Transcript No Slide Title

Buckling of Columns
A designer should satisfy
-Strength requirements (Ultimate Strength, Plasticity)
-Deflection requirements (Serviceability)
-Stability requirements
-Fatigue and Fracture Strengths
-Durability conditions
If specified conditions are exceeded the structure is assumed
to have failed and should be repaired or condemned.
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Buckling of Columns
Stability requirements can be described under
three equilibrium conditions;
Stable Equilibrium
Neutral Equilibrium Unstable Equilibrium
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Buckling of Columns
Critical Loads
P
P1
(a small load)
Deformed
state under P1
P
Practical cases: Coke cans; Lateral buckling; Truss buckling
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Buckling of Columns
If P  Pcr, when P1 is removed, the column becomes straight.
If P = Pcr when P1 is removed, the column does not return to its original position.
P = Pcr is called the critical load or the buckling load of the column
Vertical position – unstable equilibrium
Deformed position – Stable equilibrium
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Buckling of Columns
Ideal Column with Pin Supports
P = Pcr
v
L
x
y
y
x
v
P
P
P
Real columns are not always straight. Here the column is assumed to be straight,
symmetric, homogeneous and isotropic. Load is applied centrally.
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Buckling of Columns
From earlier derivations
M  E
 
I
y 
M 
EI


d 2v
EI 2
dx
  dy 
1   
  dx 
2



3/ 2
d 2v
 EI 2
dx
for small deformations.
M   Pv
d 2v
EI 2   Pv
dx
d 2v P
i.e., 2 
v0
EI
dx
d 2v
P
2
2


v

0
,
with


EI
dx 2
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Buckling of Columns
Solution to the differential equation is
v  A sin x  B cos x
Using boundary conditions
At x = 0, v = 0
0 = B
v  A sin x
At x = L, v = 0
0  A sin L
Either A = 0, or sin L  0
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Buckling of Columns
If A = 0, the column does not deform laterally, i.e., PPcr
SinL = 0, column deforms
P = Pcr
i.e., L  n , n  1, 2, .......n
2 L2  n 2 2
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Buckling of Columns
Pcr L2
 n 2 2
EI
i.e.,
n 2 2 EI
Pcr 
L2
When n = 1
Pcr 
First mode of
buckling
 2 EI
L2
v  A sin
Pcr
x
EI
When n = 2
EI
L2
 2x 
v  A sin

 L 
Pcr  4 2
Second mode
of buckling
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Buckling of Columns
 2 EI  2 EAk 2
Pcr  2 
where k radius of gyration
2
L
L
Pcr
 2E
  cr 
A
 L / k 2
cr
Y
Euler’s curve
L/k is the slenderness ratio.
(L/k)
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Buckling of Columns
Other boundary conditions for columns
Differential equation for a fixed-fixed column is,
P
d 2v
EI 2  Pv  M 0
dx
d 2 v Pv M 0


EI
dx 2 EI
M0
Point of
contraflexure
Solution is
v  A sin x  B cos x 
M0
P
M0
P
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Buckling of Columns
At x = 0, v = 0
0 = B + M0/P
B = - M0/P
At x = L, v = 0
0  A sin L  B cos L 
A sin L   B cosL 
A
M0
P
M0 M0
cosL  1

P
P
M0
cosL  1
P sin L
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Buckling of Columns
v
M0
cosL  1sin x  M 0 cosx  M 0
P sin L
P
P
At x = 0,
dv
0
dx
dv
 M0
cos L  1 cos x  M 0  sin x 
0
dx
P
 P sin L
 x 0

M0
cosL  1  0
P sin L
 cosL  1  0
i.e., cosL  1
 L  2n
i.e., 2 L2  4n 2 2
Pcr 2
L  4 n 2 2
EI
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Buckling of Columns
4n 2 2 EI
Pcr 
L2
n 2 2 EI

, n  1, 2, ......
2
L / 2
Effectivelength  L e  ( L / 2)
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Buckling of Columns
For a hinged – fixed column
P
2
M0
d v


Pv

( L  x)
2
L
dx
M0
d 2v
2


v

( L  x)
2
EIL
dx
P
where 2 
EI
EI
M0/L
L-x
v
L
x
Point of
contraflexure
x
v
M0
( x  L)  A sin x  B cosx
PL
M0/L
M0
At x = 0, v = 0
0 = M0/P + B
y
P
 B = - M0/P
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Buckling of Columns
At x = L, v = 0
0  0  A sin L  B cosL
M0
cosL
P
M cosL
A 0
P sin L
 A sin L 
M0
M  cosL 
M
( L  x)  0 
 sin x  0 cosx
PL
P  sin L 
P
M 

x  cosL
 0 1   
sin x  cosx 
P  L  sin L

v
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Buckling of Columns

dv M 0  1 
cos L

cos x   sin x 
   
dx
P  L 
sin L

At x  0,
0
dv
0
dx
M0
P
 1 

cos L




0
 L 

sin L



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Buckling of Columns
 cosL 1

sin L
L
cosL
1
i.e.,

sin L L
sin L
i.e.,
 L
cosL
i.e.,
i.e., t anL  L  0
First root is L = 1.4303
Solution is obtained as,
Pcr 
n 2 2 EI
0.7 L 2
Effective length = Le = 0.7L
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