Soi~ Dynamics and Earthquake Engineering 11 (1992) 249-253

Vertical radiation damping for a circular footing resting on a simple layered half-space

C. Navarro Departamento de Ciencia de Materiales, Escuela de Ingenieros de Caminos, Ciudad Universitaria s/n, 28040-Madrid, Spain

(Received 4 March 1991; revised version received 3 February 1992; accepted 3 February 1992)

An analytical development is carried out to roughly calculate the soil radiation damping for a rigid circular foundation subjected to the action of a vertical harmonic load of fixed amplitude. The foundation is considered to be resting on an elastic layer of constant thickness, which in turn rests on an elastic half-space (simple layered half-space). This method gives the conventional radiation damping ratio (D) as a function of that between the layer thickness and the foundation radius, Poisson ratio and layer and half-space mechanical impedances. Although some mathematics are implicit in the text, the results are presented graphically to simplify their use for design and practical purposes.

INTRODUCTION

Much effort has gone into studying the forced vibration of rigid foundations placed on an elastic or viscoelastic half-space, given the inherent applicability of results to the design of machine foundations and soil-structure interaction (SSI). The problem was first investigated by applying the half-space theory and considering only circular or strip foundations (two-dimensional situa- tions). An excellent review of work until 1967 is found in Richart et al. ~ They proposed a treatment of the problem by simulating the effect of soil on the footing by means of a set of spring and dashpot parameters for all the footing vibration modes. The main advantage of this methodology is its simplicity; all soil parameters involved are supposedly frequency independent, and thus direct application of the basic theory of the damped vibration of a system of one degree-of-freedom may be utilised for design purposes.

A number of contributions have been made on the subject in the last two decades, incorporating different effects: foundation shape, foundation flexibility, fre- quency dependence of soil behaviour, non-linear and plastic soil response, etc. A comprehensive 'state-of-the- art' in those subjects up to 1982 was published by Gazetas. 2 At present, new numerical and analytical techniques are available to solve the problem of foundation vibrations on layered half-space: finite element discretisation, 3 boundary element method, 4 or analytical developments. 5'6'7

Soil Dynamics and Earthquake Engineering 0267-7261/92/$05.00 1992 Elsevier Science Publishers Ltd.

Soil layering greatly affects the soil impedance functions s'9'1 so that they are more frequency-depen- dent than those of the half-space. From analytical results s for the problem of a footing resting on a horizontal elastic layer placed on an elastic half-space, the vertical soil impedance functions are more affected by soil layering than those of horizontal and rocking vibration modes. So, for intermediate and high contrast between layer and half-space shear wave velocities, there is a considerable reduction in the radiated energy, and thus of the radiation damping coefficient, in the low frequency range. This means that the soil impedance functions, or the equivalent frequency-independent soil spring and dashpot constants, to be used in dynamic and seismic soil-structure interaction analyses, would account for the difference.

Numerical solutions for SSI problems often need appropriate finite element or boundary element compu- ter codes, which make them unattractive for practical design purposes due to the high number of analyses required to account for the uncertainties of soil dynamic properties. Analytical approaches, based on solving the continuum mechanics equations, are difficult to use because of the complex mathematics involved. The use of SSI frequency-independent parameters i'll frees the engineer from difficult costly studies.

This paper suggests a simplified approximate method of obtaining the conventional vertical radiation damp- ing coefficient for a particular soil profile geometry: a constant thickness horizontal layer resting on a stiffer half-space (simple layered half-space). The soil beha- viour is supposedly elastic and linear, and the founda- tion is considered as circular and rigid. Mass foundation

249

250 C. Navarro

is taken into account and a vertical harmonic load of constant amplitude is supposed. The problem consid- ered presents a basic difference from the half-space theory: due to wave reflection at the layer/half-space interface, vertical radiation damping is lower than in a half-space situation. This should be borne in mind in design to avoid important errors.

ANALYTICAL DEVELOPMENT

Ps ln ~t

]mass = m

Fig. 2. Simple one degree-of-freedom model to study machine foundations.

Let there be a rigid circular foundation of mass m and radius r0 resting on a homogeneous and isotropic elastic half-space (Fig. l(a)) of density p and shear wave velocity Vs and subjected to the action of a harmonic vertical force of constant amplitude P and frequency fL As is well known, 1 the soil effect on the foundation has two parts: the elastic soil response, and the energy loss due to propagation of elastic waves from the founda- tion. These two phenomena may be taken into account in an approximate manner by considering a soil response model as depicted in Fig. 2, in which a spring constant k and a dashpot coefficient c represent the soil reactions. Rather than constant c one can use the damping ratio DHS, which is defined as the ratio between c and the critical damping Cc when resonance takes place.

In a half-space situation, the damping ratio DHS may be written 1 as:

0.425 DHS-= V~z (1)

where Bz is related to the Poisson ratio u and density p

~ _~sin t~ t I --rl

(a)

~ L'lt i

Elastic layer T H

,N\%\NNN\\\NN\\N\\\x ~ Rigid base

(b)

nf l t

/Elasticlayer pl vt i~ H

(c) Fig. 1. Different layered soil distributions.

of the soil, and to the mass and radius of the foundation. That is:

Bz - (1 - u_____)b (2) 4

In this last expression b represents the mass ratio defined as:

= m b pr- (31

Although damping effects in soils are frequency- dependent, the value of the damping ratio given by eqn (1) may be taken as constant ~ when the dimensionless frequency a0, obtained as

f~r 0 a0 = (4)

Vs

is between 0 and 1. For the case of a forced vibration of a foundation

placed on an elastic layer of thickness H resting on a rigid base (Fig. l(b)), the frequency dependence of the spring and dashpot soi~ constants becomes important. This problem was analysed by Warburton, 12 who obtained the Reissner's displacement functionsfl and f2 in terms of ao, u and H/r o. The spring k and dashpot c coefficients for this case can be given as:

Gro C2 k=GroC 1 and c - ~ (5)

where Cl and C2 may be expressed in the following form:

-f, C1 -~ (6) +f2 J f

A c2 =f? (7/

Also the critical damping Cr may be calculated as

Cr = 2 V/-~omCl (8)

The radiation damping ratio D is defined as:

D- c _ GroC2/f~ (9) Ccrit 2m(o0

where 0~0 is the natural frequency of the system. When the resonance condition is reached, the excitation

Vertical radiation damping for a circular footing resting on a simple layered half-space 251

frequency must be:

f~ = o~0X,/1 - 2D 2 (10)

From eqns (9) and (10), the following implicit equation may be obtained when resonance occurs:

C2 ~/ 1 D=~-~I ( l _2D 2) (11)

V

From eqn (11) the value of the damping ratio D may be computed. To illustrate the results, Fig. 3 shows the dependence of D/DI~s versus H/ro, for a Poisson ratio of about 0.25, the value considered by Warburton ~2 in his study, and for different values of the foundation mass ratio b.

The two types of problem treated above correspond to limit and ideal situations: a layer of either finite or infinite thickness (half-space) on a rigid base. A more common situation is given in Fig. 1 (c), in which the soil profile is an elastic layer of constant thickness, of density P] and compression wave velocity Vl, resting on an elastic half-space of density [02 and compression wave velocity v2. Waves emanating from the foundation suffer reflection and refraction when the incident waves reach the layer/half-space interface. This means that the value of the damping ratio D in the latter case, (Fig. 1 (c)), will be between those calculated above and corresponding to the limit situations (Figs l(a), (b)).

For the last situation (see Fig. 1 (c)), damping ratios may be found from a set of basic hypotheses. Let us suppose a compression wave propagating vertically and downward. On reaching the soil layer/half-space inter- face, one part, Er, of the incoming energy, El, is reflected towards the soil layer, and the remaining energy, Er, is refracted towards the half-space. This latter energy is lost because it propagates to the infinite. For this problem, and following the work of Zoeppritz (see results in Ref. 1), the refracted and incident energies are

0,3-

0.25 -

0.2-

~ 0.15-

0.1

0.05

0

I v=0.25 b=$

b=lO

b=20

b=30

. . . . . . . . I . . . . I . . . . I . . . . I . . . . I . . . . I . . . .

0.5 1 ].5 2 2.5 3 3,5 4 H/r 0

Fig. 3. D/DHs versus H/r 0 for I= 0 from Warburton 12 analysis.

related by the following expression:

~=R= 1- ( l+ I ) 1 (12)

where I represents the mechanical impedance ratio between layer and half-space. That is:

I = JPl v__ L (13) P2V2

where p and v represent the mass density and compression wave velocity, respectively, in both layer and half-space, and can be easily determined from laboratory and in situ tests.

After deriving eqn (12) the Rayleigh wave propaga- tion in the layered system should be considered. An analytical study carried out by Luco, 8 of the vibration of a rigid circular footing placed on three different soil layered systems, shows the strong dependence of the imaginary part of the soil impedance functions on the dimensionless frequency a0, in the high frequency range. However, in the lower frequency range, as is implicit in this paper, this is not so. The frequency dependence is due to the presence of surface Rayleigh waves emanating from the foundation and propagating away from it. A detailed discussion of this subject can be found in Ref. 13. For the low frequency range the Rayleigh wave length is much greater 14 than the layer thickness. As a Rayleigh wave propagates in a layered media at a velocity commensurate with that of the layer through which the major part of the wave energy is transmitted, the fundamental mode of the Rayleigh wave, in a simple layered medium, looks very similar to that obtained in a half-space situation. When the excitation frequency is very high, the wave length becomes smaller, and the soil layering affects the Rayleigh wave propagation and thus the energy transmitted. This means that no energy correction need be considered in the frequency range investigated.

Stating that:

D/DHs = e;/gi = R (14)

the value of D/DHs may be obtained for each mechanical impedance ratio L Some values of R from eqn (14) are shown in Table 1.

Coming back to the problem of the forced vibration of a rigid circular footing resting on the simple layered

Table 1.

I R

0 0 0.05 0.181 0-1 0.331 0.2 0.556 0.4 0.816 0.6 0.938 1.0 1.0

252 C. Navarre

half-space depicted in Fig. 1 (c), approximate values of D/DHs may be interpolated among those obtained from Warburton 12 (Fig. 3), for a mechanical impedance ratio of zero, and unity, corresponding to the half-space situation, with the following limit criteria:

a) For any Hire ratio and for the case of a massless footing (b = 0), the soil behaviour under vertical vibration corresponds to the vertical of a rod of elastic material fixed at the base, free at the top, and constrained laterally, which means that only vertical compression waves are produced. In this case, and for low values of the parameter b, values of the ratio D/DHs would be obtained directly from eqn (14).

b) For any value of the mass ratio b, and when H/ro becomes infinity, the ratio D/DHs reaches unity since a half-space condition appears.

From these two criteria (Figs l(a) and l(b)) an expression of D/DHs can be derived from the problem shown in Fig. l(c), corresponding to a generic mechanical impedance ratio L

Values of the ratio D/DHs may be obtained by interpolation between the value of that ratio, computed from Fig. 3 - - the case of a soil elastic layer on an

undeformable bed rock - - and unity - - the case of a half space situation - - considering the value of the parameter R, which represents the fraction of the incident wave energy refracted at the layer/half-space interface. Other ways of interpolating would be applicable but the one proposed, based on the limit situations above-mentioned, and involving the energy coefficient transmission R, seems to be reasonable from the engineering point of view.

Therefore:

DHS, ] = R + k, DHs , ]

Some results from eqn (15), for different values of the mechanical impedance ratio L are summarised graphi- cally in Fig. 4. The range of values of the ratio Hire have been limited to four since layered soil impedance functions are reasonably close to those of the half-space for values of that ratio equal to or greater than five. 8

DISCUSSION OF RESULTS

Some aspects of Figs 3 and 4 require attention. There is an important increment of D/DHs, over the correspond-

0.45 0.6

e~

0.4"

0.35"

0 .3

0.25"

0.2"

0.15

0 0.5 1 1.5 2 2.5 3 3.5

H/r o

(a)

b=5

b=10

b=20

b=30

e~

0.55

0.5

0.45

0.4

0.35

0.3 . . . . I . . . . i . . . . i . . . . i . . . . i . . . . i . . . . i ' ' '~

0 0.5 1 1.5 2 2.5 3 3.5 4

H/r e

(b)

b=5

b=10

b=20

b=30

e~

0.8

0.75 -

0 .7 -

0 .65-

0 .6 -

0.55"

0.5 . . . . [ . . . . ] . . . . [ . . . . I . . . . I . . . . I . . . . ] . . . .

0 0.5 1 1.5 2 2.5 3 3.5 4

H/r o

(e)

b=5

b=10

b=20

b=30

1

0.95

0.9

0.85

0 .8

0 .75

0.7

0 0.5 1 1.5 2 2.5

H/r o

(d)

Fig. 4. D/DHs versus H/ro for different values of the mechanical impedance ratio I.

. . . . i . . . . i . . . . i . . . . i . . . . t . . . . i . . . . i . . . .

3 3.5

b=$ b=10 b=20 b=30

Vertical radiation damping for a circular footing resting on a simple layered half-space 253

ing values for I = 0, when the mechanical impedance ratio I is as low as 0-05. As the mechanical impedance ratio I increases, the curves D/DHs versus H/ro are less sensitive to the value of the mass ratio b of the foundation. For instance, when I = 0.4, the values of D/DHs, for b = 5 and b = 30, differ in only about 4% if H/ro = 4, and 2% if H/ro = 1. In Fig. 4(d), the ratio D/DHs does not seem to depend a lot on the value of the parameter b. This means that for mechanical impedance ratios greater than 0.4, D/DHs could be considered independent of the parameter b. When I= 0.6, the curves D/Dns versus H/r o are practically coincident, so they are not included in Fig. 4, and the value reached by D/DHs is about 0.95. Since that value is very near unity (a half-space situation), it could be suggested that, for practical purposes, the ratio D/DHs be taken as unity when I is over 0-6. This last recommendation greatly simplifies the computation of the vertical damping ratios in a more complicated layered half-space.

CONCLUSIONS

As is shown in the discussion, the vertical damping ratio of a circular footing placed on a simple layered half- space is dependent on the contrast of layer and half- space wave impedances. However, when the layer/ half-space mechanical impedance ratio I is over 60%, the vertical damping ratio can be calculated directly from the half-space theory. The maximum error in this approach is less than 6%, diminishing as the mechanical impedance ratio raises above 0-6.

REFERENCES

1. Richart, F.E. Jr., Woods, R.D. & Hall, J.R. Vibration of Soil and Foundations, Prentice-Hall Inc., Englewood Cliffs, N.J., 1970.

2. Gazetas, G. Analysis of machine foundation vibrations: state-of-the-art, Soil Dyn. Earth. Eng., 1983, 2, 1-42.

3. Bayo, E. & Wilson, E.L. Numerical techniques for the valuation of soil-structure interaction effects in the time domain, Earth. Eng. Res. Center, Report No. UCB/ EERC-83/04, 1983.

4. Karabalis, D.L. Dynamic soil-structure interaction by BEM, Boundary Elements X, Vol 4, Geomechanics, Wave Propagation and Vibrations, Computational Mechanics Publications, 1988, pp. 3-28.

5. Dobry, R. & Gazetas, G. Dynamic response of arbitrary shaped foundations, Jour. Geoth. Eng., 1986, 112(2), 109- 135.

6. Chow, Y.K. Simplified analysis of dynamic response of rigid foundations with arbitrary geometries, Earth. Eng. Struct. Dyn., 1986, 14, 643-653.

7. Triantafyllidis, T. Dynamic stiffness of rigid rectangular foundations on half-space, Earth. Eng. Struct. Dyn., 1986, 14, 391-411.

8. Luco, J.E. Impedance functions for a rigid foundation on a layered medium, Nuclear Eng. Des., 1974, 31, 204-217.

9. Apsel, R.J. & Luco, J.E. Impedance functions for foundations embedded in a layered medium: an integral equation approach, Earthquake Eng. Struct. Dyn., 1987, 15, 213-231.

10. Spyrakos, C.C. & Beskos, D.E. Dynamic response of rigid strip-foundation by a time-domain boundary element method, Int. J. Num. Meth. Eng., 1986, 23, 1547-1565.

11. Kausel, E., Whitman, R.V., Morray, J.P. & Elsabee, F. The spring method for embedded foundations, Nuclear Eng. Des, 1978, 48, 377-392.

12. Warburton, G.B. Forced vibration of a body upon an elastic stratum, J. Appl. Mech,, Trans. ASME, 1957, 24, 55-58.

13. Kashio, J. Steady-state response of certain foundation systems, Ph.D. Thesis, Rice University, 1971.

14. Ewing, W.M., Jardetzky, W.S. & Press, F. Elastic Waves in Layered Media, McGraw-Hill Book Co., 1957.