## 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 |

### Bibliographical note

Funding Information:Financial support by the U.S. National Science Foundation under Grant CMS-0556323 to Northwestern University is gratefully acknowledged. The work was started during J.E.’s visiting appointment at Northwestern University, half of which was supported by a Fulbright-Masaryk grant from the Fulbright Foundation and half by the aforementioned grant. Additional financial support was provided by Ministry of Education, Youth and Sports of the Czech Republic under project number ME10030.

## Keywords

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