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SUMMARYThis paper presents the exact dynamic sti ness matrix for a circular beam with a uniform cross-section. The sti ness matrix is frequency dependent, and the natural frequencies are those that cause the matrix to become singular. Using this matrix the exact natural frequencies of circular beams with various boundary conditions are calculated and compared with available results in the literature.

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A dynamic stiffness method is presented for determining the free vibration frequencies and mode shapes of thick spherical shell segments with variable thickness and different boundary conditions. The analysis uses the equations of the two-dimensional theory of elasticity, in which the effects of both transverse shear stresses and rotary inertia are accounted for. The displacement components are taken to be sinusoidal in time, periodic in the circumferential direction, constant through the thickness, and solved exactly in the meridional direction using the exact element method. The shape functions are derived from the exact solutions for the system of the differential equation of motion with variable coefficients. The dynamic stiffness matrix is derived from the exact shape functions and their derivatives. Highprecision numerical results are presented for thick spherical shell segments with constant or linearly varying thickness and for several combinations of boundary conditions. Comparison is made with results of published research and with two-and three-dimensional finite element analyses.