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.

Reliability, Brittleness, Covert Understrength Factors, and Fringe Formulas in Concrete Design Codes

Abstract: The paper analyzes the reliability consequences of the fact that the current design codes for concrete structure contain covert (or hidden) understrength (or capacity reduction) factors. This prevents distinguishing between different combinations of separate risks due to the statistical scatter of material properties, the error of the design formula, and the degree of brittleness of failure mode, and also makes any prediction of structural reliability (or survival probability) impossible. The covert formula error factor is implied by the fact that the design formula was calibrated to pass not through the mean but through the fringe (or periphery, margin) of the supporting experimental data. The covert material randomness factor is the ratio of the reduced concrete strength required for design to the mean of the strength tests. As a remedy, the covert understrength factor of design formula should be made overt, its coefficient of variation (based on the supporting test data) should be specified, and the type of probability distribution (e.g., Gaussian or Weibull) indicated (which then also implies the probability cutoff). Alternatively, the code could give the mean formula, specify its coefficient of variation and type of distribution, and either prescribe the probability cutoff or overtly declare the understrength factor. The mean of strength tests required for quality control should be figured out from the required design strength on the basis of a specified probability cutoff and the coefficient of variation of these tests. Furthermore, it is proposed that the currently used empirical understrength factor, which accounts mainly for the risks of structural brittleness (or lack of ductility), should be based on the expected maximum kinetic energy that could be imparted to the structure. The reliability integral taking into account the randomness of both the load and structural resistance is generalized for the case of multiple (statistically independent) understrength factors. Finally, it is pointed out that the currently assumed proportionality of the tensile and shear strengths to the square root of compressive strength of concrete is realistic only for the mean, but grossly underestimates the scatter of tensile and shear strengths.

Nonlocal microplane concrete model with rate effect and load cycles. i: general formulation

Abstract: The nonlocal microplane model for concrete is improved to describe unloading, reloading, cyclic loading, and the rate effect. The differences compared to the previous formulation are: (1) Normal strain component on the microplane is not split into its volumetric and deviatoric parts—rather the normal component is made dependent on the lateral normal strains on the microplane; and (2) instead of considering on each microplane only one shear strain vector parallel to the shear stress vector, the shear strain is represented by two independent components on the microplane. To introduce rate effect, the stress-strain law for each microplane component is described by a generalized Maxwell model—a series coupling of a linear viscous element and an elastoplastic-fracturing element. Nonlinear unloading-reloading hysteresis rules with back- and objective-stresses are developed to introduce hysteresis. The model is then combined with nonlocal theory to enable describing localization phenomena and avoid spurious mesh sensitivity due to strain softening. The numerical implementation in finite-element programs is described. The study consists of two parts; part I deals with the general formulation and part II deals with experimental verification.

Smeared-tip superposition method for cohesive fracture with rate effect and creep

Abstract: A recent formulation of the smeared-tip superposition method presented by Bažant [1], which itself was a generalization and modification of an integral equation formulation with an asymptotic series solution derived by Planas and Elices [2], is further improved, generalized and adapted to an efficient finite difference solution scheme. A crack with bridging stresses is modeled as a superposition of infinitely many LEFM cracks with continuously distributed (smeared) tips having infinitely small intensity factors. Knowledge of the stress intensity factor as a function of the location of the crack tip along the crack path is all that is needed to obtain the load-displacement relation. The solution is reduced to a singular integral equation for a function describing the components of applied load associated with crack tips at various locations. The integral equation is complemented by an arbitrary relation between the bridging stress and the crack opening displacement, which can be rate-independent or rate-dependent. Furthermore, using the creep operator method, the equation is extended to aging linearly viscoelastic behavior in the bulk of the specimen. The previously presented finite difference solution is improved and generalized in a form that leads to a system of nonlinear algebraic equations, which can be solved by an optimization method. Application of the smeared-tip method to the analysis of recent measurements of the size effect in three-point-bend fracture specimens of different sizes is presented and a crack opening law that yields the main qualitative characteristics of the test results, particularly an increase of brittleness with a decreasing loading rate, is presented.

Singular History Integral for Creep Rate of Concrete

Abstract: A uniaxial constitutive equation describing the deviations from the linear principle of superposition of aging concrete at constant moisture content and temperature is presented. Both the creep increase (flow) at high stress and the stiffening (adaptation) due to low sustained compressive stress are modeled, the latter being of principal interest. The constitutive equation expresses the creep rate as a history integral with a singular kernel involving the time lag of creep strain. The integral has the property that the strain response is proportional to the stress history but depends nonlinearly on the stress history when nonproportional stress histories are superposed. The double power law for aging creep is a special limiting case. The constitutive relation is also explained in terms of the rate-process (activation energy) theory for the rate of bond ruptures causing creep. For structural creep problems, a corresponding step-by-step integration algorithm which correctly captures the asymptotic properties of the integral is also developed. A good agreement with test data from the literature is achieved.

Fracturing in concrete via lattice-particle model

Abstract: Numerical simulation is used to explore the behavior of concrete beams of different sizes and different notch lengths, loaded in three-point bending. The entire range of notch depth is studied. One limit case is type 1 fracture, which occurs when the notch depth is zero and the crack initiates from a smooth surface (this is the case of the modulus of rupture test). Another limit is type 2 fracture, which occurs for deep enough notches. Both cases exhibit very different size effects. The fracture is simulated numerically with a robust mesolevel lattice-particle model. The results shed light on the transitional behavior in which the notch depth is non-zero but not deep enough for developing the the type 2 size effect dominated by energy release from the structure. In agreement with experimental observations and theoretical predictions, the numerical results show evidence of a decreasing macroscopic fracture energy as the ligament gets very short.

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.