2.4 Veletsos Model

Impedance parameters

Figure 2.4.1a Block Foundation Equivalent Radius

Figure 2.4.1b Block Foundation Notations

This model first introduced by Veletsos in 1973 for a circular rigid disk setting on a viscoelastic half space. Further studies were carried on to account for the effect of soil material damping and the effects of the footing embedment on the soil. The impedance terms introduced are frequency dependent and can be expressed in the form,

In this expression, ${k_i}^*$ is the frequency dependent impedance, $k_i$ is the static stiffness for the i-th degree of freedom, $\omega$ is the circular frequency of the vibration, $c_i$ is the damping constant for the i-th degree of freedom, and $i$ is the complex numbers operator and is equal to $\sqrt{-1}$. Expressions for vertical, horizontal, rocking, and torsional degrees of freedom will be presented in this section.

Dimensionless frequency$\space a_{oi}$

It is very common in the literature to develop the impedance expressions in terms of the dimensionless frequency$\space a_{oi}$ defined as,

Vertical Degree of Freedom $v$

The vertical degree of freedom impedance can be expressed as,

In Eq. 2.4.2, the terms $C_{v1}$ and $C_{v2}$ can be expressed as,

$R'$ is a factor to account for material damping,

$\beta_m$ is the base soil damping ratio, and $\chi_v$ is defined as,

$\psi_v$ is defined as,

$\nu$ is Poisson's ratio, and $\gamma_i$ are numerical coefficents that depend on Poisson's ratio.

Horizontal Degree of Freedom $u$

The vertical degree of freedom impedance can be expressed as,

In Eq. 2.4.2, the terms $C_{u1}$ and $C_{u2}$ can be expressed as,

$\alpha_1$ is a numerical coefficent that depend on Poisson's ratio.

Rocking Degree of Freedom $\psi$

The vertical degree of freedom impedance can be expressed as,

In Eq. 2.4.2, the terms $C_{\psi1}$ and $C_{\psi2}$ can be expressed as,

$\chi_{\psi}$ is defined as,

$\psi_{\psi}$ is defined as,

$\beta_i$ are numerical coefficent that depend on Poisson's ratio.

Torsional Degree of Freedom $\eta$

The Torsional degree of freedom impedance can be expressed as,

In Eq. 2.4.2, the terms $C_{\psi1}$ and $C_{\psi2}$ can be expressed as,

$A$ is defined as,

$B$ is defined as,

The constants in these equations are $b_1 = 0.425$ and $b_2 = 0.687$.

Effect of Foundation Embedment

Figure 2.1.1 Effect of Embedded Foundation

Because footing are normally embedded in the soil for normal construction methods, the effect of full or partial embedment in the soil must be studdied. The effect of embedment is a very complex topic with lots od studies in the literature. The most credible and somewhat straignforward to implement method is introduced in this section. This method estimates the impedance of the side layer in a similar fashion to the estimated impedance of the base soil then add the two quantities together to obtain the adjusted impedance of the overall foundation.

The complex dynamic empedance of the side layer can be expressed using the following equations, vertical impedance,

Horizontal impedance,

Rocking impedance,

Torsional impedance,

In the above equations, the terms $S_{i1}$ and $S_{i2}$ can be determined approximately from Table 2.4.1, if the value of the dimensionless frequency $a_{oi}$ is less than 2 ($a_{oi} < 2.0$) for two categories of side layers, cohesive and granular.

For the broader range of $a_{oi}$ values, mathematical expressions involving Bessel functions of the first and second kinds are available in the literature. Due to the complexity of these expressions, they will not be introduced here, however, it is implemented in the impedance calculator provided in section 2.5 of this course.