Dynamic Response of Inclined Isotropic Elastic Damped Rectangular Mindlin Plate resting on Pasternak Foundation under a Moving Load

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1 Proceedings of he Inernaional MliConference of Engineers and Comper Scieniss 016 Vol II, IMECS 016, March 16-18, 016, Hong Kong Dnamic Response of Inclined Isoropic Elasic Damped Recanglar Mindlin Plae resing on Pasernak Fondaion nder a Moving oad Agarana M.C., Gbadean J.A., and Ajai O.O. Absrac In his aricle, he dnamic behavior of inclined damped recanglar Mindlin plae nder he inflence of moving load along he mid-plae on he plae srface is considered. A nmerical mehod is sed o solve he nondimensional form of he resling copled parial differenial eqaions. The desired solions are obained wih he aid of comper program developed in conjncion wih MATAB. I is observed ha he response amplide of he plae is affeced significanl b he fondaion modli. Also, he effecs of he shear deformaion, roaor ineria, damping and angle of inclinaion of he plae, o he horizonal, are noiceable. Inde Terms Pasernak fondaion, Damped Inclined Mindlin plae, Moving oad, Dnamic response. A I. INTRODUCTION N inclined recanglar Mindlin plae is a plae se a an angle, no perpendiclar o a horizonal plane. However, he work done is he same: Work = Force Disance, and he disance is increased, whereas he force is decreased [3,9]. In Elemenar Phsics, an objec placed on a iled srface (inclined plane) will ofen slide down he srface. The greaer he il of he srface (i.e. he angle of inclinaion), he faser he rae a which he objec will slide down i [10,11]. According o Newon s laws of moion, a moving load on an inclined plane will conine o slide down he plane if here is no applied force o balance he forces acing on i, especiall if he srface is fricionless or wih minimal fricion. There are alwas, a leas, wo forces namel: he force of gravi and he normal force, acing pon he moving load posiioned on an inclined plae [1,10]. Manscrip received November 30, 015; revised December 17, 015. This work was sppored b Covenan Universi. M. C. Agarana is wih he Deparmen of Mahemaics, Covenan Universi,Oa,OgnSae,Nigeria. michael.agarana@covenanniver si.ed.ng. J. A. Gbadean is wih he Mahemaics Deparmen. Universi of Ilorin, Kwara Sae, Nigeria. O. O. Ajai is wih he Deparmen of Mechanical Engineering, Covenen Universi Oa, Ogn Sae, Nigeria. olsei.ajai@covenanniversi.ed.ng. The force of gravi acs in a downward direcion, while he normal force acs in a direcion perpendiclar o he srface [3,8]. An inclined plane problem is in ever wa like an oher ne force problem wih he sole ecepion ha he srface has been iled. An inclined plane herefore can be ransformed ino he form wih which we are more comforable, as illsraed in figre. Afer his ransformaion, we can ignore he force of gravi since i has been replaced b is wo componens [11]. We can now solve for he ne force and he acceleraion. For a load moving p he inclined plae, he applied force ms be greaer han he componen of is weigh moving down he inclined plae, o avoid sliding down [10,11]. Gbadean and Dada eended heir works recenl b considering he dnamic response of a Mindlin elasic recanglar plae sbjeced o disribed moving load, b negleced he effec of damping [6,7]. Also mos ahor did no consider he possibili of he plae being inclined or resing on an elasic fondaion.[1,,3,6] The presen paper consider he dnamic response of damped Mindlin elasic pe of plaes resing on a Pasernak fondaion nder he inflence of a pariall niform disribed moving load [4,5,8,11,1,13]. Finie difference echniqe is sed o solve he ransformed non-dimensional form of he copled differenial eqaions governing he moion of sch plaes [13]. II. GOVERNING EQUATIONS The se of dnamic eqilibrim eqaions which governs he behavior of damped inclined Mindlin plae sppored b Pasernak fondaion, and raversed b a pariall disribed moving load can be wrien as follows [6,1,13]: 3 M M h Q 1 T h M M T T D( 1) T B M M D ( 1) T (1) ISSN: (Prin); ISSN: (Online) IMECS 016

2 Proceedings of he Inernaional MliConference of Engineers and Comper Scieniss 016 Vol II, IMECS 016, March 16-18, 016, Hong Kong N F mg sin mg mg cos Fig. 1. Diagram of moving load on an inclined plane N F11 mg sin F F1 mg cos Fig.Diagram of a ransformed inclined plane o a fla plane ISSN: (Prin); ISSN: (Online) IMECS 016

3 Proceedings of he Inernaional MliConference of Engineers and Comper Scieniss 016 Vol II, IMECS 016, March 16-18, 016, Hong Kong 3 M M h Q 1 T M M 3 h1 T T D( 1) T B 1 M M D ( 1) T () Q Q kw ( M f h) M [ g cos G 1 A M M D( 1) D( 1) Q Q W B] h M sin g Gh M M M D D D(1 ) W Q Gh Q W Gh W W W (3) (4) (5) (6) (7) (8) (9) (10) (11) where Eqaions. (4 8) are he eqaions for bending momens, wising momens and shear force, and are local roaions in he and direcions respecivel. h and h 1 are he hickness of he plae and load respecivel, is he viscos damping coefficien, is he angle of inclinaion of he plae wih he horizonal, and are he densiies of he plae and he load per ni volme respecivel. W (,, ) is he raverse displacemen of he plae a ime, g is he acceleraion de o gravi, is he angle of inclinaion of he plae, is he veloci of he load ( M ) of recanglar dimension b wih one of is lines of smmer moving along Y Y1, he plae is I b I in dimensions and, B BB where 1 H, 0 H H, B H, 0, (1) 1 1 B H H 1, 0 H ( ) 0.5, 0 0, 0 (13) (14) H( ) is called Heaviside fncion. G is he modls of rigidi of he plae, D is he fleral rigidi of he plae defined b Gh D Eh (1 ) for isoropic plae, is he 1 6(1 ) shear correcion facor and is he Poisson s raion of he plae. Since he ineria effec of he load is considered, he niform pariall disribed applied load akes on he form [6]: M dw P(,, ) g sin B M sin g A d Acceleraion dw d is defined as d W W W W d Similarl, and d d d d (15) (16) (17) (18) ISSN: (Prin); ISSN: (Online) IMECS 016

4 Proceedings of he Inernaional MliConference of Engineers and Comper Scieniss 016 Vol II, IMECS 016, March 16-18, 016, Hong Kong A. Iniial Condiions W W (,,0) (,,0) (19) T W since, Therefore, Eq. (4) becomes W and h is a mass. B. Bondar Condiions W (,, ) M (,, ) (,, ) 0, for 0 and a W (,, ) M (,, ) (,, ) 0, for 0 and b (0) III.. PROBEM SOUTION The se of parial differenial Eqaions. (1) - (11), are he parial differenial eqaions o be solved for he following eleven dependen variables M, M, M, Q, Q,,, W,, and. A nmerical procedre, finie difference mehod, can be sed o solve he ssem of Eqaions. (1) - (11). Rearranging hem in mari form resls in R S' P T S' Y S Z (1) i, j1 i, j1 i1, j1 i, j i, j i1, j i1, j k i 1,, 3 N 1; j 1,, 3 M 1 Where N and M are he nmber of he nodal poins along and aes respecivel. Z k is a mari represening he righ hand side of he ransformed Eqaions. (1) (11) defined b Q Q kw M g sin When 0, 1 Q Q W k When 30, Mg 1 Q Q W k k When 60, 3M g 1 Q Q W k k When 90, (5) (6) (7) (8) Z A, S P S G, S, D, S E k i j i, j i, j1 i, j1 i1 j i1 j i1 j1 i1, j1 1 Effec Of Angle Of Inclinaion On Deflecion Of The Inclined Plae () For he prpose of his paper le B 0, which implies B 0 and. Also, M f h M (mass); and 0. For 1, eqaion (3) becomes: Q Q W kw M h M sin g (3) W Q Q h M kw M sin g (4) W W since, and h is a mass. Therefore, Eq. (4) becomes W M g Q Q k k 1 From Eqaion. (15), if B 0, he applied load becomes (9) P(,, ) M g sin (30) When 0, P 0 (31) When 30, 1 P M g (3) When 60, 3 P M g (33) When 90, P M g (34) Now for 1, eqaion 4 becomes ISSN: (Prin); ISSN: (Online) IMECS 016

5 Proceedings of he Inernaional MliConference of Engineers and Comper Scieniss 016 Vol II, IMECS 016, March 16-18, 016, Hong Kong W Q Q M (1 ) kw M g sin (35) W Q Q M (1 ) kw M sin (36) g 1 W Q Q [ (1 ) W M M g sin ] (37) k From eqaion (37); as he angle of inclinaion of he plae,,increases and 1, he magnide of deflecion, W, increases. Each erm in Eqaions. (1) and () is an mari. IV. RESUT DISCUSSION The nmerical calclaions were carried o for a simpl sppored recanglar inclined plae resing on a Pasernak fondaion and sbjec o a moving load. Damping effec was considered. The vales of he damping raios are aken o be 0, 1, 100 and 150 respecivel. In Fig.3 he deflecion of Mindlin, non-mindlin,, a differen vales of ime and fondaion modls, were shown. I is obvios ha he maimm amplide of Mindlin plae is higher han ha of non-mindlin plae. We noice, also, from figre 4 ha he higher he damping raio, he lower he deflecion amplide, a a pariclar ime. We can dedce from eqaion (31) - (34) ha for 1, we need o appl more force o be able o pll he load phill as increases. Also, for 1, we can dedce, from eqaions (7) (9), ha he magnide of he deflecion (W) increases as he angle of inclinaion increases. This shows ha damping affecs boh he applied force and he deflecion of he inclined plae. From eqaion (37), i can be seen ha, for a pariclar vale of k, he magnide of W increases as he angle of inclinaion increases. In Fig. 6, he deflecion of he plae for differen vales of he fondaion modls (G) is presened. I is observed ha he fondaion siffness have effec on he deflecion of he plae. The highes vale of he fondaion modls, prodces he maimm response amplide. Fig. 5 shows he effec of he veloci on he deflecion of he inclined plae. From he figre i can be seen ha he higher he veloci he higher he response amplide Figre 3: Deflecion of Mindlin and non-mindlin plaes for differen vales of ime Fig.4. Effec of damping on deflecion of he plae ISSN: (Prin); ISSN: (Online) IMECS 016

6 Proceedings of he Inernaional MliConference of Engineers and Comper Scieniss 016 Vol II, IMECS 016, March 16-18, 016, Hong Kong sbgrade, on which he damped inclined Mindlin plae ress has a significan effec on he dnamic response of he plae o a pariall disribed load. The effecs of he angle of inclinaion and he damping coefficien were ver eviden. For Mindlin plae, boh he effec of roaor ineria and shear deformaion, on he dnamic response of he damped inclined Mindlin plae, o he moving load are considered. This gives a more realisic resl for pracical applicaion, especiall when sch plae is considered o res on a fondaion. Fig.5. Deflecion of plae a differen velociies () and ime(t) Fig. 6. Deflecion of plae a varios fondaion modls and differen imes V. CONCUSION The dnamic response of a damped inclined Mindlin plae, carring a niform pariall disribed moving load, sppored b a Pasernak fondaion, has been analsed. The non-dimensional eqaions of moion were ransformed ino eqivalen finie difference ones, and hen solved. Resls have been presened no onl for he deflecion b also for he effec of veloci on he deflecion of he inclined plae. Also he effecs of boh he damping and angle of inclinaion of he plae was eamined. Hence mos of he componens composing he dnamic response of he ssem have been obained. A nmerical eample of simpl sppored recanglar plae is presened. I is shown ha he elasic REFERENCES [1] M. Ab-Hilal, Dnamic response of a doble Eler-Bernolli beam de o a moving consan load. Jornal of Sond and Vibraion, Vol. 97, pp [] J.V. Amiri, A. Nikkho, M.R. Dnvoodi, and M.E. Hassanabadi, Vibraion analsis of a Mindlin elasic plae nder a moving mass eciaion b eigenfncion epansion mehod.. Thin-Walled Srcre, Vol. 6, pp 53-64, 013 [3] O. Civalek, arge deflecion saic and dnamic analsis of hin circlar plaes resing on wo parameer elasic fondaion HDQ/FO cople mehodolog approaches. Inernaional Jornal of Compaional Mechanics, Vol., no., pp 71-91, 005. [4] J.A. Gbadean and M.C. Agarana, 014. Dnamic analsis of railwa bridges sppored b Winkler fondaion nder niform pariall disribed moving railwa vehicle. WIT Transacions on The Bil Environmen, Vol.135. WIT Press, 014. [5] J.A. Gbadean, and M.S. Dada, Dnamic response of plaes on Paernak fondaion o disribed moving load.. Jornal of he Nigerian Associaion of Mahemaical Phsics, Vol. 5, pp , 001 [6] J.A. Gbadean and M.S. Dada, Dnamic response of a Mindlin elasic recanglar plae nder a disribed moving mass. Inernaional Jornal of Mechanical Science, Vol. 48 pp , 006 [7] R.D. Mindlin, Inflence of roaor o ineria and shear on fleral moions of isoropic elasic plaes. Jornal of Applied Mechanics, Vol. 18, [8] T. Ngan-Thoi, H. ong-van, P. Phng-Van, T. Rabczk, and D. Tran- Trng, Dnamic responses of composie plaes on he paernak fondaion sbjeced o a moving mass b a cell-based smoohed discree shear gap (CS-FEM-DSG3) mehod.. Inernaional Jornal of Ccomposie Maerials, Vol. 3 Isse 6A, pp [9] T. Zhang, T. and G. Zheng, Vibraion analsis of an elasic beam sbjec o a moving beam wih fleible connecions.. American Associaion of Civil Engineers, Vol. 130, no.1, pp , 010. [10] Khan Academ, 014, forces on inclined plane. [11] M. C. Agarana, J. A. Gbadean, Olasmbo O. A. T. A. Anake and O. J. Adeleke, Dnamic Response of an inclined Railwa bridge sppored b Winkler Fondaion nder a Moving Railwa Vehicle. Asralian jornal of basic and applied sciences, Vol. 9, no.11, pp , 015 [1] E.B. Are, A.S. dow and J.A. Gbadean, Vibraion of damped simpl sppored orhoropic recanglar plae resing on elasic winkler fondaion, sbjeced o moving loads Advances in Applied science research, Vol. 4 No.5, pp , 013. [13] M.C. Agarana and J.A. Gbadean, Finie difference dnamic analsis of railwa bridges sppored b Pasernak fondaion nder niform pariall disribed moving Railwa vehicle, Inernaional conference on ssems Engineering and Engineering Managemen 015, IAENG. 1-3 Ocober, 015. ISSN: (Prin); ISSN: (Online) IMECS 016