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In shear failure, reinforced concrete (RC) beams always develop, in a stable manner, a finite length crack before the maximum load is reached. Thus, the crack tip location cannot sample a large volume of material with random strength because a small region in which the crack tip can lie is fixed by fracture mechanics. Consequently, the size effect on the mean strength cannot be statistical. It must be predominantly energetic or deterministic and, thus, must follow the Type-2 size effect law. What has not yet been clarified is the size effect on the coefficient of variation (CoV) of beam strength, which is important for anchoring the probability distribution of shear strength and choosing the safety factor. In this study, we run thousands of explicit finite element simulations using Abaqus-Explicit version 6.14 with microplane model M7, each with a random input of material strength and Young's modulus for each finite element in the structure. The CoV of beam strength is found to decrease with the structure size when geometrically similar beams are compared, although the CoV tends to a constant for large sizes. This size effect on the CoV is similar to that in ductile failure governed by a Gaussian distribution of strength and contrasts with that in brittle failures following the Weibull distribution, for which the CoV is size independent. To characterize the size dependence of the strength CoV, an analytical formula is developed based on the statistics of the sample quantiles of a series of random variables. [ABSTRACT FROM AUTHOR]

Understanding the tail behavior of the probability distribution of structural strength is of paramount importance for reliability-based design of engineering structures. For brittle and quasibrittle structures, the tail distribution not only determines the design strength at a low failure probability, but also governs the functional form of the strength distribution of large-size structures. There exists clear evidence that the Weibull distribution is applicable to brittle structures. Based on the theory of extreme value statistics, the applicability of Weibull distribution suggests that the tail distribution must follow a power law. The justification of the power-law tail distribution was first proposed by Freudenthal based on an assumed type of flaw statistics and linear elastic fracture mechanics of non-interacting flaws. A series of recent studies suggested that the power-law tail can be explained by the transition rate theory governing the statistics of material failure at the nano-scale. In this study, we investigate the tail distribution by considering the randomness in both material strength and applied stress field. The present analysis adopts a nonlocal strength-based failure criterion, in which the spatial variability of the material strength is represented by an autocorrelated random field. The failure statistics of the structure is calculated as a first passage probability. We analyze this problem in 1D, 2D and 3D settings, and the results indicate that in all cases the tail distribution of the nominal structural strength follows a power law. It is shown that the power-law tail behavior of strength distribution stems from the tail distribution of material strength. The flaw statistics introduces additional randomness to the nominal structural strength, but does not dictate the power-law form of its tail distribution. [ABSTRACT FROM AUTHOR]

Anchored by the theory of extreme value statistics, Weibull distribution is the most widely used mathematical model for strength distribution of brittle structures. In a series of recent studies, a finite weakest-link model was developed for strength distribution of quasi-brittle structures, and the classical Weibull distribution was shown to represent the large-size asymptote of the model. By employing a length scale, the finite weakest-link model is capable of capturing correctly the size effects on both the probability distribution and the mean value of structural strength. However, the connection of this length scale with the basic material properties is still missing. This study investigates the relationship between the length scale of the finite weakest-link model and the material length scales by analyzing the size effect on the mean structural strength. The mathematical form of this relationship is derived through dimensional analysis. To validate the model, a set of mean size effect curves is obtained through stochastic simulations, which use a nonlinear constitutive model involving both the Irwin characteristic length and the crack band width. The internal length scale of the weakest-link model is determined by optimum fitting of the benchmark size effect curves in the small-size range. Furthermore, the effect of stress field on this internal length scale is studied by considering three different loading configurations. The present analysis reveals the importance of the mean size effect analysis for the calibration of finite weakest-link model. [ABSTRACT FROM AUTHOR]

Many modern engineering structures are made of brittle heterogenous (quasibrittle) materials, such as concrete, composites, and tough ceramics. Recent studies have shown that the strength distribution of quasibrittle structures varies from a Gaussian distribution modified by a power-law tail to a Weibull distribution as the structure size increases, and such size dependence can be well captured by a finite weakest link model. This paper investigates the implication of the size-dependent strength distribution on the reliability-based analysis and design of quasibrittle structures by considering a linear failure limit state. The size effect on the statistical properties of structural strength directly leads to a size-dependent Cornell index, which could give a reasonable estimation of the failure risk in some cases. To improve the prediction of the structural failure risk, the size dependence of the strength distribution is further incorporated into the computation of the Hasofer–Lind index, where it is shown that the transformation method for the non-Gaussian random structural strength is dependent on the structure size and geometry. By assuming that the applied load follows a Gaussian distribution, approximate size effect equations are developed for the central and nominal safety factors through asymptotic matching. The proposed formulation of the size effect on reliability indices and safety factors is applied to analyze the failure of the Malpasset arch dam. [ABSTRACT FROM AUTHOR]

The design of various engineering structures, such as buildings, infrastructure, aircraft, ships, as well as microelectronic components and medical implants, must ensure an extremely low probability of failure during their service lifetime. Since such a low probability is beyond the means of histogram testing, we must rely on some physically based probabilistic model for the statistics of structural lifetime. Attention is focused on structures consisting of quasibrittle materials. These are brittle materials with inhomogeneities that are not negligible compared with the structure size, as exemplified by concrete, fiber composites, tough ceramics, rocks, sea ice, bone, wood, and many more at the micro- or nano-scale. This paper presents a finite weakest-link model of the fatigue lifetime of quasibrittle structures that fail at the fracture of one representative volume element (RVE). In this model, the probability distribution of critical stress amplitude is first derived by assuming a prescribed number of loading cycles and a fixed stress ratio. The probability distribution of fatigue lifetime is then deduced from the probability distribution of critical stress amplitude through the Paris law for fatigue crack growth. It is shown that the present theory matches well with the experimentally measured lifetime histograms of various engineering and dental ceramics, which systematically deviate from the two-parameter Weibull distribution. The theory indicates that the mean fatigue lifetime of quasibrittle structures must strongly depend on the structure size and geometry. Finally, the present model indicates that the probability distribution of fatigue lifetime can be determined from the mean size effect analysis. [ABSTRACT FROM PUBLISHER]

This paper presents a general size effect equation for the strength of hybrid structures, which are made of two dissimilar quasibrittle materials with a thin and weak bimaterial interface. Depending on the material mismatch and structure geometry, a singular stress field could occur at the bimaterial corner. For structures with strong stress singularities, an energetic size effect is derived based on the equivalent linear elastic fracture mechanics and asymptotic matching. For structures without stress singularities, a finite weakest link model is adopted to derive the size effect. A general scaling equation that bridges the limits of strong and zero stress singularities is formulated by combining the energetic scaling of fracture of the bimaterial corner and the finite weakest link model. [ABSTRACT FROM AUTHOR]

Abstract: The statistics of structural lifetime under constant load are related to the statistics of structural strength. The safety factors applied to structural strength must ensure failure probability no larger than , which is beyond the means of direct verification by histogram testing. For perfectly brittle materials, extrapolation from the mean and variance to such a small tail probability is no problem because it is known that the Weibull distribution applies. Unfortunately, this is not possible for quasibrittle materials because the type of cumulative distribution function (cdf) has been shown to vary with structure size and shape. These are materials with inhomogeneities and fracture process zones (FPZ) that are not negligible compared to structural dimensions. A probabilistic theory of strength of quasibrittle structures failing at macro-crack initiation, which can be experimentally verified and calibrated indirectly, has recently been deduced from the rate of jumps of atomic lattice cracks governed by activation energy barriers. This paper extends this nano-mechanics based theory to the distribution of structural lifetime. Based on the cdf of strength and a power law for subcritical crack growth rate, the lifetime cdf of quasibrittle structures under constant loads is derived. The lifetime cdf is shown to depend strongly on the structure size as well as geometry. It is found that, for the creep rupture case, the mean structural lifetime exhibits a very strong size effect, much stronger than the size effect on the mean structure strength. The theory also implies temperature dependence of the lifetime cdf. For various quasibrittle materials, such as industrial ceramics and fiber composites, it is demonstrated that the proposed theory correctly predicts the experimentally observed deviations of lifetime histograms from the Weibull distribution. [Copyright &y& Elsevier]

Abstract: Ensuring a small enough failure probability is important for the design and selection of restorative dental ceramics. For this purpose, the two-parameter Weibull distribution, which is based on the weakest link model with infinitely many links, is usually adopted to model the strength distribution of dental ceramics. This distribution has been thoroughly validated for perfectly brittle materials. However, dental ceramics are generally quasibrittle because the inhomogeneity size is not negligible compared to the size of the ceramic part. For such materials, the experimental histograms of many quasibrittle materials have been shown to exhibit strong deviations from the two-parameter Weibull distribution. As a remedy, the three-parameter Weibull distribution, which has a nonzero threshold, has been proposed. However, the improvement of the fits of histograms of quasibrittle materials has been only partial. Instead of making the threshold non-zero, the correct remedy is to consider the weakest link model to have a finite number of links, each of them representing one finite-size representative volume element of material. This model has recently been justified on the basis of the probability of random jumps of atomic lattice cracks over the activation energy barriers on the free energy potential of the lattice. It is shown that, in similarity to other quasibrittle materials, this new model allows excellent fits of the experimental strength histograms of various types of dental ceramics. [Copyright &y& Elsevier]