Thin and thick cylinders

Pressure pipes, water tanks, boiler shells, penstocks and gas cylinders carry internal pressure that pushes the wall outward. If the wall is thin compared with the diameter (thickness less than about one twentieth of the diameter), stress across the wall can be taken as uniform. Cut the cylinder lengthwise and balance pressure against wall tension to get the hoop (circumferential) stress p d / (2t). Cut it across and the longitudinal stress is half that, p d / (4t). Hoop stress governs, so pipes split along their length, as anyone who has seen a frozen water pipe knows.

A thin sphere has equal stress in every direction, p d / (4t), so for the same pressure and stress a sphere needs half the thickness of a cylinder. That is why large gas holders are spherical. Joint efficiency reduces the effective thickness where seams are riveted or welded.

For thick cylinders, such as high-pressure hydraulic pipes and gun barrels, stress varies across the wall. Lamé's equations give radial and hoop stress as A ± B/r². Hoop stress is highest at the inner surface. Adding thickness has diminishing returns, so very high pressures use compound cylinders shrunk one over another, or autofrettage, to pre-compress the bore. Civil engineers meet these ideas in penstock design, pressure mains and prestressed concrete tanks.