o_04_frctr_cncpt

o_04_frctr_cncpt - 1.054/1.541 Mechanics and Design of...

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1.054/1.541 Mechanics and Design of Concrete Structures Spring 2004 Prof. Oral Buyukozturk 1 / 9 1.054/1.541 Mechanics and Design of Concrete Structures (3-0-9) Outline 4 Fracture Concepts ± Fracture mechanics: o Failure of concrete structures typically involves crack propagation and growth of large cracking zones before the maximum load is reached. Fracture mechanics, for design of concrete structures, has been introduced for a realistic prediction of crack stability. o Some reasons for introducing fracture mechanics into the design of concrete structures: 1. Energy required for crack formation, 2. The need to achieve objectivity of finite element solutions, 3. Lack of yield plateau, 4. The need to rationally predict ductility and energy absorption capability, and 5. The effect of structure size on the nominal strength, ductility, and energy absorption capability. o Fracture problem P 2 a Massachusetts Institute of Technology P
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1.054/1.541 Mechanics and Design of Concrete Structures Spring 2004 Prof. Oral Buyukozturk Outline 4 ± The focus of the problems in fracture mechanics is to study the stress distribution around the crack and the propagation of the crack. Æ Cracks are prescribed geometrically in the material. Æ Energy-based crack propagation (failure) criterion is usually applied in the approach. Æ The method allows investigation of the following phenomena during the propagation of cracks: 1. Strain-softening due to distributed cracking, 2. Localization of cracking into large fractures prior to failure, and 3. Bridging stresses at the fracture front. Æ Traditionally, the size effect has been explained by statistical theories, in terms of the randomness of strength distribution. ² Weibull’s statistical theory (Weibull’s weakest link statistics) o Weibull-type theory of failure The probability of failure of a structure under load P and the mean normal stress at failure are: () 0 , Prob 1 exp m V r Px dV P V σ ⎧⎫ ⎡⎤ ⎪⎪ =− ⎨⎬ ⎢⎥ ⎣⎦ ⎩⎭ 0 1 1P r o b N P Pd P bd bd == where N = mean normal stress, P = mean load, V = volume of structure, r V = representative volume of material, m = Weibull modulus of the material, 0 = scaling parameter, and 2 / 9
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1.054/1.541 Mechanics and Design of Concrete Structures Spring 2004 Prof. Oral Buyukozturk Outline 4 ( , Px σ ) = a function representing the stress caused by load P at point x .
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o_04_frctr_cncpt - 1.054/1.541 Mechanics and Design of...

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