Hoop Stress Calculator
Calculate hoop stress, longitudinal stress, and max shear stress in cylindrical and spherical pressure vessels. Free online engineering calculator with interactive charts and step-by-step breakdowns.
About This Calculator
The Hoop Stress Calculator is a free online engineering tool for analyzing stresses in thin-walled cylindrical and spherical pressure vessels. Engineers, designers, and students use it to determine hoop (circumferential) stress, longitudinal stress, maximum shear stress, and dimensional changes caused by internal pressure. It supports both cylinder and sphere geometries with adjustable joint efficiency and material properties.
The calculations are based on the standard thin-walled pressure vessel theory. For a cylinder, hoop stress is calculated as σ_h = P × d / (2 × t × η), and longitudinal stress as σ_l = P × d / (4 × t × η). For a sphere, hoop stress is σ_h = P × d / (4 × t × η), with no longitudinal stress component. The calculator also computes the change in diameter using the material's Young's modulus and Poisson's ratio, providing a complete stress and deformation analysis.
How to Use This Calculator
Start by selecting the shell shape (Cylinder or Sphere). Enter the vessel diameter in meters, wall thickness in millimeters, and internal pressure in megapascals (MPa). Adjust the joint efficiency to account for welded or riveted joints — use 1.0 for seamless vessels. For dimensional change calculations, provide the material's Young's modulus in GPa and Poisson's ratio. The calculator instantly returns hoop stress, longitudinal stress (cylinders only), maximum shear stress, thickness-to-diameter ratio, and the resulting change in diameter.
Applications
Hoop stress analysis is critical in designing boilers, water tanks, gas cylinders, spray cans, fire extinguishers, pipes, and industrial pressure vessels. Understanding these stresses ensures safe operation below the material's yield strength and helps determine appropriate wall thickness, material selection, and joint design for pressure-containing equipment worldwide.
Frequently Asked Questions
What is hoop stress?
Hoop stress (circumferential stress) is the tensile stress acting along the tangential direction to the circumference of a cylindrical or spherical pressure vessel. It results from internal pressure trying to expand the vessel radially and is the primary stress that determines pressure vessel wall thickness.
What is the hoop stress formula for a cylinder?
For a thin-walled cylinder, hoop stress σ_h = P × d / (2 × t × η), where P is internal pressure, d is diameter, t is wall thickness, and η is joint efficiency. Longitudinal stress is σ_l = P × d / (4 × t × η), which is exactly half the hoop stress.
What is the hoop stress formula for a sphere?
For a thin-walled spherical pressure vessel, hoop stress σ_h = P × d / (4 × t × η). A sphere has only hoop stress (no longitudinal stress) and the stress is half that of a cylinder with the same diameter, thickness, and pressure, making spheres more efficient for containing pressure.
What is the difference between hoop stress and longitudinal stress?
Hoop stress acts tangentially around the circumference of a cylinder, while longitudinal stress acts along the cylinder's length. In a thin-walled cylinder, hoop stress is exactly twice the longitudinal stress. Hoop stress causes the vessel to split into two troughs (longitudinal rupture), while longitudinal stress causes a circumferential rupture.
When is a pressure vessel considered thin-walled?
A pressure vessel is considered thin-walled when the wall thickness is not greater than one-tenth of the radius (t/d ratio ≤ 0.05). Thin-walled theory assumes stress is uniformly distributed across the wall thickness. For thicker vessels, thick-walled cylinder theory (Lame's equations) should be used instead.
Why is joint efficiency important in hoop stress calculations?
Joint efficiency accounts for the strength reduction at welded or riveted joints compared to the base material. Typical efficiencies range from 0.45 for single-riveted lap joints to 1.0 for seamless vessels. A lower joint efficiency increases the required wall thickness and reduces the allowable working pressure.
How does internal pressure change the dimensions of a pressure vessel?
Internal pressure causes the vessel diameter and volume to increase. The change in diameter depends on the hoop stress, Young's modulus, and Poisson's ratio of the material. For steel pressure vessels, the diameter change is typically very small (fractions of a millimeter) and is elastic, meaning the vessel returns to its original size when pressure is released.
Is this hoop stress calculator free to use?
Yes, this hoop stress calculator is completely free to use with no registration or hidden charges. You can calculate hoop stress, longitudinal stress, shear stress, and dimensional changes for cylindrical and spherical pressure vessels instantly.