1 Classes #9 & #10 Civil Engineering Materials – CIVE 2110 Buckling Fall 2010 Dr. Gupta Dr....

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Classes #9 & #10 Classes #9 & #10 Civil Engineering Materials – CIVE 2110Civil Engineering Materials – CIVE 2110

BucklingBuckling

Fall 2010Fall 2010

Dr. GuptaDr. Gupta

Dr. PickettDr. Pickett

Buckling = the lateral deflection of long slender members caused by axial compressive forces

Buckling of ColumnsBuckling of Diagonals

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 Column Deflections are small. Column has a Plane of Symmetry. Resultant of All Loads acts

in the Plane of Symmetry. Column has a Linear

Stress-Strain Relationship. Ecompression = Etension

σyield compression = σyield tension

σBuckle < (σyield ≈ σProportional Limit ).

Y

Y

E

σBuckle

Column Buckling Theory usesASSUMPTIONS 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, - no moment restraint Fixed Ends – no rotation allowed, - restraining moment applied

d

P P

Column Buckling Theory

An IDEAL Column will NOT buckle. IDEAL Column will fail by:

Punch thru Denting σ > σyield compressive .

Fracture

In order for an IDEAL Column to buckle

a TRANSVERSE Load, F,

must be applied

in addition to the

Concentric Uniaxial Compressive Load.

The TRANSVERSE Load, F, applied to IDEAL Column

Represents Imperfections in REAL Column

Pcr = Critical Load

Pcr = smallest load at which

column may buckle

F

P=Pcr

P=Pcr

P=Pcr

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 .

Three states of Equilibrium are possible for an Ideal Column Stable Equilibrium Neutral equilibrium Unstable Equilibrium

d

P=Pcr

P=Pcr

Column Buckling Theory – Equilibrium States

Stable Equilibrium

P<Pcr P<Pcr P<Pcr

P<Pcr P<PcrP<Pcr P=Pcr P=Pcr P=Pcr

P=PcrP=PcrP=Pcr

Neutral Equilibrium Unstable Equilibrium

P>Pcr P>Pcr P>Pcr

P>PcrP>PcrP>Pcr

Δ=small

F

F

FF

P>Pcr

P>Pcr

Δ=growsF F

Column Buckling Theory – Equilibrium States

Stable Equilibrium

P<Pcr P<Pcr P<Pcr

P<Pcr P<PcrP<Pcr P=Pcr P=Pcr P=Pcr

P=PcrP=PcrP=Pcr

Neutral Equilibrium Unstable Equilibrium

P>Pcr P>Pcr P>Pcr

P>PcrP>PcrP>Pcr

Δ=smallF

P>Pcr

P>Pcr

Δ=growsF

0Δ/L=

Pcr

P

0Δ/L=

Pcr

PIdeal Column

Real Column

Ideal Column

Real Column

0 Δ/L=

P Ideal Column

Real Column

Pcr

F

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.

Elastic Buckling Theory – Ideal Column

From Moment curvature relationship;

Pydx

ydEI

yPM

M

dx

yd

EI

M

CurvatureNegative

x

CCWatX

2

2

2

2

0

)(0

0

1

P

Pydx

ydEI

yPM

M

dx

yd

EI

M

CurvaturePositive

x

CCWatX

2

2

2

2

0

)(0

0

1

P P

P

PP

PP

P

M

y=(+)

M

x

y=(-)

M

M

x

Tension Tension

Ten

sion

Tension

Ten

sion

Ten

sio

nT

ensi

on

Co

mp

ress

ion

Co

mp

ression

TensionCompression

Compression


EI

PxBCos

EI

PxASiny

becomessolutiontheEI

PmThusEImP

mxBCosmxASinEImP

mxBCosmxASinPmxBCosmxASinEIm

Pydx

ydEIoSubstitue

mxBCosmmxASinmdx

yd

mxmBSinmxmACosdx

dy

mxBCosmxASinyAssume

Pydx

ydEICurvatureEitherFor

:

:0

)()(0

)()()()(0

0:int

)()(

)()(

)()(:

0:

2

2

2

2

2

222

2

2

2


EI

PLSinhavemustThus

AifoccursneverBuckling

EI

PLxASin

yLxat

B

BCosASinyxat

ColumnPinPinaFor

EI

PxBCos

EI

PxASiny

Pydx

ydEICurvatureEitherFor

0:

0

)(0

0

0

)0()0(000

:

0:2

2

x

y

L

x=0 y=0

x=L y=0


2

2

2

22

1,

:

.,.........3,2,,0

0

0:

:

:

L

EIP

nwhenLoadLowestP

bucklemaycolumnwhichatLoadLowestPP

shapebuckledtheinL

EInPThus

wavesSinehalfofnumbernnEI

PL

atfunctionSinThe

EI

PLSinhavemust

ColumnPinPinaFor

EI

PxBCos

EI

PxASinyisequationBucklingThe

critical

cr

crcritical

x

y

L

x=0 y=0

x=L y=0


factorlengtheffectiveK

wavesSinehalfofnumbernnK

ColumnACTUALtheas

LoadBucklingSAMEthehaving

ColumnPinPinaofLengthLKL

KLn

LLColumnofLengthEffective

KL

EI

L

EI

nL

EI

L

EInPThus

shapebuckledinwavesSinehalfofnumbern

nEI

PLColumnPinPinaFor

effective

effective

effective

critical

11

:

,...3,2,,0:

2

2

2

2

2

2

2

2

22

x

y

L

x=0 y=0

x=L y=0

nK

KL

EI

nL

EI

L

EIP

effective

critical

12

2

2

2

2

2

2

Half Sine Wave

Half Sine Wave

0.5 of Half Sine Wave


Pin

Pin

PB

22

0.1

0.10.1

A

Acr

Aeff

L

EIP

LL

Kn

ABAcrBcr

AA

Bcr

ABeff

LLPP

L

EI

L

EIP

LLL

Kn

4

4

5.0

)(5.0

5.00.2

2

2

2

2

ACAcrCcr

AA

Ccr

ACeff

LLPP

L

EI

L

EIP

LLL

Kn

2

25.0

)2(5.0

5.00.2

2

2

2

2

LA=Leff LB 0.5LB=LeffLC

0.5L

C=

Le

ff

Fixed

Fixed

Fixed

Fixed

PA

PB

PC

PAPC

0.5L

AL

A0.

5LA

nK

KL

EI

nL

EI

L

EIP

effective

critical

12

2

2

2

2

2

2

Half Sine Wave

Half Sine Wave


Pin

Pin

PD

22

0.1

0.10.1

A

Acr

Aeff

L

EIP

LL

Kn

ADAcrDcr

AA

Dcr

ADeff

LLPP

L

EI

L

EIP

LLL

Kn

2

2

7071.0

)(7071.0

7071.0414.1

2

2

2

2

AEAcrEcr

AA

Ecr

AEeff

LLPP

L

EI

L

EIP

LLL

Kn

25.0

25.0

2

)(2

0.25.0

2

2

2

2

LA=Leff

FixedFixed

Free

PA

PD

PE

PAFixed

LE =

L A

PE

Pin

LD

0.7071LD=Leff

2LE=

Le

ff

0.5

of H

alf

Sin

e W

ave


ratiosslenderneseffectiver

KL

conditionsendiousforrKL

E

LK

Er

A

I

LK

E

A

IrGyrationofRadiusDefine

KL

EI

AArea

PStress

LoadBucklingatStresseDeter

BuckleCritical

BuckleCritical

criticalBuckleCritical

var/

:

1

:min

2

2

_

22

22

22

2

_

2

2

_

x

y

L

x=0 y=0

x=L y=0

nK

KL

EI

nL

EI

L

EIP

effective

critical

12

2

2

2

2

2

2

RADIUS OF GYRATIONB&J 8th, Section: 9.5

A

P

A

Ir

A

Ir Y

YX

X

σcrσcr

KL/r

2

22

2

_

KL

rE

rKL

EBuckleCritical

A

Ir yY

A

Ir XX

ININ

IN

A

Ir

2

4

Xr

Yr = 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

= 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.

rX

rY

X

X

Y

Y

Elastic Buckling – Ideal vs. Real Column123456789

10111213141516171819202122232425262728293031323334353637383940414243444546474849505152

A B C D E F G H I J K L M N O P Q

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 F-01-5 F-01-5 F-01-5 F-01-5

STEEL: Modulus of Elasticity = 2.90E+07 PSI STEEL: Yield Stress = 36 PSI

Moment Radius Buckling Buckling Buckling % Moment Radius BucklingEnd Column Column Column Column of of Slenderness Stress Load Stress Error of of Slenderness Stress

Conditions Length Width Thickness Area Inertia Gyration Ratio Theoretical Experimental Experimental Inertia Gyration Ratio Theoretical

K L B H A I R KL/r σcr P σexp (exp-thry)/thry I R KL/r σcrY axis Y axis Y axis Y axis Y axis Y axis X axis X axis X axis X axis

Pin-Pin buckling buckling buckling buckling buckling buckling buckling buckling(in.) (in.) (in.) (in.^2) (in.^4) (in.) (in./in.) (KSI) (Lb.) (KSI) (in.^4) (in.) (in./in.) (KSI)

INPUT INPUT INPUT INPUT INPUT1.0 6.0625 0.755 0.190 0.143 0.000432 0.0548 111 23.43 2950 20.56 -12% 0.006814 0.2179 28 369.921.0 8.0625 0.757 0.190 0.144 0.000433 0.0548 147 13.25 1900 13.21 0% 0.006868 0.2185 37 210.271.0 11.0625 0.750 0.198 0.149 0.000485 0.0572 194 7.64 1100 7.41 -3% 0.006961 0.2165 51 109.631.0 15.0000 0.759 0.192 0.146 0.000448 0.0554 271 3.91 500 3.43 -12% 0.006996 0.2191 68 61.071.0 18.0000 0.757 0.193 0.146 0.000454 0.0557 323 2.74 400 2.74 0% 0.006977 0.2185 82 42.19

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 F-01-5 F-01-5 F-01-5 F-01-5

Column Buckling, Pin-Pin, Steel, F-2001-Fall

0

5

10

15

20

25

0 50 100 150 200 250 300 350

Slenderness Ratio (KL/ry)

Bu

ckli

ng

Str

ess

(KS

I)

TheoreticalExperimental


nK

KL

EI

nL

EI

L

EIP

effective

critical

12

2

2

2

2

2

2

PAcr Pcr = 4PAcr for L=LA

Pcr = 2PAcr for L=LA

Pcr=0.25PAcr for L=LA

L=LA

L=LA

LA L=LAL=LA

Leff=2L

Leff=0.5LLeff=0.7L

1 Classes #9 & #10 Civil Engineering Materials – CIVE 2110 Buckling Fall 2010 Dr. Gupta Dr....

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