modes of vibration, it is represented by the mass

the rigid circular footing resting on the surface of an

moment of inertia about the axis of rotation. For torsional

elastic semi-infinite body for each of the modes of

oscillation about the vertical axis, it is designated as Ιθ;

vibration. Introduce these quantities into equations given

whereas for rocking oscillation, it is Ιψ, (for rotation about

in paragraph 17-2 to compute resonant frequencies and

amplitudes of dynamic motions. The dimensionless

the axis through a diameter of the base of the

frequency, ao, for all modes of vibration is given as

foundation). If the foundation is considered to be a right

follows:

circular cylinder of radius ro, height h, and unit weight y,

expressions for the mass and mass moments of inertia

2πforo

p

(17-9)

are as follow:

= ωro

ao = Vs

G

π

2

ro h

y

The shear velocity, Vs, in the soil is discussed in

m= g

(17-6)

paragraph 17-5.

π ro h y

(3) Figure 17-4 shows the variation of the

4

damping ratio, D, with the mass ratio, B, for the four

Is = 2g

(17-7)

modes of vibration. Note that D is significantly lower for

the rocking mode than for the vertical or horizontal

πro hy

(

)

2

2

2

r0 +

h

(17-8)

translational modes. Using the expression M = 1/(2D)

Ιψ=

g

4

3

for the amplitude magnification factor and the

appropriate D, from figure 17-4, it is obvious that M, can

(2) Theoretical solutions describe the

become large. For example, if Bψ = 3, then Dψ = 0.02

motion magnification factors M, or ML, for example, in

and Mψ = 1/(2 x 0.02) = 25.

terms of a "mass ratio" Bz and a dimensionless

damping ratios, and spring constants corresponding to

vibrations

of

Table 17-1. Mass ratio, Damping Ration, and Spring Constant for Rigid Circular Footing on the Semi-Infinite Elastic Body

U. S. Army Corps of Engineers

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