Vertex effect in strain-softening concrete at rotating principal axes

Ferhun C. Caner*, Zdeněk P. Bažant, Jan Červenka

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

39 Scopus citations


The inelastic behavior of concrete for highly nonproportional loading paths with rotating principal stress axes is studied. Test cylinders are first loaded in compression under uniaxial stress and then torsion is applied at constant axial displacement. Proportional compressive-torsional loading tests are also carried out for comparison. The tests demonstrate that the response of concrete for load increments parallel in the stress space to the current yield surface is highly inelastic (i.e., much softer than elastic) in the peak load range and especially in the postpeak range. The classical tensorial models of plasticity type incorrectly predict for such load increments the elastic stiffness. The experiments are simulated by three-dimensional finite element analysis using the microplane model M4, in which the stress-strain relations are characterized not by tensors but by vectors of stress and strain on planes of various orientations in the material. It is shown that the observed vertex effect is correctly predicted by this model, with no adjustment of its material parameters previously calibrated by other test results. The experiments are also simulated by a state-of-the-art fracture-plastic model of tensorial type and it is found that the vertex effect cannot be reproduced at all, although an adjustment of one material parameter suffices to obtain a realistic postpeak slope and achieve a realistic overall response. What makes the microplane model capable of capturing the vertex effect is the existence of more than 60 simultaneous yield surfaces. Capturing the vertex effect is important for highly nonproportional loading with rotating principal stress axes, which is typical of impact and penetration of missiles, shock, blasts, and earthquake.

Original languageEnglish (US)
Pages (from-to)24-33
Number of pages10
JournalJournal of Engineering Mechanics
Issue number1
StatePublished - Jan 2002


  • Concrete
  • Inelastic action
  • Nonproportional loads
  • Rotation

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering


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