Thick cylinders – Maximum tangential (hoop) stress vs. internal pressure In a thick-walled cylinder subjected to internal pressure p_i only, how does the maximum tangential (hoop) stress at the inner surface compare with p_i?

Introduction / Context:
For thick cylinders, the stress distribution is non-uniform and must be found using Lame’s equations, unlike thin-wall approximations. The hoop stress peaks at the inner surface where the pressure is applied.

Given Data / Assumptions:

• Internal pressure only; no external pressure.
• Elastic, axisymmetric behavior; long cylinder (plane strain or plane stress assumptions for wall).
• Inner radius r_i, outer radius r_o.

Concept / Approach:
Lame’s formula for hoop stress is σ_θ(r) = A + B / r^2. Applying boundary conditions at r = r_i (−p_i radial) and r = r_o (0 radial) gives a maximum σ_θ at r_i. Its magnitude exceeds p_i depending on thickness ratio.

Step-by-Step Solution:
Use Lame’s relations to determine constants A and B from radial stress boundary conditions.Evaluate σ_θ at r = r_i and compare with p_i.For any finite wall (r_o > r_i), σ_θ(r_i) > p_i; the thicker the wall (smaller r_i/r_o), the larger the amplification factor.

Verification / Alternative check:
Numerical examples with r_o/r_i > 1 always yield σ_θ(r_i) greater than p_i. The thin-wall limit tends toward σ_θ ≈ p_i * r_i / t, but for thick walls, inner hoop stress is elevated further.

Why Other Options Are Wrong:
Equal to or less than p_i contradicts Lame’s solution; dependence on Poisson’s ratio is weak for principal stresses; zero at inner surface is impossible under internal pressure.

Common Pitfalls:
Applying thin-wall formulas when t is not small; ignoring radial stress boundary conditions.

Final Answer:
greater than the internal pressure