Performance Evaluation of Vibration-sensitive Equipment Foundations Under Ground-transmitted...

Performance evaluation of vibration-sensitiveequipment foundations under ground-transmittedexcitation

M. Hesham El Naggar

Abstract: The planning of foundations for equipment that is sensitive to vibrations requires a thorough dynamic inves-tigation of the proposed location of the foundation with regard to the effect of already existing or additional vibrationsources. This paper discusses the analyses performed for a number of foundations supporting vibration-sensitive equip-ment that has been subjected to ground-transmitted excitations. These analyses considered the dynamic response of thefoundations resulting from the normal operation of the supported equipment or the ground-transmitted excitations. Inone case, the foundation of the Canadian Light Source, a third generation synchrotron that will be capable of generat-ing electromagnetic radiation used in the study of the atomic and subatomic structure of materials, is examined.Another case involves the vibration analysis of a magnetic resonance imaging unit affected by traffic excitation. In thethird case, a power plant facility that is subjected to blast-induced vibration from an adjacent quarry is investigated.The last case involves the response analysis of a compressor foundation affected by the ground-transmitted vibrationfrom another compressor situated on a different foundation within the same facility. To assess the level of seismic exci-tation at the site due to traffic on an adjacent roadway in the first two cases and to blasting activity in the third case,extensive green field ground vibration-monitoring programs were carried out. The ground accelerations due to trafficand blasting were measured and recorded for three directions simultaneously: a vertical and two orthogonal horizontaldirections. The measurements with the most intense ground accelerations taken at the ground surface in the location ofthe future equipment foundation were selected as the final design acceleration time-history. A Fourier analysis approachwas used to predict the response of the foundation to the ground-induced vibrations in the first three cases, and afrequency domain analysis was used in the last case.

Key words: machine foundations, vibration, blasting, kinematic, soilstructure interaction.

Rsum : Planification de fondations pour de lquipement qui est sensible aux vibrations requiert un tude dynamiquecomplte du site propos pour la fondation en considrant leffet des sources de vibrations dj existantes ou addition-nelles. Cet article discute des analyses ralises pour un certain nombre de fondations sur lesquelles reposent des qui-pements sensibles aux vibrations et qui ont t soumises des sollicitations transmises par le terrain. Ces analysesconsidrent la rponse dynamique des fondations due lopration normale de lquipement quelles supportent ou auxsollicitations transmises par le terrain. Dans un cas, on examine la fondation du Canadian Light Source, un synchrotronde troisime gnration qui pourra gnrer des radiations lectromagntiques utilises dans ltude de la structureatomique et subatomique des matriaux. Un autre cas implique lanalyse des vibrations dune unit dimagerie dersonnance magntique dues la sollicitation du trafic. Dans le troisime cas, on tudie une centrale lectrique qui estsoumise des vibrations induites par le dynamitage dans une carrire adjacente. Le dernier cas implique lanalyse dela rponse de la fondation dun compresseur due la vibration transmise par le sol en provenance dun autrecompresseur situ sur une autre fondation lintrieur de la centrale. Pour valuer le niveau dexcitation sismique surle site, due au trafic sur une route adjacente dans les deux premiers cas, et lactivit de dynamitage dans le troisimecas, on a ralis des programmes labors de mesures des vibrations du terrain naturel. Les acclrations du terraindues au trafic ou dynamitage ont t mesures et enregistres simultanment dans trois directions: une directionverticale et deux horizontales orthogonales. Les mesures ayant les acclrations du sol les plus intenses prises lasurface du terrain sur le site de la fondation du futur quipement ont t choisies pour la conception finale de lhistoireen fonction du temps de lacclration. On a utilis une approche danalyse de Fourier pour prdire la rponse de lafondation aux vibrations induites par le terrain dans les trois premiers cas, et une analyse dans le domaine desfrquences dans le dernier cas.

Mots cls : fondations de machine, vibration, dynamitage, cintique, interaction solstructure.

[Traduit par la Rdaction] El Naggar 615

Can. Geotech. J. 40: 598615 (2003) doi: 10.1139/T03-014 2003 NRC Canada

598

Received 5 November 2001. Accepted 28 December 2002. Published on the NRC Research Press Web site at http://cgj.nrc.ca on20 May 2003.

M.H. El Naggar. Geotechnical Research Centre, Faculty of Engineering Science, The University of Western Ontario, London ONN6A 5B9, Canada. (e-mail: [email protected]).

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Introduction

The main objective when designing a foundation forvibration-sensitive equipment is to limit the response ampli-tudes of the foundation to the specified tolerance in all vi-bration modes. The tolerance is usually set by the machinemanufacturer to ensure satisfactory performance of the ma-chine. The displacement of foundations subjected to dy-namic loads depends on the type and geometry of thefoundation, the flexibility of the supporting ground, and thetype of dynamic loading.

The objective of this paper is to establish a rational ap-proach for the evaluation of the dynamic performance offoundations supporting vibration-sensitive equipment. Thisapproach incorporates the dynamic characteristics of boththe foundation system and the seismic excitation. It includesthe planning and execution of vibration monitoring programsat the location of the proposed foundations, the evaluation ofthe characteristics of the dynamic loading, the calculation ofthe foundation impedance functions, and the response of thefoundation to the dynamic loads.

Design procedure

The vibration criteria stated by the manufacturer are al-ways specified as floor vibrations. Before the facility isbuilt, no floor vibration can be directly measured. On theother hand, the ground-transmitted excitation at the site dueto external sources of vibration could be, in many cases, animportant factor for designing the facility or even in decid-ing whether or not it will be built. Therefore, establishingthe relationship between measured ground vibrations andexpected floor vibrations is the first step in the evaluationprocess. The procedure used to establish this relationshipincludes the following steps:(1) Evaluating the dynamic loads: this includes the determi-

nation of their magnitudes and characteristics, includingintensity and frequency content of the ground-transmitted vibration.

(2) Establishing the soil profile and evaluating the soilproperties required for the dynamic analysis (shearmodulus, mass density, Poissons ratio, and materialdamping ratio).

(3) Selecting the type and trial dimensions of the founda-tion based on experience.

(4) Computing the dynamic response of the trial foundationsupported by the given soil profile due to the estimatedload and comparing the response with the performancecriteria. If the response is not satisfactory, the dimen-sions of the foundation are modified and the analysis isrepeated until a satisfactory design is achieved.

The dynamic response analysis is the major componentin the design process. The analysis essentially involves thecalculation of the vibration characteristics of the machinefoundationsoil system (i.e., the natural frequencies and thevibration amplitudes due to all sources of vibration). Thecomplexity of the response analysis required depends on thetype of foundation system used. For flexible foundation sys-tems (e.g., thin mat foundations), dynamic finite elementanalysis may be necessary. For rigid foundations resting di-

rectly on the soil or supported by pile groups, simplifiedanalytical or numerical methods or both are commonly used.

The response of soils and foundations to dynamic excita-tion is frequency dependent and thus is a function of thestiffness and damping parameters of the soil and the founda-tion. Therefore, the evaluation of the appropriate stiffnessand damping parameters (impedance functions) for the foun-dation soil or pilesoil system is a key step in the analysis.

Evaluation of foundation impedancefunctions

The evaluation of the dynamic response of foundations sup-porting vibration-sensitive equipment requires that proper val-ues for the dynamic stiffness and damping of the foundation beused. The variation of these values with dynamic soil charac-teristics is usually notable, and consequently their effect on thefoundations response is important. Both shallow and deepfoundations are commonly used to support machinery.

Several approaches are available for the analysis of foun-dation systems to account for dynamic soilstructure interac-tion. The analyses used to determine the impedancefunctions of shallow and deep foundations are describedbriefly below.

Shallow foundationThe stiffness and damping constants of shallow founda-

tions resting on the surface of a linear viscoelastic halfspacecan be obtained using either three-dimensional or two-dimensional continuum approaches. The analytical solutionsinclude the contribution of many researchers: Bycroft(1956); Luco and Westmann (1971); and Veletsos and Verbic(1973). Veletsos and Wei (1971) and Luco and Hadjian(1974) introduced numerical solutions. For circular bases thecomplex stiffness Ki associated with direction i is obtainedby determining the relationship between the harmonic forceacting on a massless disc that rests on the surface of thehalfspace and the resulting displacement of the disc. Thiscomplex stiffness can be expressed in terms of the true stiff-ness constant, ki , and the damping constant, ci , as

[1] K k k a ia c ai i i i= + [ ( ) ( )]0 0 0in which ki is the static stiffness, a0 = R/Vs is thedimensionless frequency, R is the disc radius, Vs = G is the shear wave velocity of the soil, and G and are thesoil shear modulus and mass density, respectively. Theparameters ki and ci are stiffness and damping constantsnormalized as follows: =k k

kii

i, =c

Vk R

cii

is (i.e., see Veletsos

and Verbic 1973).Embedment is known to increase both stiffness and

damping, but the increase in foundation damping is moresignificant. The response of embedded footings can be ap-proximated by assuming that soil reactions acting on thebase are equal to those of a surface footing and that the reac-tions acting on the footing sides are equal to those of anindependent layer overlying the halfspace assuming planestrain conditions. Beredugo and Novak (1972) found thatthis approximate approach yields reasonable results com-pared with the finite element predictions. In this study, the

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plane strain solutions developed by Novak et al. (1978) forside reactions and the halfspace solution for base reactionsdeveloped by Veletsos and Verbic (1973) were used to eval-uate the stiffness and damping constants of the shallowfoundation.

A foundation resting on a shallow layer has increasedstiffness and reduced or even nil geometric damping (due towave propagation away from the foundation). Kobori et al.(1971) showed that geometric damping might completelyvanish if the excitation frequency is lower than the first natu-ral frequency of the soil layer. The first vertical and horizon-tal natural frequencies, v and u, respectively, of a shallowlayer are

[2]

vs

usand=

=VH

VH2

2 21 2 2( )

where H is the layer thickness, and is Poissons ratio of thesoil layer. At frequencies lower than v and u, the onlysource of damping is the material damping. For frequenciesbelow the first layer natural frequencies, it would be safe toignore geometric damping completely, and the damping canbe established as a fraction of stiffness giving c as

[3] c k k= 2

in which is the material damping ratio of the soil.Deep foundation

Stiffness and damping of piles are affected by interactionof the piles with the surrounding soil. In groups of closelyspaced piles, the character of dynamic stiffness and dampingis further complicated by interaction between individualpiles. To account for pilesoilpile interaction effects, thesuperposition approach was used in the analysis. In this ap-proach, the stiffness and damping of single piles are calcu-lated first, then group effect is accounted for using theinteraction factors.

The dynamic stiffness (impedance function) of piles canbe described as

[4] K k a i c ai i i= +( ) ( )0 0The stiffness constants, ki , and the equivalent viscous

damping constants, ci , for individual motions of the pilehead can be evaluated as a function of the pile and soil prop-erties using the approach developed by Novak and Aboul-Ella (1978) for piles in a layered medium.

The dynamic group effects can be evaluated approxi-mately using the interaction factors approach and theapproximate approach due to Dobry and Gazetas (1988) andGazetas and Makris (1991) in which the interaction problemis reduced to the consideration of cylindrical wave propaga-tion. A simplified approximate analysis for the dynamicgroup effects is formulated on the basis of dynamic interac-tion factors, , introduced by Kaynia and Kausel (1982) whopresented charts for dynamic interaction. In this analysis, theimpedance functions of single piles and the interaction fac-tors are calculated first, then the group impedance functionsare computed using the approach described in El Naggar andNovak (1995). All of the techniques used to calculate theimpedance functions for the foundation are encoded in the

computer code DYNA5 (Novak et al. 2002) that was used inthis study. The DYNA5 program is used to calculate the re-sponse of rigid foundations to all types of dynamic loads, in-cluding loads from centrifugal or reciprocating machines,shock-producing machines, earthquakes, traffic, and othersources of dynamic forces. The response to harmonic load-ing for a flexible, rectangular mat on elastic halfspace or ona group of piles can also be calculated. The stiffness anddamping constants of the foundation (needed for the analy-sis) are evaluated within the program for surface founda-tions, embedded foundations, and piles accounting for theinteraction of piles in a group. For rigid footings, all sixdegrees of freedom are considered as coupled.

Vibration monitoring

To assess the level of seismic excitation at a site causedby ground-transmitted vibration from external sources, aground vibration-monitoring program should be carefullyplanned and executed. This involves taking ground accelera-tion measurements at several stations situated across the siteprior to the start of construction. The evaluation of ground-transmitted vibration caused by the operation of vibratingequipment in an adjacent facility involves vibration monitor-ing if the facility already exists or dynamic response analysisof the proposed foundation system if it is to be constructedin the future.

Vibration monitoring equipmentComponents of the ground vibration monitoring equip-

ment included sensors, mountings for the sensors, and a dataacquisition system. The monitoring system was designedto provide the required sensitivity, minimize data samplingerrors, and achieve the robust performance necessary for theanticipated environmental conditions.

In this study, ground vibrations were measured using ICPmodel 393B31 seismic accelerometers supplied by PCBPiezotronics Inc. (Depew, New York) with a sensitivity of1.0 106g and a measurement range of 0.5 g, a frequencyrange of 0.07300 Hz (at 10% gain), and an operationaltemperature range from 18 to 65C. These accelerometerswere deemed to satisfy the stringent project requirements. Inaddition, a mounted natural frequency in the order of 1 kHzhelped to minimize measurement bias in the frequency rangeof interest.

The accelerometers were mounted directly on speciallyfabricated aluminum posts installed in the ground at themeasuring stations. Mounting arrangements enabled thesimultaneous attachment of accelerometers in three mutuallyorthogonal directions, with two oriented horizontally and thethird vertically. An embedded length of 0.6 m for the postswas selected to enhance the rigidity of the system. At thesame time it was significantly smaller than the minimumwavelength of soil vibrations for the maximum frequenciesconsidered. The sensors were protected from interferencefrom other factors such as wind, snow, and electromagneticfields.

Dynamic tests were conducted on the mounted sensorassembly using an impact hammer apparatus. It was foundthat the embedded posts exhibited a fundamental resonantfrequency ranging between 120 and 150 Hz with a single

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accelerometer mounted on it, and between 70 and 90 Hzwith three mounted accelerometers. Free vibration dampingwas observed to be approximately 810% of critical damping.

A digital data acquisition system compatible with the sen-sors was used to record the acceleration time history. Properanalog filtering ensured that no frequency interference oc-curred. The sampling frequency (1 kHz) was selected so thatthe highest frequency component of interest could be prop-erly identified. All of the measurement data were recordedusing the data acquisition system and stored in digital formfor subsequent analyses.

Vibration monitoring programsGround-transmitted vibration was the main design consid-

eration in the four projects considered in this study. In twocases, heavy traffic on a highway near the proposed facilityrepresented the external source of vibration. In the thirdcase, the vibration was caused by blasting in a quarry opera-tion near a proposed power plant. The last case involved theresponse analysis of a compressor foundation affected by itsown operating load as well as the ground-transmitted vibra-tion from another compressor situated on a different founda-tion within the same facility. In the first three cases,vibration monitoring programs were executed to assess thelevel of vibration.

Canadian Light Source (CLS)The CLS is a third generation synchrotron that will be

capable of generating electromagnetic radiation used in thestudy of the atomic and subatomic structure of materials.The accuracy required in aiming the electron beam andthe resulting radiation necessitates very stringent operationaltolerances on foundation vibrations, with peak dynamic dis-placements being limited to less than 0.35 m over the fre-quency range 050 Hz. Ground acceleration measurementswere taken at 10 stations distributed across the CLS site. Ateach station, simultaneous readings were taken in one verti-cal and two horizontal directions for the various types ofexcitation that were considered. Additional tests were alsoconducted with corresponding measurements recorded atthree different stations simultaneously.

Vibration events caused by general automobile traffic,buses, and loaded gravel trucks were recorded. Snow cov-ered the site during the measurement period, and the groundwas at least partially frozen. In these tests, gravel truckevents were found to generate the largest ground vibrationsand therefore formed the basis for subsequent analyses.Additional tests were also performed using a mechanicaltamper to estimate the correlation between stations for asomewhat uniform excitation source. The corresponding hor-izontal and vertical accelerations at three stations locatedalong a straight line were measured simultaneously to char-acterize the attenuation of ground vibrations over the foun-dation area.

To ensure that the ground vibration measurements wererepresentative of the most severe anticipated loading condi-tions, including the effects of varying weather conditionsand the potential for significant bumps on the roadway, anew set of ground vibration monitoring tests was performedin the summer. These tests featured a loaded gravel trucktraveling at 4045 km/h (the speed limit on this road is

40 km/h) that strikes a 50 mm (2 in.) bump in the road (a2 4 hollow structural steel (HSS) tube installed across thedriving lane). Several sets of measurements were taken, eachconsisting of 10 truck events (five events with the trucktraveling in each direction along the roadway). The measure-ments with the most intense ground accelerations taken atthe ground surface in the location of the future machinefoundation were selected as the final design accelerationtime history.

Magnetic resonance imaging (MRI) facilityThe MRI is an advanced medical device that represents

a significant tool in the diagnosis of ailments. The deviceis extremely sensitive to external vibrations. The vibrationtolerance specified by the manufacturer of the MRI unitconsidered in this study is given in terms of accelerationamplitudes as follows: 1 106g over the frequency range010 Hz, 10 106g for the frequency range 1022 Hz; and25 106g for the frequency range 2240 Hz.

The proposed site is situated 15 m from a busy roadway.The soil profile at the site includes a shallow layer (5 mthick) of silty sand underlain by weathered bedrock. Dryconditions were encountered in the boreholes. The proposedfoundation system includes drilled shafts bearing on the bed-rock. Therefore, the vibration-monitoring program includedvibration measurements in the soil layer and in the bedrock(1.2 m below the surface of the bedrock). The accelerationswere measured along the vertical direction and two perpen-dicular horizontal directions. The vibration measurementswere taken at the centre of the future foundation because ofthe small size of the foundation. Because of the randomnature of the traffic excitation, the vibration measurementswere represented in terms of the power spectra of the groundacceleration measured over durations of 2, 8, and 25 s.These power spectra were obtained by subjecting the mea-sured acceleration time history to a Fourier transform.

The results of the monitoring program showed that themaximum acceleration amplitudes measured in the soil layerwere 20 times greater than those measured in the bedrock.Therefore, the soil accelerations were used in the subsequentanalysis.

Blast-induced vibration on a power plantThis case study involves a power plant facility that is sub-

jected to blast-induced vibration from an adjacent quarry.The proposed facility houses three turbine generator units,each of which will be supported by an independent founda-tion. The blasting will occur up to 100 m from the proposedfoundations. Blasts occur on average two times a week andlast for about 0.5 s each time. The requirements of the facil-ity dictate that the turbines must not be damaged during theblast. The standard operational tolerances on foundationvibrations for the turbine foundation are limited to peakdynamic displacements of less than 0.075 mm.

The selected foundation option is a shallow foundationsystem that consists of a rigid concrete slab. The soil profileat the site consists mainly of extremely weathered bedrock.The native soil beneath the concrete slab would be excavatedand removed to a depth of 6 m below the existing groundsurface and an engineered fill 4 m thick would be placedprior to the foundation being constructed.

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To assess the level of seismic excitation at the site of thefuture foundation resulting from blasting activities in thequarry, an extensive ground vibration monitoring programwas carried out. This program included initial blast-monitoring of production blasts at the future turbine locationand the development of a final design acceleration time his-tory. The ground acceleration time histories were measuredat the weathered rock in the location of the subject founda-tion. The acceleration time histories were measured in thelongitudinal, transverse, and vertical directions. The resultsof the vibration-monitoring program showed the pattern andvibration characteristics of the production blasts (i.e., dura-tion of blasts, intensity of the ground vibration, and fre-quency content). The analysis of the results of monitoringindicated that there was the potential for an unacceptablefoundation response.

Multiple closely spaced machine foundationsThe last case study represents a fairly common situation.

It involves the response analysis of a compressor foundationaffected by its own operating load as well as the ground-transmitted vibration from another compressor situated on adifferent foundation within the same facility. In this case, theexpected ground vibrations are computed rather than mea-sured. The response analysis is discussed in the followingsection.

Response analysis

The basic mathematical model used in the dynamic analy-sis is a lumped mass with a spring and dashpot. If the mass,m, is able to move in only one direction, e.g., vertical, it issaid to have a single degree of freedom (SDF). The founda-tion block has six degrees of freedom, three translational andthree rotational. These are the displacements along the x, y,and z axes and rotation about the same axes.

The response of the mass depends on the nature of the soilreaction that is modeled by both the spring and the dashpot.The stiffness and damping constants are calculated for dif-ferent foundation types using the approaches described ear-lier. Due to the large resulting stiffness of the foundationblock or pile cap relative to that of the soil or the piles, thefoundation block can be assumed to vibrate as a rigid body.The equation of motion for this rigid body in one direction(i.e., SDF) when subjected to a dynamic excitation is[5] m c k P t ( ) + + =where m is the mass of the system; c and k are the dampingcoefficient and stiffness constant, respectively, of the foun-dation along the direction considered; P(t) is the loading ex-citation; and , and are the acceleration, velocity, anddisplacement of the foundation, respectively. For basic har-monic loading, the response is given by

[6]

( )( )

cos( )t Pk m c

t= +

+2 2 2 2

where is the loading frequency and = tan [ /( )]1 2c k mis the phase shift.

For ground-transmitted excitation, the forcing function,P(t), is given by {m(t)} where (t) is the absolute ground

acceleration time history measured at the location of thefuture foundation. In this case, there are two approachesused to solve for the response of the foundation. In the firstapproach, the Duhamel integral of (t) is used to calculatethe relative displacement of the foundation, i.e.,

[7]

0

( ) ( ) sin[ ( ) ]( )tD

u tt

D t=

11 2 0 e ddwhere 0 = k m/ , D c km= / 2 and d = 1 2D .

The response of the machine-foundation system is influ-enced by both its natural frequency and the frequency con-tent of loading. The traffic loading is transmitted to thefoundation as a combination of seismic waves propagatingin the ground at different frequencies. While eq. [7] impliesthat the stiffness and damping of the foundation system areconstant, they are, in fact, frequency dependent; the use ofeq. [7] to calculate the response may therefore compromisethe resulting accuracy.

Alternatively, a Fourier analysis can be used to calculatethe response of the foundation to the transient load in thefrequency domain. In this type of analysis, the load is repre-sented by the sum of a series of harmonic componentsobtained by subjecting the load time history to a fast Fouriertransform (FFT). In the FFT, the input function x(t)(i. e., mu(t)) is given as an even number, N, of equidistantpoints in the time domain. The number of frequency compo-nents is limited, and for N data points, N/2 frequency com-ponents are obtained. Thus, increased accuracy can only beobtained by increasing the number of data points.

The response of a SDF system acted on by the nth har-monic component of the load would be governed by[8] m c k x i tn + + = k ein which xk and n are the amplitude and frequency of thatharmonic component. The response of the system can berelated to the loading by[9] n n i tt H x n( ) ( )= k ewhere H(n) is a transfer function given by[10] H

i D

Hnn n

ni( ) ( )

=

+

=1

1 20

2

0

e

where |H(n)| is the modulus of the complex transfer func-tion. For the current study, |H(n)| was defined using thefoundation model described in the section titled Designprocedure. The real part of the response due to the nth har-monic component is then

[11] n n nt xk H t( ) ( ) cos( )= +k

The principle of superposition gives the total response as(t) = n(t).

Kinematic soilstructure interactionThe kinematic interaction alters the free field motion by

virtue of the relatively stiff foundation and wave scatteringeffects. When subjected to vertically propagating coherent

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shear waves, embedded foundations experience a modifica-tion in base-slab translational motions relative to the free-field, and rocking motions are introduced. These modifiedfoundation motions, named foundation input motions (FIM)are used as the excitation in the analysis of inertial interac-tion of the equipment foundations to ground-transmittedvibrations.

The significance of kinematic interaction depends on thecharacteristics of the ground-transmitted vibration and thesize, embedment, and flexibility of the foundation. Roesset(1980) suggests that these embedment effects are likely to besignificant for e/r (e is the embedded depth, r is the founda-tion radius or equivalent radius) greater than about 0.5.Analytical and empirical studies have been performed toexamine embedment effects on foundation input motions.

Analytical studies of embedment effects have focused onthe evaluation of transfer functions expressing the amplituderatio of base-slab translational and rocking motions tofree-field motions (Elsabee et al. 1977; Day 1977). Theseformulations are generally based on assumed vertically prop-agating coherent waves and the results are given in terms ofdimensionless frequency, a0. The results of these analysesindicate significant filtering of translational motions for a0 >0.5 and the development of a significant rocking componentfor a0 > 1.0. At low frequencies, a0 < 1.5, the filtering offoundation motions and magnitude of rocking motions in-crease with increasing embedment ratio, while at higher fre-quencies there is little sensitivity to this parameter. Theseresults can be contrasted with the behaviour of a surfacefoundation that would have no reduction of translational mo-tions and no rocking motions when subjected to verticallyincident coherent shear waves.

For a foundation embedded at depth e, with or withoutsidewalls, and subjected to harmonic vertical and obliquewaves, the amplification functions for translational and rock-ing components of FIM, Iu() and I () can be estimated as(Luco 1969; Elsabee et al. 1977; Tassoulas 1984)

[12] Ie

ra a a

a au

s

s

( )cos

.

=

>

0 0

0

23

0 457 23

I re

ra a a

ra a

( ).

cos

.

=

>

0 247 10 245

0 0

0

s

s

in which as = r/2e is the dimensionless shear frequency ofsoil stratum of thickness e. Empirical studies by Seed andLysmer (1980) and Chang et al. (1985) have documented re-ductions in ground motion with depth using both downholefree-field arrays and comparisons of basement and free-fieldmotions. The results of their studies indicated reductions ofpeak ground acceleration and high frequency spectral ordi-nates with depth.

Alternatively, the method proposed by Clough andPenzien (1993) can be used to estimate the effect of kine-matic interaction on the ground motion to the rigid founda-tion. In this method, the ratio of the amplitude of a harmonic

component of the rigid foundation in translation to the sameharmonic of the incident wave is given by

[13]

= 1 2 1 1 2[ ( cos )]

in which

= =

DV

DL

f fs

2, Df is the dimension of the foun-

dation in the direction of wave propagation and L is thewavelength corresponding to . The ratio varies between 1and 0 as varies from 0 to 2. This means the -factor (i.e.,kinematic interaction) could significantly lower the input ex-citation, thus reducing response accordingly.

Results of dynamic analysis

Dynamic analyses were conducted to evaluate the perfor-mance of the machine-foundation systems under theground-transmitted vibration. These dynamic analyses in-volved calculating the frequency content of the ground mo-tions, the dynamic characteristics of the foundation systems,and the dynamic response of the foundation systems to theground motions. The following sections will summarize theresults of the analysis.

Response of CLS foundationThe proposed foundation system consists of a concrete

slab 78 m 78 m and 0.35 m thick supported by 400 con-crete piles. Because of the large area of the foundation, theground vibration measurements at stations S4 (at the edge ofslab near the road) and S5 (at the centre of the slab) wereexamined as they were deemed to be most representativeof the ground vibrations that would be experienced by thefoundation. Each event included the ground vibration mea-surements that lasted 12 s. Different segments of the groundvibration time history were examined to identify the criticalloading.

Force Fourier amplitudesThe inertial force time history was calculated by multiply-

ing the measured ground acceleration time history by themass of the equipment and its supporting structure (assumedto be 1.0 106 kg). The force time history was subjected toa FFT to transfer the load into the frequency domain. Figs. 1and 2 show the variation of the force Fourier amplitudeswith frequency at stations S5 and S4 for a truck travelingeast. It can be noted from Figs. 1 and 2 that the force ampli-tudes at S4 are much higher (17 times higher) than those atS5, because of the attenuation of the ground motion betweenstations S4 and S5. It can also be seen that most of theenergy at S4 is concentrated in the frequency range 400600 rad/s (6090 Hz). The winter tests showed an energyconcentration in the frequency range 10001500 rad/s(160240 Hz). The shift in the frequency range may be at-tributed to the fact that the ground was not frozen during thisset of tests. At S5, however, the energy is concentrated in thefrequency range of 100400 rad/s (1560 Hz). This can beattributed to the fact that waves with higher frequencies attenu-ate faster than waves with lower frequencies. Also, this fre-quency range is lower than that of the winter tests, which isconsistent with the measurements at S4. It was found that the

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attenuation in the measurements taken in the summer (shownin Figs. 1 and 2) was higher than it was in the winter measure-ments. This may be attributed to the fact that the ground wasstill fully or partially frozen at the time of the winter tests.

The vibration measured at S5 (Fig. 1) was used as theinput excitation for the dynamic analysis. The vibration

measured at S4 was also used in another set of analyses, andthe wave attenuation was considered subsequently.

Stiffness and dampingThe foundation is supported by 400 prestressed concrete

piles 0.6 m in diameter and 10 m in length. The stiffness and

Fig. 1. Force Fourier amplitudes based on ground accelerations measured at S5.

Fig. 2. Force Fourier amplitudes based on ground accelerations measured at S4.



damping of the foundation were calculated over the fre-quency range of interest. Figure 3 shows the horizontal andvertical stiffness and damping of the foundation. It can benoted from Fig. 3 that the stiffness and damping of the foun-dation vary considerably with frequency, and care should beexercised in the selection of the stiffness value used in thedynamic analysis. It should also be noted that the stiffness

is very small in the frequency range 500700 rad/s (80115 Hz), but the damping increases slightly. Also, the im-portant frequency range is 400600 rad/s (6090 Hz) basedon the vibration measurements at S4 and 100400 rad/s (1560 Hz) based on the measurements at S5. Therefore, thereare no resonance conditions based on the measurements atS5 and limited resonance based on the measurements at S4.

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Fig. 3. Stiffness and damping of the CLS foundation: (a) vertical stiffness, (b) vertical damping, (c) horizontal stiffness, and (d) hori-zontal damping.



The response is governed by the stiffness of the system notby its damping characteristics.

Response of foundation system to ground motionThe pile cap was assumed to be rigid and to sit above the

ground. It was also assumed that there was no contact be-tween the bottom of the slab and the ground surface (void

form of 150 mm). In other words, the soil reactions at thebase and along the sides of the slab as well as the -factorwere neglected. The piles were assumed to be fixed in thepile cap. The inertial force due to the ground motion wasused as the dynamic excitation. Figure 4 shows that the pro-posed foundation would result in a satisfactory dynamic per-formance with maximum horizontal vibration amplitudes of

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Fig. 3 (concluded).



3.3 107 m. The vertical vibration amplitudes were foundto be less than 1 107 m, which is much lower than thespecified tolerance.

The dynamic analysis was repeated with the vibration mea-surements at station S4 used as the ground input motion. Themaximum horizontal vibration amplitude is 1.75 106 m.Although the vibration amplitudes obtained using the groundvibration measurements at S4 are higher than the specifiedtolerance value of 0.35 106 m, they represent satisfactorydynamic performance for two reasons: the -factor was ne-glected, and the ground motion was assumed to have thesame value at all points under the slab area (i.e., ignoring theattenuation effect). These two assumptions overestimated theresponse by a factor of 20.

Response of MRI foundationThe vibration measurements showed that ground motions

in the overburden soil are an order of magnitude higher thanthose measured in the bedrock. Therefore, the response anal-ysis was based on the ground motion measured in the soil.

Forcing functionBecause of the limited extent of the foundation, only the

measurements at the location of its centre were consideredin the analysis. The FFT analysis showed that the vibrationenergy was concentrated in the frequency range 100400 rad/s (1560 Hz). The natural frequency in the horizon-tal vibration mode was about 22 Hz. This represented partialresonance in that vibration mode and increased response.

Foundation stiffness and dampingThe foundation is a concrete slab 4.5 m 5.7 m 0.45 m

supported by four drilled shafts each 0.9 m in diameter and5.4 m long. The piles penetrate through a silty sand layer

with Vs = 200 m/s and rest on weathered bedrock with Vs =2000 m/s. Figure 5 shows the variation of the stiffness anddamping with frequency. Figure 5 shows that the stiffnessand damping of the foundation vary considerably and shouldbe accounted for in the analysis.

Response of the foundation system to the ground motionThe foundation was considered to be rigid. Because of the

random nature of the traffic loading, the response was calcu-lated using the random vibration approach. Figure 6 showsthe horizontal response spectrum of the foundation system.The maximum response in the frequency range 010 Hz wasfound to be 1 109 m. This corresponded to slightly lessthan 0.5 106 g, which represented a satisfactory perfor-mance. Similar results were obtained for other frequencyranges.

Response of turbine foundation to blast loadingThe proposed foundation system consisted of two rigid

concrete blocks. The first was 20.75 m 33.5 m 1.8 mand the second was 21.5 m 35 m 1.8 m. Each blockwould support a turbine generator. The inertial force due tothe blast-induced ground motion was used as the dynamicexcitation force. The ground accelerations measured at theweathered rock in the location of the proposed foundationwere designated as the design acceleration-time history.

Force Fourier amplitudesThe ground acceleration-time history with its highest

acceleration measured during the vibration monitoring pro-gram was used as the input ground motion. Figure 7 showsthe variation of the force Fourier amplitudes with frequencyfor the longitudinal (x), transverse (y), and vertical (z) direc-tions. It is noted in Fig. 7 that the force amplitudes were

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Fig. 4. Canadian Light Source vibration amplitudes based on ground vibration measured at S5.



highest in the longitudinal (x) direction and lowest in thetransverse (y) direction and that the maximum Fourier am-plitudes of the force in the longitudinal direction are almostdouble those in the vertical direction. It is also noted inFig. 7 that most of the energy is concentrated in the fre-quency range 600780 rad/s (95125 Hz) for the longitudi-nal direction and 500920 rad/s (80150 Hz) for the verticaldirection. In the transverse direction, the energy is scatteredover the frequency range 2001000 rad/s (30160 Hz) witha slightly higher concentration in the frequency range 680780 rad/s (110125 Hz).

Stiffness and dampingThe inspection of the frequency content of the forcing

function showed that it covers a wide range of frequencies.Because of the uncertainty about the shear wave velocity, Vs,of both the engineered fill after construction and the bed-rock, different soil profiles were considered in the analysis.Two main profiles were considered: a homogeneous half-space with a uniform shear wave velocity (halfspace profile);and a layer underlain by a homogeneous halfspace (compos-ite medium profile). In the composite medium profile, the

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Fig. 5. Stiffness and damping of the MRI foundation: (a) vertical stiffness, (b) vertical damping, (c) horizontal stiffness, and (d) hori-zontal damping (1 lb/ft = 14.59 N/m).

Fig. 6. Horizontal response spectrum of the MRI foundation(1 ft2 = 0.093 m2 ).



shear wave velocity of the layer is assumed to be uniformand less than the shear wave velocity of the halfspace. Forboth profiles considered, the shear wave velocity of the soilat the bottom of the foundation was assumed to be equal to180, 240, or 300 m/s (600, 800, or 1000 ft/s). In the compos-ite medium profile, the shear wave velocity of the halfspacewas assumed to be equal to 300 or 360 m/s (1000 or1200 ft/s). The soil reactions along the top 1 m (3 ft) of thefoundation sides were ignored to account for any separationbetween the foundation and soil along the sides.

The stiffness and damping of the foundation were calcu-lated over the frequency range of interest. Figure 8 showsthe stiffness and damping of the foundation for the halfspaceprofile. The strong variation of the vertical stiffness with fre-quency can be noted in Fig. 8. However, the vertical damp-ing increases rapidly with frequency to a maximum value ata frequency of about 140 rad/s. On the other hand, the hori-zontal stiffness and damping show a small variation with fre-quency in the frequency range of interest. Figure 9 showsthe stiffness and damping of the foundation for the compos-ite medium profile and Vs = 180 m/s (600 ft/s) and the shearwave velocity for the underlying halfspace is 300 m/s(1000 ft/s). It can be noted in Fig. 9 that the vertical stiffnesscould be considered constant for frequencies higher than80 rad/s. However, the vertical damping decreases with fre-quency. Figure 9 also shows that the horizontal stiffness anddamping decrease quickly with frequency. Comparing thevertical stiffness and damping from the two profiles, it canbe noted that the composite medium yielded higher verticalstiffness values (almost double) and much smaller dampingvalues than the halfspace profile. Similar observations canbe made for the horizontal direction, especially for damping.It should be noted that the stiffness assumes negative values

due to the inertial effect of the foundation. However, thetotal stiffness (stiffness and damping) of the foundation sys-tem is positive.

Response of the foundation system to the ground motionThe foundation was assumed to be rigid and embedded in

the ground. It was also assumed that there was no contactbetween the upper 1 m (3 ft) of the sides of the slab and theadjacent soil. The inertial force due to the ground motionwas used as the dynamic excitation force.

The natural frequencies of the machine-foundation systemwere found to be 6.7 and 10 Hz for the horizontal and verti-cal vibration modes, respectively. Therefore, resonance con-ditions would not occur. The response was calculated fordifferent values of shear wave velocities for both profiles.The composite medium with Vs = 240 m/s and reduceddamping resulted in a maximum vertical displacement of0.066 mm (0.00022 ft) as shown in Fig. 10.

The vibration amplitudes obtained using the ground vibra-tion measurements are very close to the specified tolerancevalue of 0.075 mm and the total vibration amplitudes (i.e.,including the vibration amplitudes due to normal machineoperation) may exceed the specified tolerance. However,they represent satisfactory dynamic performance. The analy-sis assumed that the entire foundation would vibrate in phaseunder the effect of the ground vibration introduced at thecentre of the foundation. This assumption overestimated thevibration of this specific foundation by an order of two tothree (i.e., the -factor). This factor depends on the wave-lengths of the seismic waves relative to the dimensions ofthe rigid foundation. It is expected that this effect will re-duce the vibration amplitudes by about 50% resulting in a

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Fig. 7. Fourier amplitudes for blast loading (1 lb = 4.448 N).



maximum vibration amplitude due to the blast loading ofless than 0.033 mm (0.00011 ft).

Response of two adjacent machine foundationsA common arrangement in cogeneration plants is to have

two or more units; each consisting of a generator, a com-pressor, and accessory equipment and each supported on aseparate foundation. If these units are closely spaced, the vi-brations emanating from one unit would induce additionalvibrations on the other units. The case study considered in-

volves two units. The foundation of each unit is a concreteblock 14 m 21 m 1.5 m supported by 130 concrete piles.The soil profile at the site consisted of the following layersstarting from the ground surface: 1 m of engineered fill(sand with Vs = 160 m/s), 2.7 m of sand clay (Vs = 165 m/s),8 m of clay (Vs = 185 m/s), 6.5 m of clay till (Vs = 250 m/s)and Empress dense sand (Vs = 450 m/s). The pile penetratedthe clay layers and rested on the Empress dense sand. Thetop of the concrete block is 3.5 m below the finished floorlevel. The distance between the two foundations is 3 m

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Fig. 8. Stiffness and damping of a foundation resting on a halfspace (1 lb/ft = 14.59 N/m).



edge-to-edge. The response at the edge of each foundationunder the harmonic loading from the operation of its ownsupported equipment was calculated using eq. [6]. The vi-bration amplitude r , at a distance r from the vertical axis ofthe foundation can be evaluated approximately by (Barkan1962)

[14] r (e=

00

1 20

r

r

r r )

where 0 is the foundation amplitude, r0 is the distance ofthe foundation edge from its vertical axis, and is the empir-ical coefficient ranging from 0 to 0.1 m1. The higher theability of the soil particles to slide against each other, thebetter the damping characteristics of the ground at appliedcyclic loads. Therefore, the magnitude of is higher forcohesionless soils (0.050.1 for sand), lower for cohesivesoils (0.020.04 for clay), and zero for rock. Equation [14]was derived assuming surface waves whose vertical axis isgreater than its horizontal axis. However, the physical prop-

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Fig. 8 (concluded).



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erties of the soil medium differ considerably from the idealproperties assumed in the theory. Consequently, the motionactually observed differs from the theoretical pattern(Barkan 1962). Hence, for practical calculations the horizon-tal amplitude may be considered equal to the vertical one.The vibration amplitudes at the edges and centre of the adja-cent foundation were evaluated using eq. [14] (assuming =0.05), and an average vibration amplitude, ave, in eachvibration mode was calculated.

There are two methods to account for the additional vibra-tion from adjacent foundations. The calculated averageamplitude can be used to calculate the acceleration of theground motion as

[15] = 2 aveThis acceleration amplitude would then be multiplied by

the mass of the foundation and supported equipment to cal-

Fig. 9. Stiffness and damping of foundation resting on a composite medium (1 lb/ft = 14.59 N/m).



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culate the input inertial (harmonic) force. This force wouldbe used in eq. [6] to calculate the additional vibration ampli-tudes due to the adjacent vibrating equipment. The totalvibration amplitude would be calculated as the sum of theadditional vibration amplitudes and the vibration amplitudesof the foundation under its own load.

Alternatively, an upper bound on the total vibration am-plitudes can be evaluated by superimposing the average vi-bration amplitudes directly on the vibration amplitude of

the foundation due to its own load. This solution wasadopted in the case history reported herein. The average vi-bration amplitudes due to the adjacent foundation were ap-proximately 50% of the foundation vibration amplitudesunder its own load.

Summary and conclusions

The paper presents some rational approaches for the

Fig. 9 (concluded).



response evaluation of foundations supporting vibration-sensitive equipment to ground-borne excitations, incorporat-ing the dynamic characteristics of both the foundation systemand the seismic excitation. The analysis of four different casehistories that involved the response of foundations supportingvibration-sensitive equipment to ground-transmitted excitationwas demonstrated. The following conclusions can be made:(1) When designing a foundation for vibration-sensitive

equipment, a ground vibration-monitoring programshould be carefully planned and executed to assess thelevel of seismic excitation at the proposed site causedby ground-transmitted vibration from external sources.

(2) The stiffness and damping of a shallow foundationshould be based on proper modeling of the actual soilprofile. The stiffness functions of layers differ substan-tially from those of the halfspace because the geometricdamping vanishes below the first layer resonance. Thus,the widely used halfspace model seriously overestimatesthe damping and underestimates the stiffness. The half-space model may lead to a gross underestimation of theresponse of a foundation resting on a layer of limitedthickness underlain by a hard stratum.

(3) The effect of kinematic interaction on the ground mo-tion to the rigid foundation (i.e., the -factor) should betaken into consideration in the response analysis of largefoundations subjected to ground-transmitted excitations.

(4) In cogeneration plants with two or more units, the vibra-tions emanating from one unit would induce additionalvibrations on other units. If these units are closelyspaced, the additional vibration should be taken intoconsideration when evaluating the dynamic performanceof the foundation.

Ackowledgements

The author would like to thank Dr. Bruce Sparling ofthe University of Saskatchewan, Mr. Eric Norum andMr. Dan Lowe of the CLS management group, andMr. Nizar Dhanani of UMA Engineering Ltd., Saskatoon,for their contributions in facilitating the study of the CLSfoundation.

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Fig. 10. Vibration of turbine generator foundation due to blast loading (composite medium) (1 ft = 0.3048 m).



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El Naggar, M.H., and Novak, M. 1995. Nonlinear lateral interac-tion in pile dynamics. Journal of Soil Dynamics and EarthquakeEngineering, 14(2): 141157.

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