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Hoop Stress Calculator

Calculate the hoop and longitudinal stresses in a thin-walled pressure vessel from internal pressure, inner radius, and wall thickness

Frequently Asked Questions

Why is hoop stress twice the longitudinal stress?

In a thin-walled cylinder the geometry makes the circumferential, or hoop, stress twice the longitudinal stress. That is why pressurized pipes tend to fail with a lengthwise split.

When does the thin-wall assumption apply?

These formulas assume the wall is thin relative to the radius, generally a radius-to-thickness ratio greater than ten. Thick-walled vessels need the more detailed Lame equations and a qualified engineer.

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Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

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How This Calculator Works

A thin-walled pressure vessel under internal pressure experiences two main stresses. The hoop, or circumferential, stress acts around the circumference and tries to split the vessel along its length. The longitudinal stress acts along the axis and tries to pull the ends apart. For a cylinder the hoop stress is twice the longitudinal stress, which is why pipes tend to fail with a lengthwise split.

The Thin-Wall Assumption

These simple formulas assume the wall is thin relative to the radius, generally taken as a radius-to-thickness ratio greater than ten. Within that range the stress is nearly uniform across the wall. Thick-walled vessels need the more detailed Lame equations, which account for the stress varying from the inner to the outer surface.

How To Use It

Enter the internal pressure in megapascals, the inner radius in millimeters, and the wall thickness in millimeters. The calculator returns the hoop and longitudinal stresses in megapascals.

Disclaimer

This is a planning estimate based on the thin-wall assumption and does not account for joints, corrosion, temperature, or safety factors. Always verify any pressure vessel or piping design with a qualified engineer and the relevant codes.

Thin-Wall vs. Thick-Wall Cylinders

The thin-wall (membrane) approximation assumes that stress is uniform through the wall thickness. This is accurate when the wall thickness t is less than about one-tenth of the inner radius r (t/r < 0.1). Most pipelines, gas cylinders, and storage tanks satisfy this criterion.

When the wall is thicker - as in gun barrels, hydraulic cylinders, and high-pressure chemical reactors - the Lame equations must be used. For a thick-walled cylinder under internal pressure p, the hoop stress varies from a maximum at the inner surface: σ_θ,inner = p(r_o² + r_i²) / (r_o² - r_i²), down to a minimum at the outer surface. The radial stress is compressive at the inner wall (equal to -p) and zero at the outer wall. This stress gradient means the material near the inner bore does most of the work, which is why auto-frettage (intentionally yielding the inner layer to create compressive residual stresses) is used to improve fatigue life in gun barrels and fuel injection components.

ASME Boiler & Pressure Vessel Code

For pressure vessels in commercial service, wall thickness is not determined by stress alone. ASME BPVC Section VIII Division 1 specifies a minimum required wall thickness: t = P·R / (S·E - 0.6P), where P is the maximum allowable working pressure, R is the inner radius, S is the allowable stress for the material at operating temperature (typically UTS/3.5 for carbon steel), and E is the weld joint efficiency factor, ranging from 0.70 for a single-pass fillet weld to 1.0 for a fully radiographed butt weld. A corrosion allowance, typically 1.5 to 3 mm for carbon steel in water service, is then added to the calculated thickness before selecting a standard plate thickness.

Applications Beyond Pressure Vessels

Hoop stress governs the design of many familiar objects beyond chemical process vessels.

Water and gas pipelines: Pipe schedules (Schedule 40, Schedule 80) are based on the wall thickness required to keep hoop stress below the allowable limit for the specified operating pressure. Thicker walls mean smaller bore for the same nominal diameter.
Hydraulic hose and tubing: High-pressure hydraulic lines (up to 700 bar in some mobile equipment) use multi-layer wire-braided hose or thick-walled steel tubing sized by hoop stress calculations.
Submarine hulls: Submarines experience external pressure rather than internal, creating compressive hoop stress and a risk of buckling rather than bursting. The design problem is the reverse of a pressure vessel: preventing implosion rather than explosion.
Grain silos: Grain exerts lateral pressure on the silo walls proportional to depth. At the base of a tall silo, this lateral (hoop) pressure can create significant hoop tension in the wall, particularly in cylindrical steel silos, which must be sized accordingly.
Rotating rings and flywheel rims: Centrifugal acceleration creates a body force that induces hoop stress in the rim of a rotating ring: σ_θ = ρω²r², where ρ is density and ω is angular velocity. High-speed flywheels and turbine discs are hoop-stress limited.

Failure Modes in Pressure Containment

The dominant failure mode in thin-walled cylinders under internal pressure is splitting along the longitudinal axis - the hoop stress exceeds the material's tensile strength. Because hoop stress is twice the longitudinal stress, this lengthwise fracture occurs before any end-cap failure. Pipeline failures caused by corrosion pitting follow this pattern: the pit reduces the effective wall thickness locally, raising the local hoop stress until it exceeds the reduced net-section strength.

The biaxial stress state (hoop plus longitudinal) in a pressurized cylinder means that von Mises yield criterion, rather than maximum principal stress, gives the most accurate prediction of when yielding begins. The von Mises equivalent stress for the biaxial state is σ_VM = √(σ_h² - σ_hσ_l + σ_l²), which for the case of σ_l = σ_h/2 evaluates to approximately 0.866σ_h. Yielding begins when this reaches the uniaxial yield strength.

Fatigue failure at stress concentrations is the most common mode in cycled pressure vessels. Nozzle openings, weld toes, and geometric discontinuities produce local stress concentrations that can be two to three times the nominal hoop stress. Repeated pressurization cycles initiate and grow fatigue cracks at these locations. Codes require proof testing at 1.3 to 1.5 times the design pressure, and hydrostatically tested vessels are safer than pneumatically tested ones because water releases far less energy on fracture than compressed gas.

Notable Failures

The 1919 Boston Molasses Disaster occurred when a 2.3-million-gallon molasses storage tank burst in a densely populated neighborhood. Post-accident analysis found the tank walls were significantly under-designed for the static head pressure alone, and the tank had no formal engineering review before construction. The failure produced a wave of molasses that killed 21 people, a dramatic illustration of hoop stress failure in a large storage vessel. The 2005 Buncefield oil storage fire in the UK was triggered by tank overfill, causing vapor cloud ignition; several floating-roof tanks subsequently failed with circumferential fractures consistent with hoop stress overload driven by the fire pressure loading.

Frequently Asked Questions

Why is hoop stress twice the longitudinal stress? Consider a free-body diagram of half the cylinder cut along its axis. The internal pressure acts on the projected rectangular area (2r × L), creating a net force of p × 2r × L trying to pull the two halves apart. This force is resisted by two strips of wall material, each of area t × L, so σ_h = p × 2r × L / (2tL) = pr/t. For the longitudinal direction, cut the cylinder with a plane perpendicular to its axis. Pressure acts on the circular end-cap area πr², giving force pπr² resisted by the annular wall area 2πrt, so σ_l = pr/(2t) - exactly half the hoop stress.

What safety factor does ASME require for pressure vessels? ASME Section VIII Division 1 uses an allowable stress equal to the lesser of UTS/3.5 or yield/1.5 at the design temperature, which implies a nominal safety factor of 3.5 on ultimate tensile strength. Division 2, which permits more detailed analysis, uses a lower factor (UTS/2.4) but requires more rigorous design checks, material documentation, and inspection. Both divisions add weld efficiency factors and corrosion allowances on top of the stress-based calculation.