Calculating Average Tangential Stress for Non-Uniform Rotating Disk

[xanderlinkz]

Assumed a disk loaded with external pressure Po, internal Pressure Pi and rotating at the speed ω.
I'm sure that average tangential stress for uniform thickness rotating disk can be calculated using equation below :

σ avg = (PiRi/Ro-Ri) - (PoRo/Ro-Ri) + (ρω^2) / 3(Ro^2+RoRi+Ri^2)

Ro = outer radius
Ri = inner radius
ρ= density

now if the disk have nonuniform thickness (e.g. L1,L2,...,Ln) and Rn as the radius for each interface of two segment (thickness variable).
How do we calculate the average tangential stress for this case?

I would really appreciate any kind of help, i will gladly describe further information if necessary.
and sorry for my bad grammar, English is not my native language. thank you very much.

 

[Valkarie]

The average tangential stress for a nonuniform thickness rotating disk can be calculated as follows: σ avg = (PiRi/Ro-Ri) - (PoRo/Ro-Ri) + {[ρω^2]/3 [Σ(Li/2)(Rn+1^2-Rn^2)]} + {[ρω^2]/3 [Σ(Li/2)(Rn^2-Rn-1^2)]}where Ro = outer radius Ri = inner radius ρ = density Li = segment length Rn = radius of each interface of two segments. Note: The summations are over all the segments.