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

Microplane Model for Concrete

As shown three decades ago, in situations where the initial stresses before buckling are not negligible compared to the elastic moduli, the geometrical dependence of the tangential moduli on the initial stresses must be taken into account in stability analysis, and the stability or bifurcation criteria have different forms for tangential moduli associated with different choices of the finite strain measure. So it has appeared paradoxical that, for sandwich columns, different but equally plausible assumptions yield different formulas, Engesser’s and Haringx’ formulas, even though the axial stress in the skins is negligible compared to the axial elastic modulus of the skins and the axial stress in the core is negligible compared to the shear modulus of the core. This apparent paradox is explained by variational energy analysis. It is shown that the shear stiffness of a sandwich column, provided by the core, generally depends on the axial force carried by the skins if that force is not negligible compared to the shear stiffness of the column (if the column is short). The Engesser-type, Haringx-type, and other possible formulas associated with different finite strain measures are all, in principle, equivalent, although a different shear stiffness of the core, depending linearly on the applied axial load, must be used for each. The Haringxtype formula, however, is most convenient because it represents the only case in which the shear modulus of the core can be considered to be independent of the axial force in the skins and to be equal to the shear modulus measured in simple shear tests (e.g., torsional test). Extensions of the analysis further show that Haringx’s formula is preferable for a highly orthotropic composite because a constant shear modulus of the soft matrix can be used for calculating the shear stiffness of the column, and further confirm that Haringx’s buckling formula with a constant shear stiffness is appropriate for helical springs and built-up columns (laced or battened). @DOI: 10.1115/1.1509486#

/pdf/shear-buckling-of-sandwich-fiber-composite-and-lattice-ak9fyicrr9.pdf

Shear buckling of sandwich, fiber composite and lattice columns, bearings, and helical springs: Paradox resolved

After theoretical derivation of the general form of the size effect formula for beam shear in the preceding Part I, this Part II presents experimental verification by least-square fitting of those existing individual data sets that have a broad size range. Subsequently, empirical prediction formulas for the size effect parameters, consisting of the asymptotic small-size strength v0 and the transitional size d0 , are calibrated by least-square regression of (1) a recent American Concrete Institute database with 398 data points, and (2) a combination of this database with large-scale Japanese tests and Northwestern reduced-scale model tests. Previous alternative proposals for dealing with the size effect in beam shear are also discussed.

Designing Against Size Effect on Shear Strength of Reinforced Concrete Beams without Stirrups: II. Verification and Calibration

The form of the age effect on linear creep and elasticity is established by treating the hydrating cement paste as a variable composite in which the volume fraction of solid grows while the material creeps under load. The formulation rests on three hypotheses: (1)The average strains in microelements solidified at various times are equal; (2)the properties of solidified matter (cement gel) are time-invariant; (3)the rate of migrations of solid particles causing creep in cement gel depends on the mean length of migration passages at the time of load application. The first hypothesis allows eliminating the microstresses from a system of two integral equations set up to relate stress and strain histories. A new form of the creep function is deduced, expressing the aging effect in terms of the growth of volume of cement gel and the increase of the migration passage length. It is shown that the double power law is obtained as a special case when the growth of gel volume is negligible, and a generalization consistent with growing gel volume is suggested and compared with creep data.

Viscoelasticity of Solidifying Porous Material-Concrete

The paper reviews experimental evidence on the size effect caused by energy release due to fracture growth during brittle failures of concrete structures. The experimental evidence has by now become quite extensive. The size effect is verified for diagonal shear failure and torsional failure of longitudinally reinforced beams without stirrups, punching shear failure of slabs, pull-out failures of deformed bars and of headed anchors, failure of short and slender tied columns, double-punch compression failure and for part of the range also the splitting failure of concrete cylinders in the Brazilian test. Although much of this experimental evidence has been obtained with smaller laboratory specimens and concrete of reduced aggregate size, some significant evidence now also exists for normal-size structures made with normal-size aggregate. There is also extensive and multifaceted theoretical support. A nonlocal finite element code based on the microplane model is shown to be capable of correctly simulating the existing experimental data on the size effect. More experimental data for large structures with normal-size aggregate are needed to strengthen the existing verification and improve the calibration of the theory.

/pdf/fracture-size-effect-review-of-evidence-for-concrete-k03m38s6nc.pdf

Zdenek P. Bazant

Papers

Microplane Model for Concrete

Shear buckling of sandwich, fiber composite and lattice columns, bearings, and helical springs: Paradox resolved

Designing Against Size Effect on Shear Strength of Reinforced Concrete Beams without Stirrups: II. Verification and Calibration

Viscoelasticity of Solidifying Porous Material-Concrete

Fracture size effect : review of evidence for concrete structures