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

This chapter consists of three sections. Objectives and scope of the book are given in Section 1.1. Section 1.2 outlines the organization. Notes on computer programming are included in the last section.

In order to investigate the vibration characteristics and dynamic responses of complicated shell structures with geometrical and material nonlinearities, it is essential to formulate shell finite elements that are easy to use, accurate, effective, and applicable to thin as well as moderately thick shells. This chapter presents the development of the mixed formulation or hybrid strain based three-node flat triangular shell elements, with a particular emphasis on the linear analysis of thin to moderately thick shells. Section 2.1 gives a brief introduction and an outline of the features of the shell elements. Section 2.2 deals with the derivation of consistent stiffness and mass matrices of a particular element. In Section 2.3, results and discussions pertaining to rigid-body modes, patch test, and mesh topology are presented. Concluding remarks are given in Section 2.4.

This chapter is concerned with the vibration analysis of linear plates employing the mixed formulation based three-node flat triangular shell elements in Chapter 2. Square, circular and skew plates are studied. In particular, circular plates with various aspect ratios and boundary conditions are included. As a special example to reveal the features of the membrane component of the shell elements a plane stress problem is presented.

This chapter is concerned with vibration analysis of shell structures having single curvature. The latter include cylindrical curved panel with rectangular projection, cylindrical curved panel with trapezoidal projection, the Scordelis-Lo roof, and cylindrical shell clamped at both ends. In this latter case the effect of aspect ratio is also studied.

Vibration characteristics of shell structures with double curvatures are studied in this chapter. These shell structures include the spherical caps, spherical panel of square projection, hemispherical panel, and hemispherical shell. The computed results are compared with available data in the literature whenever they are available.

Vibration characteristics of box structures, generally known as rectangular prismatic shell structures, are studied in this chapter. To limit the scope in this book, only single-cell box structure and double-cell box structures are presented. In these two types of shell structures the first ten frequency parameters or natural frequencies along with their corresponding mode shapes are included.

To further theobjective 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 threenode 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.

This chapter deals with the nonlinear dynamic responses of structures with flat mid-surfaces, such as plates and boxes by employing the mixed formulation based three-node flat triangular nonlinear shell elements presented in Chapter 7. Geometrical nonlinearity due to large deformation, material nonlinearity due to elastic-plastic material behaviour, and various loading situation including non-conservative loads, will be investigated.

This chapter is concerned with the nonlinear dynamic responses of structures of single curvature and of double curvatures, examples of which include cylindrical shells or panels, spherical caps, and hemispheres. Geometrical nonlinearity due to large deformation, material nonlinearity due to elastic-plastic material behaviour, and various loading situation including non-conservative pressure loads, will be examined.