Axisymmetric Free Vibration of Thick Orthotropic Hemispherical Shells Under Various Edge Conditions

Abstract

Abstract Axisymmetric free vibration of moderately thick polar orthotropic hemispherical shells are studied under the various boundary conditions of sliding, guided pin, clamped and hinged edges. Based on the improved linear elastic shell theory with the transverse shear strain and rotatory inertia taken into account, the dynamic equilibrium equations are formulated and transformed into the displacement form in terms of mid-surface meridian and radial displacements and parallel circle cross-section rotation. These partial differential equations are solved by the Galerkin method using proper Legendre polynomials as admissible displacement functions with the aid of the orthogonality and a number of special integral relations. Numerical results of the present theory compare well with existing data, which is available only in the isotropic theories. Good convergence is obtained for natural frequencies and mode shapes. Study of the effects of thickness and modulus ratio reveals higher frequencies for the thicker and/or stiffer shells with E\ oriented parallel to the meridians. Ranking of the natural frequencies descends in the order of guided pins, sliding, clamped and hinged edges in general. Also seen are the effects of transverse shear strain from the mode shapes with clamped and sliding edges on the slant. For the guided pin and sliding edges, frequencies increase fast as thickness increases so that new fundamental modes are generated in filling up the “frequency gap”. These are the new discoveries in the field of anisotropic shells, as a result of polar orthotropy of shell material and construction.