cy excessive dynamic motion will occur. To restrict the

introducing values from table 17-1 into equations (17-1)

dynamic oscillation to slightly larger than the static

and (17-5). (Note that equation (17-14) becomes less

useful when Dx is greater than about 0.15). The first

displacement, the operating frequency should be

mode resonant frequency is usually most important from

maintained at one half, or less, of the natural frequency

a design standpoint.

(fig 17-2).

g. Examples.

Figure 17-5, Example 1,

(2) The relative thickness (expressed by

H/ro) also exerts an important influence on foundation

illustrates a procedure for design of a foundation to

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

support machine-producing vertical excitations. Figure

on the elastic layer will have a dynamic response

17-5, Example 2, describes the analysis of uncoupled

comparable to that for a foundation on the elastic half-

horizontal and rocking motion for a particular foundation

space. For H/ro < 8, geometrical damping is reduced,

subjected to horizontal excitations.

The design

and the effective spring constant is increased. The

procedure of Example 1 is essentially an iterative

values of spring constant, k, in table 17-1 are taken as

analysis after approximate dimensions of the foundation

reference values, and table 17-2 indicates the increase

have been established to restrict the static deflection to a

in spring constant associated with a decrease in

value comparable to the design criterion.

thickness of the elastic layer. Values of the increase in

(1) In figure 17-5, Example 1 shows that

spring constants for sliding and for rocking modes of

relatively high values of damping ratio D are developed

vibration will tend to fall between those given for vertical

for the vertical motion of the foundation, and Example 2

and torsion for comparable H/ro conditions.

illustrates that the high damping restricts dynamic mo-

f.

Coupled modes of vibration. In general,

tions to values slightly larger than static displacement

vertical and torsional vibrations can occur independently

caused by the same force. For Example 2, establishing

without causing rocking or sliding motions of the

the static displacement at about the design limit value

foundation. To accomplish these uncoupled vibrations,

leads to satisfactory geometry of the foundation.

the line of action of the vertical force must pass through

(2) Example 2 (fig 17-5) gives the

the center of gravity of the mass and the resultant soil

foundation geometry, as well as the analysis needed to

reaction, and the exciting torque and soil reaction torque

ascertain whether the design criterion is met. It is

must be symmetrical about the vertical axis of rotation.

assumed that the 400-pound horizontal force is constant

Also, the center of gravity of the foundation must lie on

at all frequencies and that a simple superposition of the

the vertical axis of torsion.

singledegree-of-freedom

solutions

for

horizontal

(1) When horizontal

or

overturning

translation and rocking will be satisfactory. Because the

moments act on a block foundation, both horizontal

horizontal displacement is negligible, the rocking motion

(sliding) and rocking vibrations occur. The coupling

dominates, with the angular rotation at resonance

amounting to (Mψ x ψs) or Aψ = 5.6 x 0.51 x 10 = 2.85 x

-6

between these motions depends on the height of the

-6

center of gravity of. the machine-foundation about the

10 radians. By converting this motion to horizontal

resultant soil reaction. Details of a coupled rocking and

displacement at the machine center line, it is found that

sliding analysis are given in the example in figure 17-6.

the design conditions are met.

(2) A "lower bound" estimate of the first

(3) In figure 17-6, the foundation of

mode of coupled rocking and sliding vibrations can be

Example 2 (fig. 17-5) is analyzed as a coupled system

obtained from the following:

including both rocking and sliding. The response curve

for angular rotation shows a peak motion of Aψ = 2.67 x

1=1+1

(17-14)

-6

10 radians, which is comparable to the value found by

2

2

2

fo fx +fψ

considering rocking alone.

The coupled dynamic

response of any rigid foundation, e.g., a radar tower, can

In equation (17-14), the resonant frequencies in the

sliding x and rocking ψ motions can be determined by

Table 17-2. Values of kL/L for Elastic Layer (k from Table 17-1)

∞

H/ro

0.5

1.0

2.0

4.0

8.0

Vertical

5.0

2.2

1.47

1.23

1.10

1.0

Torsion

--

1.07

1.02

1.009

--

1.0

U. S. Army Corps of Engineers

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