Code_Aster. Element CABLE_GAINE

Transcription

1 Titre : Élément CABLE_GAINE Date : 28/07/2015 Pae : 1/12 Element CABLE_GAINE Summary: The element CABLE_GAINE presented in this document is to model cables of prestressin bein able to rub or slip into their sheaths and thus not followin completely displacements of the concrete in which they are pluned. It thus comes in complement from modelin BAR allied with the operator DEFI_CABLE_BP who allows to model cables of prestressed completely adherent with the concrete (see [R ]). This element is very stronly inspired by the element of interface describes in [R ]. It models the interface of the cable with its sheath and the cable itself.

2 Titre : Élément CABLE_GAINE Date : 28/07/2015 Pae : 2/12 Contents 1 Variational formulation Potential enery and minimization Laranian increased Characterization of the point saddles Discretization of the finite element Static condensation Interation Perfect adherence Slip without friction Slip with friction (BPEL) Discretized writin Notations Discretization of the conditions of optimality Internal forces Matrix of riidity Case adherent and slippin Rubbin case Converence of calculation: adjustment of the parameters Coefficient of penalization Converence criteria by values of reference... 11

3 Titre : Élément CABLE_GAINE Date : 28/07/2015 Pae : 3/12 1 Variational formulation 1.1 Potential enery and minimization The potential enery, whose one will seek the local minimum, is written as the sum of the elastic deformation enery (in the cable), of cohesive enery (with the interface/cable sheaths) less the work of the external efforts: E pot u=e el ue c u W ext u The expression of elastic enery is the followin one: E el u= u cable sds where is the way (1D) cable, is the density of enery elastic and u cable is the displacement of the cable. The displacement of the cable is defined as follows: u cable s=u aine ss T s where T represent the tanent with the cable, u aine the displacement of the sheath and the relative displacement of the cable compared to the sheath. There are then (mechanical cf Salençon of the continuous mediums Volume II: Elasticity curvilinear Mediums ): u cable s= du T aine T = du aine T d ds ds ds It is specified that the multiplication by the tanent vector with the cable T allows to pass from a vector lenthenin accordin to the three directions of space to a scalar lenthenin alon the dt cable. In fact to obtain this expression one nelects. ds The two followin sizes are defined: ε u aine (u aine )= d (u ) aine T and ds ε ( )= d ds Thus the deformation of the cable is written finally: ε(u cable (s))=ε uaine (u aine )+ ε () Cohesive enery (sheath/cable) as for it is written: E c u= sds where is the density of cohesive enery.

4 Titre : Élément CABLE_GAINE Date : 28/07/2015 Pae : 4/12 Takin into account nonthe derivability, one will use a method of decomposition-coordination to treat this minimization. That consists in introducin the slip, to pose s= s and to be reduced to the problem of minimization under constraint accordin to: with min Eu, u, = Eu,= u cable s ds sds W ext aine u aine W extcable That will make it possible to separately treat (decomposition) the minimization of cohesive enery at the local level (static condensation) whereas the minimization of elastic enery and work is treated at the total level. 1.2 Laranian increased With the problem of minimization under constraints is dealt by dualisation: the Laranian one is introduced increased L and the field of multipliers (coordination): Lu,,=E u, s s s ds r 2 s s² ds with r coefficient of penalization. 1.3 Characterization of the point saddles The conditions of optimality of order 1 make it possible to write: δ u aine : Γ σ :ε uaine (δ u aine (s))ds=w ext aine (δ u aine ) with = [éq. 1.1] δ : Γ σ :ε (δ (s))ds+ Γ [λ( s)+ r( (s) δ(s))] δ (s)ds=w ext cable (δ ) [éq. 1.2] : [s s] sds=0 [éq. 1.3] : [t s s r s s] s ds=0 with t [éq. 1.4] 2 Discretization of the finite element Notice preliminary: The finite element considered models the interface of the cable with its sheath and the cable itself. It would have been possible to model separately the interface cable-sheath (with a finite element of interface) and even cables it him (with a finite element of cable), nevertheless, as it thereafter will be seen the rubbin law of behavior requires the knowlede of the tension in the cable, it is thus advantaeous to have in the same element of the interface and cable. Moreover, as it below will be seen, modelin suested is pressed on quadratic elements, it is thus not possible to re-use the element BAR existin.

5 Titre : Élément CABLE_GAINE Date : 28/07/2015 Pae : 5/12 The finite element considered is linear (eometrical support in the form of sements), contrary to the elements of interface which are surface or voluminal. It is considered to represent a cable pluned in a volume (of concrete) 3D. Consequently, the derees of freedom are: displacements of the sheath (3 derees of freedom): u aine the relative lonitudinal displacement of the cable in the sheath and the multiplier of Larane. In a way similar to the elements of interface, relative slip is discretized at the points of Gauss then eliminated by static condensation. The adopted interpolation is P2 for displacements (sheath and cable) and P1 for the multiplier of Larane. The eometric standards of support will be thus SEG3. The constraint = is imposed on the weak direction. The points of selected Gauss are the same ones as for the element of interface under-interated, that is to say 2 points of interation (contrary to the elements of interface presented in [R ], the fact that the cable does not have that the lonitudinal displacements controlled by the riidity of the cable does not impose the use of an interation at 3 points of Gauss. This point was checked durin the tests). The followin fiure makes it possible to visualize all this information. u aine u aine u aine Nœuds Points de Gauss Fiure 2-1: discretization of the element CABLE_GAINE 3 Static condensation The field disappears from the total formulation thanks to static condensation. In each point of collocation s, one has (accordin to éq.1.4): t = r with t =t s, =s, =s and =s. Once the discretized problem, and will be obtained by interpolations with the functions of forms appropriate of the discretized values to the nodes of and that one notes {G} and { }. The interation of the constitutive relation (cf below) makes it possible to calculate accordin to {G} and { } that one notes : t = r =, = {G},{ }

6 Titre : Élément CABLE_GAINE Date : 28/07/2015 Pae : 6/12 4 Interation The laws of friction considered are perfect adherence, the slip without friction and a friction of Coulomb with a threshold makin it possible to find the profiles of tensions iven by the BPEL (or by the ETCC since the 2 codes use equivalent formulas). These three cases are usable with the law of behavior CABLE_GAINE_FROT. The addition of new laws of friction adapted to other reulation is possible. Of a point of Gauss iven, the interation of the laws consists in determinin. For that, one seeks the point of intersection between the line representative of the equation t = r (reen curve on the fiures Fiure 4-1, Fiure 4-2, Fiure 4-3) and the curve representative of the derivative of (blue curve on the fiures). Note: in the pararaphs which follow, one adopts a notation on displacements not to rather weih down the notations than on the increments of displacements as that is really the case. 4.1 Perfect adherence In the first case, correspondin to perfect adherence, one a: =I R + I R frot r Fiure 4-1: Graphic interpretation of the interation of the law of perfect adherence The solution is: =0 4.2 Slip without friction In the second case, slip without friction, one a: =0

7 Titre : Élément CABLE_GAINE Date : 28/07/2015 Pae : 7/12 frot r Fiure 4-2: Graphic interpretation of the interation of the perfect law of slip The solution checks: from where 0= r = r Slip with friction (BPEL) This law makes it possible to take into account rectilinear friction and curved friction. The tensions imposed by the BPEL are found by choosin the threshold of the law of Coulomb c as follows: c = f s N with the notations of [R ] which one points out: f the coefficient of friction of the cable on the partly curved concrete, in rad 1, the coefficient of friction per unit of lenth, in m 1, the cumulated anular deviation. and with N normal effort. Indeed, by considerin that the cable slips over all its lenth at the time of the settin in tension and notin s the curvilinear X-coordinate alon the cable, the balance of the cable is written: from where by interation: dn = f ds s N

8 Titre : Élément CABLE_GAINE Date : 28/07/2015 Pae : 8/12 N=N 0 exp sf s One then reconizes the expression of the tension of the cable in the presence of rectilinear friction of the BPEL ([R ] 2.2.2). frot c r To write the solution, there are 3 cases: if r c : =0 if r c : = c r c Fiure 4-3: Graphic interpretation of the interation of the law of friction BPEL if r c : = c r The case where normal effort N is neative does not have normally to occur for the problems of cables of prestressins. However it can sometimes happen to have values very slihtly neative. One makes the choice to adopt the behavior slippin into such a case. Notice 1 : by introducin such a dependence of the slip to the tension into the cable, the contribution of the variation of this tension to the slip N induced a nonsymmetrical matrix. Notice 2: So that friction curves is taken into account at the elementary level, it is imperative that the curve of the cable is retranscribed in the elements the component, i.e. the three nodes of meshs SEG3 should not be alined. 5 Discretized writin 5.1 Notations It is pointed out that the elements CABLE_GAINE have 3 nodes and that only the nodes ends have derees of freedom of Larane (cf. Fiure 2-1).

9 Titre : Élément CABLE_GAINE Date : 28/07/2015 Pae : 9/12 The functions of form of displacements are noted n m with m {1,2, 3 }. The functions of form of the multipliers of Larane are noted l p with p {1, 2}. One notes (the quantities are evaluated in each point of Gauss): s the curvilinear X-coordinate the weiht of the points of Gauss the tensor of the constraints N 3 matrix of the values of the functions of form at the point of Gauss discretizin displacements of the sheath (3 components) N 3 =n 1 s 0 0 n 2 s 0 0 n 3 s n 1 s 0 0 n 2 s 0 0 n 3 s n 1 s 0 0 n 2 s 0 0 n 3 s N 1 matrix of the values of the functions of form at the point of Gauss discretizin relative displacements of the cable (1 component): N 1 =n 1 s n 2 s n 3 s T the tanent vector at the point of Gauss N 1 the derivative of the functions of form discretizin relative displacements of the cable N 3 the derivative of the functions of form discretizin displacements of the sheath One also poses to simplify the notations: B =T T N 3 L matrix of the values of the functions of form discretizin the multiplier of Larane at the point of Gauss L =l 1 s l 2 s {U } nodal displacements (sheath + cable) {U a } nodal displacements of the sheath {G c } the nodal relative displacement of the cable { } the multiplier of nodal Larane {F a ext } the vector forces external nodal, dual displacements of the sheath c {F ext } the vector forces external nodal, dual nodal relative displacement of cable A the section of the cable 5.2 Discretization of the conditions of optimality Equations 1.1,1.2 and 1.3 (the éq. 1.4 bein treated with the level of the law of behavior) which characterizes the point saddle are discretized in the form: B T : ={F a ext } [[ N 1 ] T : [N 1 ] T [[ L ]{ }r[ N 1 c ]{G c } r {G c },{}]]={F ext } [[L ] T [[ N 1 ]{G c } {G c },{}]]={0} 5.3 Internal forces The notations are introduced: f u =B T :

10 Titre : Élément CABLE_GAINE Date : 28/07/2015 Pae : 10/12 f c =[ N 1 ] T : [ N 1 ] T [[L ]{}r [N 1 ]{G c } r {G c },{ }] f =[ L ] T [[N 1 ]{G c } {G c },{ }] Contributions to the forces, for i {1,2, 3}, m {1,2, 3 } and p {1, 2}, are written then: f u i,m = f uaine i,m f c m = f ucable m f p = f p where f u i, m is the component 3m 1i of f u, f c m is the component m of f c and f p is the component p of f 5.4 Matrix of riidity One points out the notations in a point of Gauss: the relative displacement of the cable: =N 1 {G c } deformation of the cable: =B {U a } N 1 {G c } the multiplier of Larane: = L { } While posin = r, the two pararaphs which follow ive the contributions to the tanent matrix for the cases adherent and slippin and the rubbin case Case adherent and slippin In the cases adherent and slippin, the tanent matrix is symmetrical (minimization of a point saddles): K uu = B T d d B K u c =K c u T T d = B d N K u =K u T =0 K c c = [ N 1 T d d N 1 N 1 T rn 1 N 1 T r² d N 1 d ] K c =K c T = [N 1 T L N 1 T r d d L ] K = [ L T d d L ] 1 It is specified that d d is obtained startin from the law of behavior of friction Rubbin case In the rubbin case, there is a dependence of the slip to the tension in the cable. followin expressions are some modified:

11 Titre : Élément CABLE_GAINE Date : 28/07/2015 Pae : 11/12 K = c u [ N 1 T d d B rn 1 T N A B ] K u = [ L T N A B ] K c c = [ N 1 T d d N 1 N 1 T rn 1 N 1 T r² d N 1 d N 1 T r N A N 1 ] K c = [L T N 1 L T r d N 1 d L T N A N (The other expressions are unchaned compared to 5.4.1) It is noted that the matrix is not symmetrical any more. 6 Converence of calculation: adjustment of the parameters 1 ] The converence of a calculation with the elements CABLE_GAINE is sometimes difficult to obtain takin into account different the order of manitude involved. This is why it is necessary well to choose the coefficient of penalization r and to prefer converence criteria by values of reference (keyword RESI_REFE_RELA). The two followin pararaphs ive advices of use on these points. 6.1 Coefficient of penalization At the time of the writin of Laranian increased, the coefficient r known as coefficient of penalization was introduced. To choose the value of r (PENA_LAGR in the cohesive law CABLE_GAINE_FROT), which one points out that the convered solution does not depend, or to clarify the construction of the force of reference used in the criterion RESI_REFE_RELA (see accordin to), it is necessary to carry out a rapid analyzes dimensional. For the choice of r, one sees in the precedin equations which it is judicious that it is of about size of u. u is the order of manitude of expected displacements. is the force of friction per i.e, unit of lenth the stress shear (friction) interated on the perimeter of the cable. If a constraint is taken ref typical waited in the concrete in the vicinity of the cable, will be of the order 2r cable ref. To obtain best possible converence, it is necessary to choose r of about size of 2 r cable ref u ref. 6.2 Converence criteria by values of reference The use of converence criteria per value is often necessary to o at the end of calculation. For that it is necessary to activate the keyword RESI_REFE_RELA in the keyword factor CONVERGENCE of STAT_NON_LINE or CALC_PRECONT. In the case of the element CABLE_GAINE, this keyword must be accompanied by three others: a force of reference (keyword FORC_REFE): tension expected in the cable, a displacement of reference (keyword DEPL_REFE): a typical displacement of the structure, a constraint of reference (keyword SIGM_REFE): the order of manitude of the constraints expected in the vicinity of the cable (the typical constraint of the concrete except typical case), this

12 Titre : Élément CABLE_GAINE Date : 28/07/2015 Pae : 12/12 constraint makes it possible to build a reference for by usin the section A cable in the form: ref = ref A