Behavioral Study of Finite Beam Resting on Elastic Foundation and Subjected to Travelling Distributed Masses

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1 3 Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses Absrac The classcal problem of he response characerscs of unform srucural member resng on elasc subgrade and subeced o unform parally dsrbued load s suded n hs work. The closed form soluons of he governng fourh order paral dfferenal equaons wh varable coeffcens are presened usng an elegan analycal echnque for he movng force and mass models. Varous resuls and analyses are carred ou on each of he pernen boundary condons and phenomenon of resonance s suded for he dynamcal sysem. I was found ha n all llusrave eamples consdered for he same naural frequency he crcal speed for movng dsrbued mass problem s smaller han ha of he movng dsrbued force problem. ence resonance s reached earler n movng mass beam-load neracon problem. Fnally hs work has suggesed valuable mehods of analycal soluon for hs caegory of problems for all boundary condons of praccal neres. B. Omolofe a T.O. Adeloye b a Deparmen of Mahemacal Scences Federal Unversy of Technology Akure Ngera. Emal: babaope_omolofe@yahoo.com b Deparmen of Mahemacal Scences Federal Unversy of Technology Akure Ngera. Emal: adeyel@yahoo.com hp://d.do.org/.59/ Receved In revsed form Acceped 9..6 Avalable onlne 3..6 Keywords Dynamc characerscs Resonance Subgrade Dsrbued oads Dynamcal sysem. INTRODUCTION Analyses of he dynamc characerscs of elasc srucural members resng on elasc subgrade such as ralway racks hghway pavemens navgaon locks and srucural foundaons consue an mporan par of he cvl Engneerng Mahemacal Physcs and oher relaed felds. These elasc srucures are very useful n many felds of research hus her dynamc behavours when under he acon of ravellng loads of dfferen forms have receved eensve aenon n he open leraure [ On and Omolofe 5a assan e al 6 Omolofe 3 Ismal 5]. When hese mporan engneerng srucures are resng on an elasc foundaon he srucure-foundaon neracon effecs play sgnfcan roles n her response behavour and aler he dynamc saes of

2 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses 33 he srucures from hose vbrang n he absence of foundaon [Ugurlu e al 8]. ence he dynamc behavour of srucures on elasc foundaon s of grea mporance n srucural aerospace cvl mechancal and marne engneerng applcaons. Consequenly s mporan o clarfy he nfluence of he foundaon on he behavour of elasc srucures n engneerng desgns. Furhermore o accuraely assess he dynamc response of any srucural member on elasc foundaons a mechancal model s requred o predc he neracon effecs beween such srucures and foundaons. Beams on elasc foundaon and under he acons of he movng loads have receved a consderable aenon n leraure; see for eample references [Clasornk e al 986 Thambranam and Zhuge 996 Sun and uo 8 ng e al 8] owever mos of hese works employed he smples mechancal model whch was developed by Wnkler and generally referred o as a one-parameer model. The defcency of hs model s ha assumes no neracon beween he sprngs so does no accuraely represen he characerscs of many praccal foundaons [Esenberger and Clasornk 987]. Thus o overcome he defcences nheren n Wnkler formulaon a wo-parameer foundaon models whch akes no accoun he effec of shear neracons beween sprngs has been suggesed. In general such analyses are mahemacally comple due o he dffculy ofen encouner n modelng he mechancal response of he subgrade whch s governed by many facors. When hese srucures are aced upon by movng loads he dynamc analyses of he sysem become much more complcaed. I s known from earler sudes ha he problem of assessng ransverse vbraons of elasc sold srucures subeced o movng loads has been commonly consdered for a pon-lke ype of movng load see for eamples [Gbadeyan and On 99 Sadku and epholz 989 ee 99 On and Omolofe 5b On and Awodola 5]. Whle sudes concernng a dynamcal sysem nvolvng movng dsrbued loads are no so common. owever n engneerng pracce movng loads are mos ofen n he form of dsrbued mass raher han ha of movng lumped mass. When he movng load s dsrbued he problem of nvesgang he load-srucure neracons becomes much more complcaed. Thus o sudy he dynamc characerscs of such dynamcal sysems o he degree of acepable accuracy requred and also for praccal purposes s useful o consder elasc srucural members subeced o movng dsrbued loads. Among few auhors n recen mes who made effor o ackle he problem of elasc srucures carryng dsrbued movng masses are [Esmalzadeh and Ghorash 995] who carred ou an analyss of he dynamc behavour of Bernoull-Euler beam carryng unform parally dsrbued movng masses. They solved he problem by means of convenonal analycal echnque whch s only suable for he smple horzonal beam and wll suffer much dffcul f he srucures are complcaed. In hs sudy he convecve erms whch descrbe he dynamc effecs of he movng mass were omed. Ths appromaon s no generally reasonable unless he mass moves a very low speed and may lead o sgnfcan errors n he evaluaon of he sysem response. Ohers nclude [Dada ] who worked on he vbraon analyss of elasc plaes under unform parally dsrbued loads and [Adeunde 3] who suded he dynamcal response of Raylegh beam carryng added mass and raversed by unform parally dsrbued movng loads. owever n hese sudes numercal smulaons were employed. an Amercan Journal of Solds and Srucures

3 3 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses More recenly [And 3 Ogunyeb 6 And and On ] carred ou dynamcal analyss of srucural members carryng unform parally dsrbued masses wh general boundary condons under ravellng dsrbued loads. In hese sudes versale analycal echnques were used o oban soluons vald for all varans of classcal boundary condons. Though hese auhors presened a very good analyss of he response of beams o dsrbued loads bu her sudes faled o represen he physcal realy of he problem formulaon as hey employed n her sudes a smplfed model of he dsrbued load n whch he facor ha measures he degree of he load dsrbuon was omed. Thus hs work herefore concerns he problem of he behavoral sudy of a slender member connuously suppored by elasc subgrade and subeced o unform parally dsrbued movng masses and ses a solvng hs class of dynamcal problem for all pernen boundary condons ofen encounered by praccng engneers. y M V g Fgure : Schemac dagram of a beam under parally dsrbued load.esmalzadeh and Gorash 995. TEOR AND FORMUATION OF TE PROBEM Consder he vbraon of a srucural member resng on elasc foundaon and raversed by unform parally dsrbued masses M. The mass M s assumed o srke he beam a he pon = and me = and ravels across wh a consan velocy v. The equaon of moon assumng unform cross secon s gven by he fourh order paral dfferenal equaon. EI z z z N z Kz G Q where E s he modulus of elascy I s he second momen of he beam s cross-secon s he mass per un lengh of he beam N s he aal force K s he foundaon consan G s he shear rgdy z s he deflecon of he beam measured upward from s equlbrum poson when unloaded and Q s he ravellng dsrbued load. I s remarked here ha he beam under consderaon s assumed o have smple ends a boh ends = and =. Thus he boundary condons are z z z z an Amercan Journal of Solds and Srucures

4 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses 35 an Amercan Journal of Solds and Srucures and he nal condons of he moons of he slender member s gven as z z 3 If he load nera s akng no consderaon Q can be epressed as g z Q Q f whle he movng force Q f acng on hs engneerng srucure s gven as Mg Q f 5 where s he eavsde un sep funcon wh he propery 6 and v 7 For he lmng condon as one obans v 8 where v s he Drac dela funcon. Furhermore he operaor used n s defned as 9 Consderng equaon 5 and 8 would lead o he foundaon for movng pon mass. owever n hs work s no lmed o be a small lengh. Subsung no and akng no accoun 5 and 9 afer some rearrangemens gves Mg Z Z z M Z G KZ Z z N z EI

5 36 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses whch s he fourh order paral dfferenal equaon descrbng he fleural moons of our engneerng member when raversed by unform parally dsrbued masses. In wha follows an analycal echnque ha wll be used o rea he problem above wll be dscussed. 3 SOUTION PROCEDURES To solve he problem above a versale mehod ofen used n problems nvolvng mechancal vbraons wll be adoped. Ths mehod wll be employed o remove he es n he governng dfferenal equaon and o reduce o sequence of second order ordnary dfferenal equaon wh varable coeffcens. Ths soluon echnque nvolves solvng equaon of he form. Q where = The dfferenal operaor lnear or non-lnear z = The srucural dsplacemen Q = The ransverse load acng on he srucure A sequence of lnearly ndependen funcons z U whch are he normalzed deflecon curves for he h mode of he vbrang beam sasfyng smply suppored boundary condons are chosen as U and approprae soluons sough n he form The funcons funcons sn Z U 3 are he unknown funcons of me o be deermned. Thus he unknown are obaned from he condon ha he epresson U should be orhogonal o he funcons U. In hs way we ge a se of coupled ordnary dfferenal equaons From whch we oban U Q U d. These se of coupled second order ordnary dfferenal equaons are called Galerkn s equaons. The se of coupled second order ordnary dfferenal equaons are furher reaed usng he modfed asympoc mehod of Sruble. Furhermore he followng propery of eavsde funcon v f f f... d f 5 3! 5! an Amercan Journal of Solds and Srucures

6 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses 37 an Amercan Journal of Solds and Srucures wll be useful and also he funcon Q s assumed o be epressble as U Q 6 where are unknown funcons of me. 3. Operaonal Smplfcaon I s evden ha an eac closed form soluon of he paral dfferenal equaon s mpossble. Thus subsung he epressons 3 and 5 and 6 no afer some rgorous mahemacal procedures and rearrangemens one obans Mg Sn Mv Mv Sn M Sn G Sn K Sn Sn N Sn EI 7 In order o deermne an epresson for s requred ha he epresson on he lef hand sde of equaon 7 be orhogonal o he funcon U. To hs effec mulplyng equaon 7 by U negrang from end = o end = and afer some smplfcaons and rearrangemens one obans P Sn Sn Sn v Sn v Sn Sn where G K N EI Mg P 9 Equaon 8 s he ransformed equaon governng he problem of he unform Euller- Bernoull beam resng on he elasc sub-grade and subeced o unform parally dsrbued parameer sysem. Now consderng he h parcle of he dynamcal sysem leads o

7 38 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses an Amercan Journal of Solds and Srucures P Sn Sn Sn v Sn v Sn Sn where M Equaon s he ransformed equaon governng he moon of a unform smply suppored beam resng on b-paramerc elasc foundaon and subeced o ravellng masses. In wha follows a closed form soluon of equaon s sough o hs end we shall consder wo specal cases of equaon namely he movng force and movng mass problems. 3. The Movng Force Beam-oad Ineracon Problem The second order ordnary dfferenal equaon descrbng he behavour of a hn beam resng on elasc sub-grade and under he acons of a unform parally dsrbued movng force may be obaned from equaon by seng. In hs case one obans P Sn In vew of 7 equaon can furher be wren as v P Sn o 3 Equaon 3 s a classcal case of a movng force problem assocaed wh he sysem. Equaon 3 afer some smplfcaons yelds v P v P Sn where P P o P Sn P o 5

8 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses 39 To oban an epresson for equaon s subeced o a aplace ransformaon defned as s e d 6 where s s he aplace parameer. Applyng he nal condons 3 one obans he followng algebrac equaon a s s s s P P 7 s a s a where v a 8 Equaon 7 afer some smplfcaons leads o s P s s a a P s s s a 9 Thus hs problem reduces o ha of fndng he aplace nverson of he equaon 9 bove. To hs effec he followng represenaons are adoped. a s F s G s and G s 3 s s a s a so ha he aplace nverson of equaon 9 s he convoluon of F and G s defned as FG F u G u 3 Where Thus he aplace nverson of equaon 9 s gven as P P I A I B 3 an Amercan Journal of Solds and Srucures

9 3 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses I A sn cos u cos sn u sn audu 33 I B sn cos u cos sn u cosaudu 3 Evaluang negrals 33 and 3 one obans I B I A a a sn sn a 35 a cos cos a 36 Subsung 35 and 36 no equaon 3 gves he epresson for as sn a a sn P cos a cos P 37 a Thus n vew of 3 akng no accoun and 37 gves z sn a asn P cosa cos P sn a 38 whch represens he ransverse response of Euler-Bernoull beam resng on elasc sub-grade and subec o unform parally dsrbued movng forces when he neral effec of he sysem s negleced. 3.3 The Movng Mass Beam-oad Ineracon Problem Snce he mass of he movng load s commensurable wh ha of he srucure he nera effec of he movng mass s no neglgble. Thus and one s requred o solve he enre equaon when no erm of he coupled dfferenal equaon s negleced. Ths s ermed he movng mass problem. Unlke n he case of he movng force an eac analycal soluon o hs equaon s no possble. Thus one resors o an appromae analycal echnque due o Sruble dscussed n [Gbadeyan and On 99 On and Omolofe 5b]. By hs echnque we seek he modfed frequency correspondng o he frequency of he free sysem due o he presence of he effec of he mass of he load. To hs end equaon s rearranged o ake he form. an Amercan Journal of Solds and Srucures

10 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses 3 an Amercan Journal of Solds and Srucures Sn Sn gq Sn Sn Sn Sn Sn Sn Sn Sn Sn Sn Sn Sn 39 Where v v v v Evdenly unlke he movng force problem an eac analycal soluon o equaon 39 does no es. Thus a modfcaon of he asympoc mehod due o sruble ofen used n reang weakly homogenous and non-homogenous non-lnear oscllaory sysem s resored o. By hs echnque one seeks he modfed frequency correspondng o he frequency of he free sysem due o

11 3 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses he presence of he effec of he movng mass. Followng he procedures eensvely dscussed n [On and Omolofe 5b] he homogeneous par of equaon 39 s smplfed o ake he form where d d mm 5 mm * s called he modfed naural frequency represenng he frequency of he free sysem due o he presence of he movng mass. Thus he enre equaon 39 reduces o v mm * gqo sn d d 3 whch when solved n conuncon wh he nal condons yelds an epresson for as Q sna asn Q cosa cos mm mm mm mm mm mm mm a where Q gqo cos Q gqo sn 5 Thus n vew of 3 akng no accoun and gves z Q sn aasn Q cosa cos sn a mm mm mm mm mm 6 mm mm Equaon 6 represens he ransverse response of a smply suppored Euler-Bernoull beam resng on an elasc sub-grade and subec o unform parally dsrbued movng mass. an Amercan Journal of Solds and Srucures

12 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses 33 COMMENTS ON TE COSED FORM SOUTIONS In an mporan sudy such as hs nvesgang he phenomenon of resonance s very crucal because he ransverse dsplacemen of an elasc beam may ncrease whou lm. I s seen from equaon 38 ha he smply suppored beams on elasc subgrade and under he acons of ravellng dsrbued forces reaches a sae of resonance whenever kv 7 whle equaon 6 clearly shows ha he same beam under he acons of movng dsrbued masses wll eperence resonance effecs whenever bu from equaon kv mm 8 mm 5 9 whch mples kv 5 5 I s herefore clear ha for he same naural frequency he crcal speed for he sysem conssng of a smply suppored Bernoull-Euler beam resng on elasc foundaon and under he acons of ravellng dsrbued force s greaer han ha of movng dsrbued mass problem. Thus for he same naural frequency resonance s reached earler n he movng dsrbued mass han n he movng dsrbued force sysem. 5 ANASIS OF RESUT AND DISCUSSION In hs secon he analyss proposed n he prevous secons are llusraed by consderng a homogenous beam of modulus of elascy E.9 N / m he momen of neral 9 3 I kgm he beam span. 9m and he mass per un lengh of he an Amercan Journal of Solds and Srucures

13 3 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses beam 758.9kg / m. The load s also assumed o ravel wh consan velocy V 8.8m / s. The 3 3 values of foundaon modul are vared beween N / m and N / m he values of aal force N vared beween N and. 8 N. Fgure dsplays he ransverse dsplacemen response of a smply suppored unform beam under he acon of unform parally dsrbued forces ravellng a consan velocy for he varous values of aal force N and for fed values of subgrade modul K= and shear modulus G=3. The fgures show ha as N ncreases he response amplude of he unform beam decreases. Smlar resuls are obaned when he smply suppored beam s subeced o parally dsrbued mass ravellng a consan velocy as shown n fgure 8. For varous ravellng me he dsplacemen response of he beam for varous values of subgrade modul K and for fed values of aal forcé N= and shear modulus G=3 are shown n fgure 3. I s observed ha hgher values of subgrade modul K reduce he deflecon of he vbrang beam. The same behavor characerzes he response of he smply suppored beam under he acons of unform parally dsrbued masses movng a consan velocy for varous values of subgrade modul K as shown n fgure 9. Also fgures and dsplay he deflecon profle of he smply suppored unform beam respecvely o parally dsrbued forces and masses ravellng a consan velocy for varous values of shear modulus and fed values of aal forcé N= and subgrade modul K=. These fgures clearly show ha as he value of he shear modul ncreases he deflecon of he smply suppored unform beam under he acon of boh movng forces and masses ravellng a consan velocy decreases..5. N= N= N= N=.5 Dynamc Deflecon m oad Travellmg Tme s Fgure : Transverse dsplacemen response of a smply suppored srucural members resng on elasc foundaon and under he acons of unform parally dsrbued forces for varous values of aal force N and for fed values of K = G = 3. an Amercan Journal of Solds and Srucures

14 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses 35.5 K=. K= K= K=.5 Dynamc Deflecon m oad Travellng Tme s Fgure 3: Dsplacemen response of a smply suppored srucural members resng on elasc foundaon and under he acons of unform parally dsrbued forces for varous values of foundaon modulus K and for fed values of N = G = G= G=3. G=3 G=3.5 Dynamc Deflecon m oad Travellng me s Fgure : Deflecon Profle of a smply suppored srucural members resng on elasc foundaon and under he acons of unform parally dsrbued forces for varous values of shear modulus G and for fed values of K= and N =. an Amercan Journal of Solds and Srucures

15 36 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses Fgure 5: Response Amplude of a smply suppored srucural members resng on elasc foundaon and under he acons of unform parally dsrbued forces for varous values of he load wdh and for fed values of G = 3 K= and N =.. =.5.3 =3.5 =6.96 =9.5. namc Deflecon m Oad Travellng Tme s Fgure 6: Response of a smply suppored srucural members resng on elasc foundaon o unform parally dsrbued forces for varous values of he load poson and for fed values of G = 3 K= and N =. an Amercan Journal of Solds and Srucures

16 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses 37.8 V=.5.7 V=.5 V=8.8.6 V=.5.5 Dynamc Deflecon m oad Travellng Tme s Fgure 7: Response characerscs of a smply suppored srucural members resng on elasc foundaon o unform parally dsrbued forces for varous values of he ravellng load veloces and for fed values of G = 3 K= and N =. Fgure 5 dsplays he response amplude of a smply suppored unform beam under he acon of unform parally dsrbued forces ravellng a consan velocy for varous values of load wdh and for fed values of subgrade modul K= aal force N= and shear modulus G=3. The fgure show ha as he wdh ncreases he effecs of he wdh on he response amplude of he unform beam ncreases as he load progresses on he srucure. Smlar resuls are obaned when he smply suppored beam s subeced o parally dsrbued masses ravellng a consan velocy as shown n fgure. For varous ravellng me he response of he beam for varous values of ravellng load posons and for fed values of aal forcé N= subgrade modulus K= and shear modulus G=3 are shown n fgure 6. I s observed ha he mpac of he ravellng load s greaes a he mddle of hs vbrang sold srucure. The same behavor characerzes he response of he smply suppored beam under he acons of unform parally dsrbued masses movng a consan velocy for dfferen ravellng load posons as shown n fgure. an Amercan Journal of Solds and Srucures

17 38 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses 8 6 N= N= N= N= Dynamc Deflecon m oad Travellng Tme s Fgure 8: Transverse dsplacemen response of a smply suppored srucural members resng on elasc foundaon and under he acons of unform parally dsrbued masses for varous values of aal force N and for fed values of K = G = 3. 6 K= K= K= K= Dynamc Deflecon m Oad Travellng Tme s Fgure 9: Dsplacemen response of a smply suppored srucural members resng on elasc foundaon and under he acons of unform parally dsrbued masses for varous values of foundaon modulus K and for fed values of N = G = 3. an Amercan Journal of Solds and Srucures

18 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses 39 8 G= G=3 6 G=3 G=3 Dynamc Deflecon m oad Travellng Tme s Fgure : Deflecon Profle of a smply suppored srucural members resng on elasc foundaon and under he acons of unform parally dsrbued masses for varous values of shear modulus G and for fed values of K= and N =. Fgure : Response Amplude of a smply suppored srucural members resng on elasc foundaon and under he acons of unform parally dsrbued masses for varous values of he load wdh and for fed values of G = 3 K= and N =. an Amercan Journal of Solds and Srucures

19 33 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses =.5.5 =3.5 =6.96 =9.5 Dynamc Deflecon m oad Travellng Tme s Fgure : Response of a smply suppored srucural members resng on elasc foundaon o unform parally dsrbued masses for varous values of he load poson and for fed values of G = 3 K= and N =. 3.5 V=.5 V=.5 V=8.8 3 V=.5.5 Dynamc Deflecon m oad Travellng Tme s Fgure 3: Response characerscs of a smply suppored srucural members resng on elasc foundaon o unform parally dsrbued masses for varous values of he ravellng load veloces and for fed values of G = 3 K= and N =. an Amercan Journal of Solds and Srucures

20 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses 33.5 MOVING FORCE MODE MOVING MASS MODE Dynamc Deflecon m oad Travellng Tme s Fgure : Comparson of he dynamc characersc of movng force and movng mass cases of a unform smply suppored beam for fed values of G = K= and N =. 8 MOVING FORCE MODE 6 MOVING MASS MODE Dynamc Deflecon m oad Travellng Tme s Fgure 5: Comparson of he dynamc characersc of movng force and movng mass cases of a unform smply suppored beam for fed values of G = 3 K= and N =. an Amercan Journal of Solds and Srucures

21 33 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses..5 K=G=N= K=G=3N=..5 Dynamc Deflecon m oad Travellng Tme s Fgure 6: Comparson of he deflecon profles of he smply suppored movng force unform beam for values K=G=N= versus G = 3 K= and N =. 8 K=G=N= 6 K=G=3N= Dynamc Deflecon m oad Travellng Tme s Fgure 7: Comparson of he deflecon profles of he smply suppored movng mass unform beam for values K=G=N= versus G = 3 K= and N =. Fgure 7 dsplays he response characerscs of a smply suppored unform beam under he acon of unform parally dsrbued forces for varous values of load velocy V and for fed values of subgrade modul K= aal force N = and shear modulus G = 3. The fgures show ha he hgher he velocy he larger he deflecon of he vbrang srucure. Smlar resuls are obaned when he smply suppored beam s subeced o parally dsrbued masses as dsplayed n fgure 3. Fgures and 5 depc he comparson of he response characerscs of he movng force and movng mass cases of a smply suppored unform beam raversed by a mov- an Amercan Journal of Solds and Srucures

22 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses 333 ng dsrbued load ravellng a consan velocy for fed values of N= K= G= and N= K= and G=3. From hese fgures s seen ha he dynamc deflecon of he beam under he acons of he movng load s grealy affeced when he srucural parameers N K and G are ncoporaed no he governng equaon of moon. Fgure 6 compares he deflecon profles of he movng force model of he beam for he wo se of values K= N= G= and N= K= and G=3. I s deduced from hs fgure ha he amplude of deflecon for he se of values N= K= and G= s much hgher an ha of he se of values N= K= and G=3. Smlar resul s obaned for a movng mass model of hs srucural member as shown n fgure 7. 6 CONCUDING REMARKS The classcal problem of he response characerscs of unform srucural member resng on elasc subgrade and subec o unform parally dsrbued load s suded n hs work. The closed form soluons of he governng fourh order paral dfferenal equaons wh varable coeffcens are presened for he movng force and mass models. Varous resuls and analyses are carred ou and phenomenon of resonance s suded for he dynamcal sysem. The fndngs of hs sudy ehb among ohers he followng useful and neresng feaures:. Ths sudy has provded a useful nformaon on he effec of aal force foundaon sffness load wdh load velocy and shear modulus on unform Bernoull-Euler beam resng on Pasernak foundaon and under he acon of unform parally dsrbued loads.. Resuls also show ha hgher values of he srucural parameers K N and G are requred n he case of movng mass problem han ha of he movng force beam problem.. I s found ha he dynamc sably of he elasc beam subeced o movng load s grealy enhanced wh he presence of he srucural parameers N K and G. v. Ths sudy has also provded a very useful nformaon on he condons under whch he vbrang sysem wll eperence he occurrence of undesrable phenomenon called resonance for boh he movng dsrbued force and mass problems nvolvng Bernoull Euler beams. References Adeunde I.A. 3 Dynamc analyss of Raylegh beam carryng added mass and Traversed by unform parally dsrbued movng loads. Ph.D Thess. Unversy of Ilorn Ilorn Ngera. And E.A. 3 Fleural vbraons of elasc srucures wh general boundary condons and under ravellng dsrbued loads. Ph.D Thess Federal Unversy of Technology Akure. And E.A. On S.T. Dynamc Behavour under Movng Dsrbued Masses of Nonunform Raylegh Beam wh General Boundary Condons. Chnese Journal of Mahemacs. hp://d.do.org/.55// Clasornk J. Esenberger M. ankelevsky D.Z. Adn M.A. 986 Beams on VarableWnkler elasc foundaon Journal of Appled Mechancs ASME Dada M.S. Vbraon analyss of elasc plaes under unform parally Dsrbued movng loads. Ph.D Thess Unversy of Ilorn Ilorn Ngera. Journal of Ngeran Assocaon of Mahemacal Physcs Vol Esenberger M. Clasornk J. 987; Beams on varable elasc foundaon. Journal of Engneerng Mechancs Vol. 3 No pp an Amercan Journal of Solds and Srucures

23 33 B. Omolofe and T.O. Adeloye / Behavoral Sudy of Fne Beam Resng on Elasc Foundaon and Subeced o Travellng Dsrbued Masses Esmalzadeh E. Ghorash M. 995 Vbraon analyss of beams raversed by unform parally dsrbued movng masses usng movng masses. Journal of sound and vbraon Engneerng: 8 pp -7. Gbadeyan J.A. On S.T. 99 Dynamc Response o movng concenraed masses of elasc Plae on a non- Wnkler s elasc foundaon. Journal of sound and vbraon asan O. Zek K. Bnnur G.K. 6 Dynamc Analyss of Elascally Suppored Cracked Beam Subeced o a Concenraed Movng oad. an Amercan Journal of Solds and Srucures İsmal E. 5 A new FEM procedure for ransverse and longudnal vbraon analyss of hn recangular plaes subeced o a varable velocy movng along an arbrary raecory load. an Amercan Journal of Solds and Srucures ee.p. 99 Dynamc Response of a beam wh nermedae pon consyans subec o a movng load. Journal of sound and vbraon Ogunyeb S.N. 6 Dynamcal Analyss of fne presressed Bernoull-Euler beams wh general boundary condons under ravellng dsrbued loads M. Tech. Dsseraon Federal Unversy of Technology Akure Ngera. Omolofe B. 3 Deflecon profle analyss of beams on wo-parameer elasc subgrade an Amercan Journal of Solds and Srucures On S.T. Awodola T.O. 5; Dynamc Response o movng concenraed masses of unform Raylegh beams resng on varable Wnkler elasc foundaon. Journal of he Ngean Assocaon of Mahemacal Physcs On S.T. Omolofe B. 5a Dynamc behavor of non-unform Bernoull-Euler beams subeced o concenraed loads ravellng a varyng veloces. Journal of he Ngeran Assocaon of Mahemacal Physcs Vol. 9 pp 79-. On S.T. Omolofe B 5b; Dynamc Analyss of a presressed elasc beam wh general boundary condons under movng loads a varyng veloces. Journal of Engneerng and Engneerng Technology FUTA Sadku S. epholz..e 989; On he dynamcs of elasc sysems wh movng concenraed masses Ingeneur Achves Sun. uo F. 8 Seady-Sae Dynamc Response of a Bernoull-Euler Beam on a Vscoelasc foundaon Subec o a Plaoon of Movng Dynamc oads Journal of Vbraon and Acouscs Vol. 3 pp 5-5. Thambranam D.P. Zhuge. 996; Dynamcs analyss of beams on an elasc foundaon subeced o movng loads. Journal of sound and vbraon Vol. 98 No pp Ugurlu B. Kulu A. Ergn A. Omurag M.. 8; Dynamcs of a recangular plae resng on an elasc foundaon and parally n conac wh a quescen flud. Journal of sound and Vbraon 37 pp ng J. u C.F. Chen W.Q. 8; Two-dmensonal elascy soluons for funconally graded beams resng on elasc foundaons. Compuer Srucure Vol. 8 No 3 pp 9-9. an Amercan Journal of Solds and Srucures