Hoop stress
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Hoop stress is mechanical stress applied in a direction perpendicular to the radius of the item in question. Along with axial stress and radial stress, it is a component of the stress tensor in cylindrical coordinates.
It is usually useful to decompose any force applied to an object with rotational symmetry into components parallel to the cylindrical coordinates r, z, and θ. These components of force induce corresponding stresses: radial stress, axial stress and hoop stress, respectively.
The classic example of hoop stress is the tension applied to the iron bands, or hoops, of a wooden barrel. In a straight, closed pipe, any force applied to the cylindrical pipe wall by a pressure differential will ultimately give rise to hoop stresses. Similarly, if this pipe has flat end caps, any force applied to them by static pressure will induce a perpendicular axial stress on the same pipe wall. Thin sections often have negligibly small radial stress, but accurate models of thicker-walled cylindrical shells require such stresses to be taken into account.
The classic equation for hoop stress created by an internal pressure on a thin wall cylindrical pressure vessel is:
where
- P is the internal pressure, t is the wall thickness, and r is the radius of the cylinder.
- σh is the hoop stress.
IPS units for P is pounds per square inch (PSI). Units for t, and r are inches (in). SI units for P is Pascal (Pa), while t and r are in meters (m).
Important assumptions include: the wall is significantly thinner than the other dimensions, which implies that the difference between inner and outer radius is negligible.
When the vessel has closed ends, the longitudinal stress under the same conditions is:
- σl = σh / 2
Fracture is governed by the hoop stress in the absence of other external loads since it is the largest principal stress. Yielding is governed by an equivalent stress that includes hoop stress and the longitudinal or radial stress when present.
The stresses in a thick-walled cylinder under a pressure differential are given by the Lamé Equations and are of the form
and
where
- A and B are set by the inner and outer pressures,
- σr is radial stress at d, and
- σt is the tangential stress at d.
A = (p2R22 - p1R12) / (R12 - R22)
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