Hoop stress calculator
The hoop stress calculator determines the stresses acting on a thin-walled pressure vessel. Pressure vessels are specially designed containers used to hold fluids at a pressure different from the ambient one. Due to the high internal pressure, parameters like hoop stress and longitudinal stress become crucial when designing these containers.
Knowledge of these stresses is helpful in designing the riveted or welded joints on the body. Various pressure vessels include boilers, water tanks, petrol tanks, gas cylinders, spray cans, fire extinguishers, pipes, and more. Read on to understand what hoop stress is, what longitudinal stress in a cylinder is, and more.
Stress on a pressure vessel
A pressure vessel is manufactured using rolled-up sheets welded or riveted together. In some cases, it is also forged. The manufacturing process depends on various factors such as the application and the required strength.
A pressure vessel design includes an estimation of the stresses that can cause failure. The shapes for the pressure vessel calculations are simplified as a cylinder or a sphere in most cases. The shells are classified as either thick or thin based on their dimensions. If a shell's wall thickness is not greater than one-tenth of the radius, it is regarded as a thin shell.
What are circumferential stress and longitudinal stress?
Consider a thin-walled pressure vessel. When a shell is subjected to a large amount of internal pressure, tensile stresses act along both directions. The stress acting along the tangential direction to the circumference of a sphere or cylindrical shell is known as circumferential stress or hoop stress. In a cylindrical shell, the stress acting along the direction of the length of the cylinder is known as longitudinal stress.
These stresses are vital parameters when it comes to pressure vessel design. The maximum permissible stress in the material must not exceed either the circumferential or the longitudinal stress. Failure from hoop stress results in the rupturing of a cylindrical shell into two cylinders, whereas excess longitudinal stress in the cylinder splits the cylinder into two troughs.
For a cylindrical shell having diameter d and thickness t, the circumferential or hoop stress σh is given by the hoop stress equation:
σh = p · d / (2 · t)
where p is the internal pressure. Large cylindrical shells are manufactured with joints, and when the efficiency η of the joints is taken into account, the circumferential stress equation becomes:
σh = p · d / (2 · t · η)
The longitudinal stress acting along the axis of the cylinder is half the hoop stress:
σl = p · d / (4 · t · η)
For a spherical shell, the stress is the same in all directions and equals:
σ = p · d / (4 · t · η)
Change in shell dimensions
Internal pressure not only stresses the shell but also deforms it. By entering the material's Young's modulus E and Poisson's ratio ν, this calculator can also estimate how much the shell stretches. The circumferential and longitudinal strains for a cylinder are:
εh = (σh − ν · σl) / E εl = (σl − ν · σh) / E
From these strains, the change in diameter is Δd = εh · d, the change in length is ΔL = εl · L, and the volumetric strain is ΔV/V = 2εh + εl. For a sphere, the volumetric strain is 3ε.
How to calculate hoop stress?
- Choose the vessel shape — a cylinder or a sphere.
- Select your preferred unit system: American (imperial) or metric.
- Enter the internal pressure p.
- Enter the internal diameter d and the wall thickness t.
- Optionally, adjust the joint efficiency η for riveted or welded joints.
- Read off the hoop stress, the longitudinal stress, and the diameter-to-thickness ratio instantly.
Using the hoop stress calculator
This calculator supports both the American (imperial) system — psi for pressure and stress, inches for length, and ksi for Young's modulus — and the metric system — MPa for pressure and stress, millimeters for length, and GPa for Young's modulus. It also tells you whether the vessel qualifies as a thin shell, in which case these formulas are valid. For thicker shells (wall thickness greater than one-tenth of the radius), the results are approximate and a thick-wall (Lamé) analysis should be used instead.
FAQs
What is hoop stress?
Hoop stress, or circumferential stress, is the tensile stress that acts tangentially to the circumference of a cylindrical or spherical shell under internal pressure. For a thin-walled cylinder it equals p · d / (2 · t).
Why is hoop stress twice the longitudinal stress?
In a cylinder, the pressure acts over the full diameter to split the shell along its length (hoop stress) but over a smaller projected area to push the end caps apart (longitudinal stress). The geometry makes the hoop stress exactly twice the longitudinal stress, which is why pressurized cylinders tend to burst along a line parallel to their axis.
When is a shell considered thin?
A shell is treated as thin-walled when its wall thickness is no greater than one-tenth of its radius (equivalently, the diameter-to-thickness ratio is 20 or more). The simple hoop and longitudinal stress formulas assume a thin shell.