Influence of Soil Stiffness on the Dynamic Behaviour of Bridges Traveled by High Speed Trains

Influence of Soil Stiffness on the Dynamic Behaviour of Bridges Traveled by High Speed Trains João Devesa Abstract The present paper aims to study the influence of the soil/structure interaction in the dynamic behaviour of bridges, when traveled by high speed trains. Two bridges are subjected to the HSLM-A load models, thus widening the field of study to different types of structures. Different models with soil inclusion are created in order to study, comparatively to the model with stiff supports, the results for accelerations, displacements and section forces at the structures deck. The study evidences the aggravation in the dynamic behaviour of the bridge when soil is considered, hence the importance of its simulation. Accelerations are always higher in such models, being also verified that the prescribed dispositions are not fulfilled in some models. In addition, section forces are always more severe in those models as well. The results are very coherent. Analysis to each model eigenfrequencies reveal sense in the consideration of soil stiffness, and it was verified a deep dependance between them and the effects on the structures behaviour. It is suggested one way to face, in the future, the dynamic analyses with soil simulation at the numerical models. 1 Introduction Countries like Portugal and Sweden will soon be part of the European high-speed network, for that represents an important economic development. This network is expected to reach the remarkable accomplishment of 6000 km by the end of the decade, which can be explained by the relation between covered distance and time, passenger safety and comfort and low levels of emitted pollution. The first European high-speed project was commenced in 1976 by the French government, with the 410 km line between Paris and Lyon, designed for a running speed of 270 km/h. The passenger-only concept allowed drastic reductions of time, making this project an enormous success. The X2 tilting train connects many cities in Sweden at 200 km/h, including Stockholm, Gothenburg and Malmö. There are, however, plans for an upgrade of the velocity up to 250 km/h when connecting these cities, but there are already high-speed lines under construction, such as the Trollhättan-Gothenburg and the Bothnia Railway. Ongoing agreements mention a new line connecting Stockholm and Copenhagen via Gothenburg, including a new bridge from Helsingborg to Helsingør, allowing speeds between 300-320 km/h, with non-tilting trains. The Portuguese high-speed network is highly dependable on the partnership with Spain. Therefore, during the XIX Iberian Meeting [1], the following high-speed railways were agreed: Lisboa- Madrid (by 2013), separating the two capitals by 2 hours and 45 minutes; Porto-Vigo (ready in 2009), connecting both cities by less than 45 minutes; Aveiro-Salamanca (opening in 2015); Faro-Huelva (concluded before 2018). The Lisboa-Porto connection is also planned to be ready in 2015, which will take 1 hour and 15 minutes to be covered. When the speed of the train is higher than 200 km/h, different analyses are required from those used for conventional trains. In the case of bridges and viaducts, dynamic analyses must be conducted because the risk of resonance increases and to conduct a static analysis might not be sufficient to know its real effect. This article studies the interaction between structure and soil, which is still neglected by the time of bridge design. The soil stiffness when modelling bridges for high-speed trains will be in prominence, and proposals about the importance of considering soil in the numerical analyses shall be presented. 2 Review of Previous Studies 2.1 State-of-the-art The first known study in history concerning dynamic behaviour of bridges was performed by Willis [2] in 1847, when the Queen of England gathered a commission to investigate the collapse of the Chester rail bridge over the river Dee. Although an exact solution for the problem was encountered, it should be seen only as a remarkable first approach onto the subject, due to over-simplifications such as considering a massless beam and neglecting its inertial effects. In 1928, the work of Timoshenko [3] provided a full understanding about the dynamic behaviour of prismatic bars submitted to a force traveling at constant speed. One year later, Jeffcott [4] included the inertia of the vehicle in his work, later on continued by Stanišić and Hardin [5]. In mid 1

1930 s, Inglis [6] studied the influence of damping and the vehicle s spring suspension, concluding that a moving load can induce higher bridge responses than the ones given by the static analysis. The dynamic amplification factor was included in the bridge design, which are still used today. Frýba [7] assumed a very important role to the awareness of dynamic behaviour of structures under moving loads, where the author addresses a large number of situations regarding dynamic loaded beams. Studies from one to three dimension solids submitted to constant, harmonic, continuous, with two degrees of freedom, two axle-system or varying in time loads are included in this comprehensive reference work. Moreover, the effect of speed and damping were in prominence, as well as forces moving on elastic half-space. In [8], Frýba defines the term Interoperability as either the capability of a bridge to carry a vehicle running at a certain speed, or as the technical conditions which verify that the train can move on a railway line at the designed speed, according to the point of view of bridge engineers or vehicle specialists. The critical train speed is also proposed by the author, as c cr = d f j k, (1) j = 1, 2, 3...k = 1, 2, 3,..., 1/2, 1/3, 1/4,... with d standing for the distance between axles, f j representing the natural frequencies of the bridge and k-multiple of the period of natural vibration 1/f j. Another significant contribution was given by Yang, Yau and Wu [9], where several two and three dimensional numerical models were used to study the Vehicle-Bridge Interaction (VBI). Their proposals allow, for example, the measurement of passengers riding comfort, the suppression of resonance responses by choosing appropriate span lengths or to decide for damping devices on bridges with light damping. Xia and Zhang took the next step by measuring accelerations, strain gauges and displacements at a real bridge [10], and compared them with the numerical model [11]. In this model, track irregularities were included but the elastic effects of ballast, rail pads and fasteners were neglected. The Thalys train was used for the experiments and modelled as 10 vehicles, 13 bogies and 26 wheel-sets in a total of 115 degrees-of-freedom. Moreover, it was concluded that the computer simulation had an accurate resemblance with the in-situ measured data and, therefore, the numerical model is valid. In the Iberian Peninsula, Goicolea and Galbadón [12] studied the influence of critical damping, torsional deck stiffness and cracked sections with negative moments on the dynamic response (accelerations, displacements and impact factor, Φ) of a composite viaduct. On the other hand, Delgado and Santos [13] investigated the importance of the mass and stiffness of a bridge, the stiffness of the train, bridge span and track irregularities on the structural behaviour of a bridge and on the comfort at the train. 2.2 Soil behaviour and vibrations Ground vibrations due to dynamically loaded foundations can induce, if in a high level, damage to buildings, disturbance of people and cause interference with local processes. They are originated by the occurrence of high levels of accelerations at the bridge deck. Barkan [14] was a pioneer in the studies about soil properties and dynamic soil behaviour. In his extensive work, the author formulates spring constants of rectangular footings resting on elastic half-space, addressed in section 3. Moreover, focus was given to the propagation of elastic waves in the soil, and a formulae to determine the effects caused by such vibrations was proposed as follows, based on the calculation of the maximum soil particle velocity: v max = v 0 ( r0 ) ne [ λ(r r 0)] r (2) where v o is the maximum vertical velocity of the foundation, r 0 represents the radius around the foundation, n stands for a geometrical damping factor (0.5-1.0) and λ designates a factor for the damping in the ground. The limiting criterion is normally that v max shall not exceed 0,1 mm/s at a distance of r=50 m from the track. Krylov [15] proposed measures to reduce the vibration impact on the surrounding ground, when trains exceed Rayleigh s wave velocity (around 125 m/s, thus it is still a very uncommon situation). In those cases, the waves are radiated forward with an intensity of approximately 1000 times higher when compared with a train going under that speed. Therefore, specially modified embankments or trenches are referenced as a good prevention for soil damage. Free field vibrations were measured for the passage of a Thalys train during the homologation test for the Brussels-Paris high-speed track. Degrande and Schillemans [16] created an unique data set, for they have given special attention to the influence of the train speed on the vertical velocity of the 2

ground, concluding there is a very weak dependance between them (see figure 1). K = υ 2L Φ n 0 ; (6) ϕ = a [ 100 56e ( L Φ 10 ) 2 LΦ n ) 0 + 50( 1 e ( L Φ 20 ) 2] ; 80 (7) a = min ( V 22 ; 1) ; (8) - K stands as the variable for the ϕ calculation, υ is the train speed, L Φ represents the determinant length, n 0 is the first natural frequency and V stands for the design speed in (m/s). Figure 1: Measured time histories of the free field vertical velocity at 8 m from track during the passage of a Thalys HST at varying speeds (km/h): (a) 256; (b) 271; (c) 289; (d) 300; (e) 314. (in [16]) Auersch [17] used a combined finite element and boundary element method to calculate the dynamic compliance of a track on realistic soil, and comparisons were established with results from ICE test runs. The following conclusions were drawn: the effect of sleeper passage is a harmonic force increasing with speed, until the passage frequency meets the track resonance frequency; vehicle-track eigenfrequencies become more dominant if soil damping is diminished with resource to rail pads; highfrequency excitation is influenced by the irregularities of the vehicle, and track irregularities are related to the medium-frequency ground vibrations. Other researchers have significantly contributed to the scientific community by developing boundary element models, e.g., Galvín and Domínguez [18] and Lombaert et al. [19]. 2.3 Design codes and International reports In the 1970 s, the International Union of Railways (UIC) studied the dynamic behaviour of bridges. An expression for the dynamic amplification factor, (1 + ϕ), was presented: A load model to cover a set of six different trains was created and called Load Model 71, together with SW/0 and SW/2 (representing the static effect of vertical loading on continuous beams due to normal and heavy traffic, respectively). At the same time, the dynamic or impact factor, Φ was established to be multiplied by the static response of such load models. It is taken as either Φ 2 or Φ 3 according to the quality of track maintenance (carefully of standard maintenance) as follows: Φ 2 = 1, 44 LΦ 0, 2 + 0, 82, with: 1, 00 Φ 2 1, 67 (9) 2, 16 Φ 3 = LΦ 0, 2 + 0, 73, with: 1, 00 Φ 3 2, 00 (10) Figures 2 and 3 show LM71, SW/0 and SW/2; table 1 refers the characteristic values to adopt in figure 3 (from [20]). Figure 2: Load Model 71 and characteristic values for vertical loads (in [20]). 1 + ϕ = 1 + ϕ + λϕ (3) (1 + ϕ ) is the DAF for a track without irregularities, ϕ stands for the amplifications due to track irregularities and λ is a coefficient dependant on the track level of maintenance. and ϕ = K, for K < 0, 76; (4) 1 K + K4 ϕ = 1, 325, for K 0, 76; (5) Figure 3: Load Models SW/0 and SW/2 (in [20]). With the high-speed trains appearance, a new problem arose: greater speeds induce higher dynamic effects on bridges, in such way that the DAF could no longer cover the train effects. Hence, it was agreed by the UIC, in 1996, to create the Specialist Sub- Committee D214, with the purpose of studying the 3

Table 1: Characteristic values for vertical loads for Load Models SW/0 and SW/2. Load Model q vk [kn/m] a [m] c [m] SW/0 133 15.0 5.3 SW/2 150 25.0 7.0 dynamic effects including resonance in bridges for high-speed trains up to 350 km/h [21]. One of the first steps was to create a set of new Load Models that could represent an envelope for the signatures of modern trains. Those Load Models comprises two separate Universal Trains, HSLM-A and HSLM-B. The first one gathers 10 trains with the configuration presented in figure 4. The coach length, D, varies between 18 and 27 m and point forces between 170 and 210 kn. On the other hand, HSLM-B is composed by N number point forces of 170 kn with constant spacing d [m], as defined in figure 5. This Load Model is applicable for simply supported bridges with less than 7 meter length and where the wave length (λ = υ/n 0 ) does not exceed 4.5 m. where ϕ dyn = max y dyn/y stat 1; y dyn and y stat are the maximum dynamic and static response, respectively. ϕ /2 is the increase in calculated dynamic load effects resulting from track defects and vehicle imperfections. The maximum permitted acceleration is a traffic safety issue that was deeply studied by the Committee. In-situ measurements and laboratory tests confirmed that an excess of 0.7g for deck accelerations may cause loss of ballast interlock, loss of rigidity of both track vertical support and track lateral resistance, increasing the risk of rail movement. By applying a Safety Factor of 2, the maximum permitted deck accelerations for ballasted bridges is 3.5 m/s 2, and for non-ballasted bridges, that value can go up to 5 m/s 2. Moreover, being known that very high frequency accelerations do not lead to deleterious ballast effects, the Committee recommended the following cut-off frequencies: - 30 Hz; - 1.5 to 2.0 times the frequency of the 1 st mode of vibration of the element being considered. The deflection limits (δ) for passenger comfort is established according to figure 6, as function of span length L [m], and train velocity V [km/h]. Figure 4: HSLM-A (in [20]). Figure 5: HSLM-B (in [20]). Simply conservative checks are useful to determine whether a dynamic calculation is, in fact, necessary. Nevertheless, when speed is higher than 200 km/h in structures that cannot be considered as simply supported bridges, a dynamic analysis is required (for further details see [20]). For the bridge design, taking into account all the effects of all vertical loads, the most unfavourable value of the following shall be used: or (1 + ϕ dyn + ϕ /2) (HSLM or RT ) (11) Φ (LM71 + SW/0) (12) Figure 6: Maximum vertical deformation (δ) corresponding to a vertical acceleration inside the coach of b v = 1.0 m/s 2, according to train s speed [km/h] (source - [22]). The graphic is applicable for railway bridges with three or more simply supported spans and when vertical accelerations inside the coach are up to 1.0 m/s 2 (very good comfort level). If other comfort levels are verified, the limit values of the L/δ relationship shall be divided by the respective acceleration, b v, presented in table 2. For decks with less than three spans L/δ shall be multiplied by 0.7 or, if the deck has three or more continuous spans, by 0.9. 4

Table 2: Comfort levels according to the vertical acceleration b v inside the coach. Comfort Level Vertical acceleration, b v [m/s 2 ] Very Good 1.0 Good 1.3 Acceptable 2.0 3 Theoretical Concepts The finite element program BRIGADE/Plus is based on an integrated implicit solver from ABAQUS and it was used in this article. It is divided into different modules for different tasks. The *.cae file stores all the input data provided in each module that is submitted for the numerical analysis. An output database (*.odb) file is generated which contains the results of the analysis, opened exclusively in the Visualization module. The analyses undertaken in this work followed the Mode Superposition Method, which is based on solving the second order differential equation of equilibrium: Mü(t) + C u(t) + Ku(t) = F(t) (13) Table 3: Frequencies of the simply supported beam [Hz]. Frýba Euler-Bernoulli beam Mode (Analytical model) (Numerical model) 1 11.61 11.61 2 46.45 46.45 3 104.52 104.52 4 185.80 185.82 5 290.32 290.40 6 418.06 418.28 Since the analytical solution is based on the Euler- Bernoulli s theory, quite similar results were verified, which was entirely expectable. Resonance speed of 175 km/h was adopted for the comparison of displacements [m] and accelerations [m/s 2 ] at the center of the beam and varying in time, as shown in figure 7. The graphics in red represent the response provided by BRIGADE/Plus, whilst the ones in blue are Frýba s analytical solution. The resolution of Frýba s expressions were made by using the software Mat- Lab 7.0, due to their complexity. where M, C and K are the mass, damping and stiffness matrices, respectively. u(t) is the displacement vector, assumed as u(t) = u cos(ωt ϕ). F is the vector standing for the externally applied loads. To demonstrate BRIGADE/Plus is a reliable and trustworthy finite element program, comparisons were established between a numerical model and Frýba s analytical solution [8]. Since the analytical solution is valid exclusively for simply supported beams, a 12 m span simple beam with 1 m height and 7 m width reinforced concrete bridge was used. The following properties were adopted: 1. Modulus of elasticity, E = 34 10 9 N/m 2 ; 2. Moment of inertia, I = 0.583 m 4 ; 3. Concrete density, ρ concrete = 2500 km/m 3 ; 4. Linear mass, µ = 17500 kg/m; 5. Damping, ξ = 0.015 (according to [20]). According to Frýba, the natural frequencies of a simple beam, ω j and f j, can be calculated based on: ω 2 j = j4 π 4 l 4 EI µ, f j = ω j, j = 1, 2, 3,... (14) 2π Table 3 resumes and compares the first 6 beam frequencies of the model. Figure 7: Comparison of the numerical response and Frýba s analytical solution for accelerations (left) and displacements (right) at a resonance speed of 175 km/h. Less modes of vibration are considered in the numerical model than in the analytical solution, therefore the graphics are not exactly equal. Despite that fact, one can positively state that BRIGADE/Plus provides accurate results from the numerical models. 3.1 Dynamic soil stiffness The dynamic stiffness of a soil is simulated in the program using translational and rotational elastic springs. Many researchers have proposed ways to calculate spring constants that could simulate the dynamic behaviour of soil. For rectangular base 5

foundations resting on elastic half-space, the following expressions were proposed by Barkan (1962) and Gorbunov-Possadov (1961), respectively. Table 4: Spring constants for rigid rectangular base resting on elastic half-space (from Whitman and Richart [23]). Motion Spring Constant Vertical K z = G 1 ν β z BL Rocking K φ = G 1 ν β φ BL 2 G represents the reduced shear modulus, ν is the Poisson s ratio, BL equals to the area of the footing and, finally, β z and β φ are given in figure 8, standing for the influence coefficients for vertical and rocking spring constants, respectively. Figure 9: Graphics for determining k Z. where ω [rad/s] stands for the frequency of the applied force - i.e., the axle train frequency - and V s [m/s] represents the shear wave velocity. By means of in-situ measurements, the maximum value of shear modulus, G max can be calculated as: G max = ρ V 2 s (19) where ρ is the mass density of the soil [kg/m 3 ]. The Swedish Railway Authority [25] also refers the shear modulus as a function of stress level and compacting grade, given by equation 20. Figure 8: Coefficients β z, β x and β φ for Rectangular Footings (from [23]). According to Gazetas [24], the dynamic soil stiffness for rectangular foundations can be obtained by multiplying the static stiffness (K) by a dynamic stiffness coefficient (k). Vertical static stiffness and its correspondent dynamic coefficient are given respectively as: with: G max = K 1 (σ m) C1 (20) - K 1 between 15000 and 30000, as a function of compacting dependant of the type of material; - σ m as the mean effective stress; - C 1 as an exponent varying between 0.4 and 0.6. K Z = 2GL 1 ν (0.73 + 1.54χ0.75 ), (15) with χ = A b /4L 2 kz = kz(l/b, ν, a0), (16) plotted in Graphics below. Rocking static stiffness (around x axis) as well as the respective dynamic stiffness coefficient were proposed by Gazetas as: K rx = G ( L ) 0.25 ( 1 ν I0.75 bx 2.4 + 0.5 B ), (17) B L with I bx =moment of inertia of foundation. k Z = 1 0.20a 0, (18) A b is the foundation-soil contact surface area, with equivalent rectangle 2L 2B; L > B. a 0 = ωb/v s, Figure 10: Normalized Shear Modulus according to dynamic shear strain level (from [25]). For both formulas, the dynamic shear strain must be estimated in order to define the reduced value of the shear modulus, G. The dependance of the shear modulus according to the dynamic shear strain is represented in figure 10 for different types of soil. 6

4 Case studies Dynamic analyses using HSLM-A are performed to two different reinforced concrete railway bridges. Bridge 1 is a single track bridge represented in figure 11, it has 41.5 meter of total length (15 meter for mid span). Each mid support has 2 fixed columns, and the end supports are 1.25 meter away from the end shields, with 2 neoprene bearings each. Mesh shell elements size at the deck are approximately 0.7 0.7 m 2. The dimension of the foundations are 6 3 m 2 and 7 4 m 2 for the end and mid supports, respectively. Figure 11: Bridge 1 with stiff supports. Bridge 2 is a double track short frame bridge. It is a 3.5 meter span and 11.2 meter width. The foundation slab has 4.5 m by 11.2 m, which is initially locked in all degrees of freedom (stiff model). The deck s mesh elements size are 0.3 0.7 m 2. A geometric representation of this bridge can be seen in figure 12. Figure 12: Geometric Model of Bridge 2. For both bridges, springs will replace the degrees of freedom initially locked, simulating soil stiffness. Thereby, comparisons between these models and the stiff model for each structure will be undertaken. Bridge 1 incorporates vertical and rocking springs at the supports and, for Bridge 2, a bed of vertical springs at the foundation slab is created. Based on formulas 19 and 20, G max was encountered to be 130 MPa and 210 MPa, respectively. Analyses to determine the dynamic shear strain revealed that a reduction of 20% must be performed for the soil at Bridge 1, and no reduction is necessary for Bridge 2 (details can be found in Devesa [26]). The final input spring stiffness values are given below in tables 5 and 6, calculated based on formulas by Whitman & Richart and Gazetas et al. Models 1, 2 and 3 were created for Bridge 1, and models 1 and 2 are related to Bridge 2. 1 Table 5: Input spring stiffness values [N/m] for each of the three analyses conducted for Bridge 1. Whitman & Richart Model 1 (G = 100 MP a) Vertical Rocking End Supports 1.5 10 9 1.1 10 10 Mid Supports 2.0 10 9 1.9 10 10 Gazetas et al. Model 2 (G = 100 MP a) Vertical Rocking End Supports 3.3 10 9 3.7 10 9 Mid Supports 4.1 10 9 7.8 10 9 Gazetas et al. Model 3 (G = 170 MP a) Vertical Rocking End Supports 5.6 10 9 6.3 10 9 Mid Supports 6.9 10 9 1.3 10 10 Table 6: Input spring stiffness values [N/m 3 ] for each of the three analyses conducted for Bridge 2. Whitman & Richart Model 1 (G = 130 MP a) Vertical Foundation Slab 7.0 10 7 Gazetas et al. Model 2 (G = 130 MP a) Vertical Foundation Slab 1.5 10 8 The cut-off frequency is the maximum frequency of interest for the modal analyses. The higher the selected value for the cut-off frequency is, the better the accuracy of the results is, since more eigenmodes are included in the analysis. However, there are practical limits on how high the cut-off frequency can be, as more solution steps are necessarily required for a higher cut-off frequency and the demanded computational time will increase accordingly. Cut-off frequencies shall thus be chosen in order to both minimize 1 A 3 rd model for Bridge 2 had been predicted as well (with G=210 MPa), but problems with the *.inp file of this model disabled running the respective analyses. 7

the analyses time and provide accurate and reliable results. 30 Hz was chosen for both bridges, regarding accelerations and displacements analyses [27]. In what concerns section forces, the most appropriate cut-off frequencies are 60 Hz and 160 Hz for Bridges 1 and 2, respectively [28]. Moreover, to choose an accurate time increment ( t) for the analysis is important to ensure the correct response is captured. Based on equation 21 [21], 0.0033 s was used for accelerations and displacements at both bridges, whilst for section Figure 15: Left) Maximum accelerations of node 187 forces 0.0017 s and 0.000625 s was adopted. according to speed (Model 2). Right) Time History of 1 Tmin the same node at 247.5 km/h, with HSLM-A2. = (21) t 10 10 fmax where fmax is the designated cut-off frequency. The dynamic analyses are performed by successively traveling HSLM-A1 to HSLM-A10 over the bridges, assuming velocities between 100 and 300 km/h, with a speed interval of 5 km/h (2.5 km/h around resonance peaks). For the node where the absolute highest acceleration was verified at each of the 4 models of Bridge 1, the graphics of speed plots and time histories were given (figures 13 to 16): Figure 16: Left) Maximum accelerations of node 187 according to speed (Model 3). Right) Time History of the same node at 242.5 km/h, with HSLM-A4. 1 is 6.89 m/s2 and 6.31 m/s2. Model 2 responded with accelerations within the range 5.32 m/s2 to 4.81 m/s2. Finally, model 3 had an acceleration interval of 3.36 m/s2 to 3.28 m/s2. It is clearly shown that models 1 and 2 are not in accordance with the prescribed recommendations, exceeding the admissible acceleration of 3.5 m/s2. Model 3 has Figure 13: Left) Maximum accelerations of node 189 also registered accelerations over the stiff model, according to speed (Stiff model). Right) Time History enhancing the idea that soil modelling cannot be of the same node at 170 km/h, with HSLM-A2. neglected. The analysis conducted for the displacements revealed the speed plots and time histories for the node with the highest value as shown in figures 17 to 20. Maximum downwards displacement were found out to be very similar for all models in terms of absolute results, and in accordance to figure 6. Moreover, it is possible to estimate the compression in soil based on speed plots (left figures): the downwards and upwards values are mirrored below Figure 14: Left) Maximum accelerations of node 187 0, since the downwards values are higher. Therefore, according to speed (Model 1). Right) Time History of that value corresponds to the compression in soil due to dynamic loads (admitting the stiff model the same node at 190 km/h, with HSLM-A4. displacement values are exactly mirrored in ordinate The accelerations at the stiff model vary between 0). 2.62 m/s2 and 2.82 m/s2 (downwards and upwards). On the other hand, the envelope for model The section forces of interest in this paper are 8

the shear force and bending moment over bridge deck. Envelopes of the results for the HSLMs are shown in the same graphic to compare the different models. Figure 17: Left) Maximum deflections of node 175 according to speed (Stiff model). Right) Time History of the same node at 275 km/h, with HSLM-A7. Figure 21: Shear forces along path for the 4 models - Bridge 1. Figure 18: Left) Maximum deflections of node 499 according to speed (Model 1). Right) Time History of the same node at 280 km/h, with HSLM-A9. Figure 22: Bending moments along path for the 4 models - Bridge 2. Figure 19: Left) Maximum deflections of node 499 according to speed (Model 2). Right) Time History of the same node at 280 km/h, with HSLM-A7. Figure 20: Left) Maximum deflections of node 175 according to speed (Model 3). Right) Time History of the same node at 285 km/h, with HSLM-A7. The stiff model is not able to guarantee safe side results for the entire bridge, as it is possible to observe in these graphics. For both shear force and bending moment, the extremities of the bridge are in prominence concerning dissimilar results. Model 1 registers a shear force 300% higher than the stiff model at end supports, and 350% higher for the bending moment. In pursuance of the type of results displayed for Bridge 1, accelerations, displacements and section forces were analysed for Bridge 2. The speed plot and time history of the node with the highest acceleration for each model are presented below. Stiff model accelerations vary from 0.016 m/s 2 to 0.016 m/s 2. Higher values were obtained with models 1 and 2: the first presented results in the range 4.16 m/s 2 to 4.29 m/s 2, which are already off the prescribed limits; the second replied results between 2.50 m/s 2 and 2.91 m/s 2. These extremely big differences to the stiff model are related to the absence of vertical modes of vibration below the cut-off frequency in that model (30 Hz). 9

Figure 23: Left) Maximum accelerations of node 107 Figure 26: Left) Maximum deflections of node 107 according to speed (Stiff model). Right) Time History according to speed (Stiff model). Right) Time History of the same node for speed 187.5 km/h, with HSLM- of the same node for speed 185 km/h, with HSLM-A2. A2. Figure 27: Left) Maximum deflections of node 404 Figure 24: Left) Maximum accelerations of node 21 according to speed (Model 1). Right) Time History of according to speed (Model 1). Right) Time History of the same node for speed 300 km/h, with HSLM-A1. the same node for speed 300 km/h, with HSLM-A9. Figure 25: Left) Maximum accelerations of node 401 Figure 28: Left) Maximum deflections of node 18 acaccording to speed (Model 2). Right) Time History of cording to speed (Model 2). Right) Time History of the same node for speed 210 km/h, with HSLM-A9. the same node for speed 100 km/h, with HSLM-A2. On another hand, both models 1 and 2 already include vertical modes below 30 Hz with relevant participation mass, due to the loss of stiffness caused by the springs inclusion in the models. It is, thus, considered to be essential, not only the revision of the cut-off frequency of 30 Hz established by the Swedish Railway Authority [27], but also to include soil stiffness in the models, since it provides off-recommended accelerations, not captured with the stiff model. Once again, due to the 30 Hz cut-off frequency, no vertical eigenmodes are present in the analyses undertaken for the stiff model, thereby the displacements are practically 0. Since models 1 and 2 include vertical eigenmodes, more realistic displacement values were obtained. Nevertheless, they are all in compliance with figure 6. The comparisons of the three models, in what concerns shear force and bending moment, can be seen in the following graphics. It is easily observed that models 1 and 2 replied, The analyses undertaken for the displacements at Bridge 2 caused by the dynamic loads replied almost for the entire deck, higher values when comparing to the stiff model, both on shear force and results as shown in the next series of graphics. 10

Figure 29: Shear forces along path for the 3 models - Bridge 2. Figure 30: Bending moments along path for the 3 models - Bridge 2. bending moment. The differences between models for the shear force, over deck extremities, is not significant though. Higher dissimilarities are observed for mid deck, where models with soil springs replied values between 250% and 300% higher than the stiff model. Nevertheless, the biggest differences have occurred in the comparison of bending moment: deck extremities are more affected when looking at the models with elastic foundation springs, with values of negative bending moment 200% higher than the stiff model, and with values of positive bending moment almost reaching 700% higher than the stiff model. 5 Conclusions The inclusion of springs in the model, in order to simulate soil/structure interaction, has a very significant influence in the eigenmodes and eigenfrequencies, which, at the end, are responsible for the differences in the results; Considering soil stiffness in the numerical analysis allows a reduction of the cut-off frequencies, when studying section forces. Since the overall stiffness of the structure decreases, a significant contribution of the effective mass (normally taken as > 90% of total mass) is tangible with lower eigenfrequencies. Therefore, with a cut-off frequency reduction, the time of the analysis decreases; 30 Hz as cut-off frequency for accelerations and displacements analyses seems to be extremely low for bridges similar with Bridge 2 when stiff supports are considered, since important vertical eigenmodes may be excluded. Eurocode 1 [20] presents a different perspective over choosing this limit, which seems indeed more appropriate; When taking soil stiffness into consideration, resonance can occur for a different train, different speed and different deck node when comparing to the corresponding stiff model. Therefore, neither the same train nor the same speed can be used for different models. Instead, a full analysis must be performed for each model; In what concerns soil stiffness inclusion, some models were not in compliance with the limit state proposed in Eurocode 1 [20], regarding vertical accelerations, whilst others provided admissible values. But for all models, such values were increased significantly: for Bridge 1, while model 1 and 2 provided off-limit values, model 3 gave values quite close from the recommendation - 3.5m/s 2 ; as for Bridge 2, accelerations were extremely increased by the time of soil inclusion. The appearance of important vertical eigenmodes below 30 Hz was the main reason for such event. Even though model 2 of that bridge presented tolerable values, this phenomena should not be neglected; Maximum displacements values were not affected by the dynamic analyses undertaken in the different models. Section forces can increase in some parts of the deck bridges and decrease in some other parts, when soil is included in the model. Although no guarantees can be given for all types of bridges, it can happen with two very different structures, as it was proved in chapter 4. Shear forces and bending moments are clearly increased in deck ends if soil stiffness is included in the model; It is considered to be of great importance the soil consideration in the numerical model of a bridge, in particular when studying accelerations and section forces. Ideally, two models should be adopted, one stiff and another with soil stiffness. Subsequently, an envelope of the two results should be done. 6 Acknowledgements The author wishes to acknowledge Scanscot Technology AB, located in Lund, Sweden, not only for all the support and interest in this work, but also for the kindness and friendship he was treated with. Also, to his home University, Instituto Superior Técnico, in Lisboa, where he studied 5 years of his life, missing it already. 11

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