12-1.3.5

seldom homogeneous to significant depths; thus theoretical results based on the

response of a semi-infinite elastic media must be used with caution. When soil layers

are relatively thin, with respect to foundation dimensions, modifications to the theoretical

half-space analyses must be included.

Generally, the effect of a rigid layer underlying a single elastic layer of

thickness, H, is to reduce the effective damping for a foundation vibrating at the upper

surface of the elastic layer. This condition results from the reflection of wave energy

from the rigid base back to the foundation and to the elastic medium surrounding the

foundation. For vertical or torsional vibrations or a rigid circular foundation resting on

the surface of the elastic layer, it has been established that a very large amplitude of

resonant vibrations can occur if

In equation (12-13), Vs is the shear wave velocity in the elastic layer and fo

is the frequency of footing vibrations. When the conditions of equation (12-4) occur, the

natural frequency (equation (12-1)) becomes the important design criterion because at

that frequency excessive dynamic motion will occur. To restrict the dynamic oscillation

to slightly larger than the static displacement, the operating frequency should be

maintained at one half, or less, of the natural frequency (figure 12-2).

The relative thickness (expressed by H/ro) also exerts an important

influence on foundation response. If H/ro is greater than about 8, the foundation on the

elastic layer will have a dynamic response comparable to that for a foundation on the

elastic half-space. For H/ro < 8, geometrical damping is reduced, and the effective

spring constant is increase. The values of spring constant, k, in table 12-1 are taken as

reference values, and table 12-2 indicates the increase in spring constant associated

with a decrease in thickness of the elastic layer. Values of the increase in spring

constant for sliding and for rocking modes of vibration will tend to fall between those

given for vertical and torsion for comparable H/ro conditions.

12-8

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