A new approach to reduce membrane and transverse shear locking for one‐point quadrature shell elements: linear formulation

In the last decade, one-point quadrature shell elements attracted many academic and industrial researchers because of their computational performance, especially if applied for explicit finite element simulations. Nowadays, one-point quadrature finite element technology is not only applied for explicit codes, but also for implicit finite element simulations, essentially because of their efficiency in speed and memory usage as well as accuracy. In this work, one-point quadrature shell elements are combined with the enhanced assumed strain (EAS) method to develop a finite element formulation for shell analysis that is, simultaneously, computationally efficient and more accurate. The EAS method is formulated to alleviate locking pathologies existing in the stabilization matrices of one-point quadrature shell elements. An enhanced membrane field is first constructed based on the quadrilateral area coordinate method, to improve element's accuracy under in-plane loads. The finite element matrices were projected following the work of Wilson et al. (Numerical and Computer Methods in Structural Mechanics, Fenven ST et al. (eds). Academic Press: New York, 1973; 43–57) for the incompatible modes approach, but the present implementation led to more accurate results for distorted meshes because of the area coordinate method for quadrilateral interpolation. The EAS method is also used to include two more displacement vectors in the subspace basis of the mixed interpolation of tensorial components (MITC) formulation, thus increasing the dimension of the null space for the transverse shear strains. These two enhancing vectors are shown to be fundamental for the Morley skew plate example in particular, and in improving the element's transverse shear locking behaviour in general. Copyright © 2005 John Wiley & Sons, Ltd.

[1]  T. Belytschko,et al.  Projection schemes for one-point quadrature shell elements , 1994 .

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

[3]  Patrick Pol.  Simulation du comportement élasto-plastique de coques minces par éléments finis , 1992 .

[4]  Eric P. Kasper,et al.  A mixed-enhanced strain method , 2000 .

[5]  N. F. Knight,et al.  Raasch Challenge for Shell Elements , 1997 .

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

[7]  Chang-Koon Choi,et al.  Efficient remedy for membrane locking of 4-node flat shell elements by non-conforming modes , 2003 .

[8]  Nielen Stander,et al.  An efficient 4‐node 24 D.O.F. thick shell finite element with 5‐point quadrature , 1995 .

[9]  Robert L. Taylor,et al.  A mixed-enhanced strain method: Part II: Geometrically nonlinear problems , 2000 .

[10]  Milos Kojic,et al.  Enhanced 8-node three-dimensional solid and 4-node shell elements with incompatible generalized displacements , 1994 .

[11]  M. L. Bucalém,et al.  Higher‐order MITC general shell elements , 1993 .

[12]  Jr. N. Knight.  The Raasch challenge for shell elements , 1996 .

[13]  Jeong Whan Yoon,et al.  One point quadrature shell elements for sheet metal forming analysis , 2005 .

[14]  J. C. Simo,et al.  On stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization , 1989 .

[15]  Robert L. Taylor,et al.  Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems☆ , 1993 .

[16]  J. C. Simo,et al.  A CLASS OF MIXED ASSUMED STRAIN METHODS AND THE METHOD OF INCOMPATIBLE MODES , 1990 .

[17]  R. M. Natal Jorge,et al.  On the use of an enhanced transverse shear strain shell element for problems involving large rotations , 2003 .

[18]  R. M. Natal Jorge,et al.  New enhanced strain elements for incompressible problems , 1999 .

[19]  Renato Natal Jorge,et al.  Development of shear locking‐free shell elements using an enhanced assumed strain formulation , 2002 .

[20]  Michael D. Gilchrist,et al.  Analysis of Rayleigh-Lamb wave scattering by a crack in an elastic plate , 1997 .

[21]  David J. Benson,et al.  Implementation of a modified Hughes-Liu shell into a fully vectorized explicit finite element code , 1985 .

[22]  Brian L. Kemp,et al.  A four‐node solid shell element formulation with assumed strain , 1998 .

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

[24]  Song Cen,et al.  Membrane elements insensitive to distortion using the quadrilateral area coordinate method , 2004 .

[25]  L. Morley.  Skew plates and structures , 1963 .

[26]  T. Belytschko,et al.  A stabilization procedure for the quadrilateral plate element with one-point quadrature , 1983 .

[27]  Robert D. Cook,et al.  Further development of a three‐node triangular shell element , 1993 .

[28]  Jean-Louis Batoz,et al.  Modelisation des structures par elements finis. Volume 3 , 1990 .

[29]  J. Yoon,et al.  One point quadrature shell element with through-thickness stretch , 2005 .

[30]  Anders Eriksson,et al.  An alternative assumed strain method , 1999 .

[31]  H. Parisch,et al.  An investigation of a finite rotation four node assumed strain shell element , 1991 .

[32]  J. M. A. César de Sá,et al.  Enhanced transverse shear strain shell formulation applied to large elasto‐plastic deformation problems , 2005 .

[33]  R. MacNeal,et al.  Finite Elements: Their Design and Performance , 1993 .

[34]  K. Bathe,et al.  A continuum mechanics based four‐node shell element for general non‐linear analysis , 1984 .

[35]  Hou-Cheng Huang,et al.  Static and Dynamic Analyses of Plates and Shells: Theory, Software and Applications , 1989 .

[36]  O. C. Zienkiewicz,et al.  Reduced integration technique in general analysis of plates and shells , 1971 .

[37]  Frédéric Barlat,et al.  Development of a one point quadrature shell element for nonlinear applications with contact and anisotropy , 2002 .

[38]  T. Hughes.  Generalization of selective integration procedures to anisotropic and nonlinear media , 1980 .

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

[40]  Patrick Massin,et al.  Nine node and seven node thick shell elements with large displacements and rotations , 2002 .

[41]  Jerry I. Lin,et al.  Explicit algorithms for the nonlinear dynamics of shells , 1984 .

[42]  Edward L. Wilson,et al.  A robust quadrilateral membrane finite element with drilling degrees of freedom , 1990 .

[43]  Hou-Cheng Huang,et al.  Static and Dynamic Analyses of Plates and Shells , 1989 .

[44]  Ekkehard Ramm,et al.  EAS‐elements for two‐dimensional, three‐dimensional, plate and shell structures and their equivalence to HR‐elements , 1993 .

[45]  Edward L. Wilson,et al.  Incompatible Displacement Models , 1973 .

[46]  E. Wilson,et al.  A non-conforming element for stress analysis , 1976 .