Comparison of Deflection and slope of Cantilever Beam with Analytical and Finite Element Method for Different Loading Conditions Sanjay Kumar

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1 ISS: 9-97 ISO 9:8 Ctiid Ittil Jul Egiig Si d Ivtiv Thlgy (IJESIT) Vlum, Issu, vmb Cmpis Dlti d slp Ctilv Bm ith Alytil d Fiit Elmt Mthd Dit dig Cditis Sjy Kum Abstt This pp psts th iit lmt multi tilv bm d btid dlti d slp d sults mpd ith Eul Bulli bm thy (lytil sluti). H uimly vyig ld (UV) hv b pplid, dlti d slp dtmid t dit stis bm. Th lytil sults dit ith iit lmt mthd slutis lss lmts but vgs sluti h umb lmts isd. Th tilv bm ith uimly distibutd ldig (UD) is ls lysd ith bth mthds d sults mpd t bm ith UV. Th iit lmt mthd (FEM) d lytil sluti mpd dit ldigs. H mput pgmm i mtlb is usd t bti dlti d slp Eul Bulli bms t dit ltis dit ldigs. Id Tms Ctilv, Eul Bulli bm, uimly vyig lds, FEM, Mtlb I. ITRODUCTIO Th iit lmt mthd (FEM) is usul my stutul pblm utd duig giig ppliti d pplid si ptil pblm. But sults btid thugh FEM is t libl lik th umil mthd. Th uy iit lmt mthd dpds up th distizti, itplti uti d umb lmts. Th iit lmt mthd sists th imptt stps: p-pssig, pssig d pst pssig.[] I p-pssig, iput dt i.. mtil pptis, gmty, lds d budy ditis pvidd b simulti. Mdl is distizd it lmts d mshs gtd utmtilly thugh mtlb st. I pssig stiss mti is gtd d ssmbld t glbl mti. Eul-Bulli Bm Thy: Th ssumpti sidd Eul- Bm thy is tht pl sti b d t bdig mis pl d ppdiul t is. Th tilv bm ith uimly vyig ld (UV) is sh i th igu (). H ldig () is vyig lily t id d t z t d tilv bm. At th d tilv th pit ld (F ) d mmt (M ) tig i dditi t UV d UD i igu () d igu ().Th ditil uti uth d is usd s gvig uti bm d tsvs dlti is dtd by []. d d d d H, E dts th mdulus lstiity, I is th mmt iti bm but is ppdiul t bth d z is d viti ldig ()= (-/) h is th ttl lgth bm. Th tilv bm ith uimly distibutd ld (UD) is sh i th igu () Z, () F M Fig () Ctilv bm ith Vyig distibutd ld

2 ISS: 9-97 ISO 9:8 Ctiid Ittil Jul Egiig Si d Ivtiv Thlgy (IJESIT) Vlum, Issu, vmb Fig () Ctilv bm ith uimly distibutd ld Th bm sists umbs lmts d sids th lmt d dist th ds is t dist m glbl dits s sh i igu (). Fig () Dg dm th lmt ith ll d glbl dits. Th tsvs displmt () lmt i z diti is itt i tms u stts,, d () Apply d ditis t bth ds lmt (igu ()) d u shp utis/itpltis utis (,, & ) btid s lligs. Ths utis ls k s Hmit ubi itplti utis.,,, () Fm uti (), tsvs displmt is pssd i tms itplti utis. d d d d Us Glkis pph/ightd sidul mthds t vlut stiss mti d ld vt. I this mthd itgt th ightd v dmi d utd t z. H ight uti is sidd sm s. ) ( d ERROR Elmt stiss mti (K) is btid Th lmtl mti is btid by itgtig ightd ld d btid s At itgti il lmtl ld mti bms i m lum vt[] K d 7 9 z, F M

3 ISS: 9-97 ISO 9:8 Ctiid Ittil Jul Egiig Si d Ivtiv Thlgy (IJESIT) Vlum, Issu, vmb 7 H, ttl umb lmts is t d umb ds is lv igu ().Th umb ithi th il psts th lmts d simpl umls blk slid il psts ds. Fm bv pssi (7), it b s tht ld vt UV is t stt vy lmts it hgs m lmt t lmt. Elmtl mti is ssmbld hl bm d it bms glbl mti [K]. Similly lmtl ld mti {F} is ssmbld d glbl ld mti is btid. Fig () umb lmts () d umb ds () F UD th ldig is stt thughut th lgth bm, s lmtl mti is btid ) (8 I glbl displmt is pstd by { }, th by Glki mthd, stiss, displmt, distibutd ld d pit ld ltd by lti (8) ) (9 p F F K H F p is th tl pit lds tig th ds. Apply th budy ditis t id d (d ) d sluti displmts d ttis btid by us uti (8). Mtlb d hs b ppd d usd t bti th sluti. Th bm is md stl mtils hvig dimsis d ldigs d mhil pptis llig:-lgth bm is =m, Yug mdulus E= k/m, Mmt iti but y is, I=9 - m, =k/m,pit ld t d F =k,mmt t d M =k-m [].I digm () lgth d mhil pptis bm sm but ldig ditis dit m igu(),h uimly distibutd ld(ud) k/m is II. AAYTICA METHOD Th dlti bm UV is btid by dit dubl itgti mmt t sti m id d s i uti (). Th slp bm is btid by dit itgti mmt t sti m id d s i uti () []. Th FEM d lytil sluti displmt d slps mpd i th digms ( t9). () M F () ) ( M F d

4 ISS: 9-97 ISO 9:8 Ctiid Ittil Jul Egiig Si d Ivtiv Thlgy (IJESIT) Vlum, Issu, vmb Th dlti bm UD is btid by dit dubl itgti mmt t sti m id d s i uti (). Th slp bm is btid by dit itgti mmt t sti m id d s i uti() []. ( ) F M F () M - () Th FEM d lytil sluti UV ldig ti lgth tilv bm t stp.m is sh i th tbl ().Similly UD ldig displmt d slp is btid d sh i tbl (). Tbl () Cmpis iit lmt mthd (FEM) sluti ith lytil UV. Dist Displmt (m) m id d X(m) Slp (ϴ) FEM Alytil FEM Alytil Tbl () Cmpis iit lmt mthd (FEM) sluti ith lytil UD. Dist Displmt (m) Slp (ϴ) m id d X(m) FEM Alyti l FEM Alytil III. RESUT AD DISCUSSIO Figu () d () sh th dlti d slp bm sptivly du t UV m lytil d FEM mthd. Th displmt d slp tilv bm btid m lytilly mthd ith iit lmt 8

5 ISS: 9-97 ISO 9:8 Ctiid Ittil Jul Egiig Si d Ivtiv Thlgy (IJESIT) Vlum, Issu, vmb sluti lmts du UV ldig (igu ()&()). Th dt btid m lytil sluti d FEM sluti ith hlp MATAB d []is mtid i tbl () UV ldig. Fist lum idits dist ds m id d t tilv t stp.m lgth. Sd d thid lums idits vlu displmt btid m FEM d lytil sptivly UV ldig. Th slp btid m FEM d lytilly is sh i uth d ith lum i tbl(). Fig (): Cmpis dlti btid lytilly d thugh FEM UV. Fig (): Cmpis slp btid lytilly d thugh FEM UV. Figu (7) d (8) sh th dlti d slp bm sptivly du t UD m lytil d FEM mthd ith us MATAB d []. Th displmt d slp tilv bm btid m lytilly is slightly dit ith iit lmt sluti lmts du UV ldig (igu (7)&(8)). This is lys tu bm subjtd t distibutd ldig is mdld by ubi displmt uti. Th sluti btid by bm thy is uti (uth d) plymil but i FEM lys it is ssums tht displmt is ubil plymil [].S iit lmt sluti pdits sti stutu th tul stutu. Th dt btid m lytil sluti d FEM sluti is mtid i tbl () UD ldig. Fist lum idits dist ds m id d t tilv t stp.m lgth. Sd d thid lums idits vlu displmt btid m FEM d lytil sptivly UD ldig. Th slp btid m FEM d lytilly is sh i uth d ith lum i tbl(). Fig (7): Cmpis dlti btid lytilly d thugh FEM UD. 9

6 ISS: 9-97 ISO 9:8 Ctiid Ittil Jul Egiig Si d Ivtiv Thlgy (IJESIT) Vlum, Issu, vmb Fig (8): Cmpis slp btid lytilly d thugh FEM UD. Figu (9) mp th dlti bm du t UD d UV by FEM mthd sm ldig. Th displmt tilv bm du t UD m th UV. Th is tu bus ldig i UD is uim thugh ut th ti lgth bm but i UD it is mimum t th id d tilv d dss t z t th d th bm. Fig (9): Cmpis dlti dit ldigs. Figu () shs slp bm du t UD d UV by FEM mthd sm ldig. Th slp tilv bm du t UD m th UV. Th is tu bus ldig i UD is uim thugh ut th ti lgth bm but i UD it is mimum t th id d tilv d dss t z t th d th bm It is bsvd tht dlti UD is gt th th UV sm ldig Fllig lusis btid m this sh. Fig (): Cmpis slps dit ldigs. IV. COCUSIO

7 ISS: 9-97 ISO 9:8 Ctiid Ittil Jul Egiig Si d Ivtiv Thlgy (IJESIT) Vlum, Issu, vmb Th sluti dlti d slp btid m FEM d bm thy tly mthd t lmts ith UV ldig Th sluti dlti d slp btid m FEM d bm thy slightly l t lmts ith UD ldig. Fiit lmt sluti pdits sti stutu th tul stutu. Th dlti d slp lg UD th UV d is m id d t d tilv bm. REFERECES [] Rddy, J..(),A Itduti t th Fiit Elmt Mthd, MG Hill, Eduti(Idi) Pivt imitd Dlhi. [] g, Dyl ().A ist us i th iit lmt mthd, Futh diti. THOMSO. [] R.D.Ck,Dvid.S.Mlkus, Mihl E Plsh, Rbt J.Witt ().Cpt d pplitis Fiit Elmt Alysis, Futh diti. [] Ktt,I. Pt () MATAB Guid t Fiit Elmts :A Ittiv Apph,Spig Ittil Sd Editi. [] A.J.M, Fi (8) Mtlb Cds Fiit Elmt Alysis: Slids d Stutus, Spig. AUTHOR BIOGRAPHY Sjy Kum is kig s Assistt Pss t Dlhi Thlgil Uivsity, Dlhi si Sptmb 8. H hs d B.S (Egg.) m M.I.T. Muzpu i 998 ith Ist Div, M.Th (Mhi Dsig) m I.I.T. BHU, Vsi i ith Ist Div, d pusuig Ph.D m IIT Dlhi. H is i Mmb ISTE.. H hs publishd m th Rsh Pps i vius til/itl Jul/ Sympsium.. H hs ls kd s Assistt Pss i K.I.E.T, Ghzibd 8 ys (-8). H hs guidd PG studts d m th UG studts.