Problem 12 - An Introduction to the Mechanics of Solids in SI Units

Question

Consider a thin-walled cylindrical shell of intemal radius \(r\) and thickness \(t\), with ends which will contain pressure. Show that the principal stresses in the cylinder wall are given approximately by the following when the cylinder contains an intemal pressure \(p\) : $$ \begin{aligned} &\sigma_{r}=0 \\ &\sigma_{\theta}=\frac{p r}{t} \\ &\sigma_{z}=\frac{p r}{2 t} \end{aligned} $$

Step-by-Step Solution

Answer

The given formulas for stresses under an internal pressure p in a thin-walled cylindrical shell are confirmed. Radial stress \(\sigma_{r}=0\) directly from the thin wall assumption. Circumferential stress \(\sigma_{\theta}=\frac{p r}{t}\) and longitudinal stress \(\sigma_{z}=\frac{p r}{2 t}\) can be derived from the force balance considerations in the respective directions.

Step 1: Apply condition for radial stress

For the radial stress \(\sigma_{r}\), it is straightforward that since the shell wall is thin, the radial stress is approximately zero.

Step 2: Derive circumferential stress

For circumferential or hoop stress \(\sigma_{\theta}\), apply the equilibrium condition. The forces in the cylindrical shell must be balanced. Therefore, the internal pressure force equals the force due to hoop stress. The equation reads: \(p \cdot 2 \pi r t = 2 \cdot t \cdot \sigma_{\theta} \cdot r \), from which it is easy to derive the given formula: \(\sigma_{\theta} = \frac{p r}{t}\) .

Step 3: Calculate longitudinal stress

The longitudinal stress \(\sigma_{z}\) can be derived in a similar manner, but now considering horizontal balance. The equation reads: \(p \cdot \pi r^2 = \sigma_{z} \cdot 2 \pi r t\).4Solving for \(\sigma_{z}\) yields \(\sigma_{z}= \frac{p r}{2t}\) , which is the given formula.

Key Concepts

Principal Stresses

When dealing with the stresses on a thin-walled cylindrical shell, it's essential to understand the concept of principal stresses. Principal stresses are the normal stresses that occur at a particular point on an object, measured at the orientations where the shear stress is zero. In thin-walled cylinders under internal pressure, such as pipes or tanks, we commonly find two principal stresses. These are hoop stress, which wraps around the cylinder, and longitudinal stress, which runs along the cylinder's axis.

The third principal stress, radial stress, is approximately zero in thin-walled cylinders because the wall thickness is small compared to the cylinder's radius. This assumption simplifies our calculations and is a standard approach in engineering problems. As a tip, to fully grasp the nature of principal stresses, visualize the cylindrical shell as if it's being pulled apart circumferentially and longitudinally by the internal pressure, but without bulging outwards, representing zero radial stress.

Hoop Stress

Hoop stress, often symbolized as \( \sigma_{\theta} \), is a principal stress that acts circumferentially around the shell's wall. Imagine a ring cut out from the cylinder; the hoop stress would be the force trying to expand that ring due to the internal pressure. It's the tension force per unit area along the cylinder’s circumference caused by the pressure exerted on the inner surface.

To calculate the \( \sigma_{\theta} \) in a thin-walled cylindrical shell, use the formula \( \sigma_{\theta} = \frac{p r}{t} \), where \( p \) is the internal pressure, \( r \) is the shell's internal radius, and \( t \) is the wall thickness. It's crucial to remember that hoop stress is a significant factor in the design and analysis of pressure vessels, as it has considerable implications for the material's structural integrity. For a better understanding, envision the hoop stress as the result of the cylinder trying to 'inflate' due to the internal pressure.

Longitudinal Stress

Analogous to hoop stress, longitudinal stress, denoted as \( \sigma_{z} \), occurs along the length of the cylindrical shell. This stress reflects how the internal pressure stretches the cylinder lengthwise. It can be visualized by imagining the pressure trying to pull apart the ends of the cylinder.

To determine the longitudinal stress, we use the formula \( \sigma_{z}= \frac{p r}{2t} \). The calculation considers that the force from the pressure is spread over a larger area along the length of the cylinder compared to the circumference, which accounts for the longitudinal stress typically being half of the hoop stress. When studying longitudinal stress, it's important to consider this stress in conjunction with hoop stress to ensure the structural safety of the cylinder under pressure, as both stresses contribute to the potential failure of the material.