Analysis of geometrically non-linear bending of beams and plates with mixed-type finite elements

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Analysis of geometrically non-linear bending of beams and

plates with mixed-type finite elements

Citation for published version (APA):

Menken, C. M. (1974). Analysis of geometrically non-linear bending of beams and plates with mixed-type finite

elements. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR5248

DOI:

10.6100/IR5248

Document status and date:

Published: 01/01/1974

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ANALYSIS

OF GEOMETRICALLY NON-LINEAR

BENDING OF BEAMS

AND PLATES WITH MIXED-TYPE

FINITE ELEMENTS

PROEFSCHRIFT

~er verkrij~ing van de gr~~d Van do(to~ in de

~echnis~he wetensch~ppen aan de T~chnischc Hogeschool Eindhoven. op gezag van de rector magnificu~~ prof.dr,ir, G. Vo~~eL~~ voar ~en cOmMi~Sie aangewezen door ne~ ~o11ege van

dekan@n in het openbaar te verdedigen op dinsdag 26 maart 1974 te 16.00 uur

door

Carnelis Marinus Menken

gebo~en te Haarlem

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DIT PROEFSCHRIFT IS GOWGEKEUKD DOOR Die PROMOTOR£N PROF,DR,TR, J,D, JANSSEN

en

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Aan: Riet Lieeb$th

Karin Marijl<e

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geometrically non-linear bending of beams •..•.••.••••••• 39 2.5 Correctness of the formulation •••••••••••• ,.,',... ...•.• 42 Chapter 3, Some procedures for obtain~ng finite element mooels. 45 3. I lntroduction .•..••.•• , ..••.•...•• ,... 45

3.Z The finite-element method and COnvergence criteria 46

3.3 Some alternative finite-element procedures as ~seo in tne

st~tionary potential energy approach ...••••••••••••.. 48 3.4 Alternativ~ finite-element procedures for the Herrmann ..

form1,l1ation 53

3.5 A finite-element formulation for large displacement of .• beams ..•..••.•..• , .••.•..•. , •. ,.,.,... 56 3,6 Numerical examples ... 61

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Chapter 4, Derivation of the Herrmann formulation for the

an~-lysis of finite defl~ctiong of plates. •.••• .•.... . . ••. 68 4. I Introduc tion ...•....•.•.•••••••.•.••... 68

4.2 The potential eneJ:"8Y form\llation suited to H,e trart~for-m8.tion to the lierrmann formuLation . ~... 69

4.3 Derivation of the Herrmann formulation .. ... ....• ••• 76

11,4 Corre,c:.tn-e$F.i of the formulation . . . r . . . w i . I . . . I 78

4.5 Alternative finite-element models . . . . . . 80 Chapter 5, A mixed finite element for the analysis of plate ••

bending

5.1 Introduction ..••.•••••••.•••••••••••••••••••••.•... 86

5.2 Formulation of the finitE-element model... 86 5.3 The cont.ribution of prescribed loads and rotations.... 100

5.4 The contribution of One element to the non-linear ...•. equations of the entire structure 101

5.5 The contrihution of an element to the incremental . . .

equations of the entire scructure 104

Chapter 6~ Th~ ~quation5 of the entire structur~t the CO~puter program and B nl.,lmeriC,::I:l example '" f +. I .. I I f " " I .... I .. + 1+.. • • • • • • 106

6.1 Tntroduction . . . 106 6.2 Tha ~quations of tha Qntir~ structur~ ... .•• .••.•• .•.•• 106

6.3 Ihe tomput~r program 114

0.4 Numerical example... . . . . . . ... . . . 121

S,lmmary and conclusions . . . I 129

Appendix A~ Equation~ of the finite di~placernent theory of ...

e 1 a 5 tic it y ... , . . . I • • I I • • • I .. I • • I I . . . I I 1 ."34

Appendix B, Numerical solution of set~ of non-linear ",quations 1.)8

Appendi.x C, Indentification of the multipliers occuring in the plate formulation... 142

Appendix D, Auxiliary relations referred to in Chapter 5 •• ••• 147

Notation 150

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iNTRODUCTION

During the last few ~eca~es the Einit~-element methods have proved to be useful engineering tool~ for th~ analysis of a great variety of structures.

,ni"ially the developments we~e con~ined to linea~ theo~ies, and

e~pecial1y after reCOurse had been had to the p~inciple of minimum potential ene~gy for the gene~~tion of finite-element models, the displacement method based on this principle wa~ widely accepted.

Finite elements formulated conforming to the ~equirements of this principle are said to be consistent, while in that case the dis-placement forwulation becomes a pa~ticular case of the Rit~ method so

that Lhe cOnVerSence theory of the latter can be applied~

A great number of elements has since been developed for the ana-lysis of ditferent classes of structures. In the case of plate ben-ding, however, it p~oved difficul" to formulate consistent triangula~

elements oeca~se the follo~ing requi.ements had to be fulfilled at the same time :

along ~a~h ~id~ both the lat~ral di,placement and the rotations of the normals to the middle plane eho~ld be compatible

- tne properties of the element eho~ld be independent of the choice of the coordinate system. ~~en describing th~ displac~ment

distri-b~tion by means of polynomial~, this can only be ac~ompli5hed by

~6ing complete polynomials.

Attention i5 here devoted to triangular e.l~ment;:s sipce these. permit

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CLOUGH and TOCHER I bave presented an evaluation of s\l<;h ele-ments. The major existing approaches are:

- to r"nounce the comp,qtibility of the normal rotations 2. OLIVEIRA]

proved that in that C0se no conve~gence can be obtained unless the

elements (ire so ol-tt'(inged that the nodes are all of the Eame kind; i.e. surrounded in the same way by ~lam~nt£. This restricts the applicability considerably;

- to te-establish compatibility by m~ans of correcrion functions 4. Strange enOugh, this adve~sely influenced thR spe~d of convergence

which, tifter all .. was e.xpected to -i.n.cre~"lse;

- to use nigner-order pol ynomi.als CfUth degree) for the displace-ments and introducin~ curvatures and twisting as degTees of

fre~-5 6 . . . '

dom ' . ThlS lmpbes special provisions if el,emencs with diffe-ring thicknesses are coupl~d to e<lch other and if at edges neith~r

the deflection nor the rotation are specified. Moreover, a large number of degrees oI freedom (18) is involved;

- oy d ivid i.ne the element into subregions. assuming independent poly-nomial di~placement distributions for each subelement and requiring compatibility between the 6ubelemenC$ 7

A.~ incomplete polynonlials are used, the properties of the element become dependent on the .;holce of the. coordin<lte system. CLOlJGH and TOCHER I and CLOUGH <Ind fELIPPA 8 have presented " complicated quadrilateral element the def.lection of which is expressed by expansions over twelve tri"ngutar subregions. Acoording to CLOUGH 9 the development o~ this element took seve:r;/il years.

The approaches mentioned above are not meant as a chorough evalua-tion of existing plate bending elements but merely as an indicaevalua-tion of tne difficultieB encountered when t1:ying to formulate triangular

plat~-bending element~ cOnsi5ten~ with the principle of minimum potential energy.

Another disadvantage of the latter formulation is that, apart froIII the deflections. in may ca$es one is inte1:ested in th~ b(!l'Iding

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atresseS. Since in the relevant principle the stress-strain rela-tions and the strain-displace~ent r~lationa are assumed to be known, the commonly rollowed way is to use thes~ to ~alculate the stresses from the approximated diapla~ementa. In the plate bending theory, however, this means that the bending stresses are related to the second derivatives o£ the approximated deflection and thus will be rather inaccurate, especially when using simple polynomials. More-over, these stresses are discontinuous at the element boundaries since equilibrium is only satisfied in the mean.

Other wayS of stress calculation, such as redistributing the

generali~ed

torces over the edges of the element 6 or using the theory of conjugate approximations to constru~t continuous atress approximations 10 are not widely used.

In 1965, HER~NN l I p~esented a variational printiple to COn~

3tr~ct approximat~ ~olutions for plat~ b~nding probl~m~ that offered a very attractive solution to the aforemention~d problems since both deflection ~nd moments appear a~ the unknowns in the formulation. The less int~r~Bting rotations and shear forces are related to these unknows by first derivates. At the element interfaces th~ only re~

quirement made on the unknowns i~ continuity, so that the diffi~ulty

of fulfilling rotation continuity is not ~ncountered. In consequence, simple polynomia~s c~n be used.

Although the principle was not a minimum principle, OLIVEIRA 3 has formulated the convergence c~iteria.

The simple functional form of the deflection can ~lso be used for the in-plane displacements, thus enabling the an!!lysis of folded plate structur~s to be made.

During the last few years the finite-element method has also been extended to a great variety of non-,inear problems. In this thesis attention will be giv~n to geometrically non-linear plate bending problems, which meanS that the deflections are of such mag-nitude that linearization of the strain-displacement relationship ~s

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The first work on the extension to geometrical non-line&rity

was reported by TURNER et ,,1 13, In a subsequent paper, GALLAGHER et

a1 14 outlined a conststenc procedure, based on the principle of

minimum potel1ti'~1 ellel.""gy for }'Dtrodl.J.cing the geomet;:rf~al n0l,1-1

ineari-ty in t))~ pertinent fini.te-element displ,'cement formu),ation,

MALLF:r ~n(i MAR~AL 15 have preHnted a summary of the developments in this field.

Still more than i.n th() case of the liIlear formulations the de-velopments were based on the principle of stationary potential ener-gy. probably because this displacement formulati.on had already been widel y accepted when solving linear problems. Moreover, a <) i.splace-ment formulation looks the most appropriate to iIlcorporate the geo-metrical non-linea<ity.

However. when developing consistent elements for geometrically

nDn-lin~ar plate bendiIlg along this line, the same problems as met in the linear fotmuletion appear. Thus it looked very attraetive to make use of the .jIdvantages of Herrmann's formulation also in the large de-flection range. To that aIld, Herrmann's principIa should be e~rended

to geomet<ically ncn-linear bending.

The <'''tension of othe~ variational principles being known, the extenBion of Herrm(lnn's pJ:"inciple seemed feasible. For instance, the following exten~ion~ (Ire known:

The extension of the princi.ple of minimum potential energy to 1,~rBe

deformations as given by KAPPUS 16

- RE1SSNER's variation,,! principle (IS e"tendeo by RElS5NER himself 18 - Hu-wAsH12u's principle 19 as ""tended by TEREC;ULOV 20

ZI 22

For the geometrical non~linear case ZUBOV and KOlTER have recently taken Rignificant steps towards effect~ve generalization of the principle of cDmplementary anergy.

These exten~i.oTl~ being establi~hed, the question remains how they can be obtained systematically.

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In the li~ear domain chere exists a general c~~n6formatio~

scheme, known as FRIEDRICHS'transfo~mation, which makes possible the cransformation of a potentional energy formulatio~ in co a compleme~

tary energy formulation by a set of ~ell-defined steps. Other formu-lations, su~h as R~ISSNER's, sre obtainable as intermediate steps.

Whereas the sbove publications o~ extended variational pri~ei pIes offer

~o

information on how they have

bee~

derived, wASHIZij 23 derived them by proceeding along the same lines as in the li~ear theory.

Up co nOw, however, with the ex~eption of the potential energy formulation, the extended principles seem not to have been followed up by an implementation in fi~ite-element models.

Therefore, the main obje~tives of the present thesis have been formulatea as follows:

I, TO establish the relationship of Herrmann's v~~iational prin~ipl~

with other existing variatio~al p~inciple~,

2, To use this r~lationship to extend Herrmann's variational prin-ciple to the particular cas~ where the strain-displa~ement rela-Lions are not linear,

3. To develop a computer program for the analysi.s of geometrically

no~-linear bending of plates based on this extended variational principle.

The following approach will be used:

The place of Herrmann's variational prin~iple in FriedriChs' transformation scheme will be determined. This pl~ce being known, the steps to be taken to create this principle from a ~iven

poten-ti~l energy formulation are defi~ed, This will be illustrated for the mathematically simple e~ample of a cantilever beam (Chapte~ I).

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StHting e1:"om the potential energy fOT!l)tIlation for the

small-otr<:lins large-displacement theory of elastic 1)",am~. the same steps

~r" to b~ used to generate the extended Hen-mann formulation for this problem. The Euler equatiortG of th" latter are compared with the

BoverninB field equations to prov~. the validity of the new formula-tion (Chapter 2).

Moreover, .some numet"ictil :results obtait'l.cd with a cons.istent finite element formulation will be presented (Chapter J).

By analogy, starting from th~_ potenti a1 energy formulat ion for the VOn Karman plate bending cheery, the Herrmann formulation of this th(!ory will be dedved (Chapter 4),

A tomputer program will be developed for the approximate solu-tiOn to finit" plate b"l'lding by means of the finite eleme.nt method, hased on thi. extended Herrmann formulation (Chapters 5 and 6).

Although a geometrical non-linear formulation implies the possi-bility of buckling analysis, the latter 5ubject~ will not be treated in t.hi~ thef! i_<;:,

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CHAPTER

I

The relauon

of

Herrmann's variational principle to other variational principles

1.1 Introduction

In the cl~b5ical linear theory of elasticity a general scheme

exists (e.g.2.1) making possible the generation of differen~ varia-tional principles by starting from a kno~ potential energy formu-lation and proceeding by taking ~ome well-cte£ined steps.

The aim being to formulate Herrmann'S variational prin~iple for geometrically non~linear analy~i~ of beam~ and plates and the rele-vant potential energy formulation being known, it seemed worthwile to determine the place of the existing Herrmann's principle in this scheme. When this plac@ is known, the steps to be taken to generate the Herrmann formulation from the potential energy formulation are defined for the linear case.

Subsequently, by starting from the potential energy formulation for the geometrically non~linear case and taking the same steps, the

Herr~anrt formultition extended to the geometrically non-linear case

will be obtained.

In this Chapter the relation of the existing Herrmann formula-tion with other variaformula-tional formulation~ wi~l be determined.

In SectiOn 1,2 the general transforma<lon scheme, known a~

Friedriche'transformation, will be 5umrnsri~ed,

In Section 1.3 it will be shown that Herrmann's principle is partly a potential energy prin~iple and partly a complementary prin-ciple, and ~an be s£nerat£d from a fully potential energy formula-tion by applying the Friedrichs'transformaformula-tion.

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The argvment. will b~ illu.trateO by means of the mathematically

simple example of a cantilever beam.

A general thr~e-diman.ional elaboration i . not practical since in a specifi( "ending probl~m the separate stress-displacement systems to be treateo in different ways are easy to identify.

Moreover, Herrmann's principle has been formulated to solve pro-t>lems typical of bending. 'rhis, ho"'ever, does not mean that the ap-proach ""nnot be used to QverCQIM analogous problems in othex engi-neering br~nches.

1.2 Gener~l transf.ormation scneme

We identify a p8rticle of a body by its rectangular Cartesian coordinates Xi (i=I,2.3) i.n the undeformed state. A oefQrmed confi-guration is described by th~ Cartesian coordinates xi+~i of the same pa.ticle, where ~i represents the displa~ement vector.

~8rtial differentiation with cespect to a coordinate Xj ,. denoted by a s~bscript j preceded by a ~omma:

O\.l./(lX. = u_ , .

, J 'oJ

Cartesian tensor notation will be employed, including the sUlllInation convention; the repetition of an index in a term denotes a s~mmation

with re~pect to that index over its range.

Prescribed quantities will be denoted by a supersc~ipt 0

Consider a linear elastic body with undeformed volume V under the action of body forces k~ per unit voLume and surfa~e forces p~

~ 1

per unit area acting on the 5urfa~e S • We shall assume these forces p

to he prescribed and kept unCh3nged in magnitude and dire~tion during

variatioIL

On the ~emaining part Su of the surface S the displ3cementa

are pre!3c.r~bed.

o

u.

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The eq~ation8 defining the linea~-elasticity problem are: t .. _EijkR.ekR. ~n V (1.1) ~J 1 _{in V} _(1.2) e .. _{~ 2(~i,/~j}

_,i'

lJ t. , .+k~ ~J ,,] i = 0 in V _(I_.3) = a _Sp (I .4) Pi Pi on 0 S (I .5) l,li 11- on ~ l,l where t ..

~j are the c.omponents of the symmetric stress tensot", e,d are

the components of the lineat' strain tensor .. _"ijktis the tenso~ o£

ela~tic constants giving the ~elation between the 5~X independent components t_ijand the six independent components eij' while

(I .6)

and Pi are the components of the stress vector. The stress veccor is related to the stress tensor by Pi a tijTIj where TI

j ~~e the compo-nents of the unit normal to the 8urfa~e drawn outwards.

The pertinent potential energy functional reads:

u,

E jW(e .. )dV

-lk~u.

<;IV

-1

p~u-

dS

V lJ V • • S 1 1

(1.7)

P where w(e

ij) is the stored elastic strain energy per unit voll,lme:

(1.8)

Strains and displacements satisfying the strain-displacement relations

(1.2) and the kinematical constraints (1.5) are sa~d to be kinemati-cally admissibl~.

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The principle of minimum porenti,,} energy state.s;

ilmong aU the Un<'"latic,aUy adm~:s8ible ,HspZacemen.t field" the

,,(:tl)cd dl:8p~aoementG make the potel1UaZ el1epg'Y (1.?) stO,tio)1.a,py

(m,:nimum) .

The actual displacements can thu~ h" obtained from th" minimi-zing conofttons o[ [unctional VI provided assumptions (1.2) and (1.5)

are taken as ~uhgidiary conditions.

Indeed, by cequiring

o

(1.9)

wh"re the subscript of a variation symbol indicates that the varia-tion is taken with respect to the variable indicated by the subscripts,

til'" equilibrium equarions (1.3) and houndary conditions (1.4) are

ob-tained.

The tran!=lformatiol1, of thi.s principle into the complementary principle

proceeds as follows:

- The subs id iary cond i t ions (1.2) and (I. 5) ar~ introduced in functio-nal U j by means of 1.agrange multipliers t

ij and Pi' The st(>t~on(>dty of functional U

j with conditions (1.2) and (l.S) is equiv"lent to the free .tationary hehaviour of functiOnal 02:

1I2 c fW(e,,)dV -

f

It,,{e, -

tCu, ,+u,)I]dV +

J

_v

1)

J

_v

L

1.) ~J ~d J,~

-lk~U.

dv

V 1 1

(I. lO)

The ind~pendent quant~tleg eUbject to variation dr~ eighteen i~

number, viz. e,., t " , u. and p. with no subsidiary conditiOns. By

~J 1 J 1 1

meanS of the station~ry conditions of 1I2 _{the multipliers t ij and Pi}

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sur-face streSS vector respectively. In subsequent chapters this identi-fication procedu.e will be carried out in detail. It i~ omitted here and U

2 is immediately considered to be a functional with identified multipliers. On taking variations with respect to the independent quantities, Ilqs. (1.1) to (l.S) incl. are obtained. Fo.mubtion (1.10) is sometimes named after Hu and Waahizu and can be rega.ded as a generalization of the principle of minimum potential energy, sinc~,

if EqS. (1.2) and (1.5) are taken as subsidiary conditions, U₂, is reduced again to U

l•

The 8~cond step implies variation of the strains. The condition:

& U - 0

e .. 2

l.1

leads to the stress-strain relation:

"wee .. )

t .. _ lJ

1J ~

1.1

According to (1.8) this means:

This relation can be inverted:

IntrOducing the complementary energy per unit volume;

(l.I!)

(I .12)

(1.13)

(1.14)

( I . 15)

this quantity can, by virtue of (1.14), be exp~essed ~n ~h~

stresses:

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In our case this results in:

w

c I

"2 Fijk~ti/H (1.17)

Under the a~sumpl;:iol1 (1.12) the strains can thos be diminate<J from U 2 :

-lw

(t. ·ldV

+jtcu .. +Il ..

)t .. dV V c ~J V ~ • J J • ~ ~J

-f

k~u.

dV -

r

v ~ 1

J

s

p~t.1-dS

-1

P.(u.-u~ldS

1 1 S 1 1 1 (1.18) p U

This functional is ~quivalent to that of the Hellinger-Reissner prin-d p 1 e ~ The ill.d"p"nd~nt var iab 1 ~5 are t,., u. and Pl."

'J 1

- The third ste~~ which is due to Friedrich8~ implies partial

inte-gration of the proper term containing t

ij in order to generat~ the

equilibrium equations (1.3): U 4 n

-fw

(t..)dv

-fCt. ...

k~)u.

dV .. V (: lJ V LJ,j L L

+J

_S

(p,-p~)IJ,

~ , ~ dS +

is

p,u~ ~ ~ dS (I. 19) p u

The number of independent variables can be reduced by requiring equilibrium. Stress distribution consistent with equations (1.3)

" According to TONTI 2,2 in Reissner's original formulation compati-bility of strains and displacements was assumed, as shown by the ab-sence of the last term in (1,18), 50 that stationarity with reopect to variations of tij yie~oeo the stress-st~ain re,ation, ,n O~T case the stre~s~strain relations are assumed, so that variation of the

str~ss~s r€sults in the kinematical boundary conditiOns and,

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and (1.4) Ii.e s<,-id to be "statically admissible" Under these anump~

tions functiona1 U

4 becomes, after inve~s~on of sign:

Us

Dlw

(t .. )dV

-J

Pi\l~

0:15 V ~ q S

u

(1.20)

Thus it remains ~o vary t

ij and Pi' Since Wc(tij, is positive

defi-nite, another minimum prin~iple is obtained. This p~incip1e of minimum potential energy states:

Among a~r statically admissibZe st~e88es the actuaZ stresses make the complementary energy U. 20) stationary (mhdm1fft1).

Indeeo:!, by requiring:

6 U = 0

r:J •• 5 (1 .21 )

1J

the strain-displacement relations (1.2) <'-nd kinematical constraints

(1.5) are obtained.

1.3 The relation of Herrmann's variational principle with other variatiOnal principles.

The general t.linsformation scheme being presented in Section 1.2, it remains to place Herrmann's principle ip tnis scheme. We

sh~ll re~trict ourselves to only a simple beam problem 5i~ce this will be sufficient for demonst~ating the argumentS.

Consider a slender, homogeneous, elastic beam of constant c~oss

section (Fig.l .1).

This cross-section has an axis of symmetry. The x-axis coincides witn the centroids, while the ~-a~is is pa~allel to the axi$ of symmetry of the cross-seeton. Torsion-free bending in the x-z-plane has been

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z

F-ig. 1.1 Beam, diapZaaements and ~oad8

realized by p~oper application of extern~l loads acting also in the x-z-plane.

The beam has a length ~, a cross-sectiDnal area f, a modulus of

elalticity E aad a mom~nt of ine~tia X,

(1.22) .

integration heing taken over the croSs-5ection S of the beam, At the

end x=Q the displacernent~ are prescribed:

w' (0)

,,0

o

(1.23)

(1.24)

In Eq. (1.24) and in the foUo"'ing ones a prime denotes differ~nta

tloo with respect to x, thus ( )' = d( l/dx.

The q"antiti~s M and Q (lre the bending moment and shear farce

of th~. cross-section, as shown in Fig. 1,2.

The beam is subjected to a distrihuted load qO(x) per unit length, at the end xci to an end force P:, in the direction of

a

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"(j

Q

. dx

Fig. 1.2 Positiv~ di~~~tio~s of Q and M

For this example the Herrmann functional 2.2 is:

and the subsidiary condition~ are:

H we 1:'equire:

we obtain as Euler equations:

M" q

"

M - Elw" 0 (1.25) (1.26) (1.27) (I .28) (1.29) (1.30)

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and the natural boundary conditions' M'U') - pO _{(I .31)} z9. w'(O) <to 0 (1.32)

Observation of the Herrmann [unctiOIlal (1.25) and its subsidi8ry conditions (\.26) and (1.27) give. the impression that with the

rele-vant vdri~tional prinr,iple two streSs-displacement ~ystema are treated in d differ~nt way:

o 0

- ter.ms - q wand - P

z£ w(£), together with condition (1.26) would

indicat~ that the deflection is treated according to a potential energy formulation. The pertinent stress Q(x) does not appear in the

formulation.

- terms - M2/2El; - M(O)4>° and condition (1.27) would indicate that

o

the moment M is treated according to a (negative) complementary energy formulation; th~ pertinent generalized displacement ¢(x), i.e. the ~otation of the cross-section, is absent.

This hypothesis will he proved nOw. Th~ obserations imply that if we wish to constr~et the Her.rnann formulation from a potential energy formulation, both di~placements wand • should be preBent in thQ

starting formulation.

II consistent way to formulate such a functional with its subsi-diary <;onditions is to Btart from II more gene.al bending theor.y where

the ~lop~ of the elastic ],ine and the rotations of cross-sections are

clearly "~parated, i.e. the bending th~ory accounting for shear de-[ormation.

The relevant potential energy functional r~ads:

(\ ._33)

whare ( and yare the hending str~in and the shear strain re5pecti-vely anJ G is the modulus of rigidity.

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Kinemati~ally admissible strains and displacements must fulfil the following requi~ements:

- the kinematical constraints

w(O) c wo

o o 4>(0) a ~o

- the ~train-di~placement relations:

y ~ ~ - w'

(1.34)

(I.35)

(1.36)

(1.37)

Returning to the theory of n~gligible shear (y*O), this formula-tion becomes:

(1.38)

w~th subsidiary ~onditions: (1.34), (1.35), (1.36), whereas the strain-displacement relation (1.37) becomes a kinematical constraint:

w' (1.39)

,his potential energy formulation now containing the rotation ¢, forms the starting point for applying F~iedrichs'transfo~mation.

Since w is assumed to be treated according to the potential energy formulation, the transformation is only applied to ~. This means that the first ~tep • viz. introducing the subsidiary conditions in the functional by means of Lagrange multipliers, is only taken for ~uh5i

diary ~onditiQns containing ~,i.e.(1 .35),(1.36) and (1.39). As in the pre~eding section, the procedure for identifying the multipliers is omitted here and the functional wich identified multipli~rs is

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given Be once:

(1.40)

The second step implies variation with re~pect to thl ~trains

3no elimi.nillion of the ~trains, The requirement '\.U

2 = 0 leads to:

M ~ Eh

and expressing K in M this gives for U

2:

=1

£i-M2

o~

v .. ) =

1--

+ M·t' - Q(~-w') - q w dx +

o '-

21': I (1.41 ) (1 .1,2)

The third step implies parcial integration such that the equili-brium equation appears for the stress that we want to treat comple~

mentarily (M), If,on the other hand, we ~equire this equilibrium eo be fulfilled the corresponding displacement ($) disappaer. from the functional.

Partial integration of the term M~' gives:

U

4

it

_{o lJn}

[-M? -

(M'+Q)¢ + QU' - q\,ldX ..

_J

(1.43)

It we require M to be statically admissible,

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( 1.45)

the following f~nctio~al is obtained:

(1.46)

Tbis is irtdeed the the same f~nctional as U

H(I.2$) wnile const~aints

(1.34) and (1.45) also ~orrespond with (1.26) and (1.27). Moreover, the a~~umptiona (1.41) artd (1.44) are obtaineo. This result confirms

our hypothesis, and we may conclude that He~rma~n's principle can b~ derived by starting f~om a potential energy forrn~lation5 that e~able5

a dis~rimination between two st~ess-displacernent systems to be made. When this condition is fulfilled, we apply Friedrich's transformatiOn to o~e of these systems.

It should be remarked that owing to the term M'w' we cannot speak of a mirtimum prirtcipl~. Moreove~, inversion of sign, as done in the last step of Friedricna'trartsformatiort. is useless in this mixed formula-tion.

Irt view of the a5s~mptions (1.41) and (1.44) the Euler equat~on

(1.29), being the stationary condition with respect to w, can be

in-terprete~ as the equilibrium of vertical forCeS;

o

(1.47)

wnile equation (1.30), being the st~ti,o~~ry 1;ondition with respect to M, cart be inte~preted as the strain-displacement relation;

K l1li W" (1.48)

Tne~e re~ults are also consistent with a potential energy approaen and a complemerttary energy app~oach ~espectively.

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CHAPTER 2 Extension

or

Herrmann's variational principle for the case of geometrically

non·linear bending of beams

2.1 Introduction

in thi~ cha.pCer H"rrmann· s variational pr inc iple will i)e extended to geometrically non~lin£Qr bending for the c~se of a hearn. The pri-mary aim is to apply and explore the approach and prove its validity in a mathematically simple situation. After establishing this form\l-lation it can be used 3S an an~logue for deriving the ext~nd~d varia-tional principle of n;,rnnann for the case of the geometrically nOn-lim!JI beI\diI\g of plates.

"'ben deriving tbe desi.l;ed f.ormulation, the same procedure will be followed as in tlle linear case. The beam itself, however, will be treated as a special case of the general large-dbplJcement theory of

elasticitYt because it 1~ felt ellac specializ3t~on from a general

[heory forms J soundon basis than gene~ali,"s.tion of s. more restricted theory. In particular, the symmetric Piola-Kirchhoff stress tensor s .. will bE used. This notion of stress has been flJrtller developed by

1J :1 I

KAPPUS • AppeIldix A gives a SI.ImJ1I(H·Y of the relevant eq\lstions.

In Section 2.2 we will develop the required starting potential energy formulation that takas iIlto account traIlsverse shear

defor-rnation.

In Section 2.3 this forrnuhtloIl will be used to oedve the govern-ing differential equations and Ilatural boundary conditions.

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Restricting ourselves to the ca.e of neslisible .hear deformation, the formulation will be u.ed tD derive the extended p~in~iple of Herrmann (Section 2.4). The pertinent Eu~er equations will be com-pared with the gove.ning differential equations to prove the correct-ness of this formulation.

2.2 Potential energy formulation for $eometrically non-linear hending of beams, including the effect of transverse shear deformation For the case ot neglected shear detormation the potential ene~gy

formuLation is given by KAPPUS 2.1

The procedure outlined in Chapter I, however, req~ires a starting potential energy formulation which enables ~s to distinguish two stress-displacement systems, one remaining ~nchanged, the other to be transformed according to the general transformation scheme. To that

end, a pot~ntial energy formulation. initially taking into accOunt

~hear deformation, i~ needed. This formulation will be developed now. Consider the same beam as treated in Section 1.3. Now the loads are of such magnitude that it is nO longer justifiable to use the re~

lations of the linear theory of elasticity. Instead, the

large-dis-pla~~mertt

theory of

~lasticity

must

b~ ~s~d

(see e.g. FUNG 2.2). The governing equations are summarized in Appendix A. The description we wish to follow 1. Lasrangian, i.e. with reference to material coordi-nates. We choos~ a fixed rectangular coordinate system ~,y,~, such that the a~i~ of the unloaded beam coincides with the x-axi~. Material points of the beam are identified by their ~oordinates X,Y,Z, relati~

v~ly to the x,Y,z sy~t~~, in this initial position. The value~ (X,Y,Z)

remain fixed to the material points when the beam deforms. The new position of point

(x,Y,Z)

is:

x = X + u* y = y + v, z = Z + W (2.1 )

where u,v,w represent the displacement. In the Lagrangian descrip-tiDn th~ symmetri~ Piola-Kirchhoff stress tensor will be used.

In addition to the boundary conditions pre.ented in Section 1.3, at the end XeD a displacement uO in the x-directiOn is prescribed,

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while at the end X-' a preseribed force P:

A in the x-direction is

,Hided (Fig. 2,1).

In this chapter, the components of the displac~m.nt vector will

1[1 generdl be repreB~nted by ;,n and 1:, while the "ymbols u,v and w

.are reserved for th-e digplacement5 of I?oints of the axis (Y'IIIlO.,. 7.:=0)

of the l>8~m:

u = s(X,a,O), v - ~(X,a,O) and w ~(X.().O) (2,2)

(X,D,O)

---~~-r~--~x

z

Pig. ,~.1

The following hypntl1eses will he used:

(I) The heam is slender + Tl1e di.men,sioTIs of the C.f05S-S12:LtioIl ar~ ro,\~h 5Il1al.l~r tlH'n the length ,~ of the- beam. In particular we have

(2) All strain components are small.

(3) The streSs components 8 . " and" mal' be neglBcted in

compa-yy zz

Y7-risOn with the other stress components, and we ~ay write

s _yy

=,

_'zz 8

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(4) Cross-sections whi~h were p8rp~ndicular to the a~is of the beam before bending, remain plan~ an~ suffer no deformation in their planes.

With theBe assumptions a potential energy formulation will be derived. Denoting the rotation of a cross-section by ¢(X). the relation~ be-tween the displacements become (Fig, 2.2):

;; = u - Z 8iM n = 0 - w - Z(I-cos¢) ..f.x

lrf __

---:'~~~

..

_.,t!._-~:

...

--.~~:~:---{-=-,.-:::;-~-)d

X

~z

z

B -. ,.-.. - - . - ; : ; - - - - = - 1

t----~~

Fig. 2.2 Re~ation$ be~@en displacements

(2.4) (2.S) (2.6)

w

aw

dX

ax

It must be re~arked, however, that (2.4) and (2,6) are inconsis-tent as regards the accuracies of the individual ~ontribution5 to ehe displacements ( and ~. Since the assumption that the length of AB is

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lhe sam~ in the unloaded as well .s in the loaded configuration

im-plies an inaccaracy~ it is superfluous to tak~ account of t11e

contri-hution DE t as accurately ~s given by sin$ and cost respectively. it

\ 3) wOI.th\ be more cond~t.nt to replace [or instance Z ~inq, hy Z(¢-

-g1'

.

Notwithstanding thi:!i t the trigonometric cxp~essiona are. retai:r'H.'d in

the formulations to fad litate interpretation of derived "e"pre~dons.

According tu the stress hypothasis (2.3) the specific strain

en~rgy ~an he written aij[

+ 2 s ~

Xl' xy + 2 8 X2 XZ e )

and U8ing Hooke's lay: .. we have.

w - -21 (Ee"

)oe + 4 Ge? xy + 4 Ge' ) Xl.

(2.7)

(2.8)

lis to the strains, we diffe,entiate between strains at pOints (Y-O, Z~O) or the axis of the beam and strains at other points, The l.Hter will be Overh~rred. Hor~ovRr, for the shear components we introduce

the quantities = 2

e

(y = 2e ) and

y

=

ze

_v•• (Yxz = 2e )

xy "y xy "y X? M . xz Tbe non-linear strain-displac.ement re.ltitiOns art!:

e xx 't_xy y xz

'r.

+ oX ~ C + <)y

+

[O~)'

)~ 3( -_dX+

_rx

+(Y+(YJ

ill; 3~ 3n + ~~

3Y

~~~ JX

n

oX 3Y

Introduction of Eqs.(2.4), (2.5) and (2,6) gives:

e _xx

(2.9)

(2, \ 0)

C2.\\)

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o

(2.13)

(2.14)

For sm~ll str~ins. the first expression can be 5impliti~d along the following lines.

I f w", denote the angle of shear ln ment

aX

the new

is rotated aver au angle length ds, e>lpre~~ed in ds = (1+<; )dX x According to (2,12) we have: e KX ¢+i¥ "z u,.w and the x-z-plane by ~)X2; ,

and from fig. 2.2 we

¢ is:

For small shear angles, ~ <~ I,Eq.(2. 15) can be written as:

xz

(I+£X) - {(I+u')cos¢ + ~'5in¢ + ~ {w'cos¢ - (1+~')5in4f

xz

C~mbin~tiQn of (2,17) and (2.14) give,:

{(I+u')co5¢ + w'sin~l an ele-see that (2. 15) (,.16) (2.18)

In view of (2.16) and (2,18) relation (2. 12) c~n be written as:

exx (2.19)

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With (2.8), (2.13) 3nd (2,20) th~ ~lAstie strain energy of the whole

boam becomes;

w =

1 ~f

[.!.

E(e -Z,')2 + -21 G Yxz2fd5 dX

X=O S '2 xx

Thus, fo>:" a bell'" witl' boundary conditiOIls as shown i.n Fig. 2.1, After

i,lt.,gration over che cro~~-section S of the beam, the potential .mer-gy f\l\\ctio~al h",come~:

(2.21 )

where a hending strain ~ is intToduced. Strictly speaking, this ben-ding strain is not the curvAture of the beam; only if .x

this bending "train equ~l the curvature. The subsidiary conditions are~

- I;he strain-displacement relations;

e _xx ¢ , I)' + 1 ,

"-'2"

y ~ w'cos¢ - (I+u')sin~ xZ

- the Hnematical c.onstraints:

\1(0) u 0 D w(o) ~ 0 w 0 ¢(Q)

¢~

0, dOes (2.16) (2.22) (2.14) (2.23) (2.24) (2.25)

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If shear deformation can be neglected (Yx~~O) , the strain-displace~ ~ent relatiOn (2.14) becomes a kinematical constraint:

arctg (2.26)

Again, this relation is inconsistent as regards acc~racy but will b~

retained because it facilitates interpretation of relations to be derived

2.3 Forces and moment of the cross-section, and heam equations Prior to the d~rivation of the extended formulation of Herrmann's variational principle some auxiliary equations will be derived. This is necessary since an important step in the procedure to be followed is the introduction of some subsidiary conditions into the potential energy functional with the aid of Lagrange mu"tipliers. These multi-pliers are to be identified with physical quantities and this is im-possible without knowledge of the governing beam equations. Therefore, these equations will be derived here. Moreover, they can be compared with the Euler equations of the Herrmann formulation to be obtained, thus providing ~ proof of the corre~tnees of the latter.

E>:pressiol'ls for forces and moment of the cross-section consistent with our potential energy formulation (2.21) Can be obtained by com-paring its variatiol'l with the virtual work equation, expr~ssed in quantities of the cross-section:

(2.n)

ln this expre5sion O~ and 6y

xz are variations of true deformd~ions. The true exial strain, however, i~ ~x' So the virtual work, performed

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by the forces and mottl,;,nt of the cross-section and the ext .. rnal load;;,

j,$ :

oW .

vlrt

(2.28)

Ih. relation between [ and e is (see A 9)

x xx

I + [

=.J

I + 2e

x xx (7..29)

and the relation hetween tht!:ir vilriationf! is:

(2.)0)

III 'Il ew "f tiH' assumption of small strains. EX could be neglect .. d wbert compared with I. The. term (1+ .. ,,) is retained as a label, how-ever, in order to facilitate interpret~Cation, see. e.g.(2.44).

Co~p8rinB (2.27) and (2.28) gives, In view of (2.30), the following

relationg~ N RF(I+[ )e " xx (2.31 ) M Eh (2.32) Q = GF Yxz (2.3J)

The equilibrium equations and dynamic bo~ndary conditions are obtained as the Euler equations a,td natural bounda,y <;:.onditions of

Eqs.(2.21) to (2.2:;) inel,

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With

Ii""xx 0 {(l+u'>ou' + w'liw'

I

(2.37) it follows from liUMO:

(2.38)

{EF~ 101' + GFy COS$}' + qO = 0

xX x,Z (2.39)

(2.40) (2.41 )

(2.42)

(2.43) That these are the relevant equilibrium equations can be shown by

ob-servin~ that (I+u') artd 101' of EQs.(2.38), (2.39), (2.41) artd (2.42)

are rehted to the art~le (co) between the deformed axis of the beam

and the X-~Xi5 (l'ig. 2.3) as follows:

sino. 101' J+u' w'

1+[ _X co.ea

i+7"

J( tga I+u' (;2.44) lntroducirtg (2.34),{2.3l),{2,32),{2.33) and (2.44) into the Eqs. (2.38) to (2.43) incl. gives:

/NCOSOl - Qsin<P\'

o

(2.45)

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(2.47)

(2.48)

(2.49)

1'01, = M~ (2.50)

Th"t these .He the proper equations follows from ,'ig. 2.3:

N~.

-._", •.•. , ... , .. ----~ X tv'!

~x

N+N'dX Q~Q 'dX z

Fo~ce{; a,atirtg on an dement elx

If shear deform.tio~ ~an he negl~ttQd (o=*), the equilibrium equation (2.47) h~eomes:

1'01' = - (I+',,)Q (2.51 )

Moreover, in that case the relation hetween the b~ndiDg strain K and

t.h~. displacements u en w becom.-:s by introduction of (2.26) in (2.22):

(t+UI)Wil ... w'u"

( 1 +u ' ):' + w' 2

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2.4 Derivatiort of Herrm~nn's variatiortal yrinciple in ~ase of geometri-cally non-linear bendinA of beams

The preparatory work being done in the preceding sections. the derivation of Herrmann's variational principle exten4ed to ~eometri

cally non-linear bending will now be ta~en in hand.

The potential energy functional (2.21) together with subsidiary conditions (2.14). (2.16) and (2.22) to (2.25) inc.l., form the ~tart

ing point for applying the precedure outlined in Chapter l. Since we confine ourselves to the case of negligible she~r deformation (YxzcO),

condition (2.14) must be ~eplaced by (2.26).

According to the procedu~e, only the

(M,t)

~ystem will be sub-jected to the trans£or~ation into a complementary £or~ulation. ln consequence, only sub~idiary conditions containin~ ~ will be b.ougnt

i~to the framework of the £unct~Dn~l ~y means of Lagrange mUltipliers in order to introduce the genera,ized force consistent with ¢, viz.M, into the functional.

Ihis implie~ that the non-linea~ condition (2,16) remains one of the SUbsidiary condition~ irt th~ formUlation to be obtained and does not co~plicate the transformation.

The sta~ting formulation is:

where AI(X), '2(X) and '3 a~e multiplier o • The remaining subsidiary conditions are'

e _x:': ut

+!

_u' _{2 •}

l

_Wi 2 2 u(O) u o o w(O) CI WO o (2.53) (2.54) (2.55) (2.56)

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The independent quantities subject to variation in functional (2.51)

.are Seven in number; u, lo):i ¢ II k;' and the: multi:pliers }'l ~ )'2 and \3'

In thi~ ca~e, tilis formulation is equivalent to the original one. Ih~ re4uirement DE a stationary heha.viour with respect to the

multi-plier. would give the origin~l subsidiary conditions:

,~

.

_(2.5:1)

,~

-

arctg w' (2.58)

~

:(0) ,p 0

0 (2.59)

Our 1~)t~'[u:io~L ib~ huwever, to rt'cinsfot"m rhi;s forIT'luldt.ion. Contrary

to Cl1.3pter I, the identific~tion of the multipliers will be given.

Sta~lonary behaviour of U

1 with re~pect to • and ¢ gives: 'I(X) - A I (2.60) (2.62) (2.63)

A comparison of (2.60) with (2.32), together with (2.61), gives:

)

I M (2.64)

M(O) (2.65)

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(2.66)

This means that by means of the Lagrange multipliers the shear force

Q and the moment M of the cross-section are introduced into the

functional:

(2.67)

The second step implied the requirement of a stationary behavio~r

with respect to K. Requiring 6.U₂= 0 gives:

MEl" (2.68)

~his res~lt is use~ to eliminate the de£ormation " from (2.67) by

mean.!;! of ~

(2.69)

Step three implied partial integration to gen€Ldte an equilib~iurn eql,l~tion'

If we require the moments to be in equilibrium: (2.70)

(2.71)

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the exten~ion of H~rrmannl~ variational princlple to geometrica1ly

non-linear \lending f.or the case of a slender be!'1ll is obt,,~ned:

u

=

r

\!..

F:F~

2

H

J

X~O \2 xx

M2 "I' 0

l

- ffi-

M' arctg \+1,1' - q w dX +

(2.73)

The ~1,1l)s~di.~ry conoitions are U.54), (2.55), (2 •. ')6) and (2.72). The linear formulation (1.2.5), (1.26) and (1.27) can be obtained as a special caSe \ly taking 1,1'=0 and w'« I.

The principle states:

Among aU d{~p[w:>"m"n.t8 1. and IJ IJll·/.:,:.h $a./.i.r;f:4 "the pre!;ora'ed geo

m,,-trlcJa/ VoundaI'!j oon,HMona, and amOng aU moments M whioh !;QUaf!!

"the, pl'<;w!r'':b"d (Zyn<1J1l-i(' boundal'Y ('onrlitl>Jnf3, I;;h(, Q('tuat dicpZao"m"nt$ m1d lIIolllrmt mak,! I;;hG Her"f'mann fU"l('I;;,;onat flta.i:ionary.

I t i~ noteworthy that when fully transforming the potenti.al

energy formulation of ~ geometrically non-linear problem into iC$ compl,:,mcI~tary formulatioI"l, a major problem is that after requiring equilibrium the displac~meI"lt" do not disapp~ar from the formulation, but products of stresses and displa~ement derivates remain. Both ZUBOV 2.3 and KOlTER 2.4 hava taken significsnt steps towards

eX-pre~dng the complementary energy functional solely in stresses. In our partial, transf"rm"tion of the poteutial energy form1,11ation intO the mixed He~~mann formulation this problem does not "ppe,,~; "'heu requiring the equilibrium equation (2.7]) to be fulfilled, the re-SUlting fun~ti"nal (2.73) beCOmes fUlly independent of the relevant

displ,,~ement ¢:

2.5 Correctness of the formulation

In this section it will be proved that the stationary principle presented in Section 2.4 is indeed eq1,1ivalent to the governing set of equations as derived in Section 2.3.

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The governing equations are:

- the strain-displacement relations (2.16) and (2.52)

- the equilibriu~ ~qoations (Z.45), (2.46) and (2.51) with, in view of the negl~cted shear deformation, ~c~,

- th~ ~onstitutive equations (2.31) and (2.32), - the kinematic constraints (Z.23), (2.24) and (2.25) - the dYnami~ constraints (2.48), (2.49) and (2.50).

In the Herrmann principle, equations (2.16), (2.51), (2.31),(2.32) (2.23), (2.24) and (2.50) have already been assumed. It remains to prove that tbe stationary conditions of the Herrmann functional (2.73)

are equivalent to (2.52),(2.45),(2.46),(2.25),(2.48) and (2.49).

Requiring DU_Hg 0 with respect to variations of u, wand M

res-pectivelYt gives as ~uler equdtiort5~

and as natural boundary condition:

{arctg

1:~1

}

0

4>0

_o

o

M' (I+u'_)

_}'+

qO " 0 (2.75) (I+o'}2 + w·? (2.76) (2.77) (2,78) (2.79)

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We perform the [allowing operations:

- the strain .xx is introduced by using assumption (2.16),

- the axial force N 15 introduced by using consitutive equation (2 • .11). It should be remembered that (1+,) is relac"d to u and '"

hy:

(2.80)

~ the sh"n force Q i, introduced by using assumption (2.51).

- according to (2.58) it is ju~tifiable to introduce ao Rog1e • as defined by (2.5H).

The,,, ()pH'~ t j on~ performedi it turnS out tlwt:

-

Eut~"(' equation (2.74 ) 15 equivalent to the equation expresl:iing the

equilih[,"lurn of forces in til,;: x-direction (2.45) •

Eut"r equation (2.75) is equivalent to the .. qua tion expressing the

eql.d 1 i.br il,lffi of forces in the z-direction (2.46) •

- Since the con~titutive equation (2.32) was assumed, the Euler equa-lion (2.76) is eq\lival",nt to the strain-displacement relation (;1.52)

or. expressed in @, the 5train-di$placement re"atian K~¢'.

- 1'l,lturnl l!oundHY 8ondition (2.77) is equivalent to the dynamic boundary condition (2.48),

- Natural houndary condition (2.78) i.s eq\livalent to thE: dynam\c houndary conditiOn (2.49).

- N,Hu!",," ['''llUd.uy c.ondition (2.79) is Eqlli"ale~t to the kinematic constr8int (2.2.).

Since this completes the Get of go"er~in~ equations. the Herrmann

formulation as obtained in Section 2.4 is correct.

F:quation5 (1.29) to (1.32) inci. can be obtained as a spedal case of (2.74) to (2.79) incl. by assuming u'=O and w' « I.

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CHAPTER 3

Some procedures for obtaining frnite element models

3.1 rntrodu~tion

In the lin~ar dom~in. th~ finite element method b~sed on Herrmann's variational principle offered an attractive solution to

problem~ encountered when analyzing plate bending by means of a finite-element method based on the principle of minimum potential energy. Since in the geometrically nOn-lin~ar domain the same

pro-bl~ms ere encountered, it wa~ felt that by means of an extension of Herrmann's vari~tional principle to this domain these dLfficulties could also be ev~ded. Th~ ultimate aim being the analysis of th~

geometrically non-linear bending of plates, it was decided to ex-plore the extension of Herrmann'. principle for the mathematicallY simpler case of a slender beam, loaded in a plane and deforming in

th~ same plane.

In the p~eceding chapter the pertin~nt functional together with its subsidiary conditiOns has been derived. Thus, in the present chapter, attention will be given to some procedures for obtaining finite-element mod~ls with the ext~nd~d Herrmann formulation.

Notwithstanding the fac:t that the considerations of this chapter are still related to the b~am problero, they hav~ a much wider meaning and can be generali~ed to the probl€m of the benoiog of plat~s.

Section 3.2 gives a short des~ription of the finite-~l~ment method, convergenc:e crit~ria and' the requirements to be mad~ on the approxi-mating functions.

Section 3.3 gives a summary of existinS procedures for formula-ting finite-element models as used in the analysis of the

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geometri-cally nDn-l inc<l1; bending of beams and plates by means of the

prin-ciple of st~tionary potential energy.

In Section 3.4 it will be exemplified for the beam tlH't

8n<>10-gallS procedures can be used wh~n the anaiYBis is bBS~d on Herrmann's

ptinciple. in Sec.tion 3.5 a finite-element model for beam problems

h~sed on one of these p~ocedures will be pre.ented, while Seccion l.h give5 some numerical relultY. AnOther procedure will be applied in connecLion with plate hending in thl next chapters.

3T2 The: finite-element method and COI"lVet"gence criteria

lhe finite-element method which is based on stationary conditions

of a fUl1~tional can h~ reg8rded 8S 3 particular form of the Ritz

mrtlwd 3.1 for

obt~lnin8

approximatl solutions of the Euler equations o( thut functional. The characteristic featura is that the approxi-mating functions are piecewise defined on the domain.

Thre~ st<l!?;e. (:'m b" distinguished in tile method:

(a) Sele..-:tl0f\ of ,".e finite numbet" of points in t.h-e domain of the

[unction5, to wl,i~!, riiscret~ valu~s of these functions (and

possibly o[ some of their derivatloIls) will be a5si~n"d. Thes" points are call"d nodat points, or simply nod"".

(b) Subdivision of tile dorn.~in into subdomains: the. finite. elements. hie reSIl>:O the actu<ll domain a .• an assemblage of finite elements, connected Lo~"ther appropriatelY at nodes an their boundari~Y.

(c) The functiOn iE apl'rO}(imated l.oCo.Uy within each finite element by continuOUS funclions which are uniquely defined in tarms of the values of lhe functions (and possibly of sOme deriv8livcs)

at ttle nodes of tIle elementl

An important Osp.cl of the finite-elam.nt method is that an alQ-meet can b0 regarded as disjoint from the structur~ to be analyzed when formulating its contribution to the whole structur~, sinca the

local approximating Eunction~ are expres~ed solely in values at the

nodes of that elem",nt. This implie.s tl"~t for each element a s\dtahle .ystem of independent <coordinates can be chosen: tile loc,_] coordinate

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system, For the whole structure, a common coordinate system must oe chQsen; the global coordinate system. bn advantage of the local coor-dinate system is that it facilitates expressing the local approxima-ting function. When assembling the elements, a coordinate transforma-tion may be necessary. One example is a three-dimensional framework composed of beams whose local hehaviour ~an be described by one-dimen-sional functions.

The choice of the shape of the element, the number of its nodes and its location with respect to these nodes is determined by the desired applicability of the element, the selection of the approxima-ting {unctions within the element, and the continuity requirements of the approximating functions over the whole domain,

For instance, when approximating displacement f~nction~ in case of plane stresS, ~omplicated structural shepes can better be des-cribed by means of triangular elements than by rectangular elements. when deal ins with a cho~en linear displacement distribution within each element it is only possible to obtain continuity of the

dis-plac~ent~ acro~~ element boundaries, if each element contains three nodes, each coinciding with an angular point of the triangle.

The choice of the approximating fUnction~ within each element is

restri~ted by the requirements of the variational principle and

sho~ld meet the convergence criteria.

If the variational principle concerned ,s a minimum principle and the functions meet the requirement~ of that principle, then the con-vergence criteria are those of the Rit. method. For instance, in case of two- or three-dimensional elasticity, the principle of minimum po-tential energy requires the displacement~ to be continuous over the whole domain, while within each element the strains should have bounded and continuo~s first derivatives. According to the convergence criteria of the Rit~ method, however, this i5 not a suffi~ient condition. To

ob-ta~n convergence, it i. necessary that the displacement should fulfil the ~o-cal1ed completeness criterion, which in this case means that the

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scrain~ should be ahle to assume arbitrary constant values within each element.

U the re.levant variation~l principle is not a minimum principle,

as is the case with H~rrm~ntl's v~r1ational principle, toen the

cOn-vergence ~riteria cannot b. derived in the same way ~s tho~e of the

Ritz method. Howevert ticcording to OLIVEIRA 3

1

2, even in tbi:!;l ca6e

and in cas" o[ a minimum principle with incomplete continuity of the fields between the element.~. it proved to be possibl,e to formu-late convergence cd.ted ~, Th1,l5, complete cotltinui ty. al though a J:e-quirement of the principle, is not neceRsary for convergence.

According to Oliveira, for the mixed Herrmann formlliat'lon i f she,H deformation is n"glected and if we remain in the domain of tll", linear tlleory of elasticity, convergence wi).1 be obtain"d if: - the displacements at" continuous ac);oss the dement boundaries and

tIle bending moments re.pect the equilibrium "quartans on such boun-daries; thie is a requiramQnt of the variational principle; - the compl"reness condition is respe~ted for the displacements as

well a" the bending moments, which ~n this caSe means that both the strains and th" moments can assume arbitrary constant values

within each element.

since a convergenCe theory for th", approximate solutio\l of

geo-metric~lly non-linear problems is lacking it is felt that the best we tan do to cr~ate condition. for conv~rgance, is to reopect th~

afor"mentionad critc~ia of the linear theory.

3.3 Some alternative finite-elemant procedures as u~ed ~n th~

statio-nary potential en~r~approach

For the finite-element anal1sis of the eeometrically non-linear bel1ding of beam" and plate5 by me~n~ of che principle of stationary potential energy, some alternative procedures have emerg~d (see ""g.

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The b~sic differencee are charaterized by:

- the local coordinate sy~tem that can be chosen for each individual

elemenL.

- the degree of non-linearity inco~po~atect in the ctescription of th~

individual element ~e18tive to its chosen local coordi~ate ~y~tem.

To typify the degree of ~on-linearity, the followins qualitative ranges of displacement, as used in plate theories but also applicable to beams, will b~ distinguished:

- finite d~fl~ctions, i,e. th~ displacements in th~ plan~ of the plate and their first derivatives a~e of the same order of magnitude as in linear theories. The displacements perpendicular to the plane of the plate (th~ defl~ction) and their first derivate5 roay be larger. An example of a theory cOn$truct~d On thes~ assumptions is offered by the von Karman plate bencting theo~y 3,7;

- large displacements, i.e. neither the in-plane ctisplacements no~

the cteflections remain in the ctomain or linea~ theories.

The loc~l coordinate system can either be fixed in the initial,

unloa~ect position of the element or mov~ with th~ ~lem~nt.

Considey a beam element bounded by nodes I ~nd 2 lying in ~ p1ane and deforming in the same plan~ (Fig, 3,1), To d~scribe the geometry of the ~hol~ struetur~, W~ establish a fi~~d Cart~sian global coordi-nate 5yst~m

x,z.

In thie system the coordinateS of a material point of the undeformed structure are X,Z and it. displacements u and w. X

and Z are ~al1ed material coordinate~, In addition~ we introduce for

each element a f~xed Cartesian local ~oordinate ~ystern. x' .z~, In

this system the material coordinate is x*~ whereas ZM=O. Since the

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Fig. ".1

*

:;:; z

E'1E'mE'nt ---~ x Globa? aoordl:natee InUiat taoal- aoardinates

coordinate syG(",m is det"rminell by the ~nlO,jlded, j,nitial poddon of the Qlam~~t. we call it th~ i~itial local coordinate system. The

tran"formatio~ betwae~ global coo~dinates ,jlnd initial local coordi-nates is <:ornpletely d<"!t"rrnin~d by the initi,jl~ geQmetry of the struc-tu<e and independent of the global displacements of node~ of the

el,,-ment.

Such fixed local frames of refe~ence have ~p to now only bee~

used within the range of finite deflections fOJ" the whole structure.

This implies that the finite oeflection belH!viour must he incorpo-rat"d in th" model of the indivio\)al element. U~ing the von Karman plat;: be~ding theory, this approach ",as applied hy BREll11IA and CONNOR 3.3 and by lll':RCAN 8..,Q GWUGH 3.4

In this ca$e the coordinate sy5te~ ~OVe~ ~ith the deforming ele-ment "'hen the str\)ct.l.lre deforms, bllt retai~s its initial metric (Fig.

Analysis of geometrically non-linear bending of beams and plates with mixed-type finite elements