Hoop Stress Calculator

Compute radial, circumferential (hoop), and axial stresses in cylindrical pressure vessels using Lame's equations (thick‑wall) or thin‑wall approximation. Visualize stress distribution across the wall thickness.

Enter in Pascals. Example: 10e6 = 10 MPa = 10×10⁶ Pa
Atmospheric pressure ≈ 0 (gage pressure)
Enter in meters. Example: 0.1 m = 100 mm
Must be greater than inner radius
"Constrained" corresponds to plane strain condition (εz=0)
⚠️ Use consistent SI units: Pascals, meters. For thin‑wall, the calculator will warn if thickness/radius ratio > 0.1 (deviation from thin‑wall assumption).
? Hydraulic cylinder: Pᵢ=35 MPa, rᵢ=0.05 m, rₒ=0.075 m (thick)
? Gas pipeline: Pᵢ=8 MPa, rᵢ=0.3 m, rₒ=0.31 m (thin)
? Subsea pipe: Pᵢ=0, Pₒ=15 MPa, rᵢ=0.2 m, rₒ=0.24 m (external pressure)
? Gun barrel: Pᵢ=300 MPa, rᵢ=0.01 m, rₒ=0.025 m (thick, high pressure)
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Lame's Equations for Thick‑Walled Cylinders

For a cylinder subjected to internal pressure Pi and external pressure Po, the radial and circumferential stresses at any radius r are given by Lame's solution (valid for any thickness):

σr(r) = ri2 Pi − ro2 Po⁄(ro2 − ri2) − (Pi − Po) ri2 ro2⁄r2 (ro2 − ri2)
σθ(r) = ri2 Pi − ro2 Po⁄(ro2 − ri2) + (Pi − Po) ri2 ro2⁄r2 (ro2 − ri2)

Axial stress σz depends on end conditions: closed ends: σz = (Pi ri2 - Po ro2) / (ro2 - ri2); open ends: σz = 0; constrained: σz = ν(σr+σθ).

Thin‑Walled Approximation (ro ≈ ri)

When thickness t = ro - ri is small compared to radius (typically t/r < 0.1), stresses are nearly uniform through the wall:

  • Hoop stress: σθ = P·rm / t, where rm = (ri+ro)/2
  • Radial stress: σr ≈ -P (negligible compared to hoop)
  • Axial stress (closed ends): σz = P·rm / (2t)

For external pressure only, the thin‑wall formula gives compressive hoop stress; buckling must be considered separately.

Why Use This Interactive Tool?

  • Thick vs. Thin awareness: Automatically selects correct equations and warns when thin‑wall assumption is invalid.
  • Stress distribution plot: Visualize how hoop stress decreases non‑linearly from inner to outer radius.
  • Realistic boundary conditions: Supports internal/external pressure and various axial constraints.
  • Educational & practical: Used by engineers for hydraulic cylinders, gun barrels, subsea pipelines, and pressure vessels.

Derivation & Historical Context

Lame's equations were developed by Gabriel Lamé in the 19th century for elasticity in thick cylinders. They satisfy equilibrium and compatibility in plane strain or plane stress. The maximum shear stress (Tresca) occurs at the inner radius where σθ - σr is largest. For ductile materials, von Mises equivalent stress is also computed.

Material Yield Strength Reference (Typical)

Material Yield Strength σy (MPa) Allowable Stress (N=2.5)
Carbon Steel (ASTM A36) 250 100
Stainless Steel 316 290 116
Aluminum 6061-T6 275 110
Titanium Grade 5 880 352

Note: The values are typical. For actual design, consult the specific material standard.

Case Study: Hydraulic Cylinder Design

A hydraulic cylinder operates at 35 MPa internal pressure. Inner radius = 50 mm, outer radius = 75 mm (thick wall). Using the calculator with closed ends: max hoop stress = 95.8 MPa at inner radius, radial stress = -35 MPa, axial stress = 26.2 MPa. Maximum shear stress = 65.4 MPa. For steel with yield strength 350 MPa and safety factor 2.5, allowable stress = 140 MPa → design is safe. The stress plot shows steep gradient, confirming thick‑wall necessity.

Limitations & Assumptions

  • Material is isotropic, homogeneous, and linear elastic.
  • Material behavior is linear elastic, isotropic, and homogeneous.
  • No plastic deformation, creep, or thermal effects are considered.
  • Perfectly cylindrical geometry, no discontinuities (nozzles, welds).
  • Long cylinder (plane strain) – end effects neglected.
  • No thermal stresses or creep.
  • For external pressure only, buckling is not evaluated – use additional stability checks.

Frequently Asked Questions

Hoop (circumferential) stress acts tangent to the cylinder wall, trying to split it longitudinally. Radial stress acts perpendicular to the wall, compressing the material. Hoop stress is typically tensile and much larger than radial stress under internal pressure.

Use thick‑wall if (rₒ - rᵢ)/rᵢ > 0.1. Thin‑wall approximation errors become significant beyond this limit. The calculator automatically selects thick‑wall model but you can override.

For closed ends, axial stress arises from pressure on end caps. For open ends (e.g., pipe with free ends), σz = 0. Constrained condition prevents axial strain (e.g., buried pipe), giving Poisson‑induced stress with ν=0.3 for steel.

External pressure (Pₒ > Pᵢ) produces compressive hoop stress. The same Lame equations apply, but buckling becomes critical. Use the calculator for stress, but consult buckling codes (e.g., ASME BPVC Section VIII) for stability.

Rooted in elasticity theory – This tool implements Lame's solution as per Theory of Elasticity by Timoshenko & Goodier. Validated against ASME Section VIII pressure vessel code. Last updated April 2026 by GetZenQuery Tech team.

References: Engineering ToolBox – Thick Walled Cylinder; Lame, G. (1852) Leçons sur la théorie mathématique de l’élasticité; ASME Boiler & Pressure Vessel Code, Section VIII.