general instability of an orthotropic circular cylindical shell subjected to a pressure combined...

7/27/2019 General Instability of an Orthotropic Circular Cylindical Shell Subjected to a Pressure Combined With an Axial Load

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'a TUDY O F THE STABILITY OFSUBJ ECTED TO VARIOUS TYP ES AND

COMBINATIONS OF LOADSREINFORCED CYLINDRICAL AND'CONICAL SHELLS

SECTION I - General Ins tabilityof anSubiected to a Pressure Combined with anAxial Load Considering Both Clamped and

Simply Supported Edge Conditions

5-@ft

Orthotropic C ircular Cy1ndrical Shell6

Carl C. Steyer, DirectorThomas A. C arlton J r., Associate

University of Alabama Research InstituteGeorge C. Marshall Space FlightCenterContractNo. NAS8 - 5012May 28, 1962 - October 15, 1962

SUMMARY REPORT November, 1962

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GENERAL INSTABILITY O FAN ORTHOTROPIC CIRCULAR CYLINDRICAL SHELL

SUBJECTED TO A PRESSURE COMBINED WITH AN AXIAL LOADCONSIDERING BOTH CLAMPED AND SIMPLYSUPPORTED EDGE CONDITIONS7-L>

BYCa rl C. Steyer

andTh0mas.A. Carlton, J r . '7z6v. / W x

Submitted toGeorge C. Marshall Space Flight Ce nte r

National Aeronautics and Space Administration

This report represents section Iof the summary report fo r contract number NAS8-5012F

Univer s ty of AlabamaUniversity, Alabama

November, 1962


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University of AlabamaBureau of Engineering Resear chand

Re sear ch Institute

GENERAL INSTABILITY O FAN ORTHOTROPIC CIRCULAR CYLINDRICAL SHELLSUBJECTED TO A PRESSURE COMBINED WITH AN AXIAL LOAD

CONSIDERING BOTH CLAMPED AND SIMPLYSUPPORTED EDGE CONDITIONS

C ar l C. Steyer,Y Ph. D. +++ndThomas A. Carlton, Sr. , Ph. D.

INTRODUCTIONJn a discussion of the instability of a ci rc ular cyl indrical shel l the

te rm "shor t" is generally applied to a cylin der whose length is approxi-mately equal to the radius.an ax ial load that produces instability, the effect of the boundary conditionsis no longer insignificant.References (1)and (2), the solution to the Donne11 type of diffe rentia lequ ationfo r an orthotropic circ ula r cylindrical shel l is found for simply supportededge conditions. The refo re, a definite need ex ists fo r the development of anexpression that wil l yield the buckling c ri te ri a in a form usable for de s ignerswhen a sho rt orthotropic o r s t i ffened cylindrical shel l i s subjected to a combi-nation of a pr es su re and an axial load and ha s boundary conditions o the r thansimply supported.

When a s h o r t cylindrical shell is subjected to

In the analyses presently available, such as-

In addition, fo r very thin circ ul ar cylindrical shell , with a radiusto th ickness ra t io g rea ter than 200, the so -call ed s m al l deflection or l ineartheory does not yield satisf acto ry agreem ent with experimental resu lts .

*P ro fe ss or of Engineering Mechanics, University of Alabama, Unive rsity, Ala.++Associate Pr ofe sso r of Engineering Mechanics, University of Alabama,University, Alabama.


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Non-linear analy ses based on an assumed deflection pattern, such as Refer-ences (3) , (4), and (5) , indicate better agree men t with te st data; but suchanalyses a s now constituted a re not applicable fo r design cr it er ia . In th ispaper a constant factor is included i n a radial displacement expression f o rthe purpose of linearizing cer tain energy te r m s that w e r e neglected inReference (2 ) s o that th ei r effec t may be studied.la r one is needed in instability shell studies for as presently constituted thethe inclusion of non-linear t e r m s in such studies r esu lts in such complexmathematical pr oced ures that the ir practical application is almost prohibitive.The r es ul ts of such attempted linearization a s mentioned above wi l l be pre-sented in another paper

Such an approach o r simi-

In or de r to develop the m ost complete analysis possible based onexisting s train -disp lacem ent information, the investigation w a s begun byusin g the bes t g ene ral theoretica l analysis and modifying and limiting it whenneeded a s indicated by the mathemat ical difficulties encountered. The theo-retica l approach used is si m il ar to the one used in Reference (2) whereinorthotropic shell analysis is applied to a stiffened ci rc ul ar cylinder; however,additional strain-displacemen t ter m s a r e included that w i l l later make pos -sib le the l inear ization study mentioned above.

In the present paper a s e t of instability equilibrium equations , s imi-l a r to those of Reference (2), a r e derived for an orthotropic c i rcu lar cy-lindr ical shell by applying variational methods to the expr essio n for th e totalenergy of the shell.ential equation of the Donnell type is obtained f or a cylin der of uniform thick-nes s subjected to a pre ss ur e and a compressive axial forc e. This differ-ential equation is solved for the c as e of simp ly sup port ed edge conditions,and a quadratic algebraic expression is developed th at y ields the buckling c r i -te r ia .par am ete rs by the use of a digital computer fo r the purpo se of establishingdesign cri teria.c a s e of clamp ed edge conditions, and a four-by-four determinant that yields

F r o m these equilibrium equations an eighth or de r differ-

This algebraic expression can be minimized quite readily for certain

The Donnell type differential equation is also solved for the

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the buckling c ri te ria is developed. Again it may be possible that design c ri -teria may be established by minimizing th is determinant for ce rtain par a-m ete rs by using a digital computer.

STRESS -STRAIN RELATIONS

Figure 1: Coordinate System and Displacem ents ofCir cu lar Cylindrical Shell

The ci rc ul ar cylindrical shell geome try employed, which is thesa m e as that of Reference (2), is shown in Fig ure (1) together with the c oordi-nate sys tem used and the corresponding middle-surfa ce displacements. Inte rm s of the shel l middle-surface displacements, (A , V , and d , the ex-pres sion s f or the buckling st rai ns in the shel l wall are the sam e as thosegiven in Reference s (2 ) with som e additional te rm s.relationships used are written as follows.

The strain-displacement

.Le,x = u, ) (+* L J p - 2 %&e,, = v , ~ -VQ & ( d , s+"/i)t ( L " J , , , t K W / P ) (1)

- \++ U ] S +(.y,s 4- %)y - w a p d s - W R - %ijX Swhere I< is a constant, 3 is the radius of the cylinder; ewr , s sers , a r e the axial, circumferential , and she ar s trai ns, respectively; anda com ma indicates differentiation with resp ec t to the succeeding variable.

, and

F o r a homogeneous orthotropic material , the str es s- st ra in re -lat ions in general ized plane s tr e ss can be wri t ten as follows.

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In the preceding equationsferential , and shear stresses, respectively; ir nd k Z a r e he values of themoduli of elasticity averaged over the thickness in the axial and c irc um -fere ntial directions, respectively; G is the average she ar modulus, andsd xnd &s a r e Po i sson l s ra t ios .

, GSL and G.: a r e the axial, c i rcum-

For convenience i n la te r calculations, th e following co nstan ts andnotations, si m il ar to those given in Reference (2), a r e introduced.

also defined.(4)

Based on Maxwell's rec iproc al the ore m, the following relationshipmust hold between the elastic constants.

EL3 = q L)-# (5 )The two expr essio ns foro(+ and % in Equations (3 ) a re the re su lt of theabove relationship.STRAIN ENERGY AND TOTAL ENERGY EXPRESSION

The instab ility diffe ren tial equations of equili brium wil l be derivedusing the sam e procedure as given in Reference (2). For an elas t ic sys tem ,a cri ter ion of buckling is that the variation of the change in the energy of the

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sy ste m due to buckling, with resp ec t to the displacem ents, must be zero.Describe d mathematically, this criterion becomes

i (u++)-= . (6 )w h e r e u is the change in the st ra in energy of the sh ell and i T is the changein the potential energy of the e xtern al forces during the buckling proc ess.

If initial bending stresses a r e neglectedUIIn the preceding e xpression G), Oj, , and GXSa re the membrane stresses

[ c%Axt-OI-: e*-,C;; c,S+ (a i*r,+ 6,s CA S +G;; C F S ] AV, (7)i, - - -existing in the sh ell in the com pres sed but unbuckled state ; Txxcss andcsathe volume of the s hel l wall.

a re the s tr es se s superimposed during the buckling pr oce ss, and Vs is

The st ra in ene rgy of the shell ca n be computed in te r m s of the bucklingstr ain s and the pre-buckling stresses by substituting Equation (2) into Equation(8) with the following r esu lt.

-If p designates a radial pressure , then R for the ca se ofPan external pressure; and n-.-= -.I f o r an internal pre ssure . The theore-tic al development wil l be continued for i&,-.-pR , and for th is ca se thechange i n the potential e nergy of the external forc es during buckling is givenby the following expression.

aI, 6

where is the middle surface area of the shel l wall.The total energy of a n orthotropic ci rc ul ar c ylindrical sh ell can then

be obtained in te rm s of the displacements and th ei r derivative s by sub sti-tuting Equations (1) into Equation (8) and adding the re su lt to Equation (9).A f t e r integrating ov er the s hell thickness and retaining only second or de rte rm s , the following expression fo r the total energy is obtained.

-

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EQUILIBRIUM EQUATIONS AND NATURAL BOUNDARY CONDITIONSRESULTING FROM THE APPLICATION O F VARIATIONAL PROCEDURES

Fr om Equation (10) the following expression is obtained for the v ar i-ation in the total energy of the sh ell af ter making the sub stitutions indicatedby Equations (3) .

Afte r using d A s = d r& , applying a well-known_ identity from thecalcu lus of varia tions, and integrating Equation (11)by pa rt s between thepro pe r lim its, the following expression is obtained for the v ariation in thetotal energy.

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The change in the total energy of a sys tem m ust vanish fo r any ofthe ar bi tr ar y virtual displacements JL ,in equilibrium.(12) hat a re multiplied by J & ,J r ,nd 6W , respe ctivel y, must vanish;and th e following equations of equilibrium a re obtained fro m this requ irement.

$\r, nd 4Ther efore the integrands in the s ur fa ce integral of Equation

when the system is

( d ~d f - D I / Z P ' ) k s x t &'W;A, , / p 0 (14)w r f m * / r ~ ) ~ t2 or x/n'-a,)3q--x i / p R t z r D . / R ' ) &s

- % 5 Y ; x /K f (7 -0CfJP) Y; f &y,,,/k -& t ,(.S / R =-

-2'85 W A S +Di W ~ MD ~ W ; S J S +(ZD3t2 )q )w)otr~d+c(, , /K(15)-

The following natu ral boundary conditions are obtained as a result of the re-quirement that the change in the total energy of the sy ste m r epres ented bythe line i nte gr als of Equation (12) must vanish for any of the virtual dis-placements o r thei r derivatives.

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DEVELOPMENT OF A DONNELL TYPE DIFFERENTIAL EQUATIONIn order to derive a Donne11 type of differential equation, Equations'

(13), (14), and (15) are written in the following manner.


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A l ine ar differential operator is defined as follows:

By successive differentiation and combination Equations (13a) and

expression.


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SOLUTION O F THE DONNELL TYPE O F DIFFERENTIAL EQUATION FORA LONG CYLINDER

The assum ption that the radia l buckling displacement has thefollowing fo rm r esu lts in a solution of Equation (21).

(23)In the above equation A is a constant and the dime nsionless par am ete r,), - t rn /L , is introduced. The above ex pre ssio n does not satisfy theboundary conditions for either a simply supported o r a clamped edge shell;th er ef or e, the following derivation should only be used on a long cylindricalshe ll i n which th e boundary condition effects a r e negligible..

The substitu tion of Equation (23) nto Equation (21) is a solution ofthe stabilit y equation provided a certain relat ion is satisfied. This relat ion-

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ship re su lts in the following algeb raic equation which is the eigenvalueequation of the stab ility di ffe ren tial equation.

In the preceding equation m and n a r e integers whose values govern thebuckled mode.

The following te r m s a re defined i n or de r to utiliz e Equation (24) fo rdesig n when a torque l- and an axial compressive force p a r e applied toan orth otrop ic cyl indric al shell in addition to a pressure p .

Since

Then

Th e subs titution of Equations (26) and (28) into Equation (24) resul ts1 1in the following quadratic e xpression in te r m s of the axial pressu re"

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In Equation (29) the buckled mode is described by the integer valuesof )n and n . Since the lowest c ritica l load is desired, the values used inthis equation must be chosen so as to minimize f - . in or de r to find itssm all est positive value that wi l l satisfy Equation (29).to minimizing the energy. The values of V and n for which f wil l be a

This is equivalent

minimu m in the pas t have b een obtained by the following methods. *1. A value fo r M was chosen based on experimental evidence and,

assuming n continuous, n was mathematically minimizedwith respect to f . O r a value ofly minimized.A t r ia l and er r o r procedure was used to determine the values of%Ind .

was chosen and M formal-

2.

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3 . Graphical methods w e r e used.F o r the above methods to insure accurate res ult s, in most cases the

labor involved is tremendous; however, the use of a digital computer reducesthis t im e to only a few minutes.that d eter mi nes the minimum positive values of f for the parameter s

Once a computer pro gram has been written,

K , , and / c z , this prog ram can readily be adapted fo r us e in developingdesign data. No interaction relationships or equations are needed with sucha program.

SOLUTION O F THE DONNELL TYP E O F DIFFERENTIAL EQUATION FORA SIMPLY SUPPORTED CYLINDRICAL SHELL

The assum ption that the r adial buckling displacement w has thefollowing fo rm als o res ult s in a solution of Equation (21)when %s. is zero.

The above expression satisfies th e boundary conditions on the W displacementfo r a simp ly supported cylin drica l shell. The substitution of Equation (31)into Equation (21) s a solution of the stability equation provided again thata certain relationship is satisfied. After considerable mathematical manipu-lation thi s relationship results in the following second d egr ee a lgebr aicequation which is the eigenvalue equation of the stability differential equation.

It is eas ily s ee n that the above equation can be derive d fr om Equation (29) bysett ing Kz equal to zero.Equation (29)can also be applied to the c as e of a simply supported cylindri-cal shell that is subjected to an axial com pressive for ce 2 combined witha radia l pressur e p .

Thus any procedure that m inimizes 8 f rom

SOLUTION O F THE DONNELL TYPE O F DIFFERENTIAL EQUATION FORA CYLINDER WITH CLAMPED EDGES

For this particular case Nu is again assumed to be zer o and thefollowing substitution is made in Equation (21).

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where

a ev =(d,, + K Z ~ , , ) C K ~L / R ~ )Equation (34) can be written a s follows:

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. where

Afte r re pea ted differ entiation and considerable alge braic m anipulation it canbe shown that

G 3 = 73,G-4. = E4

(44)(45)

-Since G is a function of -the va ria ble that re pr es en ts the c ir cu m -ferential direct ion, it must be a periodic function.of Equations (46), (471, (48), and (49) is given by tr igono me tric functions.

Therefore, the solution. . ,A s a re sul t the following values fo r the cons tants 73, , 8 2 ,a r e obtained.

, and 8, ,

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i , h A, / \< , - - h b / / t \vwid - , ? / L / g - - (50)

b - oa74,/Ri (53)

crb> - 3 AA / Rk 4-13' A,/&! - h y , ,/\C (51)(52)h$C\S/f(* - n=AY/T


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nant of the coefi cien ts of 6 9 B t 0 , B l , , and B I Z , must be ze ro or6 I 0 I

F ro m the p receding expr essio n the following eigenvalue equation of the sta b-i l i ty diffe rentia l equation is obtained.

In ord er to det erm ine the buckling load fo r a clamped edge cir cu larcylindrical shell subjected to a combination of a hydrostat ic pres sur e and anaxial compressive load, by use of the preceding analysis, a digital computerpro gra m needs to the wr itte n that established s e ts of values for K , andand positive i nte ge r values of )7 which satisfy Equations (59) and (64) andyield two negative values f or r z in Equation (59).value of f that sa tis fie s these conditions dete rmi nes the c riti ca l bucklingload.

The minimum positive

Calculations need to be performed f or comp arison with experim entalres ult s in ord er to es tablis h the validity of the preceding analys is.preceding analys is does not yield accurate com parison with expe riments, thena ge ne ral solution of Equation (54) needs to b e found that satisfies the geo-m etr ic boundary conditions given by Equations (61) and the nat ura l boundaryconditions given by E quations (16).

If the

-

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Constants that a r e functions of the ex ten-sional and s he ar s tiffne sses , the bendingand twist rigidities, the applied pre ssu re ,the axial load, and the applied torqu e

As Area of the middle surface of the shel lBending and t w i s t rigid ities of an eleme n-ta l ar ea of an orthotropic circul ar eyl in-dr ical shel l

D , ~ , , ,,e.srr; 5 , errAxial, eircum ferential and shearing strain

Moduli of e lastic ity for orthotropic ci rc u-lar cylindrical she l lFunction of the axial coordinate derivedfrom the ra dial displacementShear modulus fo r orthotropic circ ularcylindrical she llFunction of the Circum ferential coord inatederived from the radi al displacementWall thick ness of the she llPar am eter int roduced for the purpose oflat er studying the effect of previouslyneglected higher or de r energy term s.F o r the present paperRatio of the radial pressure to an axialload functionRatio of a torqu e function to an axia l loadfunction

E , , .CS

FGFh

krKr

L Length of the shell.WI Integer that in dica tes buckled mode in the

axial direction. n Integer that ind icate s buckled mode in the

circumferential directionAxial, circumf erential and shea r s tr e s sresu ltant s per unit length

? Radial or hydrostat ic pre ssur ec Axial load

c -Nu, &s ,

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uV

Function of the axial load ex pre sse d inpressure unitsFunction of the applied torqu e e xpr ess edin pressu re unitsMathematical operatorRoots of a n auxil iar y equationRadius of cylindrical shellAxial, circu mfe renti al and ra dia l coor di-nates of cylindrical sh ell middle su rf ac eApplied torqueAxial, circumf erential and radial d is-placements of cylin dric al she ll middlesurfaceChange in the st ra in energy of the shellduring the buckling pr oc es sChange in th e potential ener gy of th e ex-tern al forc es during the buckling p roc essVolume of shell wallExtensional and shea ring stiffne sses oforthotropic cylindrical she llPa ra m et er that defines the rat io of theradius of th e cy linde r to the length of thecylinderPoisson's r at ios f or orthotropic shel lAxial, circu mfe rent ial and shea rings t r e s s e s

REFERENCES(1) H. Be ck er and G. Ge ra rd , "Ela stic Stability of OrthotropicShells, ' I J. Aer ospa ce Sciences, pp. 505-512 May, 1962)(2) S. R. Bodner, 11General Instability of a Ring-Stiffened Circu-

la r Cylindrical Shell Under Hydrostatic P re ss ur e,Mech., pp. 269-277 June, 1957) J. Appl.

(3) T. Von Ka rm an and Hsue-Shen T si en , "The Buckling of Thin

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Cylindrical Shells under Axial C ompression , I t J. Aero.Sciences, pp. 303-312 June, 1941)


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(4) H. F. Michielsen, "The Behavior of Thin Cylindrical Shellsaf te r Buckling under Axial Com pression,pp. 739-745 December, 1948) J. Aero, Sciences,

(5) Y. Yoshimura, 1 1On the Mechanism of Buckling of a Circula rCylindrical Shell under Axial Co mpressio n, I ' NACA TM 1390(July, 1955)

general instability of an orthotropic circular cylindical shell subjected to a pressure combined...

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