Bernoulli-Euler Beam Response to Constant Bi-parametric Elastic Foundation Carrying Moving Distributed Loads

Transcription

1 Aercan Jurnal f Engneerng Research (AJER) 14 Aercan Jurnal f Engneerng Research (AJER) e-issn : p-issn : Vlue-3, Issue-1, pp Research Paper Open Access Bernull-Euler Bea Respnse t Cnstant B-paraetrc Elastc Fundatn Carryng Mvng Dstrbuted ads Ogunyeb S.N Departent f Matheatcal Scences, Ekt State Unversty, Ad-Ekt, Ekt State, Ngera. ABSTRACT: A study f the dynac respnse f Bernull-Euler bea t cnstant b-paraetrc elastc fundatn t vng dstrbuted frces s presented. The transfred equatn gvernng the syste s btaned by eans f the Galerkn s technque. The cases f the dynac respnse f the bea t dstrbuted lads f equal agntude are studed. Nuercal eaples are gven n rder t deterne the effects f varus paraeters n the respnse f the sply-supprted Bernull-Euler bea. Keywrds- Deflectn, fundatn dulus, Resnance, Shear frce, Sply supprted bea. I. INTRODUCTION Beas are f great prtance n cnstructn engneerng. Ths ay be due t ther lght weght and ths has cntrbuted t ther wdely usage. The dynac respnse f Bernull-Euler bea have been etensvely studed especally fr sply supprted beas [1], [], [3], [4], [5]. In st f the prevus wrks, the prble f assessng the dynac respnse f Bernull-Euler bea carry vng lads, has been restrcted t the case when the lads are splfed as vng cncentrated frces [6], [7], [8] and stly placed n Wnkler elastc fundatn [9], [1]. The classcal Wnkler del has varus applcatns n cnstructn engneerng and ths del suffers several crtcss as t has se shrtcngs due t the dscntnuty f the adacent dsplaceent [11]. T tackle ths defcency, a better del that ntrduced shear nteractn between adacent Wnkler sprng eleents was ntrduced [1]. Several authrs n the area f structural dynacs have thrughly nvestgated the dynacs and stablty f the Wnkler-type fundatn del by bth apprate ethds [13] and eact appraches [14]. In 1991, [15] presented se fnte eleent dels fr the statc analyss f Euler-Bernull bea restng n a Wnkler-type fundatn. In 1, Olfe [16] nvestgated the dynac respnse t vng lad f an elastcally supprted nn-prsatc Bernull-Euler bea n varable elastc fundatn and btaned analytcal slutns fr whch the nuercal slutns are dsplayed n pltted curves. In ths paper, the prble f dynacal analyss f Bernull-Euler when t s sply supprted and restng n cnstants b-paraetrc elastc fundatn under vng dstrbuted frces are presented. All the cpnents f nerta ters are cnsdered n the analyss. Whle sectn II descrbes the thery and bref descrptn f the prble under nvestgatn, sectn III fcuses n the technque nvlved n the transfratn f the furth rder partal dfferental equatn f the dynacal syste. Se rearks n the analytcal slutn s btaned are reprted n sectn IV and fnally, nuercal results are dsplayed n pltted curves n sectn V. II. THEORY AND FORMUATION The prble f the dsplaceent respnse f sply-supprted Bernull-Euler bea restng n cnstant b-paraetrc elastc fundatn carry vng dstrbuted lads s gverned by the furth rder partal dfferental equatn w w w. a e r. r g Page 11

2 Bernull-Euler Bea Respnse t Cnstant B-paraetrc 4 Y (, t ) E I Y (, t ) Y (, t ) N Q ( ) Y (, t ) M H ( c t ) P Y (, t ) P (, t ) 4 f t (.1) E s the yung dulus, Y (, t ) s the transverse dsplaceent, Q ( ) s the b-paraetrc elastc fundatn, P (, t ) s the vng lad. Als, H (, c t ) s the Heavsde functn defned as, f r < H (, ct ) 1, f r > (.) In ths paper, when the effect f the ass f the vng lad n the bea s cnsdered, P (, t ) takes the fr P (, t ) M H (, c t ) g c c Y (, t ) f t t The bundary cndtns fr ur dynacal syste are arbtrary and the ntal cndtns wthut any lss f generalty are taken t be (.3) Y (, ) Y (, ) t (.4) The relatnshp that ests between the fundatn reactn and the lateral deflectn Y (, t ) s gven by Q ( ) G Y (, t ) K Y (, t ) (.5) G and K are tw cnstant paraeter f elastc fundatn. Thus, G s the cnstant fundatn stffness and K s the varable shear dulus. T ths end, equatn (.5) can be wrtten as Y (, t ) Y (, t ) Q (, t ) G Y (, t ) K K (.6) Substtutng equatns (.) t (.6) nt equatn (.1), ne btans 4 d d E I Y (, t ) Y (, t ) N Y (, t ) G Y (, t ) K Y (, t ) M H ( c t ) 4 t d d Y (, t ) C Y (, t ) C Y (, t ) M g H ( c t ) t t III. MATHERIAS AND METHOD A clse fr slutn t equatn (.7) des nt est, s the elegant Galerkn s ethd descrbed n [1] s eplyed t tackle the furth rder partal dfferental equatn. The ethd s presented n the fr 1 (.7) Y (, t ) W ( t ) U ( ) (3.1) U ( ) s chsen such that the desred bundary cndtns are satsfed. And snce we are cnsderng sply supprted bundary cndtn, U ( ) s defned thus, sn U ( ) (3.) Substtutng equatn (3.1) nt equatn (.7), takng nte f equatn (3.) ne btans w w w. a e r. r g Page 111

3 Bernull-Euler Bea Respnse t Cnstant B-paraetrc IV E IU ( ) W ( t ) U ( ) W ( t ) G U ( ) W ( t ) K U ( ) W ( t ) K U ( ) W ( t ) 1 t M s n c s C W ( t ) U ( ) C W ( t ) U ( ) C W ( t ) U ( ) 1 M g c t c s c s (3.3) wher e H ( ct ) have been defned as (3.4) 1 t H ( c t ) sn c s C psng rthgnaly cndtn, equatn (3.3) after splfcatn beces 1 v v E IW ( t ) U ( ) W ( t ) U ( ) G W ( t ) U ( ) K W ( t ) U ( ) K W ( t ) U ( ) M c t sn c s C W ( t ) U ( ) C W ( t ) U ( ) C W ( t ) U ( ) 1 M g c t U ( ) c s c s k whch can be re-wrtten as (3.5) 1 E I S ( ) K ( ) K ( ) W ( t ) H (, k ) H (, ) (, ) (, ) (, ) ( ) 1 H k H k H k H k W t M M ct MC H (, k ) W ( t ) c s H (, k ) W ( t ) W ( t ) H (, k ) C c c M M c t M c W ( t ) H (, k ) c s H (, k ) W ( t ) W ( t ) H (, k ) W ( t ) H (, k ) 1 C c c s H (, k ) H (, k ) c s c s (3.6) 1 M c c t M g c t H (, k ) U ( ) U ( ) d ; v H (, k ) U ( ) U ( ) d k 1 k H (, k ) U ( ) U ( ) d ; H (, k ) ( ) ( ) U U d 4 k 3 k H (, k ) U ( ) U ( ) d ; H (, k ) U ( ) U ( ) d 6 k 5 k w w w. a e r. r g Page 11

4 H (, k ) U ( ) U ( ) sn d 7 k Bernull-Euler Bea Respnse t Cnstant B-paraetrc ; H (, k ) 8 U ( ) U ( ) d k H (, k ) ( ) ( ) sn 1 U U d ; H (, k ) U ( ) U ( ) d H (, k ) k 1 1 k 4 H (, k ) U ( ) U ( ) d 1 k H (, k ) U ( ) U ( ) sn d ; 13 k H (, k ) U ( ) U ( ) d (3.7) 14 k equatn (3.5) when re-arranged gves W ( t ) W ( t ) H (, k ) c s H (, k ) C H (, k ) W ( t ) c t c t C H (, k ) c s H (, k ) C H (, k ) W ( ) 9 t C ct C H (, k ) c s (, ) (, ) 1 H k C C H k M g c t c s c s (3.8) s that 1 1 8k c s n c t W ( t ) W ( t ) ( k ) ( k ) ( n k ) ( n k ) n 1 c 4 ( n k ) 4 n c W ( t ) c W ( t ) ( n k ) ( n k ) k n 1 1 8k c n c c n ( ) W t ( n k ) ( n k ) ( n k ) ( n k ) M g c t c s c s Further arrangeents f equatn (3.9) gve (3.9) M g c t H (, k ) ( ) ( ) 1 ( 1) c s (3.1) W t W t 1 w w w. a e r. r g Page 113

5 Bernull-Euler Bea Respnse t Cnstant B-paraetrc 1 E I G K K H (, k ) H (, k ) H (, k ) H (, k ) H (, k ) and by aplace ethds, ne btans (3.11) M g S ( 1) W ( t ) S W ( t ) H (, k ) S S 1 (3.1) c (3.13) then fr equatn (3.1) M g W ( t ) S T T H (, k ) a b 1 (3.14) T a S 1 S S ; T b R 1 S S and R ( 1) (3.15) S when slutns f T andt a b are substtuted nt equatn (3.14), ne btans M g c s t c s t 1 c s t W ( t) H (, k ) 1 whch when nverted gves, (3.16) s n Y (, t ) H k M g c s t c s t 1 c s t (3.17) 1 (, ) 1 whch represents the dsplaceent respnse t vng dstrbuted frce f sply supprted Bernull-Euler bea restng n cnstant b-paraetrc elastc fundatn. Net, t s pertnent t seek slutn t the vng dstrbuted ass f the prble. If s nt equal t zer n equatn (3.9), t eans that the nerta ter s retaned and t s evdent that an eact slutn t the equatn (3.9) s nt pssble. A dfcatn f Strubles technque etensvely dscussed n [1] s used t btan the dfed frequency. T ths end, equatn (3.9) s wrtten n the fr 8k c s n c t c 1 W ( ) t ( k ) ( k ) ( n k ) ( n k ) n 1 4 ( n k ) 4 n c 1 1 c W ( t ) c n ( n k ) ( n k ) k n ( n k ) ( n k ) 8k c n c M g c t W ( t ) c s c s ( n k ) ( n k ) (3.18) Further re-arrangeents gve w w w. a e r. r g Page 114

6 Bernull-Euler Bea Respnse t Cnstant B-paraetrc 4 ( n k ) 4 n c W ( t ) c W ( t ) 1 ( n k ) ( n k ) k n 8k c s n c t 1 ( k ) ( k ) ( n k ) ( n k ) n k c n c 1 c n W ( t ) ( n k ) ( n k ) ( n k ) ( n k ) M g c t c s c s (3.19) when 1,the hgenus part f equatn (3.19) can be wrtten as ( t ) W ( t ) ( t ) c s t W ( t ) (3.) and are slwly varyng functns r equvalently. By Struble s ethd, ne btans 4 k 4 c n 1 Y ( t ) c Y ( t ) Y ( t ) Y ( t ) 1 1 n ( n k ) ( n k ) k n (3.1) and whch when slved gves and c Y ( t ) Y ( t ) (3.) 1 s a cnstant. ( t) B 1 C (3.3) c ( t) t 1 Substtutng (3.3) and (3.4) nt equatn (3.), ne btans W ( t ) C B c s t (3.4) 1 (3.5) Therefre when the ass effect f the partcle s cnsdered, the frst appratn t the hgenus syste s gven as W ( t ) ( t ) c s t (3.6) w w w. a e r. r g Page 115

7 Bernull-Euler Bea Respnse t Cnstant B-paraetrc c 1 1 (3.7) s the dfed frequency crrespndng the frequency f the free syste due t the presence f the vng dstrbuted ass. Thus, M g c t W t W t ( ) ( ) c s c s (3.8) Equatn (3.8) s a prttype f equatn (3.1) and fllwng slar arguents, ne btans whch when nverted gves, M g c s t c s t c s t W ( t) 1 (3.9) Y (, t ) sn H k M g c s t c s t 1 c s t (3.3) 1 (, ) 1 whch represents the dsplaceent respnse t vng dstrbuted ass f sply supprted Bernull-Euler bea restng n cnstant elastc fundatn. IV. REMARKS ON ANAYTICA SOUTION 4.1 Effect f Resnance When an undaped syste such as ths s cnsdered, t s pertnent t eane the resnance cndtn f the structure. Fllwng [1], and fr equatn (3.17), the Bernull- Euler bea traversed by a vng dstrbuted frce wll be n a state resnance when c (4.1) whle equatn (3.3) shws that the sae Bernull- Euler bea reaches resnance effect at Thus, c (4.) C (4.3) 1 1 Clearly, 1 C 1 1 C 1 (4.4) V. NUMERICA RESUTS Fr the analytcal slutns, calculatns f practcal nterests n structural engneerng and physcs are presented n ths sectn. The sply supprted Bernull-Euler bea f length 1.13, velcty c=8.1/s, fleural rgdty c /, and fr the shear dulus K, the values are vared E J N 3 5 between N / and 5 1 N /, fr fundatn stffness G vares between N/ and 3 1 N /, aal frce N values vared between N / and 1 N / and the ass per unt length f the structure s g/. The results are shwn n the varus curves belw fr the sply supprted Bernull-Euler bea n cnstant b-paraetrc elastc fundatn. w w w. a e r. r g Page 116

8 Bernull-Euler Bea Respnse t Cnstant B-paraetrc Fgure 1and depct the fleural deflectns f the bea restng n cnstant b-paraetrc elastc fundatn at cnstant velcty fr bth vng dstrbuted frce and vng dstrbuted ass. It s clearly seen that fr fed values f the shear dulus K and fundatn stffness G, the dsplaceent respnse f the bea decreases as the values f the prestress functn ncreases. Fg.1: Dsplaceent respnse f sply supprted Bernull-Euler bea fr vng dstrbuted frce fr varus values f aal frce N and fed values f shear dulus and fundatn stffness. Fg.: Dsplaceent respnse f sply supprted Bernull-Euler bea fr vng dstrbuted ass fr varus values f aal frce N and fed values f shear dulus and fundatn stffness. Fgure 3 and 4 dsplays the deflectn prfle f the sply supprted Bernull-Euler bea restng n cnstant b-paraetrc elastc fundatn at cnstant velcty fr bth vng dstrbuted frce and vng dstrbuted ass respectvely. It s fund that the dynac deflectns f the structure decreases as the values f the shear dulus K ncreases fr fed values f the prestress functn N and fundatn stffness G. w w w. a e r. r g Page 117

9 Bernull-Euler Bea Respnse t Cnstant B-paraetrc Fg.3: Deflectn prfle f sply supprted Bernull-Euler bea fr vng dstrbuted frce fr varus values f shear dulus K and fed values f aal frce and fundatn stffness. Fg.4: Deflectn prfle f sply supprted Bernull-Euler bea fr vng dstrbuted ass fr varus values f shear dulus K and fed values f aal frce and fundatn stffness. Slarly, fgure 3 and 4 dsplays the dynac deflectn f the structure restng n cnstant bparaetrc elastc fundatn at cnstant velcty fr bth vng dstrbuted frce and vng dstrbuted ass. Fr fed values f prestress N, shear frce K and varus values f fundatn dulus G, t s fund that the dynac deflectns f the bea decreases as the values f the fundatn stffness G ncreases. Fg.5: Deflectn prfle f sply supprted Bernull-Euler bea fr vng dstrbuted frce fr varus values f fundatn stffness and fed values f aal frce and shear dulus. w w w. a e r. r g Page 118

10 Bernull-Euler Bea Respnse t Cnstant B-paraetrc Fg.6: Deflectn prfle f sply supprted Bernull-Euler bea fr vng dstrbuted ass fr varus values f fundatn stffness and fed values f aal frce and shear dulus. The cparsn f the dsplaceent respnse f the sply supprted Bernull-Euler bea restng n cnstant b-paraetrc elastc fundatn at cnstant velcty fr vng dstrbuted frce and vng dstrbuted ass. It s bserved that the vng dstrbuted frce slutn s nt an upper bund fr vng dstrbuted ass slutn as shwn n the fgure belw Fg.7: Cparsn f the dsplaceent respnse f sply supprted Bernull-Euler bea restng n b-paraetrc elastc fundatn fr fed values f fundatn stffness, aal frce and shear dulus. VI. CONCUSION The prble f dynac analyss under a vng dstrbuted lad f a sply supprted Bernull- Euler bea n cnstant b-paraetrc elastc fundatn has been slved. And n vew f the cndtn f resnance establshed n sectn VI, t s deduced that fr the sae natural frequency, the crtcal speed fr the vng dstrbuted frce prble s greater than that f the vng dstrbuted ass prble. Hence, fr the sae natural frequency, resnance s reached earler n the vng dstrbuted ass prble than n the vng dstrbuted frce. REFERENCES [1]. Fryba, Vbatn f slds and structures under vng lads (Grnnggen: Nrdhf, (196), 197). [] M. R. Shandna, F. R Rfe, M. Mfd, and B. Mehr, Perdcty n the respnse f nn lnear vng Mass, Thn-walled structures, 4,, [3] S.Tshenk, On the crrectn fr shear f the dfferental equatn fr transverse vbratn f prsatc bars, Phl. Mag. S, 6(41), l91, [4] G. Muscln and A. Paler, Respnse f beas restng n vscelastcally daped fundatn t vng Oscllatrs, Internatnal Jurnal f Slds and Structures, 44(5), 7, [5] K. Achawakrn and T. Jearsrpngkul, Vbratn analyss f Epnental Crss-Sectn Bea Usng Galerkn s Methd, Internatnal Jurnal f Appled Scence and Technlgy, (6), 1. [6] S.T. On, Respnse f a nn-unfr bea restng n an elastc fundatn t several asses, ABACUS, Jurnal f Matheatcal Asscatn f Ngera, 4(),1996. w w w. a e r. r g Page 119

11 Bernull-Euler Bea Respnse t Cnstant B-paraetrc [7] B. Olfe, S. T. On and J. M. Tlrunshagba, On the Transverse Mtns f Nn-prsatc Deep bea under the actns f varable agntude vng lads, atn Aercan Jurnal f Sld and Structures, (6), 9, [8] S. Sadku and H. H. E. ephlz, On the Dynacs f elastc systes wth vng Cncentrated asses, Ingeneur Archve, (57), 1981, 3-4. [9] S.T. On and T.O. Awdla, Dynac respnse t vng cncentrated asses f unfr Raylegh beas restng n varable Wnkler elastc fundatn, Jurnal f the Ngeran Asscatn f Matheatcal Physcs. (9), 5, [1] S. T. On and S. N. Ogunyeb, Dynacal Analyss f fnte prestressed Bernull-Euler beas wth general Bundary cndtns under travellng dstrbuted lads, Jurnal f the Ngeran Asscatn f atheatcal Physcs,(1), 8, [11] H. Tanahash, characterstcs f rgd fundatns n Pasternak del (n case f tw-densnal plane Stran cndtn), Suares f Techncal Papers f Annual Meetng, AIJ (n Japanese), 1, [1] P. Pasternak, On a new ethd f analyss f an elastc fundatn by eans f tw fundatn cnstants, Gsudarstvenne Izdatelstv Iteratur P Strtelstuve n Arkhtekture, Mscw (n Russan) [13] M.A. De Rsa, Free Vbratns f Tshenk beas n Tw-Paraeter Elastc Fundatn, Cputer and Structures, 57(1), 1995, [14] S.H. Farghaly and K.M.Zed, An eact frequency Equatn Fr an Aally laded Bea-ass-sprng Syste Restng n a Wnkler Elastc Fundatn, Jurnal Of Sund and Vbratn, 185(), 1995, [15] A.G Razaqpur, and K.R Shah, Eact analyss f Beas n Tw-Paraeter Elastc Fundatns, Internatnal Jurnal f Slds Structures,7(4), 1991, [16] B.Olfe, Dynac respnse under a vng lad f an elastcally supprted nn-prsatc Bernull-Euler bea n varable elastc fundatn, atn Aercan Jurnal f Sld and Structures,7, 1, 3-. w w w. a e r. r g Page 1