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PREDICTION EQUATIONS FOR CREEP AND DRYING SHRINKAGE
IN CONCRETE OF WIDE-RANGING STRENGTH
(Translation from Proceedings of JSCE, No.690/V-53, November 2001)
Kenji SAKATA
Tatsuya TSUBAKI
Shoichi INOUE
Toshiki AYANO
As the performance of concrete is enhanced and high-strength concrete comes into
common use, the model code for concrete is being improved to cope with the changes. Today,
prediction equations for creep and shrinkage in high-strength concrete are an essential part of the
model code. In this study, the precision of current prediction equations is examined using a
number of databases, from which data providing reliable knowledge of the effect of concrete
strength on creep and shrinkage is extracted. New prediction equations are proposed, and this
paper shows that they are very simple and applicable to a wide range of concrete strengths.
Keywords: prediction equation, creep, drying shrinkage, high-strength concrete, data base
K. Sakata, a professor in Environmental and Civil Engineering at Okayama University, Okayama,
Japan, received his Doctor of Engineering degree from Kyoto University, Japan in 1976. His
research interests include the prediction of concrete shrinkage, creep, and fatigue properties. He is
a fellow of the JSCE.
T. Tsubaki is a Professor in the Department of Civil Engineering at Yokohama National University,
Japan. He obtained his Ph.D. from Northwestern University in 1980. His research interests include
the time-dependent mechanical behavior of concrete and the analysis of concrete structures. He is
a member of the JSCE and the JCI.
S. Inoue is a Professor in the Department of Civil Engineering at Tottori university, Tottori, Japan.
He received his Doctor of Engineering Degree from Kyoto University in 1985. His research
interests include the fatigue properties, time-dependence deformation of concrete structures, and
properties of fresh concrete. He is a member of the ACI, JSMS, JCI, LPC and JSCE.
T. Ayano, an associate professor in Environmental and Civil Engineering at Okayama University,
Okayama, Japan, received his Doctor of Engineering degree from Okayama University in 1993.
His research interests include the prediction of concrete shrinkage and creep. He is a member of
the JSCE.
- 145 -
1. INTRODUCTION
Every increment in concrete creep and shrinkage strain increases the deflection of a concrete
member, affects cracking progress, and causes loss of prestress. Thus, to check the safety,
durability and serviceability of concrete structures, it is very important to have a means of
predicting creep and shrinkage over the long term.
Higher strength concrete has now been used extensively, and research into high-performance
concrete and its applications has been very active. The wide acceptability of high-performance
concrete can be attributed to the establishment of a technique for producing high-performance
concrete using silica fume and high-range water-reducing admixtures. Its use has led to the
construction of slender, lightweight concrete members. Increasing the strength of concrete may
also improve the durability and extend the service life of concrete structures. In this sense, many
hope that high-performance concrete may decrease environmental loading. On the other hand,
others point out that high-strength concrete is prone to cracking. Whatever one's perspective,
however, it is very important to precisely predict both creep and shrinkage at the design stage in
order to discuss the durability and service life of concrete structures.
Most model codes are based on prediction equations that consist of several factors relating to
concrete creep and shrinkage [1][2]. Some existing codes are applicable to high-performance
concrete. The Japan Society of Civil Engineers (JSCE) has its own prediction equations [3], but
the model is proven only for normal-strength concrete up to 60 N/mm2. It has not been clarified
whether the model is available for high-strength concrete as well.
High-strength concrete with strength at 28 days exceeding 70 N/mm2 is not unusual. Indeed, the
2002 edition of the Standard Specifications for Design and Construction of Concrete Structures
cover concrete whose 28-day strength is up to 80 N/mm2. Thus the requirement for creep and
shrinkage prediction equations suitable for a wide range of concrete strengths is increasing.
Both RILEM and JSCE are establishing databases of creep and shrinkage. In this paper, new
prediction equations that are simple and available for a wide range of strengths are proposed on
the basis of data extracted from the RILEM and JSCE databases as well as other minor sources.
2. REPRESENTATIVE PREDITION EQUATIONS
2.1 Outline of data used
The main source of data used in establishing the new prediction equations was the creep and
shrinkage data collected by RILEM and JSCE. The RILEM database holds data consisting of 512
creep records and 419 shrinkage records [4], while the JSCE database contains 259 creep records
and 219 shrinkage records [5]. The RILEM database consists mainly of data from the western
countries, while the JSCE has collected data from papers published by Japanese organizations.
Every data set was subjected to regression using the hyperbola expressed by Equation (1). The
standard divisions of the optimal value of C0 and C1 were checked. If each of them was beyond
10, the data was judged to be unreliable and was excluded from use in the study.
Creep or shrinkage =
C0 ⋅ t
C1 + t
(1)
In total, 295 creep records and 200 shrinkage records from the RILEM and JSCE database were
selected for use in the study. Not all sets of shrinkage and creep data include all information.
However, the selected data sets were obtained for concrete specimens with a wide range of
strength and dimensions. Summaries of the selected drying shrinkage and creep data are shown
- 146 -
in Figures 1 and 2, respectively. All data was measured at 20°C. Drying shrinkage was obtained
from unloaded specimens. The drying shrinkage strain of high-strength concrete was obtained by
subtracting the strain of a sealed specimen from that of an unsealed specimen. In cases where the
start of drying was more than one week after casting, the effect of autogenous shrinkage on the
strain of unsealed specimens was regarded to be negligible. Creep strain was obtained by
subtracting the strain of an unloaded specimen from that of a loaded specimen. The size and
shape of specimens under loading and unloading were the same.
100
200
Total = 318
Number of data records
Total = 497
147
150
111
100
62
61
50
31 26
17
6
15 16
5
79
75
75
60
50
30
20
25
17
12
10
0
20
40
60
80
100
120
0
2
Strength at 28 days - N/mm
20
40
60
80
100
120
2
Strength at start of drying - N/mm
(a) Compressive strength at 28 days
(b) Compressive strength at start of drying
200
500
Total = 497
Total = 497
3
2
0
0
380
400
300
200
100
71
39
7
146
150
151
100
67
78
50
27
21
5
2
0
0
40
50
60
70
80
90
100
80
100 120 140 160 180 200 220 240
Relaitve humidity - %
Water content - kg/m 3
(c) Relative humidity at atmosphere
(d) Water content of Concrete
400
Total = 318
75
75
79
60
50
30
20
25
17
10
12
2
8
2
3
100
8
2
Total = 482
282
300
200
97
100
39
0
0
0
20
40
60
80
100
120
0
200
27
11 5 6 0 0 9 0 0 0 0 6
400 600
800 1,000 1,200 1,400 1,600
V / S - mm
2
(e) Age at start of drying
(f) Specimen size
Figure 1 Summaries of data used for drying shrinkage
- 147 -
200
Total = 476
Total = 483
151
150
129
100
65
50
35
15
13
23
22
13
9
1
200
0
150
20
40
60
80
100
Strength at 28 days - N/mm
50
37
30
9
1
120
0
2
19 19
7
6
1
20
40
60
80
100
120
2
Strength at applicaton of load - N/mm
(a) Compressive strength at 28 days
(b) Compressive strength at initial load application
250
300
266
Total = 485
Total = 350
250
200
150
100
35
50
34
6
5
4
188
200
150
122
100
80
50
35
33
16
9
2
0
0
30
40
50
60
70
80
90
100
80
120
(c) Relative humidity of atmosphere
250
224
239
200
100
56
37
29
7
Total = 485
58
3~7
240
Total = 474
200
150
126
100
86
50
31
5 1
0 0
0 0
600
800
1,000 1,200 1,400
0
0
0~3
200
(d) Water content of concrete
300
59
160
Relative humidity - %
105
97
100
0
0
152
7~14 14~28 28~56 56~90 90~
0
200
400
V / S - mm
Age at application of load - days
(e) Age at initial load application of load
(f) Specimen size
Figure 2 Summaries of data used for creep
- 148 -
0 1
2.2 Applicable range of representative prediction equations
Table 1 shows the applicable range of the representative prediction equations for creep and
shrinkage. Table 2 shows the factors needed to predict creep and shrinkage using each prediction
equation, since the factors differ according to the equation selected. Model B3 [6] requires more
factors than any of the other prediction equations listed in Table 1. As for creep, model B3 predicts
the creep function J (t,t') ; models CEB [7] and GZ [8] predict the creep coefficient; and the JSCE
model [9] predicts specific creep C (t,t') . Furthermore, the creep coefficient φ 28 (t,t') predicted by
model CEB is the product of specific creep and Young’s modulus at 28 days. The creep coefficient
φ (t,t') predicted by model BZ is the product of specific creep and Young’s modulus at the
application of loading. When Young’s modulus at the age of t’ is represented by E (t') , the
relationship between J (t,t') , C (t,t') ,
J (t,t') =
φ 28 (t,t') and φ (t,t') is represented as follows:
1
+ C (t,t')
E (t')
(2)
φ (t,t')
φ (t,t')
1
1
=
+ 28
=
+
E (t') E (28) E (t') E (t')
Where, t: current age (days), t’: age at first application of loading (days)
The accuracy of creep models CEB, BZ, and the JSCE model was examined by using the specific
creep strain. The accuracy of model B3 was examined by using the creep function. In cases where
Young’s modulus was also available in the database, it was used to calculate specific creep strain.
Otherwise, Young’s modulus was calculated using the proposed equation in each model code
including the creep and shrinkage model.
The factors used as the basis for prediction are listed in Table 3; these are values for ordinary
concrete. Table 4 gives the predicted shrinkage strain and specific creep for each model based on
these factors. The values predicted by the JSCE model are a little bigger than the others.
Table 1 Applicable range of each prediction equation.
Model B3
17.2~68.95
2.5~13.5
160~721
0.35~0.85
40~100
Normal
Low heat
Rapid
Normal
hardening
Type of Cement
High
strength
Volume-surface ratio (mm)
100~300
t’≧ t０
t’ and t0
Autoclave
Water curing
Curing method
Sealed
curing
t’: Age at initial load application (days), t0: Age at start of drying (days)
Strength at 28 days(N/mm2)
Aggregate/cement ratio in weight
Unit cement content (kg/m3)
Water-cement ratio
Relative humidity (%)
JSCE Model
Less than 70
260~500
0.40~0.65
45~80
- 149 -
Model CEB
20~90
40~100
Normal
Low heat
Rapid
hardening
High
strength
-
Model GZ
20~68.95
0~0.6
40~100
Normal
Low heat
Rapid
hardening
High
strength
≧2days
-
Table 2 Necessary factors for each prediction equation
JSCE Model
Creep
Shrinkage
Strength at 28 days
Strength at t’ or t0
Aggregate/cement
○
Unit cement content
○
○
Unit water content
○
Water-cement ratio
○
○
Relative humidity
Type of cement
○
Start of drying
○
t’
○
○
Size of specimen
Shape of specimen
Curing method
t’: age at initial load application (days)
Model CEB
Creep
○
○
Shrinkage
Model B3
Creep
○
○
Shrinkage
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
Model GZ
Creep
○
○
○
Shrinkage
○
○
○
○
○
○
○
Table 3 Set values for prediction
Item
Start of drying
Age at initial load application
Relative humidity
Strength at 28 days
Strength at initial load application
Cement type
Aggregate-cement ratio
Unit cement ratio
Unit water content
Water-cement ratio
Size and shape of specimen
Input data
7 days
7 days
60 %
30 N/mm2
20 N/mm2
Normal
6
300 kg/m3
180 kg/m3
60 %
100×100×400 mm
Table 4 Predicted values
Drying shrinkage (μ)
Specific creep (μ/(N/mm2))
JSCE Model Model CEB
790
557
214
130
Model B3
441
107
Model GZ
817
134
2.3 Accuracy of shrinkage prediction equations
Figure 3 presents a comparison between experimental shrinkage data extracted from the
databases and the JSCE model predictions. Here, however, the 28-day concrete compressive
strength is less than 80 N/mm2. On the other hand, the comparison in Figure 4 is for concrete with
- 150 -
a 28-day compressive strength greater than 80 N/mm2. The start of drying is indicated by ● and
○, representing 28 days and 1 day, respectively. As is clear from these figures, the JSCE model
tends to predict larger values than the experimental data when the start of drying is late.
Figure 5 is a comparison between experimental shrinkage data and predictions by model CEB.
Here, only data within the applicable range of model CEB was used for comparison. Model CEB is
able to closely predict the experimental shrinkage data from the western database well. However,
the predicted values are smaller than the data from the JSCE database. Figure 6 compares data
outside the applicable range of model CEB. The 28-day strength expressed by ● is greater than
90 N/mm2. Model CEB underestimates the shrinkage strain of this high-strength concrete.
Figure 7 is a comparison between experimental shrinkage data and predictions by model B3. Here,
only data within the applicable range of model B3 was used for comparison. Model B3 tends to
underestimate experimental shrinkage data, even within its applicable range. As is clear from
Figure 8, which is a comparison with data outside its applicable range, model B3 underestimates
concrete shrinkage especially significantly when the 28-day strength is greater than 70 N/mm2. In
this figure, data for concrete in this strength range is indicated by ●.
Figure 9 compares experimental shrinkage data and predictions by model GZ. Here again, only
data within the applicable range of model GZ was used for comparison. The GZ model can predict
experimental shrinkage strain to an accuracy of ±40% when the data is within its applicable range.
On the other hand, Figure 10 is a comparison between predictions by model GZ and experimental
shrinkage data beyond the applicable range of the model. Symbols ○ represent shrinkage data for
concrete whose strength at 28 days is greater than 70 N/mm2. Symbols □ represent concrete
whose strength at 28 days is less than 20 N/mm2. Shrinkage data for concrete whose start of
drying is earlier than 2 days is shown by ●. Model GZ is able to predict the shrinkage strain of
high-strength concrete. However, in the case of concrete whose start of drying is earlier than 2
days, the values predicted by model GZ are larger than the experimental values.
800
0
-4
%
400
Outside limits (f'c(28) > 80MPa)
600
○ : t0＝1day
● : t0＝28days
500
+4
0%
○ : JSCE data
□ : RILEM data
Experimental shrinkage strain - µ
700
Within limits
+4
0%
1,200
400
0%
-4
300
200
100
Number of data sets = 8
0
0
0
400
800
Calculated shrinkage strain -
0
1,200
µ
100
200
300
400
500
600
Figure 3 Precision of the JSCE Model
700
µ
- 151 -
1,000
750
- 40
%
500
250
○ : JSCE data
□ : RILEM data
750
500
%
- 40
250
●: f'c(28) > 90MPa
0
0
0
250
500
750
1,000
0
1,250
500
750
1,000
µ
Figure 6 Precision of Model CEB
Experimental shrinkage strain -
1,000
750
%
- 40
500
○ : RILEM data
□ : CEB data
◇ : JSCE data
250
0
0
250
500
750
1,000
750
500
%
- 40
250
●: f'c(28) > 70MPa
0
0
1,250
250
500
750
1,000
µ
µ
+4
0
Outside limits
µ
+4
0%
µ
Within limits
%
1,000
1,250
Figure 7 Precision of Model B3
1,250
+4
0
µ
Within limits
1,000
750
- 40
500
%
○ : RILEM data
□ : CEB data
◇ : JSCE data
250
%
1,250
250
500
750
1,000
1,000
750
- 40
%
500
250
●: t0 < 2days
0
0
0
Outside limits
○: f'c(28) > 70MPa
□: f'c(28) < 20MPa
+4
0%
250
µ
0%
Outside limits
+4
+4
1,000
Within limits
µ
0%
1,250
0
1,250
250
500
750
1,000
µ
Figure 9 Precision of Model GZ
- 152 -
1,250
µ
Each of the models described above is able to predict shrinkage strain reasonably well when the
data is within its applicable range. However, out of this range, the capabilities of the models vary.
The JSCE model overestimates shrinkage for high-strength concrete when the age at the start of
drying is 28 days. On the other hand, predictions by the other models are smaller than the
experimental values in the case of high-strength concrete. Thus, prediction equations used for
normal-strength concrete cannot be applied as is to high-strength concrete.
2.4 Accuracy of creep prediction equations
Figure 11 is a comparison of experimental values of specific creep extracted from the databases
and predictions by the JSCE model. The figure includes data for concrete whose 28-day strength
is greater than 100 N/mm2. Figure 12 shows a case where the predictions by the JSCE model are
extremely unreasonable. The JSCE model takes account of factors representing the concrete
mixture, specimen size, and environmental conditions. It does not include a strength factor. Even
where other creep models that include a strength factor are able to predict creep well, the JSCE
model sometimes gives unreasonable results.
Figure 13 compares the experimental specific creep strain with predictions by model CEB. Data
within the applicable range of model CEB was used for the comparison. Figure 14 is a similar
comparison but for data outside the applicable range of model CEB. As is clear from these figures,
model CEB is able to predict creep well even outside the usual range.
○ : RILEM data
□ : CEB data
◇ : JSCE data
200
150
0%
-4
100
50
0
0
50
100
150
Calculated specific creep strain -
200
250
µ/(N/mm 2 )
500
○ : RILEM data
□ : CEB data
+4
0%
µ/(N/mm 2 )
+4
0%
250
Experimental specific creep strain -
µ/( N/mm 2 )
Figure 15 compares experimental creep data with predictions data by model B3. Data within the
applicable range of model B3 was used for comparison. The difference between experimental
values and predictions appears large even within the applicable range. Figure 16 shows a similar
comparison for data outside the applicable range of model B3. Here, data for concrete whose 28day strength is greater than 70 N/mm2 is indicated by ●. It is clear that model B3 is able to predict
the creep of high-strength concrete with accuracy.
400
300
0%
-4
200
100
0
0
100
200
300
400
500
µ/(N/mm 2 )
- 153 -
150
100
50
%
○ : RILEM data
□ : CEB data
◇ : JSCE data
0
0
50
100
150
200
µ/(N/mm 2 )
Outside limits
150
100
%
- 40
50
●: f'c(28) > 90MPa
0
0
µ/(N/mm 2 )
%
+4
0
Experimental creep compliance -
µ/(N/mm 2 )
Experimental creep compliance -
200
150
%
- 40
100
○ : RILEM data
□ : CEB data
◇ : JSCE data
0
0
50
100
150
200
150
200
µ/(N/mm 2 )
250
50
100
Within limits
50
250
Calculated creep compliance - µ/(N/mm 2 )
250
Outside limits
+4
0%
- 40
200
+4
0%
µ/(N/mm 2 )
Within limits
+4
0%
µ/(N/mm 2 )
200
200
150
- 40
%
100
50
●: f'c(28) > 70MPa
Number of data sets= 52
0
0
50
100
150
200
250
Calculated creep compliance - µ/(N/mm 2 )
Figure 17 compares experimental creep data with predictions by model GZ for data within the
applicable range of model GZ. Model GZ tends to underestimate creep even for data within the
applicable range. On the other hand, Figure 18 shows a similar comparison for data that is out of
model GZ's applicable range. Symbol ● represents creep data for concrete whose strength at 28
days is above 70 N/mm2. Symbol □ represents creep data for concrete for which drying began
within 2 days. From this figure, it is clear that model GZ tends to overestimate creep strain when
the concrete strength is high or the first application of loading is early. Despite that, the accuracy of
model GZ is as good as any of the others.
All the models are able to predict creep to an accuracy of ±40%, and it is clear that each
prediction equation offers reasonable predictions of creep from the range of concrete strength from
normal to high. However, in general, the prediction equations for creep are more complicated than
those for shrinkage strain.
- 154 -
150
- 40
%
100
○ : RILEM data
□ : CEB data
◇ : JSCE data
50
0
0
50
100
150
200
250
µ/(N/mm 2 )
250
+4
0%
µ/(N/mm 2 )
200
Within limits
+4
0%
µ/(N/mm 2 )
250
Outside limits
200
150
%
- 40
100
50
●: f'c(28) > 70MPa
□: t' < 2days
Number of data = 135
0
0
50
100
150
200
250
µ/(N/mm 2 )
3. PROPOSED NEW SHRINKAGE PREDITION EQUATION
3.1 Effect of concrete strength on ultimate shrinkage strain
Figure 19 shows the relationship between ultimate shrinkage strain and compressive strength at
the start of drying. By regression, the development of shrinkage strain with time fits the hyperbola
expressed by equation (3).
εsh (t,t 0 ) =
εsh∞ ⋅ (t − t 0 )
β + (t − t 0 )
(3)
µ
1,000
900
Ultimate shrinkage strain -
µ
Where, εsh∞ : ultimate shrinkage strain ( µ ), β : the term representing development of shrinkage
strain, t: current age of concrete (days), t0: the start of drying (days).
800
0%
+4
750
500
0%
-4
250
R.H.=60~68%
100 ×100 ×400mm prism
0
0
30
60
90
120
Figure 19 Relationship between
shrinkage and concrete strength
Gravel Type-A
(water absorption: 1.12%)
Gravel Type-B
700
600
500
Gravel Type-C
400
300
200
10
2
20
30
40
50
60
70
Water-cement ratio - %
ultimate
Figure 20 Relationship
shrinkage and W/C
- 155 -
between
ultimate
µ
Ultimate s hrinkage strain - µ
1,400
1,200
1,000
558
800
613
390
600
445
334
400
279
500
200
100
223
125
150
175
200
Cement content
(kg/m 3 )
225
250
1,000
W=220kg/・
750
W=200kg/・
W=180kg/・
500
250
275
Unit water content - kg /m 3
70N/mm 2
50N/mm 2
0
0.00
0.01
0.02
0.03
20N/mm2
0.04
0.05
(Strength at start of drying)
Figure 21 Effect of cement and water content
on ultimate shrinkage
Figure 22 Relationship between
shrinkage and concrete strength
0.06
0.07
-1
ultimate
As this makes clear, the greater the concrete strength at start of drying, the lower the ultimate
shrinkage strain. When the relationship between ultimate shrinkage strain εsh∞ and the concrete
strength at the start of drying is fitted to the hyperbola, every data point εsh∞ is within ±40% of the
value on the curve. Many conventional prediction equations make use of this fitting [10][11].
However, as is clear from the results shown in Figure 20, the effect of concrete strength on
ultimate shrinkage strain is small when the range of water-cement ratio is between 50% and 60%,
that is, when the concrete strength is low. This result is independent of the type of gravel used.
Other investigations have also found the same result. Figure 21 shows the effect of water content
on the ultimate shrinkage strain of normal-strength concrete [12]. It is clear from this figure that the
effect of water content on ultimate shrinkage strain is much higher than that of water-cement ratio.
The horizontal axis of Figure 22 is the inverse of concrete strength at the start of drying. When
concrete strength is less than 50 N/mm2, the effect of strength is very small and the effect of water
content is much greater. The results given in both Figures 21 and 22 were collected from not only
normal concrete, but also concrete containing a wide range of water contents in order to
investigate the effect of concrete mix proportion. We can conclude that the effect of water content
on ultimate shrinkage strain depends on concrete strength, and that the effect of concrete strength
on ultimate shrinkage strain is much higher than that of water content when the concrete strength
is high.
This situation makes it very difficult to establish a shrinkage prediction equation that takes full
account of all these dependencies, since neither of the databases stores information on concrete
strength at the start of drying, though the 28-day strength was collected. For this reason, a
prediction equation for ultimate shrinkage strain εsh∞ based on 28-day strength was established.
The relationship between ultimate shrinkage strain εsh∞ and age at the start of drying is expressed
by equation (4).
εsh∞ =
Where,
εshρ
1 + ϕ ⋅ t0
(4)
εshρ , ϕ : the optimal values obtained by regression, t0: start of drying (days); however,
t0=98 when t0>98.
- 156 -
1,000
ε shρ
ε sh∞ =
1+ ϕ ⋅ t 0
1000
W=200kg/・
800
Oita Univ.
Okayama Univ.
600
W=220kg/・
750
εsh ρ
µ
Ultimate shrinakge strain -
1200
500
W=180kg/・
250
70N/mm2
400
0
28
56
84
0
0.000
112
Age at start of drying - days
0.005 0.010
50N/mm2
0.015 0.020
0.025 0.030
(C ompressive strength at 28days)
Figure 23 Effect of start of drying
-1
εshρ and concrete strength
Figure 24
1,000
700
W=220kg/m 3
R.H.=68%
200 kg/m 3
750
εsh ρ
εsh ρ
650
500
W=180 kg/m 3
600
250
550
170
180
190
200
210
220
0
0.0
230
0.1
Water content - kg/ m3
Figure 25
0.2
0.3
0.4
0.5
1-R.H./100
εshρ and water content
Figure 26
εshρ and relative humidity
Figure 23 shows a regression example for the relationship between ultimate shrinkage strain
and age at the start of drying. It is clear that equation (4) expresses the relation well.
Figure 24 shows the relationship between
strength. This demonstrates that
εsh∞
εshρ in equation (4) and the 28-day compressive
εshρ can be expressed by a single curve as long as the water
content of the concrete is constant. The curves in Figure 24 were drawn using equation (5).
εshρ =
Cρ 3

Cρ 2 
1+ Cρ1 exp−

 f ' c (28)
(5)
Where, Cρ1 , Cρ 2 , Cρ 3 : the optimal value decided by regression.
Figure 25 shows the relationship between
εshρ in equation (4) and water content in the case of
normal-strength concrete whose 28-day strength is less than 40 N/mm2. Further, Figure 26 shows
the relationship between εshρ and relative humidity of the atmosphere. Both relationships can be
regarded as linear. Equation (6) is derived from this finding. The coefficients in this equation were
obtained by regression from experimental data.
- 157 -
0.14
0.011
Data: RILEM
0.12
0.010
0.009
0.08
0.008
ϕ
ϕ
0.10
0.06
0.007
0.04
0.006
0.02
0.005
0.00
75
Strength at 28 days -
Figure 27
100
0.004
170
125
N/mm 2
Figure 28
1,000
+4
0
%
Data : RILEM
750
%
- 40
250
d ata=52
= 52
NumberNumber
of dataofsets
0
0
250
500
750
1,000
Calculated ultimate shrinkage strain - µ
Figure 29 Precision of new model for ultimate
shrinkage strain
εshρ =
190
200
210
220
230
ϕ and concrete strength
500
180
1,500
ϕ and water content
Data : JSCE
Mesured in Japan
OKADAI ('80s)
+4
0%
50
Experimental ultimate shrinkage strain - µ
25
Experimental ultimate shrinkage strain - µ
Data: RILEM
1,000
%
- 40
500
NumberNumber
of dataofsets
d ata=310
= 310
0
0
500
1,000
1,500
Calculated ultimate shrinkage strain - µ
Figure 30 Precision of new model for ultimate
shrinkage strain
α (1− h)W
 500 
1+ 150exp−

 f ' c (28) 
(6)
Where, α : coefficient depending on the type of cement, h: relative humidity as a decimal, W: water
content (kg/m3), f 'c (28) : 28-day concrete strength in compressive mode. When Japanese normal
Portland cement is used, α is equal to 11.
Figure 27 shows the relationship between factor ϕ in equation (4) and 28-day concrete strength.
As is clear from this figure, the greater the concrete strength at 28 days, the higher the factor ϕ .
Factor ϕ is also affected by the water content of concrete, as shown in Figure 28, with the
relationship between the two being linear. This understanding leads to the following equation
(where the coefficients were obtained by regression from experimental data):
ϕ = 10−4 {15exp(0.007 f 'c (28)) + 0.25W }
Where, f 'c (28) : concrete compressive strength at 28 days, W: water content (kg/m3).
- 158 -
(7)
β
V/S=25mm
45
t0=14days
40
t0=28days
35
30
t0=56days
25
20
170
180
190
200
210
220
230
70
Shrinkage development term -
β
Shrinkage development term -
50
Data: RILEM
50
30
10
0
40
80
120
160
Volume-surface ratio - mm
Figure 31 β and water content
Figure 32 β and volume surface ratio
The vertical axes of both Figure 29 and Figure 30 are the experimental values of ultimate
shrinkage strain. The horizontal axes are the calculated ultimate shrinkage strain using equations
(4), (5), (6), and (7). The data in Figure 29 are from the European database, while those in Figure
30 are from the Japanese one. In equation (6), the coefficient α for the type of cement is set as
follows: Japanese normal Portland cement and slow-hardening Portland cement: α =11, Japanese
rapid-hardening cement: α =15, European early hardening and high-strength cement: α =11,
European normal and rapid-hardening cement: α =10, European slow-hardening cement: α =8.
These values of coefficient α were obtained by regression from the experimental data. In fact,
coefficient α also includes the effect of aggregate types used in different countries and other
factors in addition to the concrete type.
It is clear from the figures presented here that the difference between experimental ultimate
shrinkage strains and shrinkage strains calculated using equations (4), (5), (6), and (7) is mostly
within ±40%, and that the proposed prediction equation for ultimate shrinkage strain is very
accurate.
3.2 Development of shrinkage strain
Figure 31 plots coefficient β for the development of shrinkage strain against concrete water
content. The relationship is linear, and the effect of water content on β is small when the start of
drying is late; that is, at 56 days. On the other hand, Figure 32 plots the volume surface ratio of the
specimen against coefficient β  The greater the volume surface ratio of specimen, the bigger β 
These findings are expressed by the following equation:


C
⋅ W + Cβ 3  ⋅ (V /S ) β 4
1+ Cβ 2 ⋅ t 0

β = 
Cβ 1
(8)
Where, W: water content (kg/m3), V/S: volume surface ratio of the specimen (mm), Cβ 1 , Cβ 2 , Cβ 3 ,
and Cβ 4 : coefficients obtained by regression.
Using regression to obtain coefficient Cβ 3 yields zero. The following equation was obtained.
β=
4W V / S
100 + 0.7t 0
- 159 -
(9)
1,000
%
-20
500
-40
%
250
0
250
500
750
+4
0%
+2
0
750
0
750
500
- 40
0
0
250
500
750
1,000
µ
Figure 33 Precision of new prediction equation
for shrinkage
%
250
1,000
µ
for shrinkage
1,500
+4
0%
Data : CEB
1,000
%
- 40
500
Data : JSCE
+4
0%
1,250
µ
Data : RILEM
µ
%
+4
0%
µ
1,000
1,000
0
750
- 40
500
%
250
0
0
500
1,000
1,500
0
µ
250
500
750
1,000
for shrinkage
1,250
µ
for shrinkage
3.3 Accuracy of proposed new prediction equation for shrinkage strain
Figures 33 to 36 compare the experimental shrinkage strain with values calculated using the new
prediction equation proposed in this study, as discussed above. Overall, the discrepancy between
measured and calculated values is within approximately ± 40%, so the proposed prediction
equation is very accurate and applicable to a wide range of concrete strengths from normal to
high-strength concrete.
4. PROPOSED NEW CREEP PREDITION EQUATION
4.1 Effect of concrete strength on ultimate creep
Figure 37 shows the relationship between ultimate specific creep and compressive strength at
initial load application. By regression, the development of specific creep with time fits the
hyperbola expressed by equation (10).
- 160 -
µ/(N/mm 2 )
Data : RILEM
○ : R.H.=48~50%
□ : R.H.=65~75%
● : R.H.=99~100%
100
50
0
0
25
50
75
100
Compressive strength at loading -
125
N/mm 2
µ/(N/mm 2 )
200
+2
0%
Data : CEB
■ : Gravel Type-A
▲ : Gravel Type-B
● : Gravel Type-C
80
60
40
20
W=140kg/m 3
0
1.0
2.0
3.0
4.0
5.0
Cement-water ratio
Figure 38 Relationship between ultimate creep
and cement water ratio
150
200
Data : CEB
150
0%
-2
100
50
0
0
50
100
150
200
µ/(N/mm 2 )
Figure 39 Accuracy when fitted to hyperbola
µ/(N/mm 2 )
Figure 37 Relationship between ultimate creep
and compressive strength
100
+2
0%
150
Ultimate specific creep -
µ/( N/mm 2 )
200
0
-2
%
100
50
0
0
50
100
150
200
µ/(N/mm 2 )
Figure 40 Accuracy when fitted to logarithmic curve
Cr(t,t') =
Cr∞ ⋅ (t − t')
β cr + (t − t')
(10)
Where, Cr∞ : ultimate specific creep ( µ /N/mm2), β cr : term representing the development of specific
creep, t: current age of concrete (days), t’: time at initial load application (days).
This figure clearly shows that the greater the concrete strength, the smaller the ultimate specific
creep. The variance from average is also small at high concrete strengths. The same result was
obtained from an experiment with a different type of gravel. Figure 38 shows that the ultimate
specific creep becomes small when the water-cement ratio is small and concrete strength is high.
4.2 Development of creep
Figure 39 shows the results of fitting the specific creep to the hyperbola in equation (10). Similarly,
Figure 40 shows the results of fitting specific creep to the logarithmic equation (11).
- 161 -
20
R.H. = 60~65%
Coefficient A - µ/(N/mm 2 )
40
30
+40%
20
10
-40%
0
0
20
40
60
80
R.H. = 99~100%
15
10
100
120
Strength at application of load - N/mm 2
+40%
5
-40%
0
0
25
50
75
Strength at application of load -
Figure 41 Relationship between coefficient A
100
125
N/mm 2
Cr(t,t') = Alog e (t − t' +1)
(11)
Where, A: coefficient obtained by regression, t: age of concrete (days), t’: age at initial load
application (days).
The discrepancy between experimental values of specific creep and the hyperbolic and logarithmic
curves fitted to specific creep by regression is basically within ± 20%. However, dispersion
increases as time passes when the hyperbola is used for regression. On the other hand, when the
logarithmic curve is used, the discrepancy with experimental data is large at an early stage.
However, at later times, the regressed value fits the experimental data more closely. More
importantly, just one coefficient is required to express the development of specific creep when the
logarithmic curve is. It is thus very simple and convenient to establish a creep prediction equation.
Figures 41 and 42 show the relationship between coefficient A in equation (11) and concrete
strength at initial load application. Figure 41 is for data measured at relative humidity levels
between 60% and 65%. In Figure 42, on the other hand, the data were measured at relative
humidity levels over 99%. It is clear from these figures that the relationship between coefficient A
and concrete strength can be expressed by hyperbola. The lines fitted to the data in these figures
are based on equation (12).
A=
Cα
Cβ + f 'c (t)
(12)
Where, Cα , Cβ : coefficients obtained by regression, f 'c (t ): concrete compressive strength at
initial load application (N/mm2).
Figure 43 is a plot of coefficient A in equation (11) against the relative humidity of the atmosphere.
The relationship is clearly linear. The relationship between coefficient A and water content is also
linear, as seen in Figure 44. Equation (13) expresses this situation; when the relative humidity is
100%, then h=1 and the equation becomes that for basic creep. Thus, equation (13) separates the
development of drying creep from the phenomenon of basic creep.
A=
Cdrying ⋅ W (1− h )
ϕ drying + f 'c (t)
+
- 162 -
Cbasic
ϕ basic + f 'c (t)
(13)
25
25
20
f'c(t)=30~40 N/mm2
15
f'c(t)=50~60 N/mm2
10
f'c(t)=70~80 N/mm2
5
0
0.0
f'c(t)=100~110 N/mm2
0.1
0.2
0.3
0.4
0.5
R.H. = 60~65%
20
R.H. = 45~50%
15
10
R.H. = 70~80%
5
0
100
0.6
120
140
160
180
200
220
240
Unit water content - kg/m 3
1-R.H./100
and relative humidity
and water content of concrete
Where, Cbasic , Cdrying , ϕ basic , ϕ drying : coefficients obtained by regression, h: relative humidity as a
decimal, W: water content (kg/m3), f 'c (t ): compressive concrete strength (N/mm2).
Obtaining every coefficient in equation (13) by regression shows that
ϕ drying . The following equation can be derived as a consequence:
Equation (14) is obtained by regression using
the many data given in Figure 2. Factors with
little influence were eliminated. For example,
the last JSCE model includes the effect of
specimen size and shape. However, the new
creep prediction equation proposed in this
study does not include this influence because
regression showed it to be small. Figure 45
shows the effect of volume surface ratio of the
specimen on the predicted ultimate specific
creep by model CEB and the JSCE model.
The predicted value at a volume surface ratio
of 100 mm is just 1.2 times bigger than that for
a ratio of 1,000 mm, even when model CEB is
used. This difference is even smaller when the
last JSCE model is used. Models B3 and GZ
include the effect of specimen size in the factor
for creep development. This factor was
excluded from the new creep prediction
equation for the purpose of simplification.
µ/(N/mm 2 )
4W (1 − h ) + 350
12 + f 'c (t)
A=
ϕ basic is almost the same as
(14)
80.0
f'c(28)=50MPa, R.H.=80%
W=180kg/m 3 , W/C=40.0%
t'=3days
75.0
70.0
CEB model
65.0
JSCE model
60.0
0
250
500
750
1,000
1,250
V/S - mm
Figure 45 Relation between ultimate creep and
volume surface ratio
4.3 Accuracy of proposed new prediction equation for specific creep
Figures 46 to 49 compare experimental measurements of specific creep with calculated values
obtained by the new prediction equation proposed in this study. The discrepancy between
measured values and predictions is generally within ±40%, so the proposed prediction equation is
very accurate and applicable to a wide range of concrete strengths from normal to high. The new
- 163 -
equation is extremely simple, yet its accuracy is as good as that of the major conventional
equations.
5. NEW PREDICTION EQUATIONS PROPOSED IN THIS STUDY
To summarize, the newly proposed prediction equations for drying shrinkage are as follows:
100
%
- 40
50
0
(16)
100
150
µ/(N/mm 2 )
µ/(N/mm 2 )
250
+4
0%
Data : CEB
200
150
- 40
100
%
50
0
0
50
100
150
200
Data : RILEM
200
150
%
- 40
100
50
0
0
50
100
150
200
250
µ/(N/mm 2 )
Figure 47 Precision of proposed model
250
µ/(N/mm 2 )
µ/(N/mm 2 )
250
150
Data : JSCE
+4
0%
50
εshρ
1 + η ⋅ t0
µ/(N/mm 2 )
+4
0%
150
0
(15)
µ/(N/mm 2 )
εsh∞ =
εsh∞ ⋅ (t − t0 )
β + (t − t0 )
+4
0%
εsh (t,t0 ) =
100
%
- 40
50
0
0
50
100
150
µ/(N/mm 2 )
- 164 -
εshρ =
α (1− h)W
 500 
1+ 150exp−

 f ' c (28) 
η = 10−4 {15exp(0.007 f 'c (28)) + 0.25W }
β=
Where
4W V / S
100 + 0.7t 0
(17)
(18)
(19)
εsh (t,t 0 ): drying shrinkage strain ( µ ), t, t0: current age and age at drying (days), when age
at drying > 98 days, t0=98. f 'c (28) : compressive strength at 28 days (N/mm2) ( f 'c (28) <120
N/mm2), h: relative humidity of atmosphere (0.4<h<0.9), W: unit water content (130<W<230 kg/m3),
V/S: volume surface ratio of specimen (100<V/S<1,000mm), α : factor accounting for cement type
(Japanese data: α = 11 for normal cement and α = 15 for rapid-hardening cement; Western data:
α = 10 for normal Portland cement and α = 8 for slow-hardening cement).
The newly proposed prediction equation for creep is as follows:
Cr(t,t') =
4W (1− h ) + 350
⋅ log e (t − t'+1)
12 + f ' c (t')
(20)
Where, Cr(t,t'): specific creep ( µ /N/mm2), h: relative humidity of the atmosphere (0.4<h<0.9), W:
unit water content (130<W<230 kg/m3), t’: time at initial load application (days) (t’>1), f 'c (t ) :
compressive strength at age t (N/mm2) ( f 'c (t )<120N/mm2).
6. CONCLUSION
New prediction equations for creep and shrinkage have been proposed and their accuracy
investigated. The equations are based on a model established by statistical methods. The
investigation results demonstrate that the proposed equations are able to predict concrete creep
and shrinkage strain to a certain degree of accuracy.
The new shrinkage equation takes into account a number of influences. In particular, it takes
account of the finding that the effect of water content on ultimate shrinkage strain depends on
concrete strength, and that the effect of concrete strength on ultimate shrinkage strain is much
higher than that of water content when the concrete strength is high.
The proposed creep equation is very simple, and specific creep can be predicted on the basis of
just three items of data: compressive strength, relative humidity of the atmosphere, and water
content. Despite this simplicity, it is able to predict the specific creep of concrete from normal
strength to high strength with accuracy equivalent to that of typical conventional prediction
equations.
The proposed prediction equations offer a simple design procedure for calculating creep and
shrinkage using information available at the design stage.
References
[1] abc, edf: dog cat pen abc, abifdasd fdsafda fdsadas. Fdsafdsa fdassdf ise, dfasdfa fdasfda,
fdsafd fdsaf fdsafa fda, 2002
- 165 -
[2] ACI Committee 209: Prediction of creep and shrinkage, and temperature effects on concrete
structures, American Concrete Institute Special Publication No.76, 1982.
[3] JSCE: Standard specification for design and construction of concrete and structures, 1996.
[4] RILEM TC 107 Subcommittee 5: Database on creep and shrinkage tests - 3.0, 1999.
[5] Dilger, W. H.: Private communication.
[6] JSCE committee 308: Concrete shrinkage and creep, Concrete Engineering Series, N0.24,
pp.56~93, 1997 (in Japanese).
[7] Comite Euro-International du Beton: CEB-FIP MODEL CODE 1990 (DESIGN CODE), Thomas
Telford, pp.52~65, 1998.
[8] Bazant, Z. P. and Baweja, S.: Creep and shrinkage prediction model for analysis and design of
concrete structures - Model B3, Materials and Structures, RILEM, Paris, France, Vol. 28,
pp.357~365, 1995.
[9] Gardner, N. J.: Design Provisions for Shrinkage and Creep of Concrete, Proceedings of
International Conference on Engineering Material, Ottawa, Canada, Vol.1, pp.647~657, 1997.
[10] CEB-FIP: International Recommendations for the Design and Construction of Concrete
Structures - Principles and Recommendations, Comite Europeen du Beton - Federation
Internationale de la Precontrainte, FIP Sixth Congress, Prague, June 1970; published by Cement
and Concrete Association: London, 1970.
[11] Bazant, Z. P. and Panula, L.: Simplified prediction of concrete creep and shrinkage from
strength and mix, Structural Engineering Reports No.78-10/6405, Department of Civil Engineering,
Northwestern University, Evanston, Illinois, pp.24, Oct. 1978.
[12] Kiyoshi Okada, Toyoki Akashi, and Wataru Koyanagi, Construction Materials for Civil
Engineering, Kokumin Kagaku-sya, p.193, 1998 (in Japanese).
- 166 -

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Transcription

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