Australian Journal of Basic and Applied Sciences

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1 Ausralian Journal of Basic and Applied Sciences, 9() a 05, Pages: 55-6 ISSN: Ausralian Journal of Basic and Applied Sciences Journal home page: Dnamic Response of an Inclined Railwa Bridge Suppored b Winkler Foundaion Under a oving Railwa Vehicle ichael C. Agarana, Jacob A. Gbadean, Olasunmbo O. Agboola, Timoh A. Anake, Olawale J. Adeleke Deparmen of ahemaics, College of Science & Technolog, Covenan Universi, P..B. 0, Oa, Nigeria Deparmen of ahemaics, Facul of Phsical Science, Universi of Ilorin, P..B. 55, Ilorin, Nigeria A R T I C E I N F O Aricle hisor: Received arch 05 Acceped 8 April 05 Available online 5 a 05 Kewords: deflecion; shear deformaion; roaor ineria; inclined railwa bridge; finie difference; Winkler foundaion A B S T R A C T An invesigaion ino he dnamical behaviour of an inclined railwa bridge raversed b uniform pariall disribued moving railwa vehicle, and suppored b an elasic foundaion is carried ou. The effecs of shear deformaion and roaor ineria are aken ino consideraion. The resuling coupled pariall differenial equaions are solved using finie difference mehod. I was found ha he foundaion moduli and angle of inclinaion of he bridge have significan effec on he deflecion of he bridge. 05 AENSI Publisher All righs reserved. To Cie This Aricle:.C. Agarana, J.A. Gbadean, O.O. Agboola, T.A. Anake, O.J. Adeleke., Dnamic Response of an Inclined Railwa Bridge Suppored b Winkler Foundaion Under a oving Railwa Vehicle. Aus. J. Basic & Appl. Sci., 9(): 55-6, 05 INTRODUCTION An inclined railwa bridge is a railwa bridge se a an angle, no perpendicular o a horizonal plane. However, he work done is he same: Work = Force Disance, and he disance is increased, whereas he force is decreased [olinear e al (0), Gbadean and Agarana (04)]. In Elemenar Phsics, an objec placed on a iled surface (inclined plane) will ofen slide down he surface. The greaer he il of he surface (i.e. he angle of inclinaion), he faser he rae a which he objec will slide down i (Sofi, 0). According o Newon s laws of moion, a Railwa vehicle on an inclined plane will coninue o slide down he plane if here is no applied force o balance he forces acing on i, especiall if he surface is fricionless or wih minimal fricion. There are alwas, a leas, wo forces namel: he force of gravi and he normal force, acing upon he railwa vehicle posiioned on an inclined bridge (Gerg and Dukkipai, 984). The force of gravi acs in a downward direcion, while he normal force acs in a direcion perpendicular o he surface [olinear e al (0), Gbadean and Dada (006)]. An inclined plane problem is in ever wa like an oher ne force problem wih he sole ecepion ha he surface has been iled. An inclined plane herefore can be ransformed ino he form wih which we are more comforable, as illusraed in Figure. Afer his ransformaion, we can ignore he force of gravi since i has been replaced b is wo componens [olinear (0), Sofi (0)]. We can now solve for he ne force and he acceleraion. For a railwa vehicle mowing up he inclined bridge, he applied force mus be greaer han he componen of is weigh F ) moving down he inclined bridge, o avoid Corresponding Auhor:.C. Agarana, Deparmen of ahemaics, College of Science & Technolog, Covenan Universi, P..B. 0, Oa, Nigeria; Tel: ; michael.agarana@covenanuniversi.edu.ng ( sliding down. Problem Formulaion: A recangular inclined railwa bridge, modelled as recangular inclined indlin plae, suppored b Winkler foundaion and raversed b a pariall disribued moving railwa vehicle is considered. is he mass of he railwa vehicle of recangular dimension and, and, wih one of is lines of smmer moving along he plae is b in dimension and le u, where u is he veloci of he load. is he angle of inclinaion, F is he componen of he weigh of he railwa vehicle parallel o he inclined plane and F is he componen perpendicular o he inclined plane. Assumpions: (i) The inclined bridge is of consan cross secion, (ii) he moving railwa vehcile moves wih a consan speed, (iii) The moving railwa vehicle is guided in such a wa ha i keeps conac wih he inclined bridge hroughou he moion, (iv) The inclined bridge is coninuousl suppored b a Winkler foundaion, (v). The moving railwa vehicle is uniforml pariall disribued, (vi) The recangular

2 56.C. Agarana e al, 05 Ausralian Journal of Basic and Applied Sciences, 9() a 05, Pages: 55-6 indlin railwa bridge is elasic, (vii) No damping in he ssem, (viii) Uniform graviaional field; (i) Consan mass ( ) of he railwa vehicle moving up he inclined plane. () Consan angle of inclinaion Iniial Condiions: W W(,, 0) 0 (,, 0) Boundar Condiions: W (,, ) (,, ) (,, ) 0, for 0 and W (,, ) (,, ) (,, ) 0, for 0 and Fig. : A moving railwa vehicle on an inclined plane suppored b Winkler foundaion. Fig. : Transformed inclined plane o a fla plane. Problem Soluion: The se of dnamic equilibrium equaions which govern he behaviour of recangular inclined railwa bridge suppored b Winkler foundaion and raversed b a pariall disribued moving Railwa vehicle can be wrien as [Gbadean and Dada (006), Shadnam, (00)]; Q Q W W h KW (,, ) f P f h d Q h B d h d Q h B d where and are local roaions in he and direcions respecivel, and are bending momens in he and direcions respecivel, is he wising momen, Q and Q are he raversed shearing forces in and direcions respecivel, h and h are hickness of he plae and load respecivel, and are he H( ), for 0 H( ) H( ), for B ( ) H( ), for ( ) 0, for B H ( ) H ( ) () (4) (5) densiies of he plae and he load per uni volume respecivel. W(,, ) is he raverse displacemen of he plae a ime, g is acceleraion due o gravi, is he angle of inclinaion of he plae. The las erms in equaions (4) and (5) accoun for ineria effecs of he load in and direcions respecivel. Also, B BB, where (6) (7) H ( ) is he Heaviside funcion defined as:

3 57.C. Agarana e al, 05 Ausralian Journal of Basic and Applied Sciences, 9() a 05, Pages: 55-6, 0 H ( ) 0.5, 0 0, 0 K is he foundaion of siffness and f is he mass of he foundaion. D is he fleural rigidi of he plane. The bending momens, shearing forces and D D D( ) W Q K Gh Q K Gh (8) wising momen can be wrien as (Gbadean and Dada, 006) (9) (0) () () W and W D From equaion (), he moving load P(,, ) can be epressed as follows [Gbadean and Dada (006), Gbadean and Agarana (04)]: From equaion (4), he sraigh derivaive d d can be epressed as follows [Gbadean and Dada (006), Gbadean and Agarana (04)]: B virue of he inclined plane, he weigh of he railwa vehicle ( ) has been resolved ino is componens. The componen parallel o he plane is gsin. Therefore, equaion () becomes: Applicaion of he boundar condiions o he nondimensional form of equaions (9) (4) and () (4) ields nine equaions wih nine unknown variables: Q, Q,,,,,,,, D and W, from where he soluions are obained. () (4) A simpl suppored recangular inclined plane (plae) has been aken as an illusraive eample. If he edge 0 of he railwa bridge (modelled as a plae) is simpl suppored, he deflecion W along his edge mus be zero. A he same ime his edge roae freel wih respec o he ais, ha is, here are no bending momens ( ) along his edge [Gbadean and Dada (006), Gbadean and Agarana (04)]. A numerical procedure, he finie difference mehod, can be used o solve he ssem of equaions (7) () and (0) () (Gbadean and Agarana, 04) The resuling se of algebraic equaions o be solved for he nine dependen variables ma be wrien in mari form as: W P(,, ) g B u uv u u u u d D( v ) D( v ) d From equaion (5), he sraigh derivaive d u uv u u u u d D( v ) D( v ) W P(,, ) g cos B sin g d can be epressed as follows (Gbadean and Agarana, 04): d Subsiuing equaions (6), (7) and (8) ino equaions (), (4) and (5) respecivel, we have (9) Q Q W W W u Q h KW f g cos g sin B KGh T h u u Q h u u u B D( ) D( ) h u u Q h u u u B D( ) D( ) Equaions (9), (0) and () can be wrien as firs order parial differenial equaions as follows (5) (6) (7) (8) (0) ()

4 58.C. Agarana e al, 05 Ausralian Journal of Basic and Applied Sciences, 9() a 05, Pages: 55-6 dq dq D D D u u u Q Q DT KW f g cos u u, sin u B h g d d A D( ) D( ) Gh T h, h, u u Q u u B D( ) D( ) h, h, u u Q u u B D( ) D( ) where, and., Ai, j S' i, j Bi, js' i, j Ci, S' i, j Di, js' i, j k, i,,,, N ; j,,,, (5) where N and are he number of he nodal poins along and aes respecivel, K S S S N S P (6) k i, j i, j i, j, i, j i i, j i, j i, j Each erm in equaions (5) and (6) is a 9 9 mari. () () (4) Effecs Of The Angle Of Inclinaion On The Deflecion Of The Bridge: From equaion (8), we have W P(,, ) g cos B sin g For free vibraion, P(,, ) 0, which implies W g cos B sin 0 g W g cos B sin g W g cos B A sin 0 g W A g sin g cos B For ver small, sin and cos. Considering small inclinaion, (i.e., 0 ), equaion (0) becomes: W g B g A where A. For numerical illusraion purpose, le 0 kg, A g. If B, hen m, 9.8 W 9.8 () Tha is, if ends o 0, he acceleraion of he deflecion is approimael he acceleraion due o gravi in he opposie direcion. Inegraing boh sides of equaion () wice, we have 9.8 c k W () where c and k are consans. Considering when is no small, W g cos B sin g A which can be wrien as A W g sin g cos, B B Specificall, for 90, equaion (5) becomes A W g B So for 90 and for free vibraion and leing B, 90, A, we obain (7) (8) (9) (0) () (4) (5) (6)

5 59.C. Agarana e al, 05 Ausralian Journal of Basic and Applied Sciences, 9() a 05, Pages: 55-6 (7) g c k W where c and k are consans. Now, for 90, equaion (5) can be wrien as W g( sin cos ) For forced vibraion, we have W P g cos B sin g (9) A where P is he applied force. A W ( P g sin ) g cos (40) B For numerical illusraion purpose, le 0 kg, A m, g 9.8ms, B, 0 90, equaion (40) becomes ( 98.sin ) 9.8cos 5 P (4) W where is he acceleraion of he deflecion. From equaion (7), k an, where k B, is a consan. (8) RESUTS AND DISCUSSION The numerical calculaions were carried ou for a simpl suppored recangular inclined plae (inclined railwa bridge) resing on a Winkler foundaion and subjeced o a moving railwa vehicle (load.). Damping effec was negleced. For specific values of oher parameers, deflecion of he bridge is calculaed and ploed (in Figure ) as a funcion of ime. I shows he deflecion of he railwa bridge for various values of veloci u. I can be seen ha he response maimum ampliude of he bridge decreases as veloci decreases. In Figure 4, acceleraion of he deflecion of he bridge,, wihou an applied force, is ploed agains ime. We can see ha increases genl, hen laer sharpl wih ime, a a given value of angle of inclinaion,, as k increases. Figure 5 shows ha deflecion of he railwa bridge decreases wih ime if here is no applied load. In Figure 6 we ploed he deflecion of he bridge under an applied load agains ime. I is clear ha deflecion increases as he applied load increases. Similarl, Figure 7 shows ha he acceleraion of he deflecion of he bridge a differen values of applied load decreases as wih an increase in he applied load. Also, Figures 8 and 9 represen insananeous dnamic response of he railwa bridge, a an insan of ime and a a given angle of inclinaion, for boh forced and free vibraion cases. The figures show, respecivel, ha deflecion of he bridge decreases wih an increase in ime for forced vibraion case, while he deflecion increases wih an increase in ime for he free vibraion case, a a given angle of inclinaion. Fig. : Deflecion of he bridge a differen values of veloci and ime.

6 60.C. Agarana e al, 05 Ausralian Journal of Basic and Applied Sciences, 9() a 05, Pages: 55-6 Fig. 4: Acceleraion of deflecion of inclined plae a various values of k. Fig. 5: Deflecion of he bridge wihou applied load a given inclinaion angles. Fig. 6: Deflecion of bridge a various values of applied load and differen ime. Fig. 7: Acceleraion of deflecion of he bridge a differen values of applied load and differen angles of inclinaion.

7 6.C. Agarana e al, 05 Ausralian Journal of Basic and Applied Sciences, 9() a 05, Pages: 55-6 Fig. 8: Deflecion of he railwa bridge for forced vibraion, a an insan of ime and a a given angle of inclinaion. Fig. 9: Deflecion of he railwa bridge for free vibraion, a an insan of ime and a a given angle of inclinaion. Conclusions: The srucure of ineres was an inclined railwa bridge on Winkler elasic foundaion, under he influence of a uniform pariall disribued moving railwa vehicle). The problem was o deermine he dnamic response of he whole ssem. Finie difference echnique was adoped in solving he resuling firs order coupled parial differenial equaions obained from governing equaions for he simpl suppored bridge. The sud has conribued o scienific knowledge b showing ha he angle of inclinaion of an incline railwa bridge in addiion o he elasic subgrade on which he bridge ress, have a significance effec on he dnamic response of he bridge o a pariall disribued moving railwa vehicle. Also, he influences of he moving railwa vehicle speed and oal mass of he moving railwa vehicle on he dnamic response of he inclined bridge are significan in mos cases. ACKNOWEDGEENTS The auhors would like o hank Covenan Universi for her faciliies and for financial suppor hrough Cenre for Research, Innovaion and Discover (CUCRID). REFERENCES Civalek, O., 005. arge deflecion saic and dnamic analsis of hin circular plaes resing on wo parameer elasic foundaion HDQ/FO Couple mehodolog approaches. Inernaional Journal of Compuaional echanics, (): 7-9. Civalek, O. and A. Yavas, 006. arge deflecion saic analsis of recangular plaes on wo parameer elasic foundaion. Inernaional Journal of Science and Technolog, : Gerg, V. K and R.V. Dukkipai, 984. Dnamics of Railwa Vehicle Ssems. Academic Press (New York). Gbadean, J.A. and.c. Agarana, 04. Dnamic analsis of railwa bridges suppored b Winkler foundaion under uniform pariall disribued moving railwa vehicle. WIT Transacions on he Buil Environmen, 5: Gbadean, J.A. and.s. Dada, 006. Dnamic response of a indlin elasic recangular plae under a disribued moving mass. Inernaional Journal of echanical Science, 48: 40. indlin, R.D., 957. Influence of roaor o ineria and shear on fleural moions of isoropic elasic plaes. Journal of Applied echanics, 8. olinear, E., P. useros, and.d. arinez- Rodrigo, 0. Rerofi of eising railwa bridges of shu o medium spans for high-speed raffic using viscoelasic dampers. Engineering Srucures, 40: Nikkhoo, A. and F.R. Rofooei, 0. Parameric sud of he dnamic response of he dnamic response of heir recangular plaes raversed b a moving mass. Aca echanica, (): 5-7. Shadnam,.R., ofid,. and J.E. Akin, 00. On he dnamic response of recangular plae, wih moving mass. Thin-Walled Srucures, 9(9): Sofi, A., 0. Non-linear in-plane vibraions of inclined cables carring moving oscillaors. Journal of Sound and Vibraion, (7): 7-74.