Bölüm 10 Stabilite Burkulma
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Transcript of Bölüm 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,
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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)
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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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