NASA/TM-/'y_; _ 208121

BUCKLING ANALYSIS OF ANISOTROPIC

CURVED PANELS AND SHELLS WITH

VARIABLE CURVATURE

Navin Jaunky and Norman F. Knight, Jr.

Old Dominion University

Norfolk, VA 23529-0247

Damodar R. Ambur

NASA Langley Research Center

Hampton, VA 23681-0001

Presented at the 39th AIAA/ASME/ASCE/AHS/ASC

Structures, Structural Dymanics, and Material Conference

AIAA Paper 98-1772

Long Beach, California

April 20-23, 1998

BUCI(LING ANALYSIS ()F ANIS()Tlq()PIC

CUFI\:ED PANEL, AND SHELLS

WITH \:MqIABLI; CUlqVATURE

Navin .launk$'* and Norman F. Knigill., .lr. t

Old l)ominion |Tniversity

Norfolk. VA 7352!M)217

I)anio(lar l{. Anll)ur*

NAS:\ Langley I{ -sear(']l (!enter

]taml)l on, VA 23681 -[)001

Abstract

A Imckling formulation for anisolrol)ic

cm'v('d panels wMl variable curvature is l)r('-

sent.ed in this i)aper. The variable curvature

panel is assumed !o consists of two or niore panels

of constant but different curvatures. F/ezier func-

tions are used as Riiz functions. I)isplacenienl

((!()), and slope ((_l) continuities between s('g-

m(mt.s are imposed by manil)ulal ion of t.h(' [/ezier

COlit.r()l t)oiIIIS. A first-order shear-deforniation

theory is used hi the I)uckli,,g fl)rnit, lation, lie-

suits obtained froni the present fornil.llalion are

compar(,d wil h those from finite eleni,mt siniula-

lions and are found to tw in good agreement.

Nomenclature

u(/, t'c) displacement along axial and

i ra.nsverse direct!ions

w (lisplacelnent along radial

direction

0,,., Ov rotations of nornials tonliddle surface, or curvatures

x, g, z axial, circutnferentia], and

normal coordinales

J'i(,,u) Bezi('r polynomial of order ,

qij Bezier control points

*P,,si,'toct,:wal [lesea, rch Associate. Department ofAer_sl)aCe Engineering. Member AIAA.

t Profe.,,sor. 1)eparlnienl of Aerospace Engineering. A.',-sot!ate Fellow AIAA.

l As.,,isia.nt ftea,'l. Structural Mechanics Branch. As:.,,,-ciale l:elh,w A IAA.

l(!,,i)yrighl @1998 I)y Navin .launky. |)ublished byAIAA with l)ermission.

_., q non-dinlensional coordinah,s

L length of Sl_glllelll

,S'I widi/I of StPglllOIl| ]If iladius of Cllrval.llr('

Introduction

The use of couiposile nlalerials for aircrafl

_)riilial'y st.rucl.ures can i'eSlllt ill sigliilicanl tmn-

.Aii s Oil aircraft porf'oriiialice and sl.rlicl.llral cosl.

;gut'h al)lilicaiions of ('onlposiie inai.eriais are ex-

pocte(I to resu]l in a 30-40 l)ercent weight sav-

ings and a 10-3(l ])orclPll!, COSt. redllctioll COlll-

pared i.o COliVenlional lilelallic s! rilcl.ilreS. Shells

with variable Cllrvattlre are widdy used for air-

crafl tiiselage and wing conil)onents. The vari-

aide curvature configuration for these structures

is due to aero(lynalnic and fulwl.iona] consi(lera-

tions. |h'nce l.h(" understanding of I)uckling t)e-

hay!or of conlposite shells with variatile ctlrval ill'e

is of iinportallce for aerospace slrucl ural design.

The earliest work reliorte(I Oll tile bu('kliilg

analysis of shells with variable curvatllre was I)y

Margllerre [1] in 1951. Hi' exl>anded the curva-

lure in a Fourier series wilh resl)ect Io the cir-

CUlllferC'lll.ia] arc-length ill order I.o illOl-i:" ac('ll-

rarely represenl the varying curvature of wing

h,atlillg edge panels. His apl)roach has been lhe

basis for subsequent analyses of stlolls wii.h vary-

ing curvature for t)oi.h buckli,,g ([2]-[4]) and vi-

I)ratio,, i,,'ol,le,ns ([5]-[il]). h, these tiefere,,ces

Marguerre's approach has been used for elliptical

cylinders or panels since lhe variation of curva-

lure (l/R) (Fall Iw reliresenied accurately by ()ill-

t ernl ill the Fourier series. Reference [4] i)rovi(h's

_llloxc<,lh,xlt review for I)uckling and vilration of

shells and panels with variahh' curw_lule.

In addition to elliptical cylinders <r panels,

,,,I.'lls with varying curvature such as isotropic

conical ([12, 13])and torisphorical sh, lls ([13])

have been analyzed for t'rec vibration h'ie et

al. [12] usml the transfer matrix technique to

obtain natural frequencies of lruncated conical

shells. According to F_eference [12], the advan-

tage of tlw transfor matrix teclmique is its sim-

plicity compared t,<) other methods which require

c<msiderabh, aualyiical effort and coml.,qtationa[

tim<'. A review of t,ho literature for conical shell

sh,,ll buckling is provided in Reference [12]. Singh

([13]) used a segmenl apl>roach whcro a series of

circular arc SOgllWlllS tangential lo each other at

t.lw scg:nwnt intorfacc juncture is used Io modelt lw shell with varying cur'catm', (e.g., conical aml

t orispherical shell) withou! any appro,,+imation

in geometry. Trigonometric and quinli.' Bezier

polynomials are used to represent dislAtC,'ment

ti<'hl.', in each arc segment. Smooth d,'formed

surface, of the shell are obtained hy inil,<,sing (,II

and ('1 conliiittitics at l lie .iuncllire of l wo adja-

c<'nt shell segnienls I)y usiug l he prol>er'.i,<_s of lhe

th'zier control points. Also I_oundary comlitions

are applied hy simple ntanipltlalion of llic Bezier

conlrol points. B<_zier polynoniials I,a,,_+ heen

used in a segnienl alq)roa('h for the free v:bralion

of is<>tropic ellipl ic cylinders ([14]), coniposito el-

liptic cylinders ([15, 113]), and a laniinatcd con-

ical shell ([16]). A first-order shear-deformation

theory has h<'en used in Refi'reiices [15] and [16].

l)<'laih'd r<,vi(,ws of lh<" fro(, vibration of sh<,lls

wilh varial>le curvature are discussed in R<'Dr-

('noes [13]-[16]. Th(' Juethod l,resenl('d in R<'f-

ereii,',.s [l:',]-[l(i] cau I><' apl)lied t,o differem shell

geometries, shells with anisol t'OlfiC mat+-rial prop-

erlies, and shells with arbitrary hotmdary condi-

lions. In addilion, lho me/hod can incorporale

tirsl-ordcr shcar-deformatioll theory or ch_ssical

laniinated plate theory amt is coinl)utalioually

etficienl.

The presellt analysis method for buckliug of

atfisolropic shells with variable curvature uses

a segment approach where displacemenl fields

within each segment are represented by Bezier

polynomials aml a first-order shear-delbrmation

the,:)ry is used. In getwral, segnients can h<, used

in both axial and circulllferel+tia] direction;;, how-

ever the present iniplenientation considers only

segnlents in the circuntferential <lireclion. ('onli-

nuity of disldaconi<'nt at the ,illnClllres of ad.iaceul

.segm<,nts are inlposed using ( !0 and ( ,1 comlilions

ohlaiuiod froni l h" properties of t h, Bezi<'u' control

I>oinls ([14]). Tll,_ shell with variald<" cltrvalure is

assmu<'d to consist of two or mow curved panels

of constaiit CUl'Valure which is rel)reseiitalive of

l'uselage or wing slrll('l.llrC.s.

The present paper suniniarizes t h<, analy-

sis approach. Two structure cases wilh curva-

ture are analyzed lo dentonstrate the ca lml)ilitios

of the pres<'nt analysis al)l>roach. Result froni

l ho i)rosont alialysis are coiiiliarc<l witli t hoso ol+-

l ained [roiii tinile eiellielil aliaiyses.

Analytical Approach

The coordinate sysl.c+iii and the disl)lacomoni

directions for a uonch'cular shell is shown in Fig-

ut'o 1. Any I)oinl in the wall of the shell is speci-

fied by iiieaiis of curvilhi<,ar coordinat,e systeni .r.

!t aiid :, where x is the axial coordinate tixed to

iliid-surface..q i._ lht' circuiiiPerontial coordiliatewhich follows the iliediali line of ilie li'alisv<'rso

cross seclion, and : is (he radial coordinale 11oi'+

iiial io I_otli a' and !/. The nolicircular shell is as-

.SUllied i.o consisl of two or more .-;ogiiienls in ihecircuiuferontial direction each of colistant radhis.

The norina] alid l.aiigeill, of the two seglllelil.s al,

a jllllCltlrc' are equal as shoWll ill Figure 1, wllcl'O

ffl = if2 and /_ = l:,.

B<'zier polynoniials are iisc'd ill the axial and

circuniferentia] directions l.o r+,proseni t.he dis-

I>lacenient fields. The Bezicr l)olynoniial is given

t)y

#! !

fi( n+ u) = i/i--I (//__ 1),t-i+1

(i- l)!(n-i+l)!

(1)whore n denotes the or(h'r of the polynonlial and

0 __ v __ 1. For a Bezier polynoliiial of order ..

there are (,+ l) control poini+s. Any point Oll the

surface of t ho seglilelil is giv<'il by a parailioiricfuliclion of lhe foriii

X Y

P,.+(g,q) = ZZf,.(_)f,(,l)q,+., (2)r=l s=l

where the coordinates _ and q are defined as

= .,,+I L

,I = (v-v+) l (.q_+_ -!s+) (3)

with 0 < _, +/ < 1, X and "t are the nuliilwr

of control points in the axial and circulii['eren-

iial direction respectively, and q,.,, are t.ho Bezier

control pOilll,S or coefli('ienl.s. Tho disl)lacenient

'¢eclor ('all hi, mril.l.Oll as

[,,,, ,,,, ,,, <>.,0_ ]_' =

P,,_ 0 0 0

0 0 I% 0

0 0 0 ()

0

0

0

0 {qlrs }

q'_,rs

q3,.._

q4 r s

qs,',_ j

(,1)

where u, and t,_ at, ihc axial and transvers('

Inenlt)rane <lisl)lacemonis. rosl)oclively. ,. is lit<'

normal disl4ac('tnenl and o,. and o v are the cur-

vatures. Subscril>l j = I. 2, 3 .... (XY). Th<'

con(.rol l>oiii(s for ('ach <h'_t'<'e o[" freedoni can I)<'

used to inipos,' l)oundary cot,lit.ions on ('ach (h'-

gr('e of freedom on each s<'gnten(.

(%nl.inuil.y of (lisl)lac('n)<'n(. functions along

s(,gni(,n( .iun<'t ut'('s ilt'(, <)l)lain('d l)y using the r<4a-

t,ions l>(,lw<'<,n <'<)iil i'(>l l)<)inls of (.h(' adjacent s('g-

ni('n( l)as('d <)It ( '(> aml ('t continuities. Figur<'

;7 shows (.wo adjac('td s('_tii('nls and the cotitrol

poinls t,hat ar(' inv<)lv(.(l in l.h(' (" and (') con-

tiuuilies fc, r th(' ('as(. <)f eh'v(,)i cont.rol poin(s in

the axial dir('ctioN ainU ,-ix control points in tlt<'

(.ransvvt's<, direct ion_. i.,'...\ = II a)id _'" = d. hl

tlt<' I th segntcni. <'oil i<)l l)oini: -, q,_,; and qA-5 arc" )'('-

laled l.o control l)Oinl ,- %) aml q:, <)f' ih<" (I+ lit/,

s(,gnn'ni, wh,'r(' i. Z' = 1.2.3,1,.'),(i according to

q(i,4" 7-- ql;

.'4/ '1-,_" + :"#+1 q'_'_q.z, = (5)

,','/ + ,h)+]

wh('r(' ,k'/ and ?"/+l at'(' lit(' width of the ]th and

( 1 + 1 )th s('gtlt('lll , t'('Sl)('cl iv(.ly. USilig (hes(' COtl-

dilions (.h(' unkno,.vlm qli al),,l q'_,i at<' exl)ress('d

in l.erins of qa_. an(I q,_,. which are slaved to tit('

masl('r conlrol points qli ;lii(l q',i.

Silice i.h(" I)licMiiig, analysis involves first-

or(ler shear-deforliialion, olily (,0 ('otli, illtlily is

r('qtlh'('<t iit i,h<' variali()tial foritttlial, iOil. Ilow('v('r

l,h(' adwuliag(' of also iliil>osin g (,1 conl, inuil,y is

liOf only i,o ot>tain a tilor(' ac('iiral,<' analysis bill,

also to l'edtt<'e (.he siz(" of i.]I(' st.ill'<leSS and R('o-

tiler.rio stiff]less Iliat.t'i('('s whell a larger itlttlll)('l'

Of Segliietlt.S are tlS('d l.o r('iires('ll(, the shell thai

is I)eing analyzed. Tal)h' 1 shows the siz(' of tit('

niat, ri('es wilh lile nunll)er o[ seginenl,s for differ-

<'ill condilions of coni, inuil, i(:'s wlien ,\" = 11 and

Y = 6. If tit(" seglll('lltS aro joini'd t,o approxiniat,('

a shell, the size of i,h(' lnai, rices are less than thai

tbr a panel. The siz(' of die sl.iffnoss aiid geo-

tnei, ric sl, ift'lli'SS nlat, rices after assenil>ly is giv('n

})V

I,S'IZE" =

5xX x } × ,V,_;E(; - 110× 31JOI.V (6)

for ('() and (,i eonlinuiiies, wher(' ISIZI:" is (lie

niai,rix size, N._'I;'(; is lh(' lilltll])('l' o[ segtllelil,s,

and M,IOleV = N,b'E(;- 1 for a I)an('l and

31,]OI'V = N,S'E(; for a sh('ll, i.('.. M,IOI.V is

the nunll)er of junci, ur('s.

The [in('ar stiffness matrices are d('rived fi'om

(,lie strain energy which is given I)y

1f/,/, F't" "" (' 1. Lo 0 17<,J

(7)

wher(' Aij is l,he exlensional slill'lt('ss coeltici('ni

inairix, Hij is l,he COUl)ling siiffn<'ss coellici('n(

tnat,rix. I)ij is tit(' I)endilig si,iffness co<'t|ici<'n(

iltai.rix and ('I,<1 is t.ll(' i.raiisvers(' shear sl.i|l'll('Ss

co('fticien( niai,rix. Th(' strain vecl.or is {(} and

{<}r = ._,, <_>.(, . }I'

The strain dist)lac('ntenl relaiions ar('

_> Om)(j. --

_-).)..

I) (_)V0 tt'f)

<": = 7}u + I-_-

(1 Ou() Ovo

%'" = Otm[ + it.)---2

1;). i

77,1"

Oo v

i)fl

00.,. O0 v ("., , Ov() i)ul)

'<"" - o:/ + o.)_'.+ "7-i71_ o:/)

0tq)_() - o,- + --

/,r : --i) .r

l) <'tWo vl)_ :/ _ -_- my -F i __ ('1 -- (!))

Os/ I_

ller(" 171 and ('.) aro "'tracer" coe|ficienl.s us('(l l.o

iltilil('til('ti(, difl'<'l'<'n( st raiii-displaeenlent relal iOliS

or sh('ll ihi'ories. Accor<lingly wh(,tl ('1 = (2 i

1. tit(' first al)l)roxiination of Sanders-Koii,er shell

lh('ory [17. 18] is obl,ained and when ('1 : 1, ('.,

= 0. l,ove's sli('ll l,]teol'y [1!)] inchiding l,i'alisv('rse

sh('ar deforniaiionsisoblaine(I. Finally. when ('1

0 an(I (':_ = (), Donnell's shell theory [20] in-

cht(liltg I raliSv('rs<' shear (lefornial,ion is ol)l,ained.

Th, _ geometric stiffness malrix is derived

['rOlll [11_' work chine, (lilt), by the applied i)re-

Imckli,tg loading and is

tla = /( ._.rq,',VL + =\_V_,,X*-

+:%'.,'u%"u_\'/. )(I,I (l(I)

where l ll_' iiolllillear s[.raill ('Olll])Oll(ql[_ are

((,r)NL

( ( !1 ) N I,

(_,..,._,)x/.

= :j(_'.,5.+"',.; )

[ ,_ UO ,,

It'

= --tlll,y (t'(),y +--j_) -- I'll,. ,. /1(),,,,

+w,.,. (u',,, --_) (11)

In the present analysis the applied prelmckling

loading is prescribed as a uniform in-plane stress

stat.e. 'I'll,' linear stiffness aml geometric stiffness

matrices are developed using analyl.ical integra-

l ion rather lhan numerical inl.egration for com-

pul.al ional el[ici,'ncv. Finally, all eigenvalue prob-

lem is solved for determining the cri/.ical buckling

load.

Nulnerical Results

Resuhs are present.ed for a composite cylin-

drical panel subjected axial compression and a

COml)ositr wing leading-edge I)anel subjected to

('omlmwd axial compression and shear. ,_all(]ers-

Koiler ([17, IS]) shell theory is used in these st.ud-

ies. Buckling h)ads fi'Oln the present aualysis are

conq)ared with those ohtained from the STA(;S

([21]) ill,it,' element code. The STA(;S finile

elelll,.'lll model consist.s of the ,110 eJelllell|, and

curved surfaces are approximal.ed as all assem-

bly of flat surfaces. The nOlninal ply mechanical

properties for lhe composite material use(I are:

/'-'l 1 = 13.75 Msi: Ee'2 = 1.03 Msi; (;1._,=(;1a=(;.,3

= 0.420 Msi and ul'2 = 0.250, and l.he laminal.e

ply stacking st'(im'nce is [+,15/()/_.)0/+ 1.5]., with

equal ply thickness for dift'orent lalninate t.hick-

IleSS.

:ylimlrical Panel

The tirst structure analyzed is a semi-

circular (t_ = IS0 _-') cylindrical panel 22.0-in.

long, and with a radius of 40.0 inches, as shown

in Figure 3. The simldy-support boundary condi-

l.ions are also shown in Figure 3. The cylindrical

panel is modeled as live curved seglnents in the

present analysis while the STAGS finite elemenl

lllO(],!qed collsist.s o|' 20 alld 40 elelllellt.S ill lhe ax-

ial and transverse direction respectiwqy. Table 2

shows |he reslll{.s for the curved panel suhject.ed

to axial compression load.

The results ill Table 2 sugges! thai for I =

0.072 ill. the preSellt analysis result is 4.3 per-

cent greal.er than l.he S'I'AGS resull while for l

= 0.14,1 in. and 0.216 ill. lhe STA(_S analysis

results is 1.,1 I)ercenl helow thai of llw I)resenl.

analysis. The difference t)el.weell resuhs is due

1o using tilt' 410 shell element of STAGS which

do not. include transw'rse shear deh)rmation. The

STAGS results are above that of the presen! anal-

ysis for 1 = 0.144 in. and 0.216 in. since t,hese

two panels are thicker and trausverse shear de-

formation effects are siguiticanl. The buckling

mode shape for the curved panel oblained fronl

STA(;S analysis results is shown in Figure 4.

Wing Leading-Edge Panel

The wing leading-edge panel is shown ill Fig-

ure 5. It consists of tllree curved seglnenl.s of

radii 7)0.0 in., 6.136 in., and 50.0 in., respect.ively.

The boundary conditions of l, he panel is shown

in Figure 5 and correspond to classical simply

support conditions. Each ply of the laminate is

0.006-in. thick. Using lhe presenl analysis, the

wing leading edge was modeled as combination

of two segments for the ,r')0.0-in. radius section

and one segment, for t.he 6.13(i-in. radius sec-

t.ion. The STAGS finite e]entelll, model consists

of I he 4 I0 shell elenlelll, alld 30 × 30 elelllelll.S ill

each curved segment, and tile formulat.ion of the

410 element is based on lhe classical laminated

plate theory. Table _ shows the results obtained

from lhe present aqalysis and those from STAGS

for some select.ed combined load cases and Figure

6 shows l.he buckling load inleraction curve be-

t.ween axial compression and l)osil.ive shear load-

ing.

'File resull.s from t.he present, analyses are

aI_out 4.0 [)ercetll al)ove lhose of' STA(_S ex-

cel)l, for the case of negative shear loading where

the result from the presen! analysis is B.4 per-

cent above t.hat, of STAGS. I:_etl.er agreemenl can

be obt,ained from the present method by using

more control points in the axial direction. How-

ever this will lead to more ('Omlmtat.ional effort

and the present percentage difference between

STAGS and results from l,he present analyses is

considered acceptabh _ for a preliminary design.

The buckling mode shape for the wing leading

+'dg_+ ohlailwd fronl ST,,\(;S analysis result,_ are [3]showu ill l:igur(" 7, 8 and !1 for axial conlpres-

siou+ aud positive aud tiegalive shear, t'espec-

lix4._. The shell (h'rornialiou is uioslly in th<'

.+-){).()-ill+ I'.+l(JillS (-urvetl st'glllelltS. 12or the case of [6]

l,t>sitiv, ` ...,lit'at'. the shell tlel+Ol'lllaliOll is lllostly in

<m,' +d' th+' 7_().0-in. radius curv,,d Seglllelll,.

Concluding Remarks

.\ I',>rlLmlali(m ll;i.s bt'en d+wtqOl)ed f'<)r buck- [7]

liu_ <,t' aNi..<otrolfiC laudnal_,d _llt, lls with vari-

;ll,I,. ,'urv;Hure. The variabh, curvature is ap-

i)l',,xit,,_I,.,l by two or niore ,st'glllt'lll, o]' constaut

I>.t ,lill;'v,'lH curvat,lres. Bezier l)olyllomials are [8]u>.,.,I ;l.. I{il;] luilctioils in t ho .+.+lrHctural axial aud

Ir;,,,-x,.r..,. (lir_'c'lio]is. l)i:.;idac(,uit,tl t ((!u) and

",1,,I,, +('1) C(nltilltlily at'_' inil+O,'sed Iwlween seg-

lJpm- l_,'_llll. +- (>htaiHed rrom the forn]iflaliotJ

:it', x;ili,I;li+',l II;";ilig tillil.o el<'tilelit ,,-;iniulations. [9]

Ihickliii: I,,;.Is ohtailied |Yon)tinile eleltielil so-

lilt i,,li-- ;ilt <l+.i+'rlliiilOd to b<" roilr i)or('eiil Iowor

lit:ill Ill,,..,, _4 Ilia, lir_'selil, analysis.

Aeknowledgerrient

[i0]l h, w,,rk of t ht' tir,'-;l tw() aulhors was ,'-;UF,-

I" ,rl ,,I I,,, x_.\_A (lrani N A( l- 1-158,8, and is gral.e-

lllll_ ;i,l. it, ,w I(._l_'d.

References [1 l]

[I] K. _l;ir_llt'i'r+', "'Stability of the (_ylindri-

,;_1 _h,,ll ()1"V;triat)l+' (lurval, ul'e,'" NA_.,\ TM

I gll:L 1!)51. [17]

[7] ('h,'li. Y. N., and I{t'liil/il_'i" .]., "Largt'

l),'ll,'cl iltii or all Axially ( 1onil)l'eSsed Oval

('vliuilrical Shell," liro('l'ediug or 9 th Inter-

liiili<,llaJ (!ongress on Apllli('d Mechanics." [1:1]\lliilich. (Ii'rnialiy, l!)(j4, ,q,ln'higer-Verlag:,

It<'i'lili. I)1). 799-:I05, 1'¢)65.

[:I] .l";[l_,illtll;ill. I. alld l"i'ier, M., "'Buckling Anal-

ysi> ()r Laniinaied (!yliiidrical Shells with

.\rllil i'ill'y Noiich'cular ( os,'-; _ectioll," AIAA [14]

]o,r,aL Vo]. 37. Nu. ;L March 19.c)4, pl ). 648-

{ii;I.

[1] ('az'+d +,\. Meyers, +'An Analytical azid Ex-

I),,rilimmal luvestigation of Ellil)tical (!onl-

liosil, ' (!ylinders,'" Virginia Polytechuic In- [15]

slilul_ ' atid State lniversily, Ph. 1). disserta-

t iOit ill t']llgiiii'ol'ing ,_('it.li('+' and MeCllanic:-;,

:\pril l!)!)6.

lloyd, I). E., "'Analysis ot'Ol)_:'n Nc, n-('h'cular

('ylindrical Shell,,,," AIAA ,]0t+rna/. Vol. 7,

No. 3, 19($!t, pp. 51i:1-565.

Kurt, (!. I:7., aiid lloyd, I). li:.; +'Free Vil)ra-

lion of Non-circular (_ylindrical SIM1 St,g-

nienl.s,'" AIAA Jo.r.al. Vol. '¢1, No. _, 1971,

pp. 2:'19-'24,1.

(lull)t, rsoli, L. 1)., aiid Boyd. I). E.. "'Fret"

Vitiratious or Freely SUliport<+d Oval (!yliu-

ders," /IIAA Journal, Vol. 9, No. _. l!)Tl,

pp. 1,174-1180.

glsbernd, (',. !".. and Leissa, A. W., +q'h<'

Vibratious or ,%n-circular ,+-;llt'lls +','it h Initial

Slresst+s," ]oit+r+,,al of Hound a.d llbrallo++,

Vol. 29, No. 3, 1973, pp. 30'¢)-:12'¢),

(_heli. Y. N., alid KOliilHier, .I.. "Modal

Melhod for Freo Viliral, ion or Oval ('ylhi-

drical ,ghell,,s with ,ghnply Supported or

( _lalnped l_]iids,'" ,4,",'"lie ,7o, r#_'al of AF, pIH d

M(r/+aPt,s. Vol. 45. No. 1, l.qT& Pl). 1,12-148.

Kouniousis, V. Is:.. and Ariiienakas, A. E.,

'+lq'eo VibraliOll or Non-circular l>anels wilh

Shnply SUl)liOi'l<'d (lut'v(,d Edges," +4H('1¢

Jottrnal of ']++'.gw(+ ri,g .ll+Hta,.++,,.',, \'oI. 110,

No. 5, 1'¢)8,1, I)l ). 81()-b;27.

Koulnousi.,-;, V. Is:., and Arnioilakas, A. E.,

"l'_ree Vil)ral, iou of Siinl)ly SIIpl)ort.ed (!ylin-

drical Shells," AM<'I ,]ottrnal. Vol. :21, No.

7, l.qWl, pp. 1017-1(127.

h%, T., Yaliiada, (;., aild l(anpko. Y.. "t:r,'e

Vihralion of of a (lonical Shell with Varial)le

'l]licknes,'.¢," ,]()ttt+ttftl of ,%uud and ltbralton,

Vol. 8:2, No. 1. 1,¢t8:2, pl ). 8:l-'¢H.

Siugh, A. V.. +'()n Vii)rations of Sll_'lls

or tlevohltion ITsillg I:h,zier Polynonlials,"

A,<,'JlI7 ,]our_++a! ((f Pr¢.s.sur( l(.s,s<+/ 77ch_of

og.q, Vol. l l:l, Now'niber 1991, pp. 579-584

Kutiiar, V., and Shigh. A. V., "+Vii)ration

Aualysis o[' Nou-circular ('ylilidrical Sht'lls

U,sing Bezier Fllll('liOllS,'" ,]oitt'nfll of ,%'ottttd

(tild Yibralioll. Vol. 161, No. '2, 1'¢)'¢):L 171).

:13:1-354.

Kill)tar, V., and Siiigh. A. \*., "'Vibration of

(_onil_osile Nou-circular ('ylindrical Shells,"

A,_,'ME Journal (if Itbration al_d A+o,.st_c,s.

Vol. 117, Ocloher 1995, pp. 470-476.

[16] Kunlar, V., and Singli. A. V., "Vil_ralion of

Hber-Iteinforced Lainiliated Deep Shells,"

A/ME ,hmrTml oJ Pr_._,_ur_ Vc,_.s¢l 77Hluof

o qy, Vol. 11_, November 1996, 1)i). .107-41,1.

[17] Sanders, J. 1,..]r., "'An hnl)roved Firsl Ap-

proxinlatioll Theory for Thin Shells," NASA

lieport P/-74, 195.9.

[18] Koiier, %%'.T., "A (1onsisl,eni Pirsl Apt>rox-iniation in (tenera] Theory of Tliin Elas-

tic Sli(']l_,'" The Theory of "I'hin Elastic

Shells. t)roc('edhigs iUTAM _yliiposiuni

I)elfi, 1c)5c), lip. 17-33, 196(), Anisl,(,l'dani,

ih(' bi('lli('riands, Norlh-Holland Publishing

( _Oliilialiy.

[[!)] Love, A. E. 11., A Treaiise Oll i he

Mathenialical 'l'h('ory of Elasticity, 41h edi-

lion, New York, Dover Pubiicalion, 1977.

[20] 1,OI.), T. ']'., "'All ]_]xl,ellSiOll of l)onnell's

I']qnatic.n for (!ircular ('ylhidrical Shell."

•]oll#'ll(ll of ,'l(l'Olt(llllIctll ,S'('i£11Ct..S, VO]. 74,

1957, pp. 390-391.

[21] Brogan, F. A.. Ilankin. (' ('.. (!abiness,

It. I)., and _iYi]liani, A. I,., "S'I'A(;S User's

Manual," Lockheed Martin Missiles ,k" Space

(1olnpany hi(-., ilepori I, MMS(' 1'03759-1,Vel'SiOll 2.:i, i.{).()[J.

Table 1 Size: of slifflies.s nialrices for panel and

shell for increasing iilmiber of sC'gliielils.

N,b'E(; I,<,'I Z E l,_,'l Z f.

(l'anel) (shell)

[(,,>] [(.,,. (,,] [(,,,. (.,]

330 330

605 550

88O 77O

1155 99O

1430 1210

1705 1430

44O

660

8_(i

1100

1320

Table 2 (:onillarison of buckling loads resulls

for coniposile Clll'Ved panel.

Thickness STA(;S Present

I (hi.) analysis

(Ibslin.) (Ibs/in.)

0.(/77 :17.1.55 39l).68

0.1+1 1481.08 1,15,9.45

0.216 3328.25 327&86

Table 3 (1oinl>ari,sOli of buckling loads results

['or wing-binding e<lge t)anel.

Loading STAGS IJresent

<'ondil,ion analysis

N,. NrU (it)s/in.) (lbs/in.)

1.0 0.0

1.0 0.4

1.0 1.0

0.4 1.0

0.0 1.0

0.0 1.0

304.67 317.03

194.48 202.57

106.68 110.90

124.39 129.53

-117.73 -125.37

139.31 145.55

Figure'I:(:ooMinat_'sy_l_'mand g)_onwlryofshellwilh variahh,<'urvalur__.

q56

q66

q16

q26

Segment I

q5k

q 6k

qli

q2i

segment ( I+1 )

:>

X

q51

q61

_qll

_q21

Y

l"igur_, 2: (!o,ltrol points [br joining sh_,ll _egm<,nl._.

L = 22 in.

R = 40.0 in.

c_ = 180 o

lz V=W=0

Figure 3: (;eomelry and boundary condilions of curw,d coJlq)osilo panel.

Y

Figure 4: Buckling mode shape for curved composite panel.

R2

Z w=v=o

6.136 in._ a ____ _/__,/'_JX

16in.t__y R1 R1 W=U=O

14 in'LC)6 i_ I_ 16 i3_in._ F 14in.

R1 = 50 in. R2 = 6.136 in.

l:igur_" 5: (;_,om_q,ry and <limension,_ of COml)<)sil,e wing lea<ling-_'dge imm'l.

Z"

32o_28O

240

200

160

120

80

40

00

_-- -_=_---- l_esent analysis

STAGS

.... 210 .... 410 .... L .... I .... I .... I,,, ,___--_,-_60 80 100 120 1_0 160

Niy (Ibs/in.)

Figure 6: Buckling load inleract, ion curve for lh_' COml)OSite wing-leading edge' from'l,

Figt,re 7: Buckling mode shape for the composite wing leading-edge panel in axial conlpr('ssion

loading.

Figu,'e 8: Buckling mode shape for the composile wing leading-edge panel in negalive shear

loading.

10

I i'..:ur, !' I+u,'l.:ling Illod_' shal+_' for t.h,' composil+' wing l,'ading-<'tlg_" imm'l ill l,ositiv,' sh<'ar loading.

11