SLIDE 1
STUDY OF THERMAL CRACKS IN CONCRETE STRUCTURES USING PROBABILITY THEORY
Masami Ishikawa Tohoku Gakuin University, Japan Masahiro Yurugi Former Prof. of Hirosaki University, Japan
SLIDE 2 JSCE Standard Spec. for concrete structures
Definition of Crack Index by JSCE standard spec(2013). Icr(t) = ftk(t) / σt(t)
Crack Index Tensile strength Maximum principal tensile stress
10 20 30 40 50 60 70 80 90 100 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Crack probability (%) Crack Index: Icr (Safty factor:γcr) 50%
In case of prevent the occurrence of cracks, ensure the crack index of 1.85
5%
Description of the JSCE spec.
If the value 1.85 can not satisfied, It will allow the occurrence of cracks
Check the crack width Check the crack risk
Thermal stress, Drying shrinkage, etc.
SLIDE 3 Crack width and Crack Index
0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1 1.2 1.4 Crack width (mm) Crack Index Icr
0.25%～0.30% 0.55%～0.65% 0.85%～0.95%
Relationship between Crack index and Crack width due to steel ratio in JSCE Spec.
300 300 300 300 5000 1500 1000
C=300 C=300 C=250 C=280 p=0.25
Metropolitan Expressway company Ltd., 1984
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 This relationship is based on the results of one particular experiment
it can not say that this graph applicable for all cases
SLIDE 4 Purpose of this study
Crack Index : Icr Crack Probability Crack width
The crack width should be also discussed using probability theory The crack occurrence can be regarded as a stochastic phenomenon
800 800 6000 700 800 5000 30000 15000
（mm）
The wall-type structure for crack width calculations
A standard design from the Ministry of Land, Infrastructure and Transport of Japan.
SLIDE 5 Calculation procedure
Calculate temperature distributions Thermal stress calculation without taking account cracks Calculate the thermal stress taking crack generation into account Crack Index Crack width The relationship is obtained
Age Stresses
Tensile strength Thermal stress Cracking
Age Stresses
Tensile strength
=Crack Index
Thermal stress
SLIDE 6 Data Unit Collected data Input data Average Standard deviation Average Standard deviation Heat conduction W/(m·°C) 2.85 0.375 2.89 0.365 Heat convection W/(m2·°C) 14.0 0.7 13.94 0.61 Heat capacity kJ/(kg·°C) 1.15 0.025 1.148 0.025 Ambient Temp. (After wall placement) °C 21.05 1.06 21.24 1.05 Placing Temp. (Wall part) °C 24.05 1.06 24.23 0.94 Adiabatic heating parameter For placing
Q∞ °C 48.39 4.70 48.18 4.70 α ― 1.054 0.27 1.028 0.266 Correction value at each placing temp. Q∞ °C ― ― 48.61 4.71 α ― ― ― 1.211 0.269 Compressive strength at 28 days N/mm2 41.7 3.336 41.41 3.09 Density kg/m3 2300 2300 Parameter: d of Eqs. 8 ― 5.17 0.395 5.25 0.323 Poisson’s ratio ― 0.18 0.18 Parameter: c of Eqs. 7 ― 0.30 0.031 0.298 0.028 Thermal expansion coeff. ×10-6/℃ 9.96 0.84 10.059 1.01
Material properties
SLIDE 7 Input data sets
Ultimate adiabatic temperature increase Heat convection
etc.
41.7 Ultimate adavtic, 48.4 Heat Convection 2.85 …… …… Placing temp. 24.0
Input data set-1
39.5 Ultimate adavtic, 45.2 Heat Convection 2.93 …… …… Placing temp. 22.3
Input data set-2
40.4 Ultimate adavtic, 47.1 Heat Convection 2.65 …… …… Placing temp. 24.0
41.7 Ultimate adavtic, 48.4 Heat Convection 2.85 …… …… Placing temp. 24.0
Input data set-… Input data set-50
・Fifty values were generated by the Monte Carlo method with normally distributed random numbers ・ Selected one value from each
・ Create the fifty sets of data with fifty different combinations
SLIDE 8
Construction Schedule
Kyoto Osaka Tokyo Sendai
Concrete Structure model was assumed to be located in the Aomori city
Season Spring Autumn Bottom Sabs May 1st September 1st Wall and top Slabs May 15th September 15th End of calculation October 31
Type of cement Blast furnace B type Cement content 300 kg/m3 Water content 165 kg/m3 Water-to-cement ratio 55%
Proportions of the concrete mix. Concrete was cured for five days
SLIDE 9
Rebar Truss Element 500 Jount Element 1250 1250 1250 1250
The model comprised only one-quarter of the total shape ・ Cracks occurred at 5.0m intervals along the longitudinal direction Bond link elements ・ The tensile strength of the bond link elements was reduced by 40%
Numerical model
SLIDE 10
10 20 30 40 50 10 20 30 40 50 60 Temp.(℃） Age(days）
Upper Middle. Lower
Temperature history (September)
SLIDE 11
0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 20 30 40 50 60 Stress (N/mm2) Age(days）
Upper Middle Lower
Stress history (September)
2.25m C
L
0.5m Crack induced joint
5m 5m 5m
1.5m
Output point for crack width Evaluation point of thecrack index
SLIDE 12
20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Crack probability (%) Crack Index: Icr (Safty factor:γcr)
0.71(May) 0.78(Sept.)
Average of Crack Index Crack probability Calculation Standard Spec. 0.71(May) 0.99 0.96 0.78(Sept.) 0.93 0.84
If the crack index can be assumed to be distributed normally, the crack occurrence probability,
0.71 1.0 99% x
May
1.0 0.78 93% x
Sept.
Evaluation of crack probability
SLIDE 13 Month Output term Average Standard Devi. Variation Coeff. May Maximum temp. (℃) 52.04 3.88 0.075 Maximum stress (N/mm2) 2.74 0.41 0.15 Crack index 0.71 0.125 0.18 Crack width(mm) 0.42 0.095 0.23 Sept. Maximum temp. (℃) 43.92 4.49 0.102 Maximum stress (N/mm2) 2.58 0.385 0.15 Crack index 0.78 0.147 0.19 Crack width (mm) 0.36 0.112 0.31
Results of calculations
・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.
0.1mm 68%
0.2mm
1σ
2σ 95%
95% of the data lie within the range of deviation of ±0.2 mm.
SLIDE 14 Relationship Crack Index and Crack width
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.4 0.6 0.8 1.0 1.2 Crack width （mm) Crack Index
The regression line ： y = −0.444x + 0.734
The correlation coefficient is 0.770 To reduce crack width by 0.04 mm, the crack index should be increased by 0.1.
−0.444(Icr=0.9) + 0.734=0.334 −0.444(Icr=1.0) + 0.734=0.290
SLIDE 15 Effectiveness of rebar on crack width
0.1 0.2 0.3 0.4 0.5 0.6
0.2 0.4 0.6 0.8 1 1.2 1.4
Crack width (mm) Reinforcement ratio （％） Crack Index 0．92 Crack Index 0．70 Crack Index 0．54
・ 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
・ 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.
SLIDE 16
Conclusions
The crack index Av. Std. div. Crack probability. September : 0.71 0.125 99% May: 0.78 0.147 96%. ・ For a culvert box with a wall thickness of 800 mm, and assuming concrete with a cement content of 300 kg/m3, ・ 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%.
