DIFFERENCE EQUATION METHOD FOR STATIC PROBLEM OF INFINITE DISCRETE STRUCTURES

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1 FOUNATION OF IVIL AN ENVIRONMENTAL ENGINEERING No. Zziłw PAWLAK Jez RAKOWKI Poznn Univeit of Tehnolog Intitute of tutul Engineeing IFFERENE EQUATION METHO FOR TATI PROBLEM OF INFINITE IRETE TRUTURE In the wok n ie i peente whih enle one to eive funentl olution fo one- n two-ienionl iete te. The tti pole of infinite e n infinite giwok tip i oniee. The olution e foun uing the iffeene eqution etho. The equiliiu onition fo oth iete tutue e eive uing the finite eleent ethoolog. In eh e n infinite et of thee eqution i eple one equivlent iffeene eqution. The funentl funtion e eteine in nltil loe fo. The n e ue in the oun eleent etho pplie to iete tutue Ke wo: funentl olution finite eleent etho iffeene eqution etho iete tutue. INTROUTION The i of the ppe i to eive nltil olution fo tti pole of one- n two-ienionl infinite iete tutue. The funentl olution e foun fo egul iete e n giwok tip. Thee ontinuou te e ivie into egul eh of ientil e finite eleent. The equiliiu onition e tte uing finite eleent ethoolog n the e epee in one equivlent iffeene eqution. Thi eqution eple n infinite nue of equiliiu onition foulte on the i of eleent tiffne tie. The olution e otine in nltil loe fo fo it lo of oniee infinite iete tutue. Pulihing Houe of Poznn Univeit of Tehnolog Poznń IN -9

2 5 Zziłw Pwlk Jez Rkowki. TIMOHENKO BEAM Let u onie n infinite Tiohenko e ietize eleent with equll pe noe (Fig. ) n loe onentte foe P n oent M. M P. P k k M k Fig.. An infinite e In the lultion we ppl -of he-fleile e finite eleent (Fig. ) with et eleent hpe funtion. ϕ ϕ w w w Fig.. The finite e eleent The following eltionhip i fulfille fo thi eleent [] whee: f T { P P } T { w φ w φ } w i n P K q f (.) q φ i ϕi i M i i. ϕ e nol tnvee ipleent n totl ottion epetivel i i M i e nol foe n oent. The eleent tiffne ti K i eteine fo the epeion of tin eneg [] φ κga w U φ n it h the fo: whee K (.) [ K K ] (.)

3 iffeene eqution etho... 5 K K. The funtion of ipleent () w n o-etion ottion () φ long the e finite eleent eive in [] e of the fo w q T N () q N T φ (.) N N whee: GA κ E G e Young n Kihhoff ouli I i the oent of ineti A i the e of the e o etion κ i the he fto. Fig.. The intenl foe t noe Aeling two jent eleent ( ) n () (Fig. ) iel the following equiliiu onition fo n it e noe ujete to the nol foe P n oent M : Q Q M M P M

4 5 Zziłw Pwlk Jez Rkowki ( w w ) ( φ φ ) ( w w ) ( φ φ ) ( φ φ ) ( φ φ ) ( ) (.5) ( w w ) ( φ φ ) ( w w ) ( φ φ ) P. Intouing in (.5) the hifting opeto E E n entl iffeene opeto E E we otin two iffeene eqution with unknown φ n w : ( ) φ ( E E ) w ( ) ( E E ) w ( ) P φ. n n (.) Afte eliintion of φ in (.) we get one fouth-oe iffeene eqution with one unknown w ( E E ) w µ P µ (.7) o one eqution with unknown φ ( E E ) P φ µ µ (.7) whee µ. Let u fin the olution of (.7) fo the e loe one foe ting t the noe P P δ M ( δ i Koneke elt). Hving intoue the olution of the hoogeneou eqution n in fo: w φ w () gn φ (.8)

5 iffeene eqution etho... 5 we onie the equiliiu onition of noe n. It le to the efinition of ontnt i. utituting (.8) into (.) one otin ontnt i. Finll we get the foul: w ( P) µp( ) ( P) µp gn φ. In the e when noe p i loe ( funentl funtion: w p p P P p ( P) µp( p p ) ( P) µp ( p) gn( p) φ. (.9) δ ) eltion (.9) tke the fo of (.) Now let u fin the olution of (.7) fo the e loe onentte oent ting t noe M M δ P M. Thi e i ntieti. The olution of hoogeneou eqution e: φ φ w w. () gn The eltion etween i n i the eqution (.7) fo noe we get (.) n e foun fo (.). utituting (.) ( M ) µ( ) φ [ ] ( M ) µ ( ) gn w. Fo the e loe one oent otin the funentl funtion: w p φ p M M δ p ( M ) µ p ( p) to (.) ting t the noe p one ( M ) µ ( )( p) ( p) gn( p) ( p). (.) Auing in (.) n (.) tht iel the funentl olution fo infinite Eule-Benoulli e.

6 5 Zziłw Pwlk Jez Rkowki. GRIWORK TRIP Now let u onie the giwok tip whih onit of two et of inteeting e. The e of the length in -ietion e ipl uppote on thei en the e (with equl itne of noe ) in -ietion e infinite (ee Fig. ). M P η M P () Fig.. The infinite giwok tip The noe n e loe onentte foe P n oent M M (Fig. 5). Fo the lultion we ppl -of Eule-Benoulli e finite eleent (tnvee ipleent ngle of ottion n ngle of twit t eh en of eleent). Q Q P M M M M M M M M Q M M Q Fig. 5. The giwok noe ()

7 iffeene eqution etho Hving ele fou jent eleent in noe ( ; ) ( ; ) ( ; ) ( ; ) we get thee iffeene eqution equivlent to the FEM ti foultion []: [ w ( E E ) φ ] [ w ( E E ) φ ] P (.) GI [ ( E E ) w ] φ φ M (.) GI [ ( E E ) w ] φ φ M. (.) whee e the itne etween noe in n ietion epetivel GI n GI e the ening n toionl tiffnee of e in oth ietion. Afte eliintion of ottion φ φ in (.) we otin one ith-oe ptil iffeene eqution with one unknown [ ( )( ) µ ( ) ( )] w µ [( )( ) P ( E E )( ) M ( E E )( ) M w (.) whee µ µ µ µ µ P GI GI. Let u ue tht the tip i loe ingle foe ting t noe ( η) P. In oe to olve the eqution (.) we ue the iete Fouie δ δ η tnfotion in -ietion []:

8 Zziłw Pwlk Jez Rkowki 5 i f f f e ~ F f f f i e ~ ~ - F (.5) n the eigenfuntion tnfotion in -ietion l w w l l in ~. (.) Appling oth tnfotion (.5) n (.) to eqution (.) we otin the foul: () in in l N P W l η µ η (.7) whee l () M L l N o (.8) () () () () o o o o L () () () () () (). µ M o o o o o o o o o o The intege (.8) n e lulte nltill. Afte onfol tnfotion we get: () () () () F F F l N (.9) () () F () F () () F

9 iffeene eqution etho () o ( o() ) () o ( o() ) [ ( o() ) ] whee: () n n o ( ) ( ) o ( ) o( ) o (.) µ µ µ () o ( o() ) [ µ ( o( ) ) ] ( o() ) o ( o() ) ( o( ) ) ( o() ) () o() 7 ( o() ) ( o( ) ) [ µ ( o() ) ( o() ) ( o() o() ) ( o() ) o µ ( o() ) ( o( ) ) () o() ( o() ) ( o( ) ) µ. µ ( o() ) ( o( ) ) Fo (.) the following euent eltionhip i vli ( n k...) fo n k fo n k () n () n n ( n )! n [( 5n 5 )! ] (.) ( n ) ( n ) ( n ). The vlue of integl (.) fo n e follow:

10 Zziłw Pwlk Jez Rkowki 58 gn gn gn n (.) gn gn gn n (.) gn gn gn n (.) i i 8 8 Ω 7 Ω.

11 iffeene eqution etho The foul (.) (.) (.) inlue funtion gn () whih eteine the ign of ople nue: if Re() > Re() I() > gn() if Re() < Re() I() <. All eining vlue of unknown ipleent the iple euent eltionhip given in (.). w e lulte fo. NUMERIAL EXAMPLE The oputtion wee pefoe fo the infinite giwok tip of with 5 loe unit foe t point (). The eult of nueil lultion e peente in Tle the plot of tip efotion i hown in Fig.. Tle. The nolize ipleent W ( ) of infinite tip Fig.. The infinite giwok tip efotion

12 Zziłw Pwlk Jez Rkowki The funentl olution eive in etion n e ue to olve the tti pole of finite giwok in nlogou w in the oun eleent etho fo ontinuou te. Let u onie the iotopi giwok oniting of the et of ientil inteeting pepeniul e (Fig. 7) E I P () R Fig. 7. The iotopi giwok te The noe e egull pe in oth ietion with the itne. All e e ipl uppote t thei en. We intoue n itionl loing X k j to the given foe P δ δ in the infinite tip in oe to fulfil the equie oun onition in the noe ling on the e n 5 (Fig. 8). X - X - X - X - X - X - X - X - () () () () () () P (5) (5) X (5) (5) R X X 7 X 7 X X 7 X X 7 Fig. 8. Aitionl foe pplie to the infinite giwok tip The vlue of the itionl foe following eqution (iniet ppoh of BEM): X e eteine fo the k j

13 iffeene eqution etho... w i ( P X k j ) i ( P X k j ) w ( P ) ( P ) 5 i X k j M M 5 i X k j (.) fo i. Hving foun the foe X k j fo (.) we finll get the ipleent of the finite giwok: whee ( k j) X W ( k j) P W ( ) w k j kj (.) W i the funentl olution given in (.7). The lultion e ie out fo the following ienionle pete: E ν G 8 the e o etion: h. The plot of giwok ipleent i peente in Fig. 9. Fig. 9. The ipleent plot of finite giwok te The oniee iete giwok tip n e tete n ppoition of plte tip. The eult otine fo thi giwok oinie with one otine fo n equte Kihoff-Love plte. The eltion etween (P) (G) ipleent fo plte w n giwok w t point () ujete to the ( P) ( G) unit foe i: w w 8.

14 Zziłw Pwlk Jez Rkowki REFERENE. Bušk I.: The Fouie tnfo in the theo of iffeene eqution n it pplition Ah. Meh. to. (959).. Hinton E. Owen.R.J.: An intoution to Finite Eleent oputtion wne U.K. Pineige Pe Rkowki J.: The intepettion of the he loking in e eleent op. tutue 7 (99) Rkowki J. Świtk R.: On oe genelition of the eigenfuntion etho ue to the elti iete te Poznń Wwnitwo Politehniki Poznńkiej 98 (in Polih). Z. Pwlk J. Rkowki METOA RÓWNAŃ RÓŻNIOWYH W PROBLEMIE TATYKI NIEKOŃZONYH UKŁAÓW YKRETNYH t e z z e n i e W p pzetwiono etoę któ pozwl wznzć funkje funentlne l jeno- i wuwiowh ukłów ketnh. Rozwżono pole ttki nie-końzonh elek i niekońzonh p uztowh. Rozwiązni uzkno toują etoę ównń óżniowh. Wunki ównowgi l ou tpów ukłów ketnh wpowzono z wkoztnie eto eleentów końzonh. W kż z pzpków nieognizon ukł ównń ównowgi ł ztąpion pzez jeno ówno-wżne u ównnie óżniowe. Funkje funentlne pono w nlitznej zkniętej foie. Rozwiązni te ogą ć wkoztne w etozie eleentów zegowh toownej w ukłh ketnh. Reeive -8-8.

DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING FLUID MECHANICS III Solutions to Problem Sheet 3

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