Elements of plate: modelings DKT, DST, DKTG []2003/03/07 · 4.3.3.1For the Q4 elementsG.....30 Warning : The translation process used on this website is a "Machine Translation"

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Titre : Éléments de plaque : modélisations DKT, DST, DKTG [...] Date : 28/06/2018 Page : 1/61Responsable : KUDAWOO Ayaovi-Dzifa Clé : R3.07.03 Révision :

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Elements of plate: modelings DKT, DST, DKTG and Q4GSummary:

These modelings of finite elements of plate are intended for calculations in small deformations and smalldisplacements of curved or plane mean structures. They are finite elements plans which do not take intoaccount the geometrical curve of the mean structures, contrary with the elements of hull which are curved: itresults from it from the parasitic inflections which can be reduced by using more elements in order to be able toapproach the curved geometries correctly. The formulation is thus simplified by it and the reduced number ofdegrees of freedom. These finite elements are famous as being among most precise for the calculation ofdisplacements and the modal analysis.

For each one of these various modelings several finite elements are available, according to the meshs:• modeling DKT, according to the model of inflection of Coils-Kirchhoff, comprises the finite elements

triangular (DKT) and quadrangular (DKQ), which uses fields under-points, so for example integrating therelation of behavior in the layers constituting the thickness ;

• modeling DST, with transverse energy of shearing in elasticity, comprises the finite elements triangular(DST) and quadrangular (DSQ);

• modeling DKTG, according to the model of inflection of Coils-Kirchhoff, comprises the finite elementstriangular (DKTG) and quadrangular (DKQG), dedicated to the “total” relations of behavior, which haveonly one sleep and one point of integration in the thickness;

• modeling Q4G (named too Q4 ) with transverse energy of shearing in elasticity, but with anotherinterpolation that for modeling DST, comprises only the quadrangular finite element (Q4G).

Note: In the document [R3.07.09], one presents modeling Q4GG, dedicated to thick plates. This modelingcomprises quadrangular finite elements (Q4G) theoretical description is made in this document and thetriangular elements (T3G) .

Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2020 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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Contents1Introduction ............................................................................................................................................ 5

2Formulation ........................................................................................................................................... 6

2.1Geometry of the elements plates [1] ............................................................................................... 6

2.1.1Intrinsic reference mark ......................................................................................................... 6

2.2Theory of the plates ........................................................................................................................ 7

2.2.1Kinematics ............................................................................................................................. 7

2.2.2Law of behavior ..................................................................................................................... 8

2.2.3Taking into account of transverse shearing [2] ...................................................................... 9

2.2.3.1The theory known as of Hencky ................................................................................ 9

2.2.3.2The theory known as of Reissner (DST, DSQ and Q4 G ) ........................................... 9

2.2.3.3Equivalence of the approaches Hencky-Coils-Kirchhoff and Reissner .................... 10

2.2.3.4Remarks .................................................................................................................. 10

2.2.3.5Calculation of the shear stress and the transverse distortion in Code_hasster ....... 10

3Principle of virtual work ....................................................................................................................... 12

3.1Work of deformation ..................................................................................................................... 12

3.1.1Expression of the resulting efforts ....................................................................................... 12

3.1.2Relation efforts resulting-deformations ................................................................................ 12

3.1.3Energy interns elastic of plate ............................................................................................. 13

3.1.4Remarks .............................................................................................................................. 13

3.2Work of the forces and external couples ...................................................................................... 14

3.3Principle of virtual work ................................................................................................................. 15

3.3.1Kinematics of Hencky .......................................................................................................... 15

3.3.2Kinematics of Coils-Kirchhoff .............................................................................................. 16

3.3.3Principal boundary conditions met [1] .................................................................................. 17

4Digital discretization of the variational formulation resulting from the principle of virtual work ........... 19

4.1Introduction ................................................................................................................................... 19

4.2Discretization of the field of displacement .................................................................................... 20

4.2.1Q4 approach G ...................................................................................................................... 21

4.2.2Approach DKT, DKQ, DKTG, DKQG, DST, DSQ ................................................................ 22

4.3Discretization of the field of deformation ...................................................................................... 23

4.3.1Discretization of the membrane field of deformation: .......................................................... 24

4.3.2Discretization of the transverse distortion ............................................................................ 24

4.3.2.1For the finite elements Q4 G ..................................................................................... 24

4.3.2.2For the finite elements of type DKT, DST, DKTG .................................................... 26

4.3.3Discretization of the field of deformation of inflection: ........................................................ 30

4.3.3.1For the Q4 elements G ............................................................................................. 30




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4.3.3.2For the finite elements of type DKT, DKTG, DST: ................................................... 31

4.4Matrix of rigidity ............................................................................................................................ 33

4.4.1Elementary matrix of rigidity for the Q4 elements G ............................................................. 33

4.4.2Elementary matrix of rigidity for elements DKT, DKTG, DKQ ............................................. 34

4.4.3Elementary matrix of rigidity for elements DST, DSQ ......................................................... 34

4.4.4Assembly of the elementary matrices .................................................................................. 35

4.4.4.1Degrees of freedom ................................................................................................. 35

4.4.4.2Fictitious rotations ................................................................................................... 35

4.5Matrix of mass .............................................................................................................................. 37

4.5.1Matrix of elementary mass classical .................................................................................... 37

4.5.1.1Q4 element G ............................................................................................................ 37

4.5.1.2Elements of the type DKT, DST ............................................................................... 38

4.5.2Elementary matrix of improved mass .................................................................................. 38

4.5.2.1Elements of type DKT ............................................................................................. 40

4.5.2.2Finite elements of the DST type .............................................................................. 40

4.5.2.3Elements of the Q4 type G ........................................................................................ 42

4.5.2.4Notice ...................................................................................................................... 42

4.5.3Assembly of the elementary matrices of mass .................................................................... 42

4.5.4Matrix of “lumpée” mass ...................................................................................................... 42

4.5.5Modification of the terms of inertia ...................................................................................... 43

4.6Linear buckling .............................................................................................................................. 45

4.6.1Field of deformation ............................................................................................................. 45

4.6.2Geometrical matrix of rigidity [KG] ...................................................................................... 45

4.7Digital integration for elasticity ...................................................................................................... 46

4.8Digital integration for the matrix of mass ...................................................................................... 46

4.9Digital integration for plasticity and other nonlinear laws .............................................................. 47

4.10Discretization of external work .................................................................................................... 48

4.11Taking into account of the thermal loadings ................................................................................ 48

4.11.1Thermoelasticity of the plates ............................................................................................ 48

4.11.2Thermomechanical chaining .............................................................................................. 50

4.11.3CAS-test ............................................................................................................................. 52

5Establishment of the elements of plate in Code_Aster ........................................................................ 53

5.1Description .................................................................................................................................... 53

5.2Introduced use and developments ................................................................................................ 53

5.3Calculation in linear elasticity ....................................................................................................... 54

5.4Calculation in linear buckling ........................................................................................................ 55

5.5Another nonlinear behavior or plastic design ................................................................................ 55

6Conclusion ........................................................................................................................................... 56

Annexe 1 : Orthotropic plates ................................................................................................................ 57

Annexe 2 : Factors of transverse correction of shearing for orthotropic or laminated plates ................ 58




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7Bibliography ......................................................................................................................................... 60

8Description of the versions of the document ....................................................................................... 61




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1 Introduction

The elements of hulls and plates are particularly used to model mean structures where the relationship betweendimensions (characteristic thickness/length) is with more 1/10 . They thus intervene particularly in fields likethe civil engineer, the interns of heart REFERENCE MARK, the vibratory analysis, the analysis of buckling ofEuler, analysis of the mean elastic multi-layer composite material structures… One limits oneself to theframework of small displacements (even if it possible but is little recommended to use the reactualization‘PETIT_REAC’) and of the small deformations.

Contrary to the elements of hull, the elements of plate plans do not make it possible to take into account thegeometrical curve of the structure to be represented and induce parasitic inflections. It is thus necessary to usea large number of these elements in order to approach the geometry of the structure correctly, and this, moreespecially as it is curved. On the other hand, one gains in simplicity of formulation and the number of degreesof freedom is reduced. In addition, formulations “Discrete Shear“(DST, DSQ and Q4G) or “DiscreteKirchhoff“(DKT, DKTG and DKQ, DKQG) kinematics, with or without transverse distortion respectively, allow goodperformances in terms of displacements and modal analysis.

The meshs support of these finite elements are linear (triangles and quadrangles). The degrees of freedom asof these finite elements are the translations and rotations of the nodes tops. The characteristics that one affectsto them are: the thickness, the coefficient of shearing, offsetting,…

The way in which these elements are established in Code_hasster as certain aspects of the use are given to[§5] present documentation.

In particular these formulations are classified:

modeling finite elements Use

DKT DKT, DKQ model of Coils-Kirchhoff, dedicated to the relations of linear behaviorand nonlinear with fields under-points of integration in the thickness, cf§ 4.9.

DKTG DKTG, DKQG model of Coils-Kirchhoff, dedicated to the “total” relations of behavior,which have only one sleep and one point of integration in the thickness

DST DST, DSQ model with transverse energy of shearing in elasticity

Q4G Q4 G model with transverse energy of shearing in elasticity

For sections thin out of material composite multi-layer rubber bands orthotropic, one will defer to [R4.01.01]and [U4.42.03], where one describes how one produces in a stage of preprocessing the homogenized elasticcharacteristics, out of membrane, inflection, transverse shearing (with modeling DST or Q4G).




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2 Formulation2.1 Geometry of the elements plates [1]

For the elements of plate one defines a surface of reference, or surfaces average, planes (plan x , y forexample) and a thickness h x , y . This thickness must be small compared to other dimensions (extensions,radii of curvature) of the structure to model. [Figure 2.1-a] below our matter illustrates.

Figure 2.1-a

One attaches to average surface a local orthonormal reference mark Oxyz associated with the tangentplan of the structure different from the total reference mark OXYZ . The position of the points of the plate isgiven by the Cartesian coordinates x , y average surface and rise z compared to this surface.

2.1.1 Intrinsic reference mark

By taking the local reference mark Oxyz precedent with for origin the first top of the element and for axis Oxthe side uniting tops 1 and 2, one defines the reference mark known as intrinsic.




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2.2 Theory of the plates

These elements are based on the theory of the plates in small displacements and small deformations.

2.2.1 Kinematics

The cross-sections which are the sections perpendicular to average surface remain right; the material pointslocated on a normal at not deformed average surface remain on a line in the deformed configuration. It resultsfrom this approach that the fields of displacement vary linearly in the thickness of the plate . If oneindicates by u , v ,w displacements of a point q x , y , z according to x , y and z , there is thus thekinematics of Hencky-Mindlin:

u x x , y , z u y x , y , z u z x , y , z

=u x , y v x , y w x , y z

y x , y

− x x , y 0

=u x , y v x , y w x , y z

x x , y

y x , y 0

(1)

where u ,v ,w are displacements of average surface and x and y rotations of this surface compared to

the two axes x and y respectively. One prefers to introduce two rotations

x x , y = y x , y , y x , y =−x x , y .

The three-dimensional deformations in any point, with kinematics introduced previously, are thus given by:

xx=exxz xx yy=e yyz yy2 xy=xy=2exy2z xy2 xz= x

2 yz= y

(2)

where e xx , e yy and e xy are the membrane deformations of average surface, x and y deformations

associated with transverse shearings, and xx , yy , xy the deformations of inflection (or variations of curve)of average surface, which are written:




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e xx=∂u∂ x

e yy=∂ v∂ y

2e xy=∂ v∂ x

∂ u∂ y

xx=∂ x∂ x

yy=∂ y∂ y

2 xy=∂x∂ y

∂ y∂ x

x=x∂w∂ x

y= y∂w∂ y

(3)

Note:

In the theories of plate the introduction of x and y allows to symmetrize the formulations ofthe deformations and, we will see it thereafter, the equilibrium equations. In the theories of hullone uses rather x and y and associated couples M x and M y compared to x and y .

2.2.2 Law of behavior

The behavior of the plates is a behavior 3D in “plane constraints”. The transverse constraint zz is

worthless because regarded as negligible compared to the other components of the tensor of the constraints(assumption of the plane constraints). The most general law of behavior is written then as follows:

xx

yy

xy

xz

yz

=C e ,xxyyxyx y

=Ce zCC with e=exxe yy2exy

00

,= xx

yy

2 xy00

et =000x y

. (4)

where C e , is the matrix of local tangent rigidity combining forced plane and transverse distortion and represent the whole of the internal variables when the behavior is nonlinear.

For behaviors where the transverse distortions are uncoupled from the deformations of membrane andinflection, C e , puts itself in the form:

C=H 00 H

(5)

where H e , is a matrix 3×3 and H e , a matrix 2×2 . One will remain within the framework of

this assumption.

For an isotropic homogeneous linear behavior elastic, one has as follows:




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C=E

1−v2 1 v 0 0 0v 1 0 0 0

0 01−v

20 0

0 0 0k 1−v

20

0 0 0 0k 1−v

2

(6)

where k is factor of transverse correction of shearing whose significance is given to the following paragraph.

Note:

One does not describe the variation thickness nor that of the transverse deformation e zz that one canhowever calculate by using the preceding assumption of plane constraints. In addition no restriction ismade on the type of behavior that one can represent.

2.2.3 Taking into account of transverse shearing [2]

The taking into account of transverse shearing depends on factors of correction determined a priori by energyequivalences with models 3D, so that rigidity in transverse shearing of the model of plate is nearest possible tothat defined by the theory of three-dimensional elasticity. Two theories including the deformation due to theshearing action exist and are presented in [2].

2.2.3.1 The theory known as of Hencky

This theory as that of Coils-Kirchhoff which results from this immediately rests on the kinematics presented tothe §2.2.1. The relation of behavior is usual and the factor of correction of shearing is worth k=1 .

Note:

When one does not take into account the transverse distortions x and y in the theory of Hencky, themodel obtained is that of Coils-Kirchhoff (finite elements DKT (G) and DKQ (G)). Two rotations of averagesurface are then related to displacements of average surface by the following relation:

x=−∂w∂ x

β y=−∂w∂ y

2.2.3.2 The theory known as of Reissner (DST, DSQ and Q4G)

The second theory, known as of Reissner, is developed starting from the constraints. Variation of themembrane stresses ( xx , yy and xy ) is supposed to be linear in the thickness as in the case of thetheory of Hencky where that results from the linearity of the variation of the deformations of membrane with thethickness.

However, whereas one supposes, in the theory of Hencky which the distortion is constant in the thickness andthus stresses shear, which violates the boundary conditions xz= yz=0 on the faces higher and lower of theplate because of law of behavior stated than the §2.2.2. , one uses within the framework of the theory ofReissner the equilibrium equations to deduce the variation from it from shear stresses in the thickness of theplate, by in particular observing the equilibrium conditions on the faces higher and lower of plate. Energyinterns model obtained after resolution of the equations of balance in 3D, for inflection only, with the variation




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of the plane constraints according to z , reveals, for an elastic material, a relation between the resulting effortsand rotations and the arrow averages. It is in this relation that the factor of correction of shearing appears ofk=5/6 instead of 1 in the relation which binds the shearing action to the distortion for a homogeneous and

isotropic plate. The determination of the factors of correction of shearing for orthotropic plates or laminatedplates is described in appendix.

2.2.3.3 Equivalence of the approaches Hencky-Coils-Kirchhoff and Reissner

If one assimilates the slopes of average surface x , y with the averages of the slopes in the thickness of the

plate and the arrow w with the average arrow, the only difference between the theory of Hencky and that ofReissner are the coefficient of transverse correction of shearing in homogeneous elasticity isoptrope of 5/6instead of 1 . This difference is due to the fact that the starting assumptions are of different nature andespecially that the selected variables are not the same ones. Indeed, the arrow on average surface is not equalto the average of the arrows on the thickness of the plate. It is thus normal that relations of behavior whichutilize different variables are not identical.

The fact of having to solve on the level finite elements of the problems in displacements rather than of theproblems in constraints by interpolation of displacements leads us to use the equivalent approach indisplacements of the problem of Reissner formulated in constraints.

2.2.3.4 Remarks

Because of preceding equivalence one presents here only the model in displacement for all the elements. Inthe facts elements DKT, DKTG, DKQG and DKQ are based on the theory of Hencky-Coils-Kirchhoff andelements DST, DSQ and Q4G are based on the theory of Reissner.

The determination of the factors of correction rests within the framework of another theory, that of Mindlin, onequivalences of Eigen frequency associated with the mode of vibration by transverse shearing. One obtainsthen k=

2/12 , value very close to 5/6 for the DST elements, DSQ and Q4G in the isotropic case.

Within the framework of plasticity the problem of the choice of the coefficient of correction of transverseshearing arises because the equivalent approach in displacements of the problem of Reissner formulated inconstraints utilizes the non-linearity of the behavior. One cannot thus deduce some, as it is the case for elasticmaterials a value of the coefficient of correction of transverse shearing. Plasticity (and other nonlinearbehaviors) are thus not developed for these elements.

2.2.3.5 Calculation of the shear stress and the transverse distortion in Code_hasster

The calculation of the shear stress is carried out while considering: – equilibrium equations in constraint and generalized effort:

∂z xz=∂x xx∂ y xy

∂z yz=∂ y yy∂ x xy

; ∂xM xx∂yM xy=T x

∂ yM yy∂xM xy=T y

(7)

– conditions of free edge xz −h /2= yz h /2=0 on the higher and lower faces;

– relations connecting the plane constraints to the derivative of the moments:

∂ xσ xx=12 /h3⋅z⋅∂xMxx

∂ xσ yy=12/h3⋅z⋅∂ yMyy

∂ xσ xy=12 /h3⋅z⋅∂ yMxy

(8)

Maybe after analytical integration compared to the variable z of σxz ( z) ,σ yz ( z) and identification of thecoefficients of the primitive:

– the expression of the shear stress:




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xz=T x⋅3/2h∗1−4 z 2/ h2

yz=T y⋅3/2h⋅1−4 z 2/h2

– the expression of the shearing strain within the framework of the theory of Reissner:

x= xz⋅21 /Ek y= yz⋅21 /Ek with k the coefficient of correction in shearing

– the expression of the shearing strain within the framework of the theory of Kirchoff:

x= xz⋅21 /Ek−∂x w y= yz⋅21 /Ek−∂ yw

It is about an approximation which one also finds in [3] (eq 9).

One generalizes the expression of shear stresses by introducing a function d1iel ( z) such as:

xz=T x⋅d1iel z yz=T y⋅d1iel z

In the classical cases, one a: d1iel z =3/2h ⋅1−4 z 2/h2

.

In cases plus generals (in the presence of offsetting for example) d1iel ( z) must be modified to takeinto account the involved phenomenon. To approximate the shear stress correctly, one makes thechoice to apply a general quadratic form for d1iel z =a⋅z 2

b⋅zc such that the followingconditions are observed:

– ∫−h/2

h/2

d1iel ( z)=1 relation effort slice-constraints

– d1iel ( z=−h /2)=0 ; d1iel ( z=h /2)=0 condition of free edges

The first condition makes it possible automatically to respect the relations of balance in linear elasticitywhile the second makes it possible to find correct results on the nonconstrained edges. The extensionof these relations to plasticity is not commonplace. However, in plasticity, one chooses to keep a linearelastic description of shear stresses.

The expression of d1iel ( z) in the case of offsetting is clarified in Doc. of dedicatedreference [R3.07.06].




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3 Principle of virtual work

3.1 Work of deformation

The general expression of the work of deformation 3D for a plate is worth:

W def=∫S∫−h /2

h /2

xx xx yy yyxy xy x xz y yz dV (9)

where S is average surface and the position in the thickness of the plate varies between – h/2 and h/ 2 .

3.1.1 Expression of the resulting efforts

By adopting the kinematics of the §2.2.1, one identifies the work of the interior efforts:

W def=∫S

exx N xxe yyN yy2e xyN xy xxM xx yyM yy2 xyM xy xT x yT y dS where:

N=N xx

N yy

N xy= ∫

−h /2

h /2 xx

yy

xydz ; M=

M xx

M yy

M xy= ∫

−h /2

h /2 xx

yy

xy z dz ; T=T x

T y= ∫

−h /2

h /2

xz

yzdz (10)

N xx , N yy , N xy are the efforts resulting from membrane (in N /m );

M xx , M yy ,M xy are the efforts resulting from inflection or moments (in N );

T x ,T y are the efforts resulting from shearing or efforts cutting-edges (in N /m ).

3.1.2 Relation efforts resulting-deformations

The expression of the work of deformation is also written:

W def=∫s∫−h /2

h /2

[C e , ]dV=∫s∫−h/2

h /2

[eCez eCz Cez 2CC ]dV (11)

where C e , is the local matrix of behavior.

By using the expression obtained for W def in the preceding paragraph one finds the relation followingbetween the resulting efforts and the deformations:

N=HmeHmf

M=Hmf eH f

T=Hct

with Hm= ∫−h /2

h /2

H dz ,Hmf= ∫−h / 2

h / 2

H zdz ,H f= ∫−h /2

h /2

H z 2dz (12)

where:




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H ct=G11 00 G22

e=e xxe yy2e xy

,= xx

yy

2 xy ,= x

y

Matrices Hm ,H f et Hct are the matrices of rigidity out of membrane, inflection and transverse

shearing, respectively. The matrix Hmf is a matrix of rigidity of coupling between the membrane andthe inflection.

For an isotropic homogeneous elastic behavior of plate these matrices have as an expression:

Hm=Eh

1−v 2 1 v 0v 1 0

0 01−v

2 ,H f=

Eh3

12 1−v2

1 v 0v 1 0

0 01−v

2 ,Hct=

kEh21v 1 0

0 1 (13)

and Hmf =0 because there is material symmetry compared to the plan z=0 .

For an orthotropic material, the behavior is given in appendix.

3.1.3 Energy interns elastic of plate

Taking into account the preceding remarks, energy interns elastic plate is more usually expressed forthis kind of geometry in the following way:

int=12∫S

[ e Hm eHmf Hmf eH f H ct ]dS (14)

that one can break up in the following way:

membrane=12∫S

e Hm edS energy of membrane

flexion=12∫S

Hm dS energy of inflection

cisaillement=12∫S

Hct dS energy of shearing

couplage=12∫S

e Hmf Hmf e dS energy of coupling membrane-inflection

3.1.4 Remarks

Relations flexible Hm , H f , Hmf with H and H ct with H are valid whatever the law of

behavior: rubber band, with unelastic deformations (thermoelasticity, plasticity,….).

For a plate made up of N orthotropic layers in elasticity, matrices Hm , H f , Hmf and H ct arewritten:




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Hm=∑i=1

N

hiHi ,Hmf=∑i=1

N

hi iHi ,H f=∑i=1

N13 z i1

3− z i

3Hi ,H ct=∑

i=1

N

h iH i (15)

where: h i=z i1−z i , i=12

zi1z i and H i ,H i the matrices represent H and H for the

layer i .

The homogenisation for multi-layer hulls can lead to matrices of rigidity of membrane and inflectionnonproportional of the type:

Hm=C1111 C 1122 0

C1122 C 2222 00 0 C1212

,H f=D1111 D 1122 0

D1122 D 2222 00 0 D1212

,Hct=G11 00 G22 (16)

for which one cannot find equivalent values of the Young modulus and thickness allowing to find theclassical expressions of rigidity, cf. [4].

3.2 Work of the forces and external couples

The work of the forces and couples being exerted on the plate is expressed in the following way:

W ext=∫S∫

−h /2

h /2

Fv .U dV∫S

Fs .U dS∫C∫−h /2

h /2

Fc . U dz ds (17)

where Fv ,Fs ,Fc are the voluminal, surface efforts and of contour being exerted on the plate,

respectively. C is the part of the contour of the plate on which efforts of contour Fc are applied.With the kinematics of the §2.2.1, one determines as follows:

W ext=∫S

f x u f yv f zwc x xc y y dS∫C

x u y vzw x x y y ds

=∫S

f x u f yv f zwc y x−cx y dS∫C

x u y vz w y x−x β y ds

•where are present on the plate:

f x , f y , f z : surface forces acting according to x , y and z

f i= ∫−h /2

h /2

Fv . e i dzF s . e i where e x and e y are the basic vectors of the tangent plan and

ez their normal vector.

c x , c y : surface couples acting around the axes x and y .

c i= ∫−h/2

h/2

z ez∧Fv .e idz±h2

ez∧F s . e i where e x ,e y ,ez are the basic vectors

previously definite.•and where are present on the contour of the plate:

x , y ,z : linear forces acting according to x , y and z




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i= ∫−h /2

h /2

Fc . e idz where e x ,e y ,ez are the basic vectors previously definite.

x , y : linear couples around the axes x and y .

i= ∫−h /2

h /2

z ez∧Fc .e i dz where e x ,e y ,ez are the basic vectors previously definite.

Note:

Moments compared to z are worthless.

3.3 Principle of virtual work

He is written in the following way: W ext=W def for all displacements and rotations virtualkinematically acceptable.

3.3.1 Kinematics of Hencky

With this kinematics, it results from them after integration by parts of work of deformation theequilibrium equations static of the following plates:

•For the efforts:

N xx , xN xy , y f x=0,N yy , yN xy , x f y=0,T x , xT y , y f z=0 .

•For the couples: M xx , xM xy , y−T xc y=0,M yy , yM xy , x−T y−cx=0.

as well as the boundary conditions following on contour C of S :

N xxnxN xyn y=x

N yy nyN xy nx= y

T x nxT y ny= z ,

M xxnxM xy ny= y

M yyn yM xy nx=− x

where

u=uv=vw=w x= y y=−x

where nx and n y are the cosine directors of the normal with C directed towards the outside of the

plate.and u indicate the trace of u on C .

The physical interpretation of these efforts ( N , T and M ) starting from the preceding equations Ci- is given below:




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Figure 3.3.1-a: Efforts resulting for an element from plate

Note:

N xx , N yy represent the tractive efforts and N xy shearing plan. M xx and M yy the couples

of inflection represent and M xy the torque. T x and T y are the shearing forces transverse.

3.3.2 Kinematics of Coils-Kirchhoff

It is pointed out that within the framework of this kinematics, one has the following relation binding the

derivative of the arrow to rotations:

x=−∂w∂ x

y=−∂w∂ y

. After a double integration by parts of the work of

deformation, one obtains the following equilibrium equations static:

•For the efforts of membrane: N xx , xN xy , y f x=0,N yy , yN xy , x f y=0,

•For the transverse shearing and bending stresses:M xx , xx2M xy , xyM yy , yy f zc y , x−c x , y=0,M xx , xM xy , y−T xc y=0,M yy , yM xy , x−T y−c x=0 .

as well as the boundary conditions on contour C and at the angular points O contour C of S :

N xxnxN xy n y=x ,N yy nyN xynx= y ,

T nM ns , s=z−n , s ,

M nn= s ,M nsO−M ns O−=-[ nO−nO−].

where

u=uv=vw=w n=−w , n=s

with

T n=T x nxT yn y ,

M nn=M xxn x22Mxy nx n yM yy n y

2 ,

M ns=-M xx nx n yM xy nx2−n y

2M yy nxn y .

.




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Figure 3.3.2-a: Boundary condition with angular points for an element of plate

Note:

The kinematics of Coils-Kirchhoff implies that on the contour of the plate the transverse shearingforce is related to the torque. It is noted that the order of the equilibrium equations of inflection ishigher than with the kinematics of Hencky. Thus, to choose the kinematics of Coils - Kirchhoffamounts increasing the degree of the functions of interpolation because one needs a largerregularity for the terms of arrow compared to the terms of membrane because of presence ofderived seconds of the arrow in the expression of the work of the deformations. No element ofplate of Code_Aster uses this kinematics. One can thus have differences between the results gotwith the elements of Code_Aster and of the analytical results got by using the kinematics of Coils-Kirchhoff for structures with angular contours.

3.3.3 Principal boundary conditions met [1]

Figure 3.3.3-a: Boundary condition for an element of plate

The boundary conditions frequently met are gathered in the table which follows. They are given for thekinematics of Hencky in the reference mark defined by S and the normal external with the plate:

Embedding Simple support Free edge Symmetry by report

with an axis sAntisymetry

compared to an




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axis su=0,v=0,w=0,s=0,n=0 .

un=0,

w=0,n=0 .

un=0,s=0 .

us=0,

w=0,n=0 .

s=0, s=0.

s=0,n=0,

z=0,

s=0,n=0

s=0,z=0,

n=0.

n=0, s=0.

with:

un=unxvny ;u s=- unyvnx ,

n= xn x yn y ;s=−x ny y nx ,

n=x nx22 xy nxn y yn y

2 ,

s=−x nxn yxy nx2−n y

2 ynx n y ,

n= x nx22 xynx ny y ny

2 ,

s=− xn xn yxy nx2−n y

2 y nx ny .

Note: one has s=−n , n=s .

.




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4 Digital discretization of the variational formulation resultingfrom the principle of virtual work

4.1 Introduction

By exploiting the law of behavior, the virtual work of the interior efforts is written (with H mf =0 untilthe §4.4, which does not remove anything with the general information following results, but allows toreduce the notations):

W int=∫S

eHmeH f Hct dS

with: e=u , xv , y

u , yv , x ,=

x , x y , y

x , y y , x ,=w, x x

w, y y .It results from it that the elements of plate are elements with five degrees of freedom per node. Thesedegrees of freedom are displacements in the plan of the element u and v , except plan w and two

rotations β x and β y .

The elements DKT, DKTG and DST are triangular elements. Elements DKQ, DKQG, DSQ and Q4are quadrilateral elements. They are represented below:

Figure 4.1-a: Real elements




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The elements of reference are presented below:

Figure 4.1-b: Elements of reference triangle and quadrangle

One defines the reduced reference mark of the element as the reference mark , element of

reference. The local reference mark of the element, in its plan x , y is defined by the user.Direction X1 this local reference mark is the projection of a direction of reference d as regards theelement. This direction of reference d is chosen by the user who defines it by two nautical angles inthe total reference mark. The normal N with the plan of the element ( 12∧13 for a trianglenumbered 123 and 12∧14 for a numbered quadrangle 1234) fix the second direction. The vectorproduct of the two vectors previously definite Y1=N∧X1 allows to define the local trihedron inwhich will be expressed the generalized efforts representing the state of stresses. The user will have totake care that the selected reference axis is not found parallel with the normal of certain elements ofplate. By default, direction of reference d is the axis X total reference mark of definition of the grid.

The essential difference between elements DKT, DKQ, DKTG, DKQG on the one hand and DST, DSQ,

Q4 in addition, comes owing to the fact that for the first the transverse distortion is worthless, that

is to say still =0 . The difference enters Q4 and elements DST and DSQ comes from a choicedifferent of interpolation for the representation of transverse shearing.

4.2 Discretization of the field of displacement

If one discretizes the fields of displacement in the usual way for isoparametric elements i.e.:

u=∑i=1

N

N i u i , v=∑i=1

N

N i v i ,w=∑i=1

N

N iwi , x=∑i=1

N

N i xi , y=∑i=1

N

N i yi ,

and that one introduces this discretization into the variational formulation of the §4.1 it a blocking intransverse shearing analyzed results from it in [1] who returns the solution in inflection controlled bythe effects of transverse shearing, and not by the inflection, when the thickness of the plate becomessmall compared to its characteristic dimension.

To cure this disadvantage the variational form presented in introduction is slightly modified so that:

Wint=∫S

eHmeH f Hct dS=∫S

eHm e H f THct−1 TdS (18)




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where are deformations of substitution checking = in a weak way (integral on the sides of the

element) and such as T=Hct . One checks thus that on the sides ij element ∫i

j

s−s ds=0

with s=w , ss .

Two approaches are then possible; in the first, that of the element Q4 , one uses the bilinear

discretization of the fields of displacement and the fact that is constant on the sides of the

element. Relations on the sides ij then allow to express the values of on the sides according tothe degrees of freedom of inflection. In the second approach, which is that of the elements of the typeDKT, DKTG and DST, one uses the weak formulation of the preceding paragraph which makes itpossible to bind the inflection to the shearing forces to deduce the interpolation from it from the termsof inflection.

4.2.1 Q4 approachG

It rests on the linear discretization of the fields of displacement presented above:

u=∑i=1

N

N iu i , v=∑i=1

N

N i v i ,w=∑i=1

N

N iwi , x=∑i=1

N

N i xi , y=∑i=1

N

N i yi , (19)

where functions N i are given below.

N i ( i=1,n )

i=1 with 4

N 1 , =141−1−

N 2 , =1411−

N 3 , =1411

N 4 , =141−1

Functions N i for the elements Q4

Note:

One notes too N i , =141i 1 i with 1 ,2 ,3 ,4 =−1,1 ,1 ,−1 and

1 , 2 ,3 ,4=−1,−1,1 ,1 .




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4.2.2 Approach DKT, DKQ, DKTG, DKQG, DST, DSQ

Like T x=M xx , xM xy , y etT y=M yy , yM xy , x and M=H f one from of deduced that is

defined according to the derivative second of x and y via two equilibrium equations internal and

law of behavior in inflection. Discretization retained for x and y , such as s is quadratic on the

sides and n linear, then utilizes of the functions of incomplete quadratic forms in the form:

x=∑k=1

N

N k xk ∑k=N1

2N

P xkk , y=∑k=1

N

N k yk ∑k=N1

2N

P ykk with P xk=PkC k et P yk=P k S k

where C k and Sk are the cosine and directing sines on the side ij which the node belongs k

defined by: C k=x ji /Lk= x j−x i /Lk ; S k= y ij /Lk= y j− y i /Lk ; Lk= x ji2 y ji

2 1/2

.

Note:

To introduce the preceding discretization amounts adding like degrees of freedom to the elementof rotations k in the middle of the sides k element. Indeed, rotations s and n such as:

sn=C SS −C x y

are quadratic for s and linear for n with:

s=1−s ' sis'sj4 s' 1−s ' k ; n=1−s ' nis

'nj where 0≤s'=s /Lk≤1 .

One observes thus that: sk= s s'=

12=

12 sisj k . It is the relation ∫

i

j

s−s ds=0

with s=w , ss who will allow to eliminate the additional degrees of freedom and to express themaccording to displacements and of nodal rotations.

Figure 4.2.2-a: Variations of βs and βn .




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N i ( i=1,n ) P i ( i=n1,2n )

i=1 with 3N 1 x , ==1−−

N 2 x , =

N 3 x , =

i=4 with 6P4 , =4

P5 , =4

P6 , =4

i=1 with 4

N 1 , =141−1−

N 2 , =1411−

N 3 , =1411

N 4 , =141−1

i=5 with 8

P5 , =121−

21−

P6 , =121− 2

1

P7 , =121−

21

P8 , =121−

21

Functions N i and P i for elements DKT, DST, DKTG, DKQG, DKQ, DSQ.

4.3 Discretization of the field of deformation

The matrix jacobienne J , is:

J= x , y , x , y , =∑i=1

N

N i , xi ∑i=1

N

N i , y i

∑i=1

N

N i , xi ∑i=1

N

N i , y i = J 11 J 12

J 21 J 22 (20)

Moreover:

∂

∂ x∂

∂ y= j

∂

∂

∂

∂ avec j= j11 j 12

j21 j 22=J−1

=1J J 22 −J 12

−J 21 J 11 où J=det J=J 11 J 22−J 12 J 21 (21)

It is pointed out that the field of displacement is discretized by:

uv =∑k=1

N

N k , uk

vk and wx y=∑k=1

N

N k , w k

xk

yk[ ∑k=N1

2N 0

Pxk ,

P yk , k ] (22)

the term between hooks being present for the elements of type DKT, DKTG, DST, but not for theelements Q4 .

Note:

In Code_hasster, the Jacobienne matrix makes it possible to pass from the element of referenceto the local reference mark of the element (and not the total reference mark) because it is easierto work in this reference mark.




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4.3.1 Discretization of the membrane field of deformation:

e xx=u , x=∑k=1

N

N k , x , uk=∑

k=1

N

j11N k , j12 N k , uk ,

e yy=v , y=∑k=1

N

N k , y , vk=∑

k=1

N

j21 N k , j 22 N k , vk ,

2e xy=u , xv , y=∑k=1

N

N k , y , ukN k , x , v

k

=∑k=1

N

j21 N k , j 22N k , uk j11N k , j12 N k , v

k

(23)

Maybe in matric form:

exxe yy2exy

=∑k=1

N

Bmk Uk where Uk=ukvk is the membrane field of displacement to the node k

and:

Bmk=j 11N k , j12N k , 0

0 j 21N k , j22 N k ,

j 21N k , j22 N k , j 11N k , j12 N k ,

The matrix of passage of the membrane deformations to the field of displacement U m=u1

v1

...uNvN

in the

plan of the element is written as follows: Bm[3×2N ]=Bm1 ...BmN .

4.3.2 Discretization of the transverse distortion

4.3.2.1 For the finite elements Q4G

The field linearly is discretized constant by side so that:

= j ref

ref=

=

1−

2

12

1

2

34

1−

2

23

1−

2

41 where ref is the transverse field of distortion in the element of

reference.




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By using the relations (cf then. 4.2):

∫−1

1

−w ,

d =0 ;

∫−1

1

−w ,

d =0

for =±1=±1

,

it is established that:

ij=

12w j−w i i j ;

kp=

12w p−w k

p k ;

for ij∈12,34 and (kp)∈(23,41) .

By deferring the two results above in the expression of loc , one from of deduced that:

ref=

=B

ref u

ref

B

ref u

ref =Bref uref where uref=

w1

β1

β1

⋮w N

β N

β N

and B ref=(B1 , ... ,BN ) with

B k=N k , k N k , 0

N k , 0 k N k , , N=4 , k∈[1,N ] , k , k are defined with 4.2.1.

It is now necessary to express the rotations given here in the element of reference according to rotations in the local reference mark.

Like k

k=J k xk yk

= J 11k J 12 k

J 21k J 22 k xk yk one from of deduced that

ref=Bloculoc where

u loc=w1

x1 y1⋮wN

xN yN

and B loc=(Bloc1 ,... ,Bloc N ) with Bloc k=N k , k N k , J 11k k N k , J 12 k

N k , k N k , J 21 k k N k , J 22 k . It

will be noticed that the Jacobienne matrix J k is expressed in each point of the element.

Finally: = x

y= j 11 j 12

j 21 j 22ref=Bc uloc with B c[2×3N ]=jB loc .




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4.3.2.2 For the finite elements of type DKT, DST, DKTG

With regard to the transverse distortions one deduces fromT x=M xx , xM xy , y et T y=M yy , yM xy , x with M=H f that T= H f , xx where:

, xxT= x , xx x , yy x , xy y , xx y , yy y , xy and

H f= H11 H 33 2H13 H13 H 23 H 12H 33

H13 H 23 H 12H 33 H 33 H 22 2H23 where them H ij are the terms i , j

of H f .

βx , xx=∑k=1

N

N k , xx ( ξ ,η )β xk+ ∑k=N+1

2N

P xk , xx ( ξ , η)αk=∑k=1

N

N k , xx(ξ , η) β xk+ ∑k=N +1

N

( j112 P xk , ξξ+2 j11 j12 P xk , ξη+ j12

2 P xk , ηη)αk ,

βx , yy=∑k=1

N

N k , yy( ξ , η) β xk+ ∑k=N +1

2N

P xk , yy( ξ , η)αk=∑k=1

N

N k , yy(ξ , η )β xk+ ∑k=N+1

N

( j 212 P xk , ξξ+2 j21 j22 P xk , ξη+ j22

2 P xk , ηη)αk ,

βx , xy=∑k=1

N

N k , xy(ξ , η )β xk+ ∑k=N+1

2N

P xk , xy (ξ , η)αk

=∑k=1

N

N k , xy(ξ ,η ) βxk+ ∑k=N+1

N

( j 11 j21 P xk , ξξ+[ j11 j22+ j12 j21 ] P xk , ξη+ j 12 j 22P xk , ηη )α k ,

β y , xx=∑k=1

N

N k , xx( ξ ,η ) β yk+ ∑k=N +1

2N

P yk , xx( ξ , η)αk=∑k=1

N

N k , xx (ξ , η) β yk+ ∑k=N+1

N

( j112 P yk , ξξ+2 j 11 j12 P yk , ξη+ j12

2 P yk , ηη )α k

β y , yy=∑k=1

N

N k , yy (ξ ,η ) β yk+ ∑k=N +1

2 N

P yk , yy (ξ , η )αk=∑k=1

N

N k , yy ( ξ ,η ) β yk+ ∑k=N +1

N

( j212 P yk , ξξ+2 j21 j22 P yk , ξη+ j22

2 P yk , ηη )α k ,

β y , xy=∑k=1

N

N k , xy( ξ ,η ) β yk+ ∑k=N +1

2N

P yk , xy( ξ , η)αk

=∑k=1

N

N k , xy(ξ ,η ) β yk+ ∑k=N +1

N

( j11 j 21 P yk , ξξ+[ j11 j 22+ j12 j21 ] P yk , ξη+ j12 j 22 P yk , ηη )αk

where Pxk , P yk and k are defined into 4.2.2.




1b7b1064f823

that is to say still in matric form:

T= H f x , xx x , yy

x , xy y , xx

y , yy

y , xy

=

H f∑k=1

N 0 j11

2 N k ,2 j11 j12N k , j 122 N k ,

0

0 j212 N k ,2 j21 j 22N k , j22

2 N k , 0

0 j 11 j 21N k ,[ j 11 j 22 j12 j21 ]N k , j12 j 22N k , 0

0 0 j112 N k ,2 j11 j 12N k , j12

2 N k ,

0 0 j 212 N k ,2 j 21 j 22N k , j22

2 N k ,

0 0 j11 j21N k ,[ j11 j22 j12 j 21]N k , j12 j 22N k ,

wk

xk yk

H f∑k=1

2N

k C k j11

2 Pk ,2 j 11 j12P k , j122 P k ,

C k j212 Pk ,2 j 21 j 22Pk , j 22

2 Pk ,

C k j11 j22Pk ,[ j11 j 22 j12 j21 ]P k , j12 j22P k ,

S k j112 P k ,2 j11 j 12Pk , j12

2 P k ,

S k j212 Pk ,2 j 21 j 22Pk , j 22

2 Pk ,

Sk j11 j22P k ,[ j11 j22 j12 j 21]P k , j12 j22 P k , =

H f∑k=1

N

P f kwk

xk

yk ∑

k=N 1

2N

H f T2C k P k ,

C k P k ,

C k P k ,

S k Pk ,S k P k ,

S k P k ,

k= H f∑k=1

N

P f k U f k H f T2 ∑k=N1

2 N

Tckk= H f P f U f H f T 2T

where T=Tc N1 ...Tc2N and T2= t2 0

0 t2 with t 2=

j112 j12

2 2j11 j12

j212 j22

2 2j21 j 22

j11 j21 j12 j22 j11 j22 j12 j21 .

We use the relation then ∫i

j

s−s ds=0 with s=w , ss for each side ij of the element which

makes it possible to obtain them k since she is still written:

w j−wiLk2C k xiS k yiCk xjS k yj

23Lk k=Lk sk where:

sk=C k S k =C k S k Hct−1 T=C k S k H ct

−1 [Bc U f Bc ]

where C k , S k and Lk are defined into 4.2.2.




1b7b1064f823

Note:

Terms B c and B c correspond to the integration of the term s on each side ij element. One

evaluates the integral by using two points of Gauss of X-coordinates ±1 / 3 and of weight 1/2 in the

element of reference [−1,1] . Thus the term B c and B c can they be written:

B cα=12

[ H f T 2 PG1T α PG1 H f T 2 PG2 T α PG2 ] and

B cβ=12

[ H f P fβ PG1 H f P fβ PG 2 ] .

The relation above is still written in matric form: Aα α=AwU fβ

with: Aα=23 LN1 0 0

0 ⋱ 00 0 L2N

−LN1C N1 LN1S N1

⋮ ⋮L2NC 2N L2NS 2N

Hct−1 B cα

and:

Aw=−12

−2 LN1C N1 LN1 S N1 2 LN1CN1 LN1 S N1 0 0 0

0 0 0 −2 Lk1C k1 Lk1 S k1 2 Lk1C k1 Lk1S k1

0 0 0 0 0 0 −2 L2N−1C 2N−1 L2N−1S 2N−1

2 L2NC 2N L2N S2N 0 0 ⋯ ⋯ 0 0

⋯ 0 0 0⋯ 0 0 0⋯ 2 L2N−1C 2N−1 L2N−1 S 2N−1

⋯ −2 L2NC 2N L2N S 2N

LN1C N1 LN1SN1

⋮ ⋮

L2NC 2N L2N S 2NHct

−1 Bc

Thus =A β U fβ avecA β=Aα−1 Aw , which implies T=[ BcβBcαA β ]U fβ .

Note:

For the DST elements, this expression is simplified a little since B cβ=0 because of linearity of the

functions of form N k k=1,2 ,3 .This expression is simpler for elements DKT, DKTG and DKQ since they are without transverse distortion,

i.e. = 0, which implies Aα= 1 0 00 ⋱ 00 0 1 and




1b7b1064f823

¿

Aw=−34 −2 /LN1 C N1 S N1 2/LN1 C N1 S N1 0 0 0 ⋯

0 0 0 −2 /Lk1 C k1 S k1 2 /Lk1 C k1 S k1 ⋯

0 0 0 0 0 0 −2/L2N−1 C 2N−1 S 2N−1 ⋯

2/ L2N C 2N S2N 0 0 ⋯ ⋯ 0 0 ⋯

⋯ 0 0 0⋯ 0 0 0⋯ 2/ L2N−1 C 2N−1 S 2N−1

⋯ −2/L2N C 2N S 2N

It is also noticed that for elements DKT, DKTG the expression of the shearing actions is calculated startingfrom balance and not starting from the behavior (on the basis of the behavior one would find a value zeroof the efforts cutting-edges what would not make it possible to carry out balance!). It results from itaccording to the §3.1.1 from nonworthless stresses shear transverse in the thickness from the plate thatone is in formulation DKT or DST.




1b7b1064f823

4.3.3 Discretization of the field of deformation of inflection:4.3.3.1 For the Q4 elementsG

The relation binding the deformations of inflection to the field of displacement of inflection is written:

k xx=β x , x= j11 β x , j 12 β x ,= j11∑k=1

N

N k , β xk j 12∑k=1

N

N k , β xk ,

yy=β y , y= j21 β y , j 22 β y , = j21∑k=1

N

N k , β yk j22∑k=1

N

N k , β yk ,

2 xy=β y , xβ x , y= j11 β y , j12 β y , j 21 β x , j 22 β x ,= j21∑k=1

N

N k , β xk j22∑k=1

N

N k , β xk

j11∑k=1

N

N k , β yk j12∑k=1

N

N k , β yk .

(24)

That is to say still in matric form:

xx yy

2 xy=∑k=1

N

B fk U fk where U fk=w k

β xkβ yk

represent the field of displacement of inflection to the

node k ,

with:

B fk=0 j 11N k , j 12N k , 0

0 0 j 21N k , j22 N k ,

0 j 21N k , j 22 N k , j 11N k , j12 N k , .

The matrix of passage of the field of displacement of inflection U f=w1

β x1β y1⋮

wN

β xNβ yN

with the deformations of

inflection is written then: B f [ 3×3n ]=B f1 ,⋯,B fN .




1b7b1064f823

4.3.3.2 For the finite elements of type DKT, DKTG, DST:

The relation binding the deformations of inflection to the field of displacement of inflection is written:

xx=β x ,x= j11 β x , j12 β x ,= j11∑k=1

N

Nk ,β xk ∑

k=N 1

2N

Pxk ,

k j12 ∑k=1

N

Nk ,β xk ∑

k=N1

2N

Pxk ,

k ,

yy= β y , y= j21 β y , j22 β y ,= j21∑k=1

N

N k , β yk ∑k=N1

2N

P yk , k j22 ∑k=1

N

N k , β yk ∑k=N1

2N

P yk , k ,

2 xy=β y , xβ x , y= j11 β y , j12 β y , j21 β x , j22 β x ,=

j21 ∑k=1

N

N k , β xk ∑k=N 1

2N

Pxk ,k j 22∑k=1

N

N k , β xk ∑k=N 1

2N

P xk , k j11∑k=1

N

N k , β yk ∑k=N1

2N

P yk , k

j12∑k=1

N

N k , β yk ∑k=N 1

2N

P yk ,k .

For elements DKT, DKTG, DKQ:

In matric form the preceding relation is also written by introducing the relation =A β U fβ :

xx yy

2 xy=

j11 B x j12 B

x

j21 B y j22 B

y

j 11B y j 12B

y j 21 B x j22 B

x U f=B f [ 3×3N ] U f where U f=

w1

β x1β y1⋮

wN

β xNβ yN

represent the field of displacement in inflection for the element with:

B x =6 PN1,C N1

4LN1

−6 P2 N ,C 2N

4 L2 N

, N 1, x−34 PN1,C N1

2P2N ,C 2N

2 ,

−34 PN1,C N1S N1P2 N ,C 2N S 2N , L ,

6 PNk , C N k

4 LNk

−6 PNk−1,C Nk −1

4 LNk−1

, N k ,−34 PN k , C N k

2 PNk −1, xC Nk −12 ,

−34 PNk ,C Nk S NkPNk−1,C N k−1S Nk−1 , L

k=2, . . , N




1b7b1064f823

B x =6 PN1,C N1

4 LN1

−6P2N , C 2N

4 L2N

,N 1,−34PN1,C N1

2P2N ,C 2N

2 ,

−34 PN1,C N1S N1P2 N ,C 2N S 2N ,⋯,

6 PNk , C N k

4 LNk

−6 PNk−1,C Nk −1

4 LNk−1

,N k ,−34 PNk , C N k

2PNk−1, hC Nk−1

2 ,

−34 PNk ,C Nk S NkPNk−1,C Nk −1S Nk−1 ,⋯

k=2, . . , N

B y =6PN1, S N1

4 LN1

−6 P2N , S 2N

4 L2N

,−34PN1,C N1S N1P2 N , C 2N S 2N ,

N 1,−34PN1, S N1

2P2 N , S 2N

2 ,⋯,

6 PNk , S Nk

4 LNk

−6 PN k−1,S Nk−1

4 LNk −1

,−34 PN k , C N k S NkPNk −1, C Nk−1SNk −1 ,

N k ,−34 PNk , S N k

2PNk −1, S Nk−1

2 ,⋯

k=2, .. , N

B y =6PN1, S N1

4 LN1

−6 P2N ,S 2 N

4L2N

,−34PN1,C N1S N1P2 N ,C 2N S 2N ,

N 1,−34 PN1, S N1

2P2 N , S 2N

2 ,⋯ ,

6 PNk , S Nk

4 LNk

−6 PNk −1,S Nk−1

4 LNk −1

,−34 PNk , C N k S N kPNk −1, C N k−1S Nk −1 ,

N k ,−34PNk , S N k

2PNk−1, S Nk−1

2 ,⋯

k=2, .. , N

For elements DST, DSQ:

The relation binding the deformations of inflection to the field of displacement in inflection is alsowritten in matric form:

xx yy

2 xy=∑k=1

N

B fβk U fβk ∑k=N1

2N

B f k U f k where U fβk=w k

β xkβ yk

and U f k=k represent the field

of displacement of inflection to the node K, so that:




1b7b1064f823

B f k=0 j 11N k , j12 N k , 0

0 0 j 21N k , j22 N k ,

0 j 21N k , j22 N k , j 11N k , j12 N k , and

B f k=j11 Pxk , j12P xk ,

j 21P yk , j22 P yk ,

j 11P yk , j 12P yk , j21 Pxk , j22 P xk , .

The matrix of passage of the field of displacement of inflection U f=U f , with U f =w1

x1 y1⋮

wN

xN yN

and =

1

⋮

N with the deformations of inflection is written then:

B f [ 3×4N ]=B fβ1 ,⋯,B f N ,B f N1 ,⋯,B f 2N =B f [3× 3N ] ,B f [ 3×N ] .

4.4 Matrix of rigidity

The principle of virtual work is written in the following way: W ext=W int that is to say still in

elasticity UT KU=FU in matric form where K is the matrix of rigidity coming from theassembly in the total reference mark of the whole of the elementary matrices of rigidity.

4.4.1 Elementary matrix of rigidity for the Q4 elementsG

W inte=∫

e

[ e HmeHmf H mf eH f H ct ]dS=

∫e

U mT Bm

T H m BmU mU mT Bm

T H mf B f U fU fT B f

T H mf BmUmU fT B f

T H f B f U f

U fT Bc

T H ct BcU f dS=

U mT ∫

e

BmT H m BmdS U mU f

T ∫e

B fT H f B f dS U fdU m

T ∫e

BmT Hmf B f dS U f

U fT ∫

e

B fT H mf BmdS Um

U fT ∫

e

BcT H ct Bc dS U f=U m

T K mU mU fT K f U fU m

T Kmf U fU fT K fmU mU f

T K cU f




1b7b1064f823

with K mf=K fmT .

This is still written: W inte=U m ,U f K U m

U f where

K [5N×5N ]= K m [2N×2N ] K mf [2N×3N ]

K mf [3N×2N ]T K f [3N×3N ]K c [3N×3N ]

is the matrix of rigidity of the element.

4.4.2 Elementary matrix of rigidity for elements DKT, DKTG, DKQ

Since the relation =0 is satisfied, one can write:

δW inte =∫

e

δ e (H m e+Hmf k )+δκ(Hmf e+H f k )dS=

∫e

(δU mTBmTH m BmU m+δU m

TBmTH mf B f U f +δU f

TB fTHmf BmU m+δU f

TB fTH f B f U f )dS=

δU mT (∫

e

BmT H m Bm dS )U m+δU f

T (∫e

B fT H f B f dS )U f +δU m

T (∫e

BmT H mf B f dS )U f

+δU fT (∫

e

B fT H mf Bm dS )U m=δUm

T K mUm+δU fT K f U f +δU m

T Kmf U f +δU fT K fmU m

(25)

with K mf=K fmT .

This is still written: W inte=U m ,U f K U m

U f

where K [5N×5 N ]=( Km [2 N×2N ] K mf [2N×3N ]

K mf [3N×2 N ]

T K f [3N×3 N ]) is the matrix of rigidity of the element.

4.4.3 Elementary matrix of rigidity for elements DST, DSQ

W inte =∫

e

e H meH mf H mf eH f THct−1T dS=

∫e

U mT Bm

T H m BmU mU mT Bm

T H mf B f U fU fT B f

T H mf BmU mU fT B f

T H f B f U f

U fβT BcβT H ct

−1BcβU fβU fβ T BcβT H ct

−1Bc T Bc T H ct

−1BcβU fβT B cαT H ct

−1Bcαα dS=

U mT ∫

e

BmT H m BmdS U mU f

T ∫e

B fT H f B f dS U fU m

T ∫e

BmT H mf B f dS U fU f

T ∫e

B fT H mf Bm dS U m

U fβT ∫

e

B fβT H ct

−1Bcβ dS U fβU fβT ∫

e

B fβT H ct

−1Bcα dS αT ∫e

BcαT H ct

−1B cβdS U fβT ∫e

Bc T H ct

−1Bc dS =

U mT KmU mdU f

T K f U fU mT Kmf U fU f

T K fmU mU fβT K ββU fβU fβ

T K T K U fβ

T K c




1b7b1064f823

It is also known that U f=U f , from where it results that:

K f= K f 11 K f 12

K f 12T K22

with:

K f 11=∫s

B fβT H f B fβ dS ;

K f 12=∫s

B fβT H f B fαdS ;

K f 22=∫s

B fαT H f B fαdS .

K mf=K mf 11 K mf 12 with:

K mf 11=∫s

BmT H mf B fβ dS ;

K mf 12=∫s

BmT H mf B fαdS .

K fm=K mfT .

Using the fact that α=AβU fβ one from of deduced that:

W int=U mT K mU mU fβ

T K f' U fβU m

T K mf' U fβU fβ

T K fm' U m where:

K f'=K f 11K ββAβ

TK f 22K cα AβK f 12K βα AβAβ

TK f 12

TK βα

T

K mf'=Kmf 11K mf 12 Aβ

.

This is still written: W inte=U m ,U fβ K U m

U fβ where K [5N×5N ]= K m [2N×2N ] K ' mf [2N×3N ]

K 'mf [3N×2N ]T K f [3N×3N ]

' is the elementary matrix of rigidity for an element of plate.

4.4.4 Assembly of the elementary matrices

The principle of virtual work for the whole of the elements is written:

W int= ∑e=1

nb elem

W inte=UT KU (26)

where U is the whole of the degrees of freedom of the discretized structure and K comes from theassembly of the elementary matrices.

4.4.4.1 Degrees of freedom

The process of assembly of the elementary matrices implies that all the degrees of freedom areexpressed in the total reference mark. In the total reference mark, the degrees of freedom are threedisplacements compared to the three axes of the total Cartesian reference mark and three rotationscompared to these three axes. One thus uses matrices of passage of the local reference mark to thetotal reference mark for each element. However it was seen previously that the degrees of freedom ofthe elements of plate are two displacements in the plan of the plate, displacement except plan and tworotations. These rotations not being exactly rotations compared to the axes of the plate since

β x x , y = y x , y , β y x , y =−x x , y it is necessary to take account of it with the level of

the assembly to reveal the good degrees of freedom xi , yi .

4.4.4.2 Fictitious rotations Case general:

Rotation compared to the normal with the plate is regarded as not being a degree of freedom. Toensure compatibility between the passage of the local reference mark the total reference mark, onethus adds a degree of additional freedom local of rotation to the plate which is that corresponding torotation compared to the normal with the plan of the element. This implies an expansion of the blocks




1b7b1064f823

of dimension 5,5 matrix of local rigidity into cubes blocks of dimension 6,6 by adding a line anda column corresponding to this rotation. These additional lines and these columns are a prioriworthless. One then carries out the passage of the matrix of local rigidity extended to the matrix oftotal rigidity.

In the preceding transformation, one was satisfied to add rotations compared to the normals with theplan of the elements without modifying the deformation energy. The contribution to the energy broughtby these additional degrees of freedom is indeed worthless and no rigidity is associated for them.

The matrix of total rigidity thus obtained presents the risk however to be noninvertible. To avoid this nuisance, it is allowed to allot a small rigidity to these additional degrees of freedom onthe level of the matrix of widened local rigidity. Practically, one chooses it between 10– 6 and 10– 3

time the diagonal minor term of the matrix of rigidity of local inflection. The user can choose thismultiplicative coefficient COEF_RIGI_DRZ itself in AFFE_CARA_ELEM ; by default it is worth 10– 5 .

Typical case of the DKT: It is possible to associate a physical direction with ddl DRZ often called “drilling rotation” in referenceto a tendency of the plate to put itself in torsion. In this case, the writing of kinematics associatedwith this ddl is:

θZ=12(∂v∂ x

−∂u∂ y

) .

With this kinematics, one associates a dual quantity τ who is equivalent to the torque of the plate toobtain θZ . The difficulty to integrate this new kinematics in the classical writing of the DKT it is: - To adopt a variational framework allowing to introduce a not fictitious but real rigidity related to therotation of the plate,- To introduce a discretization of θZ and τ .

One notes by:

- ∇ZU=

12(∂ v∂ x

−∂ u∂ y

) the operator differential who allows knowing the values of membrane

displacement to calculate the kinematics of “drilling rotation”. - γ is a reality strictly positive.

Kinematics is reinforced ( ∇ZU−θZ=0 ) by one method of Lagrangian increased.

δW inte= ∫

−h /2

+ h/2

∫e

δe( H m e+H mf k )+δκ(H mf e+H f k )dS + ∫−h /2

+h/2

∫e

((∇ZU−θZ)−

12γ

τ) τ*dΩ

There is a second condition which is τ=2 γ(∇ZU−θZ) sufficient small to guarantee a weak

reinforcement of kinematics around DRZ. With this new variational writing, the classical framework ofstudy of the DKT is enriched. In particular one can expect a kinematic answer different with themodels from classical DKT which does not integrate physical rigidity around the normal. It is seen wellthat one reveals all the same a penalization of the physical kinematic condition ( τ sufficient small). Itis currently a weakness of the method: its dependence according to γ .

One interpolates from now on θZ at the points of Gauss thanks to the nodal values of θZ elementby using the functions of linear form N . In the same way the differential operator is calculated

∇ZU grace with the value nodal of u , v element by using the linear functions of form N and

incomplete polynomials P . After discretization, the following matric form is obtained:




1b7b1064f823

(K [6 N×6N ] Lτ

LτT 1

γ−1 T ) (Uτ )=(F

int

0 )

Lastly, as one does not want to reveal a degree of freedom on the ddl additional τ one will proceed bya method of condensation staitque which gives finally:

(K [6 N×6N ]−Lτ

1

γ−1T Lτ

T)U=F int

One enriched the kinematic framework with an impact on the matrix by total rigidity by the system.

4.5 Matrix of mass

The terms of the matrix of mass are obtained after discretization of the following variationalformulation:

δW massac

= ∫−h /2

+ h/2

∫S

ρ uδudzdS=∫S

ρm( uδu+ v δ v+w δw )+ ρmf ( uδβx+v δβy+ β xδu+ β yδ v )

+ρ f ( β x δβx+ β y δβy )dS

with m= ∫−h /2

h /2

dz ,mf= ∫−h /2

h /2

zdz ,et f= ∫−h/2

h/2

z 2dz .

Note:

If the plate is homogeneous or symmetrical compared to z=0 then mf =0 . One considers inthe continuation of the talk that it is always the case.

4.5.1 Matrix of elementary mass classical

4.5.1.1 Q4 elementG

The discretization of displacement for this finite element is:

u=∑k=1

N

N k ukvkw k

β xkβ yk

}k=1,⋯, N (27)

The matrix of mass, in the base where the degrees of freedom are gathered according to thedirections of translation and rotation, has then as an expression:




1b7b1064f823

M=Mm 0 0 Mmf 00 Mm 0 0 Mmf

0 0 Mm 0 0

MmfT 0 0 M f 0

0 MmfT 0 0 M f

(28)

with:

Mm=∫S

mNT N dS , Mmf=∫S

mf NT N dS and M f=∫S

f NT N dS and N= N 1⋯N k (29)

4.5.1.2 Elements of the type DKT, DST

Like wβ xβ y=∑k=1

N

N k , w k

β xkβ yk

∑k=N1

2N

0

P xk ,

P yk , k where =A U f one from of deduced

that:

wβ xβ y=∑k=1

N N k , 0 0

N kxw , N kxx , N kxy ,

N kyw , N kyx , N kyy , wk

β xkβ yk

(30)

The membrane part of the elementary matrix of mass is the same one as for Q4G with k=3 insteadof k=4 in N . The inflection part is composed of the blocks kp ( k ième line and p ième column)following:

f N kxw N pxwN kywN pyw ρmN k N p/ ρ f N kxwN pxxN kyw N pyx N kxwN pxyN kywN pyy

N kxxN pxwN kyxN pyw N kxxN pxxN kyx N pyx N kxxN pxyN kyxN pyy

N kxyN pxwN kyyN pyw N kxyN pxxN kyy N pyx N kxyN pxyN kyyN pyy

4.5.2 Elementary matrix of improved mass

As the arrow of a flexbeam can be represented by a linear approximation with difficulty, one can enrichthe functions by form for the terms of inflection. This approach is used in Code_hasster for theelements of type DKT, DST and Q4G where the functions of form used in the calculation of the matrixof mass of inflection are of order three. The interpolation for w is written as follows:

w=∑k=1

N

N k−1 N1 , w kN k−1 N2 , w , kN k−1 N3 , w , k




1b7b1064f823

where the functions of form are given for the triangle and the quadrangle in the following table:

Interpolation

for w=1−−

i=1 with 9N 1 , =32

−2 32

N 2 , =2 /2

N 3 , =2 /2

N 4 , =32−23

2

N 5 , =2−1 −

N 6 , =2 /2

N 7 , =32−2 3

2

N 8 , =2/ 2

N 9 , =2−1 −

i=1 with 12

N 1 , =181− 1− 2−

2−

2−−

N 2 , =181−1−1−

2

N 3 , =181− 1− 1−

2

N 4 , =1811−2−

2−

2−

N 5 , =−1811−1−

2

N 6 , =1811− 1−

2

N 7 , =1811 2−

2−

2

N 8 , =−18111−

2

N 9 , =−18111−

2

N 10 , =181−1 2−

2−

2−

N 11 , =181−1 1−

2

N 12 , =181−1 1− 2

Functions of interpolation for the arrow of the elements of type DKT, DST, DKTG and Q4G, in dynamics and modal.




1b7b1064f823

4.5.2.1 Elements of type DKT

It is known that in the approximation of one Coils-Kirchhoff has β x=−w , x and β y=−w , y in anypoint of the element.

Because of discretization stated above one a:

w=∑k=1

N

N k−1 N1 ,w k J 11N k−1 N2 , J 21N k−1 N3 , w , xk J 12N k−1 N2 ,

J 22 N k−1 N3 , w , yk

since: w , k

w , k= J 11 J 12

J 21 J 22 w , xk

w , yk .

This is still written:

w=∑k=1

N

N k−1 N1' , w kN k−1 N2

' , β xkN k−1 N3' , β yk

where:

N k−1 N1'

, =N k−1 N1 ,

N k−1 N2'

, =- J 11N k−1 N2 , −J 21N k−1 N3 ,

N k−1 N3'

, =- J 12N k−1 N2 , −J 22N k−1 N3 ,

.

By not taking account of the effects of inertia, the matrix of mass has the following form thus:

M=[Mm 0 0

0 Mm 00 0 M f

] where M f=∫S

mN' N' dS . (31)

4.5.2.2 Finite elements of the DST type

It is known that for these elements one has x= x−w , x and y= y−w , y where the distortion is constant on the element.

Like:

w=∑k=1

N

N k−1 N1 , w k J 11 N k−1 N2 , J 21N k−1 N3 , w , xk J 12N k−1 N2 ,

J 22 N k−1 N3 , w , yk




1b7b1064f823

one can also write:

w=∑k=1

N

N k−1 N1' , w kN k−1 N2

' , β xkN k−1 N3' , β yk

J 11 xJ 12 y SN k−1 N2 , J 21x J 22 y SN k−1 N3 ,

where:

N k−1 N1'

, =N k−1 N1 ,

N k−1 N2'

, =−J 11N k−1 N2 , −J 21N k−1 N3 ,

N k−1 N3'

, =−J 12N k−1 N2 , −J 22N k−1 N3 ,

,

∑ N k−1 N1 , =∑k=1

N

N k−1 N1 ,

∑ N k−1 N2 , =∑k=1

N

N k−1 N2 ,

∑ N k−1 N3 , =∑k=1

N

N k−1 N3 ,

and x

y=Hct

−1[BcβBc A β ]

w1

β x1β y1⋮

wN

βxNβ yN

=Tw w1

β x1β y1⋮

wN

β xNβ yN

.

One obtains the interpolation then for w :

w=∑k=1

N

N k−1 N1' '

, w kN k−1 N2' '

, β xkN k−1 N3' '

, β yk

where:

N k−1 N1' '

, =N k−1 N1'

,

J 11T w 1,k−1 N1 J 12T w 2,k−1N1 ∑ N j−1 N2 ,

J 21T w 1,k−1 N1J 22T w 2, k−1 N1∑ N j−1 N3 ,

N k−1 N2' '

, =N k−1 N2'

,

J 11T w 1,k−1 N2 J 12T w 2, k−1 N2 ∑ N j−1 N2 ,

J 21T w 1,k−1 N2 J 22T w 2,k−1 N2 ∑ N j−1 N3 ,

N k−1 N3' '

, =N k−1 N3'

,

J 11T w 1,k−1 N3 J 12T w 2,k−1 N3∑ N j−1 N2 ,

J 21T w 1,k−1 N3J 22T w 2, k−1 N3 ∑ N j−1 N3 ,




1b7b1064f823

By not taking account of the effects of inertia, the matrix of mass has the following form thus:

M=Mm 0 0

0 Mm 00 0 M f

where M f=∫S

mN' ' N' ' dS . (32)

4.5.2.3 Elements of the Q4 typeG

One proceeds in the same way that for the elements of the DST type but with:

x y=Bc

w1

β x1β y1

MwN

β xNβ yN

where Bc is the matrix established with the § 4.3.2.1.

4.5.2.4 Notice

One neglects in the form of the elementary matrix of mass the terms of inertia of rotation

∫S

f x x y y dS because the latter are negligible [5] compared to the others. Indeed a

multiplicative factor of h2/12 the dregs with the other terms and they become negligible for a

thickness report over characteristic length lower than 1/20 .

4.5.3 Assembly of the elementary matrices of mass

The assembly of the matrices of mass follows same logic as that of the matrices of rigidity. Thedegrees of freedom are the same ones and one finds the treatment specific to normal rotations withthe plan of the plate. For modal calculations utilizing at the same time the calculation of the matrix ofrigidity and that of the matrix of mass, it is necessary to take a rigidity or a mass on the degree ofnormal rotation to the plan of the plate of 103 to 106 time smaller than the diagonal minor term of thematrix of rigidity or mass for the terms of inflection. That makes it possible to inhibit the modes beingable to appear on the additional degree of freedom of rotation around the normal with the plan of theplate. By default, one takes a rigidity or a mass on the degree of normal rotation to the plan of plate105 time smaller than the diagonal minor term of the matrix of rigidity or mass for the terms ofinflection

4.5.4 Matrix of “lumpée” mass

The use of a matrix of “lumpée” mass has two advantages: it is simpler to implement numerically andit allows a better convergence. However the results are less good than with the classical diagram(consistent matrix) for which the error is minimal [6] . The matrix of lumpée mass is recommended intheory only for transitory calculations using an explicit diagram of temporal integration, which is usedalmost exclusively in fast dynamics. Contrary to calculations with the implicit schemes where witheach increment (and each iteration Newton in non-linear case) one assembles and opposite a matrix,given like a linear combination between the matrix of rigidity, the matrix of mass and the matrix ofdamping, in explicit calculations only the matrix of mass is assembled and reversed. Therefore, byusing a matrix of diagonal mass profit in term of time CPU of resolution as well as the profit in term ofthe storage of the matrices are enormous compared to the use of a matrix masses consistent.




1b7b1064f823

Code_Aster not being a code specialized in fast dynamics, this advantage of the diagonal matrix is notexploited. The option of the matrix of lumpée mass is thus only one choice of modeling, allowingpossible comparisons other computer codes. One presents here two methods for diagonaliser a matrix of coherent mass. It will be also shown inwhat the choice between the two methods is conditioned by the choice on the coherent matrixbetween the classic (§4.5.1) and improved (§4.5.2).

The technique undoubtedly simplest to obtain a lumpée matrix is to retain the diagonal value for eachdegree of freedom as the sum of the elements of the line of the coherent matrix. Moreover, it ispointed out that the most important property of a matrix of lumpée mass is that it makes it possible torepresent a movement of the rigid body correctly. It is satisfied in the following way by the methodof “summation per line”, put in equations:

M C=∑

M

.

Unfortunately the summation of the lines does not guarantee only all the terms M Cα are positive. The

negative terms appear in particular by using the matrix of improved coherent mass (mentioned in§4.5.2). This reason and for most elements in Code_Aster, one chose another approach, which had in

Hinton (see [6]), where diagonal terms corresponding to the directions x , y , z , M Hαx , M H

αy

and M Hαz , are calculated like:

M Hαx=

∫V

ρdV

∑β

M xββ M x

αα M Hαy=

∫V

ρdV

∑β

M yββ M y

αα M Hαz=

∫V

ρdV

∑β

M zββ M z

αα (33)

where indices α correspond to the numbers of nodes. Although the method of Hinton is generallymore robust, it is unsuited to the elements plates and hull, since [eq. 33] does not make it possible to

include the terms of inertia, the terms M Hα⋅ defined in [eq. 33] having inevitably the units of mass and

never of inertia.

Consequently, for the elements treated here one modifies the calculation of the matrix of consistentmass being used with calculation of the matrix as lumpée mass. One adopts the classical methoddescribed in §4.5.1 for the matrix of coherent mass, then the approach of “summation per line” for thelumping. The option impacted of Code_hasster is MASS_MECA_EXPLI and only for the elements DKTand DKTG . For the others one does not have the matrix of lumpée mass.

4.5.5 Modification of the terms of inertia

The matrix of lumpée mass described in §4.5.4 is not very effective for a calculation in explicitdynamics, where the step of time of stability is strongly penalized by a bad conditioning of the matrixof mass. Terms corresponding to rotations, i.e. the terms of inertia are the principal culprits, sincemuch smaller than the terms of translation, i.e. displacements. For this reason, one proposed in [7] amethod to modify the problematic terms while avoiding degrading the quality of the solution. Althoughold and not completely rigorous approach suggested in [7] is largely referred by the literature of thefield and was not really prone to remarkable improvements.

One focuses oneself on the terms due to the inflection, θ x , θ y and w =uz , termscorresponding to the membrane being obtained in a way classical and also applied to the elements2D. The matrix of mass Μ defined §4.5 becomes:




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Μ αβ=mαβ

h3

120 0

0h3

120

0 0 h (34)

where h is the thickness of the plate and mαβ defined like:

mαβ=∫

S

ρN αN βdS (35)

It is noted that [eq. 34] and [eq. 35] are equivalent to [eq. 2.3-1] for the kinematics of plates. In [7] oneproposes to build the matrix lumpée starting from [éq. 2.5-2] by using the squaring of Lobatto, whosealternatives are the trapezoidal diagram and the diagram of Simpson, where the points of integrationcoincide with the nodes. The construction of the matrix of mass is done through one finite element ofbeam, linear and with two nodes, by using the trapezoidal diagram, leading to:

Μ 0pout

=12ρLA

IA

0 0 0

0 1 0 0

0 0IA

0

0 0 0 1 (36)

for which the vector of degrees of freedom is written like θ1 w1 θ2 w2 T

. A , L and Iare the surface of the section, the length and the moment of inertia of the element beam, respectively.The use of the matrix [éq. 36] seeming too restrictive compared to the stability condition, oneproposes in [7] rather:

Μ pout=

12ρLA

α 0 0 00 1 0 00 0 α 00 0 0 1

(37)

where the parameter α is introduced so that its adjustment can maximize the step of time of stability.

According to [7] its optimal value would be α=18L2

. By directly applying these results by analogy to

the plates, one replaces the matrix of [éq. 34] by:

Μ αβ=mαβ

hAe

80 0

0hAe

80

0 0 h (38)




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where Ae is the surface of the element considered. In the passage of the beam to the plate onesupposed a certain equivalence between the length of the element beam and the surface of theelement plates, so that Ae≈L2 . It is pointed out that the approach suggested in [7] is not rigorousfrom the geometrical point of view and which it focuses on the maximization of the step of stability. Inthe established version, one makes sure of the desired effect of the modification of [éq. 34] with [éq.37] while using:

Μ αβ=mαβ

max h3

12,hAe

8 0 0

0 max h3

12,hAe

8 0

0 0 h (39)

because [éq 37] is not interesting that for the grids coarse, a priori more current, while [éq. 34]becomes favorable for very fine grids.

4.6 Linear buckling

Linear buckling is presented in the form of a typical case of the geometrical nonlinear problem. It isbased on the assumption of a linear dependence of the fields of displacements, strains and stressescompared to the level of load.

4.6.1 Field of deformation

From the assumption of Kirchhoff, the components of the tensor of the deformations of LagrangeGreen are related to the components of displacement in the plan of the plate in the following way:

= x y xy

=u , xv , y

u , yv , x

12 x

2

12 y

2

x x z x , x

y , y x , y y , x

(40)

That one can express in the following form:

=eLeNLz with

< eL >=<u , x v , y u , yv , x> linear deformations of membrane

< eNL>=<12 x

2 12 y

2 x y > non-linear deformations of membrane

<>=< x, x y , y x, y y, x > linear deformations of inflection

4.6.2 Geometrical matrix of rigidity [KG]

From the second variation of the internal energy of deformation and non-linear deformations, oneobtains the matrix [KG ] following:




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∫S<w , x w , x >[ N xx N xy

N xy N yy] { w , x

w , y} dAe=< un

f > [K G]{ unf } (41)

That one can write in the following form:

< unf > [KG ]{ un

f }=< unf >∫S

[BNL][N ][BNL]T dS { un

f}

With

[N ]=[ N xx N xy

N xy N yy] normal efforts

< unf > degrees of freedom of inflection

[BNL] the matrix connecting the quadratic deformations to the degrees offreedom

Note:

Linear buckling is available only for the elements DKT and DKTG with meshs TRIA3 and QUAD4 .

4.7 Digital integration for elasticity

For the triangular elements DKT, DKTG and DST the matrices of rigidity are obtained exactly with threepoints of integration of Hammer:

Cordonnées of the points Weight l

1=1/6 ;1=1/6 1/6

2=2 /3 ; 2=1/6 1/6

3=1/6 ;3=2 /3 1/6

∫0

1

∫0

1−

y , d d = ∑i=1

n

i y i , i

Formulas of digital integration on a triangle (Hammer)

For the elements quadrangles DKQ, DKQG and DSQ an integration of Gauss 2x2 is used for thematrices of rigidity.


1=1/ 3 ;1=1/ 3 1

2=1/ 3 ;2=−1/ 3 1

3=−1/ 3 ;3=1/ 3 1

3=−1/ 3 ;3=−1/ 3 1

∫0

1

∫0

1−

y , d d = ∑i=1

n

i y i , i

Formulas of digital integration on a quadrangle (Gauss)

4.8 Digital integration for the matrix of massWarning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2020 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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For the triangular elements DKT, DKTG and DST the matrices of mass are obtained with four points ofintegration.

For the elements quadrangles DKQ, DKQG and DSQ the matrices of mass are obtained with anintegration of Gauss 3x3.

4.9 Digital integration for plasticity and other nonlinear laws

Integration on the surface of the element is supplemented by an integration in the thickness of thebehavior since:

Hm= ∫−h /2

h /2

H dz ,Hmf= ∫−h /2

h /2

H zdz ,H f= ∫−h /2

h /2

H z 2dz where H is the matrix of plastic behavior

local (or other nonlinear laws).

The initial thickness is divided into N layers of thicknesses identical and there are three points ofintegration per layer (except for the elements DKTG and DKQG who have only one sleep and a point ofintegration in the layer). The points of integration are located in higher skin of layer, in the middle ofthe layer and in lower skin of layer. For N layers, the number of points of integration is of 2N1 .

One advises to use from 3 to 5 layers in the thickness for a number of points of integration being worth7.9 and 11 respectively.

For rigidity, one calculates for each layer, in plane constraints, the contribution to the matrices ofrigidity of membrane, inflection and coupling membrane-inflection. These contributions are added andassembled to obtain the matrix of total tangent rigidity. For each layer, the state of the constraints is

calculated xx , yy , xy and the whole of the internal variables, in the middle of the layer and in

skins higher and lower of layer, starting from the local plastic behavior and of the local field of

deformation xx , yy , xy . The positioning of the points of integration enables us to have the rightest

estimates, because not extrapolated, in skins lower and higher of layer, where it is known that theconstraints are likely to be maximum.


1=−1 1/3

2=0 4/3

3=1 1/3

∫−1

1

y d = ∑i=1

n

i y i , i

Digital formula of integration for a layer in the thickness

Note:

One already mentioned with the §2.2.3 that the value of the coefficient of correction in transverseshearing for the elements DST, DSQ and Q4G was obtained by identification of elasticcomplementary energies after resolution of balance 3D. This method is not usable any more inelastoplasticity and the choice of the coefficient of correction in transverse shearing is posed then.Plasticity is thus not developed for these elements.




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4.10 Discretization of external work

The variational formulation of work external for the elements of plate is written:

W ext=∫S

f x u f y v f z wmx xm y y dS∫C

f x u f y v f zwmx xmy y ds

By taking account of a linear discretization of displacements, one can write for an element:

W exte =∑

k=1

N

∫S

f x N k , duk f y N k , v k f z N k , w k

m xN k , xkm y N k , yk dS∫

C

x N k , uk yN k , v kz N k , w k

x N k , xk y N k , yk ds

=∑k=1

N

∫S

f x N k , dS∫C

x N k , ds⋮ ∫S

f y N k , dS∫C

y N k , ds⋮

∫S

f z N k , dS∫C

z N k , ds⋮ ∫S

m x N k , dS∫C

x N k , ds⋮

∫S

m yN k , dS∫C

y N k , ds⋮U ke

=∑k=1

N

FkeUk

e=FeU e

(42)

The variational formulation of the work of the efforts external for the unit of the elements is writtenthen:

W ext= ∑e=1

nbelem

W exte=FU= UT FT where U is the whole of the degrees of freedom of the

discretized structure and F comes from the assembly of the vectors forces elementary.

As for the matrices of rigidity, the process of assembly of the vectors forces elementary implies that allthe degrees of freedom are expressed in the total reference mark. In the total reference mark, thedegrees of freedom are three displacements compared to the three axes of the total Cartesianreference mark and three rotations compared to these three axes. One thus uses matrices of passageof the local reference mark to the total reference mark for each element.

Note:

The external efforts can also be defined in the reference mark user. One then uses a matrix ofpassage of the reference mark user towards the local reference mark of the element to have theexpression of these efforts in the local reference mark of the element and to deduce the vectorfrom it elementary corresponding room forces. For the assembly one passes then from the localreference mark of the element to the total reference mark.

4.11 Taking into account of the thermal loadings4.11.1 Thermoelasticity of the plates

The temperature is represented by the model of thermics to three fields according to [R3.11.01]:

T x , x3 =Tm x . P1 x3 T

s x . P2 x3 Ti x . P3 x3 (43)




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with: T m x : the temperature on the average layer

T s x : the temperature on the higher skin

T i x : the temperature on the lower skin

P j x3 : three polynomials of Lagrange in the thickness: ]−h/ 2,h /2[ :

P1 x3 =1−2x3/ h 2 ; P2 x3 =

x3

h12x3/h ; P3 x3=−

x3

h1−2x3/h (44)

WITH to leave the representation of the temperature above, one obtains:

•the average temperature in the thickness:

T x =1h∫

−h/2

h/2T x , x3 dx3=

16 4Tm x T

s x Ti x ;

•the average variation in temperature in the thickness:

T x =12

h2∫−h /2

h /2T x , x3 x3dx3=T

s x −Ti x ;

Thus the temperature can be written in the following way:

T x , x3 =T x T x . x3/h T x , x3 such as:

∫−h /2

h /2T x , x3 =0 ;∫−h /2

h/2x3

T x , x3 =0 .

If the temperature is indeed closely connected in the thickness one has, T=0 .

Code_Aster draft three different thermoelastic situations, where thermoelastic characteristics E , ,

depend only on the average temperature T in the thickness:

• the case where the material is thermoelastic isotropic homogeneous in the thickness;• the case where the plate models an orthotropic grid (concrete reinforcing steels);• the case where the behavior of the plate is deduced from a thermoelastic homogenisation, cf.

[4] .

For the elements of plate in thermoelasticity, the heating effects are taken into account via generalizedefforts, membrane and inflection. Thus, in the case of a homogeneous plate, knowing the dilationcoefficient , the generalized thermal efforts are defined starting from the plane constraints in thethickness by:

N

ther=∫

−h /2

h /2C

e

therdx3=∫−h/2

h/2C

T−T réf

dx3

M

ther=∫

−h/2

h/2x3C

e

ther dx3=∫−h /2

h /2α x3C

T−T réf

dx3

V

ther=0

(45)

Maybe in the homogeneous isotropic thermoelastic case in the thickness:




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N

ther= .C

.h . T−T réf = .

Eh1−

. T−T réf ;

M

ther= .C

.h2

12T= .

Eh2

12 1− . T

;V

ther=0 .

(46)

The thermal constraints of origin withdrawn from the usual mechanical constraints are calculated inthree positions (sup., moy. and inf.) in the thickness:

ther= . E1−

T−T réf T .x3/h (47)

In the case deduced from the thermoelastic homogenisation, cf. [4] , the generalized thermal effortsare defined by the general relation, starting from the “correctors” of membrane c , those of

inflection x , and that of dilation udil , like averages on representative ground volume (cell z ):

N

ther=⟨⟨C

. . T−T réf

T x . z 3/h T x , x3 ⟩⟩Z⟨⟨e ij c C ijkl .ekl udil ⟩⟩Z ;

M

ther=⟨⟨ z3.C

. . T−T réf

T x . z 3/h T x , x3

⟩⟩Z⟨⟨e ij x

C ijkl .ekl udil ⟩⟩Z ;

V

ther=0

In this case when one limits oneself to the orthotropic situations without coupling inflection-membrane,

one neglects the role of T x , x3 on the corrector udil , and it is thus found that the thermal efforts

which appear to the second member have as an expression:

N

ther= .H

m . T−T réf hz ;M

ther= .H

f . T ;V

ther=0

(48)

One cannot however go back to the complete three-dimensional constraints: it would be necessary toknow the “correctors” within the basic cell having been used with the determination of the coefficientsas homogenized behavior.

In the thermoelastoplastic situations, or for the hulls (elements of the family COQUE_3D), it isnecessary to evaluate the three-dimensional constraints, of which thermal stresses, in each point ofintegration in the thickness.

Note:

To go back to the complete three-dimensional constraints is not immediate for the multi-layer hulls(laminated) because it is necessary to know layer by layer the state of stress; in elasticity, thisone results from the state of deformation and the behavior on the level of each layer.

4.11.2 Thermomechanical chaining

For the resolution of chained thermomechanical problems, one must use for the thermal calculation ofthe finite elements of thermal hull [R3.11.01] whose field of temperature is recovered like input datumof Code_hasster for mechanical calculation. It is necessary thus that there is compatibility between thethermal field given by the thermal hulls and that recovered by the mechanical plates. This last isdefined by the knowledge of the 3 fields TEMP_SUP, TEMP_MIL and TEMP_INF given in skins lower,medium and higher of hull.

The table below indicates compatibilities between the elements of plate and the elements of thermalhull:




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ModelingTHERMICS

Mesh Finiteelement

to use with

Mesh Finite element ModelingMECHANICS

HULL QUAD4 THCOQU4 QUAD4 MEDKQU4MEDKQG4MEDSQU4MEQ4QU4

DKTDKTGDSTQ4G

HULL TRIA3 THCOTR3 TRIA3 MEDKTR3MEDKTG3MEDSTR3

DKTDKTGDST

Note:

The nodes of the thermal elements of hulls and mechanical plates must correspond. The grids willbe identical.The elements of thermal hulls surface are treated like elements plans by projection of the initialgeometry on the level defined by the first 3 tops.

The thermomechanical chaining with definite multi-layer materials via the order DEFI_COMPOSITE[U4.23.03] is not available in Code_hasster for the moment.

The thermomechanical chaining is also possible if one knows by experimental measurements thevariation of the field of temperature in the thickness of the structure or certain parts of the structure. Inthis case one works with a map of temperature defined a priori; the field of temperature is not givenany more by the three values TEMP_INF, TEMP_MIL and TEMP_SUP thermal calculation obtained byEVOL_THER. It can be much richer and contain an arbitrary number of points of discretization in thethickness of the hull. The operator DEFI_NAPPE allows to create such profiles of temperatures startingfrom the abundant data by the user. These profiles are affected by the order CREA_CHAMP (cf theCAS-test hpla100e). It will be noted that it is not necessary for mechanical calculation that the numberof points of integration in the thickness is equal to the number of points of discretization of the field oftemperature in the thickness. The field of temperature is automatically interpolated at the points ofintegration in the thickness of the elements of plates or hulls by the order CREA_RESU operationPREP_VRC2.

For the elements DKTG on the other hand, which does not have under-points in the thickness, oneshould not use PREP_VRC2. Three values TEMP_INF, TEMP_MIL and TEMP_SUP are assigned tovariables of the same order name, recoverable directly in the programs.




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4.11.3 CAS-test

The cas-tests for the thermomechanical chaining between thermal elements of hulls and elements ofplate are the hpla100e (elements DKT) and hpla100f (elements DKQ). It is about a heavythermoelastic hollow roll in uniform rotation [V7.01.100] subjected to a phenomenon of thermal dilationwhere the fields of temperature are calculated with THER_LINEAIRE by a stationary calculation.

Thermal dilation is worth: T −T réf =0.5T sT i2.T sT ir−R/ h

with:

• T s=0.5 °C , T i=−0.5 °C , T réf=0.° C

• T s=0.1 °C , T i=0.1°C , T réf=0. ° C

One tests the constraints, the efforts and bending moments in L and M . The results of referenceare analytical. One obtains very good performances whatever the type of element considered.




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5 Establishment of the elements of plate in Code_Aster

5.1 Description

These elements (of names MEDKTR3, MEDSTR3, MEDKQU4, MEDSQU4, MEDKTG3, MEDKQG4 andMEQ4QU4) are pressed on meshs TRIA3 and QUAD4 plane. These elements are not exact with thenodes and it is necessary to net with several elements to get correct results.

5.2 Introduced use and developments

These elements are used in the following way:

•AFFE_MODELE (MODELING = ‘DKT’.) for the triangle and the quadrangle of the type DKT•AFFE_MODELE (MODELING = ‘DST’.) for the triangle and the quadrangle of the DST type•AFFE_MODELE (MODELING = ‘DKTG’.) for the triangle and the quadrangle of the type DKTG•AFFE_MODELE (MODELING = ‘Q4G’.) for the quadrangle of the type Q4G

One calls on the routine INI079 for the position of the points of Hammer and Gauss on the surface ofthe plate and the weights corresponding.

•AFFE_CARA_ELEM (COQUE=_F (EPAISSEUR=' EP'ANGL_REP = (‘ ‘‘ ‘) COEF_RIGI_DRZ = ‘CTOR’)

To make postprocessings (forced, generalized efforts,…) in a reference mark chosen by the user who isnot the local reference mark of the element, one gives a direction of reference D defined by two nauticalangles in the total reference mark. The projection of this direction of reference as regards the plate fixesa direction X1 of reference. The normal with the plan into fixed one second and the vector product ofthe two vectors previously definite make it possible to define the local trihedron in which the generalizedefforts and the constraints will be expressed. The user will have to take care that the selected referenceaxis is not found parallel with the normal of certain elements of plate of the model. By default thisdirection of reference is the axis X total reference mark of definition of the grid.

The value CTOR corresponds to the coefficient which the user can introduce for the treatment of theterms of rigidity and mass according to normal rotation with the plan of the plate. This coefficient must besufficiently small not to disturb the energy assessment of the element and not too small so that thematrices of rigidity and mass are invertible. A value of 10−5 by default is put.

•ELAS =_F (E =YOUNG NAKED = naked ALPHA = alpha . RHO = rho .)

for a homogeneous isotropic thermoelastic behavior in the thickness one uses the keyword ELAS inDEFI_MATERIAU where the coefficients are defined E Young modulus, Poisson's ratio, thermaldilation coefficient and RHO density;

•ELAS_ORTH (_FO) =_F (E_L =ygl. E_T =ygt. G_LT =glt. G_TZ =gtz. NU_LT =nult.ALPHA_L =alphal. ALPHA_T =alphat.)




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for an orthotropic thermoelastic behavior whose axes of orthotropism are L , T and z with isotropy ofaxis L (fibres in the direction L coated by a matrix, for example) the seven independent coefficientsshould be given ygl, longitudinal Young modulus, ygt, transverse Young modulus, glt, modulus ofrigidity in the plan LT , gtz, modulus of rigidity in the plan TZ nult, Poisson's ratio in the plan LTand dilation coefficients thermal alphal and alphat for longitudinal and transverse thermal dilation,respectively.

The orthotropic elastic behavior available is only associated with the keyword DEFI_COMPOSITEwho allows to define a multi-layer composite hull.

For only one orthotropic material, one will thus use DEFI_COMPOSITE with only one sleep. If one wishesto use ELAS_ORTH for transverse shearing, it is necessarily necessary to employ DST modeling. If oneuses modelings DKT, or DKTG, the transverse energy of shearing is not taken into account.

•ELAS_COQUE (_FO) =F (MEMB_L =C1111. MEMB_LT =C1122. MEMB_T =C2222. MEMB_G_LT =C1212. FLEX_L =D1111. FLEX_LT =D1122. FLEX_T =D2222. FLEX_G_LT =D1212. CISA_L =G11…. CISA_T =G22…. ALPHA =alpha. RHO =rho.)

This behavior was added in DEFI_MATERIAU to take into account matrices of rigidity nonproportional outof membrane and inflection, obtained by homogenisation of a multi-layer material. The coefficients of thematrices of rigidity are then introduced with the hand by the user into the reference mark user defined bythe keyword ANGL_REP. The thickness given in AFFE_CARA_ELEM is only used with the density definedby RHO. alpha is thermal dilation. If one wishes to use ELAS_COQUE for transverse shearing, it isnecessarily necessary to employ DST modeling. If modeling DKT is used, transverse shearing is nottaken into account.

•DEFI_COMPOSITE _F (LAYER = THICKNESS: ‘EP’ MATER = ‘material’ ORIENTATION = (theta))

This keyword (cf [R4.01.01] and [U4.42.03]) makes it possible to define a multi-layer composite hull onthe basis of the sub-base towards the roadbase starting from its characteristics sleep by layer, thickness,type of material constitutive and orientation of fibres compared to a reference axis. The type ofconstitutive material is produced by the operator DEFI_MATERIAU under the keyword ELAS_ORTH.theta is the angle of the first direction of orthotropism (longitudinal direction or direction of fibres) in thetangent plan with the element compared to the first direction of the reference mark of reference definedby ANGL_REP. By default theta is null, if not it must be provided in degrees and must be understoodenters – 90º and 90º .

•AFFE_CHAR_MECA (DDL_IMPO =_F (DX =. DY =. DZ =. DRX =. DRY MARTINI =. DRZ =. degree of freedom of plate in thetotal reference mark.FORCE_COQUE =_F (FX =. FY =. FZ =. MX =. MY =. MZ =. ) They is the surfaceefforts (membrane and inflection) on elements of plate. These efforts can be given in the totalreference mark or the reference mark user defined by ANGL_REP.

•FORCE_NODALE =_F (FX =. FY =. FZ =. MX =. MY =. MZ =. ) They is the efforts of hullin the total reference mark.

•

5.3 Calculation in linear elasticityThe matrix of rigidity and the matrix of mass (respectively options RIGI_MECA and MASS_MECA) areintegrated numerically. It is not checked if the mesh is plane or not. Calculation takes account owingto the fact that the terms corresponding to the degrees of freedom of plate are expressed in thereference mark room of the element. A matrix of passage makes it possible to pass from the localdegrees of freedom to the total degrees of freedom.




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Elementary calculations (CALC_CHAMP) currently available correspond to the options:

•EPSI_ELNO and SIGM_ELNO who provide the strains and the stresses to the nodes in thereference mark user of the element in lower skin, to semi thickness and in higher skin ofplate, the position being specified by the user. One stores these values in the following way:6 components of strain or stresses:

•EPXX EPYY EPZZ EPXY EPXZ EPYZ or SIXX SIYY SIZZ SIXY SIXZ SIYZ•DEGE_ELNO : who gives the deformations generalized by ‘element to the nodes starting

from displacements in the reference mark user: EXX, EYY, EXY, KXX, KYY, KXY,GAX, GAY.

•EFGE_ELNO : who gives the efforts generalize by element with the nodes startingfrom displacements: NXX, NYY, NXY, MXX, MYY, MXY, QX, QY.

•SIEF_ELGA : who gives the efforts generalize by element at the points of Gauss startingfrom displacements: NXX, NYY, NXY, MXX, MYY, MXY, QX, QY.

•EPOT_ELEM : who gives the elastic energy of deformation per element starting fromdisplacements.

•ECIN_ELEM : who gives the kinetic energy by element.

Finally one calculates also the option FORC_NODA of calculation of the nodal forces for the operatorCALC_CHAMP.

5.4 Calculation in linear buckling

The option RIGI_MECA_GE being activated in the catalogue of the element, it is possible to carry outa classical calculation of buckling of Euler after assembly of the matrices of elastic and geometricalrigidity.

5.5 Another nonlinear behavior or plastic design

The matrix of rigidity is there too integrated numerically. One calls on the option of calculationSTAT_NON_LINE in which one defines in the level of the nonlinear behavior the number of layers tobe used for digital integration.

For modelings DKT, all the laws of plane constraints available in Code_Aster can be used.

For modelings DST and Q4G, only linear elasticity is usable.

For modeling DKTG, the only laws of behavior used are total laws (since there is only one point ofintegration in the thickness), connecting the deformations generalized to the generalized constraints.These laws are, in version 9.4: GLRC_DM and GLRC_DAMAGE, like their coupling with elastoplastic lawsout of membrane (KIT_DDI).

Currently available elementary calculations correspond to the options:

•EPSI_ELNO who provides the deformations by element to the nodes in the reference mark userstarting from displacements, in lower skin, with semi thickness and in higher skin of plate.

•SIGM_ELNO who allows to obtain the stress field in the thickness by element with the nodes forall the under-points (all the layers and all the positions: in lower skin, in the medium and inhigher skin of layer).

•EFGE_ELNO who allows to obtain the efforts generalized by element with the nodes in thereference mark user.

•VARI_ELNO who calculates the field of internal variables and the constraints by element with thenodes for all the layers, in the local reference mark of the element.




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6 Conclusion

The finite elements of plate plans which we describe here are used in the mean structural analyses, insmall displacements and deformations, whose thickness report over characteristic length is lower than1/10 . As these elements are plans, they do not take into account the curve of the structures, and it is

necessary to refine the grids if this one would be important.

It is elements for which the strains and the stresses in the plan of the element vary linearly with thethickness of the plate. Moreover, the distortion associated with transverse shearing is constant in thethickness of the element. Two families of finite elements of plate exist: elements DKT, DKQ (or DKTG,DKQG) for which the transverse distortion is worthless and finite elements with energy of shearingtransverse DST, DSQ and Q4G (or Q4G) for which it remains constant and nonworthless in thethickness. One advises to use the second type of elements when the structure studied has a thicknessreport over characteristic length understood enters 1/20 and 1/10 and first in the remainder of thecases. When the transverse distortion is nonworthless, the elements of DST plate, DSQ and Q4G donot satisfy the equilibrium conditions 3D and the boundary conditions on nullity with stresses sheartransverse on the faces higher and lower of plate, compatible with a constant transverse distortion inthe thickness of the plate. It results from it thus that on the level from the elastic behavior a coefficientfrom 5/6 for a homogeneous plate corrects the usual relation between the constraints and thedistortion transverses in order to ensure the equality between energies of shearing of the model 3Dand the model of plate constant distortion. In this case, the arrow w as an interpretation averagetransverse displacement in the thickness of the plate has.

The nonlinear behaviors in plane constraints are available for the elements of plate DKT and DKQonly. Indeed the rigorous taking into account of a transverse shearing constant not no one on thethickness and the determination of the correction associated on rigidity with shearing compared to amodel satisfying the equilibrium conditions and the boundary conditions are not possible and thusreturn the use of the DST elements, DSQ and rigorously impossible Q4G in plasticity.

For the elements of family DKTG, only of the total relations of behavior (membrane relations moment-curves and efforts – elongations) are available.

Elements corresponding to the machine elements exist in thermics; the thermomechanical chainingsare thus available except, for the moment, laminated materials.




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Annexe 1 : Orthotropic platesFor an orthotropic material like that represented on the figure 6-1, made up for example of fibres of directionL coated with a matrix, whose axes of orthotropism are L , T and Z with isotropy of axis L , the

expression for the matrices H and H in the reference mark of orthotropism previously definite becomes:

HL=H LL H LT 0H LT H TT 0

0 0 GLT and HL=G LZ 0

0 GTZ (49)

with

H LL=E L

1− LT TL; H TT=

ET

1− LT TL

H LT=ET LT

1− LT TL=E LνTL1− LT TL

and

GLZ=E L

2 1 LZ

GTZ=ET

21TZ

.

The knowledge of the five independent coefficients E L , E T , GLT , GTZ and νLT is sufficient to determine

the coefficients of the matrices H and H since:

TL=ET LT

E L

and GLZ=GLT .

If one indicates by the angle enters the reference mark of orthotropism and the main axis of the referencemark defined by the user by means of ANGL_REP it is established that:

H=T1T HLT1 and H=T2

T HL T2 (50)

with: T1= C 2 S 2 CSS 2 C 2

−CS−2CS 2CS C 2

−S 2 and T2= C S−S C where C=cos , S=sin and =x , L as

indicated on the figure below.

In the case of forced initial of thermal origin, we have moreover:

th=−T1T HL

L T

T T0 (51)

where L and T are the dilation coefficients thermal in the directions L and T and T temperaturevariation.


Figure 6-1: Composite plate



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Annexe 2 : Factors of transverse correction of shearing fororthotropic or laminated plates

The matrix Hct is defined so that the surface density of transverse energy of shearing obtained in the case ofthe three-dimensional distribution of the constraints resulting from the resolution of balance is equal to that ofthe model of plate based on the assumptions of Reissner, for a behavior in pure bending. One must thus findHct such as:

12∫−h/2

h/2

H

−1=

12

THct−1 T=

12Hct with = xz

yz and T= ∫

−h/2

h/2

dz=Hct (52)

To obtain Hct one uses the distribution of according to z obtained starting from the resolution of theequilibrium equations 3D without external couples:

xz=−∫−h/2

z

xx , x xy , y d ; s yz=− ∫−h /2

z

xy , x yy , y d with xz= yz=0 for z=±h /2 .

If there is no coupling membrane inflection (symmetry compared to z=0 ), constraints in the plan of the

element σ xx , σ yy and σ xy in the case of have as an expression a behavior of pure inflection:

= zA z M with A z =H z H f−1 .

If H z and H f do not depend on x and y one can determine Hct . Indeed:

z =D1 z TD2 z where T=T x

T y=M xx , xM xy , y

M xy , xM yy , y and =

M xx , x−M xy , y

M xy , x−M yy , y

M yy , x

M xx , y

like:

D1=− ∫−h /2

z

2 A11A33 A13A32

A31A23 A22A33 d ,

D2 =- ∫−h /2

z

2 A11−A33 A13−A32 2 A12 2 A31

A31−A23 A33−A22 2 A32 2 A21 d .

It results from it that 12∫−h/2

h/2

H

−1=12 T

C 11 C 12

C 12T C 22

T with:

C11= ∫−h/2

h/2

D1T H

−1 D1dz ;

C12= ∫−h/2

h/2

D1T H

−1 D2dz ;

C22= ∫−h /2

h /2

D2T

H

−1D2dz




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As in addition 12∫−h/2

h/2

H

−1=

12

THct−1 T one proposes to take Hct=C11

−1 to satisfy the two equations as

well as possible whatever T and .

While comparing Hct thus calculated with Hct= ∫−h /2

h /2

Hdz one reveals the coefficients of transverse

correction of shearing following

k1=H ct11/ H ct

11 , k12=H ct12/ H ct

12 and k 2=H ct22/ H ct

22 (53)

For a homogeneous, isotropic or anisotropic plate, one finds as follows: Hct=khH with k=5/6 .

Note:

This method is valid only when the composite plate is symmetrical compared to z=0 .

• For a multi-layer material, one establishes that:

C11=∑i=1

N h i4∑p=1

i−1

h p p A pT−

12zi

2 A iT H

−1 ∑p=1

i−1

h p pA p−12z i

2 Ai

124

z i13

− z i3[Ai

T H

−1∑p=1

i−1

h p p A p−12zi

2 A i∑p=1

i−1

h p p A pT−

12zi

2 A iTH

−1 Ai ]

1

80 z i1

5−zi

5A i

T H

−1 Ai

(54)

where: h i=z i1−z i , i=12

zi1z i and Ai represent the matrix A11A33 A13A32

A31A23 A22A33 for the layer i

.

• Validity of the choice Hct=C11−1 can be examined a posteriori when one has an estimate of the

solution (fields of displacements and plane constraints, in particular). One can then estimate thedifference between the two estimates on energy. A approach of calculation in two stages for themulti-layer plates and hulls (with Hct diagonal and two coefficients k 1 and k 2 ) was developedbesides by Noor and Burton [8] and [8] .

• In the case of an isotropic or anisotropic homogeneous plate, the equality between two energies issatisfied in a strict sense since D2=0 . The choice makes above is then valid and noexamination a posteriori is necessary.




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7 Bibliography

[1] Batoz, J. - L. and Dhatt, G., (1992). Modeling of the structures by finite elements: beams and plates.Hermes.

[2] Bui, D. (1992). Shearing in the plates and the hulls: modeling and calculation, EDF R & D - Note HI71/7784, Internal report.

[3] Taylor, Mr.; Vasiliev, V. and Dillard, D. (1997). One the preblem of shear locking in finite element basedone shear deformable punt theory, International Newspaper of Solids and structures 34, pp. 859-875.

[4] Voldoire, F. (1997). Modeling by thermal and thermoelastic homogenisation of thin mechanical components,EDF R & D - CR MMN/97/091, Internal report.

[5] Ren, J. (1986). With new theory of laminated punt, Composite Science and Technology 26, pp. 225-239.

[6] Hughes, T., (1987). The finite element method. Prentice Hall.

[7] Hughes, J.; Cohen, Mr. and Haroun, Mr. (1978). Reduced and selective integration techniques in the finiteelement analysis of punts, Nuclear Engineering and Design 46, pp. 203-222.

[8] Noor, A. and Burton, W. (1989). Composite Assessment of shear deformation theories for multilayeredpunts, ASME - Applied Mechanics Review 42, pp. 1-13.

[9] Rock'n'roll, T. and Hinton, E. (1976). With finite element method for the free vibration of transverse puntsallowing for shear deformation, Computers and Structures 6, pp. 37-44.

[10] Hinton, E.; Rock'n'roll, T. and Zienkiewicz, O. (1976). With note one farmhouse lumping and relatedprocesses in the finite element method, Earthquake Engineering and Structural Dynamics 4, pp. 245-249.




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8 Description of the versions of the document

Indexdocument

Version Aster

Author (S) Organization (S)

Description of the modifications

With 5 P. MASSIN EDF DER MN Initial text.B 9.4 X. DESROCHES, D.MARKOVIC,

EDF R & D AMAAddition of DKTG, and the matrices of lumpéesmasses.

C 13.2 D. KUDAWOO, EDF R & D /AMA Addition of the explanations of the method ofcalculating of the shear stress in the thickness.

D 14.1 F.VOLDOIRE, EDF/DR&D/ERMES Introductory paragraph on the various finiteelements of this family. Some corrections.


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Elements of plate: modelings DKT, DST, DKTG []2003/03/07 · 4.3.3.1For the Q4 elementsG.....30 Warning : The translation process used on this website is a "Machine Translation"