Bölüm 10 Stabilite Burkulma

8/9/2019 Blm 10 Stabilite Burkulma

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Mukavemet IIMukavemet II

SStrength oftrength ofMaterials IIMaterials II

Teaching Slides

Chapter 10:

Mukavemet IIMukavemet II

SStrength oftrength ofMaterials IIMaterials II

Stabilite (Burkulma)

Stabilit (Buckling)


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Stabilit (Buckling)


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Stabilit (Buckling)

Chapter OutlineChapter Outline

Stabilit of Structures !uler"s #ormula for $in%!nded Columns

!&tension of !uler"s #ormula to Columns 'ith

ther !nd Conditions !ccentric oading* the Secant #ormula

+esign of Columns under a Centric oad

+esign of Columns under an !ccentric oad


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Stabilit (Buckling)FIGURE Buckling demonstrations,


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Stabilit (Buckling)


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Stabilit (Buckling)

In #ig, a, the ball is said to be in stable equilibrium because- if it isslightl displaced to one side and then released- it 'ill mo.e back

to'ard the e/uilibrium position at the bottom of the .alle,""


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If the ball is on a perfectl flat- le.el surface- as in #ig, b-it is said to be in a neutral-equilibrium configuration, If

slightl displaced to either side- it has no tendenc to

mo.e either a'a from or to'ard the original position-

since it is in e/uilibrium in the displaced position as 'ell

as the original position,


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THE IDEAL PIN-ENDED COLUMN EULER !UC"LING LOAD

To in.estigate the stabilit of real columns- 'e begin b

considering the ideal pin-ended column, as illustrated in #ig,10:


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e make the follo'ing simplifing assumptions:

The column is initiall perfectl straight- and it is made of linearlelastic material,

The column is free to rotate- at its ends- about frictionless pins*

that is- it is restrained like a simpl supported beam, !ach pinpasses through the centroid of the cross section,

The column is smmetric about thexy plane- and an lateral

deflection of the column takes place in thexy plane,

The column is loaded b an a&ial compressi.e force P applied

b the pins,



2o'e.er- if a load P=Pcris applied to the column-the straight configuration becomes a neutral-

equilibrium configuration- and neighboring

configurations- like the buckled shape in #ig,

10,3b- also satisf e/uilibrium re/uirements,Therefore- to determine the .alue of the critical

load- $cr- and the shape of the buckled column- 'e

'ill determine the .alue of the load P such that the

(slightl) bent shape of the column in #ig, 10,3b isan e/uilibrium configuration,


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Stabilit (Buckling)

#irst- using the #B+ of #ig, 10,3c- 'e get

Ax=P (fromFx=0),

Ay=04from MB=05- and By=0,

Therefore- on the #B+ in #ig, 10,3d- 'e sho'

onl a .ertical force P acting on the pin atA- and

'e sho' no hori6ontal force V(x) at sectionx,

E#uili$rium %& the !u'kle( C%lumn


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Stabilit (Buckling)

The sign con.ention adopted for the

moment M(x) in #ig, 10,3d is a

positi.e bending moment, #rom #ig,

10,3d 'e get


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Substituting M(x) into the moment%cur.ature e/uation-

'e obtain elastic cur.e e/uation as belo':

Di&&erential E#uati%n %& E#uili$rium) an( En( C%n(iti%n*+

)()()(" xPvxMxEIv ==

v

EI

P

EI

xM

dx

vd==

)(2

2

or

or

0)()(" =+ xPvxEIv


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Stabilit (Buckling)

This is the differential equation that go.erns the

deflected shape of a pin%ended column,

It is a homogenous- linear- second%order- ordinar

differential e/uation,

The boundary conditions for the pin%ended member are

0)0( =v 0)( =Lvand


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Stabilit (Buckling)

The term !x" means that 'e cannot simpl integrate t'ice to get

the solution, hen #$ is constant- there is a simple solution to this

e/uation,

It is an ordinar differential e/uation 'ith constant coefficients, #or

the uniform column- therefore- it becomes

,%luti%n %& the Di&&erential E#uati%n+

EI

P=

2 0" 2 =+ vv 0" =+ vPvEI


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Stabilit (Buckling)

The %eneral solution to this homogeneous e/uation is

xCxCxv cossin)( 21 +=

e seek a .alue of and constants of integration &'

and &(such that the t'o boundar conditions of this

homogeneous e/uation are satisfied,


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Stabilit (Buckling)

Thus-

00)0( 2 == Cv

0sin0)( 1 == LCLv

b.iousl- if the constants &'and &(e/ual to 6ero- the

deflection (x) is 6ero e.er'here,

e 7ust ha.e the original straight configuration,


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Stabilit (Buckling)

#or an alternati.e e/uilibrium configuration- since &'can

not be 6ero- sin(8'))must be 6ero, So- the c*aracteristic

equation is satisfied 'ith &'+0,

( ) ,....2,1,0sin === nL

nL

nn


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Stabilit (Buckling)

Combining the e/uations abo.e gi.es the follo'ing

formula for the possible buckling loads:

2

22

L

EInP

n

=

EI

P=

2


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Stabilit (Buckling)

The deflection cur.e that corresponds to each load Pn

is obtained b combining the follo'ing !/uations,

( ) 0sin =Ln

xCxCxv cossin)( 21 += and


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Stabilit (Buckling)

'e get ( )

=

L

xnCxv sin

The function that represents the shape of the deflected column

is called a mode s*ape, or buclin% modeThe constant &- 'hich determines the direction (sign) and

amplitude of the deflection- is arbitrar- but it must be small,


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Stabilit (Buckling)

The e&istence of neighboring

e/uilibrium configurations is

analogous to the fact that the ball

in #ig, 10,9b can be placed atneighboring locations on the flat-

hori6ontal surface and still be in

e/uilibrium,


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The .alue of P at 'hich buckling 'ill actuall occur is

ob.iousl the smallest .alue gi.en b !/,

(n=1),Thus- the critical load is2

22

L

EInP

n

=


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The critical load for an ideal column is kno'n as the Euler

$u'klin l%a() 'ho 'as the first to establish a theor of

buckling of columns,

eonhard !uler (10;1


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Stabilit (Buckling)


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Stabilit (Buckling)

et us e&amine some important implications of the

!uler buckling%load formula, e can e&press this in

terms of 'riti'al (buckling) *tre**.

( ) 22

/rL

E

e

cr

=

r

Le=

2

2

Ecr=


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Stabilit (Buckling)

2

2

Ecr=


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Stabilit (Buckling)

#I=>?!:Critical stress .ersus slenderness ratio for structural steel columns,

#or slenderness ratios less than 100- the critical stress is not meaningful,

2

2

Ecr=


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Stabilit (Buckling)


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Stabilit (Buckling)

@ A%m%long pin%ended column of s/uare cross

section is to be made of 'ood, @ssuming #19 =$a-

all1A M$a- and using a factor of safet of A,D in

computing !uler"s critical load for buckling-

determine the si6e of the cross section if the column

is to safel support

a) a 100%kE load-

b) a A00%kE load,

E/AMPLE


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Stabilit (Buckling)

>sing the gi.en factor of safet- 'e make

a0 F%r the 122-kN L%a(.

in !uler"s formula and sol.e for I, e ha.e


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e check the .alue of the normal stress in the column:

?ecalling that- for a s/uare of side a- 'e ha.e - 'e 'rite

Since is smaller than the allo'able stress- a 100&100%mm cross

section is acceptable,


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Stabilit (Buckling)

Sol.ing again buckling e/uation for I- but making no'PcrA,D(A00)D00 kE- 'e ha.e

$0 F%r the 322-kN L%a(.

The .alue of the normal stress is


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Since this .alue is larger than the allo'able stress- the dimensionobtained is not acceptable- and 'e must select the cross section

on the basis of its resistance to compression, e 'rite

@ 190&190%mm cross section is acceptable


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Stabilit (Buckling)

E/TEN,ION OF EULER4, FORMULA TO

COLUMN, 5ITH OTHER END CONDITION,

!uler"s formula 'as deri.ed in the preceding section for

a column that 'as pin%connected at both ends,

Eo' the critical load Pcr'ill be determined for columns 'ith

different end conditions,


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Stabilit (Buckling)

In the case of a column 'ith one free end

A supporting a load P and one fi&ed end

B (#ig,%a)- it is obser.ed that the column

'ill beha.e as the upper half of a pin%

connected column (#ig,%b),

C%lumn 6ith One Free En(


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Stabilit (Buckling)

The critical load for the column of #ig,%a is

thus the same as for the pin%ended column

of #ig,%b and can be obtained from !uler"s

formula b using a column length e/ual to

t'ice the actual length ) of the gi.en

column,


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Stabilit (Buckling)

e sa that the effectie len%t* )e of the column is e/ual to A)

and substitute )eA) in !uler"s formula:

2

2

e

cr

LEIP =

( ) 2

2

2

2

42 L

EI

L

EI

Pcr

== )eA)


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Stabilit (Buckling)

The critical stress is found in a similar 'a from the formula

( ) 22

/rL

E

e

cr

=

The /uantit )e.r is referred to as the effectie slenderness ratio of

the column and- in the case considered here- is e/ual to A)Fr,

r

Le

= 22

Ecr=


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Stabilit (Buckling)

Consider ne&t a column 'ith t'o

fi&ed endsA and B supporting a

load P (#ig, 10,10),

C%lumn 6ith T6% Fi7e( En(*


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Stabilit (Buckling)

The smmetr of the supports

and of the loading about a

hori6ontal a&is through the

midpoint & re/uires that the

shear at & and the hori6ontal

components of the reactions atA

and B be 6ero (#ig, 10,11),


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It follo's that the restraintsimposed upon the upper halfA&

of the column b the support at

A and b the lo'er half &B areidentical (#ig, 10,1A),


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$ortionA& must thus be smmetric about its

midpoint /- and this point must be a point of

inflection- 'here the bending moment is

6ero, @ similar reasoning sho's that the

bending moment at the midpoint # of the

lo'er half of the column must also be 6ero

(#ig, 10,19a),


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Since the bending moment at the ends

of a pin%ended column is 6ero- it follo's

that the portion /# of the column of #ig,

10,19a must beha.e as a pin ended

column (#ig, 10,19b), e thus conclude

that the effecti.e length of a column 'ith

t'o fi&ed ends is )e =).(

2

2

e

cr

L

EIP

=

( ) 42/ 2

2

2

2

L

EI

L

EIP

cr

== )e).(


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Stabilit (Buckling)

hat is the ma&imum compressi.e load that can beapplied to an aluminum%allo compression member of

length )=G m if the member is loaded in a manner that

permits free rotation at its ends and if a factor of safet

of 1,D against buckling failure is to be appliedH

E/AMPLE


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Stabilit (Buckling)


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Stabilit (Buckling)


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Stabilit (Buckling)

$hsicall- the effecti.e length of a column is the distance


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Stabilit (Buckling)

bet'een points of 6ero moment 'hen the column is deflected in

its fundamental elastic buckling mode, #igure 10,19 illustrates the

effecti.e lengths of columns 'ith se.eral tpes of end conditions,


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Stabilit (Buckling)


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Stabilit (Buckling)


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Stabilit (Buckling)

2o'e.er- it ma happen that- as the

load is applied- the column 'ill bucle*

instead of remaining straight- it 'ill

suddenl become sharpl cur.ed(#ig, 10,A), $hoto 10,1 sho's a

column that has been loaded so that it

is no longer straight* the column has

buckled, Clearl- a column thatbuckles under the load it is to support

is not properl designed,


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Stabilit (Buckling)

EULER4, FORMULA FOR PIN-ENDED COLUMN,


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Stabilit (Buckling)

ur approach 'ill be to determine the

conditions under 'hich the

configuration of #ig, 10,A is possible,


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Bölüm 10 Stabilite Burkulma