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In this paper, a method is presented for the calculation of the vibration created in buildings by the operation of underground railways. The method is based on the sub-modelling approach which is used to couple a model of a building on a piled foundation to another model that calculates the vibration generated in the soil in underground railway tunnels. The method couples a building on a piled foundation to the soil at discrete points by satisfying equilibrium and compatibility requirements at those points. The method results in efficient numerical calculations. A two-dimensional frame made of beam elements is used to model the building and its piled foundation. The elements are formulated using a dynamic stiffness matrix which accounts for Euler-Bernoulli bending and axial behaviour. Vibrations created by a train moving in an underground tunnel are calculated using the well-known pipe-in-pipe (PiP) model. The model calculates the power spectral density (PSD) of the displacement in the soil. The excitation mechanism is the roughness of the rail and the PSD is calculated for a train moving on a floating-slab track in an underground railway tunnel for a stationary process. The current version of PiP accounts for a tunnel embedded in a half-space. The building frame is coupled in this paper at 90 to the tunnel's centreline. The main result of this paper illustrates the significant contribution of the building's dynamics to the displacement wave field received by the building. The example presented in this paper shows a decrease of more than 20 dB in the displacement PSDs at frequencies larger than 10 Hz when accounting for the change in this wave field.

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This paper explores the coupling of the Perfectly Matched Layer technique (PML) with the Thin Layer Method (TLM), the combination of which allows making highly efficient and accurate simulations of layered half-spaces of infinite depth subjected to arbitrary dynamic sources anywhere. It is shown that with an appropriate complex stretching of the thickness of the thinlayers, one can assemble a system of layers which fully absorbs and attenuates waves for any angle of propagation. An extensive set of numerical experiments show that the TLM+PML performance is clearly superior to that of a standard TLM model with paraxial boundaries augmented with buffer layers (TLM+PB). This finding strongly suggests that the proposed combination may in due time constitute the preferred choice for this class of problems.