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 classical truss model (or strut-and-tie model) for shear failure of reinforced concrete beans is modified to describe fracture phenomena during failure. The failure is assumed to be caused by propagation of a compression fracture across the concrete strut during the portion of the loading history in which the maximum load is reached. The compression fracture may consist of a band of splitting cracks that later interconnect to form a shear crack or a shear fracture band inclined to the strut. The width of the fracture band is assumed to occupy only a portion of the strut length and to represent a fixed material property independent of the beam depth. The energy release from the truss is calculated using two alternative approximate methods: (1) using the potential energy change deduced from the concept of stress relief zones; and (2) using the complementary energy change due to stress redistribution caused by propagation of the fracture band across the compressed concrete strut. Both approaches show that a size effect on the nominal strength of shear failure must exist and that it should approximately follow the size effect law proposed by Bazant in 1984. The physical mechanism of the size effect is also explained in a clear and simple intuitive manner. Finally, it is shown that the applied nominal shear stress that causes large initial diagonal cracks to form also exhibits a size effect.

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Fracturing truss model: size effect in shear failure of reinforced concrete

The approximate methods for prediction of structural effects of creep, such as the effective modulus (EMM), age-adjusted effective modulus (AEMM), rate of creep (RCM), rate of flow (RFM), and Levi's and Arutyunian's methods are all linear and satisfy the principle of superposition. Predictions of the approximate methods are compared with the exact numerical solutions based on the principle of superposition and given (undistorted) realistic unit creep curves. It is found that AEMM is in general superior to all other methods and along with EMM is also the simplest one. RFM (with effective modulus treatment of delayed elastic strain) appears as second best and should be resorted to when the table of aging coefficient required by AEMM is unavailable.

Comparison of approximate linear methods for concrete creep.

A nonlinear integral-type creep law is developed by generalizing the linear superposition integral for the creep rate rather than the total strain. At low service stress level there is a significant (though previously overlooked) nonlinearity that consists in gradual stiffening or adaptation to a sustained compressive stress. It is modeled by a stress-dependent acceleration of the age-dependence of stiffness, and by an adaptation parameter whose rate is a function of the stress and age. The high-stress nonlinearity that consists of a weakening of the stiffness, is essentially without memory and is described by an additive rate-type flow term. Its stress dependence and the flow rate decay is modeled by kinematic hardening. An extension to elevated temperatures, which agrees with recovery data, is indicated. Although uniaxial creep is of primary interest, a rational triaxial generalization involving proper stress invariants is derived.

Nonlinear creep of concrete adaptation and flow

Investigated are thermodynamic restrictions on rate-type creep laws for porous materials which slowly solidify while carrying load (aging of concrete, i.e., gradual filling of pores in concrete by hydration products) or slowly melt (gradual dehydration of concrete at high temperature). Thermodynamic potentials (Helmholtz free energy and complementary or Gibbs free energy) are determined. The chemical dissipation of elastic energy is calculated and the condition of its positiveness is proposed; this requires that elastic relations be introduced in terms of stress and strain rates for solidifying materials and in terms of stresses and strains for melting materials. Some creep laws used in practice are found to have thermodynamically inadmissible form. Creep laws of thermodynamically correct form are shown. The known forms of such laws often cannot, however, fit available long-time creep test data for concrete very well, unless some material parameters or rates are allowed to have thermodynamically inadmissible negative values for short periods of time.

Thermodynamics of Solidifying or Melting Viscoelastic Material

Proceedings of the France-US Workshop on Strain Localization and Size Effect due to Cracking and Damage, Laboratorie de Mecanique et Technologie, Cachan, France, 6-9 September 1988.

Zdenek P. Bazant

Papers

Fracturing truss model: size effect in shear failure of reinforced concrete

Comparison of approximate linear methods for concrete creep.

Nonlinear creep of concrete adaptation and flow

Thermodynamics of Solidifying or Melting Viscoelastic Material

Cracking and Damage