STUDY OF THERMAL CRACKS

IN CONCRETE STRUCTURES USING PROBABILITY THEORY

Masami IshikawaTohoku Gakuin University, Japan

Masahiro YurugiFormer Prof. of Hirosaki University, Japan

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

0

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

Cra

ck

pro

bab

ility

(%)

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 on design proccese

Thermal stress, Drying shrinkage, etc.

Crack width and Crack Index

0

0.1

0.2

0.3

0.4

0.5

0.2 0.4 0.6 0.8 1 1.2 1.4

Cra

ck

wid

th (

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=300C=300C=250C=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 experimentit can not say that this graph applicable for all cases

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 8006000

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.

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

0Age

Stresses

Tensile strength

Thermal stress

Cracking

0Age

Stresses

Tensile strength =Crack Index

Thermal stress

Data UnitCollected data Input data

Average Standard deviation Average Standard

deviationHeat conduction W/(m·°C) 2.85 0.375 2.89 0.365Heat 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 wallplacement)

°C 21.05 1.06 21.24 1.05

Placing Temp. (Wall part) °C 24.05 1.06 24.23 0.94

Adiabaticheatingparameter

For placing temp. 20 °C

Q∞ °C 48.39 4.70 48.18 4.70α ― 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 0 2300 0Parameter: d of Eqs. 8 ― 5.17 0.395 5.25 0.323Poisson’s ratio ― 0.18 0 0.18 0Parameter: c of Eqs. 7 ― 0.30 0.031 0.298 0.028Thermal expansion coeff. ×10-6/℃ 9.96 0.84 10.059 1.01

Material properties

Input data sets

Comp. strength

Ultimate adiabatictemperature increase

Heat convection

etc.

Comp. strength 41.7

Ultimate adavtic, 48.4

Heat Convection 2.85

…… ……

Placing temp. 24.0

Input data set-1

Comp. strength 39.5

Ultimate adavtic, 45.2

Heat Convection 2.93

…… ……

Placing temp. 22.3

Input data set-2

Comp. strength 40.4

Ultimate adavtic, 47.1

Heat Convection 2.65

…… ……

Placing temp. 24.0

Comp. strength 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 of the fifty values

・ Create the fifty sets of data with fifty different combinations

Construction Schedule

KyotoOsaka 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

RebarTruss 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

0

10

20

30

40

50

0 10 20 30 40 50 60

Tem

p.(℃

）

Age(days）

Upper

Middle.

Lower

Temperature history (September)

-1.0-0.50.00.51.01.52.02.53.0

0 10 20 30 40 50 60

Stress (N/mm2)

Age(days）

Upper

Middle

Lower

Stress history (September)

2.25mCL

0.5m

Crack induced joint

5m 5m 5m

1.5m

Output point for crack width

Evaluation point of thecrack index

0

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

Cra

ck

proba

bilit

y(%

)

Crack Index: Icr (Safty factor:γcr)

0.71(May)

0.78(Sept.)Average ofCrack 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%

xMay

1.00.78

93%

xSept.

Evaluation of crack probability

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 0.18Crack 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%

-0.1mm 0.2mm-0.2mm1σ-2σ -1σ 2σ

95%

95% of the data lie within the range of deviation of ±0.2 mm.

Relationship Crack Index and Crack width

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.4 0.6 0.8 1.0 1.2

Cra

ck

wid

th（m

m)

Crack Index

The regression line ： y = −0.444x + 0.734The 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=0.290-0.04

Effectiveness of rebar on crack width

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Cra

ck

wid

th(m

m)

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 selectedfrom the calculations of the 50 sets September construction cases.

・ Sensitivity analyses for eight levels of rebar were carried out.

・ In order to control the crack width to 0.3 mm, it is necessary to raisethe crack index to 0.9 or more for wall type structure with 0.13% reinforcement ratio.

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 concretewith 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 xand 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 indexto approximately 0.9 for a wall-type structure with a reinforcement ratio of approximately 0.13%.