Thick cylinder under internal pressure only: How does radial stress vary across the wall thickness?

Introduction / Context:
Design of thick-walled pressure components requires correct understanding of radial and hoop stress distributions. Errors in the sign or boundary conditions can seriously compromise safety.

Given Data / Assumptions:

• Internal pressure p acts at the bore; external pressure is zero unless stated.
• Elastic, isotropic material; Lame’s solution applies.
• Static loading; no thermal gradients.

Concept / Approach:
From Lame’s equations: sigma_r = A − B/r^2. Boundary conditions for internal-only pressure are sigma_r(r_i) = −p (compressive) and sigma_r(r_o) = 0. Thus the largest magnitude of radial stress occurs at the inner radius and decreases to zero at the outer radius.

Step-by-Step Solution:

Verification / Alternative check:
Plotting sigma_r versus r yields a monotonic curve increasing from −p to 0. Hoop stress sigma_theta remains tensile and larger than |sigma_r| near the inner radius.

Why Other Options Are Wrong:
Options (a) and (c) reverse the boundary conditions.Option (b) is incomplete; it omits that sigma_r at the outer surface equals zero when external pressure is absent.Uniform distribution is a thin-wall assumption for hoop stress, not for radial stress in thick walls.

Common Pitfalls:
Assuming thin-wall results; forgetting that radial stress at a free outer surface is zero when no external pressure acts.

Final Answer:

Maximum at the inner surface and zero at the outer surface (when no external pressure)