Vibrations of thick plates and shells

Using an asymptotic series approach, a thick shell theory is proposed for doubly curved sheIls with variable thickness. This theory includes the effects of transverse shear stresses and rotatory inertia. The displacement functions are designed to give non-zero transverse shear stresses internal to the shell, which satisfy the stress-free boundary conditions on the upper and lower surface. Use of the stress-free conditions makes the displacement functions, which vary through the thickness of the shell, dependent only on the middle surface displacements. This theory is applied to the twisted plate. A similar approach is applied to the cylindrical shell, but the effects of transverse normal stress are also included.\ud \ud The theory is applied to the problem of free vibrations of shells clamped along one edge with the other three edges free. The results obtained are compared with practical and theoretical results of other researchers, and with those obtained from thin shell theory. The twisted plate results show the answers that are expected from a thick shell theory, in that it predicts lower frequencies than thin shell theory for modes in which the wavelength/ thickness ratio is less than ten. \ud \ud The results for the cylindrical shell show that the inclusion of transverse normal stress to the order assumed is not warranted. \ud \ud The numerical techniques used for the solution of the free vibration problem are based on variational methods in which the Hamiltonian for the shell is minimised, subject to the constraints of the displacement boundary conditions