Cylinder, Vessel, and Pipe Stress Calculator

Estimate Lamé elastic stresses and diametric expansion for an open-ended thick circular cylinder under uniform internal pressure.

Engineering screening tool—verify inputs and results independently. Use at your own risk.

Open-ended thick-cylinder cross-section; internal pressure acts on the bore and the outside surface is unloaded.

Geometry, pressure, and material

Stress analysis

Maximum von Mises stress

—

MPa · maximum of inner and outer surfaces

Wall thickness

— mm

Material area

— mm²

Hoop stress · inner

— MPa

Hoop stress · outer

— MPa

Radial stress · inner

— MPa

Radial stress · outer

— MPa

von Mises · inner

— MPa

von Mises · outer

— MPa

Diametric expansion

— mm

How to use

Use this model only for an open-ended thick circular cylinder under uniform internal pressure.
Enter outer and inner diameters in mm; outer diameter must be larger.
Enter non-negative internal pressure in MPa.
Enter Young’s modulus in GPa and a dimensionless Poisson ratio.
Select Calculate to update the canonical result and surface metrics.
Select Reset to restore the steel example.
Read positive hoop stress as tensile and inner radial stress as compressive.
Apply independently selected design-code allowables, factors, and allowances before design use.
Treat expansion as an elastic model result, not an assembly clearance guarantee.
Do not use this model when closed ends, external pressure, yielding, buckling, fatigue, creep, corrosion, thermal gradients, joints, or discontinuities govern.

Equations used

Let d be outer diameter, dᵢ inner diameter, pᵢ internal pressure, E Young’s modulus, ν Poisson ratio, and D² = (d/dᵢ)².

s = (d − dᵢ)/2; A = π(d² − dᵢ²)/4

σt,i = pᵢ(D² + 1)/(D² − 1); σt,o = 2pᵢ/(D² − 1)

σr,i = −pᵢ; σr,o = 0

σe = √(σt² + σr² − σtσr)

δ = (pᵢd/E)[(D² + 1)/(D² − 1) − ν]

Positive normal stress is tensile; radial pressure is compressive. Lamé theory here assumes homogeneous, isotropic linear elasticity, axisymmetry, zero external pressure, and open ends with zero axial stress.

Reference: G. Lamé, Leçons sur la théorie mathématique de l’élasticité des corps solides (1852).

This is an elastic screening result, not pressure-vessel code certification. It excludes plasticity, fatigue, creep, buckling, thermal stress, defects, residual stress, joints, nozzles, corrosion allowance, and manufacturing tolerances.

Worked example

1. For d = 200 mm, dᵢ = 180 mm, pᵢ = 10 MPa, E = 210 GPa, and ν = 0.3, select Calculate. The expected maximum von Mises stress is 100.636 MPa, wall thickness is 10.00 mm, inner hoop stress is 95.26 MPa, and diametric expansion is 0.0879 mm.

The bore is the governing screened surface in this example. Compare the equivalent stress with a code-qualified allowable only after applying all required factors and allowances.

Theory and method

Lamé’s axisymmetric thick-cylinder solution expresses radial and hoop stress as constants plus or minus a radius-dependent term. Evaluating those fields at the bore and outside surface gives the implemented stress equations; the open-ended condition sets axial stress to zero, so the displayed equivalent stress uses the local plane-stress von Mises expression.

The displacement expression assumes small strain, homogeneous isotropic linear elasticity, uniform internal gauge pressure, zero outside pressure, and no end restraint. Validity ends when plasticity, significant geometric change, anisotropy, thermal strain, external pressure, closed-end axial load, local discontinuities, fatigue, creep, corrosion, or code-specific requirements become important.

Reference: G. Lamé, Leçons sur la théorie mathématique de l’élasticité des corps solides (1852). For higher-fidelity geometry, materials, loads, and independent numerical checks, Continue in TriboSolver.