Hoop Stress Calculator
Calculate hoop stress, longitudinal stress, and other important parameters for thin-walled pressure vessels. This calculator helps engineers analyze stresses in cylindrical pressure vessels, pipes, and tanks under internal pressure.
Material Properties
More Information
What is Hoop Stress?
Hoop stress (circumferential stress) is the stress that acts along the circumference of a cylindrical pressure vessel. It is caused by internal pressure trying to expand the vessel's diameter, creating tension in the vessel wall.
In a thin-walled pressure vessel, the hoop stress is approximately twice as large as the longitudinal stress, making it often the critical design parameter.
Hoop stress (σₕ) acts circumferentially around the vessel, longitudinal stress (σₗ) acts along the vessel length, and radial stress (σᵣ) acts through the vessel wall thickness.
Key Formulas
Hoop Stress
The circumferential stress in the vessel wall:
Where:
- P = Internal pressure
- Dm = Mean diameter (Outside diameter - t)
- t = Wall thickness
Longitudinal Stress
The axial stress along the vessel length:
Where:
- P = Internal pressure
- Dm = Mean diameter (Outside diameter - t)
- t = Wall thickness
Note: Longitudinal stress is half of hoop stress for cylindrical vessels.
Von Mises Stress
Combined stress used for yield evaluation:
Where:
- σₕ = Hoop stress
- σₗ = Longitudinal stress
- σᵣ = Radial stress
Engineering Notes
Mean Diameter Calculation
This calculator uses the mean diameter (Dm) in accordance with engineering standards for thin-walled pressure vessels. The mean diameter is calculated as:
While you input the outside diameter of the vessel, the calculations use the mean diameter for accuracy. For very thin-walled vessels, the difference between outside, mean, and inside diameters becomes minimal.
Thin-Walled Assumption
The thin-walled pressure vessel theory is valid when the vessel wall thickness is no more than:
- 1/10 of the vessel radius
- 1/20 of the vessel diameter
When the t/D ratio exceeds 0.05 (or t/r exceeds 0.1), thick-walled pressure vessel theory should be used instead, which accounts for the variation of stress through the wall thickness.
Common Applications
Hoop stress calculations are essential for designing and analyzing:
Pressure Pipelines
Oil, gas, and water transmission lines
Storage Tanks
Industrial and commercial fluid storage
Boilers
High-pressure steam generation systems
Pressure Vessels
Chemical processing and storage
Gas Cylinders
Compressed gas storage for industrial and medical use
Hydraulic Components
Cylinders and accumulators for power systems
Limitations and Assumptions
This calculator is based on thin-walled pressure vessel theory which assumes:
- Vessel wall thickness is small compared to diameter (t/D ≤ 0.05 or t/r ≤ 0.1)
- Pressure is uniformly distributed
- Material behaves in a linear, elastic manner
- No stress concentrations or discontinuities exist
- No external loads or thermal stresses are present
For thick-walled vessels (t/D > 0.05), irregular geometries, or vessels under combined loading, more detailed analysis methods like Lame's equations should be used.
Safety Considerations
When designing pressure vessels, a safety factor is typically applied to ensure safe operation:
- Safety Factor < 1.0: UNSAFE - Vessel will likely fail
- Safety Factor 1.0-1.5: MARGINAL - Not recommended for general use
- Safety Factor 1.5-2.0: ACCEPTABLE - Minimum for controlled applications
- Safety Factor 2.0-4.0: GOOD - Typical for general engineering applications
- Safety Factor > 4.0: CONSERVATIVE - Used for critical applications or where uncertainty exists