# How to apply static failure criteria for bending stress in an angled cylindrical surface?

Let’s say I have an irregularly shaped bearing with a cross section that looks like the angled surface of a conical frustum, shown below.

If a bending moment is applied which causes the inside diameter to ‘cave outwards’, how would I perform a static loading failure analysis to see at what bending moment the shape would cave outwards? How would I determine the second moment of area and distance from centroid values required to apply the bending stress formula sigma = My/I? Or is there a manufacturing method in the form of some sheet metal bending formula that would be more helpful in looking at failure mode in a cylindrical cross section instead of a planar one.

EDIT: The end goal is to have the curved surface completely vertical, as if it was the surface area of a rod. If there is an easier method of calculating the forces required, let me know!

## 1 Answer

This is a typical case of stresses on shells and plates and obviously lends itself to analyzing using polar coordinates.

If your element is a thin shell and is loaded in a way that only membrane stresses are involved there are some formulas in references such as Roark's formulas for stress and strain. Otherwise you may need to use FEM and define your loading.

• Unfortunately the cross-section shown in the image above does have a thickness and thus is not a membrane. I am trying to do a basic calculation to see what kind of force you would need to apply axially to get the element to form a 'cylindrical' interior instead of angled. I agree with the use of polar coordinates, are there any keywords or resources I should look into to get some more information? Jan 24 '18 at 22:24
• I would start by thinking of this as mainly built of thin rings placed on top of each other which need to be stretched gradually more and more to a larger radius as you go up and for the first estimate ingnor the vertical stresses. So you have a differential elemen of a ring that can easily be calculated the force needed to stretch. Then you integrate these froces. Many of the ring need to be stretched beyond yield point to conform to a cylinder's geometry. Jan 24 '18 at 22:35
• @RichardYang I'm not familiar with Roark's formulars, but I reckon at least some of them include bending properties of the membranes considered, thus allowing to incorporate thickness properties. Jan 25 '18 at 8:07
• @Robin, I know per definition a membrane can not have bending moment. I do have Roark's book and I looked quickly, did not find a case of moment in membranes. membranes could have in plane tension/compression only. shells can have moment! If I find something I edit my answer. Jan 25 '18 at 20:13
• @kamran Thanks for this clarification. I don't deal with this every day and forgot there is a difference between membranes and shells ... Jan 26 '18 at 7:57