Calculate hoop, axial, and radial stresses in thin-walled and thick-walled pressure vessels. Verify design safety factors per ASME codes.
The Cylinder Stress Calculator determines hoop (circumferential), axial (longitudinal), and radial stresses in cylindrical pressure vessels under internal and/or external pressure. The tool handles both thin-walled (r/t > 10) and thick-walled (r/t ≤ 10) cylinders using the appropriate equations.
For thin-walled vessels, the simple formulas σ_h = pr/t and σ_a = pr/(2t) apply. For thick-walled vessels, the Lamé equations account for the stress variation through the wall thickness — stress is highest on the inner surface and decreases toward the outer surface. This distinction is critical for high-pressure applications (>1000 psi) and thick-walled tubes.
Enter the vessel geometry, pressure conditions, and material yield strength to calculate all stress components, verify the safety factor, and determine minimum required wall thickness per ASME Section VIII guidelines. It gives you a fast sanity check before you move to a more detailed design review. That is useful when you want a quick pass/fail check before doing a deeper vessel analysis.
Use this calculator when you need to check hoop, axial, and radial stress in a pressurized cylinder before committing to a wall thickness. It is useful for vessel design, pipe checks, and quick ASME-style sanity checks. That is especially helpful when you want to compare a thin-wall estimate against a thick-wall case quickly.
Thin-Wall: σ_hoop = p×r/t, σ_axial = p×r/(2t). Thick-Wall (Lamé): σ_h = (p_i×r_i² - p_o×r_o²)/(r_o² - r_i²) + (p_i - p_o)×r_i²×r_o²/[r²×(r_o² - r_i²)]. σ_r = (p_i×r_i² - p_o×r_o²)/(r_o² - r_i²) - (p_i - p_o)×r_i²×r_o²/[r²×(r_o² - r_i²)]. σ_a = (p_i×r_i² - p_o×r_o²)/(r_o² - r_i²).
Result: Hoop stress = 105 MPa, Safety factor = 2.38
r/t = 100/10 = 10 (borderline thin/thick). Thin-wall: σ_h = 10×100/10 = 100 MPa. Thick-wall (at inner surface): σ_h = 105 MPa. Axial: σ_a = 50 MPa. von Mises: 91.0 MPa. Safety factor = 250/105 = 2.38.
Thin-wall theory (also called membrane theory) assumes that stress is uniform through the wall thickness. This simplification works well when the thickness is small relative to the radius (t/r < 0.1 or r/t > 10). The resulting formulas are simple: σ_hoop = pr/t, σ_axial = pr/(2t).
Thick-wall theory (Lamé equations, 1833) accounts for the variation of stress through the wall. At the inner surface, hoop stress is higher than the thin-wall prediction. At the outer surface, it's lower. The average through the thickness equals the thin-wall value. For a vessel with r_i/r_o = 0.5 (very thick wall), the inner surface hoop stress exceeds the thin-wall value by about 67%.
ASME Section VIII provides rules for unfired pressure vessels. Division 1 (design-by-rule) uses the maximum normal stress theory with safety factors applied to the allowable stress. Division 2 (design-by-analysis) allows detailed stress analysis with different limits for primary membrane, primary bending, secondary, and peak stresses. Division 2 typically results in thinner walls because it uses more accurate stress classification.
For multiaxial stress states, the von Mises criterion provides a single equivalent stress: σ_vm = √[(σ_h - σ_a)² + (σ_a - σ_r)² + (σ_r - σ_h)²] / √2. Yielding occurs when σ_vm equals the uniaxial yield strength. For internal pressure vessels, the von Mises stress is about 13% lower than the maximum principal stress (hoop stress), which provides additional design margin when using the maximum stress criterion.
Use thick-wall (Lamé) equations when r/t ≤ 10, or when the stress variation through the wall exceeds 5% (which happens when t > 0.1×r). Thin-wall formulas assume uniform stress across the thickness, which becomes inaccurate for thick walls. Always use thick-wall for high-pressure applications.
In a closed cylinder under internal pressure, hoop stress is exactly 2× axial stress because the pressure acts on a larger projected area in the circumferential direction than in the axial direction. This is why cylindrical vessels fail along longitudinal seams (hoop stress failure), not circumferential ones.
ASME Section VIII Division 1 requires: 3.5× on ultimate tensile strength, or approximately 1.5-2.0× on yield strength depending on the specific requirement. For new designs, use minimum 2.0× on yield. For critical applications, 3.0-4.0× on yield. The factor accounts for material variability, corrosion, and fatigue.
External pressure creates compressive hoop stress, which can cause buckling collapse before material yielding. Buckling is a stability failure that occurs at lower stress than yield. ASME uses an iterative chart method for external pressure design because the critical buckling pressure depends on geometry and stiffening rings.
ASME BPVC Section VIII, Division 1: t = PR/(SE - 0.6P) for hoop stress, or t = PR/(2SE + 0.4P) for axial stress, where P = design pressure, R = inner radius, S = allowable stress, E = joint efficiency (0.65-1.0 depending on radiography).
Temperature gradients through the wall create thermal stresses. If the inner surface is hotter, it tries to expand but is constrained by the cooler outer surface, creating compressive stress inside and tensile outside. Thermal stress σ_t = Eα∆T/(2(1-ν)). This adds to pressure stresses and can be significant in process equipment.