I have an axisymmetrical part that is subject to an internal (non-constant along the inner border, but constant in time) pressure (among other loads, but this is the one I am interested about) and I have to analytically evaluate the stresses across its cross section.
Due to the geometry of the part and the distribution of the load, I assumed symmetry about the central axis. I also chose to simplify the cross section like this:
The pressure can be decomposed into radial and axial pressure. I know how to deal with the hoop stresses generated by the radial pressure, I can use the thick wall model and get a good estimate from that. The axial pressure will compress the ring, this too looks like a simple calculation.
The problem comes when I want to evaluate the effects of the torque induced by the pressure about the centroid of the section.
I did some research and found this case covered in the book "Strength of Materials - Part 2" by Timoshenko, pages 138-143, but there are still some things I don't understand, because this topic is covered very briefly, and only for rectangle cross sections, so I still have some questions that I can't figure out. I've been stuck on this problem for some days, and I can't seem to find any other source that covers this particular topic in depth.
I also checked Roark for tables of various cross sections, but didn't find anything about this particular case.
I am also uncertain about how to exactly calculate the resultant forces and torques due to pressure, because the book calls this problem "Twisting of a circular ring by couples uniformly distributed along its center line", but I have the doubt that since the cross section rotates around the neutral fiber (similarly to how curved beams bend around the neutral axis) shouldn't the forces and moments be calculated about the neutral fiber ,and not about the centroid?
I also have no reason to think the centroid and the neutral fiber coincide because I am working in radial coordinates (which could generate radial offset, like in curved beams), and there's no symmetry about the horizontal plane in this case (which could generate vertical offset), like in the following picture, where G is the centroid, N is the neutral fiber:
My questions are:
Should I decompose the pressure about the centroid or the neutral fiber?
Are there manuals/handbooks/tables about this particular load that contain properties about various cross sections? I am in the process of finding these properties manually for a trapezoidal cross section, but it's quite a slow process and it's easy to miscalculate.
I could as well have misunderstood something, these topics about axial symmetry are something I am trying to study by myself, we didn't cover this at uni for now, but I need it for a student project. I am still trying to wrap my head around this topic and may have missed something obvious.
Additional information:
The ring is bolted to an external thin walled cylinder, which is also subject to internal pressure in the radial direction (this is a combustion chamber):
The cylinder provides the vertical equilibrium, and it is long enough so that what happens on the other side (top) of the cylinder won't affect what happens to the ring, apart for vertical forces that make the vertical equilibrium of the ring possible.
I chose to ignore the screw holes (8) at first to simplify the problem, I'll include their effects later after I'll manage to get a hold of the simpler case first. I also want to assume that the screws provide a distributed, rather than concentrated, reaction force. Again, I'll try to see if I can remove this simplification later in the analysis.
I am also thinking that since the ring is visibly stiffer (thicker) than the thin cylinder, I can assume that the radial component of the pressure is unloaded on the ring only, at least in this zone, so what I mean to say is that the cylinder doesn't help the ring much in resisting tangential elongation. The same can be said for the rotation that I am investigating I think, I can assume that the only thing resisting the rotation of the ring are only its fibers and not those of the thin cylinder. This also puts me in a safety condition, as the cylinder will surely help a bit somewhere.
EDIT
Alright thanks to everybody who commented, I think I am getting a hang of this now. I had to go back to basics and follow carefully the derivation of the stress formula in the case of bending of a curved beam, to better understand the logic behind choosing the centroid as the point of reference. For this case the logic is similar, I am not forced to choose the centroid, but it's useful to choose it as reference point when deriving formulas and decomposing loads because there will be other loads that will cause other deformations in this part, so it's better to use the centroid as it's a point that is easy to utilize for other loading conditions too. If say I decomposed the loads with respect to the neutral fiber rather than the centroid, I would easily calculate the rotation of this ring, but the choice of the point of reference would make it harder to calculate stresses for the other components of the given load.