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

A large series of data on carefully controlled shrinkage test of concrete involving groups of large numbers of identical specimens is reported. The data are used to compare existing shrinkage formulas in ACI, CEB-FIP, and BP models. Assuming that only the measured data for a certain initial period are known, predictions are made for long- times and are compared with the sebsequently observed shrinkage strains. In this manner, various possible statistical regression models are examined and compared. Best predictions are obtained when the shrinkage formula is fitted to test data using nonlinear optimization, then linear regression in transformed variables is used to obtain the confidence limits for long-time predictions. It is concluded that good long-time predictions of shrinkage can be obtained on the basis of shrinkage test results on 80 mm diameter cylinders for a 3-week duration. These predictions involve only the intrinsic uncertainty of the material, on which the uncertainty due to ramdom environment, curing history, and differences in concrete composition must be superimposed for practical application.

Statistical extrapolation of shrinkage data - part i: regression

The endochronic theory for inelasticity and failure, previously established, is used to predict the response of reinforced concrete beams in cyclic bending at large strains. Cross sections are assumed to remain plain and are subdivided in slices. Existing bending theories must be enhanced by inclusion of transverse normal strain as a third variable, in addition to curvature and transverse shear angle. The forces in stirrups bring concrete under confining hydrostatic pressure, and, according to endochronic theory, this greatly increases ductility and strength and suppresses strain-softening. The theory is applicable to any history of bending moment, shear force, and axial force, and allows the necessary cross-sectional area of stirrups to be calculated. It is most remarkable that a number of test data have been correctly predicted without having to adjust any of the material parameters determined previously from tests of plain concrete. The endochronic theory represents not merely a descriptive model, but a prediction method.

Prediction of hysteresis of reinforced concrete members

The results of reduced-scale failure tests of simply supported four-point-bend beams of different sizes, containing lapped bond splices of smooth (undeformed) longitudinal reinforcing bars, are reported. The tests consist of two groups, with splices located either in the midspan region with a uniform bending moment, or in one of the end regions with a uniform shear force. The specimens were made of microconcrete with a maximum aggregate size of 4.76 mm. Beams of three different heights (50 mm, 100 mm, and 200 mm) were tested. The beams were geometrically similar in three dimensions, and even the bar diameters and cover thicknesses were scaled in proportion. The reinforcement ratio was 0.31%. The results reveal the existence of a significant size effect, which can be approximately described by the size effect law previously proposed by Bazant. The size effect is found to be stronger for splices without any spiral than for splices confined by a spiral, and stronger for splices in the maximum shear force region of a beam than for splices in the maximum bending moment region. Generalization of the existing formula of Orangun and colleagues is proposed and recommended for design. Although the formula provides a safer alternative to the existing approach, further testing is needed for better calibration. The size effect on the nominal bond strength implied by the development length provisions of the current and previous American Concrete Institute code is discussed and shown to be inadequate.

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Size Effect on Failure of Bond Splices of Steel Bars in Concrete Beams

The time dependence of concrete fracture, and particularly the effect of loading rate, has so far been studied mainly in the dynamic range. The present study extends a preceding investigation of the rate effect in the static range that covered times to peak from 1 to 300,000 sec. Geometrically similar three-point-bend specimens of three different sizes are subjected to either a sudden 1000-fold increase of the loading rate or a 10-fold sudden decrease of the loading rate. It is found that the post-peak softening can be reversed to hardening, followed by a second load peak that can be either higher or lower than the previous load peak. The rise to the second peak depends on the previous post-peak load drop from the first peak load. A sudden decrease in the loading rate causes initially a steeper softening slope. The source of these time-dependent effects appears to be not only the thermally activated nature of the process of bond ruptures in the fracture process zone but also the effect of creep, both a nonlinear creep in the fracture process zone and a linear creep in the bulk of the specimen. The results of this study and a previous study suggest that there is a significant difference in fracture behavior for short-time and long-time loads. The phenomena observed are of interest, for example, for the analysis of concrete dams with cracks that evolve over many years. Mathematical modeling of the present test results is left for a subsequent study.

Softening Reversal and Other Effects of a Change in Loading Rate on Fracture of Concrete

A method of analysis of the global behavior of long curved or straight single-cell girders with or without initial stress is presented. It is based on thin-wall beam elements that include the modes of longitudinal warping and of transverse distortion of cross section. Deformations due to shear forces and transverse bimoment are included, and it is found that the well-known spurious shear stiffness in very slender beams is eliminated by virtue of the fact that the interpolation polynomials for transverse displacements and for longitudinal displacements (due to rotations and warping) are linear and quadratic, respectively, and an interior mode is used. The element is treated as a mapped image of one parent unit element and the stiffness matrix is in integration in three dimensions, which is numerical in general, but could be carried out explicitly in special cases. Numerical examples of deformation of horizontally curved bridge girders, and of lateral buckling of box arches, as well as straight girders, validate the formulation and indicate good agreement with solutions by other methods.

Zdenek P. Bazant

Papers

Statistical extrapolation of shrinkage data - part i: regression

Prediction of hysteresis of reinforced concrete members

Size Effect on Failure of Bond Splices of Steel Bars in Concrete Beams

Softening Reversal and Other Effects of a Change in Loading Rate on Fracture of Concrete

Stiffness Method for Curved Box Girders at Initial Stress