastic modulus:

1 1 E (t') - E

stat (t') =J (t' +0.1, t')

= El + ~T 1O- nT(t:- m + IX). (40) o 0

EFFECf OF COMPOSITION ON BASIC CREEP OF HEATED CONCRETE

By fitting of test data ([59], [23], [61], [63], [64], [65]~ [66], [67], [68], [69], [22], [70], [71 ], [72 D it was verified that:

Co= ~ (~)2 (~)a 8 C C 1> (41)

where a l accounts for the cement type and is the same as in equation (18) of Part II; wi c= water-cement ratio; al c= aggregate-cement ratio. An increase of creep rate with the water-cement ratio, as given by equation (41), is logical to expect. The increase of Co with the aggregate­cement ratio means that the restraining effect of aggre­gate on creep is stronger at lower temperatures. Equa-

Z. P. BAZANT - l. PANUlA

1.3~------------------,

1.2

• exp(4ooo _ 4000) 10 T. T

~ u 5

0.25 IT - 253.15)7 ST = 1. 747.(T_ 253.15/7

19.4 -1 CT = (100 3.'

1 + IT _ 253.2)

60 80 100

Temperature in·C

a 120

Fig. 26. - Coefficients CT and BT as function of temperature.

tion (41) does not involve strength, but since the strength depends on wi c and al c, the effect of strength is present indirectly.

COMPARISONS WITH TEST DATA ON HEATED SEALED SPECIMENS

Fits of numerous test data shown in figures 27-34 indicate a reasonable agreement of the present model with experiments. Basic information on the test data used is given in Appendix IV.

For some data sets important information was not reported and, therefore, has had to be assumed. e. g., for England and Ross' data it has been assumed that the heat was applied at the age of 10 days, simulta­neously with load application (i. e., no heat stabilization period before the test). Also, the initial "elastic" strains at elevated temperatures have been assumed using proportionality to the values of Marechal. For the tests of Silveira and Florentino, it has been assumed that the heat was applied three days before loading.

For Nasser and Neville's data ([68], [69 D, the sand­gravel ratio was not available, and so exponent n has been assumed. The initial elastic strains have had to be assumed also (0. 2 x 10 - 6 I psi). Papers [68] and [69] mentioned that E was not a function of temperature; therefore, the value of II Eo has been found by optimiza­tion. The E-modulus was reported to increase by 22% from t' = 14 days to t' = 365 days, and the value of J (t' +0. 001, t') has been assumed to change in pro­portion. Moreover, these data indicate, independently of curing temperature, a 22% increase of elastic modulus upon heating, which conflicts with references [61] and [62]. The deviations from test data in figures 28 and 32 must be judged in the light of the preceding remarks.

When unspecified, the unit weight of concrete has been assumed as 2,400 kglm 3

.

It is interesting to compare J (90 + 365,90) for the data of Silveira and Florentino [67], McDonald [22] and Kennedy [23]. At room temperature, the values are 0.425,0.285,0.285 (all in 10 - 61 psi), and at elevated temperatures tested (45, 65.6 and 65. 6°e respectively), the values are 0.748,0.400 and 0.445. This is a consi­derable scatter in view of the fact that the mix panl.'ineters and test conditions were quite similar (see Appendix IV).

For temperatures beyond 95°e, the present model gives only very crude estimates. Even though all

425

VOL. 11 - N° 66 - MATERIAUX ET CONSTRUCTIONS

Hannant.1967 0.9

0.8 liE. = 0.097' laO/psi lj', = 4.087 n = 0.181 0.8 m = 0.295

0.7 01 = 0.053 CT27= 0.041

0.7 CT53 = 0.551 CT57 = 0.652

0.6 Cr75= 1.123 0 0.6

6'0

0.5 0

00 °cP

0.5

0.4 • •• 00 0.4

• 0 ~1 • 0 . "' ..... 0.3 0

0 ~ ..... .. -.. 0.3 ••• •

• • (",180days 0.2 •

0.2 10 100 1000

Nishizawa and Okamura. 1970 0.9 0

1IE,,= 0.100 . 1~lpsi

(J) 'Il = 3.234 0.. n = 0.147

........ 0.8 m = 0.303 0.0

ex = 0.053 '0 GT20= - D. 023

C 0.7 CT70= 0.548 0 GTgo = 0.964

0 ~ 0 0 0 '\<:>

0 0

0.6 0

0 -,

0 0

Do 0

0.5

0.4 /"

0.3

r,28 days

10 100

McDonald, lW,5 0.40 1/E,,=0.101·10 I psi

'f, =3.272 n =0.147 m = 0.305 cx =0.059 •

CT23=-0.OO5 • '" GT66= 0.668

0.30 • i!ft'a

• ..~~

e~ • 00

~ e 0 r,90days 0.20

eo o 600 psi. ] 22..S"C • 2400 pSI

• 2400 psi

10 100

t - t' in

Arthanari and Yu. 1967

l/E. = 0.111 . lOG/psi <Ii =4.085 n = 0.167 m =0.321 0<. =0.044

CT2o =-0.032 CT40 = 0.232 CT62 = 0.711 CT80= 1.133

r, 15 days

• 000L'. uniaxial test

•

0 1.1

1.0

0.9

0.8 0

0

0.7

0.6

----

0.3

0.6

0.5

0.4

0.3

days

• ••• biaxial tost

10

England and Ross. 19&2

1IEo=D.1D7 '1D-olpsi 'i1 = 3.012 CT126 = 1.861 n =0.128 CT'140= 1.997 m = 0.317 0< = 0.055

CT20=-0.032 Cr53= 0.499 Cr77=1.063 Cr94=1.414 Cr"4=1.725

• 10

Johansen and Best, 1952

1 IE" =0.176 '10-slps; 'f, =4.628 CT2o =-0.101 n =0.120 CT12 =-0.255 m = 0.466 4t-'21 =- 0.407 0( =0.036 Cr(-2O)= -0 .412

0

10

(0/,0.2)

0

0

0

0

100

v o

100

Fig. 27. - Fits of Tests of Temperature Effect on Creep by Hannant (1967) [61], Arthanari and Yu (1967) [63], Nishizawa and Okamura (1970) [64], England and Ross (1962) [65], McDonald (1975) [22] and Johansen and Best (1962) [66]. CT optimized-solid line; subscript number refers to corresponding temperature. CT with .formula - dashed line. 1/ Eo calculated from experimental E2B or optimized from basic creep data.

426

specimens considered here were sealed, moisture may have moved out of concrete and collected under a bulged jacket. Also, rapid redistribution of moisture within the heated specimen may have had considerable effect on creep. In particular, the present model does not describe the decrease in creep rate (i. e., in CT )

Z. P. BAZANT - L. PANULA

that is sometimes observed upon passing 100°C; see the curves near 100°C in figure 28 for Nasser and Neville's data, and the reversed order of temperatures for the curves near 100°C in figure 27 for England and Ross's data.

O.SO r--------------------------,p'-----O

0.40

0.30

0.20

0.7

III a ",'- 0.6

'0

C

O.S :.... ...;

--, 0.4

York, Perry, Kennedy, 1970 /

1/Eo= 0.087· 10·/psi / If, = 3.410 / c n=0.153 / m=0:303 / 0'.=0.059 /

• CT2• = o.b .. CT>. = 0.003 / - Cra, = 0 . .257 - - CT66 = 0.667 / ./

,.

o 600 psi .J 24'C • 2400 psi

c 600ps'i J «6'C • 2400 psi =.

,. /'

/' /'

./ /'

./ ./

./

././ ,,;{. ./ ",,,.

10

/

o o

/' /'

/'

./ /'

/'

./ ./

./ ./

./

./

/ ./

/ /

/

o o

.{. ~,

o 00

o .. -'l.' •• /' /' .,. /'

/' ocP ., •• /'/'

o ./ OJoo • • ././ .. 0·· /"/

100

100

Zielinski and Sadowski, 1973

o

0.2 • • 10

CT20 =-0.029

CT60 = 0.601

20'C ----.... --.. -.-.-... t' = 123 days

100

t - t' In do ys

Fig. 28. - Fits of Tests of Temperature Effect on Creep by York, Kennedy and Perry (1970) [23], Da Silveira and Florentino (1968) [67], Nasser and Neville (1965) [68], Nasser and Neville (1967) [69], Seki and Kawasumi (1970) [70] and Sielinski and Sadowski (1973) [71]. Cy optimized-solid line, C T witb formula-dashed line_ 1/ Eo optimized from basic creep data_

427

VOL. 11 - N° 66 - MATERIAUX ET CONSTRUCTIONS

1.2 Browne, 1967

1.0

0.8

Vl .f2: 0.6 <0 10

1.0

t'~7days

lIEo~0.124 10-Glpsi 'l\,3.511 n ~0.156 m,0.307

0.8 0( ~ 0.060

0.6

0.4

t': "L8 days CT20 ~ -0.022 CT40 ~ 0.159 CTOS ~ 0.536 CT93.5~ 0.958

t', 400 days

CT20~ 0.0 CT40~0.857

o ° °

0.2 ~-:------:10~--1"'00=-----C::-' "'----'----10"------I..LOO----';>..J '-----'-----,10'-----------'-100----'-'-' "'LA~"--10L....--l..LOO--:..J <==---10'--------I..LOO-~1OOJw

t-t' in days

Fig. 29. - Fits of Tests of Temperature Effect on Creep by Browne (1967) [59]. CT optimized - solid line in figure I-j. CT with formula­solid line in figure a-e. 1/ Eo optimized from basic creep data. Experimental datas are smoothed mean values.

APPENDIX IV

Basic Information on Test Data Used Hannant's Tests of Temperature Effect on Creep (1967)

[61]. - Cylinders 4 k x 12 inch (105 x 305 mm), cured for 24 hours in molds under wet rags, then 5 months under water at 20°e. and then one month sealed in copper. Heating rates about 10°C/hr. For temperature stabilization all specimens were heated for 24 hours before loading. Stress 2,000 psi (13.8 N/mm2). Water-cement-sand-gravel ratio 0.47 : I : 1.845 : 2.655. Sulphate resisting Portland cement with Plastocrete plasticizer. Coarse aggregate limestone max. size 3/8 inch (10 mm). 28-day cube strength 9,350 psi (64.5 N/mm2).

Nishizawa and Okamura's Tests of Temperature Effect on Creep (1970) [64]. - Specimens 15 x 15 x 55 cm, sealed in copper, prestressed to compressive stress 120 kg/cm2

(11.8 N/mm2) at the age of 28 days. After 7 days of loading at 20°C, specimens exposed to temperature of 70 or 90°e. Water-cement-ratio 0.40, cement content 377 kg/m3, sand percentage 36.5%. (In calculations water-cement-sand­gravel ratio 0.40 : I : 1.85 : 3,22 was used.) Max. size of coarse aggregate = 25 mm, normal cement. Cylinder strength 459 kg! cm2 (45 N/ mm2).

McDonald's Tests of Temperature Effect on Creep (1975) [22]. - Cylinders 6 x 16 inch (152 x 406 mm) demolded after 24 hours, coated with epoxy and returned to fog room. After 24 hours another coat of epoxy, and sealed in copper. At age of 83 days, specimens recoated with epoxy, sealed in neoprene, and placed to environment of test tempe­rature, loaded at age of 90 days. Water-cement-sand-gravel ratio 0.425 : 1 : 2,03 : 2.62. Type II portland cement (404 kg/m3). Limestone aggregate, max. size 3/4 inch (19 mm). 28-day average cyl. strength 6,300 psi (43.4 N/mm2).

Arthanari and Yu's Tests of Temperature Effect on Creep (1967) [63]. - Slabs 12 x 12 x 4 inch (305 x 305 x 102 mm), cured under wet hessian for 7 days. For tests under mass­concrete conditions sealed by epoxy resin and two coats of

428

plastic emulsion paint. Loaded at age of 15 days, stress 1,000 psi (6.9 N/mm2). Heating began 1 day before loading. Water-cement-sand-gravel ratio 0.564: 1 : 1.125 : 2.625. Thames river gravel of size 3/16-3/8 inch (4.76-9.5 mm), ordinary portland cement. 28-day average cube strength 6,000 psi (41.4 N/mm2).

England and Ross' Tests of Temperature Effect on Creev (1962) [65]. - Cylinders 4.5xl2inch (114x305mm), demolded at age of I day, placed under water for additional 3 days, after which stored at 17°C and 90% R.H. until tested at age of 10 days in a sealed state. The seal was a polyester resin, with fibre glass reinforcement. Water­cement-sand-gravel ratio 0.45: I : 2 : 4. Compressive strength of 4 inch (102 mm) cubes at age of 14 days = 5,500psi (37.9 N/mm2). Elastic modulus 5 x 10· Ib/in2

04,480 N/mm2).

Johansen and Best's Tests of Temperature Effect on Creep (1962) [66]. - Cylinders lOx 30 cm and 15 x 30 cm cast in steel molds and remolded at age of 1 day, then stored at 100% reI. humidity and 20°e. At age of 42 days, specimens were sealed and moved to test environment. At the end of 3 days stabilization period specimens were loaded at their respective temperatures to 30~~ of their ultimate strength in compression as measured at 20°e. Water-cement-sand­gravel ratio 0.7 : I : 3.5 : 3.5. Normal portland cement. Max. size of aggregate 3/8 inch (9.5 mm). Average compres­sive strength 179 kp/ cm2 (I7. 6 N/ mm2) at the age of 42 days on cylinders 15 x 30 cm.

York, Kennedy and Perry's Tests for Temperature Effect on Creep (1970)[23]. - Cylinders 6 x 16 inch (152 x 406mm), removed from molds 24 hours after casting. Then epoxy coat applied and specimens stored in fog room. Next, 48 hours after casting, specimens sealed in copper and placed in test environment at 73. 4°F (2~°C). \~t age of 83 days specimens sealed in neoprene jacket, and exposed to test temperature. Loaded at age of 90 days. Water­cement-sand-gravel ratio 0.425 : I : 2.03 : 2.62. Cement type II (404 kg! m3). Limestone aggregate, max. size 3! 4 inch (19 mm); 28-day cyl. strength 6,560 Dsi (45.2 N!mm2).

.j::>. N '-0

en C-

" <I) I

0

C

0+-

J

0..8 1, ~-----:---:----------------Ui

Komendanl, Pol ivka, Pirlz, 1976

0..7

0.6

0..5

0..4

0..3

0..2

II Eo

<P,

- 0..10.3· 10.-6 Ipsi

- 3.293

- D. r51

m -0..30.3

a -0..0.66

CTn - 0..0.

- C T4 • - 0..0.1

CT7I - 0..468

D o

Cu. - -0..0.0.4

--- Cu. - 0..162

CHI -0..512 .& d

I' : 28 days (mix A)

Komendanl, Polivka, Pirlz, 1976 D.7f--

IIEo - 0..10.3,10.- 6 Ipsi

<P, - 3.235

D6f- - 0..147

m - 0..30.5

0.51-a - 0..0.65

Cu. - 0..0. Cu. - -0..0.0.4

DT - C T4 • - 0..0.86 ---C T4 • - 0..153

CHI -0..424 CT7I -0..483

0.3

• o 0 00

0.2 • I': 28 days (mix 8)

I I , , !!!! ! I "J I ! ! I , "! I J ! 'J II ! t!!! I ! I ! I ': 0.1 I ! ! !!! I ! I ! J ! I ! ! t r! I ! ! ! !, I ! ! "! ! , , [ "!~

0..7

0..6

0..5

0..4

0..3

0..2

0.6 r-

0.5 f--

0..41-

D.3f--

0..2

Komendanl, Polivka, Pirlz,

IIEo - 0..10.3' 10.- 6 Ipsi

<P, - 3.293

- 0..151

m - D. 30.3

a - 0..0.66

CTn -0..0. CT25 - -0..0.0.4

--Cu. -0..211 --- C T4 .-D.186

C T7I - 0..938 CHI -0..587

D.7f-- Komendanl, Polivka, Pirlz, 1976

IIEo -0..10.3' 10.- 6 Ipsi

D.6r- <P, - 3.235

0.51-

04f-

0..3

0..2

m

a c T23

-- CT4 !

D

•

C T7 !

- 0..147

- 0..30.5

- 0..0.65

- 0..0. Cu. - -0..0.0.4

-0..278 ---C T4• -0..175

- 0..760. CHI - 0..554

D

I ',"'! I""! I """! "'111 I ""! ""'1 0 .1 : "I!l! ,! '!l' "! !I!! ! J 11111 ! J! !'" !! [1'\

Komendanl, Polivka, Pirlz, IIEo -0..10.3'10. I psi

<P, - 3.293 D

-0..151

m - 0..30.3

a - 0..0.66

C Tn -D.O.. C T2.--D.DD5

--CTU -0..411 - -- C T4.-D.217

Cnl -1.132

D 0..6

0..5

0..4

0..3

0..2

(mix A)

Komendanl, Polivka, Pirlz, 1976 IIEo -D.ID3·ID-6 /psi

4>, -3235

- 0..147

m -0..30.5

a - 0..0.65

Cu. - 0..0.

-- CT4 • - 0..515

CHI - 1.0.0.4

a •

Cu. - -0..0.0.5

---C u .-D.2D5

'''''''' 11,,11 !!III! 1111110.1' ",,!! ,",11 ,.11,,1 'IJlII! ""II III!II" 0..1 I "I" I III" 1000 0.001 0.001 0.01 0.1 10 100 0.01 100 1000 0.1 10

t - t I in days Fig. 30. - Fits of Tests of Temperature Effect on Creep by Komendant, Polivka and Pirtz (1976) [72l- CT optimized - solid line, CT witb formula - dashed line.

1/ Eo optimized from basic creep data.

N

~

ID l> N l> z --I

: "tJ l> z c ~

VOl. 11 - N° 66 - MATERIAUX ET CONSTRUCTIONS

0.8

0.7

0.5

0.4

0.3

0.2

0.9

Vl a..

" '" 0.8 '0

C 0.7

......

...... 0.6

J

0.5

0.4

0.3

0.40

0.30

0.20

Hannan!, 1957

llEo ' 0.080 . lO-o/psi 'A ,4.087 Crv'O.041 n '0.181 CT53 'O.SS1 m '0.295 --Cr57'O.652 eX ,C.OS3 CT75 '1.123

11....-""- A -- ~ .... . --- ... . --..........-:--...... . .. - .

1',180 days •

0.9

0.8

0.7

0.6

0.5

0.4

0.3

Arthanari and Yu, 1967

lIEo ,O.lOS·'0-6/psi If, ,4.08S CT2O'-0.032 n • 0.167 CT40,O.232 m ,0.321 CT62 , 0 .711 d.. ' 0.044 CTBCf 1. 1 33

CT20 ·O.0 CT40 • 0.226 CT62 ,O.S98 CTOO' 0.858

./

,/

,/ ,/ 0

,/

•

/0

t'. 15 days

uniaxial test biaxial test

10 100 1000 0.2 LL---'----L..J-LJ-'-!-----'---'--'---LL.LLJ:::

,0-_'----'--'---,l.....1..J...J

Nishizawa and Okamura, 1970

l/Eo,O.110·1O-o/psi

'P, ,3.234 n ,0.147 m '0.303 0( , 0.063

CT20 ,O.0 - CT70 , 0.368

CT90 '0.919

CT20 '-0.023 - - CT70 ' 0.648

CT90 ' 0.964

McDonald,1975 -6

l/Eo' 0.086' 10 /psi

10

<Ii .3.272 CT22.8'O.O n '0.147 - CT65.6 '0.431 m 'O.30S <X , 0.OS9

10

~/.

,/ ,/

>f

o

./"

I

o /

9'

/ /

/

1.1

1.0

0.9

o 0.8 o

0.7

./ 0.6

O.S

1',28days

0.4

100

0.3

England and Ross, 1962 1IEo = O. 105.166 /psi

0

'1\ '3.012 CT20, -0.032 1\ '0.128 CT53 '0.499 m=0.317 CT77,1.063 ex '0.056 - -CT94 , 1.414 CT20, 0.0 CT114' 1 .725 Cp';3 = 0.698 CT12S' 1.861 Cm = 1.254 CT140, 1.997 CT94 ,1.525 CT114 ,1.864 Cn26,1.760 Cn4o,'.977

----./

./

-------

10

0.7 Johansen and Best, 1962 1IE,.'0.225·'0-6;psi

'i\ .4.628 CT20=0.0 n ,0.120 CT12 ,-0.305

m,O.466 Cn-12)' -0.312 0.6

ex .0.036 C'1(;20)'-0.314

0.5

o 600 psi 1 • 2400 psi 22.8·C

0.4

• 2400 psi

,/

./ ./

./

~ ./

100

100 0.3~-----L~--~-LLU~,0~---L~~~~\I·LU~,0~0

t- t' in days Fig. 31. - Fits of Tests of Temperature Effect on Creep by Hannant (1967) [61], Arthanari and Yu (1967) [63], Nishizawa and Okamura

(1970) [64], England and Ross (1962) [65], McDonald (1975) [22] and Johansen and Best (1962) [66]. CT and 1/ Eo both calculated with formula.

430

~ 0<>

'0

.S

....

..... -,

Z. P. BAZANT - L. PANULA

O.~r-------------------------------------------,r----,o'-' York. Per ry. Kennedy. 1970 Do Silveira and Florentino, 1968

0 1IE,= 0.100' 10-' I psi

", = 3.410 n =0.153 m =0.303 (/.. =0.05J

0.40 CT24= 0.003 c,...;= 0.&67

0.30

0.20

0.1

0.7

0.6

0.5

0.7

0.6

0.5

0.4

0.3

0.2

o o

o 600 pSi. ] 24'C • 2400ps,

o 600 pSi. ] 65.6'C • 2400 psr

! o o

l'«Isser and Neville. 1965

1IE, = 0.096·10-6/psi <ii, = 4.308 n = 0.140 m = 0.302 01. = 0.042 CT2I =-0.043 CT7I =1.7

Seki and Kawasumi,1970

I/Eo - 0.102 10-' I psi

<p, - 3.354

-0.151

m - 0.305

a - 0.063

CT"O - '0.025(" -28doys) CT.O - 0.184 (,' -29 days)

Cno -0.713 (,'- 29 days)

0.1

o

•

• • a

10

o

o o

o '00

1'(' 8

~

10

D

o

a • • 0 ••

0 • i • •

a . " .... • <SP'0

• 000

0 0 0

100

100

t - t I

0.9 1IE" = 0.111 '1(f6 /pSi 'I, = 2.987 n = 0.122

0.8 m = 0.326 ()( = O.OSO

CT2o =-0.026, \'= 2Bdays

0.7 CT20 = - 0.030, t' = 90 days CT20 =-O.036. t·= 365days CT •• = 0.445, t' = 28 days

0.6 CT •• = 0.509, t' = 90 days CT4• = 0.620. t' = 365 days

O.S 00

0 0

0 0

0.4 a

0.3

0.2 • • 1 10

Nasser and Neville, 1967

1/10,= 0.096 ·10-·/psi 0.7 '1\ =4.308

n = 0.140 m = 0.302

0.6 0( = 0.042 Cr21 =-0.068 CT46 =l.023

0.5 CT71 = 2.712 CT9!I =4.2

0.4

0

0 0

0

0 -$" #" ." 0

,:" ,~a {" oo:f' • <,:-<f> 0 0 '))~(, • l1'i~' •

~,1.'1> • • 1>.(,0 . ~~

(;#>. ' 0 0

100

.-0.3

t·, 365 days

0.2

Z ie linski and Sadowski, 1973

05 I I Eo =0108 ·10-' Ipsi

</>, = 2.989

=0.123 Cno = -0.029

m =0.326 CT60 = 0.601

0.4 a =0.055

0.3

0 • 0.2 • • ,'=123days

10 100

In days

Fig. 32. - Fits of Tests of Temperature Effect on Creep by York. Kennedy and Perry (1970) [23]. Da Silveira and Florentino (1968) [67]. Nasser and Neville (1965) [68]. Nasser and Neville (1967) [69]. Seki and Kawasumi (1970) [70]. Zielinski and Sadowski (1973) [71]. CT and 1/ Eo both calculated with formula,

431

VOl. 11 - N° 66 - MATERIAUX ET CONSTRUCTIONS

1.2 Browne, 1967 t',7days t", 28 days

lIEo' 0.102 '1O-6/psi CT20 '-0.022 1.0 ~ '3.511 CT40,0.159

n '0.156 CT65,0.536 m ,0307 CT93.5 , 0.958

0.8 0( '0.060

0.6 0 0

'Vi 0 • CT20 '-0.020 0. •

........ CT4::J' 0.144 \D CT65 , 0.487 '0

CT93.5'0.870 C 0.2

10 100 1000 10

....... 0

~ 1.0 t', 180 days 0

-, CT20' -0.027 0 CT40' 0.199

<.;c., 0

0.8 CT65' 0.673 C T93. 5' 1.203

c,"> 0 • 0 \:>"" •

0 • 0.6 •

• 0.4

0.2 10 100 1000

100 1000

r, 400 days CT20' -0.031 CT40' O. 223

0

0 0

0 0

• • 10

t", 60 days

CT20' - 0.024 CT40' 0.172 CT65 ,0.582 C193.5 ,1.040

10

1.0·~ 0 o 0

0 0

100

100 1000

0

1000

t - t' in days

Fig. 33. - Fits of Tests of Temperature Effect on Creep by Browne (1967) [59]. CT and II Eo both calculated with formula. Experimental datas are smoothed mean values.

Da Silveira and Florentino's Tests of Temperature Effect on Creep (1968) [67J. - Prisms 20 x 20 x 60 cm, in copper jackets. Heat is assumed to be applied 3 days before loading. Water-cement-sand-gravel ratio 0.5: 1 : 2.35 : 3.84. Granite aggregate, modified portland cement, similar to ASTM type II. Cement content 314.6 kg/m', 8-day cube strength 297 kp/ cm2 (29. I N/ mm2).

Nasser and Neville's Tests of Temperature Effect on Creep (1965) [68]. - Cylinders 3 x 9 ± inch (76 x 235 mm), sealed in polypropylene jackets, stored from 24 hours onwards in a water bath at the desired temperature and loaded at age of 14 days. Water-cement ratio 0.6 and aggregate-cement ratio 7.15. Max. size of aggregate 3/4 inch (19 mm). Aggregate was a mixture of dolomite and hornblende. Cement type III (320 kg/m'). Strength 5,660 psi (39 N/mm2) at 14 days measured on cylinders 3 x 9 t inch (76 x 235 mm). Stress/ strength ratio 0.35.

Nasser and Neville's Tests of Temperature Effect on Creep (1967) [69]. - Cylinders 3 x 9 tinch (76 x 235 mm), stored in water at 70°F (21°C) up to 1 week prior to appli­cation of load. Concrete I: 7. 15 mix; water-cement ratio of" 0.6. Max. size of dolomite and hornblende aggregate was 3/4 inch (19 mm), cement type III (320 kg/m'). Speci­mens loaded at age of 1 year and remained under water while loaded. Mean strength at the time of load application (determined on specimens of same size) = 7,250 psi (50 N/mm2).

Browne's Tests of Temperature Effect on Creep (1967) [59]. - Cylinders 6 x 12 inch (152 x 305 mm), sealed at casting in 1/16 inch (1 .6 mm) polypropylene jackets, cured at room temperature. Heat applied 1 day before loading. Water-cement-sand-gravel ratio 0.42: 1: 1.45: 2.95. Ordinary portland cement, crushed foraminiferal limestone, max. size 1. 5 inch (38 mm). Average 6 inch (15.2 cm) cube strength = 7,250 psi (50 N/mm2).

432

Zielinski and Sadowski's Tests of Temperature Effect on Creep (1973) [7lJ. - Cylinders 160 x 480 mm within the first 70 days stored in atmosphere 100% relative humidity and temperature 20-23°C, then sealed with rubber coat. Specimens were heated at the age of 120 days and loaded three days later. W a ter-cemen t-aggrega te ratio 0.456 : I : 4.154. Sand/cement-gravel/cement ratio assumed to be 1.9: 2.254. Cement ordinary Portland Cement type I, 450 kg/m" aggregate crushed basalt and river sand, max. size 20 mm. 120 day compressive cylinder (160 x 160 mm) strength 430 kg/cm2.

Seki and Kawasumi's Tests of Temperature Effect on Creep (1970) [70 J. - Cylinders 150 x 600 mm were cast into 0.2 mm copper jackets. Specimens loaded at room temperature (20nC) at the age of 28 and 96 days. The tempe­rature 40°C was applied at the age of 28 and 97 days and loaded at the age of 29 and 100 days. The temperature 70°C was applied at the age of 27 and 104 days and specimens loaded when they were 29 and 105 days old. Water-cement­sand-aggregate ratio 0.4: I: 1.761: 3.834. Normal Portland cement 343 kg/m', fine aggregate Fuji-Gawa river sand, coarse aggregate from the river Ara-Kawa. 28-day cylinder strength 445 kp/ cm2.

Komendant, Polivka and Pirtz's Tests of Temperature Effect on Creep (1976) [72J. - Cylinders 6 x 16 inch (152 x 406 mm) sealed with butyl rubber against moisture loss and cured at 73°F (23°C) until five days prior to the age of loading. The specimens were then heated to test temperatures 110 and 160°F (43 and 71°C) at a rate of 24°F/day (13. 3°C/day) and remained for, the ci,l,Iration of the creep test. Specimens were loaded at the age of 28, 90 and 270 days. Cement, Medusa type II. Mix A: water­cement-sand-gravel ratio 0.381: 1 : 1.734: 2.605; 28-day cylinder strength 6,590 psi (45.4 N / mm2). Cement 706 lbs/ cy (419 kg/ m'). Max size of aggregate 1.5 inch.

.j::.. w w

(/)

C-

'-'" , 0

C

"-

J

0.8

oj Komendont, Polivko, Pirtz, 1976

I I Eo -.0.101 10·o/psi

<1>, - 3.293

061- -0.151

m - 0.303

a - 0.066 05f-

C T23 - 0.00

CT •• -0.162

04 f- CT71 - 0.512

03t

00 0.2 o •

",I " 11,1 ,I

0.7f· Komendant, Polivka, Pirtz, 1976

IIEo - 0.101 10·6/psi

<1>, - 3.293

"f - 0.151

m - 0.303 0.5

a - 0.066

C T23 - 0.00 0.4 C T43 - 0.186

C T71 -0.587

n'~ 0.2 0

e

Kamendant, Polivka, P"tz, 1976 0.6f-

I I Eo -0.101 10.6 I psi

<1>, - 3.293 0.5 ~ - 0.151

m - 0.303

0.4 r a - 0.066

CT23

03 C T43

CTT' -0.685

0.2 0

0.1 0.01 0.1

03 -.... ~ I • 0 0

0

000

t' = 28 days (mIX A) 02~ 00

,I 0

0

i' 0.7

0 0

tI 0.6

a 05

0.4

0.3

t' = 90 days (mix A) 02~ 0.1

0 OOIC

0 0 0 0

06

% I.r- 0.5 "\,

n orf'rP~~ _/ . 0.4

D~_~~ 103

1

10 100

t - t' In

Komendant, Polivka, Pirtz, 1976

I I Eo - 0.101 10·6/psi

<1>, - 3.235

- 0.147

m - 0.305

a - 0.065

C123 - 0.00

CT43 - 0.153

CTTI - 0.483

• ~ • 0

Komendant, Polivka, Pirtz, 1976

II Eo - 0.101 10- 6 ; psi

<1>, - 3.235

- 0.147

m - 0.305

a - 0.065

C T2, - 0.00

C 143 - 0.175

C T71 - 0.554

0 0 0

• • 0

Komendant, Polivka, Pirtz, 1976

I I Eo - 0.101·10· 6/psi

<1>, - 3.235

- 0.147

m - 0.305

a

C T11

0

·0 ......

days

t' =28days (mixB)

rf rP

61

aI o

o

t'= 90 days (mix B)

o

r#

r9 o

Fig. 34. - Fits of Tests of Temperature Effect on Creep by Komendant, Polivka and Pirtz (1976) [72], Cy and 1/ Eo both calculated with formula.

!'I

:0 Ol ~ N ~ Z ...j

1000 :-'tJ ~ Z C r-~

VOL. 11 - N° 66 - MATERIAUX ET CONSTRUCTIONS

REFERENCES

'[59] BROWNE R. D. - Properties of concrete in reactor vessels. Proceedings, Conference on Prestressed Con­crete Pressure Vessels, Institution of Civil Engineers, London, England, group C, paper 13, 1967, pp. 11-31.

[60] BAiANT Z. P., Wu S. T. - Thermoviscoelasticity of aging concrete. Journal of the Engineering Mechanics Division, ASCE, Vol. 100, No. EM3, Proc. Paper 10621, June, 1974, pp. 575-597.

[61] HANNANT D. J. - Strain behaviour of concrete up to 95°C under compressive stresses. Group C, paper 17. Conference on Prestressed Concrete Pressure Vessels, Institution of Civil Engineers, London 1967, pp. 57-71.

[62] MAREcHAL J. C. - Variations in the modulus of elasticity and Poisson s ratio with temperature. Paper SP34-27. Concrete for nuclear reactors, Vol. I, ACI Special Publication SP-34. American Concrete Institute, Detroit, Michigan, 1972, pp. 495-503.

[63] ARTHANARI S., Yu C. W. - Creep of concrete under uniaxial and biaxial stresses at elevated temperatures. Magazine of Concrete Research, Vol. 19, No. 60, September 1967, pp. 149-155.

[64] NISHlZAWA N., OKAMURA H. - Strength and inelastic properties of concrete at elevated temperature. Paper SP34-22. Concrete for nuclear reactors, Vol. I, ACI Special Publication SP-34 American Concrete Institute, Detroit, Michigan 1972, pp. 407-421.

[65] ENGLAND G. L., Ross A. D. - Reinforced concrete under thermal gradients. Magazine of Concrete Research, Vol. 14, No. 40, March 1962, pp. 5-12.

Un modele de preVISion pratique des deformations du beton en fonction du temps. III. Fluage en sechage. -Le modele pratique de determination du jluage et du retrait expose dans les parties I et II de ce memoire est d present applique au jluage en ambiance seche et d temperature constante. £augmentation du jluage due au sechage est reliee au retrait. On donne les formules pour determiner les parametres des materiaux d partir de la resistance du betan et de la composition du ·melange, et on les verifie par des comparaisons nombreuses avec les resultats d'essai publies.

IV. Influence de la temperature sur Ie fluage de base. -Le developpement d'un modele pour Ie (luage de base

434

[66] JOHANSEN R, BEST C. H. - Creep of concrete with and without ice in the system. Bulletin RILEM, No. 16, September 1962, pp. 47-51.

[67] DA SILVEIRA A., FLORENTINO C. A. - Influence of temperature on the creep of mass concrete. Paper SP25-7. Temperature and concrete. ACI Publication SP-25. American Concrete In~titute, Detroit, Michigan, 1971, pp. 173-189.

[68] NASSER K. W., NEVILLE A. M. - Creep of concrete at elevated temperatures, Journal of the American Concrete Institute, Vol. 62, December 1965, pp. 1567-1579.

[69] NASSER K. W., NEVILLE A. M. - Creep of old concrete at normal and elevated temperatures, Journal of the American Concrete Institute, Vol. 64, February 1967, pp. 97-103.

[70] SEKI S., KAWASUMI M. - Creep of concrete at elevated temperatures. Paper SP34-32. Concrete for nuclear reactors, Vol. I, ACI Special. Publication SP-34, American Concrete Institute, Detroi( Michigan, 1972, pp. 591-638.

[71] ZIELINSKI J. L., SADOWSKI A. - The influence of moisture content on the creep of concrete at elevated temperatures, 2nd International Conference on Struc­tural Mechanics in Reactor Technology, 1973, paper H6/3, pp. 1-8.

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qui est l'objet de la deuxieme partie de ce memo ire est suivi ici par un modele de determination du jluage a differentes temperatures maintenues constantes durant Ie phenomene. Ce modele qui preserve la loi de double puissance traduit deux effets contraires de la. tempera­ture : l'augmentation de la vitesse du jluage due a la chaleur et la diminution du jluage due a l'acceleration de l'hydratation par la chaleur. L'etude comprend la determination des parametres des materiaux a partir de la composition du melange et de nombreuses compa­raisons avec les resultats d'essai indiquent une bonne concordance.

To be continued by Parts Vand n.