Calculation of Stiffness :Matl'ices forFinite Elements of Thin Shells

of Arbitrary Shape

J. T. ODEN*

University of Alabama in Huntsville, Huntsville, Ala.

Introduction

l-'HE use of finite elements in the analysis of complex shellproblems has received considerable attention' in recent

years. Several authorsl-7 haw suggested that flat-plateelements be IIsed in the analysis of certain types of shellswhereas the conclusions of Fulton, Eppink, aud Walzs indi-cate that flat clements may not always adequately representthe beha\;or of a eun'ed structw·e. Conical shell clementsfor the analysis of shells of re\'olution were proposed by l\Ieyerund FIarmon9; Grafton lind Strome1o; Popov, Penzien, IUldLull; und Percy, Pian, Ii:lein, and NItVllratllll.II IImon~others.IS-IG A survey of the anlllysis of shelL,;hy finite cle-ments is contllined in the paper by Jones and StrOlllel1 llndin the book by Zienkiewiez and Cheung,18 where IIdditionalreferences can be found on the subject. Bogner, Fox, undSchmit19 presented interpolation formulas in orthogollalcurvilinear coordinates, which they used to obtain stiffnessmatrices for flat rectangular plates. In a recent, paper,Bogner, Fox, and Scbmitlll presented a stiffness matrix for acylindrical shell clement.

Heceived June 9, 1967; revision received JRnullry 2\1, l!J68.This work was sponsored by the Structures and l\Iat.erialBLabora-tory U.S. Army Missile Command, Huntsville, Ala., under COll-t.mct AMC-14897(Z).

• Professor of Engineering :Mechanics, Department of Engi-neering, University of Alabama Research Institute. MemberAIAA.

Reprinted/rom AlAA JOURNALCopyright, 1968,by the Ameriel\1lInstitute of Aeronautics aud A:ltronautics, and reprinted by pprmi""iolluf the copyright owner

970 AIAA JOURNAL VOL. 6, NO.5

Geometric functions:

Transposition matrices:

i = DJj = [~

0 0 ~]1 12 :I() ()

(5)

k = [~ 6J (5)

Fig. 1 CUl'Vilincul' finitc elcmcnt of U thin Ilhcll. Opel'lltors:

(8)

(7)

u = l~:!d = IB di)

1I'll!Y = 'Y12'Y22

Kinematic variables:

KinClllatic Considcrations

In this Note, thc development of gcncml stiffness matriceSfor finite clcmcnt.'! of thin shells of arbit1'llry shape is con-sidered. The equatiolls governing the differential geometryare cast in matrix form and formulas arc presented which, forgiven shell geometrics, lead to the corresponding stiffnessand consistent mass matrices for a finite shell clement. B)'choosing appropriate displacement approximations, it isshown that stiffness matrices for a numbcr of types of shcllelemcnts, including the cylindrical shell elcment, 1Il can heobtained as special cases.

'YaP = !(ua;p + Up;a - 2liabaP) (2)

Xall = - HUa;p + U Il;a + b,l'w .." + ba"wp,,) (3)

where 'Yap and Xap are the membralle strain and the curvaturetensors. respectively. If it is further assumed that stretchingof the middle surface can be ignored in computing changes incurvature (Donnell theory), then it can be showll23 that 'Y a~

and Xap are related to the components of displacement vectoru of the middle surface as follows:

(12a)U2 = al + allOI + al202 + al30102

where aI, ... , ala, aI, ... , al3 ure constants, and the normalcomponent Ua is represented by a generalized hermite inter-

The strain-displaeernent and cllrvature-displaccment rela-t.ions for an arbitrary shell can now be writt.en in the form

Displaccments and Slopes

In order to develop stiffness matrices for finite elements ofarbitrary shells, a generalization of the usual procedure ofapproximating the displacement field wit.hin the elementmust now be considered. The clements !t .. of the matrixU of Eqs. (8) are not the physical components of tbe dis-placcment vector; rather, theiw functions arc the displace-ment component.<;referred to thc buse vcot.ors ai, a2, and nof coordinate lines 0, I 8,2 und z (i.e., the covariant com-ponents). To appro:";mute these functions, 11 network ofcurvilinear quadrilateral finit.e clements of the form indicatedin Fig. 1 is used. The nodal lines of the elements follow sur-face coordinate lines oa = const. The matrices y and X ofEqs. (9) and (10) involve the first partial derivatives of ltl

and U2 and the second partial derivatives of U3. To insurecomplete continuity of deformations along the nodal lines,elements ttl and U'l are reprcsented by generalized La!;l'angiuninterpolation polYllomials in the curvilinear coordinates (Jaof the form

The submatrix - b in DI rcpresents t.he efTeets of initialcurvature and normal displaccments 011 the stretching of themiddle surface whereas the term -D"u in (10) representsthe contribution of the tangential displacements to thechange in curvature of the middle surface of the shell.

y = DIU (9)

lind

X = -D"u - D3u (10)

wherein

D1 = u~1 - bJ (lIa)

D2 = j~d (1lb)

D3 = t1bdkl 0] (lle)

(1)

(4e)

(4d)

(4a)

(4b)

Wap = HUa;/l - ull;a)

btf = a"abap

Ua;1l = (c>ua!c>Of1) - u"I'atr"

Ua = - [(C>Ua/c>oa) + ba"tt,,]

in which Ua are the covariant componellts of u and

[n these equations, the functions ra/l" are the Christoffelsymbols of the second kind for the surface, aaP(aap) are con-travariant (covariant) components surface metric tensor,and ba/l(b/la) are the covariant (mixed) components associatedwith the second fundamental form of the surface. Equations(1-4) are the fundamental kinematic relations for thin shellsof arbitrary shape. To cast these relationships in a formconvenient for numerical computation, the following matricesare introduced:

To begin the investigation, it is first necessary to recordcertain relations from differential geometry2l and the kine-matics of thin shells.22,2a In the following, indicial notationand the summation convention are used, with Greek indicesranging from 1 to 2.

The geometry of a thin shell of arbitrary shape is definedby a system of curvilinear sw-face coordinates 8a(a = 1,2)embedded in the middle surface and a normal coordinate z.The surface coordinates need not, be orthogonal. If it isassumed that the displacement field of the shell is linear illthe thickness coordinate z and that z remains normal to thedeformed middle sUlface (Kirchhoff-Love theory), then thestate of strain at an arbitrary point in the deformed shellis given by

l\IAY 1968 TECHNICAL NOTES 971

Assuming that the shell is linearly elastic, the strain energyof the finite element is

polation formula,19 which can be written concisely in theequivalent form

where F is a 3 X 24 matrix whose elemcnts are functions ofna.

It is important to note tiJut the derivatives aUa/bO" do not,in general, represent chang(',q in slopes of the shell. For ele-ments of generul curvature, the slopes are functions of alla/()Oaas well as UI,U'l and the coefficients UlJa and balJ. Ncvcl·the-less, full slope continuity is insurcd since continuity of til,u',Ua, and bua/()oa is provided in Eqs. (12). Note also thatit is necessary that the noclallines coincide with the coordi-nate lillcs Oa = const in order that inlerelcmcnt clisplacementand slope continuity be provided.

(20)

(23c)

(22c)

(23b)

(2211.)

(23a)

(22b)

[TO' ~JA = 0

[ -To ~JB = 0

.' [-liro ~JB = 0

a0 To001

1 b I ()0DI = 1'2 b01 '2 002

000 b02

020 0 ii(82)'

b~D2 = 10 ()

OorD02

020 0 '0(02)2

[" 0 ]TO b i)

D, = 4- 0~2 b01

0

I 1: ;.;. dT." 1. M'T = - pu"u y = - VI V2 v " 2

where 01 = 0, the angular coordinate, and 02 = z, the coordi-nllte measured along a generator of the eylinder. By intro-ducing Eqs. (22) and (23) into (6) and (19) along with dis-placemcnt functions of the form in Eqs. (12), a stiffnessIllatrix is obtained which is simillll' to that derived by Bogner,Fox, and Schmit.20 Stiffness matrices for a number of othertypes of shell elements. including flat rectangular plates,circular plates, cones, sphere.'>. etc. may be obtained in amanner sinlilar to that outlined previou.sly. The procedureis aL'lo applicable to surfaces of irregular geometry if elementsof the matrices A and B of Eqs. (5) are approximated byappropriate interpolation formulas.

Conclusion

References

1 Greene, B. E., St.rome, D. R., and "'iekel, R C., "Applica-tion of the Stiffness Method t~. the Anlllysis of Shell Struc-tures," Proceedings of the Avwtion Conference, American Societyo( Mechanical Engineers, Paper 61-AV-58, March 1961, ASl\1E.

I Best, G. C. and aden, J. T., "Analysis of Shell-Type Struc-

so that

mass densi ty ,

By appropriately defining the basic geometric: matriceslind operators of Eqs. (5-7), the preceding equations can beused to generatc stiffness and mass matrices for a wiele varietyof shell elements. For example, in the case of a finite ele-ment of a right circular cylindrical shell of constlmt radiusTo,

where a" is the velocity component of an arbitrary point inthe shcll and M is the consistent mass matrix for thc finiteclement:

(18)

(14)

(17)

(16a)

(lUb)

(15a)

(15b)

(12b)

u = Fv

3 3L L e..n(01)"(02)n

n=O m-OUa

I ~ 'A 1IV = - y'EydY = - vTKv2 v 2

Stiffncss Rnd Mass l\Iatrice"

Substituting Eq. (14) into Eqs. (9) and (10) gives

"l' = eIlv

X = 1\:v

tI> = D,F

1ft = (D2 + D,)F

The strain at an arbitrary point in the element is then

r = (c., - z1ft)v

wherc

where V is the volrmle of the element, E is the 3 X 3 matrixof ela.'ltic constants, and K is the stiffness matrix of the ele-ment. It is found that

K = f hcIJTEtI'(U)I/2d01d02 + 1- f ha1ft'E~(a)1J2dOld02A 12 A

(19)

Here h is the thickness of the shell and a = laa/JI = det A.Equation (19) represents the stiffness matrix of a finite cle-ment of a thin shcll of arbitrary shape.

A cOllBistent mass matrix for the clement can be obtaincdin a similar manner. If T is the kinetic energy and p is the

in which C .. n are constants. The functions defined in Eqs.(12) lead to an element with 24 degrees of freedom. Thesedisplacement polynomials are applicable to any type of shellgeometry and arc not restricted to orthogonal coordinatelines. It is emphasized, however, that Eqs. (12) are citedonly as an example of gencral displacement functions whichwill provide interelement compatibility; if continuity ofslopes across nodal lines is not to be maintained, any suitablepolynomial in Oland 02 can be used instead of Eq. (12b).

Let (vli,t'2i,Va;,I',,) denote tbe values of u;(i = 1,2,3), ({Jla,(J14,{Jaa,{J ... ) denote the values of alta/ana, and ('1I,'12,'1a,'1t)denote the values of a'ua!i)0Ia02 at nodes 1,2,3, and 4 of theelements. Then the 24 X 1 m!ltrix

v = It'II,t'2I, ... , (JU,{J21, ... , '11,'12,'la, 'It I (13)

defines the generalizcd displacements of the clement. Byevaluating Eqs. (12) and their derivativcs at cach node, weobtain 24 equations in the constanta ai, ... , e... in terms ofthe generalized displaccments. Upon solving these, thematrixu can be expressed in the form

972 AIAA JOURNAL VOL. 6, NO.5

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