What is the difference between thick and thin cylinder analysis?

Thin cylinders (t/d < 1/20) have uniform hoop stress σh = pd/(2t) and longitudinal stress σl = pd/(4t). Thick cylinders have varying stresses across thickness, analyzed using Lame's equations: σr = A - B/r², σh = A + B/r². Internal pressure causes maximum hoop stress at inner surface. Pressure vessels, hydraulic cylinders, and gun barrels require thick cylinder analysis.