Hoop Stress in Pressure Vessels
Hoop (circumferential) stress resists internal pressure in cylinders and spheres. Thin-wall: σ_h = pr/t (cylinder) or pr/(2t) (sphere). Thick-wall uses Lamé equations.
Did our AI summary help? Let us know.
Thin-wall: t/r < 0.1; stress assumed uniform through wall Cylinder hoop stress is 2× longitudinal for closed ends Sphere has half the hoop stress of a cylinder at same p, r, t Lamé equations handle stress variation in thick walls
Ready to run the numbers?
Why: Hoop stress governs pressure vessel design for boilers, pipelines, and storage tanks. ASME BPVC sets design limits.
How: Enter pressure, radius, and wall thickness. Select thin or thick wall. The calculator returns hoop stress, longitudinal stress, and safety factor.
Run the calculator when you are ready.
🔩 Thin-Walled Pipe
10 MPa pressure, 500mm radius, 10mm wall
🛢️ Pressure Tank
5 MPa pressure, 1m radius, 15mm wall
⚪ Spherical Vessel
8 MPa pressure, 600mm radius, 12mm wall
🔧 Thick-Walled Cylinder
20 MPa pressure, 200mm inner, 300mm outer
🛢️ Pipeline
15 MPa pressure, 300mm radius, 20mm wall
🔥 Boiler Drum
3 MPa pressure, 800mm radius, 25mm wall
⚙️ Hydraulic Cylinder
25 MPa pressure, 50mm radius, 8mm wall
💨 Gas Storage Sphere
2 MPa pressure, 2m radius, 20mm wall
🔬 High Pressure Vessel
50 MPa pressure, 100mm inner, 150mm outer
🌊 Submarine Hull
1 MPa external pressure, 3m radius, 30mm wall
💨 Compressed Gas Cylinder
30 MPa pressure, 100mm radius, 8mm wall
🔥 Heat Exchanger Tube
5 MPa pressure, 25mm radius, 2mm wall
⚛️ Reactor Vessel
15 MPa pressure, 1.5m radius, 50mm wall
💧 Water Tower
0.5 MPa pressure, 5m radius, 20mm wall
Enter Values
Vessel Type
Pressure
Thin-Walled Geometry
Material
Analysis Options
For educational and informational purposes only. Verify with a qualified professional.
🔬 Physics Facts
Thin-wall formula σ = pr/t assumes t << r; error grows for thick walls.
— ASME
Spherical vessels are more efficient; same stress with half the wall thickness.
— Pressure vessel design
Longitudinal stress in closed cylinders is pr/(2t).
— NIST
Lamé equations (1852) solve thick-walled cylinder elasticity.
— Continuum mechanics
What is Hoop Stress?
Hoop stress (circumferential stress) is the stress acting tangentially around the circumference of a pressure vessel. It is the primary stress component in pressure vessels and is typically the maximum stress, making it critical for design. Hoop stress resists the tendency of internal pressure to burst the vessel radially outward.
Circumferential Stress
Acts tangentially around the vessel circumference, perpendicular to the longitudinal axis.
Primary Design Stress
Usually the maximum stress component, determining required wall thickness.
Pressure Vessels
Critical for boilers, tanks, pipelines, hydraulic cylinders, and pressure vessels.
Thin vs Thick-Walled Vessels
Thin-Walled (t/r < 0.1)
Stress is assumed uniform through wall thickness. Simple formulas apply:
Used for most pressure vessels, pipes, and tanks.
Thick-Walled (t/r ≥ 0.1)
Stress varies through wall thickness. Lamé equations required:
Used for high-pressure vessels, gun barrels, and heavy-walled pipes.
Stress Components in Pressure Vessels
Hoop Stress (σ_h)
Circumferential stress, usually maximum. Resists radial expansion.
Longitudinal Stress (σ_l)
Axial stress, half of hoop stress for cylinders. Resists length change.
Radial Stress (σ_r)
Through-thickness stress, varies from -p (inner) to 0 (outer).
How to Calculate Hoop Stress
Calculation Steps
- 1Determine vessel type: thin-walled cylinder, sphere, or thick-walled
- 2Check thickness ratio: t/r < 0.1 for thin-walled assumption
- 3Apply appropriate formula: σ_h = pr/t for thin cylinder, Lamé equations for thick
- 4Calculate longitudinal and radial stresses if needed
- 5Determine Von Mises stress and compare with material yield strength
Key Considerations
- • Hoop stress is typically 2× longitudinal stress for cylinders
- • Maximum hoop stress occurs at inner surface for thick-walled vessels
- • Safety factors: 2-4 for pressure vessels depending on code
- • Consider corrosion allowance in wall thickness
- • Fatigue loading requires lower stress limits
- • Temperature effects reduce material strength at high temperatures
When to Use Hoop Stress Calculations
Pressure Vessel Design
Design boilers, tanks, and pressure vessels to withstand internal pressure. Critical for safety and code compliance.
Pipeline Analysis
Calculate wall thickness requirements for oil, gas, and water pipelines. Essential for transmission line design.
Hydraulic Systems
Design hydraulic cylinders, accumulators, and high-pressure fluid systems. Determine safe operating pressures.
Design Codes and Standards
Pressure vessel design follows strict codes and standards:
ASME BPVC Section VIII
Rules for Construction of Pressure Vessels. Most common code in North America. Defines design pressure, allowable stress, and safety factors.
API 579
Fitness-for-Service assessment for existing pressure vessels. Evaluates remaining life and safe operating conditions.
EN 13445
European standard for unfired pressure vessels. Uses limit state design approach with partial safety factors.
PD 5500
British standard for unfired fusion welded pressure vessels. Similar to ASME but with some differences in approach.
Failure Modes and Safety Considerations
Burst Failure
Occurs when hoop stress exceeds ultimate tensile strength. Catastrophic failure mode. Prevented by adequate wall thickness and material selection.
Yield Failure
Occurs when stress exceeds yield strength. Causes permanent deformation. Design typically limits stress to yield strength divided by safety factor.
Fatigue Failure
Occurs under cyclic pressure loading. Requires lower stress limits than static loading. Critical for vessels with frequent pressure cycles.
Creep Failure
Occurs at elevated temperatures over time. Material deforms slowly under constant stress. Important for high-temperature applications.
Wall Thickness Design and Corrosion Allowance
Minimum Wall Thickness
For thin-walled cylinders: t_min = (p × r) / (σ_allowable × E - 0.6p) where E = joint efficiency (0.7-1.0), σ_allowable = allowable stress (yield strength / safety factor).
ASME BPVC Section VIII formula includes joint efficiency and accounts for internal pressure reduction.
Corrosion Allowance
Add corrosion allowance to minimum thickness: t_design = t_min + t_corrosion. Typical values: 1.5-3 mm for general service, 3-6 mm for corrosive service, 0-1.5 mm for non-corrosive service. Consider vessel lifetime and corrosion rate.
Manufacturing Tolerance
Account for manufacturing variations. Typical tolerance: ±12.5% of nominal thickness for rolled plate, ±10% for seamless pipe. Minimum thickness after manufacturing must meet design requirements.
Pressure Testing and Safety Factors
Hydrostatic Test Pressure
Test pressure typically 1.3-1.5× design pressure per ASME BPVC. Ensures vessel can withstand overpressure without failure. Test at room temperature to avoid thermal stress.
Where σ_test = material strength at test temperature, σ_design = strength at design temperature
Safety Factors
Typical safety factors: 2.0-4.0 for yield strength, 3.0-5.0 for ultimate strength. Higher factors for: hazardous contents, high pressure, cyclic loading, high temperature, or when failure consequences are severe. Lower factors acceptable for: low pressure, non-hazardous contents, static loading, well-characterized materials.
📚 Official Data Sources
Stress Distribution in Thick-Walled Vessels
Radial Stress Variation
In thick-walled cylinders, hoop stress is maximum at inner surface and decreases toward outer surface. Radial stress is compressive at inner surface (equal to -p) and zero at outer surface. This non-uniform distribution means inner surface is most critical.
Autofrettage
Pre-stressing technique for thick-walled vessels. Vessel is pressurized beyond yield to create beneficial residual stresses. After unloading, inner surface has compressive residual stress, reducing tensile stress under operating pressure. Increases fatigue life and burst pressure.
Transition from Thin to Thick
Thin-walled assumption (t/r < 0.1) gives uniform stress. For t/r > 0.1, stress varies through thickness. Error increases with thickness ratio. At t/r = 0.2, thin-walled formula underestimates maximum stress by ~10%. Always use thick-walled analysis for t/r ≥ 0.1.
Related Calculators
Principal Stress Calculator
Calculate principal stresses and maximum shear stress from stress tensor. Essential for failure analysis and material design.
PhysicsShear Stress Calculator
Calculate shear stress in beams, shafts, and connections. Analyze transverse and torsional shear for mechanical design.
PhysicsStress Calculator
Calculate mechanical stress from force and area. Analyze tensile, compressive, and bending stresses for structural engineering design.
PhysicsThermal Stress Calculator
Calculate thermal stress from temperature change and constraint. Analyze thermal expansion effects in structures and components.
PhysicsMohr's Circle Calculator
Calculate and visualize Mohr's circle for 2D stress states. Determine principal stresses, maximum shear, and stress transformation.
PhysicsShear Modulus Calculator
Calculate shear modulus (modulus of rigidity), shear stress, and shear strain. Essential for torsion and shear loading analysis in structural engineering.
Physics