A New Analytical Integration Expression Applied To Quadrilateral Finite Elements

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1 A New Analytcal Integraton Expresson Appled To Quadrlateral Fnte Elements D. HAMADI and A. ZATAR 2 Abstract A new analytcal ntegraton expresson s presented and used to evaluate the element stffness matrx for the quadrlateral fnte elements. For reasons of applcatons and partcularty of the stran based fnte elements (hgher order shape functons expressed n terms of ndependent strans); t s necessary to ntroduce rregular forms, whch requre a specal ntegraton technque, and a specfc classfcaton n programmng level for dfferent geometrc forms. To overcome ths geometrcal nconvenence; the paper presents the new ntegraton expresson wth valdaton tests, and t s appled to some quadrlateral fnte elements, ths wll help to extend ther applcatons for the dstorted forms and rregular structures. Index Terms Analytcal ntegraton, rregular forms, new expresson, quadrlateral element. I. INTRODUCTION Many researchers among them, Ashwell, Doud, Sabr, Salh and elarb, have developed numerous fnte elements; some of them were undertakng ther research work at Cardff Unversty n the U.K. The formulated elements are based on the stran approach, and characterzed by a regular form and approprate coordnates wth the form of the element. The stran based approach was further appled by Sabr [] to develop a new class of elements for general plane of elastcty problems n Cartesan coordnates. A smple an effcent rectangular element ncludng the n-plane rotaton s derved. Ths element was frst appled to the smple problem of cantlevers and smply supported beams, where the results for deflectons as well as stresses were satsfactory and converged to the exact soluton. Wth the contnuaton of the development of the stran based approach many elements for general plane elastcty as well as shells have been derved by Sabr [2] and elarb [7]. To model a structure whch has rregular geometrcal shape n real problem, by a lmted number of elements as cted above; s not suffcent at all. To overcome ths geometrcal nconvenence; the paper presents a new ntegraton expresson. The performance of ths new expresson s tested by applyng to the analyss of the problems used n prevous publcatons and to obtan solutons for some practcal problems n engneerng. Manuscrpt submtted for revew December 27, A New Analytcal Integraton Expresson Appled to Quadrlateral Fnte Elements. Damel. HAMADI, LESIA Laboratory, Cvl Engneerng Department, Faculty of Scences and Engneerng Scences, skra Unversty, , Algera, fax: ; (e-mal: dhamaduk@yahoo.com). Abdallah. ZATAR, Cvl Engneerng Department, Faculty of Scences and Engneerng Scences, skra Unversty, , Algera fax: ; (e-mal: abdullah_zatar@yahoo.fr). II. NUMERICAL INTEGRATION The element stffness matrx [Ke] can be calculated usng the well known equaton (). T [ e ] = [ ] [ ][ ]det. ξ. K D J d dη S () Where: []: the stran matrx [D]: the behavour matrx det J: the determnant of Jacoban matrx To carry out the ntegral, we have to choose ether numercal ntegraton (e.g Gauss ntegraton) or analytcal ntegraton. One of the dsadvantages of the numercal ntegraton s the hgh order of the monomals after the three multplcatons of ntegral matrces (), whch would sgnfy many ntegraton ponts. III. A NEW ANALYTICAL INTEGRATION The evaluaton of the element stffness matrx s summarsed wth the evaluaton of the followng expresson: T T [ K ] = [ A ] [ Q] [ D][. Q]. dx. dy [ A ] e T [ K ] = [ A ] [ K ][ A ] S 0 (2a) e (2b) T Wth: [ K 0] [ Q] [ D ][. Q]. dx dy = (2c) s [A] and ts nverse can be evaluated numercally, the evaluaton of the ntegral (2c) becomes the key of the problem. In general, the multplcaton [Q] T [D] [Q] can be done manually, we wll end up by calculatng the double ntegrals of the form: α β [ K0 ] = C. x y d x. d y I = (3) s Knowng that, for certan elements, a too great dstorton can lead to erroneous numercal results partcularly n the calculaton of the Jacoben. An expresson that s general, and easy to mplement numercally s beng formulated. It allows the evaluaton of the matrx [K 0 ] n an automatc way whatever the degree of the polynomal of the knematcs feld and the dstorton of the element (Fg..). ISN: WCE 2008

2 Y 4 y 4 I y II 2 y 3 III 3 y 2 X X4 X2 X3 Fg.. Quadrlateral element I = I + (4) In whch + II2 III3 β + 2 = 2 ( ) α β x y dxdy C k β + k= k + α k k k+ α k+ α.( a b a b )( x x ) In our case: 3 I = I = (6) The general expresson of I for a quadrlateral would be: I β + 2 C = C( k) β + k + α k k k+ α k+ α ( a b a b )( x x ). n m 3 X (5) k= (7) The calculaton of ntegral I s the prncpal problem of the calculaton of the element stffness matrx [Ke]. In a very smple and effectve manner, the ntegral s solved by the subroutne ''INTEGRATION'' [8], [9], see Appendx. h Data: Young s modulus E = 00, osson s rato v = 0 =, L=0, h=, Thckness t= Fg.2. A smple beam; Data and meshes L n Regular mesh Dstorted mesh m IV. EVALUATION OF THE ELEMENT STIFFNESS MATRIX [KE] After the programmng of the routnes whch calculate the ntegral, we can fnally carry out the calculaton of the element stffness matrx [Ke]. V. TESTS AND ALICATIONS In order to nvestgate the new ntegraton expresson s thus developed. We have chosen to examne the membrane element Q4SE [8], [0] through the followng tests. These tests are regarded as a tool to valdate of the membrane elements. A. A Smple eam A smple beam wth a length to heght aspect rato of 0 s subected to a pure bendng state. The beam s modelled by x6 meshes wth both regular and rregular elements as shown n Fg.2. Only a mnmum number of restrants are mposed to elmnate rgd body movement. The load s a unt couple appled at the free end. Ths beam s selected as a test problem by Ibrahmbegovc, Taylor and Wlson []. The results obtaned for both regular and rregular mesh are compared wth some of the results avalable n lterature, and the exact soluton gven by beam s theory. All are presented n Table I. The results obtaned for the dstorted element Q4SE are found to be more accurate than the other elements for the same fnte element mesh sze Table I. It s observed that the results show very good numercal accuracy obtaned for both regular and dstorted mesh, and confrm the good performance of the Q4SE element. TALE I: A SIMLE EAM UNDER URE ENDING Formulaton Mesh Vertcal dsplacement Mxte-type [] Reg.,50000 Mxte-type [] Dst.,485 Dspl-type [] Reg.,50000 Dspl-type [] Dst.,4045 Taylor et Smo [2] Reg.,50000 Taylor et Smo [2] Dst.,495 Q4 Reg. 0,62888 Q4 Dst. 0,26362 Q4SE* Reg.,50000 Q4SE* Dst.,50000 eam s theory,50000 * Q4SE: Stran ased Quadrlateral Element [8], [0]. Tapered anel Under End Shears Ths problem, proposed by Cook as a test for the accuracy of quadrlateral elements [3] and ergan et al. [4], s another popular test problem. A tapered panel of unt thckness wth one edge subected to a dstrbuted shear load and wth the other edge fully clamped (u=v=0) s shown n Fg.3. The panel s analyzed by usng 2x2 and 4x4 meshes ISN: WCE 2008

3 (Fgs. 3a, 3b). The normalzed vertcal deflecton Vc at pont C, maxmum prncpal stress σ maxa at pont A and mnmum prncpal stress σ mn at pont are presented n Table II. The results obtaned for the Q4SE element are compared to the other quadrlateral elements. It can be noted that the dsplacement predctons of the Q4SE are slghtly better than the other quadrlateral elements for both meshes (Table II). The results obtaned for the deflecton and prncpal stresses for the refned mesh (4x4) are very good compared to an accurate soluton gven by ergan and Felppa usng a (32x32) mesh [4] (error %). 6 n E 48 n C y E y 44 n A A D (a) 2 x 2 mesh D (b) 4 x 4 mesh Fg.3: Tapered anel Subected to End Shear; Data and Meshes y = p (Unformly Dstrbuted load) Young s modulus E = ps osson s rato ν = /3 Thckness t = n oundary condtons: U=V=0 (DE) Element model TALE II: RMALISED REDICTION FOR TAERED ANEL UNDER END SHEAR 2 x 2 mesh 4 x 4 mesh V C σ maxa σ mn V C σ maxa σ mn Q4 0,496 0,437 0,533 0,766 0,756 0,79 AQ[6] 0,890 0,780 0,900 0,965 0,936,00 Ref. [5] 0,848 0,77 0,856 0,953 0,956 0,997 S5β[7] 0,884 0,786 0,77 0,963 0,950 0,924 MAQ[8) 0,890 0,779 0,886 0,965 0,94 0,967 Ref. [4] 0,852 0,720 0,898 0,938 0,902 0,849 Ref [] 0, , β[9] 0,945 0,835,069 0,98 0,982,02 Q4SE[0],0652,508,7,0,004 0, x 32 mesh Ref. [4],000 (23,90),000 (0,236),000 (-0,20),000 (23,90),000 (0,236),000 (-0,20) AQ: Cook s quadrlateral counterpart Cook [6]. S5β: an and Sumhara s four- node fve-beta mxed element an [7]. MAQ: a mxed counterpart of AQ usng complete lnear stress modes (n term of soparametrc coordnates) for all stress components Yunus [8]. 07β: the Sze element [9]. I. CONCLUSION The effcency of the new ntegraton expresson was shown. The sgnfcance of the analytcal ntegraton to evaluate the element stffness matrx for the fnte elements wth rregular shapes was examned. The good results are partly explaned probably by the nature of analytcal. ISN: WCE 2008

4 AENDIX SUROUTINE INTEGRATION DATA IU<=49 TI(IU) = 0 TI2(IU) = 0 TI3(IU) = 0 SUROUTINE FORM_ICORD SUROUTINE COEFF dx() >small and dx(4) >small SUROUTINE EXRESS (x(icord()),x(icord(2)),a,b,a4,b4,,ti) dx(2) >small and dx(4) >small ICO = ICORD() + ICO = 5 ICO = ICORD(2) = ICO SUROUTINE EXRESS (x(icord(2)),x(icord(3)),a2,b2, a4,b4,2,ti2) ISN: WCE 2008

5 dx() >small and dx(3) >small ICO = ICORD() - ICO = 0 ICO = 4 ICORD(2) = ICO SUROUTINE EXRESS (x (ICORD(2)), x(icord(3)), a, b, a3, b3, 2, TI2) dx(3) >small and dx(4) >small ICO = ICORD() - ICO = 0 ICO = 4 ICORD(4) = ICO SUROUTINE EXRESS (x (ICORD(3)),x(ICORD(4)), a3, b3, a4, b4, 3, TI3) dx(2) >small and dx(3) >small ICO = ICORD() + 2 ICO = 5 ICO = ICO = 6 ICO = 2 ISN: WCE 2008

6 ICORD(4) = ICO SUROUTINE EXRESS (x(icord(3)),x(icord(4)),a2,b2, a3,b3,3,ti3) dx() >small and dx(2) >small ICO = ICORD() + ICO = 5 ICO = ICORD(4) = ICO ROCEDURE EXRESS (x(icord(3)),x(icord(4)),ab, a2,b2,3,ti3) SUROUTINE TI END REFERENCES [] A.. Sabr, "A new class of fnte elements for plane elastcty problems" CAFEM7 7 th Int. Conf. Struct. Mech. n reactor technology, Chcago, 983. [2] A.. Sabr, "A rectangular and trangular plane elastcty element wth drllng degrees of freedom" Chapter 9 n proceedng of the 2 nd nternatonal conference on varatonal methods n engneerng, Southampton Unversty, Sprnger-Verlag, erln, pp. 985, pp [3] A.. Sabr and F. Ramadhan "A shallow shell fnte element for general shell analyss" Varatonal methods n engneerng, roceedngs of the 2 nd Internatonal Conference, Unversty of Southampton England (985). [4] A.. Sabr and A. Sfend "Trangular and rectangular plane elastcty fnte elements "Thn-Walled Structures Vol. 2, 995, pp [5] elarb M.T. and A. Charf, "Développement d'un nouvel élément hexaédrque smple basé sur le modèle en déformaton pour l étude des plaques mnces et épasses, Revue Européenne des éléments fns, vol. 8, N 2, 999, pp [6] M.T. elarb and D. Hamad, "Ameloraton of the Sabr rectangular fnte element SRIER based on the stran model", proceedng st Internatonal Conference of Cvl Engneerng Scence, ICCES, Assut, Egypt, 2003, vol., pp [7] M.T. elarb and T. Maalem, "On mproved rectangular fnte element for plane lnear elastcty analyss", Revue Européenne des elements fns, vol. 4, N 8, [8] D. Hamad, and M.T. elarb, "Integraton soluton routne to evaluate the element stffness matrx for dstorted shapes. Asan Journal of Cvl Engneerng (uldng and Housng), vol. 7, N 5, 2006, pp [9] D. Hamad, "Analyss of structures by non-conformng fnte elements", hd Thess, Cvl engneerng department, skra Unversty, Algera, 2006, pp. 30. [0] D. Hamad, M. Mellas, R. Chebl and M. Nouaoura, "An effcent quadrlateral membrane element for cvl engneerng analyss, World Journal of Engneerng, Vol. 4 No., 2007, pp [] A. Ibrahmbegovc, R.L. Taylor and E.L. Wlson "A robust quadrlateral membrane fnte element wth drllng degrees of freedom", Internatonal Journal for Numercal Methods n Engneerng, Vol. 30, 990, pp [2] R. L. Taylor and J.C. Smo, "endng and membrane elements for analyss of thck and thn shells", J. Mddelton and G.N. ande (eds.), roceedng NUMETA 85, pp , (985). [3] R. D. Cook., "A plane hybrd element wth rotatonal d.o.f and adustable stffness", IJNME, Vol. 24, 987, pp [4].G. ergan and C.A. Felppa, "A trangular membrane element wth rotatonal degrees of freedom", "CMAME, vol. 50, 985, pp [5] D.J. Allman, "A quadrlateral fnte element ncludng vertex rotatons for plane elastcty analyss, IJNME, Vol. 26, pp , (988). [6] R.D. Cook, "On the Allman trangle and a related quadrlateral element" Comp. Struct., Vol. 22, (986). pp , [7] T.H. an and K. Sumhara, "Ratonal approach for assumed stress fnte elements", IJNME, vol. 20, 984, pp [8] S.M. Yanus, S. Sagal and R.D. Cook, "On mproved hybrd fnte elements wth rotatonal degrees of freedom", IJNME, vol. 28, 989, pp [9] K.Y. Sze, W. Chen and Y.K. Cheung, "An effcent quadrlateral plane element wth drllng degrees of freedom usng orthogonal stress modes", Comp. Struct., vol. 42, N 5, 992, pp ISN: WCE 2008

Module 6: FEM for Plates and Shells Lecture 6: Finite Element Analysis of Shell

Module 6: FEM for Plates and Shells Lecture 6: Fnte Element Analyss of Shell 3 6.6. Introducton A shell s a curved surface, whch by vrtue of ther shape can wthstand both membrane and bendng forces. A shell