STUDY OF THERMAL CRACKS IN CONCRETE STRUCTURES USING PROBABILITY - PowerPoint PPT Presentation

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). Thermal stress, Maximum principal Crack Index Drying shrinkage, etc. tensile stress I cr (t) = f tk (t) / σ t (t) 100 Tensile strength Check the crack risk 90 on design proccese 80 Crack probability (%) Description of the JSCE spec. 70 60 In case of prevent the occurrence of cracks, 50% 50 ensure the crack index of 1.85 40 30 If the value 1.85 can not satisfied, 20 5% It will allow the occurrence of cracks 10 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Check the crack width Crack Index: Icr (Safty factor:γ cr )

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. 300 0.5 p=0.25 0.25%～0.30% 0.4 0.55%～0.65% Crack width (mm) 300 300 300 0.85%～0.95% 0.3 1000 0.2 C=280 C=250 C=300 C=300 1500 0.1 0 5000 0.2 0.4 0.6 0.8 1 1.2 1.4 Metropolitan Expressway Crack Index Icr 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

Purpose of this study Crack Index : I cr The crack occurrence can The crack width should be also be regarded as a stochastic discussed using probability theory phenomenon Crack width Crack Probability 30000 15000 700 （mm） 5000 A standard design from the Ministry of Land, Infrastructure and Transport of Japan . 800 800 6000 800 The wall-type structure for crack width calculations

Calculation procedure Calculate temperature distributions Thermal stress calculation Calculate the thermal without taking account stress taking crack cracks generation into account Stresses Stresses Thermal Cracking stress Tensile Tensile =Crack Index strength strength Thermal stress 0 0 Age Age Crack width Crack Index The relationship is obtained

Material properties Collected data Input data Data Unit Standard Standard Average Average deviation deviation Heat conduction 2.85 0.375 2.89 0.365 W/(m· ° C) Heat convection W/(m 2 · ° 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 ° C 21.05 1.06 21.24 1.05 placement) Placing Temp. (Wall part) ° C 24.05 1.06 24.23 0.94 Q ∞ ° C 48.39 4.70 48.18 4.70 For placing α ― 1.054 0.27 1.028 0.266 temp. 20 ° C Adiabatic heating Q ∞ ° C ― ― 48.61 4.71 Correction parameter value at each α ― ― ― 1.211 0.269 placing temp. Compressive strength at 28 days N/mm 2 41.7 3.336 41.41 3.09 Density kg/m 3 2300 0 2300 0 Parameter: d of Eqs. 8 ― 5.17 0.395 5.25 0.323 Poisson’s ratio ― 0.18 0 0.18 0 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

Input data sets Input data set-1 Comp. strength Comp. strength 41.7 ・ Fifty values were generated by the Monte Carlo method with normally Ultimate adavtic, 48.4 distributed random numbers Heat Convection 2.85 …… …… ・ Selected one value from each Placing temp. 24.0 of the fifty values Input data set-2 Ultimate adiabatic temperature increase Comp. strength 39.5 ・ Create the fifty sets of data with fifty different combinations Ultimate adavtic, 45.2 Heat Convection 2.93 …… Input data set-… …… Placing temp. 22.3 Comp. strength 40.4 Ultimate adavtic, 47.1 Heat convection Input data set-50 Heat Convection 2.65 Comp. strength 41.7 …… …… Ultimate adavtic, 48.4 Placing temp. 24.0 Heat Convection 2.85 …… …… etc. Placing temp. 24.0

Construction Schedule Season Spring Autumn Bottom Sabs May 1st September 1st Concrete Structure model was assumed to be located in the Wall and top Slabs May 15th September 15th Aomori city End of calculation October 31 Concrete was cured for five days Proportions of the concrete mix. Type of cement Blast furnace B type Sendai Cement content 300 kg/m 3 Water content 165 kg/m 3 Water-to-cement ratio 55% Tokyo Osaka Kyoto

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 40 Middle. Temp.(℃） Lower 30 20 10 0 0 10 20 30 40 50 60 Age(days）

Stress history (September) 3.0 2.5 Upper Middle 2.0 Stress (N/mm 2 ) Lower 1.5 1.0 0.5 0.0 0 10 20 30 40 50 60 2.25m C -0.5 Crack induced joint L Evaluation -1.0 Output point point of Age(days） for crack thecrack width index 1.5m 0.5m 5m 5m 5m

Evaluation of crack probability 0.71(May) 100 Crack probability (%) Average of Crack probability 0.78(Sept.) 80 Crack Index Calculation Standard 60 Spec. 0.71(May) 0.99 0.96 40 20 0.78(Sept.) 0.93 0.84 0 If the crack index can be assumed to be 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 distributed normally, Crack Index: Icr (Safty factor:γ cr ) the crack occurrence probability, x Sept. x May 99% 93% 1.0 0.78 0.71 1.0

Results of calculations Standard Variation Month Output term Average Devi. Coeff. Maximum temp. 52.04 3.88 0.075 ( ℃ ) Maximum stress 68% 2.74 0.41 0.15 (N/mm 2 ) May Crack index 0.71 0.125 0.18 95% Crack width(mm) 0.42 0.095 0.23 -2σ -1σ 1σ 2σ Maximum temp. 43.92 4.49 0.102 -0.2mm 0.1mm -0.1mm 0.2mm ( ℃ ) Maximum stress 95% of the data lie within the Sept. 2.58 0.385 0.15 (N/mm 2 ) range of deviation of ± 0.2 mm. Crack index 0.78 0.147 0.19 Crack width (mm) 0.36 0.112 0.31 ・ 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 0.6 -0.04 −0.444(Icr=1.0) + 0.734=0.290 0.5 Crack width （mm) 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 0.5 Crack Index 0．70 Crack Index 0．54 Crack width (mm) 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/m 3 , The crack index Av. Std. div. Crack probability. September : 0.71 0.125 99% May: 0.78 0.147 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%.