Energetic-statistical size effect simulated by SFEM with stratified sampling and crack band model

Zdeněk P. Bažant*, Sze Dai Pang, Miroslav Vořechovský, Drahomír Novák

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

57 Scopus citations


The paper presents a model that extends the stochastic finite element method to the modelling of transitional energetic-statistical size effect in unnotched quasibrittle structures of positive geometry (i.e. failing at the start of macro-crack growth), and to the low probability tail of structural strength distribution, important for safe design. For small structures, the model captures the energetic (deterministic) part of size effect and, for large structures, it converges to Weibull statistical size effect required by the weakest-link model of extreme value statistics. Prediction of the tail of extremely low probability such as one in a million, which needs to be known for safe design, is made feasible by the fact that the form of the cumulative distribution function (cdf) of a quasibrittle structure of any size has been established analytically in previous work. Thus, it is not necessary to turn to sophisticated methods such as importance sampling and it suffices to calibrate only the mean and variance of this WE Two kinds of stratified sampling of strength in a finite element code are studied. One is the Latin hypercube sampling of the strength of each element considered as an independent random variable, and the other is the Latin square design in which the strength of each element is sampled from one overall cdf of random material strength. The former is found to give a closer estimate of variance, while the latter gives a cdf with smaller scatter and a better mean for the same number of simulations. For large structures, the number of simulations required to obtain the mean size effect is greatly reduced by adopting the previously proposed method of random property blocks. Each block is assumed to have a homogeneous random material strength, the mean and variance of which are scaled down according to the block size using the weakest-link model for a finite number of links. To check whether the theoretical cdf is followed at least up to tail beginning at the failure probability of about 0.01, a hybrid of stratified sampling and Monte Carlo simulations in the lowest probability stratum is used. With the present method, the probability distribution of strength of quasibrittle structures of positive geometry can be easily estimated for any structure size.

Original languageEnglish (US)
Pages (from-to)1297-1320
Number of pages24
JournalInternational Journal for Numerical Methods in Engineering
Issue number11
StatePublished - Sep 10 2007


  • Extreme value statistics
  • Quasibrittle fracture
  • Scaling
  • Size effect
  • Stochastic simulation

ASJC Scopus subject areas

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics


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