σ_θ, σ_Φ and r_θ, r_Φ definition

It might be hard to imagine how σ_θ, σ_Φ and r_θ, r_Φ are defined for a toroidal shell, but as long as you follow the convention shown in the picture below you should be able to answer toroidal shell problems.

Sign conventions

For toroidal shell problems you might be asked to find the stress in any of the 4 quadrants:

Using the same rules, we can work out the signs of r_Φ and p for quadrants III and IV:

Calculating force for [+↑ΣF_y = 0]

For the force due to pressure in the y-direction, we take the segment of the shell from the point of interest to where it ends at the minor-axis:

For σ_Φ, it acts on the thickness along the circumference with r = r_o. Since r >> t, we can approximate the area σ_Φ acts on as 2πr_ot:

Let’s look at an example now.