Report and Opinion, 1(1), 2009,
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1 Rpor and Opinion ( hp:// scincpub@gail.co O THE REPOE OF ODED EM UJECTED TO MOVIG ME D EXTER FORCE Idowu.I. Gbolagad.W Olayiwola.M.O Dparn of ahaical and physical scincs Olabisi Onabanjo Unirsiy go-iwoy igria Ws frican jabola@yahoo.co TRCT: hory dscribing h rspons of a loadd ba subjcd o oing asss and xrnal forcs is considrd. Th gorning quaion is a fourh ordr parial diffrnial quaion. Th fini Fourir ransforaion is usd o ransfor h gorning parial diffrnial quaion ino scond ordr ordinary diffrnial quaions. Th ordinary diffrnial quaions for oing forcs and oing asss ar sold wih aplac ransforaion hod. ubr xapls ar usd o donsra h fficincy of h soluion. uric analysis shows ha for a sipl suppord ba h rsonanc frquncy is lowr wih corrsponding dcras in axiu apliud whn inrial is considrd. [Rpor and Opinion. ;(:-]. (I: 3-3. yword: a Transforaion Inrial Rsonanc Classificaion: Dynaical ys. ITRODUCTIO ba or girdr bridg is h sipls kind of bridg. In h pas hy ay ha akn h for of a log across a sra bu oday hy ar or failiar o us as larg box sl girdr bridgs. Thr ar los of diffrn yps of ba. ba bridg nds o b siff. I nds o rsis wig and bnding undr load. In is os basic for a ba bridg consiss of a horizonal ba ha is suppord a ach nd h pair. Th wigh of h ba pushs sraigh down on h pairs undr load. Moing loads causs solid bodis o ibra innsily. Paricularly a high lociis. Thus h sudy of h bhaiors of bodis subjcd o oing lnds has bn h concrn of sral insigaors. ong h arlis work in his ara of sudy was h work of Tioshnko ( who considrd h probl of siply suppord Uni bas rg on an lasic Foundaion and rarsd by oing loads. In his analysis h assud ha h loads wr oing wih consan lociis along h ba. Furhror nny ( ook up h probl of insigaing h dynaic rspons of infini lasic bas on lasic foundaion whn h ba is undr h influnc of a dynaic load oing wih consan spd. i includd h ffcs of iscous daping in h gorning diffrnial quaion of oion. Mor rcnly Oni ( considrd h probl of a haronic i ariabl concnrad forc oing a a unifor lociy or a Uni dp ba. Th hods of ingral ransforaions ar usd. In paricular h Uni Fourir ransfor is usd for h lngh coordina and h aplac ransfor h i coordina. ris soluion which conrgs as obaind for h dflcion of siply suppord bas. Th analysis of h soluion was carrid ou for arious spds of h load. Oni ( usd h Galrkin hod o obain h rspons o sral oing asss of a non-unifor ba rg on an lasic foundaion. Th ffcs of h lasic foundaion on h ransrs displacn of h non-unifor ba wr analyzd for boh h oing ass and h associad oing forc probls. wodolat.o ( workd on h influnc of foundaion and axial forc on h ibraion of a siply suppord hin (rnoulli Eulr ba rg on a unifor foundaion undr h acion of a ariabl agniud haronic load oing wih ariabl lociy is insigad in h papr. Th gorning quaion is a fourh ordr parial diffrnial quaion. For h soluion of his probl in h firs insanc h fini Fourir ransforaion is usd o rduc h quaion o a scond ordr parial diffrnial quaion. Th rducd quaion is hn sold ug h aplac ransforaion. urical analysis shows ha h ransrs dflcion of h hin ba rg on a unifor foundaion undr h acion of a ariabl agniud haronic load oing wih ariabl lociy dcrass as h foundaion consan incrass. I also shows ha as h axial forc incrass h ransrs dflcion of h hin ba dcrass. Furhror Milorir anisic.m and Hardin J. C. ( dlopd a hory dscribing h rspons of a rnoulli-eulr ba undr an arbirary nubr of concnrad oing asss. Th hory is basd on h Fourir chniqu and shows ha for a siply suppord ba h rsonanc frquncy is lowr wih no corrsponding dcras in axiu apliud whn h inria is considrd. ii. Forulaion of h probl
2 Rpor and Opinion ( hp:// scincpub@gail.co Th ibraion of a uniforly siply suppord ba carrying an arbirary nubr of discr asss is considrd. This ass is assud o srik h ba a and ral across i wih lociy i. Th quaions of oion wih daping nglcd is wrin as y y y y ρ i ( u i cd EI y gf ( x i x x ( i x i Whr f (x i ( EI h flxural rigidiy of h ba f (x h ransrs dflcion of h ba ρ h ass dnsiy of h ba arial cross scional ara of h ba g h acclraion du o graiy k h foundaion consan. h lngh of h ba x ( i is h Dirac dla funcion dfin o b zro rywhr xcp x i ( x i. i ( x dx in addiion i (3 Th boundary condiions ar Y (o Y ( Y xx (o Y xx ( ( pplying h Fourir fini ransfor i. x x i and ( Y ( x dx ( y y x y x x M i ( x i dx cd dx EI dx i x y x x x y Yxx dx ρ F( x dx x ( oling quaion ( by ingraion by par w obaind diffrnc sris soluion which addd oghr. Th rsul b EI yi g i i i i i ( p ( z( M ( T δ ( x Y ( Q EI p Qz yi g i i i i ( z ( Cz ( T δ ( x y ( i C T ( ( ( iδ ( x i y ( Q Q Q i g i i Q i y h quaion w obaind h ransforaion quaion. ( ( ( (
3 Rpor and Opinion ( hp:// scincpub@gail.co ( ( ( i i ( 3. oluion of h ransford quaion. For h purpos of h soluion w considr only on ass raling wih lociy υ. Th soluions for grar nubrs of asss ay b obaind in h sa annr.eidnly h following spcial cass fro quaion follow: (a Moing forc: - If w nglc h inria r w ha h classical cas of a oing forc. Undr h abo assupion quaion b (b (a ( ( ( i i Moing Mass:-firs approxiaion only h linar inrial r is considrd Equaion b. ( 3 i ( ( ( i For oing forc ( ( ( i ow soling quaion for oing forc w ha ( ( ( i i i i This iplis ha ( c p ( For c ; quaion ( b ( ( M ( M M M ( M ; hnm y quadraic soluion b ± b ac M a b andc a ± M M Thrfor z c M ( (3
4 Rpor and Opinion ( hp:// scincpub@gail.co pplying hs boundary condiions ( o ( o or ( for (o ; and ( ( ( o ( ub ( in ( ( Thrfor or y xpansion
5 Rpor and Opinion ( hp:// scincpub@gail.co Fro c i.. c ( For p ; i g p Whr i g i p p p p p p Th quaion b
6 Rpor and Opinion ( hp:// scincpub@gail.co ( For h soluion ( c p ( x x x x ( (x ( ( 3
7 Rpor and Opinion ( hp:// scincpub@gail.co ( ( Fig. Conrgnc of cofficin for oing ass soluion y ug h sa hod for soling h quaion for oing forc i. quaion (3 for oing ass b:- ( 3 Mi ( ( ( ( i i nd l 3 Mi i ( ( (
8 Rpor and Opinion ( hp:// scincpub@gail.co Th soluion for h oing ass b (3 V( Δ Whil his quaion is rsisan o analyic chniqu i yilds radily o nurical procdurs. For z ( h soluions for ar abulad in abl blow Moing ass / ( ( Tabl. oing ass Obiously highr approxiaions ar possibl by considring or rs of h sris and following h sa procdur. Howr considring h ra of conrgnc of h lowr-ordr soluions i should no b ncssary o coninu his procss.. Rark on h soluion In a probl such as his on is inrsd in h axiu apliud of ibraion and h condiion undr which is can occur. For h classical soluion Eq. i can bn shown ha undr crain alus of h lociy h condiion of rsonanc occurs. In his cas h apliud of ibraion b a linar funcion of i. Fig.. h axiu alu of h apliud which occurs whn h ass is a h nd of h ba.
9 Rpor and Opinion ( hp:// scincpub@gail.co z ( - ( Moing Mass Moing Forc Moing Mass Fig. 3 Coparison of soluion
10 Rpor and Opinion ( hp:// scincpub@gail.co ( Fig. pliud growh undr rsonan condiions
11 Rpor and Opinion ( hp:// scincpub@gail.co. Conclusion hory is prsnd on h rspons of a loadd ba subjcd o oing asss and xrnal forc. Th hory is sipl nough o b usd in copuaion for dsign considraions. Th quaion of oion is gin in rs of δ-dirac funcions and is sold hrough h us of Fourir fini ransfors. n analyic approxiaion is obaind and copard wih h soluion for a oing forc. Th figur fi shown h apliud of h graph which growh undr a rsonanc condiions and figur wo shown h conrgnc of cofficin of oing ass soluion whil figurs 3 and shows h coparison of h soluion for oing ass and oing forc which oing ass is qual o oing forc soluion of I is found ha for a siply suppord ba h rsonan frquncy is lowr wih no corrsponding dcras in axiu apliud whn h inria is considrd. For any highr approxiaion h soluion can b obaind by ans of nurical chniqus and for fuur work; h conrgnc of h soluion can b sablishd.. RECOMMEDTIO (a This work will assis h pracicing nginr o alua h dynaic rspons of a loadd ba subjcd o oing asss and xrnal forcs. (b This work can b applid o calculaions inoling prsrssd or rinforcd bas ofn ncounrd in srucural dsign and Consrucion Copany. REFERECE. jibola. O. and Oolof. (: On h Transrs oions undr hay loads of bas wih ariabl prsrss. Journal of h igrian ssociaion of Mahaical Physics ol... wodola T.O (: Influnc of foundaion and axial forc on h ibraion of hin ba undr ariabl haronic oing load. Journal of h igria ssociaion of Mahaical Physics ol llan R.E Casi J. (: Diffrnial quadraur and long r ingraion. Journal of Mahaical nalysis and pplicaion iu F. iw.m. (: nalysis of ibraing hick rcangular plas wih ixd boundary consrains ug diffrnial quadraur ln hod. Journal of ound and Vibraion ( - 3. iu F.. iw.m. (: Vibraion analysis of disconinuous Mindlin plas by diffrnial quadraur ln hod. Journal of Vibraion and cousics -.. Mi C Dcha-Uphai.. (: fini ln hod for nonlinar forcd ibraion of rcangular plas. I Journal 3 -. Milorir M. anisic M. and Hardin J.C (: On h rspons of bas o an arbirary nubr of concnrad oing asss. Journal of h Franklin Insiu o.. Oni and. T. wodola T. O (: Dynaic rspons o oing concnrad of unifor Rayligh bas rg on ariabl winklr lasic foundaion. Journal of h igrian ssociaion of Mahaical Physics ol... Oni. T and Tolorunsagba J. M. (: Roaory Inria Influnc on h highly prsrssd orhoropic rcangular plas undr raling loads. Journal of h igrian ssociaion of Mahaical Physics ol. 3.. Graff.F. (: Wa Moion in Elasic olids. w York: Dor Publicaions Inc.. Haron R.F.. (: Th frquncy of flxural ibraion of rcangular orhoropic plas wih clabd or suppord dgs. Journal of pplid Mchanics Hurlbaus Gaui. and Wang J.T.. (: n xac sris soluion for calculaing h ignfrquncis of orhoropic plas wih coplly fr boundary. Journal of ound and Vibraion. 3. Jiya M iysii Y. M. and. O. Mohad (: Dynaic nalysis of a rnoulli-eulr ba ia h aplac ransforaion chniqu. Journal of h igrian ssociaion of Mahaical Physics ol. 3.. argarnoin M.H Younsian D. Thopson D.J Jons C.J.C (. Rspons of bas on nonlinar iscolasic foundaions o haronic oing loads Copur and rucurs 3.. nny J. ( ady a Vibraions of a ba on an lasic foundaion fir a oing load. Journal of pplid Mch. Vol. pp ancasr P Tinsky M. (: Th Thory of Marics wih pplicaions. cond d. cadic Prss. Orlando. F... Y. i Y.W. (: nalysis of nonlinar ibraion of hybrid coposi plas. Copurs and rucurs (3 3-.
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