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TitleACI 209.2R-08
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Table of Contents
	1.1— Background
	1.3—Basic assumptions for development of prediction models
		1.3.1 Shrinkage and creep are additive
		1.3.2 Linear aging model for creep
		1.3.3 Separation of creep into basic creep and dryingcreep
		1.3.4 Differential shrinkage and creep or shrinkage andcreep gradients are neglected
		1.3.5 Stresses induced during curing phase are negligible
	2.1— Notation
	3.1— Data used for evaluation of models
	3.2—Statistical methods for comparing models
	3.3—Criteria for prediction models
	3.4—Identification of strains
	3.5—Evaluation criteria for creep and shrinkage models
	4.1—ACI 209R-92 model
	4.2—Bazant-Baweja B3 model
	4.3—CEB MC90-99 model
	4.4—GL2000 model
	4.5—Statistical comparisons
	4.6—Notes about models
	5.1— Referenced standards and reports
	5.2—Cited references
	A.1— ACI 209R- 92 model
		A.1.1 Shrinkage
		A.1.2 Compliance
	A.2—Bazant-Baweja B3 model
		A.2.1 Shrinkage
		A.2.2 Compliance
	A.3—CEB MC90-99 model
		A.3.1 Shrinkage CEB MC90
		A.3.2 Shrinkage CEB MC90-99
		A.3.3 Compliance
	A.4—GL2000 model
		A.4.1 Relationship between specified and mean compressivestrength of concrete
		A.4.2 Modulus of elasticity
		A.4.3 Aggregate stiffness
		A.4.4 Strength development with time
		A.4.5 Shrinkage
		A.4.6 Compliance equations
	B.1—BP coefficient of variation
	B.2—CEB statistical indicators
		B.2.1 CEB coefficient of variation
		B.2.2 CEB mean square error
		B.2.3 CEB mean deviation
	B.3—The Gardner coefficient of variation (
	C.1—ACI 209R-92 model solution
		C.1.1 Estimated concrete properties
		C.1.2 Estimated concrete mixture
		C.1.3 Shrinkage strains εsh(t,tc)
		C.1.4 Compliance J(t,to)
	C.2—Bazant-Baweja B3 model solution
		C.2.1 Estimated concrete properties
		C.2.2 Estimated concrete mixture
		C.2.3 Shrinkage strains εsh(t,tc)
		C.2.4 Compliance J(t,to) = q1 + Co(t,to) + Cd(t,to,tc)
	C.3—CEB MC90-99 model solution
		C.3.1 Estimated concrete properties
		C.3.2 Estimated concrete mixture
		C.3.3 CEB MC90 shrinkage strains εsh(t,tc)
		C.3.4 CEB MC90-99 shrinkage strains εsh(t,tc)
		C.3.5 Compliance J(t,to)
	C.4—GL2000 model solution
		C.4.1 Estimated concrete properties
		C.4.2 Estimated concrete mixture
		C.4.3 Shrinkage strains εsh(t,tc)
		C.4.4 Compliance J(t,to)
	C.5—Graphical comparison of model predictions
		C.5.1 Shrinkage strains εsh(t,tc)
		C.5.2 Compliance J(t,to)
Document Text Contents
Page 1

ACI 209.2R-08

Reported by ACI Committee 209

Guide for Modeling and Calculating
Shrinkage and Creep

in Hardened Concrete

Page 2

Guide for Modeling and Calculating Shrinkage and Creep
in Hardened Concrete

First Printing
May 2008

ISBN 978-0-87031-278-6

American Concrete Institute

Advancing concrete knowledge

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Page 24


Table A.9—ks as function of cross section shape,
B3 model

Cross section shape ks

Infinite slab 1.00

Infinite cylinder 1.15

Infinite square prism 1.25

Sphere 1.30

Cube 1.55

Note: The analyst needs to estimate which of these shapes best approximates the real
shape of the member or structure. High accuracy in this respect is not needed, and ks
≈ 1 can be used for simplified analysis.
where m and n are empirical parameters whose value can be
taken the same for all normal concretes (m = 0.5 and n = 0.1).

In Eq. (A-40), q3 is the nonaging viscoelastic compliance
parameter, and q4 is the aging flow compliance parameter.
These parameters are a function of the concrete mean
compressive strength at 28 days fcm28 (in MPa or psi), the
cement content c (in kg/m3 or lb/yd3), the water-cement ratio
w/c, and the aggregate-cement ratio a/c

q3 = 0.29(w/c)
4q2 (A-46)


The compliance function for drying creep is defined by
Eq. (A-48). This equation accounts for the drying before

q4 20.3 10

× a c⁄( )

in SI units=

q4 0.14 10

× a c⁄( )

in in.-lb units=
Cd(t,to,tc) = q5[exp{–8H(t)} – exp{8H(to)}]
1/2 (A-48)
loading. Note that drying before loading is considered only
for drying creep
In Eq. (A-48), q5 is the drying creep compliance parameter.
This parameter is a function of the concrete mean compressive
strength at 28 days fcm28 (in MPa or psi), and of εsh∞ , the
ultimate shrinkage strain as given in Eq. (A-32)

q5 = 0.757fcm28
–1|εsh∞ × 10

6|–0.6 (A-49)

H(t) and H(to) are spatial averages of pore relative
humidity. Equations (A-50) to (A-53) and Eq. (A-36) are
H(t) = 1 – (1 – h)S(t – tc) (A-50)

H(to) = 1 – (1 – h)S(to – tc) (A-51)
(A-52)S t tc–( )
t tc–

-----------⎝ ⎠

⎛ ⎞
1 2⁄

(A-53)S to tc–( )
to tc–

-------------⎝ ⎠

⎛ ⎞
1 2⁄

required to calculate H(t) and H(to).
where S(t – tc) and S(to – tc) are the time function for shrinkage
calculated at the age of concrete t and the age of concrete at
loading to in days, respectively, and τsh is the shrinkage
A.3—CEB MC90-99 model
The CEB MC90 model (Muller and Hilsdorf 1990; CEB

1993) is intended to predict the time-dependent mean cross-
section behavior of a concrete member. It has concept similar
to that of ACI 209R-92 model in the sense that it gives a hyper-
bolic change with time for creep and shrinkage, and it also uses
an ultimate value corrected according mixture propor-
tioning and environment conditions. Unless special provi-
sions are given, the models for shrinkage and creep predict the
time-dependent behavior of ordinary-strength concrete (12
MPa [1740 psi] ≤ fc′ ≤ 80 MPa [11,600 psi]) moist cured at
normal temperatures not longer than 14 days and exposed to
a mean ambient relative humidity in the range of 40 to 100%
at mean ambient temperatures from 5 to 30 °C (41 to 86 °F).
The models are valid for normalweight plain structural
concrete having an average compressive strength in the range
of 20 MPa (2900 psi) ≤ fcm28 ≤ 90 MPa (13,000 psi). The age
at loading to should be at least 1 day, and the sustained stress
should not exceed 40% of the mean concrete strength fcmto at
the time of loading to. Special provisions are given for
elevated or reduced temperatures and for high stress levels.

The CEB MC90-99 model (CEB 1999) includes the latest
improvements to the CEB MC90 model. The model has been
developed for normal- and high-strength concrete, and
considers the separation of the total shrinkage into autogenous
and drying shrinkage components. The models for shrinkage
and creep are intended to predict the time-dependent mean
cross-section behavior of a concrete member moist cured at
normal temperatures not longer than 14 days and exposed to
a mean ambient relative humidity in the range of 40 to 100%
at mean ambient temperatures from 10 to 30 °C (50 to 86 °F).
It is valid for normalweight plain structural concrete having
an average compressive strength in the range of 15 MPa
(2175 psi) ≤ fcm28 ≤ 120 MPa (17,400 psi). The age at loading
should be at least 1 day, and the creep-induced stress should
not exceed 40% of the concrete strength at the time of loading.

The CEB model does not require any information regarding
the duration of curing or curing condition, but takes into
account the average relative humidity and member size.

Required parameters:
• Age of concrete when drying starts, usually taken as the

age at the end of moist curing (days);
• Age of concrete at loading (days);
• Concrete mean compressive strength at 28 days (MPa

or psi);
• Relative humidity expressed as a decimal;
• Volume-surface ratio or effective cross-section thickness

of the member (mm or in.); and
• Cement type.

A.3.1 Shrinkage CEB MC90—The total shrinkage strains
of concrete εsh(t,tc) may be calculated from

εsh(t,tc) = εcsoβs(t – tc) (A-54)

Page 25


where εcso is the notional shrinkage coefficient, βs(t – tc) is
the coefficient describing the development of shrinkage with
time of drying, t is the age of concrete (days) at the moment
considered, tc is the age of concrete at the beginning of
drying (days), and (t – tc) is the duration of drying (days).

The notional shrinkage coefficient may be obtained from

εcso = εs( fcm28)βRH(h) (A-55)


εs(fcm28) = [160 + 10βsc(9 – fcm28/fcm0)] × 10
–6 (A-56)


where fcm28 is the mean compressive cylinder strength of
concrete at the age of 28 days (MPa or psi), fcmo is equal
to 10 MPa (1450 psi), βsc is a coefficient that depends on the
type of cement (Table A.10), h is the ambient relative

βRH h( ) 1.55 1
-----⎝ ⎠

⎛ ⎞ 3– for 0.4 h 0.99<≤–=

βRH h( ) 0.25 for h 0.99≥=
Table A.10—Coefficient βsc according to
Eq. (A-56), CEB MC90 model

Type of cement according to EC2 βsc

SL (slowly-hardening cements) 4

N and R (normal or rapid hardening cements) 5

RS (rapid hardening high-strength cements) 8
humidity as a decimal, and ho is equal to 1.
The development of shrinkage with time is given by


where (t – tc) is the duration of drying (days), t1 is equal to 1 day,
V/S is the volume-surface ratio (mm or in.), and (V/S)o is
equal to 50 mm (2 in.).

The method assumes that, for curing periods of concrete
members not longer than 14 days at normal ambient
temperature, the duration of moist curing does not significantly
affect shrinkage. Hence, this parameter, as well as the effect of
curing temperature, is not taken into account. Therefore,
in Eq. (A-54) and (A-58), the actual duration of drying (t – tc)
has to be used.

When constant temperatures above 30 °C (86 °F) are
applied while the concrete is drying, CEB MC90 recom-
mends using an elevated temperature correction for βRH(h)
and βs(t – tc), shown as follows.

The effect of temperature on the notional shrinkage
coefficient is taken into account by

In SI units:


In in.-lb units:

βs t tc–( )
t tc–( ) t1⁄

350 V S⁄( ) V S⁄( )o⁄[ ]

t tc–( ) t1⁄+



βRH T, βRH h( ) 1

1.03 h ho⁄–
----------------------------⎝ ⎠

⎛ ⎞ T To 20–⁄

-------------------------⎝ ⎠
⎛ ⎞+=

βRH T, βRH h( ) 1

1.03 h ho⁄–
----------------------------⎝ ⎠

⎛ ⎞ 18.778 T⋅ To 37.778–⁄

--------------------------------------------------------⎝ ⎠
⎛ ⎞+=
The effect of temperature on the time development of
shrinkage is taken into account by

In SI units:


In in.-lb units:

where βRH,T is the relative humidity factor corrected by
temperature that replaces βRH in Eq. (A-55), βs,T(t – tc) is the
temperature-dependent coefficient replacing βs(t – tc) in
Eq. (A-54), h is the relative humidity in decimals, ho is equal
to 1, V/S is the volume-surface ratio (mm or in.); (V/S) is
equal to 50 mm (2 in.), T is the ambient temperature (°C or °F),
and To is equal to 1 °C (33.8 °F).

A.3.2 Shrinkage CEB MC90-99—With respect to the
shrinkage characteristics of high-performance concrete, the
new approach for shrinkage subdivides the total shrinkage
into the components of autogenous shrinkage and drying
shrinkage. While the model for the drying shrinkage component
is closely related to the approach given in CEB MC90 (CEB
1993), for autogenous shrinkage, new relations had to be
derived. Some adjustments, however, should also be carried
out for the drying shrinkage component, as the new model
should cover both the shrinkage of normal- and high-perfor-
mance concrete; consequently, the autogenous shrinkage also
needs to be modeled for normal-strength concrete.

The total shrinkage of concrete εsh(t,tc) can be calculated
from Eq. (A-61)

εsh(t,tc) = εcas(t) + εcds(t,tc) (A-61)

where εsh(t,tc) is the total shrinkage, εcas(t) the autogenous
shrinkage, and εcds(t,tc) is the drying shrinkage at concrete
age t (days) after the beginning of drying at tc (days).

The autogenous shrinkage component εcas(t) is calculated
from Eq. (A-62)

εcas(t) = εcaso( fcm28)βas(t) (A-62)

βs T, t tc–( )
t tc–( ) t1⁄

350 V
---⎝ ⎠

⎛ ⎞ V
---⎝ ⎠

⎛ ⎞


----- 20–⎝ ⎠

⎛ ⎞–
t tc–( )





βs T, t tc–( )
t tc–( ) t1⁄

350 V
---⎝ ⎠
⎛ ⎞ V

---⎝ ⎠
⎛ ⎞⁄

0.06 18.778 T

------ 37.778–⎝ ⎠

⎛ ⎞–
t tc–( )





Page 47

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Phone: 248-848-3700
Fax: 248-848-3701

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Advancing concrete knowledge

Page 48


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Guide for Modeling and Calculating
Shrinkage and Creep in Hardened Concrete

American Concrete Institute

Advancing concrete knowledge

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