Dynamic Analysis of Axially Beam on Visco - Elastic Foundation with Elastic Supports under Moving Load

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1 Dyamc Aalyss of Axally Beam o Vsco - Elastc Foudato wth Elastc Supports uder Movg oad Saeed Mohammadzadeh, Seyed Al Mosayeb * Abstract: For dyamc aalyses of ralway track structures, the algorthm of soluto s very mportat. For estmatg the mportat problems the ralway tracks such as the effects of ral jots, ral supports, ral modelg the earess of brdge ad other problems, the models of the axally beam model o the elastc foudato ca be utlzed. For studyg the effects of axally beam o the elastc foudato, partal dfferetal equatos whch represet the depedet ad process of the axally beam o the elastc foudato by cosderg the elastc supports uder movg load have bee studed ad equatos have bee aalyzed as closed form. The beam model cludes vsco elastc foudato ad elastc supports codtos. For cosderg the beam elemet, axal force has bee cosdered besde of shear ad elastc supports are derved ad the these equatos are solved parametrcally by usg separato of varables ad orthogoalty propertes of modes. Ths process ad soluto have bee preseted as closed form ths paper. Ths problem was t vestgated the techcal lterature. Ths model ca be utlzed for the most problems the ralway tracks. The advatage of ths paper s presetato of algorthm ad process of parametrc soluto for a axally beam o the vsco - elastc foudato wth elastc supports. Keywords: Ralway track, dyamc aalyss, axally beam o elastc foudato, elastc supports Correspodg author Emal:mosayeb@ust.ac.r - Assocate Professor, School of Ralway Egeerg, Ira Uversty of Scece ad Techology, Tehra, Ira - Ph.D. Studet, School of Ralway Egeerg, Ira Uversty of Scece ad Techology, Tehra, Ira 89 Iteratoal Joural of Trasportato Egeerg, Vol./ No./ Sprg 5

2 Dyamc Aalyss of Axally Beam o Vsco - Elastc Foudato wth Elastc Supports.... Itroducto Dyamc aalyss of structure has a mportat role desg of structures. The dyamc effects related to teral forces structures whle kematc effects related to dsplacemets ad deformatos caused by vbratos. Dyamc loads created by dfferet sources. Dyamc loads ca be caused by ubalaced masses vehcles, wd or earthquake, waves due to explosos, move of ral vehcles ad sea waves. I geeral, there are two types of dyamc loads that are cyclc ad ocyclc loads. The smplest type of perodc loads s harmoc loads. No-cyclc loads created by dfferet sources such as blast, wd or earthquake. May parameters ca affect o the structures aalyss. These parameters clude the mass of the structure, degrees of freedom, stffess ad dampg of structure. tos for estmatg the vbratos of a structure. These equatos ca be solved by umercal or aalytcal methods. Parameters such as dsplacemet ad stresses ca be calculated by solvg these equatos. There are three commo methods the formato of the moto equatos whch are: drect approach equlbrum, Hamlto ad agrage methods. Geerally, structures terms of degrees of freedom freedom, several degrees of freedom ad cotuous systems. the partal dfferetal equatos are used. Ths study s aalyzg the structures s the beam o the elastc fou- beam elemet o a elastc foudato by takg two spose of beam o vscoelastc foudato uder a seres of movg load. [Akour, ] studed the olear effect of beam o the elastc foudato. [Abu-Hlal, 6] vestgated the dyamc respose of Euler Beroull beam due to a costat movg load. [Yag ad Chag, 9] obtaed frequeces due to the movg load o the brdge [Clough ad Peze, 3]. Also, Chopra [995] ad Paz ad egh [] studed the dyamcs of structures ad earthquake egeerg problems. Yag [986] vestgated the radom vbrato problems. [Fryba, 999] studed the problems subjected to movg loads. alae [] studed the problem related to radom ad mechacal vbratos. Also, [Soles,997] vestgated the process of stochastc ad radom vbratos. I the most research works, the support codtos have bee cosdered as deal. Therefore the preset study, the beam o the elastc foudato wth elastc supports has bee studed by utlzg the dyamcs aalyss ad the applcato of the orthogoalty propertes of modes. For cosderg the beam elemet, axal force has bee cosdered besde of shear ad momet forces. I ths elastc foudato wth elastc supports are obtaed ad the these equatos are solved by usg separato of varables ad orthogoalty propertes of modes.. Algorthm ad Problem-Solvg Process I order to study ad aalyze the problems, the algorthm must be preseted. Ths algorthm s preseted Fgure. Ths algorthm cludes the problem modelg, free dagram of beam elemet, the dfferetal equato of beam elemet, solvg the dfferetal equato by usg separato of varables ad solvg the dfferetal equato by usg orthogoalty propertes of modes. Also, the soluto algorthm of dfferetal equato by the method of separato of varables cludes vertcal dsplacemet of beam term of two fuctos wth varables of x ad t, the equato of free vbratos wthout effect of load, the equatos term of fuctos wth two depedet varables, solvg two depedet equatos, applyg boudary codtos at begg ad ed of the beam, the determat of a matrx. Fally, the soluto steps of the dfferetal equato by usg the orthogoalty propertes of modes clude calculato of beam dsplacemet as two fuctos wth varables of x ad t, use of the orthogoalty propertes of modes, calculato of dampg as a combato of mass ad stffess ad access to the equato wth oe degree of freedom. Iteratoal Joural of Trasportato Egeerg, Vol./ No./ Sprg 5 9

3 Saeed Mohammadzadeh, Seyed Al Mosayeb the begg of beam ad ampltude of appled load s the Drac delta fucto. The dfferetal equato of beam elemet s obtaed wth cosderg the beam elemet ad free dagram of forces actg o the elemet (Fgure 3). Fgure 3. Free dagram of beam elemet Fgure. Algorthm ad process of soluto Thus, the problem ca be studed ad vestgated by usg the above algorthm. I cotue, the model of beam o the elastc foudato wth elastc supports s preseted. 3. Beam o the Elastc Foudato wth Elastc Supports For vestgatg the model of beam o the elastc foudato wth elastc supports, a model s cosdered as a Fgure. Ths model cludes stffess, dampg, mass per ut legth, elastc supports ad so o. Fgure. Beam o the elastc foudato wth elastc supports The cosdered beam s subjected to movg load P (x) as a followg equato: () I ths Fgure, V (x, t), M (x, t) ad N (x) are shear, momets ad axal forces of elemet respectvely. Also, F I, F S ad F D represet the erta force, the stffess force ad dampg force respectvely. By cosderg the free dagram of beam elemet, the equato of beam elemet restg o elastc foudato s preseted as follows: () After achevg the partal dfferetal equato of beam elemet, ths equato s solved based o the followg procedures.. Solvg the Problem by usg Separato of Varables By usg the separato of varables, the vertcal dsplacemet of beam, V (X, T), terms of the varables x ad t are cosdered as follows: (3) Free vbrato equato of the beam wthout the load effect s obtaed as follows: I ths equato, v ad P are dstace of load from 9 Iteratoal Joural of Trasportato Egeerg, Vol./ No./ Sprg 5

4 Dyamc Aalyss of Axally Beam o Vsco - Elastc Foudato wth Elastc Supports... By dvdg the free equato by equato s obtaed as follows:, the beam (5) By dvdg the above equato by EI, the followg equato s obtaed. // ( x) N ( x) K m Y( t) C Y ( t) ( x ) ( x ) EI ( x ) EI EI Y () t EI Y () t (6) The above equatos are separated as fuctos of two depedet varables. // ( x) N ( x) K m Y( t) C Y ( t) ( x) a ( x ) EI ( x ) EI EI Y () t EI Y () t (7) Thus, two equatos terms of two varables are obtaed. If the above equato s a costat, the above equato wll have a soluto. K a b EI (8) N g EI Therefore, two depedet equatos are obtaed terms of x ad t whch are: // ( x) g ( x) b ( x) my t CY t EI a Y t () () () (9) () g b A A g A b g g b A, g g,,3, b // ( x) g ( x) b ( x) / () g g b g g b Therefore, the equato of ( x ) s obtaed. ( x) A cos( x) A s( x) () (3) Boudary codtos for shear ad momet at the be- /// EI () K () EI () K () // / /// EI K ( ) ( ) EI ( ) K ( ) // / 5 / / A sh( x) A cosh( x) 3 () By applyg the boudary codtos, matrx equato s obtaed as follows: a square matrx for ths equato must be equal to zero that the frequecy equato s reached. The determat b s preseted the appedx as parametrcally (Equa- are preseted the appedx as parametrcally (Table ). The beam equato ca be calculated by usg the orthogoalty propertes of modes. The soluto process s preseted cotue. 5. Sol v g the Equat os by us g the Orthogoalty Propertes of Modes For solvg the equat o, orthogoalty propertes of K 3 3 A K EI EI EI A EI K K EI sh( ) K cosh( ) A 3 EI s( ) K cos( ) EI cos( ) K s( ) EI cosh( ) K sh( ) EI cos( ) K 5 s( ) EI s( ) K 5 cos( ) EI sh( ) K 5cosh( ) EI cosh( ) K 5sh( ) A (5) Iteratoal Joural of Trasportato Egeerg, Vol./ No./ Sprg 5 9

5 Saeed Mohammadzadeh, Seyed Al Mosayeb modes ca be utlzed. For ths purpose, dsplacemet ad t as follows: V ( x, t) ( x). Y ( t) (6) I result, the beam equato s obtaed as follows: d d ( x) d d ( x) EI Y ( t) N ( x) Y ( t) m( x) ( x) Y ( t) Cx ( ) ( xy ) () t K ( xy ) () t Pt () ( xv) (7) Also, orthogoalty propertes of modes are cosdered as follows: ( x) ( x) (8) ( x) m( x) ( x) (9) d d ( x) ( x ) EI ( ) ( ) x m x M () C am ak () Where a ad a By multplyg both sdes of equato (7) by ( x ) ad by usg the orthogoalty propertes of modes, the below equato s obtaed. By usg the orthogoalty property of modes, the below equato s obtaed as follows: (3) Y M Y M Y ( a M a M ) Y K P ( v) I these equatos, dsplacemet s cosdered as follows: V ( x, t) ( x) Y ( t) (5) Also, shear ad momet of beam are calculated as follows. v( x, t) M ( x, t) EI x 3 v( x, t) V ( x, t) EI 3 x (6) 6. Numercal Solutos the Specal Case equato as follows: (7) If t s assumed that oly k s effectve for the boudary codtos ad other parameters are cosdered as ut, the the boudary codtos are cosdered as follows: (8) By solvg the equato (7) based o the boudary codtos (8), the respose s obtaed as follows: Parameters of above equato are cosdered as follows: M m( x) ( v) ( ) ( ) K k x v P ( t) P ( x v) ( v) P ( v) () For ths, the followg parameters are cosdered. (9) d d ( x) d d( x) Y ( t) ( x) EI Y ( t) ( x) N ( x) Y ( t) ( x) m( x) ( x) Y ( t) a M a M ( x) ( x) Y ( t) K ( x) ( x) ( x) P( t) ( x v) () 93 Iteratoal Joural of Trasportato Egeerg, Vol./ No./ Sprg 5

6 Dyamc Aalyss of Axally Beam o Vsco - Elastc Foudato wth Elastc Supports... a C K (3) As observed from the Fgure, the respose of equato approaches to egatve values by creasg k rage of egatve values of x, the respose of equato approaches to postve values by creasg k. But the rage of x =, the effect of k s eglgble. If t s assumed that k s effectve the boudary codtos besde of k, the soluto of the equato s obtaed as follows: the respose of equato ca be plotted as follows: Fgure6. Respose of equato based o «x» wth cosderg k ad k Fgure. Respose of equato based o x (the horzotal axs (x) s dstace of load) the respose of beam s preseted as three-dmesoal plot as follows: Fgure 5. Respose of equato based o x ad k As observed from the Fgure, the respose of equato approaches to postve values by creasg x. Also, the respose of equato approaches to egatve values wth decreasg x. 7. Cocluso For evaluatg the mportat problems the ralway tracks such as the effects of ral jots, ral supports, ral modelg the earess of brdge ad other problems, the models of the axally beam model o the elastc foudato ca be utlzed. The soluto algorthm of the ralway track model s mportat for aalyzg these problems. For aalyzg these problems, the partal dfferetal equatos that represet the depedet varables are utlzed. I the most research works, the beam wth a deal support codtos has bee studed. Therefore ths paper, the effect of beam o the elastc foudato wth elastc supports was studed by usg the dyamcs aalyss of structures ad orthogoalty propertes of modes. The soluto algorthm ad process of beam o the elastc foudato was preseted as tos of beam restg o the elastc foudato wth elastc supports were obtaed ad the these equatos were solved as closed form by usg the separato of varables ad the orthogoalty propertes of modes. The advatage of paper s that the algorthm ad process of dyamc soluto are preseted for a beam o the elastc foudato wth elastc supports. The ma results of the paper are as follows: Iteratoal Joural of Trasportato Egeerg, Vol./ No./ Sprg 5 9

7 Saeed Mohammadzadeh, Seyed Al Mosayeb * Obtag the frequecy equato ad determat of foudato wth elastc supports as parametrcally. o the elastc foudato wth elastc supports accordg to oe factor as parametrcally. * For umercally aalyss for specal case.e. f t s assumed oly k s effectve for boudary codtos, the respose of equato approaches to egatve values wth creasg k creasg x. But the rage of egatve values of x, the respose of equato approaches to postve values wth creasg k. But the rage of x =, the effect of k s eglgble.* Also f t s assumed that k s effectve the boudary codtos besde of k, the respose of equato approaches to postve values wth creasg x. Also, the respose of equato approaches to egatve values wth decreasg x. 8. Refereces - Abu-Hlal, M. (6) Dyamc respose of a double Euler Beroull beam due to a movg costat load, Joural of Soud ad Vbrato, vol. 97, o. 3 5, pp Akour, S. N. () Dyamcs of olear beam o elastc foudato, Proceedgs of the World Cogress o Egeerg, vol. II, odo, U.K. beam o vsco-elastc foudato to movg dstrbuted load, Joural of Theoretcal ad Appled Mechacs, vol. 6, o., pp Chopra, A. K. (995) Dyamcs of structures - theory ad applcatos to earthquake egeerg, Pretce- Hall, Upper Saddle Rver, New Jersey. - Clough, R. W. ad Peze, J. (3) Dyamcs of structures, Computers ad Structures, Berkeley, USA. - Fryba,. (999) Vbrato of solds ad structures uder movg load, Telford, odo, UK. - alae, C. () Mechacal vbrato & shock, radom vbrato, III, Hermes Peto, odo. to of a geeralzed beam elemet o a two-parameter elastc foudato wth sem-rgd coectos ad rgd offsets, Computers ad Structures, vol. 8, o. 5, - Paz, M. ad egh, W. () Structural dyamcs: theory ad computato, Kluwer Academc Publshers, Norwell, MA. - Soles, J. (997) Stochastc processes ad radom vbratos: theory ad practce, Joh Wley & Sos, Chchester, UK. - Yag, C. Y. (986) Radom vbrato of structures, Joh Wley ad Sos, New York. - Yag, Y. B. ad Chag, K. C. (9) Extracto of brdge frequeces from the dyamc respose of a passg vehcle ehaced by the EMD techque, Joural of Soud ad Vbrato, vol. 3, o. -5, pp Appedx - A A A 3 A*(-k*k*k*a-- (a^3*k*b-k*e*i*b^3*k*a (-k*e*i*b^3*k*a - (k*b-k*e*i*b^3*k*a 95 Iteratoal Joural of Trasportato Egeerg, Vol./ No./ Sprg 5