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Spherical Pressure Vessel
Thin-walled pressure vessels are one of the most typical applications of plane stress.

Consider a spherical pressure vessel with radius r and wall thickness t subjected to an internal gage pressure p.

For reasons of symmetry, all four normal stresses on a small stress element in the wall must be identical. Furthermore, there can be no shear stress.

The normal stresses s can be related to the pressure p by inspecting a free body diagram of the pressure vessel. To simplify the analysis, we cut the vessel in half as illustrated.

Since the vessel is under static equilibrium, it must satisfy Newton's first law of motion. In other words, the stress around the wall must have a net resultant to balance the internal pressure across the cross-section.

Cylindrical Pressure Vessel
Consider a cylindrical pressure vessel with radius r and wall thickness t subjected to an internal gage pressure p.

The coordinates used to describe the cylindrical vessel can take advantage of its axial symmetry. It is natural to align one coordinate along the axis of the vessel (i.e. in the longitudinal direction). To analyze the stress state in the vessel wall, a second coordinate is then aligned along the hoop direction.

With this choice of axisymmetric coordinates, there is no shear stress. The hoop stress sh and the longitudinal stress sl are the principal stresses.

To determine the longitudinal stress sl, we make a cut across the cylinder similar to analyzing the spherical pressure vessel. The free body, illustrated on the left, is in static equilibrium. This implies that the stress around the wall must have a resultant to balance the internal pressure across the cross-section.

Applying Newton's first law of motion, we have,

To determine the hoop stress sh, we make a cut along the longitudinal axis and construct a small slice as illustrated on the right.

The free body is in static equilibrium. According to Newton's first law of motion, the hoop stress yields,

Remarks
  • The above formulas are good for thin-walled pressure vessels. Generally, a pressure vessel is considered to be "thin-walled" if its radius r is larger than 5 times its wall thickness t (r > 5 · t).
  • When a pressure vessel is subjected to external pressure, the above formulas are still valid. However, the stresses are now negative since the wall is now in compression instead of tension.
  • The hoop stress is twice as much as the longitudinal stress for the cylindrical pressure vessel. This is why an overcooked hotdog usually cracks along the longitudinal direction first (i.e. its skin fails from hoop stress, generated by internal steam pressure).
Glossary