APPLICATION OF A THREE-DIMENSIONAL SHELL THEORY TO THE FREE VIBRATION OF SHELLS ARBITRARILY DEEP IN ONE DIRECTION

A three-dimensional shell theory is presented which is applicable to doubly curved thick open shells which are arbitrarily deep (have a large side-length to radius of curvature ratio) in one principal direction but are shallow in the other direction. The strain–displacement equations for the proposed “deep-shallow” shell theory are expressed in Cartesian co-ordinates and the limits of applicability of these equations are discussed. These equations are then used in a Ritz variational formulation with algebraic polynomials as trial functions to solve for the natural frequencies of a number of doubly curved shell problems. A novel approach is also proposed in which penalty functions are introduced to enforce continuity of displacements at two opposite ends of a shell of rectangular platform, increasing the range of problems which can be treated to include closed shells, such as cylinders, barrels, cooling-tower-type structures, toroids, rings, etc. (a sub-class of shells of revolution).