Why Airplane Windows Are Round

Equivalent forms

K_t = 1 + \frac{2a}{b}

\sigma_{max} = \sigma\left(1 + 2\sqrt{a/\rho_c}\right)

\sigma_{hoop} = \frac{p\,r}{t}

A famous disaster distilled into one ratio: round the corner and divide the stress.

Symbol	Name	SI unit	Dimension	Range
$\sigma$	Nominal Hoop Stress Far-field skin stress from cabin pressurization	Pa	M·L⁻¹·T⁻²	1000000 – 200000000
$a$	Semi-axis Along Stress Half-width of the cutout measured along the stress direction	m	L	0.01 – 0.5
$b$	Semi-axis Across Stress Half-width perpendicular to the stress (corner sharpness)	m	L	0.005 – 0.5
$\sigma_{max}$	Peak Stressoutput Maximum stress at the cutout edge	Pa	M·L⁻¹·T⁻²	0 – 2000000000

Unit systems

Symbol	SI	CGS	Imperial
$\sigma$	Pa	ba	psi
$a$	m	cm	in
$b$	m	cm	in
$\sigma_{max}$	Pa	ba	psi

Where it holds

Inglis elastic solution for an elliptical hole in a large, thin, isotropic plate under uniform far-field tension. Quantitative below yield; near sharp corners local plasticity blunts the true peak but the design lesson — round the corners — holds.

Dimensional analysis

[\sigma(1+2a/ $b)] = (M\cdot L^{-1}\cdot T^{-2})\times$ (\text{dimensionless $}) = M\cdot L^{-1}\cdot T^{-2} = Pa \checkmark$ (stress)

Discovery

Charles Inglis / de Havilland Comet inquiry · 1913

Inglis derived the elliptical-hole stress concentration in 1913. Its deadly relevance appeared in 1954 when two de Havilland Comets — the first commercial jets — broke up in flight; the Farnborough investigation traced fatigue cracks to the near-square window and antenna cutouts, and every airliner since uses rounded windows.

Try this

Square windows doomed the first jetliner — sharp corners concentrate stress until the fuselage tears open.

A pressurized cabin skin carries a hoop stress σ. Around a sharp elliptical window of semi-axes a (along stress) and b, the peak stress is σ(1 + 2a/b). Compare a near-square corner (a/b = 5) with a circle (a/b = 1).

Research status: stable

Real-world applications

Aircraft window, door and cutout design
Pressure-vessel nozzle reinforcement
Ship hatch corners (Liberty-ship fractures)
Fillets and radii in machine parts

Common misconceptions

Windows are round just for looks or visibility — the driver is fatigue-crack prevention
Stress concentration needs a crack to matter — any sharp geometric corner concentrates stress
A bigger window is the danger — it is the corner sharpness (a/b), not size alone, that spikes $\sigma$

Experimental verification

Photoelastic stress models show fringe crowding at sharp corners exactly as predicted; the Comet wreckage and full-scale fuselage water-tank fatigue tests at Farnborough reproduced cracks initiating at the square window corners.

Derivation

Inglis solved the 2D elasticity problem for an elliptical hole of semi-axes a (parallel to the applied tension

\sigma )

and b in an infinite plate, finding the maximum edge stress

\sigma _\max = \sigma (1 + 2

a/b).

Writing it via the corner radius of curvature

\rho _c = b^{2}

/a gives

\sigma _\max = \sigma (1 + 2\surd

(a/

\rho _c))

— sharper corners (smaller

\rho _c)

spike the stress.

Limiting cases

a = b

(circle)⟶ \

\sigma _{\max } = 3

\sigmaA circular hole gives the classic stress-concentration factor of 3 — the gentlest case for a hole.

a \gg b (sharp slot)⟶ \sigma_{max} \to \inftyA long, thin crack-like opening drives the stress concentration sky-high — the Comet's near-square corners.

b \gg a⟶ \sigma_{max} \to \sigmaA hole elongated across the stress barely perturbs the field.

What if…

What if you make the window a perfect circle?

The concentration factor drops to its minimum of 3 — the gentlest possible for a hole, well below the fatigue limit.

What if a tiny scratch sharpens a corner?

$\rho _c$ plummets $and \sigma _\max$ soars; that is exactly how fatigue cracks nucleate at sharp features.

1

Square corner vs round window

Given ·

\sigma:: 50000000
a:: 0.1
b:: 0.02

Find · \sigma_{max}

Steps

Sharp corner: $K_t = 1 + 2$ (a/ $b) = 1 + 2(0.1/0.02) = 11$
$\sigma _\max = 11 \times 50\,\mathrm{MPa} = 550\,\mathrm{MPa}$ (likely past fatigue limit)
Round it $(a=b)$ : $K_t = 3$ , $\sigma _\max = 150\,\mathrm{MPa}$ — why every airliner window is curved

Result · Sharp (a/

b=5)

:

\sigma _\max = 50e_{6}(1+2\times 5) = 50e_{6}\times 11 = 550\,\mathrm{MPa}

. Circle (a/

b=1)

:

\sigma _\max = 50e_{6}\times 3 = 150\,\mathrm{MPa}

— the corner carries

\sim 3.7\times

more.

Hydrostatic Pressure→Pascal's Principle→Bernoulli's Equation→