Vibration and Nonlinear Dynamics of Plates and Shells - Applications of Flat Triangular Finite Elements

Meilan Liu, Cho W. S. To

• Publisher: Bentham Science Publishers
• ISBN:
  9781608057719
• Topic: Tecnologia e ingegneria
• Subtopic: Ingegneria civile

This e-book focuses on the vibrational and nonlinear aspects of plate and shell structure dynamics by applying the finite element model. Specifically, shell finite elements employed in the computational studies included in this book are the mixed formulation based lower order flat triangular shell finite elements. Topics in the book are covered over nine chapters, including the theoretical background for the vibration analysis of plates and shells, vibration analysis of plate structures, shells with single curvature, shells with double curvatures, and box structures (single-cell and double-cell) and the theoretical development for the nonlinear dynamic analysis of plate and shell structures. In addition to presenting the steps in the derivations of the consistent element stiffness and mass matrices, constitutive relations of elastic materials and elasto-plastic materials with isotropic strain hardening, yield criterion, return mapping, configuration and stress updating strategies, and numerical algorithms are presented and discussed. The book is a suitable reference for advanced undergraduates and post-graduate level engineering students, research engineers, and scientists working in the field of applied physics and engineering.

Mixed Formulation Based Three-Node Flat Triangular Shell Elements for Nonlinear Dynamics

Abstract

To further the objective of investigating the vibration characteristics and dynamic responses of complicated shell structures with geometrical and material nonlinearities, this chapter presents the development of mixed formulation based three-node flat triangular shell elements suitable for the general nonlinear analysis of thin to moderately thick shells. Section 7.1 gives a brief outline of the features of the shell elements. Section 7.2 presents the incremental variational principle and its linearization. The derivations of the consistent element stiffness matrices and the consistent element mass matrices are dealt with in Sections 7.3 and 7.4. The constitutive relations of elastic and elasto-plastic materials with small as well as finite strain deformations are given in Section 7.5. The last two sections, Sections 7.6 and 7.7, respectively, are concerned with configuration and stress updating, and numerical algorithms.

Keywords: : Mixed formulation, three-node shell elements, dynamics, nonlinear, elastic and elasto-plastic.

7.1.. Introduction and Overview

This chapter presents the mixed formulation based three-node triangular flat shell elements in the context of general nonlinear dynamic analysis of structures. Some of the important features of the elements are as follows.

1. The updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are employed. The independently assumed fields include the incremental displacements and incremental strains. Accordingly, the incremental second Piola-Kirchhoff stress and the incremental Washizu strain are selected as the incremental stress and strain measures.
2. Two versions of the nonlinear element stiffness matrices are developed. These are the director version and the simplified version. In the director version, it is assumed that for every node on the shell mid-surface the director can be uniquely defined. The stiffness matrices are found to be dependent of the current position of the director. Thus, it requires the updating of the director at every time step. The simplified version, on the other hand, is useful for cases where the director is not unique, or is difficult to determine. For brevity, only the derivation of director version of the matrices is presented. The simplified version can be easily deducted from the director version.
3. To be consistent with element stiffness matrices, the element consistent mass matrices have their director version and simplified version, depending on whether the director is uniquely defined. The consistent mass matrix is defined with respect to the reference configuration. Therefore, it is to be calculated at every time step, since mass density, thickness of the shell, positions of mid-surface nodes, and directors for the director version of formulation, change as the shell deforms.

(iv) Material nonlinearity is of the elasto-plastic type with isotropic strain hardening. The J₂ flow theory of plasticity, in conjunction with Ilyushin's yield criterion, is employed. The non-layered approach is adopted, in order to simplify the derivation of stiffness matrix and to facilitate the acquisition of explicit expressions for the stiffness matrices.

(v) For homogeneous, isotropic and linearly elastic materials, as well as elasto-plastic materials with isotropic strain hardening, formulations f...