The simplest mechanical model assigns all properties of a bridge structure to one simply-supported bending beam (Bernoulli-Euler beam) and idealizes the dynamic excitation from a passing high-speed train as a series of constant static axle loads moving over it (Moving Load Model). However, this model is highly conservative and often generates large calculated vibration responses, which in many cases significantly exceed the vibrations measured on real structures (e.g., in [8,9,10]).

More complex models considering the interaction dynamics between the train masses, superstructure, and supporting structure by implementing multi-body modeling of all elements allow for a more accurate computational prediction of structural vibrations while requiring a higher number of structural or train parameters in the modeling. With this, precise knowledge of coupling properties regarding the train’s suspension stages plays a central role as they significantly influence the calculated vibrations.

A large part of the current research activities on railway bridge dynamics, for instance, described in [4,6,10,11,12,13,14,15], deals with the computational vibration reduction that can be achieved by modeling the train more accurately as a multi-body system and, therefore, including interaction dynamics between the vehicle and bridge components. In practice, however, it is often complicated, if not impossible, to obtain specific information on the technical characteristics of, e.g., the trains’ wheelset and bogie suspensions since they are often kept secret due to economic interests by the vehicle manufacturers. The future introduction and admission of new high-speed trains to the existing European railway network will further aggravate this problem.

Deepening the modeling complexity of the considered bridge structures offers another approach to more realistic calculation results, whereby many different modeling alternatives with different requirements for input parameters are applicable, amongst others described in [16,17,18,19,20]. The present contribution focuses on the influence of modeling the bridge structures as coupling beams, i.e., by considering them as two vertically coupled beams representing the track (rails and sleepers) and the supporting structure. Both beams are interconnected vertically by continuously distributed Kelvin-Voigt elements, whose stiffness and damping properties reflect those of the ballasted superstructure. The equation of motion of the proposed model (explained in more detail in equations (A1-A34)) is approximated using trigonometric shape functions and solved by numerical time-step integration, whereby the system can be dynamically excited by moving load models as well as multi-body models of the train.

This bridge model has the advantage of requiring only a few additional input parameters compared to the reference model of the Bernoulli-Euler beam and still being computationally highly efficient. At the same time, it allows taking the load-distributing influence of the ballast bed as well as the energy dissipation mechanisms depending on the relative vertical movements of the track grid and bridge structure into account.

However, identifying and implementing appropriate and realistic model-related dynamic stiffness and damping parameters represents a further problem. According to the current state of research, no closed and mechanically and physically justifiable unit of calculation model and related parameters can be given. Literature references that focus on the effects of track-bridge interaction (e.g., in [16,18,19,20,21,22,23,24,25,26,27,28,29,30,31]) apply various models with different levels of complexity to idealize the many interconnected elements of the supporting structure (usually modeled as Bernoulli-Euler beam) and the components of the superstructure (rails, sleepers, and ballast bed). The idealized dynamic bridge system can consist of two layers, as in the numerical investigations described in the present contribution in Section 2, but also three or more; see schematically displayed in Figure 1 and, for instance, described in [16,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. The more layers and elements are included, the more information on the dynamic behavior and interaction of the respective elements is required, which poses a challenge, particularly regarding the complex subject of dynamic ballast bed behavior.

The applied coupling properties k_i and c_i between rails and subsoil (bridge structure) in various literature show a significant spread by a factor of 24, respectively 77 for damping or stiffness coefficients representing the interconnection of the two-layered structural model (literature data are, for example, compiled in [20]). In addition, the origin of those specific coefficients is rarely documented comprehensibly in the corresponding literature, making a plausibility check more difficult.

The influence of considering the interaction effects of the track, vehicle, and supporting structure has been investigated by several research activities in the past [4,6,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. The impact of taking vehicle-bridge interaction into account by multi-body modeling of the passing trains has shown to be dependent on many different characteristics of considered trains and bridges, mainly the bridge structure’s span, fundamental frequencies, and mass distribution, in relation to the axle distances, masses, and properties of the suspension system of the train cars (amongst others described in [10,12,15]). Research indicates that the vibrations are reduced particularly effectively in the case of resonance excitation of the bridges due to the regularly spaced train axles.

Besides explicitly modeling the vehicle and bridge structure as multi-body systems, it is also possible to consider the beneficial effects of the interaction effects by simplified methods, which is partly specified in the current standards to be applied, the Eurocode EN 1991-2:2003/AC:2010 [32]. The vibration-reducing effects from considering vehicle-bridge interaction, which primarily takes place in the primary suspension stage (the coupling of the unsprung wheelset masses and the bogies), can currently be considered by the Additional Damping Method as described in EN 1991-2 [32] chapter 8.7. An additional damping ratio representing the vibration-reducing effects of vehicle-bridge interaction in calculations without explicitly considering those effects (applying the Moving Load Model) may be assigned to the examined bridge structures. The amount of this additional damping is defined in the current specification of the EN 1991-2 [32] as only dependent on the bridge span. However, several recent research activities have revealed considerable discrepancies between the acceleration results calculated with more sophisticated multi-body models of different high-speed trains and the alternative application of the Additional Damping Method [10,11,12,13]. In many cases, the latter leads to a substantial underestimation of vehicle-bridge interaction and, therefore, uneconomical results, while in other cases, the impact is overestimated, producing unsafe results. These investigations indicate the complexity of the topic and its dependency on multiple influencing factors regarding bridge and train properties. As a result, the redesigned EN 1991-2, currently in the drafting process and available as prEN 1991-2:2021 [33], will no longer contain recommendations for additional damping. The ÖBB guideline RW 08.01.04, Appendix 1 [34], to be applied for the dynamic calculation of railway bridges in the Austrian rail network, allows implementing additional damping for train passages of the Railjet in dependency of the bridge's fundamental frequencies and distinguished by bridge type (concrete and steel structures). It was derived in [14] through numerical investigations on a parametric field of bridges.

The load-distributing effect of the ballast bed can be implemented in calculations on simple beam models of the bridge by distributing the moving axle loads, for example, by applying the normatively specified distribution pattern of 25%-50%-25% in the distance of the sleepers according to EN 1991-2 [32]. The standard recommends this distribution for calculations on bridge structures with small spans below L = 10 m, for which especially pronounced reductions of resulting accelerations can be expected. However, since the actual stiffness properties of the ballast bed under dynamic loading are not comprehensively and experimentally researched, the question of whether alternative distributions might consider the ballast bed behavior more accurately cannot be resolved at the current state.

Both simplified methods according to EN 1991-2 [32] of considering the interaction effects, the Additional Damping Method, and the load distribution pattern of 25%-50%-25% were based on numerical investigations performed by the European Rail Research Institute (ERRI) in 1999 [17], which had at the time limited computational capacities and, therefore, a limited range of investigations and multiple simplifications regarding the considered train and bridge modeling and parameters.

The numerical study on a broad range of bridge parameters described in the following sections aims to extensively investigate the influence of expanding the simplest reference modeling of bridge structures and trains utilizing the Bernoulli-Euler beam and the Moving Load Model by either including track-bridge interaction with a two-layered bridge model and/or vehicle-bridge interaction with multi-body modeling of the train. Section 2 contains a more detailed summary of the fundamental mechanical modeling and the variation of parameters of bridge structures and trains. The evaluation of results obtained using different modeling alternatives enables identifying the structural properties for which applying a more sophisticated coupling beam model of the bridge structure significantly influences the maximum accelerations, and quantifying that influence. Additionally, it is examined to what extent the influence of the coupling beam modeling depends on the chosen train model.

The analyses demonstrate the potential for obtaining lower acceleration results by considering the load-distributing impact of the ballasted superstructure in coupling beam models (facilitating verification of compliance with normative acceleration limits in EN 1990/A2 [1]). They indicate that the influence of multi-body models of the train, which consider vehicle-bridge interaction, is particularly pronounced for different structures than that of the coupling beam modeling.

These findings open up the possibility of formulating structure-dependent recommendations concerning the targeted application of more complex modeling of the structure (coupling beam model) on the one hand and train (multi-body model) on the other. Thus, they enable a realistic calculational prediction of the structural vibrations independent of vehicle information for the greatest possible share of structures.

As previously mentioned, the reference model of the bridge used in the following investigations is a simply-supported single-span shear-rigid bending beam, referred to as the Bernoulli-Euler beam (see Figure 2a at top). It is characterized by only four parameters: the span L [m], bending stiffness EI [Nm²;], mass per unit length μ [kg/m], and Lehr's damping ratio ζ [%]. Its fundamental frequency n₀, which is varied in the parameter study described in the following sections, can be recalculated from these four fundamental parameters:

Only lateral deformations in the vertical direction (bending deformations w(x,t)) are considered; longitudinal, lateral, and torsional vibrations are omitted.

The equation of motion (see equation (A1) and further) can be derived using modal analysis, utilizing the first three modal functions (i = 1, 2, 3) of the form φ_i = sin (iπx / L) as shape functions for the discretization of the bending deformation, as they are expected to govern the maximum vibration response, as, for instance, the investigations in [5,35,36] indicate, and to be sufficient for realistic representation in most cases. The damping matrix of the system is obtained by applying Rayleigh damping with ζ₁ = ζ₂ = ζ. Further information on the derivation of the system of equations of motion can be found, among others, in [7].

The multi-body modeling of the bridge structure (Figure 2a bottom) consists of two vertically coupled beams, the lower one representing the supporting structure (with the properties L_s, E_sI_s, μ_s, and ζ_s) and the upper beam the rails (with the properties L_r, E_rI_r, μ_r, and ζ_r). Both beams are vertically coupled by Kelvin-Voigt elements assuming linear spring stiffness $k_{ba}$ and viscous damping $c_{ba}$. As mentioned in Section 1, these coupling coefficients holistically represent the properties of the ballast bed, including the contributions of the ballast itself, sub-ballast mats, elastic sleeper sole pads, sleeper fasteners, and rail pads.

The beam representing the rails is being prolonged beyond the bearings of the beam representing the supporting structure into the embankment over a length L_emb = (L_r – L_s)/2. This embankment area, where the rail beam is coupled to a rigid subsoil, prevents a sudden impuls excitation of the supporting structure initiated by excessive deformations close to the rail beam bearings that can occur due to the sudden change of subsoil stiffness.

In order to make calculations with the different bridge models comparable, the properties of the overall bridge assigned to the Bernoulli-Euler beam have to be allocated to both beams of the coupling beam model. The mass distribution corresponding to the mass contribution of the ballast bed is allocated to the supporting structure beam μ_s, while the discrete sleeper masses are smeared and allocated to the rail beam μ_r. The bending stiffness of the rail beam is derived from the properties of the two rails (assumed 60E1 profiles), and its damping ratio is assumed to be zero (ζ_r = 0).

The mathematical discretization of this dynamical model for its numerical solution can be performed using, for example, finite bar elements with Hermitian shape functions of the third or fourth order, as described, among others, in [27]. However, this approach has the disadvantage of requiring a high number of bar elements to simulate the beam vibrations accurately, especially in cases where the bending stiffness of the rail beams is substantially lower than that of the supporting structure. The subsequent high number of degrees of freedom results in high demands for computational capacities. In contrast, discretizing the system by applying trigonometric shape functions of both beams leads to calculation results that converge at comparably low degrees of freedom associated with short computation times. The latter approach is extensively described, for example, in [37]. Since the trigonometric shape functions represent the modal functions in the case of two uncoupled beams, this approach can be degenerated to represent the Bernoulli-Euler beam if the input parameters are set accordingly. In the following evaluations, the number of shape functions N_s for the supporting structure beam is kept at three (i_s = 1, 2, 3) as in the calculations for the uncoupled Bernoulli-Euler beam, while the number of shape functions for the rail beam i_r = 1, 2, …, N_r is dependent on the required accuracy of the rail beam vibrations. Regarding the influence of the number of shape functions N_r on the accuracy of the predicted vibrations of the supporting structure, previous investigations and the analyses in [37] concluded that those converge if a minimum natural frequency n_r,lim of the rail beam is considered. This leads to a minimum number of considered shape functions N_r of the rail beam according to the following limitations in dependency on the uncoupled rail beams fundamental frequency n_0,r: where $n_{N_r,r}$ can be written as

With equation (2), this leads to with $n_{r,lim}$ = 200 Hz (chosen according to preliminary investigations).

The system of equations of motions is subsequently derived by applying the trigonometric shape functions to approximation methods according to Ritz and Galerkin (analogously to the approach for the Bernoulli-Euler beam) and including coupling terms in the stiffness and damping matrices of the system. In order to implement the different beam lengths into the coupling terms, it is necessary to assume discrete coupling of subsoil and rail at a limited number of points along the rail's longitudinal x-axis with a distance of e and transform the continuous coupling properties ${\overline{k}}_{ba}$ and ${\overline{c}}_{ba}$ to $k_{ba}={\overline{k}}_{ba}/e$ and $c_{ba}={\overline{c}}_{ba}/e$. In the following calculations, a continuous coupling is simulated by adjusting this distance very shortly to e = 0.01 m. Another possibility would be to use the actual sleeper distance of e = 0.6 m, which, according to preliminary investigations, has a negligible influence on the results of medium to long-span bridges. In very short-span bridges, however, it is possible to obtain slightly different results depending on the number and location of the sleepers on the bridge structure if the ratio of bridge span to sleeper distance becomes too small. Thus, the sleeper distance is set to a minimal value of e = 0.01 m for better comparability.

Both bridge models require the implementation of an external force vector as excitation, which can be obtained by applying different train models.

The Moving Load Model (MLM) is the first train model utilized in the following investigations and is the most simple one applicable (see Figure 2b top). It idealizes the train as a series of its static axle loads F_stat,j moving with constant speed v and distances over the bridge model. It can be easily implemented as a generalized force vector in the equation of motion of the bridge by performing a modal transformation with the shape functions φ_i of either the total bridge (Bernoulli-Euler beam i = 1, 2, 3) or the rail (coupling beam i = i_r = 1, 2, …, N_r) evaluated at the respective contact points of each load.

This train model considers no interaction effects between the train and the subsoil, respectively, bridge structure. In order to evaluate the impact of these interactions, it is necessary to model the train as a system of coupled masses, for instance, by applying the Detailed Interacion Model (DIM) consisting of the interconnected rigid masses of the car bodies, bogies, and wheelsets (see Figure 2b bottom). The coupling properties reflect the train's primary and secondary suspensions. Implementing this kind of train model into the dynamic equations requires extending the system of equations of motion of the Bernoulli-Euler beam or the coupled beam by additional degrees of freedom associated with the train masses, particularly the vertical displacements of car bodies, bogies, and wheelsets as well as the car bodies and bogies rotation about the transverse horizontal axis through their center of gravity. The wheelset displacements are equated with the bridge deflection, respectively rail, at the point of contact, assuming rigid and continuous contact of wheels and rails at all times. This forms the coupling condition of the train and the bridge structure and reduces the vehicle's degrees of freedom by the displacements of the wheelsets.

The composition of the system matrices necessary to implement the different mechanical models into dynamic calculation programs is summarized in equations (A1-A34) in the appendix and was derived by modifying the approach described in [27].

The equations of motion can be solved for the rail and bridge structure beam accelerations, velocities, and deflections by applying numerical time step integration methods for a specific train speed or a range of speeds. All the investigations described in this article were performed using MATLAB [38] scripts, applying the implemented differential equation solver for stiff problems (MATLAB ode15s, see [39]) based on the numerical differentiation formulas (NDFs).

In order to draw valid conclusions regarding the effects of considering track-bridge or vehicle-bridge interactions using the previously described model alternatives on a spectrum of single-span bridge structures representative of reality, calculations were conducted on a broad parameter field of bridges. The varied input parameters were the bridge's span L, mass distribution μ, Lehr's damping ratio ζ, and fundamental frequency n₀ according to equation (1). The limits of the parametric field were obtained by analyzing the characteristics of 275 existing single-span and single-track bridges in the European rail network. Information on their properties was derived from several catalogs of existing bridges, primarily located in Austria, and documented during previous research activities. Additionally, the information on 74 bridges was obtained from the ERRI (European Rail Research Institute) reports RP3 [40] and RP8 [41], on 35 bridges from Frýba [3], and on 25 bridges from Rauert [42], provided these bridges also meet the conditions of being constructed as single-span, single-track structures with ballasted beds.

The bridges are distinguished by their construction type, with 101 steel and seven composite structures being grouped in the same category as they have similar properties regarding mass distribution and span lengths. The same applies to 99 concrete and 68 filler beam structures. Measured values regarding the fundamental frequencies are available for 104 steel and composite structures and 114 concrete and filler beam structures. For the remaining 57 bridges, fundamental frequencies were recalculated by applying equation (1) and estimating the bending stiffness EI from the documented cross-sections. Furthermore, not all bridge structures have reliable information on their mass distribution available. Consequently, only the 45 steel and composite structures and 136 concrete and filler beam structures with recorded mass distributions are taken into account for determining this parameter bandwidth. The combinations of span, fundamental frequencies, and mass distribution are graphically represented in Figure 3 and Figure 4.

Figure 3 and Figure 4 also contain the parameter combinations used in the numerical study (marked by small grey, respectively blue and green circles). They are obtained by applying regression functions as upper and lower bound limits, which were derived from analyzing the property combinations of the existing bridges by Glatz et al. in [14,43]. The regression functions for the upper and lower bound limits of the span-dependent fundamental frequencies n_0,ub(L) and n_0,lb(L) have been determined in [14] as power functions as follows:

The combinations of parameters for the numerical study were obtained by evaluating the equations (5) and (6) for spans L from 4 to 40 m in 1-meter increments and choosing five equidistant fundamental frequencies ranging from n_0,ub(L) to n_0,lb(L). The linear regression functions for the span-dependent mass distributions were differentiated according to the masses of the heavier concrete and filler beam and the lighter steel and composite structures. Three mass distributions per construction type (concrete and filler beam: μ₁ to μ₃ steel and composite: μ₄ to μ₆) were chosen following equation (7) and the input parameters a and b from Table 1.

The Lehr's damping ratio of the bridges examined was chosen according to the standard EN 1991-2 [32], which must be currently applied. The given span-dependent values also differentiate between reinforced concrete and filler beam structures and steel and composite structures:

These values are designed as lower and conservative limits of the structural damping and often underestimate the actual damping ratios measured in-situ at structures, as, for instance, observed by Reiterer et al. in [8,9]. It is to be expected that future research on the complex discrepancy between measured and normatively prescribed damping ratios will ultimately lead to a revision of the related standards. To investigate whether the influence of vehicle-bridge and track-bridge interaction is independent of the damping ratio used in calculations, the numerical study was also conducted with the normatively prescribed damping value increased by the factors 1.5 and 2 (ζ₁ = ζ_EC; ζ₂ = 1.5 ζ_EC; ζ₃ = 2.0 ζ_EC).

Additionally, applying the coupling beam requires the definition of coupling properties reflecting the ballast bed under dynamic excitation. The approach to modeling the coupling in this article is based on combining all the coupling properties of the ballasted superstructure, i.e., all the individual elements such as the rail fasteners, intermediate layers, sleepers, sleeper sole pads, the sub-ballast mats, and the ballast bed itself, into a single coupling element with linear stiffness and viscous damping characteristics. The literature references dealing with this modeling show a significant scattering of both coefficients, the ballast bed stiffness ${\overline{k}}_{ba}$ and damping ${\overline{c}}_{ba}$ per unit length. The scattering of literature values was, among others, documented by Stollwitzer in [20], see Figure 5, which also displays the literature values for three- and four-layer models, including the stiffness and damping properties of the rail fasteners and pads (index rp – rail pads) and/or of the sub-ballast mat (index sb).

Since experimentally determined and verified values are scarce, the impact of applying different stiffness and damping values is investigated by varying these properties as follows:

In total, 37 different bridge spans from 4 to 40 m with five fundamental frequencies each were considered in the following study. All calculations on the therefore 185 span and fundamental frequency combinations were performed for six different mass distributions and three different structural damping ratios, leading to 3 330 parameter combinations for the reference modeling alternative (V1-B1, see Section 2.5). The calculations with the coupling beam model excited by the MLM are performed for a total of five combinations of ${\overline{k}}_{ba}$ and ${\overline{c}}_{ba}$ (${\overline{k}}_{ba,1}$ + ${\overline{c}}_{ba,2}$, ${\overline{k}}_{ba,2}$ + ${\overline{c}}_{ba,2}$, ${\overline{k}}_{ba,3}$ + ${\overline{c}}_{ba,2}$; ${\overline{k}}_{ba,2}$ + ${\overline{c}}_{ba,1}$, and ${\overline{k}}_{ba,2}$ + ${\overline{c}}_{ba,3}$), thus, in total, 16 650 parameter combinations are investigated. Preliminary results showed only a marginal influence of varying the ballast damping coefficient ${\overline{c}}_{ba}$ and the structural damping ζ on the effects of applying the coupling beam model (which will be confirmed by the numerical study, see Section 3.2.3). In order to limit the computational expenses for the most demanding model (coupling beam model and DIM), only the variation of ${\overline{k}}_{ba}$ is investigated with this modeling alternative (${\overline{k}}_{ba,1}$, ${\overline{k}}_{ba,2}$, and ${\overline{k}}_{ba,3}$ in combination with ${\overline{c}}_{ba,2}$ and ζ₁ = ζ_EC → 3 330 parameter combinations).

Not all combinations of bridge parameters are equally likely to occur in reality. For instance, steel constructions are usually applied for single-span bridges with higher spans and are simultaneously characterized by a low mass distribution. In order to draw reliable conclusions on the practical benefit of the different modeling alternatives, the results on the parametric field were evaluated while taking the statistical probability of each combination into account. This probability was determined as a multivariate normal probability density function, which can be calculated utilizing pre-implemented functions in MATLAB [38] based on the theoretical description by Kotz et al. in [44]. With this, the documented properties regarding bridge span L, fundamental frequency n₀, and the mass distribution μ served as input parameters, whereby the reciprocal of the span was used to reflect the non-linear relation of span and fundamental frequency. Further, the structural types of concrete and filler-beam structures and steel and composite structures were distinguished by performing two separate analyses with the corresponding bridge catalogs.

The matrices of mean values $\overline{m}$ (not to be confused with the mass distribution μ), and covariance Σ are as follows:

Concrete and filler beam structures:

From the first component in $\overline{m}$ follows $\overline{L}=$10.654 m.

Steel and composite structures:

From the first component in $\overline{m}$ follows $\overline{L}=$15.261 m.

This information can be utilized to calculate a probability value for each combination of span, mass distribution, and fundamental frequency of the parametric field. A graphical representation of these probabilities is displayed in Figure 6:

It can be observed in Figure 6 that, based on the information on existing bridges available to the authors, concrete and filler beam structures tend to have a high or medium mass distribution (μ₁ or μ₂), spans between L = 6 and 25 m, and corresponding fundamental frequencies of n₀ = 4 to 18 Hz. Steel and composite structures usually have a lighter mass distribution (mainly μ₅), longer spans of L = 9 to 40 m, and subsequently lower fundamental frequencies of n₀ = 2 to 14 Hz. Hence, the influence of different modeling alternatives on these most probable combinations of structural parameters will be reviewed in more detail in the following sections.

The investigated train in this paper is a locomotive-hauled Austrian Railjet in the standard configuration, which consists of seven regular passenger cars (index pc) and one front locomotive (index loc) with higher axle loads and differing axle distances. In reality, the individual passenger cars feature slightly different characteristics due to their function (i.e., the onboard restaurant) and their position in the configuration. These deviations are considered negligible in the present investigations. Thus, all seven cars are modeled with the parameters of a standard middle car. The train parameters are taken from [4] and are summarized in Table 2, using the same abbreviations as in Figure 2b bottom:

All cars are vertically decoupled from each other, which means that displacements or rotations of individual elements of one car do not affect the adjacent cars. The centers of gravity of all car bodies are assumed to be precisely in the midpoint of the bogie distances, and the centers of gravity of the bogies in the midpoint of the wheelset distances.

In order to evaluate the effect of considering vehicle-bridge or track bridge interaction, the accelerations ẅ_max occurring at the structure (either at the Bernoulli-Euler beam or the structural beam of the coupling beam model) are compared with each other in calculations with the different modeling alternatives. The ratio of the maximum accelerations in percent serves as a comparison value, whereby only maximum absolute values during the train crossings in the time domain at midspan are used. According to equation (13), negative values of η indicate that the maximum accelerations ẅ_reference obtained with a reference model are reduced by the alternative modeling to ẅ_model, while positive values of η indicate an increase in maximum accelerations.

Only peak accelerations are considered, i.e., only local maxima occurring in the defined range of train speeds for which the calculations are performed. Dynamic calculations result in exceptionally high structural vibrations, for instance, peak accelerations, whenever resonance occurs. The acceleration peaks at resonance (generally representing the maximum accelerations within a considered speed range) mainly occur at train speeds at which the regularly spaced train axles exert a periodic excitation close to a natural frequency n_i of the bridge structures. This means that if the train speed is near one resonance speed v_x,i,j according to equation (14), resonance vibrations are to be expected. with i = 1, 2, … , j = 1,2,… and x = d, r or b (see Table 2)

In this regard, the length over buffers d_pc of the passenger cars and the fundamental bridge frequency (i = 1) are of particular significance, which means that the highest computational acceleration results are obtained at v_dpc,1,1. The acceleration peak at this critical speed exceeds the accelerations at subordinate resonance scenarios without exception. The Railjet has a maximum operational speed of 230 km/h; however, the calculations for the numerical study were performed for a speed range from 100 to 420 km/h. Since the fundamental frequencies n₀ of the investigated single-span bridges are usually in the range from 2 to 40 Hz, and the passenger car length of the Railjet is d_pc = 26.5 m, the Railjet reaches only on a few bridges the critical speed v_dpc,1,1 within a realistic speed range, for example for the lowest considered fundamental frequency n₀ = 2 Hz at 190 km/h. For most bridge structures, v_dpc,1,1 lies well beyond the operational speed maximum. Subordinate critical speeds at j > 2 most likely produce the critical resonance scenario with the highest acceleration peaks for these bridge structures. The examined speed range was extended to 420 km/h to investigate the effects of considering track-bridge interaction for a representative number of bridge structures at the resonance speed v_dpc,1,1. Thus, all bridge structures with fundamental frequencies below n₀ = 4.4 Hz can reach this particular resonance scenario (48 combinations of span and natural frequency, with spans between 19 and 40 m).

It should be noted that the maximum accelerations occur at train speeds slightly deviating from v_x,i,j when vehicle-bridge or track-bridge interaction effects are considered. In the case of vehicle-track interaction, the train axle loads rigidly connected to the structure influence their modal mass matrix, which leads to a slight reduction of the natural frequencies and, therefore, the critical speeds at which resonance occurs. In the case of track-bridge interaction, the coupling between the rail beam and the structure beam increases the modal stiffness of the overall bridge, leading to a slight increase in the natural frequencies. Both effects are particularly pronounced in bridge structures with very low mass.

Additionally, another scenario can suppress resonance acceleration peaks: the cancellation effect of bridge vibrations at a specific cancellation speed according to equation (15). with i = 1, 2, … , j = 1,2, …

Resonance effects that might occur at critical speeds v_x,i,j that are close to one only span- and frequency-dependent cancellation speed v_canc,i,j are generally less pronounced or might get suppressed entirely.

All calculations are performed with four different model alternatives, which are referred to as follows:

The reference calculations with model V1-B1 are performed for the entire speed range in 1 km/h steps. The results are used to identify the highest local acceleration maxima in the chosen speed range, whereby up to five acceleration peaks are considered. The calculations with the other three modeling alternatives considering the interaction effects are performed only for the identified critical speeds with maximum accelerations and a limited speed range around them. This approach helps to significantly reduce the calculation effort for the computationally demanding modeling alternatives V2-B1, V1-B2, and V2-B2. An example can be seen in Figure 7. Acceleration peaks that occur at the boundaries of the overall speed range (close to 100 km/h or 420 km/h) are only included in the following evaluations if all modeling alternatives fully capture them. This prevents distorting the comparison results by shifts of peaks beyond the examination scope due to the increase or reduction of critical speed at which they occur due, as mentioned before, to the modal stiffness or mass increment of the bridges.

Figure 7 illustrates the speed-dependent acceleration results obtained with the four different modeling alternatives for one exemplary bridge structure representing a light concrete structure (μ₃), which is in terms of mass distribution very close to a heavy steel structure (μ₄, see Figure 4). The Lehr's damping ratio is chosen according to the normative prescriptions in equation (8). The ballast bed stiffness is ${\overline{k}}_{ba}$ = ${\overline{k}}_{ba,2}$ = 100 000 kN/m²; and the damping ${\overline{c}}_{ba}$ = ${\overline{c}}_{ba,2}$ = 60 kNs/m²;.

In Figure 7 it can be observed that there are three pronounced acceleration peaks in the examined speed range that occur close to v_dpc,1,4 = 410 km/h, between v_dpc,1,8 = 205 km/h and v_dpc,1,7 = 234 km/h and close to v_dpc,1,11 = 149 km/h. Different acceleration peaks become relevant or decisive depending on which speed range is investigated in bridge verifications under dynamic loading. This is of interest, for example, if lower operational train speeds are prescribed in certain railway sections than the maximum train speed, which may result from other routing conditions (curves, distances to railway stations, etc.). Applying the simplest structural and vehicle model, the Bernoulli-Euler beam and the MLM (V1-B1), leads to the highest acceleration results in all cases. By considering vehicle-bridge interaction (V2-B1), it is possible to obtain lower acceleration results, reduced by η = -2.4% to -12.9%, depending on the respective critical speed at which the acceleration peaks occur. Applying the coupling beam model and thus considering track-bridge interaction (V1-B2) leads to a slightly lower reduction of structural accelerations for the peak close to v_dpc,1,4, but a significantly higher reduction of the acceleration peaks at lower critical speeds. The acceleration peak at v_dpc,1,8 is damped by η = -55.9%. The acceleration peak at the even lower resonance speed v_dpc,1,11 is strongly suppressed by applying the coupling beam model so that it no longer appears as a local acceleration maximum in the speed-acceleration curve. If the acceleration at exactly the same speed at which the maximum acceleration peak occurs with the reference model is used for calculating the acceleration reduction η, then it yields a value of η = -80.6%.

Applying the DIM in combination with the coupling beam model (V2-B2) generally yields the lowest acceleration results. The acceleration peak at v_dpc,1,8 is damped to such an extent that the peak at the next higher resonance speed v_dpc,1,7 now slightly exceeds that at v_dpc,1,8 (η = -62.5% compared to the reference acceleration at v_dpc,1,8). With this model alternative, the difference compared to the results of considering only the track-bridge interaction (MLM and coupling beam model V1-B2) is relatively small and of the same order as applying vehicle-bridge interaction to calculations with the Bernoulli-Euler beam. This indicates that applying vehicle-bridge interaction might be relatively independent of the applied structural model. The specific results are summarized in Table 3.

All parameter combinations described in Section 2.3 were investigated analogously to the exemplary bridge, and the results are mainly presented graphically in the following sections.

All four computational models were applied to a parameter field of bridge structures defined by the span (L), mass distribution (μ), fundamental bending frequency (n₀), and structural damping (ζ) of the bridges. A statistical analysis was performed on the probability of occurrence of all combinations of span, mass distribution, and fundamental bending frequency within the parameter field based on measured or documented structural properties of 275 structures (see Figure 3 and Figure 4). This statistical analysis of the combinations of structural properties allows for a better assessment of the benefits of different modeling variants in terms of their practical relevance in dynamic analyses. The following combinations of structural parameters represent a particularly relevant framework for further investigations (see also Figure 6):

When analyzing the results for the specific exemplary structure (see also Figure 7, Figure 8 and Figure 9 and Table 3, Table 4, Table 5, Table 6 and Table 7), several tendencies emerge that are also confirmed when examining the results of the entire parameter field:

The results obtained from the analysis of the entire parameter field confirm the tendencies observed for the exemplary structure and provide the basis for deriving some fundamental conclusions. The evaluation focuses on examining the specific impact of the extended models (V2-B1, V1-B2, and V2-B2) on the acceleration peaks of the reference model (V1-B1), as well as determining the critical velocity v_lim, which represents the train speed at which the normative acceleration limit ẅ_max = 3.5 m/s²; is first exceeded.

The results of the parameter study highlight the significant potential of coupling beam modeling in determining lower computational accelerations compared to a simple beam model. This modeling approach requires only a few input parameters and can be universally applied, independent of the specific train model. As a result, it presents a good complement or alternative to multi-body modeling of the train that considers the vehicle-bridge interaction. This is particularly advantageous in situations where crucial input parameters for multi-body modeling of the train are not accessible. In this regard, considering the vehicle-bridge interaction as well as the track-structure interaction tends to have a particularly favorable effect on bridges with different structural parameters.

In principle, it is evident in the case of the coupling beam model that its impact on the maximum computational accelerations is influenced not only by the normalized train speed, which corresponds to the resonance scenario at which they occur, but also by the applied coupling or ballast bed stiffness. However, estimating these characteristic values solely based on existing literature sources is associated with inadequate certainty. Therefore, conducting a more comprehensive and experimentally underpinned investigation into the dynamic load-bearing behavior of the ballasted track is essential. Only through such efforts can the coupling beam model's reliability be firmly established, thus providing the necessary confidence for its widespread adoption in engineering practice.

When examining the calculation results underlying the regression functions used to describe the impact of coupling beam modeling, it is evident that some positive values are observed, indicating an increase in the maximum calculated acceleration peaks compared to the reference model. While this contradicts physical reality, it can be attributed to simplifications in modeling the coupling beam, such as the linear stiffness assumption of the coupling springs. Therefore, further investigation into these cases and the specific conditions in which they occur would be valuable for future research.

Additionally, the deviation observed in the regression functions describing the influence of the coupling beam for different vehicle models, particularly in the case of the lowest coupling stiffness, warrants more detailed investigation. It is plausible that with such low coupling stiffness, minimal or negligible acceleration peaks are formed, making a direct comparison with the acceleration peaks of the reference model less feasible and subsequently influencing the resulting regression functions.

However, the overall finding that the vehicle model does not significantly interact with the influence of the coupling beam model in other cases allows for simplifications in future investigations. It would no longer be necessary to extensively analyze different vehicle models to make universally applicable statements regarding the impact of considering track-bridge interaction through a two-layer coupling beam model, thereby significantly reducing the computational effort required for future numerical investigations. The same applies to the limited influence of coupling or ballast bed damping and structural damping as parameters with minimal impact, allowing them to be excluded.

The following equations summarize the composition of the system matrices for the dynamic calculation for the most complex applied train and bridge models (Detailed Interaction Model of the train and coupling beam model of the bridge). The modifications while applying the simpler models (Moving Load Model of the train and Bernoulli-Euler beam model of the bridge) are summarized in equation (A32 -A33).

The system of equations of motions is written as follows:

In equation (A1), all matrices – the mass matrix M, damping matrix C, stiffness matrix K, force vector p,and the displacement vector x and its time derivatives (velocities ẋ and accelerations ẍ) – are generally time-dependent and two-dimensional. The matrices are formed by combining several submatrices denoting the contributions of the train model (vehicle – index v), the rails (index r), and the supporting structure (index s), including their interactions. The derivation of the matrices was adopted as a modified version of the model described by Lou [27] and is further described in Weber [37].

The displacement vector x consists of the following subvectors: with a total number of degrees of freedom N = 6 N_v + N_r + N_s. where N_v is the number of train wagons, and each displacement vector x_vi consisting of six degrees of freedom assigned to car bodies (index c: displacement w_c and rotation φ_c) and bogies (index b: displacements w_b,1, w_b,2 and rotations φ_b,1 and φ_b,2): where N_r is the total number of considered rail modes, here limited by equation (2). where N_s is the total number of considered modes of the supporting structure, here limited to N_s = 3.

Just as the displacement vector, the mass, damping, and stiffness matrices are composed of submatrices, which then again consist of several contributions of the structural mass, damping, and stiffness properties as well as their interconnections, as follows:

The submatrices of the vehicle M_vv, C_vv, and K_vvcan be written as follows:

Each submatrix M_vicorresponding to the i^th train wagon is formed with the car body mass m_c, inertia I_c, and the bogies masses m_b,1, and m_b,2, and inertia I_b,1, and I_b,2:

The damping and stiffness matrices C_vi and K_vi of each train wagon contain coupling terms denoting the connection of car bodies and bogies with spring-damper elements in the primary (c_p and k_p) and secondary suspension stage (c_s and k_s). Also, the distances of wheelset axles to the bogies' center of gravity (L_b) and the distance of the secondary suspension to the car bodies' center of gravity (L_c) are considered in (A13).

The stiffness matrix of each wagon K_vi can be formed analogously by replacing c_p and k_p and c_s by k_s in (A13).

The submatrices of the rail M_r, C_r, and K_rare composed of several matrices containing the contributions of the rail itself, the wheelset masses m_w acting on them, and the coupling to the supporting structure:

Contribution of wheelsets to rail mass matrix: whereby x_k denotes the longitudinal distance of the k^th wheelset from the left rail end, and N_w is the total number of wheelsets N_w = 4 N_v.

Contribution of coupling to train through the primary suspension stage: where v is the train speed.

Contribution of coupling to the supporting structure beam: whereby x_p denotes the longitudinal distance of the p^th discrete coupling point from the left rail end.

Contribution of coupling to train through the primary suspension stage:

The contribution to the damping matrix of the rail denoting the coupling to the supporting structure beam (${\mathit{C}}_{\mathit{r}\mathit{r}}^3$) can be derived analogously to (A18) by replacing k_ba with c_ba.

The submatrices of the supporting structure M_s, C_s, and K_s are similarly obtained as the ones for the rail beam.

Contribution of coupling to the rail beam: whereby x_q denotes the longitudinal distance of the q^th discrete coupling point from the left bridge end.

Again, the contribution to the damping matrix of the supporting structure denoting the coupling to the rail beam (${\mathit{C}}_{\mathit{s}\mathit{s}}^2$) can be derived analogously to (A23) by replacing k_ba with c_ba.

Furthermore, the coupling between the vehicle and rail beam, as well as between the beams of the rail and the supporting structure, induce interaction submatrices C_vr, C_rv, K_vr, K_rv, C_rs, C_sr, K_rs, and K_sr.

The submatrices C_vir and C_rvi corresponding to the i^th train wagon can be written as follows: and the rotatory share of the k^th bogie

In (A26), the first two rows correspond to the degrees of freedom of the car bodies, which are not coupled to the rail. The rows associated with the translational and rotatory degrees of freedom of each car wagon (i) bogies (bk, k = 1, 2) consist of coupling terms related to both wheelsets w1 and w2 of each bogie, in which x_w1 and x_w2 denote the longitudinal distances of the wheelset to the left end of the rail beam.

The submatrices K_virandK_rviare obtained similarly to (A26) as follows: denoting the rotatory share of the k^th bogie.