@incollection{TranTranMatthiesetal.2017,
  author    = {Tran, N. T. and Tran, Thanh Ngoc and Matthies, M. G. and Stavroulakis, G. E. and Staat, Manfred},
  title     = {Shakedown Analysis Under Stochastic Uncertainty by Chance Constrained Programming},
  series = {Advances in Direct Methods for Materials and Structures},
  booktitle = {Advances in Direct Methods for Materials and Structures},
  publisher = {Springer},
  address   = {Cham},
  isbn      = {978-3-319-59810-9},
  doi       = {10.1007/978-3-319-59810-9_6},
  pages     = {85 -- 103},
  year      = {2017},
  abstract  = {In this paper we propose a stochastic programming method to analyse limit and shakedown of structures under uncertainty condition of strength. Based on the duality theory, the shakedown load multiplier formulated by the kinematic theorem is proved actually to be the dual form of the shakedown load multiplier formulated by static theorem. In this investigation a dual chance constrained programming algorithm is developed to calculate simultaneously both the upper and lower bounds of the plastic collapse limit and the shakedown limit. The edge-based smoothed finite element method (ES-FEM) with three-node linear triangular elements is used for structural analysis.},
  language  = {en}
}
@article{NguyenXuanRabczukNguyenThoietal.2011,
  author    = {Nguyen-Xuan, H. and Rabczuk, T. and Nguyen-Thoi, T. and Tran, Thanh Ngoc and Nguyen-Thanh, N.},
  title     = {Computation of limit and shakedown loads using a node-based smoothed finite element method},
  series = {International Journal for Numerical Methods in Engineering},
  volume    = {90},
  journal   = {International Journal for Numerical Methods in Engineering},
  number    = {3},
  publisher = {Wiley},
  address   = {Weinheim},
  issn      = {1097-0207},
  doi       = {10.1002/nme.3317},
  pages     = {287 -- 310},
  year      = {2011},
  abstract  = {This paper presents a novel numerical procedure for computing limit and shakedown loads of structures using a node-based smoothed FEM in combination with a primal-dual algorithm. An associated primal-dual form based on the von Mises yield criterion is adopted. The primal-dual algorithm together with a Newton-like iteration are then used to solve this associated primal-dual form to determine simultaneously both approximate upper and quasi-lower bounds of the plastic collapse limit and the shakedown limit. The present formulation uses only linear approximations and its implementation into finite element programs is quite simple. Several numerical examples are given to show the reliability, accuracy, and generality of the present formulation compared with other available methods.},
  language  = {en}
}