Non-linear strain-displacement equations exactly representing large rigid-body motions. Part I Timoshenko-Mindlin shell theory

Abstract The precise representation of arbitrarily large rigid-body motions in the displacement patterns of curved Timoshenko–Mindlin (TM) shell elements is considered. This consideration requires the development of the strain–displacement relationships of the finite deformation TM shell theory with regard to their consistency with the large rigid-body motions. For this purpose a refined TM theory of multilayered anisotropic shells undergoing finite deformations is elaborated. The transverse shear and transverse normal deformation response and bending–extension coupling are included. The fundamental unknowns consist of six displacements and 11 strains of the bottom and top surfaces of the shell, and 11 stress resultants. On the basis of this theory the simple and efficient mixed models are developed by using the incremental total Lagrangian formulation in conjunction with the Newton–Raphson method. The elemental arrays are derived applying the Hu–Washizu mixed variational principle. Numerical results are presented to demonstrate the high accuracy and effectiveness of the developed four-node shell elements and to compare their performance with other non-linear finite elements reported in the literature.

[1]  M. Crisfield.  A consistent co-rotational formulation for non-linear, three-dimensional, beam-elements , 1990 .

[2]  S. Klinkel,et al.  A finite rotation shell theory with application to composite structures , 1995 .

[3]  E. Stein,et al.  A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains , 1996 .

[4]  G. Kulikov,et al.  Investigation of Locally Loaded Multilayer Shells by a Mixed Finite-Element Method. 2. Geometrically Nonlinear Statement , 2002 .

[5]  K. Washizu.  Variational Methods in Elasticity and Plasticity , 1982 .

[6]  Yavuz Başar,et al.  Refined shear-deformation models for composite laminates with finite rotations , 1993 .

[7]  Carlos A. Felippa,et al.  A three‐dimensional non‐linear Timoshenko beam based on the core‐congruential formulation , 1993 .

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

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

[10]  Gennady M. Kulikov,et al.  Non-linear analysis of multilayered shells under initial stress , 2001 .

[11]  S. Plotnikova,et al.  Comparative analysis of two algorithms for numerical solution of nonlinearstatic problems for multilayered anisotropic shells of revolution 2. Account of transverse compression , 1999 .

[12]  D. Talaslidis,et al.  A Simple and Efficient Approximation of Shells via Finite Quadrilateral Elements , 1982 .

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

[14]  E. Ramm,et al.  Shear deformable shell elements for large strains and rotations , 1997 .

[15]  Mingrui Li,et al.  The finite deformation theory for beam, plate and shell. Part IV. The Fe formulation of Mindlin plate and shell based on Green-Lagrangian strain , 2000 .

[16]  Mikhail Itskov,et al.  Composite laminates: nonlinear interlaminar stress analysis by multi-layer shell elements , 2000 .

[17]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory , 1990 .

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

[19]  Erdogan Madenci,et al.  Nonlinear analysis of laminates through a mindlin-type shear deformable shallow shell element , 1997 .

[20]  S. W. Lee,et al.  An assumed strain finite element model for large deflection composite shells , 1989 .

[21]  E. Popov,et al.  Nonlinear finite element analysis of sandwich shells of revolution , 1972 .

[22]  Mingrui Li,et al.  The finite deformation theory for beam, plate and shell part III. The three-dimensional beam theory and the FE formulation , 1998 .

[23]  T. Hughes,et al.  Nonlinear finite element shell formulation accounting for large membrane strains , 1983 .

[24]  Ted Belytschko,et al.  Resultant-stress degenerated-shell element , 1986 .

[25]  C. W. S. To,et al.  Hybrid strain based geometrically nonlinear laminated composite triangular shell finite elements , 1999 .

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

[27]  S. Timoshenko,et al.  LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars , 1921 .

[28]  Sven Klinkel,et al.  A continuum based three-dimensional shell element for laminated structures , 1999 .

[29]  A. L. Goldenveizer.  THEORY OF ELASTIC THIN SHELLS , 1962 .

[30]  T. Belytschko,et al.  Physical stabilization of the 4-node shell element with one point quadrature , 1994 .

[31]  M. A. Crisfield,et al.  Some aspects of the non-linear finite element method , 1997 .

[32]  Mingrui Li,et al.  The finite deformation theory for beam, plate and shell Part I. The two-dimensional beam theory , 1997 .

[33]  Eduardo N. Dvorkin,et al.  A formulation of general shell elements—the use of mixed interpolation of tensorial components† , 1986 .

[34]  P. G. Bergan,et al.  Nonlinear analysis of free-form shells by flat finite elements , 1978 .

[35]  J. C. Simo,et al.  A three-dimensional finite-strain rod model. Part II: Computational aspects , 1986 .

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

[37]  Chahngmin Cho,et al.  An efficient assumed strain element model with six DOF per node for geometrically non‐linear shells , 1995 .

[38]  E. Stein,et al.  An assumed strain approach avoiding artificial thickness straining for a non‐linear 4‐node shell element , 1995 .

[39]  K. Bathe,et al.  Large displacement analysis of three‐dimensional beam structures , 1979 .

[40]  T. Hughes,et al.  Finite Elements Based Upon Mindlin Plate Theory With Particular Reference to the Four-Node Bilinear Isoparametric Element , 1981 .

[41]  Gennady M. Kulikov,et al.  Simple and effective elements based upon Timoshenko–Mindlin shell theory , 2002 .

[42]  Gennady M. Kulikov,et al.  Comparative analysis of two algorithms for numerical solution of nonlinear static problems for multilayer anisotropic shells of revolution. 1. Account of transverse shear , 1999 .

[43]  J. C. Simo,et al.  Geometrically non‐linear enhanced strain mixed methods and the method of incompatible modes , 1992 .

[44]  G. Kulikov.  Analysis of initially stressed multilayered shells , 2001 .

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

[46]  Christian Miehe,et al.  A theoretical and computational model for isotropic elastoplastic stress analysis in shells at large strains , 1998 .

[47]  M. Géradin,et al.  A beam finite element non‐linear theory with finite rotations , 1988 .

[48]  E. Ramm,et al.  On the physical significance of higher order kinematic and static variables in a three-dimensional shell formulation , 2000 .

[49]  H. Parisch.  A continuum‐based shell theory for non‐linear applications , 1995 .

[50]  Rüdiger Schmidt,et al.  Finite elements based on a first-order shear deformation moderate rotation shell theory with applications to the analysis of composite structures , 1997 .

[51]  G. Kulikov.  Refined Global Approximation Theory of Multilayered Plates and Shells , 2001 .