Dynamic analysis of railway bridges by means of the spectral method

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1 Dynamic analysis of railway bridges by means of the spectral method Giuseppe Catania, Silvio Sorrentino DIEM, Department of Mechanical Engineering, University of Bologna, Viale del Risorgimento, 436 Bologna, Italy ABSTRACT This study investigates the dynamic behaviour of railway bridges crossed by travelling trains. A simplified formulation was adopted in order to perform a direct analysis of the effects of the parameters involved in the problem. The bridge is modelled as a rectangular plate, while the trains are modelled as travelling inertial distributed loads. The formulation is accomplished by the use of the Rayleigh-Ritz method, yielding a low order model with time-dependent coefficients. Several numerical examples are presented and discussed, aimed at investigating the effects of each of the model-governing parameters. Keywords Railway bridges, moving loads, Rayleigh-Ritz method. Nomenclature Introduction C damping matrix u unit step distribution D stiffness of the plate V potential of applied loads E Young s modulus v velocity of the load g gravity acceleration w displacement velocity response function x spatial coordinate h thickness of the plate y spatial coordinate K stiffness matrix velocity ratio L length frequency parameter l width relative error M mass matrix modal matrix n modal index length ratio p load per unit area Poisson s ratio r mass ratio mass per unit area q modal coordinate n modal natural frequency t time moving coordinate U potential of strain energy damping factor In the analysis of dynamic effects of railway vehicles on bridges, simplified models are suggested and usually used, taking into account only some aspects, such as deterministic, vertical effects, and the influence of moving forces and masses [-]. Inertial effects of both bridge and vehicle can be influential, and not negligible, since the mass of the external load introduces a coupling effect between the load and the structure. Other important aspects, such as dynamic properties of travelling vehicles and track irregularities, are not considered in the present study. The railway bridge model most commonly used is a continuous Euler Bernoulli beam [], or a Timoshenko beam [3], traversed by either concentrated [4] or distributed moving loads [5]. Possible applications of lumped vibration absorbers have also been investigated [6]. owever, in the present study a homogeneous Kirchhoff plate is considered, allowing the analysis of lateral vibrations due to trains travelling on double-track bridges. Structure damping is included in the model, as it may play an important role. T. Proulx (ed.), Civil Engineering Topics, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series 7, DOI.7/ _, The Society for Experimental Mechanics, Inc.

2 The train is simply modelled by means of a continuous load in the form of a moving strip, an idealization which can be adopted when the span of the bridge is large in comparison with the distance between axles []. A constant speed of motion of vehicles along the bridge is assumed. The formulation is accomplished by the use of the Rayleigh-Ritz method [7], and the solution is expressed in terms of a linear combination of functions, which in the present study are selected as tensor products of eigenfunctions of prismatic pinned-pinned and free-free beams in flexural vibration. This approach yields a reduced order model with time-dependent coefficients, allowing a parametric analysis of plates loaded by travelling distributed masses. Different example cases are presented and discussed in detail, analyzing the effects of velocity, mass and length of the train on the plate s dynamic response with respect to the mass, stiffness and damping of the plate itself. Theoretical model A homogeneous isotropic Kirchhoff plate is considered, simply supported on two opposite sides, free on the other two sides and crossed by a travelling distributed load. The load per unit area p over the plate may be expressed as: d w( x, y, t) pxyt (,, ) t f(, y) g, x v xt, () dt where w is the vertical displacement of a point of the plate or of the load (Fig.), t is the equivalent mass per unit area of the load, g is the gravity acceleration, v x is the travelling speed in the x direction, is a moving coordinate in the same direction [] and f models the translating strip representing the instantaneous position of the load: f (, y) u( L) u( ) uy ( ) uy ( l). () t Note that within the present study t is assumed to be constant; however, piecewise-constant or other distributions t ( ) may be considered and adopted in the following developments. Equation contains the unit step distribution u(), L t and l t are the length and the width of the strip modelling the train, and is the distance between the side of the strip and the edge y = of the plate, as shown in Fig.. The second term on the right-hand side of Eq. describes the inertial action of the load. The total acceleration may be expressed in the following general form: t d w w v w v v w v w v w v w a w a w x x y y x y x y dt t x x y y x t y t x y (3) where v x, v y, a x, a y express the velocities and accelerations of the travelling load in the x and y directions respectively []. Considering a train travelling at constant speed v in the x direction, Eq. 3 reduces to: dw w v w v w dt t xt x (4) The first term of the right-hand side of Eq. 4 expresses the influence of vertical acceleration of the moving load, the second term the influence of Coriolis acceleration, and the third term the influence of track curvature []. y L b L t l t v x l b x Fig. Model scheme

3 3 The functional of the total potential energy of the coupled system can be written as the sum of a term U due to the strain energy plus a term V representing the potential of all applied loads (including the inertial forces): U V (5) The potential of the strain energy can be written in terms of second order derivatives of the out-of-plane displacement w: lb 3 Eh xx yy xx yy ( ) xy, (6) U D w w w w w dxdy D ( ) where the subscripts denote differentiation with respect to the spatial variables and D is the flexural stiffness of the plate, expressed as a function of Young s modulus E, Poisson s ratio and thickness h [7]. In the adopted formulation the inertial forces are included in the potential of applied loads V as follows: lb ( b ) (7) V ww wp dxdy where b is the mass per unit area of the plate and p is the load in Eq.. The out-of-plane displacement w is expressed by means of a linear combination of shape functions, selected as products of homogeneous uniform prismatic beam eigenfunctions : N w q (, ) w q (8) n n n where q is the generalized coordinate vector. Introducing the displacement expansion in the quadratic functional, and imposing its stationarity, yields the following algebraic eigenproblem: MrM q r C q Kr K q rgf (9) [ ] [ ] [ ] with: v D r,, () t b b where is a frequency parameter and is a dimensionless parameter depending on the speed v. The matrices in square brackets in Eq. 9 can be regarded as dimensionless quantities, and they can be computed according to the following integrals: lb lt x lt x T ( ), ( ), b ( x ) x x M dxdy M dxdy C L dxdy lb 4 T T T T T b xx xx yyyy xxyy yyxx xyxy K L [ ( ) ( )( )] dxdy lt x lt x T b xx f x x K L ( ) dxdy, dxdy () In Eq. the integration interval [x, x ] is time-dependent. Introducing the ratio between the lengths L t and L b : then x and x vary according to Tab.. Lt () L b

4 4 Table Time-dependent interval of integration case case case x vt Lt x vt x vt Lt vt x vt Lt x vt Lt x vt Lt x vt vt x x vt x vt L b x vt vt x x L vt Lt x x Lt vt Lt x vt Lt b To model energy dissipation within the structure, a dimensionless damping matrix C may be defined by means of the plate modal matrix (mass normalized) and eigenvalues n (computed from the M and K matrices), and considering a modal damping ratio equal for all modes: Introducing Eq. 3 in Eq. 9 yields: diag diag T n T n C (3) MrM q Cr C q Kr K q rgf (4) [ ] [ ] [ ] Equation 4 is a reduced order discretized model with time-dependent coefficients, which can be solved numerically. 3 Numerical results Some numerical examples are presented for studying the dynamic behaviour of the model described in section. The influence of parameters v, r,,, governing Eq. 4 is highlighted by studying time responses w(t) and dynamic response functions of the dimensionless frequency (playing the role of frequency response functions ) defined according to: max t wt ( ) ( x, y; ) (5) w where w s is the static deflection due to the load centered in L b /. Numerical solutions of Eq. 4 are computed using the Runge-Kutta algorithm, expanding the solution w (Eq. 8) with 4 beam eigenfunctions (4 pinned pinned eigenfunctions along x direction and free free eigenfunctions along y direction). Realistic values for parameter are computed by means of the empirical expression: s x, y; a L b (6) based on large collections of experimental data [], where a and are parameters depending on the kind of bridge considered, as reported in Tab.. The values (in z) of the first natural frequency and of parameter for different kinds of bridges are reported as functions of the length L b in Fig..

5 5 f [z] 5 5 General bridges Steel truss bridges Steel plate girder bridges with ballast Concrete bridges with ballast Concrete bridges without ballast [rad/s] General bridges Steel truss bridges Concrete bridges with ballast L b [m] L b [m] Effect of the speed of the load Fig. First natural frequency f [z] (left) and frequency parameter [rad/s] (right) as functions of the length L b for different kinds of bridges Table Parameters in Eq.6, as reported in [] Kind of bridge a General bridges (average case) 33.9 Steel truss bridges 37. Steel plate girder bridges with ballast 59.7 Steel plate girder bridges without ballast 8 Concrete bridges with ballast 9. Concrete bridges without ballast 5. As a reference case study, the following values for the parameters are assumed: Plate: L b = 5 m, l b = m, = 5 rad/s, =.5. Moving load: =.4, l t =.5 m, =.5 m, r =.5. Time responses w(t) are computed at coordinate x = L b /, y = l b / with speed v varying from 3 m/s to 5 m/s (8 Km/h to 8 Km/h), as shown in figure 4. Maximum deflection at different points Parameter values are assumed as in the reference case. Response functions () are computed at different points (x, y) along the structure, as reported in Fig. 3. The frequency parameter varies from. up to, i.e. v varies from 4 Km/h up to 4 Km/h. Functions (x, y; ) show a peculiar undulating trend, not significantly affected by the choice of coordinate x. Effect of the mass of the load Parameter values are assumed as in the reference case with v = 4 m/s = 44 Km/h, varying r from. to. Response functions w(t) and () are computed in x = L b /, y = l b /, as reported in Fig. 4. Effect of the length of the load Parameter values are assumed as in the reference case with v = 4 m/s = 44 Km/h, varying from. to. Response functions w(t) and () are computed in x = L b /, y = l b /, as reported in Fig. 5. Parameter (related to L t ) is able to significantly affect the behaviour of (). Note that the plots of () in the case are superimposed.

6 6 w [m].5 x x = L b /; y = l b / x = L b /4; y = l b / x = 3L b /4; y = l b / x = L b /; y = x = L b /; y = l b -.5 v = 4 m/s -3 v = 5 m/s v = 3 m/s t [s] Fig. 3 Effect of parameter v on w(t) (left); () at different points (x, y) (right) x r =. r =.8 r =.5 r =. w [m] t [s] r =.5 r =. r = Fig. 4 Effect of parameter r on w(t) (left); effect of parameter r on () (right) w [m].5 x t [m] =. =.6 =. =.4 = =. =. =.5 = Fig. 5 Effect of parameter on w(t) (left); effect of parameter on () (right)

7 = 5.5 rad/s = 3.5 rad/s = 8.5 rad/s..8.6 = =. =.5 =. =. = Fig. 6 Effect of parameter on () (left); effect of parameter on () (right) w [m].5 x = / = /6 = /48 = / = / t [s] = /48 = /48 = / Fig. 7 Effect of partially distributed load on w(t) (left); effect of partially distributed load on on () (right) Effect of structural stiffness Parameter values are assumed as in the reference case, varying from 3 rad/s to 8 rad/s. Response functions () are computed in x = L b /, y = l b /, as reported in Fig. 6. The plots are almost superimposed. Effect of structure damping Parameter values are assumed as in the reference case, varying from to. Response function () is computed in x = L b /, y = l b /, as reported in Fig. 6. Raising reduces the amplitude of oscillation of (), until its behaviour becomes monotonic (however this is not the case for real bridge structures). Effect of partially distributed load Parameter values are assumed as in the reference case, with L t = 4 m and v = 4 m/s = 44 Km/h. Different loading distributions are compared: the continuous one (as represented in Fig. ) and partial distributions consisting of two shorter sections in which the load is distributed. The assumed partial distributions are given by: Lt Lt and Lt Lt (7)

8 8 with.5 ( =.5 yields the continuous distribution). Since for the continuously distributed load it is assumed r =.5, for the partially distributed load described by Eq. 7 r increases to r = /() r. Response functions w(t) and () are computed in x = L b /, y = l b / for different values of (/6, /48, /48) as reported in Fig. 7. Load distribution variations such as that described in Eq. 7 may dramatically affect the behaviour of the response function (). Effect of time dependent matrices Parameter values are assumed as in the reference case, with v = 4 m/s = 44 Km/h. The effect of neglecting the time dependent matrices M, C, K on the solution w(t) is evaluated by introducing a relative error, according to: M M max t wt ( ) wt () wt () where [w(t)] M = refers to the solution computed assuming M = in Eq. 4. Similarly, C and K can be defined, considering C = and K =. (8) 6 x M = C = K = Exact C = K = C = C K M Tot t [s] Fig. 8 Effect of neglecting time dependent matrices M, C, K on w(t) (left) and on () (right) The error functions M, C and K are plotted versus time in Fig. 8, where Tot represents the total error, assuming M, C and K equal to at the same time. The smallest, and negligible contribution to the error Tot appears to be C, while the main contribution is due to M. Fig. 8 also shows the effect on () of neglecting the time dependent matrices M, C, K. Again, the effect of neglecting C is very small. The total error, on the contrary, can be significant. 4 Discussion Function (x, y; ) appears to be an effective tool for studying the dynamic behaviour of a structure crossed by travelling loads with constant speed, in some way equivalent to a frequency response function for time-varying coefficient systems. This function shows peculiar undulating trends (Fig. 3), influenced by the parameters governing Eq. 4. The response can be evaluated at any coordinate point (x, y) of the plate, making it possible to study the variation of structural deflection also along the y coordinate (Fig. 3). Mass parameter r can produce important shifts in magnitude, but not in shape (Fig. 4). On the contrary, length parameter controls both shape and magnitude of () (Fig. 5), but only in the case <. The damping parameter has the effect of progressively smoothing the oscillation of (), until it becomes monotonic (Fig. 6, though the latter limit case is not realistic for actual bridge structures): in general, the reduction in amplitude becomes particularly significant at high speed. Frequency parameter, within the range of real bridges, scarcely affects the behaviour of () (Fig. 6), so may be considered independent from. Changes in the spatial distribution of the load can produce dramatic variations in () (Fig. 7): this result should highlight the importance of properly modelling the ballast, directly influencing the load distribution on the actual structure. The contribution to the solution of the time dependent matrices M, C and K is globally not negligible (Fig. 8), however the effect of C is usually very small in comparison with the contributions of K, and especially of M.

9 9 5 Conclusions and future work In this study, the dynamical behaviour of railway bridges crossed by travelling trains was investigated by adopting a simplified model, i.e. a plate loaded by a travelling distributed mass, solved by means of the Rayleigh-Ritz method. The effects of each of the model governing parameters was studied introducing a dynamic function of the travelling speed, equivalent to a frequency response function for time-varying coefficient systems. This function can be an effective tool for studying the dynamic behaviour of a structure crossed by travelling loads, since the travelling speed is the most important parameter influencing the dynamic stresses in railway bridges, which in general increase with increasing speed. In particular, it was shown how different spatial distributions of the load can deeply influence the dynamic response of the structure, highlighting the importance of properly modelling the ballast. Future work will thus concern this significant problem. Acknowledgments This study was developed within the INTERMEC laboratory with the contribution of the Regione Emilia Romagna - Assessorato Attività Produttive, Sviluppo Economico, Piano telematico, PRRIITT misura 3.4 azione A Obiettivo. References [] Fryba L., Vibration of Solids and Structures under Moving Loads, 3rd edition, Telford, 999. [] Fryba L., Dynamics of Railway bridges, Telford, 996. [3] Lin Y.., Vibration analysis of Timoshenko beams traversed by moving loads, Journal of Marine Science and Technology (4), pp. 5-35, 994. [4] Stancioiu D., Ouyang., Mottershead J.E., Vibration of a continuous beam excited by a moving mass and experimental validation, Journal of Physics, Conference series 8, 999. [5] Adetunde I.A., Dynamical Behavior of Euler-Bernoulli Beam Traversed by Uniform Partially Distributed Moving Masses, Reasearch Journal of Applied Sciences (4), pp , 7. [6] Lin Y.., Cho C.., Vibration suppression of beam structures traversed by multiple moving loads using a damped absorber, Journal of Marine Science and Technology (), pp , 993. [7] Timoshenko S., Young D.., Weaver W., Vibration problems in engineering, 4th edition, Wiley, 974.

Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian

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