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The buckling of conical shells subjected to external loads is a classic problem in shell structures. Since the theoretical predications could not agree with the experimental results, a number of research works have been performed [44]. In the present study, an improvement is made by following the two suggestions in Chaps. 2 and 8: (1) treating the smaller end constraint properly and (2) taking the effect of the angle change into consideration. It is shown that with these improvements, the theoretical predictions are closer to the experimental results.

The equilibrium equations for a surface element of a conical shell in the framework of generalized plane stress of linear theory of elasticity are given in [19]. These equations may be applied to linearly varying thickness as well as to constant thickness of conical shells.

The solutions of conical shells with linearly varying thickness presented in the last chapter have suggested a new approach to obtain asymptotic solutions of conical shells with constant thickness. One may assume that the two types of conical shells would behave alike if the ratios of thickness to the radius of the smaller end of the cone is very small. Under this assumption, asymptotic solutions of conical shells with constant thickness are obtained in this chapter. Two numerical examples are given for comparison with the present solutions: the semi-circular conical segment discussed in the last chapter and a complete cone. A solution of the latter is given in [12].

The vibrations of conical shells are due to the inertial and external forces acting the transverse, circumferential and meridional directions of the cone. For small vibrations, the system is linear, thus, the effects of these three forces may be treated separately. It may be due to the mathematical dificulty, the transverse vibration, so far, is solved by numerical methods only, such as Rayleigh-Ritz method, Galerkin procedure, power series, matrix method, and the finite element method. A comprehensive literature review on the conical shell vibrations is provided in [8].