A 4-noded hybrid stress element with optimized stress for moderately thick and thin shallow shells

In this paper, a 4-noded hybrid stress finite element is developed for arbitrary plates and shallow shells. The finite element formulation is derived from the Hellinger-Reissner functional. In the process of the element derivation, the optimization principle of the numerical behaviour of multiple-variable FEM, in which an energy compatible condition is introduced to optimise the unconstrained stress trials, is employed. An appropriate unconstrained stress field is carefully chosen to validate the optimization. Because the transverse shear effect is taken into account, the element can be used for moderately thick and thin plates and shallow shells. Numerical study demonstrates that the present element passes the patch test and is of high accuracy and free of superfluous zero energy deformation modes. There is no shear locking in the limit of thin plates and shells.

[1]  M. Crisfield.  A quadratic mindlin element using shear constraints , 1984 .

[2]  E. Hinton,et al.  A family of quadrilateral Mindlin plate elements with substitute shear strain fields , 1986 .

[3]  Klaus-Jürgen Bathe,et al.  A study of three‐node triangular plate bending elements , 1980 .

[4]  Irwan Katili,et al.  A new discrete Kirchhoff-Mindlin element based on Mindlin–Reissner plate theory and assumed shear strain fields—part II: An extended DKQ element for thick-plate bending analysis , 1993 .

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

[6]  T. Pian,et al.  Rational approach for assumed stress finite elements , 1984 .

[7]  Ray W. Clough,et al.  Improved numerical integration of thick shell finite elements , 1971 .

[8]  K. Y. Sze,et al.  A quadratic assumed natural strain curved triangular shell element , 1999 .

[9]  C.-C. Wu,et al.  Dual zero energy modes in mixed/hybrid elements-definition, analysis and control , 1990 .

[10]  Rezak Ayad,et al.  A new hybrid‐mixed variational approach for Reissner–Mindlin plates. The MiSP model , 1998 .

[11]  K. Y. Sze,et al.  A stabilized eighteen-node solid element for hyperelastic analysis of shells , 2004 .

[12]  T. Pian.  Derivation of element stiffness matrices by assumed stress distributions , 1964 .

[13]  G R Heppler,et al.  A Mindlin element for thick and deep shells , 1986 .

[14]  Jean-Louis Batoz,et al.  Evaluation of a new quadrilateral thin plate bending element , 1982 .

[15]  J. A. Teixeira de Freitas,et al.  The hybrid stress model for mindlin-reissner plates based on a stress optimization condition , 1993 .

[16]  Medhat A. Haroun,et al.  Reduced and selective integration techniques in the finite element analysis of plates , 1978 .

[17]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[18]  Hai-chʿang Hu,et al.  Variational Principles of Theory of Elasticity with Applications , 1984 .

[19]  K. Y. Sze,et al.  An Explicit Hybrid Stabilized Eighteen-Node Solid Element for Thin Shell Analysis , 1997 .

[20]  Theodore H. H. Pian,et al.  Alternative ways for formulation of hybrid stress elements , 1982 .

[21]  Irwan Katili,et al.  A new discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fields. I - An extended DKT element for thick-plate bending analysis. II - An extended DKQ element for thick-plate bending analysis , 1993 .

[22]  T. Pian,et al.  A rational approach for choosing stress terms for hybrid finite element formulations , 1988 .

[23]  Y. K. Cheung,et al.  Hybrid quadrilateral element based on mindlin/reissner plate theory , 1989 .

[24]  Robert L. Spilker,et al.  A Serendipity cubic‐displacement hybrid‐stress element for thin and moderately thick plates , 1980 .

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