Thick cylinder under internal pressure p: What is the maximum radial stress at the inner surface of a thick cylindrical shell (sign convention: tensile positive)?

Introduction / Context:
Lame's theory for thick-walled cylinders provides radial and hoop stresses that vary across the thickness. Knowing boundary values, especially at the bore where stresses are critical, is fundamental for pressure vessel design and safety assessment.

Given Data / Assumptions:

• Thick-walled, cylindrical geometry; internal pressure p only.
• Elastic, isotropic material; plane stress in the cylinder wall.
• Sign convention: tensile positive, compressive negative.

Concept / Approach:
Lame's solution gives radial stress sigma_r and hoop stress sigma_theta as functions of radius. Boundary condition at the inner radius r_i is sigma_r(r_i) = -p (compressive), and at the outer radius r_o (if no external pressure) sigma_r(r_o) = 0. Thus the maximum magnitude of radial stress occurs at the inner surface and equals the internal pressure but compressive in sign.

Step-by-Step Solution:

Verification / Alternative check:
Physical reasoning: internal pressure pushes the wall inward; on the inner face the fluid contact produces compressive normal stress equal to p on the solid surface, matching sigma_r = -p.

Why Other Options Are Wrong:

• Zero or +p contradict boundary conditions.
• 2p or p/2 are not consistent with Lame's boundary values for radial stress.

Common Pitfalls:
Confusing radial stress with hoop stress (which is tensile and typically larger in magnitude); mixing sign conventions; assuming thin cylinder relations where radial stress is approximated as negligible across thickness.

Final Answer:
-p (compressive)