Computation of probability distribution of strength of quasibrittle structures failing at macrocrack initiation

Jia Liang Le, Jan Eliáš, Zdenek P. Bažant*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Scopus citations


Engineering structures must be designed for an extremely low failure probability, Pf < 10-6. 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 10-4 to 10-3. 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 Weibull-Gaussian 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.

Original languageEnglish (US)
Pages (from-to)888-899
Number of pages12
JournalJournal of Engineering Mechanics
Issue number7
StatePublished - 2012


  • Composites
  • Concrete structures
  • Finite weakest link model
  • Fracture
  • Representative volume element
  • Strength statistics
  • Structural safety

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering


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