Variational approach to probabilistic finite elements

W. K. Liu, G. H. Besterfield, T. Belytschko

Research output: Contribution to journalArticlepeer-review

62 Scopus citations


A probabilistic Hu-Washizu variational principle (PHWVP) formulation for the probabilistic finite element method (PFEM) is presented. The formulation is developed for nonlinear elasticity using the Saint Venant Kirchhoff model. The PHWVP allows incorporation of probabilistic distributions for the compatibility condition, constitutive law, equilibrium, domain, and boundary conditions into the PFEM. Solution of the three stationary conditions for the compatibility relation, constitutive law, and equilibrium yield the variations in displacement, strain, and stress. Finally, the statistics such as expectation, autocovariance, and correlation of displacement, strain, and stress are determined. Thus, a probabilistic analysis can be performed in which all aspects of the problem are treated as random variables and/or fields. The Hu-Washizu variational formulation is amenable to many conventional finite element codes, thereby enabling the extension of present codes to probabilistic problems.

Original languageEnglish (US)
Pages (from-to)2115-2133
Number of pages19
JournalJournal of Engineering Mechanics
Issue number12
StatePublished - 1989

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering


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