Shell buckling — thin-walled cylinder under axial load: Foundations

This page assumes you know nothing about the notation on the parent page. We build every letter, every ratio and every picture from the ground up, in an order where each idea leans on the one before.

1 · The object itself: the thin-walled cylinder

Before any formula, picture the thing. In s01 below, follow the magenta arrow (that is $R$ , the spoke of the round top), the orange band on the left edge (that is the wall thickness $t$ ), and the navy double-arrow on the right (that is the height $L$ ).

Figure — Shell buckling — thin-walled cylinder under axial load

2 · Squeezing it: force, stress and area

The tank is squeezed along its axis by the weight and thrust stacked above.

3 · The material: $E$ and $ν$

Two numbers describe the metal itself, independent of shape.

4 · Two failure modes: yield vs stability

The whole point of the parent page is that this cylinder does not fail the "obvious" way.

5 · The two restoring effects (the tug-of-war)

When the wall dimples inward or outward by a small radial amount $w$ , two "springs" fight back.

In s04, the top strip (small $k$ ) has a few long humps; the bottom strip (large $k$ ) has many tight humps. Same wall, two possible fold patterns.

Definition Membrane (hoop) spring constant

$K = \frac{E t}{R ^{2}}$ Pushing the wall radially by $w$ stretches or squeezes the circumference (the hoop). The stored stretch-energy fights the dimple, and its stiffness is a spring constant $K = \frac{E t}{R ^{2}} .$ Notice: no $(1 - ν^{2})$ here. The hoop stretch is a uniaxial stretch of the circumferential fibres — a single direction being lengthened — so it uses the plain axial stiffness $E t$ , not the two-directional plate stiffness. That is the honest reason $ν$ appears in $D$ but not in $K$ : $D$ is a bending (two-way plate) effect, $K$ is a stretch (one-way membrane) effect. (Fuller treatments keep a mild $ν$ correction, but at this level $K = E t / R^{2}$ is the clean number that enters $2 D K$ .)

The picture: the round hoop resisting being made smaller or bigger — it exists only because the wall is curved ( $1/ R$ ). A flat plate has no hoop, so no such term. This is the same circumferential stretch idea as Hoop stress in pressurised cylinders.

Intuition WHY the hoop energy scales as

$K / k^{4}$ Here is the subtle part. When the wall dimples in one spot, the material has to go somewhere. For a long gentle fold (small $k$ ), the wall genuinely pushes in and out radially, so the whole circumference must stretch — big hoop energy. For a tight short fold (big $k$ ), the metal can slosh sideways along the wall into the neighbouring dimple instead of stretching the hoop, so almost no circumference change happens — tiny hoop energy.

Working this through the geometry of the buckle pattern (the radial motion needed to keep the folds compatible falls off as the folds get tighter) gives a hoop energy $U_{hoop} \propto \frac{K}{k ^{4}} w_{0}^{2} .$ In words: long loose folds are expensive for the hoop (they force real circumference stretch); tight folds cheat by shuffling material sideways. That is the $1/ k^{4}$ — the exact mirror image of the bending term.

Intuition Why two springs, the optimum

$k^{*}$ , and their geometric mean Put the two costs side by side (per unit dimple depth): $U (k) = D k^{4} + \frac{K}{k ^{4}} .$ Bending is worst for tight folds (big $k$ ); hoop is worst for loose folds (small $k$ ). The wall picks the $k$ that makes the total smallest. Set the derivative to zero: $\frac{d U}{d k} = 4 D k^{3} - \frac{4 K}{k ^{5}} = 0 ⟹ k^{4} \cdot k^{4} = \frac{K}{D} ⟹ k^{*} = (\frac{K}{D})^{1/8} .$ Substituting $K = E t / R^{2}$ and $D = E t^{3} / [12 (1 - ν^{2})]$ gives the natural fold size $k^{*} \sim [12 (1 - ν^{2}) / (R^{2} t^{2})]^{1/8} \sim 1/ R t$ , i.e. wavelength $λ^{*} = 2 π / k^{*} \sim R t$ — the dimples are about $R t$ across, small compared with either $R$ or $L$ .

At $k^{*}$ the two costs are equal ( $D k^{4} = K / k^{4}$ ), and by $a + b \geq 2 ab$ (equality when $a = b$ ) the minimum value is the geometric mean $min_{k} (D k^{4} + \frac{K}{k ^{4}}) = 2 D K .$ That is why the parent's clean result reads $σ_{cr} t = 2 D \cdot \frac{E t}{R ^{2}}$ — the two springs balanced exactly, and the buckle length $L$ has vanished (the wall found its own natural fold size $\sim R t$ , independent of tube height — provided the tube is not too short, see §6).

6 · The length $L$ and the ends: local vs global buckling

We defined $L$ but it dropped out of $σ_{cr}$ . That is only true in the middle regime — the ends and the aspect ratio $L / R$ decide which kind of buckling happens.

Definition Aspect ratio

$L / R$ and the two buckling families

Local shell buckling (short/medium tube): the wall folds into little diamond dimples of size $\sim R t$ , and $σ_{cr} = 0.605 E t / R$ ignores $L$ , because the natural fold is much smaller than the tube. This is the regime the parent page solves. In s05, left panel.
Global (Euler) column buckling (very long, slender tube): the whole cylinder bows sideways like a bent straw, exactly the Euler column buckling mode, and now $L$ matters strongly — the critical load falls as $L^{2}$ grows. In s05, right panel.

Quantitative switch-over: compare the local shell stress $0.605 E t / R$ with the global Euler stress of a tube ( $σ_{Euler} = π^{2} E (R^{2} + ...) /...$ ; for a thin ring the Euler stress is $\approx \frac{π ^{2}}{2} E (R / L)^{2}$ for pinned ends). Setting the two equal, the crossover sits near $\frac{L}{R} \approx \frac{π ^{2} /2}{0.605} \cdot \frac{R}{t} \approx 2.85 R / t .$ Concrete rule of thumb: for $L / R ≲ 2.85 R / t$ (i.e. most tanks) local dimpling governs and you use $0.605 E t / R$ ; only for very slender tubes beyond this do you switch to the Euler formula. Example: $R / t = 600$ gives crossover $L / R \approx 2.85 600 \approx 70$ — a tube would have to be ~70 radii long before global bowing wins.

7 · The reality gap: knockdown $γ$ and $ϕ$

Prerequisite map

Equipment checklist

Cover the right side and see if you can state each from memory.

What does the ratio

R / t

measure, and why a ratio not raw sizes?

How "thin/flimsy" the wall is; proportion, not metres, decides buckling — same

R / t

feels equally tinny.

Definition of stress

σ

and its units?

Force per unit area,

σ = P / A

; measured in pascals (

N / m^{2}

), usually MPa.

Which area carries the axial load, and its formula?

Only the wall ring:

A = 2 π R t

What does Young's modulus

E

describe?

Material stiffness in stretch — stress needed per unit fractional stretch.

What does Poisson's ratio

ν

describe, and why does the topic need it?

Sideways thinning per lengthwise stretch; it produces the

(1 - ν^{2})

plate-stiffening factor in

D

Difference between yield and buckling failure?

Yield = material crushed (strength); buckling = shape becomes unstable (stiffness) — buckling strikes first here.

What is the radial displacement

w

How far the wall pokes in/out from perfectly round — the depth of a dimple.

What does the buckle wavenumber

k

count?

How tightly packed the folds are — big

k

many short folds, small

k

few long folds; wavelength

λ = 2 π / k

Formula and meaning of flexural rigidity

D

D = E t^{3} / [12 (1 - ν^{2})]

— resistance to curving the wall; costs

D k^{4}

for tight folds.

Why does bending energy scale as

D k^{4}

Curvature is the 2nd derivative of the fold,

\propto k^{2}

; bending energy

\propto

curvature

^{2} \propto (k^{2})^{2} = k^{4}

What is the membrane spring constant

K

(exact), and its cost?

K = E t / R^{2}

(no

ν

— it's a one-way hoop stretch, not two-way plate bending); costs

K / k^{4}

, worst for loose folds.

Why does hoop energy scale as

1/ k^{4}

Loose folds force real circumference stretch; tight folds let material shuffle sideways instead, so hoop cost falls off as folds tighten.

Where does

ν

enter the two springs, and why only there?

Only in

D

(two-directional plate bending, Poisson coupling); not in

K

(single-direction hoop stretch).

What is the optimal wavenumber

k^{*}

and fold size?

k^{*} = (K / D)^{1/8}

; wavelength

λ^{*} \sim R t

— the natural dimple size, independent of

L

Why is

σ_{cr}

a geometric mean

2 D K

Minimising

D k^{4} + K / k^{4}

balances the two springs equally at

k^{*}

;

a + b \geq 2 ab

gives the geometric mean.

Show the algebra from

2 D K

to the boxed formula.

Sub

D, K

, get

E^{2} t^{4} / [12 (1 - ν^{2}) R^{2}]

, pull out

E t^{2} / R

, divide by

t

12 = 23

cancels the 2 →

E t / [R 3 (1 - ν^{2})]

At what

L / R

does buckling switch from local to global?

Roughly

L / R \approx 2.85 R / t

; below it local dimpling governs (use

0.605 E t / R

), above it Euler bowing wins.

How do clamped vs free ends affect

σ_{cr}

Clamped = stiffest, highest

σ_{cr}

; free end = weakest (global Euler load can drop ~4×). Always check the boundary condition.

What do

γ

and

ϕ

do?

γ

knocks the perfect stress down for real dents;

ϕ = \frac{1}{16} R / t

feeds the empirical

γ = 1 - 0.901 (1 - e^{- ϕ})