### Abstract

Engineering structures must be designed for an extremely low failure probability, P_{f} < 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 language | English (US) |
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Pages (from-to) | 888-899 |

Number of pages | 12 |

Journal | Journal of Engineering Mechanics |

Volume | 138 |

Issue number | 7 |

DOIs | |

State | Published - 2012 |

### Keywords

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

### ASJC Scopus subject areas

- Mechanics of Materials
- Mechanical Engineering