Stress-Based Finite Element Methods for Dynamics Analysis of Euler-Bernoulli Beams with Various Boundary Conditions

Transcription

1 9 Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons Abstract In ths rsarch, two strss-basd fnt lmnt mthods ncludng th curvatur-basd fnt lmnt mthod (CFE) and th curvatur-drvatv-basd fnt lmnt mthod (CDFE) ar dvlopd for dynamcs analyss of Eulr-Brnoull bams wth dffrnt boundary condtons. In CFE, th curvatur dstrbuton of th Eulr-Brnoull bams s approxmatd by ts nodal curvaturs thn th dsplacmnt dstrbuton s obtand by ts ntgraton. In CDFE, th dsplacmnt dstrbuton s approxmatd n trms of nodal curvatur drvatvs by ntgraton of th curvatur drvatv dstrbuton. In th ntroducd mthods, compard wth dsplacmnt-basd fnt lmnt mthod (DFE), not only th rqurd numbr of dgrs of frdom s rducd, but also th contnuty of strss at nodal ponts s satsfd. In ths papr, th natural frquncs of bams wth dffrnt typ of boundary condtons ar obtand usng both CFE and CDFE mthods. Furthrmor, som numrcal xampls for th statc and dynamc rspons of som bams ar solvd and compard wth thos obtand by DFE mthod. Kywords Eulr-Brnoull bams, Strss-basd fnt lmnt, atural frquncy, Dynamc analyss. Majd Gholampour a Bahman our Rahmat Abad a,* Mhrdad Fard a Wllam L. Clghorn b a School of Mchancal Engnrng, Shraz Unvrsty, Shraz, Iran. b Dpartmnt of mchancal and ndustral Engnrng, Unvrsty of Toronto, Toronto, Ontaro, Canada. *Corrspondng author: E-mal: bahmannour@shrazu.ac.r Rcvd..7 Accptd..7 Avalabl onln 7..7 ITRODUCTIO Dsplacmnt-basd fnt lmnt (DFE) mthod has xtnsvly bn usd n computatonal sold mchancs. In ths mthod, th dsplacmnt and slop ar usd as th nodal valus n th modllng of bams. Th man dsadvantag of DFE s th dscontnuty n th strss dstrbuton. Furthrmor, strss boundary condtons ar not xactly satsfd whch causs th naccuracy of th approxmatd soluton. To lmnat th mntond problm, strss-basd fnt lmnt (SFE) has bn ntroducd (D Vubk, 95; D Vubk, 97). In ths mthod, strss dstrbuton s approxmatd by assumd strss functon and th transvrs dflctons and slops ar obtand by

2 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons ntgraton. Consquntly, th consdrd mthod provds th contnuts of not only transvrs dflcton but also strss at nods. Ths tchnqu was usd for analyzng dffrnt problms, such as Krchhoff plats (Morly, 98; Punch and Atlur, 98), plan lastc problms (Watwood and artz, 98; Wckowsk t al., 999) and lasto-plastc analyss (Wckowsk, 995; Kuo t al., ). Kuo t al. () ntroducd CFE mthod for Eulr- Brnoull bam. In thr work (Kuo t al., ), a cantlvr bam and a slwng bam wr studd. Aftr that, thy usd CFE (Kuo and Clghorn, ) and SFE mthod (Kuo and Clghorn, 7) to study a four-bar mchansm and a flxbl sldr crank mchansm wth small stran but larg rgd body moton, rspctvly. Latr, Fard and Clghorn () utlzd CFE mthod for th frst tm to modl th dynamcs of a sngl-flxbl-lnk spatal manpulator. Thy also obtand th dynamc quatons of planar mult flxbl-lnk manpulators and vrfd th rsults wth th dsplacmnt fnt lmnt mthod (Fard and Clghorn, ). Furthrmor, an mprovd curvatur-basd fnt lmnt mthod was dvlopd n (Chn t al., 5) for th dynamc modllng of a hgh-spd planar paralll manpulator wth flxbl lnks. Also, th mthod was usd for solvng a sldng bam problm (Kuo, 5). Th varyng-lngth bam lmnt was stablshd for solvng th consdrd problm. To th bst of our knowldg, th CFE mthod has bn usd for th analyss of th problms n whch th bams ar consdrd to b clampd-fr. Th man scop of th prsnt rsarch s to xtnd th CFE and to ntroduc CDFE mthod for vbraton analyss of Eulr-Brnoull bams wth dffrnt boundary condtons. Th papr s organzd as follows: Scton ntroducs both strss-basd fnt lmnt mthods. In scton, th shap functons of both CFE and CDFE mthods ar obtand for dffrnt boundary condtons n ordr to approxmat th dflcton n ach lmnt. In scton, usng Lagrang s quaton, quatons of moton ar obtand and th natural frquncs of bams ar obtand. Fnally, n scton 5, numrcal xampls rlatd to th statc and dynamc rsponss of som bams ar nvstgatd. STRESS-BASED FIITE ELEMET METODS In Fgur, th Eulr-Brnoull bam dvdd nto lmnt s dpctd. Th transvrs dflcton, slop and th nodal varabl at th lft nd of th th lmnt ar dsgnatd wth w, and v, whl thos at th rght nd ar shown wth, w, and v,rspctvly. Also, th th global nodal varabl, v n ach of CFE and CDFE mthods ar consdrd m and n, rspctvly. Fgur : An Eulr-Brnoull bam lmnt. Latn Amrcan Journal of Solds and Structurs (7) 9-7

3 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons In squnc, th shap functons n ach of th curvatur and th curvatur drvatv-basd fnt lmnt mthods ar obtand.. Curvatur-Basd Fnt Elmnt Mthod (CFE) Th curvatur dstrbuton n th th lmnt, m whr, S ( ) and S ( ) ar consdrd as n whch Th slop n th th lmnt, ( ) ( ), can b lnarly approxmatd as m S m S m () S ( ), S ( ) () ( x x )/( x x ) () can b obtand by ntgratng Eq. (). h m m c () whr, c s a constant. Consdrng th slop of th frst nod as, th constant can b wrttn as c h (5) Usng th contnuty of slop btwn th frst and th scond lmnt, th constant, c s drvd as c hh m hh m h () h In gnral, th constant c for th th lmnt can b obtand n a smlar way as c [ hh m ( hh h h ) m ( h h h h ) m h ( hh h h) m h hm h (7) Intgratng Eq. (), th transvrs dflcton n th th lmnt can b obtand by th followng quaton. w h m m c c (8) Latn Amrcan Journal of Solds and Structurs (7) 9-7

4 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons In Eq. (8), c s a constant paramtr dtrmnd by boundary condtons. Consdrng th contnuty of dflcton at th ntrnal nods, th constant s obtand as c [ h m ( h h ) m ( h h ) m ( h h ) m h h m h c h c h c c ] (9) Usng Eqs. (7-9), th dflcton of th th lmnt s approxmatd as w m w () as In th abov rlaton, For =, and ar th shap functons of th th lmnt obtand h (-a) For =,,, h h hh h (-b) k k For =,,, h (-c) For =,,, h h h h ( ) (-d) For =,,., For =, 5,., For,..., h hh h h h h (-) h h h h h h h h h h (-f) (-g) Latn Amrcan Journal of Solds and Structurs (7) 9-7

5 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons whr, s th total numbr of lmnts. Also, and ar drvd as h h h h (-a) (-b). Curvatur Drvatv-Basd Fnt Elmnt Mthod (CDFE) Th curvatur drvatv dstrbuton n th th lmnt, v n S( ) n S( ) n, can b lnarly approxmatd as () whr, S ( ) and S ( ) ar dfnd n Eq. (). Th curvatur dstrbuton n th bam can b obtand by ntgratng Eq. (). m () h n n c as Th slop and transvrs dflcton of th th lmnt can b obtand by ntgratng Eq. () h n n c c (5) w h n n c c c () n whch, c, c and c ar th constant paramtrs obtand by th contnuty of curvatur, slop and dflcton btwn lmnts. Th constants c and c ar smlar to th CFE mthod and th constant c s drvd gvn as h h h c n h h n [( ) hk ( ( ) ( ) ) k hk 5 h h h k k ( h h ( ) ( ) ) k hk h h n k ( h h ( ) ( ) k hk 5 h h k h h h h ( ) n n ] 8 ) n (7) Th dflcton of th th lmnt n th CDFE mthod can b wrttn as w n m w (8) Latn Amrcan Journal of Solds and Structurs (7) 9-7

6 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons n whch, th shap functons For = ar obtand as h (9-a) For =,,, h h h h ( ) h h (9-b) + + ( k ) 8 k For =,,, h (9-c) For =,,, h h h h( ) (9-d) For =,,., h h 8 h h h h h h h h (9-) For =, 5,., h h h h h h h h h h h h 8 h h h h (9-f) For,..., (9-g) Furthrmor,, and ar drvd as ( ) h h h h (-a) h h h h (-b) (-c) In th appndx, th frst fv shap functons n th CFE and CDFE mthods ar gvn. Latn Amrcan Journal of Solds and Structurs (7) 9-7

7 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons 5 BEAMS WIT DIFFERET BOUDARY CODITIOS In ths scton, th unknown constants n Eqs, () and (8) ar obtand by consdrng th boundary condtons. In CFE mthod, two of th boundary condtons ar usd to dtrmn th constants and w, th othr boundary condtons ar ncorporatd as constrants. In CDFE mthod, th constant m, and w ar obtand by usng thr boundary condtons and th othr on s mposd as constrant. Thrfor, th dflcton of th lmnts n th CFE and CDFE mthods can b wrttn n trms of nodal varabls as In what follows, th shap functons, w v () n th CFE and CDFE mthods ar obtand for dffrnt boundary condtons such as clampd-fr, pnnd-pnnd, pnnd-gudd, clampd-pnd, clampd-gudd and clampd-clampd.. Clampd Fr (CFE) For th clampd fr bam, th dflcton and slop of th frst nod ar zro and th boundary condtons ar wrttn as w () Thus, and w ar zro and th shap functon. Clampd Fr (CDFE) ar obtand th sam as For th clampd fr bam, th constants w, and m n Eq. (8), ar obtand usng th followng condtons w m. () Constants and w ar zro and th followng rlaton for m s drvd m [ n n n ] () as Thrfor, th shap functons can b prsntd n th form of Eq. (), whr,,..., s obtand (5) Latn Amrcan Journal of Solds and Structurs (7) 9-7

8 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons. Pnnd-Pnnd (CFE) In ths cas, th boundary condtons ar gvn as w w () Consdrng th frst boundary condton, constant w s zro. Incorporatng, th scond boundary condton, constant s obtand as [ m m m ] (7) By substtutng Eq. (7), to Eq. (), th dflcton of th nods s obtand n whch th shap functon, s obtand as,,..., (8). Pnnd-Pnnd (CDFE) Snc th dflcton and th curvatur at th lft sd of th bam ar zro, constants w and m ar zro. Constant can b obtand by consdrng zro dflcton at th lft sd of th bam as [ n n n ] (9) s ob- In ths cas, th dflcton of th bam can b wrttn n th form of Eq. (), whr tand smlar to th pnd-pnd bam n CFE mthod gvn n Eq. (8)..5 Pnnd-Gudd (CFE) For th pnnd-gudd cas, th boundary condton ar wrttn as w w () Consdrng th boundary condtons, th unknown paramtr, w s zro and th paramtr s drvd as m m m () Usng Eqs. (), and (), th nods dsplacmnt of th pnnd-gudd bam s drvd whr, s obtand as,,..., () Latn Amrcan Journal of Solds and Structurs (7) 9-7

9 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons 7. Pnnd-Gudd (CDFE) Consdrng th followng condtons w m w () Constants w and ar zro and m s drvd obtand as m n n n () In ths cas, th shap functons can b drvd as gvn n Eq. ()..7 Clampd-Pnnd (CFE) Consdrng zro dflcton and slop for th frst nod, th shap functons ar obtand smlar to th clampd fr bam n th CFE mthod. Th zro dsplacmnt at th rght nd s consdrd as a constrant whr can b obtand by multplyng th matrx Γ by th vctor of curvatur. Th matrx Γ s gvn as.8 Clampd-Pnnd (CDFE) Usng th followng condtons Γ (5) w w w Constants w and ar zros and m s found as () m n n n (7) By substtutng Eq. (7), to Eq. (8), th dflcton of th nods s obtand n th form of Eq. (8)..9 Clampd-Gudd (CFE) In ths cas, th shap functons ar smlar to th clampd-fr bam n CFE mthod. Also, th zros slop at th rght nd of th bam s consdrd as a constrant. In ths cas, th matrx Γ s dfnd as. Clampd-Gudd (CDFE) Γ (8) In ths cas, th constants w, and m ar obtand usng th followng condtons Latn Amrcan Journal of Solds and Structurs (7) 9-7

10 8 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons w w w Constants w and ar zro and m s drvd as (9) m n n n () Th shap functons ar smlar to Eq. ().. Clampd-Clampd (CFE) For bams wth ths boundary condton, th shap functons ar smlar to thos of th clampdfr bam n CFE mthod. Furthrmor, th constrants ar zro dsplacmnt and zro slop at th rght nd of th bam whch can b obtand by multplcaton th matrx Γ to th curvatur vctor. In ths cas, matrx Γ can b prsntd as Γ. Clampd-Clampd (CDFE) In ths cas, th condtons ar w w w w () () Constants w and ar zro and m s obtand as m n n n () Th zro slop at th rght sd of th bam s consdrd as a constrant whch, can b obtand by multplyng matrx Γ by curvatur drvatv vctor Γ () Th shap functons can b sn n Eq. (8). FREQUECY EQUATIO In ths scton, usng Lagrang s quaton and th assumd dflcton of th th lmnt n trms of nodal curvaturs and curvatur drvatvs n CFE and CDFE mthods, rspctvly, th mass matrx and stffnss matrx can b obtand.. Mass Matrx Th kntc nrgy of th th bam lmnt can b wrttn as Latn Amrcan Journal of Solds and Structurs (7) 9-7

11 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons 9 T (5) t w A ( ) d whr, th dnsty and th cross ara of th bam ar dsgnatd wth constants and A, rspctvly. Usng Eqs. () and (5), th kntc nrgy can b rwrttn as () T A jvv jd j Thus, th componnts of th th lmnt mass matrx ar m j j A d (7) Also, th kntc nrgy of a bam carryng a concntratd mass, m attachd at th th global nod s gvn as T m ( ) ( ) j vv j (8) j Thrfor, th corrspondng componnts of th th lmnt mass matrx can b obtand as mj m ( ) j ( ) (9). Stffnss Matrx Th potntal nrgy of th th lmnt of th Eulr bam can b wrttn as w U EI d (5) n whch, EI s th flxural stffnss of th th lmnt. Consdrng th transvrs dflcton of th th lmnt, th componnt of th th lmnt stffnss matrx can b obtand as whr, s th scond drvatv of j j k EI d (5). If lnar and torsonal sprngs wth stffnss k l and k t ar attachd to th th global nod, th corrspondng componnt of th stffnss matrx can b obtand as k k ( ) ( ) k ( ) ( ) (5) j l j t j Rmark: Th sz of th total mass and stffnss matrcs of th sprng-mass-bam systm s ( ) ( ). Th j mponnt of th assmbld mass and stffnss matrx s obtand by summaton of all th j componnt of lmntal mass and stffnss matrcs. Latn Amrcan Journal of Solds and Structurs (7) 9-7

12 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons. Load Vctor Th vrtual work of a dscrt load, Whl th vrtual dsplacmnt of ach nod s as F k actng at th th nod can b wrttn as W F. w (5) k k w v (5) Usng Eqs. (5) and (5), th gnralzd forc can b wrttn as f k F (55) k whr, th vctor s dfnd as (5) Th gnralzd forc vctor assocatd to a concntratd momnt, wrttn as M k at th th nod can b f k M (57) k whr, th vctor for th momnt s obtand as (58) Furthrmor, t can b shown that th gnralzd forc vctor du to a contnuous forc, fξ M ξ n th th lmnt can b obtand from Eqs. (59) and (), r- and a contnuous momnt, spctvly. f f fξ d fξ d fξ d (59) Latn Amrcan Journal of Solds and Structurs (7) 9-7

13 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons f M ξ M d M ξ d M ξ d () Th th column of th assmbld load vctors s obtand by summaton th th column of th lmnts.. atural Frquncy Usng th obtand assmbld mass and stffnss matrcs, th dynamc quaton of a bam wthout constrant can b wrttn as M v Kvf () Th natural frquncs of ths bams can b obtand from th followng gnvalu rlaton K M () For th bams wth constrants, by ncorporatng th constrants, th rsultng dffrntal algbrac quatons can b wrttn as T M v K Γ v f p Γ p () n whch, th vctor of racton forc s prsntd by p. Th natural frquncs for ths bams can b obtand by solvng th followng quaton T K Γ M () Γ 5 UMERICAL EXAMPLES In ths scton, som numrcal xampls ar prsntd and th rsults ar vrfd usng DFE mthod. For ths purpos, th bams n th prsntd xampls ar assumd to b mad of stl bar of.m.m rctangular cross scton for whch 78 kg / m and E GPA. Also, th lngth of th bam s consdrd to b m. Th frst fv natural frquncs of th bams wth dffrnt boundary condtons ar obtand wth DFE, CFE and CDFE mthods and ar shown n Tabl. Th numbr of lmnts n ach cas s dtrmnd. Latn Amrcan Journal of Solds and Structurs (7) 9-7

14 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons typ 5 Clampdfr Pnndpnnd Pnndgudd Clampdpnnd Clampdgudd Clampdclampd xact DFE() DFE() CFE() CDFE() CDFE () xact DFE() DFE() CFE() CDFE () CDFE () xact DFE() DFE() CFE() CDFE () CDFE () xact DFE() DFE() CFE() CDFE () CDFE () xact DFE() DFE() CFE() CDFE () CDFE () xact DFE() DFE() CFE() CDFE () CDFE () Tabl : atural frquncs of th dffrnt bam usng CFE, CDFE and DFE mthods. ow, two xampls for th statc analyss of bams ar prsntd. In th frst xampl, dflcton, slop and curvatur dstrbuton of a smply support bam carng a unformly dstrbutd load w K / m s obtand usng DFE, CFE and CDFE mthods wth dffrnt numbr of lmnts. Th rsults ar shown n Fgurs to. Latn Amrcan Journal of Solds and Structurs (7) 9-7

15 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons Fgur : Dflcton dstrbuton of smply support bam usng DFE, CFE and CDFE mthods. Fgur : Slop dstrbuton of smply support bam usng DFE, CFE and CDFE mthods. Fgur : Curvatur dstrbuton of smply support bam usng DFE, CFE and CDFE mthods. Latn Amrcan Journal of Solds and Structurs (7) 9-7

16 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons For a clampd-clampd bam wth unformly dstrbutd load w K/m and ts curvatur dstrbutons ar plottd n Fgurs 5 to 7., dflcton, slop Fgur 5: Dflcton dstrbuton of a clampd-clampd bam usng DFE, CFE and CDFE mthods. Fgur : Slop dstrbuton of a clampd-clampd bam usng DFE, CFE and CDFE mthods. Fgur 7: Curvatur dstrbuton of a clampd-clampd bam usng DFE, CFE and CDFE mthods. Latn Amrcan Journal of Solds and Structurs (7) 9-7

17 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons 5 It can b sn from Fgurs to 7 that th dflcton and slop dstrbuton n th DFE, CFE and CDFE mthods wth two lmnts hav th sam accuracy. Th curvatur dstrbuton n CDFE wth two lmnts s clos to th rsults of DFE mthod wth tn lmnts whch confrm th ffctvnss of th CDFE mthod n comparson wth DFE mthod. ow, th dynamc rspons of an Eulr-Brnoull bam wth CFE and CDFE mthods ar nvstgatd. In th frst xampl, mdpont dflcton of a clampd fr bam undr a suddnly appld concntratd load w K at pont x s shown n Fgur 8. Fgur 8: Mdpont dflcton of a clampd-pnd bam usng CFE mthod. Th scond xampl s rlatd to th dynamc rspons of a clampd fr bam wth a sprng at ts rght nd ( k K / m ). Th dflcton of th mdpont of th bam n th prsnc of a suddnly dstrbutd unform load w K / m s dpctd n Fgur 9. Fgur 9: Mdpont dflcton of th clampd-fr bam usng CDFE mthod. Latn Amrcan Journal of Solds and Structurs (7) 9-7

18 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons As can b sn, CFE and CDFE mthods hav th sam accuracs n comparson wth DFE mthod. Snc th numbr of nodal varabls n CFE and CDFE mthods s lss than that of DFE mthod, th computatonal cost s rducd. Thus, th proposd mthods ar mor ffcnt for dynamc analyss of bams and can b usd for th dynamc analyss of dffrnt problms n sold mchancs. COCLUSIO Ths study focusd on th dynamc analyss of Eulr-Brnoull bams usng curvatur and curvatur drvatv-basd fnt lmnt mthods. In curvatur basd fnt lmnt mthod (CFE) nstad of ntrpolatng dsplacmnt of Eulr Brnoull bam n usual dsplacmnt basd fnt lmnt mthod (DFE), scond drvatv of dsplacmnt s ntrpolatd. CFE mthod prvously was usd by a fw rsarchrs for dynamc analyss of clampd bams. In ths rsarch, CFE mthod was modfd for statc and dynamc analyss of bams wth varous boundary condtons. In addton, a nw mthod calld CDFE (curvatur drvatv-basd fnt lmnt) whch s somhow a modfcaton of CFE, was proposd. CDFE mthod, whch ntrpolats th drvatv of curvatur nstad of curvatur, was usd for bams wth dffrnt boundary condtons. Th rsults wr compard wth thos obtand by DFE mthod and th ffctvnss of th CFE and CDFE mthods was shown. In comparson wth DFE mthod, th proposd mthods hav th followng advantags: Th bndng momnt n CFE mthod and th bndng momnt and th shar strss at th ntrnal nods n CDFE mthod ar contnuous. Wth fwr numbrs of lastc dgrs of frdom, CFE and CDFE mthods ar mor accurat than DFE mthod. Rfrncs Chn, Z., Kong, M., J, C., & Lu, M. (5). An ffcnt dynamc modllng approach for hgh-spd planar paralll manpulator wth flxbl lnks. Procdngs of th Insttuton of Mchancal Engnrs, Part C: Journal of Mchancal Engnrng Scnc, 9(), -78. D Vubk, B. F. (95). Dsplacmnt and qulbrum modls n th fnt lmnt mthod. Strss analyss, 9, D Vubk, B. F., Znkwcz, O. C. (97). Stran-nrgy bounds n fnt-lmnt analyss by slab analogy. Journal of Stran Analyss, (), 5-7. Fard, M., & Clghorn, W. L. (). Dynamc modlng of mult-flxbl-lnk planar manpulators usng curvatur basd fnt lmnt mthod. Journal of Vbraton and Control, (), 8-9. Fard, M., Clghorn W. L. (). Dynamc Modlng of a Sngl-Flxbl-Lnk Spatal Manpulator Usng Curvatur Basd Fnt Elmnt Mthod. Procdngs of th Canadan Socty for Mchancal Engnrng Intrnatonal Congrss. Kuo, Y. L. (5). Strss-basd Fnt Elmnt Analyss of Sldng Bams. Appl. Math, 9(L), 9-. Kuo, Y. L., & Clghorn, W. L. (). Curvatur-and dsplacmnt-basd fnt lmnt analyss of flxbl four-bar mchansms. Journal of Vbraton and Control, 7(), Kuo, Y. L., Clghorn, W. L. (7, Jun). Applcaton of Strss-basd Fnt Elmnt Mthod to a Flxbl Sldr Crank Mchansm. In th IFToMM Congrss, Bsancon, Franc. Latn Amrcan Journal of Solds and Structurs (7) 9-7

19 M. Gholampour t al. / Strss-Basd Fnt Elmnt Mthods for Dynamcs Analyss of Eulr-Brnoull Bams wth Varous Boundary Condtons 7 Kuo, Y. L., Clghorn, W. L., & Bhdnan, K. (). Strss-basd fnt lmnt mthod for Eulr-Brnoull bams. Transactons of th Canadan Socty for Mchancal Engnrng, (), -. Morly, L. S. D. (98). Th trangular qulbrum lmnt n th soluton of plat bndng problms. Aronautcal Quartrly, 9(), 9-9. Punch, E.F., Atlur, S.. (98). Larg dsplacmnt analyss of plats by strssd-basd fnt lmnt approach. Computrs and Structurs, (), 7-7. Watwood, V. B., & artz, B. J. (98). An qulbrum strss fld modl for fnt lmnt solutons of twodmnsonal lastostatc problms. Intrnatonal Journal of Solds and Structurs, (9), Wckowsk, Z. (995). Dual fnt lmnt analyss for plastcty frcton torson of compost bar. Intrnatonal journal for numrcal mthods n ngnrng, 8(), 9-9. Węckowsk, Z., Youn, S. K., & Moon, B. S. (999). Strss basd fnt lmnt analyss of plan plastcty problms. Intrnatonal journal for numrcal mthods n ngnrng, (), APPEDIX Th frst fv Shap functons of Eulr-Brnoull bam for CFE and CDFE mthods ar prsntd n th followng tabl. Elmnt Elmnt Elmnt Elmnt Elmnt 5 h ) h ( ) h ( ) h ( ) h ( ) ( ( ) h 5 h ( ) h ( ) h ( ) h ( ) h ( ) h ( ) h ( ) h ( ) h ( 5 ) h ( ) Tabl : Shap functons (CFE). h ( ) 5 h ( 5 ) 5 h ( ) 5 ( ) h Elmnt Elmnt Elmnt Elmnt Elmnt 5 h ( ) h ( ) h ( ) h ( 5 7 ) h ( ) 8 h 7 h ( ) h ( 55 ) h ( 5 55 ) h ( ) h ( ) h ( ) h ( ) h ( ) h ( 5 5 ) h ( ) 5 h ( ) h ( 5 ) 5 h ( ) h ( ) 5 Tabl : Shap functons (CDFE). Latn Amrcan Journal of Solds and Structurs (7) 9-7