Structural Stability of Beam Columns

CVEN9802 Structural Stability

Beam Columns

Chongmin Song University of New South Wales
Page 1 CVEN 9802 Stability

Outline
• Effective Length Concept
• Beam-Column with Distributed Load • Column with Imperfection

• Southwell Plot
• Column Design Formula

Page 2

CVEN 9802 Stability

Fundamental cases of buckling

PE 

 EI
2

L

2

2 2.045 EI P  4 EI Pcr  cr 2 L2 L
2

Pcr 

 2 EI
4L
2

PE 

 2 EI
L2

Page 3

CVEN 9802 Stability

What is “effective length”

Unified expression of critical load

PE  2 EI Pcr  2  K ( KL)2 K : effective length factor
Page 4 CVEN 9802 Stability

Effective Length
• The Effective Length is the length of an

equivalent simply supported column that will buckle at the same load as in the example problem!
Pcr 

 2 EI
L2 e

Le  

EI Pcr

Page 5

CVEN 9802 Stability

Effective Length
Recall the solution for a simply supported column: x A

y(x)

B

P

EI
Buckled Shape

L

Page 6

CVEN 9802 Stability

Solution d y EI  Py  0 2 dx y 0  0 y  L  0
Unstable if det.=0
2

y  A sin  x   B cos  x  P A, B constants,   EI
2

0  sin  L    det .

  A  0       cos  L    B  0 1
 sin  L   0

 L  n
Page 7

Pcr  n

 2 EI 2
L
2

n  1, 2,3....

CVEN 9802 Stability

General Solution
Pts of inflexion d2y/dx2=0 xo A Le B

EI - constant
C

P

Le – effective length

Buckled Shape

d4y d2y EI P 0 4 2 dx dx
Page 8

No distributed load

CVEN 9802 Stability

General Solution y  A sin  x   B cos  x   Cx  D P A, B, C , D constants,   EI
2

d2y   A 2 sin  x   B 2 cos  x  dx 2 d2y 0 2 dx x  xo d2y 0 2 dx x  xo  Le
Page 9

 A 2 sin  xo   B 2 cos  xo   0  A 2 sin   xo  Le   B 2 cos   xo  Le   0
CVEN 9802 Stability

General Solution
  2 sin  xo   2  sin   xo  Le  

 2 cos  xo