Thin-Wall Pressure Vessel Stress

The engineering

Hoop stress is double the longitudinal — which is why sausages split lengthwise, why pipe bursts run along the pipe, and why filament-wound tanks put most fibers nearly circumferential. The factor of two comes straight from the geometry of what each stress has to hold together, and it decides the orientation of every failure this card describes.

The r/t gate is not pedantry: thin-wall theory assumes stress is uniform through the wall, and below r/t ≈ 10 the inner surface really carries meaningfully more than the average — the Lamé thick-wall equations take over. And nothing here covers nozzles, welds, or heads; that's why pressure vessel codes are books, not cards.

Where this math comes from

The steam century learned hoop stress by explosion: boilers burst with such regularity that the 1865 Sultana disaster — a riverboat boiler failure that killed some 1,200 people, most of them freed Union prisoners of war — remains the deadliest maritime disaster in US history. The theory already existed; Gabriel Lamé and Émile Clapeyron had published the exact thick-cylinder solution in 1833.

It took a 1905 shoe-factory boiler explosion in Brockton, Massachusetts to force the institutional fix: the ASME Boiler and Pressure Vessel Code of 1914–15, the document that turned Pr/t plus weld factors plus inspection into law. It is still the largest standards effort in mechanical engineering, and this card is its opening equation.

1. 1833Lamé & ClapeyronExact thick-cylinder stress solution published.
2. 1865Sultana boiler explosionDeadliest US maritime disaster — steam's cost made plain.
3. 1914ASMEBoiler & Pressure Vessel Code — hoop stress becomes law.

See the full timeline of the math behind every calculator →

Runs entirely in your browser — nothing you enter leaves this page. Your recent runs are stored only on your device.