1. Mechanics of fracture and progressive cracking in concrete structures.- 1.1 Introduction.- 1.2 Blunt crack band theory.- 1.3 Finite element implementation.- 1.4 Energy considerations.- 1.5 Applications and practical analysis.- 1.6 Crack development.- 1.7 General model for progressive fracturing.- 1.8 Conclusion.- References.- 2. Scale effects in fracture of plain and reinforced concrete structures.- 2.1 Introduction.- 2.2 Dimensional analysis applied to plain and reinforced concrete structures.- 2.3 Fracture stability in plain and reinforced concrete elements.- 2.4 Hysteretic behaviour of reinforced concrete elements.- 2.5 Appendix: Dimensional independence.- References.- 3. Numerical methods to simulate softening and fracture of concrete.- 3.1 Introduction.- 3.2 The behaviour of concrete in a tension test.- 3.3 A comparison between concrete and steel.- 3.4 Tensile fracture zones.- 3.5 A general model for the tensile fracture of concrete.- 3.6 Material properties.- 3.7 FEM analysis of a fracture zone: coincident with predetermined crack path.- 3.8 FEM analysis of a fracture zone: not coincident with predetermined crack path.- 3.9 Some comparisons with test results.- 3.10 Summary and conclusions.- References.- 4. Numerical modeling of discrete crack propagation in reinforced and plain concrete.- 4.1 Introduction.- 4.2 Discrete crack models for concrete.- 4.3 The linear model.- 4.4 The nonlinear model.- 4.5 Crack propagation modeling: the future.- References.- 5. Fracture of steels for reinforcing and prestressing concrete.- 5.1 Introduction.- 5.2 Fracture.- 5.3 Fracture under extreme conditions.- 5.4 Fatigue.- 5.5 Environment sensitive cracking.- References.- Author's index.

Fracture Mechanics of Concrete: Structural Application and Numerical Calculation

The paper deals with numerical investigations of a deterministic and statistical size effect in granular bodies during quasi-static shearing of an infinite layer under plane strain conditions, free dilatancy and constant pressure. For a simulation of the mechanical behaviour of a cohesionless granular material during a monotonous deformation path, a micro-polar hypoplastic constitutive relation was used which takes into account particle rotations, curvatures, non-symmetric stresses, couple stresses and the mean grain diameter as a characteristic length. The proposed model captures the essential mechanical features of granular bodies in a wide range of densities and pressures with a single set of constants. In the paper, a deterministic and statistical size effect is analysed. The deterministic calculations were carried out with an uniform distribution of the initial void ratio for four different heights of the granular layer: 5, 50, 500 and 2000 mm. To investigate the statistical size effect, the Monte Carlo method was applied. The random distribution of the initial void ratio was assumed to be spatially correlated. Truncated Gaussian random fields were generated in a granular layer using an original conditional rejection method. The sufficient number of samples was determined by analysing the convergence of the outcomes. In order to reduce the number of realizations without losing the accuracy of the calculations, stratified and Latin hypercube methods were applied. A parametric analysis of these methods was also presented. Some general conclusions were formulated. Copyright © 2007 John Wiley & Sons, Ltd.

Deterministic and statistical size effect during shearing of granular layer within a micro‐polar hypoplasticity

Reinforced concrete (RC) deep beams mainly fail in shear, brittle and sudden in nature can lead to calamitous consequences. Thence, it is critical to determine the shear characteristics of RC deep beams accurately due to involving many parameters at same time. Some of the recent researches have shown that current equations for predicting ultimate shear strength are non-conservative when applied to high strength concrete (HSC) beams as well as some of design codes provisions. There are different approaches for analyzing the behavior of beams in shear.

Statistical Prediction Equations for RC Deep Beam Without Stirrups

During the last two decades, researches on quasibrittlc failure have led to major advances in the understanding and modeling of the energetic and statistical size effects in the mean statistical sense •2• Computational approaches and simple design code formulas giving better mean predictions have becn developed. It now appears, however, that the existing design codes• and standard practice for concrete and other quasibrittle structures also necessitate major revisions with regard to the effect of structure size and, more generally, degrec of brittleness, which depend not only on the size of the structure but also on its structure geometry.

Reliability of fracturing concrete structures and challenges of stochastic finite element modeling

This final report consists of one journal article and two dissertations. The journal article (designated Part 1) was prepared by Zdenek P. Bazant and Qiang Yu and was published in the ASCE Journal of Structural Engineering, December 2005. The paper discusses the size effect on shear strength of reinforced concrete beams without stirrups. The first dissertation (designated Part 2) was prepared by Quang Yu in June 2007. It discusses incorporating fracture mechanics into design practice through the size effect. The second dissertation (designated Part 3) was prepared by Sze-Dai Pang in December 2005. It discusses the probabilistic size effect in fracture mechanics of quasibrittle materials.

Introducing Size Effect into Design Practice and Codes for Concrete Infrastructure: Part 1: Designing Against Size Effect on Shear Strength of Reinforced Concrete Beams without Stirrups; Part 2: Size Effect and Design Safety in Concrete Structures Under Shear; Part 3: Probabilistic Size Effect in Fracture Mechanics of Quasi-Brittle Materials

An overview of several recent advances in reliability of quasibrittle structures, identifYing the main challenges and pointing out opportunities for further progress, is presented. The paramount importance of reliability analysis is obvious if one notes that the load factors and understrength (capacity reduction) factors, still essentially empirical and physically poorly understood, are far larger than the typical errors of modern computer analysis of structures. An effect of particular interest for structural reliability is the transition from quasi-plastic to brittle response with increasing structure size and the effect of structure type and geometry (or brittleness). To simulate this effect in the sense of extreme value statistics, a hierarchical model of nano-scale interatomic bonds is proposed, simulating one representative volume element (RVE) of the heterogenous material. A chainof-RYEs is used to model structures larger than one RVE size and the number of RVEs in the chain is related to structure size (as well as geometry). The model shows that the distribution of structural strength exhibits, at increasing size (or brittleness), a gradual transition from Gaussian distribution (except in far-out tails) to Weibull distribution. The fact that the distribution of interatomic bond strength must be governed by Maxwell-Boltzmann distribution of atomic energies and that the activation energy barriers are modified by applied stress leads to a physical proof that perfectly brittle failures must follow Weibull distribution. It also reveals a new physical meaning of Wei bull modulus-the number of dominant bonds that must fail simultaneously to cause an RVE to fail. Observing that, for Weibull distribution of typical variance, the point offailure probability such as 10-6 (a value tolerable in design) is about twice as far from the mean than it is for Gaussian distribution of the same mean and variance, one must conclude that the safety factor cannot be size-independent (as in current codes) but must be approximately doubled as the structure size changes from very small to very large. A way to capture it through reliability indices and through understrength factors applied to computational results is suggested. The brittleness factor, which needs to be made size dependent (contrary to current design codes), the material randomness factor, and the model error factor (both covertly implied by code provisions), need to be used as random variates in multi-variate Freudenthal-type reliability integral. At the same time, the size effect hidden in excessive self-weight factors currently imposed by codes has to be eliminated. The objective is to supplant realistic reliability estimates on deterministic computational mechanics. The paper concludes with examples of probabilistic structural analysis of some major structural disasters.

Computational structural reliability - A major challenge and opportunity for concrete and other quasibrittle structures

Throughout most of the 20th century, it was widely believed that the size effect on structural strength has a purely statistical origin, explained by extreme value statistics based on the weakest link model, and described by Weibull statistical theory of random strength. However, beginning with the first suggestions made already in the early 1970s, it gradually transpired that, in quasibrittle materials (i.e. heterogeneous brittle materials with a non-negligible fracture process zone), the mean size effect is essentially deterministic, stemming from energy release caused by stress redistribution in a structure prior to maximum load. The quasibrittle energetic scaling bridges three simple asymptotic power-law scalingsthose of linear elastic fracture mechanics, plasticity, and Weibull theory. Renormalization group transformation does not suffice to handle the transitional nature of this quasibrittle size effect, often spanning several orders of magnitude of size. As is now widely accepted, quasibrittle materials including concrete, rock, tough ceramics, sea ice, snow slabs and composites exhibit major size effects on the mean structural strength that are largely or totally deterministic in nature, being caused by stress redistribution and energy release associated with stable propagation of large fractures or with formation of large zones of distributed cracking.

Sze Dai Pang

Papers

Reliability of fracturing concrete structures and challenges of stochastic finite element modeling

Computational structural reliability - A major challenge and opportunity for concrete and other quasibrittle structures

Statistical Mechanics of Safety Factors and Size Effect in Quasibrittle Fracture