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

This model accounts not only for the rapid increase of creep at high compressive stress (flow) but also for the decrease of creep observed for subsequent load increments after a long period of low sustained compressive stress (adaptation). A simply supported column is analyzed for typical load histories: sustained loading; short-time loading to failure; and sustained load followed by short-time loading to failure. Numerical results confirm that the flow nonlinearity makes the column fail in a finite time and the adaptation nonlinearity makes the response to the short-time load applied after a long period of a low sustained load stiffer. Also, there is a substantial difference in the column strength for the same load duration depending upon the loading history, especially the ratio of constant sustained load (dead load) and short-time load (live load) and their durations. This could be taken into account in setting the safety factors for columns subjected to dead and live loads of various ratios.

Nonlinear Creep Buckling of Reinforced Concrete Columns

A finite element procedure for determining the dynamic response and liquefaction susceptibility of a deposit of saturated sand subjected to an arbitrary bedrock acceleration is developed. The method is structured in accordance with the generalized Biot constitutive equations for an inelastic two-phase medium, and it includes a law of the endochronic type to model the deviatoric constitutive behavior of the sand. The approach has the capability of accounting for: (1) Strain hardening and softening; (2) densification; (3) hysteretic and viscous damping; (4) the development of pore pressure due to volume change; and (5) consolidation due to internal pressure generation. The results reported in this investigation demonstrate that "site" factors, such as the shear resistance of the overburden, seepage conditions, nature of the loading history, layer thickness, bulk compressibility, and phase coupling, greatly affect the pore pressure increase in saturated sand layers subjected to a seismic excitatin. (Authors)

Site Factors Controlling Liquefaction

Microplane model for triaxial deformation of soils

Material behavior under various types of loading.

The size effect on the nominal strength of a floating ice plate subjected to a vertical uniform line load is analyzed. The cracks produced by the load, which are parallel to the load line, are treated as softening inelastic hinges. The problem is one dimensional in the direction normal to the load line, equivalent to a beam on elastic foundation provided by buoyancy of ice in water. The softening moment-rotation diagram of inelastic hinges is simplified as linear and its dependence on structure size ~ice thickness! is based on the energy dissipated by fracture. For thick enough plates, no two hinges ~on one side of the line load! can soften simultaneously, in which case a simple analytical solution is possible. In that case, the load-deflection diagram has multiple peaks and troughs and consists of a sequence of spikes that get progressively sharper as the plate thickness increases. In terms of a dimensionless nominal strength, the effect of a finite fracture process zone at ice surface leads to an up-and-down size effect plot, such that each load peak decreases with the size at first but then asymptotically approaches a rising asymptote of the type (thickness) 1/4 ~which implies a reverse size effect, caused by buoyancy!. The energy dissipation when the crack in the hinge gets deep causes a strong monotonic size effect, such that the dimension- less troughs between two spikes, in the case of thick enough plate, decrease asymptotically as (thickness) 21/2 . For thin enough plates, more than one hinge soften simultaneously and, in the asymptotic case of vanishing ice thickness, the plasticity solution, which has no size effect, is approached. In the intermediate size range with hinges softening simultaneously, the exact solution is complicated and only approximate formulas for the size effect are possible. They are constructed by asymptotic matching.

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Zdenek P. Bazant

Papers

Nonlinear Creep Buckling of Reinforced Concrete Columns

Site Factors Controlling Liquefaction

Microplane model for triaxial deformation of soils

Material behavior under various types of loading.

Size Effect on Strength of Floating Sea Ice under Vertical Line Load