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 nonlinear triaxial behavior of plane concrete that is free of continuous cracks is modeled by an algebraic relation between total strains and stresses, analogous to deformation theory of plasticity. Unloading is not modeled. Good agreement with numerous test data is achieved. The formulation implies algebraic expressions for failure envelopes and some stress-strain curves, which make data fitting easier. The model is enhanced by corrective path-dependent terms that vanish for proportional loading so as extend it for highly nonproportional loading. The principal directions of stress and strain then cease to coincide. In contrast to previous models, the present one applies to much largest atrains, gives the peak stress points, failure envelopes, strain softening, inelastic dilatancy, etc. A tangential incremental form for structural analysis is also indicated.

Total strain theory and path - dependence of concrete

Assuming constant water content, linear stress range, and temperatures between 2°C and 80–100°C, creep of aging concrete at normal and elevated varying (or constant) temperatures, a phenomenon of importance for nuclear reactor vessels, is formulated by a rate-type creep law with hidden stresses based on Maxwell chain model of time-variable properties. Dependence of the creep rate upon temperature is expressed with the help of activation energies which, in general, may be different for the individual Maxwell units in the chain. Introducing the equivalent hydration period, acceleration of aging due to temperature rise is formulated by means of the activation energy of hydration. It is demonstrated that all of the pertinent more extensive data available in the literature can be fitted satisfactorily by the method described. However, no set of data for one concrete available at present is complete enough to assure unique solution to the problem of identification of material parameters from data.

Thermoviscoelasticity of aging concrete

The paper analyses the effect of structure size on the nominal strength of the structure that is implied by the cohesive (or fictitious) crack model proposed for concrete by Hillerborg et al. A new method to calculate the maximum load of geometrically similar structures of different sizes without calculating the entire load-deflection curves is presented. The problem is reduced to a matrix eigenvalue problem, in which the structure size for which the maximum load occurs at the given (relative) length of the cohesive crack is obtained as the smallest eigenvalue. Subsequently, the maximum load, nominal strength and load-point displacement are calculated from the matrix equilibrium equation. The nonlinearity of the softening stress-displacement law is handled by iteration. For a linear softening law, the eigenvalue problem is linear and independent of the matrix equilibrium equation, and the peak load can then be obtained without solving the equilibrium equation. The effect of the shape of the softening law is studied, and it is found that the size effect curve is not very sensitive to it. The generalized size effect law proposed earlier by Bažant, which describes a transition between the horizontal and inclined asymptotes of strength theory and linear elastic fracture mechanics, is found to fit the numerical results very well. Finally some implications for the determination of fracture energy from the size effect tests are discussed. The results are of interest for quasibrittle materials such as concrete, rocks, sea ice and modern tough ceramics.

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Eigenvalue analysis of size effect for cohesive crack model

The paper deals with the problem of initiation of thermal or shrinkage cracks from the surface of a half plane. Linear elastic fracture behavior is assumed. The initial spacing and initial stable equilibrium length of parallel equidistant cracks emanating from the surface is determined from three conditions formulated in a preceding study of penetration of sea ice plate: (1) The stress at the surface reaches a given strength limit. (2) After the initial cracks form, the energy release rate equals its given critical value. (3) The finite energy release due to the initial crack jump equals the energy needed to form the crack (according to the given fracture energy of the material or fracture toughness). The problem is reduced to a singular integral equation which is solved numerically by Erdogan's method. The results of analysis appear to be compatible with the experimental evidence on thermal cracking in glass, and appear to give also reasonable predictions for thermal cracking on top of sea ice plates and shrinkage cracking in concrete.

Initiation of parallel cracks from surface of elastic half-plane

A large series of concrete shrinkage tests which can be used for statistical purposes is reported. The series involves two groups of 36 identical cylindrical specimens, with a diameter of 83 mm for Group 1 and 160 mm for Group 2. Statistical analysis of shrinkage strains and strain rates is presented, and the goodness of fit by normal distribution, log-normal distribution, and gamma distribution is analyzed. Correlations between the values at various times are determined. The study reveals only the intrinsic randomness of the shrinkage process (along with measurement errors) and omits the superimposed uncertainties due to randomness of environment and to prediction uncertainties in the influence of material composition, curing, and other factors. It is found that the coefficient of variation of shrinkage values first decreases with time and then levels off at 6 to 9%. The normal, log-normal, and gamma distributions all give acceptable fits of the observed distributions. There are small systematic deviations from normal distribution: the distributions are slightly skewed to the right and slightly platykurtic. The skewness is more pronounced for the distribution of shrinkage rates, and asymmetric distributions such as gamma give slightly better fits. The correlation of single specimen shrinkage deviations from the group mean at various times is characterized by correlation coefficients. Their value is found to decrease slowly as the length of the time interval increases. For shrinkage rates, this decrease is much faster. The standard deviation of the single specimen shrinkage from its smoothed shrinkage curve is much less than (about one third of) the standard deviation of the single specimen relative to the mean for all specimens. The standard deviation of all single-specimen smoothed shrinkage curves at one time is almost the same as (about 95% of) the standard deviation of the original unsmoothed data for that time.

/pdf/statistics-of-shrinkage-test-data-2j0np81pl5.pdf

Zdenek P. Bazant

Papers

Total strain theory and path - dependence of concrete

Thermoviscoelasticity of aging concrete

Eigenvalue analysis of size effect for cohesive crack model

Initiation of parallel cracks from surface of elastic half-plane

Statistics of Shrinkage Test Data