Machine Foundations

Chapter 1. Soil Structure Interaction

Chapter 1
1.1 What is Soil-Structure-Interaction (SSI)?1.2 Dynamic Vs Static Impedance1.3 Notations and Sign Conventions1.4 Degrees of Freedom Animation1.5 Why Dynamic Stiffness is Frequency Dependent?1.6 Foundation and Soil Design Parameters1.7 Examples

1.5 Why Dynamic Stiffness is Frequency Dependent?

To understand and visualize why the dynamic stiffness is a frequency dependent quantity, let us consider the case of a single degree of freedom column vibrating axially under a harmonic load.

Figure 1.5.1 Effect of soil column mass on the dynamic stiffness

Assume that the time dependent displacement of the tip of the spring is expressed by,

v(t)=vocos⁡(ωt)v(t) = v_o \cos(\omega t)

Eq. 1.5.1

By modeling the column as a massless spring, there is no inertia force to oppose the excitation force, only the spring stiffness needs to be overcome,

kstvocos⁡(ωt)=Focos⁡(ωt) k_{st} v_o \cos(\omega t) = F_o \cos(\omega t)

Eq. 1.5.2

Or,

Fo=kstvo F_o = k_{st} v_o

Eq. 1.5.3

Observations

Notice that the response amplitude is independent of the frequency!

By taking into account the mass of the spring, we can lump it at the top of the spring as showin in Figure 1.5.1. The mass of the spring will create an inertia force that will oppose the excitation force, this inertia force is equal to,

mv¨(t)=−mω2vocos⁡(ωt) m \ddot{v}(t) = - m \omega^2 v_o \cos(\omega t)

Eq. 1.5.4

Assuming that there is no damping, the equation of motion can be expressed as,

Focos⁡(ωt)=kstvocos⁡(ωt)−mω2vocos⁡(ωt) F_o \cos(\omega t) = k_{st} v_o \cos(\omega t) - m \omega^2 v_o \cos(\omega t)

Eq. 1.5.5

Fo=(kst−mω2)vo=kvo F_o = (k_{st} - m \omega^2) v_o = k v_o

Eq. 1.5.6

And the dynamic stiffness kk can be represented by,

k=kst−mω2 k = k_{st} - m \omega^2

Eq. 1.5.7

In equation 1.5.7, kstk_{st} represnts the spring static stiffness while kk represents the spring dynamic stiffness. It is clear that the dynamic stiffness is frequency dependent and is lower than the static stiffness by mω2m \omega^2 ,

Observations

By inspection, the value of the dynamic stiffness may become negative for higher values of ω \space \omega \space and this can cause stability issues when trying to solve the dynamic equation. To solve this problem, we will introduce the "Added Mass" concept in the upcoming sections.

Relating this concept to foundation dynamics is by representing the soil by an impedance function that has both stiffness and damping as frequency dependent quantities. The mass of the soil column under the vibrating footing represents the mass of the spring.