15 AUGUST 2005
exciting torque and soil reaction torque must be symmetrical about the vertical axis of
rotation. Also, the center of gravity of the foundation must lie on the vertical axis of
When horizontal or overturning moments act on a block foundation, both
horizontal (sliding) and rocking vibrations occur. The coupling between these motions
depends on the height of the center of gravity of the machine-foundation about the
resultant soil reaction. Details of a coupled rocking and sliding analysis are given in the
example in figure 12-6.
A "lower bound" estimate of the first mode of coupled rocking and sliding
vibrations can be obtained from the following:
In equation (12-14), the resonant frequencies in the sliding x and rocking w motions can
be determined by introducing values from table 12-1 into equations (12-1) and (12-5).
(Note that equation (12-14) becomes less useful when Dz is greater than about 0.15).
The first mode resonant frequency is usually most important from a design standpoint.
Examples. Figure 12-5, Example 1, illustrates a procedure for design of a
foundation to support machine-producing vertical excitations. Figure 12-5, Example 2,
describes the analysis of uncouple horizontal and rocking motion for a particular
foundation subjected to horizontal excitations. The design procedure of Example 1 is
essentially an iterative analysis after approximate dimensions of the foundation have
been established to restrict the static deflection to a value comparable to the design
Figure 12-5, Example 1, shows that relatively high values of damping ratio D are
developed for the vertical motion of the foundation, and Example 2 illustrates that
the high damping restricts dynamic motions to values slightly larger than static
displacement caused by the same force. For Example 2, establishing the static
displacement at about the design limit value leads to satisfactory geometry of the
Example 2 (Figure 12-5) gives the foundation geometry, as well as the analysis
needed to ascertain whether the design criterion is met. It is assumed that the
400-pound horizontal force is constant at all frequencies and that a simple
superposition of the single-degree-of-freedom solutions for horizontal translation
and rocking will be satisfactory. Because the horizontal displacement is
negligible, the rocking motion dominates, with the angular rotation at resonance
amounting to (metersψ x ψs) or Aψ = 5.6 x 0.51 x 10-6 = 2/85 x 10-6 radians. By
converting this motion to horizontal displacement at the machine centerline, it is
found that the design conditions are met.
Figure 12-6, the foundation of Example 2 (figure 12-5), is analyzed as a coupled
system including both rocking and sliding. The response curve for angular
rotation shows a peak motion of Aψ = 2.67 x 10-6 radians, which is comparable to
the value found by considering rocking alone. The coupled dynamic response of