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CTC / MTC 222 
Strength of Materials 

Chapter 12

Pressure Vessels


Chapter Objectives 

  • Classify a pressure vessel as thin-walled or thick-walled
  • Calculate the stress (hoop stress) in a sphere subject to an internal pressure
  • Calculate the stress (hoop stress and longitudinal stress) in a cylinder subject to an internal pressure
  • Determine the required wall thickness of a pressure vessel to safely resist a given internal pressure

Thin-walled Pressure Vessels 

  • Terminology:
    • Ri , Ro, Rm, - Inside, outside and mean (average) radii
    • Di , Do, Dm, - Inside, outside and mean (average) diameters
    • t – wall thickness
  • If Rm / t �� 10, pressure vessel is considered thin-walled
    • In terms of the diameter;  Dm / t �� 20
  • Derivation of formulas for stresses in thin-walled pressure vessels are based on the assumption that the stresses are constant throughout the wall of the vessel
  • If Rm / t �� 10, pressure vessel is considered thick-walled
    • Stresses in thick-walled pressure vessels are not constant throughout the wall of the vessel
 

 


Thin-walled Spheres 

  • Internal pressure in sphere acts perpendicular to the surface
    • Uniform over the interior surface
  • Cut Free-Body Diagram through center of sphere
  • Internal forces in walls appear on FBD
    • Since FBD was cut through center of sphere, these forces are horizontal
  • For vertical equilibrium: �� FY = 0
    • Vertical components of internal pressure are equal and opposite
    • Vertical components in opposite directions cancel each other
  • For horizontal equilibrium: �� FX = 0
    • Internal force in wall must equal the resultant horizontal force due to internal pressure

 


Thin-walled Spheres 

  • Considering horizontal components of internal pressure
    • Resultant force FR = p AP
      • AP = projected area of sphere on plane cut through the diameter = �� Dm2 / 4
    • Since �� FX = 0, internal force in wall = FR
  • Stress in wall: �� = F/A = FR / AW
    • AW = Area of sphere wall
    • AW =�� Do2 / 4 - �� Di2 / 4 = �� (Do2 - Di2) / 4
  • For a thin-walled sphere
    • AW �� �� Dm t  --> the area of a strip of thickness = t and   length = average circumference (�� Dm )
  • Stress in wall of sphere
    • �� = FR / AW = p AP / AW =  p (�� Dm2 / 4) / �� Dm t = p Dm / 4 t

 


Thin-walled Cylinders 

  • Cylinders used as pressure vessels and for piping of fluids under pressure
  • Two types of stresses
    • Longitudinal stress – along the long axis of the cylinder
    • Hoop stress (tangential stress) – around the circumference of the cylinder

 


Longitudinal Stress in 
Thin-walled Cylinders 

  • Longitudinal Stress
  • Cut Free-Body Diagram through cylinder, perpendicular to longitudinal axis
  • Longitudinal internal forces in walls appear on FBD
    • Forces are horizontal
  • For horizontal equilibrium: �� FX = 0
    • Internal force in wall must equal the resultant horizontal force due to internal pressure
 
 
 
 

 


Longitudinal Stress in 
Thin-walled Cylinders 
 

  • If end of cylinder is closed, resultant force FR = p A=p �� Dm2 / 4
    • Since �� FX = 0, longitudinal internal force in wall = FR
  • Stress in wall: �� = F/A = FR / AW
    • AW = Area of sphere wall
    • AW =�� Do2 / 4 - �� Di2 / 4 = �� (Do2 - Di2) / 4
  • For a thin-walled cylinder
    • AW �� �� Dm t ,
    • The area of a strip of thickness = t and length = average circumference (�� Dm )
  • Longitudinal stress in wall of cylinder
    • �� = FR / AW = p AP / AW =  p (�� Dm2 / 4) / �� Dm t = p Dm / 4 t
    • Longitudinal stress is same as stress in a sphere
 

 


Hoop Stress in 
Thin-walled Cylinders 
 

  • Isolate a ring of length L from the cylinder
    • Cut a vertical section through ring, passing through its center
    • Draw a FBD of segment either side of section
  • Similar to analysis of sphere, resultant force FR = p AP
    • AP = projected area of ring = Dm L
  • Stress in wall: �� = F/A = FR / AW
    • AW = Cross-sectional area of cylinder wall = 2 t L
  • Hoop stress in wall of cylinder
    • �� = FR / AW = p AP / AW =  p Dm L / 2 t L = p Dm / 2 t
    • Hoop stress is twice the magnitude of longitudinal stress
    • Hoop stress in the cylinder is also twice the stress in a sphere of the same diameter carrying the same pressure
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