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Vibration of surface foundations of arbitrary shapes
Author(s) -
Liou GinShow
Publication year - 1991
Publication title -
earthquake engineering and structural dynamics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.218
H-Index - 127
eISSN - 1096-9845
pISSN - 0098-8847
DOI - 10.1002/eqe.4290201203
Subject(s) - piecewise , mathematical analysis , finite element method , mathematics , generalized coordinates , matrix (chemical analysis) , coordinate system , constant (computer programming) , cylindrical coordinate system , differential equation , geometry , physics , materials science , computer science , composite material , programming language , thermodynamics
Presented is a systematic procedure for generating impedance (or compliance) matrices for foundations with arbitrary shapes, resting on an elastic half‐space medium. A technique to decompose prescribed harmonic tractions on the half‐space medium is employed to solve analytically the differential wave equations in cylindrical coordinates. However, the interaction stresses due to the vibration of a foundation with arbitrary shape are described in rectangular coordinates, and assumed to be piecewise constant in the region of the arbitrary shape. A coordinate transformation matrix is introduced for the piecewise constant tractions in order to use the solution of the differential wave equations in cylindrical coordinates. Finite element modelling is assumed in rectangular coordinates for the foundation itself. The impedance matrix is then obtained for the finite element model, using a variational principle and the reciprocal theorem. A simple example of a rigid square plate resting on a half‐space medium and subjected to vertical excitation is used to demonstrate the efficiency and effectiveness of the procedure. Some numerical aspects are investigated and some possible extensions of the procedure are also discussed.