Thin and Thick Cylinders

Why it matters

Thin and thick cylinders are commonly used in engineering applications such as pipelines, boilers, and pressure vessels. Understanding the stress distribution in these cylinders is crucial for ensuring their structural integrity and safety under various loading conditions.

Key ideas

• Thin Cylinders: These are cylinders where the wall thickness is small compared to the diameter (typically, thickness is less than 1/20th of the diameter). Hoop and longitudinal membrane stresses are approximated as uniform; radial stress still varies from -p_i to -p_o and is usually small compared with hoop stress.
• Thick Cylinders: These have a wall thickness that is not negligible compared to the diameter. The stress distribution varies across the thickness, and more complex analysis is required.
• Stress Analysis: For thin cylinders, hoop stress and longitudinal stress are the primary concerns. For thick cylinders, radial and hoop stresses are analyzed using Lame's equations.
• Applications: The applicable model depends on geometry and required accuracy, not pressure alone.

Formulas

• Hoop Stress for Thin Cylinders: σ_h = (p·d) / (2·t)
  • σ_h: Hoop stress (Pa)
  • p: Internal minus external pressure (Pa)
  • d: Diameter of the cylinder (m)
  • t: Thickness of the cylinder wall (m)
• Longitudinal Stress for Thin Cylinders: σ_l = (p·d) / (4·t)
  • σ_l: Longitudinal stress (Pa)
• Lame's Equations for Thick Cylinders:
  • Radial Stress: σ_r = (A - B/r²)
  • Hoop Stress: σ_h = (A + B/r²)
  • A and B are constants determined by boundary conditions.
  • r: Radial distance from the center (m)

The longitudinal formula assumes closed ends whose pressure load is carried by the wall. For thick cylinders, use σ_r(r_i) = -p_i and σ_r(r_o) = -p_o to determine A and B. These are ideal stress calculations, not complete pressure-vessel design checks.

Worked example

Given: A thin, closed-ended cylindrical pipe with an internal pressure of 2 MPa, diameter of 0.5 m, and wall thickness of 0.01 m.

1. Calculate the hoop stress using the formula: σ_h = (p·d) / (2·t) σ_h = (2×10^6 Pa · 0.5 m) / (2 · 0.01 m) σ_h = 50×10^6 Pa Hoop Stress = 50 MPa

2. Calculate the longitudinal stress using the formula: σ_l = (p·d) / (4·t) σ_l = (2×10^6 Pa · 0.5 m) / (4 · 0.01 m) σ_l = 25×10^6 Pa Longitudinal Stress = 25 MPa

Common mistakes

• Confusing the formulas for hoop and longitudinal stress.
• Applying thin cylinder formulas to thick cylinders without checking the thickness-to-diameter ratio.
• Ignoring the units, leading to incorrect stress calculations.

For GATE CE

Questions often involve calculating hoop and longitudinal stresses for thin cylinders or using Lame's equations for thick cylinders. Practice problems that require identifying whether a cylinder is thin or thick and applying the appropriate formulas.

Quick check

1. What is the primary difference in stress analysis between thin and thick cylinders?
2. Write the formula for hoop stress in a thin cylinder.
3. Why are Lame's equations used for thick cylinders?

Answers: 1. Stress distribution; thin cylinders assume uniform stress, thick cylinders do not. 2. σ_h = (p·d) / (2·t). 3. To account for varying stress across the thickness.