Natural frequencies of cooling tower shells

Approximate explicit formulae are presented for (i) the fundamental natural frequency of vibration of a uniform hyperboloidal cooling tower shell mounted on a rigid base, and (ii) for the circumferential wavenumber associated with fundamental mode. These formulae agree well with results previously obtained by finite element computation and they may be adapted readily for use with cooling tower shells mounted on non-rigid supports. The simplicity of the formulae is a consequence of various approximations which are made in the analysis. “Shallow shell” equations are used, and it is assumed that the dominant structural effects in the shell are longitudinal stretching and circumferential bending. The equations are applied first to the more straightforward case of a cylindrical shell which is clamped at one end and free at the other. The various assumptions are examined systematically, and a wide domain is established in which the resulting formulae for fundamental natural frequency and circumferential wavenumber are valid. The most appropriate dimensionless form of the length of the shell in this domain is Λ=Lh 1/2 / a 3/2 , where L is length, h is thickness and a is radius. A simple scheme is presented, with justification, for the adaptation of the results to a shell which is mounted on a non-rigid base. The equations are then applied to a hyperboloidal shell, and the corresponding formulae are obtained. Again the dimensionless group Λ is appropriate, but it must now be supplemented by a second parameter which defines the hyperboloidal shape of the shell. Remarks are made concerning the design of ring stiffeners for the upper edge, the sensitivity of the fundamental frequency to small changes in any of the leading dimensions of the tower and cases in which the elastic modulus of the material may be different in different parts of the shell.