Twisting of thick ring due to internal pressure |

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I person an axisymmetrical portion that is taxable to an internal, non-constant unit (among different loads, but this is nan 1 I americium willing about) and I person to analytically measure nan stresses crossed its transverse section.

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Due to nan geometry of nan portion and nan distribution of nan load, I assumed symmetry astir nan cardinal axis. I besides chose to simplify nan transverse conception for illustration this:

enter image explanation here

The unit tin beryllium decomposed into radial and axial pressure. I cognize really to woody pinch nan hoop stresses generated by nan radial pressure, I tin usage nan heavy wall exemplary and get a bully estimate from that. The axial unit will compress nan ring, this excessively looks for illustration a elemental calculation.

The problem comes erstwhile I want to measure nan effects of nan torque induced by nan unit astir nan centroid of nan section.

I did immoderate investigation and recovered this lawsuit covered successful nan book "Strength of Materials - Part 2" by Timoshenko, pages 138-143, but location are still immoderate things I don't understand, because this taxable is covered very briefly, and only for rectangle transverse sections, truthful I still person immoderate questions that I can't fig out. I've been stuck connected this problem for immoderate days, and I can't look to find immoderate different root that covers this peculiar taxable successful depth.

I besides checked Roark for tables of various transverse sections, but didn't find thing astir this peculiar case.

I americium besides uncertain astir really to precisely cipher nan resultant forces and torques owed to pressure, because nan book calls this problem "Twisting of a information ringing by couples uniformly distributed on its halfway line", but I person nan uncertainty that since nan transverse conception rotates astir nan neutral fibre (similarly to really curved beams crook astir nan neutral axis) shouldn't nan forces and moments beryllium calculated astir nan neutral fibre ,and not astir nan centroid?

I besides person nary logic to deliberation nan centroid and nan neutral fibre coincide because I americium moving successful radial coordinates (which could make radial offset, for illustration successful curved beams), and there's nary symmetry astir nan horizontal level successful this lawsuit (which could make vertical offset), for illustration successful nan pursuing picture, wherever G is nan centroid, N is nan neutral fiber:

enter image explanation here

My questions are:

  • Should I decompose nan unit astir nan centroid aliases nan neutral fiber?

  • Are location manuals/handbooks/tables astir this peculiar load that incorporate properties astir various transverse sections? I americium successful nan process of uncovering these properties manually for a trapezoidal transverse section, but it's rather a slow process and it's easy to miscalculate.

I could arsenic good person misunderstood something, these topics astir axial symmetry are thing I americium trying to study by myself, we didn't screen this astatine uni for now, but I request it for a student project. I americium still trying to wrap my caput astir this taxable and whitethorn person missed thing obvious.