8-node quadrilateral element. Numerical integration

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1 Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag of 0 -nod quadrlatral lmnt. Numrcal ntgraton h tchnqu usd for th formulaton of th lnar trangl can b formall tndd to construct quadrlatral lmnts as wll as hghr ordr trangls. But t s connctd wth som dffcults:. h constructon of shap functons satsfng consstnc rqurmnts for hghr ordr lmnts wth curvd boundars bcoms ncrasngl dffcult.. Computatons of shap functon drvatvs to valuat th stran-dsplacmnt matr.. Intgrals that appar n th prssons of th lmnt stffnss matr and consstnt nodal forc vctor can no longr b carrd out n closd form. h -nod lmnt s dfnd b ght nods havng two dgrs of frdom at ach nod: translatons n th nodal (u and drctons (v. It provds mor accurat rsults and can tolrat rrgular shaps wthout much loss of accurac. h -nod ar wll sutd to modl curvd boundars. 7 7 (, 6 (, ξ (, 5, ( 5 5 (, ( ξ, (, (, (, (, (, (, (, (, (, ( 0, (, (,0 (, ( 0, (, (,0 (,

2 Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag of 0 N (ξ, N 6 (ξ, = = = = ( ξ, ( ξ, N ( ξ, ( ξ, N ξ ξ N( ξ, = ( ξ ( ( + ξ + N( ξ, = ( + ξ ( ( ξ + N( ξ, = ( + ξ ( + ( ξ N( ξ, = ( ξ ( + ( + ξ N5( ξ, = ( ξ ( N6( ξ, = ( + ξ ( N7( ξ, = ( ξ ( + N( ξ, = ( ξ ( Shap functons N and N 6 N 0 N 0 N 0 N 0 N5 0 N 0 = 0 N 0 N 0 N 0 N 0 N5 0 N = = [ N ] [ N ]{ }

3 Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag of 0 = = ( ξ, ( ξ, u N u = = ( ξ, ( ξ, v N v u N 0 N 0 N 0 N 0 N5 0 N 0 = u v 0 N 0 N 0 N 0 N 0 N5 0 N v u v u v u v u = v [ N ]{ q} 0 0, { ε} = [ R]{ u} = N ( ξ { q} = [ B]{ q} [ B] N N N N N N N N N N N N N N N N =

4 Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag of 0 Partal drvatvs of shap functons wth rspct to th Cartsan coordnats and ar rqurd for th stran and strss calculatons. Snc th shap functons ar not drctl functons of and but of th natural (local coordnats ξ and, th dtrmnaton of Cartsan partal drvatvs s not trval. W nd th Jacoban of two-dmnsonal transformatons that connct th dffrntals of {,} to thos of {ξ,} and vc-vrsa N N ξ ξ ξ δξ = = = N N = = = = [ J ] J ( ξ, Matr J s calld th Jacoban matr of (, wth rspct to (ξ,, whras J - s th Jacoban matr of (ξ, wth rspct to (,. J and J ar oftn calld th Jacoban and nvrs Jacoban, rspctvl. h scalar smbol J mans th dtrmnant of J: J = J =dt J. Jacobans pla a crucal rol n dffrntal gomtr. N N ξ N = + ξ N N ξ N = + ξ N ξ N N ξ [ J ] ξ N = ξ = N N N N N ξ ξ ξ = [ J ] N N = N

5 Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag 5 of 0 { ε} = B( ξ { q}, U = ε { σ} dd = q [ B] [ D][ B]{ q} dd ( (,, f (, dd = f ( ξ, dt [ J ] dξd dd = dt [ ] ( ( ξ A, A, ( ξ, ( [ ] ( ( { } U = / q B ξ, D B ξ, dt J ξ, dξd q 6 6 U = q k q [ ] { } J dξd [ ] [ ] ( [ ][ ] ( ξ, [ ] ( ξ, dt ( ξ, ξ ( ξ,, k = B D B dd = B D B J d d B 6 6 6

6 Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag 6 of 0 Nodal forcs of th lmnt quvalnt to th bod load: { } [ ]{ } { } z = = =, W u d N q d F q = [ ] F N d. F = A F/ F/ F/ F/ Work-quvalnt nodal forcs for unform constant bod load n th cas of CS lmnt and -nod quadrlatral lmnt

7 Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag 7 of 0 Fnt lmnt mthod rsults: contnuous dsplacmnt fld and dscontnous strss fld Dsplacmnt componnt (.g. u(, ntrpolaton ovr two fnt lmnts u u = u u ( ε = ( ε ( ε ( ε ( σj ( σj

8 Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag of 0 Numrcal Gauss ntgraton n FE algorhms h us of numrcal ntgraton s ssntal for valuatng lmnt ntgrals of soparamtrc lmnts. h standard practc has bn to us Gauss ntgraton bcaus such ruls us a mnmal numbr of sampl ponts to achv a dsrd lvl of accurac. hs proprt s mportant for ffcnt lmnt calculatons bcaus w shall s that at ach sampl pont w must valuat a matr product. [ ] [ ] ( ξ [ ][ ] ( (,, [ ] ( ( k = B D B dd = B ξ, D B ξ, dt J ξ, dξd 6 6 h numrcal ntrgraton hav to b also prformd for fndng th quvalnt nodal forcs. On dmnsonal ntgraton In gnral: b a n ( α ( F d = F + R = n Introducng th nw varabl - ( a + b b a b a = + d = F( b a b a b a F ( d = f ( d = f ( d h Gauss ntgraton ( ( n f d = w f + R = n R d f d n n = 0 n =a =- 5 n =b = Hr n s th numbr of spcall dfnd Gauss ntgraton ponts, w ar th ntgraton wghts, and ar samplpont abcssa n th ntrval [,]. h us of th ntrval [,] s no rstrcton, bcaus an ntgral ovr anothr rang, from a to b can b transformd to th standard ntrval va a smpl lnar transformaton of th ndpndnt varabl, as shown abov. h valus and w ar dfnd n such a wa to am for bst accurac. Indd, f w assum a polnomal prsson, t s as to chck that for n samplng a polnomal of dgr n can b actl ntgratd.

9 Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag 9 of 0 abl blow shows th postons and wghtng coffcnts for gaussan ntgraton. Abscssa and wght coffcnts of th gaussan Quadratur n ç (=,n W (=,n 0 / + / /9 /9 5/ Rmarks: h sum of wghng coffcnts s alwas h ntgraton gvs th act soluton for polnomals of n- dgr. Numrcal ntgraton rctangular rgon: n n n f ( ξ, dξd f ( ξ, w d w j w f ( ξ, j = = j= = ( ξ, ( ξ, n n m = w w f = w f j j k k k = = k = f( w = = w 5 = = w = = ξ

10 Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag 0 of 0 Eampl D FE modl of th cantlvr bam RESULS OBAINED USING -NODE ELEMENS - AVERAGING Bndng strss σ dstrbuton (lmnt soluton PRES-NORM ANSS.0 MA 00 0:9: PLO PRES-NORM NO. 5 ANSS.0 MA 00 0:0: PLO NO. 6 MA 00 PLO NO σ lmnt and nodal soluton σ lmnt and nodal soluton

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