A pressure vessel, and when the thin-wall formula stops being true

The thin-wall hoop stress is one of the most useful approximations in engineering and one of the most quietly misused. It is exact in the limit of a membrane and good to a few per cent while the wall is thin; it understates the stress at the bore of a thick one, and the error grows with the wall. This worksheet works a cylinder both ways and puts the two answers side by side, so the approximation is checked rather than assumed. The vessel: a 600 mm bore receiver at 2 MPa, 12 mm wall, S355.

use steel

Four significant figures.

digits 4

r_{i}

300 mm

t_{wall}

12 mm

p

2 MPa

r_{o}

r_{i} + t_{wall}

300 mm + 12 mm

312 mm

Is it thin?

The usual rule is that the wall is thin below a tenth of the radius. It is a rule about the answer rather than about the geometry: what it really says is that the through-thickness variation of hoop stress is small enough to ignore.

ratio

\frac{t_{wall}}{r_{i}}

\frac{12 mm}{300 mm}

0.04

check

ratio \leq 0.1

0.04 \leq 0.1

pass

Thin wall

Hoop stress from a cut along the axis, longitudinal from a cut across it. The factor of two between them is why a cylinder that fails from internal pressure splits along its length rather than round its circumference.

σ_{hoop_thin}

\frac{p \cdot r_{i}}{t_{wall}}

\frac{2 MPa \cdot 300 mm}{12 mm}

50 MPa

σ_{long_thin}

\frac{p \cdot r_{i}}{2 \cdot t_{wall}}

\frac{2 MPa \cdot 300 mm}{2 \cdot 12 mm}

25 MPa

Thick wall

Lamé's solution, evaluated at the bore where the hoop stress is largest. It makes no assumption about the wall being thin and is what the thin-wall formula approaches as the wall gets thinner.

σ_{hoop_bore}

\frac{p \cdot ({r_{o}}^{2} + {r_{i}}^{2})}{{r_{o}}^{2} - {r_{i}}^{2}}

\frac{2 MPa \cdot ({(312 mm)}^{2} + {(300 mm)}^{2})}{{(312 mm)}^{2} - {(300 mm)}^{2}}

51.02 MPa

The radial stress at the bore is not zero — it is the pressure itself, in compression — and the thin-wall treatment drops it. At this wall it is 4% of the hoop stress; at a quarter-radius wall it is not negligible at all.

σ_{radial_bore}

- p

- 2 MPa

-2 MPa

How much the approximation is out by, which is the number this worksheet exists to produce.

{error}_{thin}

\frac{σ_{hoop_thin} - σ_{hoop_bore}}{σ_{hoop_bore}}

\frac{50 MPa - 51.02 MPa}{51.02 MPa}

-0.01998

{error}_{thin}

-1.998 %

The check

Against the thick-wall stress, because it is the true one, and on the distortion-energy criterion with all three principal stresses — hoop, radial and longitudinal. Dropping the radial term is the same approximation again and this worksheet has just finished measuring it.

σ_{long}

\frac{p \cdot r_{i}}{2 \cdot t_{wall}}

\frac{2 MPa \cdot 300 mm}{2 \cdot 12 mm}

25 MPa

s1

σ_{hoop_bore}

51.02 MPa

s2

σ_{long}

25 MPa

s3

σ_{radial_bore}

-2 MPa

σ_{vm}

\sqrt{\frac{{(s1 - s2)}^{2} + {(s2 - s3)}^{2} + {(s3 - s1)}^{2}}{2}}

\sqrt{\frac{{(51.02 MPa - 25 MPa)}^{2} + {(25 MPa - -2 MPa)}^{2} + {(-2 MPa - 51.02 MPa)}^{2}}{2}}

45.92 MPa

n_{design}

1.5

σ_{allow}

\frac{{Fy}_{S355}}{n_{design}}

\frac{355 MPa}{1.5}

236.7 MPa

check

σ_{vm} \leq σ_{allow}

45.92 MPa \leq 236.7 MPa

pass

Where the approximation goes wrong

Hoop stress at the bore against wall thickness, both ways, on one plot. They agree while the wall is thin and part company as it thickens — and the thin-wall line is the lower of the two, which is the dangerous direction.

fn thin defined

fn thick defined

The legend would otherwise name the curves after the functions that drew them, which here are thin and thick. label says what they are.

axis x "Wall thickness"

axis y "Hoop stress"

label "Thin-wall", "Thick-wall (Lamé)"

plot (thin, thick, 5, 120)

What this leaves out

A plain cylinder, away from its ends. The heads, the nozzles and the joints between them are where a real vessel is governed, the stress concentrations there are not small, and a vessel is signed off against a code — ASME VIII, EN 13445 — that fixes the allowable stress, the joint efficiency and the corrosion allowance rather than leaving them to a factor of safety chosen on the page.