Maximum resonance and cancellation phenomena in orthotropic plates traversed by moving loads: application to railway bridges

The vibrational response of railway bridges is an issue of main concern, especially since the advent of High-Speed traﬃc. In the case of short-to-medium lengths and simply-supported spans excessive transverse acceleration levels may be induced at the platform, with detrimental consequences for passengers and infrastructures. The orthotropic plate has proven to be an appropriate model for the prediction of the response of certain typologies in the aforementioned cases such as multiple girder decks, solid or voided slabs or ﬁller-beam multiple-track decks. In this contribution, the vibrational response of orthotropic plates, simply and elastically supported, circulated by vertical moving loads is investigated. First, maximum free vibration and cancellation conditions are derived analytically. From these, bridge span length-characteristic distance ratios leading to maximum and minimum resonances under series of equidistant loads are depicted. Second, the applicability of these ratios in oblique decks is analysed for the most common ﬁrst three mode shapes: ﬁrst longitudinal bending, ﬁrst torsion and ﬁrst transverse bending modes, and the errors in relation to the straight reference case are bounded. To this end, an extensive bridge catalogue of girder bridges is designed in the range of lengths of interest, covering ﬂexural stiﬀnesses typical from both conventional and High-Speed railway lines. Finally, the applicability of the previous theoretical results is exempliﬁed with experimental measurements performed on a bridge from the Spanish railway network. bridges, orthotropic plates, resonance, cancellation, moving loads.


Introduction
Since the opening of the first High-Speed (HS) railway line in the 1960s in Japan, railway infrastructures have evolved dramatically in most developed countries. During the last decades, several HS lines have been constructed and many conventional lines have been upgraded to higher operating speeds. Spain is one of the leading countries in this regard with almost 2600 km of operational HS lines [1]. The increase of the design speeds brings along difficulties and new challenges. In particular, railway bridges must accomplish now strict requirements in terms of maximum vibration levels for the sake of passenger comfort and traffic safety. In this context, the dynamic behaviour of such structures has received considerable attention from the engineering and scientific community in the past decades.
This investigation focuses on the particular case of short-to-medium length (10 −25 m) simply-supported 10 (SS) bridges, which are prone to experience high transverse acceleration levels at the platform due to their usually associated low mass and structural damping [2,3,4,5]. In countries like Spain in the aforementioned range of lengths common deck typologies built in the 80's and 90's for double track platforms are voided or solid concrete slabs, pseudo-slabs and precast decks composed by double-T beams [6] (see Fig. 1). In the last years, twin cell box girder decks are becoming more common due to their enhanced dynamic performance. 15 With the exception of the lastest, the vertical dynamic response of the aformentioned structures may be adequately predicted with orthotropic plate models, showing good correspondence with experimental measurements [7]. For this reason, the response of orthotropic plates under moving loads and its application to railway bridges constitutes the main interest of this work. The basic phenomenon that governs the level of vibrations induced on a bridge by the circulation of a 20 railway convoy is the amplitude of the free vibrations that a single axle leaves on the structure after its passage. Depending on the ratio between the travelling time of the load crossing the bridge and a certain natural period of the structure, the amplification of the free vibration response in that particular mode may be maximum or may be theoretically cancelled, implying that the structure would remain at rest under certain ideal conditions (i.e. in the absence of structural damping). This basic problem has been analysed 25 in detail by authors such as Yang et al. [8], Savin [9] and Museros et al. [10] for SS beams and also for beams resting on vertical elastic supports (ES beams in what follows). The aim and novelty of this investigation is to extend this study and evaluate its validity to orthotropic plates representative of bridges with a dynamic behaviour characterized by modal contributions different from the longitudinal bending ones (i.e. first torsion, first transverse bending modes). This is the case of SS multiple track decks with 30 length to width ratios close to unity and sections similar to those included in Figs. 1(a-d). In general, these structures exhibit several natural modes below 30 Hz which may significantly contribute to the maximum vertical acceleration [7].
The interest of knowing a priori the maximum free vibration and cancellation speeds for a certain structure is diverse. First, if a train induces resonance on a bridge, the dynamic amplification will depend 35 on the closeness of the resonant velocity to these maximum free vibration or cancellation conditions. If the resonant speed coincides with a maximum free vibration condition, important amplification levels should be expected. On the contrary, if it takes place in the vicinity of a cancellation speed, the resonant peak will become imperceptible. Even though it is well known that in a real situation resonant amplitudes depend on several factors, e.g. number of convoys, vehicle-structure and soil-structure interaction, and other damping 40 and amplitude dependent mechanisms, the amplitude of the free vibrations can be understood as the basic phenomenon. Second, when amplitude dependent magnitudes must be determined from an experimental campaign performed on a bridge with a test vehicle, e.g. modal damping, the test speed should be selected such that it leads to a representative level of vibrations in the structure, avoiding close to cancellation velocities. 45 The problem of transverse vibrations of beams and plates subjected to moving forces has received considerable attention in the past due to its applications in structural engineering. Nevertheless, the number of studies on plates is rather limited compared to those devoted to beams with different boundary conditions [11]. Gbadeyan and Oni [12] formulated the dynamic problem of Rayleigh beams and plates under an arbitrary number of moving masses based on modified generalized finite integral transforms. Shadnam et al. [13] 50 introduced a new method to compute the transient response of a rectangular plate excited by relatively large masses along arbitrary trajectories. Lee and Yhim [14] analysed single and two-span continuous composite plate structures subjected to multi-moving loads taking into account third order transverse shear deformation and rotary inertia. Au and Wang [15] investigated the vibratory response of rectangular orthotropic thin plates with general boundary conditions traversed by moving loads with the aim of determining the 55 acoustic pressure distribution around the plate in the time domain. Gbadeyan and Dada [16] analysed the maximum internal forces and displacements of Mindlin elastic plates under uniform partially distributed moving masses. Malekzadeh et al. [17,18] and Ghafoori et al. [19] investigated the dynamic response of composite plates under moving loads. Zhu and Law [20] analysed the dynamic behaviour of orthotropic plates simply supported on a pair of parallel edges applying Lagrange's equations and modal superposition.

60
More recently some authors have investigated the resonance phenomena caused by the circulation of periodic forces or masses following either a semi-analytical [21,22] or a numerical approach [23, 24,7]. Neverthe-less, in the authors' opinion, regardless of the approach adopted, resonance phenomena caused by a series of periodic loads and the conditions for its maximization or cancellation depending on the free vibration amplitudes left by each single load have not been analysed in the case of orthotropic plates, especially when 65 these separate from ideal well-known conditions i.e. SS boundary conditions and non-oblique configurations.
This work constitutes a contribution in this area.
In what follows, first, the transverse vibration problem of a rectangular Kirchhoff orthotropic plate under the circulation of a single load crossing it at constant speed moving parallel to its longitudinal axis is formulated analytically, both for SS and ES boundary conditions on two opposite edges. Then, the non- three vibration modes, showing a good correspondence even in the case of structures that differ considerably from the reference straight plate. Finally, the applicability of the conclusions is depicted through the experimental response measured on a real HS railway bridge undergoing resonance.

Free vibrations of orthotropic rectangular plates under a single moving load
The partial differential equation governing the transverse oscillations w(x, y, t) of a Kirchhoff orthotropic rectangular plate under a generic load distribution q z (x, y, t) ( Fig. 2a) neglecting shear deformation and rotary inertia, and being X and Y the principal directions of orthotropy, is given by where D x and D y are the flexural rigidities per unit of length L or width B of the plate with respect to 85 the XZ and Y Z planes, respectively, 2H = D 1 + D 2 + 2(D xy + D yx ), D 1 and D 2 are the coupling flexural rigidities, D xy and D yx the torsional rigidities with respect to the same planes and ρ the uniform mass density [20,25]. In Eq. 1b, which represents the particular source of excitation of a train of N P constant moving loads P k travelling along y = y P , parallel to the X axis at constant speed V (Fig. 2b), H and δ stand for the Heaviside unit step and the Dirac Delta functions, respectively. In what follows the free vibration ,t

Plates simply-supported on two opposite borders
Applying boundary conditions corresponding to two opposite sides SS and the other two borders free, the transverse displacement of the neutral plane w(x, y, t) under the circulation of the moving loads may be expressed as a linear combination of the normal mode shapes Φ ij (x, y) where Ψ ij (t) stands for the ij-th modal amplitude, C ij depends on the mode normalization and Y ij (y) is provided in Appendix A (Eqs. A.1a-A.1e). For the details of the derivation of the natural frequencies and 95 mode shapes of a rectangular SS orthotropic plate the reader is referred to [25].
Substituting Eq. 2 in 1a, multiplying by the mn-th mode, integrating over the plate domain and in virtue of the modes orthogonality condition [25], the differential equation governing the mn-th modal amplitude in the absence of structural damping may be expressed as where M mn stands for the mn-th modal mass and ω ij is the circular frequency of the orthotropic plate (Eq.A.2a in Appendix A). If only one load crosses the plate at a constant velocity, Eq. 3 presents the following solution where K mn is the non-dimensional speed parameter defined as Notice that in the previous equations m corresponds to the number of half-sine waves travelled by the load when crossing the plate in the mn-th mode. As the load travels parallel to the X axis, the modal deflection along the load path is always a sinusoidal function. When the load exits the plate, the latter is left in free vibration. The amplitude of this free vibration stage may be obtained solving Eq. 6 with initial conditions given by Eq.
If the amplitude of the free vibrations is divided by the static solution when the load acts at (L/2, y P ), Ψ st mn , a non-dimensional amplitude in free vibration associated to a certain mode may be defined and expressed In Fig. 3 the evolution of R mn with K mn is represented. R mn curves superimpose for modes with the  Leissa's approach [26], the plate transverse displacement may be approximated as a linear combination of the product of two functions X i (x) and Y j (y).  where X i (x) and Y j (y) are approximated as the mode shapes of the associated undamped beams having the same boundary conditions as the plate in the X and Y directions, respectively. For the details of the formulation the reader is referred to Appendix A (Eqs. A.3 to A.6).
Following the same approach as in the SS case, in virtue of the modes orthogonality condition and neglecting structural damping, one concludes that Eq. 3 governs again the ES plate mn-th modal amplitude under the circulation of a single load travelling at constant speed, with the only difference that now Φ mn stands for the ES plate mode. As the load moves parallel to the longitudinal axis, admitting the approximation of the ES plate modes as the product of the corresponding beams modes, the modal load time variation is proportional to that of a ES beam subjected to the same loading conditions. The solution for the forced vibration of a Bernoulli-Euler ES beam under a single load was first presented in a compact format by Museros et al. and may be consulted in [10]. From the modal amplitude and its derivative at the time instant when the load exits the plate, the mn-th modal response in free vibration may be obtained. Finally, the modal amplitude in free vibration can be defined as in the SS plate case applying a proper normalization For clarification, an overbar is used in what follows to differentiate magnitudes corresponding to the ES plate. In Eq. 9K mn is defined as the ratio between the forcing frequency of the load and the ES plate circular frequency,ω 2 mn (Eq.A.6 in Appendix A). In order to be able to compare R mn curves for different support conditions, the following relation betweenK mn and K mn is admitted between the ES and the SS plate, even though it strictly applies to the beam case: When R mn is plotted vsK mn in the ES plate case a similar evolution to that shown in Fig. 3 is observed, 110 where cancellation and maximum free vibration conditions alternate as the velocity increases. In the next subsection these values are computed for SS and ES boundary conditions.

Non-dimensional speeds for maximum modal response and cancellation
In the previous section it was shown that the amplification of the free vibrations in SS or ES orthotropic rectangular plates after the circulation of a single load can be expressed in terms of only one parameter, the non-dimensional speed K mn . From Eqs. 7 and 9 it is possible to obtain the conditions for maximum free vibration and cancellation in each mode as In Eq. 11 K ci mn for i = 1, 2, 3, . . . stands for the values of K mn of cancellation as the velocity decreases (i.e. K c1 mn is the first cancellation taking place at the highest speed). Likewise, K max,i mn designates the non-115 dimensional speeds leading to a local maximum of the plate free vibrations, corresponding K max,1 mn to the local maximum taking place at the highest speed. In Fig. 3, first and second cancellation speeds and local maxima are indicated for the SS plate modes with m = 1, 2, 3. In Table 1 the values for the first three cancellation and maximum free vibration speeds are included for modes with m=1, κ = 0 (SS case) and Only the values associated to m = 1 are included as the main focus of this study 120 is the response of girder or slab decks associated to the first longitudinal bending, first torsion and first transverse bending modes, which presumably will have the highest contribution to the overall transverse response for frequencies up to 30 Hz. Notice that these values coincide with those of the SS and the ES beams [10] due to the expression for the load in modal coordinates. In section 4, these ratios are compared to real cancellation and maximum free vibration conditions computed numerically, and their accuracy is 125 evaluated within a catalogue of realistic structures.   The conditions for resonance in a SS (Eq. 12a) or an ES (Eq. 12b) rectangular plate can be formulated in a dimensional or non-dimensional format as where j is the resonance order, or number of oscillations that the plate undergoes in a particular mode between the passage of two consecutive loads separated a distance d k . At resonance, the free vibrations that every single axle leaves on the structure add in phase and the plate response progressively increases.
Consequently, the amplitude of the free vibrations at the particular resonant speed, will determine to an important extent the prominence of the resonant peak. When a resonant speed coincides or is close to a maximum free vibration condition, an important amplification of the response should be expected. On the contrary, as it approaches to a cancellation condition, it will become almost imperceptible. Based on the previous, the conditions for resonance and either maximum free vibration or cancellation are equated, and L/d k ratios for maximum resonance or cancellation of it are extracted both for the SS (Eq. 13) and the ES (Eq. 14) plates: In Table 2 Table 1 into Eqs. 13 and 14. These values coincide with those of SS and ES beams presented in [10], as it is justified through the derivations presented in section 2.
In section 4, the accuracy of analytical K ci 1n and K max,i 1n from Table 1 and, therefore that of the L/d k ratios included in Table 2, is evaluated in the case of 112 real bridges with skewed geometries.       Table 3 show, respectively, the cross section and main properties of the bridge catalogue.  are not always present or may not be properly incorporated; and (ii) in previous works by the authors [28] it has been shown that the effect of these elements on the first three modal shapes of this type of bridges  In Fig. 6

Numerical model description
The dynamic response of the previously described bridges under railway traffic is obtained numerically 185 in the time domain. The main features of the numerical model, sketched in Fig.7, are the following: • An orthotropic thin plate is used to calculate the deck dynamic response under railway traffic. The plate is discretised in C 1 compatible linear varying curvature triangular finite elements with 12 degrees of freedom [29].
• The weight of the dead loads (ballast, sleepers, rails, sidewalks and handrails) is uniformly distributed 190 over the plate. No additional rigidity from the rail components is considered in a first approach. The orthotropic plate constants, D x , D y , D 1 , D 2 , D xy and D yx , are computed for each of the bridges of the catalogue from its geometry and material data. The calculated values for these constants are given in Table 4. For further details on this step the reader is referred to the classical reference [31].  Table 1, and errors when using these predictions in 215 skewed and ES decks are delimited. Finally, the effect of structural damping is shown on a subset of short bridges.

Evolution of modal parameters
In Fig. 8 Table 1 to predict these phenomena with the increase of the plate obliqueness.
Finally, conclusions regarding the effect of structural damping are provided.
First, the free vibration response of the 112 bridges under study after the circulation of a single load is obtained in the absence of damping. The load travels in all the cases with the same eccentricity (1/4 of the deck width). In the SS case (κ = 0) and for the straight configurations (α = 0) this response coincides with that shown in Fig. 3(a). The maximum modal displacement in the first three modes is obtained for several travelling speeds. The range of speeds considered is such that the first cancellation and second maximum of the free vibrations are reached and a sufficiently small velocity increment is selected to obtain these two conditions with accuracy (∆V = 10 −4 m/s). The second maximum is evaluated as the first one will correspond to an unrealistic high speed with existing bridges and railway convoys. Finally the errors  with respect to the theoretical predictions given in Table 1 are obtained as per 1n,an · 100 (15) where −an indicates the analytical values provided in Table 1, and K c1 1n and K max,2 1n are computed from the actual velocities applying Eqs. 5 and 10 for the SS and the ES case, respectively.

240
In Fig. 9 the errors in the estimation of the first cancellation K c1 1n of the free vibrations (plots 9(a)-(b)-(c)) and the second maximum K max,2  (Table 1) with errors lower than 1% even for the highest levels of obliqueness considered; (ii) for the third mode, bridges with the highest skew angle (45 • ) and least flexural stiffness lead to the highest deviations, which reach 21% for the 1 st cancellation and 14% for the 250 2 nd maximum. This is due to the fact that with the increase of obliquity the third mode deflection along the load path diverges from the pure half-sine function; (iii) in the the great majority of analysed cases the errors are negative, as the real dimensionless cancellation and maximum free vibration speeds are smaller than the theoretical ones. This happens because with the plate obliquity the dimensional speeds for these two situations reduce and the natural frequencies increase in all the cases.  Table 1, the relative errors admitted are negligible for the 1 st cancellation (< 0.2%) and lower than 2% for the 2 nd maximum; (ii) as in the SS case, errors are higher than 2% only for the transverse bending mode (n = 3). In this case, errors for the 1 st cancellation 260 reach 3.3% for straight plates and diminish up to −16% for the most oblique ones. The second maximum is better approximated with a maximum error of −9.9% for plates with α = 45 • . In general these errors are higher for L/δ 2000 bridges and are not much affected by the bridge length.
From the previously presented analysis it can be inferred that for the 1 st longitudinal bending and 1 st torsion mode, the non-dimensional values for the speeds causing cancellation or maximum free vibration of 265 the orthotropic plate simply or elastically supported that are proposed in section 2 are good estimates in the case of oblique plates, even in highly skewed configurations. The previous conclusions are directly applicable to the L/d k theoretical ratios for maximized and cancelled resonances included in Table 2, in virtue of Eqs.

and 14.
Finally, the effect of damping on the free vibration response is briefly presented. From the catalogue, the (%) and E c1 1m (%), for the elastically-supported bridges κ = 0.05 of the bridge catalogue with respect to the straight reference case (κ = 0.05, α = 0 • ) bridge with L = 10 m, L/δ 2000 and α = 0 is selected as a representative example. Nevertheless the conclusions derived from it are applicable to the remaining cases. Fig. 11 shows the non-dimensional amplitude of the free vibrations for the second mode computed considering modal damping ratios ζ 1n = [0, 1, 2, 3] %. Fig.   11(a) corresponds to the SS case (κ = 0.0) and Fig. 11(b) to the ES case (κ = 0.05). R 1n is represented versus the speed parameter K 1n computed as per Eqs. 5 and 10 for n = 2. The same evolution is identified for the 275 first and third modes and only the case of the torsion mode is presented for the sake of brevity. The presence of damping has two main effects: (i) the modal amplitudes reduce, especially in the vicinity of the local maxima; (ii) the plate vibrations are not completely cancelled at the cancellation speeds. Nevertheless, and for the low levels of damping expected in railway bridges, cancellation phenomena will still be noticeable and the non-dimensional speeds for either cancellation or maximum free vibration should be adequately 280 predicted with the theoretical values provided in Table 1.

305
In 2016 the authors carried out an experimental campaign with the purpose of characterizing the structure and soil dynamic properties along with the bridge dynamic response under railway traffic. During the campaign, the vertical acceleration of the bridge deck was measured at 11 points of the lower flange lower horizontal face of the pre-stressed concrete girders (points 1 -11 in Fig. 14), and at the pile foundation upper horizontal surface, close to the central girder support (point 12 in Fig. 14). First, the modal parameters of 310 the bridge were identified from ambient vibration data by state-space models using MACEC software [32]. For the purpose of this study, the structural response under only two of these trains is presented as in these cases the travelling velocities were in the vicinity of theoretical resonant speeds. These trains are RENFE Class 103 (ICE 3 or S103) circulating at 279 km/h and RENFE Class 130 (Talgo 250 or S130) crossing the 320 bridge at 247 km/h. In Table 5 the theoretical resonant velocities associated to the first and second modes of the bridge deck (first longitudinal bending and first torsion modes) and to the length of the passengers' cars are included for the first three resonance orders. For the S103 and S130 trains these lengths are 24.8 and 13.1 m, respectively.
In Table 5 two resonant speeds are highlighted: the third resonance of the fundamental mode induced 325 by the S103 train and the second resonance of the first torsion mode induced by the S130 train. It can be  Table 5: Arroyo Bracea bridge first theoretical resonance speeds associated to S103 and S130 passengers' cars lengths.
observed that these two theoretical critical velocities are in the vicinity of the actual circulating velocities of the trains and, therefore, could lead to a certain structural amplification.
In what follows the experimental structural response is shown for these two convoys. In Fig. 15 the bridge vertical acceleration is represented at sensor number 5 located at mid-span below girder #1 for the  It can be affirmed that both trains induce resonance on the bridge to a certain extent. Nevertheless, the amplification associated to the circulation of train S103 is much more evident, especially in the frequency domain, than the one associated to train S130, even though the former is a third resonance (one loaded and two empty cycles of oscillation between consecutive loads) and the latter is a second resonance (one loaded and one empty cycle of oscillation between consecutive loads). In the view of the authors this difference in amplitude could be related, at least partially, with the level of free vibrations that every axle load leaves on the structure at the particular velocity of circulation in each mode. Eq. 16a shows how the L/d k ratio for the S103 train is close to a condition of maximum resonance for the third resonance of modes with m = 1 (see Table 2). On the other hand, Eq. 16b shows that the L/d k ratio for the S130 train is close to a cancellation condition for second resonances of also modes with m=1.  Figure 15: Vertical acceleration measured in Arroyo Bracea bridge under S103 and S130 trains.
The amplitude of the free vibrations associated to a particular speed, and the effect of it on the character of a resonant response is evidently a basic phenomenon. Other factors will affect the resonant response of the 345 bridge such as those related to vehicle-track-structure interaction or soil-structure interaction. Nevertheless, being able to predict the basic geometrical effects of the loads at resonance will permit a much better understanding of more complex coupling mechanisms in the vibrational response of railway bridges.

Conclusions
The free vibration response of simply-supported and elastically-supported orthotropic rectangular plates 350 traversed by a single load moving at constant speed is formulated in this work. Cancellation and maximum free vibration conditions alternatively take place as the speed of the load increases. Analytical non-dimensional speeds associated to these situations are proposed for a generic straight plate with simple or elastic supports.
When the plate is traversed by a sequence of equidistant loads travelling at constant speed parallel to 355 the plate free edges resonance may be induced on the former. The main application of this study is the dynamic response of railway bridges with simply-supported orthotropic decks, due to its relation with the Serviceability Limit State of vertical acceleration, which is nowadays one of the most strict requirements for structures located in High-Speed lines. When a resonant speed coincides or is close to a maximum free vibration condition, a prominent resonant peak in the response of the deck, should be expected. On the other hand, when it is close to a cancellation condition, the resonant response will be mild or may not even be perceptible for the particular mode. By equating maximum free vibration and cancellation conditions to the resonance ones, bridge length-to-characteristic distance ratios are obtained and presented for maximum or cancelled resonances for SS and ES rectangular orthotropic plates. Why these conditions coincide with those of a simple beam, in the SS case, and are well approximated by those of an ES beam, in the ES case • The first transverse bending mode, generally the third in frequency order in the bridge typologies under study, is more affected by the plate obliquity and incurred errors approach 20% for the highest obliquity levels and most flexible bridges considered.

380
• In the presence of damping the modal amplitudes reduce, specially in the vicinity of the local maxima, and the plate vibrations are not completely cancelled at the cancellation speeds. Nevertheless, and for the low levels of damping expected in railway bridges, cancellation phenomena will still be noticeable and the non-dimensional speeds for either cancellation or maximum free vibration should be adequately predicted with the analytical values provided.

385
Finally the experimental response of an existing bridge under HS railway traffic is presented. Two trains induce resonance of two different modes associated to the length of the passengers' cars. The level of free vibrations is evaluated and discussed in relation to the type of response in each case.
The amplitude of the free vibrations associated to a particular speed, and the effect of it on the character of a resonant response is evidently a basic phenomenon. Other factors will affect the resonant response of the 390 bridge such as those related to vehicle-track-structure interaction or soil-structure interaction. Nevertheless, being able to predict the basic geometrical effects of the loads at resonance will permit a much better understanding of more complex coupling mechanisms in the vibrational response of railway bridges. Appendix A.