STUDY OF THERMAL CRACKS
IN CONCRETE STRUCTURES USING PROBABILITY THEORY
Masami Ishikawa
Tohoku Gakuin University, Japan
Masahiro Yurugi
Former Prof. of Hirosaki University, Japan
JSCE Standard Spec.
for concrete structures
Definition of Crack Index by JSCE standard spec(2013).
Crack Index
Thermal stress,
Drying shrinkage, etc.
Maximum principal
tensile stress
Icr(t) = ftk(t) / σt(t)
100
Tensile strength
In case of prevent the occurrence of cracks,
ensure the crack index of 1.85
If the value 1.85 can not satisfied,
Crack probability (%)
Description of the JSCE spec.
Check the crack risk
on design proccese
90
It will allow the occurrence of cracks
80
70
60
50%
50
40
30
20
5%
10
0
0.0
Check the crack width
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Crack Index: Icr (Safty factor:γcr)
2.0
Crack width and Crack Index
JSCE recommendation
・Large crack width affects durability, water-tightness, and the aesthetics of the structure
・Predict the crack width in advance by FEM etc.
If it is difficult to calculate the crack width by simulation, the following
relation available
Relationship between Crack index and Crack width due to steel ratio in JSCE Spec.
0.5
p=0.25
0.4
0.55%～0.65%
300
300
300
C=250 C=300
C=300
0.85%～0.95%
0.3
1000
0.2
C=280
0.1
0
5000
0.2
0.4
0.6
0.8
1
Crack Index Icr
1.2
1.4
Metropolitan Expressway
company Ltd., 1984
This relationship is based on the results of one particular experiment
it can not say that this graph applicable for all cases
1500
Crack width (mm)
0.25%～0.30%
300
Purpose of this study
Crack Index : Icr
The crack occurrence can
be regarded as a stochastic
phenomenon
The crack width should be also
discussed using probability theory
Crack width
Crack Probability
30000
15000
700
（mm）
5000
800
800
A standard design from the Ministry of
Land, Infrastructure and Transport of Japan.
6000
800
The wall-type structure for crack width
calculations
Calculation procedure
Calculate temperature distributions
Thermal stress calculation
without taking account
cracks
Stresses
Tensile
strength
0
Calculate the thermal
stress taking crack
generation into account
Stresses
Thermal
stress
Cracking
=Crack Index
Thermal
stress
0
Age
Age
Crack Index
Tensile
strength
Crack width
The relationship is obtained
Material properties
Data
Unit
Collected data
Standard
Average
deviation
2.85
0.375
14.0
0.7
Input data
Standard
Average
deviation
2.89
0.365
13.94
0.61
Heat conduction
Heat convection
W/(m·°C)
W/(m2·°C)
Heat capacity
kJ/(kg·°C)
1.15
0.025
1.148
0.025
wall
°C
21.05
1.06
21.24
1.05
Q∞
α
°C
°C
―
24.05
48.39
1.054
1.06
4.70
0.27
24.23
48.18
1.028
0.94
4.70
0.266
°C
―
―
48.61
4.71
Ambient Temp. (After
placement)
Placing Temp. (Wall part)
Adiabatic
heating
parameter
For placing
temp. 20 °C
Q∞
Correction
value at each α
placing temp.
Compressive strength at 28 days
―
―
―
1.211
0.269
N/mm2
41.7
3.336
41.41
3.09
Density
Parameter: d of Eqs. 8
kg/m3
―
2300
5.17
0
0.395
2300
5.25
0
0.323
―
―
×10-6/℃
0.18
0.30
9.96
0
0.031
0.84
0.18
0.298
10.059
0
0.028
1.01
Poisson’s ratio
Parameter: c of Eqs. 7
Thermal expansion coeff.
Input data sets
Input data set-1
Comp. strength
Comp. strength
41.7
Ultimate adavtic,
48.4
Heat Convection
2.85
……
……
Placing temp.
24.0
・Fifty values were generated by the
Monte Carlo method with normally
distributed random numbers
・ Selected one value from each
of the fifty values
Input data set-2
Ultimate adiabatic
temperature increase
Comp. strength
39.5
Ultimate adavtic,
45.2
Heat Convection
2.93
……
Placing temp.
Input ……
data set-…
22.3
Comp. strength
Ultimate adavtic,
Heat convection
・ Create the fifty sets of data with
fifty different combinations
Heat Convection
……
Placing temp.
40.4
47.1
Input data set-50
2.65
Comp. strength
……
Ultimate adavtic,
24.0
Heat Convection
……
etc.
Placing temp.
41.7
48.4
2.85
……
24.0
Construction Schedule
Season
Spring
Autumn
Bottom Sabs
May 1st
September 1st
May 15th
September 15th
Wall and top Slabs
End of calculation
October 31
Concrete Structure model was
assumed to be located in the
Aomori city
Concrete was cured for five days
Proportions of the concrete mix.
Type of cement
Blast furnace B type
Cement content
300
Water content
165 kg/m3
Water-to-cement ratio
Sendai
kg/m3
55%
Osaka Kyoto
Tokyo
Numerical model
The model comprised only one-quarter of the total shape
Bond link elements
500
1250
1250
Rebar
Truss Element
Jount Element
1250
1250
・ Cracks occurred at 5.0m intervals
along the longitudinal direction
・ The tensile strength of the bond
link elements was reduced by 40%
Temperature history (September)
50
Upper
Temp.(℃）
40
Middle.
Lower
30
20
10
0
0
10
20
30
Age(days）
40
50
60
Stress history (September)
Stress (N/mm2)
3.0
2.5
Upper
2.0
Middle
1.5
Lower
1.0
0.5
0.0
-0.5 0
-1.0
10
20
30
Age(days）
40
50
60 joint
Crack induced
Evaluation
point of
thecrack
index
C
2.25m L
Output point
for crack
width
1.5m
0.5m
5m
5m
5m
Average of
Crack Index
Crack probability
Calculation
Standard
Spec.
0.71(May)
0.99
0.96
0.78(Sept.)
0.93
0.84
Crack probability (%)
Evaluation of crack probability
If the crack index can be assumed to be
distributed normally,
the crack occurrence probability,
x
Sept.
May
99%
0.71
0.71(May)
100
0.78(Sept.)
80
60
40
20
0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Crack Index: Icr (Safty factor:γcr)
x
93%
1.0
0.78
1.0
Results of calculations
Month
May
Sept.
Average
Standard
Devi.
Variation
Coeff.
52.04
3.88
0.075
2.74
0.41
0.15
68%
Crack index
0.71
0.125
0.18
95%
Crack width(mm)
0.42
0.095
0.23
43.92
4.49
0.102
2.58
0.385
0.15
Crack index
0.78
0.147
0.19
Crack width (mm)
0.36
0.112
0.31
Output term
Maximum temp.
(℃)
Maximum stress
(N/mm2)
Maximum temp.
(℃)
Maximum stress
(N/mm2)
-2σ
-1σ
-0.2mm -0.1mm
1σ
0.1mm
2σ
0.2mm
95% of the data lie within the
range of deviation of ±0.2 mm.
・The standard deviation of crack width on the wall surface is approximately 0.1 mm.
・ If a crack width of 0.3 mm was obtained from the analysis results,
the range is 0.1～0.5 mm, owing to the fluctuation of material properties.
Relationship Crack Index and Crack width
The regression line ： y = −0.444x + 0.734
The correlation coefficient is 0.770
0.7
−0.444(Icr=0.9) + 0.734=0.334
Crack width （mm)
0.6
−0.444(Icr=1.0) + 0.734=0.290
0.5
-0.04
0.4
0.3
0.2
0.1
0
0.4
0.6
0.8
1.0
1.2
Crack Index
To reduce crack width by 0.04 mm, the crack index should be increased by 0.1.
Effectiveness of rebar on crack width
・ Three cases with crack indexes of 0.5, 0.7, and 0.9 were selected
from the calculations of the 50 sets September construction cases.
・ Sensitivity analyses for eight levels of rebar were carried
out.
0.6
Crack Index 0．92
Crack Index 0．70
Crack Index 0．54
Crack width (mm)
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Reinforcement ratio （％）
・ In order to control the crack width to 0.3 mm, it is necessary to raise
the crack index to 0.9 or more for wall type structure with 0.13% reinforcement ratio.
Conclusions
・ For a culvert box with a wall thickness of 800 mm, and assuming concrete
with a cement content of 300 kg/m3,
The crack index
Av.
September : 0.71
May:
0.78
Std. div.
0.125
0.147
Crack probability.
99%
96%.
・ The standard deviation of the crack width on the concrete surface
was approximately 0.1 mm.
Therefore, 95% of the data lay within ±0.2 mm.
・ The linear regression of the relationship between the crack index x
and the crack width y (mm) was obtained.
y = −0.444x + 0.734
・ In order to limit crack width to 0.3 mm, it is necessary to control the crack index
to approximately 0.9 for a wall-type structure with a reinforcement ratio of
approximately 0.13%.