Seismic Response of Inhomogeneous Soil Deposits with Exponentially Varying Stiffness

The response of an inhomogeneous soil layer with exponentially varying stiffness with depth is explored using one-dimensional viscoelastic wave propagation theory. The governing equation is treated analytically, leading to an exact harmonic solution of the Bessel type. Both positive and negative velocity gradients are examined using a pertinent dimensionless parameter. It is shown that (1) for positive stiffness gradients, strains attenuate with depth faster than displacements that, in turn, attenuate faster than stresses; and (2) close to the soil surface, curvatures are controlled by acceleration, whereas they are controlled by strain at depth. The fundamental natural frequency of the layer compares well against approximations on the basis of the Rayleigh quotient. Novel asymptotic and ad hoc approximate solutions for the base-to-surface transfer function are proposed, providing good alternatives to the complex exact solution at both high and low frequencies. New expressions are derived relating (1) shear strain and peak particle velocity; and (2) curvature and peak ground acceleration close to the soil surface. A full-domain approximation is provided, allowing the practical implementation of the specific velocity model. Numerical examples are presented.