6. Beam and Plate Bending Problems

  1. Pavel Šolín

Published Online: 30 NOV 2005

DOI: 10.1002/0471764108.ch6

Partial Differential Equations and the Finite Element Method

Partial Differential Equations and the Finite Element Method

How to Cite

Šolín, P. (2005) Beam and Plate Bending Problems, in Partial Differential Equations and the Finite Element Method, John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/0471764108.ch6

Publication History

  1. Published Online: 30 NOV 2005
  2. Published Print: 4 NOV 2005

ISBN Information

Print ISBN: 9780471720706

Online ISBN: 9780471764106

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Summary

Chapter 6 is devoted to the derivation, analysis and finite element solution of fourth-order problems rooted in the bending of elastic beams and plates. Presented is the Bernoulli (biharmonic) beam model, its weak formulation, analysis of the existence and uniqueness of the solution, and discretization by means of lowest-order (cubic) and higher-order Hermite elements. Introduced are both the nodal and hierarchic Hermite elements, and their performance is compared. Attention is paid to the implementation aspects. Presented are the lowest-order (cubic) and higher-order nodal Hermite elements in 2D, and their conformity to the space of continuous functions is duscussed. The Reissner-Mindlin (thick plate) and Kirchhoff (thin plate) models are derived and their weak formulation is presented. Various types of boundary conditions are discussed, and the analysis of existence and uniqueness of solution is performed. The Kirchhoff model is discretized by means of the lowest-order (quintic) and higher-order nodal Argyris elements.