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.

/pdf/crack-band-theory-for-fracture-of-concrete-4hm1hrkirl.pdf

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

Deflections of partially prestressed concrete beams are analyzed on the basis of bending theory with plane cross sections, taking into account not only plasticity of steel and nonlinearity of concrete in compression, but also the tensile strain softening of concrete. However, their effect is not very large and is much less than found in a previous analysis of nonprestressed concrete beams. The effect of tensile strain softening is found to be significant only in the initial post-cracking response and vanish near the ultimate load. The effect of tension stiffening of steel due to the surrounding concrete and the associated bond slip is neglected, since the discrepancy from test data which could be ascribed to this phenomenon is quite small and adequate agreement with tests in the initial post-cracking stage is found without tension stiffening. Comparisons are also made with the I-effective method of Branson and Trost, although generally these predictions are softer than measured and become unacceptable near the ultimate load. Comparisons of calculations as well as measurements between beams with bonded and unbonded reinforcement show a relatively small difference.

/pdf/deformation-of-progressively-cracking-partially-prestressed-7xriuooj0m.pdf

Deformation of progressively cracking partially prestressed concrete beams

http://www.civil.northwestern.edu/people/bazant/PDFs/Backup%20of%20Papers/458.pdf

Postcritical imperfection sensitivity of sandwich or homogenized orthotropic columns soft in shear and in transverse deformation

Micromechanics analysis of damage in heterogeneous media and composites cannot ignore the interactions among cracks as well as between cracks and inclusions or voids. Previous investigators can to this conclusion upon finding that states of distributed (diffuse) cracking (damage) cannot be mathematically represented merely as crack systems in a homogeneous medium, even though stable states with distributed damage have been experimentally observed in heterogeneous materials such as concrete. This paper presents a method for modeling interactions between a crack and many inclusions. Based on the Duhamel‐Neuman analogy, the effect of the inclusions is equivalent to unbalanced forces acting on the contour of each inclusion in an infinite homogeneous solid. The problem is solved by superposition; it is decomposed into several standard problems of elasticity for which well‐known solutions are available. The problem is finally reduced to a system of linear algebraic equations similar to those obtained by Kachano...

Cracks Interacting with Particles or Fibers in Composite Materials

Using a uniaxial localization model, it is shown that parallel elastic restraint increases ductility, while an increase in support flexibility or length of specimen reduces ductility. Statistical macroscopic nonhomogeneity of the specimen is modeled by a system of uniaxial parallel elements of random properties following the normal distribution. The stability analysis and Monte Carlo simulations explain that in such a system an increase of length or support flexibility reduces not only ductility but also the strength of the system. The effect on strength depends on the number of elements (width of specimen), which represents a new non-classical statistical size effect, and on the standard deviation of peak stress values within the parallel system. Existence of an inflection point and the prolonged tail on the descending branch is explained by the nonhomogeneity of the specimen, and the shape of the descending branch, along with the location of the inflection point, is obtained as a function of machine stiffness, parallel elastic restraint, and specimen length and width.

/pdf/statistical-stability-effects-in-concrete-failure-1k1mr2xnc7.pdf

Statistical Stability Effects in Concrete Failure

A micromechanically based enrichment of the nonlocal operator by a term taking into account the directional dependence of crack interactions (Bazant,1992) can be expected to improve the performance of the nonlocal model. The aim of this paper is to examine this new model in the context of a simple localization problem reducible to a one dimensional description. Strain localisation in an infinite layer under plaine stress is studied using both the old and the new nonlocal formulations. The importance of renormalization of the averaging function in the proximity of a boundary is demonstrate and the differences between the localization sensitivity of the old and new model are pointed out

/pdf/nonlocal-model-based-on-crack-interactions-a-localization-31g3ry3lzz.pdf

Zdenek P. Bazant

Papers

Deformation of progressively cracking partially prestressed concrete beams

Postcritical imperfection sensitivity of sandwich or homogenized orthotropic columns soft in shear and in transverse deformation

Cracks Interacting with Particles or Fibers in Composite Materials

Statistical Stability Effects in Concrete Failure

Nonlocal Model Based on Crack Interactions: A Localization Study