49 Khan, Mushtaq a. and Herrrera, Justine (1)

8/21/2019 49 Khan, Mushtaq a. and Herrrera, Justine (1)

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ELASTIC BUCKLING OF BARS

Mushtaq A. Khan

and

Justine Herrera

Norfolk State University

700 Park Avenue

Norfolk, Virginia 23504

Tele!one 757"#23"$5%2

e"&ail &ak!an'nsu(edu


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Elasti Bu!lin" #$ Bars

B%

&r. Mushtaq Khan

&e'art(ent #$ Mathe(atis

N#r$#l! State Uni)ersit%

A*strat

)any stru*tures are *onstru*ted using girders, or +ea&s, and t!ese +ea&s defle*t ordistort under t!e influen*e of so&e eternal for*e( T!e +u*kling analysis of taered

*olu&n, in -!i*! t!e &o&ent of inertia of t!e *ross se*tion varies a**ording to a o-er

of t!e distan*e along t!e +ar, is *arried out +ot! analyti*ally and nu&eri*ally( )o&ent ofinertia ./ is aroi&ated +y a *ontinuous fun*tion of ( T!e resulting differential

e1uation is analyti*ally solved and t!e for&ula for t!e *riti*al load is derived( T!e sa&ero+le& is solved nu&eri*ally, using inite ifferen*e s*!e&e(

.t see&s to +e t!at t!e )et!od of eig!ted esiduals /) is a natural fit for t!is

ro+le&( T!e 6alerkin )et!od is used, sin*e t!is &et!od results -!en t!e -eig!ting

fun*tion is *!osen to +e t!e +asis fun*tion( T!e +ases fun*tions are for&ally re1uired to+e t!e &e&+ers of a *o&lete set of fun*tions( T!e re1uire&ent of *o&leteness allo-s

an alternative interretation of t!e 6alerkin for&ulation i(e( a *ontinuous fun*tion &ust

+e ero if it is ort!ogonal to every &e&+er of a *o&lete set( So t!e 6alerkin )et!od *an+e vie-ed as a s*!e&e in -!i*! t!e residual is for*ed to ero in t!e sense t!at it is &ade

ort!ogonal to t!e *o&lete set of fun*tions( T!is s*!e&e is used to derive t!e for&ula for

t!e *riti*al load for a taered *olu&n(

T!e results of +ot! &et!ods are *o&ared fro& a arti*ular ro+le& and it is deter&ined

t!at t!e *riti*al ressure value fro& t!e for&ula is -it!in t!e &argin of error of t!e values

of t!e finite differen*e s*!e&e( T!is 8ustifies t!e validity of t!e for&ula derived using6alerkin9s )et!od(


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Intr#+uti#n

)any stru*tures are *onstru*ted using girders, or +ea&s, and t!ese +ea&s defle*t or

distort under t!e influen*e of so&e eternal for*e( .n t!e eig!teent! *entury :eon!ard

;uler -as one of t!e first &at!e&ati*ians to study an eigenvalue ro+le& in analying

!o- a t!in elasti* *olu&n +u*kles under a *o&ressive aial for*e(


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Bars ,ith C#ntinu#usl% -ar%in" Cr#ss Seti#n

A *ase of *onsidera+le ra*ti*al i&ortan*e, in -!i*! t!e &o&ent of inertia of t!e *ross

se*tion varies a**ording to a o-er of t!e distan*e along t!e +ar, is *onsidered in t!is

reort(


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&eri)ati#n #$ Critial ressure F#r(ula Usin" Galer!in/s

Orth#"#nailit% C#n+iti#n

:et y Ax

l= = s i n

andy Ax

l2 = s i n

, +e t!e solution of t!e a+ove differential e1uations(

T!ese solutions satisfy +ot! t!e differential e1uations and all t!e +oundary *onditions(

6alerkin9s ort!ogonality *ondition for t!e -eig!t fun*tion sin// xl

xF = re1uires

t!at

0/@/A =

2

0

2

=2

=+ dxxFpydxyd

xEI

l

a/=

and

0/@/A 2

2

2

2

2

=+ dxxFpydxyd

xEIl

l

b /2

Using t!e eressions for =y , 0/xIa and 0/xF in e1uation /=, -e get

=+ 2

0

2

0

2

2

=

22

3

=2

2

0sin/sin/2

l l

dxl

x

l

EIAApdx

l

xx

l

IIEA /3

!ere using t!e integration +y arts, it is deter&ined t!at

2

222

0

2

4=%sin

ll

dxl

x

x

l

+=/4

and

=2

0

2

4sin

l

ldx

l

x /5

Using e1uations /4 and /5 in e1uation /3 and solving t!e resulting eression forp ,-e get t!e follo-ing for&ula for t!e *riti*al ressure

2

=22=

2

2

/4/

l

IIEIIEp

++=

/%

Using t!e eressions for 2y , 0/xIb and 0/xF in e1uation /2, and using t!e sa&e

ro*edure as a+ove, -e found t!e sa&e eression for t!e *riti*al ressure(


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Nu(erial S#luti#n

T!e a+ove &at!e&ati*al &odel governing t!e ro+le& is nu&eri*ally solved using

inite"ifferen*e )et!od(

0/ =2=

2

=+Pydx

yd

xEIa , or 20

l

x /7

Boundary *onditions to +e satisfied are00/ =y

02

/ = l

y

!ere

==2 /2/ Ix

l

IIxIa +

=

)et!od involving finite differen*es for solving +oundary"value ro+le&s rela*es ea*!

of t!e derivatives in t!e differential e1uation +y aroriate differen*e"1uotient

aroi&ations( T!e follo-ing aroi&ations at t!e interior &es! oints are used tosolve t!e a+ove +oundary"value ro+le&(

h

yyxy iii

2/ == +

=

and

2

== 2/h

yyyxy iiii

+ += for ni ,(((((((((2,==

T!e use of t!ese aroi&ation for&ulas in e1uation /7 results in t!e e1uation

02/2

2

===

=2 =+

+

+ + i

iiii py

h

yyyIx

l

IIE /#

A inite"ifferen*e &et!od -it! trun*ation error of order / 2hO results +y using

e1uation /# toget!er -it! a+ove +oundary *onditions(

or node ==i , -e get

( ) 02 2=== =++ yAypA /$


7/8

-!ere

+

= ==

=2

2=

/2Ix

l

II

h

EA /=0

or nodes 2=i t!roug! n , -e get02/

== =+++ + iiiiii yAypAyA /==

!ere

+

= 2

=2

2

/2Ix

l

II

h

EA ii /=2

or nt! node

( ) 022 = =++ nnnn ypAyA /=3!ere

+

= 2

=2

2

/2Ix

l

II

h

EA nn /=4

T!e resulting syste& of e1uations is eressed in t!e tridiagonal N C N Syste& -it! t!efollo-ing *oeffi*ient &atri(

+

+

+

+

+

PAA

APAA

APAA

PAA

APA

nn

nnn

2200000

20000

00020

000022

000002

===

333

22

==

T!e *riti*al ressure is t!e first ositive value of P for -!i*! t!e deter&inant of t!e*oeffi*ient &atri is ero(


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Results

Using )ale, -e evaluated t!e values of *riti*al ressure P for different n"values, to get

t!e follo-ing results for =50=l , $0= =I , =402 =I and 2$000=E (

Documents

49 Khan, Mushtaq a. and Herrrera, Justine (1)