Thick-walled cylinder under internal pressure: where along the wall thickness is the hoop (circumferential) stress the greatest—at the inner radius or elsewhere?

Introduction / Context:
In pressure vessel design, understanding how circumferential (hoop) stress varies through the wall of a thick cylinder is essential for safe sizing and material selection. Thick-cylinder behavior differs from thin-shell formulas because stresses vary significantly from the inside surface to the outside surface.

Given Data / Assumptions:

• Cylindrical vessel with inner radius and outer radius, subjected to internal pressure p only.
• Isotropic, homogeneous, linearly elastic material.
• No thermal gradients or external pressure unless stated.

Concept / Approach:
Lame’s equations govern radial and hoop stresses in thick cylinders. For internal pressure only, the radial stress is most compressive (−p) at the inner surface and approaches zero at the outer surface. The hoop stress is tensile and attains its maximum at the inner surface, decreasing towards the outer surface.

Step-by-Step Solution:

Verification / Alternative check:
Plot sigma_theta versus r from the inner to outer surface. The curve peaks at the inner wall for an internally pressurized cylinder, matching standard design charts.

Why Other Options Are Wrong:
False/conditional options conflict with Lame’s solution; dependence on Poisson’s ratio does not overturn the location of maximum hoop stress.

Common Pitfalls:
Using thin-wall formula sigma_h = p d / (2 t) when D/t is small; neglecting through-thickness stress variation and inner-wall peaking.

Final Answer:

True