Exact solutions for free vibration of circular cylindrical shells with classical boundary conditions

• Exact solutions of free vibration for cylindrical Donnell shell are presented.
• The exact solutions are in compact and well-regulated form.
• The computed results frequencies agree with those in literature.
• High accurate membrane approximate solutions are presented.

The obtaining of exact solutions for free vibration of circular cylindrical shells with classical boundary conditions, except for all edges being shear diaphragm, was known to be difficult or impossible. This work presents those exact solutions based on the Donnell–Mushtari shell theory . They are shown to be in simple and compact form, and not as complex as they have generally been regarded. The computed natural frequencies are in excellent agreement with those in literature and those of the highly accurate semi-analytical differential quadrature finite element method developed in this study. The exact solutions can be used as benchmarks for researchers to check their numerical methods and for engineers to design such shell structures. Two of the four eigenvalues are found to be much larger than the other two for some boundary conditions. High accurate membrane approximate solutions are also presented for the circular cylindrical shells.