A discontinuous quasi-upper bound limit analysis approach with sequential linear programming mesh adaptation

Abstract In this paper, a simple discontinuous upper bound limit analysis approach with sequential linear programming mesh adaptation is presented. Rigid, infinitely strong triangular elements with both linear and Bezier curved edges are considered. A possible jump of velocities is allowed at the interfaces between contiguous elements, thus allowing plastic dissipation on curved interfaces. Bezier curved edges are used with the sole aim of improving the element performance when dealing with limit analysis problems involving curved sliding lines. The model performs poorly for unstructured meshes (i.e. at the initial iteration), being unable to reproduce the typical plastic deformation concentration on inclined slip lines. Therefore, an iterative mesh adaptation based on sequential linear programming is proposed. A simple linearization of the non-linear constraints is performed, allowing to treat the non-linear programming (NLP) problem with consolidated linear programming (LP) routines. The choice of inequalities constraints on elements nodes coordinates turns out to be crucial on the algorithm convergence. Several examples are treated, consisting in the determination of failure loads for ductile, purely cohesive and cohesive-frictional materials. The results obtained at the final iteration fit well, for all the cases analyzed, previously presented numerical approaches and analytical predictions.

[1]  C. Martin,et al.  Lower bound limit analysis of cohesive‐frictional materials using second‐order cone programming , 2006 .

[2]  Scott W. Sloan,et al.  A steepest edge active set algorithm for solving sparse linear programming problems , 1988 .

[3]  L. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communications.

[4]  S. Sloan,et al.  Upper bound limit analysis using discontinuous velocity fields , 1995 .

[5]  Edmund Christiansen,et al.  Automatic mesh refinement in limit analysis , 1999 .

[6]  K. M. Reinicke,et al.  Plastic limit analysis with an anisotropic, parabolic yield function , 1977 .

[7]  David Johnson.  Yield-line analysis by sequential linear programming , 1995 .

[8]  Knud Winstrup Johansen.  Yield-line theory , 1962 .

[9]  Scott W. Sloan,et al.  Lower bound limit analysis using non‐linear programming , 2002 .

[10]  P. Lourenço.  Computational strategies for masonry structures : Proefschrift , 1996 .

[11]  Gabriele Milani,et al.  Analysis of masonry structures: review of and recent trends in homogenization techniquesThis article is one of a selection of papers published in this Special Issue on Masonry. , 2007 .

[12]  V. Leitáo,et al.  Computer Graphics: Principles and Practice , 1995 .

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

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

[15]  Francis Tin-Loi,et al.  Performance of the p-version finite element method for limit analysis , 2003 .

[16]  Francis Tin-Loi,et al.  A simple mixed finite element for static limit analysis , 2006 .

[17]  Poul Colberg Olsen.  Rigid-plastic finite element analysis of steel plates, structural girders and connections , 2001 .

[18]  P. Lourenço,et al.  Multisurface Interface Model for Analysis of Masonry Structures , 1997 .

[19]  Francis Tin-Loi,et al.  Shakedown analysis using the p-adaptive finite element method and linear programming , 2007 .

[20]  L. Damkilde,et al.  Limit state analysis of reinforced concrete plates subjected to in-plane forces , 2000 .

[21]  Wai-Fah Chen,et al.  Plasticity for Structural Engineers , 1988 .

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

[23]  Scott W. Sloan,et al.  Lower bound limit analysis using finite elements and linear programming , 1988 .

[24]  Scott W. Sloan,et al.  Upper bound limit analysis using linear finite elements and non‐linear programming , 2001 .

[25]  David Johnson.  On the safety of the strip method for reinforced concrete slab design , 2001 .

[26]  Paulo B. Lourenço,et al.  Abbreviated Title : Homogenised limit analysis of masonry , failure surfaces , 2007 .

[27]  Scott W. Sloan,et al.  Lower bound limit analysis with adaptive remeshing , 2005 .

[28]  Scott W. Sloan,et al.  A new discontinuous upper bound limit analysis formulation , 2005 .

[29]  Knud D. Andersen,et al.  Computation of collapse states with von Mises type yield condition , 1998 .

[30]  Chris Martin,et al.  Upper bound limit analysis using discontinuous quadratic displacement fields , 2007 .

[31]  M. Ferris,et al.  Limit Analysis of Frictional Block Assemblies as a Mathematical Program With Complementarity Constraints , 2001 .

[32]  D. Johnson,et al.  Analysis of practical slab configurations using automated yield-line analysis and geometric optimization of fracture patterns , 1998 .

[33]  Steven K. Feiner,et al.  Computer Graphics - Principles and Practice, 3rd Edition , 1990 .

[34]  Zhangzhi Cen,et al.  Lower bound shakedown analysis by using the element free Galerkin method and non-linear programming , 2008 .

[35]  Gabriele Milani,et al.  Homogenised limit analysis of masonry walls, Part II: Structural examples , 2006 .