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An intrinsic formulation for the nonlinear theory of shells and some approximations
AN INTRINSIC FORMULATION FOR THE NONLINEAR THEORY OF SHELLS AND SOME APPROXIMATIONS ROGERVW officeNational d’Etudes et de Recherchcs Acrospatiales(ONERA),
92320 Chatillon, France
(Receiwd 22 May 1978)
AWract-An intrinsicgeometric formulation of the nonlinear theory of sheik is proposed.Asymptotic expansions of the variational formulationis made in view of its applicationsby the finite element method. These expaasions take account for the variation of the connection in case of non-nqiiibk extension of the mean surface. A hybrid mixed principle also presented in an intrinsic formulation utiiiis the preceding asymptotic developments.
1. lNTmDucTtoN
of powerful computers and the great versabibty of the finite element method as a discretixed metbod have resulted in several consequences in structural analysis. Fust of all after a remarkable success in the linear fkld, the method has heen used since several years in the non-linear field, (non-linear deformations or non-linear constitutive laws of materials), by means of appropriate aigoritbms and at tbe cost of greater computational efforts. This complexity is evolve in particukr in buckling analysis, but leads sometimes to undue s~p~~ons. It is tben necessary to bring into tire formulation other non&tear terms which were dis-’ regarded up to that time[ 11. Secondly tbe very use of these powerful and refined means has heen responsible for a certain decline of theoretical investigations concerning sbell probkms. in effect in many applications tbe analysts did not be&ate to use ~~~en~o~, or like-looking tbreedimensional tkite ekements,wbose merit was to refer to more elementary tbeories, in particular in tbe non-linear field, to he easily programmed, tbanks to the uti&ation of isoparametric elements[2,31;and to allow for tbe analysis of very miscellaneouscases as those of multilayered shells made of anisotropic composite materials taking account for large transverse shears and thickness variations as well[4]. But tbis decline couId also be caused by the great decays of the theory wbieb can he bandkd only tbrougb drastic but not always acceptable simplifications. Nevertheless some studies of a theoretical character remain fruitful in this domain, not only for tbe sake of those who are going on applying tbe two-dimensional shell theories, but also to provide the others, we mean the adepts of tbreediiensional theories, with tbe results and approximations given and justified as consequences of shell theories. In the last resort we can argue that tbe three-dimensionaltheory appliedto shell structures does not fail to And its own limitations due to the limits of accuracy of tbe computers which are actually available: in fact smaller becomes the shell tbicbness, greater has to he tbe numericalprecision in order to avoid unacceptable quasisingukrities (ill-conditioned matrices); but this has also its liiitations, though these ones are nowadays more and more remote, as can be observed. As it is well known the non-linear tbeory of shells has
The use
heen marbed by important contributions, among which can be quoted those of Sanders[SJ,Koiter[6], Mushtari and Gabmov[‘!I. Here we propose an application of an intrinsic geometric method, worked out in tbe linear case@-111, to tbe development of tbe shell theory in tbe non-linear fkld, and in vkw of some asymptotic expansions, in a variational form&ion in order to mabe tbem workable by tbe finite element method. Tbetbcorywillbecompktedbytbeformuktionofa mixed-hybrid non-linear principk, in tbe Rian meaning, from the pmcedkg ~~1~~~. taIlEm NcnMWEuWOCaralm Tbe formuktion is developed fbst in tbe frame of tbe Kircboff-Love kinematic assumptions, tbat is to say in negkcting tbe transverse shear strain and normal strain ClWgiCS. Cdiii E,, in tbe actual state 8, tbe deformed mean ~~~~c~ktrn~~~~~~~t normal N, it is assumed that 2, is an oriented, parametric, di&mtiabk, compact surface,embedded in tbe tbreedimensional eucbdian space ES.2, is supposed to have a Rkmanian connection, and its boundary will be calkd CL,.Then there exists at each point m of X,,, a linear mt manifoldcalkd I$, also euclidian,a metric diierentiabk tensor-fteld defbmd on &, induced by tbe metric of tbe embedding space ES, and fkaliy an bermitian Projector fkk =, WtboPml GOVE, also differontkbk, wbkb applks & upon & and which defines the Rkmanian linear connection. Us&a superbar for any transpose quantity, tben n and NN are two bermitian supplementary projectors respectively upon & and N, so that
Q=*2=v
+r+NN=lE,,
(2.1)
where 113,is tbe identity operator on ES. Let f he any Minsk tangent vector&eld which vahle~is v, so that V=f(m)EEz
the covarknt differential a V = w d V, V dm E &. if m is a map of class C’ of H, defined on an open set of R2-
IS3
m=@X),
VXEOCR’
ROGER VALID
184
It is found that
S will be the natural basis, at point m, of 82, if dm =$x=SdX
S=s,
(2.6) In general any differential vector V and its derivative will be related by the formula It is remembered that dV=gdm. Sm E $, Then let YE R’ represent the vector V E 4% in the basis S, it comes: V=SY,
S[~dm]=O,
asm
m)
in respect of which any material unit normal remains a geometric unit normal. The virtual strain energy is then referred to a previous prestressed but well known state %, corresponding to the mean surface Z,,,, of generic point m. and tangent plane J3m. Using the identity operator
~dV=nd[nVI+Pd[N~Vl =~[=~+~~~v~u~~v sdm
rdV=&I+NV8m where
VdmE$
8N=-am~
(2.2)
r is the Christoffel operator, f(dX) the Cl&toffCl matrix, and l$ = r: theClxlstoffel symbols of the Rle~ conne&m of P, If the surface vector held is a threedimension vector Gelddellned on z,
VVE&(asafunctionof
V8mE&,
giVCS
d V = &[I’; d’X’Yt $Y].
V=nV+NNV,
and that the kinematic assumptionaccording to which
dV=dSY+SdY
dV=ndSY+SdY = SWXW+dY1
8N f & because g = 1,
dm am, --= am0 am
(2.2)
aN/dm isthe curvature operator of 8, at m, and
1
E2
formula (2.6) becomes by reciprocal image transformation:
&t&es
the second fundamental form of &,: d~~~ff~a~)dm. ~~~~~~~~n~~setof~~~ operators from the linear space E on the liiar space Ft, let us recall that
iii
a8N a&n aN I-t-n,-g--)dis(~)&. am0 am0
andthattbefirst~~formofZ,atmisdmdnr, the scalar square product of any dilferential tangent vector dm E K. Under that conditions the variational formulation of the shell theory can he achieved directly from the three~~~~~~~~o~tf~~~a~ ~s~p~ns, and leads to the expression of tk vimal deformation energy, as referred to previous studies[111:
Whence:
with SW=
I x,
T,(n&y+ m&r)CCE,. VSmE E3 (2.5)
where ISand m are the surface Cauchy stresses, respectively dual of the virtual surface ~f~~ns 87 and &r of extension and cmvature variation of 2,. In that formula n, 14 &y, &: are hermitian endomorphisms of the tangent plane rf, at m of t,.
tNotc
(2.7)
that BERBR),
VVE8
The
two genera&d surface stresses a’ and m’are those
of Hola-IGhhoff ; they are hermitiantoo and defined on
& in the reference state if0. The non-linear surface &f~a~~ f and K, due to the vect~~~~~ V of point m between the states I and %,are respectively defined by
An intrinsic fo~u~ation for the nonlinear theoryof sheUsand someapproximations
Now
dM=dm+dNziNdz, drmd.Cmo=fdlmo
dnram
amz--Ie [ 0 0
VdmE&
185
VrER,
then I d2m&
dMdM=[dmidNz+jjdr][dm+dNz+Ndr] dMdM=dmdmi2dmdNz+dNdNz’+dz*
Whence &
sr=i
atnamlr& -2 --_ amoamo
C
3
= i;
Z a8m aSmam =Z. --_-+-2 [ &noam0 am0am01
I
The commutation(@mo)8 = W/am,) has been utiIized everywhere, due to the fact that the virtual variation 8m is made for any given point m.on );Now
w~n~N=l,~N=O=~N. The second term of the difference is derived from the first one by putting a subscript zero at ah terms. It is seen, as it is well known, that the deformation is a function not only of the first and second form variations, but also of the third one dNdN =~~~drn %dN
,;i;l~% aN aN am o~~~~dm~ 0 0
StiIt through a simple transformation and aIlowingfor the reiation
so that the fotmuiae can aIso be written:
am am0 --_=I am0am H2 it comes:
(2.9) EdNe;i;;;.drnaN
o~amO
am dm am am am0 O
am a&aN
Then (2.8), gives: drmak: d2mo= h,mo
K=
;I;;; aN am alvo zXz&zK 0 aI d2m,
dNdN=i&
m,[j$$+]u,,-a
6KmriGaN 0 t
am
I
(2.10)
I
0
So with (2.7) it is found again the formula: SW=
I I,0
T,(nW + m’6K) dZo
(2.11)
where every stress or strain surface field in the actual state 8 is defined on &, the tangent plan of H, in the reference state go. It should be noted that the three&tensional deformation at any point M of the shelI in the state 8, of projection m upon the tangent plane, is given by the difference ~d~-~dMo. If
M=m+ z=m+Nz,
[ K+aNo ~~t~+I~~-‘~+~]drn*
(2.12)
-’ aaN --+--n~ro=lEK. am0 am0 am0 am0
asm with&N=-xNO
whence by (2.91,and (ZIO),:
As to the last term dt2 it would give the thickness ~fo~a~n whose contribution in the strain energy has been neglected. In the setting up of an elastic constitutive law, for instance, it may be used the linear law, which is written in linearized strains as functions of the displacement vector
V=m-ma, the only independent variable is the present case. But fuller expansions, up to a given order in terms of V, will take account, strictly speaking, for non-linear ~fo~~es which are also functions of the third fundamental form of Z,. If it is obvious that this term will be linked in the strain energy to the surface couple-stress because of the factor z2, the way by which this term WP contribute and be associated to the &urge of curvature will be devised only through a simiIarway as that was worked out to 8nd the Iinear law. Incidentahy it is recommended in that case to adopt Na&di’s elegant method[ll, 111,which
186
ROGER VALID
utilizes very logically the variational principle outlined by Hu-Washizu, precisely in view to establish in the mean a bi-dirnensional law. We shah not here undertake this task which is just to be modeled upon Naghdi’sone.
by the second line of (2.141,thatis to say: aru,aAN
2 ---z2 1ahNaAN
amT7ZC2dm
am
dm.
I
’
It will be remembered that (2.5) and (2.11)are correct only in involvingthe Kiichhoff-Love assumptions. If those hypotheses are left away, it is necessary to introduce in particular the transverse shear, and SN
and the transverse shear deformation would be the only term
becomes, as usual, another (virtual) independent kinematic variable. In that case let M be again a point of the shell whose projection upon the tangent plane & is m, in the actual state &?.We have again
It is incidentallyeasy to refer these terms by reciprocal image transformation to the reference state L. The
M=m+Nz,
extra-tangential deformation (with unvariable thickness) becomes: z dN aAN amzEZ+__L_Z2
whereflN=l,
am0 am0
and VdmE&,
dM=dm+dNz+Ndz,
drER.
The non-linear deformation at M wiU again be given in computing the following difference referred to the state 8.0.
am, am am0 -1 am aAN aAN am 2 +?Ko am am amOL ’
and the transverse shear strain: z[ANNo].
dMdM-dModMo, IA us call
but now dMdM=dmdm+2dmdNz+dNdNz2 + dr2 + 2dmN dz.
(2.13) then by similar arguments as for (2.12):
Let us put N=N,tAN.
with
amaN,%_ am0 dm am0
tiN=&N,=i,
z aN,aAN am -am0 am am am0
--
where N, is the geometric unit normal, and AN the extra-variation of the mate& normal, such that: AN=N-N,. It is shown that the vector AN is only responsible of the transverse shear in the shell. Then keeping in mind that the condition N=dm=O
And also: -I dm --JAN JAN am
idmoam amam,--
still holds, (2.13) becomes: dMdM=dmdm
--
1 am dAN am amoamoamdAN am =-------w2 am, am amoam am am0 am am,
t2dmdNd+dN,dN,z2+dz2
t2dmiiANzt2dN,dANz2t~dANz2 t Z&AN dz
(2.14)
All that is necessary is to subtract to (2.14) the square d&d& to obtain the three-dimensional deformation at
M. Now the first line of (2.14)givestheexact deformadon in the case of the Linemotic Rirchoff-Love assumptions. If the computation has been achieved through these hypotheses, all that is necessary to obtain the extra-strain is to add the contribution provided by the two foIIowing lines, where are the supplementary variation AN and its derivative. In case that it should be admitted that No in the state & be the geometric unit normaI of & and that the thickness variation shotdd be provisionally negIected. the extra-tangential shear strain due to AN would be given
But of course it is easier to utilize
--
I am aANaAN am ZdmodmdmYGG=2
1dbNdAN ---
am0 dmo'
Finally the components of the extra-strain
TOLL. NoNo =
D, are
(2.15)
&ANIVo. 0
If on the contrary in the state go the vector No is not orthogonal to I;,,,,,, then No = No, + ANo
187
An intrinsic formulation for the no&neartheoryof shetisand someapproximations
where Nomis the geometric unit normal of Z,. In that case it is necessary to subtract from the preceding deformation, the same quantity subscripted by zero, and the same for z, if .z# 4. It is worth noting that the virtual surface strain is also modified. In fact, (for z = z. giving a zero thickness strain energy).
am Go-m’vo=-4p,
gives a virtual displacement 6M = &r + &N#t+ SAP&, and the strain energy is supplemented in the following way in state 8:
baa -am
O~dVO 0
In
The second term of the inte~and is the pr~uct of two skew-symmet~c operators of $ and can be written as the product of two scalar quantities. It is quite easy to refer this strain energy by reciprocal images to I, in the state Pr, The theory must be completed by the research of a surface constitutive law in that case of transverse shear deformation, always assuming that the normal strain energy is negligible.This assumption will be realized in a realistic way by neglectingthe normal stress component @XX. If the constitutive faw is to be found in linear elasticity for instance in taking onfy into account the hnearized strains, then an a~roximate law will give, by similar previous ~guments, relations between the surface stresses IF,
m, xh
Q.
and the surface strains
But it is easy to outline that this s~~~~ficationis not acceptable if one wants to keep for instance in the energy expansion all terms of, say, second order, as it will be sub~quendy done in a strict sense. In the preceding method the only kinematic variables are definitively V= m - nro and AN = N-N,, and ah the deformations have to be expressed non linearly in terms of these variables. In fact this computation is very cumbersome and probably useless in general. So we intend to put forward some approximate expansions. It should be noted lastly that the preceding formulae allow to write immediately the equ~ib~~ equations, referred to the mean surface Xmcand its boundary C, in the reference state go, in a quite classical way, by means of inte~at~ons by parts and use of Stoke’s form&a in ~emanian connection. The fo~lowi~ local equ~i~um. equations are written in an intrinsic form:
*
I
-
= ph.
at k points on C,.
that formulae[lll
I (2.17)
f is a surface load density given on 2,: F is a curvifinear load density given on C,,+ C &,; rp is a coupk~om~nent load density given on C&F; $%are coupk~orn~~nt ~~ontinuit~s given on k: points of C+ to and v. are unit vectors, tangent and normal respecttvely to C, on X, 3MWPlvncgxPIWBogsAsFuNRlNcFMNs OFKINSMATICUNKNWNS
Several approximations are possible from the variationnal preceding formulae. Let us recall that according to the magmtude of the surface deikction, Koiterf61 considered 4 dasscs of problems, name&: (a) Large deflections characterized by the absence of restriction. (b) Moderate deflections ch~cte~ by small values of ah di~~~rn~t gradient comments in apex with unity, ahhough nothing is assumed in advance about their relative orders of magnitude. (c) Small finite deflections characterized by smal1displacement gradients and by rotations whose squares do not exceed the middle surface strains in order of magnC tude. (d) Infinitesimal deflections characterized by small displacement gradients, none of which exceeds the middle surface strains in order of rn~j~. goiter has so completed the three classes previously ~n~~u~ by Musbtari and GaIiiov~‘ff, by the ctass fi in which at least one component of the Robin, small in comp~~n with unity, has its square large in ~~p~son with the mean surface strains. These ro~~ons are some components of the derivative W/am, of V at & In that case ~~V~~~~~. fn considering this last assumption which has been little explored, it is possible to make limited asymptotic expansions, for instance up to the second order against V and AN, that is the full order of the non-linear purely threedimensionai theory, where many applications have &own the necessity of keeping all terms. These expansions will be applied to all terms of (2.11) or (2.16)in view of subsequent applications by the finite element method, by means of in~olations of tire variables V and AN, and pacify in the case of nonlinear bu~~rn~ in the fohowing for sake of brevity the expansion will
ROGER\IhLID
188 be
developed only in the frame of the Kirchhoff-Love /(3.2) gives then successively with CF’ =0 assumptions. In order to take account for a possibly non negligible I& dmo= 0, &NO = 1 extension of the mean surface. the expansion will be applied also to the Riemanian connection of 2, represented by projector n. The condom can tK.c~c~at~, as it is w&-known, by means of the Christoffel symbols which are given by Whence r’,, = $j%., + &.k
-
i,j,k,l=1,2,
&?kd
There gil = $S, are the components of the Gram matrix SS of S = &n/ax a natural basis relative to a map of a complete atlas of P,. It is clear that the connation in the state 8 depends on the variation of the first fun~en~ form r. But this method will certainly be most cumbersome. Another one would be to develop a expressed with the basis S by the classical formula
(3.3’ It is then possibIeto expand projector ?r such that
which gives 7r= S[$!q-‘5 but this method is also rather heavy. It has been preferred the simpler followingone: FWting
(3.4)
By means of (3.3)and (3.4).the deformations r, &I’.K, SK can then be expanded up to the second order. Thus for I’ given by (2.9), with (3.4) and with by proper expansion
n= 1&_NE;j N=No+eN,+g2N2,
-[iv,&+ No&l ,r2= -fN,& + N&+ N&l. I
WI =
with
c’=O
m=mo+ v
I
aV am --&=?rO+~~=o the first bracket of the right hand member provides the linearized extension of an element of Z, the second bracket provides the rotation of that ehment around the normal, and the iast term gives the linearized rotation of the normal itself. Then assuming[W/ad 4 1, the foliowingWUttiOnS ~dm=O,
flN=l,
VdmoElh,
one writes 1
r=p
av no+er~Ono
I
I[
av CO 'Iltr+~~~*no -5, I
and finally
can be expanded from: & dmo = 0, &No= I. gN&o$-] 0
ThUS
or I-
.sr is given by (2.9)~and by similararguments It should be noted that aV/am is supposed to be small compared with 1% extended upon & by projector ‘~0, and our intention being to keep every first and second whence order term it is useless to put: av _,s++ am,- am, tTranspose quantity.
0
+_** e
2
Z aw +amam XF av lr~C0.E2,
am0am,
0
1
0
(3.5)
An intrinsic formulation for the nonlinear theory of shells and some approximations
189
or 8 =
sr*+Bf,.
(3.6)
where admittedly It is thus revealed that the second order term SF, vanishes exactly, so that the expansion up to the first order constitutes an excellent approximationby itself. The quantities K and SK, which are given by (2.10). The fonnuhte can be developed in taking account for will be developed in the same manner, but the compu- (3.10);(or for complicated laws if other terms are introtation, though as simple as the preceding, is very much duced which arc non-linear functions of I and K), in the longer. One finds then following guidelines. (3.51,(3.7) and (3.10)give a’ and m’ in the form of: n”~+cn,+c21t2 [ m’=mo+rmt+c2m2’ (3.71
Finally (3.6) and (3.8) give
6r=ia+csr, [ SK = 6Ko + &K, + r26K2,
or K = SK, + C’KS.
f
and (3.9) becomes:
Then SK = SK, + &K, + e2SKz
SW-
with
1
TAti+ enI + 2n2wkteSr,j +bb+iml
+%lJlsK,+ear,+&v&]) a%,
(3.11)
But in (3.1I) the product of terms subscripted by zero must verify the equiIibriumin the state goo,possibly with external forces. Leaving out these terms, it remains up to the second order:
(3.81 + 2b2iwo+ ihn, +m4Ko+ mdKt + mdK& ao. (3.12)
In this formula the vector eN has been developed simi!ariy in the form of:
Now we may extend to more general problems the following admitted remark of KoiterIl9: “Finally the most important shell buckling problems are those in which the fundamental state I, (%01,is (approximately) a membrane state of stress, and the tensor of stress resuftam M”, (IIS&may then be negiected everywhere”. It shouid be also noted that a shell which would be initially built to resist to large bending couples would be certainly ill-drawnor utilized. Then ignoringthe bending term tm,,(3.12)gives: SW=
T,teftirr f =m0
+ sMo + mt8Kol + e2fa2bro
+l@r,+adKo+mt6K,])dL
(3.13)
in which the very complicated term SK2has disappeared. It is aiso powibk when the need arises, as will be remembered, to take account for extra-rotations and x (No+ tN, + e2N21. shear stresses, and linked tangantialenergies. The pre&ing asymptotic expansions are quite adapThe problem is now to calculate (2.1I), namely: ted to the use of the We eicmtat method for non-Iii curved shells with or without transverse shear, in a total T,(n’Gf t m’SK)d&. 13.9) or updated 4rarq#m procuhtre. In this fixation the =nyl constitutive law will be elastic, hypereiastic or even If we assume to simplify that a linear constitutive law hypoelastic (in particular in the case of a plastic flow law). is set down for the shell between the stresses n’, m’ and Let us also notice that all these formulae can be the deformations I and R. and if no and m. are known considerably simpliied in the case of plates by vanishing surface prestresses in the state Ed,then
I
ROGER VALID
190
In order to throw a light on this formulation in an the curvature operator <3mo, but in the natural State intrinsic manner, it seems useful to show how such a only. method can be applied to the more simple case of If the shell is submitted to external follower loads, it threedimensional structures. will be necessary to expand these loads consistently up Let us consider a three-dimensional body, made of a to the same order as the other terms. Let us consider for instance the important case of a . non-polar material, which occupies a volume & of boundary Z. in the reference state go. and which is pressure loading. The work of the pressure p in a virtual submitted to a volume density f of external load and to a displacement SV of Z, in the state 8 is developed up to surface density F of external load on a part SW of the the second order in the following way: boundary I,,. One supposes also that the displacement U is given on the complementary part Pov of X0. no is the ZpN ti = %[po + 91 + e’pz] internal unit normal of IO. x[N,+ eN,+ c*N&&+e d, + e*d&] Let ME fi be any point of the body in the actual state 8, which corresponds to M,E!?, in the reference state where the constitutive law for example CO. -In the state $ the equilibrium is given by the following principle of virtual work referred to go:
pv’ = povoy between pressure and volume allows to calculate the terms pl and p2 in terms of V. Vectors N, and N2 being already calculated, it is only necessary to expand dI asymptotically. In developing the quantity dI = N d,m x dim,
I o,,
T,(C’SD) dfio - I,~HJ
dflo - I,,
&J
&o = 0.
V 6CJC.A.
(4.1)
SU is the value of any kinematic admissible displacement field (C.A.); c’ = c’ is the Piola-Kirchhoff stress in the state g, referred to 0,: C’E &??,, &): and with M= Ma+U.
Vd,m, d2m E &
, it is found: I
1
=bEa(B
39 8,
3
(4.2) UC.A. j U = U, on Zoo 8UC.A. $8U
d):
=d&+.[&?roV+ T,(~)&,V]
= 0 on PO”.
dZ,,
It is supposed for sake of simplicity that the constitutive law is linear such that if one considers c’ and D as vectors of an euclidian six-dimensional vector-space $. C’=AQwhere A=~EZ(&,.$). T,(D,&) = &a, where Lj is the transpose of D in $. Then the strain energy volume density will be written as a quadratic form of D:
+e*[~oAdj(uo~uo)N.+~~o~~N.]d2o. In the particular case where V = No?, (T E R), dX-dPo=
-~~~+~I~+wz~*+;~~
(3*‘4)
I
a =$4D=+(D)(D).
dIo (3.15)
where pI and h are the principal curvatures of PO. It is worth noting that the Laplace law of capillarity is a law of isotropic extension of the mean surface which utilizes (3.15). Incidentally the product of &V times dZ gives the virtual variation 6~. To conchrde this paragraph one may say that the precedii developments are well adapted to derive the second derivative of the potential energy as a function of the displacement V, keeping SV constant, which derivative govcms the sfatic stability probkms of conservative stJuctures[l4, 13, 111. +DBxBoANDrrYBuDmNcDLef~
Now it is possible to take the variable D as an independent variable in the domain fl,, by means of a Lagrange multiplier t = ?E Y(&. &I and to introduce the constraint U-Ud=OonZov (4.3) in a weak form by means of another multiplier T E ES. (4.1) gives then: (-/,[a+
~,(t[f~~-~la-D])-jLi]dno
-I
PUCE,-
I,,
nu
-
U,l dz,] = 0,
%F
NON-UNEARSBHLS
An interesting application of the preceding developments is the formulation of mixed principles for nonlinear shells.
vt=ilr,~.u.
(4.4)
Hardly necessary to say that these variables must
An intrinsicfo~ulalion for the nonlineartheoryof shellsandsomeapproximations
191
belong to certain definite spaces in order that every energy which is involved in (4.4) be bounded. The Euler equations of principle (4.4)are immediately: +
1 aMaM 5 cai\l,z-‘~, div I$$ [
I =D
ZOF
in &
1
+
+f=O
0
u = Lid
ii&O-
I
on Zou.
I
aMo
ud&#
1
d&,
divC,tf,=O,
(4.8)
in&,
(4.9)
(j CIS.A.k statically admissibleinside), it remains:
+
C=AI)*D=A-‘(J’tBC’,
I
[iioC,-FJUdzot
%F
(with B = A-’ E .5?(&, $). -
In fact we shall separate as usual the prestress Co in state & from its increment C1in state 8 such that:
co+ ct
Co is then the Cauchy prestress in &. It will also be set down the strain energy in terms of stresses between go and 8: /3 = $,BC,
0
If one takes C, in a class such that
If we assume &at (43, is strictly verified in (4.4),then the m~tiplier t is precisely the local stress C’ and the variable L) must disappear from the principle. Under that condition:
=
“OCOZ
4u
+f,U,
Z&O ‘OfaMo z-t+) ‘OfaMo
C’
I [ aU +iioc,-F, I [[ aU +iioc,-Ijst u u,7-1. I I=0,ve,,
= f~(C,XG).
whence now
i’,[U-
&lldXo]=O,
I zou
hC,
u
VC,S.A.I., U, T,.
(4.IO)
At this step it would be tentatively possible to eliminate the vector T, by means of the local equation
but it would be seen that terms functions of U would not be totally eliminated in the integral on IO” as it must be. It is then necessary to keep the multiplier T,, as independant variable, or take U in the class where U = U,, on &. Then (4.10)gives finally:
D= BC,.
s
Principle (4.4) becomes then:
VCJ.A.L, W,T,.
-I
RJdZ0ZOF
I,,,F[u -
ud]
d?;O]
=
0,
vc,, u,T.
(4.6) It is quite possible to achieve the same result in starting from the so-called natural state of the material[l5], but the derivation is a little more complicated and uses Almansi’s operator of deformation between the reference and natural states. Moreover a hybrid principle can be derived from (4.6)115,161,either in assuming that (4.% is locally verified, (a method which is called consistent), or in verifying only the linear part of (4.3,. Keeping only the terms of the second order of magnitude for simplicity’s sake and taking into account the equilibrium in the state BP0 between Co, fo, Fo, To, (4.6) gives successively:
(4.1I)
It should be noted that the hybridification has always the result fortunately to keep the prestress Co by which the memory of previous states is taken into account. The application of the finite elements method to (4.11) is immediate theoretically, choosing C, statically admissible inside each element, and taking account for the fact that (4.11) needs that the fyst integrand must be square integrable, that implies that the displ~ement be continuous at the common boundary of to adjacent elements. This method was successfully applied in particular by Boland and Pian[lS, 161,and Bergan and Horrigmoe[17). and especially in the case of plates and shallow shells. We are going to show how the developments of the preceding section can be applied, in a similar way as in the three-dimensional case, to that of non-linear shells. In order to simpliiy the presentation one will accept to keep only the second order terms in the strain energy, and only in the case of Kirchhoff-Love kinematic assumptions, i.e. without transverse shear, but the method is of course more general.
ROCER VALID
192
We shall start again from the principle of virtual work, or more precisely from that of the minimum of the potential energy. But the surface deformations I’ and K will be taken as independent variables by means of Lagrange multipliers,which can be as before replaced as soon by the surface stresses I’ and m’. The strain energy densities are now the quadratic forms between % and 5%
\ +m, zg-&[giVo]])-ftV]dzo [
’
[~~tV-V~l+T,[~-~I,l]ds -I =mov
-I
c oF hQ,dy m E
or
e]ds-TQkWk]=&
.
I (4.13)
V K nt, m,, T,, 7,.
The principle (4.13) gives easily by integrations by parts and applications of Stokes’ formula. in fiemanian separating as before the prestresses and the extrastresses.
connection:
n’=l+llc m’=Ello+m,.t Let us call here f the surface density of given loads on 2, and F the curvilinear density of given loads on the part C,,,+ of the boundary C,,,,+and also w the normal component of the displacement V on C,,,,,,s the cur& inear abscissa of C,, and to, vo, as before its unit tangent and unit normal, the-latter oriented outside C, on H,, It IS assumed lastly that the displacement Vd will be given on the compiemen~y part C&v of C,, as well as the normal derivative &U/&&Jof w. From now on the subscript zero will refer to state L and the subscript one to the increments between % and Is. Under that conditions the mixed principle. in the meaning of HeiIinger-Rcissner,which takes account for (2.91,and (2.101,is found as before in the ttueedimensionai case with the Lagrange multipliers T and r.
V V, n,,
(4.14)
m,, T,, 71.
The hybrid principle can also be derived as well from (4.14)in choosing the surface stresses nl and ml among those which satisfy locally the eq~b~um, di~e~d~ the prestress, (and that at each step $1, such that: &
n,tm,$+~]+?,=O. C
on&.
(4.15)
Here again we draw attention to the fact that, as in the three-dimensional case, one has to take care not to eliminate the Lagrange multiplier T, on C,, for the integral on C,, would still contain a function of the (4.12) displacement vector V, which is unacceptable. VV.I’,i(,n,,m,,T,r. I Neverthekss it is possibk to eliminatethe m~tipiier TV In (4.12)it is remembered that V = m - ma w = i&V, by the local equation T and FE $, T E R. and that cpand Qk E R are couple(4.16) 7, = - i;dn,uo, on C&V. components given on C,, or on k points of C,, respectively. (It is also remembered that this is possible So (4.14)becomes with (4.15)and (4.16) in the case of Kirchhoff-Love assumptions[ll], and that y, and Qk are offset by internal couples). As before the strain energy due to the bending surface prestress will be disregarded. Taking account of formulae (3.5)_63.8)and for the eqoiiib~um of the prestressed state &, it remains as in the threed~mensjon~ case:
An intrinsic formulation for the non&eartheoryof shellsend someapproximations
+ atlh
I
= 0, Q V, T$, and Qa,, m,S.A.I.
(4.17)
Incidentally the local equilibrium equation on rrnO (4.151,can be satistied in breaking piI and ml into two parts, the first one satisfying the complete non-homogeneous equ~ib~um,namely II,’ and m,‘, and the second part, namely at and mt satisfying the homogeneous equilibrium.Such that: If, =n,=+nt m, =mlc+mt 1 aNo d% nlc +mlc am+ Nod%m,’ +fi=O 0 I d%
af +mTz+ NodsmT
C
I
-0.
(4.18)
I
It is just necessary to make the stresses II? and a? derive from a vectorial stress function field v*, such thatflI]:
n*= i2-~~o[~No]i,-~iz~h (4.19)
where the operator izE .%‘(&, &) applies on vectors of & a rotation of +90”. The application of the finite elements method is straight forward at least in its principle to (4.17). The interpolations will apply of course on V, V* and II,‘, rn,=. In general the given load density f will be approximate in the class of the adopted discretized fields nlc and ml’. As to the muitipiier T,, it can also be kept and discretized, or left out by ve~~cation of the kinematic condition V= Vd on C,,, taking approximately Vd in the class of the discretized field V. The formulation (4.17) which does contain second order terms has been given as an example, but can be sophisticatedeasily in adding terms of @eater order in V. It is only necessary to take account for (3.61and (3.8). It is important to notice that (3.6) and (3.8) are the very derivatives of (3.5) and (3.7) respectively, under the condition that projector n be correctly developed as was done. This sophistication may be important in some buckling analysis in particular, when the extension of the mean surface cannot be neglected anymore in certain terms. Let us add finally that the introduction of the transverse shear does not imply unexpected difficulties in principle. 5. coNcLJJStorI The intrinsic theory of shells that we have proposed in the case of non-linear deformations gives a compact and
193
hopefully clear presentation of a theory known to be complicated. It is not possible admittedly to avoid however iinai cumbersome comp~t~ns in covariant derivation, but they are automatic and rather easy to master. We have given rigourous asymptotic expansions in the case of non-negligibleextension of the mean surface. These expansions, on which it is always possible to make any legitimate specific ~pro~~ons, take account for the variation of the Riemanian connection and outline the zero-value of the second order term of the virtual extension of the mean surface. It has been shown also how these expansions could be applied to the fo~ulation of an hybrid mixed principle in the general case of non-linearshells, a principle known to be excellent to provide good stresses and good displacements as well. The introduction of possibly non-linear transverse shear may be made without overturning the developments previously obtained through the kinematic Kirchhoff-Love assumptions, a method that will reveal convenient and incidentally classical in linear theory. In the formulation of the non-linear hybrid principle a vector function has been utilized from which the surface stresses are derived through second order derivatives. These second order derivatives come, as it is recalled, from th? stress symmetry, which reaiise a real closure condition by itself, as in the three-dimensional case[ll]. This symmetry can be relaxed, according to the path previously pioneered by Fraeijs de Veubeke in the thrce~~ension~ non-linear caseIl81, a way which brings the rotation as a new independant variable into the principle. This method has been successfully applied by Sander et a/.[191and Athni et al. [20]in the case of plate elements. These new developments in the non-linear theory of shells are some of our present research subjects.
1. B. 0. AImroth and
F. A. Brogan,Bifurcation buckling for
aenerai sklis. AI~A~XE 2.
3. 4.
13th Smctnres.
Smcwal
yffemics M&&al Co@. San Antonio, Texas iApr. 1972). AMA Paper No. 72-354. B. Irons and A. Razzaque. The evolution of the isoparametric elements. World Cone. Fmite Eiew~enrMerh. Structm~l Mech. (J?diu?dby J. Robinson). Boumemouth, Dorset, England (Oct. 1975). R. D. Wood and 0. C. Ziinkiewict, G~me~~y non-linear finite element analysis of beams, frame, arches and axisymmetric shells. Campus Structures 7, 725-735(1!977). R. Giid. Elements finis isoparamstriques’de coque multicouche pour Ie calcul des struchues akrospatiales en mattriaux composites.LA Rechmhe Att~~~oti& No. B77-
2. p. 131-132.-
J. L. Sanders, Jr., Non-linear theory of thin shells. @wt. Appl. Murh. 21(l) (Apr. 1963). W. T. Koiter, Gn the non-linear theory of thin elastic shells. Pm. Physical Sci. Mech. Series B 69(l) (l%S). K. H. Mushtari and K. 2. Galimov, Non&ear theory of thin elastic shells. NASA ‘fTF62, U.S. Department of Commerce. O&e of Technical Services, Washington (1961). 8. R. Valid, Sur le calcul dcs coqueshors du domaine tlastique. C&A. SC. Paris t. 263,89-91 (1966). 9. R. Valid, La thtoric tin&ire des coques et son application aux calculs inklastiques. Th&se (1973). Publ. ONERA No. 147 (1973). Trad. ESA T.T.-109 (1975). IO. R. Valid, Conditions de compatibilitt et fonctions de contrainte dans It cas des coqk multiplement connexes. 14&ne Congds IUTAM Dclft (30 Aodt-4 Sept. 1976)-_Rech. ALrosp. (19765), 289-299.
Il. R. Valid. La M~c~~que des Mj~iea~ Conrinus et ie C&al des Stmctums. Eyrolks, Paris (1977). 12. P. M. Naghdi. Foundations of elastic sheil theory. In Progress in Solid Mechanics. Vol. 4, .DD. . l-90. North-Holland. Amsterdam ( 1%3). 13. W. T. Koiter, General equations of elastic stability for thin she&. Pmt. Symp. Theory Shek Honor Lloyd Hanrilton Awtnel. DD. 187-223. University of Houston. Texas (19671. Chr the stability of elastic equilibrium. The& 14. W. T. K&r. Delft (19431: NASA T.T.F. 10 (1%7). IS. P. L. Boknd, Larpe drfkction analysis of thin elastic structures by the assumed stress hybrid tinite ekment method. Aeroekstii and Structures Research Laboratory. Department of ~ron~ics and As~~utics, M&T. Cambridge M~hu~tts 02139 Contract of Air Force G&cc of S&ui& Research U.S.A. AFGSR.TR 76 1111. ASRL. TR IX-4 @ct. 1973).
16. P. L. Boland and T. H. H. Pian. Large deflection analysts of thin ekstic structures by the assumed stress hybrid element method. Cornput. S&ucrrtres 7. I-12 (Feb. 1977). 17. P. G. Bergan and G. Horrigmoe. instability analysis of free form shells by finite elements. 14fh Inf. Gong. 77reorerrcal Appl. Mech. HJTAM Delft. The Netherlands (30 Aug.-4 Sept. 1976). 18. B. Fraeijs de Veubeke. A new variational principle for finite elastic displacements. fnr. J. Engng Sci. 10. 745-763 ( 1972). 19. G. Sander and E. Carnoy, Equilibrium and mixed formulation in stability analysis. Int. Conf. Finite Elements NonLinear S&d Structural Mech. Geilo. Norway (29 Aug.-i Sept. 1977). 20. S. N. Athui and H. Mur~wa. On hybrid finite element models in Non-Linear Solid Mcch~ic~. Int. Conf. Finite Elements Non-Linear Solid Stmc~um~ Mechanics. Geilo. Norway (29 Aug.-I Sept.. 1977).
92320 Chatillon, France
(Receiwd 22 May 1978)
AWract-An intrinsicgeometric formulation of the nonlinear theory of sheik is proposed.Asymptotic expansions of the variational formulationis made in view of its applicationsby the finite element method. These expaasions take account for the variation of the connection in case of non-nqiiibk extension of the mean surface. A hybrid mixed principle also presented in an intrinsic formulation utiiiis the preceding asymptotic developments.
1. lNTmDucTtoN
of powerful computers and the great versabibty of the finite element method as a discretixed metbod have resulted in several consequences in structural analysis. Fust of all after a remarkable success in the linear fkld, the method has heen used since several years in the non-linear field, (non-linear deformations or non-linear constitutive laws of materials), by means of appropriate aigoritbms and at tbe cost of greater computational efforts. This complexity is evolve in particukr in buckling analysis, but leads sometimes to undue s~p~~ons. It is tben necessary to bring into tire formulation other non&tear terms which were dis-’ regarded up to that time[ 11. Secondly tbe very use of these powerful and refined means has heen responsible for a certain decline of theoretical investigations concerning sbell probkms. in effect in many applications tbe analysts did not be&ate to use ~~~en~o~, or like-looking tbreedimensional tkite ekements,wbose merit was to refer to more elementary tbeories, in particular in tbe non-linear field, to he easily programmed, tbanks to the uti&ation of isoparametric elements[2,31;and to allow for tbe analysis of very miscellaneouscases as those of multilayered shells made of anisotropic composite materials taking account for large transverse shears and thickness variations as well[4]. But tbis decline couId also be caused by the great decays of the theory wbieb can he bandkd only tbrougb drastic but not always acceptable simplifications. Nevertheless some studies of a theoretical character remain fruitful in this domain, not only for tbe sake of those who are going on applying tbe two-dimensional shell theories, but also to provide the others, we mean the adepts of tbreediiensional theories, with tbe results and approximations given and justified as consequences of shell theories. In the last resort we can argue that tbe three-dimensionaltheory appliedto shell structures does not fail to And its own limitations due to the limits of accuracy of tbe computers which are actually available: in fact smaller becomes the shell tbicbness, greater has to he tbe numericalprecision in order to avoid unacceptable quasisingukrities (ill-conditioned matrices); but this has also its liiitations, though these ones are nowadays more and more remote, as can be observed. As it is well known the non-linear tbeory of shells has
The use
heen marbed by important contributions, among which can be quoted those of Sanders[SJ,Koiter[6], Mushtari and Gabmov[‘!I. Here we propose an application of an intrinsic geometric method, worked out in tbe linear case@-111, to tbe development of tbe shell theory in tbe non-linear fkld, and in vkw of some asymptotic expansions, in a variational form&ion in order to mabe tbem workable by tbe finite element method. Tbetbcorywillbecompktedbytbeformuktionofa mixed-hybrid non-linear principk, in tbe Rian meaning, from the pmcedkg ~~1~~~. taIlEm NcnMWEuWOCaralm Tbe formuktion is developed fbst in tbe frame of tbe Kircboff-Love kinematic assumptions, tbat is to say in negkcting tbe transverse shear strain and normal strain ClWgiCS. Cdiii E,, in tbe actual state 8, tbe deformed mean ~~~~c~ktrn~~~~~~~t normal N, it is assumed that 2, is an oriented, parametric, di&mtiabk, compact surface,embedded in tbe tbreedimensional eucbdian space ES.2, is supposed to have a Rkmanian connection, and its boundary will be calkd CL,.Then there exists at each point m of X,,, a linear mt manifoldcalkd I$, also euclidian,a metric diierentiabk tensor-fteld defbmd on &, induced by tbe metric of tbe embedding space ES, and fkaliy an bermitian Projector fkk =, WtboPml GOVE, also differontkbk, wbkb applks & upon & and which defines the Rkmanian linear connection. Us&a superbar for any transpose quantity, tben n and NN are two bermitian supplementary projectors respectively upon & and N, so that
Q=*2=v
+r+NN=lE,,
(2.1)
where 113,is tbe identity operator on ES. Let f he any Minsk tangent vector&eld which vahle~is v, so that V=f(m)EEz
the covarknt differential a V = w d V, V dm E &. if m is a map of class C’ of H, defined on an open set of R2-
IS3
m=@X),
VXEOCR’
ROGER VALID
184
It is found that
S will be the natural basis, at point m, of 82, if dm =$x=SdX
S=s,
(2.6) In general any differential vector V and its derivative will be related by the formula It is remembered that dV=gdm. Sm E $, Then let YE R’ represent the vector V E 4% in the basis S, it comes: V=SY,
S[~dm]=O,
asm
m)
in respect of which any material unit normal remains a geometric unit normal. The virtual strain energy is then referred to a previous prestressed but well known state %, corresponding to the mean surface Z,,,, of generic point m. and tangent plane J3m. Using the identity operator
~dV=nd[nVI+Pd[N~Vl =~[=~+~~~v~u~~v sdm
rdV=&I+NV8m where
VdmE$
8N=-am~
(2.2)
r is the Christoffel operator, f(dX) the Cl&toffCl matrix, and l$ = r: theClxlstoffel symbols of the Rle~ conne&m of P, If the surface vector held is a threedimension vector Gelddellned on z,
VVE&(asafunctionof
V8mE&,
giVCS
d V = &[I’; d’X’Yt $Y].
V=nV+NNV,
and that the kinematic assumptionaccording to which
dV=dSY+SdY
dV=ndSY+SdY = SWXW+dY1
8N f & because g = 1,
dm am, --= am0 am
(2.2)
aN/dm isthe curvature operator of 8, at m, and
1
E2
formula (2.6) becomes by reciprocal image transformation:
&t&es
the second fundamental form of &,: d~~~ff~a~)dm. ~~~~~~~~n~~setof~~~ operators from the linear space E on the liiar space Ft, let us recall that
iii
a8N a&n aN I-t-n,-g--)dis(~)&. am0 am0
andthattbefirst~~formofZ,atmisdmdnr, the scalar square product of any dilferential tangent vector dm E K. Under that conditions the variational formulation of the shell theory can he achieved directly from the three~~~~~~~~o~tf~~~a~ ~s~p~ns, and leads to the expression of tk vimal deformation energy, as referred to previous studies[111:
Whence:
with SW=
I x,
T,(n&y+ m&r)CCE,. VSmE E3 (2.5)
where ISand m are the surface Cauchy stresses, respectively dual of the virtual surface ~f~~ns 87 and &r of extension and cmvature variation of 2,. In that formula n, 14 &y, &: are hermitian endomorphisms of the tangent plane rf, at m of t,.
tNotc
(2.7)
that BERBR),
VVE8
The
two genera&d surface stresses a’ and m’are those
of Hola-IGhhoff ; they are hermitiantoo and defined on
& in the reference state if0. The non-linear surface &f~a~~ f and K, due to the vect~~~~~ V of point m between the states I and %,are respectively defined by
An intrinsic fo~u~ation for the nonlinear theoryof sheUsand someapproximations
Now
dM=dm+dNziNdz, drmd.Cmo=fdlmo
dnram
amz--Ie [ 0 0
VdmE&
185
VrER,
then I d2m&
dMdM=[dmidNz+jjdr][dm+dNz+Ndr] dMdM=dmdmi2dmdNz+dNdNz’+dz*
Whence &
sr=i
atnamlr& -2 --_ amoamo
C
3
= i;
Z a8m aSmam =Z. --_-+-2 [ &noam0 am0am01
I
The commutation(@mo)8 = W/am,) has been utiIized everywhere, due to the fact that the virtual variation 8m is made for any given point m.on );Now
w~n~N=l,~N=O=~N. The second term of the difference is derived from the first one by putting a subscript zero at ah terms. It is seen, as it is well known, that the deformation is a function not only of the first and second form variations, but also of the third one dNdN =~~~drn %dN
,;i;l~% aN aN am o~~~~dm~ 0 0
StiIt through a simple transformation and aIlowingfor the reiation
so that the fotmuiae can aIso be written:
am am0 --_=I am0am H2 it comes:
(2.9) EdNe;i;;;.drnaN
o~amO
am dm am am am0 O
am a&aN
Then (2.8), gives: drmak: d2mo= h,mo
K=
;I;;; aN am alvo zXz&zK 0 aI d2m,
dNdN=i&
m,[j$$+]u,,-a
6KmriGaN 0 t
am
I
(2.10)
I
0
So with (2.7) it is found again the formula: SW=
I I,0
T,(nW + m’6K) dZo
(2.11)
where every stress or strain surface field in the actual state 8 is defined on &, the tangent plan of H, in the reference state go. It should be noted that the three&tensional deformation at any point M of the shelI in the state 8, of projection m upon the tangent plane, is given by the difference ~d~-~dMo. If
M=m+ z=m+Nz,
[ K+aNo ~~t~+I~~-‘~+~]drn*
(2.12)
-’ aaN --+--n~ro=lEK. am0 am0 am0 am0
asm with&N=-xNO
whence by (2.91,and (ZIO),:
As to the last term dt2 it would give the thickness ~fo~a~n whose contribution in the strain energy has been neglected. In the setting up of an elastic constitutive law, for instance, it may be used the linear law, which is written in linearized strains as functions of the displacement vector
V=m-ma, the only independent variable is the present case. But fuller expansions, up to a given order in terms of V, will take account, strictly speaking, for non-linear ~fo~~es which are also functions of the third fundamental form of Z,. If it is obvious that this term will be linked in the strain energy to the surface couple-stress because of the factor z2, the way by which this term WP contribute and be associated to the &urge of curvature will be devised only through a simiIarway as that was worked out to 8nd the Iinear law. Incidentahy it is recommended in that case to adopt Na&di’s elegant method[ll, 111,which
186
ROGER VALID
utilizes very logically the variational principle outlined by Hu-Washizu, precisely in view to establish in the mean a bi-dirnensional law. We shah not here undertake this task which is just to be modeled upon Naghdi’sone.
by the second line of (2.141,thatis to say: aru,aAN
2 ---z2 1ahNaAN
amT7ZC2dm
am
dm.
I
’
It will be remembered that (2.5) and (2.11)are correct only in involvingthe Kiichhoff-Love assumptions. If those hypotheses are left away, it is necessary to introduce in particular the transverse shear, and SN
and the transverse shear deformation would be the only term
becomes, as usual, another (virtual) independent kinematic variable. In that case let M be again a point of the shell whose projection upon the tangent plane & is m, in the actual state &?.We have again
It is incidentallyeasy to refer these terms by reciprocal image transformation to the reference state L. The
M=m+Nz,
extra-tangential deformation (with unvariable thickness) becomes: z dN aAN amzEZ+__L_Z2
whereflN=l,
am0 am0
and VdmE&,
dM=dm+dNz+Ndz,
drER.
The non-linear deformation at M wiU again be given in computing the following difference referred to the state 8.0.
am, am am0 -1 am aAN aAN am 2 +?Ko am am amOL ’
and the transverse shear strain: z[ANNo].
dMdM-dModMo, IA us call
but now dMdM=dmdm+2dmdNz+dNdNz2 + dr2 + 2dmN dz.
(2.13) then by similar arguments as for (2.12):
Let us put N=N,tAN.
with
amaN,%_ am0 dm am0
tiN=&N,=i,
z aN,aAN am -am0 am am am0
--
where N, is the geometric unit normal, and AN the extra-variation of the mate& normal, such that: AN=N-N,. It is shown that the vector AN is only responsible of the transverse shear in the shell. Then keeping in mind that the condition N=dm=O
And also: -I dm --JAN JAN am
idmoam amam,--
still holds, (2.13) becomes: dMdM=dmdm
--
1 am dAN am amoamoamdAN am =-------w2 am, am amoam am am0 am am,
t2dmdNd+dN,dN,z2+dz2
t2dmiiANzt2dN,dANz2t~dANz2 t Z&AN dz
(2.14)
All that is necessary is to subtract to (2.14) the square d&d& to obtain the three-dimensional deformation at
M. Now the first line of (2.14)givestheexact deformadon in the case of the Linemotic Rirchoff-Love assumptions. If the computation has been achieved through these hypotheses, all that is necessary to obtain the extra-strain is to add the contribution provided by the two foIIowing lines, where are the supplementary variation AN and its derivative. In case that it should be admitted that No in the state & be the geometric unit normaI of & and that the thickness variation shotdd be provisionally negIected. the extra-tangential shear strain due to AN would be given
But of course it is easier to utilize
--
I am aANaAN am ZdmodmdmYGG=2
1dbNdAN ---
am0 dmo'
Finally the components of the extra-strain
TOLL. NoNo =
D, are
(2.15)
&ANIVo. 0
If on the contrary in the state go the vector No is not orthogonal to I;,,,,,, then No = No, + ANo
187
An intrinsic formulation for the no&neartheoryof shetisand someapproximations
where Nomis the geometric unit normal of Z,. In that case it is necessary to subtract from the preceding deformation, the same quantity subscripted by zero, and the same for z, if .z# 4. It is worth noting that the virtual surface strain is also modified. In fact, (for z = z. giving a zero thickness strain energy).
am Go-m’vo=-4p,
gives a virtual displacement 6M = &r + &N#t+ SAP&, and the strain energy is supplemented in the following way in state 8:
baa -am
O~dVO 0
In
The second term of the inte~and is the pr~uct of two skew-symmet~c operators of $ and can be written as the product of two scalar quantities. It is quite easy to refer this strain energy by reciprocal images to I, in the state Pr, The theory must be completed by the research of a surface constitutive law in that case of transverse shear deformation, always assuming that the normal strain energy is negligible.This assumption will be realized in a realistic way by neglectingthe normal stress component @XX. If the constitutive faw is to be found in linear elasticity for instance in taking onfy into account the hnearized strains, then an a~roximate law will give, by similar previous ~guments, relations between the surface stresses IF,
m, xh
Q.
and the surface strains
But it is easy to outline that this s~~~~ficationis not acceptable if one wants to keep for instance in the energy expansion all terms of, say, second order, as it will be sub~quendy done in a strict sense. In the preceding method the only kinematic variables are definitively V= m - nro and AN = N-N,, and ah the deformations have to be expressed non linearly in terms of these variables. In fact this computation is very cumbersome and probably useless in general. So we intend to put forward some approximate expansions. It should be noted lastly that the preceding formulae allow to write immediately the equ~ib~~ equations, referred to the mean surface Xmcand its boundary C, in the reference state go, in a quite classical way, by means of inte~at~ons by parts and use of Stoke’s form&a in ~emanian connection. The fo~lowi~ local equ~i~um. equations are written in an intrinsic form:
*
I
-
= ph.
at k points on C,.
that formulae[lll
I (2.17)
f is a surface load density given on 2,: F is a curvifinear load density given on C,,+ C &,; rp is a coupk~om~nent load density given on C&F; $%are coupk~orn~~nt ~~ontinuit~s given on k: points of C+ to and v. are unit vectors, tangent and normal respecttvely to C, on X, 3MWPlvncgxPIWBogsAsFuNRlNcFMNs OFKINSMATICUNKNWNS
Several approximations are possible from the variationnal preceding formulae. Let us recall that according to the magmtude of the surface deikction, Koiterf61 considered 4 dasscs of problems, name&: (a) Large deflections characterized by the absence of restriction. (b) Moderate deflections ch~cte~ by small values of ah di~~~rn~t gradient comments in apex with unity, ahhough nothing is assumed in advance about their relative orders of magnitude. (c) Small finite deflections characterized by smal1displacement gradients and by rotations whose squares do not exceed the middle surface strains in order of magnC tude. (d) Infinitesimal deflections characterized by small displacement gradients, none of which exceeds the middle surface strains in order of rn~j~. goiter has so completed the three classes previously ~n~~u~ by Musbtari and GaIiiov~‘ff, by the ctass fi in which at least one component of the Robin, small in comp~~n with unity, has its square large in ~~p~son with the mean surface strains. These ro~~ons are some components of the derivative W/am, of V at & In that case ~~V~~~~~. fn considering this last assumption which has been little explored, it is possible to make limited asymptotic expansions, for instance up to the second order against V and AN, that is the full order of the non-linear purely threedimensionai theory, where many applications have &own the necessity of keeping all terms. These expansions will be applied to all terms of (2.11) or (2.16)in view of subsequent applications by the finite element method, by means of in~olations of tire variables V and AN, and pacify in the case of nonlinear bu~~rn~ in the fohowing for sake of brevity the expansion will
ROGER\IhLID
188 be
developed only in the frame of the Kirchhoff-Love /(3.2) gives then successively with CF’ =0 assumptions. In order to take account for a possibly non negligible I& dmo= 0, &NO = 1 extension of the mean surface. the expansion will be applied also to the Riemanian connection of 2, represented by projector n. The condom can tK.c~c~at~, as it is w&-known, by means of the Christoffel symbols which are given by Whence r’,, = $j%., + &.k
-
i,j,k,l=1,2,
&?kd
There gil = $S, are the components of the Gram matrix SS of S = &n/ax a natural basis relative to a map of a complete atlas of P,. It is clear that the connation in the state 8 depends on the variation of the first fun~en~ form r. But this method will certainly be most cumbersome. Another one would be to develop a expressed with the basis S by the classical formula
(3.3’ It is then possibIeto expand projector ?r such that
which gives 7r= S[$!q-‘5 but this method is also rather heavy. It has been preferred the simpler followingone: FWting
(3.4)
By means of (3.3)and (3.4).the deformations r, &I’.K, SK can then be expanded up to the second order. Thus for I’ given by (2.9), with (3.4) and with by proper expansion
n= 1&_NE;j N=No+eN,+g2N2,
-[iv,&+ No&l ,r2= -fN,& + N&+ N&l. I
WI =
with
c’=O
m=mo+ v
I
aV am --&=?rO+~~=o the first bracket of the right hand member provides the linearized extension of an element of Z, the second bracket provides the rotation of that ehment around the normal, and the iast term gives the linearized rotation of the normal itself. Then assuming[W/ad 4 1, the foliowingWUttiOnS ~dm=O,
flN=l,
VdmoElh,
one writes 1
r=p
av no+er~Ono
I
I[
av CO 'Iltr+~~~*no -5, I
and finally
can be expanded from: & dmo = 0, &No= I. gN&o$-] 0
ThUS
or I-
.sr is given by (2.9)~and by similararguments It should be noted that aV/am is supposed to be small compared with 1% extended upon & by projector ‘~0, and our intention being to keep every first and second whence order term it is useless to put: av _,s++ am,- am, tTranspose quantity.
0
+_** e
2
Z aw +amam XF av lr~C0.E2,
am0am,
0
1
0
(3.5)
An intrinsic formulation for the nonlinear theory of shells and some approximations
189
or 8 =
sr*+Bf,.
(3.6)
where admittedly It is thus revealed that the second order term SF, vanishes exactly, so that the expansion up to the first order constitutes an excellent approximationby itself. The quantities K and SK, which are given by (2.10). The fonnuhte can be developed in taking account for will be developed in the same manner, but the compu- (3.10);(or for complicated laws if other terms are introtation, though as simple as the preceding, is very much duced which arc non-linear functions of I and K), in the longer. One finds then following guidelines. (3.51,(3.7) and (3.10)give a’ and m’ in the form of: n”~+cn,+c21t2 [ m’=mo+rmt+c2m2’ (3.71
Finally (3.6) and (3.8) give
6r=ia+csr, [ SK = 6Ko + &K, + r26K2,
or K = SK, + C’KS.
f
and (3.9) becomes:
Then SK = SK, + &K, + e2SKz
SW-
with
1
TAti+ enI + 2n2wkteSr,j +bb+iml
+%lJlsK,+ear,+&v&]) a%,
(3.11)
But in (3.1I) the product of terms subscripted by zero must verify the equiIibriumin the state goo,possibly with external forces. Leaving out these terms, it remains up to the second order:
(3.81 + 2b2iwo+ ihn, +m4Ko+ mdKt + mdK& ao. (3.12)
In this formula the vector eN has been developed simi!ariy in the form of:
Now we may extend to more general problems the following admitted remark of KoiterIl9: “Finally the most important shell buckling problems are those in which the fundamental state I, (%01,is (approximately) a membrane state of stress, and the tensor of stress resuftam M”, (IIS&may then be negiected everywhere”. It shouid be also noted that a shell which would be initially built to resist to large bending couples would be certainly ill-drawnor utilized. Then ignoringthe bending term tm,,(3.12)gives: SW=
T,teftirr f =m0
+ sMo + mt8Kol + e2fa2bro
+l@r,+adKo+mt6K,])dL
(3.13)
in which the very complicated term SK2has disappeared. It is aiso powibk when the need arises, as will be remembered, to take account for extra-rotations and x (No+ tN, + e2N21. shear stresses, and linked tangantialenergies. The pre&ing asymptotic expansions are quite adapThe problem is now to calculate (2.1I), namely: ted to the use of the We eicmtat method for non-Iii curved shells with or without transverse shear, in a total T,(n’Gf t m’SK)d&. 13.9) or updated 4rarq#m procuhtre. In this fixation the =nyl constitutive law will be elastic, hypereiastic or even If we assume to simplify that a linear constitutive law hypoelastic (in particular in the case of a plastic flow law). is set down for the shell between the stresses n’, m’ and Let us also notice that all these formulae can be the deformations I and R. and if no and m. are known considerably simpliied in the case of plates by vanishing surface prestresses in the state Ed,then
I
ROGER VALID
190
In order to throw a light on this formulation in an the curvature operator <3mo, but in the natural State intrinsic manner, it seems useful to show how such a only. method can be applied to the more simple case of If the shell is submitted to external follower loads, it threedimensional structures. will be necessary to expand these loads consistently up Let us consider a three-dimensional body, made of a to the same order as the other terms. Let us consider for instance the important case of a . non-polar material, which occupies a volume & of boundary Z. in the reference state go. and which is pressure loading. The work of the pressure p in a virtual submitted to a volume density f of external load and to a displacement SV of Z, in the state 8 is developed up to surface density F of external load on a part SW of the the second order in the following way: boundary I,,. One supposes also that the displacement U is given on the complementary part Pov of X0. no is the ZpN ti = %[po + 91 + e’pz] internal unit normal of IO. x[N,+ eN,+ c*N&&+e d, + e*d&] Let ME fi be any point of the body in the actual state 8, which corresponds to M,E!?, in the reference state where the constitutive law for example CO. -In the state $ the equilibrium is given by the following principle of virtual work referred to go:
pv’ = povoy between pressure and volume allows to calculate the terms pl and p2 in terms of V. Vectors N, and N2 being already calculated, it is only necessary to expand dI asymptotically. In developing the quantity dI = N d,m x dim,
I o,,
T,(C’SD) dfio - I,~HJ
dflo - I,,
&J
&o = 0.
V 6CJC.A.
(4.1)
SU is the value of any kinematic admissible displacement field (C.A.); c’ = c’ is the Piola-Kirchhoff stress in the state g, referred to 0,: C’E &??,, &): and with M= Ma+U.
Vd,m, d2m E &
, it is found: I
1
=bEa(B
39 8,
3
(4.2) UC.A. j U = U, on Zoo 8UC.A. $8U
d):
=d&+.[&?roV+ T,(~)&,V]
= 0 on PO”.
dZ,,
It is supposed for sake of simplicity that the constitutive law is linear such that if one considers c’ and D as vectors of an euclidian six-dimensional vector-space $. C’=AQwhere A=~EZ(&,.$). T,(D,&) = &a, where Lj is the transpose of D in $. Then the strain energy volume density will be written as a quadratic form of D:
+e*[~oAdj(uo~uo)N.+~~o~~N.]d2o. In the particular case where V = No?, (T E R), dX-dPo=
-~~~+~I~+wz~*+;~~
(3*‘4)
I
a =$4D=+(D)(D).
dIo (3.15)
where pI and h are the principal curvatures of PO. It is worth noting that the Laplace law of capillarity is a law of isotropic extension of the mean surface which utilizes (3.15). Incidentally the product of &V times dZ gives the virtual variation 6~. To conchrde this paragraph one may say that the precedii developments are well adapted to derive the second derivative of the potential energy as a function of the displacement V, keeping SV constant, which derivative govcms the sfatic stability probkms of conservative stJuctures[l4, 13, 111. +DBxBoANDrrYBuDmNcDLef~
Now it is possible to take the variable D as an independent variable in the domain fl,, by means of a Lagrange multiplier t = ?E Y(&. &I and to introduce the constraint U-Ud=OonZov (4.3) in a weak form by means of another multiplier T E ES. (4.1) gives then: (-/,[a+
~,(t[f~~-~la-D])-jLi]dno
-I
PUCE,-
I,,
nu
-
U,l dz,] = 0,
%F
NON-UNEARSBHLS
An interesting application of the preceding developments is the formulation of mixed principles for nonlinear shells.
vt=ilr,~.u.
(4.4)
Hardly necessary to say that these variables must
An intrinsicfo~ulalion for the nonlineartheoryof shellsandsomeapproximations
191
belong to certain definite spaces in order that every energy which is involved in (4.4) be bounded. The Euler equations of principle (4.4)are immediately: +
1 aMaM 5 cai\l,z-‘~, div I$$ [
I =D
ZOF
in &
1
+
+f=O
0
u = Lid
ii&O-
I
on Zou.
I
aMo
ud&#
1
d&,
divC,tf,=O,
(4.8)
in&,
(4.9)
(j CIS.A.k statically admissibleinside), it remains:
+
C=AI)*D=A-‘(J’tBC’,
I
[iioC,-FJUdzot
%F
(with B = A-’ E .5?(&, $). -
In fact we shall separate as usual the prestress Co in state & from its increment C1in state 8 such that:
co+ ct
Co is then the Cauchy prestress in &. It will also be set down the strain energy in terms of stresses between go and 8: /3 = $,BC,
0
If one takes C, in a class such that
If we assume &at (43, is strictly verified in (4.4),then the m~tiplier t is precisely the local stress C’ and the variable L) must disappear from the principle. Under that condition:
=
“OCOZ
4u
+f,U,
Z&O ‘OfaMo z-t+) ‘OfaMo
C’
I [ aU +iioc,-F, I [[ aU +iioc,-Ijst u u,7-1. I I=0,ve,,
= f~(C,XG).
whence now
i’,[U-
&lldXo]=O,
I zou
hC,
u
VC,S.A.I., U, T,.
(4.IO)
At this step it would be tentatively possible to eliminate the vector T, by means of the local equation
but it would be seen that terms functions of U would not be totally eliminated in the integral on IO” as it must be. It is then necessary to keep the multiplier T,, as independant variable, or take U in the class where U = U,, on &. Then (4.10)gives finally:
D= BC,.
s
Principle (4.4) becomes then:
VCJ.A.L, W,T,.
-I
RJdZ0ZOF
I,,,F[u -
ud]
d?;O]
=
0,
vc,, u,T.
(4.6) It is quite possible to achieve the same result in starting from the so-called natural state of the material[l5], but the derivation is a little more complicated and uses Almansi’s operator of deformation between the reference and natural states. Moreover a hybrid principle can be derived from (4.6)115,161,either in assuming that (4.% is locally verified, (a method which is called consistent), or in verifying only the linear part of (4.3,. Keeping only the terms of the second order of magnitude for simplicity’s sake and taking into account the equilibrium in the state BP0 between Co, fo, Fo, To, (4.6) gives successively:
(4.1I)
It should be noted that the hybridification has always the result fortunately to keep the prestress Co by which the memory of previous states is taken into account. The application of the finite elements method to (4.11) is immediate theoretically, choosing C, statically admissible inside each element, and taking account for the fact that (4.11) needs that the fyst integrand must be square integrable, that implies that the displ~ement be continuous at the common boundary of to adjacent elements. This method was successfully applied in particular by Boland and Pian[lS, 161,and Bergan and Horrigmoe[17). and especially in the case of plates and shallow shells. We are going to show how the developments of the preceding section can be applied, in a similar way as in the three-dimensional case, to that of non-linear shells. In order to simpliiy the presentation one will accept to keep only the second order terms in the strain energy, and only in the case of Kirchhoff-Love kinematic assumptions, i.e. without transverse shear, but the method is of course more general.
ROCER VALID
192
We shall start again from the principle of virtual work, or more precisely from that of the minimum of the potential energy. But the surface deformations I’ and K will be taken as independent variables by means of Lagrange multipliers,which can be as before replaced as soon by the surface stresses I’ and m’. The strain energy densities are now the quadratic forms between % and 5%
\ +m, zg-&[giVo]])-ftV]dzo [
’
[~~tV-V~l+T,[~-~I,l]ds -I =mov
-I
c oF hQ,dy m E
or
e]ds-TQkWk]=&
.
I (4.13)
V K nt, m,, T,, 7,.
The principle (4.13) gives easily by integrations by parts and applications of Stokes’ formula. in fiemanian separating as before the prestresses and the extrastresses.
connection:
n’=l+llc m’=Ello+m,.t Let us call here f the surface density of given loads on 2, and F the curvilinear density of given loads on the part C,,,+ of the boundary C,,,,+and also w the normal component of the displacement V on C,,,,,,s the cur& inear abscissa of C,, and to, vo, as before its unit tangent and unit normal, the-latter oriented outside C, on H,, It IS assumed lastly that the displacement Vd will be given on the compiemen~y part C&v of C,, as well as the normal derivative &U/&&Jof w. From now on the subscript zero will refer to state L and the subscript one to the increments between % and Is. Under that conditions the mixed principle. in the meaning of HeiIinger-Rcissner,which takes account for (2.91,and (2.101,is found as before in the ttueedimensionai case with the Lagrange multipliers T and r.
V V, n,,
(4.14)
m,, T,, 71.
The hybrid principle can also be derived as well from (4.14)in choosing the surface stresses nl and ml among those which satisfy locally the eq~b~um, di~e~d~ the prestress, (and that at each step $1, such that: &
n,tm,$+~]+?,=O. C
on&.
(4.15)
Here again we draw attention to the fact that, as in the three-dimensional case, one has to take care not to eliminate the Lagrange multiplier T, on C,, for the integral on C,, would still contain a function of the (4.12) displacement vector V, which is unacceptable. VV.I’,i(,n,,m,,T,r. I Neverthekss it is possibk to eliminatethe m~tipiier TV In (4.12)it is remembered that V = m - ma w = i&V, by the local equation T and FE $, T E R. and that cpand Qk E R are couple(4.16) 7, = - i;dn,uo, on C&V. components given on C,, or on k points of C,, respectively. (It is also remembered that this is possible So (4.14)becomes with (4.15)and (4.16) in the case of Kirchhoff-Love assumptions[ll], and that y, and Qk are offset by internal couples). As before the strain energy due to the bending surface prestress will be disregarded. Taking account of formulae (3.5)_63.8)and for the eqoiiib~um of the prestressed state &, it remains as in the threed~mensjon~ case:
An intrinsic formulation for the non&eartheoryof shellsend someapproximations
+ atlh
I
= 0, Q V, T$, and Qa,, m,S.A.I.
(4.17)
Incidentally the local equilibrium equation on rrnO (4.151,can be satistied in breaking piI and ml into two parts, the first one satisfying the complete non-homogeneous equ~ib~um,namely II,’ and m,‘, and the second part, namely at and mt satisfying the homogeneous equilibrium.Such that: If, =n,=+nt m, =mlc+mt 1 aNo d% nlc +mlc am+ Nod%m,’ +fi=O 0 I d%
af +mTz+ NodsmT
C
I
-0.
(4.18)
I
It is just necessary to make the stresses II? and a? derive from a vectorial stress function field v*, such thatflI]:
n*= i2-~~o[~No]i,-~iz~h (4.19)
where the operator izE .%‘(&, &) applies on vectors of & a rotation of +90”. The application of the finite elements method is straight forward at least in its principle to (4.17). The interpolations will apply of course on V, V* and II,‘, rn,=. In general the given load density f will be approximate in the class of the adopted discretized fields nlc and ml’. As to the muitipiier T,, it can also be kept and discretized, or left out by ve~~cation of the kinematic condition V= Vd on C,,, taking approximately Vd in the class of the discretized field V. The formulation (4.17) which does contain second order terms has been given as an example, but can be sophisticatedeasily in adding terms of @eater order in V. It is only necessary to take account for (3.61and (3.8). It is important to notice that (3.6) and (3.8) are the very derivatives of (3.5) and (3.7) respectively, under the condition that projector n be correctly developed as was done. This sophistication may be important in some buckling analysis in particular, when the extension of the mean surface cannot be neglected anymore in certain terms. Let us add finally that the introduction of the transverse shear does not imply unexpected difficulties in principle. 5. coNcLJJStorI The intrinsic theory of shells that we have proposed in the case of non-linear deformations gives a compact and
193
hopefully clear presentation of a theory known to be complicated. It is not possible admittedly to avoid however iinai cumbersome comp~t~ns in covariant derivation, but they are automatic and rather easy to master. We have given rigourous asymptotic expansions in the case of non-negligibleextension of the mean surface. These expansions, on which it is always possible to make any legitimate specific ~pro~~ons, take account for the variation of the Riemanian connection and outline the zero-value of the second order term of the virtual extension of the mean surface. It has been shown also how these expansions could be applied to the fo~ulation of an hybrid mixed principle in the general case of non-linearshells, a principle known to be excellent to provide good stresses and good displacements as well. The introduction of possibly non-linear transverse shear may be made without overturning the developments previously obtained through the kinematic Kirchhoff-Love assumptions, a method that will reveal convenient and incidentally classical in linear theory. In the formulation of the non-linear hybrid principle a vector function has been utilized from which the surface stresses are derived through second order derivatives. These second order derivatives come, as it is recalled, from th? stress symmetry, which reaiise a real closure condition by itself, as in the three-dimensional case[ll]. This symmetry can be relaxed, according to the path previously pioneered by Fraeijs de Veubeke in the thrce~~ension~ non-linear caseIl81, a way which brings the rotation as a new independant variable into the principle. This method has been successfully applied by Sander et a/.[191and Athni et al. [20]in the case of plate elements. These new developments in the non-linear theory of shells are some of our present research subjects.
1. B. 0. AImroth and
F. A. Brogan,Bifurcation buckling for
aenerai sklis. AI~A~XE 2.
3. 4.
13th Smctnres.
Smcwal
yffemics M&&al Co@. San Antonio, Texas iApr. 1972). AMA Paper No. 72-354. B. Irons and A. Razzaque. The evolution of the isoparametric elements. World Cone. Fmite Eiew~enrMerh. Structm~l Mech. (J?diu?dby J. Robinson). Boumemouth, Dorset, England (Oct. 1975). R. D. Wood and 0. C. Ziinkiewict, G~me~~y non-linear finite element analysis of beams, frame, arches and axisymmetric shells. Campus Structures 7, 725-735(1!977). R. Giid. Elements finis isoparamstriques’de coque multicouche pour Ie calcul des struchues akrospatiales en mattriaux composites.LA Rechmhe Att~~~oti& No. B77-
2. p. 131-132.-
J. L. Sanders, Jr., Non-linear theory of thin shells. @wt. Appl. Murh. 21(l) (Apr. 1963). W. T. Koiter, Gn the non-linear theory of thin elastic shells. Pm. Physical Sci. Mech. Series B 69(l) (l%S). K. H. Mushtari and K. 2. Galimov, Non&ear theory of thin elastic shells. NASA ‘fTF62, U.S. Department of Commerce. O&e of Technical Services, Washington (1961). 8. R. Valid, Sur le calcul dcs coqueshors du domaine tlastique. C&A. SC. Paris t. 263,89-91 (1966). 9. R. Valid, La thtoric tin&ire des coques et son application aux calculs inklastiques. Th&se (1973). Publ. ONERA No. 147 (1973). Trad. ESA T.T.-109 (1975). IO. R. Valid, Conditions de compatibilitt et fonctions de contrainte dans It cas des coqk multiplement connexes. 14&ne Congds IUTAM Dclft (30 Aodt-4 Sept. 1976)-_Rech. ALrosp. (19765), 289-299.
Il. R. Valid. La M~c~~que des Mj~iea~ Conrinus et ie C&al des Stmctums. Eyrolks, Paris (1977). 12. P. M. Naghdi. Foundations of elastic sheil theory. In Progress in Solid Mechanics. Vol. 4, .DD. . l-90. North-Holland. Amsterdam ( 1%3). 13. W. T. Koiter, General equations of elastic stability for thin she&. Pmt. Symp. Theory Shek Honor Lloyd Hanrilton Awtnel. DD. 187-223. University of Houston. Texas (19671. Chr the stability of elastic equilibrium. The& 14. W. T. K&r. Delft (19431: NASA T.T.F. 10 (1%7). IS. P. L. Boknd, Larpe drfkction analysis of thin elastic structures by the assumed stress hybrid tinite ekment method. Aeroekstii and Structures Research Laboratory. Department of ~ron~ics and As~~utics, M&T. Cambridge M~hu~tts 02139 Contract of Air Force G&cc of S&ui& Research U.S.A. AFGSR.TR 76 1111. ASRL. TR IX-4 @ct. 1973).
16. P. L. Boland and T. H. H. Pian. Large deflection analysts of thin ekstic structures by the assumed stress hybrid element method. Cornput. S&ucrrtres 7. I-12 (Feb. 1977). 17. P. G. Bergan and G. Horrigmoe. instability analysis of free form shells by finite elements. 14fh Inf. Gong. 77reorerrcal Appl. Mech. HJTAM Delft. The Netherlands (30 Aug.-4 Sept. 1976). 18. B. Fraeijs de Veubeke. A new variational principle for finite elastic displacements. fnr. J. Engng Sci. 10. 745-763 ( 1972). 19. G. Sander and E. Carnoy, Equilibrium and mixed formulation in stability analysis. Int. Conf. Finite Elements NonLinear S&d Structural Mech. Geilo. Norway (29 Aug.-i Sept. 1977). 20. S. N. Athui and H. Mur~wa. On hybrid finite element models in Non-Linear Solid Mcch~ic~. Int. Conf. Finite Elements Non-Linear Solid Stmc~um~ Mechanics. Geilo. Norway (29 Aug.-I Sept.. 1977).