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...

/pdf/theoretical-stress-strain-model-for-confined-concrete-3eemamm28s.pdf

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

Closure of "Fracture Mechanics of Reinforced Concrete"

The previously formulated model of the gravity-driven collapse of the twin towers of the World Trade Center (WTC) on September 11, 2011 was shown to match all the existing observations, including the video record of the crush-down motion of the top part of tower during the first few seconds, the seismically recorded duration of collapse, the size distribution of particles caused by impact comminution of concrete floor slabs, the loud booms due to near-sonic lateral ejection velocity of air and dust, and precedence of the crush-down collapse mode before the crush-up. Nevertheless, different degrees of ductility, fracturing and end support flexibility of WTC columns could lead to an equally good match of these observations and remained uncertain, due to lack of test data. Recently, Korol and Sivakumaran reported valuable experiments that allow clarifying this uncertainty. They revealed that, under the simplifying assumptions of rigid end supports and unlimited ductility (or no fracturing) of unheated columns, the energy dissipation of the WTC columns would have been at maximum 3.5-times as large as that calculated by the plastic hinge mechanism normally considered for small-deflection buckling. This increase would still allow close match of all the aforementioned observations except for the first two seconds of the video. The proper conclusion from Korol and Sivakumaran’s tests, based on close matching of the video record, is that the fracturing of unheated columns and the flexibility of their end restraints must have significantly reduced the energy dissipation in columns calculated under the assumptions of no fracture and no end restraint flexibility.

Mechanics of Collapse of WTC Towers Clarified by Recent Column Buckling Tests of Korol and Sivakumaran

Author’s Reply to “Some comments on the paper ‘Microplane constitutive model and metal plasticity’ (Brocca M and Bažant ZP, 2000, Appl Mech Rev 53(10) 265–281),” by A Lagzdiņš and V Tamužs

The paper proposes an improved method of analysis of reinforced concrete columns in braced (no-sway) frames, which is suitable as a simple computer solution for design practice and is more realistic than the existing ACI and CBB methods. The elastic restraint provided by beams adjacent to columns is described by rotational springs. The inelastic behavior of concrete is defined by a uniaxial stress-strain curve with postpeak softening in compression and a zero strength in tension. Plasticity of reinforcement is also considered. The deflection curve is assumed to be a sine curve. The improvement consists in considering the wavelength as unknown and variable during loading. The problem is reduced to a system of seven nonlinear algebraic equations, which are easily solved for small increments of axial displacement by a standard library optimization algorithm. The convergence always occurs and is fast if the increments are small enough. The influence of various param­ eters on the load-deflection curve, the path in the diagram of axial load P versus moment M, and the failure envelope are studied. Various phenomena, such as the possibility of a concave P(M) path at constant load eccentricity, are explained. It is shown that the ACI approach is slightly conservative in most cases, although situations exist in which the ACI approach is either grossly overconservative or slightly unconservative.

Zdenek P. Bazant

Papers

Closure of "Fracture Mechanics of Reinforced Concrete"

Mechanics of Collapse of WTC Towers Clarified by Recent Column Buckling Tests of Korol and Sivakumaran

Author’s Reply to “Some comments on the paper ‘Microplane constitutive model and metal plasticity’ (Brocca M and Bažant ZP, 2000, Appl Mech Rev 53(10) 265–281),” by A Lagzdiņš and V Tamužs

Closure of "Inelastic Buckling of Concrete Column in Braced Frame"

Mathematical modeling of progressive cracking and fracture of rock.