In this paper, a dynamic interaction model of the coupled system of railway bridge, foundation, subsoil, ballast, track, and high-speed train is presented, with special emphasis on the longitudinal interaction between the track and the bridge structure, taking into account the flexibility of the ballast. After a description of the model of this interaction system, the equations of motion are given separately for each subsystem. The discretization of the bedded rails is performed by two different approaches. In the first approach, the deflection of the rails is expanded into the eigenfunctions of a finitely long bedded beam representing the rails. In the second, simplified approach, the track response is represented by a superposition of the static deflection of the infinitely long bedded beam due to a concentrated load. The coupling of the bridge structure with the track is achieved by a component mode synthesis technique, which in the first approach leads to a representation of the equations of motion in state-space. A discrete substructure technique is used to couple this subsystem with the train model. The two presented strategies are verified by comparison with results of a finite element model of this interaction system. Several application examples reveal the influence of the horizontal track–bridge interaction and other modeling parameters on the dynamic bridge response.
Hinweise
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
1 Introduction
The constantly growing demand for efficient transportation has led to a steady expansion of high-speed rail networks over the last two decades. This development makes dynamic evaluation necessary for more accurate prediction of the behavior of railway bridges. Excessive accelerations of the bridge deck in response to the passing train may lead to instability of the track ballast and consequently to misalignment of the rails, requiring shortening of maintenance intervals or reduction of train speeds. Since excessive vibrations in combination with misaligned rails can also affect the running safety of the train and, in the worst case, cause the train to derail, limits for the maximum acceleration of the bridge deck have been introduced in various design specifications. Predicting the dynamic response of railway bridges has therefore become an important task for the engineer when designing railway bridges.
Although the prediction of the dynamic response is of undisputed importance and a variety of modeling strategies have been developed to date, discrepancies between prediction and measurement are still observed [4, 20]. The reason for these discrepancies is the frequent uncertainties in mechanical modeling. While the first mechanical models summarized in the books of Frýba [8, 9] were already able to take into account the resonance excitation of the bridge structure caused by the regular axle spacing, the models often idealized the train as moving individual loads and also neglected the interaction with the substructure of the track as well as the subsoil under the foundation of the bridge. Early methods considering the dynamic interaction of bridge and train by modeling the train with so-called mass-spring-damper (MSD) systems can be found in Yang et al. [33], with simple two-dimensional and more sophisticated three-dimensional formulations. A more recent publication Stoura and Dimitrakopoulos [29] presented an approach to decoupling the vehicle-bridge interaction problem.
Anzeige
The large influence of the soil–structure interaction, frequently discussed by various authors [10, 23, 31, 34], represents an additional uncertainty factor that is often ignored by standard approaches. In an attempt to accurately represent the subsurface, discretizing the half-space with finite elements (FE) or boundary elements [10, 23, 31] often leads to a high computational cost. Therefore, recent contributions [12‐14] have developed simplified interaction models that idealize the subsoil beneath the foundation as spring-dashpot elements.
The mechanical models to represent the ballast superstructure in practical application mainly use the vertical interaction properties of the track. These models, which mostly focus on two-dimensional descriptions, vary greatly in their level of detail, ranging from a beam representing the rails connected to the ground or bridge by a single intermediate layer [3, 13, 14, 16, 21], to models with two intermediate layers consisting of springs and dashpots representing the rail pad and ballast between the rail beam and the ground or bridge. Track models that additionally represent the substructure as a viscoelastic layer can be also found in the literature [19, 35]. In addition to the vertical interaction, the consideration of the longitudinal interaction between the track and the bridge has been shown to have a significant impact on the dynamic behavior of the structure [2, 17, 22, 30], affecting both the damping and the natural frequencies of the coupled system. In these studies, the longitudinal stiffness of the ballast is usually modeled with linear-elastic or linear-elastic ideally plastic springs, often referring to the admissible horizontal stiffness and shear resistance specified in EN1991-2 [6]. While the assumption of linear or bi-linear material behavior is convenient for numerical approaches, recent experimental investigations [26, 28] provide important insights into the strictly nonlinear stiffness properties, which strongly depend on the displacement amplitude, as well as into the frequency-dependent damping mechanisms of the ballast.
With the aim to partially close the gap between modeling approaches with vertically oriented interaction models considering train, track, bridge and subsoil and modeling approaches considering longitudinal track–bridge interaction, this paper extends the vertically oriented models of König et al.[14] and König et al. [13] with longitudinal ballast interaction. Here, the involved train, track, and bridge-soil subsystems are first described individually, focusing on the longitudinal track–bridge interaction. The equations of motion of the track and bridge-soil subsystems are then coupled using component mode synthesis (CMS) [3, 14]. The paper focuses on the modal expansion of the viscoelastically bedded track based on König et al. [14]. Alternatively, a simplified but computationally more efficient approach based on a Rayleigh–Ritz approximation of the deflection of the stand-alone track subsystem proposed in König et al. [13] is also presented. By applying a discrete substructure technique (DST) to the resulting equations of the track–bridge system and the train MSD system, the full description of the coupled system is derived for both modeling approaches.
The verification of the proposed modal expansion-based approach (Approach 1) is achieved by comparing the time history results obtained with the presented method with the results of a corresponding FE model. On the basis of a numerical study, insights into the influence of track–bridge interactions in the longitudinal direction are provided. Furthermore, the second approach (Approach 2) based on the simplifying Rayleigh-Ritz approximation of the rail deformation is compared with Approach 1 and the applicability of both methods is demonstrated.
Anzeige
2 Mechanical model and fundamental equations
2.1 Modeling strategy
The planar mechanical model considered here, representing the dynamically interacting subsystems of the train moving at constant speed v, the track, and the single-span bridge resting on the subsoil, is shown in Fig. 1. The conventional train, consisting of \(N_\textrm{c}\) individual cars, each consisting of a car body, two bogies, and four axles, is represented by an MSD system.
Both the slender bridge structure and the rails are modeled as Euler-Bernoulli beams. The beams are assumed to be rigid in the axial direction, prohibiting longitudinal deformations. The origin \(x = 0\) of the axis coordinate x is located at the left end of the bridge beam, which has the length \(L_\textrm{b}\). Hinged supports are placed at both ends of the (in reality infinitely long) rail beam of length \(L_\textrm{r}\) for Approach 1. However, the length \(L_0\) of the beam section before and after the bridge structure must be chosen sufficiently large to prevent the influence of these “artificial” boundary conditions on the dynamic system response [14]. For the Rayleigh-Ritz approximation of the infinite rail beam used in Approach 2, these “artificial” boundary conditions are not required. Regardless of the modeling approach chosen, the influence of the train on the interacting system is considered only for the finite section of length \(L_\textrm{r}\) between points A and D.
The stiffness and damping of the ballast are accounted for by continuous vertical and horizontal viscoelastic bedding of the track. Experiments have shown that these assumptions are appropriate for approximating the bridge-track system response [17]. In addition, continuous vertical damping is added to the bridge beam to account for energy dissipation in the ballast due to the absolute motion of the bridge and track as proposed in Stollwitzer et al. [28].
The lumped mass at the two beam ends representing the bridge structure corresponds to the foundation mass and a proportional soil mass. The spring-damper elements below the lumped mass model the properties of the soil, which is in reality a half-space. The spring and damping parameters can be derived, for example, with the cone model of Wolf [32], requiring the soil density \(\rho _\textrm{s}\), the constrained modulus \(E_\textrm{s}\), the Poisson’s ratio of the soil \(\nu \) and foundation base area \(A_0\). Further details on the modeling strategy for the mechanical subsystems as well as a description of the involved model parameters are provided in the subsequent sections.
Fig. 1
Planar MSD model of the j-th vehicle crossing the track resting on the bridge with viscoelastic supports (modified from König et al. [14])
×
2.2 Track–bridge–soil subsystem
2.2.1 Ballast
The interaction between the rails and the bridge structure is primarily determined by the ballast bed. The ballast bed is flexible and has a major impact on the damping behavior of the system. In the model, these properties are represented by viscoelastic elements. Figure 2 shows a detail of the layerwise setup of the coupled bridge-track structure, the corresponding viscoelastic model of the ballast between bridge beam and rail beam, and a deformed element according to the subsequently described modeling assumptions. The vertical flexibility and damping properties of the ballast is accounted for by continuously distributed vertical spring-damper elements with parameters \(k_\textrm{z}\) and \(c_\textrm{z}\) between the bridge and the rail subsystem. With \(w_\textrm{b}(x,t)\) denoting the vertical displacement of the bridge and with the rail deflection \(w_\textrm{r}(x,t)\), the vertical deformation of the ballast layer \(\Delta w(x,t)\) becomes
The window function \(\Pi (x,0,L_\textrm{b}) = H(x) - H(x - L_\textrm{b})\), which is composed of the two Heaviside step functions H(x) and \(H(x - L_\textrm{b})\), defines the range of interaction between the track and the bridge of span length \(L_\textrm{b}\). The corresponding distributed vertical interaction force between the bridge beam and the rail beam
is depicted in Fig. 3, which shows the free-body diagram of the rail and the bridge-soil subsystems.
Fig. 2
Detail of the coupled rail and bridge beam system (left). Interaction model based on the longitudinal and vertical ballast parameters (center). Longitudinal displacement difference due to rotation of the cross sections and corresponding shear traction (right)
Fig. 3
Free-body diagram of the rail and bridge-soil subsystems
×
×
Experiments have shown that there is also in horizontal direction a relative displacement between the rails and the bridge structure [17, 28], which builds up over the ballast height. Since the horizontal displacement is primarily concentrated at the base of the sleepers, it is assumed that a shear plane forms at the lower edge of the sleepers [17]. Additionally, it is also assumed that the rotation of the cross-section of the rail beam is also imposed on the ballast layer between the rails and the shear plane, see Fig. 2. Accordingly, the rotation of the cross-section of the bridge beam is imposed on the ballast layer below the shear plane [17, 28]. Thus, in the model conception, the overall horizontal relative displacement \(\Delta u\) is formed in the shear plane and composed of
where the displacement component \(\Delta u_\textrm{b}\) is related to the cross-sectional rotation \(\varphi _\textrm{b}=-w_{\textrm{b},x}\) of the bridge beam and the displacement component \(\Delta u_\textrm{r}\) to the cross-sectional rotation \(\varphi _\textrm{r}=-w_{\textrm{r}, x}\) of the rail beam as follows: see Fig. 2,
In this relation, \(r_\textrm{b}\) denotes the distance from the axis of the bridge beam to the base of the sleepers (i.e., the shear plane) and \(r_\textrm{r}\) the distance from the base of the sleeper to the axis of the rail beam, see Fig. 2. The resistance to the horizontal relative displacement between the bridge structure and the rails and the corresponding damping effect is represented by continuously distributed horizontal springs and dampers with parameters \(k_\textrm{x}\) and \(c_\textrm{x}\) at the height of the shear plane, as shown in Fig. 2. Therefore, the shear traction T in the shear plane depicted in Fig. 2 is composed of the components \(T_\textrm{b}\) and \(T_\textrm{r}\),
Stollwitzer et al. [28] reports that, in addition to the ballast related damping effects due to the ballast deformation in horizontal and vertical direction, there is also a damping mechanism due to the absolute displacement of the ballast. This is accounted for in the model by a distributed viscous bedding of the bridge structure with the damping parameter \(c_\textrm{ob}\), see Fig. 1. The corresponding distributed damping force
is also depicted in Fig. 2. The mass of the ballast and the sleeper is added to the mass of the bridge beam. However, in Sect. 5.4 an example is considered where the sleeper mass is added to the mass of the rail beam, discussing the possible effects of the additional mass moving together with the rail beam.
2.2.2 Rails
The characteristic parameters of the Euler-Bernoulli beam of length \(L_\textrm{r}=2 L_0 + L_\textrm{b}\) representing the rail subsystem are given by the effective bending stiffness \(EI_\textrm{r}\) of the two rails, their mass per unit length \(\rho A_\textrm{r}\), the lateral and longitudinal bedding stiffness coefficients \(k_\textrm{z}\) and \(k_\textrm{x}\), and the lateral and longitudinal bedding damping coefficients \(c_\textrm{z}\) and \(c_\textrm{x}\). Based on the modeling assumptions, the equation of motion of the rail beam shown in Fig. 3 can be derived as follows:
It is noted that the shear traction due to the relative horizontal deformation of the ballast enters the above equation in the form \((T_{\textrm{b},x}(x,t) + T_{\textrm{r},x}(x,t)) r_\textrm{r}\), since it is applied at the distance \(r_\textrm{r}\) from the beam axis. The interaction forces resulting from the contact of train wheels (axles) with the rails read as [14]
where \(x_k^{(j)} = v t - l_k ^{(j)}\) is the axle position of the k-th axle of the j-th vehicle at time t. The variable \(l_k^{(j)}\) denotes the initial distance of the k-th axle of the j-th vehicle from \(x = - L_0\), see Fig. 3. The position of the k-th axle load of the j-th vehicle \(F_k^{(j)}\) at \(x_k^{(j)}\) is defined by the Dirac-delta function \(\delta \big (x-x_{k}^{(j)}(t)\big )\). The variable \(\Pi (t,t_{\textrm{A}k}^{(j)},t_{\textrm{D}k}^{(j)}) = H(t - t_{\textrm{A}k}^{(j)}) - H(t - t_{\textrm{D}k}^{(j)})\) denotes the time range in which the respective axle load crosses the track between point A and point D. The variables \(t_{\textrm{A}k}^{(j)}\) and \(t_{\textrm{D}k}^{(j)}\) denote the time instances at which the axle load arrives at point \(\textrm{A}\) and leaves the rail beam at point \(\textrm{D}\) (points A, B, C, D are indicated in Figs 1 and 2).
The boundary conditions representing the hinged supports at points \(\textrm{A}\) and \(\textrm{D}\), added for Approach 1, read as
With the constant flexural rigidity \(EI_\textrm{b}\), mass per unit length \(\rho A_\textrm{b}\) and the span \(L_\textrm{b}\), the equation of motion of the bridge beam subjected to the vertical interaction force F(x, t) and the horizontal shear traction T(x, t) at distance \(r_\textrm{b}\) from the beam axis (see Fig. 2) is derived as
As an example, the planar model of a conventional train described in detail in Salcher and Adam [24] is considered. In this model, each of the \(j = 1,...,N_\textrm{c}\) vehicles has ten degrees of freedom (DOFs). As shown in Fig. 1, these vehicles consist of a car body (whose variables are dentoed by the subscript“p”), two bogies (subscript“s”) and four axles (subscript“a”). Herein, four vertical translational DOFs describe the axle displacements (\(u_{\textrm{a}l}^{(j)}\), \(l = 1\),..., 4), two vertical translational DOFs represent the displacements of the bogies (\(u_{\textrm{s}1}^{(j)}\), \(u_{\textrm{s}2}^{(j)}\)) and one vertical translational DOF is used for the displacement of the car body, \(u_{\textrm{p}}^{(j)}\). The rotational DOFs considered in the present model involve the rotations of the bogies (\(\varphi _{\textrm{s}1}^{(j)}\), \(\varphi _{\textrm{s}2}^{(j)}\)) and the rotation of the car body, \(\varphi _{\textrm{p}}^{(j)}\). The corresponding masses are \(m_\textrm{a}^{(j)}\), \(m_\textrm{s}^{(j)}\) and \(m_\textrm{p}^{(j)}\). The moments of inertia are denoted by \(I_\textrm{s}^{(j)}\) and \(I_\textrm{p}^{(j)}\). Furthermore, the parameters of the spring-damper elements between the axles and boogies are the spring stiffness \(k_\textrm{a}^{(j)}\) and the damping parameter \(c_\textrm{a}^{(j)}\). Accordingly, stiffness and damping parameters of elements connecting the car body and the bogies are denoted as \(k_\textrm{s}^{(j)}\) and \(c_\textrm{s}^{(j)}\), respectively. The geometrical parameters defining the positions of the spring-damper elements are \(h_\textrm{a}^{(j)}\) and \(h_\textrm{s}^{(j)}\). The equations of motion of the complete train [24]
result from adding the equations of motion of all vehicles. Consequently, the mass, damping and stiffness matrices \(\textbf{M}_\textrm{c}\), \(\textbf{C}_\textrm{c}\), \(\textbf{K}_\textrm{c}\) and the vector of degrees of freedom \(\textbf{u}_\textrm{c}\) are composed of the corresponding matrices/vectors of the individual vehicles \(\textbf{M}_\textrm{c}^{(j)}\), \(\textbf{C}_\textrm{c}^{(j)}\), \(\textbf{K}_\textrm{c}^{(j)}\) and \(\textbf{u}_\textrm{c}^{(j)}\), \(j = 1,...,N_\textrm{c}\), as
The vector of interaction forces \(\textbf{F}_\textrm{c}\) between the train axles and the rails is also composed of the individual vectors of interaction forces \(\textbf{F}_\textrm{c}^{(j)}\) of all vehicles,
The system matrices, vector of degrees of freedom and interaction force vector for the j-th vehicle can be found in Salcher and Adam [24]; Hirzinger et al. [12].
The state-space representation of Eq. (13), needed for the coupling procedure based on König et al. [14], is given by (e.g., Hirzinger et al. [12])
In the following, two different strategies are used for coupling the subsystems. In the first approach, hereafter referred to as Approach 1, the modally expanded rail and bridge subsystems are coupled with a CMS procedure as described in König et al.[14]. The second approach, hereafter referred to as Approach 2, involves a simplified description of track deflection based on a Rayleigh-Ritz approximation, similar to König et al. [13]. Since these coupling strategies are already described in detail in the aforementioned publications König et al. [14] and König et al. [13], the procedure will be described only briefly, pointing out the main differences with these references.
3.1 Modal series expansion of the bridge-soil response
Both strategies require the approximation of bridge displacement \(w_\textrm{b}(x,t)\) by a complex modal series expansion in \(N_\textrm{b}\) modes in the form of
where \(y_\textrm{b}^{(m)}(t)\) is the m-th modal coordinate of the non-classically damped bridge-soil subsystem and \(\Phi _{\textrm{b}}^{(m)}(x)\) denotes the corresponding complex eigenfunction. The variables \(\bar{y}_{\textrm{b}}^{(m)}(t)\) and \(\bar{\Phi }_{\textrm{b}}^{(m)}(x)\) represent the corresponding complex conjugate counterparts. For the derivation of the underlying modal properties of the stand-alone bridge-soil subsystem based on its homogeneous equation of motion, given by
which includes the damping parameter \(c_\textrm{ob}\) capturing the damping effect due to the absolute motion of the ballast, also present in Eq. (19). The parameter \(\lambda _{\textrm{b}}^{(m)}\) corresponds to the m-th eigenvalue [14]. The generalized modal mass \(M_\textrm{b}^{(m)}\), derived from the orthogonality relations of the non-classically damped bridge-soil system reads as [7, 15]
The normalizing constants \(a_\textrm{b}^{(m)}\) and \(b_\textrm{b}^{(m)}\) of the associated orthogonality relations are related to the generalized modal mass in the form of [7, 15]
The first contribution to the rail deflection \(w^{(\textrm{b})}_\textrm{r}(x,t)\) corresponds to the rail response due to interaction forces resulting from the ballast deformation from the bridge displacement \(w_\textrm{b}(x,t)\). It represents therefore the coupling term between the rail and the bridge-soil subsystems. According to Biondi et al. [3] and König et al. [14], \(w^{(\textrm{b})}_\textrm{r}(x,t)\) is approximated as
where the modal coordinates \(y_\textrm{b}^{(m)}(t)\) and \(\bar{y}_\textrm{b}^{(m)}(t)\) are multiplied by a corresponding shape function \(\Psi _\textrm{r}^{(m)}(x)\) and its complex conjugate \(\bar{\Psi }_\textrm{r}^{(m)}(x)\), respectively. This shape function is the solution of an ordinary differential equation resembling the quasi-static version of the equation of motion Eq. (8) [3],
The solution is found numerically, taking into account the boundary conditions of the rail beam (see Eq. (10)).
The second part \(w^{(\textrm{f})}_\textrm{r}(x,t)\) to the response is the deflection of the stand-alone track in response to the axle loads. Depending on the coupling strategy, \(w^{(\textrm{f})}_\textrm{r}(x,t)\) is approximated differently, as discussed below.
3.2.1 Modal series representation of the rail response for Approach 1
As shown in König et al. [14], the response \(w^{(\textrm{f})}_\textrm{r}(x,t)\) of the stand-alone rail bream can be approximated by the complex modal series expansion in \(N_\textrm{r}\) modes,
where \({\Phi }_{\textrm{r}}^{(n)}(x)\) and \(\bar{\Phi }_{\textrm{r}}^{(n)}(x)\) represent the n-th complex eigenfunction and its complex conjugate counterpart, and \(y_{\textrm{r}}^{(n)}(t)\) and \( \bar{y}_{\textrm{r}}^{(n)}(t)\) are the n-th modal coordinate and its complex conjugate counterpart, respectively, of the rail beam. Since the rail beam is simply supported (see Fig. 1) with constant bedding stiffness and damping values, the general form of the n-th eigenfunction given in König et al. [14] simplifies to a pure sine function,
where \(\lambda _{\textrm{r}}^{(n)}\) is the eigenvalue of the n-th mode and \(C^{(n)}\) is an arbitrary scaling factor. The eigenfunctions are real-valued because the eigenvalue \(\lambda _{\textrm{r}}^{(n)}\) is also real-valued. However, the natural frequency \(s_\textrm{r}^{(n)}\) is complex valued, and related to the corresponding eigenvalue \(\lambda _{\textrm{r}}^{(n)}\) as
3.2.2 Rayleigh-Ritz approximation of the rail response for Approach 2
In the alternative solution strategy, the deflection \(w^{(\textrm{f})}_{\textrm{r}}(x,t)\) is approximated by a Rayleig-Ritz approach, where the static displacement of the infinitely long, elastically bedded beam due to a concentrated load serves as a shape function [13],
Here, \(x_p(t)\) refers to the position of the p-th axle of the \(N_\textrm{a}\) axles of the whole train (\(p = 1,...,N_\textrm{a}\)), \(y_{\textrm{r}p}(t)\) denotes the vertical displacement at the p-th axle and \(\varphi _\textrm{r}^*(x - x_p(t))\) is the shape function corresponding to the static deflection centered around the p-th axle. With the local spacial coordinate \(\hat{x}_p = x - x_p (t)\), the static deflection of the vertically bedded rail beam due to a concentrated load reads as [11, 13]
As can be seen, with these symmetric shape functions the influence of the longitudinal bedding of the rail beam with stiffness \(k_\textrm{x}\) is not taken into account. Therefore, this approximation cannot be considered as a full replacement of Approach 1, but it represents a very efficient alternative if appropriately verified in the parameter space of interest.
3.3 Component mode synthesis in Approach 1
Application of CMS on the rearranged equations of motion of the bridge-soil and track substructures (Eq. (11) and (8)),
as described in König et al. [14] ultimately leads to the equations of motion of the coupled track–bridge–soil subsystem in terms of modal coordinates in state-space [14],
Herein the vectors \(\textbf{h}_\textrm{B}\) and \(\textbf{f}_{\textrm{B}}\) denote the vector of modal coordinates and the force vector, respectively. These vectors are specified in König et al. [14]. The system matrices \(\textbf{A}_\textrm{B}\) and \(\textbf{B}_\textrm{B}\) are given by
and are composed of individual sub-matrices as shown in König et al. [14]. The sub-matrices are listed in Appendix A.
Similar to the coupling procedure in state-space used in König et al. [14], a representation closer related to the formulation used in König et al. [13] can be derived in the form of
It is emphasized that both formulations of Eq. (37) and (39) can be used interchangeably. However, the present implementation of Approach 1 is based on the formulation of Eq. (37) and the subsequent coupling procedure of König et al. [14], while Eq. (39) has been added only to illustrate the similarities and differences with Approach 2, which is described below.
3.4 Component mode synthesis in Approach 2
In contrast to the state-space representation of the coupled track–bridge–soil subsystem equations in Approach 1 (Eq. (37)), in Approach 2 the coupled system equations are provided in standard representation as [13]
The system matrices \(\widetilde{\textbf{M}}_\textrm{B}(t)\), \(\widetilde{\textbf{C}}_\textrm{B}(t)\) and \(\widetilde{\textbf{K}}_\textrm{B}(t)\) are composed of sub-matrices
given in Appendixes A and B. The vector \(\widetilde{\textbf{h}}_\textrm{B}(t)\) comprises both the modal coordinates of the bridge beam and the generalized coordinates of the rail beam (see König et al. [13]). The force vector \(\widetilde{\textbf{f}}_\textrm{B}(t)\) is given in König et al. [13]. The difference in the approximation of the rail deflection component \(w_\textrm{r}^{(f)}\) in the two approaches leads to differences in the sub-matrices in Eq. (38) (or Eq. 40) and Eq. (42), in the coupling matrices between the rail and the bridge (subscripts “\(\textrm{br}\)”and “\(\textrm{rb}\)”), and in the matrices involving the time-dependent coordinates of the rail (subscript “\(\textrm{r}\)”). For these subsystem matrices, please refer to Appendix B.
4 Coupled equations of the complete system
The coupling of the train with the track–bridge–soil subsystem is based on the assumption of a rigid contact between wheel and rail according to the so-called corresponding assumption [36].
Fig. 4
Corresponding assumption coupling axle and rail displacements (modified from Hirzinger et al. [12])
×
As shown in Fig. 4, also an irregularity profile function \(I_{\textrm{irr}}(x)\) can be added to the track deflection, to account for random track irregularities in vertical direction, i.e., deviations from the perfectly straight and smooth track.
4.1 Coupled equations of motion in Approach 1
In analogy to König et al. [14], a DST is applied to condense the equations of motion Eq. (13) of the train subsystem and the equations of motion Eq. (37) of the track–bridge–soil subsystem to a coupled set of equations of motion, with the result [14]
where \(\textbf{A}(t)\) and \(\textbf{B}(t)\) are the time-dependent system matrices of the fully coupled system of train, track, bridge and subsoil. The vectors \(\textbf{x}(t)\) and \(\textbf{f}(t)\) denote the vector of system coordinates and the force vector, respectively. Due to the coupling procedure based on the corresponding assumption, the DOFs related to the displacements of the train axles are condensed into the modal coordinates of the rail beam. Thus, the vector \(\textbf{x}(t)\) is composed of the modal coordinates of the bridge-soil subsystem, the modal coordinates of the rail beam, and the DOFs of the train related to the displacements and rotations of the car bodies and bogies of the individual vehicles. For a more detailed description of the coupling procedure, reference is made here to König et al. [14]. This set of equations is solved numerically by application of the Runge–Kutta method.
4.2 Coupled equations of motion in Approach 2
Application of DST as described in König et al. [13] to the equations of motion Eq. (13) of the train subsystem and the equations of motion Eq. (41) of the track–bridge–rail subsystem according to Approach 2 finally yields the single set of coupled equations
where \(\textbf{M}(t)\), \(\textbf{C}(t)\) and \(\textbf{K}(t)\) denote the time-dependent mass, damping and stiffness matrices of the coupled system. The vectors \(\widetilde{\textbf{x}}(t)\) and \(\widetilde{\textbf{f}}(t)\) are the vectors of the system coordinates and forces of Approach 2. Similar to Approach 1, the coupling procedure of Approach 2 condenses the axle DOFs of the train into DOFs of the rail beam. However, the difference is that the rail DOFs are now represented by the time-dependent coordinates of the Rayleigh-Ritz approximation instead of modal coordinates. To solve Eq. (44), the Newmark-\(\beta \) method has proven to be efficient.
5 Computations
5.1 Verification
Considering the additional approximations introduced in Approach 2, the verification of the presented modeling strategies is divided into two parts. In this section, a rigorous verification of Approach 1 is performed by comparing the results of the developed MATLAB [18] code with the outcomes of finite element (FE) analyses using the software suite ABAQUS [1]. Approach 2 is verified in the subsequent section by comparison with representative results from Approach 1.
The bridge considered in all computational examples is the same steel bridge as considered in König et al. [13]. The parameters of the bridge beam as well as the parameters of the rail beam consisting of two 60E1 (UIC60) rails are listed in Table 1.
Table 1
Parameters of the bridge beam [13] and the rail beam [21]
Bridge parameters
Value
Rail parameters
Value
Unit
\(L_\textrm{b}\)
17.5
\(L_0\)
30
m
\(EI_\textrm{b}\)
\(1.356 \cdot 10^{10}\)
\(EI_\textrm{r}\)
\(12.831 \cdot 10^6\)
Nm\(^2\)
\(\rho A_\textrm{b}\)
7830
\(\rho A_\textrm{r}\)
120.73
kg/m
For the vertical stiffness and damping coefficients of the ballast, the fixed values \(k_\textrm{z} = 104 \cdot 10^6\) N/m\(^2\) and \(c_\textrm{z} = 50\) kNs/m\(^2\) are chosen, as given in Yang et al. [33]. The eccentricities for the longitudinal interaction model are \(r_\textrm{b} = 0.7\) m for the bridge beam and \(r_\textrm{r} = 0.3\) m for the rail beam. The base values of the horizontal ballast stiffness and damping coefficients are \(k_\textrm{x} = 10.4 \cdot 10^6\) N/m\(^2\) and \(c_\textrm{x} = 50\) kNs/m\(^2\), but are also varied to verify the model for a wider range of parameters. Model verification involving the soil–structure interaction of the bridge and underlying soil have already been performed in König et al. [14] and König et al. [13]. Since the focus is on verifying the track–bridge interaction model in the longitudinal direction, the influence of underlying soil is therefore omitted and the bridge is considered as a beam on rigid supports.
The described track–bridge system is crossed by a simple MSD system representing the interaction model with respect to one axle of the ICE 3 train model moving at a constant speed of \(v = 70\) m/s. This interaction model, shown in Fig. 5, is characterized by the parameters given in Table 2. Figure 5 also shows the vertical irregularity profile superimposed to the perfectly straight rail geometry. This irregularity profile was randomly generated to represent a track of poor quality, based on the methodology presented in Claus and Schiehlen [5].
Fig. 5
Two degree of freedom vehicle and irregularity profile superimposed to the rail surface
Table 2
Parameters of two degree of freedom vehicle model [14, 19], for the variables see Fig. 5
Parameter
Value
Unit
\(m_\textrm{s}\)
15125
kg
\(m_\textrm{a}\)
1800
kg
\(k_\textrm{a}\)
\(1.764 \cdot 10^6\)
N/m
\(c_\textrm{a}\)
\(4.800 \cdot 10^4\)
Ns/m
×
Fig. 6
Schematic of FE model with used element types
×
The FE model represents an equivalent interaction system using the same input parameters and is visualized in Fig. 6. Both, the bridge and the rail beam are modeled using Euler-Bernoulli beam elements (B23) with a uniform mesh-length of 6.125 cm. The continuous vertical bedding is idealized by discrete spring and dashpot elements at each node of the beam elements. The eccentricities for the longitudinal interaction are modeled with rigid massless beam elements (RB2D2), equally spaced at a distance of 6.125 cm, and connected with horizontally oriented spring and dashpot elements representing the longitudinal bedding. This configuration is shown in Fig. 6, where it should be noted that the longitudinal offset has been added for illustration purposes only and the respective element nodes of the rail and bridge beams are situated at the same positions in the longitudinal direction. Hence, in the actual FE model, the nodes of the rigid beam elements of the rail beam and bridge beam, which are connected by longitudinal spring and damper elements, are situated at the same positions inside the shear zone. Furthermore, the two DOF system is modeled with rigid body lumped masses (Mass) connected by a spring and a damper element, respectively. In the FE model, rigid contact between the moving axle and the rail beam was implemented using ABAQUS tube-to-tube contact elements (ITT21). The deviation of the rail geometry from a perfectly straight line is accounted for by offsetting the rail beam nodes perpendicular to the straight beam axis by the amount of the irregularity profile of Fig. 5. To keep Fig. 6 relatively simple, this deviation of the rail nodes in the vertical direction is not illustrated. It is also noted that the relatively small element size of 6.125 cm was used to sufficiently approximate the irregularity profile along with the continuous bedding. In the verification examples, a constant time increment of \(\Delta t = 10^{-4}\) s is used.
5.2 Variation of the horizontal stiffness
To verify the prediction of the system response, several computations were performed with different horizontal stiffness coefficients. In the computations with the proposed approach (Approach 1), the bridge displacement is approximated by \(N_\textrm{b} = 6\) modes. The deflection of the stand-alone track is approximated by \(N_\textrm{r} = 100\) modes, emphasizing the need to include such a large number of track modes to adequately predict the system response (see also König et al. [14] and König et al. [13]). In Fig. 7, the bridge response at the quarter point (Fig. 7a, b) and the center point (Fig. 7c, d), predicted by the semi-analytical approach (“Approach 1; ref. sol.”), is compared with the results of the FE model (“FEM; ref. sol.”). The dashed vertical lines indicate the time instants at which the MSD system arrives at the left bridge support (\(t_\textrm{B}\)) and leaves the bridge at the right support (\(t_\textrm{C}\)). As can be seen, the bridge displacements computed with the proposed approach perfectly match those of the FE model, see Fig. 7a, b). This excellent agreement is also observed for the accelerations (Fig. 7c, d).
To further verify the modeling approach, another set of computations was performed in which the horizontal ballast stiffness was set to ten times its original value (\(10 k_\textrm{x} = 104 \cdot 10^6\) N/m\(^2\)). Comparing the respective lines representing the results of the proposed approach and those of the FE model, again very good agreement is observed in the bridge displacements and accelerations. The comparison of the two sets of computations with different horizontal stiffness values clearly illustrates the stiffening effect related to the longitudinal interaction in the reduced displacement amplitude as well as in the reduced period of the free vibration response after the MSD system has left the bridge.
Fig. 7
Approach 1 versus FE analysis. Two different values of horizontal ballast stiffness \(k_\textrm{x}\). a Bridge displacement and b acceleration at \(x = L_\textrm{b}/4\). Second row: c Displacement and d acceleration at \(x = L_\textrm{b}/2\)
×
5.3 Variation of the damping
The verification of the response prediction taking into account the damping due to the longitudinal track–bridge interaction is done in the same manner as for the horizontal stiffness. The results of the computations with the base value parameter set already shown in Fig. 7 are shown again in Fig. 8 as a reference. Since the track irregularities were found to have a greater impact on the bridge response at the quarter point, Fig. 8 shows results at this position only. The base value of horizontal damping was also multiplied by ten (\(10 c_\textrm{x} = 500\) kNs/m\(^2\)). Again, very good agreement is observed of the predicted bridge displacement (Fig. 8a) and acceleration (Fig. 8b), further verifying the proposed approach.
Fig. 8
Approach 1 versus FE analysis. Two different values of horizontal ballast damping \(c_\textrm{x}\). a Bridge displacement and b acceleration at \(x = L_\textrm{b}/4\)
×
Next, the consideration of the additional damping parameter \(c_\textrm{ob}\), which is the last ballast-related damping mechanism reported in Stollwitzer et al. [28], is verified. While this parameter was omitted in the above computations, Fig. 9 also shows results where this damping parameter is set to \(c_\textrm{ob} = 8.7\) kNs/m\(^2\) (“with \(c_\textrm{ob}\)”), again showing excellent agreement between the proposed Approach 1 and the FE model.
Fig. 9
Approach 1 versus FE analysis. Consideration of ballast damping \(c_\textrm{ob}\). a bridge displacement and b acceleration at \(x = L_\textrm{b}/4\)
×
From the verification examples presented here, it can be concluded that the contribution of the additional damping mechanisms in the horizontal and vertical directions has also been successfully integrated into the semi-analytical modeling approach.
5.4 Application problems
Considering the large uncertainties in the ballast parameters in combination with the highly nonlinear properties reported in the literature, a thorough parametric study is presented for the application example, where the influence of the model parameters is further discussed. For this purpose, the dynamic response of the example bridge, which was also used for verification, is further analyzed. The base parameters of the bridge and track are the same as in Sect. 5.1. However, taking into account structural damping of the bridge, the modal damping coefficient of \(\tilde{\zeta }_\textrm{b}^{(m)} = 0.8125 \%\) according to [6] for a steel bridge of the given length is assigned to each mode of the rigidly supported bridge. This is achieved by modulating the complex natural frequency \(s_\textrm{b}^{(m)}\) as described in König et al. [14]. In contrast to the simple MSD system used for the verification in 5.1, a more thorough representation of the \(N_\textrm{c} = 8\) vehicles of the ICE 3 train model is considered. Each vehicle is represented by the same ten DOF system with parameters (compare Fig. 1), which include the masses (\(m_\textrm{p}\), \(m_\textrm{s}\), \(m_\textrm{a}\)), moments of inertia (\(I_\textrm{p}\), \(I_\textrm{s}\)), stiffness coefficients (\(k_\textrm{s}\), \(k_\textrm{a}\)), damping coefficients (\(c_\textrm{s}\), \(c_\textrm{a}\)), and dimensions (\(h_\textrm{s}\), \(h_\textrm{a}\)), given in König et al. [14] and Nguyen et al. [19]. All analyses discussed below involve individual time history analyses of train crossings at various train speeds from 10 to 90 m/s at uniform intervals of \(\Delta v = 0.5\) m/s. The bridge response is evaluated at 101 uniformly distributed points along the track, and representative maximum absolute bridge displacements (\(\textrm{max} \vert w_\textrm{b} \vert \)) and maximum absolute bridge accelerations (\(\textrm{max} \vert \ddot{w}_\textrm{b} \vert \)) are computed for each considered train speed. These response quantity are then presented in the form of response spectra in Figs 10 to 14 for a variety of input parameter configurations.
5.4.1 Variation of horizontal ballast parameters
As in the verification examples, the horizontal stiffness and damping parameters \(k_\textrm{x}\) and \(c_\textrm{x}\) are varied. The parameter variation is performed in the range of 0.5 to 10 times the respective base value. In this section, the influence of track irregularities is not considered. Response spectra of maximum absolute bridge deflection (\(\max \vert w_\textrm{b} \vert \)) and maximum absolute bridge acceleration (\(\max \vert \ddot{w}_\textrm{b} \vert \)) resulting from variation of \(k_\textrm{x}\) are depicted in Fig. 10. Here, the results derived from computations with the base values of the input parameters are represented by the black lines and the blue lines represent results with different horizontal stiffness. In Fig. 10 (a), the increase in the system stiffness of the complete coupled system associated with the increase in \(k_\textrm{x}\) is clearly visible, both in the decrease in vertical displacements and in the shift of response peaks toward higher train speeds.
Fig. 10
Approach 1. Different values of horizontal ballast stiffness \(k_\textrm{x}\). Response spectra. a Maximum absolute displacement and b maximum absolute acceleration
×
The peaks in the response spectra apparent in this figure are related to critical train speeds, at which the system is excited to resonance by the crossing train. One reason of resonance is the uniform axle spacing of the individual vehicles of the train crossing the bridge at constant speed. The corresponding so-called second-order resonance speeds \(v_i^{(m)}\) are defined as follows for a bridge without a track [33]:
For the considered ICE 3 train model, the regular spacing is \(d_\textrm{c} = 24.775\) m. Estimating the natural frequencies of the track–bridge model with the natural frequencies of the stand-alone simply supported bridge beam [9],
reveals that the most pronounced response peak in Fig. 10 is related to the first mode (\(m = 1\)) at the resonant speed of \(v_3^{(1)} = 58.61\) m/s. Therefore, the shift in response peaks with increasing horizontal ballast stiffness \(k_\textrm{x}\) is due to the increased natural frequencies of the coupled track–bridge system compared to the natural frequencies of the stand-alone bridge from Eq. (46). In both cases, the displacement response quantity \(\max \vert w_\textrm{b} \vert \) and the acceleration response quantity \(\max \vert \ddot{w}_\textrm{b} \vert \), a decrease in the overall response is observed. However, this effect is less pronounced in the acceleration response.
Similar to the results with variation of the horizontal stiffness in Fig. 10, the results with variation of the damping parameter \(c_\textrm{x}\) are shown in Fig. 11. Here, the decrease in resonance peaks with increasing horizontal ballast damping is clearly visible, while, as expected, no significant changes of the respective resonance speeds are observed.
Fig. 11
Approach 1. Different values of horizontal ballast damping \(c_\textrm{x}\). Response spectra. a Maximum absolute displacement and b maximum absolute acceleration
×
5.4.2 Effect of track irregularities
In view of the results of the previous section derived from the variation of the horizontal ballast parameters, track irregularities are considered in the following computations in the form of the irregularity profile \(I_\textrm{irr}(x)\) depicted in Fig. 5. Since the influence of ballast parameters is likely to be even more pronounced when considering the imperfect track geometry, different values of \(k_\textrm{x}\) and \(c_\textrm{x}\) are also considered here. However, computations are only performed using the base values of \(k_\textrm{x}\) and \(c_\textrm{x}\) and ten times these base values, respectively. The corresponding results are shown in Fig. 12 for the different values of \(k_\textrm{x}\). In addition to the results derived with Approach 1, Fig. 12 also contains results derived with the simplified Approach 2 to also verify the latter method for cases where the influence of the longitudinal track–bridge interaction is most pronounced. Examining Fig. 12 (a), effects of the horizontal stiffness parameter are similar to Fig. 10 (a) and only a small effect of the track irregularities on the maximum absolute displacement \(\max \vert w_\textrm{b} \vert \) is visible when comparing the corresponding lines in Figs 10 (a) and 12 (a).
However, inspection of Fig. 12 (b) shows the significant influence of track irregularities on the acceleration response quantity \(\max \vert \ddot{w}_\textrm{b} \vert \). Moreover, the tendency observed in Fig. 10 (b) for the acceleration response peaks to decrease with increasing horizontal ballast stiffness is present only for response peaks below \(v=50\) m/s, and the most prominent response peak increases rather dramatically when a high ballast stiffness of \(k_\textrm{x}\) is considered. Apart from that, an overall good agreement between the results of Approach 1 and Approach 2 is observed in Fig. 12. However, at higher train speeds, minor deviations between the two approaches are observed. This may be attributed to the simplifications of the Rayleigh-Ritz approximation introduced in Approach 2, as well as to a larger influence of the track model on the bridge response at higher speeds when track irregularities are considered. This slight disagreement in results was also observed for the purely vertically oriented model in König et al. [13].
Fig. 12
Approach 1 versus Approach 2. Track irregularities considered. Two different values of the horizontal ballast stiffness \(k_\textrm{x}\). Response spectra. a Maximum absolute displacement and b maximum absolute acceleration
×
It is also evident from Fig. 12 (b) that, compared to Fig. 10 (b), an increased influence of the longitudinal stiffness parameter on the computed bridge accelerations is observed when additional track irregularities are considered. However, the bridge displacements appear to be affected by the longitudinal stiffness parameter to a similar extent with or without track irregularities, see Figs 10 (a) and 12 (a). This may be due to the fact that the vertical bridge accelerations are generally more sensitive to amplifications due to track irregularities than the vertical displacements (see, e.g., Salcher and Adam [25]). Together with the additional amplified relative movement between bridge and track, this is likely the cause of the overall greater influence of the track parameters on the acceleration response of the bridge in the presence of track irregularities.
For completeness, Fig. 13 includes results of analyses with ten times the base value of the longitudinal ballast damping \(c_\textrm{x}\), where the effect of the additional damping becomes visible again. For all examples in Figs 12 and 13, Approach 2 agrees quite well with the values derived with Approach 1, justifying the simplifications introduced in Approach 2.
Fig. 13
Approach 1 versus Approach 2. Track irregularities considered. Two different values of the horizontal ballast damping \(c_\textrm{x}\). Response spectra. a Maximum absolute displacement and b maximum absolute acceleration
×
5.4.3 Influence of soil–structure interaction
The last example comprises the full interaction model considering the soil–structure interaction based on the same soil and foundation parameters also used in König et al. [13]. These are the constrained modulus \(E_\textrm{s} = 2.5\cdot 10^{8}\) N/m\(^{2}\) of the soil, the soil density \(\rho _{\textrm{s}} = 2300\) kg, Poisson’s ratio \(\nu = 0.28\), base area of the foundation \(A_0 = 40\) m\(^2\) and the foundation mass \(m_\textrm{b} = 2.5 \cdot 10^5\) kg. Application of the cone model of Wolf Wolf and Deeks [32] yields the following equivalent subsoil coefficients, i.e., the stiffness coefficient \(k_\textrm{b} = 1.514 \cdot 10^9\) N/m and the damping coefficient \(c_\textrm{b} = 3.033 \cdot 10^7\) Ns/m [13].
Considering the frequency- and displacement-dependent characteristics of the longitudinal ballast properties reported from experiments [26, 28], the problem of appropriate selection of equivalent linear model parameters \(k_\textrm{x}\) and \(c_\textrm{x}\) remains. For the damping parameter, it is easy to argue that a minimum value should be chosen to avoid unrealistically high damping, which subsequently underestimates the predicted vibration response of the bridge. Such a value is also recommended in Stollwitzer et al. [26] and given as \(c_\textrm{x} = 62\) kNs/m\(^2\). However, the selection of a suitable value for \(k_\textrm{x}\) is not as straight forward. In addition to the relative displacement amplitude within the assumed shear plane in the ballast and the excitation frequency, the loading by the axle loads of the train and the ballast condition (frozen or unfrozen) also strongly influence the longitudinal stiffness of the ballast. Therefore, the computational example presented here covers only the case of unfrozen ballast, where the longitudinal stiffness parameter is estimated based on measurements on a loaded track. For this case, Stollwitzer et al. [26] proposes the following regression formula for the estimation of \(k_\textrm{x}\),
where \(u_0\) denotes the relative displacement amplitude in longitudinal direction. Additional regression formulas for unloaded and frozen states are also specified in [26]. Since \(u_0\) is not constant over the length of the bridge, an equivalent displacement amplitude \(u_\textrm{0eq}\) must be used. In the same context, such an equivalent displacement amplitude is given in Stollwitzer and Fink [27] as
based on the fundamental mode shape in form of a sine half wave and strain energy considerations. Herein \(w_\textrm{b0}\) denotes the bridge deflection at the center and \(e_{\textrm{b}}\) the additional support eccentricity of the bridge. Since the displacement amplitude varies quite significantly for different values of \(k_\textrm{x}\), the deflection at the center is simply estimated by taking the average of the peak values of the maximum absolute deflection of the two response spectra shown in Fig. 10 with the lowest and highest horizontal stiffness, resulting in \(w_\textrm{0eq} \approx 6 \cdot 10^{-3}\) m. Since no additional support eccentricity \(e_{\textrm{b}}\) is considered in the present model, the equivalent displacement amplitude is obtained as \(u_\textrm{0eq} \approx 0.76 \cdot 10^{-3}\) m. Consequently, substituting \(u_\textrm{0eq}\) into Eq. (47) leads to an estimated longitudinal stiffness coefficient \(k_\textrm{x} = 26.40 \cdot 10^{6}\) N/m\(^2\).
In addition to the vertical and horizontal ballast coefficients, ballast-related damping due to absolute movement of the ballast can be accounted for in the form of the damping parameter \(c_\textrm{ob}\). Based on a regression for the damping coefficient \(c_\textrm{ob}\), which is also frequency dependent [27], and the first natural frequency of the stand-alone rigidly supported bridge \(f_\textrm{b,rigid}^{(1)} = 7.10\) Hz, the damping parameter is estimated as \(c_\textrm{ob} = 8.7 \cdot 10^3\) Ns/m\(^2\).
Given the detailed interaction model and the damping mechanisms involved due to soil and ballast model, applying the full structural damping of \(\tilde{\zeta }_\textrm{b}^{(m)} = 0.8125 \%\) could result in an uncertain response prediction because the system damping is overestimated. Therefore, only the span-independent amount of structural damping, given in EN1991-2 [6] as \(\tilde{\zeta }_\textrm{b}^{(m)} = 0.5 \%\), is assigned to the modes of the bridge beam.
Figure 14 shows the bridge response spectra of the full model, including the track irregularity profile. Here, the solid lines represent the results without and with considering the damping parameter \(c_\textrm{ob}\) (“wo. \(c_\textrm{ob}\)”and“w. \(c_\textrm{ob}\),” respectively). To facilitate visual comparison of these results with the previous results for the rigidly supported bridge, Fig. 14 (a) shows the maximum absolute value of the bridge displacement relative to the straight bridge axis \(\max \vert w_\textrm{b, rel} \vert \) defined as [13]
In the underlying model of these two spectral representations, as in all previous computations, the sleeper mass is included in the bridge mass per unit length \(\rho A_\textrm{b}\). However, considering all the modeling details of the presented ballast interaction model, the obvious problem of the mechanical model idealizing the entire track as a continuous system must also be addressed. Although the presented model does not include a detailed description of the discrete support of the rails on the sleepers, it can still be argued that, at least in the case of a relatively stiff rail-sleeper connection, the sleeper mass can contribute significantly to the dynamic response of the rail beam. Therefore, the results of two additional computations are included in Fig. 14 that add a distributed mass per unit length \(\delta \rho A_\textrm{r} = 500\) the kg/m to the rail beam instead to the bridge (“with sleeper mass”).
In Fig. 14, the results denoted by “wo. \(c_\textrm{ob}\); without sleeper mass,” were obtained using a similar horizontal damping parameter as in the reference solutions (“ref. sol.”) of Figs 12 and 13, without applying the additional damping \(c_\textrm{ob}\) and sleeper mass \(\delta \rho A_\textrm{r}\). Therefore, a comparison of these results is possible, giving an indication of the influence of the additionally considered soil–structure interaction on the results of Fig. 14. Comparing the response peaks of these results for both the displacement in Fig. 14 (a) and the acceleration in Fig. 14 (b) with the response peaks of the reference solution of Figs 12 and 13, significantly lower values are observed when the soil–structure interaction is considered. This indicates an increased system damping associated with the soil–structure interaction.
Fig. 14
Response spectra—full interaction model. a maximum absolute deflection and b maximum absolute acceleration
×
Moreover, the inclusion of the additional damping parameter \(c_\textrm{ob}\) also contributes significantly to the system damping, which is evident from the reduced response peaks. The consideration of the sleeper mass as a distributed mass for the rail girder has no significant effect on the relative bridge deflection. However, as the train speed increases, the bridge acceleration tends to be greater when the sleeper mass is added to the rail beam. Considering the observation that track irregularities significantly affect the bridge acceleration at train speeds above 55 m/s (see Figs 12 and 13), it can be concluded that the increase in bridge acceleration in Fig. 14, results from the increased effect of rail vibration on the bridge due to the additional mass of the rail beam combined with the additional excitation mechanism due to track irregularities.
6 Summary and conclusions
In more complex mechanical models of track–bridge interaction systems, the properties of the rails, rail pads, sleepers, and ballast are accounted for by multiple layers of vertically oriented continuously distributed spring-damper elements. However, recent experimental studies have shown that also the longitudinal ballast interaction may have a significant impact on the system behavior. Effects such as an increase in system stiffness and additional damping have been reported. Therefore, in the present contribution, modeling approaches were developed to address the complex nature of longitudinal track–bridge interaction in response prediction of high-speed railway bridges. These approaches involve discretization of the track in two different forms. While the first approach uses a modal series expansion of the rail deflection, the second approach adopts a Rayleigh-Ritz approximation of this response quantity. The coupling of the track and bridge-soil subsystems is achieved by variations of the component mode synthesis. The application of a discrete substructuring technique to the track–bridge–soil subsystem and the mass-spring-damper model of the high-speed train leads to the equation of motion of the entire interaction system. The resulting models allow the analysis of several interaction effects that affect the prediction of the dynamic system response. In addition to the longitudinal track–bridge interaction, these include the vertical track–bridge interaction, the train-track interaction, and the soil–structure interaction of the bridge.
The two modeling approaches were developed with the aim of providing an accurate description of the model considering the longitudinal track–bridge interaction and an efficient computation. Based on the theory and the implementation of the two approaches, it can be concluded that Approach 1 provides a more accurate description of the model taking into account the longitudinal interaction, but at the cost of higher computational times due to the large number of rail modes to be considered. In contrast, Approach 2 uses a simplified description of the track, resulting in a much lower computational cost, but with the possible loss of accuracy. However, for the parameters considered in this publication, Approach 2 provided reliable results for the predicted bridge response.
Using the models based on the modeling approaches presented, the results of a variety of response analyses were shown, providing insight into the influence of longitudinal bridge-track interaction effects on the system response. In addition, the response of the model with realistic parameter configurations was shown, including the soil–structure interaction of an example bridge. From these results, the following conclusions can be drawn for the train-bridge interaction systems considered, taking into account the modeling assumptions:
The longitudinal track–bridge interaction leads to stiffening and damping effects, which become visible in the predicted displacement and acceleration response of the bridge.
The dynamic response prediction is strongly affected by the presence of track irregularities, which in turn strongly depends on the longitudinal interaction parameters.
The additional damping effect of the ballast, related to the absolute motion of the ballast, leads to a significant reduction of the displacement and acceleration peaks at resonant speeds.
Consideration of sleeper mass in the rail beam leads to prediction of higher accelerations when track irregularities are present.
Soil–structure interaction has a strong influence on the predicted dynamic response, i.e., the response peaks due to the additional damping are reduced.
The presented models provide improved simulation tools for more realistic estimation of the dynamic response of the dynamic train-bridge interaction system, which can be used to conduct extensive parameter studies in the future.
Acknowledgements
The computational results presented have been achieved (in part) using the HPC infrastructure LEO of the University of Innsbruck.
Declarations
Competing interests
The authors have no relevant financial or non-financial interests to disclose.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This Appendix specifies system matrix components in the equations of motion of the coupled track–bridge–soil subsystem (Eqs (37) and (38)) according to Approach 1. The diagonal matrices \(\textbf{S}_\textrm{b}\) and \(\textbf{S}_\textrm{r}\) containing complex natural frequencies of the bridge beam and the rail beam read as [14]
The matrices \(\textbf{A}_\textrm{b}\) and \(\textbf{B}_\textrm{b}\) containing the normalizing constants for the modal equations of the bridge beam, specified in Eq. (23), read as [14]
The matrices \(\textbf{A}_\textrm{r}\) and \(\textbf{B}_\textrm{r}\) containing the normalizing constants for the modal equations of the rail beam, given in Eq. (33), read as [14]
The matrices \(\Delta \textbf{M}_{\textrm{b}}\), \(\Delta \textbf{C}_{\textrm{b}}\) and \(\Delta \textbf{K}_{\textrm{b}}\) differ from those specified in König et al. [14], because in the present model the longitudinal stiffness and damping of the ballast are considered. For the present model, they are derived as
with the vectors of eigenfunctions of the bridge beam \(\mathbf {\Phi }_{\textrm{b}}\) and the vector of shape functions \(\mathbf {\Psi }_{\textrm{r}}\) [14],
Note that for \(\Delta \textbf{K}_{\textrm{b}}\) in Eq. (A4) the identity given by Eq. (26) has been considered. With the vector of eigenfunctions of the rail beam \(\mathbf {\Phi }_{\textrm{r}}\) given by [14]
Here \(\textbf{K}_{\textrm{rb}} = \textbf{0}\) follows directly from Eq. (26). The expression for \(\textbf{K}_{\textrm{br}}\) was simplified by integrating by parts for terms containing spatial derivatives, together with the identity of Eq. (26).
Appendix B Sub-matrices - Approach 2
In the following, the sub-matrices in the equation of motion of Approach 2 (Eqs (41) and (42)) are specified. The sub-matrices related to the modal coordinates of the bridge (subscript “\(\textrm{b}\)”) are the same as for Approach 1 and can be taken directly from Appendix A. However, the matrix components related to the coupling of the substructures of the rail beam and bridge-soil beam differ from Approach 1. With the definition of the vector of rail shape functions \(\varvec{\Phi }^*_{\textrm{r}}\),
the coupling matrices of Eq. (42), denoted by the subscripts “\(\textrm{br}\)” and “\(\textrm{rb}\)”, are obtained. Simply replacing \(\varvec{\Phi }_{\textrm{r}}\) by \(\varvec{\Phi }^*_{\textrm{r}}\) in the expressions of Eq. (A7) results in the equivalent representations of the sub-matrices \(\widetilde{\textbf{M}}_{\textrm{br}}(t)\), \(\widetilde{\textbf{M}}_{\textrm{rb}}(t)\), \(\widetilde{\textbf{C}}_{\textrm{br}}(t)\), \(\widetilde{\textbf{C}}_{\textrm{rb}}(t)\), \(\widetilde{\textbf{K}}_{\textrm{br}}(t)\) and \(\widetilde{\textbf{K}}_{\textrm{rb}}(t)\). Here the time dependency of the matrices is a direct result of the time dependency of Eq. (B8).
The matrix \(\widetilde{\textbf{K}}_{\textrm{br}}(t)\) can be simplified in the same manner as \({\textbf{K}}_{\textrm{br}}\) in Eq. (A7), i.e., It can be shown that
if \(L_0\) is sufficiently large. A similar simplification can also be found in König et al. [13].
The remaining sub-matrices of Eqs (41) and (42) related to the time-dependent coordinates of the rail beam (subscript“\(\textrm{r}\)”) can be found using the same procedure as in König et al. [13]. Herein the dynamic response of the rail due to an individual axle load \(F_p(t)\) is considered at first. In this regard, the equation of motion of the stand-alone rail beam approximated by the p-th Rayleigh-Ritz shape function can be derived from Eq. (8) as
As a shape function for the Rayleigh-Ritz approximation of the rail beam deflection, the deflection of the infinitely long elastically bedded beam due to a single load is employed (Eq. (35)). However, also in Approach 2 the influence of the train on the interacting system is only considered for a finite section of the rail beam. Choosing the considered section of the track in same interval as in Approach 1, \(-L_0 \le x_p \le L_\textrm{b} + L_0\), the time window in which the p-th axle crosses the track between points A and D is defined by \(\Pi (t,t_{\textrm{A}p},t_{\textrm{D}p})\) [13]. As shown in König et al. [13], the equation of motion of a single degree of freedom (SDOF) oscillator, equivalent to the bedded rail beam under point load, can be found by pre-multiplying Eq. (B10) by \(\varphi _\textrm{r}^*(\hat{x}_p)\) and integrating from \(-\infty \) to \(\infty \), resulting in
As in König et al. [13], the \(N_\textrm{a}\) SDOF systems that capture the response of the rail beam to the individual axle forces are considered to be decoupled because the deflection shape around the axle load is very isolated. This consideration, together with Eq. (B10), leads to the diagonal matrices \(\textbf{M}_\textrm{r}\), \(\textbf{C}_\textrm{r}\), \(\textbf{K}_\textrm{r}\), which represent the \(N_\textrm{a}\) decoupled SDOF oscillators,
Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.
