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

Metal-composite joints are common in composite structures made up of conventional steel parts and advanced polymer composite parts, e.g., in hybrid ship hulls. Correct prediction of the size effect on the strength of the joints is important for the extrapolation of the small-scale laboratory testing results to full size joints, for ensuring an efficient and reliable design. In this study, the size effect on the strength of metal-composite joints is analyzed theoretically, numerically and experimentally. The analytical formulation of the size effect is asymptotically anchored at the large size limit by the linear elastic fracture mechanics (LEFM), which exhibits a singularity with a complex exponent. Numerical analysis with cohesive fracture model is used to design the specimens to be tested, and also to analyze the experimental results to be reported at the conference. Preliminary results from the ongoing experimental studies are also presented. Both computational and analytical methods indicate that the fracture must initiate from the inner steel-composite edges. The experimental findings confirmed that indeed the fracture initiates in the inner steel-composite edges. To determine the size effect law that spans all sizes, the energy release rate is determined numerically and used in the sense of equivalent LEFM. Initial findings indicate that the size effect on the strength of hybrid joints is of the LEFM type.

Size Effect on Strength of Hybrid Metal-Composite Joints

The paper reviews the modeling of damage by the microplane model and presents extensions required for cyclic loading. The general microplane constitutive model is implemented in a three-dimensional finite element code and used in damage and bifurcation analysis of various structures. The results of the analysis are compared with test results and some of the comparisons are shown in the present paper.

Nonlinear analysis of strain-softening damage under monotonic and cyclic loading

A two-dimensional finite element program using triangular plane strain elements has been developed to determine the stress and displacement fields that result in an earth dam subjpcted to earthquake loadinq, The governing ',"luations are derived fro!:l the principle of vlrtual work, and they account for the exi stence of two phases (solid and fluid) with coupling and daJll['inq due to relatlve accelerations and velocities. Endochronic (nonlinear inelastic) and linear elastic constitutive relationships are used to describe the soil behavior, and a step-by-step integration in ti",e is l,erformed by an implicit methoJ. Two Jifferent earthquake acceleration records with ,lrI,ruximately the same maximum acceleration were employed in the endochronic analysis. Cc11culated results demonstrate the importan~" of utilizing appropriate constitutive relationships and the effect caused by tho nature of the input earthquake record, Endochronic theory and the concept of soil as a two-phase medium are combined to conduct a dynamic finite element analysis of an earth dam. Calculations are made (a) with both endochronic and linear elastic constitutive relationships using a modified version of the N-S component of the E1 Centro accelerogram from the 1940 earthquake and (b) with an endochronic constitutive relationship using a dec onvoluted and enriched version of the Pacoima accelerogram from the 1971 San Fernando 559 earthquake. Treatment of the soils as twophase media accounts for the coupling between phases and allows the stress distribution to be determined in both the solid and fluid phases; as such, it is a more accurate and realistic representation of the actual behavior of soils. Virtually all of the available models that have been applied to this problem are based on the concept of a one-phase medium, and the pore pressure response is determined explicitly. ENDOCHRON Ie CONSTITUTIVE LAW FOR SOIL.S Meaningful advances in our analytical ability to solve field problems in soil dynamics requires, among other things, the formulation of more realistic and generally applicable constitutive relationships for soi Is. Toward this end endochronic theory, which was first applied to describe the mechanical behavior of metals (24), has been extended to characterize the response of concrete (3) and soils (2, 4,11,15). In endochronic theory the notion of an intrinsic time scale is formulated to account for the independent dissipative effects of accumulated strain and external time. The intrinsic time parameter, z, depends on external time (strain rate) only for rate-dependent materials, such as cohesive soils, and it should be a monotonically increasing function of strain and external time. Assuming that the development of inelastic strains is gradual, the intrinsic time increment, dz, can be expreased as a crement, dC ij , function of the strain inand the external time incre-

Seismic analysis of an earth dam based on endochronic theory.

Consequences of fracture mechanics for size effect, crack spacing and crack width in concrete pavements

The paper reviews a series of studies at Northwestern University which led to the establish- ment of a theory of probability distributions of short-time strength, residual strength after static preload and lifetime of structures made of quasibrittle materials such as concrete, fiber composites and tough ceramics. The theory is based on the frequency of probability of interatomic bond breaks on the atomic scale and on the multi-scale transition of power-law probability tail. The conclusion is that if the failure is not perfectly brittle, the probability distribution of strength and lifetime is a graft of Gaussian and Weibull distributions and varies from nearly Gaussian at the scale of one RVE to Weibullian for very large struc- tures consisting of many RVEs. As a consequence, the safety factors should depend on structure size. Numerous experimental comparisons and computational simulations are given.

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Zdenek P. Bazant

Papers

Size Effect on Strength of Hybrid Metal-Composite Joints

Nonlinear analysis of strain-softening damage under monotonic and cyclic loading

Seismic analysis of an earth dam based on endochronic theory.

Consequences of fracture mechanics for size effect, crack spacing and crack width in concrete pavements

Nanomechanics Based Theory of Size Effect on Strength, Lifetime and Residual Strength Distributions of Quasibrittle Failure: A Review