Thin and Thick Cylinders

Why it matters

Thin and thick cylinders are fundamental in engineering applications such as pipelines, boilers, and pressure vessels. Understanding the stress distribution in these structures ensures their safe and efficient design, preventing catastrophic failures.

Key ideas

• Thin Cylinders: Assumes the wall thickness is small compared to the diameter (typically less than 1/20th). The stress distribution is uniform across the thickness.
  • Hoop Stress: Circumferential stress due to internal pressure.
  • Longitudinal Stress: Axial stress due to internal pressure.
• Thick Cylinders: Wall thickness is not negligible compared to the diameter. Stress varies across the thickness.
  • Radial Stress: Varies from internal to external surface.
  • Hoop Stress: Maximum at the inner surface, decreases outward.
  • Lame's Equations: Used for calculating stresses in thick cylinders.

Boundary conditions and scope

Take tension positive and pressure positive in compression. For inner radius a, outer radius b, internal pressure p_i, and external pressure p_o, the Lamé constants in the convention below are A = (p_i a² − p_o b²)/(b² − a²) and B = (p_i − p_o)a²b²/(b² − a²). They satisfy σ_r(a) = −p_i and σ_r(b) = −p_o.

These elastic axisymmetric formulas apply away from end effects. With internal pressure only, hoop stress is greatest at the bore. External pressure changes the distribution and can introduce buckling concerns. The thin-wall longitudinal formula assumes closed ends carrying the pressure thrust. Use differential pressure relative to the outside, with the diameter convention appropriate to the thin-wall approximation.

Formulas

• Thin Cylinder Hoop Stress: σ_h = (p·d) / (2·t)
  • σ_h: Hoop stress (Pa)
  • p: Internal pressure (Pa)
  • d: Diameter of the cylinder (m)
  • t: Wall thickness (m)
• Thin Cylinder Longitudinal Stress: σ_l = (p·d) / (4·t)
  • σ_l: Longitudinal stress (Pa)
• Thick Cylinder Hoop Stress (Lame's Equation): σ_h = (A + B/r²)
  • A, B: Constants determined by boundary conditions
  • r: Radius at the point of interest (m)
• Thick Cylinder Radial Stress: σ_r = (A - B/r²)

Worked example

Given: A thin, closed-ended cylindrical vessel with gauge internal pressure p = 2 MPa, diameter d = 1 m, and wall thickness t = 10 mm.

1. Calculate Hoop Stress

   • Formula: σ_h = (p·d) / (2·t)
   • Calculation: σ_h = (2×10^6 Pa · 1 m) / (2 · 0.01 m)
   • Result: σ_h = 100 MPa

2. Calculate Longitudinal Stress

   • Formula: σ_l = (p·d) / (4·t)
   • Calculation: σ_l = (2×10^6 Pa · 1 m) / (4 · 0.01 m)
   • Result: σ_l = 50 MPa

Final Answer: Hoop Stress = 100 MPa, Longitudinal Stress = 50 MPa

Common mistakes

• Confusing hoop and longitudinal stresses.
• Applying thin cylinder formulas to thick cylinders.
• Ignoring units or incorrect unit conversions.

For GATE ME

Questions often involve calculating hoop and longitudinal stresses in thin cylinders or using Lame's equations for thick cylinders. Practice problems on stress distribution and boundary conditions.

Quick check

1. What is the primary assumption for thin cylinders?
2. How does hoop stress vary in a thick cylinder?
3. What formula is used for radial stress in thick cylinders?

Answers: 1. Wall thickness is small compared to diameter. 2. Maximum at the inner surface, decreases outward. 3. σ_r = (A - B/r²).