buckling of elliptical plates under uniform pressure

8/17/2019 BUCKLING OF ELLIPTICAL PLATES UNDER UNIFORM PRESSURE

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Visvesvaraya Technological University

Technical seminar on

BUCKLING OF ELLIPTICAL PLATES UNE! UNIFO

P!ESSU!E


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CONTENTS

• Introduction• Objective of work

• Literature review

• Methodology

•

Results and conclusion• References


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Intro#$ction

• The buckling of elliptical plates under uniformcompression has so far been studied for clamped Prompted by the lack of studies on such plate shapaper attempts to provide further new buckling reelliptical plates without and with internal line/curvsupports

• The method employed views the entire elliptical pa single supercontinuum element, even with the pof complicated internal curved supports, by incorpthe support equations into the Ritz function.


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• The elastic buckling of clamped elliptical plates undercompression was rst solved by Rayleigh!Ritz energy

and de"ection surface was appro#imatedwith hyperbotrigonometric series in elliptical coordinates. $ome upbuckling solutions were given for different aspect ratiwhere a and b are the half!lengths of the ma'or and m

• To nd the solution procedure for determining the e#aloads of clamped elliptical plates. (nly an appro#imat

giving the relationship between the buckling load andeccentricity, e = is presented. The formula predicts asolutions when the values of eccentricity are small) ththe plate is close to a circular shape.

•


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O%&ectives o' the (or)

• To present buckling solutions for simply supported ellipticalto provide simple formulas for both clamped and simply sup

plates.

• To present new buckling solutions for elliptical plates with inlinecurved supports! the plate periphery may be either simpl

supported or clamped.


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Literat$re !evie(

*+ C+ "+ ,ang- an# K+ "+ Lie( .

•.The recently developed pb"# Rayleigh"Rit$ method %Liew an'((#) '((*+ will be used for the buckling analysis.

•.The method is designated as such because) for the Rit$ functi

employs a two"dimensional polynomial function %p"#+ and a function %b+ defined by the product of the internal support e,and boundary e,uations in which the latter are raised to poweor # corresponding to a free) simply supported) or clamped edrespectively.


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• s an appro/imate continuum method) it0

i. 1liminates the need of discreti$ation of support condi

means there are no 2boundary or support2 losses+

ii. 3oes not re,uire any mesh generation.

iii. reduces the number of degrees of freedom re,uired fo

solution by a considerable amount) making the methofor usage in a microcomputer.


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"etho#ology

• 4onsider a flat) thin) isotropic and elastic elliptical plate of cthickness h and aspect ratio a/b, as shown in 5ig. I. The platesubjected to in"plane uniform pressure 6) and is internally su

by some linecurved supports whose e,uations are specified. problem is to determine the buckling load of the elliptical platotal potential energy of the plate is given by0


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5I7. '. 8uckling of Internally 9upported 1lliptical :late 9ubjected to ;niform 6


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The transverse displacement surface may be parameteri$ed by0

the subscript r given by0


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To ensure direct satisfaction of the geometric boundary condit

preceding basic function ɸ* /0- y1 is taken as the product of th

boundary e,uation and the internal linecurved support e,uatio

where n < number of internal linecurved supports! j < e,uation

internal support! =< ' for a simply supported edge! and = < # for

edge. 5or generality and convenience) the coordinates will be nor


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pplying the Rayleigh"Rit$ method0

nd substituting %'+ "%>+ into %?+ yields0


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&here @


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!es$lt

•

8uckling factors @


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FIG+ 2+ Variation o' B$c)ling Factors- 3- (ith !es4ect to

As4ect !atios- 5- 'or Si64ly S$44orte# an# Cla64e#

Elli4tical Plates+


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• The preceding formula for damped elliptical plate is more gethe one given by 9hibaoka %'(D>+) which is only restricted towith ' A E A '.#D. The results furnished by 9hibaokaFs formu

by at most *C %at E < '.#D+ from the results given by %'>+ folimited range of validity.

• 5igs. *%a+ and *%b+ show the variations of the buckling factorrespect to different aspect ratios for simply supported and claelliptical plates) respectively) with an internal line support at

distance of '*b away from the major a/is. Interestingly) the bfactors of the clamped plates are appro/imately double that osimply supported counterparts.


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FIG+ 7/%1+ B$c)ling Factor

Elli4tical Plate (ith Interna

Parallel to "a&or

FIG+ 7/a1+ B$c)ling Factors 'or Si64ly S$44orte#

Elli4tical Plate (ith Internal Line S$44ort Parallel to

"a&or A0is


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• &hen the internal support lies on the major a/is %i.e.) G < -+of the buckling factors is indeed very close to two. It can als

that the buckling factor increases with respect to the aspect rdecreases as the internal line support moves away from the m

• 5igs. B%a+ and B%b+ give the solutions for elliptical plates witconcentric elliptical ring support whose aspect ratio is HE. T

buckling factors show sensitivity to the si$e of the elliptical

support. 5uture study may consider the optimi$ation of the rsupport aspect ratio so as to ma/imi$e the buckling capacity


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FIG+ 8/a1+ B$c)ling Factors 'or Si64ly S$44orte#

Elli4tical Plate (ith Concentric Internal Elli4tical

!ing S$44ort+

FIG+ 8/%1+ B$c)ling Factors 'or C

Plate (ith Concentric Interna

S$44ort+


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CONCLUING !E"A!KS

•

The pb-2 Rit$ functions for appro/imating the deflection shape make tapplication of Rayleigh"Rit$ method relatively easy for buckling analyelliptical plates with various edge conditions and any number and formlinecurved supports.

• The method is very useful when performing an optimi$ation study on tshape and location of internal supports for ma/imum buckling capacitydue to the simplicity of the method in catering for the internal curved ssince there is no need to bother with the coincident of nodes along the curves as re,uired in discreti$ation methods.

• The simple buckling formulas developed for simply supported and clamelliptical plates and the buckling design curves for internally supportedshould be useful to designers.


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!e'erences

'. Laura) :. .) and 9hahady) :. %'(>(+. 24omple/ variable theory andstability problems.2 J. Engrg. Mech. Div., 941) (D%'+) D(">?.

#. Liew) . M.) and &ang) 4. M. %'((#+. 21lastic buckling of rectangu

with curved internal supports.2 J. Struct. Engrg., 941) ''J%>+) 'B

*. Liew) . M.) and &ang) 4. M. %'((*+. "pb-2 Rayleigh"Rit$ method

plate analyses.2 Engrg. Struct., 'D%I+) DD">-.

B. 9hibaoka) K. %'(D>+. 2On the buckling of an elliptic plate with clam

I.2 . Phy. S!c. Japan, ''%'-+) '-JJ"'-('.

D. &oinowsky"rieger) 9. %'(*?+. 2The stability of a clamped elliptic

uniform compression.2 J.pp#.Mech.,B%B+)'??"'?J.


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T6 KO;N

buckling of elliptical plates under uniform pressure

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