Abstract :
Previously published analytical solutions of dynamic problems for continuously nonhomogeneous
bases concerned one-dimensional, plane or axisymmetric cases. In this paper a
solution in cylindrical co-ordinates is presented for an arbitrary angle distribution in the horizontal
plane. The medium is assumed as isotropic, continuously non-homogeneous in the depth direction
and homogeneous in the horizontal direction. Poissonʹs ratio is adopted as constant. For each angle
component of the solution including cos(ng) or sin(n~), the problem is reduced to three ordinary
differential equations (or two for the axisymmetric case where n = 0) ; two of them are coupled.
Corresponding boundary conditions are formulated for given stresses or displacements at planes
z = const. An example of non-homogeneity where shear modulus increases exponentially with depth,
G(z) = G(O) exp(z/zo), is considered (z0 is a constant). The solution for the half-space subjected to
a surface load is represented in the form of integrals including Bessel functions and suitable solutions
of above-mentioned ordinary differential equations. At low frequencies the integrands have no
singularities on the real axis of the complex plane ; then, beginning from a definite value of the
frequency (cutoff frequency), poles of integrands appear on the real axis and energy can be passed
to the half-space. At some frequencies (resonance frequencies) there are double poles on the real
axis leading to infinite amplitudes in the non-dissipative case. For calculations, shear modulus was
treated as a complex quantity (G(0) = G0(1 +ie)), where e is a small positive constant. Results of
calculations for surface displacements induced by vertical and horizontal acting point forces on the
surface of the half-space are presented for static and dynamic problems, and comparison with results
for the homogeneous half-space is demonstrated. © 1997 Elsevier Science Ltd.
