where

a. Foundations on elastic half-space. For very

me = the total rotating mass

small deformations, assume soils to be elastic materials

e = the eccentricity

with properties as noted in paragraph 3-8. Therefore,

theories describing the behavior of rigid foundations

second

resting on the surface of a semi-infinite, homogeneous,

isotropic elastic body have been found useful for study of

e. The ordinate Mz. (fig 17-2(b)) relates the

the response of real footings on soils. The theoretical

dynamic displacement, Az, to me e/m. The peak value of

treatment involves a circular foundation of radius, ro, on

the response curve is a function of the damping ratio and

the surface of the ideal half-space. This foundation has

is given by the following expression:

six degrees of freedom: (1-3) translation in the vertical

1

(z) or in either of two horizontal (x and y) directions; (4)

Mz(max) or Mz = 2D√1-D

2

torsional (yawing) rotation about the vertical (z) axis; or

(17-4)

(5-6) rocking (pitching) rotation about either of the two

horizontal (x and y) axes. These vibratory motions are

For small values of D, this expression becomes 1/2D.

illustrated in figure 17-3.

These peak values occur at frequency ratios of

(1) A significant parameter in evaluating

fo

fn=√1-D

2

the dynamic response in each type of motion is the

(fig. 17-2a)

inertia reaction of the foundation. For translation, this is

or

(17-5)

simply the mass, m = (W/g); whereas in the rotational

1

fo

fn= √1-2D

2

(fig. 17-2b)

(Courtesy) of F. E Richart, Jr., J R. Hall. Jr., and R. D.

Woods, Vibrations of Soils and Foundations, 1970, p 316.

Reprinted by permission of Prentice-Hall, Inc., Englewood

Cliffs, N. J.)

Figure 17-1. Response spectra for tibraton limits.

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