Radiation damping for rigid foundations. Approximate expressions

1. Introduction

The design of foundations to support heavy machinery that could induce vibrations was already recognized as an important practical problem in the 1930s giving rise to the field of Soil Dynamics. In the thirties Reissner [3, 4] derived the first analytical solution for the vertical displacements on the surface of a linear elastic, homogeneous, and isotropic half space subjected to a harmonic normal stress uniformly distributed over a circular area. The solution for the vertical case was followed immediately by a solution for torsional vibrations. Work along these lines was continued in the following years by Reissner and Sagoci [5] and by Shekhter [6], who used the average of the displacements at the center and at the edge of the loaded area to obtain curves of dynamic amplification as a function of a dimensionless frequency and a mass ratio. These curves were widely used.

The fifties saw a significant increase both in the number of researchers engaged in this area and in the number of related publications. The solution of the true mixed boundary value problem, where stresses are specified along the surface of the soil, outside the area of the foundation (stress free surface) while displacements are imposed at the base of a rigid and massless body, was addressed by Borodatchev in 1964 [7] for the vertical case. A rigorous solution for the dynamic stiffness of a rigid and massless circular foundation on the surface of a linear elastic, homogeneous and isotropic half space was presented in graphical and tabular form over an extended range of frequencies by Veletsos and Wei [1] for the coupled horizontal and rocking vibrations, in 1971 an independent solution was published by Luco and Westmann [8] almost at the same time. Additional results for vertical and torsional excitations and for viscoelastic or hysteretic media were obtained by Veletsos and Verbic [2].

At the same time that the analytical formulations were developed, Reissner explored the possibility of reproducing his results with a lumped parameter model consisting of a mass, a spring and a dashpot, but he concluded that their values would have to be functions of frequency, and he could not find simple expressions for them. Whitman and Richart in 1967 [9] used again lumped parameter models with springs, masses and dashpots to provide graphical results for the amplitudes of vibration of a machine foundation as a function of frequency in terms of a mass ratio. In the 70s Veletsos and Wei [1] and Veletsos and Verbic [2] found approximate but accurate expressions for the foundation stiffness terms of a rigid circular mat on the surface of an elastic, homogeneous and isotropic half space as function of the dimensionless frequency and these are still the best simplified formulas available to date. This work presents approximate explicit expressions for the natural frequencies and effective damping in each mode of a rigid mass on the surface of an elastic half space subjected to both vertical and coupled horizontal/rocking harmonic excitations. These formulas are intended for preliminary estimates when designing machine foundations.

2. Formulation

It is considered a rigid cylindrical mass representing a machine foundation resting on the surface of an elastic half space. Calling $M$ the total mass, $I$ the mass moment of inertia with respect to the center of gravity, $H$ the height of this point, $R$ the radius, $c_s$ the shear wave velocity of the soil, $G$ its shear modulus, ${\rho }_s$ its mass density and $\nu$ Poisson’s ratio, one can consider both vertical and coupled horizontal-rocking vibrations. A schematic representation of the mass is shown in Fig. 1.

Fig. 1Rigid mass on elastic halfspace system

3. Results

The explicit expressions for the natural frequency and damping for the vertical case were already included with the formulation. Fig. 2 compares the results using the exact expressions with those provided by the suggested approximate formulas. The results show that the approximate expressions provide a very good accuracy for values of the mass ratio larger than 10.

For the horizontal/rocking case, the variation of the 2 dimensionless natural frequencies $a_{01}$, $a_{02}$ and the corresponding estimated values of the effective damping at these frequencies as a function of the mass ratio $\mu$ are shown in Figs. 3 and 4 for aspect ratios of 1, 1.5, 2 and 2.5, as well as for Poisson’s ratios of 1/3 and 1/2. These results show that the Poisson ratio has a negligible effect on the solution, although the damping ratio shows larger differences, these still are negligible.

Fig. 2Dimensionless frequencies and damping for the vertical case

a)$\nu =\mathrm{}$1/3

b)$\nu =$ 1/2

The dimensionless frequencies and the associated values of damping for the 2 modes can be expressed in the form:

where the coefficients $b$ and $a$ are functions of the Poisson’s ratio and the slenderness ratio $H/R$. The exponent $a$ is relatively insensitive to the values of the Poisson’s ratio and $H/R$ (except for the lower values of $H/R$, the second mode and in particular the damping in that mode). It is approximately 0.5 as for the approximate expressions proposed earlier for the vertical case. The factor b is also relatively insensitive to Poisson’s ratio but depends strongly on $H/R$. It decreases with increasing $H/R$ for the first mode (natural frequency and associated damping) and it increases for the second mode. Fitting the exact solutions, we propose the following expressions to evaluate de dimensionless frequencies and the damping ratios as a function of the aspect ratio and independently of the Poisson’s ratio:

These expressions were obtained fitting the numerical results obtained for values of $H/R$ between 0.5 and 2.5. In the limit as $H/R$ tends to 0 there would be only one natural frequency and the average expressions for a Poisson’s ratio between $1/3$ and $1/2$ would be $a_1=\text{2.25}$ and ${\zeta }_1=\text{0.725}$. The first is about 10 % higher than the value provided by the suggested formula while the second is almost exactly the same. Figs. 3 and 4 show a comparison of the results obtained with the approximations and the exact solutions. The results for an aspect ratio of 0.5 are not presented for lack of space. These results show that the proposed formulas are quite accurate for $H/R$ less than 2.5.

Fig. 3Exact vs. approximate solutions for a01 and a02

Fig. 4Exact vs. approximate solutions for ζ1 and ζ2

4. Conclusions

A series of approximate expressions have been presented to estimate the natural frequencies and associated damping of rigid masses on the surface of an elastic half space. These expressions provide in general a reasonable accuracy and should be of value for preliminary estimates at the early stages of a design. The results indicate that the dimensionless frequencies are essentially proportional to the radius to the exponent 3/2. This implies that the actual frequencies in rad/sec or Hz would be proportional to the square root of the radius and the square root of the soil modulus as would have been predicted using the static values of the foundation stiffness terms.