Buckling Phenomena

Stability of Structures

P<

P<

Elastic Hinge Analysis of Column

tan then 0, If

)2/(2 LkPcr

)2/(tan2 LkkFP

4

kLPcr

(independent of )(How to find k?)

Euler’s Formula for Pin-ended Columns(Eigen-value Problem)

EI

Pv

EI

M

dx

vd

2

2

02

2

vEI

P

dx

vd

xEI

PCx

EI

PCv cossin 21

0 0)0( 2 Cxv

.....3 ,2 ,1 , 2

22

nL

EInP

xL

CxvL

EIP

sin)( 12

2

cr

0sin 0)( 1 LEI

PCLxv

0sin ' ,0 1 LEI

PsolutiontrivialsitthenCIf

Critical Stress in Pin-ended Columns

2

2cr

cr AL

EI

A

P

A

Ir where,y I 22 ArdA

Gyration of Radius theis

/ 2

2

2

22

crrL

E

L

Er

Note: L/r is Slenderness Ratio

Consideration in Section Property

2

2cr

cr AL

EI

A

P

Note: In the above formula, I is a minimum on the cross-section. Therefore, the buckling of

columns has a selective direction.

y y y

z z z

I is isotropic Iy = Iz Iy < Iz

Extension of Euler’s Formula

2

2

2

2

2

2

cr 4

1

2 L

EI

L

EI

L

EIP

e

2

2

2

2

cr/4

/ rL

E

rL

E

e

Euler’s Formula for Clamp-ended Columns

2

2

creL

EIP

2

2

2/L

EI

2

24 L

EI

Effective Length of Various Columns

L