Thin Walled Cylinders Under Pressure.The three principal Stresses in the Shell are the Circumferential or Hoop Stress; the Longitudinal Stress; and the Radial Stress. If the Cylinder walls are thin and the ratio of the thickness to the Internal diameter is less than about 1/20 then it can be assumed that the hoop and longitudinal stresses are constant across the thickness. It may also be assumed that the radial stress is small and can be neglected. In point of fact it must have a value equal to the pressure on the inside surface and zero at the outside surface. These assumptions are within the bounds of reasonable accuracy.

The above cylinder has an internal diameter d and a wall thickness of t. If the applied internal pressure is p then the Hoop stress is and the Longitudinal stress is . Sectioning the above cylinder through a diametral plane and consider the equilibrium of the resulting half cylinder. acts upon at area of . The resultant vertical pressure force is found from the projected horizontal area . Clearly the Pressure force upwards is equal to the hoop Stress times the area over which it acts. Hence (1)

(2) Now consider the equilibrium of a section cut by a transverse plane. The longitudinal Stress acts on an area of approximately ( Note the diameter taken should really be the mean diameter) and the pressure p acts upon a projected area of actual shape of the end of the cylinder) Equating longitudinal Forces:( Note this is true no matter the

(3)

(4) Where long cylinders or tubes are braced or carried in brackets, the longitudinal Stress may be much less than given by equation (4) and is sometimes neglected.