σ_θ and σ_Φ – perpendicular stresses in shell structures

Shell structures experience loadings such as the internal pressure or the weight of the fluid it holds. These "action" loadings are resisted by the stresses that act along the thin-walled shells of the structure.

The general way of denoting the stresses along the shells is by the perpendicular σ_θ and σ_Φ, which act in the “θ” and “Φ” direction. We use a simple sphere to illustrate how σ_θ and σ_Φ acts and how r_θ and r_Φ are measured:

Note:

• r_θ - radius of the arc that "sweeps" along the local latitude direction
• r_Φ - radius of the arc that "sweeps" along the longitudinal direction
• r_o - radius from the major-axis to the point of interest. The direction of the radius is always perpendicular to the major axis of the geometry.
• σ_θ - stress of the spherical shell acting in the local latitude direction from the point of interest
• σ_Φ - stress of the spherical shell acting in the longitudinal direction from the point of interest
• Φ measures the vertical angle along the longitudinal line, from the point of interest to the equator

We will be looking at how to define the directions of σ_θ, σ_Φ and r_θ, r_Φ for other shapes later.

Membrane stress equation

Based on the σ_θ and σ_Φ that we’ve just defined, the formula to solve for the stress is as follows:

This equation is usually used to solve for σ_θ. For σ_Φ, we solve it by using [+↑ΣF_y = 0], which includes other y-forces such as pressure in the vessel and weight of the fluid contained.

We’ll see how this formula is used in the example shortly.

σ_θ, σ_Φ and r_θ, r_Φ for other shapes

Here we look at the σ_θ, σ_Φ and r_θ, r_Φ for shell structures with other shapes:

Procedure to solve

Problems from this chapter can be tricky, but if you follow the procedure below you can solve the problems easily and systematically:

1. Get values for r_o, r_Φ, r_θ, angle Φ, p, t (all input parameters)*
2. [+↑ΣF_y = 0], find σ_Φ
3. Solve membrane stress equation, find σ_θ
4. Final answers for σ_θ, σ_Φ, σ_r (principal stresses), or find p_max or t_min (depending on the question)

*Note: Sometimes pressure p or thickness t is what you need to find as the answer of the question.

Lots of theory covered so far! Let’s look at an example now.