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 size effect on the nominal strength of quasibrittle structures failing at crack initiation, and particularly on the modulus of rupture of plain concrete beams, is analyzed. First, an improved deterministic formula is derived from the energy release caused by a boundary layer of cracking (initiating fracture process zone) whose thickness is not negligible compared with beam depth. To fit the test data, a rapidly converging iterative nonlinear optimization algorithm is developed. The formula is shown to give an excellent agreement with the existing test data on the size effect on the modulus of rupture of plain concrete beams. The data range, however, is much too limited; it does not cover the extreme sizes encountered in arch dams, foundations, and retaining walls. Therefore, it becomes necessary to extrapolate on the basis of a theory. For extreme sizes, the Weibull type statistical effect of random material strength must be incorporated into the theory. Based on structural analysis with the recently developed statistical nonlocal model, a generalized energetic-statistical size effect formula is developed. The formula represents asymptotic matching between the deterministic-energetic formula, which is approached for small sizes, and the power law size effect of the classical Weibull theory, which is approached for large sizes. In the limit of infinite Weibull modulus, the deterministic-energetic formula is recovered. Data fitting with the new formula reveals that, for concrete and mortar, the Weibull modulus is approximately equal to 24 rather than 12, the value widely accepted so far. This means that, for extreme sizes, the nominal strength (modulus of rupture) decreases, for two-dimensional (2D) similarity, as the -1/12 power of the structure size, and for 3D similarity, as the -1/8 power (whereas the -1/4 power has been assumed thus far). The coefficient of variation characterizing the scatter of many test results for one shape and one size is shown not to give the correct value of Weibull modulus because the energetic size effect inevitably intervenes. The results imply that the size effect at fracture initiation must have been a significant contributing factor in many disasters (for example, those of Malpasset Dam, Saint Francis Dam and Schoharie Creek Bridge.)

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Energetic-Statistical Size Effect in Quasibrittle Failure atCrack Initiation

The near-tip angular variations of elastodynamic stress and displacement fields are investigated for rapid transient crack propagation in isotropic and orthotropic materials. The 2-dimensional near-tip displacement fields are assumed in the general form r/sup P/ T(t, c) K(theta, c), where c is a time-varying velocity of crack propagation, and it is shown that p = 0.5. For isotropic materials, K(theta, c) is determined explicitly by analytical considerations. A numerical procedure is employed to determine K(theta, c) for orthotropic materials. The tendency of the maximum stresses to move out of the plane of crack propagation as the speed of crack propagation increases is more pronounced for orthotropic materials, for the case that the crack propagates in the direction of the larger elastic modulus. The angular variations of the near-tip fields are the same for steady-state and transient crack propagation, and for propagation along straight and curved paths, provided that the direction of crack propagation and the speed of the crack tip vary continuously.

Elastodynamic Near-Tip Stress and Displacement Fields for Rapidly Propagating Cracks in Orthotropic Materials

The nonlocal probabilistic theory developed in Part I is applied in numerical studies of plain concrete beams and is compared to the existing test data on the modulus of rupture. For normal size test beams, the deterministic theory is found to dominate and give adequate predictions for the mean. But the present probabilistic theory can further provide the standard deviation and the entire probability distribution (calculated via Latin hypercube sampling). For very large beam sizes, the statistical size effect dominates and the mean prediction approaches asymptotically the classical Weibull size effect. This is contrary to structures failing only after the formation of a large crack, for which the classical Weibull size effect is asymptotically approached for very small structure sizes. Comparison to the existing test data on the modulus of rupture demonstrates good agreement with both the measured means and the scatter breadth.

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Probabilistic nonlocal theory for quasibrittle fracture initiation and size effect. ii: application

A recently proposed microplane model, which describes not only cracking but also general nonlinear triaxial response, is extended to cyclic loading and the rate effect, and is implemented in a three‐dimensional finite element code. The material properties are characterized separately on planes of various orientations within the material, called the microplanes, on which no tensorial invariance requirements need to be observed. The state of each microplane is described by normal deviatoric and volumetric strains, and by shear strain. To avoid spurious localization instabilities due to strain softening and the consequent mesh‐sensitivity problems, the concept of nonlocal continuum with local strain is adopted. The rate effect is introduced by combining the damage model on each microplane with the Maxwell rheologic model. The results of finite element analysis of some basic cases on the material level, as well as of plain concrete specimens loaded in bending and compression, are demonstrated. The calculated ...

Microplane Model for Cyclic Triaxial Behavior of Concrete

This paper addresses the problem of selecting the most realistic creep and shrinkage prediction model, important for designing durable and safe concrete structures Statistical methods of standard and several nonstandard types and a very large experimental database have recently been used to compare and rank the existing prediction models, but conflicting results have been obtained by various investigators This paper attempts to overcome this confusion It introduces data weighting required to eliminate the bias due to improper data sampling in the database, and then examines Bazant and Baweja's Model B3, the ACI model, the CEB model, and two of Gardner's models The statistics of prediction errors are based strictly on the method of least squares, which is the standard and the only statistically correct method, dictated by the maximum likelihood criterion and the central limit theorem of the theory of probability, as well as the requirement of noncorrelation of errors Several nonstandard statistical methods that have recently been invented to evaluate creep and shrinkage models are also examined and their deficiencies are pointed out The ranking of the models that ensues from the least-square regression statistics is shown to be quite different from the rankings obtained by the nonstandard statistics

/pdf/unbiased-statistical-comparison-of-creep-and-shrinkage-3epa7hsbeq.pdf

Zdenek P. Bazant

Papers

Energetic-Statistical Size Effect in Quasibrittle Failure atCrack Initiation

Elastodynamic Near-Tip Stress and Displacement Fields for Rapidly Propagating Cracks in Orthotropic Materials

Probabilistic nonlocal theory for quasibrittle fracture initiation and size effect. ii: application

Microplane Model for Cyclic Triaxial Behavior of Concrete

Unbiased Statistical Comparison of Creep and Shrinkage Prediction Models