A transfer function method to predict building vibration and its application to railway defects

A transfer function method to predict building vibration and its application to railway defects D. L´opez-Mendozaa, D.P. Connollyb, A. Romeroa, G. Kouroussisc, P. Galv´ın∗,a aEscuela T´ecnica Superior de Ingenier´ıa, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain bInstitute for High Speed Rail and Systems Integration, School of Civil Engineering, University of Leeds, UK cFaculty of Engineering, Department of Theoretical Mechanics, Dynamics and Vibrations, Universit´e de Mons, Belgium Abstract This work presents a simplified method to evaluate building shaking due to arbitrary base excitations, and an example application to railway problems. The model requires minimal computational effort and can be applied to a wide range of footing shapes, thus making it attractive for scoping-type analysis. It uses the soil excitation spectrum at the building footing location as it’s input, and computes the building response at any arbitrary location within it’s 3D structure. To show an application of the model versatility, it is used to compute building response due to a variety of singular railway defects (e.g. switches/crossings). It is however suitable for more general applications including railway problems without defects. The approach is novel because current railway scoping models do not use soil-structure transfer functions combined with free-field response to estimate building vibration by railway defects. First the soil-structure interaction approach is outlined for both rigid and flexible footings. Then it is validated by comparing results against a comprehensive fully-coupled 3D FEM-BEM model. Finally, it is used to analyse the effect of a variety of variables related to railway defects on building response. Local track defects are shown to have a strong influence on building vibrations. Further, vibration levels close to the threshold of human comfort are found in buildings close to the railway line. Overall the new approach allows for the computation of building vibrations accounting for soil-structure interaction, floor amplification and the measured/computed free-field response due to railway traffic using minimal computational effort. Key words: Ground-borne vibrations, Railway traffic, High speed rail, Building vibrations, Structural vibration, Environmental Impact Assessment (EIA), Railway singular defects 1. Introduction The response of structures to ground-borne waves induced by blasting, earthquakes, roads and railway traffic, are examples of where soil-structure interaction (SSI) is an important issue [1, 2]. ∗Corresponding author. Tel.: +34 954487293 Email address: [email protected] (P. Galv´ın) Preprint submitted to Construction and Building Materials November 1, 2019 This was addressed by Wu and Hao [3, 4] who investigated blast-induced ground excitation. They proposed a numerical model to predict surface ground motion due to underground blasting. The freefield response was used as an input to obtain the building response using a simple approach, where the source (blasting)-receiver (building) interaction was neglected. Alternatively, Bayraktar et al. [5] developed a detailed nonlinear dynamic model to simulate concrete and masonry structures using an hybrid approach. Ground excitations due to blasting were measured and combined with a numerical building model, informed using experimental dynamic characteristics. Dogan et al. [6] also presented a combined experimental/numerical procedure to obtain building response due to blasting. Ground motion was measured while building vibration was computed using a 3D model ignoring SSI. A comparison between underground and surface blasting was made and it was found that vibrations were lower for the underground case. Alternatively, the effect of SSI on the seismic response of buildings in soft layered soils was analysed by Savin et al. [7], using a detailed 3D model. Also, Gatti et al. [8] presented a complete approach to model the full path from the source (earthquake) to a nuclear reactor. To do so, wave-motion was used as an input for a SSI boundary element (BEM)- finite element (FEM) model. Alternatively, simplified procedures [9–11] have been proposed to model SSI for seismic applications. Numerical models to compute building vibrations due to road traffic include Pyl et. al [12, 13] who presented a coupled BEM-FEM methodology to analyse the road-soil-structure system. Alternatively, Fran¸cois et al. [14] studied dynamic building behaviour considering the relative stiffness between the building and the soil, with the aim of simplifying soil-structure interaction. Regarding the rail sector, the growth of urban railway track infrastructure has led to an increase in the number of properties affected by ground-borne railway vibrations [15–19]. The negative effects caused by railway traffic are more prominent in the presence of local irregularities [20] and are addressed in international standards [21–23]. Thus, it is desirable to estimate the potential increase in vibrations levels in nearby buildings. To do so, a variety of numerical models have been proposed to compute building induced vibrations due to railway traffic. Prior to the construction stage of a new railway project or the construction of a building near an existing line, a detailed design is required [23] possibly using comprehensive 3D models with high computational cost. These include Fiala et al. [24] who developed a comprehensive BEMFEM model to calculate building vibration and indoor noise. Alternatively, Galv´ın et al. [25] presented a coupled train–track–soil-structure 3D BEM-FEM model formulated in the time domain where the nonlinear behaviour of structures could be also considered. Moreover, the problem of vibration in bridges was studied using comprehensive models as shown in [26, 27]. Coulier et al. [28] studied the source (track) and receiver (building) interaction in order to determine the uncertainty of using uncoupled approaches. It was concluded that for a ballasted track the assumption of decoupling was acceptable for distances from the track greater than six times the Rayleigh wave length. 2 Uncoupled simplified procedures are normally used at early stages of railway line development [23]. These represent attractive tools due to their low computation times. Two such methods to evaluate building vibrations due to a train passage have been proposed by the Federal Railroad Administration (FRA) and the Federal Transit Administration (FTA) of the U.S. Department of Transportation [29–31]. R¨ucker et al. [32] developed a simplified prediction tool to evaluate free-field and building vibrations. Auersch [33] analysed building vibration in inhomogeneous soils and proposed a simplified methodology to consider SSI in layered ground. He studied building induced vibrations using a simple soil-wall-floor model based on an empirical transfer function obtained from the characteristics of the structure [34]. Moreover, this also included a simple method to estimate vibration in buildings on pile foundations [35]. Hussein et al. [36] developed a sub-modelling method where a train-track-soil 3D model was coupled with a 2D building approach based on beam elements. Hussein also presented a 3D model to calculate vibrations in a piled foundation building due to railway traffic from a nearby underground tunnel [37]. Later, Kouroussis et al. [38] proposed a decoupled FE model to predict building vibrations due to tramway traffic with local irregularities. Also a hybrid numerical/experimental model to assess ground and building vibration was presented [39, 40]. In this a vehicle-track numerical approach which simulated vibration generation due to a variety of railroad artefacts was combined with a experimental procedure based on multiple single source transfer mobilities that modelled the transmission mechanism between rail and nearby structures. Lopes et al. [41, 42] developed an uncoupled model to evaluate building vibrations induced by railway traffic in tunnels. Free-field response was computed using a 2.5 D FEM-Perfectly Matched Layers (PML) model and combined with a 3D FEM model to evaluate the building response. Connolly et al. [43, 44] proposed a scoping model to predict vibrations and in-door noise in buildings due to railway traffic. A wide range of soil vibration records generated by a 3D FEM model was used to build a machine learning approach. This procedure was combined with empirical factors [31] to compute building vibrations. L´opez-Mendoza et al. [45] presented a scoping model based on modal superposition analysis. The free-field vibration was discretised into the frequency range corresponding to the modes of the structure. Kuo et al. [46] presented a hybrid model that combined recorded data and numerical predictions considering the definitions proposed by the FRA [31]. The source, propagation and receiver mechanisms were uncoupled. Recently Connolly et al. [47] presented a decoupled procedure to analyse soil-building vibrations due to railway irregularities. A 2.5D time-frequency domain model to compute soil vibrations was combined with a 3D FEM procedure to obtain building vibrations induced by railway defects. This paper uses a simple procedure where the source (ground motion) and the receiver (building) are uncoupled. It is focused on the receiver model and proposes soil-structure transfer functions considering SSI. These soil-structure transfer functions are combined with free-field vibrations to compute building vibration with low computational effort. The model is numerically verified by comparing with a comprehensive BEM-FEM model. Finally, the proposed model is used to analyse building vibrations due to railway local 3 irregularities. 2. Methodology ISO 14837-1 standard [23] defines the magnitude of building vibration A(f) in the frequency domain f as a function of the source S(f), the propagation P(f) and the receiver R(f). Considering the assumption that all the three terms are uncoupled (Figure 1), the magnitude of the building vibration A(f) is expressed as: A(f) = S(f)P(f)R(f) (1) The procedure developed by the Federal Railroad Administration (FRA) [31] to estimate building response u u0 ug ug u u0 Step 1 - Free field response Step 2 - Simplified building-soil coupling model Figure 1: (left) Coupled problem and (right) scheme of decoupled model. due to railway traffic proposes two factors influencing the receiver: 1) the floor-to-floor attenuation, and, 2) the amplification due to the resonance of floors, walls and ceilings. The present work includes these factors defining the floor amplification Faas the increment in the building response uwith respect to the foundation response u0(Figure 1). The floor amplification is computed as: Fa(f) = u(f)/u0(f) (2) Also, the effect of the building foundation should be considered using the coupling loss Cl[31]. The coupling loss is related to the soil-foundation interaction. Therefore it is the ratio between the building foundation response u0and the free-field vibration ug(Figure 1). In this work, the coupling is evaluated as: Cl(f) = u0(f)/ug(f) (3) 4 The following expression to calculate the building response ucan be obtained by combining Equations (2) and (3): u(f) = Fa(f)Cl(f)ug(f) (4) Comparing Equations (1) and (4), it can be seen that the source S(f) and the propagation P(f) terms are included in the free-field vibration ug, whereas the receiver term R(f) is part of the floor amplification Fa and the coupling loss Cl. In this approach, the problem is decoupled in two steps (Figure 1): i) computing or measuring the free-field vibration ugand ii) the structural response induced by an incident wavefield considering a simplified building model. The main novelty of this work is applying the soil-structure transfer functions u/ug=Fa(f)Cl(f) depending only on the receiver, to predict building vibration by railway traffic. The soil-structure transfer function represents the building response due to a displacement impulse applied at the building foundation. A key advantage of this approach is the computational efficiency arising because the soil-structure transfer function is computed only once for a soil-building subsystem and later it is combined with a wide range of free-field vibration data to analyse multiple scenarios. These low requirements mean the approach is well-suited to early stage railway projects. On the other hand, although this work is focused on the application of soil-structure transfer functions to assess building vibrations by railway traffic, these soil-structure transfer functions can be used to predict building vibration due to diverse sources (e.g. construction, earthquake, road traffic, blast) where the free-field vibration spectrum is known. 2.1. Step 1 - Free field response ug Numerical methods or Measurements Figure 2: Step 1 - Free field response. This work uses the methodology presented in Reference [47] to model the source-propagation subsystem (S(f), P(f)). It has been extended and validated, as discussed in [47], for the computation of free-field vibrations due to railway defects (Figure 2) , so is not repeated here. The free-field response is required as the input for Step 2, and can be either computed or measured using in-situ tests. Once it is known for the source-propagation subsystem, it is used in Step 2 to compute the building vibration A(f). To do so, the building foundation is excited by the free-field response ug(f). The SSI is integrated in the proposed 5 methodology using a simplified method. Below, it will be related the procedure to model the receiver soil-structure subsystem (R(f)). 2.2. Step 2 - Simplified building-soil coupling model The simplified method is a 3D time domain FEM model. The dynamic equilibrium equation of a structure can be written as: M¨ u(t) + C˙ u(t) + Ku(t) = F(5) where M,Cand Kare the mass, damping and stiffness matrices, respectively. u(t), ˙ u(t), and ¨ u(t) are the building displacement, velocity and acceleration, respectively, while Frepresents the external force. The FEM equation is solved at each time step following an implicit time integration GN22 Newmark method [48, 49]. Structural damping is considered using a Rayleigh model [50], where the damping matrix Cis proportional to the the mass Mand stiffness Kmatrices: C=dmM+dkK. Constants dmand dkare chosen depending on the modal damping of the structure. The simplified method is based on recommendations from the National Institute of Standards and Technology (NIST) [51]. This proposes to integrate SSI by adding spring-damper elements to the foundation of the building. As the building is not embedded in the soil, expressions to define horizontal spring-damper elements are discarded. The formulation to calculate vertical spring-damper elements is below. To explain the simplified model, consider a rectangular building with floor plan dimensions 2L×2B, where L≥B(Figure 3). Note that in the following formulation, the sub-indices x,yand zare related to the translation along the respective axis. Also the sub-indices xx and yy refer the rocking about the xand yrespectively, whereas sub-index zz is related to the torsion about the zaxis (Figure 3). Hereafter, the formulation considers the xaxis to be the largest dimension of the foundation (2L). zz z 2B 2L x xx yyy Figure 3: Scheme of the plan geometry of the building foundation. A spring-damper system is added to the foundation allowing it to be modelled as rigid or flexible. For the rigid case, a single spring-damper element defined by its stiffness kzand the dashpot coefficient cz. On the other hand, flexible foundations are simulated using spring-damper elements (ki z,ci z) spread across the foundation area, where ki zand ci zare the properties of the ith spring-damper element. 6 (a) us 0 us kz, cz ug (b) us 0 us k1 z, c1 zkn z, cn z ug Figure 4: Step 2 - Simplified building-soil coupling model: (a) rigid foundation and (b) flexible foundation. 2.2.1. Rigid foundation If the foundation is rigid, it can be represented by a single spring-damper element (kz,cz). The vertical stiffness of the full system kz, is evaluated using the formulation presented in Reference [52]: kz=Kz,surf αz(6) where Kz,surf is the vertical static stiffness of the surface foundation and αzis the dynamic stiffness modifier. The vertical static stiffness Kz,surf is obtained from the shear modulus Gand Poison’s ratio of the soil ν, and the foundation dimensions Land B, using: Kz,surf =GB 1−ν 3.1L B0.75 + 1.6!(7) Also the dynamic stiffness modifier αzdepends on the structural properties and is evaluated as: αz= 1 −0.4 + 0.2 L Ba2 0 10 1+3(L B−1)+a2 0 (8) where a0is the dimensionless frequency computed from the S-wave velocity csand the angular frequency of the first bending mode ω1(discarding SSI), as: a0=ω1B cs (9) Once the vertical stiffness kzis obtained, the dashpot coefficient of the full foundation czcan be computed using [51]: cz= 2kz βz+βs ω1 (10) where βsis the damping ratio of the soil and βzis the radiation damping ratio obtained as [52]: βz=4ψL B Kz,surf GB a0 2αz (11) 7 where ψ=p2 (1 −ν) (1 −2ν), limited to ψ≤2.5. Following the same procedure to calculate the vertical stiffness kz(Eq. (6)), the rocking stiffness can be obtained as [52]: kyy =Kyy,surf αyy kxx =Kxx,surf αxx (12) The rocking static stiffnesses (Kxx,surf ,Kyy,surf ) and the dynamic stiffness modifiers (αxx,αyy) are evaluated as: Kyy,surf =GB3 1−ν 3.73 L B2.4 + 0.27!Kxx,surf =GB3 1−ν3.2L B+ 0.8(13) αyy = 1 −0.55a2 0 0.6 + 1.4 (L B)3+a2 0 αxx = 1 −0.55 + 0.01qL B−1a2 0 2.4−0.4 (L B)3+a2 0 (14) In the same way as for the vertical dashpot (Eq. (10)), the rocking dashpot is computed as [52]: cyy = 2kyy βyy +βs ω1 cxx = 2kxx βxx +βs ω1 (15) where the radiation damping ratios βxx and βyy are calculated as: βyy = 4ψ 3L B3a2 0 Kyy,surf GB31.8 1+1.75(L B−1)+a2 0 a0 2αxx βxx = 4ψ 3 L Ba2 0 Kxx,surf GB32.2−0.4 (L B)3+a2 0 a0 2αyy (16) 2.2.2. Flexible foundation Equations 6 and 10 compute the spring-damper element (kz,cz) properties for rigid foundations. However, in order to consider the effect of a flexible foundation, the NIST proposes smeared spring and damper elements. To do so, the vertical values kzand czare normalized by the foundation area to obtain the stiffness intensity ˜ ki z=kz/4BL and dashpot intensity ˜ci z=cz/4BL. Then, the stiffness ki zand dashpot ci z of a vertical spring-damper element in the interior of the foundation can be computed as: ki z=˜ ki zdAici z= ˜ci zdAi(17) where dAiis the individual area for the ith spring-damper element (Figure 5). If these expressions (Eq. (17)) are used across the full foundation, the rotational stiffness would be underestimated and the rotational damping would be overestimated [51]. To correct these effects, factors Rkand Rcare applied to the spring-damper elements along a strip area on the foundation edge. To do so, the stiffness and damping of a vertical spring-damper element at the foundation edge are estimated as: ki z=Rk˜ ki zdAici z=Rc˜ci zdAi(18) 8 2B 2L dAi ki z, ci z Figure 5: Individual area dAifor the ispring-damper element. The width of the foundation edge strip is computed from the foundation end ratio Reas ReLand ReBfor the xand yaxes, respectively. A value in the range from 0.3 to 0.5 is usually selected for the foundation end ratio Re. In this work an end ratio Re= 0.5 is used. Figure 6 shows the spring-damper element properties (ki z, ci z) depending on the position of the ith element across the foundation. 2B 2L 1 3 3 4 2 4 2 4 4 x y ReB ReL 1ki z=˜ ki zdAici z= ˜ci zdAi 2ki z=Rk,yy ˜ ki zdAici z=Rc,yy ˜ci zdAi 3ki z=Rk,xx ˜ ki zdAici z=Rc,xx˜ci zdAi 4ki z=Rk,xx +Rk,yy 2 ˜ ki zdAici z=Rc,xx +Rc,yy 2˜ci zdAi Figure 6: Spring-damper element properties across the foundation. Regarding the estimation of the correction factors Rkand Rc, the following expressions are proposed by NIST: Rk,yy = 3kyy 4˜ ki zBL3−(1 −Re)3 1−(1 −Re)3Rk,xx = 3kxx 4˜ ki zB3L−(1 −Re)3 1−(1 −Re)3(19) Rc,yy = 3cyy 4˜ci zBL3 Rk,yy 1−(1 −Re)3+ (1 −Re)3Rc,xx = 3cxx 4˜ci zBL3 Rk,xx 1−(1 −Re)3+ (1 −Re)3(20) where kxx and kyy are the rotational stiffnesses about the xand yaxes respectively, considering a rigid foundation. Also the dashpot coefficients cxx and cyy represent the rotational damping about the xand y axes respectively. 9 excitation, in the low frequency range. However, at higher frequencies, the center of the foundation presents the highest level according to mode shape deformation. It should be noted that the simplified model is composed of spring-damper elements which have modified properties depending upon location (Figure 6). Nevertheless, the spring-dampers at the center of the foundation are unmodified and thus the stiffness and damping values are not overestimated. The comparison between the results computed from both methodologies shows that the simplified methodology yields higher attenuation factors compared to the ’reference’ model. This is because the simplified approach models the soil-flexible foundation system using a set of spring-damper elements, meaning it has softer properties than the actual soil. Finally, it should be noted that References [29–31] show that heavier buildings result in greater coupling loss. The proposed model is consistent with this finding. The floor amplification (Equation 2) is shown in Figure 11 from the first to fourth floors, for the simplified and reference models, and the case of ignoring SSI. Overall the floor amplification increases with storey level in the low frequency range, while the excitation is filtered at higher frequencies according to the modal parameters of the buildings. The floor amplification yields maximum values at the floor local resonance frequencies. At frequencies higher than the fundamental bending mode shapes of the floors, the floor amplifications have negative values. Regarding the ’reference’ model, the simplified methodology underestimates the floor amplification at higher frequencies, while the solution ignoring SSI overestimates the response. According to References [29– 31], the amplification due to the resonances of floors is +6 dB, but lower near the wall-floor and wall-ceilings intersections. In the present analysis, the maximum value obtained is 10.5 dB from the simplified model and 9.2 dB from the ’reference’ model. The model ignoring SSI provides the highest floor amplification. Figure 12 presents the soil-structure transfer function u(f)/ug(f) for all 3 models. This soil-structure transfer function is obtained from the receivers terms u(f)/ug(f) = Fa(f)Cl(f) (Equation (4)) and relates the structural response to the load. It can be concluded that the shape and magnitude of the response from the simplified method match reasonable well with those obtained from the ’reference’ model, although the results from the simplified method are underestimated. This is due to the high attenuation of the ground-building coupling shown in Figure 10. Therefore the response ignoring SSI overestimates the results at frequencies higher than the floor resonance. The global effect of SSI on the building response is shown in Figure 13, which displays the ratio ∆us (Equation (22)) for all building floors . These ratios represent the changes in the building response due to SSI, computed using the proposed methodologies. Also, it superimposes the coupling loss factor Cs lto evaluate the accuracy of the approximation ∆us≈Cs l, defined above. Although there are amplifications at low frequencies, particularly at the floors closest to the ground, it can be seen that there are attenuations in the mid and high frequency ranges due to SSI. It is observed that the effect of SSI is weakly dependent on 16 1 2 4 8 16 31.5 63 Frequency [Hz] -40 -20 0 20 Fa [dB] (a) 1 2 4 8 16 31.5 63 Frequency [Hz] -40 -20 0 20 Fa [dB] (b) 1 2 4 8 16 31.5 63 Frequency [Hz] -40 -20 0 20 Fa [dB] (c) 1 2 4 8 16 31.5 63 Frequency [Hz] -40 -20 0 20 Fa [dB] (d) 1 2 4 8 16 31.5 63 Frequency [Hz] -40 -20 0 20 Fa [dB] (e) 1 2 4 8 16 31.5 63 Frequency [Hz] -40 -20 0 20 Fa [dB] (f) 1 2 4 8 16 31.5 63 Frequency [Hz] -40 -20 0 20 Fa [dB] (g) 1 2 4 8 16 31.5 63 Frequency [Hz] -40 -20 0 20 Fa [dB] (h) 1 2 4 8 16 31.5 63 Frequency [Hz] -40 -20 0 20 Fa [dB] (i) 1 2 4 8 16 31.5 63 Frequency [Hz] -40 -20 0 20 Fa [dB] (j) 1 2 4 8 16 31.5 63 Frequency [Hz] -40 -20 0 20 Fa [dB] (k) 1 2 4 8 16 31.5 63 Frequency [Hz] -40 -20 0 20 Fa [dB] (l) Figure 11: One-third octave band center frequency of the floor amplification due to an incident wave field, at the observation point Aof the (a,d,g,j) four-storey, (b,e,h,k,) six-storey and (c,f,i,l) twelve-storey buildings, at the (a,b,c) first, (d,e,f) second, (g,h,i) third and (j,k,l) fourth floors, from the (black line) SSIFiBo toolbox, the (green line) simplified methodology and (red line) ignoring SSI. 17 1 2 4 8 16 31.5 63 One-third octave band center frequency [Hz] 10-4 10-2 100 102 u/u g [-] (a) 1 2 4 8 16 31.5 63 One-third octave band center frequency [Hz] 10-4 10-2 100 102 u/u g [-] (b) 1 2 4 8 16 31.5 63 One-third octave band center frequency [Hz] 10-4 10-2 100 102 u/u g [-] (c) Figure 12: One-third octave band center frequency of the soil-structure transfer function due to an incident wave field, at the observation point Aof the (a) four-storey, (b) six-storey and (c) twelve-storey buildings, at the top floor, from the (black line) SSIFiBo toolbox, the (green line) simplified methodology and (red line) ignoring SSI. 1 2 4 8 16 31.5 63 One-third octave band center frequency [Hz] -40 -20 0 20 u [dB] (a) 1 2 4 8 16 31.5 63 One-third octave band center frequency [Hz] -40 -20 0 20 u [dB] (b) 1 2 4 8 16 31.5 63 One-third octave band center frequency [Hz] -40 -20 0 20 u [dB] (c) Figure 13: One-third octave band center frequency of the ratio ∆usat the observation point Aof the (a) four-storey, (b) six-storey and (c) twelve-storey buildings. SSI attenuation from the (darkest line) first floor to the (red line) top floor. Superimposed is the (green line) coupling loss computed from the simplified method. storey level, with the greatest effect evident on the tallest building. This analysis leads to the possibility of studying of the structural response using a building model where the soil is not considered, and modelling SSI as an attenuation of the computed response, as suggested in some standards. This attenuation can be estimated using the coupling loss factor obtained from the proposed methodology because it represents an approximate envelope as shown in Figure 13. 18 4. Case study: railway track defects The proposed methodology is now used to analyse building vibration due to local track defects. The building response is calculated by combining the free-field ugresponse due to railway traffic (Figure 2) with the soil-structure transfer function u(f)/ug(f) = Fa(f)Cl(f) (Equation (4)) due to an incident wave field computed in the previous section using the simplified model shown in Figure 4. The free-field response is calculated using a methodology experimentally validated in Reference [47]. Therefore this paper aims to use these excitation time histories as example datasets, rather than to validate or address the assumptions associated with the underlying excitation generation model. Moreover, it is focused on the effect of local track defects on the response of building using the proposed numerical method, rather than the analysis of track forces due to defects. Using a pre-determined excitation facilitates reduced computational effort because the soil-structure transfer function u(f)/ug(f) does not depend on train passage and is thus only evaluated once. Therefore, the proposed methodology presents a major computational advantage compared to existing numerical procedures where the track-soil-building system is recomputed every time the train passage changes. The proposed approach is thus similar to the FTA standard: the soil-building system is modelled using coupling loss and floor amplification factors (step 2), while the excitation is measured or computed (step 1). Once ug(f), Cl(f) and Fa(f) are obtained, the structural response is computed with minimal effort using multiplication (linear scale) or addition (logarithm scale). A sensitivity analysis of the effect of defect type and size is presented. Figure 14 shows the singular defects considered in the analysis, where v0is the train speed, hthe defect height and lthe defect length. The defect size influence on building vibrations is analysed considering several defect lengths l={80,110,140,170,200}mm. h v0 h v0 l h v0v0 l Figure 14: Local defect shape (from left to right: step up, step down, positive pulse, negative pulse). Vibrations are obtained for the six storey building analysed previously in Section 3 (Figure 8). The midpoint of the building foundation is located at a distance, d= 30 m, from the track centreline and observation point B(Figure 8) is selected because it presents the highest response. Also, building vibrations are calculated considering a single point response (SPR) excitation model, where the free-field vibration is transmitted simultaneously to the whole building foundation. The SPR model is used for obtaining the maximum response level threshold, accounting for the minimum distance from the building foundation to the 19 track (i.e. 20 m). Alternatively, changing the implementation to a multiple point response (MPR) excitation model would also be straightforward but is not considered here. The soil was considered as a homogeneous half-space with: E= 100 MPa, ν= 0.35, ζ= 0.05, and ρ= 2000 kg/m3. The shear wave velocity was Vs= 136.1 m/s. Two vehicle configurations were analyse: i) an AM96 intercity train (Figure 15) travelling on a ballasted track (Table 3), and ii) a classic tram (Figure 16) on an urban slab track (Table 4). The vehicles were modelled using a detailed multibody vehicle approach [60]. The AM96 and classic tram properties are shown in Table 2. Figure 15: AM96 train dimensions. Table 2: AM96 train properties. mcIcmbIbmwk1d1k2d2 [kg] [kg m2] [kg] [kg m2] [kg] [MN/m] [kNs/m] [MN/m] [kNs/m] AM 96 HVB 25200 1.26 ×1066900 1.52 ×1031700 1.3 3.7 0.69 22.6 HVADX 28900 1.45 ×1067050 1.58 ×1031700 1.3 3.7 0.69 22.6 HVBX 25930 1.3×10611800 2.6×1031700 1.81 1.14 0.69 14 Classic tram 7580 8.75 ×1043530 6.0×102160 5.876 6 0.96 56.25 Lc= 7800mmLc= 7800mm Lb= 850mmLb= 850mm Ld= 1130mmLd= 1130mm Lm= 570mmLm= 570mm 8300kg8300kg 3450kg3450kg 4250kg4250kg Figure 16: Geometrical configuration of the classic tram. Figure 17.(a) shows the free-field acceleration agconsidering the passage of the AM96 train at v0= 120 km/h, over a ballasted track, in the presence of a negative pulse defect. Dominant frequencies are located at mid frequencies due to dynamic excitation. Also, the soil-structure transfer function a/agin the frequency domain is shown for the building in Figure 17.(b). The soil-structure transfer function shows 20 Table 3: Ballasted track properties. Ballast track properties (2 rails) Track gauge [m] 1.435 Rail 2nd moment of area m43.09 ×10−5 Rail Young’s modulus hN/m2i2.1×1011 Rail density hkg/m3i7850 Railpad stiffness per unit length (2 rails) hN/m2i6.15 ×108 Railpad damping per unit length (2 rails) hNs/m2i1.2×104 Sleeper spacing [m] 0.65 Sleeper mass per unit length [kg/m] 461.5 Ballast stiffness hN/m2i1.3×108 Ballast damping hNs/m2i1.3×105 Ballast density hkg/m3i1700 Ballast height (below sleeper) [m] 0.3 Ballast cross-sectional area m20.59 Ballast Poisson’s ratio 0.3 amplifications at low and mid frequencies up to 30 Hz, whereas the response is damped at high frequencies. These amplifications are concentrated at 20 Hz, corresponding to the natural frequencies of the building (Figure 9) and, therefore, where the function u/ug=Cl(f)Fa(f) presents maximum values. Moreover, this frequency range matches with a maximum in the excitation. Building vibration is obtained by combining the free-field response agand the soil-structure transfer function (Figure 17). Figure 18 shows the frequency content[58] and the running RMS value of the building response. The running RMS value is computed using a time window of 1 s as prescribed by ISO 2631 [21]. In addition to the dominant frequencies indicated above, in the soil-structure transfer function a/ag(Figure 17. (b)), the additional frequencies due to the source are significant. As previously mentioned, the proposed methodology is straightforward and it allows for the computation of the building response by summing the results shown in Figure 17. The structural response in the low frequency range is amplified in relation to the source due to resonances at the dominant natural frequencies of the building. However, in the high frequency range, the excitation is attenuated according to the transfer function (shown in Figure 17.(b)). The results of the test case yield running RMS values lower than the threshold for harmful effects to humans according to ISO 2631 [21]. As an example, the Spanish standard [61] uses a maximum threshold 21 Table 4: Slab track properties. Slab track properties (2 rails) Track gauge [m] 1.435 Rail 2nd moment of area m43.09 ×10−5 Rail Young’s modulus hN/m2i2.1×1011 Rail density hkg/m3i7850 Railpad stiffness per unit length (2 rails) hN/m2i4×108 Railpad damping per unit length (2 rails) hNs/m2i1.2×104 Slab thickness [m] 0.3 Slab width [m] 2.5 Slab stiffness (concrete)hN/m2i3×1010 Slab 2nd moment of area m45.63 ×10−3 Slab density (concrete) hkg/m3i2500 Slab Poison’s ratio (concrete) 0.2 1 2 4 8 16 31.5 63 125 One-third octave band center frequency [Hz] 0 20 40 60 80 100 Acceleration [dB, ref 10 -6 m/s 2] (a) 1 2 4 8 16 31.5 63 125 One-third octave band center frequency [Hz] 10-4 10-3 10-2 10-1 100 101 a/ag [-] (b) Figure 17: (a) One-third octave band center frequency of the vertical acceleration of the free field response agat 20 m to the ballasted track due to a AM96 train passage at v0= 120 km/h and (b) soil-structure transfer function a/agat the top floor of a six-storey building. running RMS value of 72 dB or 75 dB depending on the intended use of the building. The frequency content of the response is also below, k = 1.4, that is the limit for residential buildings during the night [59]. Next, the influence of defect type on building vibrations is analysed for both the AM96 train and the classic tram considering several defect types (Figure 14). Figure 19 presents the building response due 22 1 2 4 8 16 31.5 63 125 One-third octave band center frequency [Hz] 0 20 40 60 80 100 Acceleration [dB, ref 10 -6 m/s 2] k = 1 k = 1.4 k = 2 k = 4 k = 8 (a) 0.5 1 1.5 2 2.5 3 3.5 4 Time [s] 0 20 40 60 80 100 Running RMS value [dB, ref 10 -6 m/s 2] (b) Figure 18: (a) One-third octave band center frequency of the zvertical velocity and (b) running RMS value of the weighted acceleration at the top floor of the six-storey building due to a AM96 train passage at v0= 120 km/h. The (dashed line) vertical acceleration of the free field response (Figure 17.(a)) and the k-curves [59] are superimposed in (a). to the AM96 passage over a ballasted track. Overall it can be observed the step up joint induces higher vibrations at low frequencies, whereas the maximum values at mid and high frequencies are due to the positive pulse. Also the defect type influences the amplitude of the response more dominantly than the shape (Figure 19.(a)). These results are consistent with the corresponding free field responses due to local defects, accounting for the fact that the building transfer function is the same in all cases. The running RMS values (Figure 19.(b)) have the following maximum amplitudes: 61.8, 61.9, 62.9 and 64 dB for negative pulse, positive pulse, step down and step up, respectively. The maxima do not show a clear dependency on the defect type because the responses have similar amplitudes. The building vibrations induced by the classic tram passage over a slab track can be observed in Figure 20. The frequency responses show a similar tendency for all defect types. However, it is more clearly shown in the running RMS curves (Figure 20.(b)) that the highest responses are found for the positive pulse and the step up joint. This is because the response (Figure 20.(a)) for both defect types yields similar magnitudes in the dominant frequency range from 8 Hz to 20 Hz. In this case, the free-field vibration is concentrated at the natural frequencies of the structure, but the vibration level doesn’t exceed the limits indicated in standards. The maximum RMS value reaches 70 dB and the frequency content is below the k = 1.4 curve. In addition, the effect of defect size on building vibrations was studied for negative pulse defect lengths spanning l={80, 110, 140, 170, 200}mm. Figure 21 shows the building response due to the classic tram passing over a slab track. The correlation between the response amplitude and the defect size is clearly observed in both the frequency and time domain curves. The level of vibration increases significantly with the defect size from l= 80 to l= 140 mm, whereas the building response increases slightly for higher values of defect size. Although the thresholds for human comfort are not exceeded, the levels are appreciable 23 1 2 4 8 16 31.5 63 125 One-third octave band center frequency [Hz] -20 0 20 40 60 80 Acceleration [dB, ref 10 -6 m/s 2] k = 1 k = 1.4 Negative Pulse (l=125 mm , h=1 mm) Positive Pulse (l=125 mm , h=1 mm) Step Down (h=1 mm) Step Up (h=1 mm) (a) 0.5 1 1.5 2 2.5 3 3.5 4 Time [s] -20 0 20 40 60 80 Running RMS value [dB, ref 10 -6 m/s 2] Negative Pulse (l=125 mm , h=1 mm) Positive Pulse (l=125 mm , h=1 mm) Step Down (h=1 mm) Step Up (h=1 mm) (b) Figure 19: (a) One-third octave band center frequency of the zvertical velocity and (b) running RMS value of the weighted acceleration at the top floor of the six-storey building for several defect types due to a AM96 train passage at v0= 120 km/h. The k-curves [59] are superimposed in (a). 1 2 4 8 16 31.5 63 125 One-third octave band center frequency [Hz] -20 0 20 40 60 80 Acceleration [dB, ref 10 -6 m/s 2] k = 1 k = 1.4 Negative Pulse (l=125 mm , h=5 mm) Positive Pulse (l=125 mm , h=5 mm) Step Down (h=5 mm) Step Up (h=5 mm) (a) 0.5 1 1.5 2 2.5 3 3.5 4 Time [s] -20 0 20 40 60 80 Running RMS value [dB, ref 10 -6 m/s 2] Negative Pulse (l=125 mm , h=5 mm) Positive Pulse (l=125 mm , h=5 mm) Step Down (h=5 mm) Step Up (h=5 mm) (b) Figure 20: (a) One-third octave band center frequency of the zvertical velocity and (b) running RMS value of the weighted acceleration at the top floor of the six-storey building for several defect types due to a tram passage at v0= 40 km/h. The k-curves [59] are superimposed in (a). showing the importance of considering local track defects in addition to quasi-static and dynamic loads when computing building vibrations due to railway traffic. These cases could be studied by railway administrators in real cases to guarantee to inhabitants live comfortably. This section has shown the application of the proposed method to analyse building vibration due to railway defects. The performance of the transfer function method to predict building vibration provides a simple assessment once the ground borne vibration is obtained. 24 1 2 4 8 16 31.5 63 125 One-third octave band center frequency [Hz] -20 0 20 40 60 80 Acceleration [dB, ref 10 -6 m/s 2] k = 1 k = 1.4 l=80 mm l=110 mm l=140 mm l=170 mm l=200 mm (a) 0.5 1 1.5 2 2.5 3 3.5 4 Time [s] -20 0 20 40 60 80 Running RMS value [dB, ref 10 -6 m/s 2] l=80 mm l=110 mm l=140 mm l=170 mm l=200 mm (b) Figure 21: (a) One-third octave band center frequency of the zvertical velocity and (b) running RMS value of the weighted acceleration at the top floor of the six-storey building for several negative pulse defect sizes due to a tram passage at v0= 40 km/h.The k-curves [59] are superimposed in (a). 5. Conclusions Building vibration induced by railway traffic is a problem that requires study during project planning and development phases. To do so, fast running methods are useful to assess building vibration quickly, considering multiple scenarios. In this work, a simplified method is presented to do this. It consists of a decoupled model, where the free-field vibration (source and propagation path) and the building vibration (receiver) are computed independently. The proposed model uses a soil-structure transfer function to characterise the structure and soil properties. This transfer function is then combined with the free-field response to obtain the building induced vibration in a computationally efficient manner. The proposed method comprises four parts: 1) determination of the free-field vibration either numerically or experimentally (Figure 2), 2) construction of a soil-building model that approximates SSI using a set of spring-damper elements (Figure 4), 3) computation of the coupling loss and floor amplification factor of the building due to the incident wavefield, and finally, 4) summation of the results from parts 1) and 3) to compute the building response. The new model inherently characterises building response using both its transfer function and the free-field vibration. The transfer function depends upon the coupling loss and the floor amplification factors. The coupling loss induces an attenuation in the structural response in relation to the free-field response, that is dependent on frequency because attenuation is greater at higher frequencies. However, this factor can be approximated as a constant value, as used in standards [29–31] which propose an attenuation ranging from −13 dB (large masonry on spread footings) to 0 dB (foundation in rock). The attenuation computed from the proposed model is frequency dependent and lower than that in the standard, so should be used with care. In contrast, floor amplification can be approximated using a building model 25