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©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 1 INTRODUCTION TO COLUMN BUCKLING

Introductio to Column Buckling

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INTRODUCTION TO COLUMN BUCKLING.

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Page 1: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 1

INTRODUCTION TO COLUMN BUCKLING

Page 2: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 2

INTRODUCTION TO COLUMN BUCKLING

•Introduction•Elastic buckling of an ideal column•Strength curve for an ideal column•Strength of practical column•Concepts of effective lengths•Torsional and torsional-flexural buckling•Conclusions

Page 3: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 3

INTRODUCTION

• Compression members: short or long

• Squashing of short column

• Buckling of long column

• Steel members more susceptible to

buckling compared to RC and PSC

members

Page 4: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 4

INTRODUCTION

A “long” column failsby predominant buckling

A “short” column fails by compression yield

Buckled shape

Fig 1: “short” vs “long” columns

Page 5: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 5

ELASTIC BUCKLING OF EULER COLUMN

Assumptions:

• Material of strut - homogenous and linearly elastic

• No imperfections (perfectly straight)

• No eccentricity of loading

Page 6: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 6

The governing differential equation is

02

2

yEI

P

dx

yd cr .

x

y

Pcr

ELASTIC BUCKLING OF EULER COLUMN

Page 7: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 7

2

2

EI

Pcr

Lowest value of the critical load

ELASTIC BUCKLING OF EULER COLUMN

Buckling load Vs Lateral deflection Relationship

9

4

1

Unstable buckling modes

crPP /

Page 8: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 8

Conclusions of Euler buckling analysis:

• Strut can remain straight for all values of P

• When P = Pcr the strut buckles in the shape

of a half sine wave

• At higher values of loads, other sinusoidal buckled shapes are possible. It is seen that for for higher values of Pcr , the column is in

unstable equilibrium.

ELASTIC BUCKLING OF EULER COLUMN

Page 9: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 9

Mean compressive buckling stress, cr is given by

ELASTIC BUCKLING OF EULER COLUMN

2

2

2

2

2

22

2

2

)/(

E

r

ErE

A

IE

A

P

cr

cr

cr

Page 10: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 10

Elastic buckling stress(cr) defined by (2E/2 )

= /r

cr

Fig. 4 Euler buckling relation between cr and

ELASTIC BUCKLING OF EULER COLUMN

Page 11: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 11

STRENGTH CURVE FOR AN IDEAL STRUT

fy

y

Yield plateau

Fig. 5 Idealized elastic-plastic relationship for steel

Page 12: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 12

Strength curve for an axially loaded initially straight pin-ended column

B1f

f y A

c = /r

Plastic yield definedby ff = y

Elastic buckling (cr )defined by 2 E/ 2

AC

B

STRENGTH CURVE FOR AN IDEAL STRUT

Page 13: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 13

• Column fails when the compressive stress is greater than or equal to the values defined by ACB.

• AC Failure by yielding (Low slenderness ratios)

• CB Failure by bucking ( c )

STRENGTH CURVE FOR AN IDEAL STRUT

Page 14: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 14

f /fy

1.0

= (fy/cr)1/2

1.0

Elastic buckling

Plastic yield

Strength curve in a non-dimensional form

STRENGTH CURVE FOR AN IDEAL STRUT

Page 15: Introductio to Column Buckling

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FACTORS AFFECTING STRENGTH OF A COLUMN IN PRACTICE:

• Effect of initial out of straightness

• Effect of eccentricity of applied

loading

• Effect of residual stress

• Effect of a strain hardening and the

absence of clearly defined yield

point

• Effect of all features taken together

Page 16: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 16

Effect of initial out of straightness

F

y x

y0

a0

Pin-ended strut with initial imperfection

P

x

ay

sin00

P enhances the deflection by the factor

)(1

1

crPP

Page 17: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 17

O O1

P

Pcr

PpPy

Pf C

D

Actual elastic-plastic response

Curve A

Curve B

Initial imperfection (a0)

Load deflection responseof a strut with initial imperfection

Ideal bifurcation type

buckling

Effects of imperfection(elastic behaviour)

Strength(plastic unloading curve)

Stress distribution at D

M

M

Stress distribution at C

fy

Stress distributions at C and D

Page 18: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 18

Strength curves for strut with initial imperfection

Lower bound curve

Data from collapse tests (marked x)

Elastic buckling curve

f

fy

= /r

X X

X X

X

X X X

X

X X

XX

X X X

X X X

Strut

P

P

Low slenderness ratios effect of initial imperfections is negligible

Intermediate slenderness ratios lower bound curve is below fy and cr curves

High slenderness ratios lower bound curve is close to the cr curve

Page 19: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 19

Effect of eccentricity of applied loading

X

X X X X

X X

X

X

X X

X

X X

Strength curve for eccentrically loaded columns

Deflectedshape afterloading

P

P

Axis ofthe column

e

Data from collapsetests

Elastic bucklingcurve

Lower bound curve

f

fy

Behaviour is similar to that of initial out of straightness

Difference is noticed in the reduction of load carrying capacity for stocky members even for low values of

Page 20: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 20

Effect of residual stress b b

a a

b b

a a b b

aa

Various stages of rolling a steel girder

(a) (b) (d)(c)

• Residual stress differential heating and cooling during rolling and forming

• Self equilibrating system of stresses

Page 21: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 21

Residual stresses in flanges

Residual stressesin web

Residual stresses distribution (no applied load)

Residual stresses in anelastic section subjectedto a mean stress a

(net stress = a +r)

The influence of residual stresses

Page 22: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 22

a

fy

p

r

Stub column yieldswhen a = fy

av

Mean axial stress vs mean axial strainin a stub column test

Page 23: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 23

f

fy

fy - r

Elastic critical buckling

Columns with residual stresses

(E/fy)1/2 = /r

Buckling of an initially straight columnhaving residual stresses

The difference between buckling and plastic squash load is most pronounced when

21

yf

Er

Page 24: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 24

Effect of strain hardening and absence of clearly defined yield point

Strain hardening athigh strains

fy

Stress-strain relationship for Steels exhibiting strain hardening

• Ignoring the effect of strain-hardening provides a margin of safety

Page 25: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 25

fy

p

a

fy

p

a 0.2% proof stress

0.2%

Lack of clearly defined yield Lack of clearly defined yield with strain hardening

• Above >p, the material exhibits non-linear

behaviour• When the yield point is not defined, the yield

stress is generally taken as 0.2% proof stress

Page 26: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 26

Effect of all features taken together

a

fy

/r

Data from collapse tests

Theoretical elastic buckling

Lower bound curve

(E/fy)1/2

Page 27: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 27

CONCEPT OF EFFECTIVE LENGTHS

Po in t o f in flection

l

B u ckled m o d e fo r d ifferen t en d co nd itio ns

2 l

l/2 2/l

Page 28: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 28

Effective lengths in different planes (No sway and sway columns)

Columns with partial rotational restraint

P

e

P

P

e

P

P

e

P

e

(a) (b) (c) (d)

No swaye always

Swaye always

Page 29: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 29

ACCURACY IN USING EFFECTIVE LENGTHS

Page 30: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 30

Torsional and Torsional-Flexural Buckling of columns

Flexural buckling Torsional buckling

Folded plate twists under axial loadPlate with unsupported edges

Twisted position

Original position

Page 31: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 31

+C1

X1

O

Y1

+ CX

O

Y

Y0

X0

u

v

Torsional -flexural buckling deformations.

C’

Shear centre

Rayleigh-Ritz energy method is used to obtain the-Ritz energy method is used to obtain thecritical loadcritical load

Page 32: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 32

BUCKLING MODES

In general there are 3 buckling loads, i.e. Euler

buckling about x and y axes and flexural

torsional buckling loads

Doubly symmetric section

• Buckling about x and y axes (One of

these is lowest)

• Flexural torsional buckling load (we

disregard this)

Page 33: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 33

BUCKLING MODES

Singly symmetric sections

• Euler buckling load about weak axis

• Flexural Torsional buckling load

Page 34: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 34

CONCLUSIONS

• Elastic buckling load dependent on the

slenderness ratio

• Factors affecting column strength ( viz. residual

stresses etc.) considered in design practice

• ‘Effective length’ concept of columns

• Elastic torsional and torsional-flexural buckling

Page 35: Introductio to Column Buckling

©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 35

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