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Showing papers by "Zdenek P. Bazant published in 1995"


01 Jul 1995
TL;DR: In this paper, a model for the characterization of concrete creep and shrinkage in the design of concrete structures is recommended, which is simpler, agrees better with the experimental data and is justified better theoretically than the previous models.
Abstract: A model for the characterization of concrete creep and shrinkage in the design of concrete structures is recommended. It is simplier, agrees better with the experimental data and is justified better theoretically than the previous models. The model complies with the general guidelines recently formulated by RILEM TC 107. Justification of the model and various refinements are to be published shortly in two parts.

623 citations


Journal ArticleDOI
TL;DR: In this article, a simple formula describing the size effect of the beam is derived, which is explained by the fact that distributed micro-cracking and slips with strain softening occur in the boundary layer of a beam before the maximum load is reached.
Abstract: The modulus of rupture, which characterizes the apparent tensile strength of unreinforced concrete beams, is known to depend on the size of the beam. Since no large stable growth of a crack exists before the maximum load is reached, the size effect cannot be explained by energy release due to fracture. Rather, it is explained by the fact that distributed microcracking and slips with strain softening occur in the boundary layer of the beam before the maximum load is reached. The beam has actually failed before any macroscopic cracks are formed. A simple formula describing the size effect is derived. Asymptotic analysis of the strain softening in the boundary layer reveals that the excess of the modulus of rupture over the direct tensile strength is inversely proportional to the beam depth and proportional to the thickness of the boundary layer, which is nearly proportional to the maximum aggregate size. The proposed formula agrees with existing experimental data. The formula, further generalized, describes the effect of the gradient of normal strains near the concrete surface. Also, approximate analysis of the size effect by linear elastic fracture mechanics yields similar formulas. Determining the effect of beam size on the modulus of rupture by this short formula is simpler than would be possible with a finite element solution.

106 citations


Journal ArticleDOI
TL;DR: In this article, a discrete element method was used to simulate fracture of an ice floe during impact on a rigid obstacle, and the effect of the floe size and its initial velocity on the failure pattern and the history of the contact force.
Abstract: Fracture of quasibrittle materials with a large zone of distributed cracking is simulated by the particle model (discrete element method). The particles at the microlevel interact only by central forces with a prescribed force-displacement or stress-strain relation, which exhibits postpeak softening and is characterized by microstrength and microfracture energy. It is shown that a regular lattice, even though capable of closely approximating isotropic elastic properties, exhibits strong directional bias favoring propagation along a few preferred directions. A randomly generated particle model has no such bias. With a proper choice of the microlevel constitutive law, it can realistically simulate fracture of an ice floe during impact on a rigid obstacle. Explicit integration of the equations of motion is used to simulate the impact process and to explore the effect of the floe size and its initial velocity on the failure pattern and the history of the contact force.

99 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered a cohesive crack model in which the cohesive (crack-bridging) stress is a specified decreasing function of the crack-opening displacement, and the conditions of stability loss of a structure with a growing cohesive crack are obtained from the condition of vanishing of the second variation of the complementary energy or the potential energy.
Abstract: The paper deals with a cohesive crack model in which the cohesive (crack-bridging) stress is a specified decreasing function of the crack-opening displacement. Under the assumption that no part of the crack undergoes unloading, the complementary energy and potential energy of an elastic structure which has a cohesive crack and is loaded by a flexible elastic frame is formulated using continuous influence functions representing compliances or stiffnesses relating various points along the crack. By variational analysis, in which the derivatives of the compliance or stillness functions with respect to the crack length are related to the crack-tip stress intensity factors due to various unit loads, it is shown that the minimizing conditions reduce to the usual compatibility or equilibrium equations for the cohesive cracks. The variational equations obtained can be used as a basis for approximate solutions. Furthermore, the conditions of stability loss of a structure with a growing cohesive crack are obtained from the condition of vanishing of the second variation of the complementary energy or the potential energy. They have the form of a homogeneous Fredholm integral equation for the derivatives of the cohesive stresses or crack opening displacements with respect to the crack length. Loadings with displacement control, load control, or through a flexible loading frame are considered. Extension to the analysis of size effect on the maximum load or maximum displacement are left to a subsequent companion paper.

60 citations


Journal ArticleDOI
TL;DR: In this paper, the size effect in the pullout strength of reinforcing bars embedded in concrete is investigated. But the authors focused on failures due solely to interface slip, with no cracking in the surrounding concrete, and the results of tests of geometrically similar specimens show that interfacial shear fracture causes a size effect on the nominal strength in pullout.
Abstract: Test results on the size effect in the pullout strength of reinforcing bars embedded in concrete are presented. Attention is focused on failures due solely to interface slip, with no cracking in the surrounding concrete. This type of failure is achieved by using smooth round bars and a sufficiently large ratio of bar diameter to embedment length. Elimination of cracking in the surrounding concrete makes it possible to study the characteristics of the interfacial shear fracture between steel and concrete. The results of tests of geometrically similar specimens show that interfacial shear fracture causes a size effect on the nominal strength in pullout. The size effect is found to be transitional between plastic failure (the current approach of concrete design codes, for which there is no size effect) and linear elastic fracture mechanics (for which the size effect is the maximum possible). This transitional size effect can be approximately described by the size effect law proposed by Bažant in 1984 for qua...

41 citations


Journal ArticleDOI
TL;DR: In this paper, a review of recent results on scaling offailure in structures made of quasibrittle materials, characterized by a large fracture process zone, and examining the question of possible role of the fractal nature of crack surfaces in the scaling is presented.
Abstract: The paper represents an extended text of a lecture presenting a review of recent results on scaling offailure in structures made ofquasibrittle materials, characterized by a large fracture process zone, and examining the question of possible role of the fractal nature of crack surfaces in the scaling. The problem of scaling is approached through dimensional analysis, the laws of thermodynamics and asymptotic matching. Large-size and small-size asymptotic expansions of the size effect on the nominal strength of structures are given, for specimens with large notches (or traction-free cracks) as well as zero notches, and simple size effect formulas matching the required asymptotic properties are reported. The asymptotic analysis is carried out, in general, for fractal cracks, and the practically important case of nonfractal crack propagation is acquired as a special case. Regarding the fractal nature of crack surfaces in quasibrittle materials, the conclusion is that it cannot play a signification role in fracture propagation and the observed size effect. The reason why Weibull statistical theory of random material strength does not explain the size effect in quasibrittle failures is explained. Finally, some recent applications to fracture simulation by particle models (discrete element method) and to the determination of size effect and fracture characteristics of carbon-epoxy composite laminates are briefly reviewed.

39 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the time dependence of concrete fracture, and particularly the effect of loading rate, and 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.
Abstract: 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.

33 citations


Journal ArticleDOI
TL;DR: In this paper, the authors extended the analysis of size effect on strength and ductility of structures to the case of geometrically similar structures of different sizes, and the criterion of stability limit was transformed to an eigenvalue problem for a homogeneous Fredholm integral equation, with the structure size as the eigen value.
Abstract: The preceding paper is extended to the analysis of size effect on strength and ductility of structures. For the case of geometrically similar structures of different sizes, the criterion of stability limit is transformed to an eigenvalue problem for a homogeneous Fredholm integral equation, with the structure size as the eigenvalue. Under the assumption of a linear softening stress-displacem ent relation for the cohesive crack, the eigenvalue problem is linear. The maximum load of structure under load control, as well as the maximum deflection under displacement control (which characterizes ductility of the structure), can be solved explicitly in terms of the eigenfunction of the aforementioned integral equation. crack model is a nonlinear theory of fracture me­ chanics in which the condition of stability limit is expressed in terms of the singularity condition of the second variation of the energy potential with respect to cohesive stresses or crack­ opening displacements. Although the criterion of stability limit can also be formulated in terms of energy variation with respect to the crack length, the resulting equation is not very useful, since the energy release rate in the cohesive crack model de­ pends on the cohesive stresses or crack-opening displacements. For a given structure, the criterion of stability limit leads to a highly nonlinear equation for crack length. However, when a class of geometrically similar structures of different sizes is considered and the relative crack length is given, the criterion of stability limit can be treated as an equation for the structure size at which the stability limit occurs at the given relative crack length. In this manner, the criterion of the stability limit is transformed into an eigenvalue problem, with the structure size as the eigenvalue. In the special case of linear softening, the eigenvalue problem is linear. It can be solved independently of the solution of the cohesive crack model. Furthermore, the corresponding maximum value of the load or loading parameter can be expressed explicitly in terms of the eigenfunction. In this way, the size effect curve can be obtained readily, without having to calculate the load-deflection curves for structures of various sizes.

33 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of initiation of thermal or shrinkage cracks from the surface of a half plane is considered. But the authors assume linear elastic fracture behavior and assume that the stress at the surface reaches a given strength limit and after the initial cracks form, the energy release rate equals its given critical value.
Abstract: The paper deals with the problem of initiation of thermal or shrinkage cracks from the surface of a half plane. Linear elastic fracture behavior is assumed. The initial spacing and initial stable equilibrium length of parallel equidistant cracks emanating from the surface is determined from three conditions formulated in a preceding study of penetration of sea ice plate: (1) The stress at the surface reaches a given strength limit. (2) After the initial cracks form, the energy release rate equals its given critical value. (3) The finite energy release due to the initial crack jump equals the energy needed to form the crack (according to the given fracture energy of the material or fracture toughness). The problem is reduced to a singular integral equation which is solved numerically by Erdogan's method. The results of analysis appear to be compatible with the experimental evidence on thermal cracking in glass, and appear to give also reasonable predictions for thermal cracking on top of sea ice plates and shrinkage cracking in concrete.

26 citations


01 Jan 1995
TL;DR: BaZant et al. as mentioned in this paper presented a new mathematical model for propagation of part-through bending cracks in floating sea ice plate, which is a problem of considerable practical importance, for example, the load carrying capacity or penetration through the ice plate.
Abstract: AMD-Vol. 207, Ice Mechanics ASME 1995 PART-THROUGH BENDING CRACKS IN SEA ICE PLATES: MATHEMATICAL MODELING Zdenek P. BaZant, J.J.H. Kim and Yuan N. U Department of Civil Engineering Northwestern University Evanston, Illinois The paper presents a new mathematical model for propagation of part-through bending cracks in floating sea ice plate, which is a problem of considerable practical importance, for example, the load carrying capacity or penetration through the ice plate. After reviewing the previous work on propagation of through-cracks due to transverse loads, the three-dimensional problem of part-through cracks is simplified as two-dimensional using the well known approximation by line springs. These nonlinear springs describe the relation of rotation and additional in-plane expansion due to part-through crack to the bending moment and normal force transmitted through the crack. The problem of several radial cracks emanating from a small loaded area is analyzed. The bending and in-plane elastic responses of the floating plate are described by compliance functions. It is shown that the rotations across the crack cause the compression resultant in the plates and the neutral axis of the stress to shift above the mid-thickness of plates. This represents a dome effect which carries a significant part of the load. The profile of the crack depth propagating upward and the shape of the dome are calculated. A study of the failure loads and the size effect is left for a subsequent paper. INTRODUCTION Sea ice plates under vertical loading from above or from below often fail by propagation of radial cracks from the loaded area (Fig. 1). The maximum or failure load is reached when circumferential cracks start to form from the radial cracks. This type of failure is important for many commercial as well as defense applications. such as a submarine sail penetrating through the ice or an airplane landing on the ice. Due to these practical needs, this problem has been studied extensively for a long time. However, due to the relatively recent initiation of fracture mechanics of ice, the problem has been solved using a strength criterion or plastic limit analysis. Obviously, since ice is a quasibrittle material, the plastic limit analysis is unrealistic and more importantly, it does not capture the size effect on the nominal strength. In early studies of the penetration problem. the load capacity of a floating ice plate was determined by the tensile strength criterion (e.g. Bernstein. 1929). Nevel (1958) analyzed the strength of the ice plate assuming that the number of radial bending cracks is very large and that the ice plate is thus split into wedges of very small angle, which can be treated as beams of variable cross section. An excellent review of the early studies of the load capacity of the floating ice plate was given by Kerr (1975).

14 citations


01 Jan 1995
TL;DR: Bazant and Ozbolt as discussed by the authors reviewed two recent developments, one dealing with the formulation and application of a special new type of non local model for materials such as concrete, which is derived physically from microcrack interactions and numerical application of this model in finite element analysis.
Abstract: As it is now generally accepted, finite element analysis of distributed softening damage in quasi-brittle structures such as concrete structures cannot be based on a classical, that is, local, constitutive model of the material. Such a model Introduces incorrect excessive localizations, spurious size effect and spurious mesh sensitivity in finite element computations. To overcome these problems, the constitutive model mllst be supplemented with some sort of the so-called localization limiter. One effective type of the localization limiter is the nonlocal continuum. The present paper reviews two recent developments, one dealing with the formulation and application of a special new type of non local model for materials such as concrete, which is derived physically from microcrack interactions (Bazant, 1994; Bazant and Jirasek, 1994), and numerical application of this model in finite element analysis (Bazant and Ozbolt, 1994). The paper also briefly summaries a new nonlinear triaxial damage model based on the microplane concept utilizing the new idea of stress-strain boundaries (Bazant,