TM 5-818-1 / AFM 88-3, Chap. 7
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
17-7
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