1. Recap

We had derived the following simplified form of equilibrium equations when the cylinder is in static equilibrium and not subjected to any body force:

(1) (2)

In this section, we introduced the key equilibrium equations that describe a hollow cylinder's behavior when it is not influenced by any external forces. These equations are crucial for understanding how the cylinder maintains its shape and responds to internal stresses. We emphasize that these equations only apply when the cylinder is at rest (static equilibrium).

Imagine a tall building standing still without any wind or earthquakes affecting it. The equilibrium equations are similar to the calculations engineers perform to ensure that the building remains stable and does not collapse under its own weight or any other forces.

We had also obtained the following expressions of stress components in terms of displacement components:

(3) (4) (5) (6) (7) (8)

Here, we presented key relationships between stress and displacement in the hollow cylinder. Stress components (like radial and hoop stresses) are connected to displacement components (how much the cylinder stretches or compresses). Understanding these relationships is crucial for engineers to determine how changes in force affect material deformation.

Think of a rubber band. When you stretch it, there is a relationship between how far you stretch it and the tension (stress) inside it. Similarly, the equations show how internal stresses change as the cylinder is subjected to different forces or internal pressures.

From equation (2), we can infer that σ does not depend on z. Also, σ does not have any term that has dependence on θ. Thus, it is a function of r only, i.e.,

(10)

This piece elaborates on how certain stress components are independent of the axial position (z) and the angular position (θ). This simplification allows us to focus only on the radial position (r), making the calculations for stress more manageable. It's important to know the conditions under which specific stress components apply, as it helps engineers design systems more effectively.

Imagine a bicycle tire. The pressure within the tire affects how firm it feels (which relates to stress), but this pressure doesn't change based on how you tilt the bike or where you measure it along the height of the tire. It is mostly dependent on how 'thick' the tire is (the radial distance).

Accordingly, the term dependent on z in the LHS must be a constant or u’ must be a constant. As u’ denotes longitudinal strain in axial direction, we will denote it by ε, an unknown but constant parameter.

In this section, we discussed how certain strain measures can be treated as constants for simplification. Specifically, we denote the longitudinal strain as ε, which helps streamline our equations by assuming that the deformation does not vary along the length of the cylinder. This assumption is often valid for engineering problems involving uniform materials and loading.

Consider stretching a straight piece of taffy. If you pull evenly from both ends, the stretching (strain) is uniform along its length. Engineers often simplify and assume conditions like this to make complex calculations easier and more reliable.

--