Identification of Energy Loss Due to Soilstructure Interaction from the Recorded Vibrations of Buildings

Abstract

The vibrations of a building are due to the waves that travel up and down the building. When the ground below the foundation is not rigid (e.g., hard rock), a portion of the downgoing waves from the lowest story is transmitted back into the ground. This transmitted portion corresponds to the energy loss due to soil-structure interaction. This paper presents an approach to identify such energy losses from the recorded vibrations of buildings. The approach is based on the discrete-time formulation of wave propagation in multi-story buildings subjected to vertically propagating shear waves. The wave propagation in each story is characterized by the upgoing and downgoing waves at the top of the story. We derive equations for the 2x2 transfer matrices that relate the upgoing and downgoing waves in one story to those in the adjacent stories. The downgoing waves at the soil-foundation interface gives a measure of the energy loss due to soil-structure interaction. Starting from the top story, and by consecutive use of story transfer matrices, we can relate the upgoing and downgoing energies at the soil foundation interface to those at the top story. This relationship is also represented by a 2x2 matrix. The matrix is basically the product of 2x2 story transfer matrices from the foundation to the top story. The inspection of the elements of the matrix indicates that this relationship is equivalent to an IIR (Infinite Impulse Response) filter, which can be represented by an AR (Auto-Regressive) model between the foundation response and the top story response