# Free Vibration of Cylindrical and Hyperboloidal Cooling-Tower
Shells

**by Mohey E-D El-Mously**

The focus of this dissertation is the small free vibration of thin
elastic hyperboloidal shells of revolution, which are often used as cooling
towers in thermal power stations. The aim of the present study is to obtain
a comprehensive understanding of such phenomena, in order to develop design
charts that map the different regimes of structural behaviour of these
shells.
The work begins with the simpler case of the cylindrical shell and encompasses
tests on a range of simplified models that have been proposed in the literature,
but which have not been assessed thoroughly hitherto. A first-order approximation
theory for the analysis of cylindrical shells has been developed. The proposed
theory aims to bridge the gap between the classical Love-Kirchhoff theory,
which permits an accurate description of the shell behaviour, and the simplified
version of Vlasov which only considers the effects of longitudinal stretching
and circumferential bending. A direct analogy is established between the
shell behaviour and the behaviour of a Timoshenko beam mounted on a Pasternak
foundation. Approximate explicit formulae are derived for the fundamental
frequency of the beam-foundation system, and by analogy for the shell.
Criteria defining the domains of validity for the proposed theory are established.

Attention is then turned to shells with slightly curved meridians (waisted,
nearly-cylindrical shells) and finally to hyperboloidal shells, both of
which exhibit various complex degenerate-case effects. A simple model of
waisted, nearly-cylindrical shells demonstrates cross-over phenomena when
geometric parameters are varied. Further investigation shows that these
phenomena are replaced by curve-veering when a more accurate shell model
is used. Hyperboloidal shells predominantly demonstrate curve-veering phenomena,
however there are particular geometries at which cross-over occurs. Most
of the above phenomena are dominated by membrane effects. The sensitivity
of these phenomena to the shell geometric parameters is examined and explained
in simple terms.

An experimental investigation, using a small-scale silicone rubber model
shell, confirms some main points of the analysis.

[Cambridge University | CUED
| Structures Group
| Geotechnical Group]

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(last update 17 November 1998)