Buckling of Elliptical Plates Under

8/17/2019 Buckling of Elliptical Plates Under

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BUCKLING OF ELLIPTICAL PLATES UNDERUNIFORM PRESSURE

CONTENT

CONTENT................................................................................................................... 1

LIST OF FIGURES........................................................................................................ 2

NOTATIONS................................................................................................................ 3

1.0 Introducton......................................................................................................... !

1.1. G"n"r#$............................................................................................................!

1.2. T%" o&'"ct(") o* t%" +or, #r"-....................................................................... !

2. Lt"r#tur" r"("+....................................................................................................• L"+/ K. M./ #nd #n/ C. M. 124. 5E$#)tc &uc,$n o* r"ct#nu$#r 6$#t")

+t% cur("d nt"rn#$ )u66ort).5..............................................................................

• S%o,#/ 7. 184. 9On t%" Buc,$n o* #n E$$6tc P$#t" +t% C$#:6"d Ed" I;

• Liew, K. M., and Wang, C. M. (1993). “pb


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LIST OF FIGURES

1.3.1. FIG. 1. Buc,$n o* Int"rn#$$= Su66ort"d E$$6tc#$ P$#t" Su&'"ct"d to Un*or:

Nor:#$ Pr"))ur"..............................................................................................

1.3.2. FIG. 2. #r#ton o* Buc,$n Factors/ / +t% R")6"ct to A)6"ct R#to)/ / *or

S:6$= Su66ort"d #nd C$#:6"d E$$6tc#$ P$#t")..............................................

1.3.3. FIG. 3#4. Buc,$n F#ctor) *or S:6$= Su66ort"d E$$6tc#$ P$#t" +t% Int"rn#$

Ln" Su66ort P#r#$$"$ to M#'or A)................................................................10

1.3.!. FIG. 3&4. Buc,$n F#ctor) *or C$#:6"d E$$6tc#$ P$#t" +t% Int"rn#$ Ln"

Su66ort P#r#$$"$ to M#'or A)........................................................................111.3.. FIG. !#4. Buc,$n F#ctor) *or S:6$= Su66ort"d E$$6tc#$ P$#t" +t%

Conc"ntrc Int"rn#$ E$$6tc#$ Rn Su66ort....................................................12

1.3.8. FIG. !&4. Buc,$n F#ctor) *or C$#:6"d E$$6tc#$ P$#t" +t% Conc"ntrc

Int"rn#$ E$$6tc#$ Rn Su66ort......................................................................13

GLOBAL ACADEMY OF TECHNOLOGY. 2


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NOTATIONS



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1.0 Introduction

1.1. General

The elastic bucli!" o# cla$%e& elli%tical %lates u!&er u!i#or$ co$%ressio! 'as #irst

sol(e& b) *oi!o(s)+,rie"er -13/0. He use& the a)lei"h+it e!er") $etho& a!&

a%%roi$ate& the &e#lectio! sur#ace 'ith h)%erbolic+tri"o!o$etric series i! elli%tical

coor&i!ates. 4o$e u%%er+bou!& bucli!" solutio!s 'ere "i(e! #or &i##ere!t as%ect ratios 5 6 a/b,

'here a a!& b are the hal#+le!"ths o# the $a7or a!& $i!or aes8 res%ecti(el). About t'o &eca&eslater8 4hibaoa -19:0 &iscusse& the solutio! %roce&ure #or &eter$i!i!" the eact bucli!" loa&s

o# cla$%e& elli%tical %lates. Ho'e(er8 o!l) a! a%%roi$ate #or$ula "i(i!" the relatio!shi%

bet'ee! the bucli!" loa& a!& the elli%se ecce!tricit)8 e = √ (1−α ) 8 'as %rese!te&. The

#or$ula %re&icts accurate solutio!s 'he! the (alues o# ecce!tricit) are s$all; that is8 'he! the

%late is close to a circular sha%e. Laura a!& 4haha&) -1:0 use& the co!#or$al $a%%i!"

tech!i


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2. Literature review

• Liew, K. M., and an!, ". M. #1$$2%. &'lastic b(cklin! of

rectan!(lar )lates with c(rved internal s())orts.&

This %a%er co!si&ers the elastic bucli!" o# recta!"ular %lates 'ith cur(e& i!ter!al su%%orts.

@! co!trast to co$$o!l) use& &iscretiatio! $etho&s8 the bucli!" a!al)sis is %er#or$e& o! a

co!ti!uu$ %late &o$ai!8 i!clu&i!" its su%%orti!" e&"es. This is $a&e %ossible b) usi!" the

a)lei"h+it $etho&8 'hich e$%lo)s the !e'l) %ro%ose& %b+2 it #u!ctio!s. The %b+2 it

#u!ctio!s co!sist o# the %ro&uct o# a t'o+&i$e!sio!al %ol)!o$ial #u!ctio! -%+20 a!& a basic

#u!ctio! -b0. The basic #u!ctio! is &e#i!e& b) the %ro&uct o# e


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the ecce!tricit) o# the elli%tic %late is &eri(e& correct to the #ourth %o'er o# the ecce!tricit)8

'hich $a) be a&e


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3. ET!O"OLOG#

pb-2 7+L'3G?673T@ M'T?/>


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#C, 8% is tae! as the %ro&uct o# the elli%tical bou!&ar) e


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3.5.1.


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3.5.5.


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3.5..


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3.5.-.


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$.0 RESULTS

Bucli!" #actors 6 Na2 /D #or si$%l) su%%orte& a!& cla$%e& elli%tical %lates 'ith as%ect ratios

ra!"i!" #ro$ 1 I a 6 a/b ≤ = 'ere obtai!e& usi!" the pb-2 a)lei"h+it $etho& -tai!" % 6

130. Fi". 2 sho's the (ariatio!s o# the #actors 'ith res%ect to the as%ect ratios. Jsi!" the least+

s


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"/;"L>3;G 7'M7K*

The pb-2 it #u!ctio!s #or a%%roi$ati!" the &e#lectio! sha%e $ae the a%%licatio! o#

a)lei"h+it $etho& relati(el) eas) #or bucli!" a!al)sis o# elli%tical %lates 'ith (arious e&"e

co!&itio!s a!& a!) !u$ber a!& #or$ o# i!ter!al li!e?cur(e& su%%orts. The $etho& is (er) use#ul

'he! %er#or$i!" a! o%ti$iatio! stu&) o! the best sha%e a!& locatio! o# i!ter!al su%%orts #or

$ai$u$ bucli!" ca%acit). This is &ue to the si$%licit) o# the $etho& i! cateri!" #or the

i!ter!al cur(e& su%%ort8 si!ce there is !o !ee& to bother 'ith the coi!ci&e!t o# !o&es alo!" the

su%%orti!" cur(es as re


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)robles.& J. Engrg. Mech. Div., *"', $#1%, $6-.

Liew, K. M., and an!, ". M. #1$$2%. &'lastic b(cklin! of rectan!(lar )lates with

c(rved internal s())orts.& J. Struct. Engrg., *"', 11H#-%, 1FHI61F$5.

Liew, K. M., and an!, ". M. #1$$5%. "pb-2 7a8lei!h67it9 ethod for !eneral

)late anal8ses.& Engrg. Struct., 1#3%, 6-I.

*hibaoka, +. #1$-%. &/n the b(cklin! of an elli)tic )late with cla)ed ed!e 3.& J.

Phys. Soc. Japan, 11#1I%, 1IHH61I$1.

oinowsk86Krie!er, *. #1$5%. &The stabilit8 of a cla)ed elli)tic )late (nder (nifor

co)ression.& J.Appl.Mech.,F#F%,161H.



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Buckling of Elliptical Plates Under