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G3/39 14555

CN IIYBRID AND MIXES FIi^ITE ELE!L'ENT IJ!ETHOLS

T . H. H. Tian

'p^4iO9 Massachusetts Institute of Technology

j. , l_ ^Fj``^^s U . A. N Cambridge, I^.aUSa^^uSe V VS, S.

Three versions of the assumed stress h.,-

brid model in finite element methods and the

corresponding variational prircipies for the

formulation are presented. Examples of rank deco

ficiency for sti_'.nesr matr_ces by the hybrid

stress model are giver. and their correspordir.F,

kine:-atic defor^-,ation modes ure identified. A

discussion of the derivation of general semiLoof elements for plates aid shells by the hybrid stress method is given. It is shown that

the equilibrium model T)y Fraeijs de Veubeke can

be derived by the approach of the hybrid stress

model as a special case of semi-Loof elements.

1. INTRODUCTION

It is generally realized that an assumed stress

hybrid element is based on the complementury enemy

principle using equiiibratin6 stress field in the interior of.the element and independent displacements along

the element boundary. However, when a compatible displacement field can be constructed it is also possible

till

,, o

.x

.o
A

to use the Hellii;;;er- eissner principle for the formulation. Indeed, it may not be generally knovrn Ill that

the original derivation of the assumed stress hybrid

element was mode by using the helli nger-Reissner principle. In that derivation the assumed stresses happened

to satisfy the equilibrium equations and the resulting

element stiffness matrix_ is identical to that by the

complementary ener.-y principle, i. e. by a model which

was later named hybrid model. Hybrid o.nd miz ed models,

thus, are not mutually exclusive.

In this pa p er we discuss several problems associated with '.', i s hybrid/.mixed method:

1) different formulations of the assumed stress

hybrid/mixed elements.

2) :tank deficiency in the assumed stress hybrid

elements.

3) :;emi-Loof elements for p1F.tes and -s hells by the

hybrid formulation and formulation of the equilihri lm element by the procedure of the hybrid

stress model.

2 . ^;^r,"^i i. ":.I^, I:°, rjF iI' 7 F'' E I i.^CXE ELE,.'.ENTS

Three different versions of the hybrid element

formulftion are presented here. The first one is by the

HellinL;er-Reissner principle which can be expressed as:

77, = E(f"(

- 1

0 TS u 

+Q T

(D U)) dV-fs,, T q dS)

= Stationary

where

o = stresses

u = displacements

2

.^-. - T>tc . r i
S = elastic compliance matrix

D = matrix of differential operators that defines

the strain displacement relations e = L' u

T = vector of surface tractions - -

V = volume of the nth element

n

S o = boundary of the nth element over which tractions T are prescribed

Here 7 denotes su; nation over all elements, end for

simplicity in the present illust: • ation, bod; forces are

considered absent and displacements u a-e assured to

satisfy the prescribed bo::ndar;{ conditions.

'..hen the state of stress o is in erui1 4 5rium, i.e.

D T c = 0 in VR : and v'o =T on ar'n (2)

where D l represents matrix of differential operators

and v represents the directional cosine of the surface

nor:ial, then by the divergence theorem the variational

functional becomes

17 = -1 ( 2 fv„ c'.; o dv- fdv, T'5 dS tj, T u dS)=-7,nC

(3) fCn

where aVn is the entire boundary of the nth element,

a y, d the displacements along the boundary are now devoted by u. It is noted that the above expression would

be identical to -r. mc where n me is the functional corresponding to the modified complementary energy principle associated with the assumed stress hybrid modelILl

When the body force term is included, a mixed variational principle results. This also suE;Cests that body

force can be distributed rationally based on independently assumed displacement functions.:

/ 3

`

till
Ii;

In case that both the equilibrium and compatibility conditions are satisfied both stresses and displacements can be expressed in terms of the same functions. The volume integral in Eq. (3) can be reduced

into a sarface integ:-ul along; the bounde..y and another

modified complementary energy principle can be written

as follows '>>.

TTmc = 7 ^av„(^ T Tu-Tru)dSt^^so T u dS]

n

=stationa_-y (4)

In the above expression the tractions T and displacements u arrive from the same stresses a

The steps to be taken in formulating the assured

stress hybrid eiemcnts by these three variational principles are indicated in Table 1. Here the approximation

of the stresses in the interior of the element is a

common step for all methods, and although many terms in

these variational functional are different, the resulting n-functions are of the same form, i.e. in terms of

stress parameters B and nodal 'displacements q. Since

the stresses are independent for different elements,

one can take variation with respect to the 6's in the

element level and obtain expressions of g in terms of

q. The final expression for n is now in terms of only

nodal displacements q as unknowns. The matrix k is the

element stiffness matrix given by

k =G T H -1 G l5)

Matrices k can be assembled to form the stiffness matrix K for the global system. TekinL variation of ri with

respect to unrestrained nodal displacements leads to

4

I
y

the system of matrix equations of the finite element

sr

method:

^y

An investigation of the relative efficiency between the formulations of the element stiffness matrices by ra p and TT for rectangular block elements with

mc

8-nodes was conducted (4) and the results indicated s

k= :

clearly that the computing time required for the nh

formulation is shorter than that by the n formulame

Lion. The reason for this is that in the evaluation of

each element of the matrix G in Table 1, the formula- J^

tion by 7 involves a single volume integral while for f'

the formulation of

TT Mc

the correspondin- surface inte- T ''

gral must be divided into six separate ones for the six A

individual faces. s^.}.•_ 

The formulation of the assumed stress hybrid element by rrH has also been extended to the general 8-node j

' 3► hexahedron elements which have straight edges but may

have non-flat faces. In formulating such a finite element- by rr, it is required to introduce over the element a set of curvilinear coordinates in the

sane way as the isoparametric element in the conventional assumed displace:ier.t method.. It is well known

that for conventional isoparametric elements, it is, in , 

general, not possible to evaluate, in closed-form, the 

integrals required in the element stiffness matrix, ;i

hence numerical integration procedures are required.

However, in the formulation of such elements by the

assumed stress hybrid r.iodels the matrix _H can be evaluated in closed form. By recognizing identical elements

in the H matrix, it is possible to optimize the com- ti w

h^iitt

'"

puting effort and as a result, the computer time needed
to evaluate the .stiffness matrix by TT R is only a few

percent hi7her than that for the conventional isoparametric element 151. Discussions on the derivation

of hybrid stress elements by a have also been given in

Refs (b-1 1) .

A typical application of the variational function

*r is in two-dimensional linear fracture mechanics.

me

Standard technique of complex stress functions in plar_e

elasticity problems permits a proper approximate solution which satisfies the equilibrium and compatibility

equations as well as the stress free boundary conditions at the surface of the crack. A super-element

which contains an embedded crack can thus be derived

to be used Jointly t%ith conventional finite elements

for the analysis of elastic stress intensity factors[3J.

3. Rn;,^; L=:r I..IEr+„Y I ti.,^liI.1LD Z'S S •Ir..S^ ^ iiY:^FtID ELEMENTS

In Table 1 the relation bet:.een the stress parameters Land the modal displacement q is governed by

l.' P =. G q

It has been pointed out [2,12] that if m is the number

of stress parameters and n is the number of generalized

displacements of which r nodal displacements must be

restrained to prevent rigid body motion, then when m is

smeller than (n-r), kinematic deformation modes will

appear and the rank of the stiffness matrix will be

less than (n-r). It is well known that the condition;

M > (n - r) (8)

is only a necessary condition for preventing any kinematic deformation modes,and an element may still be rank

6

i
f

N

C

C

IL

r••f

N

b

A

L

W

c

v

u

C

Gi

49

m

4i

O

C

J

•N

r-I

7

0

r

d

A

H

* U

^E

C

^

c0

m! m) 0- 1

MI 5T J!

11 II 4

7i h) 171

ac I

M)

^ 1

'sl

c

ro

1

II

21

J 1

OI

^ QI

C 2' 1 O! QI I

f- > L7 ! Q! Q f .o

t

?` vl II S! ~Q!

ro l'J•

f (7 f II ^IN

C Q1/ O

m !

u

U 41 II

ml II

p E 0 vl m1 i•-.

} C QI of J) dl Of

H 11 Y1

I I m ! H qa Q! 7 ! h- 1 —I N ro

^, I m cr

H c

Cl. l YI

^I

II NI ~Q1

k d 1 .SIN o

I I

c

v

t

> w c o

^ I+

j II II

S ! t+ ^

C

m)

a!

Qt

PI m! 'C

t

Q 1 II ~

II 7 t II

7 1 O 1 4,7 !

4.

r

i

L

r

6

f

V 

i.

I}

I

A.

7
deficient even if the above inequality is satisfied.

A kinematic mode corresponds to nodal and boundary displacements for which no work is done by the assumed streso distribution, hence is also called zero

energy mode. This means that

R z H -1

G9

= 0 or G9 =0 (9)

Since the assumed stresses are equilibrating, all riCid

i body modes will involve no external work. A kinematic

deformation mode refers to ele ment bourdsry displacements which indicate element deformation but involve

no work from the assumed stresses.

One of the cases that kinematic deformation modes

appear when the condition is satisfied is a twelve degree of freedom rect%ngular plate element under Kirchhoff assumption 1131 . It is derived by assuming linear

distributions in stress couples (Y X , My , N:Xy ) within

the element, and cubic distribution in lateral displacement (<<;) and linear distribution in normal slopes

(w n ) along each edge. The condition of Eq. (b) is satisfied. The element has three riLid body degrees of

freedom yet the stiffness matrix has five zero eigenvalues. One can verify that the boundary displa,-.ements

depicted in Pie. 1 and represented by the following

equations will yield no work due to linear distributions of moments N. n and Itns and hence uniform distribution in Kirchhoff shear V along the boundary:

along y = b, w=—x(a2-x1); w.y= 0

along x = +a , w = 0; w X = 2a'y/b

w=0; w.X =2a1y1b

e

#lei
a

Another kinematic mode involves similar displacements

with zionzero values of w and w y along x =± a, and y =

} -+ b respectively. To suppress these kinematic modes it

is only necessary to add stress terms such as MX =91GI4

and My 1 1 xy or Mxy = 12x2 + E3 13y2 which will lead

to linear distribution of

uX

r

r`

f '^ r

j' }

Fig. 1 Kinematic deformation mode of a rectangular plate element under lirear

moment distribution

shear force along the edges. For an irregular shaped

quadrilateral plate element, the stiffness matrix obtained by linear distribution of 14 X , My and LIXy , would,

in -eneral, not be rank deficient.

Another 12 DOF plate elements derived by assuming

linear distributions in stress-couple is a triangular

element with w, w and w at each corner and w at

' x ,y . n

' 9

c ^^ 1
w

W = 0 w

2

^ = 2

W x = Ci

w'), ° C2

--- x

W=0

w,x 

= C,

W y = C2

y

W=0

W x = C,

W y =C2

=_ C, S i n az ^r COS a1

the midpoint of each side as nodal dieplacements^14,15)

as shorin in Fig. 2. For the twelve nodal displElcements

shown in the figure, the distribution of normal rotation V/0 Y1 along all edges are in the form of

win = t(s) = C(1-6(SIP) + 6(s/Q) 1 )

(II)

F''ic. 2 Kinem-tic defurru ticn movies of c 12-

LOP plate element under linear moment

distribution

where s is the coordinate along the ed.-e and I is the

len,;th of the edge. Eere f(s) is symmetric about the

midside point and its integral over p vanishes, thus

the integral over the product of f(s) and any linear

function in s will be zero. Also, along till edges the w

distribution is cubic and is antisyi,jnetric t+bout their

I

10

^.......r...^.^.•^
r►,

r.^

s

midpoints. Now under linear distributions in stress

couples, the bending moment 14 is linear along s and

the Ki: • chhoff shear V11 is censtart. They should do no

veork under the boundary of displacements described

above. Since w is zero et all corners, the corner

forces also do no wore.. '.-he deformation pattern shown

in Fie. 2 represents the combination of two independent

kinemL. tic modes and the r-:ik of the stiffness ma t

rix is

seven.

Finally, a remark -, hould be made about the rectanlur rnembrc.ne elenei:t derived by usirz 5 p terms which

was used as an exa.:.ple v.her the assumed stress hybrid

elernent v,as first introduced. The element has eivht

de,;rees of freedom and three rigid body :nodes, hence,

Ea. (r3) is satisfied End for d;enerul arranCement of the

reference axes the resulting element stiffness matrix

..ill have e ra::k of fi : e. :iovrever, , for a square element

if the diaronEl lines are --sed as the reference axes as

shown in Fib.;. 3, t ,-.e :esultinr- stiffness matrix will

have a rank of only three. If the stress assumption is

C, - Br +!>4y

0Y = 92 + Qs X (121

TXy =Q,

then in the & - q coordinate system the correspondin6

assumed stress is

04 =(1 1 +04 4 +a; q

Or =Q1 +a4 (+aSq

Txy = aJ -a4 F1 - a s ^

i

i

a

i
4

Fig. 3 :quc:re r.:e^tiZr_no el•:?:aent v.ith raf+:rerce

exes for stresses r:long the diaL-;on,..s

In this case the fcllowing two nodal displc.ce:aent patteens are kinemetic moc.s:

(a ) UE _ - U = UE ^ _ - U{4 = I

and U"^ = U^ 1 - U^3 = U^ 4 = 0

( b) Ufa - Li (2 - u(j = u^ 4 =0

and -U9 = U 92 = - U9 = U94

Similarly, if the x-axis coincidee- with one- dic.gonul of

e generc:l rectangular membrune element, the resulting;

element stiffness rlatrix by 5-P assumption will have

a rank of only four.

Spilkev et ul. (11) have studied the pi• ebleia cf

runk deficiencU c: the 5-p square element by first

identifying c.11 independent deformation modes that can

uniquely define uny displece.ments of the element end

then fub,titutin,^ into Eq . (9) to see whether C9 q

vanishes. By chungin^ ti:e angle 0 between x znd E axes

12
they wer,, able to detect the t%:, o kinematic deformation

!c

. at e = 45 . This example cn t. c- membrane element also

j. illustrates the danczer of rot ta::in L; complete poly=

nonials in the stress terms in the hybrid element formulation. In that case, the resultinC element stiff-

; ness matrix will be lack of invariance on and, indeed,

it may be rank deficiency under certain reference coordinates.

t For any newly devQloped 'hybrid stress ele:-ento it

ill

is suggested. in addition to rater-test, an ei`erVLlua

[. survey should be conducted in order to detect any kinematic deformation modes. if invariance of the stif--

^. ness matrix c eJiot L, , mainta .ted, such is sure;;, will

hLve to be made for different coordinate systems used

^- for the stresses. It should be noted to-t i:inenctic

deforma.' in modes can always be suppressed by addinC,

` appropriate stress terms.

y_.

;x. 4. £EIi.I-L(,0: L;' Liril:T.i FGR Pik TLS t.1vD 3;IELLS

^'. BY THE riYBRID 1'UR11UiA IW4

For the analyses of plates and shells, one of the

difficult tasks is to matc`l the compatibility at a node

f at which the reference planes of the elements are not

coplanar. At such nodes all six degrees of fr-edom

should be ccnsidered but for plate and s:.ell el_lznents

s a maximum of five degrees of freedom can be used at a

node. Irors (10 has suggested the une of the so-called

semi-Loof clement for which the normal L•otutions alone

F each edge are defined at nodes which are not located at

cornern of the element. By the conventional assumed

displacement method it is still a difficult task to

construct 4hape functions fora semi-Loof element. But,

i

A

s'

r

^a

f'

13
it is a more or less routine procedure to formula's

such an element for plates and for simple shells by the

assumed stress hybrid method.

The equilibrium model of Praeijs de Veubeke09^

always results to semi-Loof elements. It has been shown

i that this model and the assumed stress hybrid model can

both be derived by the modified complementary energy

principle ( <j . The only difference is that for the

former, tractions along each boundary t y re represented

by generalized loading parameters and the corresponding

fictitious nodal displacements are vjei6hted integrals

of t o boundary displacements, while for the latter,

boundary displacements are interpolated in terms of

nodal displacements and the corresponding nodal forces

ere obtained frcm the variational sense. The expression

for stiffness matrices are the same for both models except the G matrices in Eq. (5) are derived differently

for the two models.

.:e can nov, silo ,:: that the equilibrium mode]. can

also be forr.;ulated by boundary displacement interpolation in the same way as for rr mc in Table 1. Consider

a problem for which certain nodal displacements are of

the Semi-Loop type, hence, the corresponding boundary

displacements are independent from one edge to the

other. This mecns that for each elerient the boundary

displacement continuity is not maintained at the corners.

For simplicity let us consider a straight boundary

O< s<Q, with its traction T (s) represented by a polynomial of order m.

Such traction distribution can be represented by

m+1 nodal forces Q

1'" . Qm+1 at arbitrary locations

s 1 ' "' ,

sm+1 When the following conditions are satisfi14

^, I
ed: 

mrl

Q; 5n = foT (5)s n ds, for n =0,---, m (14)

=1

We can easily show that if the boundary displacement

u(s) is also a polynomial of order m then

I

T(5) u,'s) ds = 

m±l 

o Q u

;=1 I I

where u is the nod,:l dicplE-cement at s i . Conversely,

if' the boundary displacement is inter ol®ted ti:roL.L;h

the nodal displacements at the m-r1 nodes elon^; the

boundary, the correspon in;; nodal forces Q obtained by

Sq. (15) :ill satisfy the conditions Ziven by Eq. (14),

r.

hence, will be equipollent to the bounder;; traction

distribution ^(s). Ahen the boundaries of tv:o neijiboring; elements ure approximated by polynomials of the

same order, a compatibility of the nodal displacements

will then necessarily guarantee the reciprocity of the

nodal forces and, hence, the pointwise interelement

equilibrium condition is maintained. The equilibrium

F model by Fraeijs de Veubeke, thus, is a special semi4 Loof element by the assumed stress hybrid model when

the number of nodes along; a boundary exceeds by one the

order of the polynominal which represents the corre

e spondint; tract i on. An exception in thi° case is that

y for const ant traction distribution there is only one

node which should be located at the midpoint of the

y edge. For example, for the linear moment triangular

plate element by Fraeijs de Veubeke (17) , the nodal disk

placements are lateral displacements w at the three

corners and the three midside points and two normal

(!S)

I

L 

r•

Y:i

15
4' •. .r

1 ^i .r ^I M !,•^ 5

5.. R^• 1 .^: _. ^^ ^ ,

^•

,.,+i ski A?f.. }. 

R' '.

_ ^ • !Y• ^.f. ^ n i -1'•f 't :f ., 1

h ^J^. 

t 2^y-"y",-nf *1 7 ^'^:i +^.=^^r!^ '"`_'y-(•

^' :/^ J^^

^k h 1

^Ga .•'^^,1Y7 r. 7}'

S!^ ^`,, ^P ^. ^^. s^ Y^t ~^, T.i ^^^ ^^^'+,,^L"C: ^j'1Y.'l ar^..^^"-:e:. a'+^^r^.i =J''.^.e. '^^ '^. ,.^,N `.^'s: '...Y^< t.^^+,^^' ,. .:^•^'•_ ^•^C. +. ^, ^

J ^

y . Jt• 

r,

r q '

..

v .4

V'^

5. CG.JCLL'LINC T:,'EX.!ti:S

For the formulation of the assumed stress hybrid

finite elements there are different methods of approach

to choose from and the selection of the appropriate

stress fields requires sufficient physical insi6ht. It

is not likely that, in the future, h;;brid stress elements can be formulated as simply as the use shape

function routines and numerical integration techniques

in the conventional assumed displacement method. But

I abundant knowledge about this method has been collected

^i and future development of Hybrid stress . eleinents should

be and can be made fool proof and should be implemented according to the most efficient method of approach.

f

16

rotations orlon- each edt-,e. Here, the normal bending

moment M

n 

is linear along,; each edge, hence, by using

two corresponding; nodal displace. ents w alonr; the

,n

ed=e, the equilibrium conditions are satisfied exactly.

I It should be remarked that in this case the locations

of the semi-Loof nodes may be arbitrary.

t

For the equilibrium model, kinematic deformation

modes often exist in one element or in a group of elements and appropriate superelement technique is often

re q uired to su p press the mecharisms [`CA . On the other

hand, for the same c oice of nodal displacements, the

r_umber of stress terms can be increased by the assumed

stress Hybrid model, hence, the possibility of any

i kinematic deforriction modes can al'.eays be eli-in-ted.

Thus, the assumed stress hybrid method is a most promising approach for the semi-Loof element.
AC III;UW LEIU GF..YIhNT

The author wishes to thank Professor

Chen for his constructive comments.

This paper was prepared under the sui

Grant NLG 3-33, by the IIASA-Levis research

La- Peng

y

port of h..'.SA j

1

Center.

REFER E.:'C E J

(11 1-ian, T.i:. W . , I?ow _id I .•slink of the t_ss,wmed Stress

rybrid ',:et:.od, quoted in Vuriational rrinciples in ^

Elasticity, by Wasl.izu, k., Series in Computer

Criented Structural Enoineerin,7,, Vol. II-3-A, 

Z-aifuken :ublishin,^ Co. , ;oi:;; o, 1U0 -1^3, 1972.

[2j Pian, T.H.H. and Tons, P., Baeis of Finite Llement

Methods for Solid Continua, Ir,t. J. I y u:.7. itiSetl:. Y

[3) Ton" P. Iiun, T.h.:?. and Lasr,,, S.J., A _9%b-idclement Approach to Crack Problems in Plane Elasticity, Int. J. Num. Yeth. En6ng, 7, 297-3uE, 1?73. ,.

(4) Peterson, K., Lerivation of Stiff,iess ','ntrix for

Hexahedron Elements by the Assumed Stress hybrid

blethod, M. S. Thesis, 1.1essachusetts Institute of

Te(-hnolo fy , 1972.

(5) Lee, S. W. , An Assumed Stress Hybrid Finite Element

for Three Dimensional Elastic Structural Anulyses, i

M.S. Thesis, Massachusetts Institute of lechr_oloLy,

1974.

(6) opilker, I.L. , A hybrid-Stress Finite Element Formulation for Thick blulti-Layer Laminates, Computers

and Structu:^es, 11, 5U7 - 514, 1960.

17
(7) Spilker, R.L., Improved Axisymmetric Hybrid-Stress

Elements Including behavior for Dearly Incompressible IMaterials, Int. J. Iium. Yeth. En6nE. (to

appear).

(8) Spilker, R. L. and .'.'urir, N.I., The Ilybrid-tress

1.ode1 for Tiffin :later, Int. J. Dori. i;:eth. En;nj;. ,

15, 1239-1266, 1960.

(9; Spilker, R.L. and Yunir, II. I. , A Ilybrid-Stresc

:;u_•drutic Displacement Mindlin Plate Bending Element, Computers and Structures, 12, 11-21, 1960.

(1;) Spilker, ii. L. and L:unir, h.I., A Serendipity Cubic

r :c for

7 A.-oplacement hybrid-Stress Element nor Thin and

;:oder,.tely Thick. rl4tes, Int. J. Lum. Meth. L- nLn tg. ,

15, 1261-1270, l!"B.;.

h1 pilker, c. L., I,:askeri, S.Y. and nania, E., Plane

Iseparametric hybrid-Stress Elements: Invariance

and upti.mal SamplinE; (to be published) .

(12) Frueijs de Veuber;e, E.iti:. , i3end^nt and Stretcnint;

of elates, Special .yodels for Upper and Lover i

'-cur-ds, in ire. Conf. on Yatrix :,lethods in Structural Yecrarics, AFr LL-Tr- 6G - c0, 863-bL-6, 1966.

(1 3? I iar., T.H.H. and Mau, S.T. , Some Recent Studies in

Assu;ied Stress Hybrid Models, in Advances in Compututional Methods in Structural Mectianics and

Design, Edited by Cden, J.T., Clough, R.W. and

Yamamoto, Y., Urli Fress, Huntsville, Alabama, 87-

%6, 1972.

h4) Allman, L.J., Triang•.ilar Finite Elements for Elate

3endin L; with Constant and Linearly Varying Bending

Moments, in high 'Speed Computing, of Elastic Structures, Edited by b. Frueijs de Veubeke, Univ. Liege,

105-136, 1971.

h 5) Irons, is. arid Ahmad, S. , Tecliniques of Finite Ele18
a:r

w'

t

ments, Halsted Press, 198G.

(161 Allman, D.J., A Simple Cubic Displacement Element

for Flate Bending, Int. J. hum. a?eth. ih c-nC, 10,

263-281, 1976.

(17) Batholomew, P., Comments on hybrid Finite Elements,

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[18) Irons, B., The Semi-Loof Snell Element, in Finite

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