Point loading for a circular ring -- Stress calculation

[prince1989]

I need to find a suitable mathematical method of calculating the stresses at multiple points of a circular ring subjected to point loading. I'll calculate the rest myself- any suggestions?

 

[pongo38]

You need to distinguish in-plane loading from out-of-plane loading. Do you mean normal stresses, shear stresses, or a combintaion?

 

[prince1989]

Ok, thank you for replying. How do I go about that for a ring? I've only looked at beams before. I've been given some specifications as well as FEA analysis of normal (circumferential) stress, strain and displacement. The ring is being compressed between two plates and the force is applied in line with the centre. I need to validate the FEA results using some kind of mathematical method. I've been looking at Castigliano's method, but can only seem to evaluate the maximums. Is there a better approach that might be able to provide values at any given point?

 

[prince1989]

This is one of the FEA results:

 

[pongo38]

It's still not clear from your description 'force in line with the centre' whether the force is in the plane of the ring or perpendicular to it. If it is in the plane of the ring, does this mean there are just two point loads, diametrically opposite each other? Or are there other possible configurations of three or more point loads?

edit. sorry. just seen your figure. It's in-plane loading. I suggest you look at one of Timoshenko's books where arcate beams are treated well

 

[pongo38]

As a rough check the combined normal stress formula N/A+-M/Z could be used on half the ring, splitting the load into two. So you get a semi-circle with encastre ends (i.e. moment-fixed). If the ends were not fixed, but pinned, the bending moment diagram would be the shape of the arc itself, with the base line being the straight between the points of application of the load. But, because the ends are fixed, the base line moves to reduce the bending moment at the outside, and to increase it on the line of action of the force. How much it moves I leave you to ponder...

 

[prince1989]

Hi, thank you so much- you're a life saver! I've come across Timoshenko before while looking through Roark for a solution, so that probably will be a good place to look. I've checked the library as well and there are a few books so I'm sure that one will have what I need in it.

One last question though, if you don't mind:
The combined stress formula used in the way that you explained- is this something that I can find in a Timoshenko book? Or is this found elsewhere?

 

[pongo38]

Many books on strength of Materials will cover the combined stress formula, in which: N is the normal force (the force at right angles to the cross-section), A is the cross-sectional area, M the bending moment, and Z the elastic section modulus (not to be confused with elastic modulus E). Z is commonly known as I/y in the bending stress formula. You will find civil engineering references to its applications in prestressed concrete, foundations for retaining walls, and many others. I dare say Timoshenko has it in one of his books, but I don't have my copy any more.

 

[MichaelDA]

Another reference is Advanced Strength of Materials by JP Den Hartog where on page 238 he summarizes a von Karma derivation that shows the strain for a ring being compressed in its own plane by a point load as being
e_1 = - (3u_o/r_o^2) * cos 2 (theta) y
with the thin-walled tube assumption, as well as a sinusoidal deflection. Best to look up the derivation yourself to understand the parameters.

 

[SteamKing]

Hi, thank you so much- you're a life saver! I've come across Timoshenko before while looking through Roark for a solution, so that probably will be a good place to look. I've checked the library as well and there are a few books so I'm sure that one will have what I need in it.

One last question though, if you don't mind:
The combined stress formula used in the way that you explained- is this something that I can find in a Timoshenko book? Or is this found elsewhere?

It's usually developed by analyzing the bending stress of curved beams. There are certain differences between the analysis of straight prismatic beams and curved beams, chiefly in how one locates the centroid of the cross section of the beam and the neutral plane.

https://www.clear.rice.edu/mech400/Winkler_curved_beam_theory.pdf

 

[MichaelDA]

Yes, thank you for this point, my misunderstanding. The derivation is part of a solution for a moving neutral plane such as in the case for a curved tube; there should be a simpler derivation available for the case where the ring is not subjected to such a moment.