MEC-33

Thin pressure-vessel hoop

σh = p r / t, σℓ = p r /(2 t). Membrane theory, t/r < 0.1.

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Pressure vesselsBarlowASME VIII

Governing equation

\sigma_h=\dfrac{pr}{t},\quad\sigma_\ell=\dfrac{pr}{2t}

$p \in \left[0.1, 20\right]$

where

p: Internal pressure (MPa)
r: Mean radius (mm)
t: Wall thickness (mm)
\sigma_h: Hoop stress (MPa)
\sigma_\ell: Longitudinal stress (MPa)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\sigma_h=\dfrac{pr}{t},\quad\sigma_\ell=\dfrac{pr}{2t}

$p \in \left[0.1, 20\right]$

$p = 0.1$ $\sigma_h = 3.125$

Along the curve

②

Domain of the variable — blue

p_{\min} = 0.1,\quad p_{\max} = 20

$p \in \left[0.1, 20\right]$

$p = 0.1$ $p = 0.1$

Along the curve

③

Response curve — green

\sigma_h=\dfrac{pr}{t},\quad\sigma_\ell=\dfrac{pr}{2t}

$p \in \left[0.1, 20\right]$

$p = 0.1$ $\sigma_h = 3.125$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\sigma_h}{\mathrm{d}p}

$p \in \left[0.1, 20\right]$

$p = 0.225$ $\sigma_h^{\prime} = 31.25$

Along the curve

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-33 — Thin pressure-vessel hoop) is the form associated with Barlow · ASME VIII. Working symbols: $p$ , $r$ , $t$ $\rightarrow$ $\sigma_h$ , $\sigma_\ell$ . A longitudinal cut balances 2 σh t L against p (2 r L). Closed-end force is carried at half the hoop.

Purpose

Purpose: compute

\sigma_h

\sigma_\ell

from

p

r

t

in Mechanical via

\sigma_h=\dfrac{pr}{t},\quad\sigma_\ell=\dfrac{pr}{2t}

σh = p r / t, σℓ = p r /(2 t). Membrane theory, t/r < 0.1. 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 = 2.000\,\mathrm{MPa}

r = 250.000\,\mathrm{mm}

t = 8.000\,\mathrm{mm}

, the governing relation

\sigma_h=\dfrac{pr}{t},\quad\sigma_\ell=\dfrac{pr}{2t}

yields

\sigma_h = 62.50\,\mathrm{MPa}

\sigma_\ell = 31.25\,\mathrm{MPa}

. A cylinder, a hoop band, a longitudinal fibre. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Internal pressure p2 MPaMean radius r250 mmWall thickness t8 mm

Outputs

Hoop stress \sigma_h62.50 MPa
Longitudinal stress \sigma_\ell31.25 MPa

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Narration of this film

A cylinder, a hoop band, a longitudinal fibre.

A longitudinal cut balances 2 σh t L against p (2 r L). Closed-end force is carried at half the hoop.

Reading speed

Watch on YouTube