Structure–soil–structure coupling in seismic excitation and “city effect”

We study the interaction between the buildings of a city through a fully coupling structure–soil–structure model. The main issues are to emphasize the collective behavior of the buildings during a seismic excitation (city effect) and to compute the global response of the city.The anti-plane shearing of an elastic half space with a part of its boundary formed by the rigid foundations of the buildings is considered. The buildings are modeled as elastic springs which relate two concentrated masses. The associated mathematical problem is a partial differential equation coupled with an ordinary differential equation through a special class of boundary conditions. To focus on the coupling structure–soil–structure (city frequencies) the associated spectral problem is rewritten in terms of a nonlinear scalar equation involving the eigenvalues of a symmetric matrix. A numerical approach, involving an integral method, is proposed to numerically investigate the city effect.The asymptotic limit of the principal eigenvalue with respect to the ratio between the width of the building and the width of the city points out the principal frequency of the city. This value does not depended on the number of buildings which compose the city but it is specific to each city via the specific properties of the buildings. The collective behavior of the buildings during a seismic excitation is emphasized through the shape of the first eigenfunction. Finally a numerical illustration of a the propagation in the city of the impact on the top of one of the buildings (a 9/11 type event) is presented.