Disclaimer:
Bazant voted against the statistical
model comparisons in this guide
and believes them to be misleading.
His name appears since this was
mandatory for committee members.
ACI 209.2R08
Guide for Modeling and Calculating Shrinkage
./
and Creep in Har~ened
Concrete
Reported by ACI Committee 209
Akthem A. AIManaseer
Zdenek P. Bazant
J eff~ey J. Brooks
Ronald G. Burg
Mario Alberto Chiorino
Carlos C. Videla'
Chair
MaI'\van A. Daye
Walter H. Dilger
Noel J. Gardner'
Will Hansen
Hesham Marzouk
*Members of the subcommittee that prepared this guide.
This guide is intended for the prediction of shrinkage and creep in
compression in hardened concrete. It may be assumed that predictions
apply to concrete under tension and shear. It outlines the problems and
limitations in developing prediction equations jor shrinkage and compressive
creep of hardened concrete. It also presents and compares the prediction
capabilities offour different numerical methods. The models presented are
valid jar hardened concrete moist cured for at least 1 day and loaded after
curing or later. The models are intended jar concretes with mean compressive
cylindrical strengths at 28
days within a range oj at least 20
to 70
MPa
(3000 to 10,000 psi). This document is
addressed to designers who wish
to predict shrinkage and creep in concrete without testing. For structures
that are sensitive to shrinkage and creep, the accuraCj of an individual
model's predictions can be improved and their applicable range
expanded if the model is calibrated' with test data oj the actual concrete
to be used in the project.
Keywords: creep; drying shrinkage; prediction models; statistical indicators.
ACI Committee Reports, Guides, Manuals, Standard
Practices, and Commentaries are intended for guidance in
planning, designing, executing, and inspecting construction.
This document is 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.
The American Concrete Institute disclaims any and all
responsibility for the stated principles. The Institute shall not
be liable for any loss or damage arising therefrom.
Reference to this document shall not be made in contract
documents. If items found in this document are desired by the
ArchitectiEngineer to be a part of the contract documents, they
shall be restated in mandatory language for incorporation by
the ArchitectlEngineer.
Domingo J. Carreira'
Secretary
David B. McDonald'
Harald S. Mueller
Ham H. A. Nassif
Lawrence C. Novak
Klaus Alexander Rieder
Ian Robertson
Kenji Sakata
K. Nam Shiu
W. Jason Weiss
CONTENTS
Chapter 1lntroduction and scope, p. 209.2R2
1. IBackground
1.2Scope
1.3Basic assumptions for development of prediction
models
Chapter 2Notation and definitions, p. 209.2R3
2. INotation
2.2Defmitions
Chapter 3Prediction models, p. 209.2R5
3 . IData used for evaluation of models
3 .2Statistical methods for comparing models
3 .3Criteria for prediction models
3.4Identification of strains
3 .5Evaluation criteria for creep and shrinkage models
Chapter 4Model selection, p. 209.2R7
4.lACI 209R92 model
4.2BazantBaweja B3 model
4.3CEB MC9099 model
4.4GL2000 model
4.5Statistical comparisons
4.6Notes about models
ACI 209.2R08 was adopted and published May 2008.
Copyright © 2008, American Concrete Institute. ' .
!. ,.
All rights reserved including rights of reproduction and use in any form or by any
means, including the making of copies by any photo process, or by electronic or
mechanical device, printed, written, or oral, or recording for sound or visual reproduction
or for use in any knowledge or retrieval system or device, unless permission in writing
is obtained from the copyright proprietors.
209.2R·1
209.2R2
ACI COMMITIEE REPORT
Chapter 5References, p. 209.2R13
5.1Referenced standards and reports
5.2Cited references
Appendix AModels, p. 209.2R16
A.1ACI 209R92 model
A.2BazantBaweja B3 model
A.3CEB MC9099 model
A.4GL2000 model
Appendix BStatistical indicators, p. 209.2R28
B.IBP coefficient of variation (til
Bp%) method
B.2CEB statistical indicators
B.3The Gardner coefficient of variation (IDG)
Appendix CNumeric examples, p. 209.2R30
C.lACI 209R92 model solution
C.2BazantBaweja B3 model solution
C.3CEB MC9099 model solution
C.4GL2000 model solution
C.5Graphical comparison of model predictions
CHAPTER 1INTRODUCTION AND SCOPE
1.1Background
To predict the strength and serviceability of reinforced and
prestressed concrete structures, the structural engineer requires
an appropriate description of the mechanical properties of the
materials, including the prediction of the timedependant
strains of the hardened concrete. The prediction of shrinkage
and creep is important to assess the risk of concrete cracking,
and deflections due to strippingreshoring. As discussed in
ACI 209.lR, however, the mechanical properties of concrete
are significantly affected by the temperature and availability of
water during curing, the environmental humidity and temper
ature after curing, and the composition of the concrete,
including the mechanical properties of the aggregates.
Among the timedependant properties of concrete that are of
interest to the structural engineer are the shrinkage due to
cement hydration (selfdesiccation), loss of moisture to the
environment, and the creep under sustained loads. Drying
before loading significantly reduces creep, and is a major
complication in the prediction of creep, stress relaxation, and
strain recovery after unloading. While there is a lot of data on
shrinkage and compressive creep, not much data are available
for creep recovery, and very limited data are available for
relaxation and tensile creep.
Cre~p under variable stresses and the stress responses
under constant or variable imposed strains are commonly
determined adopting the principle of superposition. The
limitations of this assumption are discussed in Section 1.3.
Further, the experimental results of Gamble and Parrott
(1978) indicate that both drying and basic creep are only
partially, not fully, recoverable. In general, provided that
water migration does not occur as in sealed concrete or the
interior of large concrete elements, superposition can be
used to calculate both recovery and relaxation.
The use of the compressive creep to the tensile creep in
calculation of beam's timedependant deflections has been
successfully applied in the work by Branson (1977), Bazant
and Ho (1984), and Carreira and Chu (1986).
The variability of shrinkage and creep test measurements
prevents models from closely matching experimental data.
The withinbatch coefficient of variation for laboratory
measured shrinkage on a single mixture of concrete was
approximately 8% (Bafant et al. 1987). Hence, it would be
unrealistic to expect results from prediction models to be
within plus or minus 20% of the test data for shrinkage. Even
larger differences occur for creep predictions. For structures
where shrinkage and creep are deemed critical, material testing
should be undertaken and longterm behavior extrapolated
from the resulting data. For a discussion of testing for
shrinkage and creep, refer to Acker (1993), Acker et al. (1998),
and Carreira and Burg (2000).
1.2Scope
This document was developed to address the issues related
to the prediction of creep under compression and shrinkage
induced strains in hardened concrete. It may be assumed,
however, that predictions apply to concrete under tension and
shear. It outlines the problems and limitations in developing
prediction equations, presents and compares the prediction
capabilities of the ACI 209R92 (ACI Committee 209 1992),
BazantBaweja B3 (Bafant and Baweja 1995, 2000), CEB
MC9099 (Muller and Hillsdorf 1990; CEB 1991, 1993,
1999), and GL2000 (Gardner and Lockman 2001) models, and
gives an extensive list of references. The models presented are
valid for hardened concrete moist cured for at least 1 day and
loaded at the end of 1 day of curing or later. The models
apply to concretes with mean compressive cylindrical
strengths at 28 days within a range of at least 20 to 70 MPa
(3000 to 10,000 psi). The prediction models were calibrated
with typical composition concretes, but not with concretes
containing silica fume, fly ash contents larger than 30%, or
natural pozzolans. Models should be calibrated by testing
such concretes. This document does not provide information
on the evaluation of the effects of creep and shrinkage on the
structural performance of concrete structures.
1.3Basic assumptions for development
of prediction models
Various testing conditions have been established to stan
dardize the measurements of shrinkage and creep. The
following simplifying assumptions are normally adopted in
the development of prediction models.
1.3.1 Shrinkage and creep are additiveTwo nominally
identical sets of specimens are made and subjected to the same
curing and environment conditions. One set is not loaded and is
used to determine shrinkage, while the other is generally loaded
from 20 to 40% of the concrete compressive strength. Load
induced strains are determined by subtracting the measured
shrinkage strains on the nonloaded specimens from the strains
measured on the loaded specimens. Therefore, it is assumed
that the shrinkage and creep are independent of each oft.1er.
Tests carried out on sealed specimens, with no moisture
movement from or to the specimens, are used to determine
autogenous shrinkage and basic creep.
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R·3
1.3.2 Linear aging model for creepExperimental
research indicates that creep may be considered approxi
mately proportional to stress (L'Hermite et al. 1958; Keeton
1965), provided that the applied stress is less than 40% of the
concrete compressive strength.
The strain responses to stress increments applied at
different times may be added using the superposition principle
(McHenry 1943) for increasing and decreasing stresses,
provided strain reversals are excluded (for example, as in
relaxation) and temperature and moisture content are kept
constant (Le Camus 1947; Hanson 1953; Davies 1957; Ross
1958; Neville and Dilger 1970; Neville 1973; BaZant 1975;
Gamble and Parrot 1978; RlLEM Technical Committee TC69
1988). Major deviations from the principle of superposition
are caused by the neglect of the random scatter of the creep
properties, by hygrothermal effects, including water diffusion
and time evolution of the distributions of pore moisture
content and temperature, and by material damage, including
distributed cracking and fracture, and also frictional
microslips. A comprehensive summary of the debate on the
applicability of the principle of superposition when dealing
with the evaluation of creep structural effects can be found
in the references (BaZant 1975, 1999, 2000; CEB 1984;
RILEM Technical Committee TC1 07 1995; Al Manaseer et
al. 1999; Jirasek and BaZant 2002; Gardner and Tsuruta
2004; Bazant 2007).
1.3.3
Separation of creep into basic creep and drying
creepBasic creep is measured on specimens that are sealed
to prevent the ingress or egress of moisture from or to its
environment. It is considered a material constitutive property
and independent of the specimen size and shape. Drying creep
is the strain remaining after subtracting shrinkage, elastic, and
basic creep strains from the total measured strain on nominally
identical specimens in a drying environment. The measured
average creep of a cross section at drying is strongly size
dependant. Any effects of thermal strains have to be removed
in all cases or are avoided by testing at constant temperature.
In sealed concrete specimens, there is no moisture movement
into or out of the specimens. Lowwatercementratio
concretes selfdesiccate, however, leading to autogenous
shrinkage. Normalstrength concretes do not change volume at
relative humidity in the range 95 to 99%, whereas samples
stored in water swell (L'Hermite et al. 1958).
1.3.4 Differential shrinkage and creep or shrinkage and
creep gradients are neglectedThe shrinkage strains deter
mined according to ASTM
C157/C157M are measured along
the longitudinal axis of prismatic specimens; however, the
majority of reported creep and shrinkage data are based on
surface measurements of cylindrical specimens (ASTM
C512). Unless fmite element analysis (BaZant et al. 1975) or
equivalent linear gradients (Carreira and Walser 1980) are
used, it is generally assumed that shrinkage and creep strains
in a specimen occur uniformly through the specimen cross
section. Kristek et al. (2006) concluded that for box girder
bridges, the classical creep analysis that assumes the shrinkage
and creep properties to be uniform throughout the cross section
is inadequate. As concrete ages, differences in strain gradients
reduce (Carreira and Walser 1980; Aguilar 2005).
1.3.5 Stresses induced during curing phase are negligible
Most test programs consider the measurement of strains
from the start of drying. It is assumed that the restrained
stresses due to swelling and autogenous shrinkage are
negligible because of the large creep strains and stress
relaxation of the concrete at early ages. For restrained
swelling, this assumption leads to an overestimation of the
tensile stresses and, therefore, it may be an appropriate basis
for design when predicting deflections or prestress losses.
For predicting the effects of restrained autogenous shrinkage
or relaxation, however, the opposite occurs. Limited testing
information exists for tensile creep.
CHAPTER 2NOTATION AND DEFINITIONS
2.1Notation
a,b
=
a
=
Co(t,to) =
Cl.t,to,te) =
c
=
d=4V/S =
E
=
Eem
=
Eem28
=
Eemt
=
Eemto
=
e=2V1S =
fem
=
fern28
=
fernt
=
fernte
=
femto
=
constants used to describe the strength gain
development of the concrete, ACI 209R92
and GL2000 models
agfregate content of concrete, kg/m
3
or lb/
yd ,B3 model
compliance function for basic creep at
concrete age
t when loading starts at age
to'
B3 model
compliance function for drying creep at
concrete age
t when loading and drying starts
at ages
to and
te, respectively, B3 model
cement content of concrete, kg/m3 or Ib/yd3,
ACI 209R92 and B3 models
average thickness of a member, mm or in.,
ACI 209R92 model
modulus of elasticity, MPa or psi
mean modulus of elasticity of concrete, MPa
or psi
mean modulus of elasticity of concrete at
28 days, MPa or psi
mean modulus of elasticity of concrete at age
t, MPa or psi
mean modulus of elasticity of concrete when
loading starts at age
to' MPa or psi
effective cross section thickness of member
or notional size of member according to B3 or
CEB MC90 and CEB MC9099 models,
respectively, in mm or in.; defined as the
crosssection divided by the semiperimeter
of the member in contact with the atmo
sphere, which coincides with the actual thick
ness in the case of a slab
concrete mean compressive cylinder strength,
MPaorpsi
concrete mean compressive cylinder strength
at 28 days, MPa or psi
concrete mean compressive cylinder strength
at age
t, MPa or psi
,
lit
concrete mean compressive cylinder strength
when drying starts at age
te, MPa or psi
concrete mean compressive cylinder strength
when loading starts at age
to' MPa or psi
209.2R4
fd
=
H(t)
=
h
=
J(t,to)
=
J(to,to)
=
kh,13RJlh)
or
13(h)
=
ks
=
ql
=
S(t 
te),
13s<t 
te)
or
13(t 
te)=
s
=
T
=
=
t te
=
te
=
to
=
VIS
=
w
=
Cl
=
Cli or
k
=
Cl2
=
Clas' Cldsl
and
Clds2 =
13as(t!
=
13e(t 
to) =
13ds(t 
te) =
13e
13RH,T
=
ACI
COMMITTEE REPORT
concrete specified cylinder strength at 28 days,
MPa or psi
spatial average of pore relative humidity at
concrete age
t, B3 model
relative humidity expressed as a decimal
compliance at concrete age
t when loading
starts at age
to' IlMPa or lIpsi
elastic compliance at concrete age
to when
loading starts at age
to' IIMPa or l/psi
correction term for effect of humidity on
shrinkage according to B3, CEB MC90 and
CEB MC9099, or GL2000 models, respec
tively
crosssection shape factor, B3 model
inverse of asymptotic elastic modulus, IIMPa
or lIpsi, B3 model
correction term for effect of time on
shrinkage according to B3, CEB MC90, or
GL2000 models, respectively
slump, mm or in., ACI 209R92 model. Also,
strength development parameter, CEB
MC90, CEB MC9099, and GL2000 models
temperature, DC, OF, or OK
age of concrete, days
duration of drying, days
age of concrete when drying starts at end of
moist curing, days
age of concrete at loading, days
volumesurface ratio, mm or in.
water content of concrete,
kg/m3 or Ib/yd3,
B3 model
air content expressed as percentage, ACI
209R92 model
shrinkage constant as function of cement
type, according to B3 or GL2000 models,
respectively
shrinkage constant related to curing conditions,
B3 model
correction coefficients for effect of cement
type on autogenous and drying shrinkage,
CEB MC9099 model
function describing time development of
autogenous shrinkage, CEB MC9099 model
correction term for effect of time on creep
coefficient according to CEB MC90 and
CEB MC9099 models
function describing time development of
drying shrinkage, CEB MC9099 model
factor relating strength development to
cement type, GL2000
correction coefficient to account for effect of
temperature on notional shrinkage, CEB
MC90modei
~sc
= correction coefficient that depends on type of
cement, CEB MC90 model
correction coefficient to account for effect of
temperature on time development of
shrinkage, CEB MC90 model
autogenous shrinkage strain at concrete age
t,
mm1mm or in.lin., CEB MC9099
drying shrinkage strain at concrete age
t since
the start of drying at age
te, mm1mm or in.lin.,
CEB MC9099 model
Eeso
= notional shrinkage coefficient,
mm1mm or
in.lin., CEB MC90 model
Eeasoifem2S) = notional autogenous shrinkage coefficient,
mm1mm or in.lin., CEB MC9099 model
Eedsoifem2S)= notional drying shrinkage coefficient,
mmI
mm or in.lin., CEB MC9099 model
Esh(t,te) = shrinkage strain at concrete age
t since the
start of drying at age
te, mm1mm or in.lin.
Eshu or
Eshoo= notional ultimate shrinkage strain,
mm1mm
IP(t, to)
IP2S(t, to)
IPo
IPRJlh)
'tsh
=
=
=
=
=
=
=
or in.lin., ACI 209R92 and GL2000 models
and B3 model, respectively
creep coefficient (dimensionless)
28day creep coefficient (dimensionless),
CEB MC90, CEB MC9099, and GL2000
models
notional creep coefficient (dimensionless),
CEB MC90 and CEB MC9099 models
correction term for effect of relati ve humidity
on notional creep coefficient, CEB MC90
and CEB M9099 models
correction term for effect of drying before
loading when drying starts at age
te, GL2000
model
ultimate (in time) creep coefficient, ACI
209R92 model
unit weight of concrete,
kg/m3 or Ib/ft
3
shrinkage and creep correction factor, respec
tively; also used as product of all applicable
corrections factors, ACI 209R92 model
shrinkage halftime, days, ACI 209R92 and
B3 models
= ratio of fine aggregate to total aggregate by
weight expressed as percentage, ACI 209R92
model
2.20efinitions
autogenous shrinkagethe shrinkage occurring in the
absence of moisture exchange (as in a sealed concrete
specimen) due to the hydration reactions taking place in the
cement matrix. Less commonly, it is termed basic shrinkage
or chemical shrinkage.
basic creepthe timedependent increase in strain under
sustained constant load of a concrete specimen in which
moisture losses or gains are prevented (sealtid sp~~men).
compliance
J(t,to)the total load induced strain (elastic
strain plus creep strain) at age
t per unit stress caused by a
unit uniaxial sustained load applied since loading age
to'
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R5
creep coefficientthe ratio of the creep strain to the initial
strain or, identically, the ratio of the creep compliance to the
compliance obtained at early ages, such as after 2 minutes.
28day creep coefficientthe ratio of the creep strain to
the elastic strain due to the load applied at the age of 28 days
(cP2s(t,to) =
cI>(t,to) .
Ecm2SIEcmto)·
creep strainthe timedependent increase in strain under
constant load taking place after the initial strain at loading.
drying creepthe additional creep to the basic creep in a
loaded specimen exposed to a drying environment and
allowed to dry.
drying shrinkageshrinkage occurring in a specimen
that is allowed to dry.
elastic compliance or the nominal elastic strain per unit
stress
J(to,to)the initial strain at loading age
to per unit
stress applied. It is the inverse of the mean modulus of elasticity
of concrete when loading starts at age
to'
initial strain at loading or nominal elastic strainthe
shortterm strain at the moment of loading and is frequently
considered as a nominal elastic strain as it contains creep that
occurs during the time taken to measure the strain.
loadinduced strainthe timedependent strain due to a
constant sustained load applied at age
to'
shrinkagethe strain measured on a loadfree concrete
specimen.
specific creepthe creep strain per unit stress.
total strainthe total change in length per unit length
measured on a concrete specimen under a sustained constant
load at uniform temperature.
CHAPTER 3PREDICTION MODELS
3.1Data used for evaluation of models
In 1978, BaZant and Panula started collecting shrinkage
and creep data from around the world and created a comput
erized databank, which was extended by Muller and Panula
as part of collaboration between the ACI and the CEB
established after the ACICEB Hubert Rusch workshop on
concrete creep (Hillsdorf and Carreira 1980). The databank,
now known as the RILEM databank, has been extended and
refined under the sponsorship of RILEM TC 107CSP,
Subcommittee 5 (Kuttner 1997; Muller et al. 1999).
Problems encountered in the development of the databank
have been discussed by Muller (1993) and others (AlMana
seer and Lakshmikantan 1999; Gardner 2000). One problem
involves which data sets should be included. For example,
some investigators do not include the lowmodulus sandstone
concrete data of Hansen and Mattock (1966), but do include
the Elgin gravel concrete data from the same researchers. A
further problem is the data of some researchers are not inter
nally consistent. For example, the results from the 150 mm.
(6 in.) diameter specimens of Hansen and Mattock are not
consistent with the results from the 100 and 200 mm (4 and
8 in.) diameter specimens. Finally, it is necessary to define the
relative humidity for sealed and immersed concrete specimens.
A major problem for all models is the description of the
concrete. Most models are sensitive to the type of cement
and the related strength development characteristics of the
material. Simple descriptions, such as ASTM C150 Type I,
used in the databank are becoming increasingly difficult to
interpret. For example, many cements meet the requirements
of Types I, II, and III simultaneously; also, the multiple
additions to the clinker allowed in ASTM C595 or in other
standards are unknown to the researcher and designer.
N ominaIly identical concretes stored in different environments,
such as those tested by Keeton (1965), have different
strength development rates. If this information exists, it
should be taken into account in model development.
In addition, cement descriptions differ from country to
country. The data obtained from European cement concretes
may not be directly compared with that of United States
cement concretes. Some researchers have suggested that
correlation should only be done with recent and relevant data
and that different shrinkage and creep curves should be
developed for European, Japanese, North American, and
South Pacific concretes (McDonald 1990; McDonald and
Roper 1993; Sakata 1993; Sakata et al. 2001; Videla et al.
2004; Videla and Aguilar 2005a). While shrinkage and creep
may vary with local conditions, research has shown that
shortterm shrinkage and creep measurements improve the
predictions regardless oflocation (Bazant 1987; Bazant and
Baweja 2000; Aguilar 2005). For this reason, the committee
recommends shortterm testing to determine the shrinkage,
creep, and elastic modulus of the concrete to improve the
predictions of the longterm deformations of the concrete.
Other issues include:
• The databank does not include sufficient data to validate
modeling that includes drying before loading or loading
before drying, which are common occurrences in practice;
• Many of the data sets in the databank were measured
over relatively short durations, which reduces the
usefulness of the data to predict longterm effects; and
• Most of the experiments were performed using small
specimens compared with structural elements. It is
debatable if the curing environment and consequent
mechanical properties of concrete in the interior of
large elements are well represented by small specimen
experiments (BaZant et al. 1975; Kristek et al. 2006).
Despite these limitations, it is imperative that databanks
such as the RILEM databank are maintained and updated as
they provide an indispensable source of data in addition to a
basis for comparing prediction models.
3.25tatistical methods for comparing models
Several methods have been used for the evaluation of the
accuracy of mOdels to predict experimental data. Just as a
single set of data may be described by its mean, mode,
median, standard deviation, and maximum and minimum, a
model for shrinkage or creep data may have several methods
to describe its deviation from the data. The committee could
not agree on a single method for comparison of test data with
predictions from models for shrinkage and creep. Reducing
the comparison between a large num~r of experimental
IJ·
results and a prediction method to a single number is fraught
with uncertainty. Therefore, the committee strongly recom
mends designers to perform sensitivity analysis of the
response of the structure using the models in this report and
209.2R6
ACI COMMITTEE REPORT
to carry out shorttenn testing to calibrate the models to
improve their predictions. The summary of the statistical
indicators given in Section 4 provides the user with basis for
comparison without endorsing any method.
One of the problems with the comparison of shrinkage and
creep data with a model's prediction is the increasing
divergence and spread of data with time, as shown in the
figures of Section 4. Thus, when techniques such as linear
regression are used, the weighting of the later data is greater
than that of the earlier data (Bazant 1987; BaZant et al. 1987).
On the contrary, comparison of the percent deviation of the
model from the data tends to weight earlyage data more than
laterage data. The divergence and spread are a measure of
the limitation of the model's capabilities and variability in
the experimental data.
Commonly used methods for detennining the deviation of
a model from the data include:
• Comparison of individual prediction curves to individual
sets of test data, which requires a casebycase evaluation;
Comparison of the test data and calculated values using
linear regression;
• Evaluation of the residuals (measuredpredicted value)
(McDonald 1990; McDonald and Roper 1993; Al
Manaseer and Lakshmikantan 1999). This method does
not represent leastsquare regression and, if there is a
trend in the data, it may be biased; and
• Calculation of a coefficient of variation or standard
error of regression nonnalized by the data centroid.
In the committee's opinion, the statistical indicators available
are not adequate to uniquely distinguish between models.
3.3Criteria for prediction models
Over the past 30 years, several models have been proposed
for the prediction of drying shrinkage, creep, and total strains
under load. These models are compromises between accuracy
and convenience. The committee concludes that one of the
primary needs is a model or models accessible to engineers
with little specialized knowledge of shrinkage and creep.
Major issues include, but are not restricted to:
• How simple or complex a model would be appropriate,
and what input infonnation should be required;
• What data should be used for evaluation of the model;
• How closely the model should represent physical
phenomena/behavior;
• What statistical methods are appropriate for evaluating
a model.
Th~re is no agreement upon which infonnation should be
required to calculate the timedependent properties of
concrete; whether the mechanical properties of the concrete
specified at the time of design should be sufficient or if the
mixture proportions are also required.
At a minimum, the committee believes that shrinkage and
creep models should include the following infonnation:
• Description of the concrete either as mixture propor
tions or mechanical properties such as strength or
modulus of elasticity;
• Ambient relative humidity;
• Age at loading;
Duration of drying;
• Duration of loading; and
• Specimen size.
. Models should also:
• Allow for the substitution of test values of concrete
strength and modulus of elasticity;
• Allow the extrapolation of measured shrinkage and
creep compliance results to get longtenn values; and
• Contain mathematical expressions that are not highly
sensitive to small changes in input parameters and are
easy to use.
As described in ACI 209.1R, it has long been recognized
that the stiffness of the aggregate significantly affects the
shrinkage and creep of concrete. Some models account for
the effect of aggregate type by assuming that the effects of
aggregate are related to its density or the concrete elastic
modulus. Models that use concrete strength can be adjusted
to use a measured modulus of elasticity to account for aggregate
properties. Models that do not use the mechanical charac
teristics of the concrete and rely on mixture proportion
information alone may not account for variations in behavior
due to aggregate properties.
Until recently, autogenous shrinkage was not considered
significant because, in most cases, it did not exceed 150
microstrains. For concretes with watercement ratios
(w/c)
less than 0.4, mean compressive strengths greater than 60 MPa
(8700 psi), or both, however, autogenous shrinkage may be
a major component of the shrinkage strain.
Some models consider that basic creep and drying creep
are independent and thus additive, while other models have
shrinkage and creep as dependent, and thus use multiplicative
factors. The physical phenomenon occurring in the concrete
may be neither.
3.4ldentification of strains
Equations (31) and (32) describe the additive simplification
discussed in 1.3.1
total strain = shrinkage strain + compliance x stress (31)
r
(elastic strain + basic creep + drying creep) (32)
comp lance =
stress
The total and shrinkage strains are measured in a creep and
shrinkage test program from which the compliance is deter
mined. Errors in the measured data result in errors in the
compliance. The elastic strain is detennined from earlyage
measurement, but as discussed previously, it is difficult to
separate earlyage creep from the elastic strain. Thus, the
assumed elastic strain is dependent on the time at which the
strain measurement is made and, therefore, on the ignored
early creep.
Basic creep and drying creep are detennined from the
compliance by subtracting the elastic strain, which may have
implicit errors, from the strains measured on dfNjng and
nondrying specimens. Errors in the measured elastic strain
used to determine the modulus of elasticity (ASTM C469),
in the measured total strain, or in the measured shrinkage
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R7
strain, are all reflected in the calculated creep strain, the
compliance, and creep coefficient.
For sealed specimens, the equations for compliance and
total strain simplify significantly if autogenous shrinkage is
ignored as in Eq. (33) and (34)
total strain = compliance x stress
(33)
compliance = (elastic strain + basic creep)
(34)
stress
3.5Evaluatlon criteria for creep
and shrinkage models
In 1995, RILEM Committee TC 107 published a list of
criteria for the evaluation of shrinkage and creep models
(RILEM 1995; BaZant 2000). In November 1999, ACI
Committee 209, which has a number of members in common
with RILEM TC 107, discussed the RILEM guidelines and
agreed on the following:
• Drying shrinkage and drying creep should be bounded.
That is, they do not increase indefinitely with time;
• Shrinkage and creep equations should be capable of
extrapolation in both time and size;
• Shrinkage and creep models should be compared with
the data in the databank limited by the conditions of
applicability of the model(s). That is, some experi
mental data, such as those with high watercement
ratios or lowmodulus concrete, may not be appropriate
to evaluate a model;
• Equations should be easy to use and not highly sensitive
to changes in input parameters;
• The shape of the individual shrinkage and creep curves
over a broad range of time (minutes to years) should
agree with individual test results;
• Creep values should be compared as compliance or
specific creep rather than as the creep coefficient. The
immediate strain/unit stress and the modulus of elasticity
are dependent on the rate of loading; however, for
developing the creep equations to determine longterm
deformations, this effect should not playa major role;
• Creep expressions should accommodate drying before
loading. Results by Abiar reported by Acker (1993)
show that predried concrete experiences very little
creep. Similarly, the very lateage loaded (2500 to 3000
days) results of Wesche et al. (1978) show reduced
creep compared with similar concrete loaded at early
ages. The effect of predrying may also be significantly
influenced by the size of the specimen;
• Shrinkage and creep expressions should be able to
accommodate concretes containing fly ash, slag (Videla
and Gaedicke 2004), natural pozzolans (Videla et al.
2004; Videla and Aguilar 2005a), silica fume and
chemical admixtures (Videla and Aguilar 2005b);
• The models should allow for the effect of specimen
size; and
• The models should allow for changes in relative humidity.
Success in achieving the following guidelines is consequent
to the method of calculation; that is, if the principle of super
position is valid and if the model includes drying before
loading, and how they are considered under unloading:
• Recovery of creep strains under complete unloading
should not exceed the creep strain from le>ading, and
should asymptotically approach a constant value; and
• Stress relaxation should not exceed the initially
applied stress.
Yue and Taerwe (1992, 1993) published two related
papers on creep recovery. Yue and Taerwe (1992)
commented, "It is well known that the application of the
principle of superposition in the service stress range yields
an inaccurate prediction of concrete creep when unloading
takes place." In their proposed twofunction method, Yue
and Taerwe (1993) used a linear creep function to model the
timedependent deformations due to increased stress on
concrete, and a separate nonlinear creep recovery function to
represent concrete behavior under decreasing stress.
CHAPTER 4MODEL SELECTION
There are two practical considerations in the models for
prediction of shrinkage and creep, namely:
• Mathematical form of their time dependency; and
• Fitting of the parameters and the resulting expressions.
If the mathematical form of the model does not accurately
describe the phenomena, extrapolations of shrinkage and
creep results will deviate from reality. After the mathematical
form has been justified, the fit of the prediction to measured
results should be compared for individual data sets.
The models selected for comparison are the ACI 209R92
(ACI Committee 209 1992), the BazantBaweja B3 devel
oped by BaZant and Baweja (1995,2000), the CEB Model
Code 199099 (CEB MC9099) (Muller and Hillsdorf 1990;
CEB 1991, 1993, 1999), and the GL2000 developed by
Gardner and Lockman (2001). Table 4.1 lists the individual
model's applicable range for different input variables
(adapted from AIManaseer and Lam 2005). Comparison of
models with experimental data is complicated by the lack of
agreement on selection of appropriate data and on the
methods used to compare the correlation. Descriptions of the
ACI 209R92, BaZantBaweja B3, CEB MC9099, and
GL2000 models are given in Appendix A. Kristek et al.
(200 1) and Sassone and Chiorino (2005) developed design
aids for determination of shrinkage, compliance, and relax
ation for ACI 209R92, BaZantBaweja B3, CEB MC9099,
and GL2000 models.
Figures 4.1 through 4.8 (Gardner 2004) compare the
predicted values for two sets of input information for
RILEM data sets extending longer than 500 days, concrete
28day mean cylinder strengthsfcm28 between 16 and 82 MPa
(2320 and 11,890 psi), watercement ratios between 0.4 and
0.6, duration of moist curing longer than 1 day (possibly
biased against ACI 209R92 because this model was
developed for standard conditions considering 7 days of
moist curing and 7 days of age at loading), age of loading
greater than the duration of moist curing, and volume
surface ratios
VIS greater than 19 mm
(3/4 in.). The humidity
range for compliance was 20 to 100%, and below 80% for
209.2R8
ACI COMMITTEE REPORT
Table 4.1Parameter ranges of each model
Input variables
ACI209R92
BazantBaweja B3
fcm28 , MPa (psi)

I7 to 70
(2500 to 10,000)
ale

2.5 to 13.5
Cement content,
279 to 446
160 to 720
kg/m3 (lb/yd3)
(470 to 752)
(270 to 1215)
w/c

0.35 to 0.85
Relative humidity, %
40 to 100
40 to 100
Type of cement,
RorRS
R,SL, RS
European (U.S.)
(I or III)
(I, II, III)
tc (moist cured)
~ I day
~ I day
tc (stearn cured)
1 to 3 days

to
~ 7 days
to ~
tc
shrinkage. Consequently, swelling was not included even
if some specimens were initially moist cured.
Two sets of comparisons are shown in each figure. One
set, identified as
"fem only," assumes that only the measured
28day strength
fem is known. The second set, identified as
"all data," uses the
fem calculated as the average of the
measured
fem' and that back calculated from the measured
Eem using the elastic modulus formula of the method and
mixture proportions if required by the model. Calculated
compliance is the calculated specific creep plus calculated
elastic compliance for the
fem graphs and the calculated
specific creep plus measured elastic compliance for the all
data graphs. The reported mixture composition was used for
ACI 209R92 and BazantBaweja B3. It was assumed that if
mixture data were available, the strength development data
and elastic modulus would also be available. Cement type was
determined by comparison of measured strength gain data with
the GL2000 strength gain equations. The same cement type
was used for predictions in all methods. For CEB MC9099,
ASTM C 150 Type I was taken as CEB Type N cement,
Type III as CEB Type R, and Type II as CEB Type SL.
It should be noted that each model should use an appropriate
value of elastic modulus for which the model was calibrated.
Therefore, for CEB, the elastic modulus was taken as
Eem =
9500(fem)1I3 in MPa
(262,250[fem]1I3 in psi). For Bazant
Baweja B3, using the shape factor
ks = 1.00 in
's (the
shrinkage time function) improved the results of the statistical
analy~is, and
all concretes were assumed moist cured; that is, u2
= 1.20 for calculations using the BaZantBaweja B3 model.
To calculate a coefficient of variation (Gardner 2004), the
durations after drying or application of load were divided into
seven halflog decade intervals: 3 to 9.9 days, 10 to 31 days,
32 to 99 days, 100 to 315 days, 316 to 999 days, 1000 to
3159 days, and greater than 3160 days. That is, each duration is
3.16 times the previous halflog decade; these are similar to
the CEB ranges. The root mean square (RMS) (calculated
observed) was calculated for all comparisons in each halflog
decade. The coefficient of variation was the average
RMSI
average experimental value for the same halflog decade.
Model
CEBMC90
CEB MC9099
GL2000
20 to 90
15 to 120
16 to 82
(2900 to 13,000)
(2175 to 17,400)
(2320 to 11,900) .








0.40 to 0.60
40 to 100
40 to 100
20 to 100
R, SL,RS
R,SL,RS
R,SL,RS
(I, II, III)
(I, II, III)
(I, II, III)
< 14 days
< 14 days
~ 1 day



> 1 day
> 1 day
to ~
tc ~ I day
4.1ACI 209R92 model
The model recommended by ACI Committee 209 (1971)
was developed by Branson and Christiason (1971), with
minor modifications introduced in ACI 209R82 (ACI
Committee 2091982). ACI Committee 209 incorporated the
developed model in ACI 209R92 (ACI Committee 209
1992). Since then, it has not been revised or updated to the
RlLEM databank, and it is compared with very recent
models. This model, initially developed for the precast
prestressing industry (Branson and Ozell 1961; Branson
1963, 1964, 1968; Branson et al. 1970; Meyers et al. 1970;
Branson and Kripanayanan 1971; Branson and Chen 1972),
has been used in the design of structures for many years.
Advantages of this model include:
• It is simple to use with minimal background knowledge;
and
•
•
It is relatively easy to adjust to match shortterm test
data simply by modifying ultimate shrinkage or creep
to produce the best fit to the data.
Its disadvantages include:
It is limited in its accuracy, particularly in the method
of accommodating member size when its simplest form
is used. This disadvantage, however, can be overriden if
the methods provided for accommodating the shape
and size effect on the timeratio are applied; and
It is empirically based, thus it does not model shrinkage
or creep phenomena.
At its most basic level, the ACI 209R92 method only
requires:
Age of concrete when drying starts, usually taken as the
age at the end of moist curing;
• Age of concrete at loading;
• Curing method;
• Relative humidity expressed as a decimal;
• Volumesurface ratio or average thickness; and
• Cement type.
This model calculates the creep coefficient rather than the
compliance, which may introduce problems due to the
assumed value of elastic modulus. Figures 4.1 and 4.2 show
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R9
the calculated and measured shrinkages and compliances,
respectively. The comparison of shrinkage data in Fig. 4.1
clearly shows that the ACI 209R92 model overestimates
measured shrinkage at low shrinkage values (equivalent to
short drying times) and underestimates at high shrinkage
values (typical of long drying times). This result indicates the
limitation of the model's equation used to predict shrinkage.
The ACI 209R92 compliance comparison is rather insensitive
to using all of the available data, including mixture proportions,
compared with just using the measured concrete strength.
4.2BazantBaweja B3 model
The BazantBaweja B3 model (Bazant and Baweja 1995,
20(0) is the culmination of work started in the 1970s (Bazant
et al. 1976, 1991; Bazant and Panula 1978, 1984; Jirasek and
Bazant 2(02), and is based on a mathematical description of
over 10 physical phenomena affecting creep and shrinkage
(Bazant 2(00), including known fundamental asymptotic
properties that ought to be satisfied by a creep and shrinkage
model (Bazant and Baweja 2000, RILEM Technical
Committee TC 107 1995). This model has been found to be
useful for those dealing with simple as well as complex
structures. The BazantBaweja B3 model uses the compli
ance function. The compliance function reduces the risk of
errors due to inaccurate values of the elastic modulus. The
model clearly separates basic and drying creep.
The factors considered include:
• Age of concrete when drying starts, usually taken as the
age at the end of moist curing;
Age of concrete at loading;
Aggregate content in concrete;
Cement content in concrete;
• Cement type;
Concrete mean compressive strength at 28 days;
Curing method;
Relative humidity;
• Shape of specimen;
• Volumesurface ratio; and
• Water content in concrete.
Both BazantBaweja B3 shrinkage and creep models may
require input data that are not generally available at time of
design, such as the specific concrete proportions and
concrete mean compressive strength. Default values of the
input parameters can be automatically considered if the user
lacks information on some of them. The authors suggest
when only
/cm28 is known, the watercement ratio can be
determined using Eq. (41), and typical values of cement
content and aggregate cement ratio should be assumed
wlc =
((fcm28!22.8) + O.535r
l
in SI units
I
wlc =
[(fcm2s133OO)+0.535]
inin.Ibunits
(41)
Equation (41) represents the bestfit linear regression
equation to the values reported in Tables A1.5.3.4(a) and
A6.3.4(a) of ACI 211.191 (ACI Committee 211 1991) for
nonairentrained concretes made with Type 1 portland
cement; for airentrained concretes, similar equations can be
.,
Q
~
.. .,
.:!
'tI
S ..
'S
u
'ii
(J
1500 ......... .,...,...,...,,.
t. fem only CV = 34%
• All data CV = 41%
1250 I++++ift
1000
750
500
250
0
0
250
500
750
1000
1250
1500
Measured Soh X 10";
Fig. 4.IACI 209R92 versus RILEM shrinkage databank
(Gardner 2004).
.. CI.
~
Of
Q
..
i ...
:::;
'tI
S ..
'S
u
'ii
(J
300 r~__,__,...."
A fem only CV = 30%
• All data CV = 30%
250 ~t_t__t___Ir_7I"f
200
150
100
50
AC1209R·92
0
0
50
100
150
200
250
300
Measured J(t,to) x 10"; (1/MPa)
Fig. 4.2ACI 209R92 versus RILEM compliance databank
(Gardner 2004).
derived by regression analysis of the reported values on ACI
211.191. For other cement types and cementitious materials,
ACI 211.191 suggests that the relationship between water
cement or watercementitious material ratio and compressive
strength of concrete be developed for the materials actually
to be used.
Figures 4.3 and 4.4 show the comparison between the calcu
lated and measured shrinkages and compliances, respectively.
The shrinkage equation is sensitive to the water content.
The model allows for extrapolation from shortterm test
data using shortterm test data and a test of shortterm moisture
content loss.
4.3CEB MC9099 model
In 1990, CEB presented a model for the prediction of
shrinkage and creep in concrete developed by Muller and
209.2R10
ACI COMMITTEE REPORT
" co
..
><
.c
J
"'CI
.s to
:;
u
iii
CJ
1500 r,...,...,...~
I:> fom only CV= 31%
• All data CV = 20%
1250 Irr+,,+~>l"~
1000
750
500
250
0
0
250
500
I:>
I:>
750
1000
Measured Esh x 10"
B3
1250
1500
Fig. 4.3BaZantBaweja B3 versus RILEM shrinkage
databank (Gardner 2004).
ii'
Do
~
" co
.. Ie
S
::!
.
"'CI
.s
.!!
:::I
U
iii
CJ
300 r,...,...,...~
I:> fom only CV = 29%
• All data CV = 27%
250 Irr~++_.>l"~
200
150
100
50
B3
0
0
50
100
150
200
250
300
Measured J(t,to) x 10" (lIMPa)
Fig. 4.4BaZantBaweja B3 versus RILEM compliance
databank (Gardner 2004).
Hilsdorf (1990). The model was revised in 1999 (CEB 1999)
to include normal and highstrength concretes and to separate
the total shrinkage into its autogenous and drying shrinkage
components, and it is called CEB MC9099. While the
revised models for the drying shrinkage component and for the
compliance are closely related to the approach in CEB MC90
(MUller and Hilsdorf 1990, CEB 1993), for autogenous
shrinkage, new relations were derived, and some adjustments
were included for both normal and highstrength concrete.
For these reasons, the CEB 1990 and the revised CEB 1999
models are described in Appendix A. Some engineers
working on creep and shrinkagesensitive structures have
accepted this model as preferable to the ACI 209R92 model
(based on the 1971 Branson and Christiason model). The CEB
models do not require any information regarding the duration
" co
..
><
"Ii
'" "'CI
.s to
:;
u
iii
CJ
1500 .,,.,..,.,.."""?I
I:> fom only CV = 32%
• All data CV = 25%
1250 1+tfi,.t"1
1000
...
750
500
250
0="1.''...... '
o
250
500
750
1000
1250
1500
Measured Esh x 10"
Fig. 4.5CEB MC9099 versus RlLEM shrinkage databank
(Gardner 2004).
300 r.,..z...,....,.
I:> fom only CV = 37%
••
• All data CV = 29%
250 I+++.~+.ifi
ii'
Do
~ 200
1:>1:>
CEB MC9099
o ~~~~~~~
o
50
100
150
200
250
300
Measured J(t,to) x 10" (lIMPa)
Fig. 4.6CEB MC9099 versus RlLEM compliance databank
(Gardner 2004).
of curing or curing condition. The duration of drying might
have a direct impact on the shrinkage and creep of concrete,
and should not be ignored when predicting the shrinkage and
compliance. The correction term used for relative humidity in
the creep equation is extremely se!lsitive to any variation in
relative humidity. Figures 4.5 and 4.6 compare the calculated
and measured shrinkages and compliances, respectively.
The method requires:
• Age of concrete when drying starts, usually taken as the
age at the end of moist curing;
• Age of concrete at loading;
• Concrete mean compressive strength at ~8 da)!s;
Relative humidity expressed as a decimal;
• Volumesurface ratio; and
• Cement type.
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R11
GL2000
a ~~~~~~~
a
250
500
750
1000
1250
1500
Measured &"h x 10'"
Fig. 4.7GL2000 versus RILEM shrinkage databank
(Gardner 2004).
Using only the data with reported concrete strength, the
model generally underestimates the shrinkage of North
American concretes, and substantially underestimates the
shrinkage of concretes containing basalt aggregates found in
Hawaii, Australia, and New Zealand (McDonald 1990;
McDonald and Roper 1993; Robertson 2000). The main
reason is that primarily European concretes (lower cement
content and other types of cement) were considered when
optimizing the model. The shrinkage model does not
respond well to earlyage extrapolation using the simple
linear regression method suggested by Bazant (1987);
however, the creep model does (Robertson 2000).
4.4GL2000 model
The GL2000 model was developed by Gardner and
Lockman (2001), with minor modifications introduced by
Gardner (2004). The model is a modification of the GZ
Atlanta 97 model (Gardner 2000) made to conform to the
ACI 209 model guidelines given in Section 3.5. Except for
the concrete compressive strength, the model only requires
input data that are available to engineer at time of design.
Figure 4.7 and 4.8 compare the calculated and measured
shrinkages and compliances, respectively.
The method requires:
Age of concrete when drying starts, usually taken as the
age at the end of moist curing;
Age of concrete at loading;
Relative humidity expressed as a decimal;
Volumesurface ratio;
• Cement type; and
• Concrete mean compressive strength at 28 days.
4.5Statistical comparisons
As stated previously, there is no agreement as to which
statistical indicator(s) should be used, which data sets should
be used, or what input data should be considered. To avoid
revising any investigator's results, the statistical comparisons of
'i'
~
)(
300 ,.rr.,.,,."
" fem only CV = 26%
• All data CV = 22%
250 1,.,.++.,.,jL~
200 1++:.L',O7i"'',rt''+~
! 150 I+~
::;
"\:J
S
~
100 I+_,
::I
U
ii
t)
50
o ~____ ~__ ~____ ~____ ~____ ~____ _J
o
50
100
150
200
250
300
Measured J(t,to) x 10'" (l/MPa)
Fig. 4.8GL2000 versus RILEM compliance databank
(Gardner 2004).
BaZant and Baweja (2000), AIManaseer and Lam (2005), and
Gardner (2004) are summarized in Table 4.2 for shrinkage and
in Table 4.3 for compliance. As the statistical indicators
represent different quantities and the investigators used
different experimental results, comparisons can only be made
across a row, but cannot be made between lines in the tables.
Descriptions of the statistical indicators are given in Appendix B.
AIManaseer and Lam (2005) noted that careful selection
and interpretation of concrete data and the statistical
methods can influence the conclusions on the performance
of model prediction on creep and shrinkage.
Brooks (2005) also reported the accuracy of five prediction
models, including ACI 209R92, BazantBaweja B3, CEB
MC90, and GL2000 models, in estimating 30year deformation,
concluding that most methods fail to recognize the influence of
strength of concrete and type of aggregate on creep coefficient,
which ranged from 1.2 to 9.2. Brooks (2005) also reported
that shrinkage ranged from 280 to 1460 x 1O{i, and swelling
varied from 25 to 35% of shrinkage after 30 years.
4.6Notes about models
The prediction capabilities of the four shrinkage and
compliance models were evaluated by comparing calculated
results with the RlLEM databank. For shrinkage strain
prediction, BazantBaweja B3 and GL2000 provide the best
results. The CEB MC9099 underestimates the shrinkage.
For compliance, GL2000, CEB MC9099, and Bazant
Baweja B3 give acceptable predictions. The ACI 209R92
method underestimates compliance for the most of the
RlLEM databank. It should be noted that for shrinkage
predictions, BazantBaweja B3 using Eq. (41) instead of
experimental values for water, cement, and aggregate
masses provides less accurate, but still acceptable, results.
Except for ACI 209R92, using more inf~rmation improved
the prediction for all other methods. The predictions from the
CEB, GL2000, and BazantBaweja B3 models were signifi
cantly improved by using measured strength development
209.2R12
ACI COMMITTEE REPORT
Table 4.2Statlstical indicators for shrinkage
Indi
ACI
Investigator cator 209R92
BaZant and
Baweja
•
TJJBP
(2000)
VCEB
•
AI
FCEB
•
Manaseer
and Lam
MCEBt
(2005)
•
TJJBP
Gardner
(2004),
•
fcm only
roo
Gardner
(2004),
•
roo
all data
'Perfect correlation = 0%.
tPerfect correlation = 1.00.
55%
46%
83%
1.22
102%
34%
41%
Model
BaZant
CEB
CEB
BawejaB3 MC90 MC9099 GL2000
34%
46%


41%
52%
37%
37%
84%
60%
65%
84%
1.07
0.75
0.99
1.26
55%
90%
48%
46%
31%

32%
25%
20%

25%
19%
Table 4.35tatistical Indicators for compliance
Indi
ACI
Investigator cator 209R92
BaZantand
Baweja
· 58%
(2000),
TJJBP
basic creep
BaZantand
Baweja
•
(2000),
TJJBP
45%
drying
creep
VCEB · 48%
AI
FCEB · 32%
Manaseer
and Lam
MCEBt
0.86
(2005)
· 87%
TJJBP
Gardner
(2004),
• 30%
roo
fcmonly
Gardner
(2004),
• 30%
all data
roo
'Perfect correlation = 0%.
tPerfect correlation = 1.00.
Model
BaZant
CEB
CEB
BawejaB3 MC90 MC9099 GL2000
24%
35%


23%
32%


36%
36%
38%
35%
35%
31%
32%
34%
0.93
0.92
0.89
0.92
61%
75%
80%
47%
29%

37%
26%
27%

29%
22%
and measured elastic modulus of the concrete to modify the
concrete strength used in creep and shrinkage equations.
It should be noted that the accuracy of the models is
limi~ by the many variables outlined previously and
measurement variability. For design purposes, the accuracy
of the prediction of shrinkage calculated using GL2000 and
BazantBaweja B3 models may be within ��20%, and the
prediction of compliance ��30%. Parametric studies should be
made by the designer to ensure that expected production
variations in concrete composition, strength, or the environ
ment do not cause significant changes in structural response.
The coefficients of variation for shrinkage measured by
BaZant et al. (1987) in a statistically significant investigation
were 10% at 7 days and 7% at 1100 days, and can be used as
a benchmark for variations between batches. A model that
could predict the shrinkage within 15% would be excellent,
and 20% would be adequate. For compliance, the range of
expected agreement would be wider because, experimen
tally, compliance is determined by subtracting two measured
quantities of similar magnitude.
.
There is not an accepted sign convention for stress and
strain. In this document, shortening strains and compressive
stresses are positive. For all models, it is necessary to estimate
the environmental humidity. The Precast/Prestressed
Concrete Institute's
PCI Design Handbook (2005) gives
values of the annual average ambient relative humidity
throughout the United States and Canada that may be used as
a guide. Care should be taken when considering structures,
such as swimming pools or structures near water. Although
the models are not sensitive to minor changes in input values,
the effect of air conditioning in moist climates and exposure
to enclosed pool in dry climates can be significant. Therefore,
the effects of air conditioning and heating on the local envi
ronment around the concrete element should be considered.
Relaxation, the gradual reduction of stress with time under
sustained strain, calculated using ACI 209R92, BaZant
Baweja B3, CBB MC9099, and GL2000, agreed with
Rostasy et al.' s (1972) experimental results indicating that the
principle of superposition can be used to calculate relaxation
provided that calculations are done keeping any drying before
loading term constant at the initial value (Lockman 2000).
Lockman (2000) did a parametric comparison of models
based upon the work of Chiorino and Lacidogna (1998a,b);
see also Chiorino (2005). CBB MC90 and ACI 209R92
underestimate the compliance compared with the GL2000
and BaZantBaweja B3 models using the same input param
eters. Relaxations calculated by BaZantBaweja B3 are
significantly different than those calculated for the three
other models. The elastic strains, calculated at 30 seconds
after loading, for the BaZantBaweja B3 model are very
different from those calculated by the other three models.
The method of calculating the elastic strain is unique to this
model, and the initial stresses of relaxation differ radically
from other models .
For all ages of loading, especially in a drying environment,
BaZantBaweja B3 predicts more relaxation than the other
models. Unlike the other models, BaZantBaweja B3 uses an
asymptotic elastic modulus (fast rate of loading), and not the
conventional elastic modulus, which typically includes a
significant earlyage creep portion. The use of a larger
asymptotic elastic modulus explains the comments about
relaxation curves obtained from the BaZantBaweja B3
model. For early ages of loading, the relaxations calculated
using CBB MC9099 and ACI 209R92 are nearly 100% of
the initial stress, with residual stresses close to zero.
For creep recovery, GL2000 and BazantBaweja B3 are
the only models that predict realistic recoveries by super
position. For partial creep recovery, that is, superposition not
assumed, with complete removal of the load, no model provides
realistic results. Calculating recovery by superpoltition is
subject to more problems than calculating relaxation by
superposition. If recovery is to be calculated by superposition,
both basic and drying creep compliance functions have to be
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R13
parallel in time to give a constant compliance after
unloading. As drying before loading reduces both basic and
drying creep, it is not yet possible to determine a formulation
that permits calculating recovery by superposition in a
drying environment. Experimental evidence (Neville 1960)
is inconclusive on whether either drying creep or basic creep
is completely recoverable.
Highstrength concretes with watercement ratios less
than 0.40 and mean concrete strengths greater than 80 MPa
(11,600 psi) experience significant autogenous shrinkage.
The magnitude of the autogenous shrinkage also depends on
the availability of moisture during earlyage curing.
Concretes containing silica fume appear to behave differently
from conventional concretes. Few data on such concretes are
held in the databank and hence, caution should be exercised
using equations justified by the databank for such concretes.
The models, however, can be used in such circumstances if
they are calibrated with test data.
CHAPTER 5REFERENCES
5.1Referenced standards and reports
The latest editions of the standards and reports listed
below were used when this document was prepared. Because
these documents are revised frequently, the reader is advised
to review the latest editions for any changes.
American Concrete Institute
116R
Cement and Concrete Terminology
209.1R
Report on Factors Affecting Shrinkage and
Creep of Hardened Concrete
ASTM International
C150
Specification for Portland Cement
C595
Specification for Blended Hydraulic Cements
C157
Test Method for Length Change of Hardened
Hydraulic Cement, Mortar, and Concrete
C512
Test Method for Creep of Concrete in Compression
C469
Test Method for Static Modulus of Elasticity and
Poisson's Ratio of Concrete in Compression
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209.2R14
ACI COMMITIEE REPORT
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British Standards Institution, 1985, "BS 8110: Part 2:
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The Adam Neville Symposium: Creep
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RIC Beams,"
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CEB, 1984, "CEB Design Manual on Structural Effects of
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P. Napoli, F. Mola, and M. Koprna, eds.,
CEB Bulletin
d'lnformation No.
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SaintSaphorin, Switzerland, 391 pp. (See also: Final Draft,
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CEB, 1991, "Evaluation of the Time Dependent Properties
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Bulletin d'lnformation No. 199, Comite Euro
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Lausanne, Switzerland, 201 pp.
CEB, 1993. "CEBFIP Model Code 1990,"
CEB Bulletin
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2131214, Comite EuroInternational du
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CEB, 1999, "Structural ConcreteTextbook on Behaviour,
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fib Bulletin 2, V. 2, Federation Inter
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Creep of
Concrete, SP227, N. J. Gardner and J. Weiss, eds., American
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Unified Approach for Analysis of Concrete Structures:
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Davies, R. D., 1957, "Some Experiments on the Appli
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MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R1S
Gamble, B. R., and Parrott, L. J.,1978, "Creep of Concrete
in Compression During Drying and Wetting,"
Magazine of
Concrete Research, V. 30, No. 104, pp. 129138.
Gardner, N. J., 2000, "Design Provisions for Shrinkage
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The Adam Neville Symposium:
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AIManaseer, ed., American Concrete Institute, Farmington
Hills, MI, pp. 101134.
Gardner, N. J., 2004, "Comparison of Prediction Provisions
for Drying Shrinkage and Creep of Normal Strength
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Canadian Journalfor Civil Engineering, V. 31,
No.5, Sept.Oct., pp. 767775.
Gardner, N. J., and Lockman, M. J., 2001, "Design
Provisions for Drying Shrinkage and Creep of Normal
Strength Concrete,"
ACI Materials Journal, V. 98, No.2,
Mar.Apr., pp. 159167.
Gardner, N. J., and Tsuruta, H., 2004, "Is Superposition of
Creep Strains Valid for Concretes Subjected to Drying
Creep?"
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pp.409415.
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Size and Shape on the Shrinkage and Creep of Concrete,"
ACI JOURNAL,
Proceedings V. 63, No.2, Feb., pp. 267290.
Hanson, J. A., 1953, "A 10Year Study of Creep Proper
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APPENDIX AMODELS
A.1ACI209R·92 model
This is an empirical model developed by Branson and
Christiason (1971), with minor modifications introduced in
ACI 209R82 (ACI Committee 2091982). ACI Committee
209 incorporated the developed model in ACI 209R92 (ACI
Committee 209 1992).
The models for predicting creep and shrinkage strains as a
function of time have the same principle: a hyperbolic curve
that tends to an asymptotic value called the ultimate value.
The form of these equations is thought to be convenient for
design purposes, in which the concept of the ultimate (in
time) value is modified by the timeratio (timedependent
development) to yield the desired result. The shape of the
curve and ultimate value depend on several factors, such as
curing conditions, age at application of load, mixture propor
tioning, al.l1bient temperature, and humidity.
The design approach presented for predicting creep and
shrinkage refers to standard conditions and correction
factors for otherthanstandard conditions. The correction
factors are applied to ultimate values. Because creep and
shrinkage equations for any period are linear functions of the
ultimate values, however, the correction factors in this
procedure may be applied to shortterm creep and shrinkage
as well.
The recommended equations for predicting a creep coefficient
and an unrestrained shrinkage strain at any time, including
ultimate values, apply to normal weight, sand lightweight,
and all lightweight concrete (using both moist and steam
curing, and Types I and III cement) under the standard
conditions summarized in Table A.l.
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);
Curing method;
Ambient relative humidity expressed as a decimal;
Volumesurface ratio or average thickness (mm or in.);
Concrete slump (rum or in.);
Fine aggregate percentage (%);
Cement content (kg/m3 or Ib/yd3);
Air content of the concrete expressed in percent (%);
and
... Cement type
A.I.1
ShrinkageThe shrinkage strain
ssh(t,tc) at age of
concrete
t (days), measured from the start of drying at
tc
(days), is calculated by Eq. (AI)
(AI)
where
f (in days) and
a are considered constants for a given
member shape and size that define the timeratio part,
sshu is
the ultimate shrinkage strain, and
(t 
tc) is the time from the
end of the initial curing.
For the standard conditions, in the absence of specific
shrinkage data for local aggregates and conditions and at
ambient relative humidity of 40%, the average value
suggested for the ultimate shrinkage strain
£shw is
£shu = 780 x 10
6
mmfmm (in.!in.)
(A2)
For the timeratio in Eq.
(AI), ACI 209R92 recommends
an average value for
f of 35 and 55 for 7 days of moist curing
and 1 to 3 days of steam cUling, respectively, while an
average value of 1.0 is suggested for a (flatter hyperbolic
form). It should be noted that the timeratio does not
differentiate between drying, autogenous, and carbonation
shrinkage. Also, it is independent of member shape and size,
because
f and a are considered as constant.
The shape and size effect can be totally considered on the
timeratio by replacing a = 1.0, andfas given by Eq. (A3), in
Eq. (AI), where
VIS is the volumesurface ratio in mm or in.
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R17
Table A.1Factors affecting concrete creep and shrinkage and variables considered in recommended
prediction method
Factors
Variables considered
Standard conditions
Cement paste content
Type of cement
Type I and III
Watercement ratio
Slump
70 mm (2.7 in.)
Air content
:5:6%
Concrete composition
Mixture proportions
Fine aggregate percentage
50%
Aggregate characteristics
279 to 446 kg/m3
Cement content
Concrete
Degrees of compaction
(470 to 752 Ib/yd3
)
(creep and shrinkage)
Moist cured
7 days
Length of initial curing
Steam cured
1 to 3 days
Initial curing
Moist cured
23.2 ��2 ��C
Curing temperature
(73.4�� 4 oF)
Steam cured
:5: 100 ��C (:5: 212 oF)
Curing humidity
Relative humidity
~95%
Concrete temperature
Concrete temperature
23.2 �� 2 ��C
Environment
(73.4�� 4 oF)
Member geometry and
Concrete water content
Ambient relative humidity
40%
environment (creep and
shrinkage)
Volumesurface ratio
VIS = 38 mm (1.5 in.)
Geometry
Size and shape
or
minimum thickness
150 mm (6 in.)
Moist cured
7 days
Concrete age at load application
Steam cured
I to 3 days
Loading history
During of loading period
Sustained load
Sustained load
Loading (creep only)
Duration of unloading period


Number of load cycles


Type of stress and distribution
Compressive stress
Axial compression
Stress conditions
across the section
Stress/strength ratio
Stress/strength ratio
:5:0.50
2
f =
26.0e{1.42 x 10
(VIS)}
in SI units
(A3)
The ambient relative humidity coefficient
Ysh,RH is
f =
26.0e{O.36(VIS)}
in in.Ib units
(A7)
For conditions other than the standard conditions, the
average value of the ultimate shrinkage
Eshu (Eq. (A2))
needs to be modified by correction factors. As shown in
Eq. (A4) and (A5), ACI 209R92 (ACI Committee 209
1992) suggests multiplying
Eshu by seven factors, depending
on particular conditions
_ {1.40 
1.02h for 0.40 S
h S 0.80
Ysh, RH 
3.00 _
3.0h
for 0.80 S
h S 1
where the relative humidity
h is in decimals.
For lower than 40% ambient relative humidity, values
higher than 1.0 should be used for shrinkage Y
sh,RH' Because
Ysh,RH= 0 when
h = 100%, the ACI method does not predict
swelling.
Eshu =
780ysh x 106
mmlmm (in.lin.)
(A4)
with
Ysh =
Ysh,teYsh,RHYsh,vsYsh,sYsh,IjIYsh,eYsh,a
(A5)
where Y
sh represents the cumulative product of the applicable
correction factors as defined as follows.
The initial moist curing coefficient Y
sh,te for curing times
different from 7 days for moistcured concrete, is given in
Table A.2 or Eq. (A6); for steam curing with a period of 1
to 3 days,
Ysh,te = 1.
The
Ysh,ep correction factors shown in Table A.2 for the
initial moist curing duration variable can be obtained by
linear regression analysis as given in Eq. (A6)
Ysh,te = 1.202 
0.233710g(te)
R2 = 0.9987
(A6)
Coefficient Y
sh, vs allows for the size of the member in
terms of the volumesurface ratio, for members with
volumesurface ratio other than 38 mm (1.5 in.), or average
thickness other than 150 mm (6 in.). The average thickness
d
of a member is defmed as four times the volumesurface
ratio; that is
d =
4V/S, which coincides with twice the actual
thickness in the case of a slab
 1 2
{O.00472(VIS)}
Ysh, vs 
.
e
 1 2
{O.12(VIS)}
Ysh, vs 
.
e
in SI units
(A8)
in in.Ib units
where
V is the specimen volume in mm
3
or in.3, and S the
specimen surface area in mm
2
or in2.
Alternatively, the method also allows the use of the
averagethickness method to account for the effect of member
size on
Eshu' The averagethickness method tends to compute
209.2R18
ACI COMMITTEE REPORT
Table A.2Shrinkage correction factors for
initial moist curing,
'Y sh,te' for use in Eq. (A5),
ACI 209R92 model
Moist curing duration
tc' days
Y5h,Ie
1
1.2
3
1.1
7
1.0
14
0.93
28
0.86
90
0.75
correction factor values that are higher, as compared with the
volumesurface ratio method.
For average thickness of member less than 150 mm (6 in.)
or volumesurface ratio less than 37.5 mm (1.5 in.), use the
factors given in Table A.3.
For average thickness of members greater than 150 mm
(6 in.) and up to about 300 to 380 mm (12 to 15 in.), use
Eq. (A9) and (AIO).
During the first year drying,
(t 
tc) $ 1 year
Ysh,d = 1.23 
0.0015d
Ysh,d =
1.230.006(VIS)
Y sh, d = 1.23 
0.038d
Ysh,d = 1.23 
0.152(V IS)
For ultimate values,
(t 
tc) > 1 year
Ysh,d =
1.170.00114d
Y sh, d = 1.17  0.00456(
VIS)
Y sh, d = 1.17 
0.029d
Ysh,d =
1.230.116(VIS)
in SI units
in in.Ib units
in SI units
in in.Ib units
(A9)
(AIO)
where
d =
4V/S is the average thickness (in mm or in.) of the
part of the member under consideration.
For either method, however,
Ysh should not be taken less
than 02. Also, use
YshEshu ~ 100 x 106
mmlmm (in.lin.) if
concrete is under seasonal wetting and drying cycles and
YshEshu ~ 150 x 106
mmlmm (in.lin.) if concrete is under
sustained drying conditions.
The correction factors that allow for the composition of
the concrete are:
• Slump factor
Ysh,s' where
s is the slump of fresh
concrete (mm or in.)
Ysh,s = 0.89 +
0.00161s in SI units
Ysh,s = 0.89 +
0.041s in in.Ib units
(All)
Table A.3Shrinkage correction factors for
average thickness of members,
'Ysh,d, for use
in Eq. (A5), ACI 209R92 model
•
•
•
Average thickness of Volume/surface ratio
VIS,
Shrinkage factor
Y slr,d
member
d, mm (in.)
mm (in.)
51 (2)
12.5 (0.50)
1.35
76 (3)
19 (0.75)
1.25
102 (4)
25 (1.00)
1.17
127 (5)
31 (1.25)
1.08
152 (6)
37.5 (1.50)
1.00
Fine aggregate factor
Ysh,IjI' where", is the ratio of fme
aggregate to total aggregate by weight expressed as
percentage
Ysh,\jJ = 0.30+0.014", for",$50%
Ysh,\jJ = 0.90+0.002", for",>50%
(A12)
Cement content factor ~h,c> where
c is the cement
content in kg/m
3
or lb/yd
Ysh,e = 0.75 +
0.OOO61c in SI units
Y sh, e = 0.75 +
0.OOO36c in in.Ib units
(A13)
Air content factor
Ysh,a' where a is the air content in
percent
Ysh,a = 0.95 +
0.OO8a ~ 1
(A14)
These correction factors for concrete composition should
be used only in connection with the average values
suggested for
Eshu = 780 x 106
mmlmm (in.lin.). This
average value for
Eshu should be used only in the absence of
specific shrinkage data for local aggregates and conditions
determined in accordance with ASTM C512.
A.l.2
ComplianceThe compliance function
J(t,to) that
represents the total stressdependent strain by unit stress is
given by
(A15)
where
Ecmto is the modulus of elasticity at the time of loading
to (MPa or psi), and
eIl(t,to) is the creep coefficient as the ratio
of the creep strain to the elastic strain at the start of loading
at the age
to (days).
a)
Modulus of elasticityThe secant modulus of elasticity
of concrete
Ecmto at any time
to of loading is given by
Emero =
0.043y!·5 Jfemto (MPa) in SI units
Emeto =
33y;.5 Jfemto (psi) in in.Ib units
(A16)
".
where
Y c is the unit weight of concrete (kg/m
3
or Ib/ft3), and
fcmto is the mean concrete compressive strength at the time of
loading (MPa or psi).
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R19
The general equation for predicting compressive strength
at any time
t is given by
fcmt =
[a: btJfcm28
(A17)
where
fcm28 is the concrete mean compressive strength at
28 days in MPa or psi,
a (in days) and
b are constants, and
t
is the age of the concrete. The ratio
alb is the age of concrete
in days at which one half of the ultimate (in time) compressive
strength of concrete is reached.
The constants
a and
b are functions of both the type of
cement used and the type of curing employed. The ranges of
a and
b for the normal weight, sand lightweight, and all light
weight concretes (using both moist and steam curing, and
Types I and III cement) are:
a = 0.05 to 9.25, and
b = 0.67 to
0.98. Typical recommended values are given in Table A.4.
The concrete required mean compressive strength
fcm28
should exceed the specified compressive strengthf~ as required
in Section 5.3.2 of ACI 318 (ACI Committee 3182005).
b)
Creep coefficientThe creep model proposed by ACI
209R92 has two components that determine the asymptotic
value and the time development of creep. The predicted
parameter is not creep strain, but creep coefficient
<I>(t,to)
(defined as the ratio of creep strain to initial strain). The latter
allows for the calculation of a creep value independent from
the applied load. Equation (A18) presents the general model
(AI8)
where
<I>(t,to) is the creep coefficient at concrete age
t due to
a load applied at the age
to; d (in days) and", are considered
constants for a given member shape and size that define the
timeratio part;
(t 
to) is the time since application of load,
and
<l>u is the ultimate creep coefficient.
For the standard conditions, in the absence of specific
creep data for local aggregates and conditions, the average
value proposed for the ultimate creep coefficient
<l>u is
<l>u = 2.35
(AI9)
For the timeratio in Eq. (A18), ACI209R92 recom
mends an average value of 10 and 0.6 for
d and", (steeper
curve for larger values of
(t 
to»' respectively.
The shape and size effect can be totally considered on the
timeratio by replacing", = 1.0 and
d =
f as given by Eq. (A
. 3), in Eq. (AI8), where
VIS is the volumesurface ratio in
mm or in.
For conditions other than the standard conditions, the
value of the ultimate creep coefficient
<l>u (Eq. (A19» needs
to be modified by correction factors. As shown in Eq. (A20)
and (A21), ACI 209R92 suggests multiplying
<l>u by six
factors, depending on particular conditions.
<l>u= 2.35yc
(A20)
Yc =
Yc.loYc,RHYc,vsYc,sYc,IjIYsh,a
(A21)
Table A.4Values of the constant a and b for use
in Eq. (A17), ACI 209R92 model
Type of
Moistcured concrete
Steamcured concrete
cement
a
b
a
b
I
4.0
0.85
1.0
0.95
III
2.3
0.92
0.70
0.98
where Yc represent the cumulative product of the applicable
correction factors as defined as follows.
For ages at application of load greater than 7 days for moist
cured concrete or greater than I to 3 days for steamcured
concrete, the age of loading factor for creep Y
c,lo is estimated
from
Y = 1
25t D. lIS for moist curing
C,lo
.
0
(A22)
Y
= 1
13t 0.094 for steam curing
c,lo
.
0
(A23)
where
to is the age of concrete at loading (days).
The ambient relative humidity factor
Yc,RH is
Yc,RH = 1.27 
0.67h for
h ~ 0.40
(A24)
where the relative humidity
h is in decimals.
For lower than 40% ambient relative humidity, values
higher than 1.0 should be used for creep
Yh'
Coefficient Y c,
vs allows for the size of the member in terms
of the volumesurface ratio, for members with a volume
surface ratio other than 38 mm (1.5 in.), or an average thickness
other than 150 mm (6 in.)
= ~(1 I 13
{O.0213(VIS)})
Yc
vs
+ ..
e
,
3
in SI units
(A25)
= ~(I 1 13
{O.S4(V IS)})
Yc
vs
+ ..
e
,
3
in in.Ib units
where
V is the specimen volume in mm
3
or in
3
, and S the
specimen surface area in mm2 or in2
.
Alternatively, the method also allows the use of the
averagethickness method to account for the effect of
member size on
<l>u' The averagethickness method tends to
compute correction factor values that are higher, as
compared with the volumesurface ratio method.
For the average thickness of a member less than 150 mm
(6 in.) or volumesurface ratio less than 37.5 mm (1.5 in.),
use the factors given in Table A.5.
For the average thickness of members greater than 150 mm
(6 in.) and up to about 300 to 380 mm (12 to 15 in.), use
Eq. (A26) and (A27).
During the first year after loading,
(t 
to) :=; 1 year
Y c,
d = 1.14 
0.00092d
Yc,d = 1.14 
0.00363(V IS)
Yc,d= 1.140.023d
Yc,d =
1.140.092(V/S)
in SI units
".
in in.Ib units
(A26)
209.2R·20
ACI COMMITTEE REPORT
Table A.5Creep correction factors for average
thickness of members, Yc,d, for use in Eq. (A21),
ACI 209R92 model
Average thickness of Volume/surface ratio
VIS,
member
do mm (in.)
mm (in.)
51 (2)
12.5 (0.50)
76 (3)
19 (0.75)
102(4)
25 (1.00)
127 (5)
31 (1.25)
152 (6)
37.5 (1.50)
For ultimate values,
(t 
to) > 1 year
Ycod :::: 1.10  0.00067
d
Yc,d:::: 1.100.00268(VIS)
Ycod:::: 1.100.017d
Yc.d :::: 1.10 
0.068(V IS)
Creep factor Y
c.d
1.30
1.17
1.11
1.04
1.00
in SI units
(A27)
in in.Ib units
where
d:::: 4(VIS) is the average thickness in mm or inches of
the part of the member under consideration.
The correction factors to allow for the composition of the
concrete are:
• Slump factor
Ye,s' where
s is the slump of fresh concrete
(mm orin.)
•
Y c,
s = 0.82 +
0.00264s in SI units
Y
c. s :::: 0.82 + 0.067
s in in.Ib units
(A28)
Fine aggregate factor
Ye,,!,, where 1jI is the ratio of fine
aggregate to total aggregate by weight expressed as
percentage
Ye.'!' = 0.88 +
0.00241j1
(A29)
Air content factor
Ye,a' where a is the air content in
percent
Ye,a = 0.46 + 0.09a ;::: 1
(A30)
These correction factors for concrete composition should
be used only in connection with the average values
suggested for cJ>u:::: 2.35. This average value for cJ>u should be
used only in the absence of specific creep data for local
aggregates and conditions determined in accordance with
ASTMC512.
A.2BaiantBaweja B3 model
The BazantBaweja (1995) B3 model is the latest variant
in a number of shrinkage and creep prediction methods
developed by Bazant and his coworkers at Northwestern
University. According to Bazant and Baweja (2000), the B3
model is simpler and is better theoretically justified than the
previous models. The effect of concrete composition and
design strength on the model parameters is the main source of
error of the model.
The prediction of the material parameters of the B3 model
from strength and composition is restricted to portland
cement concrete with the following parameter ranges:
0.35 ~
w/e ~ 0.85;
• 2.55: ale '5. 13.5;
• 17 MPa ~
fem28 ~ 70 MPa (2500 psi ~
fem28 ~ 10,000
psi); and
• 160
kglm
3 ~ e ~ 720
kglm
3 (270 Ib/yd3 ~ e ~ 1215 Ib/yd3
)
where
fem28 is the 28day standard cylinder compression
strength of concrete (in MPa or psi), w/c is the watercement
ratio by weight, e is the cement content (in
kg/m
3 or Ib/yd3),
and
ale is the aggregatecement ratio by weight. If only
design strength is known,
thenfem28 ::::
f; + 8.3 MPa
(fcm28 =
f; + 1200 psi).
The BazantBaweja B3 model is restricted to the service
stress range (or up to about
0.45fem28)' The formulas are
valid for concretes cured for at least 1 day.
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);
• Aggregate content in concrete
(kg/m
3
or Ib/yd
3
);
Cement content in concrete
(kglm
3 or Ib/yd3);
• Water content in concrete (kg/m3 or Ib/yd3
);
• Cement type;
• Concrete mean compressive strength at 28 days (MPa
or psi);
• Modulus of elasticity of concrete at 28 days (MPa or
psi);
Curing condition;
Relative humidity expressed as a decimal;
Shape of specimen; and
Volumesurface ratio or effective crosssection thickness
(mm orin.).
A.2.t ShrinkageThe mean shrinkage strain
Esh(t,te) in
the cross section at age of concrete
t (days), measured from
the start of drying at
tc (days), is calculated by Eq. (A31)
(A31)
where
Eshoo is the ultimate shrinkage strain,
kh is the humidity
dependence factor (Table A.6),
Set 
tc) is the time curve, and
(t 
te) is the time from the end of the initial curing.
The ultimate shrinkage
Eshoo is given by Eq. (A32)
(A32)
where Esoo is a constant given by Eq. (A33), and
Eem607/
Ecm(tc+'tsh) is a factor to account for the time dependence of
ultimate shrinkage (Eq. (A34»
S,OO = a,a2[O.019w2Ifc~O~2S+ 270] x 106
S,OO = a,a2[0.02565w2Ifc~o;~+ 270] x 106
in SI units
in in. lb units
(A33)
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R21
and
E E (
t
)
emt 
em28 4 +
0.85t
(A34)
where w is the water content in kglm
3
or
Ib/yd
3
,fem28 is the
concrete mean compressive strength at 28 days in MPa or
psi, and 0.1 and 0.2 are constants related to the cement type
and curing condition. (Note: The negative sign is the model
authors' convention.) The values of 0.1 and 0.2 are given in
Tables A.7 and A.8, respectively. This means that
&shoo =
&soo
for
te = 7 days, and
'tsh = 600 days.
The time function for shrinkage
S(t 
te) is given by Eq.
(A35)
(A35)
where
t and
te are the age of concrete and the age drying
commenced or end of moist curing in days, respectively,
and
'tsh is the shrinkage halftime in days as given in Eq.
(A36).
The size dependence of shrinkage is given by
T,h =
0.085tcO,08/cm28 O,2s[2k,(V IS)]2 in SI units
T,h =
190.8tcO·08/cm28 O,2S[2k,(V IS)]2 in in.Ib units
(A36)
where
ks is the crosssection shapecorrection factor
(Table A.9), and
VIS is the volumesurface ratio in mm or in.
A.2.2
ComplianceThe average compliance function
J(t,to) at concrete age
t caused by a unit uniaxial constant
stress applied at age
to' incorporating instantaneous defor
mation, basic and drying creep, is calculated from
(A37)
where ql is the instantaneous strain due to unit stress (inverse
of the asymptotic elastic modulus) that is, in theory,
approached at a time of about 10
9
second;
Co(t,to) is the
compliance function for basic creep;
Cj..Mo,te) is the additional
compliance function for drying creep; and
t, te, and
to are the
age of concrete, the age drying began or end of moist curing,
and the age of concrete loading in days, respectively.
The instantaneous strain may be written ql =
lIEo, where
Eo is the asymptotic elastic modulus. The use of
Eo instead
of the conventional static modulus
Ecm is convenient
because concrete exhibits pronounced creep, even for very
short loads duration.
Eo should not be regarded as a real
elastic modulus, but merely an empirical parameter that can
be considered age independent. Therefore, the instantaneous
strain due to unit stress is expressed in Eq. (A38)
ql =
0.61Ecm28
(A38)
where
Table A.6Humidity dependence k", 83 model
Relative humidity
h~0.98
h=1.00
D.2
0.98 <
h < 1.00
Linear interpolation: 12.74 12.94h
Table A.7a.1 as function of cement type,
83 model
Type of cement
Type I
1.00
Type II
0.85
Type III
1.10
Table A.~
as function of curing condition,
83 model
Curing method
Steam cured
0.75
Cured in water or at 100% relative humidity
1.00
Sealed during curing or normal curing in air
with initial protection against drying
Eem28 = 4734
Jfem28
in SI units
Eem28 =
57,OOOJfem28
in in.Ib units
1.20
(A39)
According to this model, the basic creep is composed of
three terms: an aging viscoelastic term, a nonaging
viscoelastic term, and an aging flow term
where
q2Q(t,to) is the aging viscoelastic compliance term.
The cement content c (in kglm3 or Ib/yd3
) and the concrete
mean compressive strength at 28 days
fem28 (in MPa or psi)
are required to calculate the parameter
q2 in Eq. (A41)
185 4 10
6
0.51"
0.9
q2 =
. x
C J
em28
8 4 0
6 0.5
0.9
q2 = 86. 1 x 1 c
fem28
in SI units
(A41)
in in.Ib units
Q(t,to) is an approximate binomial integral that must be multi
plied by the parameter
q2 to obtain the aging viscoelastic term
(A42)
Equations (A43) to (A45) can be used to approximate the
binomial integral
(A43)
(A44)
(A45)
209.2R22
ACI COMMITIEE REPORT
Table A.9ks as function of cross section shape,
B3 model
Cross section shape
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
k,
" 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. (A40),
q3 is the non aging 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
f cm28 (in MPa or psi), the
3
I~h
.
cement content c (in kg/m or Ib/yd' ), the watercement ratIo
wlc, and the aggregatecement ratio
ale
(A46)
6(
)07
q4 = 20.3 x 10
alc .
in SI units
(A47)
6
07
q4 = 0.14 x 10
(alc)'
in in.lb units
The compliance function for drying creep is defined by
Eq. (A48). This equation accounts for the drying before
loading. Note that drying before loading is considered only
for drying creep
In Eq. (A48),
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
Eshoo' the
ultimate shrinkage strain as given in Eq. (A32)
(A49)
HU) and
HUo) are spatial averages of pore relative
humidity. Equations (A50) to (A53) and Eq. (A36) are
required to calculate
H(t) and
H(to).
H(t) ::: 1  (1 
h)S(t 
tc)
(A50)
(A51)
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
'tsh is the shrinkage
halftime
(A52)
[(
t 
t) 1/2J
S(to 
tc) = tanh ~
'tsh
(A53)
A.3CEB MC9099 model
The CEB MC90 model (Muller and Hilsdorf 1990; CEB
1993) is intended to predict the timedependent mean cross
section behavior of a concrete member. It has concept similar
to that of ACI209R92 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
timedependent behavior of ordinarystrength concrete (12
MPa 1I740 psi] ~f; ~ 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 oF).
The models are valid for normaIweight 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 MC9099 model (CEB 1999) includes the latest
improvements to the CEB MC90 model. The model has been
developed for normal and highstrength 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 timedependent mean
crosssection 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 OF).
It is valid for normal weight plain structural concrete having
an average compressive strength in the range of 15 MPa
(2175 psi) ~fcm2'8 ~ 120 MPa (17,400 psi). The age at loading
should be at least 1 day, and the creepinduced 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;
Volumesurface ratio or effective crosssection thickness
of the member (mm or in.); and
• Cement type.
A.3.1
Shrinkage CEB MC90The total shrinkage strains
of concrete
Esh(t,tc) may be calculated from
\I'
(A54)
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R23
where
Eeso is the notional shrinkage coefficient.
i3s(t 
te) is
the coefficient describing the development of shrinkage with
time of drying,
t is the age of concrete (days) at the moment
considered,
te is the age of concrete at the beginning of
drying (days), and
(t 
te) is the duration of drying (days).
The notional shrinkage coefficient may be obtained from
with
6
Eifem28) = [160 +
lOi3se(9 
fem28IfemO)] x 10
i3RH(h) = 1.55[1(~YJ for0.4~h<0.99
i3RH(h) = 0.25 for
h ~ 0.99
(A55)
(A56)
(A57)
where
fem28 is the mean compressive cylinder strength of
concrete at the age of 28 days (MPa or psi),
femo is equal
to 10 MPa (1450 psi),
i3se is a coefficient that depends on the
type of cement (Table A. 10),
h is the ambient relative
humidity as a decimal, and
ho is equal to 1.
The development of shrinkage with time is given by
where
(tte) is the duration of drying (days),
tl is equal to 1 day,
VIS is the volumesurface ratio (mm or in.), and
(VIS)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. (A54) and (A58), the actual duration of drying
(t t
e)
has to be used.
When constant temperatures above 30��C (86 oF) are
applied while the concrete is drying, CEB MC90 recom
mends using an elevated temperature correction for
i3dh)
and i3s(t 
te), shown as follows.
The effect of temperature on the notional shrinkage
coefficient is taken into account by
In SI units:
(A59)
In in.Ib units:
_
[ ( 0.08
)(18.778'TlTo37.77~]
~RH,T  ~RH(h) I + 1.03 _
hlho
40
)
Table A.1OCoefficient i3scaccording to
Eq. (A56), CEe MC90 model
Type of cement according to Ee2
SL (slowlyhardening cements)
Nand R (normal or rapid hardening cements)
RS (rapid hardening highstrength cements)
~sc
4
5
8
The effect of temperature on the time development of
shrinkage is taken into account by
In SI units:
(A60)
In in.Ib units:
where
i3RH T is the relative humidity factor corrected by
temperatur~ that replaces
i3RH in Eq. (A55),
i3s,T(t 
te) is the
temperaturedependent coefficient replacing
i3s(t 
te) in
Eq. (A54),
h is the relative humidity in decimals,
ho is equal
to 1,
VIS is the volumesurface ratio (mm or in.);
(VIS) is
equal to 50 mm (2 in.), Tis the ambient temperature eC or oF),
and
To is equal to 1 ��c (33.8 OF).
A.3.2
Shrinkage CEB MC9099With respect to the
shrinkage characteristics of highperformance 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 highperfor
mance concrete; consequently, the autogenous shrinkage also
needs to be modeled for normalstrength concrete.
The total shrinkage of concrete
Esh(t,te) can be calculated
from Eq. (A61)
(A61)
where
Esh(t,te) is the total shrinkage,
Eeas(t) the autogenous
shrinkage, and
Eeds(t,te) is the drying shrinkage at concrete
age
t (days) after the beginning of drying at
te (days).
The autogenous shrinkage component
Eeas(t) is calculated
from Eq. (A62)
(A62)
209.2R24
ACI COMMmEE REPORT
where
Eeaso(fcm28) is the notional autogenous shrinkage coeffi
cient from Eq. (A63), and
f3as(t) is the function describing the
time development of autogenous shrinkage from Eq. (A64)
a
) _
(
fem28
1
femo )2.5 x 106 (A63)
EeasoVem28 
aas 6 +/.
If.
em28 emo
(A64)
wherefcm28 is the mean compressive strength of concrete at an
age of 28 days (MFa or psi),fcmo = 10 MPa (1450 psi),
t is the
concrete age (days),
tl = 1 day, and
aas is a coefficient that
depends on the type of cement (fable A.II).
The autogenous shrinkage component is independent of
the ambient humidity and of the member size, and develops
more rapidly than drying shrinkage.
The drying shrinkage
Eeds(t,te) is calculated from Eq. (A65)
(A65)
where
Eedso(fcm28) is the notional drying shrinkage coefficient
from Eq. (A66),
f3RH<h) is the coefficient that takes into
account the effect of relative humidity on drying
shrinkage from Eq. (A67), and
f3ds(t 
te) is the function
describing the time development of drying shrinkage from
Eq. (A68)
(A67)
f3R1lh) = 0.25 for
h ~ 0.99f3s1
(A69)
where adsl and
ads2 are coefficients that depend on the type of
cement (fable A.II), f3s1 is a coefficient that takes into
account the selfdesiccation in highperformance concrete,
h
is the ambient relative humidity as a decimal,
ho = I,
VIS is the
volumesurface ratio (rom or in.),
(VIS)o = 50 rom (2
in.),fcmo
= 10 MPa (1450 psi),
te is the concrete age at the beginning of
drying (days), and
(t 
te) is the duration of drying (days).
According to Eq. (A67) for normalstrength concretes,
swelling is to be expected if the concrete is exposed to an
ambient relative humidity near 99%. For higherstrength
grades, swelling will occur at lower relative humidities
Table A.11Coefflcients according to Eq. (A63)
and (A66), CEB MC9099 model
Type of cement according to EC2
a.as
a.ds!
a.ds2
SL (slowlyhardening cements)
800
3
0.13
Nor R (normal or rapid hardening cements)
700
4
0.12
RS (rapid hardening highstrength cements)
600
6
0.12
because of the preceding reduction of the internal relative
humidity due to selfdesiccation of the concrete.
A.3.3
ComplianceThe compliance function
J(t,to) that
represents the total stressdependent strain by unit stress is
given by
J(t, to) = ~[l1(to) + «P28(t,
to») =
f+ «P~(t,
to)
em28
cmlo
em28
(A70)
where
TJ(to) =
Eem281Eemto' Eem28 is the mean modulus of
elasticity of concrete at 28 days (MPa or psi),
Eemto is the
modulus of elasticity at the time of loading
to (MPa or psi),
and the dimensionless 28day creep coefficient
(1)28(t,to)
gives the ratio of the creep strain since the start of loading at
the age
to to the elastic strain due to a constant stress applied
at a concrete age of 28 days. Hence,
I/Eemto represents the
initial strain per unit stress at loading.
The CEB MC9099 model is closely related to the CEB
MC90 model; however, it has been adjusted to take into
account the particular characteristics of highstrength
concretes.
a)
Modulus of elasticityFor the prediction of the creep
function, the initial strain is based on the tangent modulus of
elasticity at the time ofloading as defined in Eq. (A7I) and
(A72).
The modulus of elasticity of concrete at a concrete age
t
different than 28 days may be estimated from
(A71)
where
Eem28 is the mean modulus of elasticity of concrete at
28 days from Eq. (A72); the coefficient
s depends on the
type of cement and the compressive strength of concrete and
may be taken from Table A.12; and tl = I day.
The modulus of elasticity of concrete made of quartzitic
aggregates at the age of 28 days
Eem28 (MPa or psi) may be
estimated from the mean compressive strength of concrete
by Eq. (A72)
Eem28 = 2I,500~/.em28 in SI units
emo
Eem28 = 3'118,31O~em28 in in.Ib units
femo
(A72)
where
fcm28 is the mean compressive cylinder strength of
concrete at 28 days (MPa or psi), and
fema = 10 MPa (1450 psi).
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R25
For concrete made of basalt, dense limestone, limestone,
or sandstone, CEB MC90 recommends calculating the
modulus of elasticity of concrete by multiplying
Ecm28 (MPa
or psi) according to Eq.
(A72) with the coefficients
(J.E from
TableA.13.
The mean compressive cylinder strength of concrete (MPa
or psi) is given by Eq.
(A73)
fcm28 =
f; + 8.0 in SI units
fcm28 =
f; + 1160 in in.Ib units
(A73)
wheref; is the specified/characteristic compressive cylinder
strength (MPa or psi) defined as that strength below which
5% of all possible strength measurements for the specified
concrete may be expected to fall.
b)
Creep coefficientWithin the range of service stresses
(not larger than 40% of the mean concrete
strengthfcmto at
the time of loading
to)' the 28day creep coefficient
CP28(t,to)
may be calculated from Eq.
(A74)
(A74)
where CPo is the notional creep coefficient,
!3c(t 
to) is the
coefficient that describes the development of creep with time
after loading,
t is the age of concrete (days) at the moment
considered, and
to is the age of concrete at loading (days),
adjusted according to Eq.
(A81) and
(A87).
The notional creep coefficient CPo may be determined from
Eq.
(A75) to
(A81)
with
_ [3.5f cmoJO.7
(J.I
 
fcm28
_ [3.5f cmoJO.2
(J.2   
fcm28
(A75)
(A76)
(A77)
(A78)
(A79)
(A80)
where
fcm28 is the mean compressive strength of concrete at
the age of 28 days (MPa or psi),fcmo = 10 MPa (1450 psi),
h
is the relative humidity of the ambient environment in decimals,
ho = 1,
VIS is the volumesurface ratio (mm or in.),
(VIS)o =
Table A.12coefflcient s according to Eq. (A71),
CEB MC90 and CEB MC9099 models
Icm28
Type of cement
RS (rapid hardening highstrength cement)
:560 MPa (8700 psi) Nor R (normal or rapid hardening cements)
SL (slowlyhardening cement)
>60 MPa (8700 psi)' All types
'Case not considered in CBB MC90.
Table A.13Effect of type of aggregate on
modulus of elasticity, CEB MC90 model
Aggregate type
Basalt, dense limestone aggregates
1.2
Quartzitic aggregates
1.0
Limestone aggregates
0.9
Sandstone aggregates
0.7
s
0.20
0.25
0.38
0.20
50 mm (2 in.),
tl = 1 day,
to is the age of concrete at loading
(days) adjusted according to Eq.
(A81) and
(A87), and
(J.l
and
(J.2 are coefficients that depend on the mean compressive
strength of concrete
«(J.l =
(J.2 = 1 in CEB MC90).
The effect of type of cement and curing temperature on the
creep coefficient may be taken into account by modifying
the age at loading
to according to Eq.
(A81)
to =
to, r[ 9
12+ 1 r ~ 0.5 days
(A81)
2 +
(to, Tltl, T) .
where
to,Tis the age of concrete at loading (days) adjusted to
the concrete temperature according to Eq. (A87) (for
T =
20��C [68 oF],
to,Tcorresponds to
to) and
tl,T= 1 day.
(J. is a
power that depends on the type of cement;
(J. = 1 for slowly
hardening cement;
(J. = 0 for normal or rapidly hardening
cement; and
(J. = 1 for rapid hardening highstrength cement.
The value for
to according to Eq.
(A81) has to be used in Eq.
(A78).
The coefficient
!3c(t 
to) that describes the development
of creep with time after loading may be determined from
Eq. (A82) to (A84)
(A82)
with
13H = 150[1 + (1.2 .
hlho)18](VIS)/(VIS)0 + 2500.3 ~ 15000.3
(A83)
_ [3.5f cmoJO.5
(J.3   
fcm28
(A84)
where
tl = 1 day,
ho = 1,
(VlS)o = 50 mm (~in.), and
(J.3 is a
coefficient that depends on the mean compressi ve strength of
concrete
«(J.3 = 1 in CEB MC90).
209.2R26
ACI COMMITTEE REPORT
The duration of loading
(t 
to) used in Eq. (A82) is the
actual time under load.
Temperature effectsThe effect of elevated or reduced
temperatures at the time of testing on the modulus of elasticity
of concrete, at an age of 28 days without exchange of moisture,
for a temperature range 5 to 80��C (41 to 176 OF), may be
estimated from
(A85)
Ecm28 (D =
Ecm28 [1.06 
0.OO3(18.778T 
600.883)ITo] in in.Ib units
where
T is the temperature eC or oF), and
To = 1 ��c (33.8 oF).
Equation (A85) can also be used for a concrete age other
than
t = 28 days.
The 28day creep coefficient at an elevated temperature
may be calculated as
(A86)
where
<Po is the notional creep coefficient according to
Eq. (A75) and temperature adjusted according to Eq. (A90),
/3c(t 
to) is a coefficient that describes the development of
creep with time after loading according to Eq. (A82) and
temperature adjusted according to Eq. (A88) and (A89),
and
Ll<PT,trans is the transient thermal creep coefficient that
occurs at the time of the temperature increase, and may be
estimated from Eq. (A92).
The effect of temperature to which concrete is exposed
before loading may be taken into account by calculating an
adjusted age at loading from Eq. (A87)
toJ =
��t.tieXP[13.65 
4000
1 in SI units
i= I
273 +
T(Mi )
To
(A87)
to.T =
��t.tieXP[13.65
4000
1 inin.Ibunits
i = I
273 +
(l8.778T(M,)  600.883)
To
where
to,T is the temperatureadjusted age of concrete at
loading, in days, from Eq. (A81),
T(Lltj) is the temperature
(OC or OF) during the time period
Lltj, Mj is the number of
days where a temperature
T prevails,
n is the number of
time intervals considered, and
To = 1 ��c (33.8 oF).
The' effect of temperature on the time development of
creep is taken into consideration using
PH,T (Eq. (A88»
(A88)
with
I3T = exp[
1500
 5.12J in SI units
(273 +
TlTo)
(A89)
13 = exp[
1500
 5.12J in in.Ib units
T
[273 +
(18.778T 
600.883)ITo]
where
/3H,T is a temperaturedependent coefficient that
replaces
/3H in Eq. (A82),
/3H is a coefficient according to
Eq. (A83),
T is the temperature (OC or OF), and
To = 1 ��c
(33.8 oF).
The effect of temperature conditions on the magnitude of
the creep coefficient
<Po in Eq. (A74) and (A75), respec
tively, may be calculated using Eq. (A90)
(A90)
with
q,T =
exp[0.015(T ITo  20)] in SI units
(A91)
q,T =
exp[0.015[(18.778T 
6oo.883)/To 20]] in in.Ib units
where
<PRH,T is a temperaturedependent coefficient that
replaces
<PRH<h) in Eq. (A75),
<PRH<h) is a coefficient
according to Eq. (A76), and
To = 1 ��c (33.8 OF).
Transient temperature conditions, that is, an increase of
temperature while the structural member is under load,
leads to additional creep
Ll<PT,trans that may be calculated
from Eq. (A92)
t.q,T"wo, =
O,OOO4(TITo 20)2 in SI units
(A92)
t.q,T, "00' =
0.0004[(l8,778T 
6oo,883)ITo  20]2 in in,Ib units
Effect afhigh stressesWhen stresses in the range of 40
to 60% of the compressive strength are applied, CEB MC90
99 (CEB 1993, 1999) recommends using a high stress
correction to the notional creep
<Po as shown in Eq. (A93)
<Po,k = <poexp{
1.5(kcr  0.4)}
(A93)
where
¢lo,k is the notional creep coefficient that replaces
<Po in
Eq. (A74), and kcr is the stressstrength ratio at the time of
application of the load.
A.4GL2000 model
The model presented herein corresponds to the last version
of the GL2000 model (Gardner 2004), including minor
modifications to some coefficients and to the strength
development with time equation of the original model
developed by Gardner and Lockman (2001). It is a modified
Atlanta 97 model (Gardner and Zhao 1993), which itself was
influenced by CEB MC90. It presents a designoffice procedure
for calculating the shrinkage and creep of normalstrength
concretes, defined as concretes with mean compressive
strengths less than 82 MPa (11,890 psi) that do not experience
selfdesiccation, using the information available at design,
namely, the 28day specified concrete strength, the concrete
strength at loading, element size, and relative humidity.
According to Gardner and Lockman (2001), the method can
be used regardless of what chemical admixtures or mineral
byproducts are in the concrete, casting terpperature, or
II'
curing regime. The predicted values can be improved by
simply measuring concrete strength development with time
and modulus of elasticity. Aggregate stiffness is taken into
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R27
account by using the average of the measured cylinder
strength and that backcalculated from the measured
modulus of elasticity of the concrete. The compliance
expression is based on the modulus of elasticity at 28 days
instead of the modulus elasticity at the age of loading. This
model includes a term for drying before loading, which
applies to both basic and drying creep.
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);
• Concrete mean compressive strength at loading (MPa
or psi);
• Modulus of elasticity of concrete at 28 days (MPa or psi);
• Modulus of elasticity of concrete at loading (MPa or psi);
• Relative humidity expressed as a decimal; and
• Volumesurface ratio (mm or in.).
A.4.1
Relationship between specified and mean compressive
strength of concreteIf experimental values are not available,
the relationship between the specified/characteristic
compressive strengthf; and the mean compressive strength
of
concretefcm28 can be estimated from Eq. (A94)
fcm28 = 1.1f; + 5.0 in SI units
fcm28 = 1.1f; + 700 in in.Ib units
(A94)
Equation (A94) is a compromise between the recommended
equations of ACI Committee 209 (1982) and ACI
Committee 363 (1992). It can be noted thatEq. (A94) does not
include any effects for aggregate stiffness or concrete density.
Instead of making an allowance for the density of the
concrete, it is preferable to measure the modulus of elasticity.
If experimental values are not available, the modulus of
elasticity
Ecmt and the strength development with time
fern!
can be calculated from the compressive strength using
Eq. (A95) and (A96).
A.4.2
Modulus of elasticity
E cmt = 3500 + 4300
JJ:::.t in SI units
Ecmt = 500,000 + 52,OOOJI:: in in.Ib units
(A95)
A.4.3
Aggregate stiffnessAggregate stiffness can be
accommodated by using the average of the measured
cylinder strength and that backcalculated from the
measured modulus of elasticity using Eq. (A95) in the
shrinkage and specific creep equations. Effectively, Eq. (A95)
is used as an indicator of the divergence of the measured
stiffness from standard values.
A.4.4
Strength development with time
(A96)
where
(A97)
where
s is a CEB (1993) style strengthdevelopment parameter
(Table A.l4), and
/3e relates strength development to cement
type. Equation (A96) is a modification of the CEB strength
development relationship.
A single measured value of
s permits values of
k in the
shrinkage equation to be interpolated, where
k is a correction
term for the effect of cement type on shrinkage (Table A. 14).
If experimental results are available, the cement type is
determined from the strength development characteristic of
the concrete, regardless of the nominal designation of the
cement. This enables the model to accommodate concretes
incorporating any chemical or mineral admixtures.
A.4.5
ShrinkageCalculate the shrinkage strain
Esh(t,tc)
from Eq. (A98)
(A98)
where
Eshu is the ultimate shrinkage strain,
/3(h) is a correction
term for the effect of humidity, and
/3(t 
tc) is a correction
term for the effect of time of drying.
The ultimate shrinkage
Eshu is given by
Eshu =
9OOk( 30'1
112
x 10
6
in SI units
fcm21
900k(435~ 112
106 . .
lb .
Eshu = 
E
X
m m. umts
:Jcm2
(A99)
where
fcm28 is the concrete mean compressive strength at
28 days in MPa or psi, and
k is a shrinkage constant that
depends on the cement type (Table A.l4).
If test results for strength development are available, the
shrinkage term can be improved by interpolating
k from
Table A.14 using the experimentally determined cement
type/characteristic.
The correction term for effect of humidity
/3(h) is given by
(A100)
Note that for a relative humidity of 0.96, there is no
shrinkage. At a higher relative humidity, swelling occurs.
The time function for shrinkage
/3(t 
tc) is given by
[
(tt)
]112
/3(t 
tc) =
c
in SI units
(ttc )+0.12(V/S)2
(AWi)
[
(tt)
]1/2
/3(t  tJ =
c
in in.Ib units
(t 
tc) +
77(V /
S)2
where
t and
tc are the age of concrete and the age drying
starts or end of moist curing in days, respectively, and
VIS is
the volumesurface ratio in mm or in.
A.4.6
Compliance equations~The "'compliance is
composed of the elastic and the creep strains. The elastic
strain is the reciprocal of the modulus of elasticity at the age
209.2R28
ACI COMMITTEE REPORT
Table A.14Parameters sand k as function of
cement type, GL2000 model
Cement type
s
k
TypeJ
0.335
1.0
Type II
0.4
0.75
Type III
0.13
1.15
of loading
Eemto' and the creep strain is the 28day creep
coefficient
<P2S(t,to) divided by the modulus of elasticity at
28 days
Ecm28 as in Eq. (A102). The creep coefficient
<P28(t,to) is the ratio of the creep strain to the elastic strain due
to the load applied at the age of 28 days
(AI02)
The 28day creep coefficient
<P28(t,to) is calculated using
Eq. (A103)
In SI units:
(AI03)
In in.Ib units:
The creep coefficient includes three terms. The first two
terms are required to calculate the basic creep, and the third term
is for the drying creep. Similar to the shrinkage Eq. (AI 00), at
a relative humidity of 0.96, there is only basic creep (there is
no drying creep).
<D(te) is the correction term for the effect of
drying before loading.
If
to =
te
(A104)
<D(t ) = 1 _
0
C
[ (
(
t 
t )
J 0.5] 0.5
c
(totc)+0.12(V/S)2
in SI units
(AlOS)
<D(t) = [1 (
(to 
tc)
J0.5]0.5 in in.Ib units
c
(totc) +
77(V/S)2
To calculate relaxation,
<D(te) remains constant at the
initial value throughout the relaxation period. For creep
recovery calculations,
<D(te) remains constant at the value at
the age of loading.
APPENDIX BSTATISTICAL INDICATORS
B.1BP coefficient of variation (tlJ8P%) method·
Developed by Bazant and Panula (1978), a coefficient of
variation
IDBP is determined for each data set. Data points in
each logarithmic decade, 0 to 9.9 days, 10 to 99.9 days, and
so on, are considered as one group. Weight is assigned to
each data point based on the decade in which it falls and
number of data points in that particular decade. The overall
coefficient of variation
(IDB3) for all data sets is the root
mean square (RMS) of the data set values
IDij =
where
n
N
(B1)
(B2)
(B3)
(B4)
= number of data points in data set number
j;
= sum of the weights of all data points in a data set;
= number of data points in the
kth decade;
= number of decades on the logarithmic scale
spanned by measured data in data setj;
= number of data sets;
Oij
= measured value of the shrinkage strain or creep
compliance for the ith data point in data set
numberj;
Cij
= predicted value of the shrinkage strain or creep
compliance for the ith data point in data set
numberj;
Cij 
Oij= deviation of the predicted shrinkage strain or
creep compliance from the measured value for the
ith data point in data set number,j;
IDij
= weight assigned to the ith data point in data set
numberj;
IDj
= coefficient of variation for data set number j; and
IDB3
= overall coefficient of variation.
B.2CEB statistical indicators
The CEB statistical indicators: coefficient of variation
V CEB, the mean square error
F CEB' and the mfian de;v.iation
MCEB were suggested by Muller and Hilsdorf (1990). The
indicators are calculated in six time ranges: 0 to 10 days, 11
to 100 days, 101 to 365 days, 366 to 730 days, 731 to 1095
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R29
days, and above 1095 days. The final values are the RMS of
the six interval values.
B.2.1
CEB coefficient o/variation
n

1
O· = ~
(0·.)
I
n~
I)
(B5)
j = 1
n
Vi = 1 _1_ ~ (c.
o.l
O.
n 1 ~ I)
I)
)
j = 1
(B6)
(B7)
where
n
= number of data points considered;
N
= total number of data sets considered;
Vj
= coefficient of variation in interval
i; and
V CEB = RMS coefficient of variation.
B.2.2
CEB mean square errorThe mean square error
uses the difference between the calculated and observed
values relative to the observed value
(B8)
FJ?n
2
F. = ~J;
I
nl~)
j=l
(B9)
(B1O)
where
Ij
= percent difference between calculated and
observed data pointj; and
F CEB = mean square error, %.
B.2.3
CEB mean deviationThe CEB mean deviation
MCEB indicates systematic overestimation or underestimation
of a given model
! ~
Cu
Mi = ~'4
nO·.
j = 1 I)
(Bll)
(B12)
where
Mj
= ratio of calculated to experimental values in time
range
i;
mean deviation;
N =
number of values considered in time interval; and
total number of data sets considered.
B.3The Gardner coefficient of variation (OlG)
Developed by Gardner (2004), the mean observed value
and RMS of the difference between calculated and
observed values were calculated in half logarithmic time
intervals: 3 to 9.9 days, 10 to 31.5 days, 31.6 to 99 days,
100 to 315 days, 316 to 999 days, 1000 to 3159 days, and
above 3160 days. That is, the duration of each time
interval is 3.16 times the previous value. To obtain a crite
rion of fit, the average values and RMSs were averaged
without regard to the number of observations in each half
decade. A coefficient of variation is obtained by dividing
the average RMS normalized by the average value. It is
necessary to emphasize that this is not the conventional
definition of the coefficient of variation
_
1
n
O.=~(O
.. )
)
n~
I)
i= 1
n
1
L
2

(C.O··)
n  1
I)
I)
_
1
n _
O=~(O.)
N~
)
j= 1
n
i = 1

1
RMS =  ~ (RMS.)
N~
J
j = 1
RMS
OlG = ~
o
(B13)
(B14)
(B15)
(B16)
(B17)
209.2R30
ACI COMMITTEE REPORT
APPENDIX CNUMERIC EXAMPLES
Find the creep coefficients and shrinkage strains of concrete at 14, 28, 60, 90, 180, and 365 days after casting, from the
following information: specified concrete compressive strength of 25 MPa (3626 psi), 7 days of moist curing, age of loading
to =
14 days, 70% ambient relative humidity, and volumesurface ratio of the member = 100 mm (4 in.).
Problem data
Concrete data:
Specified 28day strength
f~
=
iAmbient conditions:
Relative humidity
h=
Temperature
T=
Specimen:
Volumesurface ratio
VIS =
Shape
Initial curing:
Curing time
te =
Curing condition
Concrete at loading:
Age at loading
to =
Applied stress range
ks =
C.1ACI209R92 model solution
C.I.I
Estimated concrete properties
Mean 28day strength
fem28 =
Mean 28day elastic modulus
Eem28 =
C.I.2
Estimated concrete mixture
Cement type
Maximum aggregate size
Cement content
c=
Water content
w=
Watercement ratio
wlc=
Aggregatecement ratio
alc=
Fine aggregate percentage
\jI=
Air content
a=
Slump
s=
Unit weight of concrete
Ye=
"Table A1.5.3.7.l and 6.3.7.1 of ACI 211.191.
C.I.3
Shrinkage strains Gsh(t,tc)
Nominal ultimate shrinkage strain
Moist curing correction factor
SI units
in. lb units
25MPa
3626 psi
0.7
20��C
68 OF
100mm
4 in.
Infinite slab
7 days
Moist cured
14 days
40%
SI units
in.lb units
33.3 MPa
4830 psi
Table 5.3.2.2 ACI 31805
28,178 MPa
4,062,346 psi
(A16)
SI units
in.lb units
I
I
20mm
3/4 in.
409 kglm
3
6901b/yd
3
205 kglm
3
3451b/yd3
Table 6.3.3 ACI 211.191
0.50
(41)
4.23
40%
2%
Table 6.3.3 ACI 211.191
75mm
2.95 in.
2345 kglm
3
39531b/yd3
146* Ib/ft3
SI units
I
in.lb units
Eshu = 780 x 106
(A2)
Ysh,te = 1.202 
0.23371og(te) = 1.005
(A6)
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R·31
Ysh,RH= lAO 1.02h if
004 :$;
h:$; 0.8
(A·7)
Ambient relative humidity factor
Ysh,RH = 3.00 
3h if 0.8 <
h :$; 1
(A·7)
Ysh,RH = 0.686
(A·7)
Volumetosurface ratio factor
y
 1
2e[{).00472(VlS)]
sh,vs  .
(A8)
Y
= 1
2e[{).12(VIS)]
sh,vs
.
(A8)
Ysh,vs = 0.749
(A8)
Ysh,vs = 0.743
(A8)
Ysh,s = 0.89 +
0.00161s
(All)
Ysh,s = 0.89 +
0.041s
(All)
Slump of fresh concrete factor
Ysh,s = 1.011
(AII)
Ysh,s = 1.011
(A11)
Ysh,IV = 0.30 + 0.014", if ",:$; 50%
(A12)
Fine aggregate factor
Ysh,IV = 0.90 + 0.002", if '" > 50%
(A12)
Ysh,IV = 0.860
(A12)
Ysh,e = 0.75 + 0.00061c
(A13)
Ysh,e = 0.75 + 0.00036c
(A13)
Cement content factor
Ysh,e = 0.999
(A13)
Ysh,e = 0.998
(A13)
Ysh,a = 0.95 +
0.008a ~ 1
(A14)
Air content factor
Ysh,a = 1.000
(A14)
Ysh =
Ysh,teYsh,RJl'fsh,vsYsh,sYsh,IVYsh,eYsh,a
(A5)
Cumulative correction factor
Ysh = 0.448
(A5)
Ysh = 0.444
(A5)
Eshu =
780ysh x 106
(A4)
Ultimate shrinkage strain
Eshu = 350 x 106
Eshu = 347 x 10
6
(A4)
(A4)
Shrinkage time function
f(t,te) =
[(t 
te)a/(f +
(t 
te)a)]
Shrinkage strains
Esh(t,tc) =
[(t 
tc)a/(f +
(t 
tc)a)]Eshu
(AI)
a=l
t, days
f(t te)
Esh(t,te), X 106
t, days
f(t te)
Esh(t,te), X 106
7
0.000
0
7
0.000
0
14
0.167
58
14
0.167
58
28
0.375
131
28
0.375
130
f= 35 days
60
0.602
211
60
0.602
209
90
0.703
246
90
0.703
244
180
0.832
291
180
0.832
288
365
0.911
318
365
0.911
316
Note that the 365day shrinkage strain reduces to 268 x 106 when the effect of the volumesurface ratio on the shrinkage
time function is considered, that is,
iff =
26eo.0142(VlS) = 108 days
(f= 26e��.36(VlS) = 110 days).
C.l.4 Compliance J(1,1o)
a)
Elastic compliance J(1o,to)
SI units
in.Ib units
I
Cement type
a=4
(Table A.4)
b=0.85
(Table A.4)
Mean strength at age
to
femto =
[ti(a +
bto)]fem28
(A17)
femto = 29.3 MPa
(A17)
fcmto = 4253 psi
(A17)
Mean elastic modulus at
Ecmto =
0.043Ye
15
fcmto
(A16)
Eemto =
33y/
5
femto
(A16)
age
to
Eemto = 26,441 MPa
(A16)
Eemto = 3,811,908 psi
(A16)
J(to,to) =
lIEemto
(A15)
Elastic compliance
J(to,to) = 37.82 x 106 (l/MPa)
(A15)
J(to,to) = 0.262 x 106 (l/psi)
(A15)
209.2R32
ACI COMMITTEE REPORT
b)
Creep coefficient ~(t,to)
SI units
in.Ib units
Nominal ultimate creep coefficient
<l>u = 2.35
(A19)
"f
 1
25t {l.118
(A22)
Age application of load factor
e,to .
0
"fe, to = 0.916
(A22)
Ambient relative humidity factor
"f e,RH = 1.27  0.67
h if
h ~ 0.4
(A24)
"fe,RH=0.801
(A24)
Volumetosurface ratio factor
"fevs = 2/3[1 +
1.13e(D.0213(V/S)] (A25)
"fe,vs = 2/3[1 +
1.13i{)·54(VIS)]
(A25)
"fe,vs = 0.756
(A25)
"fe,vs = 0.754
(A25)
"fe,s = 0.82 + 0.OO264s
(A28)
"fe,s = 0.82 + 0.067 s
(A28)
Slump of fresh concrete factor
"fe,s = 1.018
(A28)
"fe,s = 1.018
(A28)
Fine aggregate factor
"fe, IV = 0.88 + 0.0024",
(A29)
"fe, IV = 0.976
(A29)
Air content factor
"f e,a. = 0.46 + 0.09<1 ~ 1
(A30)
"fe,a. = 1.000
(A30)
Cumulative correction factor
"fe =
"fe,to"fe,RJlYe,vs"fe,s"fe,IV"fsh,a.
(A21)
"fe = 0.551
(A21)
"fe = 0.549
(A21)
<l>u =
2.35"fe
(A20)
Ultimate shrinkage strain
<l>u = 1.29
(A20)
<l>u = 1.29
(A20)
Creep coefficient time function
f(t 
to) =
[(t 
to)IVJ(d +
(t 
to)IV)]
Creep coefficients
<I>(t,to) =
[(t 
to)IVJ(d +
(t 
to)IV)]<I>u
(A18)
'" = 0.6
t, days
f(t te)
<I>(t,to)
t, days
f(t te)
<I>(t,to)
d= 10 days
14
0.000
0.000
14
0.000
0.000
28
0.328
0.424
28
0.328
0.423
60
0.499
0.646
60
0.499
0.643
90
0.573
0.742
90
0.573
0.740
180
0.682
0.883
180
0.682
0.880
365
0.771
0.998
365
0.771
0.995
c)
Compliance J(t,to)=
l/Ecmto+ «t,to}/Ecmto
SI units
in.Ib units
t, days
J(to,to)' x 106
<I>(t,to)JEemto' x 10
6
J(t,to) (lIMPa), x 106
J(to,to)' x 10
6
<I>(t,to)JEemto' x 106
J(t,to) (lJpsi), x 106
14
37.82
0
37.82
0.262
0
0.262
28
37.82
16.04
53.86
0.262
0.111
0.373
60
37.82
24.42
62.24
0.262
0.169
0.431
90,
37.82
28.08
65.90
0.262
0.195
0.457
180
37.82
33.41
71.24
0.262
0.231
0.493
365
37.82
37.75
75.58
0.262
0.261
0.523
Note that when the effect of the volumesurface ratio is considered in the time function of the creep coefficient as
d =
26eo.0142(VIS) = 108 days
(f= 26e��.36(VIS) = 11Odays) and", = 1, the creep coefficient and the compliance rate of development
are initially smaller than when the effect of the volumesurface ratio is not considered; however, after 365 days under load,
they are similar.
,
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
C.2BazantBaweja B3 model solution
C.2.1
Estimated concrete properties
SI units
in.Ib units
209.2R33
Mean 28day strength
fcm28 =
33.3 MPa
4830 psi
Table 5.3.2.2 ACI 31805
Mean 28day elastic modulus
ECm28 =
27,318 MPa
3,961,297 psi
(A39)
C.2.2
Estimated concrete mixture
SI units
in.Ib units
Cement type
I
Maximum aggregate size
20mm
3/4 in.
Cement content
c=
409kglm
3
690lb/yd3
Water content
w=
205 kglm3
345lb/yd
3
Table 6.3.3 ACI 211.191
Watercement ratio
w/c=
0.50
(41)
Aggregatecement ratio
alc=
4.23
Fine aggregate percentage
'If =
40%
Air content
U=
2%
Table 6.3.3 ACI 211.191
Slump
s=
75mm
2.95 in.
Unit weight of concrete
Yc=
2345 kglm3
3953lb/yd3
1461b/ft3*
·Table A1.5.3.7.l and 6.3.7.1 of ACI 211.191.
C.2.3
Shrinkage strains &sh(t,tJ
SI units
in.Ib units
kh = 0.2 if
h = 1
(Table A.6)
Ambient relative
kh = 12.74 
12.94h if 0.98 <
h < 1
(TableA.6)
humidity factor
kh = 1 
h
3 if
h ~ 0.98
(Table A.6)
kh = 0.657
(Table A.6)
Cement type factor
ul = 1.000
(Table A.7)
Curing condition factor
u2 = 1.000
(Table A.8)
Esoo =al~[0.01~.Ifcm28"'()28
Esoo = alU2[0.02565~·1cm28 "'().28
Nominal ultimate
+ 270] x 106
(A33)
+ 270] x 106
(A33)
shrinkage
Esoo = 780 x 106
(A33)
E
= 781 x 106
(A33)
SOC)
Member shape factor
ks = 1.000
(Table A.9)
't  0
085t O.08!. 0.25
[2k (VIS)]2(A36)
't = 190
8t O.08!. 0.25
[2k (V/S)]2
(A36)
Shrinkage halftime
sh· c
cm28
s
sh
·c
cm28
s
'tsh = 1211.323
(A36)
'tsh = 1253.630
(A36)
Ecmfl.)7IEcm(tc+'tsh) =
1.167421[(tc +
'tsh)/(4 +
0.85(tc +
'tsh))]
(A32) & (A34)
Time dependence factor
Ecm607IEcm(tc+'tsh) = 0.996 (A32) & (A34)
Ecm607IEcm(tc+'tsh) = 0.996
(A32) & (A34)
Eshoo =
EsooEcm607IEcm(tc+Tsh)
(A32)
Ultimate shrinkage strain
Eshoo = 777 x 106
Eshoo = 778 x 106
(A32)
(A32)
Shrinkage time function
S(t 
tc) =
tanh[(t 
tc)/'tsh]0.5
(A35)
Shrinkage strains
Esh(t,tc) =
Eshookhtanh[ (t 
tc)/'t sh]0.5
(A31)
209.2R34
ACI COMMITTEE REPORT
t, days
S(t 
tc)
Esh(t,tc)' x 106
t, days
7
0.000
14
0.076
28
0.131
60
0.206
90
0.256
180
0.361
365
0.496
C.2.4 Compliance J(t,ta) = q] + Co(t,to) + Cd(t,t(Ytc)
a)
Instantaneous compliance q]=
O.6/Ecm2B
SI units
Instantaneous
0
7
39
14
67
28
105
60
131
90
184
180
253
365
I
qI =
lIEo =
0.6/Ecm28
S(t 
tc)
Esh(t,tc)' x 10
6
0.000
0
0.075
38
0.129
66
0.203
104
0.252
129
0.355
182
0.489
250
in.Ib units
(A38)
compliance
qI = 21.96 x 106(lIMPa)
I
qI = 0.152 x lO6(l/psi)
b)
Compliance function for basic creep CoCt,to) = q2Q(t,ta) +
q3ln[I +
(ttalJ +
q4In(tlto)
Aging viscoelastic tenn
q2Q(t,to)
SI units
in.Ib units
q = 185 4 x 1O6cO.5f,
0.9
2 ·
cm28
(A41)
q = 86 814 x lO6c0.5f,
D.9
2 ·
cm28
q2 = 159.9 x 106 (lIMPa)
(A41)
q2 = 1.103 x 106 (l/psi)
Qt(to) =
[0.086(to)2/9 +
1.21(to)4/9rI
Qt(to) = 0.246
m =0.5
n = 0.1
r(to) =1.7(to)O.l2 + 8
r(to) = 10.333
Aging viscoelastic tenn
Aging viscoelastic tenn
Z(t,to) =
(tormln[1 +
(t 
to)n]
Q(t,to) =
Qt(to)[1 +
{Qt(to)/Z(t,to)}',lolrIlr(tol
q2Q(t,to) (l/MPa),
t, days
Z(t,to)
Q(t,to)
x 106
t, days
Z(t,to)
Q(t,to)
14
0.000
0.000
0
14
0.000
0.000
28
0.223
0.216
34.59
28
0.223
0.216
60
0.241
0.228
36.41
60
0.241
0.228
90
0.249
0.232
37.02
90
0.249
0.232
180
0.262
0.236
37.78
180
0.262
0.236
365
0.275
0.240
38.30
365
0.275
0.240
Nonaging viscoelastic tenn q3ln[1 +
(t 
to)n]
SI units
I
in.Ib units
q3 =
0.29(w/c)4q2
q3 = 2.924 x 106 (lIMPa)
(A46)1
q3 = 0.020 x 106 (l/psi)
n = 0.1
(A41)
(A41)
(A43)
(A43)
(A45)
(A45)
(A44)
(A42)
q2Q(t,to) (l/psi),
x 106
0
0.239
0.251
0.255
0.261
0.264
(A46)
(A46)
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R35
Nonaging viscoelastic term
Nonaging viscoelastic term
t, days
In[ 1 +
(t 
to)n] q3ln[1 +
(t 
to)n] (llMPa), x 106
t, days In[1 +
(t 
to)n] I
q31n[1 +
(t 
to)n] (l/psi), x 106
14
0.000
0
14
0.000
0
28
0.834
2.44
28
0.834
0.017
60
0.903
2.64
60
0.903
0.018
90
0.933
2.73
90
0.933
0.019
180
0.981
2.87
180
0.981
0.020
365
1.029
3.01
365
1.029
0.021
SI units
in.lb units
q4 = 20.3 x 1O6(alcrO·
7
(A47)
q4 = 0.14 x
106(alc){)·7
(A47)
q4 = 7.396 x 106 (llMPa)
(A47)
q4 = 5.106 x 10
8
(l/psi)
(A47)
Aging flow term
Aging flow term
t, days
In(t,to)
q41n(tlto) (llMPa), x 106
t, days
In(t,to)
I
q41n(tlto) (l/psi), x 106
14
0.000
0
14
0.000
0
28
0.693
5.13
28
0.693
0.035
60
1.455
10.76
60
1.455
0.074
90
1.861
13.76
90
1.861
0.095
180
2.554
18.89
180
2.554
0.130
365
3.261
24.12
365
3.261
0.167
SI units
in.Ib units
Co(t,to) =
q2Q(t,to) +
q31n[1 +
(t 
to)n] +
q4ln(tlto)
(A40)
q4ln(tlto)'
Co(t,to)
q41n (tlto),
Co(t,to)
t, days
q2Q(t,to) q3ln[1 +
(t 
to)n] x 106 (llMPa), x 106
t, days
q2Q(t,to) q31n[1 +
(t 
to)n]
x 106 (l/psi), x 106
14
0
0
0
0
14
0
0
0
0
28
34.59
2.44
5.13
42.15
28
0.239
0.Q17
0.035
0.291
60
36.41
2.64
10.76
49.81
60 0.251
0.Q18
0.074
0.344
90
37.02
2.73
13.76
53.51
90 0.255
0.019
0.095
0.369
180 37.78
2.87
18.89
59.54
180 0.261
0.020
0.130
0.411
365 38.30
3.01
24.12
65.42
365 0.264
0.021
0.167
0.451
c)
Compliance function for drying creep Cit, to. tc) = C!5[
exp{ ~H(t)}  exp(~H(to)}
f5
SI units
I
in.lb units
q5 =
0.757fcm28IIEshoo x 106~.6
(A49)
q5 = 419.3 x 106 (llMPa)
(A49)1
q5 = 2.889 x 106 (l/psi)
(A49)
S(to 
tc) =
tanh[(to 
tc)/T.
sh]O.5
(A53)
S(to 
tc) = 7.587 x 102
(A53)1
S(to 
tc) = 7.459 x 102
(A53)
H(to) = 1  (1 
h)S(to 
tc)
(A51)
H(to) = 0.977
(A51)]
H(to) = 0.978
(A51)
S(t 
tc) = tanh[(t 
tc)hsh]o.5
(A52)
209.2R36
ACI COMMITTEE REPORT
H(t) = 1  (1 
h)S(t 
tc)
(A50)
f(H) = [exp{
8H(t)}  exp{
8H(to)} ]0.5
Cjt,to,tc) = q5[exp{
8H(t)}  exp{
8H(to)} ]0.5
(A48)
f(H),
Cjt,to,tc) (1lMPa),
f(H),
Cjt,to,tc) (1/psi),
t, days
Set tc)
H(t)
x 102
x 106
t, days
Set tc)
H(t)
x 102
x 106
14
0.076
0.977
0
0
14
0.075
28
0.131
0.961
0.754
3.16
28
0.129
60
0.206
0.938
1.216
5.10
60
0.203
90
0.256
0.923
1.475
6.19
90
0.252
180
0.361
0.892
1.988
8.34
180
0.355
365
0.496
0.851
2.646
11.10
365
0.489
SI units
J(t,to) =
q1 +
Co(t,to) +
Cjt,to,tc)
q1'
Co(t,to)' Cjt,to,tc)' J(t,to) (1IMPa),
q1'
t, days x 10
6
x 106
x 106
14
21.96
0
0
28
21.96
42.15
3.16
60
21.96
49.81
5.10
90
21.96
53.51
6.19
180
21.96
59.54
8.34
365
21.96
65.42
11.10
C.3CEB
MC9O99 model solution
C.3.1
Estimated concrete properties
Mean 28day strength
Strength constant
Mean 28day elastic modulus
C.3.2
Estimated concrete mixture
Cement type
Maximum aggregate size
Cement content
Water content
Watercement ratio
Aggregatecement ratio
Fine aggregate percentage
Air content
Slump
Unit weight of concrete
'Table A1.5.3.7.l and 6.3.7.1 of ACI 211.191.
x 106
t, days x 10
6
21.96
14
0.152
67.27
28
0.152
76.87
60
0.152
81.66
90
0.152
89.84
180
0.152
98.48
365
0.152
SI units
fcm28 =
33.0 MPa
fcmo =
lOMPa
Ecm28 =
32,009 MPa
SI units
N
20mm
c=
406 kg/m3
w=
205 kg/m
3
wlc=
0.504
alc =
4.27
"'=
40%
a=
2%
s=
75mm
Yc=
2345 kg/m3
C.3.3
CEB MC90 shrinkage strains csh(t,tc)
SI units
I
Cement type factor
~sc = 5
0.978
0
0
0.961
0.746
0.022
0.939
1.202
0.035
0.925
1.458
0.042
0.893
1.964
0.057
0.853
2.613
0.076
in.Ib units
(A37)
Co(t,to)'
Cjt,to,tc)' J(t,to) (1/psi),
x 106
x 106
x 106
0
0
0.152
0.291
0.022
0.464
0.344
0.035
0.530
0.369
0.042
0.563
0.411
0.057
0.619
0.451
0.076
0.678
in.Ib units
4786 psi
(A73)
1450 psi
(A72)
4,642,862 psi
(A72)
in.Ib units
3/4 in.
6851b/yd
3
3451b/yd
3
Table 6.3.3 ACI 211.191
(41)
Table 6.3.3 ACI 211.191
2.95 in.
39531b/yd
3
146* Ib/ft3
in.lb units
(Table A.I0)
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R·37
Concrete strength
Es(fcm2S) = [160 +
I013sc(9 
fcm2Slfcmo)] x lO6
(A·56)
factor
Es(fcm2S) = 445 x lO6
(A·56)
13RIIh) = 1.55[1
(hlho)3] for 0.4 ~
h < 0.99
(A57)
Ambient relative
13RIIh) = 0.25 for
h ~ 0.99
(A57)
humidity factor
ho = 1
f3RIIh) = 1.018
(A57)
Notional shrinkage
Ecso =
Es(fcm2s)13RIIh)
(A·55)
coefficient
Ecso = 453 x lO6
(A55)
Ecso = 453 x lO6
(A55)
13it 
tc) = [{
(t 
tc)/td/{350([(VIS)/(VIS)o]2 +(t 
tc)td]O.5
(A58)
Shrinkage
tl = 1 day
time function
(VIS)o = 50 mm
(VIS)o = 2 in.
Shrinkage strains
Esh(t,tc) =
Ecso13it 
tc)
(A54)
t, days
13it 
tc)
Esh(t,tc)' x 106
t, days
13it 
tc)
Esh(t,tc)' x 106
7
0.000
14
0.071
28
0.122
60
0.191
90
0.237
180
0.332
365
0.451
C.3.4
CEB MC9099 shrinkage strains GSh(t,tc)
a)
Autogenous shrinkage &cas(t)
Cement type factor
0
32
55
87
lO7
150
205
SI units
7
0.000
0
14
0.071
32
28
0.122
55
60
0.191
87
90
0.237
107
180
0.332
150
365
0.451
205
in.Ih units
aas= 700
(Table A. 11)
Notional
Ecaso(fcm2S) =
aas[(fcm2Slfcmo)/{ 6 +
(fcm2S/fcmo)} ]2.5 x 106
(A63)
autogenous shrinkage
Ecaso(fcm2S) = 52.5 x 106
(A63)
Ecaso(fcm2S) = 52.5 x 106
(A63)
Autogenous shrinkage time function
f3as(t) = 1 
exp[O.2(tlti)O.5]
(A64)
tl = 1 day
Autogenous shrinkage strains
Ecas(t) =
Ecaso(fcm2S)13as(t)
(A62)
t, days
13as(t)
Ecas(t), X lO6
t, days
13ait)
Ecas(t), X lO6
0
0.000
0
0
0.000
0
7
0.411
22
7
0.411
22
14
0.527
28
14
0.527
28
28
0.653
34
28
0.653
34
60
0.788
41
60
0.788
41
90
0.850
45
90
0.850
45
180
0.932
49
180
0.932
49
365
0.978
51
365
0.978
51
b)
Drying shrinkage &cdS<t,tc)
SI units
I
in.Ib units
adsl = 4
. fl·
(Table A. 11)
Cement type factors
ads2 = 0.12
(Table A. 11)
209.2R38
ACI COMMITTEE REPORT
Notional drying shrinkage coefficient
Eedso(fem28) = [(220 +
l1Oudsl)exp(uds2fem28ifemo)] x 106
(A66)
Eedso(fem28) = 444 x 10
6
(A66)
Eedso(fem28) = 444 X 106
(A66)
ho= 1
Pool =
[3.5femdfem28]O.l S; 1.0
(A69)
Pool = 1.000
(A69)
Pool = 1.000
(A69)
Ambient relative humidity factor
PRH(h) = 1.55[1 
(h/ho)3] for 0.4 S;
h <
0.99Psl
(A67)
PRH(h) = 0.25 for
h ~
0.99Psl
(A67)
PRH(h) = 1.018
(A67)
PRH(h) = 1.018
(A67)
Pdit 
te) = [{
(t 
te)/t 1 }/ {350([
(V/S)/(V/S)o]2 +
(t 
te)/ti} ]0.5
(A68)
Drying shrinkage time function
tl = 1 day
(V/S)o = 50 mm
(V/S)o = 2 in.
Drying shrinkage strains
Eedit,te) =
Eedso(fem28)PRH<h)Pds(t 
te)
(A65)
t, days
Pds(t 
te) Eeds(t,tc)' X 106
t, days
Pds(t 
tc) f.cdit,tc)' x 106
7
0.000
0
7
0.000
0
14
0.071
32
14
0.071
32
28
0.122
55
28
0.122
55
60
0.191
86
60
0.191
87
90
0.237
107
90
0.237
107
180
0.332
150
180
0.332
150
365
0.451
204
365
0.451
205
c)
Total shrinkage strains csh(t,tc)
SI units
in. lb units
f.sh(t,tc) =
Ecait) +
Ecds(t,tc)
(A61)
t, days
f.cas(t), X 106
f.cdit,tc)' x 106
Esh(t,tc)' x 106
t, days
Ecait), x 106
Ecds(t,tc)' x 106
Esh(t,tc)' x 106
0
0

0
0
0

0
7
22
0
22
7
22
0
22
14
28
32
60
14
28
32
60
28
34
55
89
28
34
55
89
60
41
86
127
60
41
87
128
90
45
107
152
90
45
107
152
180
49
150
199
180
49
150
199
365
51
204
255
365
51
205
256
C.3.S Compliance J(t,lo)
a)
Elastic compliance J(lo,lo)
SI units
I
in.Ib units
N
Cement type
s = 0.25
(Table A.12)
Pe =
exp[s/2{ 1 
(28/to)0.5}]
(A97)
Mean strength at age
to
Pe = 0.950
(A97)
fcmto =
P/fcm28
(A96)
fcmto = 29.8 MPa
(A96)1
fcmto = 4315.1 psi
(A96)
Mean elastic modulus at age
to
Eemto =
Ecm28exP[s/2{ 1 
(28/to)0.5)]
(A71)
Ecmto = 30,394 MPa
(A71)1
Ecmto = 4,408,587 psi
(A71)
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R·39
J(to,to) =
lIEcmto
(A·70)
Elastic compliance
J(to,to) = 32.90 x 106 (l/MPa) (A·70)
J(to,to) = 0.227 x 106 (llpsi) (A70)
Ecm2S(1) =
Ecm2S(1.06 
0.003TITo) (A85)
Ecm2S(1) = Ecm2S(1.06  0.003
. [18.778T 
6OO.883]1To) (A85)
Effect of temperature
Ecm2S(1) = 32,009 MPa
(A85)
Ecm2S(1) = 4,642,853 psi (A85)
on modulus of elasticity
Ecmti1) =
EcmtoO.06 
0.003TITo) (A85)
Ecmto(1) =
Ecmto(1.06  0.003
. [18.778T 
600.883]ITo) (A85)
Ecmti1) = 30,394 MPa
(A85)
EcmtO<1) = 4,408,579 psi (A85)
J(to,to) =
11Ecmto
(A70)
Elastic compliance temperature adjusted
J(to,to) = 32.90 x 106 (l/MPa) (A70)
J(to,to) = 0.227 x 106 (llpsi) (A70)
b)
Creep coefficient ¢2S< t, to)
SI units
in.Ib units
0.1 =
[3.5fcmJfcm2S]0.7
(A79)
Compressive strength factors
0.2 =
[3.5fcmJfcm2S]0.2
(A79)
0.1 = 1.042
(A79)
0.1 = 1.042
(A79)
0.2 = 1.012
(A79)
0.2 = 1.012
(A79)
tPREih) = [1 + {(l 
hlho)al/(O.l (VIS)/(VIS)0}]a2
(A·76)
Ambient relative humidity and
ho = 1
volumesurface ratio factor
(VIS)o = 50 mm
(VIS)o = 2 in.
tPREih) = 1.553
(A76)
tPREih) = 1.553
(A76)
Concrete strength factor
!3(fcm2S) =
'5.3/(fcm2S/fcmo)0.5
(A77)
!3(fcm2S) = 2.918
(A77)
!3(fcm2S) = 2.917
(A77)
to,T = ~At,exp[13.65 
40001
to,T= ~At,exp[13.65
40001
{273 +
(T(AtITo))}]
(A87) {273 +
(18.778T(Atj) 
6OO.883ITo)}] (A87)
To = 1��C
To = 33.8 OF
Temperatureadjusted
to,T= 14.0 days
(A87)
to,T= 14.0 days
(A87)
age of loading
to =
to,rl9/{2 
(to,Tltl,T)1.2} + 1]1l ~ 0.5 days
(A81)
0.=0
tl,T= 1 day
to = 14.0 days
(A81)
Adjusted age of loading factor
!3(to) =
1/[0.1 +
(tJt 1)0.2]
(A78)
!3(to) = 0.557
(A78)
!3(to) = 0.557
(A78)
tPo =
tPREih)!3(fcm2S)!3(to)
(A75)
Notional creep coefficient
tPo= 2.524
(A75)
tPo= 2.524
(A75)
0.3 =
[3.5fcmJfcm2S]0.5
(A84)
0.3 = 1.030
(A84)
0.3 = 1.030
(A84)
Creep coefficient time function
!3H = 150[1 +
(1.2hlho)IS](VIS)/(VlS)0 +
2500.3:5; 1500a.3
(A83)
!3H= 570.470
(A83)
!3H = 570.445
(A83)
!3c(t to) =
[(t to)/tl/{!3H+(t to)/t!l]O.3
(A82)
Creep coefficients
tP2S(t,to) =
tPo!3c(t 
to)
(A74)
209.2R40
ACI COMMITTEE REPORT
t, days
f3c(t 
to)
$28(t,to)
t, days
f3c(t 
to)
$28(t,to)
14
0.000
0.000
14
0.000
0.000
28
0.326
0.824
28
0.326
0.824
60
0.459
1.159
60
0.459
1.159 '
90
0.526
1.328
90
0.526
1.328
180
0.640
1.614
180
0.640
1.614
365
0.749
1.890
365
0.749
1.889
SI units
in.lb units
$T= exp[0.015(T1To  20)]
(A91)
$T = exp[0.015
{(18.778T 
600.883)ITo  20}] (A91)
$T= 1.000
(A91)
$T= 1.000
(A91)
Effect of temperature
$RH,T= $T+ fiRH<h) 1]$/2
(A90)
conditions
$RH,T = 1.553
(A90)
$RH,T = 1.553
(A90)
$0 =
$RH,rf3(fcm28)f3(to)
(A75)
$0 = 2.524
(A75)
$0 = 2.524
(A75)
Effect of high stresses
$o,k =
$oexp[1.5(ka  0.4)]
(A93)
$o,k = 2.524
(A93)
$o,k = 2.524
(A93)
Notional creep
$0 =
$ck
coefficient temperature
$0 = 2.524
$0= 2.524
and stress adjusted
~T=
exp[1500/(273 +
T1To)  5.12]
(A89) ~T= exp[1500/{273 +
(18.778T 
6OO.883)ITo}) 5.12] (A89)
f3T= 0.999
(A89)
f3T= 0.999
(A89)
f3H,T =
f3Hf3T
(A88)
Effect of temperature
f3H,T= 570.159
(A88)
f3H,T= 570.128
(A88)
conditions on creep
coefficient time function
~$T.trans =
0.0004(T1To  20)2 (A92) ~$T.trans =
0.0004[(18.778T 
600.883)ITo  20]2(A92)
~$T.trans = 0.000
(A92)
~$T.trans = 0.000
(A92)
f3c(t 
to) =
[(t 
to)lt11 {f3H +
(t 
to)lttJ ]0.3
(A82)
Creep coefficients
temperature and stress
$28(t,to,1) =
<Pof3c(t 
to) + ~$T.trans
(A86)
adjusted
t, days
f3c(t 
to)
$28(t,to,1)
t, days
f3c(t 
to)
$28(t,to,1)
14
0.000
0.000
14
0.000
0.000
28
0.326
0.824
28
0.326
0.824
60
0.459
1.159
60
0.459
1.159
90
0.526
1.328
90
0.526
1.328
180
0.640
1.615
180
0.640
1.614
365
0.749
1.890
365
0.749
1.890
c)
Compliance J(t,1o)=
llEcmto+ ¢2B<'t,lo)IEcm28
'.
SI units
in.lb units
J(t,to) =
llEcmto +
$28(t,to)IEcm28
(A70)
$28(t,to)IEcm28' J(t,to) (1IMPa),
$28(t,to)IEcm28' J(t,to) (l/psi),
t, days
J(to,to)' x 106
x 106
x 106
t, days
J(to,to)' x 106
x 106
x 106
14
32.90
0
32.90
14
0.227
0
0.227
28
32.90
25.74
58.65
28
0.227
0.178
0.404
60
32.90
36.20
69.10
60
0.227
0.250
. 0.47,6
90
32.90
41.49
74.39
90
0.227
0.286
0.513
180
32.90
50.44
83.34
180
0.227
0.348
0.575
365
32.90
59.04
91.94
365
0.227
0.407
0.634
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R41
Compliance temperature and stress adjusted
SI units
in.Ib units
J(t,to) =
lIEemto(T) + ~28(t,to)/Eem28(T)
(A70)
tp28(t,to)/Eem28, J(t,to) (llMPa),
~28(t,to)/Eem28'
J(t,tJ (l/psi),
t, days
J(to,to)' x 106
x 106
14
32.90
0
28
32.90
25.75
60
32.90
36.21
90
32.90
41.50
180
32.90
50.44
365
32.90
59.04
C.4GL2000 model solution
C.4.1
Estimated concrete properties
Mean 28day strength
fcm28 =
Mean 28day elastic modulus
Ecm28 =
C.4.2
Estimated concrete mixture
Cement type
Maximum aggregate size
Cement content
c=
Water content
W=
Watercement ratio
wlc=
Aggregatecement ratio
ale =
Fine aggregate percentage
'If =
Air content
0.=
Slump
s=
Unit weight of concrete
Ye=
'Table AI.5.3.7.1 and 6.3.7.1 of ACI 211.191.
C.4.3
Shrinkage strains Esh(t,tJ
Cement type factor
x 106
t, days
J(to,to)' x 106
x 106
x 106
32.90
14
0.227
0
0.227
58.65
28
0.227
0.178
0.404
69.11
60
0.227
0.250
0.476
74.40
90
0.227
0.286
0.513
83.34
180
0.227
0.348
0.575
91.94
365
0.227
0.407
0.634
SI units
in.Ib units
32.5 MPa
4689 psi
(A94)
28,014MPa
4,060,590 psi
(A95)
SI units
in.Ib units
I
20mm
3/4 in.
402kglm
3
676lb/yd
3
205 kglm
3
345lb/yd
3
Table 6.3.3 ACI 211.191
0.510
(41)
4.33
40%
2%
Table 6.3.3 ACI211.191
75mm
2.95 in.
2345 kglm
3
3953lb/yd
3
1461b/ft3*
SI units
in.Ib units
k=1.ooo
(Table A.14)
Ultimate shrinkage strain
&shu =
9OOk[30ifcm28]o.5 x 106 (A99)
&shu =
9OOk[4350ifem28]0.5 x 106 (A99)
&shu = 865 x 106
(A99)
&shu = 867 x 10
6
(A99)
Ambient relative humidity factor
/3(h) = (1 
1.18h4)
(A1 00)
/3(h) = 0.717
(A1 00)
Shrinkage time function
J3(t
te) =
[(tte)/{ t 
te +
O.12(Vlst n��.5(AIOl)
J3(t 
te) =
[(t 
te)!{t 
te +
77(VIS)2} ]0.5
(AlOI)
Shrinkage strains
&sh(t,te) =
&shu/3(h)/3(t 
te)
(A98)
t, days
/3(t 
te) &sh(t,te), x 106
t, days
/3(t 
te)
&sh(t,te), x 106
7
0.000
0
7
0.000
0
14
0.076
47
14
0.075
47
28
0.131
81
28
0.129
80
60
0.206
128
60
0.203
". 126
90
0.254
158
90
0.251
156
180
0.355
220
180
0.351
218
365
0.479
297
365
0.475
295
209.2R42
ACI COMMITTEE REPORT
C.4.4 Compliance J(t,to)
a)
Elastic compliance J(ta,ta)
SI units
in.Ib units
Cement type
s = 0.335
I
(Table A.14)
f3e =
exp[s/2{ 1 
(28/to)0.5}]
(A97)
Mean strength at age
to
f3e = 0.933
(A97)
fcmto =
f3/fcm28
(A96)
fcmto = 28.3 MPa
(A96)
fcmto = 4081.1 psi
(A96)
Mean elastic modulus at
Ecmto (MPa) = 3500 +
4300(fcmto)0.5
(A95)
Ecmto (psi) = 500,000 +
52,000(fcmto)0.5 (A95)
age
to
E cmto = 26,371 MPa
(A95)
E cmto = 3,821,929 psi
(A95)
J(to,to) =
lIEcmto
(A102)
Elastic compliance
J(to,to) = 37.92 x 106 (l/MPa)
(A102)
J(to,to) = 0.262 x 106 (l/psi)
(A102)
b)
Creep coefficient ¢2sCt,to)
SI units
in.Ib units
J(t,to) =
lIEcmto + ~28(t,to)/Ecm28
(A102)
Effect of drying before loading factor
Effect of drying before loading factor
<P(tc) = 0.961
(AI04) & (A105)
<P(tc) = 0.962
(AI04) & (A105)
Basic creep coefficient
1st term
2[(tto)0.3/{(tto)0.3 + 14}]
2nd term
[7
/to]0.5[ (t 
to)/ {(t ~
to) + 7} ]0.5
t, days
1st term 2nd term Basic creep coefficient
t, days
1st term 2nd term Basic creep coefficient
14
0.000
0.000
0.000
14
0.000
0.000
0.000
28
0.272
0.577
0.850
28
0.272
0.577
0.850
60
0.368
0.659
1.026
60
0.368
0.659
1.026
90
0.415
0.677
1.092
90
0.415
0.677
1.092
180
0.497
0.693
1.190
180
0.497
0.693
1.190
365
0.586
0.700
1.286
365
0.586
0.700
1.286
Drying creep coefficient
Ambient
2.5(1 
1.086h2)
relative humidity factor
1.170
Time function
fit, to) =
[(t 
to)/{ (t to) +
0.12(V/S)2} ]0.5
Time function
fit, to) =
[(t 
to)/{ (t to) +
77(V/S)2}]0.5
f(t,to)
Drying creep coefficient
f(t,to)
Drying creep coefficient
t, days
3rd term
t, days
3rd term
14
0.000
0.000
14
0.000
0.000
28
0.107
0.126
28
0.106
0.124
60
0.192
0.225
60
0.190
0.222
90
0.244
0.285
90
0.241
0.282
180
0.349
0.408
180
0.345
0.403
365
0.476
0.556
365
0.471
0.551
Creep coefficient
~28(t,to) =
<P(tc) x [basic + drying creep]
(A103)
t, days
Basic + drying creep
~28(t,to)
t, days
Basic + drying creep
~28(t,to)
14
0.000
0.000
14
0.000
0.000
28
0.975
0.937
28
0.974
0.936
60
1.251
1.203
60
1.248
1.201
MODELING AND CALCULATING SHRINKAGE AND CREEP IN HARDENED CONCRETE
209.2R43
90
1.377
1.324
90
1.374
1.321
180
1.598
1.536
180
1.593
1.532
365
1.843
1.771
365
1.837
1.767
SI units
in. Ib units
J(t,to) =
1!Ecmto+ 'i>2s(t,lo)/Ecm2S
(A102)
J(t,to) (l/MPa),
I, days
J(lo,to)' x 106
'i>2S(I,to)/Ecm2S' x 106
x 106
J(I,to) (1!psi),
I, days
J(tooto)' x 106
'i>2s(t,to)/Ecm2S' x 106
x 106
14
37.92
0
37.92
14
0.262
0
0.262
28
37.92
33.46
71.38
28
0.262
0.231
0.492
60
37.92
42.93
80.85
60
0.262
0.296
0.557
90
37.92
47.25
85.17
90
0.262
0.325
0.587
180
37.92
54.82
92.74
180
0.262
0.377
0.639
365
37.92
63.22
101.1
365
0.262
0.435
0.697
209.2R44
ACI COMMITTEE REPORT
C.SGraphical comparison of model predictions
C.S.1
Shrinkage strains Gsh(t,tc)
v~
o .
II GL2000 Model
o
50
100
150
200
250
300
350
Concrete age t (days)
Fig. C.lShrinkage strain predictions.
C.S.2
Compliance l(t,to)
110
Ii 100
IL
~ 90
~ 80
II)
e 70
u
��.
60
S
e 50
..,
CD
40
u
c
_EI .
.
..t:
:::.. ~
 
..
~

~  
~~
~
r ,,",
, .;~
J~~
.....
'I
:ll!
30
Do
E
0
20
(J
10
o
14
42
70
98
126 154 182 210 238 266 294 322 350
Concrete age t (days)
1
400
0.70
n
o
0.60 3
'g,
AI
0.50 S
c..
0.40 ~
"3
:::: i
j
0.10 .::
0.00
¢AC1209 Model (1IMPa)
...•.. ACI209 Model (I/psi)
_83 Model (lM'a)
 _  83 Model (Ilpsi)
 + CEB 1.C9099Model (1IMPa) ____ GL2000 Modal (1IMPa)
"* CEB 1£9099 Model (Ilpsi)  <>  GL2000 Modol (1/psl)
Fig. C.2Compliance predictions.
".