Thick-walled cylinder under internal pressure: Where is the maximum tensile (hoop) stress located?

Introduction:
Pressure-vessel integrity depends on the distribution of stresses through the wall. In thick cylinders, elastic stress variation is non-uniform. Identifying where the maximum hoop (circumferential) stress occurs is essential for sizing and for assessing crack initiation risk.

Given Data / Assumptions:

• Elastic, isotropic, homogeneous cylinder.
• Internal pressure only; no external pressure.
• Closed-end effects on axial stress neglected for the location question.

Concept / Approach:
Lame’s equations give radial and hoop stresses that vary with radius. For internal pressure, the hoop stress is highest at the inner radius and decreases toward the outer radius. The radial stress is compressive and drops from internal pressure at the bore to approximately zero at the outer surface (if unpressurized externally).

Step-by-Step Solution:
Use Lame: sigma_hoop(r) = A + B/r^2 with constants from boundary conditions.At r = inner radius, sigma_hoop is maximum.Thus, maximum tensile hoop stress occurs at the inner surface.

Verification / Alternative check:
Standard plots of hoop stress vs. radius for thick cylinders confirm the inner wall maximum for internal pressure only.

Why Other Options Are Wrong:

• Mid-thickness/outer surface: hoop stress monotonically decreases outward in this case.
• “None of these” is incorrect because the inner surface is the well-known maximum location.

Common Pitfalls:
Confusing thin-wall assumptions (nearly uniform hoop stress) with thick-wall distributions.

Final Answer:
At the inner surface