10. Thin Cylinders and Spherical Shells free study notes

1. What is Thin Cylinders and Spherical Shells ?

Thin cylinders and spherical shells are pressure vessels used to store fluids (liquids or gases) under pressure. Common examples include boilers, gas cylinders, pipelines, and storage tanks.

When a vessel is subjected to internal pressure, stresses develop in its walls. If the wall thickness is small compared to its diameter, it is treated as a thin shell.

2. What is Thin Shell?

A thin cell (thin-walled shell) is a structure in which the wall thickness is very small compared to its diameter, so that the stress across the thickness can be assumed uniform. A cylinder or sphere is considered thin if:$$\frac{t}{d} \leq \frac{1}{10}$$

Where:

• $t$ = thickness of shell
• $d$ = diameter

Assumptions for Thin Shell Analysis:

1. Stress is uniformly distributed across thickness
2. Radial stress is negligible
3. Material is homogeneous and isotropic
4. Deformations are small

3. Thin Cylindrical Shell (Pressure Vessel)

3.1 Types of Stresses in Thin Cylinder

When a thin cylinder is subjected to internal pressure $p$, two main stresses develop:

Key Observation

$$\sigma_h = 2 \sigma_l$$

Hoop stress is twice the longitudinal stress, so failure usually occurs along the length.

3.2 Strains in Thin Cylinder

(a) Circumferential Strain

$$\epsilon_c = \frac{\sigma_h}{E} – \mu \frac{\sigma_l}{E}$$

(b) Longitudinal Strain

$$\epsilon_l = \frac{\sigma_l}{E} – \mu \frac{\sigma_h}{E}$$

Where:

• $E$ = Young’s modulus
• $\mu$ = Poisson’s ratio

3.3 Change in Dimensions

(a) Change in Diameter

$$\Delta d = d \cdot \epsilon_c$$

(b) Change in Length

$$\Delta L = L \cdot \epsilon_l$$

3.4 Change in Volume of Cylinder

$$\frac{\Delta V}{V} = 2\epsilon_c + \epsilon_l$$

4. Thin Spherical Shell

4.1 Stress in Spherical Shell

Unlike cylinders, spherical shells have only one type of stress (same in all directions):

$\sigma = \frac{p d}{4 t}$

4.2 Important Notes

• Stress is uniform in all directions
• No distinction between longitudinal and circumferential stress
• Stronger than cylindrical shells for the same thickness

4.3 Strain in Spherical Shell

$$\epsilon = \frac{\sigma}{E}(1 – \mu)$$

4.4 Change in Diameter

$$\Delta d = d \cdot \epsilon$$

4.5 Change in Volume

$$\frac{\Delta V}{V} = 3 \epsilon$$

5. Comparison: Cylinder vs Sphere

Feature	Thin Cylinder	Thin Sphere
Number of stresses	Two (Hoop & Longitudinal)	One
Maximum stress	Hoop stress	Uniform stress
Strength	Less	More
Stress formula	$\sigma_h = \frac{pd}{2t}$	$\sigma = \frac{pd}{4t}$
Efficiency	Lower	Higher

Conclusion: A spherical shell is stronger than a cylindrical shell under the same internal pressure.

6. Applications

• Boilers
• LPG gas cylinders
• Pressure tanks
• Submarine hulls
• Pipelines
• Storage vessels

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