Geometric nonlinear analysis of plates and cylindrical shells via a linearly conforming radial point interpolation method

In this paper, the linearly conforming radial point interpolation method is extended for geometric nonlinear analysis of plates and cylindrical shells. The Sander’s nonlinear shell theory is utilized and the arc-length technique is implemented in conjunction with the modified Newton–Raphson method to solve the nonlinear equilibrium equations. The radial and polynomial basis functions are employed to construct the shape functions with Delta function property using a set of arbitrarily distributed nodes in local support domains. Besides the conventional nodal integration, a stabilized conforming nodal integration is applied to restore the conformability and to improve the accuracy of solutions. Small rotations and deformations, as well as finite strains, are assumed for the present formulation. Comparisons of present solutions are made with the results reported in the literature and good agreements are obtained. The numerical examples have demonstrated that the present approach, combined with arc-length method, is quite effective in tracing the load-deflection paths of snap-through and snap-back phenomena in shell problems.

[1]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[2]  Hirohisa Noguchi,et al.  Element free analyses of shell and spatial structures , 2000 .

[3]  K. Y. Dai,et al.  A radial point interpolation method for simulation of two-dimensional piezoelectric structures , 2003 .

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

[5]  R. Hauptmann,et al.  A SYSTEMATIC DEVELOPMENT OF 'SOLID-SHELL' ELEMENT FORMULATIONS FOR LINEAR AND NON-LINEAR ANALYSES EMPLOYING ONLY DISPLACEMENT DEGREES OF FREEDOM , 1998 .

[6]  Chang-Koon Choi,et al.  An effective four node degenerated shell element for geometrically nonlinear analysis , 1996 .

[7]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part II: the linear theory; computational aspects , 1989 .

[8]  J. L. Sanders,et al.  NONLINEAR THEORIES FOR THIN SHELLS , 1963 .

[9]  Guangyao Li,et al.  A linearly conforming point interpolation method (LC‐PIM) for three‐dimensional elasticity problems , 2007 .

[10]  T. Belytschko,et al.  Nodal integration of the element-free Galerkin method , 1996 .

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

[12]  W. Hao,et al.  Numerical simulations of large deformation of thin shell structures using meshfree methods , 2000 .

[13]  Sivakumar Kulasegaram,et al.  Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations , 2000 .

[14]  K. Y. Dai,et al.  A LINEARLY CONFORMING RADIAL POINT INTERPOLATION METHOD FOR SOLID MECHANICS PROBLEMS , 2006 .

[15]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model , 1990 .

[16]  Guirong Liu,et al.  A point interpolation meshless method based on radial basis functions , 2002 .

[17]  T. Belytschko,et al.  Analysis of thin shells by the Element-Free Galerkin method , 1996 .

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

[19]  Anthony N. Palazotto,et al.  Nonlinear Analysis of Shell Structures , 1992 .

[20]  J. N. Reddy,et al.  Large deformation analysis of functionally graded shells , 2007 .

[21]  R. L. Harder,et al.  A proposed standard set of problems to test finite element accuracy , 1985 .

[22]  A. B. Sabir,et al.  The applications of finite elements to large deflection geometrically nonlinear behaviour of cylindrical shells , 1972 .

[23]  Wing Kam Liu,et al.  Stress projection for membrane and shear locking in shell finite elements , 1985 .

[24]  M. A. Crisfield,et al.  Non-Linear Finite Element Analysis of Solids and Structures: Advanced Topics , 1997 .

[25]  K. Y. Sze,et al.  Popular benchmark problems for geometric nonlinear analysis of shells , 2004 .

[26]  K. Y. Sze,et al.  An eight‐node hybrid‐stress solid‐shell element for geometric non‐linear analysis of elastic shells , 2002 .

[27]  K. Y. Sze,et al.  A hybrid stress nine-node degenerated shell element for geometric nonlinear analysis , 1999 .

[28]  P. Bergan,et al.  Nonlinear Shell Analysis Using Free Formulation Finite Elements , 1986 .