multifibre element of beam (right)

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Titre : Élément de poutre multifibre (droite) Date : 13/05/2020 Page : 1/28Responsable : FLÉJOU Jean-Luc Clé : R3.08.08 Révision :

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Element of beam multifibre (right-hand side) POU_D_EM

Summary :

This document presents the elements of multifibre beam based on a resolution of a problem of beam for whicheach section of a beam is divided into several fibres. Each fibre behaves then like a beam of Euler. Severalmaterials can be affected on only one support finite element (SEG2) what avoids having to duplicate the meshs(steel + concrete, for example).

The beams are right (element POU_D_EM). The section can be of an unspecified form, described by a “fibregrid”, to see [U4.26.01].The assumptions selected are the following ones:• assumption of Euler: transverse shearing is neglected (this assumption is checked for strong twinges),• the elements of multifibre beam take into account the effects of thermal dilation, drying and the hydration

(terms of the second member) and in a simplified way torsion. The effort-normal coupling inflection istreated naturally, by integration in the section of the uniaxial answers of the models of behavior associatedwith each group with fibres. An enrichment of the axial deformation, solved by local condensation in thecase of nonlinear behaviors, allows digital good performances, whatever the evolution in the section of thecentre of gravity matériau of the section.

Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in partand is provided as a convenience.Copyright 2021 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 Element of theory of the beams (recalls) .............................................................................................. 6

3 Equations of the movement of the beams ............................................................................................ 6

4 Element of right beam multifibre ........................................................................................................... 8

4.1 Element beam of reference ............................................................................................................ 8

4.2 Determination of the matrix of rigidity of the multifibre element ...................................................... 9

4.2.1 Case general (beam of Euler) ............................................................................................... 9

4.2.2 Case of the multifibre beam ................................................................................................ 10

4.2.3 Discretization of the fibre section – Calculation of ............................................................... 12

4.2.4 Integration in the linear elastic case (RIGI_MECA) ............................................................. 13

4.2.5 Integration in the non-linear case (RIGI_MECA_TANG) ..................................................... 13

4.3 Determination of the matrix of mass of the multifibre element ..................................................... 14

4.3.1 Determination of .................................................................................................................. 14

4.3.2 Discretization of the fibre section - Calculation of ............................................................... 15

4.4 Calculation of the internal forces .................................................................................................. 16

4.5 Formulation enriched in deformation ............................................................................................ 17

4.5.1 Method of the incompatible modes ..................................................................................... 17

4.5.2 Digital establishment ........................................................................................................... 18

4.5.3 Taking into account of offsetting .......................................................................................... 19

4.6 Nonlinear models of behavior usable ........................................................................................... 20

5 Element skeleton of assembly ............................................................................................................ 21

5.1 Kinematics of the finite element ................................................................................................... 22

5.2 Determination of the matrix of rigidity ........................................................................................... 24

5.3 Internal stress analysis of the element ......................................................................................... 25

6 Case of application ............................................................................................................................. 28

7 Bibliography ........................................................................................................................................ 28




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Notations

One gives the correspondence between the notations of this document and those of the documentationof use.

DX ,DY ,DZ and DRX ,DRY ,DRZ are in fact the names of the degrees of freedom associated withthe components with displacement u ,v ,w ,θx , θy , θz .

E Young modulus E

ν Poisson's ratio NU

G module of Coulomb = E

2.(1+ν)G

I y , I z geometrical moments of inflection compared to the axes y , z IY , IZ

J x constant of torsion JX

K matrix of rigidity

M matrix of mass

M x ,M y ,Mz moments around the axes x , y , z MT ,MFY ,MFZ

N normal effort with the section N

S surface of the section A

u ,v ,w translations on the axes x , y , z DX ,DY ,DZ

V y ,V z efforts cutting-edges along the axes y ,z VY ,VZ

ρ density ρ

θx ,θy ,θz rotations around the axes x , y , z DRX ,DRY ,DRZ

qx , qy , qz external linear efforts




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1 IntroductionThe analysis of the structures subjected to a dynamic loading requires models of behavior able torepresent non-linearities of material.

Many analytical models were proposed. They can be classified according to two groups:• detailed models founded on the mechanics of the solid and their description of the local behaviorof the material (microscopic approach)• models based on a total modeling of the behavior (macroscopic approach).

In the first type of models, we can find the models classical with the finite elements as well as “thefibre” models type (having an element of type beam how support).

While the “classical” models with the finite elements are powerful tools for the simulation of thenonlinear behavior of the complex parts of the structures (joined, assemblies,…), their applicationto the totality of a structure can prove not very practical because of a prohibitory computing time or sizememory necessary to the realization of this calculation. On the other hand, a modeling of typemultifibre beam [Figure 1-a], has the advantages of the simplifying assumptions of a kinematics of typebeam of Euler - Bernoulli while offering a practical solution and effective for a nonlinear analysiscomplexes composite elements of structures such as those which one can meet for example out ofreinforced concrete.

Moreover, this “intermediate” modeling is relatively robust and inexpensive in time calculation becauseof use of nonlinear models of behavior 1D.

Figure 1-a : Beam reinforced concrete with frameworks and reinforcements.




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Figure 1-b : Modeling of a beam reinforced concrete by a multifibre beam.

Figure 1-c : Support finite element of a modeling of type multifibre beam




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2 Element of theory of the beams (recalls)One takes again here the elements developed within the framework of the elements of beam of Euler(see [R3.08.01]).

A beam is a solid generated by a surface of surface S of which the geometrical centre of inertia Gfollowed a curve C called the average fibre or neutral fibre. The surface S is the cross-section (crosssection) or profile, and it is supposed that if it is evolutionary, its evolutions (size, form) are continuousand progressive when G described the average line.

For the study of the beams in general, one makes the following assumptions:• the cross-section of the beam is indeformable,• transverse displacement is uniform on the cross-section.

These assumptions make it possible to express displacements of an unspecified point of the section,according to displacements of the point corresponding located on the average line, and according to anincrease in displacement due to the rotation of the section around the transverse axes.

The discretization in “exact” elements of beam is carried out on a linear element with two nodes and sixdegrees of freedom by nodes. These degrees of freedom are the three translations u ,v ,w and threerotations θx , θy ,θz [Figure 2-a].

u1

v 1

w1

θ x1

θ y1

θ z1

}

{u 2

v 2

w 2

θx 2

θy 2

θz2

FigurE 2-a : Element beam.

Waited until the deformations are local, it is built in each top of the grid a local base depending on theelement on which one works. The continuity of the fields of displacements is ensured by a basicchange, bringing back the data in the total base.

In the case of the right beams, one traditionally places the average line on axis X of the local base,transverse displacements being thus carried out in the plan (y , z ) .

Finally when we arrange sizes related to the degrees of freedom of an element in a vector or anelementary matrix (thus of dimension 12 or 122 ), one arranges initially the variables for the top 1then those of the top 2 . For each node, one stores initially the sizes related to the three translations,then those related to three rotations. For example, a vector displacement will be structured in thefollowing way:

u1 ,v 1 ,w1, θx 1, θy 1

, θz1⏟sommet 1

, u2 , v 2 ,w2 , θx2, θy 2

, θz2⏟sommet 2

3 Equations of the movement of the beamsWe will not include in this document all the equations of the movement of the beams. For morecomplements concerning this part one can refer to documentation concerning the elements POU_D_Eand POU_D_T .




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4 Element of right beam multifibreOne describes in this chapter obtaining the elementary matrices of rigidity and mass for the element ofright beam multifibre, according to the model of Euler. The matrices of rigidity are calculated with theoptions ‘RIGI_MECA’ or ‘RIGI_MECA_TANG’, and matrices of mass with the option ‘MASS_MECA’for the coherent matrix, and the option ‘MASS_MECA_DIAG' for the matrix of diagonalized mass.

We present here a generalization [feeding-bottle 3] where the reference axis chosen for the beam isindependent of any geometrical consideration, inertial or mechanical. The element functions for anunspecified section (heterogeneous is without symmetry) and is thus adapted to a nonlinear evolutionof the behavior of fibres.

One also describes the calculation of the nodal forces for the nonlinear algorithms: ‘FORC_NODA’ and‘RAPH_MECA’.

4.1 Element beam of reference[the Figure 4.1-a] the change of variable realized to pass from the real finite element [Figure showsus 2-a] with the finite element of reference.

Figure 4.1-a : Element of reference vs real Élément.

One will then consider the continuous field of displacements in any point of the average line comparedto the field of displacements discretized in the following way:

U s=[N ]. {U} [éq 4.1-1]

The index s indicate the quantities attached to average fibre.




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By using the functions of form of the element of reference, the discretization of the variablesus ( x ) , vs ( x ) ,w s ( x ) ,θ sx ( x ) ,θsy (x ) , θsz ( x ) becomes:

(us ( x )

vs ( x )

ws ( x )

θ sx ( x )

θ sy ( x )

θ sz (x ))=(

N1

0 0 0 0 0 N2

0 0 0 0 0

0 N 3 0 0 0 N 4 0 N 5 0 0 0 N 6

0 0 N 3 0 −N 4 0 0 0 N 5 0 −N 6 0

0 0 0 N 1 0 0 0 0 0 N 2 0 0

0 0 −N 3, x 0 N 4, x 0 0 0 −N 5, x 0 N 6, x 0

0 N 3, x 0 0 0 N 4, x 0 N 5, x 0 0 0 N 6, x

)⋅(u1v1

w1θ x 1θy 1θ z 1u2

v2w2

θ x 2θ y 2

θ z 2

)

[éq 4.1-2]

With the following functions of interpolation, and their derivative useful:

N1=1− xL

; N 1,x=−1L

N 2=xL

; N2, x=1L

N 3=1−3 x 2

L2+2 x3

L3; N3, xx=−

6

L2+12 x

L3

N 4=x−2 x2

L+

x 3

L2; N 4, xx=−

4L+6 x

L2

N 5=3x2

L2−2

x3

L3; N5,xx=

6

L2−12

x

L3

N 6=−x 2

L+

x 3

L2; N 6, xx=−

2L+6

x

L2

[éq 4.1-3]

4.2 Determination of the matrix of rigidity of the multifibre element

4.2.1 Case general (beam of Euler)

Let us consider a beam Euler, line, directed in the direction x , subjected to distributed effortsqx , qy , qz [Figure 4.2.1-a].

Figure 4.2.1-a : Beam of Euler 3D.




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The fields of displacements and deformations take the following shape then when one writes thedisplacement of an unspecified point of the section according to displacement (U s) and of rotationθs line of average:

u ( x , y , z)=us(x)− yθsz (x)+ z θ sy( x) [éq 4.2.1-1]v (x ,y , z )=v s(x ) [éq 4.2.1-2]w (x ,y , z )=w s (x ) [éq 4.2.1-3]ε xx=ux

' ( x)− yθ sz' ( x)+ z θsy

' ( x) [éq 4.2.1-4]εxy=εxz=0 [éq 4.2.1-5]

Note:Torsion is treated overall by admitting an elastic assumption, except for, one does not calculate ε yz

here. f ’ (x ) indicate the derivative of f (x ) compared to x .

By introducing the equations [éq 4.2.1-4] and [éq 4.2.1-5] in the principle of virtual work one obtains:

∫V0σxx . δε xx dV 0=∫0

L

(δus (x ) q x+δvs ( x ) q y+δws ( x ) qz ) dx [éq 4.2.1-6]

qx ,q y ,qz indicating the linear efforts applied. What gives by using the equation [éq 4.2.1-1]:

∫0

L

(N δus' ( x )+M x δθ sx

' ( x )+M y δθsy' (x )+M z δθ sz

' (x ) )dx

=∫0

L

(qx δus ( x )+q y δv s (x )+qz δw s (x ))dx[éq 4.2.1-7]

with:

N=∫SσxxdS ; M y=∫ S

zσxx dS ; M z=∫S− yσxxdS [éq 4.2.1-8]

Note:Torque M x is not calculated by integration but is not calculated directly starting from the stiffness intorsion (see [éq 4.2.2-4 ]). The theory of the beams associated with an elastic material gives: σ xx=E ε xx

4.2.2 Case of the multifibre beam

We suppose now that the section s is not homogeneous, materials with different mechanicalcharacteristics.Without adopting particular assumption on the intersection of the axis x with the section s or on theorientation of the axes Y ,Z , the relation between the “generalized” constraints and deformations“generalized” Ds becomes [bib2]:

Fs=Ks⋅Ds [éq 4.2.2-1]with:

Fs=(N , M y , M z , M x )T

Ds=(u s' ( x ) , θ sy

' (x ) , θ sz' ( x ) , θ sx

' (x ))T [éq 4.2.2-2]

The matrix K s can then put itself in the following form:

Ks=(K s11 K s12 K s13 0

K s 22 K s 23 0

K s33 0

sym K s44) [éq 4.2.2-3]




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with:

K s11=∫SEdS ; K s12=∫S

Ezds ; K s13=−∫SEyds

K s 22=∫SEz2dS ; K s23=−∫S

Eyzds ; K s33=∫SEy2ds

[éq 4.2.2-4]

where E can vary according to y and z . Indeed, it may be that in modeling section planes, severalmaterials cohabit. For example, in a concrete section reinforced, there are at the same time concreteand reinforcements.The discretization of the fibre section makes it possible to calculate the integrals of the equations[éq 4.2.2-4]. The calculation of the coefficients of the matrix Ks is detailed in the paragraph [§4.2.3]according to.

Note:The term of torsion K s44=GJ x is given by the user using the data of J x , using the orderAFFE_CARA_ELEM .

The introduction of the equations [éq 4.2.1-1] with [éq 4.2.2-4] in the principle of virtual work leads to:

∫0

LδDs

T . K s .D s dx−∫0

L

(δus ( x ) qx+δv s (x ) q y+δw s ( x ) qz ) dx=0 [éq 4.2.2-5]

The generalized deformations are calculated by ( DS is given to the equation [éq 4.2.2-2]):

Ds=B {U } [éq 4.2.2-6]

With the matrix B following:

B=⌊N 1, x 0 0 0 0 0 N 2, x 0 0 0 0 0

0 0 −N 3, xx 0 N 4, xx 0 0 0 −N 5, xx 0 N 6, xx 0

0 N 3, xx 0 0 0 N 4, xx 0 N 5, xx 0 0 0 N 6, xx

0 0 0 N 1, x 0 0 0 0 0 N 2, x 0 0⌋

[éq 4.2.2-7]

Discretization of space [ 0,L ] with elements and the use of the equations [éq 4.2.2-5] the equation [éqreturns 4.2.1-6] equivalent to the resolution of a classical linear system:

K .U=F [éq 4.2.2-8]

The matrix of rigidity of the element [Figure4.2.2-a] and the vector of the efforts results are finally givenby:

Kelem=∫0

LBT .K s .B dx

F=∫0LNT .Q dx

[éq 4.2.2-9]




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Figure 4.2.2-a : Multifibre beam – Calculation of Kelem

With the vector Q who depends on the external loading: Q=(q x q y qz 0 0 0)T

.

If we consider that the distributed efforts qx ,q y , qz are constant, we obtain the vector nodal forcesaccording to:

F=( Lqx2

Lqy2

Lq z2

0 −L2q z12

L2q y12

Lqx2

Lqy2

Lq z2

0L2 qz12

L2 q y12 )

T

[éq 4.2.2-10]

4.2.3 Discretization of the fibre section – Calculation of K s

The discretization of the fibre section makes it possible to calculate the various integrals whichintervene in the matrix of rigidity, and the other terms necessary.

Geometry of fibres gathered in groups of fibres, via the operator DEFI_GEOM_FIBRE [U4.26.01])contains in particular the characteristics (Y, Z, SURFACE) for each fibre. One can envisage with morethe 10 groups of maximum fibres by element beam.

Thus, if we have a section which comprises n fibres we will have the following approximations of theintegrals:

K s11=∑i=1

n

E i S i ; K s12=∑i=1

n

E i z i S i ; K s13=∑i=1

n

E i y iS i

K s 22=∑i=1

n

E i zi2Si ; K s 23=−∑

i=1

n

E i yi zi S i ; K s33=∑i=1

n

E i yi2S i

[éq 4.2.3-1]

with E i the initial or tangent module and S i the section of each fibre. The state of stress is constantby fibre.Each fibre is also located using yi and z i coordinates of the centre of gravity of fibre compared tothe axis of the section defined by the keyword ‘COOR_AXE_POUTRE' (see the orderDEFI_GEOM_FIBRE [U4.26.01]).The classification of fibres depends on the choice of the keyword ‘FIBRE' or ‘SECTION' (see theorder DEFI_GEOM_FIBRE [U4.26.01]).




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4.2.4 Integration in the linear elastic case (RIGI_MECA)

When the behavior of material is linear, if the element beam is homogeneous in its length, theintegration of the equation [éq 4.2.2-9] can be made analytically.One obtains the matrix of following rigidity then:

Kelem=(Ks11L

0 0 0

Ks12L

Ks13L

−Ks 11L

0 0 0

−Ks12L

−Ks13L

12Ks33

L3

−12Ks23

L3

0

6Ks23

L2

6Ks33

L2

0

−12 Ks 33

L3

12 Ks 23

L3

0

6 Ks 23

L2

6 Ks33

L2

12 Ks22

L3

0

−6Ks22

L2

−6Ks23

L2

0

12 Ks 23

L3

−12 Ks 22

L3

0

−6 Ks 22

L2

−6Ks23

L2

Ks44L

0 0 0 0 0 −

Ks 44L

0 0

4Ks22L

4Ks23L

−Ks12L

−6 Ks 23

L2

6 Ks 22

L2

0

2Ks 22L

2Ks23L

4Ks33L

−Ks 13L

−6Ks33

L2

6Ks 23

L2

0

2Ks 23L

2Ks33L

Ks11L

0 0 0

Ks 12L

Ks13L

12 Ks33

L3

−12 Ks23

L3

0

−6Ks 23

L2

−6Ks33

L2

SYM

12 Ks 22

L3

0

6Ks 22

L2

6Ks 23

L2

Ks44L

0 0

4Ks 22L

4 Ks 23L

4Ks33L

)

[éq 4.2.4-1]

with the following terms K s11 , K s12 , K s13 , K s22 , K s33 , K s23 , K s44 given to the equation [éq 4.2.2-4].

Note:The matrix of rigidity presented above does not take into account a possible eccentricity of thereference axis compared to the elastic center, not to weigh down the presentation. However theadditional terms are well taken into account in the programming (see §4.5.3).

4.2.5 Integration in the non-linear case (RIGI_MECA_TANG)

When the behavior of material is nonlinear, to allow a correct integration of the internal efforts (seeparagraph [§4.4]), it is necessary to have at least two points of integration along the beam. We choseto use two points of Gauss.

The integral of Kelem [éq 4.2.2-9] is calculated under digital form:

Kelem=∫0LBT .Κ s .B dx= j∑

i=1

2

wiB (xi )T .K s (xi ) . B ( xi) [éq 4.2.5-1]

• where x i is the position of the point of Gauss i in an element of reference length 1, i.e.:(1±0,57735026918963)/2 ;

• wi is the weight of the point of Gauss i . One takes here wi=0,5 for each of the 2 points; j isJacobien. One takes here j=L , the real element having a length L and the function of form to

pass to the element of reference being xL

.




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K s is calculated using the equations [éq 4.2.1-4], [éq4.2.2-4] (see paragraph [§4.2.3] for the digital

integration of these equations).

The analytical calculation of B ( xi )T .K s ( xi ) . B ( x i) give:

(B12Ks 11

−B1B2Ks 13

B1B2Ks 12

0 − B1B3Ks 12

−B1B3Ks13

−B12Ks11

B1B2Ks13

−B1B2Ks 12

0 −B1B4Ks 12

−B1B4Ks 13

B22Ks 33

B22Ks 23

0 B2B3Ks 23

B2B3Ks33

B1B

2Ks 13

−B22Ks 33

B22Ks 23

0 B2B4Ks 23

B2B4Ks 33

B22Ks 22

0 −B2B3Ks 22

−B2B3Ks 23

−B1B2Ks 12

B22Ks 23

− B22Ks 22

0 − B2B4Ks 22

− B2B4Ks 23

B12 Ks 44

0 0 0 0 0 −B12 Ks 44

0 0

B32Ks22

B32 Ks 23

B1B3Ks 12

− B2B3Ks23

B2B3Ks 22

0 B3B4Ks 22

B3B4Ks23

B32 Ks 33

B1B3Ks 13

− B2B3Ks33

B2B3Ks 23

0 B3B4Ks 23

B3B4Ks33

B12Ks11

−B1B2Ks13

B1B2Ks12

0 B1B4Ks12

B1B4Ks13

B22 Ks33

− B22Ks 23

0 − B2B4Ks 23

− B2B4Ks 33

B22 Ks 22

0 B2B4Ks 22

B2B4Ks 23

B12 Ks 44

0 0

B42 Ks 22

B42 Ks 23

B42 Ks 33

) [éq 4.2.5-2]

where them B i are calculated with the X-coordinate x i element of reference with:

B1=−N 1, x=N 2, x=1L

B2=−N 3, xx=N 5, xx=−6

L2+

12 xiL2

B3=N 4, xx=−4L+

6 xiL

B4=N 6, xx=−2L+

6 xiL

[éq 4.2.5-3]

4.3 Determination of the matrix of mass of the multifibre element

4.3.1 Determination of Melem

In the same way, the virtual work of the efforts of inertia becomes [bib2]:

W inert=∫0L∫S ρ(δu (x , y )

d 2u (x , y )

dt2+δv ( x , y )

d2 v ( x , y )

dt2+δw (x , y )

d 2w (x , y )

dt2 ) dS dx=∫0

LδUs .Ms .

d 2Us

dt2dx

[éq 4.3.1-1]

with Us the vector of “generalized” displacements.What gives for the matrix of mass:

Ms=(M s11 0 0 0 M s12 M s13

M s11 0 −M s12 0 0

M s11 −M s13 0 0

M s22+M s33 0 0

M s22 M s23

sym M s33

) [éq 4.3.1-2]




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with:

M s11=∫Sρds ; M s12=∫S

ρ zds ; M s13=−∫Sρ yds

M s22=∫Sρ z2ds ; M s 23=−∫S

ρ yzds ; M s33=∫Sρ y2ds

[éq 4.3.1-3]

with ρ who can vary according to y and z .

As for the matrix of rigidity, we take into account the generalized deformations and the discretization ofspace [0,L ] . What gives finally for the elementary matrix of mass of dimension 12x12 :

M elem1 =[ LM s11

3

−M s13

2

M s12

20LM s12

12

LM s13

12

LM s11

6

M s13

2

−M s12

20−L M s12

12

−L M s13

12 ]

M elem2 =[ sym13 LM s 11

35+

6M s33

5 L

−6M s 23

5 L

−7 LM s12

20

M s23

10

11 L2M s11

210+M s 33

10

−M s13

2

9 LM s11

70−

6M s33

5 L

6M s23

5 L

−3LM s12

20

M s23

10

−13 L2M s11

420+M s33

10 ]

M elem3 =[ symsym13 LM s11

35+

6M s22

5 L

−7L M s13

20

−11 L2M s11

210−M s22

10

−M s 23

10

M s12

2

6M s23

5L

9LM s11

70−

6M s22

5 L

−3LM s13

20

13L2 M s11

420−M s22

10

−M s23

10 ]

M elem4 =[ symsymsym LM s22+LM s33

3

L2 M s13

20

−L2 M s12

200−3LMs 12

20

−3LM s13

20

LM s22+ LM s33

6

−L2M s13

30

L 2M s 12

30 ]

M elem5 =[ symsymsymsym L3M s 11

105+

2 LM s 22

15

2LM s23

15

−LM s 12

12

−M s23

10

−13 L2M s11

420+M s22

10

L 2M s13

30

−L3M s 11

140−LM s22

30

−LM s 23

30 ]

M elem6 =[ symsymsymsymsym L3M s11

105+

2 LM s 33

15

−LM s13

12

13L2 M s11

420−M s 33

10

M s23

10

−L 2M s 12

30

−LM s23

30

−L3 M s11

140−LM s33

30 ]

M elem7 =[ sym symsymsymsym sym LM s11

3

M s13

2

−M s 12

20LM s12

12

LM s13

12 ]

M elem8 =[ symsymsymsymsym symsym13LM s11

35+

6M s33

5 L

−6M s23

5L

−7 LM s12

20

−M s23

10

−11 L2M s11

210−M s33

10 ]

M elem9 =[ symsymsymsymsym symsymsym13L M s11

35+

6M s 22

5 L

−7 LM s13

20

11L2 M s11

210+M s22

10

M s23

10 ]

M elem10 =[ symsymsymsymsym symsymsymsym LM s22+LM s33

3

−L2M s13

20

L 2M s 12

20 ]

M elem11 =[ symsymsymsymsym symsym symsym sym L

3M s11

105+

2LM s22

15

2 LM s 23

15 ]

M elem12 =[ symsymsymsymsym symsymsymsym symsym L

3 M s11

105+

2 LM s33

15 ]

with the following terms: M s 11 ,M s12 ,M s13 ,M s 22 ,M s33 ,M s 23 who are given to the equation [éq 4.3.1-3].

Note:The matrix of diagonal mass is reduced by the technique of the concentrated masses[R3.08.01] . This matrix of diagonal mass is obtained by the option ‘MASS_MECA_DIAG’ of theoperator CALC_MATR_ELEM [U4.61.01].

The additional terms in the event of eccentricity of the axes are not presented here but are well takeninto account (see §4.5.3).

4.3.2 Discretization of the fibre section - Calculation of M s

The discretization of the fibre section makes it possible to calculate the various integrals whichintervene in the matrix of mass. Thus, if we have a section which comprises n fibres we will have thefollowing approximations of the integrals:

M s11=∑i=1

n

ρi S i ; M s12=∑i=1

n

ρi z i S i ; M s13=−∑i=1

n

ρi y iS i

M s22=∑i=1

n

ρ i zi2Si ; M s 23=−∑

i=1

n

ρi yi zi S i ; M s33=∑i=1

n

ρ i yi2S i

[éq 4.3.2-1]

with ρi and S i density and the section of each fibre. yi and z i are the coordinates of the centre ofgravity of fibre defined as previously.




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4.4 Calculation of the internal forcesThe calculation of the nodal forces Fint had in a state of internal stresses given is done by the integral:

F int=∫0

LBT .F s dx [éq 4.4-1]

where B is the matrix giving the generalized deformations according to nodal displacements[éq 4.2.2-6] and where Fs is the vector of the generalized constraints given to the equation [éq 4.2.2-2],

Figure 4.4-a : Multifibre beam – Calculation of Fint

FsT=(N M y M z M x ) [éq 4.4-2 ]

Normal effort NR and bending moments M y and M z are calculated by integration of the constraintson the section [éq 4.2.1-8].

Behaviour in torsion being supposed to remain linear, the torque is calculated with nodal axial rotations:

M x=GJ xθ x 2−θ x 1

L [éq 4.4-3 ]

The equation [éq 4.1-1] is integrated numerically:

Fi=∫0LBT .Fs dx= j∑

i=1

2

wiB ( xi )T .Fs ( xi ) [éq 4.4-4 ]

The positions and weights of the points of Gauss as well as Jacobien are given in the paragraph[§4.2.5].

The analytical calculation of B ( xi )T .F s (xi ) give:

[B ( xi )T . F s ( xi ) ]

T= [−B1N B2M z −B2M y 0 B3M y B3M z

B1 N −B2M z B2M y 0 B4M y B4M z ][éq 4.4-5]

where them B i are given to the equation [éq 4.2.4-1].




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4.5 Formulation enriched in deformationWith the interpolations of displacements of the equation [éq 4.1-1], the axial generalized deformation isconstant and the curves are linear (see equations [éq 4.2.2-6], [éq 4.2.2-7] and [éq 4.2.5-3]):

{εs( x )=

u 2−u1

L

χ ys( x)=−(− 6

L2 +12 x

L3 )w1+(6x

L2−4L )θ y1−(−

12 x

L3 +6

L2 )w2+(6x

L2−4L )θ y2

χzs( x )=(− 6

L2+

12 x

L3 )v1+(6x

L2−

4L )θ z1+(−

12 x

L3+

6

L2 )v2+( 6x

L2−

4L )θ z2

[éq 4.5-1 ]

If it there has not coupling between these two deformations (elastic case, with the average line ofreference which passes by the barycentre of the section), that does not pose problems. But in thenonlinear case general, there is a shift of the neutral axis, and the terms K s 12 and K s 13 of K s

(equations [éq 4.2.1-4] and [éq 4.2.2-4]) are not worthless, there is coupling between the moments andthe normal effort. There is then an incompatibility in the approximation of the axial deformations of afibre:

ε=εs(x )− y χ

zs( x )+z χ

ys( x) [éq 4.5-2 ]

A means of eliminating this incompatibility is to enrich the field by axial deformation:

ε s( x) ↦ ε s( x)+ε̃s (x ) ; ε̃s( x )=α .G( x) ; G (x)=4L−

8 x

L2 [éq 4.5-3 ]

for x∈[−L2 , L2 ]

where G (x ) is an enriched deformation which derives from a function “bubble” in displacement andα the degree of freedom of enrichment. The variational base of such an enrichment is provided by theprinciple of Hu-Washizu [bib5] which can be presented same manner as the method of theincompatible modes [bib6].

4.5.1 Method of the incompatible modes

The regular field of generalized displacements Us is defined by the equation [éq 4.1-1]. Generalizeddeformations Ds and generalized constraints Fs by the equation [éq 4.2.2-2].

The principle of Hu-Washizu consists in writing the weak form of the equilibrium equations, but also ofthe calculation of the deformations and the law of behavior, in projection on the three virtual fields(generalized displacements Us

* , generalized deformations Ds* and generalized constraints Fs

* ):

{∫

0

L dU s*

dx⋅F s dx−W ext=0

∫0

L

F s*⋅( d U s

dx−Ds )dx=0

∫0

L

Ds*⋅(F s−K s⋅D s )dx=0

[éq 4.5.1-1 ]

One introduces the enrichment of the real deformations, and one chooses to break up the virtual fieldof deformations into a “regular” part exit of the virtual field of displacements and an enriched part:

Ds=dU s

dx+ D̄s Ds

*=dU s

*

dx+ D̄s

* [éq 4.5.1-2]

One defers [éq 4.5.1-2has] in [éq 4.5.1-1B], which justifies “enrichment” by orthogonality:




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∫0

LFs

*.Ds dx=0 [éq 4.5.1-3]

The equation [éq 4.5.1-1C] breaks up into two since one has two independent virtual fields in [éq 4.5.1-2B]:

∫0

L dU s*

dx. (F s−K s .Ds ) dx=0 ∫0

LDs

*⋅(Fs−K s⋅D s) dx=0 [éq 4.5.1-4]

Lastly, the method of the incompatible modes consists in choosing the orthogonal space of theconstraints to the space of the enriched deformations, so that [éq 4.5.1-3] is automatically checked and[éq 4.5.1-4B] thus gives simply:

∫0

LD̄s

* .K s .Ds dx=0 [éq 4.5.1-5]

If one returns to the strong formulation of the law of behavior in [éq 4.5.1-4has] and [éq 4.5.1-5], thesystem [éq 4.5.1-1] becomes:

∫0

L dU s*

dx.F s dx−W ext=0 ∫0

LDs

*⋅Fs dx=0 Fs=K s⋅D s [éq 4.5.1-6]

Note:Here one enriches only the axial deformation by an element of beam of Euler-Bernoulli, with a

continuous function, therefore D̄=(ε̄s 0 0 0)T

.

4.5.2 Digital establishment

From the finite elements point of view, one can write displacements and the deformations in matricform, with the enriched part:

Bs=N .(U)+Q .(α) D=B .(U)+G .(α) [éq 4.5.2-1]where N and B are the classical matrices of the functions of interpolation and their derivative (see[éq 4.1-1] and [éq 4.2.2-7]) and:

Q=(4 xL−

4 x2

L20 0 0)

T

and G=( 4L−

8 x

L20 0 0)

T

[éq 4.5.2-2]

Note: G was selected so that the element always passes the “patch test” (worthless deformation energy

for a movement of solid) : ∫0LG ( x )dx = 0 [éq 4.5.2-3]

After classical handling of passage of continuous to discrete, the system of equations [éq 4.5.1-6],written for the whole of the structure, approximates itself by:

{ Ae=1N elem (F int−F ext )=0

he=0 ∀ e∈[1, N elem ] [éq 4.5.2-4]

with:

{Fint=∫0

LBT .Fs dx=∫0

LBT .K s . (B .U s+G .α ) dx

Fext=∫0

LNT .f dx

he=∫0

LGT .Fs dx

[éq 4.5.2-5]

Ae=1

Nelem indicate the assembly on all the elements of the grid; f is the axial loading distributed on the

element beam. The system of equations [éq 4.5.2-4] is nonlinear, it is solved in an iterative way (seeSTAT_NON_LINE).




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With the iteration (i+1) , with ΔU( i )=U( i+1)−U( i ) and Δα( i )=α( i+1 )

−α( i ) , the linearization of thesystem gives (iterations of correction of Newton):

{Ae=1

N elem((Fint( i+1)−Fext

(i+1))+Ke(i ). ΔU(i )+Xe

(i )Δα(i ))=0

he(i+1)+Xe

(i )T . ΔU(i )+H e(i )Δα(i )=0 ∀ e∈[1,. .. , N elem ]

[éq 4.5.2-6]

with:

{K e( i )=∫L B

T .K s( i ).B dx

Xe( i )=∫L B

T .K s( i ).G dx

H e( i )=∫L G

T .K s( i).G dx

[éq 4.5.2-7]

The second equation of the system [éq 4.5.2-6] is local. It makes it possible to calculate the degree offreedom of enrichment α independently on each element. One calculates it by a local iterativemethod (iterations ( j) for a displacement d( i )=ΔU(i ) fixed):

α( j+1)(i )

=α( j )( i )−(H e( j )

( i ) )−1he ( j )( i )

[éq 4.5.2-8]

Thus, when one converged at the local level, one a:

he (d( i) , α( i))=0 [éq 4.5.2-9]

And one can operate a static condensation to eliminate α at the total level.

Ke( i)=Ke

( i )−Xe

( i )(H e( i ))

−1Xe( i )T [éq 4.5.2-10]

From a practical point of view, this technique makes it possible to treat enrichment at the elementarylevel without disturbing the number of total degrees of freedom. It is established with the level of theelementary routine charged to calculate the options FULL_MECA, RAPH_MECA and RIGI_MECA_TANG.

Note:• in the typical case exposed here, H e

(i) is a reality, therefore very easy to reverse.• in the same way, he and α are also realities.

• The calculation of Ke( i ) is explained in the paragraph §4.2.5, the other sizes of the equation

[éq 4.5.2-8] are calculated according to the same technique.• In the same way the calculation of Fint is explained in the paragraph § 4.4, he in the equation

[éq 4.5.2-5] is calculated according to the same technique.

4.5.3 Taking into account of offsetting

For an elastic behavior, the enrichment of the axial deformations makes it possible to take accountcorrectly coupling between the normal effort and the bending moments, and to return the answer of thebeam independent of the position chosen for the reference axis (see keyword COOR_AXE_POUTRE inthe operator DEFI_GEOM_FIBRE, [U4.26.01]). It thus makes it possible to treat the case of the offsetbeams.

The digital establishment of the enrichment of the axial deformations presented in paragraph 4.5.2relates to nonlinear calculations with STAT_NON_LINE or DYNA_NON_LINE, for the optionsFULL_MECA, RAPH_MECA and RIGI_MECA_TANG. Indeed, determination of α is done by localiterations at the elementary level [éq 4.5.2-8].




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In the case of calculations with the option RIGI_MECA (MECA_STATIQUE, calculation of modes,calculations nonlinear with prediction ‘ELASTIC’), one can determine α explicitly according tooffsettings of the elastic center compared to the reference axis:

e y=∫S

Ey dS

∫SE dS

=K S13

K S11

and e z=∫S

Ez dS

∫SE dS

=K S12

K S11

One can then treat static condensation to eliminate α on the level of the elementary matrix of rigidity.These calculations were done analytically and it matrix is established explicitly in Code_Aster(modification of about twenty terms if e y and/or e z are not worthless).

Since displacement is enriched, the matrix of mass (see §4.3) is modified. As for the matrix of rigidity,the terms modified if e y and/or e z are not worthless, were calculated analytically and programmedexplicitly. The enrichment of the deformations also modifies the calculation of the options DEGE_ELNO [éq 4.5.2-1] and EPSI_ELGA [éq 4.5-2].

4.6 Nonlinear models of behavior usableThe supported models are on the one hand the relations of behavior 1D of type VMIS_ISOT_LINE,VMIS_CINE_LINE, VMIS_ISOT_TRAC, CORR_ACIER and PINTO_MENEGOTTO [R5.03.09] forsteels, in addition the model MAZARS_GC [R7.01.08] dedicated to the uniaxial behavior of theconcrete into cyclic. One can thus have several materials by multifibre element of beam.

In addition, if the behavior used is not available in 1D, one can use the other laws 3D using the methodof R.De Borst [R5.03.09]). For example, one can treat: GRAN_IRRA_LOG, VISC_IRRA_LOG.However in this case, one can treat one material by multifibre element of beam.

Note:The internal, constant variables by fibre, are stored in the under-points attached to the point ofintegration considered. The access to the postprocessing of the sizes defined in the under-points is done via formatMED3.0, of Salomé.




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5 Element skeleton of assembly

The element skeleton of assembly is a generalization of the element of beam multifibre presented tothe preceding section. This element consists of an element having kinetics of beam, but where batchesof fibres are gathered in bolsters who represent the guide tube. These bolsters are themselvesmultifibre, as shown with the figure 5-a. This element is only accessible within the framework fromkinematics ofS beams of Euler.

The assumptions selected are the following ones:• assumption of Euler: transverse shearing is neglected (this assumption is checked for strongtwinges);• the element is carrying the degrees of freedom of usual displacement and rotation the beams likethree additional degrees of freedom modelling the rotation of the grid binding the tubes guideskinematically;• rotations in the whole of the beams are all the same ones;• the element skeleton takes in account effects of thermal dilation, gravity and in a simplified waytorsion.

Figure 5-a : Example of a section multi-beamS constituted by 4 bolsters described by a

rectangular section. In the example the bolsters consist of 4 fibres.

Quantity Description In code_aster

E Young modulus E

ν Poisson's ratio NAKED

G module of Coulomb G= E2(1+ν)

G

I y I z geometrical moments of inflection compared to the axes y and z

IY IZ

J x constant of torsion JX

K matrix of rigidity

M matrix of mass

M x M y M z moments beams around the axes x , y and z MX MY MZ




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M g , x M g , y M g , z moments grids around the axes x , y and z MGX MGY MGZ

N normal effort with the section FX

S surface of the section With

u v w translations on axes x , y and z DX DY DZ

V y V z efforts cutting-edges according to axes y and z FY FZ

ρ density RHO

θx θy θz Rotations beam around axes x , y and z DRX DRY MARTINI DRZ

ωx ωy ωz Rotations grid around the axes x , y , z DRGX DRGY DRGZ

qx q y qz external linear efforts

5.1 Kinematics of the finite elementOne considers here an element of structure describing the skeleton of a fuel assembly made up byN p guide tubes here considered as three-dimensional beams. These guide tubes are kinematically

dependent enters them by the grids of the fuel assembly, here considered as a rigid body. From thesetwo assumptions, it is possible to describe all the kinematics of this assembly of beams using ninedegrees of freedom, as indicated on the Figure 5.1-a :

• three degrees of freedom representing the displacement of the grid: (ux , uy ,uz)

• three degrees of freedom describing rotations of the guide tubes: (θx ,θ y ,θz)

• tr hear degrees of freedom modelling the rotation of the grid binding the guide tubes:

(ωx ,ω y ,ω z)

Figure 5.1-a : Diagram of the combustible fastener

The position of a guide tube i compared to the center of the skeleton of the assembly is described

using a couple of realities (Y i , Z i) . By employing the degrees of freedom (three displacements and




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three rotations) usual of a beam of Euler in three dimensions and supposing that rotations in all thebeams are the same ones, the kinematics of the tube guides i is written:

uxi=u x (x )−Y i .ωz ( x )+Zi .ω y (x )

u yi=u y (x )−Zi

ωx ( x ) uz

i=uz (x )+Y i

ωx (x ) θx

i=θx (x)

θ yi=θ y (x)

θzi=θz (x)

(1)

D E more, in optics to use beams with a non-linear behavior, the section of the beams itself isdescribed using fibre S (numbered 1 with N f ) and the theory of the multifibre beams is here

employee e. Thus, for each beam i ( varying of 1 with N p ), the axial displacement of its fibre j isfunction of the average displacement of the section of the beam, to which one adds the displacementimposed by the rotation of the section. Since the theory of the beams imposes that the section of thebeams remains right, the field of axial displacement of fibre j of a guide tube i is written:

uxi, j=ux

i−( y i , j

−Y i ) .θz ( x )+ (z i , j−Zi ) .θ y ( x ) (2)

By injecting the first equation there (1):

uxi, j=ux (x )−Y i .ωz (x )+Zi .ωy (x )−( y i , j

−Y i ) .θz ( x )+(z i , j−Z i ) .θ y ( x ) (3)

Where quantity ( y i , j , z i , j) corresponds to the position of fibre j of a beam i compared to the center

of the assembly of beams. The axial field of deformation E fibre is determined thereafter by:

εxxi, j=

∂uxi , j

∂ x=ux

' (x )−Y i.ωz' (x )+Zi .ωy

' (x )−( yi , j−Y i) .θ z' (x )+( zi , j−Zi ) .θ y

' (x ) (4)

The principle is of bases itselfR on the beams multifibre existing while passing degrees of freedomelement skeleton of assembly towards i fields of “generalized displacements of beam multifibre” thecomponent, while using (1). While thus proceeding (i.e by treating all the beams individually), onedetermines a “field of local generalized displacement” U i with each beam i . One brings back oneselfthus, for each beam, with the determination of the fields using the six degrees of “classical” freedomalong the element.

Thus, one to determine the generalized deformations Di on each beam i with Di=B .U i (cf[R3.08.08]). Once that, one can:

• to determine the deformations uniaxial on fibres in the same way that for the beams multifibreof Euler;

• to obtain L be forced by the law of behavior; • to determine the efforts generalized on the beam i as for a beam multifibre of Euler starting

from the constraints;• to assemble the elementary matrix of the skeleton of assembly starting from the matrices of

each bolster;• finally it is necessary to build the generalized efforts of the fastener starting from the efforts

generalized on the beams i . IL will act to separate the moments caused by rotation from thebeams and those caused by rotation from the grid.

•

The implementation of this finite element wants to be like a encapsulation of the model of beam ofEuler multifibre.Ldiscretization has element skeleton be carried out on a linear element with two nodes and ninedegrees of freedom per node. These degrees of freedom are the three translations (u , v ,w) and

them six rotations relating to the beams (θx ,θ y ,θz) and with the grids (ωx ,ω y ,ω z) .




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{u1

v 1

w 1

θ x 1

θ y 1

θ z1

ωx1

ωy 1

ωz 1}

{u2

v 2

w 2

θx 2

θy 2

θz 2

ωx 2

ωy 2

ωz2}

FigurE 5.1-b : Element skeleton.

Waited until the deformations are local, it is built in each top of the grid a local base depending on theelement on which one works. The continuity of the fields of displacements is ensured by a basicchange, bringing back the data in the total base. In the case as of right beams, one traditionally placesthe average line on the axis x local base, transverse displacements being thus carried out in the plan

( y , z ) .

Finally when we arrange sizes related to the degrees of freedom of an element in a vector or anelementary matrix (thus of dimension 18 or 182 ), one arranges initially the variables for the top 1then those of the top 2 . For each node, one stores initially the sizes related to the three translations,then those related to six rotations. For example, a vector displacement will be structured in thefollowing way:

u 1 ,v 1 ,w 1 ,θx1,θy 1

,θz 1,ωx 1

,ωy 1,ωz1⏟

sommet 1

, u2 ,v 2 ,w 2 ,θx2,θy 2

,θz2,ωx2

,ωy 2,ωz2⏟

sommet 2

5.2 Determination of the matrix of rigidityThe matrix of rigidity of the element skeleton rests directly on an assembly of the elementary matricesof each beam of the beam of the assembly. One will thus refer to documentation related with theelement of right beam multifibre concerning the calculation of these matrices. The matrices of rigidityare calculated with the options ‘RIGI_MECA' or ‘RIGI_MECA_TANG'.

The element skeleton is carrying two nodes (subscripted 1 and 2) provided each of nine degrees offreedom describing the three fields (U ,θ ,ω) . The whole of these degrees of freedom come togenerate displacements/rotations on the bolsters multifibre who have for their part twelve degrees offreedom. For différenC1st them degrees of freedom element skeleton and of the bolsters, one employsrespectively majuscules/tiny. For displacements:

ux1=U x1

−ωz1Y i+ωy1

Zi and ux2

=U x2−ω z 2

Y i+ω y2

Zi

u y1=U y1

−ωx1Zi

and u y2=U y2

−ωx2Z i

uz 1=U z1

+ωx1Y i

and uz 2=U z2

+ωx2Y i

(5)

Pour rotations:θx1=Θx1 and θx2

=Θx2

θ y1=Θy1 and θ y2

=Θy2

θz 1=Θ z1 and θz 2

=Θ z2

(6)

If one notes K i the matrix of size 12x12 corresponding to the beam i , one can calculate for example

the normal effort f xi on the beam in this way:




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f xi=K 1,1

i .ux1+K1,2

i .u y1+K 1,3

i .uz1

K1,4i .θx1

+K1,5i .θ y1

+K1,6i .θz1

+

K1,7i .ux2

+K1,8i .u y2

+K1,9i .uz2

K 1,10i .θx2

+K1,11i .θ y2

+K1,12i .θz2

(7)

This effort is expressed according to degrees of freedom “local”, one can also express it according todegrees of freedom “total” i.e. those of the element skeleton:

f xi=K1,1

i . [U x1−ωz1

Y i+ω y1

Z i ]+K1,2

i . [U y1−ωx1

Z i ]+K1,3

i . [U z1+ωx1

Y i ]+K 1,4

i .θx1+K1,5

i .θ y1+K1,6

i .θz1+

K1,7i . [U x2

−ωz2Y i+ωy2

Z i ]+K1,8

i . [U y2−ωx2

Z i ]+K1,9

i . [U z2+ωx2

Y i ]+K1,10

i .θx2+K1,11

i .θy2+K1,12

i .θz2

(8)

The efforts generalized on the element are defined for the normal effort by simple summation :

F x=∑i=1

N p

f xi (9)

The tangent matrix is obtained by deriving the efforts by reports with degrees of freedom. One thushas, for example:

∂Fx

∂U x1

=

∂(∑i=1

N p

f xi )

∂U x1

=∑i=1

N p ∂ f xi

∂U x1

=∑i=1

N p

K1,1i

(10)

And:

∂ Fx

∂ωx1

=

∂(∑i=1

N p

f xi )

∂ωx1

=∑i=1

N p ∂ f xi

∂ωx1

=∑i=1

N p

(−Z i. K1,2i+Y i . K1,3

i ) (11)

It is thus seen that the first term of this matrix of the element skeleton is not other than the sum of thefirst terms of the matrix of the “penny S-beams “composing the assembly. More generally, the elementary matrix skeleton is a linearcombination of the terms of the tangent matrices of the bolsters.

We will not detail here the contents of the whole of the matrix (18x18) the stored useful part contains171 terms. The scheduling of degrees of freedom is the following:

(U x1,U y1

,U z1,θx1

,θ y1,θ z1

,ωx1,ω y1

,ωz 1,U x2

,U y2,U z 2

,θ x2,θy2

,θz2,ωx2

,ω y2,ωz 2

) (12)

And that efforts: (Fx1

,F y1,F z1

, Mx1, M y1

,M z 1,MG x1

, MG y1,MG z1

, Fx2, F y2

, F z2, M x2

, M y2, M z 2

,MG x2,MG y2

, MGz 2) (13)

5.3 Internal stress analysis of the element

A law of behavior is employed, like within the framework of the theory of the multifibre beams, in orderto determine constraints on fibres. One thus has:




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σxxi, j=f (εxx

i , j ,α) (14)

Où i is a functionnelle deformation of fibre and variables internal α Dwriting the law of behavior.The normal effort in each fibre is obtained thereafter by integration of the constraints on the section.Since these last are constant by fibre:

f xi , j=σ xx

i , j . s i , j (15)

Où s i , j is the surface of fibre j of a beam N p . One can thus rebuild the generalized efforts of a beam multifibre usual starting from the efforts in fibresconstituting it, as carried out for the elements of beams multifibre [R3.08.08]. For the normal efforts i :

N i=∑

j=1

N f

f xi , j (16)

and for the efforts cutting-edges i :

V yi=∑

j=1

N f

f yi , j and V z

i=∑

j=1

N f

f zi , j (17)

Bending moments of the beam i are written:

M yi=∑

j=1

N f

f xi , j. (z i , j

−Z i ) and Mzi=−∑

j=1

N f

f xi , j .( yi , j

−Y i ) (18)

Lastly, torque Mxi=Gx

i .θxi

is calculated directly starting from the stiffness Gx on the assumption of apurely elastic behavior. These efforts thus make it possible to establish a balance in the various guidetubes constituting the skeleton. By introducing the equation ( 4 ) in the principle of virtual work, one obtains:

∫V 0

σ xxi , j .δεxx

i , j .dV 0=∫0

L

(N i .δux' (x )+M y

i .δ θ y' ( x )+M z

i .δθ z' (x )+M g y

i .δω y' (x )+Mg z

i .δ ωz' (x ) ) .dx (19)

With the normal effort louseR the beam i :

N i=∫S

σxxi , j.dS=∑

j=1

N f

σxxi , j . s i , j

=∑j=1

N f

f xi , j (20)

Bending moment compared to y :

M yi=∫S

σ xxi , j. (z i , j

−Z i ) .dS=∑j=1

N f

σxxi , j . (z i , j

−Zi ). si , j=∑

j=1

N f

f xi , j . (z i , j

−Zi ) (21)

And bending moment compared to z :

Mzi=−∫S

σxxi , j . ( yi , j

−Y i ) .dS=−∑j=1

N f

σxxi , j .( yi , j

−Y i ) . si , j=−∑

j=1

N f

f xi, j . ( yi , j

−Y i) (22)

The contribution of the beam i at the bending moment of the grid compared to y is written:

Mg , yi=∫S

σxxi , j .Zi .dS=∑

j=1

N f

σxxi , j . s i , j .Zi

=∑j=1

N f

f xi , j .Zi

=N i.Z i (23)

And bending moment of the grid compared to z is written:

Mg , zi=−∫S

σxxi , j .Y i .dS=−∑

j=1

N f

σxxi , j . si , j .Y i

=−∑j=1

N f

f xi, j .Y i

=−N iY i (24)

These the last two terms feed the generalized efforts of grid clarified hereafter.

The efforts in the beams are assembled in order to describe the total effort in the element skeleton. Thenormal effort and the efforts cutting-edges are obtained by summoning the individual efforts on thebeams: these efforts are calculated by summation of the efforts on each beam of the beam ofassembly.

There are thus the efforts normal and cutting-edges telq that:




41892bc08b70

N=∑i=1

N p

N i V y=∑i=1

N p

V yi V z=∑

i=1

N p

V zi (25)

And and inflection torques of the beams:

Mx=∑i=1

Np

M xi M y=∑

i=1

N p

M yi Mz=∑

i=1

N p

M zi (26)

Lbe beams being rigidly dependent enters they by the grids, the normal efforts and cutting-edges in thebeams generate, because of the offsetting of the guide tubes by raphoweverT in the center of theassembly of beam, the bending moments. The balance of the efforts having to be respected within themeaning of the skeleton of the assembly, three new generalized efforts are thus introduced and consistinto cubes moments applied to the rigid body. They represent the dual one degrees of freedomadditional presented in the description of the kinematics of the structure. Moment of torsion and inflection of the grids are thus worth :

Mg , x=∑i=1

Np

Fzi .Y i

−∑i=1

Np

F yi .Z i Mg , y=∑

i=1

N p

N i .Zi Mg , z=−∑i=1

N p

N i .Y i (27)

Thus, it is possible to describe balance in the skeleton using these nine generalized efforts.




41892bc08b70

6 Case of applicationOne will be able usefully to consult the cases following tests:

• ssll111a: Static response of a reinforced concrete beam (section in T) to linear behaviorthermoelastic, [V3.01.111].

• ssls143a: Beam cantilever with offset heart, [V3.03.143].• sdll130b: Seismic response of a reinforced concrete beam (rectangular section) to linear behavior,

[V2.02.130].• sdll132a : Clean modes of a frame in multifibre beams; [V2.02.132].• sdll150a: clean modes of a beam with offset heart, [V2.02.150].• ssnl119a, ssnl119b: Static response of a reinforced concrete beam (rectangular section)

to nonlinear behavior, [V6.02.119].• sdnl130a: Seismic response of a reinforced concrete beam (rectangular section) to nonlinear

behavior, [V5.02.130].• ssll102j: Fixed beam subjected to unit efforts, [V3.01.102].• ssnl106g, ssnl106h: Elastoplastic beam in traction and pure inflection, [V6.02.106].• ssnl122a: Beam cantilever multifibre subjected to an effort [V6.02.122].• ssnl504a: Beam of beams multifibre [V6.02.504].• ssnl504B : Element skeleton subjected to imposed displacements [V6.02.504].• ssnl504C : Element skeleton subjected to loadings forces, moments, temperature and gravity

[V6.02.504].• ssnl123a: Buckling of a beam multifibre [V6.02.123].

7 Bibliography[1] J.L. BATOZ, G. DHATT: Modeling of the structures by finite elements - HERMES.

[2] J. WOADS, P. PEGON & A. PINTO: With fibre Timoshenko beam element in CASTEM 2000 –Ispra, 1994.

[3] P. KOTRONIS: Dynamic shearing of reinforced concrete walls. Simplified models 2D and 3D –Doctorate of the ENS Cachan – 2000.

[4] O.C ZIENKIEWICZ and R.L TAYLOR. The Finite Method Element. Butterworth-Heinemann,Oxford, the U.K., 5th ED. Zienkiewicz and Taylor – 2000.

[5] A. IBRAHIMBEGOVIC and E.L. WILSON. With modified method of incompatible modes.Commum. Numer. Methods Eng., 7:187-194 – 1991.


multifibre element of beam (right)

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