A continuum-based mixed shell element for shakedown analysis

Abstract The objective of this paper is to present a new triangular shell element for shakedown analysis. The element is based on a mixed variational formulation with displacement/velocity and stress fields independently interpolated. We also employ the continuum based (CB) approach which allows the definition of the yield function at the continuum level without any approximation in terms of generalized stresses. The formulation of the proposed element is presented in details after a brief review of relevant classical shakedown theory concepts. The performance of the element is assessed by means of a set of selected representative examples. The numerical tests include: (i) analyses of thin and thick-walled straight pipes under combined loads, (ii) the shakedown analysis of a pipe bend under internal pressure and in-plane bending and (iii) limit analysis of a cylinder–cylinder intersection subjected to bending and internal pressure.

[1]  Manolis Papadrakakis,et al.  Postbuckling performance of the TRIC natural mode triangular element for Isotropic and laminated composite shells , 1998 .

[2]  H. Nguyen-Dang,et al.  A primal–dual algorithm for shakedown analysis of structures , 2004 .

[3]  Manolis Papadrakakis,et al.  Nonlinear dynamic analysis of shells with the triangular element TRIC , 2003 .

[4]  E. A. de Souza Neto,et al.  A continuum-based mixed axisymmetric shell element for limit and shakedown analysis , 2014 .

[5]  Hany F. Abdalla,et al.  Shakedown Limit Loads for 90 Degree Scheduled Pipe Bends Subjected to Steady Internal Pressure and Cyclic Bending Moments , 2011 .

[6]  Klaus-Jürgen Bathe,et al.  A triangular six-node shell element , 2009 .

[7]  J. Simon,et al.  Numerical lower bound shakedown analysis of engineering structures , 2011 .

[8]  Thomas J. R. Hughes,et al.  Isogeometric shell analysis: The Reissner-Mindlin shell , 2010 .

[9]  Manfred Staat,et al.  Upper bound limit and shakedown analysis of shells using the exact Ilyushin yield surface , 2008 .

[10]  Castrenze Polizzotto.  On the Conditions to Prevent Plastic Shakedown of Structures: Part II—The Plastic Shakedown Limit Load , 1993 .

[11]  Thomas J. R. Hughes,et al.  Nonlinear finite element analysis of shells: Part I. three-dimensional shells , 1981 .

[12]  J. L. Silveira,et al.  Quadratic velocity‐linear stress interpolations in limit analysis , 2014 .

[13]  Phill-Seung Lee,et al.  Measuring the convergence behavior of shell analysis schemes , 2011 .

[14]  Donald Mackenzie,et al.  Shakedown and limit analysis of 90° pipe bends under internal pressure, cyclic in-plane bending and cyclic thermal loading , 2011 .

[15]  Quoc Son Nguyen,et al.  Stability and Nonlinear Solid Mechanics , 2000 .

[16]  Leone Corradi,et al.  A triangular finite element for sequential limit analysis of shells , 2004 .

[17]  G. Maier,et al.  3.12 – Direct Methods of Limit and Shakedown Analysis , 2003 .

[18]  Jacov A. Kamenjarzh,et al.  Limit Analysis of Solids and Structures , 1996 .

[19]  Manolis Papadrakakis,et al.  Elasto-plastic analysis of shells with the triangular element TRIC , 2002 .

[20]  K. Bathe,et al.  Development of MITC isotropic triangular shell finite elements , 2004 .

[21]  Jose Luis Silveira,et al.  An algorithm for shakedown analysis with nonlinear yield functions , 2002 .

[22]  W. Han,et al.  Plasticity: Mathematical Theory and Numerical Analysis , 1999 .

[23]  W. T. Koiter.  General theorems for elastic plastic solids , 1960 .

[24]  D. Owen,et al.  Computational methods for plasticity : theory and applications , 2008 .

[25]  Manfred Staat,et al.  Probabilistic limit and shakedown analysis of thin plates and shells , 2009 .

[26]  José Herskovits,et al.  An iterative algorithm for limit analysis with nonlinear yield functions , 1993 .

[27]  Antonio J. Gil,et al.  Analysis of a continuum-based beam element in the framework of explicit-FEM , 2009 .

[28]  Manfred Staat,et al.  Numerical methods for limit and shakedown analysis : deterministic and probabilistic problems , 2003 .

[29]  Leone Corradi,et al.  Post-collapse analysis of plates and shells based on a rigid-plastic version of the TRIC element , 2003 .

[30]  J. Argyris,et al.  The TRIC shell element: theoretical and numerical investigation , 2000 .

[31]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[32]  Manfred Staat,et al.  LISA : a European project for FEM-based limit and shakedown analysis , 2001 .

[33]  J. C. Simo,et al.  On stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization , 1989 .

[34]  Manfred Staat,et al.  Analysis of pressure equipment by application of the primal-dual theory of shakedown , 2006 .

[35]  E. Ramm,et al.  Shell theory versus degeneration—a comparison in large rotation finite element analysis , 1992 .

[36]  N. Zouain.  Shakedown and Safety Assessment , 2004 .

[37]  O. C. Zienkiewicz,et al.  Analysis of thick and thin shell structures by curved finite elements , 1970 .

[38]  A. V. Lyamin,et al.  Bounds to Shakedown Loads for a Class of Deviatoric Plasticity Models , 2007 .

[39]  J. Z. Zhu,et al.  The finite element method , 1977 .

[40]  E. Christiansen.  Limit analysis of collapse states , 1996 .

[41]  K. Bathe.  Finite Element Procedures , 1995 .

[42]  H. Nguyen-Dang,et al.  Direct Finite Element Kinematical Approaches in Limit and Shakedown Analysis of Shells and Elbows , 2000 .

[43]  David W. Murray,et al.  Nonlinear Finite Element Analysis of Steel Frames , 1983 .