a method of function space for vertical impedance function of a circular rigid foundation on a transversely isotropic ground

this paper is concerned with investigation of vertical impedance function of a surface rigid circular foundation resting on a semi-infinite transversely isotropic alluvium. to this end, the equations of motion in cylindrical coordinate system, which because of axissymmetry are two coupled equations, are converted into one partial differential equation using a method of potential function. the governing partial differential equation for the potential function is solved via implementing hankel integral transforms in radial direction. the vertical and radial components of displacement vector are determined with the use of transformed displacement-potential function relationships. the mixed boundary conditions at the surface are satisfied by specifying the traction between the rigid foundation and the underneath alluvium in a special function space introduced in this paper, where the vertical displacements are forced to satisfy the rigid boundary condition. through exercising these restraints, the normal traction and then the vertical impedance function are obtained. the results are then compared with the existing results in the literature for the simpler case of isotropic half-space, which shows an excellent agreement. eventually, the impedance functions are presented in terms of dimensionless frequency for different materials. the method presented here may be used to obtain the impedance function in any other direction as well as in buried footing in layered media.