On the frequencies of inhomogeneous soil strata: Dobry's paradox

This brief article elaborates on some clearly unlikely predictions made some four decades ago by R. Dobry within the context of his doctoral dissertation, which concern the resonant frequencies of soil strata whose stiffness starts at zero at ground level and then increases continuously as some power of the depth. Although the problem was eventually traced to subtle changes in the boundary conditions which take place when the soil parameter exceeds a threshold value, and the discrepancy was ultimately fully resolved, Dobry initially countered to the technical objections of the writer by presenting an extremely elegant counter‐proof based on dimensional analysis whose strength emanated from the fact that it was free from the issue of the boundary conditions, and thus forced the writer to seek alternative explanations for the contradictory results. It is this fruitful exchange of ideas which provides the motivation for this brief technical note expounding on an apparent paradox, which may prove useful to the soil dynamics community for its potential didactical value, not to mention as an outright tool. The Problem In section 2.4 of his rather massive doctoral dissertation at MIT concerned with soil properties and one‐dimensional wave amplification in soil deposits, R. Dobry considered the amplification of shear waves in a soil stratum of finite thickness h on rigid rock, where the mass density  of the stratum is uniform but the shear wave velocity is zero at the ground surface and increases steadily with depth in the following form: /2 /2 p p s h z V Az V h         , /2 p h Ah V  (1) where , A p are constants, h V is the shear wave velocity at the base of the stratum of depth h , and z is the depth measured down from the free surface. Although Dobry suggested that typical values for p lie in the range 0.33 1 p   for most normally consolidated soils at small strains, his formulation was quite general and included other values of p . The special case 1 p  , in which the shear modulus increases linearly with depth, is often referred to as a Gibson soil. After solving in closed form the differential equation with a boundary condition of zero strain at the surface, Dobry concluded —correctly— that the fundamental period of the inhomogeneous stratum is proportional to the depth raised to some power, namely (p. 98)   1 2 1 1 ~ p T h  (2) Professor of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 Soil Dynamics and Earthquake Engineering 47 (2013) 38–40