MACHINE FOUNDATIONS/nNo of mechine/foundationLength(ft)Width(ft)Depth(ft) (min)
aas: aas:M15 OR M20Depth(ft) (max)
aas: aas:M15 OR M20Unit wt. of Concrete (ib) wt./machine(lbs)foundn wt. (Ibs )machine wt/foundn.(lbs)SELECT Machine type :
aas: aas:1 25 KG/M3 IMPACT2 25 KG/M3 RECIPROCATING3 50 KG/M3 ROTARY4 100 KG/M3 STEAM for multi scylinder engine : 40% to 60% increase in case of single cylindergas engine 725 kg foundation per each break HPdizel engine 565 kg steam engine 225 kgHP ? SPEED ?IMPACTROTARY RECIPROCATINGSTEAM TURBO GENERATORTRIAL( REINF PROVIDED)SPEED?HP?Reinf top & bottom IN KG
aas: aas:1 25 KG/M32 25 KG/M33 50 KG/M34 100 KG/M3for multi scylinder engine : 40% to 60% increase in case of single cylindergas engine 725 kg foundation per each break HPdizel engine 565 kg steam engine 225 kg% OF REINF MIN% OF REINF MAXREINF SPACING IN FTREINF BARNO OF BARNO OF BARTOTAL KG1118.008.005.609.27150
aas: aas:M15 OR M2039600120960.0011880002AEPRECIPROCATING571.31
aas: aas:MIN945.38
aas: aas:MAX103.91 %171.94 %0.5123616470.612314.003.8644.0773.38150
aas: aas:M15 OR M20
aas: aas:1 25 KG/M3 IMPACT2 25 KG/M3 RECIPROCATING3 50 KG/M3 ROTARY4 100 KG/M3 STEAM for multi scylinder engine : 40% to 60% increase in case of single cylindergas engine 725 kg foundation per each break HPdizel engine 565 kg steam engine 225 kgHP ? SPEED ?39600357210.6035640002AEPIMPACT1687.14
aas: aas:MIN2809.35
aas: aas:MAX103.91 %173.02 %0.512288980.363417.004.5041.5169.12150
aas: aas:M15 OR M2039600476347.5047520003 ATSIMPACT2249.84
aas: aas:MIN3746.12
aas: aas:MAX103.91 %173.01 %0.5123491134.754817.009.8637.9063.10150
aas: aas:M15 OR M2039600952914.3095040003ATSRECIPROCATING4500.71
aas: aas:MIN7493.27
aas: aas:MAX103.91 %173.00 %0.51234201477.095612.509.8638.6664.36150
aas: aas:M15 OR M20
aas: aas:1 25 KG/M32 25 KG/M33 50 KG/M34 100 KG/M3for multi scylinder engine : 40% to 60% increase in case of single cylindergas engine 725 kg foundation per each break HPdizel engine 565 kg steam engine 225 kg39600714648.7571280003ATSIMPACT3375.36
aas: aas:MIN5619.78
aas: aas:MAX103.91 %173.00 %0.51225201213.446312.504.5042.3470.50150
aas: aas:M15 OR M20
aas: aas:MIN39600357243.7535640003ATSIMPACT1687.30
aas: aas:MIN2809.51
aas: aas:MAX103.91 %173.02 %0.512259903.337417.683.8646.5277.47150
aas: aas:M15 OR M20
aas: aas:MAX
aas: aas:MIN39600476223.6747520001 SRMIMPACT2249.25
aas: aas:MIN3745.53
aas: aas:MAX103.91 %173.03 %0.5123581235.668416.003.5056.6794.39150
aas: aas:M15 OR M20
aas: aas:MAX
aas: aas:MIN39600476040.004752004RMIMPACT2248.38
aas: aas:MIN3744.66
aas: aas:MAX103.91 %173.06 %0.5123271320.999816.006.5061.02101.64150
aas: aas:M15 OR M20
aas: aas:MAX
aas: aas:MIN39600951960.009504004RMRECIPROCATING4496.20
aas: aas:MIN7488.76
aas: aas:MAX103.91 %173.07 %0.51232131715.981028.163.4456.5394.15150
aas: aas:M15 OR M20
aas: aas:MAX
aas: aas:MIN39600238021.062376004RMIMPACT1124.20
aas: aas:MIN1872.34
aas: aas:MAX103.91 %173.06 %0.512167766.451148.166.5059.8399.65150
aas: aas:M15 OR M20
aas: aas:MAX
aas: aas:MIN39600475995.604752004 RmIMPACT2248.18
aas: aas:MIN3744.45
aas: aas:MAX103.91 %173.06 %0.51216131055.581215.502.5057.7096.10150
aas: aas:M15 OR M20
aas: aas:MAX
aas: aas:MIN39600119006.25118800tollingIMPACT562.08
aas: aas:MIN936.15
aas: aas:MAX103.91 %173.06 %0.512115530.851315.514.9229.3048.76150
aas: aas:M15 OR M20
aas: aas:MAX
aas: aas:MIN39600119206.88118800miller machineIMPACT563.03
aas: aas:MIN937.10
aas: aas:MAX103.91 %172.94 %0.5121110393.1814212.204.5928.3647.19150
aas: aas:M15 OR M20
aas: aas:MAX
aas: aas:MIN
aas: aas:MAX
aas: aas:MIN
aas: aas:MAX
aas: aas:MIN
aas: aas:MAX
aas: aas:MIN
aas: aas:MAX
aas: aas:MIN
aas: aas:MAX39600238440.88237600GrinderRECIPROCATING1126.18
aas: aas:MIN1874.32
aas: aas:MAX103.91 %172.94 %0.512249627.2528689.1547766.7013825.54

DOCFOUNDATIONS FOR VIBRATING MACHINESSpecial Issue, April-May 2006, of the Journal of Structural Engineering, SERC, Madras. INDIASHAMSHER PRAKASH VIJAY K. PURIABSTRACTThe paper discusses the methods of analysis for determining the response offoundations subjected to vibratory loads. The design of a machine foundation isgenerally made by idealizing the foundation- soil system as spring-mass dashpotmodel having one or two degrees of freedom. Most machine foundations aretreated as surface footing and the soil spring and damping values are determined usingthe elastic-half space analog. The spring and damping values for response of embeddedfoundations can also be determined from the elastic half space concept as per Novakswork. The soil spring and damping values can also be obtained following the impedancecompliance function approach. The paper also presents a brief discussion of thepredicted and observed response of machine foundations .INTRODUCTIONMachine foundations require a special consideration because they transmitdynamic loads to soil in addition to static loads due to weight of foundation, machine andaccessories. The dynamic load due to operation of the machine is generally smallcompared to the static weight of machine and the supporting foundation. In a machinefoundation the dynamic load is applied repetitively over a very long period of time but itsmagnitude is small and therefore the soil behavior is essentially elastic, or elsedeformation will increase with each cycle of loading and may become unacceptable.The amplitude of vibration of a machine at its operating frequency is the most importantparameter to be determined in designing a machine foundation, in addition to the naturalfrequency of a machine foundation soil system.There are many types of machines that generate different periodic forces. Themost important categories are:1. Reciprocating machines: The machines that produce periodic unbalanced forces (suchas steam engines) belong to this category. The operating speeds of such machines areusually less than 600r/min. For analysis of their foundations, the unbalanced forcescan be considered to vary sinusoidally.2. Impact machines: These machines produce impact loads, for instance, forginghammers. Their speeds of operation usually vary from 60 to 150 blows per minute.Their dynamic loads attain a peak in a very short interval and then practically die out.3. Rotary machines: High-speed machines like turbogenerators or rotary compressorsmay have speeds of more than 3,000r/min and up to 12,000r/min.A suitable foundation is selected, depending upon the type of machine. Forcompressors and reciprocating machines, a block foundation is generally provided(Fig.1a). Such a foundation consists of a pedestal resting on a footing. If two or moremachines of similar type are to be installed in a shop, these can profitably be mounted onone continuous mat.A block foundation has a large mass and, therefore, a smaller natural frequency.However, if a relatively lighter foundation is desired, a box or a caisson type foundationmay be provided. (Fig.1b) The mass of the foundation is reduced and its naturalfrequency increases. Hammers may also be mounted on block foundations, but theirdetails would be quite different than those for reciprocating machines.Steam turbines have complex foundations that may consist of a system of wallscolumns, beams and slabs. (Fig.1c) Each element of such a foundation is relativelyflexible as compared to a rigid block and box or a caisson-type foundation.The analysis of a block foundation is relatively simple as compared to a complexfoundation. There are several methods of analysis for both the block and the complexfoundations. The criteria for designing machine foundations shall be discussed firstfollowed by the methods of analysis.Figure 1. Types of Machine Foundations (a) Block foundations. (b) Box or caissonfoundations. (c) Complex foundationsCRITERIA FOR DESIGNA machine foundation should meet the following conditions for satisfactory performance:*******Static loads1. It should be safe against shear failure2. It should not settle excessivelyThese requirements are similar to those for all other foundations.*******Dynamic loads1. There should be no resonance; that is, the natural frequency of the machinefoundation-soil system should not coincide with the operating frequency of themachine. In fact, a zone of resonance is generally defined and the natural frequencyof the system must lie outside this zone. The foundation is high tuned when itsfundamental frequency is greater than the operating speed or low tuned when itsfundamental frequency is lower than the operating speed. This concept of a high orlow tuned foundation is illustrated in Fig..2.2. The amplitudes of motion at operating frequencies should not exceed the limitingamplitudes, which are generally specified by machine manufacturers. If the computedamplitude is within tolerable limits, but the computed natural frequency is close to theoperating frequency, it is important that this situation be avoided.3. The natural frequency of the foundation soil system should not be whole numbermultiple of the operating frequency of the machine to avoid resonance with the higherharmonics.4. The vibrations must not be annoying to the persons working in the shops or damagingto the other precision machines. The nature of vibrations that are perceptible,annoying, or harmful depends upon the frequency of the vibrations and the amplitudeof motion.The geometrical layout of the foundation may also be influenced by the operationalrequirements of the machine. The failure condition of a machine foundation is reachedwhen its motion exceeds a limiting value which may be based on acceleration , velocityor amplitude. . Richart (1962) defined the failure criteria in terms of limitingdisplacement amplitudes at a given frequency. The limiting or permissible amplitudes canbe established from Fig. 3 (Blake, 1964), who also introduced the concept of servicefactor.Figure2. Tuning of a foundationFigure 3. Limiting amplitudes of vibrations for a particular frequency. (Blake, 1964)Criterion for vibration of rotating machinery. Explanation of classes :AA Dangerous. Shut it down now to avoid dangerA Failure is near. Correct within two days to avoid breakdown.B Faulty. correct it within 10 days to save maintenance dollars.C Minor faults. Correction wastes dollars.D No faults. Typical new equipment.This is guide to aid judgment, not to replace it. Use common sense. Take account of alllocal circumstances. Consider: safety, labor costs, downtime costs. (after Blake, 1964.)Reproduced with permission from Hydrocarbon Processing, January 1964.The service factor indicates the importance of a machine in an installation. Typical valuesof service factors are listed in Table1. Using the concept of service factor, the criteriagiven in Fig. 3 can be used to define vibration limits for different classes of machines.Also, with the introduction of the service factor, Fig. 3 can be used to evaluate theperformance of a wide variety of machines. The concept of service factor is explained bythe following examples.A centrifuge has a 0.01 in (0.250 mm) double amplitude at 750 rpm. The value of theservice factor from Table 1 is 2, and the effective vibration therefore is 2X 0.01 = 0.02 in(0.50 mm). This point falls in Class A in Fig. 3. The vibrations, therefore, are excessive,and failure is imminent unless the corrective steps are taken immediately. Anotherexample is that of a link-suspended centrifuge operating at 1250 rpm that has0.00.30 in(0.075mm) amplitude with the basket empty. The service factor is 0.3, and the effectivevibration is 0.00090 in (0.0225mm). This point falls in class C (Fig. 3) and indicates onlyminor fault.General information for the operation of rotary machines is given in Table 2 (Baxter andBernhard 1967).These limits are based on peak-velocity criteria alone and are represented by straightlines in Fig. 3a Effective vibration - measured single amplitude vibration, in inches multiplied by the service factor. Machine toolsare excluded. Values are for bolted-down equipment; when not bolted, multiply the service factor by 0.4 and use theproduct as the service factor. Caution: Vibration is measured on the bearing housing except, as stated.b Horizontal displacement basket housing.Table 2. General Machinery Vibration Severity Criteria (Baxter and Bernhart, 1967)Horizontal Peak Velocity(in/sec)Machine Operation0.630 Very roughTable 1. Service Factors aSingle-stage centrifugal pump, electricmotor, fan1Typical chemical processing equipment,noncritical1Turbine, turbogenerator, centrifugalcompressor1.6Centrifuge, stiff-shaft b; multistagecentrifugal pump2Miscellaneous equipment, characteristicsunknown2Centrifuge, shaft-suspended, on shaft nearbasket0.5Centrifuge, link-suspended, slung 0.3DEGREES OF FREEDOM OF A RIGID BLOCK FOUNDATIONA typical concrete block is regarded as rigid as compared to the soil over which it rests.Therefore, it may be assumed that it undergoes only rigid-body displacements androtations. Under the action of unbalanced forces, the rigid block may thus undergodisplacements and oscillations as follows (Fig. 4)1. translation along Z axis2. translation along X axis3. translation along Y axis4. rotation about Z axis5. rotation about X axis6. rotation about Y axisAny rigid-body displacement of the block can be resolved into these sixindependent displacements. Hence, the rigid block has six degrees of freedom and sixnatural frequencies.Of six types of motion, translation along the Z axis and rotation about the Z axiscan occur independently of any other motion. However, translation about the X axis (or Yaxis) and rotation about the Y axis (or X axis) are coupled motions. Therefore, in theanalysis of a block, we have to concern ourselves with four types of motions. Twomotions are independent and two are coupled. For determination of the naturalfrequencies, in coupled modes, the natural frequencies of the system in pure translationand pure rocking need to be determined. Also, the states of stress below the block in allfour modes of vibrations are quite different. Therefore, the corresponding soil-springconstants need to be defined before any analysis of the foundations can be undertaken.Figure 4.Modes of vibration of a rigid block foundationINFORMATION NEEDED FOR DESIGNThe following information is required and must be obtained for design of amachine foundation:1. Static weight of the machine and accessories.2. Magnitude and characteristics of dynamic loads imposed by the machine operationand their point of application3. The soil profile of the site and dynamic soil properties such as dynamic shearmodulus and damping4. Trial dimensions of the foundation. These are generally supplied by the manufacturer.This will give the total static weight.5. An acceptable method of analysis i.e., a mathematical model to determine theresponse of the foundation-soil system6. A criteria for adequate designThe above items are briefly discussed below:Dynamic Loads: The information on dynamic loads and moments may be available fromthe manufacturer of the machine. It may be possible to determine the dynamic loads andmoments for design of a machine foundation in some simple cases such as for the case ofreciprocating and rotary machines.SOIL PROFILE AND DYNAMIC SOIL PROPERTIESSatisfactory design of a machine foundation needs information on soil profile, depth ofdifferent layers, physical properties of soil and ground water level. This information canbe obtained by usual sub-surface exploration techniques. In addition, one must determinedynamic shear modulus, material damping, poisons ratio and mass density of soil fordynamic analysis of the machine foundation. Dynamic shear modulus of a soil isgenerally determined from laboratory or field tests. Material damping can be determinedfrom vibration tests on soil columns in the laboratory. The values of dynamic shearmodulii and damping may be estimated from empirical estimations for preliminarydesign purposes. Geometrical damping is estimated from elastic half-space theory andappropriate analogs. Detailed discussion of determination of dynamic soil properties andinterpretation of test is beyond the scope of this paper and a reference may be made toPrakash (1981) and Prakash and Puri (1981, 1988)TRIAL DIMENSIONS OF THE FOUNDATIONThe trial dimensions of the machine foundation are selected based on the requirements ofthe manufacturer, the machine shop and the machine performance and experience of thedesigner. These trial dimensions of the foundation are only the first step in the design andmay need alteration after the analysis.METHODS OF ANALYSISThe analysis of machine foundation is usually performed by idealizing it as a simplesystem as explained here. Figure 5 shows a schematic sketch of a rigid concrete blockresting on the ground surface and supporting a machine. Let us assume that the operationof the machine produces a vertical unbalanced force which passes through the combinedcentre of gravity of the machine-foundation system. Under this condition, the foundationwill vibrate only in the vertical direction about its mean position of static equilibrium.The vibration of the foundation results in transmission of waves through the soil. Thesewaves carry energy with them. This loss of energy is termed geometrical damping. Thesoil below the footing experiences cyclic deformations and absorbs some energy which istermed material damping. The material damping is generally small compared to thegeometrical damping and may be neglected in most cases. However, material dampingmay also become important in some cases of machine foundation vibrations.The problem of a rigid block foundation resting on the ground surface, (Fig. 5a)may therefore be represented in a reasonable manner by a spring-mass-dashpot systemshown in Fig. 5b. The spring in this figure is the equivalent soil spring which representsthe elastic resistance of the soil below the base of the foundation. The dashpot representsthe energy loss or the damping effect. The mass in Fig. 5b is the mass of the foundationblock and the machine. If damping is neglected, a spring-mass system shown in Fig. 5cmay be used to represent the problem defined in Fig. 5a. Single degree of freedommodels shown in Fig. 5 b and c may in fact be used to represent the problem of machinefoundation vibration in any mode of vibration if appropriate values of equivalent soilspring and damping constants are used. For coupled modes of vibration, as for combinedrocking and sliding, two degree-of-freedom model is used as discussed later in the paper.Figure 5. Vertical Vibrations of a Machine Foundation (a) Actual case, (b) Equivalentmodel with damping (c) Model without dampingAll foundations in practice are placed at a certain depth below the ground surface.As a result of this embedment, the soil resistance to vibration develops not only below thebase of the foundation but also along the embedded portion of the sides of the foundation.Similarly the energy loss due to radiation damping will occur not only below thefoundation base but also along the sides of the foundation. The type of models shown inFig. 5 b and c may be used to calculate the response of embedded foundations if theequivalent soil spring and damping values are suitably modified by taking into accountthe behavior of the soil below the base and on the sides of the foundation.Several methods are available for analysis of vibration characteristics of machinefoundations. The commonly used methods are1 Linear elastic spring method,2 Elastic half-space analogs method, and3 The impedance function method.1. The Linear Elastic Spring method (Barkan, 1962) treats the problem offoundation vibrations as spring- mass model , neglecting damping in the soil. The soildamping can be included if desired.2. The Elastic Half Space Analogs: The elastic half space theory can be used todetermine the values of equivalent soil springs and damping then make use of theory ofvibrations to determine the response of the foundation. These are known as the theelastic half space analogs. They can be used for surface as well as embeddedfoundations. It may be mentioned here that the equivalent soil spring and damping valuesdepend upon the ; Soil stiffness and dampingmmPz Sint Pz Sin t Pz Sin tkzkcz zm(a) (b) (c)(i) type of soil and its properties,(ii) geometry and layout of the foundation, and(iii) nature of the foundation vibrations occasioned by unbalanced dynamic loads.3. The Impedance Function Method: They also provide vales of soil spring and dampingfor surface and embedded foundations.The solutions based on the elastic half space analog are commonly used for machinefoundation design and are discussed first followed by the impedance function method.Elastic-half space -analogsSurface FoundationsVertical vibrations: The problem of vertical vibrations is idealized as a single degreefreedom system with damping as shown in Fig. 13.15b. Hsieh (1962) and Lysmer andRichart (1966) have provided a solution .The equation of vibration is:s P tvGrGzvrmz zo o sin1413.4 2 1Where ro = radius of the foundation (For non-circular foundations, appropriateequivalent radius may be used, see Eqs. 40-42).The equivalent spring for vertical vibrations is given byvGrk oz 142And the damping z c is given byGvrc oz .13.43The damping constant for vertical vibrations z is given byzzB0.4254In which z B is known as the modified mass ratio, given by3 .41oz rv mB 5The undamped natural frequency of vertical vibrations may now be obtained using Eqs. 6and 7.mkznz 6mkf znz 217In which nz = the circular natural frequency (undamped) of the soil foundation systemin vertical vibration (rad/sec) and nz f = natural frequency of vertical vibrations (Hz).The amplitude of vertical vibration is obtained as:2-Jan2 2 2 2 2 2 1 2 1 / 2 / z nz z nzzz zzzkPk r rPA 8Sliding vibrationsThe equation of the analog for sliding is (Fig. 6)mx c x k x P t x x z sin 9Figure 6. Sliding Vibrations of a Rigid Block (a) Actual case (b) Equivalent modelHall (1967) defined the modified mass ratio for sliding as:3 32 17 8ox prmvvB 10where ro = radius of the foundation .m mPxSintPxSintSoil stiffnessand dampingcxkxbaThe expressions for the equivalent spring and damping factors are as follows:The equivalent springx o Grvvk7 832 111And the equivalent dampingr Gvvc x o27 818.4 112The damping ratio x is given bye xxxc Bc 0.287513The undamped natural frequency of sliding vibration may be obtained as follows:mkxnx 14amkf xnx 2114bIn which nx = the circular natural frequency (undamped) in sliding vibrations and nx f =natural frequency of sliding vibrations (Hz).The damped amplitude in sliding is obtained as:2221 2nxxnxxxxkPA 15Rocking Vibrations: A rigid block foundation undergoing rocking vibrations due to anexciting moment M t y sin is shown in Fig. 7.Hall (1967) proposed an equivalent mass-spring-dashpot model that can be used todetermine the natural frequency and amplitude of vibration of a rigid circular footingresting on an elastic half-space and undergoing rocking vibrations (Fig.7). The equivalentmodel is given in equation 16M c k M t mo y sin 16In which k = spring constant for rocking, c = damping constant and mo M = massmoment of inertia of the foundation and machine about the axis of rotation through thebase.2 M M mL mo m 17Where m M = mass moment of inertia of foundation and machine about an axis passingthrough the centroid of the system and parallel to the axis of rotation and L = the heightof the centroid above the base.The terms k and c can be obtained as follows:vGrk o3 18 318Andv Br Gc o1 10.8 419in which 0 r = radius.B in Eq. 19 is known as the modified inertia ratio which obtained as follows:5 83 1omorv MB 20Figure 7. Rocking vibrations of a rigid block under excitation due to an applied momentThe damping factor is given byc B Bcc 10.1521The undamped natural frequency of rockingrad / secMkmon 22Damped amplitude of rocking vibrations A is given by Eq. 232221 2n nykMA 23Torsional vibrations: A block foundation undergoing torsional vibrations is shown inFig.8. Non-uniform shearing resistance is mobilized during such vibrations. The analogsolution for torsional vibrations is provided by Richart et al, (1970).Figure 8. Torsional vibrations of rigid block: (a) Block subjected to horizontal moment.(b) Development of nonuniform shear below the baseThe equation of motion isi tmz zM C k M e 24In which mz M = mass moment of inertia of the machine and foundation about the verticalaxis of rotation (polar mass moment of inertia). The spring constant k and the dampingconstant c are given by (Richart and Whitman, 1967):3316o k Gr 25Br Gc o11.6 426where ( ) o o r r = equivalent radius..The undamped natural frequency n of the torsional vibrations is given byrad / secMkmzn 27The amplitude of vibration A is given by2221 2n nzkMA 28In which the damping ratio is given by1 2B0.529The modified inertia ratio B is given by5omzrMB 30Combined rocking and sliding: The problem of combined rocking and sliding is shownschematically in Fig. 9. The equations of motion are written as:i tx x x x xmx c x k x Lc Lk P e 31i tm x x x x yM c L2C k L2k Lc x Lk x M e 32The undamped natural frequencies for this case can be obtained from Eq. 33.0. 2 222 24 nx nnnx nn 33In whichmomMM34Figure 9. Block subjected to the action of simultaneous vertical Pz(t), horizontal Px(t)forces and moment My(t)The damping in rocking and sliding modes will be different. Prakash and Puri (1988)developed equations for determination of vibration amplitudes for this case. Dampedamplitudes of rocking and sliding occasioned by an exciting moment y M can be obtainedas follows:22-Jan2 2 2 2 . .LMMA nx x nxmyx 35622-Jan2 2 2 2 2 nx x nxmyMMA 36The value of 2 is obtained from Eq. 382-Jan22 2 2 222 2 2 22 4 244nxnnnxxn nx x nx n nx n37Damped amplitudes of rocking and sliding occasioned by a horizontal force x P are givenby Eqs.38 and 3922-Jan22 2 2 2 2 1 M k L k 4 k M L k mmMPAm x mo x xmxx 38And2212 2 4 nx nx xmxMP LA 39In case the footing is subjected to the action of a moment and a horizontal force,the resulting amplitudes of sliding and rocking may be obtained by adding thecorresponding solutions from Eqs.35, 36, 38 and 39.Effect of shape of the foundation on its response: The solutions from the elastic halfspacetheory were developed for a rigid circular footing. The vibratory response for noncircularfoundations may be obtained using the concept of equivalent circular footing.The equivalent radius of the foundation for different modes of vibration is not the same.For vertical and sliding vibrations:2-Janabr r ro oz ox 40For rocking vibrations4-Jan33bar ro o 41Special Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras19For torsional vibrations4-Jan2 26ab a br ro o 42.Foundations on elastic layer: The elastic half-space solution is based on the assumptionof a homogenous soil deposit. In practice soils are layered media with each layer havingdifferent characteristics. An underlying rock below a soil layer may cause largemagnification of amplitude of vibration because of its ability to reflect wave energy backinto the soil supporting the foundation. Special care should be taken during design toovercome this effect.Embedded FoundationsThe embedment of the foundation results in an increased contact between the soiland the vertical faces of the foundation. This results in increased mobilization of soilreactions which now develop not only below the base of the foundation but also along thevertical sides of the foundation in contact with the soil. The overall stiffness offered bythe soil therefore increases. Similarly, more energy is carried away by the waves whichnow originate not only from the base of the foundation but also from the vertical faces ofthe foundation in contact with the soil. This results in an increased geometrical damping.The elastic half-space method for calculating the response of embedded foundations wasdeveloped by Novak and Beredugo (1971, 1972), Beredugo (1976), Novak and Beredugo(1972) and Novak and Sachs (1973) by extending the earlier solution of Baranov (1967).The solution is based upon the following assumptions:1) The footing is rigid.2) The footing is cylindrical.3) The base of the footing rests on the surface of a semi-infinite elastic half-space.4) The soil reactions at the base are independent of the depth of embedment.5) The soil reactions on the side are produced by an independent elastic layer lyingabove the level of the footing base.6) The bond between the sides of the footing and the soil is perfect.Based on the above assumptions, the expressions for equivalent spring and dampingvalues for different modes of vibrations were obtained. The soil properties below the baseof foundation were defined in terms of the shear modulus G, Poissons ratio v and themass density of the soil . The properties of the soil on the sides of the foundation weresimilarly defined in terms of shear modulus s G , the Poissons ratio s v and the massdensity s . The values of equivalent spring and damping for vertical, sliding, rocking andtorsional modes of vibrations were then obtained. The values of spring and damping werefound to be frequency dependent. However, it was found that within the range ofpractical interest, the equivalent spring and damping may be assumed to be frequencySpecial Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras20independent. This range was defined using a dimensional frequency ratio o a . Thedimensional frequency ratio is defined as:soo vra 43in which = operating speed of the machine in rad/sec.The values of equivalent frequency-independent spring and damping for theembedded foundation for the vertical, sliding, rocking and torsional modes are given inthe Tables 3 and 4. The vibratory response of the foundation may then be calculatedusing the appropriate equations as for the elastic half-space analog for the surfacefoundations after replacing the spring stiffness and damping values with thecorresponding values for the embedded foundations.The response of a foundation undergoing coupled rocking and sliding vibrationsmay similarly be calculated. However, some cross-coupling stiffness and damping termsappear in the analysis of embedded foundations according to the elastic half-spacemethod (Beredugo and Novak, 1972). The necessary equations for calculating thestiffness, damping, natural frequencies and amplitude of vibrations are summarized inTable 5.For a given size and geometry of the foundation, and the soil properties, the stiffness anddamping values for an embedded foundation are much higher than those for a surfacefoundation. The natural frequency of an embedded foundation will be higher and itsamplitude of vibration will be smaller compared to a foundation resting on the surface.Increasing the depth of embedment may be a very effective way of reducing the vibrationamplitudes. The beneficial effects of embedment, however, depend on the quality ofcontact between the embedded sides of the foundation and the soil. The quality of contactbetween the sides of the foundation and the soil depends upon the nature of the soil, themethod of soil placement and its compaction, and the temperature. Reduced values of soilparameters should be used for the soil on the sides of the foundation if any gap is likelyto develop between the foundation sides and the soil, especially near the ground surface.Impedance Function Method(Surface and Embedded Foundations)The dynamic response of a foundation may be calculated by the impedance functionmethod. (Gazettas 1983, 1991a, b, Dobry and Gazettas 1985) This method is brieflydiscussed here. The geometry of rigid massless foundation considered by Gazettas(1991b) is shown in Fig.10a for a surface foundation in Fig.10b for an embeddedfoundation. The response of this foundation due to a sinusoidal excitation can be obtainedfollowing theory of vibration after the appropriate dynamic impedance functions Sfor the frequencies of interest have been determined.The dynamic impedance is a function of the foundation soil system and the natureand the type of exciting loads and moments. For each particular case, of harmonicexcitation, the dynamic impedance is defined as the ratio between force (or moment) Rand the resulting steady-state displacement (or rotations) U at the centriod of the base ofthe massless foundation. For example, the vertical impedance is defines bySpecial Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras21U IR ISzzz 44In which R I R i t z z exp and is the harmonic vertical force; and U I U i t z z expharmonic vertical displacement of the soil-foundation interface. The quantity z R is thetotal dynamic soil reaction against the foundation and includes normal traction below thebase and frictional resistance along the vertical sides of the foundation.The following impedances may similarly be defined: y S = lateral sliding orswaying impedance (force-displacement ratio), for horizontal motion in the y- direction;x S = longitudinal swaying or sliding impedance (force-displacement ratio), for horizontalmotion along x-direction; rx S = rocking impedance (moment-rotation ratio), for rotationalmotion about the centroidal x-axis of the foundation base.Special Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras22Table 3. Value of equivalent spring and damping constants for embedded foundations (Beredugo and Novak 1972, Novakand Beredugo 1972, Novak and Sachs 1973)The values of frequencyindependent parameters sfor the elastic space aregiven in Table 4.The values of frequencyindependent parameters sfor the elastic space aregiven in Table 4.ro and h refer to radius anddepth of embedment of thefoundation respectivelyDampingratioEquivalentDamping constantEquivalentspringSpecial Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras23Mode ofVibrationVerticalSlidingRockingTorsionalor YawingSpecial Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras24Table4. Values of elastic half-space and side layer parameters for embedded foundations(Beredugo and Novak 1972, Novak and Beredugo 1972, Novak and Sachs 1973)Mode ofvibrationPoissonsratio vElastic half-spaceSide layerFrequencyindependentconstantparameterValidity rangeFrequencyindependentconstantparameterValidity rangeVertical 0.00.250.53.90 1 C3.50 2 C5.20 1 C5.00 2 C7.50 1 C6.80 2 C0 1.5 0 a(for all valuesof v)2.70 1 S6.7 2 S(for allvalues of v)0 1.5 0 a(for all valuesof v)Sliding 00.250.40.54.30 x1 C2.70 x2 C5.10 x1 C0.43 x2 C0 2.0 0 a0 2.0 0 a3.60 x1 S8.20 x2 S4.00 x1 S9.10 x2 S4.10 x1 S10.60 x2 S0 1.5 0 a0 2.0 0 a0 1.5 0 a0 2.0 0 a0 1.5 0 aRocking 0 2.50 1 C0.43 2 C0 1.0 0 a 2.50 1 S1.80 2 S(for anyvalue of v)0 1.5 0 aTorsionalor yawingAny value 4.3 1 C0.7 2 C0 2.0 0 a 12.4 1 S10.2 1 S2.0 2 S5.4 2 S0 2.0 0 a0.2 2.0 0 a0 2.0 0 a0.2 2.0 0 aSpecial Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras25Table5 Computation of response of an embedded foundation by elastic half-space methodfor coupled rocking and sliding (Beredugo and Novak 1972)ItemEquationStiffness insliding 1 x1osxe o x SrhGGk Gr CStiffness inrocking 2 2 12221 12133 xo o o ososxoe o SrhLrLrhrhGGSrhGGCrLk Gr CCrosscouplingstiffness1 1 2 xosx e o x ShLrhGGk Gr LCDampingconstant insliding2 22xs soxe o x SGGrhc Gr CDampingconstant inrocking2 2 22222 22243 xo o os so oxoe o SrhLrLrhSGGrhrhCrLc Gr CCrosscouplingdamping2 222 xs sox e o x ShLGGrhc Gr LCFrequencyequation0 2 2 2xe n e m n x e k m k M kAmplitude insliding(damped)22212221xe x A PAmplitude inrocking(damped)22212221e y A MVariousterms inequations forxe e A and Ax exye m kPMk M 21x exye cPMc 2x eyxxe kMPk m 21 x eyxxe cMPc 24 2 2 21 m e m xe xe e x e xe e x e mM mk M k c c c k k ke m xe xe e e xe x e x e mc M c c k c k 2c k 32The values of parameters 1 2 1 2 1 2 1 2 C ,C ,C ,C , S , S , S , and S x x x x are given in Table 4.L is the height of the centre of gravity above the base.The horizontal force x P and the moment y M act at the centre of gravity of the foundation.The equations given in this table are used for coupled rocking and sliding of embedded foundations only.Special Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras26ry S = rocking impedance (moment-rotation ratio), for rotational motion about the shortcentroidal axis (y) of the foundation basement; and I S = torsional impedance (momentrotationratio), for rotational oscillation about the vertical axis (z).In case of an embedded foundations, horizontal forces along principal axes inducerotational (in addition to translational) oscillations; hence two more cross-couplinghorizontal-rocking impedances exist x ry y rx S and S . They are negligible for surface andshallow foundations, but their effects may become significant as depth of embedmentincreases.Material and radiation damping are present in all modes of vibration. As a result Ris generally out of phase with U. It has become traditional to introduce complex notationand to express each of the impedances in the formS K i c 45in which both K and C are functions of the frequency . The real component, K is thedynamic stiffness, and reflects the stiffness and inertia of the supporting soil. Itsdependence on frequency is attributed solely to the influence that frequency exerts oninertia, since soil properties are practically frequency independent. The imaginarycomponent, C , is the product of the (circular) frequency times the dashpotcoefficient, C. C is the radiation and material damping generated in the system (due toenergy carried by waves spreading away from the foundation and energy dissipated in thesoil by hysteric action, respectively).Equation 45 suggests that for each mode of oscillation an analogy can be madebetween the actual foundation-soil system and the system thats consists of the samefoundation, but is supported on a spring and dashpot with characteristic moduli equalto K and C , respectively.Gazettas (1991a, b) presented a set of tables and figures for determination ofdynamic stiffness and damping for various modes of vibration of a rigid foundation asshown in Tables 6 and 7 and Figs. 11 and 12.Table 6 and Fig 11 contain the dynamic stiffness (springs), K K for surfacefoundations. Each stiffness is expressed as a product of the static stiffness, K, times thedynamic stiffness coefficient k k .K K.k 46Table 7 and Fig. 12 similarly give the information for an embedded foundation. Tables 6and 7 and Figs. 11 and 12 contain the radiation damping (dashpot) coefficients,C C . These coefficients do not include the soil hysteric damping . To incorporatesuch damping, one may simply add the corresponding material dashpotconstant 2K / to the radiation C value.Ktotal C radiation C247Special Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras27Gazettas (1991a, b) has also illustrated the procedure for calculating the response of thefoundation using the impedance method. The solutions have also been developed for arigid footing resting or partly embedded into a stratum (Gazettas, 1991a).Figure10 Foundations of arbitrary shape (a) surface foundation, (b) embeddedfoundation (Gazettas 1991b)Special Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras28Table 6. Dynamic stiffness an damping coefficients for foundation of arbitrary shape resting on the surface ofhomogeneous half-space (Gazettas 1991b)Radiation dashpot coefficient, c-4whereis plotted in Fig. 11cwhereis plotted in Fig.11dwhereis plotted in Fig.11ewhereis plotted in Fig.11fwhereis plotted in Fig. 11gEquivalent spring for the surface footing for any mode of vibration can be obtained by multiplying he values of K in col. 2 with the corresponding values of k incol. 3.Values of K in col. 2 and k in col.3 of this table are for calculating the equivalent soil springs by the impedance method only.L, B and Ab are defined in Fig. 10. Ibx,, Iby and Ibz represent the moment of inertia of the base area of the foundation about x, y and z-axis respectively.is the apparent velocity of propagation of longitudinal waves.Dynamic stiffness coefficient, k-3is plotted inFig. 11ais plotted inFig. 11bStatic stiffness, k-2WithSpecial Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras29Vibrationmode (1)Vertical (z)Horizontal (y)(lateraldirection)Horizontal (x)(longitudinaldirection)Rocking (rx)about thelongitudinalaxis, x-axisRocking (rx)about thelateral, y-axisTorsion (t)Special Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras30Table 7. Dynamic stiffness and damping coefficients of foundations of arbitrary shape embedded in half-space (Gazettas 1991b)Radiation dashpot coefficient,-4Where is obtained formTable 4 and the associated chart of Fig. 11whereareobtained from Table 6 and the associatedchart of Fig. 11whereCry,emb is similarly evaluated from Cryafter replacing x by y and interchanging Bwith L in the foregoing two expressions.In both casesDynamic stiffness coefficient,-3:Fully embedded:In a trench::Fully embedded with L/B = 1 2:fully embedded with L/B > 3:in a trench:from table 6All v, partially embedded: interpolatecan be estimated in terms ofL/B, D/B and d/b for each ao value of fromthe plots in Fig 12The surface foundation krx and kry areobtained from Table 6Static stiffness,-2Where is obtained fromTable 4. =actual sidewall-soil contactarea; for consultant effective contactheight, d, along the perimeter:whereand areobtained from Table 6whereand areobtained from Table 6Special Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras31Vibrationmode-1Vertical(z)Horizontal(y) and (x)Rocking(rx) and(ry)Special Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras32Table 7 continuedwhere is obtained from Table 6 andFigure 11Equivalent soil spring for the embedded foundation for any mode of vibration is obtained by multiplying the values of kemb in col.2 with the corresponding values of kemb in col.3The Kemb and kemb given in cols. 2 and 3 respectively in this table are for calculating the equivalent soil springs by the impedance method only.L, B, D, d, Ab and Aw are define in Figure 10Ibx, Iby and Ibz represent the moment of inertia of the base area of the foundation about x, y and z axis respectively.is the apparent velocity of propagation of longitudinal waves.where isobtained from Table 6Swayingrocking(x-ry) x (yrx)Torsion (r)Special Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras33Figure 11. Dimensionless graphs for determining dynamic stiffness and damping coefficients ofsurface foundations (accompanying Table6) (Gazettas, 1991b)Special Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras34Figure 12. Dimensionless graphs or determining dynamic stiffness coefficients of embeddedfoundations (accompanying Table 7) (Gazettas, 1991b)Special Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras35COMPARISON OF PREDICTED AND OBSERVED RESPONSEVery little information is available on comparison of measured response of machine foundationswith theory. Such comparisons will increase the confidence of the designer. Richart andWhitman (1967) compared model footing test data with calculated values using the spring anddamping obtained from the elastic half-space analog. The computed amplitudes of verticalvibrations were in the range of 0.5 to 1.5 times the observed values. Prakash and Puri (1981),however found that somewhat better agreement between computed and observed amplitudes ispossible if the soil properties are selected after accounting for the effect of significant parameterssuch as mean effective confining pressure and strain amplitude.. Based on the results of thesmall-scale field experiments, Novak (1985) pointed out that the elastic half-space theoryoverestimates damping. Variation of soil properties and the presence of a hard stratum alsoinfluence the response of the footing. Adequate geotechnical investigations are necessary beforemeaningful comparisons of computed and predicted response can be made (Dobry and Gazettas1985, Novak 1985).Prakash and Puri (1981) compared the observed and computed response of areciprocating compressor foundation which was undergoing excessive vibrations. The analysis ofthe compressor foundation was performed using the linear weightless spring method and also theelastic half-space analogs using soil properties for the as-designed condition and correspondingto the observed vibration amplitudes. The computed amplitudes by both the methods were far inexcess of the permissible amplitudes as per manufacturers specifications. The computed naturalfrequencies were found to be within about 25% of the observed natural frequencies in horizontalvibrations. Adequate soil exploration and a realistic determination of soil constants play animportant role in the design of machine foundations.Dobry et al, (1985) compared the observed response of model footing of different shapeswith predictions made using the method proposed by Dobry and Gazettas (1985) for dynamicresponse of arbitrarily shaped foundations. They observed a strong influence of the footing shapeon the stiffness and damping values. Gazettas and Stokoe (1991) compared results of 54 freevibration tests of model footing embedded to various depths in sand with theory. The modelfooting had rectangular, square and circular shapes. They observed that for the case of verticalvibrations and coupled rocking and sliding vibrations, the theory predicts reasonable values ofdamped natural frequencies provided the effective shear modulus is realistically chosen.Manyando and Prakash (1991) reanalyzed the earlier data on circular footings ( Fry 1963)considering nonlinearity of soil that is by using the values shear modulus corrected the effectmean effective confining pressure and shear strain induced in soil by the footing. Their analysisis essentially based on the concept of elastic-half space analogs with modifications made fornonlinearity of soil.The shear strain z induced in the soil due to vertical vibrations was defined asbelow:BA2max 48in which, Amax = amplitude of vertical vibrations and B= width of the foundationSpecial Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras36Shear strain for torsional vibrations was considered to be equal to the rotational displacement atthe edge of the base of the surface footing divided its radius. The shear strain for coupledrocking and sliding vibrations was considered as the rotation about the lateral axis of vibrationthrough the combined center of gravity of the machine foundation system. The response of thesurface footings was then predicted using equations 7,8, 14,15, 22,23,27 and 28 depending on theappropriate vibration mode and following an iterative procedure to account for the nonlinearityof soil. The effect of damping was also included in computations.Typical results comparing the predicted and observed response of foundations for vertical,torsional and coupled rocking and sliding modes of vibrations are shown in Figs 13,14 and 15respectively.Figure 13. Measured and predicted response of vertical vibration for different values ofeccentricity (a) e = 0.105 and (b) e = 0.209 inches, Eglin, base 1-1 ( Manyando & Prakash1991)Special Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras37Figure 14. Measured and predicted response of torsional vibrations for different values ofeccentricity (a) e = 0.105 and (b) e = 0.209 inches, Eglin, base 1-1( Manyando & Prakash 1991)Figure 15. Measured and predicted response of couples rocking and sliding vibrations fordifferent values of eccentricity (a) e = 0.105 and (b) e = 0.209 inches, Eglin, base 1-1( Manyando& Prakash 1991 )Figure 13 presents a comparison of the measured and computed response for the case of verticalvibrations . The general trend of the measured and computed response curves in Fig 13 ( a,b) issimilar. The predicted natural frequency of vertical vibration for the foundation under discussionshows good agreement with the measured natural frequency. Similar trend of data is observed forthe case of torsional (Fig. 14, a and b) and for coupled rocking and sliding (Fig. 15, a and b). Thecomputed amplitudes in all the cases are within about 20 to 50 % of the measured amplitudes.Special Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras38Manyando and Prakash 1991) also investigated the role of geometrical and material damping onthe comparison between measured and computed response. Based on their study it seems thatnatural frequencies are reasonably predicted by their model but more work is needed as for asprediction vibration amplitude is concerned. Prakash and Puri (1981) made a similar observation.Prakash and Tseng (1998) used frequency dependent stiffness and damping values to determinethe response of vertically vibrating surface and embedded foundations. They compared thecomputed response with the reported data of Novak(1970). They observed that the radiationdamping obtained from the elastic half space theory is generally over estimated and suggestedfactors for modification radiation damping..SUMMARYThe methods for determination dynamic response of machine foundations subjected to harmonicexcitation have been presented. Analogs based on the elastic half-space solutions are commonlyused for their simplicity. The soil stiffness is generally considered frequency independent fordesign of machine foundations. Observations by several investigators have shown that the elastichalf-space analog generally overestimates radiation damping. The impedance function method isa recent addition to the approaches available for design of machine foundations. The embedmentof a foundation strongly influences its dynamic response.REFERENCES:Baranov, V.A. (1967). On the calculation of excited vibrations of an embedded foundation (inRussian) Vopr, Dyn. Prochn., 14:195-209Bakan, D.D.,(1962) Dynamics of Bases and Foundations. McGraw Hill NYBeredugo, Y.O. (1976). Modal analysis of coupled motion of horizontally excited embeddedfootings, Int. J. Earthquake Engg. Struct. Dyn., 4: Q3-410Beredugo, Y.O. and M. Novak. (1972). Coupled horizontal and rocking vibrations of embeddedfootings, an. Geotech. J., 9(4):477-497Blake, M.P.(1964), New Vibration Standards for Maintenance. Hydrocarbon ProcessingPetroleum Refiner, Vol.43 , No.1, pp 111-114.Dobry, R. and G. Gazettas. (1985). Dynamic stiffness and damping of foundations by simplemethods., Proc. Symp, Vib, Probs, Geotech. Engg., ASCE Annu. Conv., Detroit, pp. 75-107Dobry, R., G. Gazettas and K.H. Stokoe, (1985). Dynamic response of arbitrary shapedfoundations: Experimental verification. ACSE, J. Geotech. Engg. Div., 112(2): 126-154Fry, Z.B.(1963) Development and Evaluatin of Soil Bearing Cpacity, Foundations forStructures, Field Vibratory Test Data, Technical Research Report #3-632, U.S. Army EngineerWaterways Experiment Station, Test Report #1, Vicksburg, Mississippi, JulySpecial Issue, April-May 2006Of the Journal of Structural Engineering,SERC, Madras39Gazettas, G. (1983). Analysis of machine foundation vibrations, state of art. Soil Dyn.Earthquake Engg., 2(1):2-42Gazettas, G. (1991a). Foundation vibrations. In Foundation Engineering Handbook, 2nd ed.,Chap. 15, Van Nostrand Reinhold, NewYork, pp. 553-593Gazettas, G. (1991b). Formulas and charts for impedances of surface and embedded foundations.ACSE, J. Geotech. Engg. Div., 117(9):1363-1381Hall, J.R. (1967). Coupled rocking and sliding oscillations of rigid circular footings. Proc. Int.Symp. Wave Propag. Dyn. Prop. Earth Matter, University of New Mexico, Albuquerque,New Mexico, pp. 139-148Hsieh, T.K. (1962). Foundation vibrations. Proc. Inst. Civ. Eng., 22:211-226., M.S. Snow, N. Matasovic, C. Poran and T. Satoh (1994). Non-intrusive Rayleigh waveLysmer, L. and F.E. Richart, Jr. (1966). Dynamic response of footing to vertical loading. J. SoilMech. Found. Div., ACSE, 92(SM-1):65-91 Major, A. (1980). Dynamics in CivilEngineering, Vol. I-IV, Akademical Kiado, Budapest.Manyando, George, M.S. and S. Prakash. (1991). On prediction and performance of machinefoundations. 2nd Int. Conf. on Recent advances in Soil Dynamics, St. Louis, University ofMissouri-Rolla. Vol. 3, pp. 2223-2232Novak, M. (1985). Experiments with shallow and deep foundations. Proc.Symp. Vib. Probl.Geotech. Eng. ACSE, Annu. Conv. Detroit (ACSE New York), pp. 1-26Novak, M. and Y.O. Beredugo. (1971). Effect of embedment on footing vibrations, Proc. An.Conf. Earthquake Eng. 1st, Vancouver, (Con. Soc. Eq. Engg., Ottawa), pp. 111-125Novak, M. and Y.O. Beredugo. (1972). Vertical vibration of embedded footings, J. Soil Mech.Found. Div., ACSE, 98(SM-12): 1291-1310Novak, M. and K. Sachs. (1973). Torsional and coupled vibrations of embedded footing. Int. J.Earthquake Engg. Struct, Dyn., 2(1): 11-33Prakash, S. (1981). Soil Dynamics. Mc.Graw-Hill Book Co., New York and SP Foundation,Rolla, MOPrakash, S. and V.K. Puri. (1981). Observed and predicted response of a machine foundation.Proc. 10th Int. Conf. Soil. Mech. Found. Eng., Stockholm (Sweden), Vol. 3, pp. 269-272,A.A. Balkema, RotterdamPrakash, S. and V.K. Puri. (1988). Foundations for machines: Analysis and design. John Wileyand Sons, New YorkPrakash, S. and Tseng, Y., (1998), Prediction of Vertically Vibrating Foundation Responsewith Modified Radiation Damping, Pro. 4th Int. Conf. on Case Histories inGeotechnical Engineering, St. Louis, Mo, pp. 630-648.Richart, F.E., Jr. (1962). Foundation Vibrations. Transactions ASCE, Vol. 127, Part 1,pp. 863-898.Richart, F.E., Jr. (1977). Dynamic stress-strain relations for soils, state-of-the-art report. Proc.Int. Conf. Soil. Mech. Found. Eng., 9th, Tokyo (Jap. Soc. SMFE, Tokyo), Vol.2, pp. 605-612Richart, F.E., Jr., J.R. Hall and R.D. Woods. (1970). Vibrations of Soils and Foundations.Prentice Hall, Englewood Cliffs, New Jersy.Richart, F.E., Jr. and R.V. Whitman. (1967). Comparison of footing vibrations tests with theory., J. Soil Mech. Found. Div., ACSE, 93(SM-6): 143-168

INFO****INFORMATION NEEDED FOR DESIGNThe following information is required and must be obtained for design of a machine foundation:1. Static weight of the machine and accessories.2. Magnitude and characteristics of dynamic loads imposed by the machine operationand their point of application3. The soil profile of the site and dynamic soil properties such as dynamic shearmodulus and damping4. Trial dimensions of the foundation. These are generally supplied by the manufacturer.This will give the total static weight.5. An acceptable method of analysis i.e., a mathematical model to determine theresponse of the foundation-soil system6. A criteria for adequate designfrequency within .40 to 1.57 TYPE OF MACHINE :amplitude withinallowable limits 1. IMPACT 2. RECIPROCATING 3. ROTARY 4. STEAM TURBO GENERATOR****The failure condition of a machine foundation is reached when its motion exceedsa limiting value which may be based on acceleration , velocity or amplitude. **** if the frequency ratio is managed to remain outside the critical range of 0.4 and 1.5 and if the amplitude lies within the allowance limits then damaging/harmful effects may be successfully eliminated, especially when the system us damped. Transfer or transmission of vibration may be controlled and damaging effects can be successfully reduced to considerable amounts by isolation of the source called active isolation or by protecting absorber or receiver called passive isolation.Figure 3. Limiting amplitudes of vibrations for a particular frequency. (Blake, 1964)****Criterion for vibration of rotating machinery. Explanation of classes :AAAA Dangerous. Shut it down now to avoid dangerAA Failure is near. Correct within two days to avoid breakdown.BB Faulty. correct it within 10 days to save maintenance dollars.CC Minor faults. Correction wastes dollars.DD No faults. Typical new equipment.This is guide to aid judgment, not to replace it. Use common sense. Take account of alllocal circumstances. Consider: safety, labor costs, downtime costs. (after Blake, 1964.)Reproduced with permission from Hydrocarbon Processing, January 1964.To have a perfect and reliant designs and constructions of machine foundation one has to fulfill certain specific criteria listed as follows:*Similar to ordinary/regular type foundation, the machine foundation have to be safe and secure against the shear failure resulted from superimposed loads and also the settlement should remain within the safer limits. The soli pressure strictly should not cross the permissible pressure for static loading.*The possibility of resonance shall not be tolerated or encountered. The natural frequency of foundation needs to be either more than or lesser than the operating/working frequency of the machine.*The amplitudes under service condition have to remain within the allowable limits for the machine.*The combination/summation of centre of gravity of the machine and the foundation should remain on the vertical line passing through the center of gravity of the base plane.*The machine foundation has to be laid lower than the level of the foundation of adjacent building and have to be separated properly.*The vibrations occurred or produced strictly should not annoy/trouble or distract to the persons nearby not detrimental to other objects/structures nearby. For this one can use the Richarts charts, which was developed during 1962 to determine the design limits of vertical vibrations. It is used as guideline to determine the numerous limits of frequency and amplitudes.*The presence of ground-water table has to be at minimum of one-fourth of the foundation width below the base plane.

Sheet1Vibrationsare responsible for causing harmful effects to the adjoining structures, foundation and machines leading to devastating damages. Additionally, the dynamic vibrations also cause annoyance or distraction to the people working or accessing near the machine operation area. However, if the frequency ratio is managed to remain outside the critical range of 0.4 and 1.5 and if the amplitude lies within the allowance limits then damaging/harmful effects may be successfully eliminated, especially when the system us damped. Transfer or transmission of vibration may be controlled and damaging effects can be successfully reduced to considerable amounts by isolation of the source called active isolation or by protecting absorber or receiver called passive isolation.The measures employed are as follows;By placement or location of machine foundation far away from adjoining structures, the process is called as geometric isolation. Mechanism is that with the increase in distance between machine foundation and adjoining structure the amplitude of surface waves (R-waves) gets reduced. For this follow the ratio of foundation mass to engine Extra masses called as dampers may be attached/added to the foundation encountering higher frequency machines in order to form a multiple degree freedom system plus for altering the natural frequency. mass. The specific decreasement in the amplitude can be achieved by placing the foundation to a greater depth since the R-waves tends to reduce successfully due to increase in depth. In case of reciprocating machines, considerable amount of vibrations can be reduced by counterbalancing the exciting forces through means of attachment of counterweights at the crank sides.The placement or use of absorbers helps to reduce the vibrations considerably. Absorbers may be rubber mountings, felts and corks applied amid the base and machine. Another technique can be by attaching an auxiliary mass along with a spring to the machine foundation such that the system gets transformed to two-degree-freedom system. Usable when the system is at resonance. The strength of soil may be improved via chemical or cement stabilization technique, which enhances the natural frequency of system. This technique is preferable in machines with operating frequencies. The propagation or fluctuation of waves can also be lowered by use of sheet piles, screens and trenches.

Sheet2To know the efficient constructions of machine foundations it is very essential to have brief, effective ideas regarding the types of machines based on speed. Principally there are three sorts of machines:Machines producing periodic unbalanced force, like the reciprocating engines and compressors. The speed of these machines is usually lower than 600 r.p.m. In these types of machines the rotary motion of the crank is transformed into the translatory motion, here the unbalanced force varies sinusoidally.Machines producing impact loads, like the forge hammers and punch presses. In such machines the dynamic force/vibrations attains a peak limit in very short instant and then expires out steadily/gradually. Here the response is a pulsating curve which vanishes before occurrence of next pulse. The speed seen is generally between 60-150 blows per minute.Lastly the high speed machines, like turbines and rotary machines. These machine rotating speeds are very high sometimes even higher than 3000 r.p.m.