Mixed finite element models for plate bending analysis theory

Abstract The theoretical background of mixed finite element models, in general for nonlinear problems, is briefly reexamined. In the first part of the paper, several alternative “mixed” formulations for 3-D continua undergoing large elastic deformations under the action of time dependent external loading are outlined and are examined incisively. It is concluded that mixed finite element formulations, wherein the interpolants for the stress field satisfy only a part of the domain equilibrium equations, are not only consistent from a theoretical standpoint but are also preferable from an implementation point of view. In the second part of the paper, alternative variational bases for the development of thin-plate elements are presented and discussed in detail. In light of this discussion, it is concluded that the “bad press” generated in the past concerning the practical relevance of the so-called assumed stress hybrid finite element model is not justified. Moreover, the advantages of this type of elements as compared with the “assumed displacement” or alternative mixed elements are outlined.

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

[2]  S. Kelsey,et al.  Triangular Plate Bending Elements with Enforced Compatibility , 1970 .

[3]  D. A. Kross,et al.  Finite-Element Analysis of Thin Shells , 1968 .

[4]  Richard M. Barker,et al.  Finite Element Bending Analysis of Reissner Plates , 1970 .

[5]  Satyanadham Atluri,et al.  On the hybrid stress finite element model for incremental analysis of large deflection problems , 1973 .

[6]  Ahmed K. Noor,et al.  Mixed models and reduced/selective integration displacement models for nonlinear shell analysis , 1982 .

[7]  William Prager.  Variational principles for elastic plates with relaxed continuity requirements , 1968 .

[8]  D. Karamanlidis,et al.  Large Deflection Finite Element Analysis of Pre- and Postcritical Response of Thin Elastic Frames , 1981 .

[9]  R T Severn,et al.  THE FINITE ELEMENT METHOD FOR FLEXURE OF SLABS WHEN STRESS DISTRIBUTIONS ARE ASSUMED. , 1966 .

[10]  Geometrisch nichtlineare Berechnung von ebenen Stabwerken auf der Grundlaǵe eines ǵemischt-hybriden Finite-Elemente-Verfahrens , 1981 .

[11]  S. T. Mau,et al.  Derivation of geometric stiffness and mass matrices for finite element hybrid models , 1974 .

[12]  Yoshio Ando,et al.  Some finite element solutions for plate bending problems by simplified hybrid displacement method , 1972 .

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

[14]  G. T. Will,et al.  A quintic conforming plate bending triangle , 1978 .

[15]  O. C. Zienkiewicz,et al.  Analysis of thick and thin shell structures by curved finite elements , 1970 .

[16]  Leonard R. Herrmann,et al.  Finite-Element Bending Analysis for Plates , 1967 .

[17]  Abigail F. Davis,et al.  IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY , 1984 .

[18]  K. Bell.  A refined triangular plate bending finite element , 1969 .

[19]  Geir Horrigmoe,et al.  Finite element instability analysis of free-form shells , 1977 .

[20]  Richard H. Gallagher,et al.  A critical assessment of the Simplified Hybrid Displacement Method , 1977 .

[21]  Hans Rudolf Schwarz,et al.  Methode der finiten Elemente , 1984 .

[22]  T. Pian,et al.  Large deflection analysis of thin elastic structures by the assumed stress hybrid finite element method , 1977 .

[23]  Satya N. Atluri,et al.  On Some New General and Complementary Energy Theorems for the Rate Problems in Finite Strain. Classical Elastoplasticity , 1980 .

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

[25]  Isaac Fried,et al.  Residual energy balancing technique in the generation of plate bending finite elements , 1974 .

[26]  H. Parisch,et al.  A critical survey of the 9-node degenerated shell element with special emphasis on thin shell application and reduced integration , 1979 .

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

[28]  Yoshio Ando,et al.  A new variational functional for the finite-element method and its application to plate and shell problems , 1972 .

[29]  H. Bufler.  Die verallgemeinerten Variationsgleichungen der dünnen Platte bei Zulassung diskontinuierlicher Schnittkräfte und Verschiebungsgrößen , 1970 .

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

[31]  Ian M. Smith.  A finite element analysis for “moderately thick” rectangular plates in bending , 1968 .

[32]  Pin Tong,et al.  New displacement hybrid finite element models for solid continua , 1970 .

[33]  R. Schröder,et al.  Schalenelemente in gemischer Darstellung: Theorie — Kritik — Beispiele , 1978 .

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

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