Introduction

Most of studies on large-amplitude (geometrically nonlinear) vibrations of circular cylindrical shells used Donnell's nonlinear shallow-shell theory to obtain the equations of motion, as shown in Chapter 5. Only a few used the more refined Sanders-Koiter or Flügge-Lur'e-Byrne nonlinear shell theories. The majority of these studies do not include geometric imperfections, and some of them use a single-mode approximation to describe the shell dynamics.

This chapter presents a comparison of shell responses to radial harmonic excitation in the spectral neighborhood of the lowest natural frequency computed by using five different nonlinear shell theories: (i) Donnell's shallow-shell, (ii) Donnell's with in-plane inertia, (iii) Sanders-Koiter, (iv) Flügge-Lur'e-Byrne and (v) Novozhilov theories. These five shell theories are practically the only ones applied to geometrically nonlinear problems among the theories that neglect shear deformation. Donnell's shallow-shell theory has already been used in Chapter 5, and the numerical results presented there are used for comparison. Shell theories including shear deformation and rotary inertia are not considered in this chapter. The results presented are based on the study by Amabili (2003).

Energy Approach

The elastic strain energy of the shell is given by equation (1.141), in which the expressions of the middle surface strain-displacement relationships and changes in curvature and torsion must be inserted according to the selected nonlinear shell theory.