##### 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

Copyright by the American Concrete Institute, Farmington Hills, MI. All rights reserved. This material

may not be reproduced or copied, in whole or part, in any printed, mechanical, electronic, film, or other

distribution and storage media, without the written consent of ACI.

The technical committees responsible for ACI committee reports and standards strive to avoid ambiguities,

omissions, and errors in these documents. In spite of these efforts, the users of ACI documents occasionally

find information or requirements that may be subject to more than one interpretation or may be

incomplete or incorrect. Users who have suggestions for the improvement of ACI documents are

requested to contact ACI. Proper use of this document includes periodically checking for errata at

www.concrete.org/committees/errata.asp for the most up-to-date revisions.

ACI committee documents are intended for the use of individuals who are competent to evaluate the

significance and limitations of its content and recommendations and who will accept responsibility for the

application of the material it contains. Individuals who use this publication in any way assume all risk and

accept total responsibility for the application and use of this information.

All information in this publication is provided “as is” without warranty of any kind, either express or implied,

including but not limited to, the implied warranties of merchantability, fitness for a particular purpose or

non-infringement.

ACI and its members disclaim liability for damages of any kind, including any special, indirect, incidental,

or consequential damages, including without limitation, lost revenues or lost profits, which may result

from the use of this publication.

It is the responsibility of the user of this document to establish health and safety practices appropriate to

the specific circumstances involved with its use. ACI does not make any representations with regard to

health and safety issues and the use of this document. The user must determine the applicability of all

regulatory limitations before applying the document and must comply with all applicable laws and regulations,

including but not limited to, United States Occupational Safety and Health Administration (OSHA) health

and safety standards.

Order information: ACI documents are available in print, by download, on CD-ROM, through electronic

subscription, or reprint and may be obtained by contacting ACI.

Most ACI standards and committee reports are gathered together in the annually revised ACI Manual of

Concrete Practice (MCP).

American Concrete Institute

38800 Country Club Drive

Farmington Hills, MI 48331

U.S.A.

Phone: 248-848-3700

Fax: 248-848-3701

www.concrete.org

Page 24

Untitled

209.2R-22 ACI COMMITTEE REPORT

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)

(A-47)

The compliance function for drying creep is defined by

Eq. (A-48). This equation accounts for the drying before

q4 20.3 10

6–

× a c⁄( )

0.7–

in SI units=

q4 0.14 10

6–

× a c⁄( )

0.7–

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–

τsh

-----------⎝ ⎠

⎛ ⎞

1 2⁄

tanh=

(A-53)S to tc–( )

to tc–

τsh

-------------⎝ ⎠

⎛ ⎞

1 2⁄

tanh=

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

half-time

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

Untitled

MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE 209.2R-23

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)

with

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

–6 (A-56)

(A-57)

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

h

ho

-----⎝ ⎠

⎛ ⎞ 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

(A-58)

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:

(A-59)

In in.-lb units:

βs t tc–( )

t tc–( ) t1⁄

350 V S⁄( ) V S⁄( )o⁄[ ]

2

t tc–( ) t1⁄+

------------------------------------------------------------------------------------

0.5

=

βRH T, βRH h( ) 1

0.08

1.03 h ho⁄–

----------------------------⎝ ⎠

⎛ ⎞ T To 20–⁄

40

-------------------------⎝ ⎠

⎛ ⎞+=

βRH T, βRH h( ) 1

0.08

1.03 h ho⁄–

----------------------------⎝ ⎠

⎛ ⎞ 18.778 T⋅ To 37.778–⁄

40

--------------------------------------------------------⎝ ⎠

⎛ ⎞+=

The effect of temperature on the time development of

shrinkage is taken into account by

In SI units:

(A-60)

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

S

---⎝ ⎠

⎛ ⎞ V

S

---⎝ ⎠

⎛ ⎞

o

⁄

2

0.06

T

To

----- 20–⎝ ⎠

⎛ ⎞–

t tc–( )

t1

----------------+exp

---------------------------------------------------------------------------------------------------------------------

0.5

=

βs T, t tc–( )

t tc–( ) t1⁄

350 V

S

---⎝ ⎠

⎛ ⎞ V

S

---⎝ ⎠

⎛ ⎞⁄

2

0.06 18.778 T

To

------ 37.778–⎝ ⎠

⎛ ⎞–

t tc–( )

t1

----------------+exp

-------------------------------------------------------------------------------------------------------------------------------------------------

0.5

=

Page 47

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38800 Country Club Drive

Farmington Hills, MI 48331

U.S.A.

Phone: 248-848-3700

Fax: 248-848-3701

www.concrete.org

American Concrete Institute

®

Advancing concrete knowledge

Page 48

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www.concrete.org

Guide for Modeling and Calculating

Shrinkage and Creep in Hardened Concrete

American Concrete Institute

®

Advancing concrete knowledge

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

Copyright by the American Concrete Institute, Farmington Hills, MI. All rights reserved. This material

may not be reproduced or copied, in whole or part, in any printed, mechanical, electronic, film, or other

distribution and storage media, without the written consent of ACI.

The technical committees responsible for ACI committee reports and standards strive to avoid ambiguities,

omissions, and errors in these documents. In spite of these efforts, the users of ACI documents occasionally

find information or requirements that may be subject to more than one interpretation or may be

incomplete or incorrect. Users who have suggestions for the improvement of ACI documents are

requested to contact ACI. Proper use of this document includes periodically checking for errata at

www.concrete.org/committees/errata.asp for the most up-to-date revisions.

ACI committee documents are intended for the use of individuals who are competent to evaluate the

significance and limitations of its content and recommendations and who will accept responsibility for the

application of the material it contains. Individuals who use this publication in any way assume all risk and

accept total responsibility for the application and use of this information.

All information in this publication is provided “as is” without warranty of any kind, either express or implied,

including but not limited to, the implied warranties of merchantability, fitness for a particular purpose or

non-infringement.

ACI and its members disclaim liability for damages of any kind, including any special, indirect, incidental,

or consequential damages, including without limitation, lost revenues or lost profits, which may result

from the use of this publication.

It is the responsibility of the user of this document to establish health and safety practices appropriate to

the specific circumstances involved with its use. ACI does not make any representations with regard to

health and safety issues and the use of this document. The user must determine the applicability of all

regulatory limitations before applying the document and must comply with all applicable laws and regulations,

including but not limited to, United States Occupational Safety and Health Administration (OSHA) health

and safety standards.

Order information: ACI documents are available in print, by download, on CD-ROM, through electronic

subscription, or reprint and may be obtained by contacting ACI.

Most ACI standards and committee reports are gathered together in the annually revised ACI Manual of

Concrete Practice (MCP).

American Concrete Institute

38800 Country Club Drive

Farmington Hills, MI 48331

U.S.A.

Phone: 248-848-3700

Fax: 248-848-3701

www.concrete.org

Page 24

Untitled

209.2R-22 ACI COMMITTEE REPORT

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)

(A-47)

The compliance function for drying creep is defined by

Eq. (A-48). This equation accounts for the drying before

q4 20.3 10

6–

× a c⁄( )

0.7–

in SI units=

q4 0.14 10

6–

× a c⁄( )

0.7–

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–

τsh

-----------⎝ ⎠

⎛ ⎞

1 2⁄

tanh=

(A-53)S to tc–( )

to tc–

τsh

-------------⎝ ⎠

⎛ ⎞

1 2⁄

tanh=

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

half-time

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

Untitled

MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE 209.2R-23

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)

with

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

–6 (A-56)

(A-57)

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

h

ho

-----⎝ ⎠

⎛ ⎞ 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

(A-58)

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:

(A-59)

In in.-lb units:

βs t tc–( )

t tc–( ) t1⁄

350 V S⁄( ) V S⁄( )o⁄[ ]

2

t tc–( ) t1⁄+

------------------------------------------------------------------------------------

0.5

=

βRH T, βRH h( ) 1

0.08

1.03 h ho⁄–

----------------------------⎝ ⎠

⎛ ⎞ T To 20–⁄

40

-------------------------⎝ ⎠

⎛ ⎞+=

βRH T, βRH h( ) 1

0.08

1.03 h ho⁄–

----------------------------⎝ ⎠

⎛ ⎞ 18.778 T⋅ To 37.778–⁄

40

--------------------------------------------------------⎝ ⎠

⎛ ⎞+=

The effect of temperature on the time development of

shrinkage is taken into account by

In SI units:

(A-60)

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

S

---⎝ ⎠

⎛ ⎞ V

S

---⎝ ⎠

⎛ ⎞

o

⁄

2

0.06

T

To

----- 20–⎝ ⎠

⎛ ⎞–

t tc–( )

t1

----------------+exp

---------------------------------------------------------------------------------------------------------------------

0.5

=

βs T, t tc–( )

t tc–( ) t1⁄

350 V

S

---⎝ ⎠

⎛ ⎞ V

S

---⎝ ⎠

⎛ ⎞⁄

2

0.06 18.778 T

To

------ 37.778–⎝ ⎠

⎛ ⎞–

t tc–( )

t1

----------------+exp

-------------------------------------------------------------------------------------------------------------------------------------------------

0.5

=

Page 47

As ACI begins its second century of advancing concrete knowledge, its original chartered purpose

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spreading knowledge.” In keeping with this purpose, ACI supports the following activities:

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members receive discounts of up to 40% on all ACI products and services, including documents, seminars

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As a member of ACI, you join thousands of practitioners and professionals worldwide who share a

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Phone: 248-848-3700

Fax: 248-848-3701

www.concrete.org

American Concrete Institute

®

Advancing concrete knowledge

Page 48

The AMERICAN CONCRETE INSTITUTE

was founded in 1904 as a nonprofit membership organization dedicated to public

service and representing the user interest in the field of concrete. ACI gathers and

distributes information on the improvement of design, construction and

maintenance of concrete products and structures. The work of ACI is conducted by

individual ACI members and through volunteer committees composed of both

members and non-members.

The committees, as well as ACI as a whole, operate under a consensus format,

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Members are encouraged to participate in committee activities that relate to their

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www.concrete.org

Guide for Modeling and Calculating

Shrinkage and Creep in Hardened Concrete

American Concrete Institute

®

Advancing concrete knowledge