Buckling does not vary linearly with load • it occurs suddenly and is therefore dangerous

Elastically Elastically Stable Unstable

P < 2e

2

cr LEIP π

= P > 2e

2

cr LEIP π

=

E …modulus of Elasticity I …moment of inertia (smallest value)

Axis of least I and ρ becomes neutral bending axis

Le…equivalent column length (length between zero bending)

Le=L

Both ends hinged

fixed fixed

fixed

0.707L 0.8 L

0.5 L 0.65 L

L 1.2 L

2 L 2.1 L

Elastic Instability - Buckling

Elastically Unstable

P > 2e

2

cr LEIP π

= with I = Aρ2 ( )2

e

2

crL

ESρ

π=

ρ …radius of gyration Le/ρ…slenderness ratio

Euler Column Buckling (1744)

Valid for all materials !

J.B.Johnson Parabola Euler Buckling

P > 2

e2

2y

ycrL

E4S

SS ⎟⎟⎠

⎞⎜⎜⎝

⎛ρπ

−= ( )2e

2

crL

ESρ

π=

ρ …radius of gyration Le/ρ…slenderness ratio

Euler Column Buckling

Johnson Parabola 2

Sy

21

y

2

SE2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ π

(1900)

Excentric Loaded Column

e=Le/400 or ec/ρ2=0.025 is recommended

Equivalent Column Stress

Scr=Sy/α α…stress multiplier

Euler Buckling

( )2e

2

crL

ESρ

π=

J.B.Johnson Parabola

P > 2

e2

2y

ycrL

E4S

SS ⎟⎟⎠

⎞⎜⎜⎝

⎛ρπ

−=

Other Forms of Buckling