Computation of Probability Distribution of Strength of Quasibrittle Structures Failing at Macrocrack Initiation

Abstract

Engineering structures must be designed for an extremely low failure probability, P f < 10 À6 . To determine the corresponding structural strength, a mechanics-based probability distribution model is required. Recent studies have shown that quasibrittle structures that fail at the macrocrack initiation from a single representative volume element (RVE) can be statistically modeled as a finite chain of RVEs. It has further been demonstrated that, based on atomistic fracture mechanics and a statistical multiscale transition model, the strength distribution of each RVE can be approximately described by a Gaussian distribution, onto which a Weibull tail is grafted at a point of the probability about 10 À4 to 10 À3 . The model implies that the strength distribution of quasibrittle structures depends on the structure size, varying gradually from the Gaussian distribution modified by a far-left Weibull tail applicable for small-size structures, to the Weibull distribution applicable for large-size structures. Compared with the classical Weibull strength distribution, which is limited to perfectly brittle structures, the grafted Weibull-Gaussian distribution of the RVE strength makes the computation of the strength distribution of quasibrittle structures inevitably more complicated. This paper presents two methods to facilitate this computation: (1) for structures with a simple stress field, an approximate closed-form expression for the strength distribution based on the Taylor series expansion of the grafted WeibullGaussian distribution; and (2) for structures with a complex stress field, a random RVE placing method based on the centroidal Voronoi tessellation. Numerical examples including three-point and four-point bend beams, and a two-dimensional analysis of the ill-fated Malpasset dam, show that Method 1 agrees well with Method 2 as well as with the previously proposed nonlocal boundary method.

Effect of Stress Singularity Magnitude on Scaling of Strength of Quasi-Brittle Structures

“…(8) does not apply as D → 0, which makes sense because the continuum model breaks down for structures of very small size. A recent study (Le et al 2012) has shown that for small and intermediate-size structures the size effect derived from this finite weakest-link model with the use of elastic stresses agrees well with the prediction from the nonlinear deterministic calculation. This is because the mean size effect behavior for small-size and intermediate-size structures is mainly caused by the operative stress redistribution mechanism, which can be well predicted by a nonlinear deterministic calculation.…”

“…This is because the mean size effect behavior for small-size and intermediate-size structures is mainly caused by the operative stress redistribution mechanism, which can be well predicted by a nonlinear deterministic calculation. At the same time, this mechanism can also be captured by the finite weakest-link model, where the statistical multiscale transition model used for the formulation of the cdf of RVE strength consists of statistical bundles and chains that represent the damage localization and load redistribution mechanisms at different scales (albeit only the elastic stresses are used) (Le et al 2011(Le et al , 2012. For large-size structures, the zone of stress redistribution is negligible compared with the structure size, and the size effect is mainly caused by the randomness of material strength, which cannot be captured by the deterministic calculation.…”

Design of quasibrittle materials and structures to optimize strength and scaling at probability tail: an apercu

“…The slip led to a vertical crack in the 8 m thick shell of the arch of the dam, and the statistical Type 1 size effect must have come into play. The conclusion from the analysis in [29,125] (improving on [126]) is that the tolerable abutment displacement should have been one half of that considered in design, which was at a time when the size effect was unknown.…”

Mechanistic Modeling of Strength Distribution of Quasibrittle Structures and Its Implication for Structural Reliability

“…The failure state for the arch can simply be written as σ N − σ L < 0. Previous studies [11,25] led to the following parameters for the RVE strength distribution of concrete: m = 24, s 0 = 2.12 MPa, µ G = 2.91 MPa, δ G = 0.44 MPa, ω = 0.15, and a RVE size of 280 mm. However, the cdf of σ L , which is determined by the statistics of abutment movement, is currently not available.…”