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.