Effect of soil properties on the dynamic response of simply-supported bridges under railway traffic through coupled boundary element-finite element analyses

Effect of soil properties on the dynamic response of simply-supported bridges under railway traffic through coupled boundary element-finite element analyses M.D. Mart´ ınez-Rodrigoa,∗, P. Galv´ ınb, A. Dom´ enecha, A. Romerob aUniversitat Jaume I, Department of Mechanical Engineering and Construction, Avda. Sos Baynat s/n, 12071 Castell´on, Spain bUniversidad de Sevilla, Escuela T´ecnica Superior de Ingenier´ıa, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain Abstract Railway induced vibrations on short-to-medium span simply-supported (SS) bridges is addressed in this contribution. Such structures may experience high levels of vertical acceleration at the platform, leading to adverse consequences such as a premature degradation of the ballast layer and passenger discomfort. In the present study, the evolution of the bridge dynamic response when soil-structure interaction (SSI) is taken into account is investigated. To this end a coupled three-dimensional (3D) Boundary Element-Finite Element model (BEM-FEM) formulated in the time domain is implemented to reproduce the soil and structural behaviour, respectively. First, a set of soil-bridge systems of interest is defined, covering a wide range of lengths and natural frequencies for the structures, and an interval of expectable elastic properties and damping levels for the soil. Then, different types of analyses are performed on the soil-bridge systems extracting conclusions regarding the effect of including SSI in numerical models for predicting the bridge behaviour under railway traffic. In particular natural frequencies and modal damping levels are identified, and the structure amplification after the passage of a moving load in free vibration is investigated. Conclusions regarding how resonance and cancellation conditions may be affected by soil properties are extracted. Finally, the dynamic response of a real bridge, belonging to the Spanish railway network, is evaluated under the circulation of trains that induce second and third resonances of the bridge fundamental mode. The effect of the soil flexibility, soil material damping and the bridge resonance order are evaluated. Conclusions regarding the appropriateness of the results provided by common models which do not include SSI effects are extracted. Keywords: Railway bridges, soil-structure interaction, resonance, cancellation, moving loads, BEM-FEM coupled models 1. Introduction The development of modern, efficient and operational transport systems is essential for a sustainable economic development. In this context the construction of new High-Speed railway lines and upgrading of conventional lines for higher operating speeds, has become a trend in Asian and European countries in the last decades. Railway infrastructures and, in particular, railway bridges, are expected to exhibit an adequate performance under these new traffic requirements guaranteeing traffic safety, passengers comfort, structural integrity and acceptable environmental conditions in terms of sound and vibration transmitted amplitudes. The level of vibrations induced on bridges due to the circulation of railway convoys has become an issue of concern among the scientific and engineering community, due to the periodic nature of the vehicles axles and the operating speeds approaching and exceeding 300 km/h in many lines. The periodic nature of the axle transmitted forces may excite important transverse vibration levels in the structures, particularly under resonant conditions [1, 2]. Especially critical in this regard are shortto-medium span bridges composed by SS decks with usually ∗Corresponding author. Tel.: +34 964387473; fax: +34 964728106 Email address: [email protected] (M.D. Mart´ ınez-Rodrigo) associated low masses. This problem aggravates for low structural damping levels, typical in the aforementioned constructions [1]. Figure 1 shows two examples of such structures, belonging to the Spanish railway network, with decks composed by concrete slabs resting on series of pre-stressed concrete girders. Even though this typology is not common in High-Speed lines of new construction, due to its poor dynamic performance [3], these beam-type bridges do exist in former conventional lines upgraded for High-Speed. Resonance in railway bridges may lead to adverse consequences such as ballast destabilization, passenger discomfort, a general degradation of the track and a raise in the maintenance costs of the line [1, 4]. For this reason, according to standards, the maximum deck acceleration must be checked at the Serviceability Limit State for the prevention of track instability, and regarded as a traffic safety requirement [5]. Resonance takes place when the excitation period of the axles, i. e., the ratio between a characteristic, or many times repeated, distance and the train speed is a multiple of one natural period of the structure. When this occurs, the free vibration oscillations induced by every load accumulate, and the transverse response of the bridge progressively increases, leading to a substantial amplification if the number of axles is sufficient. In short to medium span bridges with nowadays maximum train speeds, the characteristic distance associated with detrimental Preprint submitted to Engineering Structures August 29, 2017 Figure 1: Railway bridges in Spanish lines composed by simply-supported bays of short-to-medium span levels of transverse accelerations due to resonance usually corresponds to the length of the passengers’ coaches. Therefore, the dynamic amplification of beams or bridges at resonance depends both on the periodicity of the loads and on the amplitude of the free vibrations left by every single load. Under ideal SS conditions and in the absence of damping, the amplification of the free vibrations left by every load depends on the ratio between the structural periods and the travelling time of the load. As indicated in [6], depending on this ratio the beam may experience substantial levels of free vibrations (maximum free vibrations) or these may practically cancel (cancellation of free vibrations). If the limits on the bridge deck acceleration cannot be met in an existing structure, strengthening measures may be applied in order to modify its dynamic properties and, consequently, its dynamic behaviour [7]. Passive control techniques could also provide cost-effective solutions increasing the overall damping levels of the structure and reducing the deck vibrational response at resonance [8]. In either case of new or existing structures, it is essential to develop accurate numerical models, able to realistically predict the vibration levels for the expected traffic conditions in order to make the best decision in the design stage or when a line is upgraded for higher operating speeds. According to some authors [9], the choice of boundary conditions for dynamic analyses appears to constitute a group of very sensitive parameters which have a considerable influence on the dynamic response of certain bridge types. The phenomena of resonance and cancellation experienced by beams or bridges under the circulation of moving loads has been studied by several researchers [6, 10–17]. Nevertheless in the previous works, soil-structure interaction is always disregarded and classical boundary conditions are assumed for the bridge deck. According to some authors, in certain soil environments an increase in the fundamental natural periods of moderately flexible structures due to SSI may have a detrimental effect on the structural behaviour [18]. The work presented herein arises in this context. Only a few authors have investigated the dynamic response of beams or bridges and, in particular, the conditions of resonance and cancellation phenomena taking into account the wave propagation in the soil. Lu et al. [19] prove numerically the occurrence of resonance and cancellation in a periodic viaduct subject to moving loads considering pile-soilstructure interaction. Wu and Yang [20] apply a semi-analytical approach to analyse ground vibrations induced by trains moving over elevated bridges. The authors use impedance functions to represent the foundation-soil interaction and an elastic half space model for the soil wave propagation problem. In [21] and [22] the authors investigate ground vibrations induced by High-Speed trains crossing continuous girder bridges and rigid-frame viaducts, respectively. In both contributions the ground response is calculated by applying reaction forces on a 3D FEM with artificial viscous boundaries. Takemiya and Bian [23] investigate numerically the waves generated in the soil near a Japanese Shinkansen multi-span viaduct. The authors also present field tests measurements on the foundations and in the ground far field, showing frequency contents related to train axle distances and structure natural periods. In [9] ¨ UlkerKaustell et al. present a qualitative analysis of the dynamic SSI phenomenon on a portal frame railway bridge based on dynamic stiffness functions. The authors conclude that the contribution of the coupled soil-bridge system to the modal damping ratios is substantial, especially for the lower range of the soil elastic modulus. Most of the previous works focus on the level of vibrations transmitted through the soil along the track, rather than on the bridge behaviour itself. In the opinion of the authors of this contribution, there is a need to investigate how soil properties, in terms of flexibility and material damping, may affect the dynamic response of short SS bridges susceptible to experience excessive accelerations at the deck level. If this kind of analysis is performed with generality, i. e., considering expectable ranges of variation of structural and soil properties, interesting conclusions could be extracted regarding the appropriateness of the numerical models usually used by engineers when it comes to assess the performance of new structures, or that of existing structures subjected to more demanding operating conditions. In this study the authors complete the investigation initiated 2 in reference [24], extending the analysis to several soil types with different levels of material damping, and particularizing the conclusions extracted to the case of a real structure. In what follows a comprehensive ensemble of soil-bridge systems is defined covering typical lengths and structural typologies of short to medium span SS railway bridges, and a wide range of variation of soil flexibilities and material damping values. A sensitivity analysis is conducted on this ensemble and the evolution of the bridges natural frequencies and structural damping levels is evaluated with the properties of the soil. The amplification of the bridge dynamic response in free vibration under a single moving load (SML) is then presented, and, based on this analysis, conclusions regarding the evolution of the resonant and cancellation phenomena induced by multiple moving loads (MML) with soil properties is discussed. Finally, the dynamic response of a real bridge belonging to the Spanish railway network is analysed under railway traffic. The evolution of the structure response under different order resonant conditions and under not resonant conditions with the flexibility and damping of the surrounding soil is evaluated. Finally conclusions are extracted regarding the adequacy of numerical models that disregard SSI effects. 2. Formulation and approach adopted 2.1. Approach of the investigation The numerical model implemented for the investigation has been previously presented in [24] and its main features are summarized herein. It is a fully coupled 3D BEM-FEM model integrated in the time domain. The SSI problem is analysed by domain decomposition in the soil and structure sub-domains, represented with the BEM and the FEM, respectively. BEMFEM coupling is performed directly. A scheme showing the main parts of the model is represented in Figure 2. The main features of the BEM-FEM model are: •A beam FEM is used to represent the deck flexural behaviour under moving loads, therefore assuming that the maximum transverse response of the structure is mainly governed by its longitudinal bending deformation. This decission is justified by the facts that: (i) according to previous studies [6, 25], short to medium span SS railway decks are expected to exhibit maximum vertical acceleration levels at mid-span; (ii) in reinforced concrete slabs or prestressed concrete girder decks, usual typologies for the range of lengths under consideration, resonances of the first torsion mode are usually not determinant in the assessment of the Ultimate Limit State of vertical acceleration [25]; (iii) the objetive of this investigation is to evaluate SSI effects on the main resonant problem that railway decks may experience under railway traffic. •The beam bridges are idealised as Bernoulli-Euler (BE) beams in a finite element context. The beam is discretized using two node beam elements with tension, compression, torsion (not excited considering the 2D nature of the applied loads), and bending capabilities. The choice of the Figure 2: Schematic representation of the 3D BEM-FEM coupled model Bernoulli-Euler theory is well suited for the analysis of railway bridges in this study due to the slenderness ratios of typical railway decks [1, 26]. Moreover, the frequency range of interest in the study is low (under 30 Hz) and mismatches between Bernoulli-Euler and Timoshenko beams are expected to be relevant above 50 Hz [27]. •The influence of the track and the ballast, which can also affect the dynamic behaviour of the bridge [28–32], has been taken into account only by means of the associated dead masses. A detailed vehicle idealisation, that would cause a reduction in the vibration levels of the bridge [33, 34] at resonance and of the ground [35], is also disregarded, and a moving load model has been used during the investigation. These simplifications, consistent with common design practices, have also been adopted in prior investigations of the resonance and cancellation phenomena in railway bridges [6, 11, 16, 17], and it has been considered convenient in a first approach to the problem. Additionally, as it will be shown in what follows, vehiclebridge interaction and SSI will both lead to a reduction of the deck acceleration at resonance. On the authors opinion it is essential to separate both effects in order to capture the effects caused by the soil separately and be able to extract conclusions in this regard. •The railway excitation is introduced as a sequence of moving loads travelling at constant speed, therefore neglecting vehicle-structure interaction effects. The gradual nature of the wheel loads application process close to the abutments due to the distributive effect of rails, sleepers and ballast must be simulated in order to avoid unrealistic high frequency modal contributions. To this end, a load distribu3 tion function based on the Zimmerman-Timoshenko solution for an infinite beam on Winkler foundation, is applied to the axle load modulus in the abutments proximities. Details of the formulation may be found in [8]. •The beam end sections are connected through kinematic constraints to two rigid plates representing the lower surface of shallow foundations at the abutments. These plates are coupled to the boundary elements simulating the interaction with the soil. With this simple idealization, the essence of the wave propagation problem is isolated from the foundations geometry, and its influence is evaluated considering only the bridge vibration response [36]. •Regarding the soil treatment, a homogeneous soil with constant properties is admitted. The Green’s function for an elastic half-space is used as the fundamental solution for displacements and tractions in the BEM [37]. Therefore, the boundary element discretisation is limited to the interface between the soil and the plates. The soil is discretised using nine node rectangular quadratic boundary elements. •Coupling of the BEM and FEM equations is carried out by imposing equilibrium and compatibility conditions at the soil-structure interface. Both systems of equations are assembled into a single system, together with the equilibrium and compatibility conditions [38]. The described model is implemented in the SSIFiBo toolbox for MATLAB previously developed by coauthors of this contribution Galv´ ın and Romero [39–41]. The FEM module of the toolbox does not include any pre-processor. Instead, a gateway for commercial software allows importing directly the structure model. Using this model, SSI effects on the transverse response of beams traversed by moving loads at constant speeds are studied by means of the following complementary steps: 1. First (section 3.1), a preliminary analysis is presented based on the frequency response function (FRF) of a soilbridge system under impulse excitation, with the aim of anticipating the influence of the soil properties on the bridges response in the frequency domain. This issue is related with the relative values of the Rayleigh and beam bending wavelengths. 2. Second (section 3.2), the variation of modal parameters (fundamental frequency and modal damping) of the bridges under study considering SSI is analysed. It should be remarked that, as explained in section 2.3, the bridges and soil properties have been selected covering a wide range of realistic combinations in the design of short SS railway bridges. 3. Third (section 3.3), the maximum response of the structures under the circulation of a single moving load in terms of the uniform speed is presented, and the conditions for maximum response and cancellation during the free vibration phase (once the load has left the structure) are shown. General conclusions regarding the influence that soil properties may have on resonant speeds and associated amplitudes are extracted from these results. 4. Finally (section 4), the dynamic response of a real SS railway bridge belonging to the Spanish railway network is analysed under the circulation of a train of moving loads exciting two relevant resonant situations in the range of speeds considered. The influence of the soil flexibility and material damping is investigated when the bridge undergoes resonances of different order and at non-resonant conditions. 2.2. BEM-FEM mathematical formulation The BEM is based on a time marching procedure to obtain the time variation of the boundary unknowns; i. e., displacements and tractions. The k−th component for displacements and tractions over the boundary is approximated from the nodal values jat each time step m,um j kand pm j k, using the space interpolation functions φj(r) and ψj(r), for tractions and displacements, respectively. After interpolating the boundary variables, the integral representation of the displacement uat a point ion the boundary becomes [40]: ci lkui k(xi,t)= n X m=1 Q X j=1    ZΓj Unm lk ψjdΓpm j k −ZΓj Pnm lk dτφjdΓum j k     (1) where Qis the total number of boundary nodes and Γjrepresents the elements to which node jbelongs. Time kernels Unm lk and Pnm lk are respectively computed through the fundamental solution for displacements and tractions due to a point load acting at xiin the ldirection. These kernels are analytically integrated by parts using constant and linear piecewise time interpolation functions for tractions and displacements [37], respectively. Eq. (1) may be written in a more compact form as: ci lkuni k= n X m=1 Q X j=1hGnmi j lk pm j k−b Hnmi j lk um j ki(2) Once the integral-free term ci lk is included in the system matrix, the integral representation for point iat time t=n∆tbecomes: Hnnun=Gnnpn+ n−1 X m=1Gnmpm−Hnmum(3) where Hnmi j lk collects for ci lk when i=jand n=m. The FEM equation at time step nis defined as [42]: M¨ un+C˙ un+Kun=fn(4) where M,CyKare the mass, damping, and stiffness matrices, respectively. un,˙ uny¨ unrepresent nodal displacement, velocity, and acceleration vectors, respectively, and fnis the load vector 4 including the effect of the constant moving load at each timestep. Equation 4 is solved using an implicit time integration GN22 Newmark method [42, 43]. An equivalent dynamic stiffness matrix is defined: Dun=fn+fn−1(5) Coupling of BEM and FEM equations (Eqs. (3) and (5)) is carried out by imposing equilibrium and compatibility conditions at the soil-structure interface. Both systems of equations are assembled into a single global system, together with the equilibrium and compatibility equations [44]. As the plate foundations have been defined as rigid bodies in a first approach through kinematic constraints, the BEM Eq. (3) is expressed in terms of the kinematic constraint matrix L relating the displacements and tractions of the central point of the plate, u0and p0, respectively, with any other point for each foundation: HnnLun 0=GnnLTpn 0+ n−1 X m=1hGnmLTpm 0−HnmLum 0i(6) where equilibrium of forces at the interface Γis fulfilled integrating nodal tractions according to the element shape function matrix N: f=ZΓ NTpN dΓ = Tp (7) The time step ∆tfor the analysis is set sufficiently small to properly integrate the structure dynamic response and load excitation. This may be expressed as: ∆t=min 2π ω1kω ,L Vkv!(8) where ω1corresponds to the fundamental frequency of the beam, Lis the beam length, and Vthe load speed. Parameters kωand kvdefine time discretizations for the structure fundamental period and the load passage time, respectively. The chosen time step determines the spatial boundary element discretization according to the stability parameter β= cs∆t/∆l, where ∆lis the distance between two nodes of a boundary element, and csis the shear wave propagation velocity in the soil. In this work, a stability parameter β=0.5 has been considered. The finite element representation is determined by the bridge bending wavelength discretization. Minimum wavelength is defined by the maximum frequency range and the phase bending wave propagation velocity in the fundamental mode cb1= 4 qω2 1EIz/mb, where EIzis the beam cross-section bending stiffness and mbis the beam mass per unit length. This work considers 20 elements for the minimum wavelength. 2.3. Definition of an ensemble of soil-bridge systems In this section the ensemble of soil-bridge systems investigated in the sensitivity analysis included in section 3 is presented. Bridge beam models of lengths ranging from 12.5 to 25 m in increments of length of 2.5 m are considered, covering the typical span lengths susceptible to experience high deck vertical accelerations under resonant conditions. The range of fundamental frequencies realistic for each span is selected from the band prescribed by Eurocode 1 [5] for the application of simplified methods (see Figure 3). Therefore, the vast majority of existing and potential SS bridges fundamental frequencies are expected to fall within these limits. Three evenly-spaced sample values between 0 % and 70 % of Eurocode 1 upper frequency limit have been analysed. These frequencies are referred to as f1,000,f1,035 and f1,070 in what follows. As can be extracted from the studies presented in [33], the majority of railway bridges for conventional and High-Speed lines fall within the selected range. In a first approach, a single value of mass per unit length is assigned to each beam, in particular mb=L(m)·1000 kg/m2. The mass of the structure will affect the level of vertical acceleration at resonance, but it has not been selected as a parameter for the sensitivity study as it does not govern the maximum free vibration and cancellation conditions in the absence of SSI effects [6]. Structural damping is not assigned to the beams in section 3 in order to isolate the effects of SSI on the modal parameters of the bridges under study. Regarding the substructure, identical 5m ×5m foundation plates are considered in all the cases to represent the soil-substructure interaction surface. As per the soil properties, three homogeneous soil types are defined with flexibilities covering the AASHTO classification [45]. In particular shear (s) and dilatation (p) wave velocities of cs={150,220,365}m/s and cp=2csare considered, admitting a Poisson’s ratio ν=1/3 for the soil. Soil density has been set equal to 1800 kg/m3in all the cases. Soil material damping levels of ζs={0,2.5,5}% are considered for each shear wave velocity. Therefore 180 BEM-FEM models are evaluated in the following sections (18 bridges ×9 soil types plus 18 bridges with infinitely rigid soil conditions). 3. Sensitivity analysis 3.1. SSI effect on the bridges behaviour in the frequency domain. Preliminary analysis In order to get some insight regarding how SSI may affect the bridges dynamic response depending on the frequency range, a preliminary analysis is included in this subsection. First, beams and soil wavelengths are computed and represented in order to estimate the frequency range in which the interaction between the structure and the soil could be appreciable. Second, the impulse response of a particular beam is presented and the effect of the soil flexibility and damping are shown in the frequency domain. Figure 4 represents the Rayleigh wavelength of the soil λR= cR/fand the beam bending wavelength λb=cb/fin terms of the frequency f.cRstands for the Rayleigh wave propagation velocity in the soil, approximated as in [46], and cbfor the beam bending wave propagation velocity: 5 ζs=5.00 % ζs=2.50 % 5 10 15 f1,000 f1,035 f1,070 f1,100 ζs=0.00 % cs=inf f1(Hz) ζs=5.00 % ζs=2.50 % f1,000 f1,035 f1,070 f1,100 ζs=0.00 % cs=150 m/s ζs=5.00 % ζs=2.50 % 510 15 20 25 30 5 10 15 f1,000 f1,035 f1,070 f1,100 ζs=0.00 % cs=220 m/s L(m) f1(Hz) ζs=5.00 % ζs=2.50 % 510 15 20 25 30 f1,000 f1,035 f1,070 f1,100 ζs=0.00 % cs=365 m/s L(m) Figure 3: Ensemble of soil-bridge systems under study. ◦Frequency and span length of analyzed bridges for particular soil properties (cs, ζs) cR=0.87 +1.12ν 1+νcscb=4 qω2EIz/mb(9) Both wavelengths have been normalised with respect to the beam wavelength in its fundamental mode for SS conditions (λ1,ss =2L). In Figure 4 each graph corresponds to a set of bridges with natural frequencies in the SS case corresponding to levels f1,000,f1,035,f1,070 and f1,100 in the Eurocode frequency band. f1,100 is considered only in this subsection for comparison purposes, as it does not represent common structures of the typologies of interest. In each plot, different curves associated to the same soil type (same gray colour traces) correspond to different span lengths L=12.5 m to L=25 m. Moreover, in the horizontal axis the frequency has been normalised by the fundamental frequency of the SS beam ( f/f1,ss). This normalisation allows to represent all the beams with a single curve given that λb/λ1,ss =(f2/f2 1,ss)0.25. Two frequency regions may be distinguished for each soil-beam system: (i) a region where the Rayleigh wavelength of the soil is higher than the beam bending wavelength (low frequency range); and (ii) a region where the Rayleigh wavelength of the soil is lower than the wavelength of the beam. The cut-offfrequency between both regions may be easily obtained equating both wavelengths (λR=λb), entailing that cR=cb. Notice that the beam bending wavelength starts to exceed the soil Rayleigh wavelength at a frequency that increases with the soil stiffness and the beam fundamental period. It should be therefore expected SSI to be more perceptible at low frequencies on the bridge response in the case of more flexible soils (lower values of cs) and of beams with higher fundamental frequencies. Moreover, higher modal contributions of the beams should be more affected by SSI effects than the response associated to the fundamental mode. As an example, Figure 5 shows the frequency response function at L/4 for a bridge span L=15 m, considering four bridge frequencies covering the complete Eurocode range for that particular length, and different soil conditions (wave propagation velocities and damping ratios). The FRF is computed loading the structure with an impulsive force acting on the same section. The FRF shows peaks at the frequencies corresponding to the first three bending modes of the bridge. The bridge frequencies and the peaks amplitudes move toward lower values as the soil becomes softer, and SSI effects become more important. Also, bridges with higher natural frequencies are most affected by SSI. Moreover, it can be concluded from this analysis that the influence of the soil material damping is almost imperceptible at the fundamental frequency of the structure and it is much more noticeable in the frequency range above the aforementioned cut-offfrequency. In the following sections modal properties of the bridge catalogue under study are identified, and SSI effects on the conditions for maximum free vibration and cancellation of the bridges under a SML are evaluated. 3.2. Identification of modal parameters In view of the results of the previous section and following the approach in [24], a parameter κ=EIzπ3/(KvL3) is defined as the ratio of the flexural rigidity of the bridges to the vertical 6 12345678910 f/f1,ss 0 0.5 1 1.5 2 2.5 3 λ/λ1,ss (a) f1,000 12345678910 f/f1,ss 0 0.5 1 1.5 2 2.5 3 λ/λ1,ss (b) f1,035 12345678910 f/f1,ss 0 0.5 1 1.5 2 2.5 3 λ/λ1,ss (c) f1,070 12345678910 f/f1,ss 0 0.5 1 1.5 2 2.5 3 λ/λ1,ss (d) f1,100 Figure 4: Rayleigh wavelength (λR/λ1,ss) for different bridge spans (12.5 to 25 m) and soil properties: cs=150 m/s, cs=220 m/s and cs=365 m/s. Beam bending wavelength (λb/λ1,ss) stiffness of the soil-foundation supports under static loading, Kv. In Figure 6 the values of κfor the soil-bridge systems under study are represented (notice that soil material damping does not affect this parameter). κ=0 corresponds therefore to an infinitely rigid soil. The natural frequency and modal damping associated to the fundamental mode are obtained from the beam response subjected to an impulse load for the complete ensemble of bridges under study (108 BEM-FEM models). The variation of these two modal parameters in terms of κfor all the soil-bridge systems are included in Figures 7 and 8, respectively. 3.2.1. Effect of soil properties on identified natural frequencies In Figure 7 the fundamental frequency variations, with respect to infinitely rigid soil conditions, experienced by the bridges are represented with circles. These results are calculated using the BEM-FEM model described in section 2.2. Figures in the same row correspond to the same soil shear-wave velocity, while figures in the same column correspond to the same value of soil material damping. In all the plots, the analytical solution for the fundamental frequency variation of an elastically supported (ES) Bernoulli-Euler beam with identical elastic supports of Kvvertical stiffness has been represented in thick black trace [6]. Finally, in each graph different frequency bands are distinguished in shaded areas, and circle sizes are proportional to the lengths of the bridges. The vertical flexibility of the soil-foundations leads to a reduction in the fundamental frequency of the bridges under study. This reduction is more evident in the case of bridges with higher natural frequencies (and therefore, higher values of κ). This is consistent with the results presented in section 3.1. For each frequency group, bridges with longer spans are most affected by soil conditions. This is due to the fact that longer bridges present higher κvalues [24]. From the analysis of the results presented it may be concluded that: (i) the frequency variation experienced by the structures when SSI is included follows the general trend shown by the ES BE beam in terms of the static relative stiffness parameter κ. The frequency dependence of the soil-foundation stiffness is not relevant, especially for low κvalues and long bridges; (ii) soil-bridge systems with similar κvalues show similar frequency variations, independently of the soil properties and the beam natural frequency in the absence of SSI; (iii) for each soil type and frequency band, shorter bridges show a slightly higher deviation with respect to the analytical solution of the ES beam; (iv) these tendencies take place for different soil material damping levels, and the influence of this parameter is almost negligible regarding the variation of the bridges fundamental frequency. This issue was anticipated in section 3.1 for low frequency ranges. 7 12345678910 f/f1,ss 10−12 10−10 10−8 10−6 FRF (m/N) (a) f1,000 12345678910 f/f1,ss 10−12 10−10 10−8 10−6 FRF (m/N) (b) f1,035 12345678910 f/f1,ss 10−12 10−10 10−8 10−6 FRF (m/N) (c) f1,070 12345678910 f/f1,ss 10−12 10−10 10−8 10−6 FRF (m/N) (d) f1,100 Figure 5: FRF of a bridge of L=15 m and soil properties: cs=150 m/s, cs=220 m/s and cs=365 m/s. Bridge response in SS case. Considering the following damping ratios: ζs=0.000 (solid lines), ζs=0.025 (dashed lines) and ζs=0.050 (dotted lines) f1,000 f1,035 f1,070 0 0.2 0.4 0.6 25.0 m 22.5 m 20.0 m 17.5 m 15.0 m 12.5 m cs=150 m/s cs=220 m/s cs=365 m/s f1,band κ Figure 6: Dimensionless ratio κfor the soil-bridge systems under study 3.2.2. Effect of soil properties on identified modal dampings In Figure 8 the values of structural damping in the fundamental mode identified from the bridges response for the ranges of soil properties under consideration are represented with circles. The structural damping ratio including SSI effects (ζ1) is identified from the free damped response of the bridges through Logarithmic decrement. As in Figure 7, different frequency bands are distinguished in shaded areas, and circle sizes are proportional to the bridges lengths. The modal damping measured from the bridge response strongly depends on the value taken by the relative flexibility κ. As the flexibility of the soil increases (higher κlevels for the same structure), so does the identified damping due to the wave radiation through the soil. Again, for the same soil properties, structures with higher fundamental frequencies in the absence of soil exhibit higher increments of structural damping when SSI is considered. This is again consistent with the analysis presented in section 3.1. On the other hand, the influence of the soil material damping on the identified structural damping is minimal in the fundamental mode. As it was exposed in the previous section, soil damping only modifies the structural response during a short transient, due to the relative wavelengths of the soil and the bridges. Therefore, the influence of soil damping is expected to be significant only at higher frequencies than the bridge fundamental one. 8 0.7 0.8 0.9 1.0 f1,070 f1,035 f1,000 ζEC 0=1.00 % f1/f1,SS ζs=0.00 % f1,070 f1,035 f1,000 ζs=2.50 % f1,070 f1,035 f1,000 ζs=5.00 % 0.7 0.8 0.9 1.0 f1,070 f1,035 f1,000 f1/f1,SS f1,070 f1,035 f1,000 f1,070 f1,035 f1,000 0.1 0.30.5 0.7 0.8 0.9 1.0 f1,070 f1,035 f1,000 κ f1/f1,SS 0.1 0.30.5 f1,070 f1,035 f1,000 κ 0.1 0.30.5 f1,070 f1,035 f1,000 κ 0.1 0.30.5 cs=365 m/s cs=220 m/s cs=150 m/s Figure 7: Bridge identified fundamental frequency vs. κ. Analytical ES beam case 3.3. Free vibration response under a SML According to [6] the dynamic amplification of a SS or ES beam at resonance caused by the circulation of MML is closely related to the free vibrations that the same beam experiences after the passage of each single load travelling at the same speed. The load travelling at certain speeds, induces on the beam a remarkably high response (maximum free vibrations) and, at some other speeds, the oscillations when the load leaves the beam are almost negligible (cancellation of free vibration). These two phenomena are independent of the periodicity of the loads, and take place for a single moving load. The aim of this section is to evaluate how SSI affects these two conditions. In [24] preliminary results were presented in this regard. Now the response of the complete ensemble of soil-bridge systems defined in section 2.3 is obtained in free vibration after the circulation of a SML in a wide range of speeds. Let us define a dimensionless speed KS S 1referred to the bridges fundamental frequency in the absence of soil, KSS 1=ΩSS 1 ωSS 1 =πV ωSS 1L(10) In Eq. (10), ωSS 1is the fundamental circular frequency of the bridge with SS boundary conditions, while ΩSS 1=πV/Lis used to represent the forcing frequency of the SML. The bridges under analysis are those indicated in Figure 3, considering soil shear wave velocities cs= {150,220,365,∞}m/s. Structure and soil material damping is neglected in this study. For each soil-bridge system, 70 evenly spaced values of KS S 1have been selected between 0.1 and 0.5. As detailed in [6] this range suffices to cover the circulation speeds expected in nowadays railway systems. In order to accurately capture the variation of the cancellation conditions when SSI is included, twenty additional speeds are computed within the ranges [0.85,1.15]KS S 1,ci, where KS S 1,ci represents the ith nondimensional cancellation speed of the first mode in the SS case. For each circulation speed, the maximum vertical displacement at the bridge mid-span section is computed, once the load has left the structure. This result, divided by the static displacement, leads to the dimensionless quantity Rrepresented in Figure 9. As the response is obtained at mid-span and due to the time-step used in the numerical integration, Rbasically corresponds to the contribution of the fundamental mode of the beam to the total response. In Figure 9 all the curves obtained for the 72 soil-bridge systems under study (18 bridges ×4 soil types) are plotted simultaneously. The curves are distinguished using a colour code based on the value of parameter κ. From the analysis performed, the following can be concluded: (i) maximum free vibration and cancellation conditions alternate with the increase of the load speed, in the same way that happens in the absence of soil; (ii) far from cancellation conditions, models without SSI always predict a higher response than those including SSI; (iii) as κ increases for softer soils and bridges with higher natural frequencies, the amplification reduces between two cancellation conditions; (iv) the cancellation speeds slightly decrease as the relative stiffness κincreases. This variation is associated with the descent of the structure fundamental frequency with the soil flexibility. An accurate prediction of the cancellation speeds is crucial, for instance, when planning an experimental test with the aim of measuring structural parameters i. e., damping. In Fig9