## Cylinder Stress Element

pd change of internal volume of cylinder under pressure = โ 4v] V

change of volume of contained liquid under pressure = โ

where K is the bulk modulus of the liquid.

For thin rotating cylinders of mean radius R the tensile hoop stress set up when rotating at co rad/s is given by aH = pa>2R2.

For thin spheres: , circumferential or hoop stress aH = โ

Effects of end plates and joints-add "joint efficiency factor" ri to denominator of stress equations above.

### 9.1. Thin cylinders under internal pressure

When a thin-walled cylinder is subjected to internal pressure, three mutually perpendicular principal stresses will be set up in the cylinder material, namely the circumferential or hoop stress, the radial stress and the longitudinal stress. Provided that the ratio of thickness to inside diameter of the cylinder is less than 1/20, it is reasonably accurate to assume that the hoop and longitudinal stresses are constant across the wall thickness and that the magnitude of the radial stress set up is so small in comparison with the hoop and longitudinal stresses that it can be neglected. This is obviously an approximation since, in practice, it will vary from zero at the outside surface to a value equal to the internal pressure at the inside surface. For the purpose of the initial derivation of stress formulae it is also assumed that the ends of the cylinder and any riveted joints present have no effect on the stresses produced; in practice they will have an effect and this will be discussed later (ยง9.6).

### 9.1. J. Hoop or circumferential stress

This is the stress which is set up in resisting the bursting effect of the applied pressure and can be most conveniently treated by considering the equilibrium of half of the cylinder as shown in Fig. 9.1.

Total force on half-cylinder owing to internal pressure = p x projected area = pxdL Total resisting force owing to hoop stress oH set up in the cylinder walls

circumferential or hoop stress aH = ^

9.1.2. Longitudinal stress

Consider now the cylinder shown in Fig. 9.2.

Total force on the end of the cylinder owing to internal pressure nd2

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