Shell structures are important engineering constructs, as they use minimum material and can maximise the volume they contain. Even the fuselages of aircrafts are shell structures; they maximise the cargo and passengers that each flight can take.
In Chapter 6.1 of Solid Mechanics I, we looked at shell structures in the form of cylinders and spheres. But what about other kinds of shells: conical, egg-shaped, a cone-cylinder hybrid; how do we analyse the stresses in these kinds of geometry?
Well, we use the membrane stress equation. But before we present the formula, we need to explain some background concepts first:
Shell structures are important engineering constructs, as they use minimum material and can maximise the volume they contain. Even the fuselages of aircrafts are shell structures; they maximise the cargo and passengers that each flight can take.
In Chapter 6.1 of Solid Mechanics I, we looked at shell structures in the form of cylinders and spheres. But what about other kinds of shells: conical, egg-shaped, a cone-cylinder hybrid; how do we analyse the stresses in these kinds of geometry?
Well, we use the membrane stress equation. But before we present the formula, we need to explain some background concepts first:
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:
We will be looking at how to define the directions of σ_{θ}, σ_{Φ} and r_{θ}, r_{Φ} for other shapes later.
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.
Here we look at the σ_{θ}, σ_{Φ} and r_{θ}, r_{Φ} for shell structures with other shapes:
Problems from this chapter can be tricky, but if you follow the procedure below you can solve the problems easily and systematically:
*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.
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:
We will be looking at how to define the directions of σ_{θ}, σ_{Φ} and r_{θ}, r_{Φ} for other shapes later.
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.
Here we look at the σ_{θ}, σ_{Φ} and r_{θ}, r_{Φ} for shell structures with other shapes:
Problems from this chapter can be tricky, but if you follow the procedure below you can solve the problems easily and systematically:
*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.