Thin-Walled Pressure Vessel (Cylinder)

stressmass-cost

Verified build 4 relations · 5 identities proven · 0 modeling steps · 6 parity samples

Pump pressure into a closed tube and the wall goes into tension two ways at once: around the circumference (hoop stress) and along the axis (longitudinal stress). Two free-body cuts — one lengthwise, one crosswise — give both stresses in three lines of statics, and the punchline is geometric, not material:

$\sigma_h = \frac{pr}{t} = 2\,\sigma_l$

The hoop direction is always twice as loaded. That single factor of 2 explains why an overcooked sausage splits lengthwise (the hoop stress lets go first, and the crack runs along the axis), why wound-filament tanks wrap most of their fiber circumferentially, and why a burst pipe shows a long axial slit rather than a clean ring.

This THING is also the portal’s clearest demonstration that relations are undirected. The analyze configuration takes geometry and pressure in, stresses and safety factor out. The design configuration takes the same four relations and runs them the other way: declare the safety factor you want, and the required wall thickness $t = \mathrm{SF}\cdot pr/\sigma_y$ comes out — along with the mass you’ll pay for it. Switch to the design configuration and swap materials: a stronger alloy doesn’t change the stress at all (that’s set by $p$ , $r$ , $t$ ), it changes how little $t$ you need. Strength buys thinness, and thinness times density is the real currency, which is why the materials page plots $\sigma_y$ against $\rho$ .

The thin-wall formulas live inside an envelope: they pretend stress is uniform through the thickness, which is honest while $r/t \ge 10$ and increasingly wrong below that. The widget will warn you as you leave the envelope and refuse beyond $r/t < 5$ — the thick-wall (Lamé) analysis is a different THING.

Try it

Configuration

Material

T3, bare flat sheet 0.010-0.128 in. thick, AMS 4037 / AMS-QQ-A-250/4 (MIL-HDBK-5J Table 3.2.3.0(b1), p. 3-71)

Bound properties of 2024-T3 aluminum sheet (bare)
sigma_y	47 ksi	design min.	mil-hdbk-5j
rho	0.1 lb/inch**3	typical	mil-hdbk-5j

3 materials in the database are not listed here: no published value in our cited sources for every property this THING needs.

Materials modeled here: 2024-T3 aluminum sheet (bare) 304 stainless steel 6061-T6 aluminum 7075-T6 aluminum AISI 1045 medium-carbon steel AISI 4340 low-alloy steel (Ni-Cr-Mo) ASTM A36 structural steel (hot-rolled) C26000 Cartridge Brass (70/30) Nylon 6/6 (PA66), unfilled Ti-6Al-4V

Governing relations

\sigma_h = \frac{p\,r}{t}

Assumes: thin wall (t ≪ r), so the stress is uniform through the thickness; internal gauge pressure; external pressure is atmospheric; away from the end caps and any nozzles or welds (Saint-Venant) · Valid while: r/t has dropped below 10 — outside the usual thin-wall envelope. The true maximum stress (at the inner surface, by Lamé's thick-wall analysis) is several percent above pr/t and climbing as the wall thickens. r/t is below 5: this is a thick-walled vessel, and the thin-wall formula is no longer an approximation but a wrong answer. Lamé's thick-wall analysis is required.

Source: Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §8.2–8.3 (thin-walled spherical and cylindrical pressure vessels, including the r/t ≥ 10 thin-shell criterion and the hoop/longitudinal 2:1 ratio).

\sigma_l = \frac{p\,r}{2\,t}

Assumes: closed ends — the caps put the pressure load into the wall as axial tension; thin wall, away from the caps themselves

\mathrm{SF} = \frac{\sigma_y}{\sigma_h}

Assumes: hoop stress governs — it is exactly twice the longitudinal stress; safety factor against first yield (not against burst; ductile vessels carry beyond yield)

m = 2\pi\,r\,t\,L\,\rho

Assumes: thin-shell mass of the cylindrical barrel only; end caps excluded; mid-surface circumference taken at the inner radius (error of order t/r)

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.

2 L \sigma_{h} t = 2 L p r

1. Slice the cylinder in half lengthwise and balance the half-shell: the pressure pushing on the flat projected area (2r by L) must be carried by the two cut walls, each of area t by L, in circumferential tension. Curvature does the projection for you — only the projected area matters. — statics: half-shell free body

\sigma_{h} = \frac{p r}{t}

2. Divide through. Stress scales with r and inversely with t: big vessels need disproportionately thick walls, which is why pressure vessels want to be small or spherical. — solve for the hoop stress

2 \pi r \sigma_{l} t = \pi p r^{2}

3. Now cut the cylinder crosswise: the pressure on the circular end cap (area πr²) is carried by the thin ring of wall (circumference 2πr times thickness t) in axial tension. — statics: end-cap free body

\sigma_{l} = \frac{\sigma_{h}}{2}

4. Compare the two: the longitudinal stress is exactly half the hoop stress, independent of every dimension. The wall is twice as close to yielding around the circumference as along the axis — which is why an overcooked sausage always splits lengthwise. — compare the two stresses

SF = \frac{\sigma_{y} t}{p r}

5. The safety factor compares yield strength to the governing (hoop) stress. Read it as a design equation: for a required SF, the wall must be t = SF·p·r/σ_y — that inversion is exactly what the "design" configuration runs. — definition of safety factor against yield

How it fails

The safety factor here guards first yield in the hoop direction. Real vessels have meaner ways to let go:

Ductile burst. Past yield, a ductile wall thins where it stretches, which raises the local stress, which thins it further — the instability ends in a longitudinal split (the hoop direction governs, so the crack runs along the axis). Design codes rate vessels against burst with margins well beyond first yield.
Brittle fragmentation. A brittle wall (cast iron, low-toughness steel in the cold, a flawed weld) doesn’t bulge and tear — it shatters, and the stored energy of the compressed gas turns the wall into shrapnel. This is why hydrostatic (water-filled) proof tests exist: water stores almost no energy, so a failure during test is a leak, not a bomb.
Stress concentrations at openings. Every nozzle, manway, and fitting hole multiplies the membrane stress locally — a small round hole in a biaxial field roughly doubles it, and sharp corners are worse. Codes require reinforcement around openings; the clean $pr/t$ field on this page exists only away from them.
Weld seams. The longitudinal seam carries the full hoop stress across it — twice the load of the circumferential seam. Seam efficiency factors and radiographic inspection exist because the governing stress crosses the most defect-prone feature.
Fatigue. Pressure cycles (fill–empty, heat–cool) grow cracks from surface flaws long before static yield is approached. Cyclic service is rated entirely differently from static.
Corrosion thinning. The wall the formula sees is the wall you have today. A corrosion allowance — extra millimeters sacrificed over the service life — is standard practice, and inspection intervals exist to verify the remaining $t$ .

Everything above is why real vessels are governed by detailed design codes (in the US, the ASME Boiler and Pressure Vessel Code) rather than by one formula — this page teaches the membrane-stress idea those codes are built on, nothing more.

And when the widget’s envelope pushes back because the wall is too thick — a caution below $r/t = 10$ , outright refusal below $r/t = 5$ — that is not a failure of the vessel but of the model: the exact theory for any wall thickness is the thick-walled cylinder, which picks up precisely where this page stops.

Compound Cylinder (Shrink Fit)
Where the monobloc wall gave up: shrink a jacket over a liner and the interference squeezes the bore into hoop compression before the pressure ever arrives. Service tension must spend that compression first — and at the balanced fit with the interface at √(r_i·r_o), the elastic pressure ceiling approaches DOUBLE the one no solid wall could pass.
- stress
- mass-cost
Rotating Disk with a Central Bore
Drill the smallest possible shaft hole through a spinning disk and the peak stress exactly doubles — not "roughly increases": doubles, in the limit of a vanishing bore. The solid flywheel's optimistic numbers meet the hole every real rotor needs.
- stress
- energy-storage
- mass-cost
Thick-Walled Cylinder (Lamé)
Where the thin-wall pressure vessel hands off: the exact elastic field for any wall thickness. The stress piles up at the bore and decays as 1/r² — and the bore shear always exceeds the pressure, so past p = σ_y/2 no amount of thickness can contain it elastically.
- stress
- mass-cost
Cantilever Beam (End Load)
A beam fixed at one end, loaded at the other — the fruit-fly of structures. One widget shows why stiffness (E) and strength (σ_y) are independent axes: swap steel for titanium and deflection goes UP while the safety factor also goes up.
- stress
- mass-cost
Composite Bar (Core + Sleeve)
A solid core inside a concentric sleeve, bonded between rigid end plates and pushed by a centric axial load. The two materials must stretch together, so the load splits in proportion to each member's axial stiffness A·E — and the build solves that coupled 2×2 share exactly. Swap the sleeve's metal and watch the load migrate to the stiffer member.
- stress
- mass-cost
Eccentric Column (Secant Formula)
Load a column even slightly off-axis and the clean buckling story dissolves: it bows from the first newton, stress grows faster than load, and the Euler limit survives only as the asymptote the deflection chases. Because nothing here is linear, the safety factor must be taken on the LOAD — the page solves that transcendental equation live, by bracketed root-finding.
- stability
- stress
- mass-cost

Chains with

Outputs whose SI dimension and quantity kind match another THING's input — the only wires the planner's connectionLegal accepts (invariant 2, computed at build time, not hand-listed). Wire these on the chaining demo.

Composite Bar (Core + Sleeve)
- t L
Impact Loading (Falling Mass, Energy Method)
- m_shell m
- t b
- t d
- t h
- t L
Symmetric Two-Bar Truss
- t d
- t L
Thermal Assembly (Two-Segment Bar Between Rigid Walls)
- t L_1
- t L_2
Cantilever Beam (End Load)
- t b
- t h
- t L
Circular Plate under Uniform Pressure (Clamped vs Simply Supported)
- sigma_h q
- sigma_h sigma_allow
- sigma_l q
- sigma_l sigma_allow
- t a
- t t
Curved Beam in Bending (Winkler — Crane Hook, C-Clamp, Press Frame)
- t r_i
- t r_o
- t w
Fixed-Fixed Beam (UDL)
- t b
- t h
- t L

+ 24 more THINGs its outputs can legally feed (showing the first 8 in course order).

Sources

Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §8.2–8.3 (thin-walled spherical and cylindrical pressure vessels, including the r/t ≥ 10 thin-shell criterion and the hoop/longitudinal 2:1 ratio).