Thick cylindrical shell under internal pressure For a thick-walled cylinder subjected to an internal pressure p, how does the circumferential (tangential/hoop) stress vary across the wall thickness?

Introduction / Context:Design of pressure vessels and gun barrels requires understanding the stress distribution in thick cylinders. Unlike thin shells where hoop stress is assumed uniform, thick cylinders exhibit a radial variation governed by Lame’s equations.

Given Data / Assumptions:

• Cylinder subject to internal pressure p only (no external pressure).
• Axisymmetric, long cylinder; plane strain or generalized plane stress idealization.
• Linear elastic, homogeneous, isotropic material.

Concept / Approach:Lame’s theory states that for a thick cylinder the radial stress σ_r and hoop (tangential) stress σ_θ vary with radius r as σ_r = A − B/r^2 and σ_θ = A + B/r^2. Constants A and B are found from boundary conditions (σ_r = −p at inner surface, σ_r = 0 at outer surface).

Step-by-Step Solution:Apply boundary conditions to determine A and B in σ_θ = A + B/r^2.Because of the +B/r^2 term, σ_θ increases as r decreases toward the inner surface.Therefore hoop stress is highest at the inner radius and reduces toward the outer radius.

Verification / Alternative check:Plotting σ_θ(r) from typical dimensions confirms a monotonic decrease from inner to outer surface when only internal pressure acts.

Why Other Options Are Wrong:Uniform stress is a thin-shell assumption; options implying zero hoop stress at one surface contradict equilibrium and elasticity solutions.

Common Pitfalls:Confusing radial stress (which is zero at the outer surface) with hoop stress; applying thin-wall results to thick-wall situations.

Final Answer:Maximum at the inner surface and minimum at the outer surface