How to Determine the Ratio of Inner to Outer Diameter in a Stressed Pipe?

[hellfish]

Basically, my problem is small part of a bigger assignment, but I'm completely stuck at this point. This is really a math problem, but I figured I'd post it here anyway considering I might have misunderstood the physics.

Homework Statement

A vertical pipe is subjected to a axial force of 1089N and an Mb of 600Nm. The maximum stress in the pipe is not to be larger than 150MPa.

I am to find the relation between di and do (di/do), where di is the inner- and do is the outer diameter of the pipe (which is part of a bigger construction.)

Homework Equations

von Mises theorem, but there are no sheer stresses in the pipe.

Also,

sigma=F/A+Mb/Wb+Mv/Wv. However Mv is zero in this case.

The Attempt at a Solution

sigma=F/A+Mb/Wb

-> 150e6=1089/[(pi/4)*(dy^2-di^2)+600/(pi/32dy)*(dy^4-di^4)

I can't seem to find a method to isolate di/dy from this equation, mainly because of the single dy in the Wb for circular pipes. Any help would be appreciated. If something is unclear don't hesitate to ask.

 

[anathema]

I can see that you are facing some difficulty in finding the relation between di and do in this problem. It is important to note that in order to solve this problem, you will need to use the equations that you have mentioned correctly.

As you have correctly stated, the von Mises theorem can be used to calculate the maximum stress in the pipe. However, in this case, there are no shear stresses present, so the equation can be simplified to:

σ = F/A + Mb/Wb

Where σ is the maximum stress, F is the axial force, A is the cross-sectional area, Mb is the bending moment, and Wb is the section modulus for bending.

Now, in order to find the relation between di and do, we need to use the equation for section modulus for bending, which is:

Wb = π/32 * (do^4 - di^4)

Substituting this into the equation for maximum stress, we get:

σ = F/A + Mb/(π/32 * (do^4 - di^4))

Rearranging this equation, we get:

(do^4 - di^4) = Mb/(σ * π/32) - F/A

(do^2 + di^2) * (do^2 - di^2) = Mb/(σ * π/32) - F/A

(do^2 - di^2) = (Mb/(σ * π/32) - F/A)/(do^2 + di^2)

Now, we can use the given values for Mb, F, and σ to find the value of do^2 + di^2. Once we have that, we can rearrange the equation to get the relation between di and do.

I hope this helps you in solving your problem. If you have any further questions, please do not hesitate to ask. Good luck with your assignment!