63.5 rad/s and the damping constant from 8.5 to 60.5. 1 /s, the shapes of .... 80.0 m2 under a forge hammer with falling parts of 7.1 kN was installed on a site with ...
A NOVEL APPROACH FOR ESTIMATING NATURAL FREQUENCIES OF FOUNDATION VIBRATIONS
Mark R. Svinkin Consulting Engineer 13821 Cedar Road #205 Cleveland, Ohio 4411 8
U.S.A.
ABSTRACT A method is presented for obtaining a pre-construction estimation of the natural frequencies of damped vertical vibrations of foundations which will be built at a specific site for machines with dynamic loads. This method pertains principally to the relationship between foundation frequencies and the natural frequencies of the soil profile at a specific construction site. It is shown the major contribution of soil conditions at a site to the formation of ground vibrations excited by operating machines installed on foundations. Good agreement is found between predicted and measured frequencies.
a foundation are dominant parameters in design and specification of foundations for machinery having dynamic loads. Knowledge of the reliable values of natural frequencies allows more accurate computation of vibration amplitudes and enables engineers to avoid the conditions of resonance of foundation vibrations. The natural frequency of foundations can be calculated on the basis of dynamic features of the foundation-soil system: mass, stiffness, and damping. There is no singular, generally accepted opinion on the nature of these features. However, there have been many theoretical and experimental approaches for determination of the characteristics of the foundation-soil system 11.2. 31 , but the application of the various methods often does not give consistent results.
NOMENCLATURE Z(t) IF
displacement as function of time impulse force transmitted from machine to foundation A contact area between foundation and soil M mass of foundation and machine kz spring constant for the vertical mode of foundation vibrations k; coefficient of vertical subgrade reaction f nz natural frequency of vertical vibrations of foundation f nd natural frequency of vertical damped vibrations of foundation a effective damping constant cp modulus of damping hz(x,y,t-r) impulse response function at the output point under consideration; r variable of integration
The effect of soil conditions is very important. Most analytical methods currently used in assessing machine foundation vibrations require results of in-situ and/or laboratory dynamic soil tests 141 • The effect of local soil conditions on the amplitude and frequency content of earthquake motions has been considered during the past three decades 15 •6.71• These studies underlined the important role of fundamental site period. This paper reveals that the contribution of soil stratification to ground vibrations is much more significant than the contribution of the machine foundations themselves and shows that natural frequencies of vertical foundation vibrations can be obtained using the relationship between measured natural foundation frequencies and the natural frequencies of the soil profile at a construction site.
1 INTRODUCTION
CONTRIBUTION OF SOIL CONDITIONS TO GROUND VIBRATIONS
On many machines, especially large ones, the soil base and foundation are paramount to successful dynamic behavior. It is well known that the natural frequencies of
The machine foundation-soil system consists of two components: the machine foundation installed on a soil base and the soil through which waves propagate from a
1633
J ·--
•
records from operating response function.
BUILDING
Output I
Output 2
Output 3
--~~v~ui'~ SOIL MEDIUM
Figure 1 : Experimental functions
determination
of impulse response
machine,
and
also impulse
It can be seen that the IRFs have strong resemblance to the actual vibrations recorded from the operating machine, reflecting a very close correlation to record shapes of the measured vibrations. Thus, soil strata are influencing ground motion not only as the medium where waves originate at the vibrating machine foundation, but also the soil strata through which ground vibrations travel between source and receiver. Predicted time-history records were calculated using the following Duhamel's integral 18 •91 •
dynamic source. For demonstration of the effect of soil conditions at a site on soil and structural vibrations, this paper uses the method founded on utilization of the impulse response function (IRF) technique for predicting complete vibration records on existing soils, buildings and equipment prior to installation of construction and industrial vibration sources 18 •91 • Impulse response functions of the considered dynamic system are determined by setting up an experiment (Figure 1). Such an approach (a) does not require routine soil boring, sampling, or testing at the site where waves propagate from the vibration source, (b) eliminates the need to use mathematical models of soil bases and structures in practical applications, and (c) provides the flexibility of considering heterogeneity and variety of soil and structural properties. Unlike analytical methods, experimental IRFs reflect real behavior of soil and structures without investigation of the soil and structure properties. Because of that, the suggested method has substantially greater capabilities in comparison with other existing methods. In this study, IRFs were measured on the ground surface at the moment of the impact onto the ground at the place for installation of the machine foundation; after installation of the machine foundation, ground vibrations were also measured at the same locations from the operating machine on the foundation. To assess the soil contribution to ground vibrations excited by vibrating machine foundations, consider very long wavelength ground vibrations measured in-situ at a great distance from the source. According to the principle of Saint-Venant, the effect of the loaded area is negligible on stresses at distances from the load which are larger in comparison with loaded area dimensions. Figure 2 displays time-history displacement records of vertical and horizontal ground vibrations at distance of 266.0 m from the foundation for a powerful drop hammer at a site with the Rayleigh-wave velocity of 270 m/sec. The falling mass of the drop hammer was 1 5.0 tonnes and the maximum drop height was 30.0 m. For both vertical and horizontal displacement components, three records of ground vibrations are depicted: predicted and measured
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( 1)
with
k,
kz'A,·
a
= -2-, cpf;,
fznd= fzn,-a; z
fznz
k,
(2)
M
To evaluate the effect of foundation-soil system parameters on ground vibrations, time-history records were computed using equation (1) with various values of initial parameters (Table 1). Results are presented in Figure 3. Correlation of predicted and measured vibration displacements is quite satisfactory. The differences between the highest calculated and measured amplitudes of oscillations are 1 6 and 30 % for horizontal and vertical components, respectively. In spite of the change of the natural foundation frequency in the range of 23.863.5 rad/s and the damping constant from 8.5 to 60.5 1 /s, the shapes of measured and predicted records are almost the same and their spectra show the same predominant vibration frequency. Spectra of these oscillations show a stability of frequency composition for even very long soil oscillations. Thus, variations of predicted soil oscillations do not exceed measurement errors even with a 2. 7 times difference in the natural frequency of the foundation.
Table 1. Parameters of Foundation-Soil System Record No.
k' z (kN/m 3 )
a (1 /sl
qJ (s)
fnz (rad/s)
M t
Experimental damped sinusoid
2
3
34433
8.5
0.03
23.8
9614
4
67885
60.5
0.03
63.5
2649
5
39240
35.0
0.03
48.3
2649
On the one hand, the effect of foundation-soil system parameters is negligible beyond certain distance from the
a 0.2 s
c::::::J IRF
cU ()
:e Q)
>
;~
Predicted
2
E
:I.
N~~~~~~~~-t~~-+-t~L-~-,~~--~[_~~--~M.easured
b
IRF
cU
c 0
N
·;;::
0
I
!,Q
(\
0
A A1\ /\ /'..
l 'Z7\J V"l)\(\}
Measured
-v
_)
Figure 2: Displacement records of vertical (a) and horizontal (b) ground vibrations at distance of 266.0 m from the drop hammer foundation {m = 15 tonnes, h = 30 m): 1 -impulse response function, 2 - predicted record, 3 - measured record, after [8]
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Vertical
Horizontal
,......
~
~i :o l§j ,. '
rod/sec
Figure 5: Frequency of foundation under press-drop hammer; foundation vibrations (a) and their spectrum (c), ground vibrations (b) and their spectrum (d)
chosen location 1 for future installation of a machine foundation. For this purpose, the use of a steel weight and a bridge or mobile crane would be practical. Magnitudes of the impacts 2 should correspond to the values of known operative dynamic loads on the designated machine foundation. Impacts of a certain magnitude are made. While impacting the soil base, soil vibrations 4 are measured nearby the contact area, but beyond the zone of plastic soil deformations. Records 5 are made during such measurements. Then spectrum analyses 6 are performed on the records of measured vibrations. The dominant frequency 7 of the soil profile is found from the spectrum of these soil vibration records. The dominant frequency of spectra of soil vibrations at the location for installation of the machine foundation is an estimate of the natural frequency 8 of vertical damped vibrations of the specified machine foundation. It is preferable that the magnitude of the impact corresponds to somewhat less than the vertical impulse force generated by the machine and applied to the foundation to be constructed at the site. As used herein, "soil base" includes the soil stratifications immediately below ground surface or existing excavation. The vibration input and output locations are within a limited area for future machine foundation installation, but the output locations are beyond the zone of plastic deformations of the soil base caused by an impact. A new approach is illustrated by case histories 1 and 2 in Figures 5 and 6, respectively. Case History 1 . A foundation with footprint area of 12.3 m 2 under a press-drop hammer with falling parts of 3.9 kN was placed on a site with the following soil
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Figure 6: Frequency of foundation foundation vibrations (a) and their vibrations (b) and their spectrum (d)
\v 100
!"--"--
200
(a)
rad/sec
under forge hammer; spectrum {c), ground
conditions. The soil at the site consisted of about 1 .5 m fill followed by about 8 m brown-yellow, moist, medium density, fine sand. The impulse of vertical force for impact on the soil was equal to 2.4 kN-sec in the test and was 34.7 kN-sec during machine operations. The ratio of these impulses was 14.5. Soil vibrations (b) and foundation vibrations (a) have spectra {d) and {c) respectively with the same frequency maximum. Case History 2. A foundation with footprint area of 80.0 m 2 under a forge hammer with falling parts of 7.1 kN was installed on a site with the following soil conditions. Fill material had a depth of 2 m. The native soil was brown-yellow, loess loam with solid consistency but slump-prone properties extending from below the fill to between 5-6 m. Fine sand existed thereunder to 7-9 m. Another layer of yellow-brown, solid consistency loam was under the sand. There was no ground water identified to a depth of 10m. The impulse of force for the impact on the soil was equal to 1 .9 kN-sec in the test and was 78.5 kN-sec during machine operations. The ratio of impulse values was 41 .3. Soil vibrations {b) and foundation vibrations {a) have spectra {d) and (c) respectively with the same frequency maximum. In general, soil profiles are nonlinear systems and the dominant frequency of soil profiles depends on the applied impact. Over a certain range, however, the system behavior may be linear and if the system is restricted to this range it is possible to safely use the linear approach. This is the reason why the magnitude of impact on soil can be an order of magnitude less than the value of operating machine impulse for the permissible vibration level. Still, this ratio might be more than ten for some soil conditions as it was shown in the above presented examples where it could be seen the good coincidence of natural frequency of vertical damped foundation vibrations
with the corresponding profile.
natural frequency of the soil
On the first review, the concept presented here may seem contrary to known concepts of soil dynamics. These concepts affirm that natural frequencies of foundation vibrations depend not only on soil properties but on foundation dimensions, masses, end embedment as well. However, there are no contradictions. Heterogeneous soil profiles with various stratifications can have unequal dominant frequencies of natural vibrations for different locations on an industrial site. Machine foundations with different parameters, mounted on like soil profiles, will have unequal natural frequencies of their vertical damped vibrations. In the case when the soil profile of a site is a homogeneous medium with one dominant frequency of natural vibrations, all foundations with different parameters, installed on this site, will have the same natural frequency of their vertical damped vibrations. Similar case histories are observed in practice.
ACKNOWLEDGEMENTS The writer is thankful to Dr. Richard D. Woods, professor of civil engineering at the University of Michigan at Ann Arbor, for valuable comments.
REFERENCES
It makes sense to suggest an analogy with a simplysupported beam. A small lump mass connected with the beam does not change the fundamental beam frequency. However, a large mass added to the beam could considerably affect the fundamental frequency of the new dynamic system. A similar situation is for a foundation installed onto the soil profile. The soil profile is a physical body with own natural frequencies of soil layers. Added foundation mass is relatively small compared to the soil mass involved in vibrations and therefore the foundation will vibrate with frequency of the soil profile. A substantially large foundation mass could affect the natural soil profile frequency, but probably this case is beyond common used foundation sizes for machines.
[1]
Barkan, D.O., Dynamics of bases and foundations, McGraw Hill Co., 1962.
[2]
Richart, F.E., Hall, J.R. and Woods, R.D., Vibrations of soils and foundations, Prentic-Hall, Inc., 1 970.
[3]
Gazetas, G., Foundation vibrations, Foundation Engineering Handbook, 2nd Ed., H.Y.Fang, ed, Van Nostrand Reinhold, pp. 553-593, 1994.
[4]
Woods, R.D. and Stokoe, K.H., Shallow seismic exploration in soil dynamics, Richart Commemorative Lectures, ASCE, pp. 120-156, 1985.
[5]
Dobry, R., Oweis, I., and Urza, A., Simplified procedures for estimating the fundamental period of a soil profile, Bulletin of the Seismological Society of America, Vol. 66, No. 4, pp. 1293-1324, 1976.
[6]
Roesset, J.M., Soil amplification of earthquakes, Numerical Methods in Geotechnical Engineering, Desai, C.S. and Christian, J.T., eds., McGraw-Hill Book Co., pp. 639-682, 1977.
[7]
Singh, Y. and Nagpal, A.K., Estimating fundamental period of soil profiles, Geotechnical Engineering, Journal of Southeast Asian Geotechnical Society, AIT, Vol. 24, No. 2, pp. 167-174, 1993.
[8]
Svinkin, M.R., Overcoming soil uncertainty in prediction of construction and industrial vibrations, Proceedings, Uncertainty in the Geologic Environment: From theory to Practice, ASCE, Geotechnical Special Publications No. 58, 2, pp. 1178-1194,1996.
[9]
Svinkin, M.R., Numerical methods with experimental soil response in predicting vibrations from dynamic sources, Proceedings of the Ninth International Conference of International Association for Computer Methods and Advances in Geomechanics, China, Vol. 3, pp. 2263-2268, November 1997.
CONCLUSIONS The effect of foundation-soil system parameters on ground vibrations is negligible at certain distance from the machine foundation. Actually, only the impulse force transmitted from a machine to a foundation affects ground vibrations. Local soil conditions make the major contributions to ground vibrations. A new method has been suggested for pre-construction determination of the natural frequency of damped vertical vibrations of foundations under machinery with known vertical impact loads which will be installed on the soil The method is founded on the relationship profile. revealed between the natural frequencies of damped vertical vibrations of rigid bodies on soil profiles and the natural frequencies of the soil profiles. The use of the method provides a means for predicting by essentially accurate estimates the natural frequencies for machine foundations prior to their erection at a specific site.
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[ 1 OJ Svinkin, M .R., Discussion of 'Impact of weight falling onto the ground' by Roesset, J.M. eta!., Journal of Geotechnical Engineering, ASCE, Vol. 122, No. 5, pp. 414-415,1996. [11] Svinkin, M.R., A method for estimating frequencies of machine foundations. U.S. Patent No. 5,610,336 issued March 11, 1 997.
