A stress‐strain model is developed for concrete subjected to uniaxial compressive loading and confined by transverse reinforcement. The concrete section may contain any general type of confining steel: either spiral or circular hoops; or rectangular hoops with or without supplementary cross ties. These cross ties can have either equal or unequal confining stresses along each of the transverse axes. A single equation is used for the stress‐strain equation. The model allows for cyclic loading and includes the effect of strain rate. The influence of various types of confinement is taken into account by defining an effective lateral confining stress, which is dependent on the configuration of the transverse and longitudinal reinforcement. An energy balance approach is used to predict the longitudinal compressive strain in the concrete corresponding to first fracture of the transverse reinforcement by equating the strain energy capacity of the transverse reinforcement to the strain energy stored in the concret...

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Theoretical stress strain model for confined concrete

http://mech.spbstu.ru/images/b/bd/Potyondy_Cundall_2004_A_bonded-particle_model_for_rock.pdf

A bonded-particle model for rock

A fracture theory for a heterogenous aggregate material which exhibits a gradual strain-softening due to microcracking and contains aggregate pieces that are not necessarily small compared to structural dimensions is developed. Only Mode I is considered. The fracture is modeled as a blunt smeard crack band, which is justified by the random nature of the microstructure. Simple triaxial stress-strain relations which model the strain-softening and describe the effect of gradual microcracking in the crack band are derived. It is shown that it is easier to use compliance rather than stiffness matrices and that it suffices to adjust a single diagonal term of the complicance matrix. The limiting case of this matrix for complete (continuous) cracking is shown to be identical to the inverse of the well-known stiffness matrix for a perfectly cracked material. The material fracture properties are characterized by only three parameters—fracture energy, uniaxial strength limit and width of the crack band (fracture process zone), while the strain-softening modulus is a function of these parameters. A method of determining the fracture energy from measured complete stres-strain relations is also given. Triaxial stress effects on fracture can be taken into account. The theory is verified by comparisons with numerous experimental data from the literature. Satisfactory fits of maximum load data as well as resistance curves are achieved and values of the three material parameters involved, namely the fracture energy, the strength, and the width of crack band front, are determined from test data. The optimum value of the latter width is found to be about 3 aggregate sizes, which is also justified as the minimum acceptable for a homogeneous continuum modeling. The method of implementing the theory in a finite element code is also indicated, and rules for achieving objectivity of results with regard to the analyst's choice of element size are given. Finally, a simple formula is derived to predict from the tensile strength and aggregate size the fracture energy, as well as the strain-softening modulus. A statistical analysis of the errors reveals a drastic improvement compared to the linear fracture theory as well as the strength theory. The applicability of fracture mechanics to concrete is thus solidly established.

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Crack band theory for fracture of concrete

A plastic-damage model for concrete

National Institute of Standards and Technology における超伝導研究及び生活

The results of an experimental study of fatigue fracture of geometrically similar high-strength concrete specimens of very different sizes are reported and analyzed. Three-point bend notched beams were subjected to cyclic loading. The number of cycles to failure ranged from 200 to 41,000. It was found that Paris law for the crack length increment per cycle as a function of the stress intensity factor, which was previously verified for normal concrete, is also applicable to high-strength concrete. However, for specimens of different sizes, an adjustment for the size effect needs to be introduced, of a similar type as previously introduced for normal concrete. A linear regression plot estimating the size-adjustment parameters is derived. An LEFM (linear elastic fracture mechanics)-type calculation of the deflections under cyclic loading on the basis of the size-adjusted Paris law yields correct values for the terminal phase but grossly underpredicts the initial deflections. Overall, the results underscore the importance of considering fatigue fracture growth in the case of high-strength concrete structures subjected to large, repeated loads, and taking into account the very high brittleness under fatigue loading.

Fatigue Fracture of High-Strength Concrete and Size Effect

Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories

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Size effect in punching shear failure of slabs.

The size effect in the split cylinder (Brazillian) tensile test is studied experimentally and analyzed theoretically. Tests of a very broad size range, 1:26, were conducted on cylindrical discs of constant thickness made from concrete with aggregate of a maximum size of 5 mm. The results confirm the existence of size effect and show that up to a certain critical diameter d, the curve of nominal strength versus diameter approximately agrees with the law proposed by Bazant for the size effect caused by energy release due to fracture growth. For larger sizes, there appears to be a deviation from the size-effect law. These and other study findings are discussed.

Size Effect in Brazilian Split-Cylinder Tests: Measurements and Fracture Analysis

Fracture mechanics of brittle failures of concrete structures due to the propagation of a cracking zone indicates that the nominal stress at failure should not be constant but should decrease as the structure size increases. This size effect may be described by a simple size-effect law recently derived by dimensional analysis and similitude arguments. A formula for diagonal shear failure based on this size-effect law was published, verified, and calibrated by a large set of data from the literature. The present paper improves this previous formula in two respects: fJ) the effect of maximum aggregate size, distinct from the effect of the relative beam size, is incorporated and (2) the formula is extended to cover the effect of stirrups on the shear capacity of concrete. The new formula is verified and calibrated according to a larger data set than before, consisting of several hundred test results compiled from the literature. [t is shown that the new formula achieves an appreciable reduction in the coefficient of variation of the deviations of the measured data from the prediction formula.

Zdenek P. Bazant

Papers

Fatigue Fracture of High-Strength Concrete and Size Effect

Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories

Size effect in punching shear failure of slabs.

Size Effect in Brazilian Split-Cylinder Tests: Measurements and Fracture Analysis

Size Effect in Diagonal Shear Failure: Influence of Aggregate Size and Stirrups