ESDU 81018 provides graphical data and a computer program for the estimation of the lower natural frequencies of thick-walled isotropic@ circular cylinders based on the three-dimensional theory of elasticity for hollow cylinders. The data on both flexural and torsional vibrations are applicable to initially unstressed@ isotropic shells having shear diaphragm end conditions. The geometric parameters and mode numbers for which the differences between natural frequency given by the three-dimensional elastic theory and thin-shell theory are more than 1 per cent and 5 per cent are indicated. General information is given on the flexural and torsional modes of free vibration of thick-walled shells. The thickness modes@ which are not included in the model when shells are modelled as thin-walled@ are discussed. The variations of displacements and stresses through the wall thickness@ obtained using the computer program provided@ are discussed and the effects of Poisson's ratio on the natural frequencies are considered. To give guidance on the desirability of modelling shells as either thin-walled or thick-walled@ the range of applicability of thin-shell theories is discussed and specific information is given for a wide range of shell geometries. A worked example@ which illustrates the application of these data@ is given as well as the computer output for this example. The program is available as ESDUpac A8118.