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Structural Engineering and Mechanics
  Volume 55, Number 1, July10 2015, pages 029-46

Vibrations of truncated shallow and deep conical shells with non-uniform thickness
Jae-Hoon Kang

    A three-dimensional (3-D) method of analysis is presented for determining the natural frequencies of a truncated shallow and deep conical shell with linearly varying thickness along the meridional direction free at its top edge and clamped at its bottom edge. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components ur, uo and uz in the radial, circumferential, and axial directions, respectively, are taken to be periodic in o and in time, and algebraic polynomials in the r and z directions. Strain and kinetic energies of the truncated conical shell with variable thickness are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated. The frequencies from the present 3-D method are compared with those from other 3-D finite element method and 2-D shell theories.
Key Words
    truncated conical shell; vibration; variable thickness; three-dimensional analysis
Department of Architectural Engineering, Chung-Ang University, 221 Heuksuk-Dong, Dongjak-Ku, Seoul, 156-756, Republic of Korea

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