[Material Science] Thin Walled Cylinders & Yield Criterion

[MarleyDH]

Homework Statement

A long cylindrical boiler shell is 1.75 m in diameter and has a wall thickness of 12 mm. Treating the boiler as a thing shell, predict what internal pressures will produce yield in the shell according to the: (a) Rankine, (b) Max. Principal Strain, (c) Tresca and (d) Von Mises Criteria.

σ_yield = 300 MPa, E = 200 GPa, v = 1/3)

Homework Equations

The only issue I am having here is with the Von Mises criteria, which states for plane stress:

σ₁² + σ₁·σ₂ + σ₂² = σ_yield²

σ_hoop = P·D/2·t
σ_axial = P·D/4·t

The Attempt at a Solution

So all good and well, I attempt the solution but get a value of 3.11 MPa, which is incorrect according to the answer. (4.75 MPa)

I then rechecked my results and did a google search. This lead me to a site which boiled the math down to a similar problem to:

σ_yield = sqrt(3)·(P·D/4·t)

When I use this equation, I get the correct answer.

I then reattempted my algebra and get to this:

(P·D/2·t)² + (P·D/2·t)·(P·D/4·t) + (P·D/4·t)² = σ_yield²

What am I missing here?

Any help would be appreciated.

Thank you,

- MarleyDH

 

[nvn]

σ₁² + σ₁·σ₂ + σ₂² = σ_yield²

Wrong. Read more carefully.

 

[MarleyDH]

Ah right. Saw my mistake, the cross terms should be a preceded with a minus sign. Thanks for the assistance once again nvn. Now its on to failure criterion!