Thick-Walled Pressure Vessel — Lamé Stresses

Exact Lamé solution for through-wall stress distribution in a thick-walled cylinder or sphere under internal or external pressure

Thick-Walled Pressure Vessel — Lamé Stresses & Deformations

This calculator gives the exact elastic stress distribution and deformations for a thick-walled cylinder or sphere under internal or external pressure, using classical thick-wall (Lamé) elasticity theory. Unlike thin-shell membrane theory — which assumes stress is uniform across the wall — the Lamé solution shows that stresses vary nonlinearly through the wall, with the highest values at the bore (inner surface). This distinction matters whenever the wall is not thin: a common rule of thumb is that thick-wall theory should be used when the radius ratio a/b > 1.1 (i.e., wall thickness greater than about 10% of the inner radius), where a is the outer radius and b the inner radius.

The three-dimensional stress state at any radius r (b \le r \le a) consists of:

• Hoop (circumferential) stress \sigma_2(r) — highest at the bore, decreases toward the outer wall.
• Radial stress \sigma_3(r) — equals -q (compressive) at the loaded surface and zero at the free surface.
• Longitudinal (axial/meridional) stress \sigma_1 — zero for open-ended cylinders; a uniform tensile (internal) or compressive (external) membrane stress for capped cylinders; equal to \sigma_2 for a sphere (biaxial symmetry).

Six load cases: For a cylinder: (1a) internal pressure, open ends; (1b) internal pressure, capped; (1c) external pressure, open ends; (1d) external pressure, capped. For a sphere: (2a) internal pressure; (2b) external pressure. The case selection drives the stress formula family and the deformation expressions for \Delta a (outer-radius change), \Delta b (inner-radius change), and \Delta l (length change, cylinders only).

For spheres, the calculator also reports a reference first-yield pressure q_{yield} (Tresca criterion at the inner surface) — this is informational only, not a pass/fail check.

Model limits: isotropic linear-elastic material; uniform wall thickness; axisymmetric geometry and loading; small deformations. For cylinders: the Lamé solution applies to the body of a long cylinder, away from the end caps — end/discontinuity effects are not included. No thermal, body-force or non-axisymmetric loads. The outputs are structural response quantities only — there is no built-in safety check against yield or fracture.

Covered cases of this table:

• Thick-walled cylinder: internal or external pressure, open or capped ends (1a–1d)
• Thick-walled sphere: internal (2a) or external (2b) pressure
• Not included: temperature and rotation cases of Table 13.5

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