CHAPTER 9 THIN CYLINDERS AND SHELLS Summary The stresses set up in the walls of a thin cylinder owing to an internal pressure p are: circumferential or hoop stress o#= pd 2t pd longitudinal or axial stress 4t where d is the internal diameter and t is the wall thickness of the cylinder. Then: longitudinal strain ELGL-van] hoop strain & ELon-vaL] change of interal volume of cylinder under pressurep 4E[5-4p]r change of volume of contained liquid under pressure- pV K 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 ωrad/s is given by H=p02R2. For thin spheres: cirumferentialor hoop stressd 4t change of volume under pressure 3pd 4E1-y Effects of end plates and joints-add "joint efficiency factor"n 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 198CHAPTER 9 THIN CYLINDERS AND SHELLS Summary The stresses set up in the walls of a thin cylinder owing to an internal pressure p are: circumferential or hmp stress aH = Pd Pd longitudinal or axial stress aL = - 4t where d is the internal diameter and t is the wall thickness of the cylinder. longitudinal strain cL = - [aL - VaH] 1 E 1 E Then: hoop strain cH = - [aH - vaL] Fd 4tE PV change of volume of contained liquid under pressure = - K change of internal volume of cylinder under pressure = - [ 5 - 4v] V 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 For thin spheres: w rad/s is given by GH = po2R2. Pd circumferential or hoop stress aH = - 4t 3Pd change of volume under pressure = - [ 1 - v] V 4tE Eflects of end plates and joints-add “joint efficiency factor” ‘1 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 198