Engineering
Lame Thick-Walled Cylinder Hoop & Radial Stress (Pressure Vessels)
Calculate inner-bore peak tensile hoop stress sigma_theta = P_i * (r_o^2 + r_i^2) / (r_o^2 - r_i^2) and outer-surface hoop stress in hydraulic cylinders, gun barrels, and autoclaves where t / r_i > 0.10.
Why thin-wall P*r/t underpredicts thick hydraulic-cylinder bore stress by 67%!
Undergraduate mechanics teaches the thin-wall hoop stress shortcut sigma = (P * r_i) / t—which assumes uniform tensile stress across the wall! Once wall thickness exceeds 10% of inner radius (t / r_i > 0.10, such as hydraulic actuator barrels, high-pressure hydrogen vessels, and shrink-fit hubs), inner fibers stretch far more than outer fibers! French mathematician Gabriel Lamé (1852) derived the exact elasticity solution for internal gauge pressure P_i (with external P_o = 0):
Peak Inner-Bore Hoop Stress (at r = r_i): sigma_bore = P_i * (((r_o ^ 2) + (r_i ^ 2)) / ((r_o ^ 2) - (r_i ^ 2))), while Inner Radial Stress equals -P_i (compressive) and Outer-Wall Hoop Stress (at r = r_o) drops to (2 * P_i * (r_i ^ 2)) / ((r_o ^ 2) - (r_i ^ 2))! On an 80 MPa cylinder with r_i = 50 mm and r_o = 100 mm (t = 50 mm), naive thin-wall P*r/t predicts 80.00 MPa—whereas true Lamé bore hoop stress is 133.33 MPa (+66.7% higher)!
| Stress Location (P_i = 80 MPa, r_i = 50 mm, r_o = 100 mm) | Formula | Stress Value (MPa) | Comparison vs Thin-Wall |
|---|---|---|---|
| Inner Bore Max Hoop Stress | Pi * (ro^2 + ri^2) / (ro^2 - ri^2) | 133.33 MPa (Peak Tensile!) | +66.7% above thin-wall! |
| Outer Surface Hoop Stress | 2 * Pi * ri^2 / (ro^2 - ri^2) | 53.33 MPa | 2.5x lower than inner bore |
| Naive Thin-Wall P*ri / (ro-ri) | 80 * 50 / 50 | 80.00 MPa (Unsafe!) | Only valid when t/ri <= 0.10 |
Lamé Peak Inner-Bore Hoop Stress sigma_theta,max (MPa)
Evaluates exact bore circumferential stress for internal gauge pressure B2 (MPa), inner radius B3, and outer radius B4.
=IF(OR(B3 <= 0, B4 <= B3), "Outer Radius ro Must Exceed ri", ROUND(B2 * (((B4 ^ 2) + (B3 ^ 2)) / ((B4 ^ 2) - (B3 ^ 2))), 2))Lamé Outer-Wall Hoop Stress (Strain-Gauge Measurable Surface Stress)
Computes external surface hoop stress (since strain gauges are mounted on the outside r = r_o of live vessels!).
=ROUND((2 * B2 * (B3 ^ 2)) / ((B4 ^ 2) - (B3 ^ 2)), 2)Try the example
Use the original values or change them to test your understanding.