Hoop stress calculation in grooves in thin shell vessels

[Surfex]

HI,

I have an application where I have a tube of a certain wall thickness, internal diameter and internal pressure with ends capped.

I also have a square groove of a certain width and depth machined in the inside diameter of my tube.

I would like to know how to calculate the hoop stress in the internal groove and also if the width of the groove has an effect on this hoop stress?

Thanks a lot for your responses.

 

[jte]

Surfex-

A post to the Boiler and Pressure Vessel engineering board might get some additional responses.

The hoop stress formula won't change between the thick part of your tube and the thin part, assuming you're dealing with tube geometry which can be described with thin wall theory. If your diameter is approaching 10 times the wall thickness, you need to consider going to thick wall theory. Depending on which code you are using, it will be some variant of S=PR/t. Now, you have a situation where if you have a very thin groove, the adjacent thicker wall will provide some reinforcement. On the negative side, the profile of the groove may introduce some stress concentrations. The simplest way to evaluate the situation if you have the tools is to run an axisymmetric FEA. It's an easy model to build and if done right will account for the size and profile of the groove in one calc. Another approach may be to look into API-579 part 5 which will provide some guidance for assessing a groove like flaw. Only in your case, the groove is intentional.

jt

 

[prex]

If you have to comply with a pressure vessel or piping code (e.g.ASME VIII Div.1) you'll normally need to consider the thickness at the bottom of the groove as the governing thickness for code calculations.
However this is a quite conservative approach for your situation (that's why you will hardly find a pressure vessel out there with a groove) as noted also by jte: if you are not under the coverage of a code you could go with a more realistic approach.
If the groove is quite short (perhaps up five bottom thicknesses, but this point would require a check) then you could forget about the hoop stress in the groove (but don't forget the longitudinal stress!).
If you want to go with a deeper analysis, besides FEM you could use the forms for shell calculations in the site below under Pipes -> Axisymm.loads -> Guided-guided (be warned though that the guided-guided condition is an approximation).

prex

http://www.xcalcs.com

Online tools for structural design

 

[Cockroach]

Hoop stress alone will not cut it for you. There are two other principal stresses on a wall element: radial, longitudinal.

Radial stress is simply the reaction to internal pressure or load itself. Longitudinal stress is induced by the reactions of the end caps to that internal pressure. With hoop stress then, you have a triaxial loading.

Fortunately for pressure vessels, this is fairly well researched. One of the most useful of models is that of Von Mises-Hencky. Using the above mentioned principal stresses for pressure vessel theory, the Von Mises-Hencky Equation simply states that twice the dot product of wall element stress (i.e. twice the magnitude of stress squared) equals the stress gradient PLUS three times the shear stress squared. Typically the shear stress may be neglected if permissable from outside boundary conditions imposed on the geometry. This reduces the equation somewhat, the gradient pressure is solved by cross product of three dimensional stress vector or by cyclic permutation of vector basis.

Your getting into some very pretty mathematics. Grinding the geometric algebra and a case of beer later, you would arrive at the result:

S = sqrt(3) P [R^2 / (R^2 - 1)] where R=D/d

and D=OD vessel, d=ID vessel, P=internal pressure. This is the triaxial stress state imposed on an element representative of wall for a pressure vessel subject to internal loading, P.

The results, you will find, are extremely accurate. Variations of the Von Mises-Hencky Model are typically given in advanced textbooks dealing with Strength of Materials. I have played with various boundary condition loading, application of a longitudinal force that would counter internal pressure for example, and obtained strain gauge readings under 5% error. Sure wish every engineering problem was so well defined.

Good luck with it.

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada

 

[Surfex]

Thanks Cockroach, prex and jte,

You were all very helpful to my understanding of the problem. I completed your information with some notions of thin wall pressure vessels that I found in the Roark's Formulas for stress and strain by Warren Young.

Thanks again.