Thick cylinder under internal pressure – Minimum hoop (tangential) stress at outer surface\nA thick cylindrical shell with inner radius r_i and outer radius r_o is subjected to internal pressure p only. What is the tangential (hoop) stress at the outer surface?

Introduction / Context:
In thick-walled cylinders, stresses vary across the wall thickness. Lame’s equations provide closed-form expressions for radial and hoop stresses. The outer-surface hoop stress is the minimum tangential stress for internal-pressure-only loading.

Given Data / Assumptions:

• Internal pressure p; no external pressure.
• Inner radius r_i, outer radius r_o.
• Elastic, isotropic, axisymmetric conditions.

Concept / Approach:
Lame’s solutions: σ_r = A − B / r^2 and σ_θ = A + B / r^2. Apply boundary conditions σ_r(r_i) = −p and σ_r(r_o) = 0 to find constants A, B, then evaluate σ_θ at r = r_o.

Step-by-Step Solution:
From σ_r(r_o) = 0 ⇒ A = B / r_o^2.From σ_r(r_i) = −p ⇒ B / r_o^2 − B / r_i^2 = −p ⇒ B = p r_o^2 r_i^2 / (r_o^2 − r_i^2).Thus A = B / r_o^2 = p r_i^2 / (r_o^2 − r_i^2).Hoop stress at outer surface: σ_θ(r_o) = A + B / r_o^2 = 2 p r_i^2 / (r_o^2 − r_i^2).

Verification / Alternative check:
Compare with inner-surface hoop stress σ_θ(r_i) = 2 p r_o^2 / (r_o^2 − r_i^2), which is larger; hence σ_θ decreases from inner to outer surface, matching expectations.

Why Other Options Are Wrong:
(a) corresponds to a different combination; (c) hoop stress is not zero at the outer surface (radial stress is zero there); (d) and (e) are not consistent with Lame’s boundary conditions.

Common Pitfalls:
Confusing radial stress with hoop stress; sign conventions; mixing r and r^2 in algebra.

Final Answer:
σ_θ(r_o) = 2 p r_i^2 / (r_o^2 − r_i^2)