r_θ and r_Φ definition

Let’s look at the cross-section of a tori-spherical end cap:

The hardest part for tori-spherical questions is usually working out the r_o, r_Φ and r_θ. Once these are conquered, the rest of the question is usually a breeze.

Stresses at different regions

For tori-spherical shells, here’s how you calculate the stress at different regions:

• spherical end-cap – pressure vessel relations: σ_θ = σ_Φ = pr/2t, with r_sphere as the r
• knuckle joint – calculate using the membrane stress method with the r_θ and r_Φ you found
• cylinder region – σ_hoop = σ_θ = pr/t, σ_long = σ_Φ = pr/2t, with r_cylinder as the r

Let’s look at an example now.