Dynamic Analysis of a Bridge Supported With Many Vertical Supports Under Moving Load

The authors investigate dynamic analysis of a bridge supported under a moving load, by many vertical supports. Modeling of each vertical support was done both as a linear damper and a linear spring. The Euler-Bernoulli beam theory forms the basis of the analysis. The concepts of damping force, spring force, and distributed moving load are utilized in the present concept, while the use of matching conditions is avoided. Obtaining an exact solution is highly simplified, when these forces are expressed in terms of the problem's unknown function. In order for an exact solution to be reached, there is utilization of an important Dirac delta distribution function. The supported bridge response is plotted and compared to different parameter values. Results are compared with previous papers in the case of an undamped bridge with no supports.


Introduction
Dynamic response of uniform beams, rods and bridges under moving loads has received considerable attention within the framework of Euler-Bernoulli beam theory. Especially, dynamic behaviour of railways of infinite lengths under moving loads has been extensively investigated by means of integral transformations. However, these methods are not suitable for analysing beams of finite length under moving loads, and involve complex algebra. Analysis by methods for suspension bridges (Grigorjeva et al. 2006;Idnurm 2006) and vibration analysis of beams under travelling loads were made (Ayre et al. 1950;Fryta 1972;Green, Cebon 1994;Inglis 1934;Inman 1994) and many other papers whose bibliographical account will not be made here.
In case of a concentrated force moving with a constant velocity along the beam, neglecting damping forces, Timoshenko (1927) found a solution and gave an expression for the critical velocity. Stanisic and Hardin (1969) also studied the same problem for the simply supported beam carrying a moving mass. Esmailzadeh and Ghorashi (1992;1995) investigated the behaviour of a finite beam carrying moving point masses. They have also dealt with the vibration analysis of beams due to partially distributed masses moving with con-stant velocity. Garinei (2006) examined the vibrations of simple beam like modelled bridge under harmonic moving loads. The effects of combined loads transmitted both by a single axle (constant component + harmonic components) and by multiple axle subsystems equally spaced have been studied. Using modal analysis, the deflection of a beam subjected to an axial tensile force N and a moving vertical force P, has been determined by Dahlberg (2006). This solution was exploited in a study of the deflections and wave propagations that occur in the contact wire of a railway overhead a catenary system.
In order to reduce the amplitude of vibrating beam, vertical supports can be used. The behaviour of the system in this case is practically very important and needs to be analysed in detail. However, since the region of solution is divided into many parts. In case of several vertical supports and ( ) (here z is the number of vertical supports) boundary and matching conditions are produced in the study, the analysis results in a determinant with large dimensions and complex eigen values. This situation requires the use of pocket programs such as Matlab, Mathematica etc. Since our aim here is to obtain explicit expressions for the dynamic response of the mechanical system, a rather different but easily applicable method will ISSN 1822-427X print / ISSN 1822-4288 online 14 http://www.bjrbe.vgtu.lt be developed. As it is shown in the following part, this method avoids dealing with large matrices, determinants or their calculations, thus yielding formulae for everyday use without a computer. The second main difference of the method is that it is developed for beams (or bridges) of a finite length.
In this paper, as a different technique, the response of a simply supported finite bridge which is subjected to vertical supports and carries a moving force is investigated by transforming singular moving force, spring forces and damping forces into continuously distributed force fields by means of the property of Dirac delta distribution function. This enables us to study with eigenvalues of hingedhinged finite beam.

Analysis
The system under consideration is shown in Fig 1. The uniform bridge is simply supported, bending rigidity EI and moving force F moves at a constant velocity V 0 . Each support is modelled by a linear spring ( ) , where c is the damping coefficient.
According to the Euler-Bernoulli beam theory, the governing partial differential equation describing the transverse vibration of the bridge carrying the time-varying force ( ) t x F , per unit length is (Stanisic, Hardin 1969): where E -the modulus of elasticity, N/m 2 ; I -the second moment of area for the cross-section of the bridge, m 4 ; m -the mass per unit length of the bridge, kg/m; c -the damping coefficient of the bridge, N·s/m; y -the deflection of the bridge measured downwards from its equilibrium position when unloaded, m.
In the present work, in contrast to the approach in (Gürgöze, 1997;Gürgöze, Mermertas 1998), we do not divide the domain of solution into parts. Instead, both damping force and the moving load are considered as distributed loads. Through this approach, matching conditions at x i = η i L (i = 1, 2,…, z) do not have to be used. As clarified later, the calculation of eigenvalues in this situation becomes very simple.
Let the transverse displacement of the bridge be expressed by ( ).
,t x y Thus the boundary conditions for the hinged-hinged beam (bridge) are: We assume that the displacement ( ) t x y , in the forced vibration of the bridge has the form where ( ) t a n -unknown functions to be determined; ( ) x X n -the eigenfunctions of the same bridge in free vibration. In order to determine ( ) x X n , we consider the free vibration of the system and write Eq (1) as: where ω -the transverse displacement of the bridge in free vibration.
Using the method of separation of variables, and thus assuming (4) can be separated into the following equations: The solution of Eq (5a) has the form: ( ) . (6) Applying boundary conditions Eqs (2a) into Eq (6) gives . Other conditions (Eqs 2b) result in: Thus after some simple calculations, nontrivial solution of eigenvalues is obtained: Since the bridge is hinged at both ends, eigenfunctions must be taken as: In the present case, substituting Eq (9) by Eq (3) yields: Before we place Eq (10) into Eq (1), moving force F, spring forces ( ) i Fy and the viscous damping forces ( ) i Fd must be expanded into Fourier sinus series to avoid the matching conditions at η It is a simple matter to show that the moving force is expressed as: Forces produced by each set of damper and spring can be expanded into Fourier sinus series as before. We start with the first group: where ( ) ( ) Substituting the expressions for ( ) 1 η , y L t into Eqs (14, 15) yields: In the same way, the other groups can be written: are expressed in term of unknowns ( ) t a n and ( ) t a n 's respectively. Inserting Eqs (10), (11) and (19) into Eq (1) and arranging terms, we obtain: We introduce the following abbreviations: Now, Eq (20) reads: The solution of homogenous part of Eq (24) is (Pala 2006): where 2 1 ,b b -constants are to be determined. The roots 2 1 , s s -given by: It is clear that three cases are valid. In examining these cases, it is convenient to define the modified critical damping coefficient, ncr D , by: where ω mn -the un-damped natural frequency corresponding to n mode. Furthermore, the non-dimensional number ς called the modified damping ratio, defined by: can be used to characterise the three types of solutions of the characteristic equation. Rewriting the roots given by Eq (26) yields: Modified damping ratio ς determines whether the roots are complex or real.
In case of modified damping ratio greater than ς 1 > , the discriminant is positive, resulting in a pair of distinct roots. The solution of the homogeneous part of Eq (24) which represents a non-oscillatory response. In Eq (30) 2 1 ,b b are determined by the initial conditions. In case of 0 ς 1, < < discriminant is negative, resulting in a complex conjugate pair of roots. These are: The solution of the homogeneous part of Eq (24) in this case is: where 2 1 ,b b -determined by the initial conditions. Recall that this solution will die out in time due to the forcing terms F and d F . Here: In case of ς 1, = which we call the modified critically damped case, the solution takes the form For a proper solution of Eq (24), let us assume: ( ) γ sin ω ξ cos ω .
Substitution of Eq (35) into Eq (24) gives: Combining Eqs (35) and (30) For simplicity, we assume that the initial conditions are in the form: This assumption is not a restriction, and the initial conditions may be taken as non-zero when desired. Using these conditions in Eq (38), we can write: Applying these conditions to Eq (37), yields: If the roots 2 , 1 s s -complex, then The problem is now completely solved.

Results and discussions
In order to prove the validity of the method, taking the values N·s/m), we consider 3 cases: a) bridge with no support, b) bridge with unique support at the middle and c) bridge with equivalent supports at points L x 25 , What is clearly seen is that the deflections decrease with the number of supports.
The variation of the displacement y versus x for in- s is shown in Figs 5a, 5b and 5c, respectively. As expected, it is clearly seen that the vibration amplitude of the bridge decreases with the number of supports. Figs 6a and 6b compare the deflection y of the bridge at the midpoint for various locations of unique support. It is observed that the deflection amplitude of the bridge decrease when the support approaches the midpoint. Thus we conclude that the optimal solution for the location of the unique support is L x 5 , 0 1 = at which the vibration amplitude becomes minimum.
An important problem arising in the analysis is that the behaviour of the bridge for large values of time t may deviate from the actual curve, the reason of which is the structure of Eq (24). Indeed, when the coefficient of n a becomes too large, then the solution starts deviating from the actual curve. Therefore for large values of t, one must use the solution carefully. This point could be partially prevented by including damping property of bridge itself into Eq (1). This is the reason why Eq (1) in the present analysis involves the damping property of the bridge itself. On the other hand, time interval in the figures must not exceed the value of time necessary for the moving load to pass across the bridge. After this moment, the moving load starts again to be applied to the bridge for the second time.
The present method can be generalised to involve several moving loads. If, for example, we have N forces, thus they can be described as:

Conclusions
In this work, dynamic analysis of a bridge supported by many vertical supports under moving load has been performed using Fourier sinus series approach. In this way, a more realistic bridge model has been formed and analytically analysed. Depending on the constants involved in the problem, whether or not a damped bridge vibrates can be explained in a systematically way. In case of a bridge with no support, the results have been exemplified, and deflection curves have been obtained for some values of parameters involved in the problem. Calculations have revealed that the contributions made by the first three modes are dominant in the deflection of the bridge. Since the results explicitly depend on the number of modes, the present formulation best explains their effects on the solution. The results of the method have been compared with those of Esmailzadeh and Ghorashi for the case of undamped beam (bridge) with no support, and found that they are in great agreement. However, the present method does not require the solution of coupled approximate dynamical equations as the work of Esmailzadeh and Ghorashi does. The present method seems easy especially, when several moving loads and supports are considered, while the other methods would require the numerical solution of coupled differential and algebraic equations. The method is even easier since no matching condition is utilised.
According to the present method, matching conditions do not constitute a problem in the solution, and frequency analysis is readily carried out.
As an extension of the method, one can also investigate the effect of curvature on the response of the bridge without making much change since the curved shape can be easily expanded into Fourier sinus or cosines series. When other methods are used, one cannot readily obtain analytical results and is forced into using numerical techniques.