Beam element with 7 degrees of freedom for the taking into account of Summarized

Transcription

1 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : /5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 Beam element with 7 degrees of freedom for the taking into account of ummaried warping: This document presents the element POU_D_TG which is a finite element of straight beam with taking into account of the warping of the sections. It allows the computation of the beams mean cross-sectional areas and opened profile, with constrained or free torsion. With regard to bending, the normal force and edges, this element is based on the element POU_D_T, which is a beam element right with transverse shears (models of Timoshenko). For the element POU_D_TG, the section is supposed to be constant (of an unspecified form) and the material is homogeneous and isotropic, of linear elastic behavior. Element POU_D_TGM is to be used for the nonlinear behaviors. This documentation of reference leans on the general documentation of reference of the beams, in linear elasticit [R3.8.]. It describes specificities of the beam element right with warping POU_D_TG. Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

2 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : /5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 Contents Field of utilisation3... Notations Kinematics specific to torsion with gauchissement Beam element right with warping: stiffness matries and of masse Traction and compression the degree of freedom are or Bending in the plane G, degrees of freedom:, there or D, Dr Bending in the G plane, degrees of freedom:, Z or D, Dr Torsion and warping, degrees of freedom: X, X, X or Dr, Gr Eccentring of the ais of torsion compared to the ais neutre... 5 tiffness geometrical - tructure précontrainte4... Chargements.... Distributed loadings, options: CHAR_MECA_FRDD and CHAR_MECA_FFDD.... oading of gravit, option: Thermal... CHAR_MECA_PEA_R.3 oading, option: CHAR_MECA_TEMP_R....4 oading per imposed strain, option: CHAR_MECA_EPI_R... 7 Torsor of the forces - nodal Forces and réactions Options disponibles3... the 7. torsor of the efforts generalied Forces, option: EFGE_ENO generalied Forces, option: IEF_EGA Computation of the nodal forces and the réactions nodal Forces, option: FORC_NODA nodal Reactions, option: REAC_NODA Bibliographie Description of the versions of the document5... Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

3 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : 3/5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 Field of application the development of the beam elements of Timoshenko with warping (modeliation POU_D_TG) in Code_Aster was carried out initiall with an aim of calculating the behavior of the plons. It was mainl a question of calculating formed structures b beams with open mean profile (corner), for which warping is important. The nonlinear behaviors are to be used with element POU_D_TGM (beam multifibers). These nonlinear behaviors relate onl to the tension, bending. The shears due to the shears, as well as warping and the bi--moment (force related to warping) remain dependant b an elastic behavior, fault of being able to epress a nonlinear behavior on these quantities. The description of torsion with warping is valid for the use of elements POU_D_TG and POU_D_TGM with the linear operators (MECA_TATIQUE, DYNA_INE_TRAN, ) or not linear (TAT_NON_INE, DYNA_NON_INE, ). Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

4 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : 4/5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 Notations the notations used here correspond to those used in [R3.8.] and [R3.8.3]. One gives here the correspondence between this notation and that of the documentation of use. DX, DY,DZ, DRX,DRY, DRZ et GRX are the names of the degrees of freedom associated with the components with displacement u, v, w,,,,,. The are epressed in total reference, ecept the degree of freedom associated with the warping GRX, which is epressed in local coordinate sstem. Notation used Meaning Notation of the documentation of area use of the section A I, I geometrical moments of bending compared to the aes and. IY, IZ C constant of torsion JX I warping constant JG k, k shear coefficients AY AZ e, e eccentring of the center of torsion/shears compared to the center of gravit of the crosssection EY,EZ N normal force to the section N V, V shears along the aes and VY, VZ M, M, M moments around the aes, and MT,MFY, MFZ M bi--moment BX u, v, w translations on the aes, and DX DY DZ,, rotations around the aes, and DRX DRY DRZ, rotar derivative of torsion according to GRX E Young modulus E Poisson's ratio NU G= E = modulates of Coulomb (identical to the coefficient G of amé) Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

5 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : 5/5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : Kinematical specific to torsion with warping the kinematics used to represent the displacement of the sections of beam is identical to that of the straight beams of Timoshenko [R3.8.] with regard to the traction and compression, and bending - shears. Onl torsion here is detailed. Two possibilities are to be considered for the modeliation of the behavior in torsion of the noncircular sections [feeding-bottle], which alwas produces a warping of the cross-section. Torsion is free (torsion of aint-coming) : the warping of the cross-sections is non-ero (it can even be important for an open mean section), but it is independent of the position on the ais of the beam, (constant according to ) and there is no aial stress which had with torsion. Torsion is constrained (Vlassov): warping is non-ero, and moreover of nonuniform aial stresses (from which the force resulting bi--moment is called) eist in the beam. Element POU_D_TG makes it possible to treat these two configurations: torsion can be free or constrained. The user will have access to warping in both cases, on the other hand the bi--moment will be non-ero onl in the case of constrained torsion. It should be noted that at the place of the connection of the beams, the transmission of warping depends on the tpe of connection. In general, torsion in an assembl of beams is constrained. Warping can then be blocked at the points of connection. Note: With elements without modeliation of warping ( POU_D_T, POU_D_E ), one can treat the case of free torsion (displacements other than warping will be correct), but not the case of constrained torsion. One can uncouple the effects of torsion and bending in a local coordinate sstem (relocated principal reference of inertia) having for origin the center of torsion. The center of torsion is the point which remains fied when the section is subjected to the onl twisting moment. It is also called shear center because a force applied in this point does not produce rotation around. Displacements in the plane of the section will thus be epressed in this reference. Aial displacements remain epressed in the principal reference of inertia related to the center of gravit G, to keep a decoupling of displacements of bending and traction and compression. The displacement of an unspecified point of the cross-section is written then in general form (free or constrained torsion): {u,, v,, w,, }={ug }{ w }{ v }{,, } c c Displacement = membrane + inflection + inflection + torsion with warping the components are epressed in the principal reference of inertia (centered in G ): is directed along the ais of the beam, and are the two other principal aes of inertia. The term,, represents aial displacement due to the warping of the cross-section., is the function of warping (epressed in m, but which does not have obvious phsical interpretation). The strains of an unspecified point of the section are then: Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

6 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : /5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 {,,,,,, =v, =w, }={ug, }{, }{, }{,,, c,, c, } Déformation=membranefleion/ fleion/ torsion avec gauchissement The term,, is null in the case of free torsion: one has indeed, =, since warping is independent of. It is considerable in the case of constrained torsion. The isotropic elastic constitutive law is written (b making the assumption of the plane stresses in the directions and ): }={ {,, E.,, },, G.,,,, G.,, The forces generalied in the section are epressed according to the stresses for a homogeneous section b [feeding-bottle]: N =,, ds V = V = M = M = M = M =,, ds,, ds.,, ds.,, ds c.,, c.,, ds.,, ds Normal force hears following hears according to Bending moment around Bending moment around Twisting moment Bi--moment (associate with warping) Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

7 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : 7/5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 M represents the generalied force associated with warping. It is epressed in N.m. One can give of it an illustration as in [feeding-bottle] for a beam to section in I (the bi--moment acts here according to onl): For an isotropic and homogeneous elastic behavior in the section, the generalied forces are thus epressed directl according to displacements b the following relations: N =E..u, V =Gk v, V =Gk w, M =E. I, M =E.I, M =G. J., M =E.I., where k, k are the shear coefficients. Warping does not intervene on the level of the shears, because those are epressed in the reference related to the shear center. Indeed, the function of warping is such as:, ds=., ds=., ds= And the warping constant is epressed according to b:, ds=i Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

8 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : 8/5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : Beam element right with warping: stiffness matries and of mass the elementar matries of stiffness and mass for element POU_D_TG are identical to those of the beam element right of Timoshenko (POU_D_T) with regard to the terms of traction and compression and bending - shears [R3.8.]. The approach is identical, one recalls simpl result. This implies that, in the case of free torsion, one preserves the properties of eactitude of the solution at the nodes for the degrees of freedom of bending and traction and compression. On the other hand, we will see that with regard to obstructed torsion, one carries out an approimation which does not make it possible to find this propert in the general case. The stiffness matries are alwas calculated with option RIGI_MECA, and the mass matries with option MA_MECA. But option MA_MECA_DIAG (diagonalied mass matri) was not carried out for this element (this option is especiall useful for the fast problem of dnamics, which is not the preferential field of application of this element). The degrees of freedom of the element are those of the beams of Timoshenko, plus a degree of freedom per node making it possible to calculate the terms relating to warping: In each of the two nodes of the element, the degrees of freedom are: u, v, w translations on the aes,, DX DY DZ,, rotations around the aes,, DRX DRY DRZ, rotar derivative of torsion according to GRX the local coordinates are epressed in the principal reference of inertia. Element POU_D_TG thus comprises 4 degrees of freedom. The element of reference is defined b: 4. Traction and compression the degree of freedom are u or DX the stiffness matri of the element is: K = E [ ] The mass matri (coherent) is written: M = Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

9 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : 9/5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : Bending in the G plane, degrees of freedom:, or D, Dr Martini the stiffness matri is written for the motion of bending in the principal plane of inertia G : K = EI 3 4 m 4 The transverse shears are taken into account b the term: = EI k G For the mass matri, w, t and, t are discretied on the basis of function tests introduced for the computation of the stiffness matri, that is to sa: w, t = w t + t + 3 w t 4 t, t = 5 w t + t 7 w t 8 t The interpolation function used for the translations ( with 4 ) are polnomials of Hermit of degree 3, that which are used for rotations ( 5 with 8 ) are of degree : for, the are defined b [R3.8.]: = [ = [ 3 = 4 = [ [ ] 5 = ] = [ ] 7 = ] 8 = [ the form of the mass matri is: [ ] 3 4 [ ] 3 ] ] [] Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

10 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : /5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 M= I 5 sm sm Bending in the G plane, degrees of freedom:, or D, Dr Of the same, for the motion of bending around the ais G, in the principal plane of inertia G, the stiffness matri is written: 4 K = EI 3 4 sm The transverse shears are taken into account b the term: = EI k G To compute: the mass matri, v, t and, t are discretied b: v, t = v t t + 3 v t 4 t, t = 5 v t t 7 v t 8 t Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

11 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : /5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 We obtain the following mass matri then: 3 M= I 35 7 sm 5 sm Torsion and warping, degrees of freedom:,, X or Dr, Gr With regard to torsion, the formulation is obviousl different from that of the beams without warping of the reference [R3.8.]. The virtual wor of the internal forces is written for torsion [feeding-bottle]: W int = o, * *.G. J.,,. E. I., d The interpolation functions of the rotation of torsion must be of class C, since the must make it possible to interpolate derivative second of rotation. B means of the balance equations, one shows in [feeding-bottle] that the analtical solution utilies interpolation function hperbolic in. This then makes it possible to get eact results with the nodes. It is not the choice made for Code_Aster : one chose, b preoccupation with a simplicit for numerical integration like avoiding the numerical problems of evaluating of the function hperbolic, of the polnomials of degree 3 of tpe Hermit, of the same kind as those used for bending [éq]. One writes them here on the element of reference [-,] according to [feeding-bottle] (instead of previousl): N = 4 =, N = 8 N 3 = 4 N 4 = 8 The interpolation for the rotation of torsion and its derivative is: N N 3 N 4 =N, = N,, N, N 3, N 4,,,, Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

12 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : /5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 The reference [feeding-bottle] notes that this approimation corresponds to a borderline case of the hperbolic interpolation, obtained for GJ. However, this parameter not being without EI dimension, it is difficult to define a priori the values for which the approimation is acceptable. The numerical tests carried out show that one converges quickl towards the solution when the sie of the elements decreases. The stiffness matri corresponding to this approimation is written then: K=K T K = GJ 4 3 EI sm 4 sm 4 The mass matri can be obtained in several was [feeding-bottle]: the most complete method would consist in calculating the terms of inertia with the interpolation functions above, b taking account of the additional term: W iner = o *..I,., d in Code_Aster, the simplest method was selected: the mass matri is identical to that of element POU_D_T. One preserves the alread definite terms for the traction and compression and bending - shears and one uses a linear approimation for torsion. The coefficients of the mass matri associated with warping are null with this approach. 4.5 Eccentring of the ais of torsion compared to the neutral ais At the center of torsion C, the effects of bending and torsion are uncoupled, one can thus use the results established in the preceding chapter. The coordinates of the point C are with being provided to AFFE_CARA_EEM : one gives the components of the vector GC ( G being the center of gravit of the cross-section) in the principal reference of inertia: GC = e e One can numericall determine them starting from the plane mesh of the section using operator MACR_CARA_POUTRE [R3.8.3]. Once the determined C point, one finds as in [R3.8.] the components of displacement at the center of gravit G b considering the relation of rigid bod: ug = uc + GC with = vector rotation, { u G = u C v G = v C +e w G = w C e. Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

13 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : 3/5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 the change of variables is written in the same wa that for POU_D_T, with additional degrees of freedom: u c u c u c e c e c u u c, c =, u c u u e c u e u c u c c c, c, P From the elementar matries of mass and stiffness calculated previousl in the reference C,,, where motions of bending and torsion are decoupled, one obtains these matries in the reference related to the neutral ais G,,, b the following transformations: K=P T K c P M= P T M c P. u Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

14 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : 4/5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : Geometrical stiffness - prestressed tructure This matri is calculated b the option RIGI_GEOM. It is used for dealing with problems of buckling or prestressed structure vibrations. In the case of a prestressed structure, therefore subjected to initial forces (known and independent of time), one cannot neglect in the balance equation the terms introduced b the change of geometr of the unconstrained state in a prestressed state. This change of geometr modifies the balance equation onl b the addition of a function term of displacements and prestressing with which the matri associated is called geometrical stiffness matri and who epresses himself b: where ij o W G = V o 3D u k i v 3D o k ij j the tensor of prestressing indicates. This term appears naturall if one introduces the tensor of the strains of GREEN-AGRANGE into the virtual wor of the strain: E = = u 3D E = = u 3D E = = u 3D [ u 3D u 3D u 3D u 3D [ u 3D 3D u [ u 3D dv u 3D u 3D ] u 3D u 3D u 3D 3D u u 3D 3D u ] u 3D 3D u ] In the statement of these strains, the quadratic terms u 3D, u 3D 3D u et u 3D 3D u are neglected here, according to the assumption usuall carried out b most authors [feeding-bottle3]. For a model of beam, the stress tensor initial is reduced in the local aes of the beam to the components, and. One uses the kinematics introduced with [ ]: N = 3D,, = u G,, 3D u,, = v C c 3D u,, = w C c {u and the statement of the forces generalied according to the stresses: o ds V = o ds V = o ds M = ds M It is supposed, moreover, that N V, V = ds are constant in the discretied element (what is inaccurate for eample for a vertical beam subjected to its inertia loading). The moments are supposed to var linearl: M = M M +M M V = M =M M +M M V = Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

15 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : 5/5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 These assumptions make it possible to epress W G for a straight beam with warping in the following wa:,, W G = o N v, v, w, w, N I I A c c c N v,,, v, c N w,,, w, M w,,, w, M v,,, v, c V c V,, V w, w, V v, v I r, c I M I r c I M I r I with the terms: dm I r d I dm d,, I r = I r = ds ds,, who represent it not - smmetr of the section. If the section has two aes of smmetr (thus C is confused with G ), these terms are null. Attention, these terms (which name IYR and IZR in the command AFFE_CARA_EEM ) are not currentl calculated b MACR_CARA_POUTRE. The user must thus inform them from tubulate values for each tpe of section (corner, right-angled, ). Moreover, to be able to deal with the problems of discharge of thin beams, requested primaril b bending moments and normal force, it is necessar to add the assumption of rotations moderated in torsion [feeding-bottle], [feeding-bottle3]. This results in the following modification of the field of displacements (onl for the computation of the geometrical stiffness): u 3D,, = u G,, The origin of this statement cannot be here detailed. It is the object of the thesis of TOWN OF GOYET [feeding-bottle] on the buckling of the beams with open mean sections. The assumption of rotations of torsion moderate (and not infinitesimal) makes it possible to correctl model the discharge of a thin beam of section in torsion (coupling torsion - bending). The assumption of moderate rotations results in adding in W the term W G G : W G = o M o,, M o,, V o V o Finall, one obtains the geometrical stiffness matri while discretiing W G =W G W G using the same interpolation functions as the stiffness matri of [ 4.4]. For having calculated these matries, it is necessar to carr out a change of reference as with [ 4.5]. One then obtains a geometrical stiffness matri of the form: K G = A A A A 3 Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

16 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : /5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 The blocks of the matri are clarified hereafter. One uses to simplif the statements: N e =. N o e N e =. N o e M = M M M = M M M = M M M = M M o k=n I I e e I = I r e I I = I r e I Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

17 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : 4/4/ Page : 7 / 5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 8939 A v 3 w 4 5 q 7 q, v. N o M N e. M N o e N M M 3 w. N o M N e. M N o e N M M 4. k M I M I e N M M e N M M km I M I e I r M I e I r M I M M 5 N e 5 N 3M 5 3 sm N e 5 N 3M 5 3 M M 7, 4 k I 3 M M 3 I 3 M M 3 Manuel de référence Fascicule r3.8 : Éléments mécaniques à fibre moenne Document diffusé sous licence GNU FD (

18 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : 4/4/ Page : 8 / 5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 8939 A 9 v : w : : 3: 4:, v. N o N o 3 w 4 5 N o 7, e N M M. N o M N e. M N o e N M M M N e. M M N e. M N o - A (4,4) e N M M e N M M e N M M k M I M I N o 3 e N M 3 M e N M M N o 3 e N M 3 M e N M M e N M M k M I M I e N M 3 M e N M 3 M k M I 3 M I 3 Manuel de référence Fascicule r3.8 : Éléments mécaniques à fibre moenne Document diffusé sous licence GNU FD (

19 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : 4/4/ Page : 9 / 5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 8939 A3 9: v : w : : 3: 4:, v. N o N o 3 w 4 5. N o M N e. M M N e. M. k. M e I r I M I M e I r I M I N o e N M M M N e N M M M e N M M e N M M k M I M I 5 e N M 5 3 N 5 e N M 5 3 7, m 4 k I M 3 I M 3 3M 3M 3M 3M Manuel de référence Fascicule r3.8 : Éléments mécaniques à fibre moenne Document diffusé sous licence GNU FD (

20 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : /5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 oadings the various tpes of loading available for element POU_D_TG are: Tpes or options CHAR_MECA_FRDD distributed loading b actual values CHAR_MECA_FFDD distributed loading b function CHAR_MECA_PEA_R loading due to thermal gravit CHAR_MECA_TEMP_R loading CHAR_MECA_EPI_R loading b imposition of a strain (of standard thermal stratification) the loadings are in the same wa calculated that for the elements without warping [R3.8.]. There is thus nothing in particular to element POU_D_TG. The other tpes of loading described in [R3.8.] are not available for this element. With regard to warping, it is possible to give boundar conditions utiliing the degree of freedom GRX (what makes it possible to model constrained torsion: GRX = ), but on the other hand, nothing is designed to affect a loading of tpe bi--moment, whose phsical interpretation is difficult to establish. In connection with connection between elements, the transmission of warping is an open-ended question as the reference [feeding-bottle ] it: the continuit of the variable GRX from one element to another (on which warping depends directl) depends b wa of technolog on connection between the various beams (weld in the ais, in which case warping can be transmitted completel, connection b gusset, ). For an assembled structure such as a truss, it seems more reasonable to suppose than torsion is obstructed, therefore that warping is null at the ends. To determine the influence of this assumption, one will be able to refer to the test (beam of corner section) whose modeliations C and D use element POU_D_TG, with free torsion for the modeliation C, and torsion obstructed for the modeliation D [V3..B]. It is noted that for the loading of bending, the variation on displacement is weak (.5%), but for a loading in torsion, one obtains for this section a non-ero side displacement (discharge) from which the value differs notabl according to the assumption taken: u =. 5 for free torsion and u =. 5 constrained torsion. In the same wa, rotation strongl varies: =3,79 4 for free torsion and =,39 4 constrained torsion (GRX is null at the ends).. Distributed loadings, options: CHAR_MECA_FRDD and CHAR_MECA_FFDD the loadings are given under ke word FORCE_POUTRE, either b actual values in AFFE_CHAR_MECA (option CHAR_MECA_FRDD), or b functions in AFFE_CHAR_MECA_F (option CHAR_MECA_FFDD). The loading is given onl b distributed forces, not b moments distributed. The second associated member with the distributed loading of traction and compression is: with f = f et d [ f f ] f = f et For a loading constant or varing linearl, one obtains: d F = n 3 n, F = n n 3. Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

21 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : /5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 n and n are the components of the aial loading as in points and coming from the data of the user replaced in the local coordinate sstem. If t, t, t and t are those of the shears, one a: F = 7t 3t M = t t 3 F = 3t 7t M = t 3 t, F = 7t 3t, M = t t 3, F = 3t 7t, M = t 3 t.. oading of gravit, option: CHAR_MECA_PEA_R the force of gravit is given b the modulus of acceleration g and a normalied vector n indicating the direction of the loading. Remarks (simplifing assumption) : The shape functions used for this computation are those of the model Eulerian-Bernoulli. The approach is similar to that used for the distributed forces, on condition that transforming initiall the vector loading due to gravit in the local coordinate sstem with the element. One obtains in the local coordinate sstem of beam: F i = i o g d =, = from where: F = g 3 F = g 3 au point, au point Bending in the plane G : M F F M = g 7 3 = g 3 = g 3 7 = g 3 Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

22 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : /5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 Bending in the plane G : F =g 7 3 M =g 3 F =g 3 7 M = g 3.3 Thermal loading, option: CHAR_MECA_TEMP_R to obtain this loading, it is necessar to calculate aial displacements induced b the difference in temperature T T référence : u = T T référence u = T T référence ( : thermal coefficient of thermal epansion) Then, one calculates simpl the induced forces par. F = K u As K is the local stiffness matri with the element, one must then carr out a change of reference to obtain the values of the components loading in the total reference..4 oading b imposed strain, option: CHAR_MECA_EPI_R One calculates as for elements POU_D_T the loading from a strain state (this option was developed to take into account the thermal stratification in the pipework). The model takes into account onl one work in traction and compression and pure bending (not of shears, not of twisting moment). The strain is given b the user using ke word PRE_EPI in AFFE_CHAR_MECA. While being given u, loading: and on the beam, one obtains the second elementar member associated with this with node : F = E u, M = E I, M = E I, with node : F = E u, M = E I, M = E I Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

23 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : 3/5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : Torsor of the forces - nodal Forces and reactions 7. Options available the various options of postprocessing available for element POU_D_TG are: Tpes or options EFGE_ENO torsor of the forces to the nodes of each element IEF_EGA field of forces necessar to the computation of the nodal forces (option FORC_NODA ) and of the reactions (option REAC_NODA ). FORC_NODA nodal forces epressed in total reference REAC_NODA nodal reactions 7. the torsor of the forces 7.. generalied Forces, option: EFGE_ENO One seeks to calculate with the two nodes of each element beam constituting the mesh of studied structure, the forces eerted on the element beam b the rest of structure. The values are given in the local base of each element. B integrating the balance equations, one obtains the forces in the local coordinate sstem of the element: e e e R OC = K OC u OC M OC ü OC f OC where: R OC = N, V Y, V Z, M T, M Y, M Z, M, N, V Y,V Z, M T, M Y, M Z, M e K OC e M OC e f OC u OC ü OC elementar matri of stiffness of the element beam, elementar matri of mass of the element beam, vector of the forces distributed on the element beam, vector degree of freedom limited to the element beam, vector acceleration limited to the element beam. One changes then the signs of nodal efforts. Indeed, b taking for eample the case of the traction and compression, one shows [R3.8.] that the forces in the element (option EFGE_ENO) are obtained b: [ N o ] = [ K ] [ u o ] N u [ f ] f 7.. Generalied forces, option: IEF_EGA option IEF_EGA is established for reasons of compatibilit with other options. It is used onl for computation of the nodal forces. It produces fields of forces b elements. It is calculated b: e R OC = K OC u OC 7.3 Computation of the nodal forces and the reactions 7.3. nodal Forces, option: FORC_NODA This option calculates a force vector nodal on all structure, epressed in total reference. It produces a field at nodes in command CAC_CHAMP b assembl of the elementar terms. For this computation, one uses the principle of the virtual wors and one writes [R5.3.]: F=Q T Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

24 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : 4/5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : 577 the matri associated with the operator divergence represents smbolicall. For an element, one writes the work of the strain field virtual: Q T u * = u u * u * kinematicall admissible where Q T For the beam elements, one calculates simpl the nodal forces b assembl of the elementar nodal forces calculated b option IEF_EGA, which are epressed b: [ F OC ]=[K OC ] [ U OC ] 7.3. Nodal reactions, option: REAC_NODA This option, called b CAC_CHAMP, makes it possible to obtain the reactions R to the bearings, epressed in the total reference, starting from the nodal forces F b: R=F F char F iner F char et F iner being nodal forces respectivel associated with the loadings given (specific and distributed) and with the forces with inertia. Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (

25 Titre : Élément de poutre à 7 degrés de liberté pour la pr[...] Date : /3/3 Page : 5/5 Responsable : Jean-uc FÉJOU Clé : R3.8.4 Révision : Bibliograph [] J.. BATOZ, G. DHATT. Modeliation of structures b finite elements - HERME. [] V. OF TOWN OF GOYET. Nonlinear static analsis b the finite element method of formed spatial structures b beams with asmmetric sections Thesis of the Universit of iege [3] J. IMI. imulation of failure of plon Ratio ERAM N 4.33, ENAM June Description of the versions of the document Author () Description of the modifications Aster Organiation () 4/7/9 J.. FEJOU, J.M. PROIX (EDF-R&D/AMA) //3 J.. FEJOU Addition remarks on POU_D_TGM. Warning : The translation process used on this website is a "Machine Translation". It ma be imprecise and inaccurate in whole or in part and is icensed under the terms of the GNU FD (