Thick-Walled Cylinder (Lamé)

stressmass-cost

Hydraulic cylinders, gun barrels, waterjet intensifiers running at 4,000 bar, high-pressure chemistry vessels — anything that holds serious pressure stops being “thin-walled” fast. The thin-wall vessel page refuses to answer below $r/t = 10$; this page is where that refusal hands off. Lamé’s exact elastic field works for any wall:

$\sigma_{\theta,i} = p\,\frac{k^2+1}{k^2-1}, \quad k = \frac{r_o}{r_i} \qquad \tau_{max} - p = \frac{p\,r_i^2}{r_o^2 - r_i^2} \;>\; 0 \qquad r_o = r_i\sqrt{\frac{\sigma_y}{\sigma_y - 2\,\mathrm{SF}\,p}}$

Three things to notice, one per formula:

• The stress is material-blind again. Like the torsion shaft (and unlike the flywheel, where density is the load), the Lamé stresses are pure geometry × pressure. Swap materials in the widget: $\sigma_{\theta,i}$ refuses to move while the margin and the mass shuffle around it. And the field decays as $1/r^2$ — the metal near the bore does almost all the work, so doubling an already-thick wall buys surprisingly little.
• The bore shear always exceeds the pressure. $\tau_{max} - p > 0$ for every finite wall, shrinking to zero only as $r_o \to \infty$. That innocent-looking inequality is a hard ceiling: first yield (Tresca) needs $2\tau \le \sigma_y$, so no thickness whatsoever elastically contains $p \ge \sigma_y/2$. That is about 124 MPa for A36 — and even the quenched-and-tempered 4340 in the dropdown, the strongest material in our database, caps near 750 MPa, so a 600 MPa waterjet intensifier can never see a margin better than about 1.25 elastically, however thick its wall. Such machines exist anyway because engineers cheat (see How it fails).
• The design formula has a wall built into it. In the design configuration, watch $r_o$ climb gently as you raise the pressure — then diverge as $2\cdot\mathrm{SF}\cdot p$ approaches $\sigma_y$. The widget refuses rather than print a 40-meter wall. The rate configuration runs the same relations the third way: given a cylinder and a margin, what pressure is it good for?

The thin-wall page isn’t wrong, just licensed: as $k \to 1$ the bore hoop stress converges to $pr/t$ (the build verifies that limit symbolically). By $k = 1.5$ — the default here — the true bore stress $p(k^2+1)/(k^2-1) = 2.6\,p$ already runs 30% above the thin-wall $p\,r_i/t = 2\,p$, on the dangerous side. Envelopes are not pedantry.

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Governing relations

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

How it fails

The widget’s margin is first yield at the bore under steady internal pressure, for a plain monolithic wall. Real high-pressure hardware fails differently — and the most interesting “failures” are the ones designers inflict on purpose:

• Autofrettage — yielding the bore on purpose. Since the elastic ceiling $p < \sigma_y/2$ cannot be out-thickened, makers of gun barrels and waterjet intensifiers pre-pressurize the cylinder until the bore yields plastically, then release. The elastic outer wall squeezes the yielded core, leaving the bore in residual compression — the next pressurization must first cancel that compression before the bore feels any tension. The same trick wearing different hats: shrink-fitted compound cylinders (now a THING of their own, margins and balanced fit included), wire-wound vessels, and the 19th-century built-up cannon. The widget’s first-yield warning marks exactly the line autofrettage deliberately crosses.
• Fatigue at the bore. Pressure equipment is cycled — every stroke of an intensifier is a full stress cycle at the point the field concentrates. Cracks start at the bore (or at any cross-bore: a side port roughly triples the local stress, far worse than the bare Lamé number) and grow radially. Autofrettage earns most of its keep here: residual compression slams the crack-driving cycle shut.
• Leak-before-burst vs. fragmentation. A tough wall lets a fatigue crack tunnel through and leak — annoying, visible, survivable. A brittle or thick-and-cold wall can instead reach critical crack size before breakthrough and burst into fragments. Pressure-vessel codes are organized around forcing the first outcome; it is the difference between a puddle and a bomb.
• The ends are not this page. Lamé describes the long barrel away from closures. End caps, threads, flanges, and nozzles add axial and discontinuity stresses that routinely govern — high-pressure closures are their own discipline (and the usual leak path).
• External pressure is not the mirror image. Vacuum jackets and submersible hulls fail by buckling — stability, not strength. A wall with a comfortable Lamé margin can still collapse inward like a drink can; see the Euler column for the same lesson in one dimension.
• Temperature steals the margin. Creep at high temperature and brittleness at low temperature both erode $\sigma_y$-based margins silently; thermal gradients through a thick wall add their own Lamé-like stress field on top of the pressure’s.

Sources

• Timoshenko, S. P., & Goodier, J. N., Theory of Elasticity, 3rd ed., McGraw-Hill, 1970 — ch. 4 (two-dimensional problems in polar coordinates), §28 "Stress Distribution Symmetrical about an Axis": the hollow cylinder under uniform pressure — the Lamé field σ = A ∓ B/r², boundary conditions, bore maximum.
• Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §3-14 (stresses in pressurized cylinders) and §5-4 (maximum-shear-stress theory, S_sy = 0.5 S_y).
• Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — prismatic members and section properties (annular area, mass per length); thin-walled pressure vessels (the k → 1 limit this THING hands off to).