Upper bound limit analysis of plates utilizing the C1 natural element method

In this article, a novel numerical solution procedure is proposed to evaluate the upper bound limit load multipliers for thin plate problems, which incorporates the C1 natural element method (C1NEM) with a direct iteration algorithm. Due to its remarkable interpolation property to the nodal function and the nodal gradient values, the C1NEM with the C1- continuous trial function is used here to deal with the upper bound limit analysis problem of perfectly rigid-plastic plates. The relevant discrete mathematical programming formulation is established based on the kinematic theorem of plastic limit analysis, and a direct iteration algorithm with the advantages of simple solution formula and easy procedure implementation is presented to solve it. Several representative examples governed by the von Mises yield criterion are investigated. The numerical solutions obtained in this paper are reasonable and satisfactory, and are in good agreement with the previously reported results.

[1]  Matthew Gilbert,et al.  Limit analysis of plates using the EFG method and second‐order cone programming , 2009 .

[2]  H. G. Hopkins,et al.  Load-carrying capacities for circular plates of perfectly-plastic material with arbitrary yield condition , 1955 .

[3]  Guirong Liu,et al.  A point interpolation method for two-dimensional solids , 2001 .

[4]  B. Moran,et al.  Natural neighbour Galerkin methods , 2001 .

[5]  P. Fischer,et al.  On the C1 continuous discretization of non‐linear gradient elasticity: A comparison of NEM and FEM based on Bernstein–Bézier patches , 2010 .

[6]  C. Duarte,et al.  A Ck continuous generalized finite element formulation applied to laminated Kirchhoff plate model , 2009 .

[7]  R. M. Haythornthwaite,et al.  Limit Analysis of Rotationally Symmetric Plates and Shells , 1963 .

[8]  Ted Belytschko,et al.  Numerical Methods for the Limit Analysis of Plates , 1968 .

[9]  E. H. Mansfield.  Collapse pressures for rhombic plates , 2000 .

[10]  Jiun-Shyan Chen,et al.  A stabilized conforming nodal integration for Galerkin mesh-free methods , 2001 .

[11]  B. Moran,et al.  C1natural neighbor interpolant for partial differential equations , 1999 .

[12]  Hwang Kehchih,et al.  A mathematical programming algorithm for limit analysis , 1991 .

[13]  M. Gilbert,et al.  Adaptive element-free Galerkin method applied to the limit analysis of plates , 2010 .

[14]  Gerald E. Farin,et al.  Surfaces over Dirichlet tessellations , 1990, Comput. Aided Geom. Des..

[15]  M. Życzkowski,et al.  Combined Loadings in the Theory of Plasticity , 1981 .

[16]  Paul Steinmann,et al.  Natural element analysis of the Cahn–Hilliard phase-field model , 2010 .

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

[18]  T. Belytschko,et al.  Finite element derivative recovery by moving least square interpolants , 1994 .

[19]  C. Duarte,et al.  p-Adaptive Ck generalized finite element method for arbitrary polygonal clouds , 2007 .

[20]  Charles Massonnet,et al.  Plastic analysis and design of plates, shells and disks , 1972 .

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

[22]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[23]  E. N. Fox.  Limit analysis for plates: the exact solution for a clamped square plate of isotropic homogeneous material obeying the square yield criteron and loaded by uniform pressure , 1974, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[24]  M. Vicente da Silva,et al.  A finite element formulation of Mindlin plates for limit analysis , 2011 .

[25]  Hung Nguyen-Xuan,et al.  Upper and lower bound limit analysis of plates using FEM and second-order cone programming , 2010 .

[26]  Nguyen Dang Hung.  Direct limit analysis via rigid-plastic finite elements , 1976 .

[27]  Oden,et al.  An h-p adaptive method using clouds , 1996 .

[28]  Emilio Turco,et al.  Elasto-plastic analysis of Kirchhoff plates by high simplicity finite elements , 2000 .

[29]  Antonio Capsoni,et al.  Limit analysis of plates-a finite element formulation , 1999 .

[30]  Matthew Gilbert,et al.  Limit analysis of plates and slabs using a meshless equilibrium formulation , 2010 .

[31]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[32]  Michael L. Overton,et al.  Computing Limit Loads by Minimizing a Sum of Norms , 1998, SIAM J. Sci. Comput..

[33]  Chen Xuefeng,et al.  A new wavelet-based thin plate element using B-spline wavelet on the interval , 2007 .

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

[35]  Eugenio Oñate,et al.  Recent developments in finite element analysis , 1994 .

[36]  T. Belytschko,et al.  THE NATURAL ELEMENT METHOD IN SOLID MECHANICS , 1998 .

[37]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[38]  R. Melosh.  BASIS FOR DERIVATION OF MATRICES FOR THE DIRECT STIFFNESS METHOD , 1963 .

[39]  Miguel Ángel Martínez,et al.  Overview and recent advances in natural neighbour galerkin methods , 2003 .

[40]  Pasquale Vena,et al.  Limit analysis of orthotropic plates ? ? Partial results were presented at the Plasticity '00 Sympos , 2003 .