Structural Stability

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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    A 0      2  cos   xo  Le    B  0  det . 

sin  xo  cos   xo  Le   cos  xo  sin   xo  Le   0 sin  Le   0

 Le  n
Page 10

Pcr  n

 2 EI 2
L
2 e

n  1, 2,3....

CVEN 9802 Stability

Effective Length
1.5071 2 EI   2 EI P   cr 2 L (0.8146L) 2

B

C

P

L/2

L

Page 11

CVEN 9802 Stability

Effective Length
B

C

P

kL

L

k keff_long keff_short

0.5 0.4 0.8146 0.7895 1.6292 1.97375

0.3 0.7662 2.554

0.2 0.7439 3.7195

0.1 0.7219 7.219

0.05 0.7106 14.212

Page 12

CVEN 9802 Stability

Problems with Effective Length
d4y d2y EI P 0 4 2 dx dx

• Assumes Le is a geometric property. But Le is

a function of the loading as well. • EI is assumed constant. • What if the axial force is distributed? Self Weight

Page 13

CVEN 9802 Stability

Beam-Column with Distributed Load

x

y(x) Deflected Shape B

A

P

wL/2

w

wL/2

L

EI - constant

Page 14

CVEN 9802 Stability

Freebody - Equilibrium
M N S y(x) B w Deflected Shape

P

NP S  w L  x  M  w L  x 2
2

L-x
wL 2 wL  Py  x    L  x 2
CVEN 9802 Stability

wL/2

Page 15

Equilibrium – Differential Equation
(small deflection theory) d y M  EI 2 dx
2

 M   Py 

wx  L  x  2

d2y  EI 2 dx

wx  x  L  d2y EI  Py  2 dx 2 wx  x  L  d2y 2  y  2 dx 2 EI Second Order, Differential Equation
Solution requires two boundary conditions!
Page 16 CVEN 9802 Stability

Particular Solution
y p  c0  c1 x  c2 x 2 y ' p  c1  2c2 x y '' p  2c2 wx  x  L  2c2  c0  c1 x  c2 x  2 EI w w c2 2   c2  2 EI 2 EI  2 wL wL 2 c1    c1   2 EI 2 EI  2 w 2c2  c0 2  0  c0   EI  4 2 2 2 2

Page 17

CVEN 9802 Stability

Solution
w wL w yp    x x2 EI  4 2 EI 2 2 EI 2 w  (2   2 x ( L  x )) 2 EI  4 w 2 y  A sin  x   B cos  x   (2   x ( L  x )) 4 2 EI P A, B constants,   EI 2...
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