Classes #9 & #10
Civil Engineering Materials – CIVE 2110
Buckling
Fall 2010
Dr. Gupta
Dr. Pickett
1
Buckling = the lateral deflection
of long slender members
caused by axial compressive forces
Buckling
of
Diagonals
Buckling of Columns
Buckling of Beams
Column Buckling Theory uses
ASSUMPTIONS OF BEAM BENDING THEORY








Column Length is Much Larger Than
Column Width or Depth.
so most of the deflection is caused by bending,
very little deflection is caused by shear

E
Column Deflections are small.

Column has a Plane of Symmetry.


Resultant of All Loads acts
Y
in the Plane of Symmetry.
σBuckle
Column has a Linear

Stress-Strain Relationship.
Ecompression = Etension
Y
σyield compression = σyield tension
σBuckle < (σyield ≈ σProportional Limit ).

Column Buckling Theory uses
ASSUMPTIONS OF BEAM BENDING
THEORY







Column Material is Homogeneous.
Column Material is Isotropic.
Column Material is Linear-Elastic.
Column is Perfectly Straight,
Column has a Constant Cross Section
(column is prismatic).
Column is Loaded ONLY by a
Uniaxial Concentric Compressive Load.
Column has Perfect End Conditions:
 Pin Ends – free rotation allowed,
P
- no moment restraint
 Fixed Ends – no rotation allowed,
- restraining moment applied

d
P
P=Pcr
P=Pcr
Column Buckling Theory


An IDEAL Column will NOT buckle.
IDEAL Column will fail by:



Punch thru
Denting
Fracture
F
σ > σyield compressive .

P=Pcr
P=Pcr
In order for an IDEAL Column to buckle
a TRANSVERSE Load, F,
Pcr = Critical Load
must be applied
Pcr = smallest load at which
column may buckle
in addition to the
Concentric Uniaxial Compressive Load.

The TRANSVERSE Load, F, applied to IDEAL Column
Represents Imperfections in REAL Column
P=Pcr
Column Buckling Theory

Buckling is a mode of failure
caused by Structural Instability
due to a Compressive Load
at no cross section of the member
is it necessary for
σ > σyield .

d
P=Pcr

Three states of Equilibrium are possible for an Ideal Column



Stable Equilibrium
Neutral equilibrium
Unstable Equilibrium
Column Buckling Theory – Equilibrium States
Stable Equilibrium
Neutral Equilibrium
Unstable Equilibrium
F
F
F
P<Pcr P<Pcr
P<Pcr
P=Pcr
F
P<Pcr P<Pcr
P=Pcr
F
P<Pcr
P=Pcr
P=Pcr
P=Pcr
P>Pcr P>Pcr
Δ=small
P=Pcr
F
P>Pcr
P>Pcr
Δ=grows
P>Pcr
P>Pcr
Column Buckling Theory – Equilibrium States
Stable Equilibrium
P
Ideal Column
Pcr
Neutral Equilibrium
P
Pcr
Ideal Column
Real
Column
Real Column
P<Pcr
0
P<Pcr
Δ/L=
P<Pcr
0
P=Pcr
F
P<Pcr P<Pcr
P=Pcr
Real Column
Δ/L=
P=Pcr
F
P<Pcr
Unstable Equilibrium
Ideal Column
P
Pcr
P=Pcr
P=Pcr
Δ/L=
0
P>Pcr P>Pcr P>Pcr
Δ=small
P=Pcr
F
P>Pcr
Δ=grows
P>Pcr
P>Pcr
Deflection - BEAM BENDING THEORY

When a POSITVE moment is applied, (POSITIVE Bending)
TOP of beam is in COMPRESSION
BOTTOM of beam is in TENSION.
NEUTRAL SURFACE:
- plane on which
NO change
in LENGTH occurs.
Cross Sections
perpendicular to
Longitudinal axis
Rotate about the
NEUTRAL (Z) axis.
P
Tension
Tension
P
Elastic Buckling Theory – Ideal Column
Compression
From Moment curvature relationship;
M
Negative Curvature
Positive Curvature
Compression
P
y=(+)
P
M P
Tension
P
M
d2y

 2

EI
dx
1
P
y=(-)
P
M
 CCW
M atX
0
  CCW
M atX
0
0   M x  P( y )
0   M x  P( y )
d2y
0  EI 2  Py
dx
d2y
0  EI 2  Py
dx
Compression
P
Compression
Tension
Tension
x
Tension
x
M
d2y

 2

EI
dx
1
M
Tension
Tension
Elastic Buckling Theory – Ideal Column
d2y
For Either Curvature:
0  EI 2  Py
dx
Assum e:
y  ASin(m x)  BCos(m x)
dy
 m ACos(m x)  m BSin(m x)
dx
d2y
2
2


m
ASin
(
m
x
)

m
BCos(m x)
2
dx
d2y
Substitue int o :
0  EI 2  Py
dx
0   m 2 EI  ASin(m x)  BCos(m x)  P ASin(m x)  BCos(m x)


0  P  m 2 EI ASin(m x)  BCos(m x)
0  P  m 2 EI
Thus : m  P
EI
the solution becom es:


P 
P 
y  ASin x
  BCos x

EI
EI




Elastic Buckling Theory – Ideal Column
x=L y=0
d2y
For Either Curvature:
0  EI 2  Py
dx


P 
P 
y  ASin x
  BCos x

EI
EI




L
x
For a Pin  Pin Colum n:
at x  0
y  0 0  ASin(0)  BCos(0)
0B
at x  L
y0

P 
0  ASin( x  L)

EI


Buckling never occurs if A  0
Thus m ust have:

P 
0  Sin L

EI


y
x=0 y=0
Elastic Buckling Theory – Ideal Column
The Buckling equation is :
x=L y=0


P 
P 
y  ASin x

BCos
x



EI
EI




For a Pin  Pin Colum n:

P 
m ust have: 0  Sin L

EI


The Sin function  0 at
P
 0,  ,2 ,3 ,..........n
n  num ber of half Sine waves
EI
n 2 2 EI
Thus : P 
in the buckled shape
2
L
Pcritical  Pcr  Lowest Load at which colum n m ay buckle
L
x
y
L
Pcr  Lowest Load, when n  1
Pcritical 
 2 EI
L2
x=0 y=0
Elastic Buckling Theory – Ideal Column
For a Pin  Pin Colum n:
n  num ber of
x=L y=0
P
 0,  ,2 ,3 ,...n
EI
half Sine waves in buckled shape
L
n 2 2 EI  2 EI
 2 EI
 2 EI
Thus : Pcritical 
 2


2
2
L 2 Leffective KL 2
L
n
L
Effective Length of Colum n Leffective   KL
n
KL  Leffective  Length of a Pin  Pin Colum n
 
1
K 
n
having the SAME Buckling Load
as the ACTUAL Colum n
1
n  num ber of half Sine waves
K  effective length factor
L
x
y
x=0 y=0
effective
 n
L2
2
KL 2
PB
PA
Half
Sine
Wave
LA=Leff
Pin
PA
LB
Fixed
Fixed
0.5 of
Half Sine
Wave
Fixed
Pin
PC
1
n
0.5LB=Leff
0.5LA
2
K
Half
Sine
Wave
LC
0.5 of
Half Sine
Wave
PB
PC
Fixed
n  1.0 K  1.0
n  2 .0 K  0 . 5
n  2 .0 K  0 . 5
Leff  1.0 LA
Leff  0.5( LB  LA )
Leff  0.5( LC  2 LA )
PAcr 
 2 EI
LA 
2
PBcr 
 2 EI
0.5LA 
PBcr  4 PAcr
2
4 2 EI

LA 2
LB  LA
PCcr 
 2 EI
0.52 LA 
PCcr  PAcr
2

 2 EI
LA 2
LC  2 LA
LA


 2 EI
0.5LA
L

 2 EI
0.5LC=Leff
Pcritical 
 2 EI
Pcritical 
 2 EI
L

2
effective

 2 EI
L n 
2
2

 2 EI
K
KL 
2
1
n
PE
Free
Pin
Half
Sine
Wave
LD
2LE=Leff
Half
Sine
Wave
LA=Leff
0.7071LD=Leff
LE =LA
Pin
0.5 of Half
Sine Wave
PD
PA
PE
Pin
PA
Fixed
0.414 of
Half Sine
Wave
PD
Fixed Fixed
n  1.0 K  1.0
n  1.414 K  0.7071
n  0.5 K  2.0
Leff  1.0 LA
Leff  0.7071( LD  LA )
Leff  2( LE  LA )
PAcr 
 2 EI
LA 
2
PDcr 
 2 EI
0.7071LA 
PDcr  2 PAcr
2
2 2 EI

LA 2
PEcr 
LD  LA
PEcr  0.25PAcr
 2 EI
2LA 2
0.25 2 EI

LA 2
LE  LA
Elastic Buckling Theory – Ideal Column
Pcritical 
 2 EI
L

2
effective

 2 EI
L n 
2
2

 2 EI
K
KL 2
x=L y=0
1
n
Deter min e Stress at Buckling Load :
L
x
Pcritical  1   2 EI
Stress   Critical _ Buckle 
 
Area  A  KL 2
Define: Radius of
 Critical _ Buckle 
Gyration  r 
I
A
y
 2 E  I   2 Er 2
  2 2
K L  A K L
 2E
 Critical _ Buckle 
for various end conditions
2
KL / r 
2 2
KL
 effective slenderness ratio
r
x=0 y=0
RADIUS OF GYRATION
B&J 8th, Section: 9.5
σ
σcrcr

 Critical _ Buckle 
 2E
 r
KL
2
 r 
  2 E

KL


2

P
A

BUCKLING= 
KL/r
rX 
IX
A
rY 
IY
A
r
I IN 4

 IN
A IN 2
rX = distance away from X-axis,
that an equivalent area should
be placed, to give the same
second moment of area ( Ix )
about X-axis, as the real area.
rY = distance away from Y-axis,
that an equivalent area should
be placed, to give the same
second moment of area ( Iy )
about Y-axis, as the real area
rX
I
rX  X
A
Y
X
X
Y
rY 
Iy
A
rY
Elastic Buckling – Ideal vs. Real Column
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
A
B
C
D
E
F
G
H
I
J
K
L
F-01-5
F-01-5
F-01-5
F-01-5
F-01-5
F-01-5
F-01-5
F-01-5
F-01-5
F-01-5
F-01-5
F-01-5
STEEL: Modulus of Elasticity = 2.90E+07 PSI
End
Column
Conditions Length
K
Column Column Column
Width Thickness Area
Moment
of
Inertia
STEEL: Yield Stress = 36
Radius
of
Slenderness
Gyration
Ratio
N
O
P
Q
F-01-5
F-01-5
F-01-5
F-01-5
PSI
Buckling
Buckling
Buckling
Stress
Load
Stress
Theoretical Experimental Experimental
%
Error
Moment Radius
Buckling
of
of
Slenderness
Stress
Inertia Gyration
Ratio
Theoretical
σcr
Y axis
buckling
(KSI)
P
σexp
(Lb.)
(KSI)
0.0548
0.0548
0.0572
0.0554
0.0557
111
147
194
271
323
23.43
13.25
7.64
3.91
2.74
INPUT
2950
1900
1100
500
400
20.56
13.21
7.41
3.43
2.74
-12%
0%
-3%
-12%
0%
0.006814
0.006868
0.006961
0.006996
0.006977
F-01-5
F-01-5
F-01-5
F-01-5
F-01-5
F-01-5
F-01-5
B
Y axis
H
Y axis
A
(in.)
(in.)
(in.)
(in.^2)
I
R
Y axis
Y axis
buckling buckling
(in.^4)
(in.)
INPUT
1.0
1.0
1.0
1.0
1.0
INPUT
6.0625
8.0625
11.0625
15.0000
18.0000
INPUT
0.755
0.757
0.750
0.759
0.757
INPUT
0.190
0.190
0.198
0.192
0.193
0.143
0.144
0.149
0.146
0.146
0.000432
0.000433
0.000485
0.000448
0.000454
F-01-5
F-01-5
F-01-5
F-01-5
F-01-5
F-01-5
Pin-Pin
M
KL/r
Y axis
buckling
(in./in.)
L
(exp-thry)/thry
KL/r
X axis
buckling
(in./in.)
σcr
X axis
buckling
(KSI)
0.2179
0.2185
0.2165
0.2191
0.2185
28
37
51
68
82
369.92
210.27
109.63
61.07
42.19
F-01-5
F-01-5
F-01-5
I
R
X axis
X axis
buckling buckling
(in.^4)
(in.)
Column Buckling, Pin-Pin, Steel, F-2001-Fall
Buckling Stress (KSI)
1
2
3
4
5
6
7
8
9
25
20
15
Theoretical
Experimental
10
5
0
0
50
100
150
200
Slenderness Ratio (KL/ry)
250
300
350
Elastic Buckling Theory – Ideal Column
Pcritical 
 2 EI
L

2
effective
PAcr

 2 EI
L n 
2
2
Pcr = 4PAcr

 2 EI
KL 2
for L=LA
K
Pcr=0.25PAcr for L=LA
1
n
Pcr = 2PAcr
for L=LA
L=LA
Leff=2L
Leff=0.7L
LA
L=LA
Leff=0.5L
L=LA
L=LA