Zdenek P. Bazant

Bio: Zdenek P. Bazant is an academic researcher from Northwestern University. The author has contributed to research in topics: Creep & Fracture mechanics. The author has an hindex of 82, co-authored 301 publications receiving 20908 citations. Previous affiliations of Zdenek P. Bazant include École Polytechnique Fédérale de Lausanne & Rensselaer Polytechnic Institute.

Nonlocal model for size effect in quasibrittle failure based on extreme value statistics

Abstract: A nonlocal generalization of Weibull theory is presented to predict the probability of failure of unnotched structures that reach the maximum load before a large crack forms, as is typ- ical of the test of modulus of rupture. The probability of failure at a material point is assumed to be a power function of the average strain in the neighborhood of that point. For normal sizes, the deterministic theory is found to dominate the mean response and govern the size effect. But the probabilistic theory can provide the entire probability distribution. For extremely large beam sizes, the statistical size effect dominates and the mean prediction approaches asymptotically the classical Weibull size effect. This fundamental feature is discussed in relation to the existing stochastic finite element models. Comparison to the existing test data demonstrates a good agreement with the theory. As became clear in the early 1980's, the size effect on the nominal strength of quasibrittle structures is in most instances predominantly deterministic. It is caused by stress redistributions and energy release associated with either the growth of a large fracture process zone (FPZ) or a long stable crack (Bazant 1984, Bazant & Chen, 1997; Bazant & Planas 1998). Since the material properties represent a random field, some aspects of the size effect should nevertheless be probabilistic. Presenting a new combined energetic-probabilistic theory and exploring where the statistical aspect is important for the mean, variance and probability distribution is objective of this paper. A combined energetic-probabilistic theory, having the classical Weibull probabilistic theory of failure as one limit and the deterministic energetic theory as another limit, can be now developed on the basis of the available evidence. Since the Weibull theory (Weibull 1939), based on the weakest link model, deals with structures that fail before a large (macroscopic) crack can form, a nonlocal generalization of Weibull statistical theory may be developed to predict the probability of failure of unnotched structures that reach the maximum load before a large crack forms, as is typical of the test of modulus of rupture (flexural strength). The tail of the cumulative probability distribution of material failure at one point is assumed to be a power function (characterized by Weibull modulus and scaling parameter )o f the average inelastic (or damage) strain in a neighborhood the size of which is characterized by the material char- acteristic length. The averaging indirectly imposes spatial statistical correlation. The deterministic size effect is automatically exhibited as the limit case of such a formulation for .

Experimental-analytical . size-dependent prediction of modulus of rupture of concrete

Abstract: The procedure for experimental-analytical prediction of flexural strength is reviewed. It allows consideration of size effect phenomena with only minor modification of standard laboratory test procedure. Two possibilities are suggested: testing with one size only and testing with two sizes. The application of this procedure to real case, a massive concrete subway tunnel constructed recently in Prague, is shown. This real case study demonstrates feasibility of the proposed approach. The recently accumulated extensive test data on the modulus of rupture, analytical stud­ ies and numerical simulations, all clearly indicate that the flexural strength of concrete, called the modulus of rupture, significantly decreases with increasing size of the beam, Bazant and Novak (2000a,b). The present paper describes a method to incorporate the size effect into the existing test standards. The proposed method is based on a recently established size effect formula that describes both the deterministic-energetic size effect caused by stress redistribution within the cross section due to finite size of the boundary layer of cracking near at the tensile face of beam, and the classical Wei bull-type statistical size effect due to randomness of the local strength of material, Bazant and Novak (2000c), Bazant and Novak (2001). Two alternatives of the test procedure are formulated. In the first alternative, beams of only one size are tested, as in the current standards, and the size effect on the mean modulus of rupture is approximately predicted on the basis of the existing information for all concretes on the average. In the second alternative, beams of two sufficiently different sizes are tested. The latter is more tedious but gives a much better prediction of size effect for the concrete. The aim of this paper is to present fundamentals of both alternative for practical us­ age and to show the application for size-dependent prediction of modulus of rupture of concrete in case of real concrete structure - large subway tunnel under Vltava river in Prague. The modification of standard testing procedure according to ASTM C78-94 and C293-94 has been recently proposed and the whole procedure can be naturally applied also for other standards for testing of plane concrete beams in three-point and four-point bending. Note, that testing of flexural strength in Czech laboratories follows CSN-ISO 4013 and it is in principle same like above mentioned ASTM standards.

Nonlocal microplane concrete model with rate effect and load cycles. II: Application and verification

Abstract: In the first of this 2-part paper, various improvements and extensions of a previously developed microplane model for concrete were presented. This, the second part of the paper, demonstrates its capability to simulate pertinent test data. The details of the study and its results are presented and discussed. It is shown that by contrast with the previous microplane model for concrete, the normal strain component on the microplane is not split into its volumetric and deviatoric parts; instead the lateral strains are considered. The stress-strain relations for a material point were simulated and then the tests were studied in finite element analyses with nonlocal calculation.

Finite Element for Buckling of Curved Beams and Shells with Shear

Abstract: Inclusion of shear deformations allows the bending theory to be extended to relatively thick beams and shells and, at the same time, simplifies the finite element formulation for both thick and thin beams because monotonic convergence may be achieved without ensuring continuity of displacement derivatives between adjacent elements. Consequently, on may use low order interpolation polynomials, including linear ones. This is particularly useful in the case of curved beams because with higher order interpolation polynomials it is very difficult to satisfy exactly the conditions of no self-staining at rigid body rotations and of availability of all constant strain states, while with linear displacement interpolation polynomials and a straight shape of the element these requirements are easily met.

Theoretical stress strain model for confined concrete

Abstract: 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...

Crack band theory for fracture of concrete

Abstract: 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.