A FOUR-NODED PLANE ELASTICITY ELEMENT BASED ON THE SEPARATION OF THE DEFORMATION MODES

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1 A FOUR-NODD LAN LASICIY LMN BASD ON H SARAION OF H DFORMAION MODS A. DÓSA D. RADU Abstract: his paper presents a for-noded qadrilateral finite element with translational degrees of freedom for plane elasticit problems, based on the displacement separation method, which is an nsmmetric stress-hbrid formlation. he element can reprodce eactl all the constant strain modes, has correct rank, is of high coarse mesh accrac and has a ver low shape distortion sensitivit. Ke words: qadrilateral element, nsmmetric formlation, distortion sensitivit.. Introdction he for-noded plane elasticit elements with translational degrees of freedom at nodes ehibit a high shape sensitivit in special to bending. According to MacNeal s theorem [], a for-noded trapezoidal element cannot have locking-free in-plane bending and constant strain deformation capabilit, while simltaneosl satisfing the reciprocit and compatibilit of the nodal displacements. In the literatre several efforts have been made to improve the performances of qadrilaterals. Man of the approaches are based on relaing some of the terms of MacNeal s theorem. For eample K.Y. Sze [] introdces a selective scaling procedre that jdiciosl redces the stiffness arising from the two bending stress/strain modes in the fornode element. Another wa to improve the bending capabilit of distorted qadrilateral elements is given b X.M. Chen et al. b sing the qadrilateral area coordinate method. he reslting elements fail to pass the strict patch test, bt pass patch tests in a weak sense []. he QMS qadrilateral element presented in this paper is based on the method of separation of displacement modes, which is an nsmmetric formlation and hence relaes the reciprocit reqirement. An nsmmetric formlation given b S. Rajendran and K.M. Liew [9],.. Ooi, S. Rajendran and J.H. Yeo [5] led to the development of sccessfl 8-node plane elements and -node heahedron elements with high distortion tolerance and ver good performances. In their formlation, the athors se two different sets of fnctions, one set as weighting fnctions to enforce compatibilit and one as shape fnctions for the completeness of the displacement field. he formlation presented in this paper, instead of the second set of shape fnctions, ses displacement modes. he elements based on this approach are capable to reprodce eactl the nodal forces corresponding to an of the displacement modes sed in the formlation of the element. As a reslt, the Faclt of Civil ngineering, ransilvania Universit of Braşov.

2 5 A For-Noded lane lasticit lement Based on the Separation of the Deformation Modes patch tests are passed withot an other special measres. Also the elements give eact answers for constant bending corresponding to the aes of the elements. Becase there is no contamination between modes, no parasitic energ arises in the elements and hence high coarse mesh accrac reslted for all the nmerical test problems.. he Displacement Mode Separation Method he virtal work principle for a linear elastic bod in internal eqilibrim nder the action of srface forces and in the absence of the bod forces can be written as: δd σ d δd tˆd =. () n t Here δd is a virtal displacement field, compatible with the displacement conditions on the bondar, σ n are the srface tractions derived from the displacement field and tˆ are the prescribed bondar tractions on t. Relation () works correctl onl if the displacement field flfills the continit reqirements. If the bod is divided in finite elements, the involved fields can be approimated at element level. he virtal displacements are δd = δa, where is a matri of conforming shape fnctions and δa is the vector of virtal nodal displacements. he real displacements are = N q, where N is the corresponding matri of interpolation and q is the vector of modal displacements. he internal stress field is: σ = C D = C D N d q d = Sq d, () where C is the elastic compliance, D is the smmetric gradient operator, N d and q d are the partitions of N and q respectivel corresponding to the deformation modes. he bondar tractions σ n are compted from the interior stress field, sing the traction-stress matri bilt with the components of the normal to the bondar of the element σ n = σ. he q d deformation mode vector can be obtained from the nodal displacement vector a of the element qd = H d a. he derivation of H d matri involves the constrction of the transformation a = q, b evalating the displacement modes at the nodal points. If matri is nonsinglar, inversion gives H =. H d reslts b etracting from H the lines corresponding to the deformation modes. Inserting () in () reslts: δa L H a δa f =. () d In () L is the leverage or connection matri, which links the stresses to the nodal forces and f is the vector of nodal loadings: L = Sd, e e () f = tˆ d. (5) Relation () mst hold for an virtal nodal displacement vector δa. he element eqilibrim eqation reslts L H d a = f, where k = LH d, is the stiffness matri of the element.. he QMS lement he reference sstems sed in this paper are shown in Figre. he element has constant thickness h. he material is assmed to be homogenos and isotropic with modle of elasticit and transversal

3 Blletin of the ransilvania Universit of Braşov Vol. (7) of the nodes. he two local rectanglar sstems have their origin in the point: X A D α C ( X, Y ) = (6) = ( X+ X + X + X, Y+ Y+ Y+ Y ), Y B and are oriented with the, respectivel aes on the median lines of the element. Fig.. For-node plane element contraction coefficient v. he eight degrees of freedom of the element are the translations.. he Internal Stress Field he stress field is based on the eight displacement modes of the element []. Relation = N q has the form (7): = v v ( + v) ( + v) ( + v ) cosα + ( sin α ( q + v )sin α q + v )cosα M q 8 where α is given in Figre. he first three modes characterized b the amplitde parameters q, q and q are rigid bod modes and do not introdce strains and stresses in the element. he net three modes are constant stress modes, or c-modes. he c-modes are conforming displacement modes. he are displacement patterns that prodce constant stress states in the element. he last two modes are higher order modes, or h-modes. h-modes are nonconforming modes and correspond to constant bending of the aes and. he are illstrated in Figre, for the rectanglar case. he c- and h-modes together form the d-modes (deformation modes). he reslting stress field (relation ()) becomes: σ cos α q σ = σ = sin α M = Sqd. (8) τ sin cos q α α 8.. he Bondar Displacement Field he bondar displacements on a given side of the element are linear fnctions of the displacements of the adjacent nodes... Finite lement qations he connection matri L can be evalated in the following form: L = [L L ]. (9) b h

4 5 A For-Noded lane lasticit lement Based on the Separation of the Deformation Modes = q7 ; = ( + ν ) q7; σ = q7, = q8 ; = ( + ν ) q8; σ = q8. b b a a δ d = q7 ; δd = ( a + νb ) q7, δ d = q8 ; δd = ( b + νa ) q8. Fig.. he h-modes (higher order modes) for the -node plane stress rectangle Matri L b concentrates at the nodes the bondar tractions corresponding to constant stresses: Y Y Y Y h L b = X X X X, () X Y X Y X Y X Y where X ij = X i X j and Y ij = Y i Y j. Matri L h concentrates at the nodes the bondar tractions corresponding to linear (higher order) stresses: L h h ( ) = ( + ) [ c [ c s s c c s s c c s s c c s] s ] 6, () where: c = cos(x, ), s = sin(x, ), c =.cos(x, ), s = sin ( X, ). Matri has the following form ():

5 Blletin of the ransilvania Universit of Braşov Vol. (7) = ν ν ν ν ν ν ν ν ( + ν) ( + ν) ( + ν) ( + ν) ( + ν) ( + ν) ( + ν) ( + ν) ( ( ( ( + ν ) + ν ) + ν ) + ν ) cos α + sin α cos α + sin α cos α + sin α cos α + sin α ( ( ( ( ( ( ( ( + ν ) sin α + ν ) cos α + ν ) sin α + ν ) cos α + ν α. ) sin + ν ) cos α + ν ) sin α + ν ) cos α In order to redce the effort of nmerical comptation, matri H d can be obtained following an idea of Bergan and Felippa [] sed for the inverse of a similar matri: = + R R R R, () where: R =. has the form: = ( ν ) ( ν ) ( + ν) [ [ [ v [ v v v v ] v], () ] ] where: ijk i i = det j j. (5) k k Since is obtainable in closed form and R is of small dimension, the comptation of H d is of low cost.. Some Remarks on the Isse of Inter- lement Continit A more complete introdction to the herein formlation can be done with the help of the stress-hbrid principle [6], [7], []: Π( σ, d) = σ C σdω + d σ n d d tˆd stat. (6) Ω t

6 56 A For-Noded lane lasticit lement Based on the Separation of the Deformation Modes Here σ is an assmed stress field in internal eqilibrim in the domain Ω in the absence of bod forces, σ n is the srface traction on the bondar, C is the elastic compliance, d is the bondar displacement field, and tˆ stands for the prescribed bondar tractions on t. If σ is a displacement derived stress field σ = σ = C D, where D is the smmetric gradient operator, and is the assmed internal displacement field, eqation (6) can be rewritten as: Π(,d) = σ + d n d σ n d d tˆ d stat. (7) t Rendering fnctional (7) stationar gives: δσ n ( d )d =, (8) δd σ d δd tˆd =. (9) n t Relation (8) enforces continit on the bondar in a weak sense and mst hold for an admissible δσ virtal stress field. Relation (9) is identical with (), introdces the static eqilibrim conditions and mst hold for an δd virtal bondar displacement field. Using relations (8) and (9) at finite element level leads to smmetric stresshbrid elements. In this case displacements d are linked to the nodal displacements and are conforming, while displacements have free parameters and are nonconforming. nforced at finite element level, relation (8) becomes a weak conformit condition. It is eas to observe that for rectangles and parallelograms relation (8) has no effect, since the bondar integral is zero for δσ n in an mode and (d-) in an mode. In these cases free incompatible displacements can develop. Unfortnatel in the case of distorted shapes this relation is too restrictive and introdces ecessive energ. If the nonconforming displacements are linked to the nodal displacements, neglecting condition (8) leads to an nsmmetric formlation. In this case the weak conformit condition is replaced b a bilt-in nodal continit condition and the inter-element continit is no more strictl controlled. However de to the less restrictive conditions, in some cases, like in Figre a, even pointwise inter-element continit is possible, while the corresponding smmetric formlation leads to poor reslts. a/b a b a) b) a/b Fig.. Deformed adjacent elements: a) constant bending; b) linear bending

7 Blletin of the ransilvania Universit of Braşov Vol. (7) Nmerical periments 5.. Mesh Distortion est for Beam Bending In the Figre is shown a cantilever beam modeled b two elements. he distortion of the elements is characterized b the eccentricit e. he two load cases correspond to constant bending and linear bending. he displacements of node A are given in able. he reference nodal displacements a RF sed in the table are compted according to the beam theor. he relative error norm is ε r = a a QMS a RF RF, where is the clidian norm. It can be observed that there is no distortion sensitivit in the constant bending case. Case B 5 e Case A Case B A Fig.. Cantilever beam (t =, = 5, ν = /) able ip displacements and relative error norms for the cantilever beam Case A Case B e v A ε r v A ε r eact vales Cook s Membrane roblem possesses high coarse mesh accrac. he he elements sed in this eample are: the standard biliniar isoparametric displacement element Q, he enhanced strain element QM6 of alor, Beresford and Wilson [] and the enhanced mied element Q b iltner and alor [8]. A, 6 6. Conclsions QMS, a for node qadrilateral plane elasticit element has been sggested, which 8 Fig. 5. Cook s membrane problem (t =, =, ν = /)

8 58 A For-Noded lane lasticit lement Based on the Separation of the Deformation Modes able Displacements v A for Cook s membrane problem Mesh Q QM6 Q QMS element is based on the deformation mode separation method, which is an nsmmetrical stress-hbrid formlation. he element passes the constant stress patch tests and presents high coarse mesh accrac and low mesh distortion sensitivit. However frther tests and theoretical investigations are necessar to establish the limits of applicabilit of the element and to get fll confidence in this formlation. References. Bergan,.., Felippa, C.A.: fficient Implementation of a rianglar Membrane lement with Drilling Freedoms, Finite lement Methods for late and Shell Strctres. ineridge ress, Chen, X.M., Cen, S., Long, Y.Q., Yao, Z.H.: Membrane lements Insensitive to Distortion Using the Qadrilateral Area Coordinate Method. In: Compters and Strctres 8 (), p. 5.. Felippa, C.A.: Advanced Finite lement Method. Sorce: (5).. MacNeal, R.H.: A heorem Regarding the Locking of apered For-Noded Membrane lements. In: Int. J. Nmer. Meth. ngng. (987), p Ooi,.., Rajendran, S., Yeo, J.H.: A -Node Heahedral lement with nhanced Distortion olerance. In: Int. J. Nmer. Meth. ngng. 6 (), p ian,.h.h.: Derivation of lement Stiffness Matrices b Assmed Stress Distribtions. In: AIAA Jornal (96), p.. 7. ian,.h.h.: Some Notes on the arl Histor of Hbrid Stress Finite lement Method. In: Int. J. Nmer. Meth. ngrg. 7 (), p iltner,., alor, R.L.: A Qadrilateral Mied Finite lement with wo nhanced Strain Modes. In: Int. J. Nmer. Meth. ngng. 8 (995), p Rajendran, S., Liew, K.M.: A Novel Unsmmetric 8-Node lane lement Immne to Mesh Distortion nder a Qadratic Displacement Field. In: Int. J. Nmer. Meth. ngng. 58 (), p. 7.. Sze, K.Y.: On Immnizing Five-Beta Hbrid-Stress lement Models From rapezoidal Locking In ractical Analses. In: Int. J. Nmer. Meth. ngng. 7 (), p alor, R.L., Beresford,.J., Wilson,.L.: A Non-Conforming lement for Stress Analsis. In: Int. J. Nmer. Meth. ngng. (976), p..