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

the mass and mass moments of inertia are as follows:

m = _π ro2hγ_

(12-7)

g

Iθ = _π ro4hγ_

(12-8)

2g

Iψ = _π ro2hγ_ o __ro2__

___h2___ p

+

g

4

3

(12-9)

Theoretical solutions describe the motion magnification factors Mz or Mż,

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

ao = _ 2π fo ro__ = ω ro w(ρ/G)

(12-10)

Vs

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.

12-4

Integrated Publishing, Inc. |