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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:
- R
_{i },_{ }R_{o}, R_{m}, - Inside, outside and mean (average) radii - D
_{i },_{ }D_{o}, D_{m}, - Inside, outside and mean (average) diameters - t – wall thickness
- If R
_{m}/ t �� 10, pressure vessel is considered__thin-walled__ - In terms
of the diameter; D
_{m}/ 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 R
_{m}/ 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: �� F
_{Y}= 0 - Vertical components of internal pressure are equal and opposite
- Vertical components in opposite directions cancel each other
- For horizontal
equilibrium: �� F
_{X}= 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 F
_{R}= p A_{P} - A
_{P}=__projected__area of sphere on plane cut through the diameter = �� D_{m2}/ 4 - Since ��
F
_{X}= 0, internal force in wall = F_{R} - Stress in
wall: �� = F/A = F
_{R}/ A_{W} - A
_{W }= Area of sphere wall - A
_{W }=�� D_{o2}/ 4 - �� D_{i2}/ 4 = �� (D_{o2 }- D_{i2}) / 4 - For a thin-walled sphere
- A
_{W}�� �� D_{m}t --> the area of a strip of thickness = t and length = average circumference (�� D_{m}) - Stress in wall of sphere
**��**= F_{R}/ A_{W }= p A_{P }/ A_{W}= p (�� D_{m2}/ 4) / �� D_{m}t =**p D**_{m}**/ 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: �� F
_{X}= 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 F
_{R}= p A=p �� D_{m2}/ 4 - Since ��
F
_{X}= 0, longitudinal internal force in wall = F_{R} - Stress in
wall: �� = F/A = F
_{R}/ A_{W} - A
_{W }= Area of sphere wall - A
_{W }=�� D_{o2}/ 4 - �� D_{i2}/ 4 = �� (D_{o2 }- D_{i2}) / 4 - For a thin-walled cylinder
- A
_{W}�� �� D_{m}t , - The area
of a strip of thickness = t and length = average circumference (��
D
_{m}) - Longitudinal stress in wall of cylinder
**��**= F_{R}/ A_{W }= p A_{P }/ A_{W}= p (�� D_{m2}/ 4) / �� D_{m}t =**p D**_{m}**/ 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 F
_{R}= p A_{P} - A
_{P}=__projected__area of ring = D_{m}L - Stress in
wall: �� = F/A = F
_{R}/ A_{W} - A
_{W }= Cross-sectional area of cylinder wall = 2 t L - Hoop stress in wall of cylinder
**��**= F_{R}/ A_{W }= p A_{P }/ A_{W}= p D_{m }L / 2 t L =**p D**_{m}**/ 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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