$9.4 Thin Cylinders and Shells 203 Fig.9.4.Half of a thin sphere subjected to internal pressure showing uniform hoop stresses acting on a surface element. nd2 px 4=0HXπd or 0H= Pd 4t pd i.e. circumferential or hoop stress (9.8) 4t 9.3.1.Change in internal volume As for the cylinder, change in volume original volume x volumetric strain but volumetric strain sum of three mutually perpendicular strains (in this case all equal) =38p=38H 3 =EoH-G小 -- change in internal volume= 3pd [1-v]v (9.9) 9.4.Vessels subjected to fluid pressure If a fluid is used as the pressurisation medium the fluid itself will change in volume as pressure is increased and this must be taken into account when calculating the amount of fluid which must be pumped into the cylinder in order to raise the pressure by a specified amount,the cylinder being initially full of fluid at atmospheric pressure. Now the bulk modulus of a fluid is defined as follows: bulk modulusK=volumetric stress volumetric strain59.4 Thin Cylinders and Shells 203 .. or Fig. 9.4. Half of a thin sphere subjected to internal pressure showing uniform hoop stresses acting on a surface element. nd 4 p x - = CTH x ndt Pd bH=- 4t Pd circumferential or hoop stress = - 4t i.e. 9.3.1. Change in internal volume As for the cylinder, change in volume = original volume x volumetric strain but volumetric strain = sum of three mutually perpendicular strains (in this case all equal) = 3ED = 3EH .. 3Pd change in internal volume = - [ 1 - v] Y 4tE (9.9) 9.4. Vessels subjected to fluid pressure If a fluid is used as the pressurisation medium the fluid itself will change in volume as pressure is increased and this must be taken into account when calculating the amount of fluid which must be pumped into the cylinder in order to raise the pressure by a specified amount, the cylinder being initially full of fluid at atmospheric pressure. Now the bulk modulus of a fluid is defined as follows: volumetric stress volumetric strain bulk modulus K =