15 AUGUST 2005
rocking (pitching) rotation about either of the two horizontal (x and y) axes. These
vibratory motions are illustrated in figure 12-3.
A significant parameter in evaluating the dynamic response in each type of
motion is the inertia reaction of the foundation. For translation, this is simply the mass,
m = (W/g), whereas in the rotational modes of vibration, it is represented by the mass
moment of inertia about the axis of rotation. For torsional oscillation about the vertical
axis, it is designated as Io, whereas for rocking oscillation, it is Iψ (for rotation about the
axis through a diameter of the base of the foundation). If the foundation is considered
to be a right circular cylinder of radius, ro, height, h, and unit weight, γ, expressions for
the mass and mass moments of inertia are as follows:
m = _π ro2hγ_
Iθ = _π ro4hγ_
Iψ = _π ro2hγ_ o __ro2__
Theoretical solutions describe the motion magnification factors Mz or Mż,
for example, in terms of a "mass ratio" Bz and a dimensionless frequency factor ao.
Table 12-1 lists the mass ratios, damping ratios, and spring constants corresponding to
vibrations of the rigid circular footing resting on the surface of an elastic semi-infinite
body for each of the modes of vibration. Introduce these quantities into equations given
in paragraph 12-1.2 to compute resonant frequencies and amplitudes of dynamic
motions. The dimensionless frequency, ao, for all modes of vibration is given as follows:
ao = _ 2π fo ro__ = ω ro w(ρ/G)
The shear velocity, Vs, in the soil is discussed in paragraph 17-5.
Figure 12-4 shows the variation of the damping ratio, D, with the mass
ratio, B, for the four modes of vibration. Note that D is significantly lower for the rocking
mode than for the vertical or horizontal translational modes. Using the expression
meters = 1/(2D) for the amplitude magnification factor and the appropriate Dψ from
figure 12-4, it is obvious that metersψ can become large. For example, if Bψ = 3, the Dψ
= 0.02 and metersψ = 1/(2 x 0.02) = 25.