Help Understanding Roark Formula: Stress in a Circular Ring

[Mark172]

I'm trying to find the yield load of a circular ring under tensile loading, given by Roark 6th Edition, Ref. number 5 in Table 17.

The Roark equation shows load forces (W) compressing the ring towards the ring center. I'm trying to find the opposite - the ring will be in tension (it is the corner fastening ring for a racing yacht's sail).

I'll be taking the maximum moment and using the standard MC/I to calculate stress. Theta as defined by Roark is 50 degrees. I'm confused as to how to find Max -M. Finding Max +M is fairly straight forward, and I intuitively conclude that the Max M will be +M at M_c (using the correct conventions for applying the load), but I'm just trying to cover my bases and make sure that -M won't result in a greater absolute value.

I can upload a picture later if needed.

 

[desertfox]

Hi Mark172

Scroll down to page 18 of this link it describes stresses in a closed ring under tension.
Also however it only deals with a single tensile force and not with two external loads as per you're post but I'm not sure if the anchor ring for a sail is loaded as the example shown in the link.

http://www.scribd.com/doc/33449829/Design-of-Curved-Beams

desertfox

 

[Mark172]

Ok, thanks!

 

[JStephen]

I don't have Roark's book here in front of me. He has a load case with N equally-spaced radial loads acting on a circular ring, and I assume that's the case you're referring to.

In that event, theta can't be 50 degrees. It would be 45 degrees for 4 loads.

The load case is appliable for either positive or negative loads.

The maximum moment should be at the loads. Either in that load case, or in the introduction to the whole circular-ring section, it should give you the equation for the moment as a function of X, and you can confirm it with that.

Note that the stress in the ring is combination of bending and hoop stresses, not just bending stresses, so don't just use Mc/I for the stress.

 

[Mark172]

Thank you for your helpful response, JStephen. The load case I'm referring to is the one attached. This actually doesn't describe the load case perfectly - in reality, a sail corner ring will have anywhere from ~5 to 20 tensioned fibres across 100 degrees of the ring (so it's closer to a distributed load), with one main tie at the opposite end of the ring from the distributed load.

I'm new to stress analysis in cylindrical coordinates, so I'm just coming up to speed on how to think about hoop (tangential) and radial stress. If I'm calculating the total stress of an element in the ring (to find breaking strength), do I simply superimpose the two stress values, much the same as I would do with, say, a beam under moment and shear loading?

Your advice is greatly appreciated - I'm in a foreign country now, and I'm the only person on our team with structural analysis experience... therefore the "expert" - at times a scary position to be in for a young engineer.

 

[Mark172]

Ah, I think it is clear to me now - Roark gives formulas for moment, hoop load, and radial shear (in the heading of table 17, not seen in the picture above). Stress will result from each of these kinds of loading, and these stresses must be added to obtain the absolute elemental stress. It is this absolute value that will enable me to find the breaking strength, by comparing it to the material yield and ultimate strengths... correct?