Shakedown analysis of modified Bree problems involving thermal membrane stress and generalized loading conditions

Hongchen Bao, Jun Shen, Yinghua Liu, Haofeng Chen

Research output: Contribution to journalArticlepeer-review


For industrial components such as pressure vessels and piping systems, it is important to determine the shake- down domains of structures under complex variable thermo-mechanical loads to avoid low cycle fatigue due to alternating plasticity or incremental plastic collapse caused by ratcheting. In this paper, the interaction among three common types of stresses are considered based on a plane model, namely, mechanical membrane stress, thermal membrane stress and thermal bending stress. Strict shakedown analysis is performed based on the Linear Matching Method under multiple variable mechanical and thermal loads. Three-dimensional shakedown do- mains for three types of modified Bree problems involving thermal membrane stress and generalized loading conditions are given for the first time, and the three-dimensional shakedown boundaries are expressed as two- dimensional parametric equations by introducing a new parameter called "secondary membrane bending ratio" R. By comparing the 3S criterion plane with the newly obtained 3D shakedown boundaries, the conservatism and non-conservatism of the 3S criterion are discussed under different loading paths. As an extension of the 3S criterion, a new and economical criterion on elastic shakedown assessment is proposed for generalized thermo-mechanical loading. The proposed shakedown boundary parametric equations and shakedown checking method can provide guidance for engineering design and safety assessment.
Original languageEnglish
Article number104432
Number of pages9
JournalInternational Journal of Pressure Vessels and Piping
Early online date28 Apr 2021
Publication statusPublished - 31 Aug 2021


  • elastic shakedown
  • two-plane model
  • linear matching method
  • 3S criterion
  • thermal membrane and bending stress

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