Final answer:
Hoop stress is a mechanical stress that occurs in thick-walled cylinders, directly related to internal pressure and cylinder geometry. It acts circumferentially and is highest at the inner surface, decreasing towards the outer surface. Calculating this stress is crucial for the design and safety of pressurized structures.
Step-by-step explanation:
Hoop Stress in Thick-Walled Cylinders
Hoop stress is a type of mechanical stress experienced by the material of a thick-walled cylinder subjected to internal pressure. The hoop stress acts circumferentially (parallel to the circular cross-section) and is perpendicular to the axis of the cylinder. It varies across the thickness of the cylinder wall, being maximum at the inner surface and decreasing towards the outer surface. This form of stress is critical when analyzing the structural integrity of pressurized containers such as pipes, boilers, and tanks.
The magnitude of the hoop stress is directly related to the internal pressure and geometry of the cylinder in question. As the internal pressure increases, the hoop stress also increases. The effect of geometry is captured through the relationship between the internal and external radii of the cylinder. For a given internal pressure, cylinders with thinner walls experience higher hoop stress.
The formula for calculating hoop stress in a thick-walled cylinder is a function of the internal pressure, the internal radius, and the external radius of the cylinder. The calculation of hoop stress is essential in designing and ensuring the safety of the structures that operate under pressure