Lame’s theory is primarily associated with stress distribution in thick cylindrical shells (pressure vessels with significant wall thickness).

Introduction / Context:
Lame’s theory provides the fundamental solution for radial and hoop (circumferential) stresses in thick-walled cylinders under internal and/or external pressure. Unlike thin-shell formulas that assume uniform membrane stress through the thickness, thick shells exhibit strong stress gradients that must be captured for safe design.

Given Data / Assumptions:

• Cylinder with inner and outer radii; wall thickness is not negligible.
• Material assumed homogeneous, isotropic, and linearly elastic.
• Axisymmetric loading due to internal pressure (and optionally external pressure).

Concept / Approach:
Lame's equations resolve the stress field as functions of radius r. The general forms are sigma_r = A − B / r^2 and sigma_theta = A + B / r^2, where constants A and B are determined from boundary conditions at the inner and outer surfaces. This captures maximum hoop stress near the inner wall and decreasing magnitude toward the outside surface.

Step-by-Step Solution:

Verification / Alternative check:
Compare to thin-shell formula sigma_h = p d / (2 t). For small t relative to diameter, Lame’s solution approaches the thin-shell result; for larger thickness, the gradient predicted by Lame is essential.

Why Other Options Are Wrong:
Thin cylindrical shells: use membrane theory; not Lame's full-field solution.Direct and bending stresses: beam theory, not pressure vessels.'None of these' is incorrect because thick cylinders are explicit domain of Lame's theory.

Common Pitfalls:
Using thin-wall equations for thick vessels; ignoring inner-wall peak hoop stress; neglecting external pressure effects.

Final Answer:

thick cylindrical shells