2. Mathematical form of u

As u and u are functions of z and r respectively, we can rewrite equation (5) as ...

In this chunk, we are establishing that the functions u (axial displacement) and u (radial displacement) depend solely on the variables z (axial position) and r (radial position), respectively. This means that we can simplify the expressions related to these functions in our calculations. By acknowledging their dependence on different variables, we can address them separately in our equations, which will help us derive further relationships more easily.

Imagine a tall glass filled with water. The height of the water (how full it is) depends on how much water you pour into it (the axial position or z), while the radius around the glass defines how far from the center you measure (the radial position or r). Just as the glass's height doesn't rely on its radius, our functions behave similarly.

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.

Here, we learn about how the stress components σ behave in relation to the variables we identified earlier. Specifically, we discover that the stress σ does not change in the axial direction (z), nor does it vary with the angle θ; it is solely a function of the radial distance (r). This simplification allows us to focus our analysis on how stress changes only with respect to radial displacement, making our equations less complex.

Consider squeezing an inflatable balloon. The pressure inside (analogous to stress σ) will vary depending on how far you are from the center (radius r), but not with the height of the balloon (which could represent z). This helps us understand how stress is distributed in cylindrical objects.

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 conclude that any term in our equations that depends on the variable z must remain constant throughout our analysis. This is crucial because it means that the longitudinal strain (denoted by u’) is not changing within the locations we are considering—it’s a constant value represented by ϵ. This idea simplifies our analysis and computation of strains and stresses further as we are now treating this term as a known constant in our equations.

Think of a rubber band stretched between two fingers. If you hold it taut (axial displacement), the amount it stretches (longitudinal strain) stays the same no matter how far apart your fingers are—this helps illustrate that the strain is constant under certain conditions.

Integrating (15) twice, we finally get ...

This segment shows that we are integrating the equation derived from earlier steps to find a solution that represents our stress in a holistic manner across the cylindrical structure. The process of integration allows us to find the total effect of stress along with the equations moving forward. It is essential for determining how stress varies throughout different sections and provides the groundwork for solving real-world engineering problems.

Imagine you have a stack of books on a shelf. If you calculate the total weight acting on the shelf (like integrating stress), you consider each book's individual weight and how they influence each other when stacked, leading to a comprehensive understanding of the total load.

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