Monday, February 20, 2006

Size Effect on Probability Distribution of Structural Strength

By Zdenék P. Bažant

Recently we have demonstrated the need for a fundamental revision of reliability concepts and design codes for quasibrittle heterogeneous structures, such as concrete structures failing due to concrete fracture or crushing (rather than reinforcement yielding), or large load-bearing fiber-composite structures for ships or aircraft, sea ice plates, etc. While ductile failure occurs simultaneously along the failure surface and is characterized by absence of size effect and Gaussian distribution of structural strength, quasibrittle failures propagates, exhibits a strong size effect and follows at large sizes extreme value statistics of weakest-link model, which leads to Weibull distribution of structural strength (provided that failure occurs at macro-crack initiation). Based on small- and large-size asymptotic properties recently deduced from cohesive crack model and nonlocal Weibull theory, the transition of cumulative probability distribution function (cdf) of structural strength from small to large sizes is modeled by a chain of fiber bundles, in which each fiber with Weibull type tail of strength probability corresponds to one dominant micro-bond within a representative volume element (RVE) in a brittle lower-scale microstructure. The cdf of each fiber (or micro-bond) properties can be deduced from Maxwell-Boltzmann distribution of the atomic thermal energies, which brings about the rescaling of cdf according to temperature, load duration and moisture content. A fascinating by-product of the analysis, with physical implications, is that the Weibull modulus is equal to the number of dominant (simultaneously failing) micro-bonds in an RVE. The structural strength distribution is based on chain-of-bundles model, for which a composite cdf with a Weibull tail grafted on a Gaussian core is proposed. For the smallsize limit, the core is totally Gaussian, and for the large-size limit totally Weibull. In between, the grafting point moves right as the Gaussian core shrinks with increasing size. This causes that the distance from the mean to a point of tolerable failure probability (such as 0.00000001) nearly doubles as the size of quasibrittle structure increases. Consequently, the understrength factor in design codes must be made size dependent. So must the Cornell and Hasofer-Lind reliability indices. Their reformulation (implying replacement of FORM with ‘EVRM’) is proposed. Inseparable from these effects are further problems due to ‘covert’ understrength factors implied in brittle failure provisions of concrete design codes, as well as an irrational hidden size effect implied by excessive load factor for self weight acting alone. To improve design safety and efficiency, experts in statistical reliability and fracture mechanics will need to collaborate to tackle these problems comprehensively. (Read more …)


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