In a thin-walled cylinder, what exactly is Hoop Stress vs Radial Stress?

If I have a thin-walled and not very stiff cylinder, such as a rubber-band, and I put it between two pins and pull it apart radially, is the force required considered the hoop force or radial force? What would the difference be between the two? I have read various derivations for hoop force/stress but I am still confused on if my above example is describing hoop or radial force.

• A rubber band is not a good example of a "cylinder" in my opinion. Cylinders generally have uniform pressure over the entire surface area. The rubber band is more like a wire, string, rope, or cable that resists tension only and has point forces. Equations for forces on cables would be more reasonable. Jun 5, 2023 at 17:44
• When Dia/t > 20, only the hoop stress is meaningful.
– r13
Jun 5, 2023 at 23:28

In a thin walled vessel it is simple. A cylinder uses coordinates $$(r, \theta, z)$$.

Hoop forces mean azimuthal forces (in the $$\theta$$ direction) that attempt to increase the circumference of the vessel. These are the highest-magnitude forces cause by internal pressure in a cylinder, and this is why tanks and pressure vessels often fail by the skin splitting, like so:

Even overcooked hot dogs tend to split this way.

Radial forces are those trying to increase or decrease the wall thickness of the vessel ($$r$$ direction). These are usually very small in a thin-walled vessel, and are neglected in first order thin-wall analysis, because the wall is so thin there is no material there to become stressed, with the radial reaction forces essentially being distributed in the hoop direction.

Longitudinal or axial forces are those trying to increase (or decrease) the length, in the $$z$$ direction.

I will add: in your rubberband example, it's not quite the same as a pressure vessel, but the stress you are describing is primarily hoop ($$\theta$$ direction) stress.

• That is an excellent explanation, thank you. As an enthusiastic consumer of hot dogs I also appreciate the parallel Jun 6, 2023 at 18:49
• @ColtonCampbell Germans even call their cylindrical shell design formula for internal pressure "Bockwurst Formel" meaning "sausage formula". Jun 10, 2023 at 7:34