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

A thermodynamically consistent approach to microplane theory. Part I. Free energy and consistent microplane stresses

The formulation of microplane model M4 (in Parts I and II) is extended to rate dependence. Two types of rate effect in the nonlinear triaxial behavior of concrete are distinguished: (1) rate dependence of fracturing (microcrack growth) associated with the activation energy of bond ruptures, and (2) creep (or viscoelasticity). Short-time linear creep (viscoelasticity) is approximated by a nonaging Maxwell spring-dashpot model calibrated so that its response at constant stress will be tangent to the compliance function of model B3 for a time delay characteristic of the problem at hand. An effective explicit algorithm for step-by-step finite-element analysis is formulated. The main reason that the rate dependence of fracturing must be taken into account is to simulate the sudden reversal of post-peak strain softening into hardening revealed by recent tests. The main reason that short-time creep (viscoelasticity) must be taken into account is to simulate the rate dependence of the initial and unloading stiffnesses. Good approximations of the rate effects observed in material testing are achieved. The model is suitable for finite-element analysis of impact, blast, earthquake, and short-time loads up to several hours duration.

Fracturing Rate Effect and Creep in Microplane Model for Dynamics

The classical applications of Weibull statistical theory of size effect in quasi-brittle structures such as reinforced concrete structures, rock masses, ice plates, or tough ceramic parts are being reexamined in light of recent results. After a brief review of the statistical weakest-link model, distinctions between structures that fail by initiation of macroscopic crack growth (metal structures) and structures that exhibit large macroscopic crack growth prior to failure (quasi-brittle structures) are pointed out. It is shown that the classical Weibull-type approach ignores the stress redistributions and energy release due to stable large fracture growth prior to failure, which causes a strong deterministic size effect. Further, it is shown that, according to this classical theory, every structure is equivalent to a uniaxially loaded bar of variable cross section, which means that the mechanics of the failure process are ignored. Discrepancies with certain recent test data on the size effect are also pointed out. Modification of the Weibull approach that can eliminate these shortcomings is left for a subsequent paper.

Statistical Size Effect in Quasi-Brittle Structures: I. Is Weibull Theory Applicable?

Simplified approximate analysis of the overall collapse of the World Trade Center towers is presented. It is shown that if prolonged heating causes the majority of columns of a single floor to lose their load-carrying capacity, the whole tower is doomed.

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Why did World Trade Center collapse? - Simple analysis

The role that plastic-frictional energy dissipation in the fracture process zone plays in the work­ of-fracture method for measuring the fracture energy of concrete or other quasibrittle materials is analyzed, and a possible improvement of this method is proposed. It is shown that by measuring the unloading compliance at a sufficient number of states on the post-peak descending load-deflection curve, it is possible to calculate the pure fracture energy, representing the energy dissipated by the fracture process alone. However, this value of fracture energy is pertinent only if the material model (constitutive law and fracture law) used in structural analysis takes into account separately the fracture-damage deformations and the plastic-frictional deformations. Otherwise, one must use the conventional fracture energy, which includes plastic-frictional energy dissipation. Either type of fracture energy should properly be determined by extrapolation to infinite specimen size. Further, it is shown that the unloading compliancies to be used in the calculation of the pure fracture energy can be corrected to approximately eliminate the time-dependent effects (material viscoelasticity) and reverse plasticity. Finally, it is proposed to improve the work-of-fracture method by averaging the work done by fracture over only a central portion of the ligament. However, experiments are needed to check whether the specimen size required for this improved method would not be impracticably large.

/pdf/analysis-of-work-of-fracture-method-for-measuring-fracture-lvz633t7t3.pdf

Zdenek P. Bazant

Papers

A thermodynamically consistent approach to microplane theory. Part I. Free energy and consistent microplane stresses

Fracturing Rate Effect and Creep in Microplane Model for Dynamics

Statistical Size Effect in Quasi-Brittle Structures: I. Is Weibull Theory Applicable?

Why did World Trade Center collapse? - Simple analysis

Analysis of work-of-fracture method for measuring fracture energy of concrete