Consider an initially straight beam as shown: E= MPa P2 P3. We would like to bring the free ends of the beam together to form a ring, as shown:

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1 Poblem decipion Poblem 58: Bending of a beam ino a Möbiu ip uing an alignmen elemen Conide an iniially aigh beam a hown: All lengh in mm E= MPa Co-ecion (no dawn o cale) P1 P2 P We would like o bing he fee end of he beam ogehe o fom a ing, a hown: Iniial configuaion: z P1 P2 P3 y Ring: (end epaaed fo claiy) z P2 x y P1 P3 ADINA R & D, Inc. 58-1

2 Then we would like o wi he fee end of he beam o fom a Möbiu ip, a hown: Möbiu ip: (end epaaed fo claiy) z x P2 y P1 P3 (Noice ha he and axe a P1 have been oaed elaive o he and axe a P3.) Boh of hee ak can be done uing an alignmen elemen. In hi poblem oluion, we will demonae he following opic ha have no been peened in peviou poblem: Uing alignmen elemen Selecing he nonymmeic pae olve Befoe you begin Pleae efe o he Icon Locao Table chape of he Pime fo he locaion of all of he AUI icon. Pleae efe o he Hin chape of he Pime fo ueful hin. Thi poblem can be olved wih he 900 node veion of he ADINA Syem. Much of he inpu fo hi poblem i oed in he following file: pob58_1.in. You need o copy hi file fom he folde ample\pime ino a woking diecoy o folde befoe beginning hi analyi. If you ae unfamilia wih he alignmen elemen, we ecommend ha you ead Secion 2.12 of he ADINA Sucue Theoy and Modeling Guide. In paicula, you hould familiaize youelf wih he concep of he alignmen elemen iad. Invoking he AUI and chooing he finie elemen pogam Invoke he AUI and e he Pogam Module dop-down li o ADINA Sucue ADINA Pime

3 Foming he ing Model definiion Poblem 58: Bending of a beam ino a Möbiu ip uing an alignmen elemen We have pepaed a bach file (pob58_1.in) ha define he following iem: Poblem heading Conol daa, including oluion oleance. The ATS mehod i uned off (ee he noe a he end of he poblem decipion). Geomey poin and line Co-ecion Bounday condiion. One poin ha boh anlaion and oaion fixed, in ode o emove igid-body moion. Elemen goup 1, which i a lage diplacemen Hemiian beam elemen goup. The elemen goup conain 20 equally paced beam elemen. Chooe File Open Bach, navigae o he woking diecoy o folde, elec he file pob58_1.in and click Open. The gaphic window hould look omehing like hi: TIME Z X Y B 12 B U 1 U 2 U B ADINA R & D, Inc. 58-3

4 Alignmen elemen definiion Now we will define he alignmen elemen. The alignmen elemen will connec he end-node of he beam model (node 1 and 21), a hown: Node 1 Local node 1 Node 21 Local node 2 Alignmen elemen Click he Elemen Goup icon field o Veboe and click OK., add goup 2 and e he Type o Alignmen. Se he Save Chooe Mehing Elemen Elemen Node, and, in he fi ow of he able, e he Alignmen Elemen # o 1, Node 1 o 1, Node 2 o 21 and click OK. Ploing he alignmen elemen iad Click he Modify Meh Plo icon, click he Node Depicion... buon, check he A iad field and click OK wice o cloe boh dialog boxe. Then, in he Model Tee, expand he Zone field, igh-click on 3. EG2 and chooe Diplay. The gaphic window hould look omehing like he op figue on he nex page. The diplay how he alignmen elemen, along wih he A iad of he node. We ee ha he A iad coincide wih he beam elemen axe (ha i, he A1 diecion coincide wih he diecion, he A2 diecion coincide wih he diecion, he A3 diecion coincide wih he diecion). Now click he Modify Meh Plo icon, click he Node Depicion... buon, uncheck he A iad field, check he B iad field and click OK wice o cloe boh dialog boxe. The gaphic window hould look omehing like he boom figue on he nex page ADINA Pime

5 TIME Z X Y 1 21 U U U No bounday condiion aigned TIME Z X Y 1 21 U U U No bounday condiion aigned Now he B iad of he node ae diplayed. The B iad coincide wih he A iad by defaul. ADINA R & D, Inc. 58-5

6 Now click he Modify Meh Plo icon, click he Node Depicion... buon, uncheck he B iad field, check he C iad field and click OK wice o cloe boh dialog boxe. The gaphic window hould look omehing like hi: TIME Z X Y 1 21 U U U No bounday condiion aigned Now he C iad of he node ae diplayed. The C iad coincide wih he B iad by defaul. We hu have, a each alignmen elemen local node, he following iad definiion:, a3, b3, c3, a1, b1, c1, a2, b2, c ADINA Pime

7 Align-anlaion and align-oaion definiion Now we can pecify he elaive anlaion and oaion of he alignmen elemen node. When he ing i fomed, he alignmen elemen node and iad will appea a in he following figue (iad epaaed fo claiy): b2 b3 b2 b1 Local node 2 b3, c3 b1 Local node 1 We ee ha he iad oigin hould coincide, and ha he iad B axe diecion hould alo coincide. Of coue, he iad B axe diecion iniially coincide, bu we noe ha if we oae he iad B axe of local node 2 by 360 degee abou he c3 axi of local node 1, he iad B axe diecion will again coincide, and ha he poce of oaing he iad B axe will bend he beam ino a ing. Thu, we pecify he elaive oaion of he alignmen elemen node uing an align-oaion definiion. Chooe Model Alignmen Popeie Roaion Alignmen, add Roaion Alignmen 1, e he Angula Uni o Degee, ene he following infomaion in he able and click OK: Time Opion Angle Axi c3 1 Angle Angle Angle Angle Thu, fo example, a ime 1, we pecify ha he elaive oaion of he iad i 90 degee abou he axi given by componen (c1, c2, c3) = (0,0,1); hee componen ae in he c iad yem of local node 1, o, in ohe wod, he axi of oaion i he c3 axi of local node 1. Now, afe hi oaion, he iad diecion ae foced o coincide, bu he iad oigin ae ill fee o anlae elaive o each ohe. So we need o enfoce he condiion ha he iad oigin coincide. Thi i done uing an align-anlaion definiion. Chooe Model Alignmen Popeie Tanlaion Alignmen, add Tanlaion Alignmen 1, ene he following infomaion in he able and click OK: Time Opion 5 Faco 6 Aligned ADINA R & D, Inc. 58-7

8 Befoe ime 5, he anlaion alignmen i no acive. A ime 5, he anlaion alignmen i acivaed wih Opion=Faco. Since he defaul Faco i 1.0, hi mean ha he pecibed alignmen fo ime 5 i equal o he cuen alignmen, whaeve ha alignmen migh be. A ime 6, he pecibed anlaion alignmen caue he wo iad oigin o coincide. Elemen goup definiion: Click he Elemen Goup icon, make ue ha he Goup Numbe i 2, e he Tanlaion Alignmen o 1, he Roaion Alignmen o 1 and click OK. Time epping: Chooe Conol Time Sep, e he Numbe of Sep o 6 in he fi ow of he able and click OK. Geneaing he ADINA Sucue daa file, unning ADINA Sucue, loading he pohole file Click he Save icon and ave he daabae o file pob58. Click he Daa File/Soluion icon, e he file name o pob58, make ue ha he Run Soluion buon i checked and click Save. You can ignoe he "Model may be unable" o "Siffne maix no poiive definie" meage ha appea in he ADINA Sucue window. When ADINA Sucue i finihed, cloe all open dialog boxe. Se he Pogam Module dop-down li o Po-Poceing (you can dicad all change), click he Open icon open pohole file pob58. and Po-poceing Click he Io View 1 icon, hen click he Modify Meh Plo icon, click he Elemen Depicion... buon, check he Diplay Beam Co Secion field and click OK wice o cloe boh dialog boxe. The gaphic window hould look omehing like he op figue on he nex page. Now click he Fi Soluion icon, ue he Pick icon and he moue o eize he meh plo ino he gaphic window, click he Modify Meh Plo icon, click he Node Depicion... buon, check he B iad field and and click OK wice o cloe boh dialog boxe. The gaphic window hould look omehing like he boom figue on he nex page ADINA Pime

9 TIME Z X Y TIME Z X Y We ee ha a ime ep 1, he elaive oaion of he B iad i 90 degee. ADINA R & D, Inc. 58-9

10 When you examine he ohe oluion, you will noice ha he oluion fo ime 4 o 6 ae nealy idenical. Liing he alignmen: Le' li he alignmen, o veify ha he pecibed alignmen ae in fac applied o he model. Chooe Li Value Li Zone, e Vaiable 1 o (Alignmen: ALIGN_ROT_ACTUAL_MAGNITUDE), Vaiable 2 o (Alignmen: ALIGN_TRANS_ACTUAL-C1) and click Apply. We ee ha up o ime 4, he acual oaion magniude i he pecibed value, and ha a ime 4, he oaion magniude i vey mall (coeponding o he diecion of he iad axe coinciding). A ime 5, he anlaional alignmen i abou -1.7E-8 (mm), and a ime 6, he anlaional alignmen i zeo (coeponding o he iad oigin coinciding). Click Cloe o cloe he dialog box. Foming he Möbiu ip Now we will fom he Möbiu ip. Se he Pogam Module dop-down li o ADINA Sucue (you can dicad all change) and chooe daabae file pob58.idb fom he ecen file li nea he boom of he File menu. The addiional ep equied in foming he Möbiu ip i o wi he wo end of he ip 180 degee. In em of he alignmen elemen iad, hi appea a follow: Local node 2 b3 b2 b1 b2 b1, c1 b3 Local node 1 Evidenly we need o pecify a elaive incemenal oaion of 180 degee abou he c1 axi of local node 1. Chooe Model Alignmen Popeie Roaion Alignmen, edi he able o ead a follow and click OK: Time Opion Angle Axi c1 Axi c3 1 Angle Angle Angle Angle Same a Peviou 24 Incemenal Angle We ee ha he oaion abou axi c1 occu beween ime 7 and 24, o ha he incemenal oaion i 10 degee pe ime ep ADINA Pime

11 The ime 6 ow i neceay fo he following eaon. Suppoe ha he ime 6 ow i no eneed. Then, fo oluion ime beween 4 and 24, he pogam inepolae daa fom he ime 4 and 24 ow of hi able, hence an incemenal oaion i aleady pecibed a oluion ime 6. Wih he ime 6 ow eneed, fo oluion ime beween 4 and 6, he pogam inepolae daa fom he ime 4 and 6 ow, and fo oluion ime beween 6 and 24, he pogam inepolae daa fom he ime 6 and 24 ow. Time epping: Chooe Conol Time Sep, e he Numbe of Sep o 24 in he fi ow of he able and click OK. Geneaing he ADINA Sucue daa file, unning ADINA Sucue Click he Save icon, click he Daa File/Soluion icon, e he file name o pob58, make ue ha he Run Soluion buon i checked and click Save. The model doe no convege fo ep 8 (he econd ep in which he incemenal oaion i applied). Cloe all open dialog boxe. Specifying he equaion olve We noe ha he incemenal oaion caue oion in he beam elemen, hu he beam elemen ae unde combined bending and oion. Fo lage diplacemen beam elemen unde combined bending and oion, he nonymmeic equaion olve i fequenly effecive. Chooe Conol Soluion Poce, e he Equaion Solve o Nonym. Spae and click OK. Geneaing he ADINA Sucue daa file, unning ADINA Sucue, loading he pohole file Click he Save icon, click he Daa File/Soluion icon, e he file name o pob58, make ue ha he Run Soluion buon i checked and click Save. Thi ime he model un fo all 24 ep. When ADINA Sucue i finihed, cloe all open dialog boxe. Se he Pogam Module dop-down li o Po-Poceing (you can dicad all change), click he Open icon and open pohole file pob58. ADINA R & D, Inc

12 Po-poceing Click he Modify Meh Plo icon, click he Elemen Depicion... buon, check he Diplay Beam Co Secion field, click he Advanced ab, e he # Segmen fo Neual Axi o 4 and click OK wice o cloe boh dialog boxe. The gaphic window hould look omehing like hi: TIME Z X Y Evidenly he incemenal oaion ha caued he end of he beam o wi ino he Möbiu ip hape. Liing he alignmen elemen foce and momen: Chooe Li Value Li Zone, e Vaiable 1 o (Alignmen: ALIGN_TRANS_FORCE-C1), Vaiable 2 o (Alignmen: ALIGN_TRANS_FORCE-C2), Vaiable 3 o (Alignmen: ALIGN_TRANS_FORCE-C3) and click Apply. We ee he foce needed o mainain he anlaional alignmen inceae aing a ime 7, and ha a ime 24, he foce ha componen ( E-02, E-01, E-07) (N); hee componen ae meaued in he C iad diecion of local node 1. (You foce componen migh have lighly diffeen numeical value.) Now e Vaiable 1 o (Alignmen: ALIGN_ROT_MOMENT_MAGNITUDE) and click Apply. A ime 24, he momen ha magniude E+01 (N-mm). Click Cloe o cloe he dialog box. Exiing he AUI: Chooe File Exi o exi he AUI. You can dicad all change ADINA Pime

13 Noe We uned off he ATS mehod in hi poblem, o ha we can demonae he ue of he nonymmeic pae olve. If he ATS mehod i ued (he defaul), hi poblem can be olved wih he defaul pae olve. Howeve he ATS mehod will pefom cuback. ADINA R & D, Inc

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