CONCRETE LIBRARY OF JSCE-NO. Z1. JUNE I993
A NON—-LINEAR CREEP PREDICTION EQUATION FOR CONCRETE
(Translation from paper in Proceedings of JSCE, no.451 V-17, Aug. 1992)
Kenji SAKAEA
Toshiki AYANO
SYNOPSIS
Concrete is considered to be an aging linear visco-elastic material. Hence, the
creep behavior of concrete under constant sustained stress
is represented by
either the creep coefficient or specific creep, both of which are based on the
assumption of a linear relationship between creep strain and externally applied
stress. What is in doubt, however, is the upper limit of this assumption. In
terms of the stress/strength ratio, an upper limit between about 0.23 and 0.75
has been observed. The purpose of this study is to clarify the non-linearity of
creep strain of concrete under constant sustained stress. We verify that there
is a significant difference among the concrete creep coefficients under various
levels of constant stress. Furthermore, we propose a non-linear creep prediction
equation which can accurately represent the results of creep experiments.
Keywords : creep strain,
stress/strength ratio,
non-linear, creep compliance,
modified Bailey equation
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K. Sakata, professor of civil engineering at Okayama University, Okayama, Japan,
received the
1976.
‘Doctor of Engineering’
degree from Kyoto University,
His research interests include prediction of concrete
Japan in
shrinkage,
creep,
and fatigue properties. I-Ie was awarded a JSMS prize in 1984 for a study of
concrete fatigue under variable repetitive compressive loading, and a CAJ prize
in 1992 for a study of non—linear properties of concrete creep strain. He is a
member of JSCE, JSMS, JCI, AC1 and JSDE.
.
_
T.
_____
Ayano,
Okayama,
____
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research
Japan,
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associate
received
the
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civil
‘Doctor
_____ _
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engineering
of
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at
Engineering‘
____
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Okayama
degree
"irvrrn
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University,
from
Okayama
University, Japan in 1993. His research interests include prediction of concrete
shrinkage and creep. He was awarded. a CAJ prize in 1992 for a study of
non—linear properties of concrete creep strain. He is a member of JSCE and JCI.
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1
.
INTRODUCTION
The relationship
between creep strain
linear.
Therefore,
the creep prediction
linear
creep compliance.
and stress
equations
of concrete
now in use
is aEiSumed to be
are based on the
Many investigations
have been performed
to clarify
the
validity
of this
assumption.
But the applicable
upper limit of this assumption
is in dispute'
In
tens
of the stress/strength
ratio,
upper limits
between 0.23 and 0.75 have been
observed
(1]. Thus, this
upper limit
varies
according
to t'he researcher.
It
seems that there iS no definite
upper limit for this linearity.
However, the linearity
of creep to stress/strength
ratio
up to 0.4 is defined
in
the standard
specification
for design
and construction
of concrete
structure
published
by JSCE concrete
corEmittee [2] and in the CEB/FIT model code 1978 [3].
when the application
service
loads exceed the valid upper limit
for linearityI
creep prediction
equations
based on the linear
assumption
cannot
give correct
results.
It is also impossible
to predict
the creep strain
over a long period of
time.
As creep
strain
ordinarily
s6ems
to
occur
under
very
low stress,
a lower
limit,
where the asstmption
of the linearity
of creep to stress
is accurate,
has been
rarely
discussed.
Most of the investigations
tO COnfim the proportionality
of
creep tO Stress
Were Carried
out by using concrete
which had reached
hygra1
equilibrium
with surrounding
medium prior to the application
of load
[4]
- [6].
In those cases,
the regression
line which represents
the relationship
betwe&n
creep
strain
and stress
almost cross;eS the origin.
But as pointed
in the
investigation
by L'Hermite
f7], if concurrent
shrinkage
occurs during the period
under loading,
the value obtained
from the curve to represent
the relationship
between time-deformation
(creep-plus-shrinkage)
and stress,
is smaller
than the
shrinkage
of an unloaded companion specimen at zero stress.
This suggests
that
the approximate
line which does not intersect
the vertical
axis is inadequate
for representing
the relationship
between creep strain
and stress.
The creep prediction
equation
modeled by CEB/FIT in 1990 [8] has taken the
non-1inearity
of creep strain
into account when the applied
stress
exceeds 40%
strength.
However, when the applied stress
is below 40%, the linear
assumption
has remained in this prediction
equation
as before.
=f the upper or the lower
limit occurs at aL stress
value smaller than 40% strength,
the non-linearity
of
creep strain
must be considered
even in cases where the stress
is below 40%
strength.
As mentioned
earlier,
the linear
asstmption
between creep strain
and stress
applies approximately
to concrete without reliable
evidence.
The purpose of this
study is to clarify
experimentally
the non-1inearity
of creep strain
of concrete
under constant
stress
and to establish
a non-linear
creep prediction
equation'
2.
THE APPLICABILITY
OF LINEAR
CREEP
PREDICTION
In order to express
the proportionality
of creep strain
to stress,
the creep
strain
of concrete
under constant
stress
is represented
by specific
creep or
c!reep coefficient.
The specific
creep i.s defined as the ratio
of creep strain
to
sustained
Stress,
as follows,.
- 94-
0
= 6
I
ia) 6.0
/
/
I)
/
O
/
t•`
t>
/
/
/
/
/
S 4.5
0
,I
a
a_
q)
/
/
I)
e3
O
e3
OF
O
/
--+
/
/
/
/
//
I)
nl
q)
a
3.0
4.5
6.0
a 0.Oo
.0
1.5
Pre[dfcted creep coeffTctenf
iq) 6.0
O
L
+-
t•`
tP
S 4,5
a 3.0
O
O
/l /
//
+U
//
a
/
0
L
1.5
3.O
?
Creep prediction
6. 0
b
+-I)
4.5
/
/
/
a o.oo0
6.0
O
/
,I
?I)
6.0
O
t>
tr-
i e TE _E A -F i E I 9 E
3.0
4.5
6.0
by CEB/FIT-90
Pkxie1.
i
o,I
q
bv Sc]-kT6f6
/
/
,'C)
S 4.5
O
a
CL
I)
e3
O
/
0
-c e - - --
o'
/
+O
/
1 5
/
/
/
-/ - /
0
//
a 3.0
O
+O
/
C
q)
0
O ,I
.E
L
/
L
O
a
a 0.Oo0
D_
BO
1.5
.E
a)
1
.0
o .o
JJ
1.5
3.0
i.i
i
4.5
6.0
Cmep prediction
by CEB/FIT-70
!We1.
1 .5
3.0
4.5
6.0
Predfcted creep coeffTcTenf
Pr8dTcted crccp cooffTcTenf
Fig.3
1.5
+-
O
C
I)
B
Fig. 5 Creep prediction
5
S4.
/
/
Predfcted creep coeffTcfent
dP /
+th
tr-
/
/
/
,
/
lI
//
/
1.5
L
q)
CL
Py Bazant ZWe1.
-=
n P - 7 0
C E B
/
1
Pr8dTcted creep coeffTcfent
Fig.2
r
/
/
//
.E
A
i
-
+U
O
a
a 0.Oo0
---
Z43de1.
, ,, ,' '
i
/
O
b y _ B i Tz c ln t
O
oJ
_
C
q)
L
o
e3.0
1.5
.E_
by CEB/FIT-78
S 4.5
//
･I--∼
./
- _- _ - 0
prediction
t>
t•`
O
aO
1.5
i 6.0
•`I)O
O
O
Ji J_Jbp3.0 f EBjE4.5 ff
fLf ej6.0
0o
PredTcted creep coeffrcfent
o ,I//
,0
O
0
BO
Fig. 4 Creep
ty mI Fkxie1.
/
/
/
.E
L
+-
/
0
=
q) 1 5
/
.E
L
Fig.1 Creep prediction
/
/
+U
/
lr
/
51.5
/
/
/
+O
/
/
U
O
C
I)
O
/
/
_i_
/
5
S4
/
/
C )
O
t>
L•`
/
I-
/
+-
/
+-
Fig.6
- 95-
creep prediction
by Sakata ZWe1.
+
5
C
(1)
C(t,t')=ecr(t,tr)/6.
+-I)
O
+•`
'Or-
tP
where,
c(t,t')
t•`
q)
0
O
i specific
Ecr(t,t')
..
qo
t
-.
tr
creep
Creep Strain
sustained
stress
age of the concrete
age at the first
application
of load
The creep
coefficient
the ratio of the creep
strain,
as follows;
is
strain
i
O
L
O
/
/
O
/
0
"A•`
?
C)
2 0
+O
defined
as
to elastic
/
C
I)
E1
O
0
a)
L
0
(2)
a)
C)
+-
a
BO
4(t,t')-Ecr(t,tl)/E.i(t,t')XE(t,)
I
LJd4
34.loo
[L
aq)
_
/
/
O
0
0 .0
1.0
2 .0
5.0
4.0
]
5.0
Predfcted creep coefffcTenf
where,
4(t,t.)
E(tr)
: creep coefficient
: modulus of elasticity
eo = elastic
strain
Fig. 7 Creep prediction
by CEB/PIP-90
Akxie1.
In general
prediction
equations
for creep,
the creep coefficient
or specific
creep
is given in terms of the basic
properties
of the concrete.
In this
section,
we investigate
the applicability
of usual creep prediction
equations
in
order to confirm the suitability
of linear
creep prediction.
Figures
1 - 6 show the comparison
data
by the
ACI-209
model
[9],
CEB/FIT
1978 model
[3],
CEB/FIT
b
etween
experimental
Bazant
model
[10],
1990 model
[8]
and
data
[13]
CEB/FIT
1970
the
authors7
and predicted
model
[11],
model
[12],
was 104.
The
respectively.
The total
number of specimens used in this experiment
horizontal
axis in these figures
shows the creep coefficient
predicted
by each
model. The vertical
axis shows the experimental
creep
coefficient.
The broken
1ine shows 40% variation
of the predicted
data from the experimental
data.
As is
evident
from these figures,
we can predict
the creep accurately
enough by any of
these model. However, the scatter
between the predicted
and experimental
data
becomes larger
with the lapse of time.
i
-
--
-
l-
Figure 7 shows the comparison
between the prediction
by CEB/FIT 1990 model and
the
experimental
data
[14]
used for the
establishment
of this
prediction
equation.
The broken line has the same meaning as above.
The predictions
are
scattered
within
i40% around the experimental
data.
It is generally
acknowledged that linear
creep prediction
has the error of 40%
due to the linear
asstmption
of the relationship
between creep
strain
and
applied
stress.
3.
TfIE NOW-LINEARITY
OF CREEP
STRAIN
OF CONCRETE
In this
section,
we examine the non-linearity
of creep in the
the creep coefficients
yielded
by Stresses
of various
magnitude.
3 .1.
Experiment
difference
among
outline
The type of cement used was normal
portland
cement
(specific
gravity
; 3.15).
The fine
aggregate
was river
sand (specific
gravity
: 2.60,
water
absorption
:
2.08,
F.M. ; 3.10),
and the coarse
aggregate
was crushed
stone
(specific
gravity
I 2.74,
water
absorption
: 1.14,
F.M. : 6,55).
The strength
of the concrete
- 96-
after
curing
proportion
Table
was
of the
25.1.MPa.
The
concrete
is
mix
Table
E;hown in
1 Mix proportion
of concrete.
1.
Lla x s iz e S lu m p A i r Y / C s / a
Un it y c ig h t( kg / n 3 )
The size
of the
prism
specimen
for
( c m ) (% ) ( % ) ( % ) W
C
S
C
(m m )
measuring
creep strain
was 10cmX10cmX
20
4 - 5 0 .7 6 6 . 1 4 4 , 0 l 8 5 2 8 0 8 0 8 lO 83
38cm. The sis:e Of the prism
specimen
for measuring shrinkage
strain was 10cmX
10cmX4Ocm. At about 24 hours after
casting,
the specimens were removed from the
mold and cured in water for two days. After that,
the specimens
were cured for
25 days in a constant
temperature
and constant
relative
htmidity
room at 20j=1
oc, 68+_5%. The total
curing period was 28 days. Two pairs
of point
gauges were
put on each surfacer
except
for the treated
surface
and the side opposite
the
treated
Surface.
MeasurementS Were made of strain
by a Whittemore
strain
meter
with minimum divisions
0f 1/1000rnm. The experiment
was performed
in a constant
temperature
and constant
relative
humidity
room at 20j=1cCl 68i5%.
Stresses
of
10%, 20%, 30%, 40% and 50% strength
were applied
to each specimen.
The specimen
used for measuring
concrete
strength
had the same Shape and size as the specimen
used for measuring
shrinkage
Strainr
and was cured in the same method as the
specimen
used for measuring
c!reep strain.
The mean value
obtained
from 3
of the concrete.
The strength
of concrete
specimens was regarded as the Strength
The total
number of the Specimens subjected
was
25.1
MPa.
to each stress
was 3,
circumstances.
In
16,
3,
15 and
18, respectively.
This was due to experimental
the stress-strain
curve,
too.
From this
we obtained
measuring
the strength,
stress-strain
curve,
we determined
the
strain,
which was yielded
when the
strain
yielded
in the specimen
required
stress
was applied.
To make this elastic
uSerd for measuring creep strain,
we judged that the required
stress
was applied.
we applied
stresses
of 0.2 - 0.3 MPa per Second to specimens
for measuring
concrete
strength
and creep strain.
Because Of the loss
of prestress
due to
shrinkager
creep and relaxation
of Steel which occurs
with time,
each 5PeCimen
was prestressed
again on the 3rdr loth and 30th day from the first
application
of load. The pemissible
error of applied
stress
was 2%.
by the stress
of 50% strength
In accordance
with Eq. (3), the loss of prestress
is calculated
as follows,.
loss
of 1.ll
Mpa occurs
in the period
between
the
day'
1.15 Mpa
first
application
and 3rd day' o.86 Mpa between 3rd day and loth
between loth day and 30th day' o.60 Mpa between 30th day and 49th day. The loss
among the losses
of
of prestress
by the stress
Of 50% Strength
is the largest
loss of prestress
is
prestress
by the other stress
level.
However, the largest
level
4.6% in terms of stress/strength
ratio
and is less than half of the stress
10%.
Therefore,
constant
the
experiment
A
A,.
orc-ix
A
A
C
P
:
:
Ep:
i C :
i
A
have been
performed
under
..
P
ec
:
(3)
E,.ii.AEc
lp
Ac
where ,
to
can be considered
stress.
area of the concrete
area of the prestressing
steel
modulu s of elasticity
of the prestressing
length of the concrete
length of the prestressing
steel
both creep strain
and shrinkage
strain
between repeated prestressings
- 97-
steel
which
occurs
in
the period
2 .50
Sfr8S8/Sfrcngth
3 .0
rcTlto..
o lox
Tfme under loQd - 49dQyS
2.8
E=
q}
O
'G 1.50
t>
0
¥•`
+ SOX
tl
8
0
C
.6
q} 2
O
Jr 4OX
O
O
O
t•`
0
O
O
50%
00
0
0
Q0
0
L 2.0
0
O
U o.50
0
1.8
a
4D
os
0
0
1
SOX
0.00
%o
oQ
(Strc99/Strength rctTo)
40%
507E
1.6
10
100
10.0
20.O
30.0
Time under JoQd - dQyS
Fig.8
( a)
o
0
odd
2 .4
2.2
a
1.00
C}
O
L
3.2.
0
EZ20X
2.00
¥+
Results
of creep
test.
Fig.9
40.0
50.0
70.0
60.0
EfQSt7c sfrQin x10-5
of creep coefficients.
Scatter
Results
Investigation
Figure
of non-1inearity
8 shows the
change
over
of creep
time
strain
of creep
coefficient
for each stress.
The
values
of the creep
coefficients
subjected
to stresses
of 10%, 20%, 30%, 40% and 50% in terms of
stress/strength
ratio(
respectively.
=f the relationship
between creep
strain
and stress
of concrete
under constant
stress
were a linear
phenomenonr the
change over time of creep coefficients
should be represented
by only a single
curver irrespective
of the magnitude of applied
stress.
But, as is evident
from
Fig.
8, the
change over time of creep
coefficient
for each stress
do not
coincide.
In
Particular,
the creep coefficient
for the stress
of 50% strength
is
much larger
than
others.
Also,
the
difference
between
creep
coefficients
produced
by stresses
of 10% and 40% strength
is constant
during
the applied
symbols
"0",
"D",
"0",
"JL.f
and
"+,.
are
the
mean
J.-
-
I
-
I-llllll
period.
Figure 9 shows the relationship
between the creep coefficient
and elastic
strain
at the 49th day after the first
application
of load. =f the relationship
between
creep
strain
and stress
of concrete
under constant
stress
were
a linear
phenomenon, the creep coefficient
must be constant
for any elastic
strain.
But,
as is obvious from Fig. 9, the larger
the elastic
strain,
the larger
the creep
coefficient.
Furthermore,
it is also clear that not only the mean but also the
scatter
of the creep coefficients
by the stresses
larger
than 40% of strength
i$
larger than the others.
Figure 10 shows the comparison of creep coefficients
obtained
by two approaches.
The creep coefficients
represented
by the horizontal
axis in this figure
are the
optimal slopes obtained
by regression
of the line expressed
by Bq. (4) at every
measuring
period.
The creep coefficients
represented
by the vertical
axis are
the creep strain
divided by elastic
strain.
E
where,
Ecr= Creep
Eo : elastic
4 : creep
cr=4.
e.
Strain
strain
coefficient
- 98-
(4)
The
of
dot-dasih
the
lines
optimal
show. 40% variation
slope
from
the
strain
divided
by elastic
strain.
the broken lines
Show 20% variation
the optimal
slope from the creep
creep
And,
of
strain
divided
by elastic
strain.
Figure
10
Sugge5tS that the confidence
limit
of
linear
creep
compliance
is
40%, and
shows that
the creep
COefficients
by
the optimal
slopes
of regression
line
result
in an overestimation
compared to
the experimental
data when applied stress
is small
and an underestimation
when
applied
stress
is large.
+
C
I)
O
4.00
;.-iT / / '7
T
+I+
t•`
[p:i-ETs$3
X
S 3.00
/1,i:
X
O
/
a
I)
/
_
a 2.00
O
U
.C/'
_tT<
Ih
51.0
0
> 1,NQ1'= --
e
.E
L
/
'
I)
go.o 0
LJ
(b) Non-linear
creep
compliance
0 .00
1.00
2.00
3.00
4.00
CQtCutQtedcreep coefffcrent
Figure ll shows the relationship
between
Fig. 10 The ccnpison
tntnveen calculated data
creep strain
and elastic
strain
at the
ty linear creep ccqliaLrlCe end test
28th day from the first
application
of
data.
load. The solid curve in this
figure
is
obtained
by regression
of the curve expressed
by Eq. (5), which is called
Bailey
equation.
For steel
creep,
the Bailey
equation
iE; Often used to represent
the
relationship
between creep strain
and elastic
strain.
b
Ecr=aeo
(5)
creep strain
elastic
strain
a and b i indeterminate
coefficients
obtained
by a non-linear
least
squares
method. In this
study, we use the hybrid method [15] derived
from
the combination
of the Gauss-Newton
method and the steepest
descent
wherer
ecr
:
Eo
I
method.
From Fig. ll, it appears that the regressed
values by the Bailey equation
are in
good agreement with experimental
data when the stresses
are 20%, 30% and 50% in
terms
of stress/strength
ratio.
However, the regressed
values
by the Bailey
equation
are less than the experimental
data when the stresses
are 10% in terms
of stress/strength
ratio and larger
than the experimental
data when the Stresses
are
40%. This is confined
at another
measuring
period.
Namely,
the curve
regressed
by the Bailey equation
can represent
the nonlinearity
of creep strain
accurately
enough.
12 are the
The creep coefficients
represented
by the horizontal
axis in Fig.
strain.
The
calculated
creep strain
by the Bailey equation
divided
by elastic
optimal
value of the indeterminate
coefficients
in the Bailey
equation
are
obtained
by regression
based on experimental
data at every measuring period.
The
creep coefficients
represented
by the vertical
axis
are experimental
creep
strain
divided
by elastic
strain
subjected
to each stress
at every measuring
from experimental
ones in
period.
The variation
of calculated
creep coefficients
Fig. 12 is smaller
than that
in Fig. 10. Therefore,
it is clear
that the Bailey
and elastic
strain
equation can represent
the relationship
between creep strain
more precisely
by taking into account the nonlinearity
of creep strain.
However,
when applied
stress
is 10% and 20% in terms of stress/strength
ratio,
the creep
coefficients
calculated
by the Bailey equation
exceed the confidence
limit
of
40%.
- 99-
160.0
16O.0
140.0
tO
TTme under
Ioc[d = 28dc]ys
140.0
0
120.0
I
O
LO
f
T7me under
Tocd I 28dQYS
120.0
0
r
=
100.O
L
8O.0
+
y)
C)
CL
I)
0
L
X
i
C
U
@
L
+
a
a)
60.0
q)
L
+0.0
lO%
3 %
400/o
T
500/o
0.0
0.0
0.0
10.0
20.0
30.0
40.0
50.0
EIQStfc strczh
60.0
70.0
0.0
4 .00
e
U
JIth
t•`
q)
ED
C
3 .00
A
//
t•`
t•`
I)
O
U
/
/
/
/
/
/
40%
, , 'lLSLh
/eoo.
?;'a3
2 .OO
00
/
.E
L
i
i?1-ui-
/
xn-o
J>
3 .00
1 O%,lox
b
SOX
A
40%
O
/
e 2.00
;-:oqJ;
O
C
q)
-
/
/
+O
1.00
/
._E
-,- -7/-
L
q)
0
LJJ
60.070.0
/
:oo{"b
-/-
40,050.0
CL
q)
not
+U
30.0
500/o
I
4.00
O
+-
5O!=
a
I)
/
SOX
C
Fig. 13 The relationship
btween elastic
strain and creep strain by th&
Wfied
Bailey equation.
+
/
10%,2O%
40%
c1/
E]QStic strqfn x10-5
/
+I-
20% I 30%
10.020.0
x10-5
Fig. ll Ttk relationShi.p tx2tWeen elastic
strain and creep strain by the
Bailey equation.
+
51
(Stress/strengthratio)
20.0
(Stress/strengthratio)
C)
20%
L
O
/
/
10%
20.0
O
O
a)
60.0
O
40.0
O
O
i
C)
80.0
y7
U
C
I)
100.0
C
U
1
00
X
0 .00
1.OO
2.00
3.00
LJJ
4.00
0 .00
1.00
2.00
3.00
4.00
CcT(CutQtedcreep coofffcrent
Cc)fcutclfed creep coefffcfenf
Fig. 12 The ccnpiex)n
0 .00
tetDQeen Calculated
Fig. 14 The cQrrPaLrisontntween calculated
data by the Wfied
Bailey
eqpation and test data.
data ty the Bailey eqpation
ard test data.
In order to express
the turning
point of the relationship
between creep strain
and elastic
strain
occuring
at the stress
of 40%, a new non-li.near
creep
compliance
which introduces
indeterminate
coefficients-cl
and c2 into the Bailey
equation
i.s proposed.
This new non-linear
creep compliance
is shown in Eqs. (6)
and (7) and referred
to as the Modified Bailey equation
in this paper.
- 100-
3.0
ne 55[
xN50E
I
2.5
U
:: 45E
z 2.0
i
odTfted Boley equctlon
L.
F 40E
+E F
¥S 35EL
f•`
C
c) CI
O
O
C) C2
1.5
C)
0
th
•`
L
I)
0
Ls
SOP
0
O
M
+
1.0
U
r'
i
25[
2.L=
0
LLuJ.Jil_JJ
7
JWL
14
21
Time under
Fig.15
In
*=-L-i;
0.5
the
The optimal
case
O.O
WW
28
35
42
0
49
E.
loc)d - dclyS
Inthe
case
e.
21
28
35
42
49
ZoQd - doys
value
of coefficient-a.
< C2,'
a(i5fi
that
-Q----C
under
Fig. 16 The optiml
1.5
ecr-
14
The
value of coefficient-c1,c2.
that
7
,:S=-n-
eo)i
(6)
/
-&---e)---A
.-q----y---i(-
E a
A
2 c2;
E
1.4
BQtleyequctlon
1.3
(
Ecr-
a(
eO-C1)t
(7)
The solid
curve
in Fig.
13 is drawn
according
to the Modified Bailey equation
whose
indeterminate
coefficients
are
calculated
by the
regression.
The
relationship
between creep strain
and
elastic
strain
shown in Fig.
13 is
first
obtained
at 28th day from the
As is
evident
application
of load.
from
Fig.
13,
the
Modified
Bailey
+
C
O
O
t•`
t•`
D
O
U
equation
proposed in this
paper can
accurately
model the actual relationship
between creep strai.n and elastic
strain
1.2
Ei,,p,A
1.1
0
1
.oE oo
Mod7f7ed Bot[ey 8qUQtlon
o.9E
o.8L
0
,__LW
7
14
21
Time under
Fig. 17 The optimal
value
28
35
42
food - dQyS
of coefficient-b.
Figure
14 shows the comparison
between the creep coefficients
given by the
Modified
Bailey
equation
and the experimental
creep
coefficients.
The optimal
value of indeterminate
coefficients,
such as a, b , cl and c2, in the Modified
Bailey equation
are obtained
by regression
based on experimental
data at every
of the data
measuring
period.
In Fig.
14, the variation
calculated
by the
data
is within
20% of the
Modified
Bailey
equation
from the
experimental
confidence
limit.
we can therefore
characteristics
of
say that the nonlinear
creep
strain
are accounted
for in the Modified
Bailey
equation
by the new
coefficients
cl
and
c2.
Figure
15 shows the optimal
values
of indeteminate
coefficients
cl and c2 in
the Modified Bailey equation.
The horizontal
axis in this
figure
shows the time
under load. These optimal
values
are nearly
constant
irrespective
of time.
The
optimal
value
of coefficient
c2 corresponds
to the elastic
strain
produced
by
the stress
of 40% strength.
In other words, it is verified
that the coefficient
c2 Which appears
in the Modified Bailey equation
can repreE;ent
the turning
point
- 101-
49
2 .50
Stress/Strength
2 .50
rQt7o:
Stress/strength
+
+ 10K
2.00
a 20X
¥•`
C
q}
O
G
1.50
t0
0
O
a) SOX
I lox
DP
I
0
O
+ SOX
O
+-
10
40%
Figure
relationship
between creep
in terms of stress/strength
16 shows
the
optimal
by
of
>
o.51 0
0 .00
1
10
100
The under loQd - dQyS
Fig. 19 The creep coefficients
calculated
Wfied
Bailey eqpation.
strain
ratio.
values
*
Y
+
100
Time under loc]d - dc[ys
Fig. 18 Tin cmep cafficients
calculated
Bailey eqpation.
of
,t'
L
0 .00
of the
40X
0
ll.0
0
0
L
/+/
1
,,;
a)
+ I
i.
*
0 SOX
t•`
-yJ.^/
__+/
E9 20X
C
0
O
t= 1.5 0
I/
+ SOX
o.50
I
,+
2 .00
+
,. ,L=+
,i
a_ 1.00
O
0
L
O
rQtTo:
+ 103r
and elastic
strain
lying
at
the
by
stress
the
indeterminate
coefficient
a which
equation.
Figure
17 shows the
axis in these figures
shows
the time under
load.
The optimal
coefficient
a which appears
in the Bailey
equation
is smaller
than that
apper5
in the Modified
Bailey
equation.
The
optimal
coefficient
b which appears
in the Bailey equation
varies
with time,
whereas that appears in the Modified Bailey equation
is constant
independent
of
time. And, the optimal
coefficient
b which appears
in the Bailey
equation
is
larger than that appers in the Modified Bailey equation.
appears in the Bailey equation
and Modified Bailey
opt ima1 values of the coefficient
b. The horizontal
Figure 18 shows the creep coefficients
calculated
by the Bailey equation
whose
indeterminate
coefficients
are the optimal
values determined
by regression
of
the experimental
data. As is obvious from this figure,
the relationships
between
the creep coefficients
calculated
by the Bailey equation
and time under load are
as Shown
logarithmic
and considerably
different
from that obtained
by experiment
in Fig.
8. This
means that
the
error
between
the
calculated
data
and
experimental
data becomes larger
with time. On the other hand, Fig. 19 shows the
creep coefficients
calculated
by the Modified Bailey equation whose indeterminate
coefficients
are the optimal
values
determined
by regression
of experimental
data.
The relationships
between
the creep
coefficients
calculated
by the
Modified
Bailey
equation
and time
under
load
are
approximated
by power
expression
and is very similar
to that obtained
by experiment
as shown in Fig.
8
of 50% strength
are considerably
larger
. The creep coeffic!ients
for the stress
than the others.
However, i.n the calculated
results,
the difference
between the
creep coefficients
for the stress
of 10% strength
and the creep coefficients
for
the stress
Of 40% strength
is larger
with time,
which disagrees
with
the
experimental
observations.
Except for this
single
inconsistency,
however,
creep
phenomena based on the Modified
Bailey
equation
agree suffici.ently
with
the
results
of the experiment.
Therefore,
it is clear that Modified Bailey equation
for nonlinear
creep prediction
is accurate
enough to describe
the nonlinear
behaviour
of creep strain
of concrete
under constant
stress.
- 102-
4
. PROPOSITION
OF A WON-LINE3n
CREEP
PREDICTION
EQUATION
By using
the Modified
Bailey
equation,
it
is possible
to represent
the
relationship
between the creep strain
and elastic
strain
of concrete.
But the
Modified Bailey equation
will not be useful unless
it is a function
of timer
especially
when it comes to calculating
the creep strain
of concrete
under
variable
stress.
Furthermore,
it may be meaningless
to represent
the relationship
if the
between creep strain
and elastic
strain
by the Modified
Bailey equation
to the elastic
coefficient
c2 appearing
in this
equation
always corresponds
service
loa.d is
strain
produced
by the stress
of 40% strength
as the applied
usually
below 40% in terms of stress/strength
ratio.
In this
section,
we establis;h
a more
general
creep prediction
equation
to
incorporate
the effects
of the age at
the first
application
of load, drying
time and the period of the application
of
load
into
the
Z4odified
Bailey
equation.
Furthemore,
we investigate
the effects
0f both the water curing
period
and the drying
period
on the
nonlinearity
of creep of the concrete.
Table 2 Ph pmphion
(m m )
U n i t W e i gh t( kg / m 3)
(c m ) (%) ( % ) ( % )
W
C
S
G
9 - l2 1. 2 6 0. 0 47 . 7 2 0 0 3 3 3 8 4 2 9 6 3
20
Table 3 Water curing
priod
euld drying time.
A g e a t the lo a d ing (da ys )
3
7
14
28
56
P i
'=3 TdO
O v
Bi B
outline
4 .1
of concrete.
M ax s i ze S l um p A i r Y / C s / a
3
7
14
28
56
10
24
52
94
2l
35
63
lO 5
}d i
63 7 7 10 5
of
cement
used
was
normal
The type
The fine aggregate
was river
sand
portland
cement
(specific
gravity
.' 3.15).
F.M.
: 2.81),
and
the
coarse
(specific
gravity
i 2.62, water absorption
= 1.78,
was crushed
stone
(specific
gravity
: 2.73,
water
absorption
; 0.76,
aggregate
F
.M. = 6.68).
The mix proportion
of the concrete
is shown in Table
2
The experiment
was perfo-ed
in
a constant
temperature
and constant
relative
humidity
room at 20+_1Qc.,
68j=7%. The size
of the prism specimen
used for
measuring
creep strain
was 10cmX10cmX38cm. The size of the prism specimen used
for measuring
shrinkage
strain
was 10cmX10cmX40cm.
two pairs
of point
gauges
were put on each surfaceT
except for the treated
surface
and the side opposite
the treated
surface.
MeasurementS
of strain
were made by using
a Whittemore
strain
meter with minimum divisions;
Of 1/1000rnm. The measuring
period
was 200
days. Because Of the loss of prestresS
due to shrinkage,
creep and relaxation
of
the steel
which occurs with time, each specimen was prestressed
again on 3rd,
loth,
30th and 70th day from the first
application
of load.
The permissible
and the
error
of applied
stress
was 2%. Table 3 shows the water curing period
number of 5PeCimens subjected
to
age at the firEit application
of load. The total
each
stress
of
10%,
20%,
30%,
40% and
50% were
1,
2,
2,
2 and
2,
respectively.
But when the strength
of concrete
exceeded 30 Mpa, a 'speCimen subjected
to the
stress
of 30% strength
was added. And when the strength
of concrete
was below 30
Mpa, a Specimen subjected
to the stress
of 50% strength
was added.
4
.2 Results
Figures
20 and 21 show the relationship
between creep strain
and elastic
Strain
of concrete,
with the stress
being applied
irEmediately
after
water curing
for 3
days and 56 days, respectively.
Figure 22 shows the relationship
between creep
strain
and elastic
strain
of concrete
whose water curing
period
is 3 days and
whose age at the first
application
of load is 94 days.
Figure
23 shows the
- 103-
180.0
F Age
E
^g8
160.0
i
i
1 40.0
Lr)
f
O
I
120.0
X
C
O
loo.0
:i 8dfpcp7TcOQf..odn"I.n,g
,o=Q.3d=cy3Sdqy.
c=1.ll
b=1.25
o1=12.7
c2=41.7(Sfn,w
E
=0.71
.1o2oO.loo
i b=1.25
q
Ln
I
O
f4VeT-5O%)
r
E
L
X
q)
q)
L
c2=36.1
(Sfr6S9
C)
[eVCI-SOX)
80.0
4-
D
L
60.0
a.
O
O
L
40.0
+
m
80.0
a
a)
c1=12.7
C
C)
O
+L
y)
2s: :tt :I,qp7'coQf.I.dn"Tonf9.o=od3d=Qy,8.dcys
Oo
6O.0
O
O
40.0
20.0
20.0
C2
0.0
-JIIL}-JJJ-I
O.0
I-Ll-J
10.0
2O.0
LJ-i_I
30.0
u
IJ_LLL_LJ
40.0
50.0
C2
LJ.I
60.0
I_i_LJ__J
J_)
0.
7O.0
0
EklSffc sfrcffn x10-5
Fig.20 The zdationship
tx2tXQ7Een
Creep
strain ard elastic strain.
60.0
E
E
^^:: :i :tpQ,7'cocf..odnqf.np
,o=Qd56=d
Qg63.Qy.
60.0
i
50.0
tL?
o2=28.0(Sfr.$3
50.0
I
O
t8V8l-25%)
P
T-
X
.E
U
L
X
10.0
70.0
O
i
4O.0
c=1.21
b=1.O5
o1=9.6
c2=24.7(StrBJ8
7eyet-2O%)
i
J-
U
i: 30.0
u7
[
V)
a_ 30.0
q)
O
L
U
20.0
60.0
C
40.0
+
JnJ
40.050.O
2;: :i 8Qt,C,77cO.f".dn"'.n.a..=c.56=dQhs5dcyq
Q=1.56
b=1.05
c1=9.6
LO
30.0
E[clSffc sfrclh x10-5
The relationship
tx2tWeen Creep
strain End elastic
strain.
Fig.22
7O.0
1O.020.0
E
0
a_
I)
L
E
e 20.O
O
(
i.•`r_L1_i_I_
0
10.0
10.0
/
C)
/
/
/
I
I
/cl
0
/
O
C2
J _I_i
20.0
i
30.O
i_L
l'
Ll.I__l__I_1JJ
40.O
O.0
EIQSffc strclTn x10-5
Fig.21 The relationship tx2tXdeenCreep
strain and elastic: strain.
C2
C1
JII
_J,_rlmJ_
0 .O
5O.0
10.0
20.0
30.0
40.0
50.0
E[c[stfc sfrclfn x10-5
Fig. 23 The relationship
between creep
strain and elastic strain.
relationship
between creep strain
and elastic
strain
of concrete
whose water
curing period is 56 days and whose age at the first
application
of load are 105
at 30th day after
days.
The data in these
figures
are obtained
the
first
application
of load.
As is evident
from these
figuresr
the Modified
Bailey
equation
represents
the experimental
results
very well.
From the optimal value of indeterminate.
coefficient
b shown inFigs.
20 -23,
it
is clear that the longer the water curingr the smaller
the nonlinearity
of creep
is same, even
strain.
And, it is confi-ed
that the nonlinearity
of creep strain
if the age at the firs;t
application
of load is differentl
if the duration
of
water curing is the same.
- 104-
+
C
I)
4.00
7
' JO C f T *
+J-
L?
O
42.0
t>
e 40.0
X
f1
Lh
38.0
S 3.00
Water curingI 3days
O
O
I 36 0
0
Aircuring= 0 days
a 2.00
/
- 1/ '
S Lk1
/
I / / LN
:a
0
0
S30
o28
. ,,v 3i
O
4l
32
t
a..
+c 3+ 0
.9
t•`
t•`
/ ' ,
+-
water curing= 14days
I
A
00
v
/
/
/
.E
L
Watercuring = 56days
26.0
+
51
+
0
U
0
24.0
0
20
+0
60
A80
1 00
120
xho 00
LL}
0
.00
1.00
2.00
3.00
4.00
Age Qt QPPZ7cQt7on of [oQd - dQyS
Cc)Icuk)ted creep coefffcfent
Fig.24 The change of c2 with curing pried.
Fig.25 The ccnpison
txtween calculated data tn'
the present non-linear creep prediction
equation and test data.
The Modified
Bailey
equation
which incorporates
the effects
of the age at the
first
application
of load,
drying time and time under load iS given by Eqs. (8)
and (9). The coefficients
involved
in EqS. (10) - (13) are calculated
by the
hybrid
Table
in
the
method
from the
experimental
case
that
eo
the
of concrete
cured
in accordance
with
< c2(t',tO),.
Ecr(eO,t,t,,tO)
in
data
3.
case
that
eo
ecr
I
a
(t,t,,tO)
cit , ,td)-cito)
cit,
,to)
-)dto'
(8,
2; c2(t-,tO),.
(eo,t,tf,to)
= a
(9)
(t,t',t.)X(eo-c1(t.))b(to'
inwhich,
a (t,t,A)
- 2.64 too.114( o.oo2 (tf-to).
(1O)
1 )-2.9 (#r434
(ll)
b(t.)=0.285exp(-0.047to)+1
a (to) i Bf.
c2(t'
where'
(12)
9.81)XlO-5
,t.)=47.1X10-5(logo(t.+1))-0'372exp(-o.o55(t'
Ecr(t,t',to):
t
:
t'
i
to:
eo=
the
the
the
the
the
virgin
age of
age at
age at
elastic
(13)
-t.)0'214)
creep strain
(X10-5)
the concrete
(days,
t2;t')
the first
application
of load (days,
the start
of drying
(days)
strain
produced by stress
( X10-5)
t72;t.)
The optimal values of coefficients
b, cl and c2 Calculated
by the hybrid method
all
are regarded
as constants
independent
of time.
The drying
time affects
optimal indeterminate
coeffic!ients
in the Modified
Bailey equation,
and the age
- 105-
at the
first
application
of i.oad influences
both
coefficients
a and c2.
Figure 24 shows the relationship
between the optimal
value of coefficient
c2 and
the curing
period.
From this
figure,
the longer
curing
period,
the smaller
the
optimal value of coefficient
c2. The optimal
values 0f coefficient
c2 Shown in
Fig.
24 never
correspond
to the elastic
strain
produced
by the 5treS$
Of 40%
strength,
and are smaller
than those
shown in Fig.
15. This means that
the
optimal values of coefficient
c2 are affected
by curing period,
curing
methodr
mix proportion
of the concrete
and so onr and that the turning
point
of the
relationship
between creep strain
and elastic
Strain
of concreteT udder constant
stress
exists
below the stress
level of 40% in terms of stress/strength
ratio.
Figure 25 shows the comparison
between the experimental
creep coefficients
and
the creep coefficients
calculated
by the Modified Bailey equation
given by Eqs.
(8) and (9),
The total
number of experimental
data points
is 3,200.
It is
evident
that Eqs. (8) and (9) represent
the experimental
result
a5 Precisely
as
the case shown in Fig. 14, in spite of the fact that Eqs. (8) and (9) involve
the age "tH, age at the first
application
of load "tand age at the start
of
drying
5
"to''.
. CONCLUSION
It
has been confirmed
that
an explicit
upper
limit
and lower
limit
of
PrOPOrtionality
of concrete
creep strain
to stress
do not exist.
It is also
clear that the creep strain
of concrete
does not increase
uniformly
with stress,
but rather
that there is turning
point in the relationship
between creep strain
and stress.
In representing
such a nonlinear
phenomenon of creep
strai.n
of
concrete
by the Modified Bailey equation presented
by the authors,
the variation
of the calculated
data from the experimental
data is within
20%, which is half
the range covered by the linear
creep compliance'
The Modified
Bailey
equation
may be imperfect,
since
it does
not involve
Properties
of concrete
such
as strength,
mix proportionr
curing
method,
envirormental
conditions
and so on. Howeverr by investigating
the effect
of
basic properties
on the coefficients
in the Modified Bailey equationr
we will be
able to develop
a creep prediction
equation
which is more precise
than the
linear
creep prediction
equation.
ACKNOWLEDGMENT
A part
Research
Science
of this
research
was Supported
by the
Grant-in-Aid
for
Scientific
(c) in 1992 from Ministry
of Education
and by Okayama Foundation
for
and Technology.
References
[1] Freudentha1,
A. M. and Roll,
F. , Creep and Creep
High Compressive
streets,
ACE Journalr
54, pp. 1111
[2] Concrete
Cornmittee of JSCEr lTStandard Specification
of Concrete
Structures.I,
concrete
Library
special
-
f3]
35,
1986
Recovery
of Concrete
under
- 1142, 1958
for Design and Construction
publication,
No. 1, pp. 32
CEB-FIT, Z4odel Code for Concrete
Structurest
vol.
==, International
System
of Unified Standard Codes of Practice
for structures,
[comite Euro-International
du Beton-Federation
International
de la precontrainte1
, 1978
- 106-
[4]
[5]
L'Hemiter
A. G., Vohme
Changes
of Concrete,
Proc.
Fourth
Int.
Symp.
the Chemistry
of Cement,
Vo1. 2, Washington
DC, pp. 659 - 694, 1960
Sheikin,
A. A. and
Baskakov,
N.
J.,
The
Influence
of Mineralogica1
on
Composition
[6]
[7]
[8]
[9]
t10]
of Portland
Cement on the Creep of Concrete
in Compression,
Translation
No. 236,
Stroite1'naya
Promyshlennost,
No. 9, pp. 39 - 40, 1955,
1956
Department
of Scientific
and Industrial
Research,
Klieger,
p.,
Early
High-strength
Concrete
for Prestressing,
proc.
world
1957
Conf. on Prestressed
Concrete,
Ban Francisco,
pp. A5-1 - A5-14,
LTHermite,
A. G. and Mamillan,
A., Further
Results
of Shrinkage
and Creep
Tests,
Proc.
Int.
Conf.
on the
Structure
of Concrete,
Cement and Concrete
Association=
London,
pp. 423 - 433,
1968
du Beton-Federation
CEB-FIT,
Z4odel Code 1990,
[Comite
Euro-Internatina1
International
de la Precontrainte]
, (Draft) , 1990
ACI Cornmittee
209,
Prediction
of Creep,
Shrinkage
and Temperature
Effects
on Concrete
Structures,
ACI-SP-76,
1982
Bazant,
Z. P. and Panula,
L., Simplified
Prediction
of Concrete
Creep and
Shrinkage
from Strength
and Mix, Structural
Engineering
Report No. 78-10/6405,
Department
of Civil Engineering,
Northwestern
University,
Evanston,
Illinois,
pp.
[11]
[12]
[13]
[14]
[15]
24,1978,
Oct.
CEB-FIT,
Internatinal
Recommendations
for the Design
and Construction
of
Concrete
Structures,
[Comite
Europeen
du Beton-Federation
Internatinale
de
la precontrainte]
I 1970
Sakata,
K., Ayano T. and Hiromura
0.,
Prediction
of
Creep
of
Concrete,
Proceedings
of the Japan Concrete
Institute,
vo1. 10-2, pp. 271 - 276, 1988
(in Japanese)
Sakata,
A. and Kohno,
I.,
Prediction
of Shrinkage
and Creep of Concrete,
Memoirs of the School of Engineering,
okayama University,
vo1. 21-1,
1986,
pp. 57 - 80
bilger,
W. H, Private
communication
Nakagawa,
T. and Royanagi,
y.,
"The Analysis
of Experimental
Data by Least
Squares
Method,"
Tokyo Univ. press,
1989, pp. 110 - 119. (in Japanese)
- 107-