Limit analysis of plates and slabs using a meshless equilibrium formulation

A meshless Element‐Free Galerkin (EFG) equilibrium formulation is proposed to compute the limit loads which can be sustained by plates and slabs. In the formulation pure moment fields are approximated using a moving least‐squares technique, which means that the resulting fields are smooth over the entire problem domain. There is therefore no need to enforce continuity conditions at interfaces within the problem domain, which would be a key part of a comparable finite element formulation. The collocation method is used to enforce the strong form of the equilibrium equations and a stabilized conforming nodal integration scheme is introduced to eliminate numerical instability problems. The combination of the collocation method and the smoothing technique means that equilibrium only needs to be enforced at the nodes, and stable and accurate solutions can be obtained with minimal computational effort. The von Mises and Nielsen yield criteria which are used in the analysis of plates and slabs, respectively, are enforced by introducing second‐order cone constraints, ensuring that the resulting optimization problem can be solved using efficient interior‐point solvers. Finally, the efficacy of the procedure is demonstrated by applying it to various benchmark plate and slab problems. Copyright © 2010 John Wiley & Sons, Ltd.

[1]  Rudolf Wolfensberger.  Traglast und optimale Bemessung von Platten , 1964 .

[2]  B. D. Veubeke.  Displacement and equilibrium models in the finite element method , 1965 .

[3]  O. Zienkiewicz,et al.  Strain-energy bounds in finite-element analysis by slab analogy , 1967 .

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

[5]  E. Anderheggen,et al.  Finite element limit analysis using linear programming , 1972 .

[6]  H. Chan,et al.  The collapse load of reinforced concrete plate , 1972 .

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

[8]  E. Faccioli,et al.  A finite element, linear programming methods for the limit analysis of thin plates , 1973 .

[9]  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.

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

[11]  J Munro,et al.  YIELD LINE METHOD BY FINITE ELEMENTS AND LINEAR PROGRAMMING , 1978 .

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

[13]  Steen Krenk,et al.  Limit Analysis and Optimal Design of Plates with Equilibrium Elements , 1994 .

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

[15]  Satya N. Atluri,et al.  A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method , 1998 .

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

[17]  Ted Belytschko,et al.  Regularization of material instabilities by meshfree approximations with intrinsic length scales , 2000 .

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

[19]  O. C. Zienkiewicz.  Displacement and equilibrium models in the finite element method by B. Fraeijs de Veubeke, Chapter 9, Pages 145–197 of Stress Analysis, Edited by O. C. Zienkiewicz and G. S. Holister, Published by John Wiley & Sons, 1965 , 2001 .

[20]  Marc Duflot,et al.  Dual analysis by a meshless method , 2002 .

[21]  Kristian Krabbenhoft,et al.  Lower bound limit analysis of slabs with nonlinear yield criteria , 2002 .

[22]  Joost Meyboom,et al.  Limit analysis of reinforced concrete slabs , 2002 .

[23]  Kristian Krabbenhoft,et al.  A general non‐linear optimization algorithm for lower bound limit analysis , 2003 .

[24]  Ni Sheng,et al.  Stabilized conforming nodal integration: exactness and variational justification , 2004 .

[25]  Dongdong Wang,et al.  Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation , 2004 .

[26]  B. Moran,et al.  Stabilized conforming nodal integration in the natural‐element method , 2004 .

[27]  Gui-Rong Liu,et al.  An Introduction to Meshfree Methods and Their Programming , 2005 .

[28]  Azim Eskandarian,et al.  Meshless Methods in Solid Mechanics , 2006 .

[29]  S. Sloan,et al.  Formulation and solution of some plasticity problems as conic programs , 2007 .

[30]  Chris Martin,et al.  Upper bound limit analysis using simplex strain elements and second‐order cone programming , 2007 .

[31]  Jaime Peraire,et al.  Mesh adaptive computation of upper and lower bounds in limit analysis , 2008 .

[32]  Mikael W. Braestrup,et al.  Yield line theory and concrete plasticity , 2008 .

[33]  Zhangzhi Cen,et al.  Lower‐bound limit analysis by using the EFG method and non‐linear programming , 2008 .

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

[35]  Mogens Peter Nielsen,et al.  Limit Analysis and Concrete Plasticity , 2010 .