A class of strain-displacement elements in upper bound limit analysis

Athanasios Makrodimopoulos*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


The main restriction in applying the upper bound theorem in limit analysis by means of the finite element method is that once the displacement field has been discretized, the flow rule and the boundary conditions should be satisfied everywhere throughout the discretized structure. This is to secure mathematically that the result will be a rigorous upper bound of the exact limit load multiplier. So far, only the linear and the quadratic displacement elements with straight sides, which result in constant and linear strains, respectively, satisfy this requirement. In this article, it is proven both theoretically and practically that there is a general class of strain-displacement triangular elements with straight sides which provide rigorous upper bounds. The interpolation scheme is based on the Bernstein polynomials, and there is no upper restriction of the polynomial order. Both continuous and discontinuous displacements can be considered. The efficiency is examined through plane strain and Kirchhoff plate examples. The generalization to 3D (i.e., tetrahedral elements with plane faces) and other structural conditions is straightforward.

Original languageEnglish
Pages (from-to)3681-3712
Number of pages32
JournalInternational Journal for Numerical Methods in Engineering
Issue number16
Publication statusPublished - 30 Aug 2022


  • Bernstein polynomials
  • limit analysis
  • upper bound


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