Upper Bound Shakedown Analysis of Plates Utilizing the C $$^{1}$$ 1 Natural Element Meth
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ISSN 1860-2134
Upper Bound Shakedown Analysis of Plates Utilizing the C1 Natural Element Method Shutao Zhou1
Yinghua Liu2 Binjie Ma1 Chuantao Hou1 Bing Wu1 Kelin Rong1
Yatang Ju1
(1 Beijing Institute of Structure and Environment Engineering, Beijing 100076, China) ( Department of Engineering Mechanics, AML, Tsinghua University, Beijing 100084, China) 2
Received 12 July 2020; revision received 10 September 2020; Accepted 15 September 2020 c The Chinese Society of Theoretical and Applied Mechanics 2020
ABSTRACT This paper proposes a numerical solution method for upper bound shakedown analysis of perfectly elasto-plastic thin plates by employing the C1 natural element method. Based on the Koiter’s theorem and von Mises yield criterion, the nonlinear mathematical programming formulation for upper bound shakedown analysis of thin plates is established. In this formulation, the trail function of residual displacement increment is approximated by using the C1 shape functions, the plastic incompressibility condition is satisfied by introducing a constant matrix in the objective function, and the time integration is resolved by using the K¨ onig’s technique. Meanwhile, the objective function is linearized by distinguishing the non-plastic integral points from the plastic integral points and revising the objective function and associated equality constraints at each iteration. Finally, the upper bound shakedown load multipliers of thin plates are obtained by direct iterative and monotone convergence processes. Several benchmark examples verify the good precision and fast convergence of this proposed method.
KEY WORDS Shakedown analysis, Kinematic theorem, Thin plate, C1 natural element method, Direct iterative algorithm
1. Introduction Thin plate structures are widely used in aerospace and pressure vessel engineering. Studying the load-carrying capacity and revealing the safety margin are very important issues for ensuring the safety and reliability of such plates. When the actual engineering structures made up of perfectly elasto-plastic materials are subjected to cyclic loads, cumulative plastic damage or alternating plastic damage will occur [1]. Shakedown analysis is an effective and direct method to investigate the structural plastic failure behavior under cyclic loads, which is built on Melan’s theorem (static or lower bound theorem) [2] and Koiter’s theorem (kinematic or upper bound theorem) [3]. The maximum shakedown load is obtained by optimizing the statically admissible residual stress field in the lower bound shakedown theorem, and the minimum shakedown load is evaluated by revising the supposed kinematically admissible strain field in the upper bound shakedown theorem. Plastic shakedown analysis has the advantage of strong practical applicability and maneuverability and plays an important role in the design and analysis of engineering structure [1, 4].
Corresponding author. E-mail: [email protected]
ACTA MECHANICA SOLIDA SINICA
Shakedown analysis is an extension of limit analysis, and the inve
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