Note of use of the elements plates, hulls, [] · Note of use of the elements plates, hulls, grids...

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Titre : Notice d’utilisation des éléments plaques, coques,[...] Date : 28/04/2020 Page : 1/50Responsable : ABBAS Mickaël Clé : U2.02.01 Révision :

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Note of use of the elements plates, hulls, grids and membranes

Summary:

This document is a note of use for modelings plates, hulls, grids and membranes.

The elements of hulls and plates play a part in the digital modeling of the mean structures on average surface,planes (modeling plates) or curve (modeling hulls). Modelings of grids intervene for the digital modeling of thereinforcements and the cables of prestressed in the reinforced concrete structures. The modeling of membranemakes it possible to model mean structures whose rigidity of inflection is negligible.

Mechanics :Models DKT are most complete because they are able for mean or slightly thick structuresits ( h/L<1/20 ) torepresent a mechanical large range of phenomenaS linear or not linear while remaining performingS. Forproblems of great transformations with following pressure and not of taking into account of contact, it is betterto prefer the models of COQUE_3D. So besides the aspect hull and of the phenomenonS non-linear (contact,following pressure), one is not interested with the part inflection (inflatable balloon for example), it is better touse the models of MEMBRANE in great transformations. If the structure is thick and that one needs to properlytreat transverse shearing and the pinching it is better to prefer models of DST in linear mechanics and to preferthe model 3D into non-linear. Lastly, for a geometry of revolution and with a loading remaining of revolution, itis to better prefer COQUE_AXIS into linear as into non-linear.

Thermics : There exists three models of hull in code_aster able to treat it thermal: HULL (surface) COQUE_PLAN andCOQUE_AXIS (1D).

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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Contents

1 Introduction ........................................................................................................................................... 4

2 Mechanics ............................................................................................................................................ 5

2.1 Capacities of modeling .................................................................................................................. 5

2.1.1 Recall of the formulations ..................................................................................................... 5

2.1.2 Formulation of the elements plates, voluminal hulls and hulls ............................................. 7

2.1.3 Comparison enters the elements .......................................................................................... 9

2.2 Orders to be used ......................................................................................................................... 11

2.2.1 Space discretization and assignment of a modeling: operator AFFE_MODELE ................ 11

2.2.2 Elementary characteristics: AFFE_CARA_ELEM ............................................................... 13

2.2.3 Materials: DEFI_MATERIAU ............................................................................................... 16

2.2.4 Limiting loadings and conditions: AFFE_CHAR_MECA and AFFE_CHAR_MECA_F ........ 17

2.3 Resolution .................................................................................................................................... 22

2.3.1 Linear calculations: MECA_STATIQUE and other linear operators .................................... 22

2.3.2 Nonlinear calculations: STAT_NON_LINE and DYNA_NON_LINE .................................... 22

2.4 Additional calculations and postprocessings ................................................................................ 26

2.4.1 Elementary calculations of matrices: operator CALC_MATR_ELEM ................................. 26

2.4.2 Calculations by elements: operators CALC_CHAMP and POST_CHAMP ......................... 27

2.4.3 Calculations with the nodes: operator CALC_CHAMP ....................................................... 29

2.4.4 Calculations of quantities on whole or part of the structure: operator POST_ELEM .......... 29

2.4.5 Values of components of fields of sizes: operator POST_RELEVE_T ............................... 29

2.4.6 Impression of the results: operator IMPR_RESU ............................................................... 30

2.5 Examples ..................................................................................................................................... 31

2.5.1 Linear static analysis .......................................................................................................... 31

2.5.2 Modal analysis in dynamics ................................................................................................ 34

2.5.3 Static analysis nonlinear material ....................................................................................... 35

2.5.4 Geometrical nonlinear static analysis ................................................................................. 36

2.5.5 Analysis in buckling of Euler ............................................................................................... 37

2.5.6 Connections hulls and other machine elements ................................................................. 38

3 Thermics ............................................................................................................................................. 41

3.1 Definition of the problem .............................................................................................................. 41

3.1.1 Space discretization and assignment of a modeling: operator AFFE_MODELE ................ 41

3.1.2 Elementary characteristics: AFFE_CARA_ELEM ............................................................... 42

3.1.3 Materials: DEFI_MATERIAU ............................................................................................... 42

3.1.4 Limiting loadings and conditions: AFFE_CHAR_THER and AFFE_CHAR_THER_F ......... 43

3.2 Resolution .................................................................................................................................... 44

3.2.1 Transitory calculations: operator THER_LINEAIRE ............................................................ 44

3.3 Additional calculations and postprocessings ................................................................................ 44

3.3.1 Calculations in postprocessing ........................................................................................... 44Warning : 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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3.4 Examples ..................................................................................................................................... 44

4 Thermomechanical chaining ............................................................................................................... 45

4.1 Formalism .................................................................................................................................... 45

5 Conclusion and advices of use ........................................................................................................... 47

6 Bibliography ........................................................................................................................................ 49




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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 much lower than 1/10 (thin hulls) or about 1/10 (thick hulls).For the very thin hulls, one advises not to exceed a report 1/500 in order to avoid problems of digital lockingwhich will underestimate displacements and will thus give false results.

These modelings are usable in linear and non-linear mechanics, under assumptions of small deformations andof small displacements or many assumptions of great displacements and great rotations, according tomodelings. A mean modeling of hull is also available in transitory linear thermics.

Two categories of thin elements of structures are described in this document:• Elements of plates , which is plans. L has curve of the structure to represent is not taken into account

and it is thus necessary to use a large number of elements in order to approach correctly the geometryof the structure (aspect facets).

• Elements of hulls , which is curved.

The elements of plates triangle plans and quadrangle are described in reference material [R3.07.03]), theyunderstand following modelings:

• DKT : mesh TRIA3 element DKT , mesh QUAD4 elements DKQ (linear geometrical); • DKTG : mesh TRIA3 element DKT , mesh QUAD4 elements DKQ (linear or not linear geometrical); • DST : mesh TRIA3 element DST , mesh QUAD4 element DSQ (linear geometrical); • Q4G : mesh QUAD4 element Q4G (linear geometrical). • Q4GG : mesh TRIA3 element T3G , mesh QUAD4 elements Q4G (linear geometrical)

The elements of curved hulls resulting from models 3D with a kinematics of hull are gathered under modelingsfollowing:

• COQUE_3D : mesh TRIA7 and QUAD9 , structure 3D with unspecified geometry ([R3.07.04] into lineargeometrical, [R3.07.05] nonlinear geometrical and [R3.03.07] with following pressures);

• COQUE_AXIS : mesh SEG3 , hulls with symmetry of revolution around the axis 0Y ([R3.07.02] intolinear geometrical);

Éléments of membrane classics are gathered under modeling MEMBRANE and can have as a geometricalsupport of meshS TRIA3, QUAD4, TRIA6, QUAD8, SORTED7 and QUAD9. They are elements of membrane witha simple membrane rigidity (not of degree of freedom of rotation). There is no offsetting. One can also use themodels of membrane into large transformations.There exist éléments of specific plates to represent the tablecloths of reinforcement:

• GRILLE_EXCENTRE : mesh S TRIA3 or QUAD4 (linear geometrical). This modeling C orrespond withelements of plates DKT orthotropic to 3 or 4 nodes offset compared to the average concrete layer, (onlyone direction of reinforcement) . The concrete can be modelled by elements of plates DKT or DST with3 nodes. The reinforced concrete structure is then represented by the superposition of modelingsGRILLE_EXCENTRE and of that used for the concrete ( DKT or DST ).

• GRILLE_MEMBRANE : meshs TRIA3 , QUAD4 , TRIA6 or QUAD8 (Doc. [R3.08.07]): they are elements ofreinforcement (only one direction of reinforcement) working only out of membrane (not of degree offreedom of rotation). There is no offsetting.




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2 Mechanics2.1 Capacities of modeling2.1.1 Recall of the formulations2.1.1.1 Geometry of the elements plates and hulls

For the elements plates and hulls one defines a surface of reference, or surfaces average, planes (plan xy forexample) or curve ( x and y a set of curvilinear coordinates define) and a thickness h x , y . This thicknessmust be small compared to other dimensions of the structure to model. The figure below illustrates thesevarious configurations.

2.1.1.1 figure - has: Assumption in theory of the plates and the hulls

One attaches to average surface a local reference mark Oxyz different from the total reference mark

OXYZ . The position of the points of the plate or the hull is given by the curvilinear coordinates 1,2

average surface and rise 3 compared to this surface. For the plates the curvilinear frame of reference is alocal Cartesian frame of reference.




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2.1.1.1 figure - B: Definition of an average surface

To represent hulls with symmetry of revolution around an axis (COQUE_AXIS), the knowledge of thesection of revolution or the trace of average surface is sufficient, as the figure [2.1.2.1 Figure - has]shows it to us. These hulls rest on a linear grid and in a point m average surface one defines a local

reference mark n , t , ez by:

t=Om , s

∥Om , s∥, n∧t=ez

When one wishes to model a solid of an unspecified form (not plan), one can use elements of hulls togive an account of the curve, or many elements of plates. In this last case, the geometry isapproximated by a network of facets.

2.1.1.1 figure - C: Modeling of an unspecified solid 3D by elements of plates or hulls

For the definition of the intrinsic reference mark, one returns in [U4.74.01] paragraph 3.4.




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2.1.2 Formulation of the elements plates, voluminal hulls and hulls 2.1.2.1 Formulation into linear geometrical

In this formulation, it is supposed that displacements are small, one can thus superimpose the initialgeometry and the deformed geometry. These elements are based on the theory of the hulls accordingto which:

• the cross-sections which are the sections perpendicular to the surface of reference remainright; the material points located on a normal at not deformed average surface remain on aline in the deformed configuration. It results from this approach that the fields ofdisplacement vary linearly in the thickness of the plate or the hull . If one indicates by u, v and w displacements of a point q x , y , z according to x , y and z , one has asfollows:

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

u x , y v x , y w x , y

} z{x x , y

y x , y

0}

The associated tensor of deformation is written then:

x , y , z =e x , y x , y z. x , y .

The first term e understands the deformations of membrane (for an element of plate theyare the deformations in the plan of the element), the second those of transverse

shearing, and the third z . deformations of inflection, associated with the tensor of curve

. For the thick plates or hulls transverse shearings are taken into account according tothe formulation suggested by Reissner, Hencky, Bollé, Mindlin. This formulation includes theapproach without transverse shearing, where the tensor null, is developed by Kirchhofffor the thin plates or hulls according to which material points located on a normal n on thenot deformed average surface remain on the normal on the deformed surface.

• The transverse constraint zz is worthless because regarded as negligible compared tothe other components of the tensor of the constraints (assumption of the plane constraints).

• One does not describe the variation thickness nor that of the transverse deformation zz thatone can calculate by using the preceding assumption of plane constraints.

• The taking into account of transverse shearing depends on factors of correction determined apriori by energy equivalences with models 3D, so that rigidity in transverse shearing of themodel of plate is nearest possible to that defined by the theory of three-dimensional elasticity.For the homogeneous plates, the factor of transverse correction of shearing based on thismethod is k=5/6 .

• Shear stresses of the elements of plate (DKT/Q, DST/Q, Q/T3G) take account of the possibleoffsetting of the elements (free assumption of edge in particular and quadratic distribution inthe thickness). However, for two plates identical from the material point of view and stuckone on the other with offsetting of their half-thicknesses, the assumption design of shearstresses does not make it possible to compare the solution in shearing with a model made upof only one plates. Indeed, in the case of the plate alone, one will obtain a parabola in thethickness cancelling itself at the ends, in the case of the two plates one will have twoparabolas, cancelling oneself at the ends of each plate. To compare two plates stuck with onlyone plates, a recourse to DEFI_COMPOSITE will be necessary.

Note:




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The determination of the factors of correction rests for Mindlin on equivalences of Eigenfrequency associated with the mode of vibration by transverse shearing. One obtains then

k=2 /12 , value very close to 5/6 . These elements utilize locally:

• Five variable kinematics for the unspecified elements plates and hulls; displacements ofmembrane u and v in the datum-line z=0 , transverse displacement w and rotations

x and y normal on the average surface in the plans yz and xz respectively.

• Three variable kinematics for the linear elements; displacements u and v in the datum-line

z=0 and rotation n normal on the average surface in the plan xy .

Figure 2.1.2.1 - has: Variable kinematics for the various elements of plates and hulls

• Three efforts resulting from membrane noted Nxx , Nyy , Nxy and three noted bendingmoments Mxx , Myy , Mxy whatever the element of plate or hull; two efforts noted cutting-edges Vx and Vy in the case of elements of plates and unspecified hulls.

2.1.2.1 figure - B: Efforts resulting for an element from plate or hull




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2.1.2.2 Formulation into non-linear geometrical, Flambement of Euler

In the formulation in non-linear geometrical, one is in the presence of great displacements and ofgreat rotations, one cannot superimpose the initial geometry and the deformed geometry.

The formulation, described in the reference material [R3.07.05], is based on an approach ofcontinuous medium 3D, degenerated by the introduction of the kinematics of hull of the type Hencky -Mindlin - Naghdi in plane constraints into the weak formulation of balance. The measurement of thedeformations selected is that of Green-Lagrange, vigorously combined with the constraints of Piola-Kirchhoff of second species. The formulation of balance is thus a total Lagrangian formulation.Transverse shearing is treated same manner as in the linear case [R3.07.04].

The element retained into non-linear is a voluminal element of hull (COQUE_3D) of average surfacecurves as presented to the preceding paragraph, whose meshs supports are QUAD9 and of TRIA7.

It is possible to apply to these elements of the following pressures, whose formulation is described inthe reference document [R3.03.07]. This loading with the characteristic to follow the geometry of thestructure during its deformation (for example, the hydrostatic pressure remains always perpendicularto the deformed geometry).

Linear buckling called too buckling of Euler, described in the reference materials [R3.07.05] and[R3.07.03], is presented in the form of a typical case of the geometrical non-linear problem. It is basedon a linear dependence of the fields of displacements, strains and stresses compared to the level ofload.

The element retained in linear buckling are:• the element of plate (DKT) of plane average surface as presented to the preceding paragraph,

whose meshs supports are QUAD4 and of TRIA3.• the voluminal element of hull (COQUE_3D) of average surface curves as presented to the

preceding paragraph, whose meshs supports are QUAD9 and of TRIA7.

2.1.3 Comparison enters the elements 2.1.3.1 Differences between the elements plates and hulls

The elements of hull are curved elements whereas the elements of plates are plans. The variation ofmetric of the geometry (i.e. its radius of curvature) according to its thickness is taken into account forthe elements of hulls but not for the elements of plates. This variation of metric implies a couplingbetween the effects of membrane and inflection for nonplane structures which cannot be observedwith elements of plate plan for a homogeneous material (see [bib1]).

The choice of the functions of form for the discretization of these elements is different because thecurved elements of hulls have a more significant number of degrees of freedom. Thus, the elementsof plates are linear elements out of membrane whereas the elements of hulls are quadratic.

2.1.3.2 Differences between the elements plates

One distinguishes the elements with transverse shearing (DST, DSQ and Q4G) elements withouttransverse shearing (DKT and DKQ). Elements DST and DKT have triangular meshs support with 3nodes ( 3×5=15 degrees of freedom) and elements DKQ, DSQ and Q4G quadrangular meshssupports with 4 nodes ( 4×5=20 degrees of freedom).

Notice important :

For the elements of plate with 4 nodes ( DSQ , DKQ and Q4G ), the 4 nodes must be coplanarso that the theory of the plates can be validated. This checking is carried out systematicallyby Code_Aster, and the user is alarmed if one of the elements of the grid does not observethis condition.




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In the case of the elements with transverse shearing, to avoid the blocking of the elements intransverse shearing (over-estimate of rigidity for very low thicknesses), a method consists in buildingfields of constant shearing of substitution on the edges of the element, whose value is the integral ofshearing on the edge in question. In Code_Aster, the elements of plate and hull with transverseshearing use this method in order not to block in transverse shearing. This blocking in shearing comesowing to the fact that the elastic energy of shearing is a term proportional to h ( h being thethickness of the plate or hull) much larger than the elastic term of energy of inflection which isproportional in h3 . When the thickness becomes low in front of the characteristic length (the report

h /L is lower than 1/20 ), for certain functions of form, the minimization of the dominating term inh led to a bad representation of the modes of pure inflection, for which the arrow is not correctly

calculated any more (see [bib1] page 295 with h /L=0.01 ).

The element Q4G is a quadrilateral element with four nodes without blocking in transverse shearing,

with bilinear functions of form in x and y to represent w , x and y .

The principal difference between modeling Q4G and DST comes owing to the fact that one uses for the

latter of the functions of form including a quadratic term to discretize rotations s where s=x , y .

The consequence for the element Q4G is a constant approximation per pieces of the curves whichimplies to net sufficiently fine in the directions requested in inflection.

Notice on the element DST :

Problems were highlighted on the triangular element DST in the case of loading in inflection. Theresults with grid are equivalent are definitely less good than those of DSQ and strongly dependon the nature of the grid (see CAS-test ssls141 modelings E and F V3.03.141). One advises withthe users: ◦ to prefer the element DSQ with the element DST ◦ on DST to prefer the free grids with the regulated grids (better results) ◦ to avoid the dissymmetrical grids compared to the loading (if there exist symmetries)

2.1.3.3 Differences between the elements hulls

One distinguishes the elements from hulls axisymmetric COQUE_AXIS elements of COQUE_3D.

The first are used to model structures of revolution of axis Oy and seconds in all the other cases. Forthese elements of hulls, the meshs supports are linear with 3 nodes. The number of degrees offreedom of these elements is of 9.

Unspecified elements of hulls COQUE_3D have triangular with 7 nodes or quadrangular meshs supportwith 9 nodes:

• In the case of triangular meshs, the number of degrees of freedom for the translations is 6(the unknown factors are displacements with the nodes tops and on the mediums on thesides of the triangle) and that of rotations is 7 (the unknown factors are 3 rotations at thepreceding points and the center of the triangle). The number of degrees of freedom total ofthe element is thus of Nddle=3×63×7=39 .

• In the case of quadrangular meshs with 9 nodes, the number of degrees of freedom for thetranslations is 8 (the unknown factors are displacements with the nodes tops and on themediums on the sides of the quadrangle) and that of rotations is 9 (the unknown factors are 3rotations at the preceding points and the center of the quadrangle). The number of degrees offreedom total of the element is thus of Nddle=3×83×9=51 . These elements thushave about twice more degrees of freedom than the elements of plate of the family DKTcorrespondents. Their cost in time, with equal number, in a calculation will be thus moreimportant.

Elements of COQUE_3D automatically take into account the correction of metric between averagesurface and the surfaces upper and lower. For the linear elements, this correction must be activated




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by the user (see the paragraph 13). The correction of metric makes a contribution in h /L with the

constraint and in h /L2 in displacement (see [V7.90.03]). For the plates this correction is withoutobject.

For the elements of hulls the coefficient of correction of shearing k in isotropic behavior can bemodified by the user. This coefficient of correction of shearing is given in AFFE_CARA_ELEM under thekeyword A_CIS. By default, if the user does not specify anything in AFFE_CARA_ELEM that amountsusing the theory with shearing of REISSNER; the coefficient of shearing is then put at k=5/6 . If thecoefficient of shearing k 1 one is worth places oneself within the framework of the theory of

HENCKY-MINDLIN-NAGHDI and if it becomes very large ( k106.h /L ) one approaches the theoryof LOVE-KIRCHHOFF.

In practice it is advised not to change this coefficient. Indeed, these elements provide a physicallycorrect solution, whether the hull is thick or thin, with the coefficient k=5/6 .

All elements of hull present in Code_Aster are protected from locking in shearing like the elementsfrom plates (see §2.1.3.2). One uses for that a selective integration [R3.07.03].However this property is lost if one projects a field of displacement on an element of hull starting froma calculation on an element of plate (with the order PROJ_CHAMP). One should not be astonished bythe shear stresses obtained then by a linear calculation starting from displacements (for lowthicknesses).

2.2 Orders to be used

2.2.1 Space discretization and assignment of a modeling: operator AFFE_MODELE

In this part, the choice is described and the assignment of one of modelings plates or hull as well as thedegrees of freedom and the associated meshs. Most described information are extracted from documentationsof use of modelings ([U3.12.01]: Modeling DKT - DST - Q4G, [U3.12.02]: Modelings COQUE_AXIS). 2.2.1.1 Degrees of freedom

The degrees of freedom of discretization are in each node of the mesh support the components ofdisplacement to the nodes of the mesh support, except indication.

Modeling Degrees of freedom (with each

node)Remarks

COQUE_3D DX DY DZ DRX DRY MARTINIDRZDRX DRY MARTINI DRZ with thecentral node

The nodes belong to the average layerof the hull

DKT, DKTG DX DY DZ DRX DRY MARTINIDRZ

The nodes belong to the tangent facetwith the average layer of the hull

DST DX DY DZ DRX DRY MARTINIDRZ


Q4G DX DY DZ DRX DRY MARTINIDRZ


Q4GG DX DY DZ DRX DRY MARTINIDRZ


COQUE_AXIS DX DY DRZ The nodes belong to the averagesurface of the hull

GRILLE_EXCENTRE DX DY DZ DRX DRY MARTINIDRZ

The nodes belong to the facet tangentwith the average layer of the hull.

GRILLE_MEMBRANE DX DY DZ Degree of freedom to all the nodes. MEMBRANE DX DY DZ Degree of freedom to all the nodes.

2.2.1.2 Meshs support of the matrices of rigidityWarning : 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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Modeling Mesh Finite element RemarksCOQUE_3D TRIA7

QUAD9MEC3TR7HMEC3QU9H

Meshs not presumedly plane

DKT, DKTG TRIA3QUAD4

MEDKTR3MEDKQU4

Plane meshs

DST TRIA3QUAD4

MEDSTR3MEDSQU4

Plane meshs

Q4G, QUAD4 MEQ4QU4 Plane meshsQ4GG QUAD4

TRIA3MEQ4GG4MET3GG3

Plane meshs

COQUE_AXIS SEG3 MECXSE3 Meshs not presumedly planeGRILLE_EXCENTRE TRIA3,

QUAD4MEGCTR3MEGCQU4

Plane meshs

GRILLE_MEMBRANE TRIA3,QUAD4,TRIA6,QUAD8

MEGMTR3MEGMQU4MEGMTR6MEGMQU8

Unspecified surface meshs

MEMBRANE TRIA3,QUAD4,TRIA6,QUAD8,SORTED7, QUAD9

MEMBTR3MEMBQU4MEMBTR6MEMBQU8MEMBTR7MEMBQU9

Unspecified surface meshs

Modeling GRILLE_EXCENTRE used to model the reinforced concrete structures has the samecharacteristics of grid as modeling DKT

Note:

In a grid, to transform meshs TRIA6 in meshs TRIA7, or QUAD8 in meshs QUAD9 , one canuse the operator MODI_MAILLAGE .

2.2.1.3 Meshs support of the loadings

All the loadings applicable to the facets of the elements used here are treated by direct discretizationon the mesh support of the element in displacement formulation. The surface pressure and the otherforces as well as gravity are examples of loadings applying directly to the facets. No special mesh ofloading is thus necessary for the faces of the elements of plates, of hulls.

For the applicable loadings on the edges of the elements, one a:

Modeling Mesh Finite element RemarksCOQUE_3D SEG3 MEBOCQ3DKT, DKTG SEG2 MEBODKT

DST SEG2 MEBODSTQ4G SEG2 MEBOQ4GQ4GG SEG2 MEBOQ4G

COQUE_AXIS POI1 Meshs supports stub to 1 pointGRILLE_EXCENTRE,GRILLE_MEMBRANE,

MEMBRANE

Pas d' element of affected edge by thesemodelings.

The forces distributed, linear, of traction, shearing, the bending moments applied to the edges ofstructures hull are included in this category of loadings.

2.2.1.4 Model: AFFE_MODELE

The assignment of modeling passes through the operator AFFE_MODELE [U4.41.01].




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AFFE_MODELE RemarksAFFE

PHENOMENON ‘MECHANICAL’MODELING ‘COQUE_3D’

‘DKT’‘DST’‘DKTG’‘Q4G’‘Q4GG’‘COQUE_AXIS’‘GRILLE_MEMBRANE’‘GRILLE_EXCENTRE’‘MEMBRANE’

Note:

It is advisable to check the number of affected elements.

2.2.2 Elementary characteristics: AFFE_CARA_ELEM

In this part, the operands characteristic of the elements of plates and hulls are described. Thedocumentation of use of the operator AFFE_CARA_ELEM is [U4.42.01].

AFFE_CARA_ELEM COQUE_3D DKT DKTG DST Q4G Q4GG COQUE_AXISHULL • • • • • • •

THICK • • • • • • •/ ANGL_REP/ VECTOR

• • • • • •

A_CIS • •COEF_RIGI_DRZ • • • • • • •OFFSETTING • • • • •INER_ROTA • • • • •MODI_METRIQUE • •COQUE_NCOU • • •

AFFE_CARA_ELEM GRILLE_EXCENTRE GRILLE_MEMBRANEGRID • •

SECTION • •/ ANGL_REP/ AXIS,ORIG_AXE

• •

OFFSETTING •COEF_RIGI_DRZ

AFFE_CARA_ELEM MEMBRANEMEMBRANE

THICK •/ ANGL_REP/ AXIS, ORIG_AXE

•

N_INIT •

The characteristics being able to be affected on the elements of plate or hull are:




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• The thickness THICK constant on each mesh, since the grid represents only the averagelayer.

• The coefficient of correction of transverse shearing A_CIS for the isotropic curved hulls.

• The taking into account of the correction of metric MODI_METRIQUE between averagesurface and the surfaces upper and lower (effective only for COQUE_AXIS).

• A direction of reference allowing to define a local reference mark in the tangent plan in anypoint of a hull. The construction of the local reference mark is done is using the two“nautical” angles and (provided in degrees) which defines a vector v whose

projection on the tangent level with the hull fixes the direction x l . maybe, if the key wordVECTOR is present, by the 3 components of the vector v One can define a single vector Vfor all the structure, or one by zone (keywords GROUP_MA / MESH). The construction of thelocal reference mark is defined in AFFE_CARA_ELEM [U4.42.01]. This local reference markthereafter will be called reference mark user. The definition of this reference axis is useful also to lay down the direction of fibres of amulti-layer or orthotropic hull (cf operator DEFI_COMPOSITE [U4.42.03]).

• The number of layers COQUE_NCOU used for integration in the thickness of the hull, theoperators STAT_NON_LINE and DYNA_NON_LINE (modelings DKT, COQUE_3D,COQUE_AXIS).

• A functionality of DEFI_GROUP allows automatically to create a group of meshs whosenormal is understood in a given solid angle, of axis the direction of reference. This order can be used in preprocessing to affect nonisotropic data material or inpostprocessing after a calculation of hull.

• Offsetting (constant for all the nodes of the mesh) OFFSETTING of each one of themcompared to the mesh support. This distance is measured on the normal of the meshsupport. In the excentré case inertias of rotation are obligatorily taken into account andINER_ROTA is put at YES.

• COEF_RIGI_DRZ a coefficient of fictitious rigidity defines (necessarily small) on the degreeof freedom of rotation around the normal in the hull. It is necessary to prevent that the matrixof rigidity is singular, but must be selected smallest possible. For the DKT, if one choosesCOEF_RIGI_DRZ negative, one reinforces by a variational writing the kinematics of rotation

of the element plates around his normal. One advises a value of 10−8 .

Figure 2.2.2-a: Total reference mark and tangent plan

For modelings GRILLE_EXCENTRE and GRILLE_MEMBRANE, The following geometrical data are necessary to model the tablecloth of reinforcements:




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• SECTION = S1 : section of the reinforcements in direction 1. The section is given per linearmeter. It thus corresponds to the section cumulated over a width of 1 meter. If there is a sections all them 20.0cm , the cumulated section is 5.s .

• The orientation of the reinforcements is defined is by • ANGL_REP, to define a vector project on the element• maybe in the case of a cylindrical hull, by ORIG_AXE, AXIS to define the angle of

the reinforcements, constant in a cylindrical reference mark.• Offsetting (constant for all the nodes of the mesh) of the tablecloth of reinforcements compared

to the mesh support (distance measured on the NOR onemale of the mesh support), (modelingGRILLE_EXCENTRE only).

To define a grid or if the section of the reinforcements in the longitudinal direction and the transverse oneare different, it is necessary to create two layers of elements (order CREA_MAILLAGE, keywordCREA_GROUP_MA), a layer of element for the longitudinal direction and a second layer of elements forthe transverse direction:

For modeling MEMBRANE into linear, only the orientation of the behavior of the membrane is necessary. Itis defined maybe by:

• ANGL_REP , to define a vector project on the element • maybe in the case of a cylindrical hull, by ORIG_AXE , AXIS to define the angle of the

reinforcements, constant in a cylindrical reference mark. If one uses MEMBRANE into nonlinear it is then necessary to inform:

• THICK , L ‘thickness membrane. • ANGLE_REP, the reference mark of posting of the constraints ( only one isotropic behavior is

possible ). • N_INIT , one initial prevoltage who has the dimension of an effort per unit of length and which

disappears after S increment S of Newton of the first step of time. This prevoltage is useful onlyfor the convergence of the first step of time.

Notice important:

The direction of the normals to each element is a recurring problem concerning the use of this kind ofelement, for example when loadings of type pressure are applied, either to define a offsetting or alocal reference mark.

By default for the surface elements the orientation is given by the vector product 12^13 for anumbered triangle 123 (DKT,…) or 1234567 (COQUE_3D) and 12^14 for a numbered quadrangle1234 (DKQ,…) or 123456789 (COQUE_3D). For the linear hulls n is given by the formula of theparagraph Error: Reference source not found with t directed in the direction of course of the mesh onthe level of the grid.

Generally, these data are accessible while looking in the file from grid, which is not very practical forthe user. Moreover, it is necessary that it checks the coherence of its grid and to make sure that all themeshs have the same orientation well.

The user can automatically modify the orientation of the elements of the grid by imposing a directionof normal, for a grid or part of using grid of modelings of hull and whatever the type of modeling. Thereorientation of the elements is done by the means of the operator ORIE_NORM_COQU orderMODI_MAILLAGE [U4.12.05]. The principle is the following: one defines under ORIE_NORM_COQU adirection by the means of a vector and a node pertaining to the group of meshs to be reorientated. Ifthe introduced vector is not in the plan of the mesh selected by MODI_MAILLAGE, one from ofautomatically deduced a direction from normal obtained like the vector less given its projection asregards the mesh. All the related meshs of the group to those initially selected will then have the sameorientation of normal automatically. In addition an automatic checking of the same orientation of therelated meshs is carried out by the means of the operator VERI_NORM order AFFE_CHAR_MECA[U4.25.01].




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2.2.3 Materials: DEFI_MATERIAU

The definition of the behavior of a material is carried out using the operator DEFI_MATERIAU[U4.43.01].

DEFI_MATERIAU COQUE_3DDKT,DST,Q4G

DKTG,Q4GG COQUE_AXIS GRILLE_EXCENTRE,

GRILLE_MEMBRANE MEMBRANE

ELASTICITY.LINEAR

ELAS • • • • • •ELAS_ORTH ELAS_ISTR

• •

ELAS_COQUE •ELAS_MEMBRANE •

DEFI_MATERIAUCOQUE_3

DDKT

DSTQ4G

DKTG

Q4GGCOQUE_AXIS

GRILLE_EXCENTREGRILLE_MEMBRANE

MEMBRANE

BY LAYERbehaviorsavailable inC_PLAN

• • •

behaviorsavailable in1D

•

GLRC_DAMAGE • •GLRC_DM,KIT_DDI

•

MEMBRANE into linear is used with ELAS_MEMBRANE and into nonlinear with ELAS.

The materials used with the whole of the elements plates or hulls can have elastic behaviors in planeconstraints whose linear characteristics are constant or functions of the temperature. All nonlinear behaviors in plane constraints (either directly, or by the method of Borst [R5.03.03]) areavailable pour modelings DKT and hulls. For more information on these nonlinearities one can refer tothe paragraph [§22]. All the nonlinear behaviors in 1D (either directly, or by the method of Borst) are available formodelings GRILLE_EXCENTRE and GRILLE_MEMBRANE.

The mean composite material structures can be treated currently only by modelings plates, whileusing DEFI_COMPOSITE with homogenized material characteristics. One can also directly introducethe coefficients of rigidity of the matrices of membrane, inflection and shearing with ELAS_COQUE.These coefficients are given in the reference mark user of the element defined by ANGL_REP. Itshould be noted that the terms of shearing are not taken into account with the behavior ELAS_COQUEthat for the elements DST and Q4G. They are not taken into account with the elements DKT.

In order to facilitate comprehension, we represented on the figure below the various reference marksused.




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Figure 2.2.3-a: Reference marks used for the definition of material

The following example is extracted from the CAS-test SSLS117B and the syntax illustrates ofDEFI_COMPOSITE :

MU2=DEFI_COMPOSITE (COUCHE=_F (EPAIS=0.2, MATER=MAT1B, ORIENTATION=0.0,),);

In this example, one defines a multi-layer composite thickness 0.2 , the material being defines byMAT1B, and the angle of the 1st direction of orthotropism (longitudinal direction or direction of fibres)being null. One will refer to documentation [U4.42.03] for more details concerning the use ofDEFI_COMPOSITE. (see also [R4.01.01].

2.2.4 Limiting loadings and conditions: AFFE_CHAR_MECA and AFFE_CHAR_MECA_F

The assignment of the loadings and the boundary conditions on a mechanical model is carried outusing the operator AFFE_CHAR_MECA, if the loadings and the boundary conditions mechanical on asystem are actual values depending on no parameter, or AFFE_CHAR_MECA_F, if these values arefunctions of the position or the increment of loading.

The documentation of use ofAFFE_CHAR_MECA and AFFE_CHAR_MECA_F is [U4.44.01].

2.2.4.1 List of the keyword factor of AFFE_CHAR_MECA




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AFFE_CHAR_MECA COQUE_3DDKT,DKTG

DSTQ4G,Q4GG

COQUE_AXIS GRILLE_*

DDL_IMPO • • • • • •FACE_IMPO • • • • • •LIAISON_DDL • • • • • •LIAISON_OBLIQUE • • • • • •LIAISON_GROUP • • • • • •CONTACT • • • • • •LIAISON_UNIF • • • • • •LIAISON_SOLIDE • • • • • •LIAISON_ELEM • • • •LIAISON_COQUE • • • • • •FORCE_NODALE • • • • • •

DDL_IMPO Keyword factor usable to impose, with nodes or groups of nodes, one ormore values of displacement.

FACE_IMPO Keyword factor usable to impose, with all the nodes of a face defined by amesh or a group of meshs, one or more values of displacements (or certainassociated sizes).

LIAISON_DDL Keyword factor usable to define a linear relation between degrees of freedomof two or several nodes.

LIAISON_OBLIQUE Keyword factor usable to apply, with nodes or groups of nodes, the samecomponent value of displacement definite per component in an unspecifiedoblique reference mark.

LIAISON_GROUP Keyword factor usable to define linear relations between certain degrees offreedom of couples of nodes, these couples of nodes being obtained whileputting in opposite two lists of meshs or nodes.

CONTACT Keyword factor usable to notify conditions of contact and friction between twowhole of meshs.

LIAISON_UNIF Keyword factor allowing to impose the same value (unknown) on degrees offreedom of a set of nodes.

LIAISON_SOLIDE Keyword factor allowing to model an indeformable part of a structure.LIAISON_ELEM Keyword factor which makes it possible to model the connections of a hull

part with a beam part or of a hull part with a pipe part (see paragraph2.2.4.5).

LIAISON_COQUE Keyword factor making it possible to represent the connection enters of thehulls by means of linear relations.

FORCE_NODALE Keyword factor usable to apply, with nodes or groups of nodes, nodal forces,definite component by component in the reference mark TOTAL or in anoblique reference mark defined by 3 nautical angles.




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AFFE_CHAR_MECAindividuals

COQUE_3DDKT,DKTG

DST,Q4G

Q4GG COQUE_AXIS GRILLE_*

FORCE_ARETE • • • •FORCE_COQUE total near tangentroom

•••

•••

•••

•••

•••

GRAVITY • • • • • •PRES_REP • • • • •ROTATION • •

PRE_EPSI • • • •

FORCE_ARETE Keyword factor usable to apply linear forces to an edge of an element of hull.For the linear elements the equivalent amounts applying a nodal force to thenodes supports of the element. There is thus no particular dedicated term.On the other hand, it requires elements of edges.

FORCE_COQUE Keyword factor usable to apply surface efforts (pressure for example) toelements defined on all the grid or one or more meshs or of the groups ofmeshs. These efforts can be given in the total reference mark or a referencemark of reference defined on each mesh or groups meshs; this referencemark is built around the normal with the element of hull and of a directionfixes (see paragraph 2.2.2).

GRAVITY Keyword factor usable for a loading of type gravity.PRES_REP Keyword factor usable to apply a pressure to one or more meshs, or of the

groups of meshs.ROTATION Keyword factor usable to calculate the loading due to the rotation of the

structure.PRE_EPSI Keyword factor usable to apply a loading of initial deformation.

Note:

The efforts of pressure being exerted on the elements of plates can apply is byFORCE_COQUE ( near ) maybe by PRES_REP . The user will have to thus pay attention notto twice apply the loading of pressure for the elements concerned, especially whenevermodelings of plates would be mixed with other modelings using PRES_REP .

In addition it should be noted that efforts of pressure, that it is with FORCE_COQUE (near) orPRES_REP are such as a positive pressure acts in the contrary direction with that of the normal to theelement. By default, this normal is dependent on the direction of course of the nodes of an element,which is not always very easy for the user.

Moreover it is necessary that this one makes sure that all these elements are directed same manner.One thus advises to impose the orientation of these elements by the means of the operatorORIE_NORM_COQU order MODI_MAILLAGE (see paragraph [§2.2.2]).

2.2.4.2 List of the keyword factor of AFFE_CHAR_MECA_F

The keyword factor generals of the operator AFFE_CHAR_MECA_F are identical to those of theoperator AFFE_CHAR_MECA presented above.

AFFE_CHAR_MECA_Findividuals

COQUE_3DDKT,DKTG

DSTQ4G,Q4GG

COQUE_AXIS

FORCE_ARETE • • • •Total FORCE_COQUEneartangent room

•••

•••

•••

•••

•••




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The loadings of pressure functions of the geometry can be indicated by the means of FORCE_COQUE(near).

2.2.4.3 Application of a pressure: keyword FORCE_COQUE

The keyword factor FORCE_COQUE allows to apply surface efforts to elements of type hull (DKT, DST,Q4G,…) defined on all the grid or one or more meshs or of the groups of meshs. According to thename of the operator called, the values are provided directly (AFFE_CHAR_MECA) or via a conceptfunction (AFFE_CHAR_MECA_F).

AFFE_CHAR_MECAAFFE_CHAR_MECA_F

Remarks

FORCE_COQUE: •

Totalreference mark

ALL: ‘YES’MESHGROUP_MA

• Place of application of the loading

FXFYFZMXMYMZ

• Provided directly for AFFE_CHAR_MECA,in the form of function forAFFE_CHAR_MECA_F

PLAN ‘MOY’‘INF’‘SUP’‘E-MAIL’

• Allows to define a torque of efforts onthe average, lower, higher level or ofthe grid (elements DKT and DST)

Localreference mark

NEAR F1F2F3MF1MF2

• Provided directly for AFFE_CHAR_MECA,in the form of function forAFFE_CHAR_MECA_F

We return in the paragraph corresponding to the keyword FORCE_COQUE document of use of theoperators AFFE_CHAR_MECA and AFFE_CHAR_MECA_F.

2.2.4.4 Limiting conditions: keywords DDL_IMPO and LIAISON_*

The keyword factor DDL_IMPO allows to impose, with nodes introduced by one (at least) of thekeywords: ALL, NODE, GROUP_NO, MESH, GROUP_MA, one or more values of displacement (or certainassociated sizes). According to the name of the operator called, the values are provided directly(AFFE_CHAR_MECA) or via a concept function (AFFE_CHAR_MECA_F).

Operands available for DDL_IMPO, are listed below:

DX DY DZ Blocking on the component of displacement in translationDRX DRY MARTINIDRZ

Blocking on the component of displacement in rotation

2.2.4.5 Connections hulls with other machine elements

These connections must meet the requirements established in [bib4] and that one finds in particular inthe connection 3D-POUTRE in [R3.03.03].

The connections available with the elements of plates and hulls are the following:




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• Connection Beam-hull : it is a question of establishing the connection between a node endof an element of beam and a group of meshs of edge of elements of hulls. The theories ofbeam and plate know only normal cuts with fibre or the average surface. The connections cantake place only according to these average fibres or surfaces. The connection beam-hull isrealizable for beams whose neutral fibre is orthogonal with the normals with the facetsof the plates or the hulls. To extend to other configurations (a beam arriving perpendicularto the plan of a plate for example) asks a feasibility study because the elements of plate orhull do not have rigidity associated with a rotation in the plan perpendicular to the normal toaverage surface. The connection is usable by using the keyword LIAISON_ELEM :(OPTION: ‘COQ_POU’) of AFFE_CHAR_MECA.

• Connection Hull-pipe : it is a question of establishing the connection between a node end ofa pipe section and a group of mesh of edge of elements of hulls. The formulation of theconnection hull-pipes is presented in the reference document [R3.08.06]. The theories of pipeand plate, know only normal cuts with fibre or the average surface. The connections can takeplace only according to these average fibres or surfaces. The connection hull-pipe isrealizable for pipes whose neutral fibre is orthogonal with the normals with the facets of theplates or the hulls. The connection is usable by using the keyword LIAISON_ELEM:(OPTION: ‘COQ_TUYAU’) of AFFE_CHAR_MECA.

2.2.4.5 figure - has: Connections hulls with other machine elements

• Connection Hull – massive 3D : the connection hull-3D solid mass is under investigation butit will be limited initially to the cases where the normal with the solid is orthogonal with thenormal with the one of the facets of the element of plate or hull (see [bib4]).

• Connection between elements of Hulls : to connect two elements of hulls between them,the keyword is used LIAISON_COQUE ofAFFE_CHAR_MECA (_F) (documentation[U4.44.01]). This connection is carried out by means of linear relations. The classicalapproach admits that 2 plans with a grid in hulls are cut according to a line which belongs tothe grid of the structure. In order to prevent that the volume which is the intersection of thetwo hulls is counted twice, one stops the grid of a hull perpendicular to a hull given to thelevel of the higher or lower skin of the latter. On [2.2.4.5 Figure - B], the link between the 2hulls is made by connections of solid body between the nodes in with respect to the segmentsA1 A2 and B1 B2 .




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Figure 2.2.4.5 - B: Connection between elements of hulls

CAS-tests making it possible to validate these connections are available in the section examples.

2.2.4.6 Variables of orders

The variables of orders taken into account by various modelings are listed here:

Variablesof orders

COQUE_3D

DKT,DST,Q4G

DKTGQ4GG COQUE_AXIS

GRILLE_MEMBRANEGRILLE_EXCENTRE

MEMBRANETEMP • • • • •others:SECH,HYDR, etc.

2.3 Resolution

2.3.1 Linear calculations: MECA_STATIQUE and other linear operators

Linear calculations are carried out in small deformations. Several linear operators of resolution areavailable:

MECA_STATIQUE : resolution of a problem of static mechanics linear([U4.51.01]);

MACRO_ELAS_MULT : calculate linear static answers for various loadingcases or modes of Fourier. ([U4.51.02]).

CALC_MODES : calculation of the values and clean vectors bymethods of subspaces or iterations opposite.([U4.52.02]).

MODE_ITER_CYCL : calculation of the clean modes of a structurewith cyclic symmetry ([U4.52.05]);

DYNA_LINE_TRAN : calculation of the transitory dynamic response to anunspecified temporal excitation ([U4.53.02]);

DYNA_TRAN_MODAL : calculation is carried out by modal superposition orunder-structuring ([U4.53.21]);

2.3.2 Nonlinear calculations: STAT_NON_LINE and DYNA_NON_LINE

2.3.2.1 Behaviors and assumptions of deformations available

Following information is extracted from the documentation of use of the operator STAT_NON_LINE :[U4.51.03].

COQUE_3D

DKT DKTG DST,Q4G

DKTG Q4GG GRILLE_*

COQUE_AXIS




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BEHAVIOR(smalldeformations)

RELATION All relations availablein plane constraints

• • •

The relations 3D byusing the method ofBorst

• • •

All relations availablein 1D

•

DEFORMATION:‘GROT_GDEP‘

Coque_3D in greatdisplacements andgreat rotationsavailable withnonlinearincrémentauxbehaviors, but insmall deformations

•

DEFORMATION:‘SMALL’ (orGROT_GDEP)

In small or greatdisplacementsavailable withnonlinearincrémentauxbehaviors, but inweak rotations andsmall deformations

• • • •

DEFORMATION:SMALL orGROT_GDEP

In small or greatdisplacements

•

RELATION GLRC_DAMAGE • • •RELATION GLRC_DM, KIT_DDI • •

BEHAVIOR(large displacements,great rotations)

RELATION ELAS •DEFORMATION:‘GROT_GDEP‘

•

TYPE_CHARGE:‘SUIV’

Following pressure •

All the mechanical nonlinear behaviors of plane constraints of the code are accessible. The relation ofbehavior connects the rates of deformation to the rates of constraints.

For modelings GRILLE_EXCENTRE and GRILLE_MEMBRANE, for reinforced concrete structures, thenonlinear behaviors 1D correspond to particular incrémentaux behaviors in STAT_NON_LINE(BEHAVIOR) :

• GRILLE_ISOT_LINE for plasticity with isotropic work hardening,• GRILLE_ISOT_CINE for plasticity with kinematic work hardening linear Bi,• GRILLE_PINTO_MEN for the behavior of Pinto Menegotto.

Modeling MEMBRANE is implemented for behavior S rubber band S in small deformations and smalldisplacements, usable with COMPORTEMENT=' PETIT' and RELATION=' ELAS' , but also forbehaviors very-rubber bands in great deformations and great displacements with DEFORMATION = 'GROT_GDEP' and RELATION=' ELAS_MEMBRANE_SV' or RELATION= ‘ELAS_MEMBRANE_NH’ .

The behaviors 3D can also be used using the method of Borst [R5.03.09].

When the elements of the type plates or hulls become coplanar, it is advisable to control the problemof rigidity around the normal. Indeed, by construction, these elements do not have rigidity in thisdirection: DRZ is one ddl fictitious who avoids the singularity of matrices DKT in total referencemark.




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The coefficient COEF_RIGI_DRZ in the operator AFFE_CARA_ELEM allows to modify this value. Thevalue by default is sufficient in most case, but, in certain situations like:– thin elements of sections used with a great thickness (DKT for example);– a non-linear calculation (STAT_NON_LINE or DYNA_NON_LINE);– elements of the type DKT on quadrangles bored in general ;– COQUE_3D with kinematics of great transformations;– Plate subjected to a rotation around its normal.

Conseil for the DKT on COEF_RIGI_DRZ: A bad value penalizes the convergence of calculations. Theresult, if it is converged, remains just, but the speed of convergence is decreased. For example, anelastic linear calculation asks for more than one iteration of Newton to converge. It is then necessaryto change the value of COEF_RIGI_DRZ and to decrease it. Nevertheless, attention with the choice ofthis coefficient because a too low value can make the matrix singular. It is also possible to enrichmodels DKT with the taking into account by rigidity according to DRZ. With this intention, it is enoughto take negative COEF_RIGI_DRZ. In this last case, it is specified that one does not have all made amodel of DKT such as it is formulated in the literature but one has a model of DKT enriched by akinematics in DRZ: DRZ becomes ddl physics.

Conseil for the COQUE_3D: A contrario for the COQUE_3D, it is to better privilege increasingly largeCOEF_RIGI_DRZ to avoid the singular matrices.

The concept RESULT of STAT_NON_LINE/DYNA_NON_LINE contains fields of displacements,constraints and variables internal at the points of integration always calculated at the points of gauss:

• DEPL : fields of displacements.

• SIEF_ELGA : Tensor of the constraints by element at the points of integration (COQUE_3D andDKT) in the reference mark user. For each layer, one stores in the thickness and for eachthickness on the points of surface integration. Thus if one wants information on a constraintfor the layer NC, on the level NCN (NCN = -1 so lower, NCN = 0 if medium, NCN = +1 sohigher) for the surface point of integration NG, it will be necessary to look at the value givenby the point defined in the option NOT such as: NP = 3* (NC-1) *NPG+ (NCN+1)*NPG+NG where NPG is the full number of points of surface integration of the element ofCOQUE_3D (7 for the triangle and 9 for the quadrangle) and of the element DKT. Formodelings GRILLE_EXCENTRE,GRILLE_MEMBRANE, one stores simply a value by point ofintegration: the component SIXX in the direction of the reinforcements. For modelings DKTGand Q4GG, SIEF_ELGA contains the 6 efforts generalized (membrane efforts, bendingmoments, efforts cutting-edges) by point of Gauss. In the case general, SIEF_ELGA is astress field of Cauchy but for COQUE_3D, it is about a stress field of the Piola-Kirchoff type ofsecond species.

• VARI_ELGA : Field of internal variables (DKT and COQUE_3D) by element at the points of

surface integration. For each point of surface integration, one stores information on the layerswhile starting with the first, level ‘INF’. The number of variables represented is thus worth2*NCOU*NBVARI where NBVARI represent the number of internal variables.

It can be enriches by the following fields, calculated in postprocessing by the operator CALC_CHAMP :

• EFGE_ELNO : activate the calculation of the tensor of the efforts generalized by element withthe nodes (membrane efforts, bending moments, efforts cutting-edges), in the reference markuser (defined in the paragraph [§2.2.2]).

• VARI_ELNO : activate the calculation of the field of internal variables by element with thenodes in the thickness (by layer SUP/MOY/INF in the thickness except indication).

2.3.2.2 Detail on the points of integration




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The classification of the surface points of Gauss for the elements COQUE_3D is given in R3.07.04§4.7.1. Attention with the order of the points of Gauss for the formula at 9 points, which is not thesame one as that adopted for the isoparametric elements.

For modelings DKT, COQUE_3D, COQUE_AXIS, in the case of non-linear calculations, the method ofintegration for the elements of plate and hulls is a method of integration by layers, of which thenumber is defined by the user. For each layer, except modeling GRID, one uses a method of Simpsonat three points of integration, in the middle of the layer and in skins higher and lower of layer. For Nlayers the number of points of integration in the thickness is of 2N1 .

To treat non-linearities material, one advises to use from 3 to 5 layers in the thickness for a number ofpoints of integration being worth 7.9 and 11 respectively. For tangent rigidity, one calculates for eachlayer, in plane constraints, the contribution to the matrices of rigidity of membrane, inflection andcoupling membrane-inflection. These contributions are added and assembled to obtain the matrix oftotal tangent rigidity. For each layer, one calculates the state of the constraints and the whole of theinternal variables, in the middle of the layer and in skins higher and lower of layer. This information isavailable in VARI_ELGA and SIEF_ELGA. One can in postprocessing reach the values of constraintand transverse shearing strain obtained starting from the derivative of the moments. With thisintention one assumes even into nonlinear an elastic relation on the transverse behavior.

For modelings GRILLE_EXCENTRE and GRILLE_MEMBRANE reinforced concrete structures, there isonly one point of integration per layer.

2.3.2.3 Geometrical non-linear behavior

Calculations into non-linear geometrical (great displacements and great rotations), available withmodeling COQUE_3D, are realized using the operator STAT_NON_LINE, while using, under thekeyword BEHAVIOR, DEFORMATION = ‘GROT_GDEP‘.

Calculations into non-linear geometrical (great displacements and great rotations), available withmodeling MEMBRANE, are realized using the operator STAT_NON_LINE, while using, under thekeyword BEHAVIOR, DEFORMATION = ‘GROT_GDEP‘. Calculations into non-linear geometrical (great displacements and small deformations), available withmodeling exce , are realized using the operator STAT_NON_LINE , while using, under the keywordBEHAVIOR , DEFORMATION = ‘GROT_GDEP’. It is possible to apply to the elements of COQUE_3D following pressures. This loading has thecharacteristic to follow the geometry of the structure during its deformation (for example: thehydrostatic pressure remains always perpendicular to the deformed geometry). To take into accountthis kind of loading, it is necessary to specify in the operator STAT_NON_LINE following information:

STAT_NON_LINE ( EXCIT =_F (LOAD = near TYPE_CHARGE = ‘SUIV’) ) The geometrical non-linear behavior of the structures can have instabilities (buckling, snap -through/snap-back…). The determination and the passage of these boundary points, cannot beobtained by imposing the loading, however the options of piloting of the loading ‘DDL_IMPO’ or‘LONG_ARC’ of the operator STAT_NON_LINE allow to cross these critical points.

L ‘use of L ‘element MEMBRANE into non-linear geometrical can be delicate because of strong non-linearities and its absence of bending stiffness. The following points will be retained:

Use of the prevoltage: L ‘use D’ an initial prevoltage via N_INIT in the operatorAFFE_CARA_ELEM [U4.42.01] is a characteristic specific to the elements of structure




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without flexural strength, one finds it in code_aster for the elements of cable. Indeed, amembrane subjected to a force perpendicular to its initial surface, as it is the case with apressure, will undergo a movement of rigid body what causes the stop of calculation. A meansof circumventing this problem is to create a rigidity initial, known as “geometrical”, to allow theconvergence of the first increment. It is then normal that the time of convergence of this firststep of time is rather long, one thus should not hesitate to allow an iteration count of highNewton (>100) and to play over the value of the initial tension N_INIT . L prevoltage has is a

generalized effort and is written in the form N init=hσ0 with h the thickness and σ0

prestressing , one council to start by taking a value of pre low constraint (for example 1.E- 3

) then to increase this value by multiplying it by 10 until obtaining convergence. One will prefermoreover to take h and

Object 156

like parameters independent then to return hσ0 like

argument under the keyword N_INIT . The value σ0 is sometimes difficult to find but it isuseful only for the first step of time, one can thus test it quickly.

Initial force and rigidity: L very strong non-linearity of the element has implies that convergencecan be bad when the force applied is too weak compared to the rigidity of material. It is difficultto quantify this report but the user should not hesitate to put a force raised at the first step oftime, since it is necessary to activating the step division of time via the orderDEFI_LIST_INST . This remark is particularly true when a pressure is applied.

Linear research (and contact): U N average to improve in a notable way convergence is toactivate linear research (keyword RECH_LINEAIRE in the order STAT_NON_LINE[U4.51.03] ). This one is sometimes essential to the convergence of calculations, inparticular at the time of the first steps of time. Linear research is unfortunately not compatiblewith the problems of contact in code_aster, that can bring to have to cut out the problems intwo parts: a first without contact with linear research and a second with only the contact.

Piloting: S I after having varied the value of the initial tension and the value of the initial forceone does not manage to make converge calculation with the first increment, it is also possibleto use piloting (keyword PILOTING in STAT_NON_LINE [U4.51.03] ). Piloting is alsoincompatible with the contact.

2.3.2.4 Linear buckling

Calculations in linear buckling are similar in search of Eigen frequencies and of modes of vibration.The problem has to solve is expressed in the form:

To find , X ∈ℝ ,ℝN such as AX= BX

where

A is the matrix of rigidity

B is the geometrical matrix of rigidity (calculated with the option RIGI_GEOM ofCALC_MATR_ELEM), available for modelings DKT, DKTG, COQUE_3D

is the critical load

X is the mode of associated buckling has the critical load

The operator CALC_MODES [U4.52.02] beT used to determine the critical load and the mode ofassociated buckling.

2.4 Additional calculations and postprocessings

2.4.1 Elementary calculations of matrices: operator CALC_MATR_ELEM

The operator CALC_MATR_ELEM (documentation [U4.61.01]) allows to calculate elementary matrices,which are then gatherable by the order ASSE_MATRICE (documentation [U4.61.22]).

Elementary options of the operator CALC_MATR_ELEM are described below:




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CALC_MATR_ELEM COQUE_3DDKT,DKTG

DSTQ4G,Q4GG

COQUE_AXIS GRILLE_*

‘AMOR_MECA’ • • • • • •‘MASS_MECA’ • • • • • •‘MASS_INER’ • • • • • •‘RIGI_GEOM’ • •‘RIGI_MECA’ • • • • • •‘RIGI_MECA_HYST’

• • • • • •

• AMOR_MECA : Matrix of damping of the elements calculated by linear combination of rigidityand the mass.

• MASS_MECA : Matrix of mass. • MASS_INER: calculation of the inertial characteristics (mass, centre of gravity)• RIGI_GEOM : Geometrical matrix of rigidity (for great displacements). • RIGI_MECA : Matrix of rigidity of the elements. • RIGI_MECA_HYST : Hysteretic rigidity (complex) calculated by the product by one coefficient

complexes structural damping of simple rigidity.

2.4.2 Calculations by elements: operators CALC_CHAMP and POST_CHAMP

One presents hereafter the options of postprocessing for the elements of plates and hulls. Theycorrespond to the results which a user can get after a thermomechanical calculation (internalconstraints, displacements, deformations, variables, etc…). For the structures modelled by elementsof hulls or beams, it is particularly important to know how the results of constraints are presented inorder to be able to interpret them correctly. Approach adopted in Code_Aster consist in calculating theconstraints in the reference mark “user” defined in the operator AFFE_CARA_ELEM.If one wishes to strip his results in another reference mark which the reference mark user, should beused the order MODI_REPERE . When a postprocessing relates to only one “under-point”, the user has the keywords NUME_COUCHEand NIVE_COUCHE keyword factor EXTR_COQUE order POST_CHAMP.

Keywords under EXTR_COQUE are described in the following table:

OPTIONS COQUE_3D DKT DST,Q4G

DKTG,Q4GG

COQUE_AXIS GRILLE_*

NUME_COUCHE • • •NIVE_COUCHE • • • •

• More precisely, in the case of a multi-layer material (multi-layer hull defined byDEFI_COMPOSITE), or of an element of structure with local nonlinear behavior, integrated bylayers, NUME_COUCHE is the whole value ranging between 1 and the number of layers,necessary to specify the layer where one wishes to carry out elementary calculation. Byconvention, layer 1 is the sub-base (in the direction of the normal) in the case of the elementsof hull.

• For the layer digital defined by NUME_COUCHE, allows to specify the ordinate where onewishes to carry out elementary calculation: INF/MOY/SUP correspond to the point ofintegration located in skin interns/average/external layer.

keyword factor COQU_EXCENT allows to modify the plan of calculation of the generalized efforts(options EFGE_ELNO and EFGE_ELGA) for a model with elements of plates (DKT, DST, Q4G, DKTG) bytaking account of offsetting (MODI_PLAN=' OUI').

The options of postprocessing available are:

OPTIONS COQUE_3D DKT DST,Q4G

DKTG,Q4GG

COQUE_AXIS GRILLE_*

‘ENEL_ELGA’ • • •




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‘ENEL_ELNO’‘ENEL_ELEM’‘ENER_ELAS’ • • •‘EPSI_ELGA’ •‘SIEQ_ELGA’‘DEGE_ELGA’ • • • • •‘DEGE_ELNO’ • • • • •‘ECIN_ELEM’ • • • • •‘EFGE_ELGA’ • • • • •‘EFGE_ELNO’ • • • • •‘EPOT_ELEM’ • • • • •‘EPSI_ELNO’ • • • • •‘SIEQ_ELNO’ • • •‘SIEF_ELGA’ • • • • • •‘SIEF_ELNO’ • • • • • •‘SIGM_ELNO’ • • • • •‘VARI_ELNO’ • • • •

• SIEF_ELGA : Calculation of the efforts generalized by element at the points of integration ofthe element starting from displacements (use only in elasticity). Reference mark user.

• SIGM_ELNO : Calculation of the constraints by element to the nodes. Reference mark user.They is the constraints of Cauchy.

• SIEQ_ELNO : Constraints equivalent to the nodes, calculated in a point thickness fromSIGM_ELNO :

VMIS : Constraints of Von Mises.VMIS_SG : Constraints of Von Mises signed by the trace of the constraints.PRIN_1, PRIN_2, PRIN_3 : Principal constraints.

These constraints are independent of the reference mark.• EFGE_ELGA : Calculation of the efforts generalized by element at the points of integration of

the element starting from displacements (use only in elasticity). Reference mark user. • EFGE_ELNO : Calculation of the efforts generalized by element with the nodes starting from

displacements (use only in elasticity). Reference mark user. • EPSI_ELNO : Calculation of the deformations by element to the nodes starting from

displacements, in a point the thickness (use only in elasticity). Reference mark user. • EPSI_ELGA : Calculation of the deformations by element at the points of integration starting

from displacements, in a point the thickness (use only in elasticity). Reference mark intrinsic. • DEGE_ELGA : Calculation of the generalized deformations by element at the points of

integration of the element starting from displacements. Reference mark user. • DEGE_ELNO : Calculation of the deformations generalized by element with the nodes starting

from displacements. Reference mark user. • EPOT_ELEM : Calculation of the linear elastic energy of deformation per element starting

from displacements. • ENER_TOTALE : calculation of the total deformation energy integrated on the element• ENER_ELAS : calculation of the elastic deformation energy integrated on the element• ENEL_ELGA/ENEL_ELNO : elastic energy at the points of integration or the nodes• ENEL_ELEM : elastic energy on the element • ECIN_ELEM : Calculation of the kinetic energy by element. • EFGE_ELNO : Option of activation of the calculation of the tensor of the efforts generalized

(see paragraph [§Nonlinear calculations: STAT_NON_LINE and DYNA_NON_LINE ]) byelement with the nodes, in the reference mark user, by integration of the constraintsSIEF_ELGA (into non-linear).

• EFGE_ELGA : Option of activation of the calculation of the tensor of the efforts generalized(see paragraph [§ Nonlinear calculations: STAT_NON_LINE and DYNA_NON_LINE ]) byelement at the points of gauss of the element, in the reference mark user, by integration ofthe constraints SIEF_ELGA (into non-linear).

• VARI_ELNO : Option of activation of the calculation of the field of internal variables byelement and layer with the nodes. For each point of surface integration, one storesinformation on the layers while starting with the first, level ‘INF’. The number of variables




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represented is worth thus 3*NCOU*NBVARI where NBVARI represent the number of internalvariables.

• NUME_COUCHE : Dyears the case of a multi-layer material (composite or hull in plasticity),whole value ranging between 1 and the number of layers, necessary to specify the layerwhere one wants to carry out elementary calculation.

• NIVE_COUCHE : For the layer n , one can specify the ordinate where one wishes to carry outelementary calculation. A calculation in internal skin is indicated by ‘INF’, in external skin by‘SUP’ and on the average layer by ‘MOY’ (according to the direction of the normal).

2.4.3 Calculations with the nodes: operator CALC_CHAMP

OPTIONS COQUE_3D DKT DST Q4G COQUE_AXIS GRID‘FORC_NODA’ • • • • • •‘REAC_NODA’ • • • • • •_NOEU • • • • • •

For the elements of plates and hulls, the operator CALC_CHAMP (documentation [U4.81.04]) only thecalculation of the forces and reactions (calculation of the fields allows the nodes by moyennation,option _NOEU):

• starting from the constraints, balance: FORC_NODA (calculation of the nodal forces startingfrom the constraints at the points of integration, element per element),

• then by removing the loading applied: REAC_NODA (calculation of the nodal forces of reactionto the nodes, the constraints at the points of integration, element per element):

• REAC_NODA = FORC_NODA - loadings applied ,• useful for checking of the loading and calculations of resultants, moments, etc.

2.4.4 Calculations of quantities on whole or part of the structure: operatorPOST_ELEM

The operator POST_ELEM (documentation [U4.81.22]) allows to calculate quantities on whole or part ofthe structure. The calculated quantities correspond to particular options of calculation of affectedmodeling.

OPTIONS Operator COQUE_3D DKT DST Q4G COQUE_AXIS GRID‘MASS_INER’ POST_ELEM • • • • •‘ENER_POT’ POST_ELEM • • • • •‘ENER_CIN’ POST_ELEM • • • • •‘ENER_ELAS’ POST_ELEM • • •

• MASS_INER : calculation of the geometrical characteristics (volume, centre of gravity, matrix

of inertia) for the elements plates and curves.• ENER_POT : calculation of the potential energy of deformation due to balance starting from

displacements in linear mechanics of the continuous mediums (2D and 3D) and in linearmechanics for the elements of structures, or the energy dissipated thermically with balance inlinear thermics starting from the temperatures (cham_no_TEMP_R).

• ENER_CIN : calculation of the kinetic energy starting from a field speed or a field ofdisplacement and of a frequency (only for the elements of structure and the elements 3D).

• ENER_ELAS : calculation of the elastic deformation energy. 2.4.5 Values of components of fields of sizes: operator POST_RELEVE_T

The operator POST_RELEVE_T (documentation [U4.81.21]) allows, on a group of nodes, to extractfrom the values or to carry out calculations:

• to extract from the values of components of fields of sizes;• to carry out calculations of averages and invariants:

•Averages,




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•Resultants and moments of vector fields,•Invariants of tensorial fields,•Directional trace of fields,•Of expression in the reference marks TOTAL, ROOM, POLAR, USER or CYLINDRICAL

The produced concept is of type counts. To use POST_RELEVE_T, it is necessary to define three concepts:

• one place : the option NODE (example: N01 N045) or the option GROUP_NO (example:SUPPORT);

• one object : with the choice, the option RESULT (SD result: EVOL_ELAS,…) or the optionCHAM_GD (CHAM_NO : DEPL,… or CHAM_ELEM : SIGM_ELNO,…) ;

• one nature : with the choice, the option ‘EXTRACTION’ (value,…) or the option ‘AVERAGE’(average, maximum, mini,…).

Notice important: If one comes from an interface with a maillor, the nodes are arranged by digital order. It isnecessary to reorder the nodes along the line of examination. The solution is to use theoperator DEFI_GROUP with the option NOEU_ORDO . This option makes it possible to createone GROUP_NO ordered containing the nodes of a set of meshs formed by segments ( SEG2or SEG3 ).

An example of extraction of component is given in CAS-test SSNL503 (see description in theparagraph [§2.5.3] page 35):

TAB_DRZ=POST_RELEVE_T (ACTION=_F ( GROUP_NO = ‘OF, ENTITLE = ‘TB_DRZ’, RESULT = RESUL, NOM_CHAM = ‘DEPL’, NOM_CMP = ‘DRZ’, TOUT_ORDRE = ‘YES’, OPERATION = ‘EXTRACTION’ ) )

The purpose of this syntax is:

to extract: OPERATION = ‘EXTRACTION’on the group of nodes D : GROUP_NO = ‘OF

the component DRZ displacement: NOM_CHAM = ‘DEPL’, NOM_CMP = ‘DRZ’,for every moment of calculation: TOUT_ORDRE = ‘YES’

2.4.6 Impression of the results: operator IMPR_RESU

The operator IMPR_RESU allows to write the grid and/or the results of a calculation on listing with theformat ‘RESULT’ or on a file in a displayable format by external tools for postprocessing to Aster:format RESULT and ASTER (documentation [U4.91.01]), format CASTEM (documentation [U7.05.11]),format IDEAS (documentation [U7.05.01]), format MED (documentation [U7.05.21]) or format GMSH(documentation [U7.05.32]).

This procedure makes it possible to write with the choice:• a grid,• fields with the nodes (of displacements, temperatures, clean modes, static modes,…),• fields by elements with the nodes or the points of GAUSS (of constraints, generalized efforts,

internal variables…).The elements of plate and hull being treated same manner that the other finite elements, we return thereader to the notes of use corresponding to the format of exit which it wishes to use.




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2.5 Examples

The cas-tests selected here are classical CAS-tests resulting from the literature and which are usuallyused to validate this kind of elements.

It is pointed out that modelings DKT correspond to the theory of Coils-Kirchhoff and modelings DST,Q4G with the theory with transverse energy of shearing (Reissner). Results for modeling COQUE_3Dare presented only for one theory with transverse energy of shearing.

2.5.1 Linear static analysis

SSLS20

Title:: Cylindrical hull pinch on free board

Documentation V: [V3.03.020]

Modelings:SSLS20A DKTSSLS20B COQUE_3D MEC3QU9HSSLS20C COQUE_3D MEC3TR7H

SSLS100

Title: Embedded circular plate subjected to a uniformpressure.


Modelings:SSLS100K COQUE_3D MEC3QU9HSSLS100L COQUE_3D MEC3TR7HSSLS100B DKTSSLS100E DKQSSLS100F DSTSSLS100G DSQSSLS100H Q4GSSLS100I, J COQUE_AXIS

SSLS101

Title: Circular plate posed subjected to a uniformpressure.


Modelings:SSLS101J COQUE_3D MEC3QU9HSSLS101I COQUE_3D MEC3TR7HSSLS101B DKTSSLS101E DKQSSLS101F DSTSSLS101G DSQSSLS101H Q4G




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SSLS104

Title: Cylindrical hull pinch with diaphragm.


Modelings:SSLS104B COQUE_3D MEC3QU9HSSLS104C COQUE_3D MEC3TR7HSSLS104A DKT

SSLS105

Title: Doubly gripped hemisphere.


Modelings:SSLS105A DKTSSLS105B COQUE_3D MEC3QU9H

SSLS107

Title: Cylindrical panel subjected to its own weight.


Modelings:SSLS107A COQUE_3D MEC3QU9HSSLS107B COQUE_3D MEC3TR7H

SSLS108

Title: Helicoid hull under concentrated loadings.


Modelings:SSLS108A COQUE_3D MEC3QU9HSSLS108B COQUE_3D MEC3TR7H

Note:Disadvised use with DKT/DKQ, without transverseshearings.




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Ssls120: cylinder under pressure This test shows that the grids triangles are much moresensitive than the grids quadrangles.

Grid triangles with an amplified deformation

Grid quadrangles with an amplified deformation

Other CAS-tests are more briefly described in the following table:

Name Modeling Remarkshpla100a hpla100b

hpla100c

hpla100d hpla100e

hpla100f

2D_AXIS COQUE_AXIS

COQUE_3D

COQUE_3D HULL

HULL

Title: Heavy thermoelastic hollow roll in uniform rotation. Documentation V: [V7.01.100]

The purpose of this test is to test the second members correspondingto the effects of gravity and an acceleration due to a uniform rotation.

Analytical solutions for COQUE_3D include the variation of metric inthe thickness of the hull. The analytical solutions for the plates arewithout correction of metric

hsls01a

hsls01b

DKT/DST/Q4G

COQUE_3D

Title: Embedded thin section subjected to a heat gradient in thethickness.


hsns100a

hsns100b

COQUE_3D/DKT

COQUE_3D/DKT

Title: Plate subjected to a variation in temperature in the thickness.


This CAS-test makes it possible to test two ways of imposing thethermal field. The got results has some and B must be identical, butthe reference solutions obtained are digital.




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ssls114a

ssls114b

ssls114c ssls114d

ssls114i

COQUE_3D

COQUE_3D

DKT/DST DKQ/DSQ

COQUE_AXIS

Title: Setting under pressure of a cylindrical quarter of binding ring.


Analytical reference solution. Allows to test the term of pressure andthe orientation of the normals. One tests the results in radialdisplacement and radial constraints.

2.5.2 Modal analysis in dynamics

Name Modeling Remarks

sdls01a

sdls01b

sdls01c

sdls01d

sdls01e

sdls01f

sdls01g

sdls01h

DKT

DKT

DKT

DKT

COQUE_3D

COQUE_3D

COQUE_3D

COQUE_3D

Title: Thin square plate free or embedded on an edge

Documentation V: [V2.03.001] It is of a modal calculation and a harmonic calculation of answer.For modal calculation, it is a question of calculating the cleanmodes of inflection of a thin square plate free or embedded on anedge. has - Edges of the plate directed according to the axes of thereference mark.b - Unspecified orientation of the plate and harmonic answer for theembedded plate.c - Modal calculation by classical and cyclic dynamic under-structuring.d - Modal calculation following a under-structuring of Guyan.e - Edges of the plate directed according to the axes of thereference mark.f - Edges of the plate directed according to the axes of the referencemark.g - Unspecified orientation of the plate and harmonic answer for theembedded plate.h - Unspecified orientation of the plate and harmonic answer for theembedded plate.

• For has and B the precision on the Eigen frequencies islower than 1% until the sixth mode of inflection

• For C under-structuring, the quality of the results can beimproved by the use of a finer grid of substructure.

• For D, it is necessary in order to obtain an accuracy of 1%on the Eigen frequencies to also condense on thenodes medium of the edges.

• For E, F, G and H, the precision on the Eigen frequencies islower than 1% until the sixth mode of inflection for theelements quadrangle and lower than 2% for elementstriangles.

The element of hull MEC3QU9H is performing compared to theelement DKT who is itself more performing that the elementMEC3TR7H.




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sdls109a

sdls109b, C

sdls109d, E

DKQ ( MEDKQU4 )and DSQ ( MEDSQU4 )

DKT ( MEDKTR3 )and DST (MEDSTR3 )

COQUE_3D(MEC3QU9H andMEC3TR7H )

Title: Eigen frequencies of a thick cylindrical ring.


This test is inspired by a vibratory study carried out on collector VVPof the N4 slices. This collector is thick and present a maximumreport thickness on average radius of 0.13. This value, being able tobe typical of an industrial structure, is slightly higher than the limitingvalue of validity usually recognized for the plates and hulls. In thisstudy, the modeling of the collector in hulls is then evaluated bycomparison with a voluminal model on a ring.

This test makes it possible to evaluate the algorithm of search foreigenvalues CALC_MODES [U4.52.02] with the matrices of rigidityand mass.

2.5.3 Static analysis nonlinear material

SSNL501

Title: Fixed beam subjected to a uniform pressure.


Modelings:SSNL501E COQUE_3D MEC3QU9HSSNL501D COQUE_3D MEC3TR7HSSNL501B DKTSSNL501C DKQ


Name Modeling Remarksssnp15a

ssnp15b

ssnp15c

ssnp15d

3D

C_PLAN

DKT

COQUE_3D

Title: Square plate in traction-shearing - Von Misès (isotropic workhardening).


A plate, made up of a plastic material with linear isotropic workhardening, is subjected to a tractive effort and a shearing force. Evenif it test validates the law of behavior rather that the elements towhich it applies, it makes it possible to test the values of theconstraints, the efforts and the deformations in the reference markdefined by the user (ANGL_REP).

ssnv115a ssnv115b ssnv115c ssnv115d

ssnv115e

D_PLAN

DKT DKT COQUE_3D

COQUE_3D

Title: Corrugated iron in nonlinear behavior.

Documentation V: [V6.04.115] This test validates the nonlinear behaviors in modelings of plates orthin hulls. Modeling A (2D D_PLAN) is useful of reference. The valuesof displacements are tested.




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2.5.4 Geometrical nonlinear static analysis

SSNV138

Title: Plate cantilever in great rotations subjected to onemoment.


Modelings:SSNV138 COQUE_3D MEC3QU9HSSNV138 COQUE_3D MEC3TR7H

Note:Greatest rotation reached is slightly lower than . The gotresults are very satisfactory, the maximum change islower than 0.01%. It is necessary to increase the value ofCOEF_RIGI_DRZ (10th-5 by default) to 0,001 in order tobe able to increase the value of the swing angle which onecan reach.

SSNV139

Title: Plate skews.


Modelings:SSNV139 COQUE_3D MEC3QU9HSSNV139 COQUE_3D MEC3TR7H

SSNL502

Title: Beam in buckling.


Modelings:SSNL502 COQUE_3D MEC3QU9HSSNL502 COQUE_3D MEC3TR7H

SSNS501

Title: Great displacements of a cylindrical panel.


Modelings:SSNS501 COQUE_3D MEC3QU9HSSNS501 COQUE_3D MEC3TR7H


Name Modeling Remarksssnv140a

ssnv140b

COQUE_3D

COQUE_3D

Title: Embedded cylindrical panel subjected to a surface force.


This force is constant for modeling has and following in modeling B. Thegoal of this CAS-test is to check modeling COQUE_3D non-lineargeometrical by using the algorithm of update of large rotations 3DGROT_GDEP of STAT_NON_LINE and to check the treatment of thefollowing pressures. The data of this problem correspond to a thin hull




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h /L=0.625% what is severe for the finite element triangle MECQTR7H(case of blocking to transverse shearing).

ssnv141a COQUE_3D Title: Segment of a sphere pinch.


The data of this problem correspond to a thin hull h /L=0.4% what issevere for the finite element triangle MECQTR7H (case of blocking totransverse shearing). It is necessary to increase the value ofCOEF_RIGI_DRZ who allots a rigidity around the normal of the elements

of hull which is worth by default 10−5 the smallest rigidity of inflectionaround the directions in the plan of the hull in order to be able to increasethe value of the swing angle which one can reach. Values of thiscoefficient until 10−3 remain licit.

ssnv144a COQUE_3D Title: Elbow in cross-bending, elastic, embedded on with dimensions andsubjected to a linear force equivalent to one bending moment.


The goal of this CAS-test is to check that, for the elements COQUE_3D,quasi-static solutions into linear geometrical (VMIS_ISOT_LINE inSTAT_NON_LINE) and into nonlinear geometrical ( GROT_GDEP inSTAT_NON_LINE) are close to the linear static solution(MECA_STATIQUE) in the field of the small disturbances.

ssnv145a

ssnv145b

COQUE_3D

COQUE_3D

Title: Plate cantilever in great rotations subjected has a followingpressure.


The goal of this CAS-test is to check modeling COQUE_3D (mesh TRIA7,QUAD9) in the presence of pressure of a following type.

2.5.5 Analysis in buckling of Euler

SSLS110

Title: Stability of a compressed square plate.


Modelings:SSLS110 COQUE_3D MEC3QU9HSSLS110 COQUE_3D MEC3TR7HSSLS110 DKT MEDKQU4SSLS110 DKT MEDKTR3




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SDLS504

Title: Side buckling of a beam (discharge).


Modelings:SDLS504 COQUE_3D MEC3QU9HSDLS504 COQUE_3D MEC3TR7H

SDLS505

Title: Buckling of a cylindrical envelope under externalpressure.


Modelings:SDLS505 COQUE_3D MEC3QU9HSDLS505 COQUE_3D MEC3TR7HSDLS505 DKT MEDKTR3SDLS505 DKT MEDKQU4

2.5.6 Connections hulls and other machine elements

SSLX100

Title: 3D-Hull-Beam mixture in inflection.


Modelings:SSLX100A 3D

1 MECA_HEXA20DKT

4 MEDKTR3POU_D_E

2 POU_D_E

SSLX100B 3D1 MECA_HEXA20 DKT

4 MEDKTR3 POU_D_E

2 POU_D_E

One tests the axial arrows, constraints, deformationsand bending moments in 4 points of the axis of thebeam.




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SSLX102

Title: Piping bent in inflection.


Modelings:SSLX102A DKT and PIPE (connection COQUE_TUYAU)

SSLX102D HULL and BEAM

SSLX101A

Title: Pipe right modelled in hulls and beams[V3.05.101].


Modelings:SSLX101A DIS_TR

POI1 DKT

MEDKQU4 POU_D_E

2 SEG2

Embedding of the hull on the edge P1 . Inflectionand traction in x1 . Variation from 3 to 5% ondisplacements and rotations in P2 with the analyticalsolution, due to the network hull with elements plans.

SSLX101B DKTMEDKQU4, METUSEG3

PIPEMETUSEG3, MEDKQU4

DIS_TRPOI1

This modeling aims to test the connection hull pipe inthe presence of unit loadings: traction, inflection andof torsion. The reference solution is analytical (RDM).The variation with the digital solution is explained bythe fact why the grid in hulls actually consists ofelements plans (facets). The geometry of the pipe isthus itself approximate.

SSLX102A DKTMEDKQU4, METUSEG3

PIPE MEDKQU4, METUSEG3

Modeling A utilizes the connection coque_tuyau, thesolution obtained ( 2.7% of variation in cross-bending, and 0.4% in inflection except plan,compared to the reference: grid any hulls (modelingD) makes it possible to test the good performance ofconnection between hull and pipe.




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Ssns115a

Swelling of a flexible membrane The objective of this test is to validate the operationof the element MEMBRANE in great deformations fortwo hyperelastic laws of behavior and differentstandard from meshs (linear, quadratic and bi-quadratic). One thus considers the swelling of a discsubjected to a following pressure and one comparesthe results with solutions drawn from the literature.

Ssls108

Test with taking into account of a physical rotationaround the normal COEF_RIGI_DRZ=-1.E-8.




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3 Thermics

For the resolution of chained thermomechanical problems, one must use for the thermal calculation ofthe finite elements of thermal hull [R3.11.01]. These elements are elements plates, or linear in thecase of structures of revolution or invariant structures along an axis. The curve of the structure is nottaken into account in itself. The metric one of the tangent plan of each element is calculated bysupposing that all the tops are coplanar. These elements suppose a distribution a priori parabolic oftemperature in the thickness, which results from an asymptotic development in linear thermics for alow thickness of hull, when the temperature variations are not too important. It is it should be notedthat a model based on a development of the richer field of temperature in the thickness sees its termsof a nature higher than two converging towards zero when the hull is thin. One cannot thus deal withthe thermal problems of shocks with strong variation of the profile of temperature in the thickness withthese hulls. The methods of use of these elements are presented in [U1.22.01].

3.1 Definition of the problem

3.1.1 Space discretization and assignment of a modeling: operator AFFE_MODELE

3.1.1.1 Degrees of freedom

The degrees of freedom are the temperatures TEMP_MIL (temperature on the average surface of thehull), TEMP_INF (temperature on the lower surface of the hull), and TEMP_SUP (temperature on theupper surface of the hull).

3.1.1.2 Meshs support of the matrices of rigidity

Modeling Mesh Nature of the mesh Finiteelement

Remarks

HULL QUAD9QUAD8QUAD4TRIA7TRIA6TRIA3

planeplaneplaneplaneplaneplane

THCOQU9THCOQU8THCOQU4THCOTR7THCOTR6THCOTR3

nodes with 3 coordinates x , y , z

COQUE_PLAN SEG3 not presumedly plane THCPSE3 nodes with 2 coordinates x , yCOQUE_AXIS SEG3 not presumedly plane THCASE3 nodes with 2 coordinates x , y

For THCOTRi, only the three tops are exploited to define the local geometry (tangent plan, normal).For THCOQUi, it is considered that the element is plan and its tangent plan is defined by default by 3of the 4 tops of the element.




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3.1.1.3 Meshs support of the loadings

Modeling Mesh Finite element RemarksHULL SEG2 THCOSE2 with TRIA3 and QUAD4HULL SEG3 THCOSE3 with TRIA6,7 and QUAD8, 9

All the loadings applicable to the facets of the elements of hull are treated by direct discretization onthe mesh support of the element in temperature formulation. No mesh of loading is thus necessary forthe faces of the elements of hulls.

For the applicable loadings on the edges of the elements of hull, a mesh support of the type SEG2(element THCOSE2) or SEG3 (element THCOSE3) must be used.

For the imposed temperatures the meshs support are meshs reduced to a point.

3.1.1.4 Model: AFFE_MODELE

The assignment of modeling passes through the operator AFFE_MODELE [U4.41.01].

AFFE_MODELE RemarksAFFE

PHENOMENON: ‘THERMAL’MODELING ‘HULL’

‘COQUE_PLAN’‘COQUE_AXIS’

3.1.2 Elementary characteristics: AFFE_CARA_ELEM

In this part, the operands characteristic of the elements of plates and hulls in thermics are described.The documentation of use of the operator AFFE_CARA_ELEM is [U4.42.01].

AFFE_CARA_ELEM HULL COQUE_PLAN COQUE_AXIS RemarksHULL

THICK • • •

The characteristics assigned to materials are the same ones as for a mechanical calculation. It is itshould be noted that it is not useful to define a particular reference mark for the analysis of the resultsof thermal calculation because those are limited to the fields of temperature, scalar size, independentof the reference frame used.

3.1.3 Materials: DEFI_MATERIAU

DEFI_MATERIAU HULL COQUE_PLAN COQUE_AXIS RemarksTHER • • •THER_FO • • •

The materials used with elements plates or hulls in thermics can have linear characteristics thermalconstant or dependent on the increment of loading.




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3.1.4 Limiting loadings and conditions: AFFE_CHAR_THER and AFFE_CHAR_THER_F

The assignment of the loadings and the boundary conditions on a thermal model is carried out usingthe operator AFFE_CHAR_THER, if the loadings and the boundary conditions mechanical on a systemare actual values depending on no parameter, or AFFE_CHAR_THER_F, if these values are functionsof the position or the increment of loading.

The documentation of use ofAFFE_CHAR_THER and AFFE_CHAR_THER_F is [U4.44.02].

3.1.4.1 List of the keyword factor of AFFE_CHAR_THER

The affected values of the loadings are real and do not depend on any parameter.

AFFE_CHAR_THERgenerals

HULL COQUE_PLAN COQUE_AXIS Remarks

TEMP_IMPO • • •

ParticularAFFE_CHAR_THER


FLUX_REP • • • on the faces and the edgesof the surface elements

EXCHANGE • • • on the faces and the edgesof the surface elements

• TEMP_IMPO : Keyword factor usable to impose, on nodes or groups of nodes, a temperature.

• FLUX_REP : Keyword factor usable to apply normal flows to a face of thermal hull defined byone or more meshs or of the groups of meshs of type triangle or quadrangle.

• EXCHANGE : Keyword factor usable to apply conditions of exchange with an outsidetemperature with a face of hull, defined by one or more meshs or groups of meshs of typetriangle or quadrangle.

3.1.4.2 List of the keyword factor of AFFE_CHAR_THER_F

The affected values of the loadings can be a function of the total coordinates and time, or thetemperature in nonlinear thermics (except in hulls).

AFFE_CHAR_THER_Fgenerals


TEMP_IMPO • • •

ParticularAFFE_CHAR_THER_F


FLUX_REP • • • on the faces and the edgesof the surface elements

EXCHANGE • • • on the faces and the edgesof the surface elements




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3.2 Resolution

3.2.1 Transitory calculations: operator THER_LINEAIRE

Transitory option ofcalculation


CHAR_THER_EVOL •

It is here the treatment of the problems of thermics evolutionary.

3.3 Additional calculations and postprocessings

3.3.1 Calculations in postprocessing

One presents hereafter the options of postprocessing for the elements of plates and hulls

OPTIONS elementary


‘FLUX_ELNO’ •‘FLUX_ELGA’ •

• FLUX_ELNO : This option carries out the calculation of heat flow to the nodes starting fromthe temperature.

• FLUX_ELGA : This option carries out the calculation of heat flow at the points of integrationstarting from the temperature.

3.4 Examples

One gives here the list of the CAS-tests available for the thermal hulls. They are CAS-tests ofstationary thermics. The results are correct for the whole of these CAS-tests, whatever the elementused.

Name Modeling Element Remarkstplp301a HULL THCOTR3 Title: Plate with imposed temperature distributed

sinusoïdalement on a side.

Documentation: [V4.05.301]

tplp302a HULL THCOTR6 Title: Rectangular plate with temperatureimposed on the sides.


tpls100atpls100b

HULLCOQUE_PLAN

THCOTR6/THCOTR3THCPSE3

Title: Infinite plate subjected to a couple ofstationary antisymmetric heat flows on its twohalf-faces.

Documentation: [V4.03.100] Conduction is linear, homogeneous and isotropic.

tpls101atpls101btpls101ctpls101dtpls101e

HULL THCOTR6/THCOSE3THCOQU4/THCOSE2THCOQU8/THCOSE3THCOQU9/THCOSE3THCOTR7/THCOSE3

Title: Infinite plate subjected to a couple ofthermal conditions with outside, symmetricalcompared to the average layer.





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Conduction is linear, homogeneous and isotropic.

tpls302atpls302btpls302ctpls302d

HULL THCOQU8/THCOSE3THCOQU4/THCOSE2THCOQU9/THCOSE3THCOTR7/THCOSE3

Title: Rectangular plate with convection andimposed temperature


4 Thermomechanical chaining

4.1 Formalism

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 _Aster 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 hulls. This last isdefined by the knowledge of the 3 fields TEMP_INF, TEMP_MIL and TEMP_SUP given in skins lower,medium and higher of hull. The table below indicates these compatibilities:

ModelingTHERMICS

Mesh Element Mesh Element ModelingMECHANICS

HULL QUAD9 THCOQU9 QUAD9 MEC3QU9H COQUE_3DHULL QUAD8 THCOQU8HULL QUAD4 THCOQU4 QUAD4 MEDKQU4

MEDSQU4MEQ4QU4

DKTDSTQ4G

HULL TRIA7 THCOTR7 TRIA7 MEC3TR7H COQUE_3DHULL TRIA6 THCOTR6HULL TRIA3 THCOTR3 TRIA3 MEDKTR3

MEDSTR3DKTDST

COQUE_PLAN SEG3 THCPSE3COQUE_AXIS SEG3 THCASE3 SEG3 MECXSE3 COQUE_AXIS

Note:

• The nodes of the thermal elements of hulls and plates or mechanical hulls mustcorrespond. The grids for thermics and mechanics will thus have the same number andthe same type of meshs.

• The elements of thermal hulls surface are treated like elements plans by projection ofthe initial geometry on the level defined by the first 3 tops. For the chaining ofcalculations with mechanical curved elements it is thus necessary that the geometry ofthe plate is not too distant from that of the hull. When the structure is curved, that thusrequires for thermal calculation to net it in a sufficiently fine way in order to have correctresults in preparation for the mechanical part. Only the linear elements of thermics areperfectly associated with the corresponding linear elements in mechanics becausefascinating of account the curve of the structure with a grid.

• The chaining with multi-layer materials is not available for the moment.• The thermomechanical chaining is also possible if one knows, analytically or by

experimental measurements, the variation of the field of temperature in the thickness ofthe structure or certain parts of the structure. In this case one works with a map oftemperature defined a priori; the field of temperature is not given any more by the threevalues TEMP_INF , TEMP_MIL and TEMP_SUP thermal calculation obtained byEVOL_THER . The operator DEFI_NAPPE allows to create such profiles of temperaturesstarting from the abundant data by the user. These profiles are affected by the orderCREA_CHAMP and CREA_RESU (cf the CAS-test hsns100b ). It will be noted that it isnot necessary for mechanical calculation that the number of points of integration in thethickness is equal to the number of points of discretization of the field of temperature in




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the thickness. The field of temperature is automatically interpolated at the points ofintegration in the thickness of the elements of plates or hulls.

• The thermal evolution which one can associate with the material field byAFFE_MATERIAU/AFFE_VARC must be ready to be used by the finite elements of themechanical model. A problem arises for the elements of type hull or pipe which use atemperature varying in the thickness on the various layers. For these elements, it isnecessary to prepare the calculation of the temperature on the layers upstream of theorder AFFE_MATERIAU. For that, the user must use the order CREA_RESU with one ofthe operations PREP_VRC1 or PREP_VRC2 Variables of Order”): - OPERATION=' PREP_VRC1' : calculation of the temperature in the layers of a hull onthe basis of a temperature TEMP= F (THICK, INST)- OPERATION=' PREP_VRC2‘: calculation of the temperature in the layers of a hull onthe basis of a temperature calculated by Code_Aster with a model of hulls(TEMP_MIL/TEMP_INF/TEMP_SUP).




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HPLA100

It is a question of studying a thermal phenomenon ofdilation where the fields of temperature are calculatedwith THER_LINEAIRE by a stationary calculation:

- thermal dilation:

T −T ref =0.5T sT i2.T sTi r−R/ h

with: T s=0.5° C ,T i=−0.5 °C ,Tref =0°C

T s=0.1 °C ,T i=0.1 °C ,T ref =0 ° C

One tests the constraints, the efforts and bendingmoments in L and M . The results of reference areanalytical. For modelings COQUE_3D one takes intoaccount the variation of metric with the thickness of thehull. Very good performances whatever the type ofelement considered.

Title: Heavy thermoelastic hollow roll in uniformrotation


Modelings:HPLA100AThermics PLAN 32THPLQU8Mechanics AXIS 32MEAXQU8

HPLA100BThermics COQUE_PLAN 10 THCPSE3Mechanics COQUE_AXIS 10 MECXSE3

HPLA100CThermics HULL 32THCOQU9Mechanics COQUE_3D 32MEC3QU9H

HPLA100DThermics HULL 64THCOTR7Mechanics COQUE_3D 64MEC3TR7H

HPLA100EThermics HULL

200 THCOQU4Mechanics HULL

200 MEDKQU4

HPLA100FThermics HULL

400 THCOTR3Mechanics HULL

400 MEDKTR3

5 Conclusion and advices of use

In the following table, a summary of the opportunities given by modelings plates and hulls aredescribed.

Modeling DKTDST,Q4G

DKTG,Q4GG

COQUE_3D

COQUE_AXIS GRILLE_*

Scope of application Linear statics: Isotropic material

X X X X X X

Orthotropic material,composite

X X

Non-linear statics material X X X X X




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Geometrical non-linearstatics

X

Dynamic analysis X X X X X XBuckling of Euler X X X

On the figure below the field of application of the plates and hulls is schematized.

h/L

0.1 (1/10) 0.05 (1/20)

DKT, DKQ, SHB8 DST, DSQ , Q4G, SHB8

COQUE_3D, COQUE_AXIS, SHB8

COQUE_C_PLAN, COQUE_D_PLAN

Éléments plans

Éléments courbes

Coques minces Coques épaisses

Figure 5-a: Fields of application of the plates and hulls

Some recommendations concerning the field of application of these elements:

• Mean structures : for these structures, whose h/L report is lower than 1/20, the effects oftransverse shearing can be neglected and the theory of Kirchhoff applies. One advises to use forthis kind of structure of the elements plates DKT-DKQ or of the elements of hull curves(COQUE_3D, _AXIS). It is advised to use the elements preferably DKT and DKQ who givevery good performances on displacements and more approximate on the constraints (to berecommended for the vibratory analyses). Even if one must use a large number of theseelements, the execution times remain reasonable compared with those of the curved elements.One advises not to exceed a h/L=1/500 report in order to avoid problems involved in digitallockings.

• Thick structures: for these structures, elements of plate will be used DST, DSQ and Q4G who take

account of transverse shearing with a factor of correction of shearing k=5/6 (theory ofReissner) or preferably elements of curved hull. The factor of correction of shearing makes itpossible to pass from a theory of Hencky-Mindlin-Naghdi for k=1 , with a theory of Reissner fork=5/6 . The coefficient of shearing is adjustable only for the elements of curved hull but it is

advised not to modify its value by default.When modeling Q4G was privileged, it is necessary to carry out a small study of sensitivityto the grid. The tests show indeed that this modeling requires a sufficiently fine grid in thedirections requested in inflection to obtain weak errors.

Elements DKT, DKQ, DST, DSQ and Q4G are elements plans, they do not take into account the curve ofthe structures, it is thus necessary to refine the grid if the curve is important if one wants to avoid theparasitic inflections.

The variation of metric of the geometry (i.e. its radius of curvature) according to its thickness is takeninto account:

• automatically for modeling COQUE_3D.




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• defined by the user for modeling COQUE_AXIS.

The optimal machine element in statics according to the whole of the CAS-tests of theparagraph [§2.5] is the element of hull with 9 nodes MEC3QU9H, which makes it possible to obtaingood displacements and good constraints thanks to its P2 interpolation out of membrane. It is ageneral-purpose element which can be at the same time used to represent very mean structures (h /L≤1/100 ) or thicker. Like, in addition, the element of hull with 7 nodes MEC3TR7H is less

performing, it is advised with the user to net his structure in hulls with the greatest possiblenumber of quadrangles.

• Non-linearity material: the nonlinear behaviors (plasticity, etc) in plane constraints are availablefor the elements of hull curves (COQUE_3D, COQUE_AXIS) and elements plates DKT-DKQ only.The plastic behavior does not take the terms of transverse shearing which are treated in an elasticway, because transverse shearing is uncoupled from the plastic behavior. For a goodrepresentation of the progression of plasticity through the thickness, one advises to use for digitalintegration 3 to 5 layers in the thickness for a number of points of gauss being worth 3.5 and 11respectively.

• Geometrical non-linearity : notlinearities geometrical (great displacements, great rotations) in

plane constraints are available for the elements of curved hull COQUE_3D only.

• Buckling of Euler : this kind of analysis is available with the elements of curved hull COQUE_3Dand elements of plates DKT and DKTG.

Elements corresponding to the machine elements exist in thermics; the thermomechanical couplingsare thus available at the same time for the elements of plates and hulls. For the moment thesecouplings are not possible for multi-layer materials.

6 Bibliography

1 J.L. Batoz, G. Dhatt “Modeling of the structures by finite elements: beams and plates”Hermes, Paris (1990).

2 J.L. Batoz, G. Dhatt “Modeling of the structures by finite elements: hulls” Hermes, Paris(1992).

3 D. Bui “Evolution of AFFE_CARA_ELEM “CR MMN/97/004.

4 S. Andrieux “Connections 3D/poutre, 3D/coques and other imaginations”.

5 E. Lorentz “Great plastic deformations. Modeling in Aster by PETIT_REAC“.EDF/DERCRMMN 1536/07.

6 Non-linear p. Kinematic Jetteur “of the hulls”. Report SAMTECH resulting from contractPP/GC - 134/96.

7 J. Argyris, P. Dunne, C. Malejannakis, E. Schelkie “with simple triangular facet Shell elementwith application to linear and not linear equilibrium and elastic stability problem”.Comp. Meth. Appl. Mech. Eng., flight 11.1977.




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8 F. Frey “Non-linear static analysis of the structures by the finite element method and itsapplication to the metal structure”. Doctorate, Liege, 1978.

9 A.B. Pidgin and A.C. Lock “The application of finite elements to the broad deflectiongeometrically and not linear behavior of cylindrical shells” Variational methods inEngineering, edited by Brebbia and Tottenham, Southampton, 1972.

10 G.S. Dhatt “Instability of thin shells by the finite elements method”. Proc. IASS Symp. , Vol1,Vienna 1970, pp1-36.

11 Connection 3D-Beam [R3.03.03].

12 Following pressure for the voluminal elements of hulls [R3.03.07].

13 Axisymmetric thermoelastic hulls and 1D [R3.07.02].

14 Elements of plate DKT, DST, DKQ, DSQ and Q4 [R3.07.03].

15 Finite elements of voluminal hulls [R3.07.04].

16 Voluminal elements of hulls into nonlinear geometrical [R3.07.05].

17 Model of thermics for the thin hulls [R3.07.11].

18 Finite elements of right pipe and curve with ovalization, swelling and warping inelastoplasticity [R3.08.06].

19 Model of thermics for the thin hulls [R3.11.01].

20 Integration of the elastoplastic relations [R5.03.02].

21 Relation of nonlinear elastic behavior [R5.03.20].

22 Static and dynamic modeling of the beams in great rotations [R5.03.40].

23 Operator DEFI_MATERIAU [U4.23.01].

24 Operator DEFI_COMPOSITE [U4.23.03].

25 Operator AFFE_CARA_ELEM [U4.24.01].

26 Operator AFFE_CHAR_MECA and AFFE_CHAR_MECA_F [U4.25.01].

27 Operator AFFE_CHAR_THER and AFFE_CHAR_THER_F [U4.25.02].

28 Operator STAT_NON_LINE [U4.32.01].

29 Operator CALC_MATR_ELEM [U4.41.01].

30 Operator CALC_CHAMP [U4.81.04].

31 BELYTSCHKO T. and BINDEMAN L.P.: “Assumed strain stabilization of the eight nodehexahedral elements”, Methods Computer in Applied Mechanics and Engineering,vol. 105,225-260, 1993.

32 FLANAGAN D.P. and BELYTSCHKO T.: “With uniform strain hexahedron and equilateral withorthogonal hourglass control”, International Newspaper for Numerical Methods andEngineering, vol. 17,679-706, 1981.

33 RIKS E.: “Incremental year approach to the solution of snapping and buckling problems”,International Newspaper of Solids and Structures, vol. 15,524-551, 1979.


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