Mixed Variational Formulation Of Large Deformation Analysis

Appl.ied Mathematics and Mechanics (English Edition, Vol. 10, No. 7, July 1989)

Published by SUT, Shanghai, China

A M I X E D V A R I A T I O N A L FORMULATION FOR LARGE D E F O R M A T I O N A N A L Y S I S OF PLATES* S. Dost (Department of Mechanical Engineering, University of Calgary, Calgary. Alberta. Canada)

B. Tabrrok (Department of Mechanical Engineering, University of 7bronto, Toronto, Ontario, Canada)

(Received Jan. 4, 1988; Communicated by Chien Wei-zang) Abstract A mixed rariationalformuhttion./br ha'ge d~/'ormation anal)'sLvqfplates is intror In thisJbrmulation the equilibrhtm and compatibility equations are satisfied identicall)' b), means of stress Jiawtions attd eh:rplacement components, respectivel),, attd the constitu, ire equations are sati.~Jied ht a/east square sense. An example is soh'ed attd the results arc compared with those available ht the literature. Further. the functional is particularizedJbr buckling analysis of plates attd a shnple example is soh,ed to illustrate the theory.

I.

Introduction

The essential equations of elastostatics may be grouped into the following three sets: (I) equilibrium equations in terms of stress variables, (ii) compatibility equations in terms of strain variables, (iii) constitutive equations relating the stresses to the strains. The force and kinematic boundary conditions may be considered as subsets of equilibrium and compatibilityequations respectively. The relations amongst various variables of elastostatics is depicted schematically in Figure I. In this figure the differential operators are denoted by 0, Thus Ot acting on the stress components yields the equilibrium equations whilst 0z , operating on the strains, yields the compatibility equations. For the exact solution, the equilibrium, compatibility and constitutive equations must be satisfied. Conventionally one introduces a set ofcontinuous displacement functions, u,, in terms of which the compatibility equations will be automatically satisfied. In this case one aiso satisfies the constitutive equations explicitly and then the equilibrium equations become the system equations in terms of displacemen.t. An alternative appro.ach is to satisfy the equilibrium equations identically through the invocation of stress functions ~b~ . Here again one satisfies the constitutive equ~ttions explic!tly and it remains to express the compatibility equations as the system equations, in terms of ff~. In both of those approaches the constitutive equations are satisfied explicitly. *The results presented here were obtained in tile course of research sponsored by tile Natural Scic0cesand Engineering Research Council of Canada Grant No. A-1628. 611

612

S. Dost and B. Tabarrok

From a variational point of view the displacement and the stress function approaches are associated with the principles ofminimuln tol".i potential energy and the minimum complementary energy. By relaxing some of the admissibility requirements for the computing functions it is possible to generalise those variational principles and derive some mixed variational principles, e.g. the Hellit~ger-Reissner principle and Hu-Washizu principleU-2L In these classical variational formulations SOlne form of a potential energy is always introduced. The determination of such a scalar potential function requires the explicit satisfaction of the constitutive equations. Accordingly in all the classical variational principles of elastostatics, the constitutive equations are always explicitly satisfied.U

t ~

9Eqlm Eqs.

Compat. Eqs.

Force B.C.

Kinematic B.C.

-

Differential operators

Fig. I

~ Stress functions

Relationships amongst the variables

It is of interest to develop variational functionals for which the computing functions satisfy the equilibrium a n d the compatibility equations identicallyt3-sL "In such a functional the errors of approximation will be placed in the constitutive equations. This interest is motivated, at least in part, by computational considerations. Of the equation sets (i), (ii) and (iii) it can be seen that the first two are mathematical in nature and are therefore exact. By contrast the set (iii) is physical in nature and is then approximate, at least to the extent of experimental determination of material. parameters. It seems therefore more logical to place the computational errors in the set ofequations I) In tile lqu-Washizu Principle the constittttive equations are satisfied explicitly in the determination of thestrai,~ energy density function,A(e,,, ewr,'"). However, as extremum conditions a set of relations amongst the Lagi'ange nlullipliers o'zz,o'~uetc.and A are ob.tained as ( 0 ~ l - o-,,) ~0 etc. These relations require tllat the true stre'~s components be eqt,al to the corresponding Lagra!~ge multipliers. In the Hellinger-Reissner Principle the determination of the complementary strain energy density function B(~r,,, o'~,...) requires the explicit satislaction of the constitutive eqt, ations. Howe~r, 's extrumum conditioJls a ~t of relations a,'e obt~finedalnongst the h,grange multipliers e,,, e ~ . ele. and B are obtained as ~----e,, ~Oelc. These relations require that the true strain components be equal to the corresponding Lagrange Multipliers.

A Mixed Variational Formulation for Large Deformation of Plates

613

which are inherently approximate, rather than equations which are exact. Furthermore, from a comput.ational viewpoint it is preferrable to satisfy the d i f f e r e n t i a l equations of equilibrium and compatability explicitly and allow the errors to fall within the a l g e b r a i c constitutive equations.We shall also see that in the functional to be developed, the constitutive equations will be qatisfied globally in a least squares sense. In this functional the order of the derivatives is not higher than those that appear in the functionals for the minimum total potential energy and the minimum complementary enr~gy. This point is also of interest for development of finite element models. To illustrate the application of the present functional, a large deflection analysis is carried out Ibr a simply supported square plate, under uniform lateral loads, and the results obtained are compared with those in the literaturet2.6aJ. Further, the functional is then particularized for buckling analysis of plates, and in a simple example it is shown that the extremisation of the functional yields decoupled eigenvalue equations which correspond to those fron- the minimum potential energy and the complementary energy formulations.

II.

Basic Equations

The basic equations for the nonlinear plate theory may be written as follows: Strain, curvatures-displacements relations

e..=u,.q---~w, a. , (2.1) K wl ~t~ 'rl

elr~ppl~ I t o ,2 2e,w-----u,i + v , , + w , ~to, w ,

Equilibrium equations N,,,,+N.,,,=0,

N.,,.+N,,,,=0.

M..,,.+2M,,,,r+M,,.,,+N.,w,.. }

(2.2)

+ N,,w,,, + 2N.,w , . , = - ~

Constitutive equations

N..=C(e..+ve,,), N,.=C(e,,+ve..),

M..=--D(K.. +vK.)

N.,=C(1 -v)e.,,

M.,= -D(1--v)K.,

M,,=--D(K,,+vK..) }

(2.3)

C o m p a t i b i l i t y equations K.,.--K.~,~=O, e.. , . -

K.;,~--K.,,.=O

2e., , .~ + e . , .. + K . . w , ,, -- 2K.,w , .r

+K,rw,

,,

+ t o 2 , , _ w , ,,to, ~,----"0

where

C=-

Eh l__v2

,

Eh 3 D =-- 12(l__vz)"

and various symbols have their conventional meanings.

}

(2.4)

614

S. Dost and B. Tabarrok

Consider next" the boundary conditions, if along part of the boundary of the plale, denoted by S,,, kinematic conditions are prescribed, and along the remaining part S~, force quantities are specified, then the boundary conditions of the plate may be stated as follows: A l o n g Su u=~i, v=~, w=W, w,.=fO,. (2.5) A l o n g $~ N.. =N..,

N., =/q.,,

V . = V., M . . = M , , .

(2.6)

where V, is the effective shear force gix en by

V . = M . , , . + M . j , I + M . . , . +w, . N . . +w, IN., At c o r n e r s

M.,

IS+ a- =17.

(2.7)

where the last condition refers to the corner forccs / 7 . . A detailed account of the derivation of the above equations and their in'herent simplifying asst, mptions can be found in [I,7,8]. In the light of the above conditio,ls, we now examine a mixed variational formulation. Ill.

A Mixed V a r i a t i o n a l F o r m u l a t i o n

In this formulation- the c9mpatibility equations (2.4) are satisfied by the use of kinematic relations (2.1). The equilibrium equatons (2.2) will be satisfied by the following stress function: N..=~,,,,

N,,=,/,,,., /V..,=--4,,.,

(3.1)

and

M..=t/-,,-tor M,~=U~'.-wr

(3.2)

M,, = -.-~-(U,, +//', .) +w'b,., where we have assumed ~ is given through a po.tential

~

as follows

i0=~,,,+,0,,,

(3.3)

The function t~ is the Airy stress ftmction while U and V are the Southwell stress functions~.9j We now introduce a functional which will satisfy the constitutive equations (2.3) globally in a weighted least squares sense. This functional has the following form

it(M, N, K, ~)= .-~-.1 [~,,,~-'tKJ-+t,,~'-lcJ-]dX where we have defined tile following residual quantities

{m}=

M,, 2M=,

+D

[!.o ji ..j I 0 0 2(t-v)

Kv, K,,

(3.4)

A Mixed Variatiolml ' Formulation for Large Deformation of Plates

{n}=

IN..! i lvo N,,. 2N.,

--C

v 1 0 0

0

2(l--v)

e,,

-- {N} - C [ C l

{e}

615

(3.5)

9 b2st$,

IK} =~-[C]" l,,,}, {~}=u-[C]I , 1,,} The explicit form of

H ( M , N, K, e) is

f{ , 1 [M:,--2vM.,M,+M:, II(M, N, K , e ) - - , l2f JJ D ( 1 - - v z) "+2(1 + v ) M . ' , ] + C ( 1 -1- v ' )

[N2 __2vN.,JV,,+N2 (3.6)

+ 2 ( l + v ) N , ' , ] + 2[ M . . K . . + M , , K ,, + ~-M,,tC.,] -- 2.[ N . . e . . + N u , e , , + 9-N.,e,,]

+D[K'-.+2~K"K,,+K~,+2(1--v)K;,] I + C[e,, + 2ve,,e,, + e,,I + 2 (1 --v)e',,) }dxdtj

Now invoking the stress functions defined in Eqs. (3.1) and (3.2) and the strain, curvaturedisplacement relations in Eqs. (221), we may express H in terms of equilibrating stress resultants and compatible displacements. The extremum conditions of H(U, V, ~, u, v, w) then emerge a s

6U. -

1 ,[(M,,-vM.. ) -(l+v)-M'.,,,]=0 -D(l_vZ) ,"

(3.7)

!

1

~Y" -- D(l--v:) ''[(M..--vM,,) , , - - (.I+v)M.,..] =0 0r

1

-- D O - v : ' )

(3.8)

[ (wM..-- vtuM,,). ,, + (wM,,--vwM..)...

1 - 2 (1 + v ) ( w M . , ) , . ] + C ( 1 - v z ) '" [ ( N . - v N . )

,.,

+ (N,,--vN..) , . . - - 2(1 +v)N.,, .,] --w,,,w,,, +w, ~.,=O C[ (e.. + ve,) . . + (1--v)e.,.,]-----0 C[ ( e . +re..), w+ (1 --v)e.,,.] = 0 D[K.. + v K . ) ,.. + D ( K . + vK..) , . + 2 D ( 1 - - v ) K . , . 1

D ( 1 - - v 2)

(3.9) (3.10)

(3.11)

{( M . . - v M , , ) I V . . + ( M . - - v M . . ) N .

+ 2 (1 + v) M . , N . } -- ( N . . w , .. + N . w , .

+ 2N.w, .)

--C(e..+ve,,)w,..-C(e,,+ve..)tu,,,--2(1--v)Ce.,w,.,+~=O (3.12) In Eqs. (3.7) -(3. I I), the forces and the moments will be expressed in terms of the stress functions , U and Y and the displacement w through Eqs. (3, I) and (3.2), and correspondingly the strains and the curvalures will be Written in terms of the displacements u.uand w through the kinematical relations (2. I ). Eqs. (3.7) - (3. I i) in terms ol" U. tt, ~ , u, v. and w will then bc the system equations

616

S. Dost and B. Tabarrok

of the present formulation. It is noteworthy that these six equations represent tile computational errors in the constitutive equations mid in Ihct they imnaediately reduce to the compatibility equations (2.4) and the equilibrium tyquations (2.2), respectively, as soon as the constitutive equations (2.3) are satisfied explicitly. By means of a simple example we now illustrate tile appiication of tile present fornaulation and compare the results obtained with those derived by other formulations. Consider a rectangular plate of dimensions a and b which is simply supported along its four edges. For tile pertinent variables of 17 we take the following admissible funct'i~ons: u--u~sin 2~.x a U=Uteos

n'x t/

P LI~"~-"u l s 1. I. 12z'y ~

.

sm~

z'y

Y=Vtsin

/ uz tl

. cos

~ y

b

( (3.13)

a

sin-b--- , ~ b = r

a

sin

tt

Note that tile above functions satisfy all the boundary conditions of the plate. For the sake of simplicity and for purposes of comparison with published results, the functional in Eq. (3.6) was evaluated for the case o f a = b = 1, and v-----0.3 *~ . Minimization of the resulting functional yields six simultaneous equations for U,, Vpu,,v,, 4h andw,. In these equations all the parameters except w, appear linearly, and as such they may be eliminated, leaving the following nonlinear equation in terms o f w t :

h~EZ(5.44h 3 + 12. 347w, - t u ~ )to t -Fh~E[ -- 7. 895t01 (Ehaw ~ -t" 22.2 lwt,Q l) (14.06ha--I - 18.27w~ ) -i-I-29.71.,Qt ] - 8 7 . 6 7 ~ , (EhZw~ q-22.21w~t) (14.06h a-S 18.27w3, ) -' -b72.13wt(Ehaw~ +22.21w~t)~(14.06hZq-18,27w~)-Z=O

(3.14)

Note that in Eq. (3.14) some coeffic.ients have dimensions. By taking h--------'0.I, numerical solutions were obtained for w t and are depicted in Fig. 2. For purposes ofcomparison the results obtained by Levyl6i DonneiltTl and Tabarrok-'Dostl2t are also shown in Fig. 2. Levy employed von Karmfin's formulation and used a large number of terms in a set of series. As such his results may be regarded as exact for most practical purposes. Donnell on the other hand used first terms of a set ofseries for the same variables and deri(,ed a simplified theory. It can be seen from Fig. 2 that various results are generally in good agreement. If one were to obtain the internal Ibrces, then in principle it is possible to calculate those quantities via wt as well as urvr.Th'ses force quantities would generally violate the equilibrium equations. However, in the present formulation we also have the choice of calculating the internal forces IYom the stress functions'~/, I." and ~/~ , and the displacement w. These lattei" forces will satisfy the equilibrium equations. The difference between the l.wo sets of for: e quantities is an indication of the errors in the cons6tutive equations. 1) All tile algebraic maniputatL;,~s here were carried out using a symbolic comptltcr progrmu called "MACS.Y MA"

A Mixed Variatiomd Formulation for Large Deformation of Plates

617

1201 i _v~ )tr~rt t f2/81,)h,~ 100

Tabarrok.DostlZl

Domlel1171

Levy[~'~l

6(

/\ Present formulation

40

2C Linear theory

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 r

Fig. 2 Lateral displacement in a square plate

IV.

L i n e a r S t a b i l i t y of Plates

In order to obtain the stability condition of plates we consider some equilibrium positions ill the neighborhood of an initial equilibrium position. To this end we express the displacements, the moments and the in-plane forces in two parts as follows:

{u}={U}o-I-{u}t, {N}={N}o+{N}t,

W=Wo-t-wt

(4. la, b) (4.2)

{M}={M}o-~{Mh

where {u},t and wo are the displacements corresponding to the initial position while {U}l and wt are the incre,nental displacements. {N}o and {M}• are also defined as the initial in-plane forces and moments while {N}, and {M}l denote the incremental quantities. Using the definitions (4.16) and (4.2) in the equilibrium equations (2.2) and eliminating the nonlinear qt, antities we obtain Ar,,~,, + / , , , , , = 0 ,

Nt,,,,+/,,,,,=O

Ml,,,f2 + 2Ml,w,,r +Miw ~,rl_St', _Slw,~= 0

}

(4.3)

where we have assumedwo,, and wo.I are negligibly small and the initial force field {N}o and { M } , are in self-equilibrium, i.e. No:,,,+No,~,,=0

No,~,f+No~w,r'-O } ,, = -- ~

Mo . . . . -I- 2Mo,~, ,, -t-Mo,,, and where

(4.4)

618

S. Dost and B. Tabarrok S,, - - / ' V o . w l , , --/'Vo.t,wl,

(4.5)

S,, = --/Vo,,wt,,--No,,W,,, Recognizing that S,,and S,,are force quantities we can see that we may express the out-of-plane equilibrium equation entirely in terms of forces. In so doing we must also account for the relations between S,,and S~,, andwt, ,andw,, ,in Eqs. (4.5). Since these equations establish a relationaship between force and kinematic quantities, we may, for analysis purposes, classify Eqs. (4.5) as a new set ot~,albeit artificial, constitutive equations. We shall see that in this way one can obtain decoupled sets of governing equations from our functional in Eq, (4.13). It is of interest to note that a similar approach was used by Stumpf for developing a complementary, energy formulation for large deformation analysis of platesU~ Using Eqs. (4.1) and (4.2); and neglecting nonlinear terms, / / may be expressed as follows:

/7=/7~

(4.6)

where

nof•2)

[{m}r{K}o+{n} {a}o]dA

//t-----lI~[ {m}r {K}, + {m}r {K}, + {.} ro{e},+ {n}r {8},]dA

(4.7) (4.8) (4.9)

and where

{P},--{S},+[N]o{w},,

{q}t= [N]o' {p}t

(4.10)

and

{S} r,=ESL, St,], .[w} r~=Fw,,. w,,,]

(4.11)

~Nb,, /V~

[N]'=I N,,,

Since the constitutive equations in the initial equilibrium position are satisfied. (][I~ leaving / p extremised, i.e. ~177= 0

' --.0 (4.12)

This is our variational formulation for the stability of plates. The explicit form of /72 is u =-~-jjUO(l_v2) +2(1-~-,)M:,]+

[M'.,~2vM,M,+M~, 1 C(l_v2)

[N'-.--2vN,.N,+N'w,

+-2(1 -l-v)iN:,] +2[.M,,K,, + M , , f f . + 2 M . , K . , ] -- 2 [N..e.. + N . e . + 2N.,e.]

+D[K;, + 2vK,,K,,+ K ~ , + 2 ( 1 - v ) K ; , ] + C[ e~., + 2ve,,e,, + e;, + 2 ( t - - v)e2-, ] 1)From now on,. we will omit inder I for simplicity.

(4.13)

A Mixed Variatiouai Formulation for Large Deformation of Plates

619

0 ~ '0 2 0 +'Iv* m* --IV* ~ [S . 2.N,, ,2S.S,N.,+S,N..] *" Sl 4"

lW

*" mJ

+ 2 I S , w , , + S , w , , ] + [N:,w, : + 2NO,w, ,w, ,+N*,,w, :]}.dxdy where

1 i+u ,.) 1" e.,=u,., e,,=v,,,.e.,=~(u, K.,=w,.., Ir K.,=,w,.,

('4.14)

We now consider the classical buckling of a simply .supported rectangular plate. In this case

No,,=--N,

(4.15)

N.,=N,.,=0

Equilibrium equations in Eqs. (4.3) are satisfied identically by

N..=N,,=N.,=O

(4.16)

and

M..=V,,+IS.dx;

M , , = U ,,, 2M.,=--U,,--V,.

We now choose

w=w~sin;txsinpy U =U leoslxsinpy V=VisinAxeosl~y S,=S,eoslxsinpy

(4.17)

} (4.18)

where ~=

m.,'T

a

tl.,-t

' /~=

b

(m,

n=l,

2,

"")

Note that expressions (4.18) satisfy all the boundary conditions. By using these functions in functional (4.13) we obtain

tL--r +

l+v

+ av,)' } +

(4.19)

Extremisation of 175 yields I z-l- l-l-v #z

---'T-

l--v ii'

U,

2 /z --/-

p , ..~-.--...[-l + l ' 2z I

7 Sym

D(I-), z)

N

0

VI

0

S,

D(A'+/z')'--A'N"

tt,t

- 0 (4.20)

620

S. Dost and B. Tabarrok

which is an eigenvalue problem for N. It is interesting to see that the eigenvalue equations for toj, and Ibr U~, V~, and S~, are decoupled, i.e,

[D(A'+M~)2-A'N]wI=O

(4.21)

and 2z

l+r +T

~

#

1--v ~,/z 2 lq-a' :2 t?+~-

Sym

r,'].

U!

--t~

VI

1--

D ( 1 - v z ) 2z N "

=0

(4.22)

_ Sl

Both Eqs. (4.21) and (4.22) yield the same exact buckling load since the exact admissible functions were used:

N=+(2z--t-#2) z

(4.23)

Eq. (4.21) correspopds to the result obtained from the minimum potential energy principle while Eqs. (4.22) represent the eigenvalue problem obtained from the complementary formulation given in [2]. For admissible functions other than the exact solution, two .different approximate buckling loads, corresponding to the results from the minimum potential energy and the complernentary formulations, will be obtained from the present procedure. It can be shown that both of these Will be bounded below by the exact eigenvalueslSL Depending upon the admissible functions used and the boundary conditions of the plate, one of the eigenvalues will be closer to the exact value. V.

Conclusions

In this paper a mixed variational formulation for large deformation analysis of plates has been introduced. In this formulation tile constitutive equations are satisfied globally in a least square sense, while the equilibriunl equations are satisfied identically by ineans of stress functions and the compatibility conditions are satisfied by the use of continuous displacements. The extremisation yields the system equations in terms of the stress functions and displacements representing the errors in the constitutive equations and they reduce to the equilibrium and compatibility equations if the constitutive equations are satisfied explicitly. Since in this fol'mulation equilibrating stress resultants constitute part of tile prhnary variable, they are expected to be more accurate than the non-equilibrating stress resultants conventionally obtained by differentiating the approximate displacement field. The proposed formulation was illustrated by a large deflection analysis of a square plate and the results obtained were found to be in good agreement with those in the literature. In the second part of this paper, the stability analysis oi" plates was examined. The wlriational statement of the stability of plates was obtained by linearizing the functinal in the neighborhood of an "initial equilibrium position. The presence of the lateral displacement in the out-of-plane equilibrium equation was eliminated by introducing a new set of force quantities, which were added to the functional as an artificial set ofconstitutive equations. The introduction of this new force field in the analysis made it possible to obtai, dccoupled eigenwdue equations in terms of the stress

A Mixed Variational Formulation for Large Delbrmation of Plates

621

functions and the lateral displacement. These two sets ofeigenvalue equations correspond to those obtained from the complementary energy al'ld the minimum potential energy formulations, respectively. When exact admissihlc l't'mctions arc tlscd, the two sets of cigenvalucs coincide. The decoupling of these equations can be considered the main feature of the present formulation. For Ioadings and boundary conditions tbr which exact admissible functions cannot be obtained, our functional will yield two different approximate solutions for the buckling loads and eigenmodes, which will correspond to solutions obtained from the minimum potential energy and the complementary energy formulationslSJ. References

[ I ] Washizu, K., Variational Method~' hz Eht.~lk'it.v and PlasticiO,. Pergamon Press, 3rd Edilion (1982). [~2 1 Tabarrok, B. and S. Dost, Some variational [ormuhltions for large dclbrmation-analysis of plates, Coo.tpt. Meth. hi Appl. Mech. Engg. 22. (1980), 279. [ 3 ] Chien, W.Z., Further study on generalized variational principles in elasticity, Paper presented at 16th International Congress of Theoretical and Applied Mechanics, 19-25 August, Lyngby, Denmark (1984). [ 4 ] Qing, J. and H. Li, New wlriational principles in linear and non-linear theories of elasticity, thermoelasticity and viscoelasticity, Paper presented at 16th Internati6nal Congress of Theoretical and Applied Mechanics, 19-25 August, Lyngby, Denmark (1984). [.5 ] Tabarrok, B. and L. Assamoi, A new variational principle in elastod~mmics, J. Compt. Meth. Appl. Mech. Engg. (to appear) [ 6 ] Levy, S., Bending of rectangular plates with large deflections, NAC'A Tech. Note, No.846 (1942). [ 7 ] Donnell, L.H., Beams, Plates, and Shells, Mc-Graw Hill (1976). [ 8 ] Fung, Y.C.,Foundations of Solid Mechanics, Prentice-Hall (1965). [ 9 ] Southwell, R.V., On the analogues relating flexure and extention of flat plates, Q. J. Mech. Appl. Math.. 3 (1950), 257-270. [10] Stumpf, H., Die Extremalprinzipe der Nichtlinearen Plattentheorie, ZAMM, /15 (1975), 110-112.