A TRIANGULAR FINITE ELEMENT FOR PLANE ELASTICITY WITH IN- PLANE ROTATION Dr. Attia Mousa 1 and Eng. Salah M. Tayeh 2

Transcription

1 A TRIANGLAR FINITE ELEMENT FOR PLANE ELASTICITY WITH IN- PLANE ROTATION Dr. Atti Mous nd Eng. Slh M. Teh ABSTRACT In the present pper the strin-bsed pproch is pplied to develop new tringulr finite element for the generl plne elsticit. This tringulr finite element hs three degrees of freedom two generl eternl degrees of freedom nd the in-plne rottion t ech of the three corner nodes. This element is bsed on liner nd qudrtic vritions of the three components of strin. This element is used to obtin solutions to two dimensionl elsticit problems where the contribution of the sher stress on deformtion cn be significnt. The performnce of the element is investigted b ppling it to two well-known plne elsticit problems; cntilever bem loded t the free end nd simpl supported bem loded b point lod t the midspn. Convergence curves re plotted for verticl deflection t the points of ppliction of lod s well s for bending stress t points t the upper tension fibre nd shering stress t internl points in ech problem. It is found tht the solutions obtined using this element re stisfctor for results of deflection bending stress nd the shering stress even when onl smll number of elements is used in the finite element solution. KEYWORDS: strin bsed pproch tringulr element plne elsticit in-plne rottion. INTRODCTION The development of displcement fields b the use of strin-bsed pproch ws first pplied to curved elements. It ws reveled tht to obtin stisfctor converged results the finite elements bsed on independent polnomil displcement functions require the curved structures to be divided into lrge number of elements [] []. This work hs shown tht there re two essentil components to n displcement field. The first of which reltes to rigid bod modes of displcements while the second component is due to the strining of the element nd these re pproimtel represented b ssumed independent polnomil terms for the vrious component of strins in so fr s it is llowed b the comptibilit equtions. In reference [] it ws shown tht the bove pproch is not confined to the development of curved elements onl but lso to plne elements for the nlsis of plne elsticit problems. Severl such rectngulr elements were developed notbl rectngulr element bsed on liner vrition of the direct strins nd constnt shering strin. This element hs the two essentil degrees of freedom t ech of the four corner Den of Fult of Engineering Islmic niversit of Gz Gz Strip Plestine Structurl Engineer Engineering & Mngement Consulting Center EMCC Gz Strip Plestine C-

2 nodes nd it ws shown to produce converged results when pplied to severl elsticit problems without the use of lrge number of elements. In reference [] Sfendji developed rectngulr element bsed on liner vrition of ll the three strin components. It hs two degrees of freedom t ech comer node nd t n internl node. A rectngulr finite element with in-plne rottion ws developed in reference []. In this pper new tringulr element with in-plne rottion is developed nd its performnce is compred to tht of the well know Constnt Strin Tringle.. ANALYTICAL CONSIDERATIONS. Displcement Fields for Strin Bsed Tringulr Element with In-plne Rottion In this section the strin pproch is used to derive the displcement fields for strin bsed tringulr element with in-plne rottion. The element hs thee degrees of freedom t ech of its three corner nodes shown in Figure. v u v φ u v φ u v φ Figure : Coordintes nd nodl points of tringulr element with in-plne rottion DOF per node The displcement fields re required to stisf the requirement of strin-free rigid bod mode of displcement in ddition to the strining within the element. To get the first prt of the displcement fields corresponding to the first mode we begin b writing the strin/displcement reltionships nd mke them equl to zero. B integrtion we obtin the epressions for the displcements tht give zero strins s follows. ε > f ε > f C- u γ > f' f' where f nd f re constnts tht cn be tken s

3 f ' f ' thus > > f f Eq. The in-plne rottion is clculted using the reltion φ Eq. where : displcements in the nd directions respectivel ε ε : direct strins in the nd directions respectivel γ : sher strin : trnsltions of the element in the & directions respectivel. : rigid bod in-plne rottion of the element. The displcements within the element hve to be defined b nine constnts through no. of constnts should equl the no. of nodes times the no. of DOF per node. Three constnts & hve lred been defined while the remining si hve to be used to describe the deformtion strining of the element. A first ttempt to do so is to ssume tht ε ε γ 6 Eq. This rrngement of strins does not contin constnt term in the epression for γ nd it is epected tht it wouldn t give good results. Also it ws found tht it leds to singulr displcement trnsformtion mtri nd hence it cn t be used to derive stiffness mtri. Severl other rrngements were tried to void this. A good rrngement tht gives none-singulr displcement trnsformtion mtri is found to be s follows: ε ε 6 Eq. γ The constnts 6 nd re the terms corresponding to stte of constnt strin tht ensures the convergence of the solution with mesh refinement. The constnts nd re the terms corresponding to liner strin behviour within the element. We observe tht if the terms of this eqution re twice differentited the stisf the generl comptibilit eqution of strins nmel: ε ε γ Eq. C-

4 C- To get the second prt of the displcement fields we first integrte the fist two equtions s follows. f f 6 Eq. 6 To get the functions f nd f we substitute their derivtives in the sher strin eqution then seprte the resulting epressions for nd respectivel s follows: f f γ ' ' Eq. ]d [ f ] d [ f Eq. Now f nd f re substituted in. B dding the epressions for & nd & then clculting the in-plne rottion Eq. the complete epressions for the displcement fields re obtined s: / Φ 6 Eq. It is noted tht we obtined qudrtic terms & without incresing the number of nodes beond the three corner nodes. This is not chieved in the known constnt strin tringulr element. It is epected tht this increse in the degree of the polnomils will result in ccurte solutions using this element s will be shown in the subsequent discussion. Hving obtined the displcement fields the stiffness mtri of ech tringulr element cn be evluted using the generl epression [ ] [ ] [ ] [ ][ ] [ ][ ] T T T e C dvol B D B C K Eq. where the trnsformtion mtri [C] is clculted s

5 C- [ ] Φ @ C nd the strin mtri [ ] B for this element is [ ] - - B nd [D]is the rigidit mtri given b [ ] E D for the stte of plne stress nd [ ] - - E D for the stte of plne strin. The performnce of the element derived bove is investigted b ppling it to two of the fmous plne elsticit problems s described below. The results obtined b the developed element re compred to those given b the well-known constnt strin tringulr nd the nlticl vlue for deflection [] nd stresses.. DEEP CANTILEER BEAM LOADED AT THE FREE END. Problem Description The first problem is deep cntilever bem loded b point lod t the free end. The bem hs length Lm height Hm nd thickness t.6m. The mteril properties; modulus of elsticit nd Poisson s rtio re tken s E KP nd. respectivel. The point lod t the free end of the cntilever is tken s P KN. In order to chieve full fiit t the built-in end of the cntilever ll the nodes occurring t tht end re ssumed to be restrined in both the nd directions s well s the in-plne rottion. The loctions of the investigted points within the cntilever bem re shown in Figure. A smple mesh is shown in Figure.

6 P B A C H m t L m Figure : Dimensions of the cntilever bem problem Figure : Smple mesh of the cntilever bem problem. Convergence Results of the Deep Cntilever Bem Tble shows summr of the used mesh sizes in the solution nd results for verticl deflection bending stress shering stress nd the in-plne rottion t the specified points within the deep cntilever bem. Figures show grphicl comprison between the results obtined b ech of the the constnt strin element nd the ect nlticl solution. Tble : Results obtined b the nd for the cntilever bem problem Mesh Size No. of Elements No. of Nodes erticl Deflection t A mm Bending Stress t B KP Shering Stress t C KP In-Plne Rottion t A Rd Ect Solutions. 6.6 C-

7 . erticl Deflection mm Anlticl Solution Figure : erticl deflection t "A" mm in the cntilever bem Bending Stress KP Anlticl Solution Figure : Bending stress t "B" KP in the cntilever bem C-

8 6 Shering Stress KP Anlticl Solution Figure 6: Shering stress t "C" KP in the cntilever bem..6 In-plne rottion Rd Anlticl Solution Figure : In-plne rottion t "A" Rd in the cntilever bem C-

9 . SIMPLY SPPORTED BEAM. Problem Description The second problem is simpl supported bem loded b point lod t the middle of its upper fce. The bem hs length Lm height Hm nd thickness t.m. The mteril properties re tken s E KP nd. respectivel. The point lod is tken s P. KN. The loctions of the investigted points within the simpl supported bem re shown in Figure below. P.m B A C.m.m H. t L m Figure : Dimensions of the simpl supported bem problem. Convergence Results of the Simpl Supported Bem Tble shows summr of the used mesh sizes in the solution nd results for verticl deflection bending stress nd shering stress t the specified points within the simpl supported bem. Figures - show grphicl comprison between the results obtined b ech of the the constnt strin element nd the ect nlticl solution. Tble : Results obtined b the nd for the simpl supported bem problem Mesh Size No. of Elements No. of Nodes erticl Deflection t A mm Bending Stress t B KP Shering Stress t C KP Ect Solutions C-6

10 erticl Deflection mm Anlticl Solution Figure : erticl deflection t "A" mm in the simple bem.. Bending Stress KP..... Anlticl Solution Figure : Bending stress t "B" KP in the simple bem C-

11 Shering Stress KP Anlticl Solution Figure : Shering Stress At "C" KP in the simple bem. CONCLSIONS The solutions obtined using the proposed element re stisfctor for results of deflection bending stress shering stress nd the in-plne rottion even when onl smll number of elements re used in the finite element solution. The new element gives good results nd convergence to the nlticl solution. Also it hs fewer discontinuities in the corner stresses thn the constnt strin element 6. REFERENCES [] Ashwell D. G. Sbir A. B. nd Roberts T. M. Limittions of certin finite elements when pplied to rches Int. J. Mech. Sci. ol.. [] Sbir A. B. THE NODAL solution routine for lrge number of liner simultneous equtions in finite element nlsis of pltes nd shells Finite elements for thin shells nd curved members Ashwell & Gllgher Wille 6. [] Sbir A. B. Stiffness mtrices for generl deformtion out of plne nd inplne of curved bems bsed on independent strin functions The mthemtics of finite elements nd pplictions II. Acdemic press. [] A. Sfendji Finite elements for plne elsticit problems. M. Sc. Thesis niversit of Wles.K.. [] A. I. Mous nd S. M. Teh A Rectngulr Finite Element For Plne Elsticit With In-Plne Rottion 6th Int. Conf. on Computtionl Structures Technolog nd rd Int. Conf. on Engineering Computtionl Technolog Prgue Czech Republic -6 September. [6] Timoshenko S. nd Goodier J. N. Theor of Elsticit Third Edition McGrw Hill New York. C-