DYNAMIC ANALYSIS OF BEAM-LIKE STRUCTURES SUBJECT TO MOVING LOADS
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1 DYNAMIC ANALYSIS OF BEAM-LIKE STRUCTURES SUBJECT TO MOVING LOADS Ivaa Štimac 1, Ivica Kožar 1 M.Sc,Assistat, Ph.D. Professor 1, Faculty of Civil Egieerig, Uiverity of Rieka, Croatia INTRODUCTION The vehicle-iduced vibratio of bridges has bee subect of iterest for more tha oe ad a half ceturies. Due to tremedous growth of traffic, most of bridges are owadays heavily loaded. The icrease i traffic itesity ad speed requires a more complex aalysis of structures tha i the past. The simplest case of a movig-load (dyamic) aalysis is the case of a simply supported beam over which a cocetrated load is movig. The problem is represeted with a 4 th order partial differetial equatio (PDE), which is owadays usually solved umerically. While we use fiite elemets for discretizatio i space, the discretizatio i time is more coveietly hadled by fiite differeces. Fiite differece method coverts the procedure of solvig a differetial equatio ito a procedure of solvig a system of liear equatios. The results are the displacemet, velocity ad acceleratio for every fiite elemet ode at every time step. DIRECT NUMERICAL INTEGRATION Problem of a massless load movig o a beam is described by the well-kow partial differetial equatio [1] 4 u( x. u( x, u( x, EI + m + c P( x, = (1) 4 x t t u(x, represets the displacemet of the beam, x represets the travellig directio of the movig load, ad t represets time. Also, EI is flexural rigidity of the beam, E is Youg s modulus, I is secod momet of area of the beam, m is mass of the beam per uit legth, c is a viscous dampig coefficiet ad P(x, is the applied exteral force. The easiest way to solve the partial differetial equatio is by umerical itegratio i cotrast to modal aalysis, which applies oly to liear aalysis, the direct umerical itegratio ca be used for both liear ad oliear problems. Discrete form of equatio (1) i matrix otatio is M D & + CD& + SD = A( () M is the mass matrix, C is the dampig matrix, S is the stiffess matrix, D, D & ad D & are the displacemet, velocity ad acceleratio vectors, ad A is the exteral force.
2 Direct liear extrapolatio procedures may be devised for both the average ad liear acceleratio methods. We choose the average acceleratio method which is ucoditioally stable although the liear acceleratio method is somewhat more accurate. Writig equatio () for two cosecutive time steps ad subtractig the results we obtai the followig icremetal form: M D& + C D& + S D = A (3) is -th time step, D & ad displacemet vectors ad, D &, D, are the icremetal acceleratio, velocity A is the icremetal exteral force. I this equatio, we have ukow icremetal acceleratios, velocities ad displacemets. Itroducig the assumptio that the acceleratio is costat withi a time iterval, their value ca be deduced ad substituted ito eq. (3) to obtai: 4 M D Q + C D R + S D = A ) ) (4) Q R 4 = t D& + D (5) = &. (6) D We rewrite this equatio i the form S D = A (7) S 4 = S + M + C ad A = A + MQ + CR. (8) ) t Thus, the pseudostatic equatio (7) is to be solved for the icremetal displacemets at each time step. The icremetal velocities ad acceleratios may the be foud usig equatios (9) ad (1): D& = D R (9) ) D& 4 = D Q (1) )
3 Fially, the total values of D + 1, D & + 1 ad D & are: D + D & = D& + D& +1 (1) D & = D&& + D& +1. (13) + 1 = D D (11) COMPARISON BETWEEN NUMERICAL AND ANALYTICAL SOLUTION Numerical solutio is coded i MathCad 1 usig the direct liear extrapolatio method with assumptio of average acceleratio withi a time step. Numerical aalysis has bee preformed of udamped or damped simply supported beams. The supports have bee modelled as pied ad movable or as udamped or damped sprigs. SIMPLY SUPPORTED BEAM WITHOUT DAMPING P=1 kn v=, m/s EI=1 knm ρ=1 t/m x L=1 m m=5; t=l/m*v; =5; x=l/ Fig. 1 Beam model The umerical solutio for the total beam displacemet i time is show i Fig.. z t g x Fig. Beam displacemet (z-displacemet, x-positio o the beam, t-time) 3D picture i Fig. shows the beam displacemet i every fiite elemet ode ( = -5) i every time step (m = -5). The aalytical solutio [5] is represeted by
4 5 P 1 t α = si ρ L ω( 1) k= 1 k ( k α ) L k x L z ( x, si k π v si( ω( k) k π (14) EI k π ρ ω ( k) = ; L v α = π (15) L ω(1) Compariso of the results betwee the umerical ad the aalytical solutio is show i D plot (Fig. 3). As it ca be see, there is excellet agreemet betwee the two solutios umerical solutio aalytical solutio Fig 3. Mid-poit displacemet i time SIMPLY SUPPORTED BEAM WITH SPRINGS AT SUPPORTS The bridge superstructure is owadays usually placed o the eopree bearigs. The eopree is a flexible structure, therefore it is ot correct to eglect its vibratio, ad impact to the whole structure. The eopree bearigs are modelled as damped liear elastic sprigs. The problem is desiged i Fig. 4. P=1 kn v=, m/s EI=1 knm ρ=1 t/ḿ x C;k L=1 m C;k C - dampig matrix k - sprig stiffess m=5; t=l/m*v; =5; x=l/ C, =, C, =, C i,i = Fig. 4 Beam model
5 If the sprigs are very stiff, there is little betwee the model with sprigs ad the model with fixed supports. (Fig. 5) t without sprigs with sprigs k=5 kn/m k=sprig stiffess Fig 5. 5 The impact of a moderate sprig stiffess to mid-poit displacemet ca be see from the diagrams i Fig zz,.15 g, t without sprigs with sprigs k=5 kn/m 5 zz, g, t without sprigs with sprigs k=1 kn/m 5 Fig. 6 SIMPLY SUPPORTED BEAM WITH STRUCTURAL DAMPING AND SPRINGS AT THE SUPPORTS Structural dampig is icorporated i umerical procedure ad it is represeted by matrix C1. C1 = αm M + αk S ; (αm=,1; αk=,) (16) The vibratio of the damped structure is reduced i compariso to the udamped model. It ca be see especially from the right part of the Fig. 7 (whe movig load is out of the structure) damped udamped t Fig. 7 1
6 The ifluece of the sprig calculated from k, k at the mid-poit displacemet ca be easily, zz, + zz, = (17) zz, ad zz, are the displacemet of the sprigs. Thus, p, = zz k (18),, zz, is the total mid-poit deflectio ad p, due to the movig load without deflectio due to the sprigs. (Fig. 8). is the mid-poit deflectio.4. p,. k,.1 zz, t Fig t 1 CONCLUSION As ca be see from the examples, the agreemet betwee umerical approach ad aalytical solutio is excellet. The procedure based o average acceleratio is robust. Further beefit of the umerical formulatio is that various boudary coditios ad dampig ca all be easily take ito aalysis. Based o the above procedure, a existig D fiite elemet computer program DARK has bee exteded to accommodate the movig-load aalysis. DARK calculates the eigefrequecies ad the mode shapes as well as the displacemets, velocities ad acceleratios due to movig load at ay costructio poit at every time step. These results eable a straight forward calculatio of strais ad stresses. Refereces [1] Iglis, C.E.: Mathematical Treatase o Vibratio i Railway Bridges, Cambridge Uiversity Press, Lodo UK, [] Weaver, W., Johsto,P. R.: Structural Dyamics by Fiite Elemets, Pretice-Hall, Ic., Eglewood Cliffs, New Jersey [3] Štimac, I., Aaliza mostovskih kostrukcia pobuđeih pokretom masom, Magistarska rada, Zagreb, 3.
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