5 This paper proposes a new efficient model for the prediction of low-amplitude ground-6 borne vibrations in buildings. The model takes into account the three-dimensional nature of 7 the building structure by analytical and semi-analytical means, making it ideal for performing 8 parametric studies or large-scale vibrations predictions. Its formulation assumes that the 9 principal component in floor vibrations is the vertical one and assumes that the vibrations 10 are transmitted to the various

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... loors through the building columns. The correctness of the 11 model is tested by comparing, in two three-story building examples, its results with those 12 obtained using a numerical model. Results regarding the isolation efficiency of implementing 13 a thicker lower floor or columns with a larger cross-section are also presented. The building-14 soil coupling is formulated considering piled foundations in a stratified soil. To ensure the 15 computational efficiency of the calculations, the piles' response to an incident wavefield is 16 modeled considering the Novak pile model for a layered half-space. Finally, a study of the 17 importance of the soil mechanical parameters in the considered problem is conducted using 18 the building-soil coupled model. 19 Ground-borne vibrations, such as the ones induced by construction works or by the 23 passing of trains and traffic, can cause annoyance to nearby building dwellers and, in the 24 case of large strains, structural damage to building structures. A correct prediction of these 25 vibrations allows deciding whether or not a particular vibration mitigation countermeasure 26 should be applied either on the generation source, in the transmission path or on the receiver. 27 These countermeasures have to take into account the frequency range of interest which ISO 28 2631-2 (2003) defines as 1-80 Hz, for the case of ground-borne vibrations in buildings, and 20-29 250 Hz, when information regarding the reradiated noise is also considered. The prediction 30 of building vibrations has been attempted using different types of models, usually classified 31 into three groups: empirical, numerical and (semi)analytical models. 32 Empirical models predict the ground-borne induced vibrations using simple decaying 33 laws usually combined with experimental data from previous measurements. In the case 34 of railway-induced vibrations, several empirical methods have been developed following the 35 stage differentiation defined by ISO 14837-1 (2005). An example is the method presented 36 by Kuppelwieser and Ziegler (1996), who developed a three-parts computer program with 37 two prediction models -one rather simple and the other much more detailed -supplied by a 38 database of vibration and noise measurements. A similar separation has been also proposed 39 in the empirical models presented by Madshus et al. (1996) and by Hood et al. (1996), 40 both also based on experimental measurements. A significant number of soil attenuation 41 laws have been discussed in the works of Auersch (2010a, 2010b). An empirical model that 42 also considers the building response was presented by Kurzweil (1979). A comprehensive 43 review of prediction models for railway-induced vibrations, especially focused on empirical 44 ones, has been recently presented by Lombaert et al. (2015). Empirical models, despite 45 being computationally very efficient, exhibit limited accuracy. 46 Prediction models based on numerical methods are currently the only option for dealing in 47 detail with the complex geometry of a building-soil system. Because the main disadvantage 48 2 of these models is their high computational cost, different types of approaches have been 49 proposed in order to improve their efficiency. A two-dimensional (2D) Finite Element-50 Boundary Element (FE-BE) model for understanding the surface excitation of a building wall 51 and foundation was proposed by Jean and Villot (2000). The model has been later extended 52 to a two-and-a-half dimensional (2.5D) formulation and used to evaluate the influence of 53 several building parameters (Villot et al. 2011). A three-dimensional (3D) FE-BE model was 54 used by Fiala et al. (2007) to predict the ground-borne noise generated by surface rail traffic. 55 The model, which solved the full problem as three weakly coupled subproblems, was later 56 used to consider the noise generated by underground railways (Fiala et al. 2008). Lopes et al. 57 (2014) have recently presented a complete building-soil model for the prediction of railway 58 induced vibrations using submodeling techniques. However, despite the rapid increase of 59 computers' computational power, numerical models are still not feasible for performing large-60 scale vibration predictions or certain parametric studies. In these cases, the use of a more 61 efficient but still realistic type of prediction model is necessary. 62 A wide variety of analytical models (where the response is given by closed-form solutions) 63 and semianalytical models (which require the numerical computation of complex analytical 64 expressions) have been proposed to predict the building response to incident vibrations. 65 Following the analytical approach proposed by Hudson (1956) in the field of seismology, 66 Waller (1969) represented the building as a 1-degree of freedom (DOF) system. A similar 67 type of building model has been more recently proposed by Auersch (2008), who studied the 68 vertical building resonance adding the mass of the building to the dynamic stiffness of the 69 ground. Newland and Hunt (1991) illustrated that this kind of approach was insufficient for 70 accurately representing the complex response of a continuous structure and proposed the 71 use of columns of infinite length for modeling the building. This type of approach has been 72 recently used by Sanayei et al. (2012), who modeled the dynamic behavior of the building 73 floors by adding the impedance of infinite thin plates to a finite column model. Cryer and 74 Hunt (1994) considered the building as a 2D framed structure composed of horizontal and 75 3 vertical beams joined at their ends. The model was later used to perform predictions of 76 the building response to railway induced vibrations (Hunt 1995). Talbot (2001) coupled the 77 building model to a 3D BE model of the piled foundations for studying the dynamic response 78 of base-isolated buildings. More recently, a prediction model that couples the previous frame 79 structure to a layered soil and computes the response of this soil using the thin-layer method 80 has been presented by Hussein et al. (2013). The authors are not aware of the existence of 81 any 3D analytical model for the prediction of building vibrations. 82 The present work proposes a new efficient three-dimensional model for predicting low-83 amplitude ground-borne vibrations in buildings. The proposed model combines an acceptable 84 accuracy of the predicted results with a small computational cost, making the model highly 85 suitable for performing a wide variety of parametric studies and also large-scale vibration 86 predictions. The model takes into account the 3D nature of a multi-story building and 87 considers that the vibration is transmitted to the different floors through the building's 88 columns. The model also considers that the building has piled foundations buried in a 89 stratified soil. 90 The paper is structured as follows. First, the analytical formulation of the proposed 91 model is described. Then, a numerical validation of the building model and the results 92 obtained in the range of frequencies of interest in building vibrations are presented. Finally, 93 the main conclusions of this work are highlighted. 94 BUILDING-SOIL MODEL FORMULATION 95 This section develops the formulation of the proposed building-soil model. A 3D diagram 96 of the considered problem is presented in Fig. 1 (a) , which represents a two-story building 97 with six columns in each floor built on soil composed of two layers over a half-space. In 98 general, the building-soil model considers a (N s + 1)-story building (N s stories over a ground 99 level) with rectangular floors supported by a distribution of round columns. The building is 100 constructed on stratified soil, which is assumed to be composed of N L horizontal layers over 101 a half-space. Piled foundations are considered for modeling the building-soil interaction. 102 4 The piles are buried in one or more of the soil layers and the whole system is excited by 103 an external circular surface or buried vertical harmonic load F out . The soil layers and the 104 building floors, columns and stories considered in this work are listed in Fig. 1 (b) . 105 The dynamic response of the building floors is obtained once the coupling forces between 106 the different parts of the complete system are determined. The coupling forces are considered 107 to be vertical forces because it is assumed that the vertical component of the building 108 vibrations is significantly more important than the horizontal ones. This assumption is 109 in agreement with experimental measurements found in the literature (Sanayei et al. 2013; 110 Athanasopoulos and Pelekis 2000; Crispino and D'Apuzzo 2001). 111 Floor and columns models 456 predict the vertical component of the building vibration, which is, for the type of induced 457 vibrations considered, the main component of the floor's response. 458 CONCLUSIONS 459 This paper presents a computationally efficient three-dimensional building-soil coupled 460 model for predicting the response of a building to ground-borne vibrations. The model 461 considers that the vibration is transmitted to the floors through the building columns and 462 that the vertical component of this vibration is the dominant one. The coupling of the 463 building and the soil has been performed considering piled foundations and a horizontally 464 stratified soil. They have been modeled using the Novak pile model and the SMM for a 465 layered half-space, respectively. The use of a simplified soil-pile model ensures the simplicity 466 and efficiency of the model and allows to take into account the effect of soil stratification. 467 The presented model is highly suitable for performing many types of parametric studies 468 and large-scale vibration predictions due to its small computational cost and considerable 469 accuracy. 470 The response of the building model has been compared to the one obtained using a FE 471 model of two different three-story building examples, finding a good agreement between both 472 in the range of frequencies of interest in building response to ground-borne vibrations. The 473 building model has been used to study the isolation efficiency of considering, first, a thicker 474 lower floor and, second, columns with a larger cross-section. The results at the center of each 475 floor show that, although both modifications can be considered as vibration countermeasures, 476 the use of columns with a larger cross-section is significantly more effective for the isolation 477 of the building's upper floors. 478 The building-soil coupled model is finally used to study the effect that the soil strati-479 fication has in the vibration response of the building floors. The results obtained for two 480 different soils show clear differences in all the frequency range studied. This result justifies 481 the importance of having the soil elastic parameters correctly characterised and highlights 482 22 the importance of considering a stratified soil model in nonhomogeneous soils. 483 The proposed building-soil coupled model is able to consider many features of the dynamic 484 response of a N-story building in a layered half-space with a very small computational cost. 485 This advantage can be used for obtaining the building response to a wide variety of excitations 486 in cases where the mechanical parameters of the problem are not accurately defined and/or 487 where the results at large areas or for a large set of buildings are required. 488 ACKNOWLEDGEMENTS 489 The results presented have been obtained in the frame of ISIBUR project TRA2014-490