naggar trenches

NOTE / NOTE

Vibration barriers for shock-producing equipment

M. Hesham El Naggar and Abdul Ghafar Chehab

Abstract: Most modern manufacturing facilities have hammers or presses in addition to precision cutting equipment astheir production machinery. Foundations supporting hammers and presses experience powerful dynamic effects. Theseeffects may extend to the surroundings and affect labourers, other sensitive machines within the same facility, or neigh-bouring residential areas. To control vibration problems, wave barriers may be constructed to isolate vibrations propa-gating to the surroundings. This paper examines the efficiency of both soft and stiff barriers in screening pulse-inducedwaves for foundations resting on an elastic half-space or a layer of limited thickness underlain by rigid bedrock. Theeffectiveness of concrete, gas-cushion, and bentonite trenches as wave barriers is examined for different cases of soillayer depth, trench location, and embedment of the foundation. The model was formulated using the finite elementmethod, and the analysis was performed in the time domain. The efficiency of different types of wave barriers in vibra-tion isolation for shock-producing equipment was assessed and some guidelines for their use are outlined.

Key words: hammer foundation, impact load, gas-cushion trenches, concrete trenches, soil–bentonite trench, finiteelement modeling.

Résumé : La plupart des équipements modernes de manufactures comprennent des marteaux ou des presses en plusd’installations de coupe de précision dans leur machinerie de production. Les fondations qui supportent les marteaux etles presses subissent de puissants effets diynamiques. Ces effets peuvent se faire sentir dans les environs et affecter lestravailleurs, ou d’autres appareils sensibles à l’intérieur du même complexe, ou les aires résidentielles environnantes.Pour contrôler les problèmes de vibrations, des écrans contre les ondes peuvent être construits pour isoler les ondes quipeuvent se propager dans les environs. Cet article examine l’efficacité des écrans souples ou rigides pour tamiser lesondes induites par pulsations dans les fondations reposant sur un demi espace élastique ou sur une couche d’épaisseurlimitée reposant sur un lit rocheux rigide. On a examiné l’efficacité du béton, d’un coussin de gaz, et de tranchées debentonite comme écrans d’ondes pour différents cas de profondeurs de couches de sol, de localisation des tranchées, etd’enfouissements de la fondation. La formulation du modèle a été faite par la méthode d’éléments finis et l’analyse aété réalisée dans une plage temporelle. L’efficacité de différents types d’écrans d’ondes pour isoler des vibrations leséquipements produisant des chocs a été évaluée et des règles ont été énoncées pour leur utilisation.

Mots clés : fondation de marteaux, charge d’impact, tranchées de coussins de gaz, tranchées de béton, tranchée desol-bentonite, modélisation par éléments finis.

[Traduit par la Rédaction] El Naggar and Chehab 306

Introduction

Large hammers, presses, and mills produce excessive vi-brations that travel long distances through the soil. In manycases, a manufacturing facility would include a hammer orpress and a vibration-sensitive piece of equipment (e.g.,lathe) housed in the same building. The vibration emanatingfrom the hammer foundation may affect the performance ofthe sensitive equipment, which leads to defective production

and substantial financial losses. In other cases, the vibrationsdue to a hammer operation could annoy neighbouring resi-dential areas, which may dictate reduced working hours or,in the case of severe vibrations, a complete shut down of thefacility.

Wave barriers are used to isolate ground-borne vibrationsand in practice include soft barriers (e.g., open trenches ortrenches filled with bentonite slurry) and stiff barriers (e.g.,sheet piling or concrete walls). Massarsch (1991) introducedan innovative gas-cushion screen installed in a deep trench,which is then filled with a self-hardening cement–bentonitegrout. He conducted field tests and numerical analyses to ex-amine the effectiveness of gas cushions and open trenches invibration isolation and concluded that their performance iscomparable.

Many experimental and analytical studies have examinedusing wave barriers to minimize vibrations emanating fromcentrifugal and reciprocating machines, which are character-

Can. Geotech. J. 42: 297–306 (2005) doi: 10.1139/T04-067 © 2005 NRC Canada

297

Received 28 February 2003. Accepted 1 June 2004. Publishedon the NRC Research Press Web site at http://cgj.nrc.ca on1 March 2005.

M.H. El Naggar and A.G. Chehab. Geotechnical ResearchCentre, Department of Civil and Environmental Engineering,Faculty of Engineering, The University of Western Ontario,London, ON N6A 5B9, Canada.

1Corresponding author (e-mail: [email protected]).

ized by periodic, high-frequency but low-amplitude excita-tions. Most studies considered vibrations in a homogeneoushalf-space. Woods (1968) performed field tests to investi-gate vibration isolation using open trenches close to the vi-bration source, where the body waves are dominant, and farfrom the vibration sources, where Rayleigh waves dominate.Based on the experimental results, Woods established designguidelines that can achieve a reduction in ground vibrationamplitudes by up to 75%. Waas (1972) used a frequency-domain finite element method to study the screening of hori-zontal shear waves using trenches. Aboudi (1973) employedthe finite difference method to evaluate the effect of a thinbarrier on the ground response. Haupt (1978) employed thefinite element method to investigate the effect of the sizeand shape on the trench efficiency in vibration isolation andverified the results using model experiments.

Beskos at al. (1985, 1986a, 1986b) employed the bound-ary element method in the frequency domain to study vibra-tion screening using open and infilled trenches. Dasgupta etal. (1986, 1988) used three-dimensional boundary elementmodels to investigate the screening effectiveness of opentrenches. Ahmad and Al-Hussaini (1991) and Al-Hussainiand Ahmed (1991) used the boundary element method toperform a parametric study on the efficiency of open andinfilled trenches in isolating harmonic vibrations. They stud-ied the effect of depth, width, and material within the trenchand proposed a simplified design procedure for infilled andopen trenches. Al-Hussaini and Ahmed (1996) used a three-dimensional boundary element algorithm to investigatevibration reduction by installing infilled walls around amoderate- to high-frequency load source. They outlined theeffect of the footing (load source) radius, location, depth,and width; the material comprising the barrier; and the soilproperties on vibration isolation effectiveness.

A few studies examined vibration barriers in layered soilprofiles. Segol et al. (1978) used a two-dimensional (2D),plane-strain, finite element model with nonreflecting bound-aries to study vibration screening by open and infilledtrenches in layered soils. They found that the barriers aremore effective in isolating the vertical component of the mo-tion than the horizontal component. Fuyuki and Matsumoto(1980) used a finite difference method to investigateRayleigh-wave scattering by rectangular open trenches anddemonstrated the effect of width for shallow open trenches.May and Bolt (1982) used a 2D finite element model tostudy the effectiveness of vibration screening using singleand twin open trenches in a two-layered soil medium.

Ground-borne vibrations originating from traffic activitiesare transient, with a significant low-frequency content. Yangand Hung (1997) investigated the effectiveness of open andinfilled trenches in isolating ground-borne vibrations, due tothe passage of trains, using a finite element model with infi-nite elements at the boundaries to allow for wave radiation.They examined the efficiency of the barriers for a range ofload frequencies and concluded that all the trenches investi-gated are not suitable for low frequencies. Kattis et al.(1999a, 1999b) compared the effectiveness of open and in-filled trenches and pile barriers (concrete and hollow piles)in screening vertical vibrations using a boundary elementmodel in the frequency domain. It was found that trenchesare more effective than pile barriers, except for vibrations

with large wavelengths where deep barriers are needed and,consequently, pile barriers are more practical.

There has been very limited effort dedicated to evaluatingthe performance of wave barriers in situations involvingtransient loading and almost none for pulse loading fromshock-producing equipment. The objective of this study isto examine the efficiency of both soft and stiff barriers inscreening pulse-induced waves for foundations resting on anelastic half-space or a layer of limited thickness underlain byrigid bedrock. The foundation, soil medium, and wave barri-ers are modeled using 2D finite elements in the time domainemploying the computer program ANSYS 5.7 (ANSYS Inc.2000).

Problem definition

A hammer foundation is founded on a soil layer of a lim-ited thickness underlain by a hard stratum at a depth D. Thefoundation width is w and its embedment depth is l, asshown in Fig. 1. A wave barrier with vertical walls, width b,and depth d is constructed at a distance Xt from the founda-tion edge. The soil layer has a uniform shear wave velocityof Vs, Poisson’s ratio νs, density ρs, and material damping δs.The open trench (i.e., gas cushions), infilled trench (concreteor soil–bentonite mixture) are considered. The properties ofthe material within the trench are as follows: shear wave ve-locity Vt, Poisson’s ratio νt , density ρt , and material dampingratio tan δt .

Two-dimensional finite element plane-strain models werebuilt and transient (time domain) analyses were performedconsidering different values of foundation embedment, soillayer thickness, trench material, and trench location. The ap-plied load is a transient, short-duration (pulse) load similarto that resulting from the normal operation of a hammer.

Unless otherwise specified, the shear wave velocity of thesoil is taken as 150 m/s, density as 1800 kg/m3, Poisson’s ra-tio as 0.3, and material damping as 0.03. The concrete shearwave velocity is taken as 2041 m/s, density as 2400 kg/m3,Poisson’s ratio as 0.25, and material damping as 0.02. Thesoil–bentonite shear wave velocity is taken as 30 m/s,density as 1500 kg/m3, Poisson’s ratio as 0.3, and materialdamping 0.02. The trench width is taken as one tenth of thetrench depth, i.e., b = d/10.

Assumptions and justification

The analysis considers 2D plane-strain conditions. Thisassumption may overestimate the efficiency of the trench be-cause it neglects waves traveling around the sides of thetrench. Al-Hussaini and Ahmed (1996) stated, however, thatthe 2D plane-strain assumption is sufficiently accurate forwave barrier analysis, especially for passive isolation. More-over, the plane-strain approach considers only 2D wave pro-pagation, one vertical and one horizontal direction, andneglects the other horizontal direction (i.e., a circular wavefront rather than a spherical one). This assumption overesti-mates the wave propagation in the directions considered andthus may underestimate the efficiency of the trench, whichfurther compensates for the effects.

The problem is symmetrical, therefore only one half ofthe actual model is considered in the analysis. The soil is as-sumed to be linear isotropic and viscoelastic. Thus, plastic

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298 Can. Geotech. J. Vol. 42, 2005

deformations and soil yielding at the load source, if any, areneglected. Hammer foundations are always designed, how-ever, such that the foundation soil remains in the elasticrange (i.e., plastic deformations are not expected). It is alsoassumed that full contact exists between the foundationblock and the soil and between the soil and the trench mate-rial. This assumption is justified for the level of deforma-tions allowed in machine foundations.

The hard stratum (the base of the model) is considered tobe very rigid compared to the soil layer. The soil propertiesare considered to be uniform throughout the depth of thelayer, which is a common assumption in soil dynamics.Lastly, only the dynamic response to the impact load is con-sidered. This means the static response due to the weight ofthe machine, the foundation, and the soil is not considered.

Finite element model

Figure 2 shows the finite element model used in the analy-sis. Two-dimensional plane-strain finite elements are used torepresent the soil–trench system.

Modeling soil and trenchThe soil medium is modeled using six-noded triangular el-

ements. The triangular element has six nodes, at the cornersand mid-edges, with two degrees of freedom in the x and ydirections at each node, as shown in Fig. 3. The soil andtrench elements are connected (i.e., no slippage or separationallowed) to ensure displacement compatibility at the nodesand thus along the soil–trench interaction boundaries. To en-

sure accuracy, the element size was kept at less than oneeighth to one fifth the shortest possible Raleigh wavelengthλr (Kramer 1996).

The soil shear wave velocity considered in the analysis is150 m/s. The vibration frequency depends on the natural pe-riod of the soil, which is a function of the thickness of thesoil layer. The pulse duration is 0.01 s, however, and thehighest vibration frequency would occur in the vicinity ofthe foundation and would be 50 Hz. Thus, the shortest wave-length was calculated to be 3 m. Therefore, an element sizeof 0.5 m (λr /6) is used for elements adjacent to the founda-tion. The effective node spacing (note that the triangular ele-ments have mid-edge nodes) in this case is less than onesixth of the wavelength. The vibration away from the loadsource depends mainly on the natural period of the soillayer. The shortest vibration period is about 0.13 s, andtherefore the shear wavelength would be about 19 m and theRayleigh wavelength would be about 18 m. The maximumallowable element size in this case is 2.3 m, which is main-tained everywhere in the model. The aspect ratio is kept be-low 2 for all the elements, and abrupt change in the elementsize is avoided.

Foundation modelThe hammer foundation block is modeled using frame

elements. The 2D frame element has two nodes with threedegrees of freedom at each node: translations along x and ydirections and rotation about z (in-plane rotation within thex–y plane). A hammer foundation block is usually rigid andmoves as one rigid body. The impact load of a hammer is

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El Naggar and Chehab 299

Fig. 1. Wave barrier used for vibration isolation.

usually concentric (eccentricity is not allowed). Thus, thefoundation displacement is vertical (similar to the load).Therefore, very high stiffness is assigned to the frame ele-ments and the first frame element is restrained against hori-zontal displacement and rotation (as shown in detail A inFig. 2) to ensure that the foundation moves in the verticaldirection only as a rigid body. The frame elements are gluedto the soil elements at the nodes, assuming full contact alongthe foundation–soil interface.

Boundary conditions

Left boundarySymmetry boundary conditions are applied along the axis

of symmetry by restraining the displacement in the x direc-tion as shown in Fig. 2. This symmetric condition impliesthat there is another trench on the other side, which mayreflect back some vibrations, reducing the calculated effi-ciency of the trench under consideration. Therefore applyingthe symmetry assumption is conservative.

Model baseThe hard stratum underlying the soil layer could be bed-

rock, hard clay till, or very dense sand. A hard stratum thatis much stiffer than the overlying soil would practically re-flect all incident waves. Therefore, it is reasonable to assumethat this stratum represents a rigid boundary and the base ofthe model is assumed to be fixed as shown in Fig. 2.

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Fig. 2. Finite element model of the problem. P(t), transient load applied to the foundation due to the hammer operation.

Right boundaryThe infinite extension of the soil (in the x direction) is

represented in dynamic problems using boundary conditionsthat allow for wave propagation by preventing reflection ofwaves back into the model (box effect). The location of theartificial boundary depends on the level of the materialdamping of the soil, the frequency range of interest, thewave velocity, and the duration of excitation (Wolf 1985).

Different types of boundaries can be used to model thesoil continuity. Wolf and Song (1996) proposed local bound-aries represented by dashpots whose damping constant perunit area (c) is calculated as

[1a] cn = ρVp (normal direction)

[1b] ct = ρVs (tangential direction)

where Vp and Vs are the compression and shear wave veloci-ties, respectively. Therefore, for waves traveling normal tothe edge of the elements at the boundary, the damping con-stant of the dashpot, applied at a specific boundary nodeparallel to the wave propagation direction, is obtained bymultiplying the damping constant given by eq. [1a] by thetributary area for that node. Similarly, the damping constantof the dashpot applied normal to the wave propagation direc-tion is obtained from eq. [1b]. To represent the stiffness ofthe soil at the boundary, consistent boundaries can be used.In this case, all degrees of freedom on the boundaries arecoupled and the force–displacement relationship is fre-quency dependent (Wolf 1985). Properly designed consistentboundaries can perfectly absorb all incident waves. There-fore, they can be applied very close to the load source and,if the structure response is the only concern, they can beplaced directly on the soil–structure interface. Consistentboundaries are not suitable for time-domain analysis, how-ever, because they are frequency dependent.

Transmitting boundaries are used in the current study. Apair of Kelvin elements, one in the vertical direction and onein the horizontal direction, is attached to each boundarynode. The Kelvin element consists of a linear spring and adashpot arranged in parallel as illustrated in Fig. 4. The con-

stants of the spring and the dashpot (stiffness and damping)are calculated using the solution of Novak and Mitwally(1988), i.e.,

[2a] KGr

S a iS ar r r= +s

oo s s o s s[ ( , , ) ( , , )]1 2ν δ ν δ

for the radial direction

[2b] KGr

S a iS ay y y= +s

oo s s o s s[ ( , , ) ( , , )]1 2ν δ ν δ

for the vertical direction

where Kr and Ky are the complex stiffnesses in the radial andvertical directions, respectively; Gs is the soil shear modulus;ro is the distance from the source of the disturbance to the fi-nite element boundary; and Sr1, Sr2, Sy1, and Sy2 aredimensionless parameters that depend on the dimensionlessfrequency a r Vo o s= ω / (where ω is the excitation frequency),the soil Poisson’s ratio νs, and damping tanδs. The stiffnessand damping per unit area are calculated as the real andimaginary parts of eqs. [2a] and [2b], i.e.,

[3a] k = GS1/ro

[3b] c = GS2/ωro

The constants of the spring and dashpot of the Kelvin ele-ment are obtained by multiplying k and c, respectively, bythe area of the element (normal to the direction of wavepropagation). The constants k and c can be considered fre-quency independent (El Naggar and Bentley 2000). In thepresent model, being symmetrical in geometry and loading,ro is taken as the distance (in plan) from the centre of thefooting to the node on the boundary where the Kelvin modelis attached. It was conceived that this distance approximatelyrepresents the corresponding radial distance in a cylindricalmodel as assumed by Novak and Mitwally (1988).

Applied load and analysisA concentric short pulse load was applied at the centre of

the foundation as shown in detail A in Fig. 2. The load isconsidered to be a rectangular pulse with amplitude of P =2 MN and duration tp of 10 ms, which is representative ofhammer loading. Linear dynamic full transient analyses areperformed to calculate the response of the model (with andwithout the trench) to the applied load. The time history ofthe vertical vibration at prespecified representative surfacenodes is obtained.

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Fig. 3. Two-dimensional six-noded triangular element used in thefinite element mesh. u, v, the displacements in the x and y direc-tions, respectively.

Fig. 4. Kelvin element used for the model boundary.

Model verification

To check the sensitivity to the element size in the model, afiner mesh was used and similar results were obtained. Themodel is verified by analyzing the response to a harmonicexcitation, and the results are compared with those obtainedby Ahmad and Al-Hussaini (1991) and Beskos et al. (1986a)for an open trench. The trench is one Rayleigh wavelength( )λ r deep and is located at a distance of five times the Ray-leigh wavelength from the applied harmonic load. Figure 5shows a reasonable agreement between the current analysisand the boundary element solutions, thus verifying themodel.

Results and discussion

For the sake of generalization, the geometric parametersand the results are presented in a dimensionless form. Thegeometric parameters are normalized by the trench depth.Unless otherwise specified, the trench depth d = 6 m and thefoundation half-width w/2 = 3 m. The time histories wereobtained of the vertical displacement for several nodes atdifferent locations x on the soil surface, in front of andbehind the trench. The maximum vibration amplitude wasfound from each time history. The vibration amplitudes fordifferent nodes were tabulated for the with-trench and with-out-trench cases. The with-trench amplitudes were normal-ized by the amplitudes of the without-trench case and wereplotted as a vibration-reduction factor versus the distance,normalized by d, from the foundation edge. The same proce-dure was repeated for different trench location, thicknessof the soil layer, embedment depth of the foundation, andtrench material.

The cases where the layer depth is 20 and 10 times thetrench depth were considered. The results from these caseswere identical and are similar to results of wave barriers in ahalf-space because the time required for the wave to travelfrom the source of the disturbance to the bottom of the

model (the rigid boundary) and back to the surface is muchgreater than the duration of the load pulse. The reflectedwave is attenuated (i.e., substantially reduced amplitude) andarrives at the ground surface much later after the originalpulse has expired. It was found that no benefit is realized byusing any type of wave barrier in these cases. As a matter offact, some vibration amplification was observed at nodes im-mediately in front of the barrier (due to wave reflection andstanding wave phenomena). A similar observation was madeby Yang and Hung (1997) in their study of train-induced vi-brations. Therefore, the case of a layer with large thicknessis not pursued any further.

Figure 6 shows the vibration reduction ratios using a gascushion (modeled as empty trench) wave barrier for vibra-tion isolation of a foundation sitting on the surface of a soillayer with varying thickness. The results for the case D =5.0d are shown in Fig. 6b, where the vibration reduction isless than 20% behind the wave barrier, whereas there issome vibration magnification observed in front of the bar-rier. In this case, the wave barrier is ineffective regardless ofthe location of the barrier from the edge of the foundation.

Figure 6b shows that for the case of D = 2.0d, a wave bar-rier located at a distance Xt/d = 0.5 can reduce the vibrationbehind the barrier by 50% or more but will increase the vi-bration in front of the barrier by up to 60%. For a barrierwith Xt/d = 1.0, the vibration reduction is about 50% withina distance d behind the barrier and there is no vibration mag-nification in the immediate vicinity of the foundation (thevibration magnification is confined to a distance 0.5d infront of the barrier and is less than 30%). For a barrier withXt/d = 1.5 or 2.0, the vibration is reduced by up to 80% inthe immediate vicinity of the barrier, but no significant re-duction is obtained away from the trench.

The results for D = 1.5d are shown in Fig. 6c. Figure 6cshows that the wave barrier reduced the vibration amplitudesby 50% or more (up to 90% right behind the trench) for allbarrier locations but is most efficient (up to 80% vibration

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Fig. 5. Comparative study for vibration screening of an open trench.

reduction) when it is located at a distance Xt/d = 0.5, al-though it causes 65% vibration amplification in front of thetrench. Based on these observations, it can be concluded thatgas cushions (or empty trenches) can be used as effectivewave barriers if their depth is more than one half the thick-ness of the soil layer, especially if they are located at a dis-tance Xt/d of from 0.5 to 1.0. A gas-cushion trench at adistance Xt = 0.5d, however, can only be implemented forsituations with embedded footings such that the bottom ofthe trench is no deeper than Xt below the base of the footing.

Figure 7 shows the vibration reduction ratios using soil–bentonite trenches for vibration isolation of a surface foun-dation in a soil layer with varying thickness. ComparingFigs. 6 and 7, it can be noted that the performance of soil–

bentonite trenches is comparable to that of empty trenches.The vibration reduction behind the trench and the vibrationmagnification in front of the trench, however, are slightlyless than those of the empty trench.

Figure 8 shows the vibration reduction ratios usingconcrete-infilled trenches for vibration isolation of a surfacefoundation in a soil layer with varying thickness. The figureshows that the concrete trench reduces the vibration ampli-tudes in the vicinity of the trench (within 0.5d behind thetrench). Its effect deteriorates quickly, however, with dis-tance behind the trench. On the other hand, no vibration am-plification occurs in front of the trench, in general, becausethe concrete can transmit a significant part of the incidentwaves. It can be concluded that concrete trenches can be

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Fig. 6. Effect of layer depth on the isolation efficiency of a gas-cushion trench (surface foundation): (a) layer depth 5.0d; (b) layerdepth 2.0d; (c) layer depth 1.5d.

used to minimize the vibration experienced by other founda-tions (receiver) if they are located within 0.5d behind thetrench (passive isolation).

Other foundation embedment depths (l = d/6, l = d/3, andl = d/2) were considered for all trench types investigatedin this study, and fairly similar results were obtained. Thisindicates that the effect of the embedment depth on theperformance of the trench is insignificant, provided thatthe contact between the soil and the foundation is main-tained.

Conclusions

The efficiency of both soft and stiff barriers in screeningpulse-induced waves for foundations resting on an elastic

half-space or a layer of limited thickness underlain by rigidbedrock is examined. Two-dimensional finite element mod-els are used to analyze the problem in the time domain em-ploying the computer program ANSYS 5.7. The transientload applied in the analysis is similar to pulses produced byhammers and presses during their normal operations. The ef-ficiency of the wave barrier is evaluated in terms of the ratioof the vibration amplitudes with and without the wave bar-rier. The following conclusions are drawn:

(1) Wave barriers are not effective for vibration isolation forhammer foundations founded on half-space soil.

(2) The effectiveness of the wave barrier increases as thetrench depth increases relative to the thickness of thesoil layer, i.e., the effectiveness increases as the ratio ofthe trench depth to the wavelength increases.

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Fig. 7. Effect of layer depth on the isolation efficiency of a soil–bentonite trench (surface foundation): (a) layer depth 5.0d; (b) layerdepth 2.0d; (c) layer depth 1.5d.

(3) Soft wave barriers (gas cushions, empty trenches, orsoil–bentonite trenches) are more effective than stiffwave barriers (concrete-infilled trenches).

(4) Soft wave barriers are more effective in vibration isola-tion if their depth is more than one half the thickness ofthe soil layer, especially if they are located at a distanceof 0.5–1.0 times their depth (d) from the edge of thefoundation. A trench at a distance Xt = 0.5d, however,can only be implemented for situations with embeddedfootings such that the bottom of the trench is no deeperthan Xt below the base of the footing.

(5) Stiff barriers can be used to minimize the vibration ex-perienced by other foundations (receiver) housed in thesame building as the vibration source if the receiver is

located within a distance behind the barrier equal to halfits depth (passive isolation).

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Fig. 8. Effect of layer depth on the isolation efficiency of a concrete trench (surface foundation): (a) layer depth 5.0d; (b) layer depth2.0d; (c) layer depth 1.5d.

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