motion. These changes are a function of the type of

motion and the embedment ratio d/ro.

(1) For vertical vibrations, both analytical

and experimental results indicate an increase in the

static spring constant with an increase in embedment

depth. Embedment of the circular footing a distance d/ro

< 1.0 produces an increase in the embedded spring

constant kzd' which is greater than kz (table 17-1) by kzd/kz

≅ (1 + 0.6 d/ro). An increase in damping also occurs, i.e.,

Dzd/Dz (1 + 0.6 d/ro). These two approximate relations

lead to an estimate of the reduction in amplitude of

motion because of embedment from Azd / Az = 1 / Dzd / Dz

x kzd/kz). This amount of amplitude reduction requires

complete soil adhesion at the vertical face, and test data

have often indicated less effect of embedment. Test

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

data indicate that the resonant frequency may be

Woods, Vibrations of Soils and Foundations, 1970. p

increased by a factor up to (1 + 0.25 d/ro) because of

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

embedment.

Englewood Cliffs, N. J.)

(2) The influence of embedment on

coupled rocking and sliding vibrations depends on the

Figure 17-4. Equivalent damping ratio for oscillation of

ratio Bω/Bx (table 17-1). For Bω/Bx ≅ 3.0, the increase in

rigid circular footing on elastic half-space.

natural frequency due to embedment may be as much as

(1 + 0.5 d/ro). The decrease in amplitude is stongly

b. Effects of shape of foundation.

The

dependent upon the soil contact along the vertical face of

theoretical solutions described above treated a rigid

the foundation, and each case should be evaluated on

foundation with a circular contact surface bearing against

the basis of local soil and construction conditions.

the elastic half-space. However, foundations are usually

e. Effect of finite thickness of elastic layer.

rectangular in plan.

Rectangular footings may be

Deposits of real soils are seldom homogeneous to

converted into an equivalent circular footing having a

significant depths; thus theoretical results based on the

radius ro determined by the following expressions:

response of a semi-infinite elastic media must be used

For translation in z- or x-directions:

with caution. When soil layers are relatively thin, with

ro = 4cd

(17-10)

respect to foundation dimensions, modifications to the

π

theoretical half-space analyses must be included.

For rocking:

(1) Generally, the effect of a rigid layer

3

ro =4 16cd

(17-11)

underlying a single elastic layer of thickness, H, is to

3n

reduce the effective damping for a foundation vibrating at

For torsion:

the upper surface of the elastic layer. This condition

2

2

ro =4 16cd(c + d )

(17-12)

results from the reflection of wave energy from the rigid

6π

base back to the foundation and to the elastic medium

In equations (17-10), (17-11), and (17-12), 2c is the

surrounding the foundation. For vertical or torsional

width of the rectangular foundation (along the axis of

vibrations or a rigid circular foundation resting on the

rotation for rocking), and 2d is the length of the

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

foundation (in the plane of rotation for rocking). Two

very large amplitude of resonant vibrations can occur if

values of ro are obtained for rocking about both x and y

axes.

Vs

c. Computations. Figure 17-5 presents

> 4H(17-13)

examples of computations for vertical motions (Example

fo

1) and rocking motions (Example 2).

In equation (17-13), V, is the shear wave velocity in the

d. Effect of embedment.

Embedment of

elastic layer and fo is the frequency of footing vibrations.

foundations a distance d below the soil surface may

When the conditions of equation (17-4) occur, the natural

modify the dynamic response, depending upon the soil-

frequency (equation (17-1)) becomes the important

foundation contact and the magnitude of d. If the soil

design criterion because at that frequen-

shrinks away from the vertical faces of the embedded

foundation, no beneficial effects of embedment may

occur. If the basic evaluation of foundation response is

based on a rigid circular footing (of radius ro) at the

surface, the effects of embedment will cause an increase

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