6259 December, 1968

Journal of the

ENGINEERING MECHANICS DIVISION

'2f;j

EM6

'fiJf

Proceedings of the American Society of Civil Engineers

FINITE-ELEMENT ANALYSIS OF THIN SHELLS

By Gerald A. Wempner/ J. Tinsley Oden,z A. M. ASCEand Dennis A. Kross3

INTRODUCTION

The piecewise approximations of displacement fields within finite elementsmust be continuous at inter-element boundaries (1-8).4 Moreover, such ap-proximations must include rigid-body motions and must lead to possibleuniform strain states. If these continuity conditions are fulfilled, then the ap-proximation converges to the continuous field as the network is refined.

The foregoing criteria are easily satisfied in the analysis of three-dimensional and plane bodies, but they complicate the analysis of Kirchhoffplates and shells. Inter-element boundaries of a plate or shell have the req-uisite continuity only if the normals defining such boundaries remain continu-ous. Since the Kirchhoff hypothesis constrains the normal to remain straightand normal to the middle surface, the continuity can be achieved only if thedeformed surface is smooth, i.e., without corners at inter-element boundaries.It follows that an acceptable approximation of the transverse displacementmust have continuous derivatives at the inter-element boundaries. A polyno-mial approximation in a rectangular element requires, at least, a six-degreepolynomial and the physical element has 24 degrees-of-freedom (8). Simplerapproximations can be used for triangular elements.

Despite these difficulties, an extensive literature exists on applications ofthe finite element method to Kirchhoff shells. Several authors have suggestedthe use of networks of flat plate elements to analyze certain types of shells(5,9-12); conical shell elements for the analysis of shells of revolution werepresented byMeyer and Harmon (13), Grafton and Strome (14), Popov, Penzien,Lu (15), and others (17-19); Bogner, Fox and Schmit (20) presented a stiffness

Copyright 1968 by the American Society of Civil Engineers.Note.-Discusslon open until May I, 1969. To extend the closing date one month, a

written request must be flied with the Executive Secretary, ASCE. This paper is part ofthe copyrighted Journal of the Engineering Mechanics DIvision, Proceedings of theAmerican Society of Civll Engineers, Vol. 94, No. EM6, December, 1968. Manuscriptwas submitted for review for possible publication on February 12, 1968.

1Prof. of Engrg. Mech., Dept. of Engrg .. Univ. of Ala., Huntsville, Ala.2 Prof. of Engrg. Mech., Dept. of Engrg., Univ. of Ala., Huntsville, Ala.3 Grad. Research Asst., Dept. of Engrg., Univ. of Ala., Huntsville. Ala.4Numerals in parentheses refer to corresponding items in Appendlx l.-Heferences.

1273

1274 December, 1968 EM 6 EM 6 FlNITE-ELEMENT ANALYSIS 1275

KINEMATICS

A particle, initially located by the position vector, r, is located by a positionvector, Ii, in the deformed configuration; similarly, the position vector, r, of a

Vector fields associated with the motion of the shell are to be referred tothe triad, ~, or the reciprocal triad, ai, in which

at = r,t .. (20)

and af . aj .: 6J . . . . . . . . . . . . . . . . . . . .. .. (2b)

Here 6J denotes the Kronecker delta and the comma denotes partial differen-tiation with respect to xi(e.g., r,j = arjaxi). Th~ vectors, aa(a = 1,2) are

tangent to the coordinate lines, xa. Scalar products formed from these basisare given by

aa . a3 = 0 .. . . . . . . . . . . . . . . . . . . . . (30)

a3 • a3 = 1. . . . . . . . . . . . . . . . . . . . . (3b)

acr . ap = aap . . . . . . . . . . . . . . . • . (3e)

a cr • aP = aa {3. •.• • • . • • •. •••••••.••• (3d)

in which aCiP and aap = respectively, the covariant and contravariant compo-nents of the metric tensor of the undeformed surface coordinates; the functions,aa{3 = the coefficients of the first fundamental form of the undeformed surface,

Coefficients of the second fundamental form of the undeformed surface aredenoted b ap, and are given in terms of the vectors, ai, by the formulas

b a fJ = a3 . a a • p = - aa . a3, (3 = - a fJ • a3, a . . . . . . . . . . . . . . . (4)The quantities, b ap, describe the initial curvature of the middle surface of theshell. The mixed and contravariant components are

b~ = aa A b AfJ • . . • . • • . • • • • . • • (5a)

bap = OPA b~ . . . . . . . (5b)

It follows from Eqs. 3, 4 and 5 that

a3, a = -b~ aA ., ••.••••••••••.••••••••••••••••••• (6)

The metric tensor, g ij, of the initial coordinates, xi, is defined by

... (7)

(80)

(8b)

(8e)

. (1)

term

r = r + x3 a3

gij = r,t . r'j

It follows from Eqs. 3-7 that

gap = aap - 2x3 bap + (X3)2 baA b~ . .

g a3 = 0 . . . . . . .

g33 = 1 . . . .

If the shell thickness Is small in comparison with the curvature, thecontaining (x3)2 in Eq. 8a is generally small and can be neglected.

SHELL GEOMETRY

matrix for a cylindrical shell element. Detailed references to previous workin this area canbe foundin thebook of Zienkiewicz and Cheung (6), the volumesedited by Przemieniecki (21), de Veubeke (22), and Zienkiewicz and Hollster(23), and in the papers by Argyris (24), Clough and Tocher (7), and Jones andStrome (25).

The literature cited previously deals with the development of stiffnessmatrices for plates and shells of specific geometric shapes. Apparently, thecomplexity involved in using high-order polynomials to satisfy continuity re-quirements has hampered the development of finite element models for arbi-trary geometries. Irons andDraper (4)point out that the source of this difficultyis the Kirchhoff hypothesis; if it is abandoned, transverse shear deformationsare admitted, and the rotation of the normal is no longer the rotation of themiddle surface. With this in mind, Melosh (5) approximated the transversedisplacement by linear functions and developed a stiffness matrix for a flattriangular plate. Utku(26)and Utku and Melosh (27),followingthe earlier work(5),developedstiffness matrices for thin triangular elements of shallow shelis.In the latter papers (26,27),the middle surface of the shell is approximated bya shallow quadratic surface.

Herein attention is directed to the development ofconsistent finite elementsfor the analysis of thin shells of arbitrary shape. A linear theory for the de-formation of thin shells, including transverse shear deformations, is derivedin terms of the displacements of points on the middle surface and the rotationsof normals to the middle surface. The displacements and rotations within acurvilinear quadrilateral element are represented by simple bilinear p,olyno-mials in the curvilinear surface coordinates for the shell. By matching dis-placements androtations at four node points onthe boundary of an element, thecomplete continuity of displacements is achieved. A discrete equivalent of theKirchhoff hypothesis is introduced to assure that the approximation approachthe Kirchhoff theory as the finite element network is refined. This result is asimple, convergent finite element representation for thin shells of arbitraryshape.

Thepaper begins with thedevelopment ofa linear shell theory which includestransverse shear deformations. For brevity, the kinematic relations are castin matrix notation. These are then used to obtain the total energy of an elasticelement in terms of the generalized nodal displacements. Equations of motionfor a general shell element are then derived which involvethe stiffness matrixand the consistent mass matrix of the element. Numerical examples of a flatplate and a cylindrical shell are included to illustrate the general theory.

In the following, certain geometrical relations are given for thin shells ofarbitrary shape. With a few exceptions, the notations follow Green and Zerna(28).

The location ofa partiCle of the shell at time, t, is speCified by curvilinearcoordinates Xl, x2, x3, in which XCi = constant(a = 1,2), x3 = 0 are curvilin-ear coordinate lines embedded in the middle surface of the shell (surface co-ordinates) and x3 is normal to the undeformed middle surface.

The position vector ofa pointonthe middle surface of the undeformed shellis denoted r and that of an arbitrary point is denotedr. Thus, if a3 is a unitvector normal to the undeformed surface

FIG. I.-MOTION OF SHELL ELEMENT

point onthe undeformed middle surface becomes R in the deformed state. If A3denotes a unit vector tangent to the deformed x3 coordinate line at the middlesurface, and if the line remains straight and unextended, then

R = R + x3 A3 ~ (9)

It should be noted that A3 is, in general, not normal to the deformed middlesurface.

The displacement vector, u, of an arbitrary point in the shell is defined byu = R - r .. , , (10)

and the displacement vector, u, of a point on the middle surface is given by

u = R - r (11)

The vectors, u and u, differ by a vector, t:, which represents the displace-

Substituting Eq. 8 and Eq. 20 into Eq. 23 it is found that

1277FINITE-ELEMENT ANALYSISEM 6

- 1Yap = Yap + x3 Xap - '2 (x3)2 (b~ Aa . aA + b~ AfJ . aA) ..• (240)

- 1Yij .. '2 (Gij - gij) , . . . .. (23)

and 8 = 8i at = 6i:it (17)in which u3 = u3 is the component of displacement in the direction of the normala3 and 63 = (J3 is the component of rotation about a3.

From Eq. 15

R,a = aa + u,a - x3 (b~ aA - 11.0') (l8a)- -..fii AR'3 - a3 + a e>.µ fIIJ. a (18b)

in which Aa" (8 x a3),a = ra eAµ (fllJ.ja aA + 6µ b~ a3) , (18c)

In Eqs. 18, a = laapl, eµ1\. is the permutation symbol, and the semicolondenotes covariant differentiation with respect to the undeformed surfacecoordinates.

The deformation of a typical shell element is indicated in Fig. 1.According to Eq. 7, the components of the metric tensor Gij of the deformed

shell are given by

GaP = R,a . R,p (19a)

Ga3 = R,a . A3 (19b)

Substituting Eqs. 18 into Eqs. 19 and neglecting terms which are nonlinear inthe components of u, 8, u,a, and 8,0'

Gap = gaP + 2yap + 2.x3 Xap - (X3)2 (b~ Aa . aA + b~ AfJ . aAJ (20a)

Ga3 = u3•a + b~ uA + ra eO'A 611. (20b)

in which 2y 0'13 = aO' . u,p + ap . u,a (21)

and 2Xap = aa . A(3 + ap . AO' - b~ aA • u,fJ - b} aA • u,a (22)

The strain tensor is defined by

EM 6December, 1968

x' a.

1276

The quantity yO' fJ is a component of strain of the middle surface and Xa p is arate-of-rotation of the normal. By means of Eq. 16 and Eq. 17 into Eq. 21 andEq. 22, Yct{J and Xap are obtained in terms of the components of rotation anddisplacement

In the case of thin shells, the term containing (x3)2 is negligible in comparisonwith the term containing X3. Then Eqs. 24 reduce to

YaP = Ya(3 + x3 Xap , (25a)

- _1 A r- AYO'3 - '2 (U3• a + bO' uA + va eaA 8) .. . . . . . . . . . . . . . .. (25b)

ment caused by the rotation of a3 into A3' If rotations of the normal are suf-ficiently small, they can be represented by a rotation vector, 8. Then

t: = 8 X (x3 a3) . . . . . • • • . . . . . . . . . . . . . . . . . . . . • . . . •• (12)

A3 = a3 + 8 x a3 (13)

and u = u + 8 x (x3a3) ., . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (14)

These vectors are indicated in Fig. 1.By substituting Eq. 11 and Eq. 13 into Eq. 9, it is found that

R = r + u + x3 (~ + 8 ~ a3) •••••••••••••••••••••••• (15)

The vectors u and 6 are given in terms of their components with respect tothe basis at and at by the formulas

u = Ut ai = ui at (16)

- 1Y0'3 = '2 U3,a + b~ UA + ra eO'1\. 811. . (24b)

Eqs. 24 through 28 are the basic kinematic relations between the strains,rotations, and displacements.

In the case of shallow shells, the second parenthetical term in Eq. 27 can beneglected and Xap acquires the form

XafJ = i Va (eaA e~p + epA e~) (28)

FINITE-ELEMENT ANALYSISEM 61279

The transformation defined in Eq. 29 is said to establish the connectivity of thediscrete system. Mathematically, it establishes the required dependencies be-tween local andglobalvalues of the field v; physically, it connects the elementstogether at their nodes to form the complete shell.

The vector field, ve, to be approximated within element, e; if the subscripte is temporarily omitted then

v = vi ai = Vi af .•.•.....•..•.••..•••.••..•••.•• (31)

It is possible to approximate the covariant components, va of v, over thefinite element e by bilinear functions of the form

Vi ,., bi + Cij xj + (if Xl x2 •.•••.•.•••.•••..•••.••.• (32)

in which bi, Cij, and (if = undetermined constants. By choosing bilinear ap-

,..,

EM 6

. (26)

December, 1968

1'Yap = '2 (Ua;p + up; a - 2us baP)

Xap = i Va (eaA e~p + epA e~a)

1 A A- '2 (ba uA;p + bp uA;a - 2us b~ bAP) (27)

1278

FINITE ELEMENTREPRESENTATIONS

The finite element model of a continuous shell shown in Fig. 2(a) is an as-sembly ofa finite number, E, of small shell elements, as indicated in Fig. 2(b).Ideally, the dimensions of an element are small enough that the displacementfie ldwithinthe element is adequatelyapproximated by simple functions (usuallypolynomials) of the surface coordinates xa. In the following,the elements arequadrilaterals whose sides coincide with the surface coordinates xa [Fig.2(b)]. The vertices of these quadrilaterals are selected as the node points ofthe elements.

A vector field, vCr), defined throughout a domain, :/J, occupied by the con-tinuous shell, identifies with every point, P, in:/J a vector, v(P). In the finiteelement representation of :lJ, the field, v, is replaced by a finite set of quan-tities which represent the values of v at each nodal point, the values of v atother pointsbeing given by appropriate interpolation formulas. Thus, if thereare m nodal points in the network of elements, and if VNdenotes the value ofvat node, N, then the set, VN (N = 1,2, ... , m), is referred to as the globalrepresentation of the field, v. On the other hand, if, of the totality of E finiteelements, element e is isolated and examined independently of the other ele-ments, then the field, v, is characterized within e by the set, vMe (M = 1, 2, 3,4), in which vMe (M = 1, 2, 3, 4; e = 1, 2, ... , E) are referred to as localrepresentations of v corresponding to elements e = 1, 2, ... , E.

This distinction between global values, VN, and local values, VMe, of thesame field, v, is introduced for convenience. The behavior of a typical finiteelement can be described in terms of local values, temporarily disregardingthe behavior of adjacent elements. Subsequently, the local and global valuesare then related through transformations of the form

(0 )

FIG. 2.-FINITE ELEMENT MODEL OF SHELL

Eroximations of this type, it is possible to express the coefficient bi, Cij, anddi in terms of vNi, the four nodal values of Vi' Moreover, Eq. 32 is linear inxi along each nodal line. This means that when elements are connected to-gether by matchingnodalvalues of v at nodes commonto two or more elements,

.... (29)

.... , .. (30)if otherwise

nMN. ={:vMe = nMNe VN .

in which M = 1, 2, 3, 4; N = 1, 2, , m; e = 1, 2, , E, and

if node M of element, e, is identical tonode N of the assembled system

1280 December, 1968 EM 6 EM 6 FlNITE- ELEMENT ANALYSIS 1281

APPROXIMATINGDEFORMATlON OF SHELL ELEMENT

Returning now to the fields u and e which describe the deformation of theshell, are approximated as follows

ui = f N uNi (39a)

(]a = f N e~ (39b)

.................................. (42h)

1. Metric, Curvature, and Transposition Matrices.

A = [aaP] (42a)

B = [b a fJ] ......................•.......••.•.• (42b)

13 = A -1 B = [bg] (42e)

b = {bw 2b12, b2J (42d)

b = {bt bu + bf b12, b~ bu + b~ b12, b~ b21 + b~ b:/2} (42e)

e, = [~ n q • • • • q •• (42/)

e, = [~ :].. q • • • • • •• (42<)

= [0 -IJ€ 1 0

2. Operators.

. [a a] [rtl rfl]d1 = diag 8xI' 8xI - r~l 111 • . • . . • • . . • . • . • . • .. (43a)

. [a a] [ rt2 rf2]~=dlag ax2'ox2 - r1 ~ •..•.•....••....•. (43b)22 22

d = ~o~l' ~ f (43e)

D1 = e1 d1 + e2 ~ ..•....•..•..••..••.....•••.•• (43d)

D2 = +(e1 € d'[ + e2 € d'[) (43e)

in which the element identification index, e, has been temporarily dropped forsimplicity. Here uNi and efS = the values of ui and ea at node N of a typicalelement.

If UMi and e~ (M = 1, 2, ... , mn) denote global values of Ui and ea atnodeM of the assembly of finite elements, then, according to Eq. 29 and Eq. 38, thecomplete description is obtained of the finite element representation of u and e

uie = f N uNie (40a)

e~ =fNe%e . (40b)

uNie = flNMe UMi . . . . . _ . . . . . . . . . . . . (41a)

e~e = flNMe eR1 . . (41b)

with i = 1, 2, 3; CJ = 1, 2; N = 1, 2, 3, 4; and M = 1, 2, , mn-At this pOint, it is convenient to cast the kinematic relations derived pre-

viously in matrix notation. For this purpose, we introduce the followingmatrices:

(34b)

(34a)

. . .. (35)

.....................

.................................

.....................................

k 1 1 2N = C eNMRS xM xR YS

[

1 X: X~ Yl]1 x2 x2 Y2

C = det 1 1 2X3 X3 Y3

1 x~ x~ Y4It is understood that the sum from 1 to 4 is to be taken on all repeated nodalindices in Eqs. 33 and 34.

The local approximation for a typical finite element follows from Eqs. 32and 33

Vi = f N vNi (36a). iv~ = fN vN (36b)

in which f N = kN + cfJN xP + dN Xl X2 ., .••••••.•••••.•••• (37)

and vNie and vile = respectively, the covariant and contravariant componentsof the field at node N of element, e. It follows that

ve = IN vNie at = fN vke ai (38)

Eq. 29 and Eq. 38 define the finite element modal of the vector field, v.

1 . XefJN = C eXfJ eNMRS zM xR Ys

and dN = ~ eNMRS iM ~ x~ - (34e)

In Eqs. 34, eNMRS is the four-dimensional permutation symbol, x~ = the sur-face coordinate of node N, Ys = the value of Xl X2 at node S, iM = unity forall M, and

the approximate fields will be continuous at every pointon the boundary of theelement. According to Melosh (1), this is necessary if the finite element rep-resentation of ve is to converge monotonically to the actual field as the finiteelement network is refined.

The evaluation of Eq. 32 at each of thefour node points of the elements leadsto eightsimultaneot;s equations in the eight constants ba, cap, and da· Solvingthese it is found that

ba = kN vNa caP = epN vNa da = dN vNa (33)in which vNa (N = 1,2,3,4; CJ = 1,2; e = 1, 2, ... E) = the covariant com-ponents at node N of element, e

and, for the bilinear approximationsu = F v (48u)

9 = Fa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (48b)

u3

= f w (48c)

in which F = [~ :]................................ (49u)

f = [f u f 2' f 3' f 4 ] ••••••••••••••••••.••••••••••• (49b)

Here the f N = the bilinear functions defined by Eq. 37.Substituting Eqs. 48 into Eq. 45, the strains and curvatures in terms of the

nodal displacements are obtained')' = ljJ v - b f w (50u)

X = .fa ¢ a - lfI1

v + b f w (50b J'YS = 1 w + B F v - .fa E: Fa .......•............ , (50c)

1283

... (56b)

El1Z2]E1Z22E2222 ••••••.•••••••••••••••••• (57b)

FINITE-ELEMENT ANALYSIS

Ell1Z

E1212

E2212[

E1111

E = E12U

EzzU

n = J (RT ~ + ru3) dT + J (TT ~ + tits) dsor 5

EM 6

Then 6e = ne t. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (54)

in which ne = a 12 by mn incidence matrix, mn being the number of nodes inthe assembled system, with elements taken from the array nNMe defined inEq.30.

in which ~ = Dl F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (51a)

'" = D2 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (51b)

"'1 = D3 F (51c)

I = d f (5Id)

The assembly of elements and the transformation of local generalizedforces and displacements can also be cast in matrix notation. For a typicalelement, e

6e = {v (e), w (e), a (e)} ............................' (52)

and for the global system

t. = {Uw U2U ... Umn3' at, ... , &mn} •...•......•..•. (53)

EQUA TIONS OF MOTION

To derive, the equations of motion of a typical finite element, an energybalance is considered for isothermal deformations of the form

K + i; = n (55)

in which K = the kinetic energy, U = the internal energy, n = the power ofthe external forces, the superposed dot· indicates differentiation with respectto time, t. By definition

1 J -=-T -'- •K = 2" p (u u + u;) dT ••••••••••••••••.•••••.•• (56a)

or

in which p = the mass density, T = the volume of the element,s = the surfacearea, R = vector of tangential components of the body force, r = the transversecomponent of body force, T = vector of tangential boundary tractions, and t =transverse boundary traction. Assuming that the element is homogeneous andperfectly elastic, the internal energy is of the form

_IJ-T - IJ TU - 2" ')' E 'Y dT + '2 'Y S G 'Ys dT . . . . . . . . . . .. (56c)or or

in which E and G are matrices of elastic properties appearing in the constitu-tive relations for the material. For isotropic materials

G = 2µ A -1 . . • . • . . • . . . • . . . . • • • • . . . • • . • . . • . . . .• (57a)

and

(44a)

(44b)

(44c)

(44d)

(44e)

(441)

(44g)

EM6December, 1968

D3 = e1

B d1

+ e2

B dz ..•....•.................. , (431)

3. Kinematic Variables.- J7. --}')' = l')' 11' 2')'12' ')'zz .••••.•..••..•.•...•••.•••••••

'Y = {'Y lU ,2'Y12, ')'zz} .

X = {Xu 2X12' X22} .')'s = G13' Y23} .u = {itu uz} .u = {uu u

2} •••••••••••••••••••••••••••••••••••

9 = {9'-, If} .

With this notation, Eqs. 25-27 assume the form

Y = ')' + x3 X ....................•....•......• (45a)

'Y = D1

U - u3

b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (45b)

X = ra D2

9 - D3 u + u3 b (45c)

')'5 = d U3

+ aT u - .fa E: 9 (45d)

Likewise, the tangential displacement components ua of an arbitrary point inthe shell are given by

u = u - x3 .fa E: 9 (46)

To express u, 9, and u3 in terms of their nodal values the matrices are in-troduced

v = {uu, u2U

u3U

u4U U12, U22' U32, u42} ••••••••••••••••• (47a)

a = {9t, 9i, 9~, 9~, 9~, 9~, 9;, 9n (47b)

w = {U13

, U23

' U33' U43} •••••••••••••••••••••••••••• (47 c)

1282

1284 December, 1968 EM 6 EM 6 FINITE-ELEMENT ANALYSIS 1285

e=l

Substituting Eq. 54 and Eq. 60 into Eq. 62, the equations of motion of the as-sembly of finite elements are

e=l

From Eq. 54, it follows that

Ee

P = L O~) Pe (62)

.. (63)

TOle) me Ole) (64a)Ee

L

M to + K to = pet)

wherein M

Eeo = AT P = L 61 Pe (61)

Substituting Eqs. 58 into Eq. 55 and noting that the result must hold forarbitrary nodal velocities, the equations of motion are obtained for a finiteelement of an arbitrary shell

me ~e + ke Oe = Pe (t) (60)

To obtainthe global equations of motionfor the entire assembly ofelements,let P denote the mn x 1 vector of generalized forces corresponding to to ofEq. 53. Then, due to the invariance of n

AI·

:.

TABLE I.-STIFFNESS. MASS, AND LOAD MATRICES

in which Ea{J),p = µ (aa). a{Jp. + aap. aP). + 1 ~v v aap a),p) .... (57c)

in which µ is the shear modulus and v is Poisson's ratio.Introducing Eqs. 50 and 56, we find that for finite element, e

. -'T .. (K - .oe me 0e 580)

D. = 61 ke 0e " . . . . . . . . . . . . .. (58b)

n = 01 Pe (58 c)

in which oe is defined in Eq. 52,me = diag [mv(e), mw(e), ma(e)] (59a)

[

kvv(e) kvw(e) kva(e)]ke = kwv(e) kww(e) kwa(e) (59b)

kav(e) kaw(e) kaa(e)

and Pe = hie, qe, me} (59i)

Here me = the consistent mass matrix, ke = the stiffness matrix, and Pe =the consistent load matrix for finite element, e. The submatrices in Eqs. 59are defined in Table 1. In the case of shallow shells, the underlined terms inthe table 'can be omitted .

e=l

h '" Thickness ds '" .fa d x' d x· s '" Middlesurfacearea c '" Boundarycurveof s

kvw(e) = ~"(e) '" -h JrJlT E b f ds + h J FT iF Gf ds - h12'J $T E b f dss s s

kvv(e) '" h J rJlT E 1/1 ds + h J FT liT GIi F ds + ~; J fiT E fI, dss s s

h3 I,

kaa(c) '" 12 J q,T E fI a ds + h j FT £G £ Fadss s

EeL 0 7e) ke n(e) (64b)

KIRCHHOFFCONSTRAINTSFOR FINITE ELEMENTS

e=l

K

Applications of the stiffness matrices developed previously to specific plateand shell problems indicate that the simple bilinear approximation lead torather stiff shell elements, and the rate of convergence of the finite elementmodel is accordingly slow. The slow convergence results because the ap-proximation Eq. 39 is a crude description of flexure. Indeed, the middle sur-face of an element can only twist and stretch, and a coordinate line on thesurface deforms to a succession of similar, but rotated segments, as depictedin Fig. 3. The line AB, for example, is rigidly displaced to A*B*. Nonethe-less, the assembly flexes by virtue of the relative rotation ofadjacent elements.However, such rotations are accompanied by transverse shear strain, e.g.,at A * and B* of Fig. 3. These shear strains are necessarily large if the ele-ments are large and, if the material resists shear deformations, the finite-element Is unduly stiff.

The shortcoming can be eliminated in two steps: First, since the approxi-mation of Eq. 43 is intended to be used for thin shells, constraints analogousto the Kirchhoff hypothesis are imposed. Specifically, it is required that acomponent of the transverse shear vanishes at the midpOintof each elementaledge; the shear i'a3 at C* of Fig. 3. The number of degrees-of-freedom Is

~'1

"' Pie) '" h J FT R ds + J FT T de<l> S Cui::g q (e) '" h J fT r ds .. J fT I de

s c'"Cl

'".3 iii (e) '" - ~; (f FT £ R ds - f FT £ TT de)

mw(e) '" h J P fT f dsS

III

:!J I h'~ male) = 12 J P a FT F ds

s

III Imv(el= h J P FT F ds~ sE'"e

III<l>

~ - h'1:: kva(e) = k7;v(e) '" - h J FT BT G £ Fads - 12 J q,"[ E q, .fa ds~ s S

~ kww(e) '" h J fT bT E b f ds .. h ffT Gf ds .. ~; J 'fiTfT E Ii f ds::.:: s .~ s

Ii) J h'kwode) = k~u'(el '" -h (T G £ F Iii ds .. 12 J fT j)T E II> a dss s

1286 December, 1968 EM 6 r EM 6 FINITE-ELEMENT ANALYSIS 1287

thereby reduced from 20 to 16. Thus, a simple model is now made even sim-pIer, and now approaches the Kirchhoff theory in the limit. Second, since thetransverse shear nowvanishes in the limit, shear effects can be dismissed atthe onset. More specifically, the strain energy attributed to transverse shearcan be omitted and all terms containing G in Table 1 can be suppressed. Thisleads to a more flexible element and to much faster convergence. Thus, theprincipal shortcoming of the bilinear approximation is overcome, and the re-sulting discrete model is simple and surprisingly good.

ment fields. A severe test of a given shell finite element, then, is its capacityto predict bending satisfactorily. With this in mind, the shell element developedpreviously was degenerated first to a flat plate element andapplled to a num-ber of plate bending problems.

Four different plates are considered. These involvetwo types of edge sup-port conditions and twoloading conditions. The geometry of the plate andcasedesignations are indicated in Fig. 4. Because of the symmetry, only one

xo(

o"2 ,

L~~

~

THICKNESS = h

.(.

X

A

FIG. 3.-DISCRETE KIRCHHOFF HYPOTHESIS

In principle, the Kirchhoff-type constraints could beused to reduce the num-ber of coordinates for each element. However, in practice, it is often moreexpedient to impose the equations of constraint in lieu ofa corresponding num-ber of continuity conditions which are no longer required by the less flexiblesystem. For example, having imposed the condition of vanishing shear at aDlidpointof an elemental boundary, it is no longer necessary to match all ro-tations at nodalpoints. Thenthe use of the discrete Kirchhoff-type constraintswould not reduce the total number of equations.

NUMERICAL RESULTS

Numerical results have indicated that membrane strains can be predictedwith reasonable accuracy by simple bilinear approximations of the displace-

I

t

EDGES: SIMPlY SUPPORTED (5S) OR CLAMPED (C)

LOADING: UNIFORMLY LOADED (U) OR CONCENTRATED (C)

CASE EDGES LOAD

a C U

b C C

c 55 U

d 55 C

FIG. 4.-SQUARE THIN PLATE

quadrant of the plate was consideredinthe analysis. An indication of the con-vergence of each finite element representation was obtained by consideringdifferent mesh sized in the analysis ofeach plate. Fig. 5 illustrates the mesharrangements studied.

Three types of elements were used in each plate problem. These are des-ignated Cases '1, IT,and III and are defined as follows:

1288 December, 1968 EM 6 EM 6 F1NITE- ELEM ENT ANALYSIS 1289

Z'_rnr.l-E-<0<~o"'Zu<Zr.lenDen~'"r.l~~E-<OrnE-<U Z1..::Ir.l.<::E

~~r.l.E-<UDZ<-",..::I"'Ull.

1>-rn..J"'::EE-<~<0~ ...tilzU::Jz-I:l~DE-<~~r.l0r.l:>§: ....z::J<oenp;U>-oI. ..J '"

0>ll.0.::;;<

D .....O~rn..::l

rn

'"..::I~o~ll.

ZoE-<U

"''''..::IE-<...<:.;p;°0Ir.l

oll..... ::;;.<

$2..::1...u

da•·!

·i•·i

00

00

{

'.,11+' ~f

1*0

•

cb>-",..JE-<::E<~~o...r.l-uZz::J",. 3DEl~E-< ~"'~ 9;>or.l ezll.E-< =oll.j Bu~ll. u

1>-0.0..::1'"

g~~"'rn..::l

cb°",'"E-<0«~o:.;..::1

~>-t:J=D~~o", ...;>zz::Jo .UOIr.l.ll. '"CO::EE-<.«

$2..::1..::1"'Ull.

1°rn",

"'0!;;<~or.l..::lu>-Z..J"'..JD<~~r.l~;>r.lZUo .uOI.~",

"'::EE-<.«$2..::1..::1...Ull.

u·

0°

r

.ia•.,

\."" #II

'.,00"'>1

5~~i~

!I

~

~

~!e~~

J~~~~ I

. I~

NE=9

NE = 25

,

FIG. 5.-MESH SIZES

NE= 4

NE=16

The numerical results obtained are presented in Fig. 6 through Fig. 11. InFig. 6 to Fig. 9, the nondimensionalcentral deflection is presented as a func-

1. Case I-Straight bilinear approximation-Kirchhoff hypothesis aban-doned, nonzero transverse shear stiffness.

2. Case II-Bilinear approximation-discrete equivalent of Kirchhoff hy-pothesis (average transverse shear strain vanishes)-strain energy due totransverse shear vanishes in limit.

3. Case ill-Same as Case II, except that shear energy is suppressed atonset, thereby giving element greater flexibility.

tion of the number of elements in a quadrant of the plate. For the case ofuniformly loaded plates, the dimensionless central deflection coefficient, a =Dw/qa4 in whichD = the flexural rigidity and q = the load intensity. In thecases of the concentrated load plates, the deflection coefficient {3 = Dw/ pa2,

P being the concentrated load. Exact solutions for these problems are foundin Timoshenko and Woinowsky-Krieger (29). Since Cases I and II include bothbending stiffness, which is proportional to the cube of the thickness, and shearstiffness, which is proportional to the thickness, the results for these cases

1290 December, 1968 EM 6 EM 6 FINITE-ELEMENT ANALYSIS 1291

are dependent on the thickness to length ratio (h/ a). In this analysis, the ratioh/ a was set equal to 0.1.

It is seen that the simple bilinear approximation of Case I yields an elementwhich is too stiff. This representation causes an unrealistic and excessiveshear energy. Usually, in the analysis of thin plates, the shear energy is neg-ligible relative to the energy inbending. Therefore, the approximations whichinclude the effects of shear, such as Case I and II, result in an element whichIs too stiff. These approximations are more realistic for thick plates andnumerical results indicate that for Cases I and II the greater the thickness thefaster the convergence.

mal stress, II, at the center of a uniformly-loaded, simply-supported squareplate. The nondimensional plots show the central stress, a/ao, and displace-ment, w/wo, versus the number of elements, in which 110 and Wo represent theexact solutions obtained from (29).

As a final example, the cylindrical shell in Fig. 12 was analyzed for the lineloading shown.

. Since the preceeding plate examples suggest that elements based on thediscrete Kirchhoff hypothesis without shear energy (Case ill) yield, consis-tently, good results, this type of cylindrical shell element was used. Fig. 13

ACKNOWLEDGMENTS

contains a plot of the transverse displacement profile compared with thatgivenin (29). Again, good results are obtained with relatively few finite elements.

1.0

O."i~o.~

X/L

FIG. 13.-DEFLECTION PROFILE OF CYLINDRICAL SHELL

0.2

o

-0.2

The reported results were obtained during the course of a research pro-gram carried out at the University of Alabama Research Institute, supportedby the United States Army Missile Command through AMC-14897(Z).

o.~

0.8

1.0

0.6

~t12

FIG. 12.-CYLINDRICAL SHELL UNDER LINE LOADING

In the Case II representation, the shear is constrained to behave in a man-ner whichis more realistic for thin plates than Case I. Since the shear energyis retained throughout the analysis, however, these constraints tend to makethe element stiffer. The Case ill approximation, which simply drops the shearstiffness from the Case II analysis, yields excellent results. These results areeven more surprising because this finite element has only 8 degrees of freedomas compared with the 16 degrees of freedom usually required to maintain com-plete continuity in Kirchhoff plate elements.

Fig. 10 indicates the deflection profiles ofa uniformly-loaded, clamped platefor various mesh sizes, and Fig. 11 indicates the convergence rate of the nor-

APPENDIXI.-REFERENCES

I.Melosh. R. J., "Basis for Derivation of Matrices for the Direct Stiffness Method," A1AA Jour·nal. Vol. I,July.1963. pp. 1631-1637.

2. Key. S. W.. "1\ Convergence Investigation of the Direct Stiffness Method." thesis pre-sented 10 the University of Washington. at ~attle. Wash .. in 1966. in partial fulfillmentof the requirements for the degree of Doctor of Philosophy.

3. Synge. J. L, Th~ Hypm:ircle in MaThemaTical Physic.\', Cambridge University Press. London.1957. pp. 209-213.

4. Irons, B. M .• and Draper. K. J .. "Inadequacy of Nodal Connections in a Stiffness Solution forPlate Bending," AI.4A Journal. Vol. 3. May. 1965. p. 61.

5. Melosh, R. J .. "A Flat Triangular Shell Element Stiffness Matrix." Matrix MeThod.\' inSTrUCTural Mechanics, J. S. Przemieniecki, et al.. eds .. Air Force Flight Dynamics Labora.tory TR-66-80. Dayton. Ohio. Dec.. 1965, pp. 503-514.

6. Zienkiewicz, O. c., and Cheung. Y. K.. The FiniTe Element MeThod in StruCTUral and Con-Tinuum Mechanics. McGraw-Hili Publishing Co. ltd .. London. 1967. pp. 21-25. 89-147.

7.C1ough. R. W .. and Tocher, J. L. "Finite Element Stiffness Matrices for Analysis of PlateBending," MaTrix Method.\' in StrucTural Mechanics. 1. S. Przemieniecki, et al.. eds ..Air Force F1ighl Dynamics Laboratory, TR.66-80. Dayton. Ohio. Dec., 1965, pp. 515-546.

8. Bogner. F. K .. Fox. R. L. and Schmit. L. A .. Jr .. "The Generation of Inter-c:lement. Com-patible Stiffness and Mass Matrices by the Use of Interpolation Formulas," MaTrix Method.f inSTrucTural Mechanics. J. S. Przemienieki. et al.. eds., Air Force Flight Dynamics Laboralory.TR-66-80. Dayton. Ohio. Dec .. 1965, pp. 397-443.

9. Zienkiewicz. O. C .. and Cheung. Y. K .. "Finite Element Method of Analysis for Arch DamShells and Comparison with Finite Differences," Proceedings. Symposium on Theor.1' of ArchDams. Soulhampton University. Pergamon Press. New York. 1965.

10.Clough. R. W .. and Tocher. J. L.. "Analysis of Thin Arch Dams by the Finite ElementMethod," Proceedings. Symposium on Thl!ory of Arch Dam.f. Southampton University.Pergamon Press. New York. 1965.

II. Argyris, J. W .. "Matrix Displacement Analysis of Anistropic Shells by the Finite ElementMethod." Journal of The Royal Al!ronauTieal Soeil!Ty. Vol. 69. Nov .. 1965. pp. 801-805.

12.Zienkiewicz. O. C .. "Finite Element Procedures in the Solution of Plate and Shell Problems,"Stre.I:. Anu~l·si.l. O. C. Zienkiewiez. and G. S. Holister. eds .. John Wiley & Sons. London.1965.pp.120-144.

13. Meyer, R. R .. and Harmon. B. S .. "Conical Segment Method for Analyzing Open CrownShells of Revolution for Edge loading," .1f11f'fkan {n.H/TUtl!(~f Aeronautics and Astronal/tlnJournal. Yol. I. No.4. 1963. pp. 886-891.

14. Grafton. P. E .. and Strome. D. R., "Analysis of Axisymmetric Shells by the Direct Stiff.ness Method." American Inflitute of Aeronautic.\' and A.I·tronautics Journal, Yol. I. No. 10.Oct .. 1963. pp. 2.342-2,347.

15. Popov. E. P .. Penzien. J.. and Lu. Z. A .. "Finite Element Solution for Axisymmetric Shells:'.Journal (!! The EngineeriUK Mechanic .• Dil'i .•ion. ASCE. Yol. 90. No. EM5. Proc. Paper 4085.Oct.. 1964. pp. 119-145.

16. Percy. J. H .. Pian. T. HOoKlein. S .. and :-.iavaratna. D. R .. "Application of Matrix Displace-ment ~ethod to Linear Elastic Shells of Revolution." American ftwiTuTe of Al'rona/ITiCJ alld.1.Hrol/uI/TinJournal. Yol. J. 1965, pp. 2.138-2.145.

17. Klein. S .. and Sylvester. R. JOo"The Linear Elastic Dynamic Analysis of Shells of Revolulionby the Matrix Displacement Method." MaTrix MeThods in STrUCTural Mechanic.t. J. S.Przemieniecki. et aI., ed .. Air Force Flight Dynamics Laboratory. TR-66-80. Dayton. Ohio.Dec.. 1965. pp. 299-328.

18. Klein. S .. "A Sludy of the Matrix Displacement Method as Applied to Shells of Revolu.tion:' MaTrix M.'ThodJ in STruCTural MechanicJ. J. S. Przemieniecki, et al.. ed .. Air ForceFlight Dynamics Laboratory. TR-66-80, Dayton, Ohio. [)ec .. 1965. pp. 275-298.

19. Green. B. E.. Strome. D. ROoarid Weikel. R. C., "Application of the Stiffness Method to theAnalysis of Shell Structures:' Proceeding. of the A viation COl/ferencI.'. American Society ofMechanical Engineers. Paper .\'0. 61-A V-58. Los Angeles. Calif., Mar., 1961.

20. Bogner. F. K.. Fox. R. L.. and Schmit. L. A.. "A Cylindrical Shell Discrete Element:'American ftrJTiTuTe of A.'ronauTiCJ and A.fTronautic' Journal. Vol. 5. No. 4. 1967. pp.745-750.

21. "Malrix Methods in Structural Mechanics." J. S. Przemieniecki, R. M. Bader. W. F.Bozich. J. R. Johnson. and W. J. Mykytow. ed .. ProceedingJ of /he Conference on MatrixMeThodJ ill STn/Clural Mechanic.\'. Wr~~hT PaTTer.mll Air Force Ba.ll'. Oel .. {9f15, AirForce Flight Dynamics Lahoratory TR-66-80. Dayton. Ohio. Dec .. 1965.

APPENDIX n.-NOTATION

The following symbols are used in this paper:

22. MaTrix MeThod.f of SIn/cTural AnalrJi.f. B. F. De Veubeke, cd .. Pergamon Press, New York1964.

23. STreJ.I Ana/l·si.f. O. C. Zienkiewicz, and G. S. Iiolister, eds .. John Wiley & Sons. London.1965.

24. Argyris. J. H .. "Matrix Displacement Analysis or Plates and Shells:' lngenieur-Archil', XXXV,1966. pp. 102-142.

25. Jones. R. E.. and Strome. D. R .. "A Survey of Analysis of Shells by the DisplacementMethod:' MaTrix MeThod.! in StruCTural MechalliCJ. J. S. Przemieniecki, et a1.. ed ..Air Force Flight Dynamics Laboratory, TR-66-80. Dayton. Ohio. Dec.. 1965. pp. 205-229.

26. Utku. S.. "Stiffness Matrices for Thin Triangular Elements of Nonzero Gaussian Curva-ture:' American Institute of Aeronautics and Astronautics 4th Aerospace Sciences Meel-ing. Paper .Yo. 6f1-530. Los Angeles. Calif .. June 27-29. 1966.

27. Utku. S .. and Melosh. R. J.. "Behavior of Triangular Shell Element Stiffness Matrices Asso-cialed with Polyhedral Deflection Distributions," Americall IlIsTilllTe I~f AeronaUTics 01/1/

.·'-'TrollauTic.15Th·/l'ro.ft'ace Scil!nct''' Ml'eTing. Paper .vo. /57-114. New York, Jan. 23-27.1967.

28. Green, A. 1:: .. and Zerna. W .. TIII'ore/leal Ela.HiciTr. Oxrord University Press. London.1954. pp. 31-39.

29. Timoshenko. S. P .. and Woinowsky·Krieger. S .• Theon of PlaTes and Shel/.t. 2nd cd..McGraw-Hili Book Co .. New York. 1959. pp. 113-118, 141. 142, 197-205.471-475.

1293FINITE- ELEMENT ANALYSIS

natural base vectors associated with surface coordinates;

covariant, contravariant coefficients of the first funda-mental form of the undeformed middle surface;covariant, mixed, contravariant components of the secondfundamental form of the undeformed middle surface;coefficients in bilinear field approximation;

array of material constants;

permutation symbols;nodal functions;metric coefficients associated with deformed state;

metric coefficients associated with undeformed state;position vectors ofpoints in undeformed, deformed shell;

position vectors of points on undeformed, deformed mid-dle surface;components of displacement of node, N,of global system;

components of displacement of node, N, of element, e;

displacement, velocity, acceleration fields;

surface coordinates;u,u,u

xi

CaN, dN, kNEijkl

at. at

aaP' aafJ

bafJ, b~, baP

UNa, U»

UNa (e), U~(e) =

eap, eijk, eMNRS

INGij, Gij

gij, gij

r,Rr, R

EM 6EM 6December, 19681292

1294 December, 1968 EM 6

r~p = Christoffel symbols of the second kind for the middlesurface;

Yij, Yap = strain tensors;i J0ij, OJ. o~ = Kronecker deltas;

e~,eNa = components of the rotation vector at node, N, of the globalsystem;

9~(e), 9Ncx(e) = components of the rotation vector of node, N, of element,e',

Xap = change-of-curvature tensor; andnMNe = mapping function.

,:-