Examples

Example 2 A thin cylinder 75 mm internal diameter, 250 mm long with walls 2.5 mm thick is subjected to an internal pressure of 7 MN/m 2. Determine the change in internal diameter, the change in length, hoop stress & longitudinal stress (E = 200 GN/m 2 &  =0.3).

Solution: For thin cylinders:

For thin cylenders:

For thin cylinders:

Example 3 A cylinder has an internal diameter of 230 mm, has walls 5 mm thick and is 1 m long. It is found to change in internal volume by 12.0 x m 3 when filled with a liquid at a pressure p. If E = 200GN/m 2 and  = 0.25, and assuming rigid end plates, determine:

(a) the values of hoop and longitudinal stresses; (b) the modifications to these values if joint efficiencies of 45% (hoop) and 85% (longitudinal) are assumed; (c) the necessary change in pressure p to produce a further increase in internal volume of 15 %. (The liquid may be assumed incompressible).

Solution: Check first type of cylinder: d / t = 230 / 5 = 46 More than 20, then the cylinder is considered: Thin cylinder

Second step look for the pressure. If the pressure is unknown then try to get it by the available data. In this example the pressure is unknown but the change of volume is given. So that you have to use the available data to get the unknown parameters.

For thin cylinders: The original volume V is not given but you can calculate it from the given dimensions of the cylinder.

Calculating the original volume V:

For thin cylinders:

b: hoop stress acting on the longitudinal joints:

Longitudinal stress (acting on the circumferential joints)

(c) Since the change in volume is directly proportional to the pressure, the necessary 15 %, then increase in volume is achieved by increasing the pressure also by 15 %.

Necessary increase in pressure: increase in p = 0.15 x 1.25 x 10 6 = 1.86 MN/m 2

Example (4) (a) A sphere, 1 m internal diameter and 6 mm wall thickness, is to be pressure-tested for safety purposes with water as the pressure medium. Assuming that the sphere is initially filled with water at atmospheric pressure, what extra volume of water is required to be pumped in to produce a pressure of 3 MN/m 2 ? For water, K = 2.1 GN/m 2.

(b) The sphere is now placed in service and filled with gas until there is a volume change of 72 x m 3 Determine the pressure exerted by the gas on the walls of the sphere.

Solution Where:

Then:

b- Change of volume of the sphere: volume change is given of 72 x10 -6 m 3

The required pressure:

Problems