Limit and Shakedown Analysis of Circular Tube Containing External Pit

Xianfeng Wang*, 1, Feng Xing1, Mingde Xue2, Bingye Xu2
1 College of Civil Engineering, Shenzhen University, Shenzhen 518060, PR China
2 Department of Engineering Mechanics Tsinghua University, Beijing 100081, PR China

© 2012 Wang et al.

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: ( This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

* Address correspondence to this author at the College of Civil Engineering, Shenzhen University, Shenzhen 518060, P.R. China; Tel: +86-755- 26534712(office); Fax: +86-755-26534021; E-mail:


The circular tubes containing external pit defects are analyzed in a lower-bound finite element computational form based on the static shakedown theorem. The shakedown analysis has not been commonly used in the engineering due to the large amount of computations. To overcome the numerical difficulties, a temperature parameter method is used, in which a pseudo-temperature field is applied to the structure and the resulting self-equilibrium thermoelastic stress is treated as the residual stress field which is used in the analysis. The pseudo temperature is assumed as a harmonic function satisfying the uniqueness theorem, therefore the nodal temperature matrix of the whole structure can be expressed by the boundary nodal temperature matrix. The nonlinear yield condition is piece-wise linearized so that the shakedown analysis is transformed into a linear programming problem in which the strategic variable is boundary nodal temperature and objective variable is the loading multiplier. The relations of limit and shakedown pressures to geometric parameters of various defects are presented.

Keywords: Shakedown analysis, thermo elastic stress, residual stress, circular tube, mathematical programming.