Lesson 2.6: Thin & Thick Cylinders

Thin and thick cylinders are fundamental in Pressure Vessel Design, which is highly important for mechanical engineers appearing in GATE & PSU exams.

🔹 1. Introduction

• Cylinders: Hollow circular structures subjected to internal/external pressure.

• Applications: Boilers, pressure vessels, hydraulic cylinders, pipes.

• Types:

  1. Thin Cylinder: Wall thickness $t\le 0.1D$ (D = internal diameter)

  2. Thick Cylinder: Wall thickness $t>0.1D$

🔹 2. Thin Cylinders

• Assumptions:

  • Stress across thickness is uniform.

  • Neglect radial stress (σr≈0\sigma_r \approx 0σr​≈0).

• Stresses:

1. Circumferential (Hoop) Stress:

σh=prt\sigma_h = \frac{p r}{t}σh​=tpr​

2. Longitudinal Stress:

σl=pr2t\sigma_l = \frac{p r}{2 t}σl​=2tpr​

Where:

• $p$ = internal pressure

• $r$ = internal radius

• $t$ = wall thickness

Applications: Boilers, pipelines, gas storage cylinders.

Example:
Thin-walled cylinder, r = 0.1 m, t = 5 mm, p = 2 MPa.

• Hoop stress: σh=2∗0.1/0.005=40 MPa\sigma_h = 2*0.1/0.005 = 40 \text{ MPa}σh​=2∗0.1/0.005=40 MPa

• Longitudinal stress: σl=2∗0.1/(2∗0.005)=20 MPa\sigma_l = 2*0.1/(2*0.005) = 20 \text{ MPa}σl​=2∗0.1/(2∗0.005)=20 MPa

🔹 3. Thick Cylinders

• Assumptions:

  • Stress varies across thickness.

  • Radial stress significant.

• Lame’s Equations (Radial & Hoop Stress):

σr=piri2−poro2ro2−ri2−(pi−po)ri2ro2(ro2−ri2)r2\sigma_r = \frac{p_i r_i^2 – p_o r_o^2}{r_o^2 – r_i^2} – \frac{(p_i – p_o) r_i^2 r_o^2}{(r_o^2 – r_i^2) r^2}σr​=ro2​−ri2​pi​ri2​−po​ro2​​−(ro2​−ri2​)r2(pi​−po​)ri2​ro2​​ σθ=piri2−poro2ro2−ri2+(pi−po)ri2ro2(ro2−ri2)r2\sigma_\theta = \frac{p_i r_i^2 – p_o r_o^2}{r_o^2 – r_i^2} + \frac{(p_i – p_o) r_i^2 r_o^2}{(r_o^2 – r_i^2) r^2}σθ​=ro2​−ri2​pi​ri2​−po​ro2​​+(ro2​−ri2​)r2(pi​−po​)ri2​ro2​​

Where:

• ri,ror_i, r_ori​,ro​ = inner & outer radius

• pi,pop_i, p_opi​,po​ = internal & external pressure

• Maximum Stress: Occurs at inner surface.

Applications: High-pressure vessels, steam boilers, nuclear reactors.

🔹 4. Thin vs Thick Cylinder Comparison

🔹 5. Solved Examples (PYQ Style)

Example 1 (GATE ME 2018):
Thin-walled cylinder, r = 0.15 m, t = 5 mm, p = 1.5 MPa. Find hoop & longitudinal stresses.

• Hoop: σh=1.5∗0.15/0.005=45 MPa\sigma_h = 1.5*0.15/0.005 = 45 \text{ MPa}σh​=1.5∗0.15/0.005=45 MPa

• Longitudinal: σl=1.5∗0.15/(2∗0.005)=22.5 MPa\sigma_l = 1.5*0.15/(2*0.005) = 22.5 \text{ MPa}σl​=1.5∗0.15/(2∗0.005)=22.5 MPa

Example 2 (PSU Exam):
Thick cylinder, r_i = 0.1 m, r_o = 0.2 m, p_i = 5 MPa, p_o = 0. Find radial & hoop stresses at inner surface.

• Use Lame’s equations → σθ≈15 MPa,σr=−5 MPa\sigma_\theta \approx 15 \text{ MPa}, \sigma_r = -5 \text{ MPa}σθ​≈15 MPa,σr​=−5 MPa

🔹 6. Practice Exercises

1. Thin cylinder, r = 0.2 m, t = 8 mm, p = 2.5 MPa. Find hoop & longitudinal stress.

2. Thick cylinder, r_i = 0.05 m, r_o = 0.1 m, p_i = 4 MPa, p_o = 1 MPa. Find maximum hoop stress.

3. Compare stresses in thin vs thick cylinder for same internal pressure.

4. Design a thin-walled cylinder to withstand p = 3 MPa with allowable stress 60 MPa.

🔹 7. Summary

• Thin Cylinder: Uniform stress across wall, simple formulas

• Thick Cylinder: Radial & hoop stress varies, Lame’s equations

• Key Parameters: Internal/external pressure, radius, wall thickness

• Applications: Pressure vessels, boilers, pipelines, mechanical cylinders