MEC-34

Lamé thick-wall hoop

σθ,i = p (ro²+ri²)/(ro²−ri²) at the inner face of a pressurised cylinder.

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Pressure vesselsLamé 1852

Governing equation

\sigma_{\theta,i}=p\dfrac{r_o^2+r_i^2}{r_o^2-r_i^2}

where

p: Internal pressure (MPa)
r_i: Inner radius (mm)
r_o: Outer radius (mm)
\sigma_{\theta,i}: Inner hoop (MPa)
\sigma_{r,i}: Inner radial (MPa)

Lecture brief

Historical brief

Machine design grew from Coulomb torsion and Hertz contact (1881) through Soderberg fatigue and the heat-engine cycle. The lab writes shaft, bearing, contact and thermodynamic limits in SI. This sheet (MEC-34 — Lamé thick-wall hoop) is the form associated with Lamé 1852. Working symbols: $p$ , $r_i$ , $r_o$ $\rightarrow$ $\sigma_{\theta,i}$ , $\sigma_{r,i}$ . Lamé solved the axisymmetric elastic cylinder. Radial and hoop stresses vary as A − B/r².

Purpose

Purpose: compute

\sigma_{\theta,i}

\sigma_{r,i}

from

p

r_i

r_o

in Mechanical via

\sigma_{\theta,i}=p\dfrac{r_o^2+r_i^2}{r_o^2-r_i^2}

σθ,i = p (ro²+ri²)/(ro²−ri²) at the inner face of a pressurised cylinder. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

p = 20.000\,\mathrm{MPa}

r_i = 40.000\,\mathrm{mm}

r_o = 70.000\,\mathrm{mm}

, the governing relation

\sigma_{\theta,i}=p\dfrac{r_o^2+r_i^2}{r_o^2-r_i^2}

yields

\sigma_{\theta,i} = 39.39\,\mathrm{MPa}

\sigma_{r,i} = -20.00\,\mathrm{MPa}

. A thick ring, an inner pressure, two stress ticks. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Internal pressure p20 MPaInner radius r_i40 mmOuter radius r_o70 mm

Outputs

Inner hoop \sigma_{\theta,i}39.39 MPa
Inner radial \sigma_{r,i}-20.00 MPa

Reading speed

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Narration of this film

A thick ring, an inner pressure, two stress ticks.

Lamé solved the axisymmetric elastic cylinder. Radial and hoop stresses vary as A − B/r².

Reading speed

Watch on YouTube