Cylinder stresses
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[edit] 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.
[edit] Thin-walled assumption
For the thin-walled assumption to be valid the vessel must have an inner diameter / wall thickness ratio of at least 10 (often cited as 20). 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
- d is the inside diameter of the cylinder.
is the hoop stress.
The hoop stress equation for thin shells is also approximately valid for spherical vessels, including plant cells and bacteria in which the internal turgor pressure may reach several atmospheres.
Inch-pound-second system (IPS) units for P are pounds-force per square inch (psi). Units for t, and d are inches (in). SI units for P are pascals (Pa), while t and d are in meters (m).
When the vessel has closed ends the internal pressure acts on them to develop a force along the axis of the cylinder. This is known as the axial stress and is usually less than the hoop stress.
Though this may be approximated to
Also in this situation a radial stress 
is developed and may be estimated in thin walled cylinders as:
[edit] Thick-walled vessels
When the cylinder to be studied has a d/t ratio of less than 10 the thin-walled cylinder equations no longer hold since stresses vary significantly between inside and outside surfaces and shear stress through the cross section can no longer be neglected.
In order to calculate the stresses and strains here a set of equations known as the Lamé equations must be used.
where
- A and B are constants of integration, which may be discovered from the boundary conditions
- r is the radius at the point of interest (eg at the inside or outside walls)
A and B may be found by inspection of the boundary conditions. For example, the simplest case is a solid cylinder:
if 
then 
and a solid cylinder cannot have an internal pressure so 
[edit] Effects of Pressure Vessel Stresses
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.
[edit] References
"Thin-walled Pressure Vessels," engineering fundamentals, June 19, 2008 - http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/pressure_vessel.cfm
