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# Hoop stress

Hoop stress is mechanical stress defined for rotationally-symmetric objects being the result of forces acting circumferentially (perpendicular both to the axis and to the radius of the object). 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:

$sigma_h = Pr/t$
where

• P is the internal pressure, t is the wall thickness, and r is the inside radius of the cylinder.
• $sigma_h !$ is the hoop stress.

Inch-pound-second system (IPS) units for P are pounds-force per square inch (psi). Units for t, and r are inches (in). SI units for P are pascals (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 small. The general rule of thumb is that r / t must be at least 5.

When the vessel has closed ends, the longitudinal stress under the same conditions is:

$sigma_l = sigma_h / 2$
The stresses in a thick-walled cylinder under a pressure differential are given by the Lamé Equations and are of the form

$sigma_r = A - B/r^2 = -P_r$

$sigma_h = A + B/r^2$
and
$sigma_l = \left(P_1r_1^2 - P_2r_2^2 \right) / \left(r_2^2 - r_1^2 \right)= A$
$B = \left(r_1^2r_2^2 \left(P_1 - P_2 \right)\right) / \left(r_2^2 - r_1^2 \right)$

where

• A and B are constants, given the values of $P_1, P_2, r_1, r_2$.
• r is the variable, as a plot is usually needed of the stresses from $r_1$ to $r_2$.
• $sigma_r$ is radial stress at r, and
• $sigma_h$ is the hoop stress at r.

Where r1 is the inner radius of the thick walled cylinder and P1 is the pressure at that radius and r2 and P2 are the outer radius and pressure.

Fracture is governed by the hoop stress in the absence of other external loads since it is the largest principal stress. Note that since the hoop stress is largest when r is smallest, cracks in pipes should theoretically start from inside the pipe. This is why pipe inspections after earthquakes. usually involve sending a camera inside a pipe to inspect for cracks. Yielding is governed by an equivalent stress that includes hoop stress and the longitudinal or radial stress when present.

## References

"Thin-walled Pressure Vessels," engineering fundamentals, June 19, 2008 - http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/pressure_vessel.cfm

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