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NATIONAL ADVISORYCOMMITTEEFOR AERONAUTICS

TECHNICALNOTE4237

GENERALINSTABILITY OF STIFFENED CYLLNDERS

ByHerbert BeckerNewYorkUniversity

WashingtonJuly1958

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NATIONAL ADV IS ORY COMM IT TEE FOR AERONAUTICSTECHN ICAL NOTE 11.237

GENERAL IN STABIL IT Y O F ST IF FENED CY L IND ER SBy HerbertBecker

SUMMARY

Theoretical buckling stresses are determined in explicit form forcircularcylinderswith circumferential and axial stiffening. Thelo adin gs are axi al com pres sio n, radial pre ssure, hydrostati cpressure,and tors ion . Ana lys es were co nfined to moderat e-length and long cylin-ders. The investigationwas basedupon the use of a form of Donnefltsequati on derived by Taylor which is applicable to ortho tropiccyl inders.The derivation of thisequ ation is pres ente d in this rep ort .

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2 NACA TN .237

cylinders, for which there isno boundaryinfluence.Detailsof bound-ary conditions for thedifferent loadings arediscussed in the pertinentsections below.

The inves tiga tio n is restr icte d to elas tic buckli ng. Lin ear th eoryis used inal l cases . Sect ion proper tie sof the frame s, sti ffeners , an dshee t ar e nomi na l, as de pict ed in figu re 1. Theac tu al pr op ert ie s tobe use d in des ig n are di sc uss ed in part VI of the Han dbo ok of Stru ct ur alSt abil it y(ref. 5). This han dboo k co nt ains a cri tical revie w of thefie ld of gen era l ins ta bi lit y. It pres en ts co mpa ri so ns of th eo ry an d. te stda tawh ich del ine ate the ut il it y both of the th eo re tic al re su lts pre sen tedhe re in an d of ot her the ori es wh ic h do not empl oy the ort ho tr opi c-she lltheory.

Asu mmar y of th e resu lts of th e inv est igat ion s ap pea rs in tabl e 1.Short di sc us sio ns of the an aly se s of ea ch cas e in ves tig at ed are als oincluded. The de ri vat ion s of th e gen er al in st abi lity st res ses are pr e-sented. in appendixes A to D.

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NACA TN 11 .2 373I.bending moment of inertia of frame including effective sheet, in.Is bending momentof inertia of stiffener including effective sheet,

J = J1 + J5Jr

distributedtorsional moment of inertia of frame, Ji/d, cu in.distributed torsional moment of inertia of stiffener,Cuin.

torsionalmoment of inertia of stiffener,Js torsional moment of inertia of frame, fl)k

bucklin coefficie nt f or unstif

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NACA TN231

p pressureloadingon cylinder, psiQx,Qy tran sverse shearson shellelement, lbR radius of cylinder, in.T torquelo adingon cylinder wall, in.t thi cknessof cylinderwall, in.t effective thickness ofcylinderwallin shear-buckledstate

tf distributed ar ea of fra me, A1/d, in.t5 distribute darea ofstiffener, As/b, in.u,v,w displacem entsin x-,y-,andz-directions, in.

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NACA TN 4-2377circumferential-compressive-buckling stress; also, generalcircumferential normal stress , psi- r

sh ear-buckling stress; al so, general shear stress, psi

HISTORICAL BACKGROUND

Early theoretical investigations into the buckling of stiffenedci rcular she lls wer e performed by FlUgge (ref. 1), Dschou (ref. 3), andTaylor (ref. 14.). Fili gge deriv ed the trio of linear equilibrium equa-tionsanalogous tothose for isotropic cylinders whichhave been utilizedby many inv estigators. Dschou solved these equations for stiffenedcircularcylinders under axial load.

Taylor derive d a differential equation for axially loaded ortho -tropic circular cylinders utilizing the seine approach as did Do nnellin obt ainin his well-known ei ht h-order artial di

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6N A C A T N ! i . 2 3 7THEO1ETICAL RESULTSThe theoretical buckling stresses for orthotropic cylinders under

various loadings are shown in table 1. Derivations of these expressionsare presented in the appendices. Buckling stresses were determined onthe assumption that the spacings of the longitudinal stiffeners and. thecircumferential frames were small enough to consider the cylinder to actas a uniform orthotropic shell. Effects of boundary conditions on thegeneral instability stresses for the various loadings are discussed inthe following paragraphs.

Axial LoadThe solution chosen by Taylor for the case of axial loading repre-sents the waveform assumed in the classical solution of simply supportedisotropic cylinders, in which linear theory is used. It is applicable

to both moderate-length and. long cylinders.

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N A C A T N 1 4 . 2 3 77th is len gthrangea satisfactory so lut ion to the pr oblem may be obta ine dal tho ugh boun dar y co nd itio ns arenot sati sfi ed for is otro pic cyli nde rs.The sa me situ at ion was as su med to app ly to sti ffe ned cy lin der s.The moderate-length solution satisfiesthe requirement thatw = 0on the bo und arie s but doe s not cor res pon d to vanis hin g mom ent sat th eseloca tio ns. How ever, the buc kli ng stressfor an isotropic mod era te-l eng thcyl inder in tor sio n is re lat ively insensiti ve to spe cif icat ion s on the selatter qu ant iti es, sin ce the re is less tha n 10-p erce nt dif fer ence betweenthe theoreti cal bu ckl ing str ess es fo r simp ly sup port ed andclamped edg es.The gene ral in sta bility behavio r of ortho tro pic cylinders is con-sidered to par all el the buc kli ng behavior of mo derate- len gth and lo ngisotropiccircularshells. Consequently, the same solution to the buck -ling equ ation and thesame simplifications in the mathematicsare assumed.to be applic able. The ma th ema tic s was simp lif iedbyassum ing th at theratio of circtmferential to axial wave length wasnegligible compared.with unity.

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8 NACA TN 11.2 37

APPENDIX A

DERIVATIONOF DIFFERENTIALEQUATION

ThederivationofTaylor'sdifferentialequationinwhichtheeffect s ofsheararid circumferentialnormal str ess are also includedispresent edbelow. Tbegeometricpropertiesofastiffenedcylinderareportraye dschematica llyinfigur e1.

Taylorderivedtheequilibriumequationfor anorthotropiccylinderbycombining thecompatibilityequationfor forces intheplaneofaplateelement withthat for equilibriumof forces normaltoa plateele-ment with initial curvatureinone direction. These twosituations aredepicted infigure2.

For equilibriuminthe"plane" of theelement

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N A C A T N 1 4 . 2 3 7 9aN-_u=_==X!E Et1'

Y

y B

I==G Gt

y

x

(3a)

(3c)

(3d)

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10 N A C A T N 1 1 . 2 3 7

If th e Airy str ess fun ction F is intr oduce d

=

(6)

thenequation (5) becomes

1 F 1 F 1wftf = Rx2 (7)

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M A C A T N ! i - 2 3 7II(

E

)[

GJ______+If-++

+

-S

+

1 i.

l( 2

____

E X

+2N

+ I ( w ) + .1 _.i = oR2 (9)

if itis assumed that p is constant. For an isotropiccylinder thiswil l reduce to Donnell' s equation for v = 0.

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12 NACA TN 237

AP PENDIXB

G EN ER A L IN ST A B IL IT Y U N D ER A )C EA L C O M PRE SSIO N

Taylor utilized, as the solution toequation (9),

W =Wmn Slfl mx sin fly

for theanalysisofaxialcompressivebucklingofanorthotropiccircularcylinder. For this case

=

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equation (12) becomesWhen t = tf = t = t a n d . 15 = If = t3/l2, then

1 and

N A C A T N 4 - 2 3 713wheretf I - t5If=

' f tSIf - (G2-tJ/E2)

- t5 tf 15 - (G2J/E2)- c

- (G2 J /E2)

The half-wave lengths are It/rn and. it/n in the longitudinal andcircumferential directions, respectively.

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N A C A T N

A PPEN D IX C

G EN ERA L IN ST A BILITY UN D E R E'ERN AL PRE SSUR E

For external-pressure loading, eq.uation (9) becomes

E _____1 [ GJ_____\tf Gt L axk E

a 2 a 2 \ Ix+NyW=0 (16)

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N A C A T N 2 37

and. consequently the pressure is found. In the form

pRn2 = I5rn1 +

rn2n2 + Ifn1 + /

2n2 +

R2a+ G

t5)

E

tf

or, with

= n/rn, the expression for the pressure becomes

15

(21)

-1= rn2/". +

+ If t32) +E

E (R2) (2 +

+ ts (22)

Moderate-Ingth Cylinder Under Radial Pressure

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16 NA CA TN 237

Actually m= and . n = where ). is thehalf-wave lengthinthe circumferential direction. Then equation (22) becomes

2fpR(at\(s G7- r) + +If)+ + E 2+\LJ\tf G t5 (25)For themi nim um integ ra lva lue of a in the solutio n ofeq ua-

tion (2 5), a = 1. Thenwith3 >>1 (where 3= L/?),which assumesthat the ax ial wave length ismu ch larg er than the circu mfe rent ial wav elength,

pR -- - + [R2(\2 p jU t6 (26)

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N A C A T N 1 4 . 2 3 717or= 5.51

1 / 1 4 .3

1

1 / 1 1 .

3/2

If Il\

5.53()

(Pf\ (RL()

) j

tf

J

For isotropic cylinders,3/14.

\1/2

E/1\ (t3ay =5.51 \12)

\R)

3/2

or

(30)

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18 NAC TN #237

(2 +132)2 + 12 ZL2m= + 132)2

where m = i i t / L . For hydrostatic pressure,2kp= (ffi2+132) 12ZL2m+2 2- 1 3 (2 + + 1 3 2 )2

When =1and2

ky=kp=13+j6

(35)

(36)

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NACA TN23719Long Cy linder Under Radial Pressure

For a long cylinder under radial pressure, the buckling mode corre-sponds to that of a ri ng, for which Lev y obtained the resul t

P = 3EIf./R3

(39)

With

pR

it follows, fromequations (39) an d (4-o), that

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23 N A C A T N 1 4 . 2 3 1

APPENDIXD

BUCKL ING OFC IRCULAR CYL INDERSUNDER TORSION

Moderate-Length CylindersFor shear loading of moderate-lengthcircular cylinders

N =tT=T/2R2l (2)Nx=Ny=O

Aswas shownby Becker and Gerard for isotropic cylind ersofmoderatelength, a useful solution for Tcr is obtained when the expression

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NACATN1423721whic h yields7 =

146(t5 \\3/81 I5/ 8 1/ 2

- )

)

( 1 4 5 )

This a gr eeswi ththe da ta of Stein, Sa nders, a nd Cra te for ring-stiffenedc ylindersintorsion( ref. 8 ) ,a tla rge va luesofZL .It reduc es to theisotropic solution for v = 0:

= 0.731(t/)5(R/L)h/2

The result ofequa tion ( 1 1 . 6 ) is 5.7 perc ent lower tha n the exa c t isotropicsolution obta ined by Ba tdorf for v = 0.3 13.

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22 NACA TN 11.2 37

27t - (:tf(n\2 + 3(tE \mR I \ ) \nL)

Upon minimization of r with respect to m and employing n = 2 forth e long cylinder,

3/11. 1/11.

T l. 754E( -L-(5\R2) rjThis reduces to the isotropicresult, with v = 0,

= O.272E( t/ R )3/ 2

( 1 1 . 9 )

(50)

(i)

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NACA TN 1 1 . 2 3 723REFERENCES

1. FlUgge, W.: Di e St abi lit t d.er Kreiszylind.erschal e. Ing .-Archi v,B d . . III, Heft5,Dec. 1932, pp. 11.63 -5 06 .

2. Donnell, L. H.: Stability of Thin-Walled Tubes Under Torsion.NACARep.11.7 9,19 33.3 . Dschou, Dji-DjU.n: Die Dru ckf estigkeit vers tei fte r zylin dri sch erSc hal en. Luftfalirtforsc hung, Bd. 11,Nr. 8, Feb.6,19 3 5,p p . 2 2 3 - 2 3 1 1 - .1 4 . Taylor,J. L. : Stabilityofa MonocoqueinCompression. R. & M.No. 167 9 , Brit ish A.R.C., 1935.5. Becker, Herbert: Han dbo ok of Stru ctu ral Stability. Part VI - StrengthofStiffened Curved Platesand Shells.NACA TN 3 7 86, 1958.

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TABLE1

THEORECALGENEALNSABILTYSE

FOR

ORTHOTOPCCIRCULARCYLNDES

Maeehcyns

Lcyns

12

+(G2E

a

r2+(G2E1

=

[2s+(E2G

Rs2s+E2G

11

32'fP/R\

=55E

)

)

ay3EPR2

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26 NACATN 237

N