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Transcript of Foundation 8
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O R I G I N A L P A P E R
Dynamic Response of Machine Foundation on Layered Soil:
Cone Model Versus Experiments
P. K. Pradhan A. Mandal D. K. Baidya D. P. Ghosh
Received: 21 May 2006 / Accepted: 24 February 2008 / Published online: 22 March 2008
Springer Science+Business Media B.V. 2008
Abstract This paper presents the experimental
validation of analytical solution based on cone model
for machine foundation vibration analysis on layered
soil. Impedance functions for a rigid massless circular
foundation resting on a two layered soil system
subjected to vertical harmonic excitation are found
using cone model. Linear hysteretic material damping
is introduced using correspondence principle. The
frequency-amplitude response of a massive founda-
tion is then computed using impedance functions. To
verify the solution field experiments are conducted intwo different layered soil systems such as gravel layer
over in situ soil and gravel layer over concrete slab
(rigid base). A total 72 numbers of vertical vibration
tests on square model footing were conducted using
Lazan type mechanical oscillator, varying the influenc-
ing parameters such as depth of top layer, static weight
of foundation and dynamic force level. The frequency-
amplitude response in general and in particular the
resonant frequencies and resonant amplitudes predicted
by cone model is compared with the results of
experimental investigation, which shows a close agree-
ment. Thus the cone model is reliable in its application
to machine foundation vibration on layered soil.
Keywords Cone model In-situ test Layered soil Machine foundation Resonant amplitude Resonant frequency Wave propagation
Notations
a0 Dimensionless frequency (xr0/cs)
B Nondimensional modified mass ratio
b0 Nondimensional mass ratio
c, c0 Appropriate wave velocity in top and bottom
soil layers respectively
c(a0) Normalized damping coefficient
cp, c0p Dilatational wave velocity in top and bottom
soil layers, respectively
cs c0s Shear wave velocity in top and bottom soil
layers respectively
d Depth of the soil layer
G, G0 Shear modulus of top and bottom soil layers
respectively
K Static stiffness coefficient on homogeneous
half-space
P. K. Pradhan (&)
Department of Civil Engineering, University College
of Engineering, Burla 768018, India
e-mail: [email protected]
A. Mandal D. K. Baidya D. P. GhoshDepartment of Civil Engineering, Indian Instituteof Technology, Kharagpur 721302, India
A. Mandal
e-mail: [email protected]
D. K. Baidya
e-mail: [email protected]
D. P. Ghosh
e-mail: [email protected]
123
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DOI 10.1007/s10706-008-9181-8
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"Ka0 Dynamic impedancek(a0) Normalized stiffness coefficient
m Mass of the foundation or total vibrating
mass (mass of foundation plus machine) in
case of machine foundation
me Unbalanced mass (on machine)
Dm Trapped mass
P0 Harmonic interaction force
Q Harmonic force on foundation
|Q| Force amplitude on the foundation
r0 Radius of circular foundation or radius of
equivalent circle for non circular foundation
u Harmonic displacement for the layered soil
at depth z
u0 Harmonic surface displacement for the
layered soil
"u Harmonic displacement at depth z for
homogeneous half-space
"u0 Harmonic surface displacement for
homogeneous half-space
|u0| Displacement amplitude for the layered soil
Greeks
h Angle for setting eccentricity in the oscillator
x Circular frequency of excitation
l Trapped mass coefficient
n,
n0Hysteretic material damping ratio of top and
bottom soil layers respectively
q,q0
Mass density of top and bottom soil layersrespectively
m, m0 Poissons ratio of top and bottom soil layers
respectively
1 Introduction
The determination of resonant frequency and reso-
nant amplitude of foundations has been a subject of
considerable interest in the recent years, in relation to
the design of machine foundations. One of the keysteps in the current methods of dynamic analysis of a
foundation soil system to predict resonant frequency
and amplitude under machine type loading is to
estimate the dynamic impedance functions of an
associated rigid but massless foundation, using a
suitable method of dynamic analysis. Over the years a
number of methods have been developed for foun-
dation vibration analysis, the extensive reviews of
which are presented in Gazetas (1983).
The cone model was originally developed by
Ehlers (1942) to represent a surface disk under
translational motions and later for rotational motion
(Meek and Veletsos 1974; Veletsos and Nair 1974).
By comparison to rigorous solutions, the cone models
originally appeared to be such an oversimplification
of reality that they were used primarily to obtainqualitative insight. For example, the surprising fact
that the cones are dynamically equivalent to an
interconnection of a small number of masses, springs,
and dashpots with frequency-independent coefficients
encouraged a number of researchers to match discrete
element representation of exact solutions in fre-
quency domain by curve fitting (Veletsos and Verbic
1973; Wolf and Somaini 1986; de Barros and Luco
1990). Proceeding in another direction, Gazetas
(1987); Gazetas and Dobry (1984) employed wedges
and cones to elucidate the phenomenon of radiationdamping in two and three dimensions. Later Meek
and Wolf (1992a) presented a simplified methodol-
ogy to evaluate the dynamic response of a base mat
on the surface of a homogeneous half-space. The
cone model concept was extended to a layered cone
to compute the dynamic response of a footing or a
base mat on a soil layer resting on a rigid rock, Meek
and Wolf (1992b) and on flexible rock, Wolf and
Meek (1993). Meek and Wolf (1994) performed
dynamic analysis of embedded footings by idealizing
the soil as a translated cone instead of elastic half-space. Wolf and Meek (1994) have found out the
dynamic stiffness coefficients of foundations resting
on or embedded in a horizontally layered soil using
cone frustums. Also Jaya and Prasad (2002) studied
the dynamic stiffness of embedded foundations in
layered soil using the same cone frustums. The major
drawback of cone frustums method as reported by
Wolf and Meek (1994) is that the damping coefficient
can become negative at lower frequency, which is
physically impossible. Pradhan et al. (2003, 2004)
have computed dynamic impedance of circular foun-dation resting on layered soil using wave propagation
in cones, which overcomes the drawback of the above
cone frustum method. The details of the use of cone
models in foundation vibration analysis are summa-
rized in Wolf (1994) and Wolf and Deeks (2004).
During the last 30 years significant developments
has been made in the analytical solutions to the
problems of foundation vibration. But the experi-
mental verification of such theories remains essential
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prerequisite for their adoption and reliable applica-
tion in practice. Attempts have been taken in the past
to verify theoretical solutions by conducting labora-
tory or field tests (Sridharan et al. 1990; Crouse et al.
1990; Gazetas and Stokoe II 1991; Baidya and
Muralikrishna 2001; Baidya and Sridharan 2002;
Mandal and Baidya 2004; Baidya and Rathi 2004).Gazetas and Stokoe II (1991) have stated different
types of experimental investigation citing their
advantages and limitations. In the above paper the
researchers have recommended to use the results of
case studies and field experiments for the purpose
taking in to consideration the complexities of the soil
medium.
For foundation vibration analyses simple models,
which fit the size and economics of the project and
require no sophisticated computer code are better
suited. For instance the cone models, which provideconceptual clarity with physical insight and is easier for
the practicing engineers to follow. To the best of
authors knowledge no literature is available with
regard to the experimental verification of cone model
for its reliable application to the analysis of foundation
vibration. Hence in the present study it is proposed to
verify the applicability of cone model for layered soil to
the problem of machine foundation vibration. A total
72 numbers of field tests are conducted on two different
layered soil systems with variation of influencing
parameters. The model predicted frequency-amplituderesponse is thoroughly compared with the results of
field tests. In particular, the predicted resonant fre-
quencies and resonant amplitudes are compared
quantitatively with experimental results.
2 Problem Statement
A rigid massless circular foundation of radius r0resting on a two-layered soil system is addressed for
vertical degree of freedom (Fig. 1). The top layerwith depth dhas the shear modulus G, Poissons ratio
m, mass density q and hysteretic damping ratio n. The
underlying half-space has the shear modulus G0,
Poissons ratio m0, mass density q0 and hysteretic
damping ratio n0. The interaction force P0 and the
corresponding displacement u0 are assumed to be
harmonic. The layer interface can also be considered
fixed. The dynamic impedance of the massless
foundation (disk) is expressed by
"Ka0 P0
u0
Kka0 ia0ca0 1
where "Ka0 dynamic impedance, k(a0) = normal-ized spring coefficient, c(a0) = normalized damping
coefficient, a0 = xr0/cs, dimensionless frequency
with cs ffiffiffiffiffiffiffiffiffi
G=qp
, shear wave velocity of the top
layer and K= 4Gr0/(1-m), static stiffness coefficient
of the disk on homogeneous half-space with material
properties of the top layer.
Using the equations of dynamic equilibrium, the
dynamic displacement amplitude of the foundation
with mass m and subjected to a vertical harmonic
force Q is expressed as
u0j j Q
Kka0 ia0ca0 Ba20
2
Where |u0| = dynamic displacement amplitude under
the foundation resting on layered soil, |Q| = force
amplitude, B 1m4
b0 , the modified mass ratio with
b0 m
qr30
, the mass ratio.
In general, |Q| can be assumed to be constant or
equal to meex2 which is generated by the eccentric
rotating part in machine, where me is the eccentric
mass, e is the eccentricity and x is the circularfrequency.
3 Wave Propagation in Cones
Figure 2a shows wave propagation in cones beneath
the disk of radius r0 resting on a two-layered soil
under vertical harmonic excitation, P0. The dilata-
tional waves emanate beneath the disk and propagate
G d
0r Massless circularfoundation
0u
0P
G
Half-space
Fig. 1 Massless foundation on layered soil under vertical
harmonic interaction force
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at velocity c equal to the dilatational wave velocity cpfor m B 1/3 and twice the shear wave velocity, cs for
1/3\ m B 1/2. These waves reflect back and forth at
layer interface and free surface, spreading and
decreasing in amplitude. Let the displacement of
the (truncated semi-infinite) cone be denoted as "u
with the value"u0 under the disk Fig. 2b, modeling a
disk with same load P0 on a homogeneous half-space
with the material properties of the top layer. The
parameters of cone model shown in Fig. 2b are given
in Table 1. This displacement "u0 is used to generate
the displacement of the layer u with its value at
surface, u0. Thus, "u0 can also be called as the
generating function. The first downward wave prop-
agating in a cone with apex 1 (height z0 and radius of
base r0), which may be called as the incident wave
and its cone will be the same as that of the half-space,
as the wave generated beneath the disk does not knowif at a specific depth an interface is encountered or
not. Thus the aspect ratio defined by the ratio of the
height of cone to the radius of the disk (z0/r0) is made
equal for cone of the half-space and first cone of the
layered soil. Since the incident wave and subsequent
reflected waves propagate in the same medium (top
layer), the aspect ratio of the corresponding cones
will be same. Thus knowing the height of the first
cone, from the geometry, the height of other cones
corresponding to subsequent upward and downward
reflected waves are found as shown in Fig. 2a. Thedisplacement amplitude of the incident wave propa-
gating in a cone with apex 1, which is inversely
proportional to the distance from the apex of the cone
and expressed in frequency domain as
"uz; x z0
z0 zeix
zc "u0x 3
The displacement of the incident wave at layer
interface equals
"
ud; x
z0
z0 deixd
c "
u0x 4
Enforcing a reflection coefficient a(x) at the inter-
face, the displacement of the first reflected upward
wave propagating in a cone with apex 2 (vide Fig. 2a)
equals
az0
z0 2d zeix
2dzc "u0x 5
At the free surface the displacement of the upward
wave derived by substituting z = 0 in Eq. 5 equals
az0
z0 2deix
2dc "u0x 6
Enforcing compatibility of the amplitude and of
elapsed time of the reflected waves displacement at
the free surface, the displacement of the downward
wave propagating in a cone with apex 3 is obtained as
az0
z0 2d zeix
2dzc "u0x 7
In this pattern thewavespropagatein their own cones and
their corresponding displacements are found. The result-
ing displacement in the layer is obtained by superposing
all the down and up waves (up tojth impingement at layer
interface) and is expressed in the following form
uz; x z0e
ixzc
z0 z"u0x X
1
j1
aj
z0eix 2
jdzc
z0 2jd z
z0eix 2
jdzc
z0 2jd z
" #"u0x 8
At the free surface the displacement of the foundation
is obtained by setting z = 0 in Eq. 8 as
u0x uz 0; x
"u0x 2X1j1
aj
1 2jdz0
eix2jd
c "u0x 9
u0x X1j0
EFj eix2jd
c
"u0x 10
with EF0 1 11a
and for j ! 1; EFj 2aj
1 2jdz0
11b
EjF
can be called as echo constant, the inverse of
sum of which at x = 0 gives the static stiffness of the
layered soil normalized by the static stiffness of the
homogeneous half-space with material properties ofthe top layer.
3.1 ReflectionRefraction at Layer Interface
The waves occurring at layer interface are addressed
in Fig. 3. In the frequency domain the incident wave
f(x) propagating downwards in the cone with apex 1
(material properties of top layer: c appropriate wave
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velocity, and q mass density), yields a reflected wave
g(x) propagating upwards in cone segment with apex
3 (same material properties of top layer c, q) and a
refracted wave h(x) propagating downwards in the
cone with apex 2 (material properties of lower half-
space c0, q0). Based on wave propagation in beams
with varying area reflection coefficient a(x) for the
translational cone is given by
ax gx
fx
qc2
z0d q
0 c02
z00
ixqc q0c0
qc2
z0d q
0 c02
z00
ixqc q0c012
z0+(2j-1)d
z0+ 2jd
2j
2j+1
z0+ d
z0+ 3d
d
4
2
1
3
z0
z0+ 2dr0
P0
u0u
z
z u0
u
P0
r0z0
1
(a)
(b)
Fig. 2 (a) Wave
propagation in cones for
layered soil, (b) Cone
model for the half-space
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where; hx 1 axfx 13
Under special case when the layer interface isfixed, i.e. the lower layer is perfectly rigid, no
refracted wave is created, and reflected wave is equal
to the incident wave with a change in sign. Thus,
setting c0 = ? in Eq. 12 yields
ax 1 14
which leads to
gx fx 15
hx 0 16
Analogously, when the interface corresponds to a
free surface (c0 = 0)
ax 1 17
leading to
gx fx 18
3.2 Dynamic Impedance
The interaction force displacement relationship for a
massless disk resting on homogeneous half-space
using the cone model can be written as
P0x K Dmx2 ixC"u0x 19
where, K- D
mx
2=
spring coefficient andC= dashpot coefficient Dm is the trapped mass and
is given by
Dm lqr30 20
with trapped mass coefficient l, the values of which
recommended by Wolf (1994) are given in Table 1.
The trapped mass Dm is introduced in order to match
the stiffness coefficient of the cone model with
rigorous solutions for incompressible soil i.e., 1/
3\ m B 1/2, Wolf (1994). After simplification Eq. 19
reduces to the form
P0x K 1 l
p
z0r0
c2x2 ix
z0
c
"u0x 21
Using Eq. 10 in Eq. 21, the interaction force
displacement relationship for the layered soil system
reduces to
P0x K1 l
pz0r0
c2x2 ix z0
cP1j0 E
Fj e
ix 2jd
c u0x 22
Substituting echo constant given by Eq. 11 in Eq. 22,
the dynamic impedance equals
"Kx P0x
u0x K
1 lp
z0r0c2
x2 ix z0c
1 2P1
j1a
j
12jd
z0
eix2jd
c 23
In the expression of the dynamic impedance "Kxgiven by Eq. 23, the summation of series over jis
worked out up to a finite term as the displacement
amplitude of the waves vanish after a finite number of
Table 1 Parameters of semi-infinite cone modeling a disk on
homogeneous half-space under vertical motion, Wolf (1994)
Cone parameters Parameter expressions
Aspect ratio z0r0
p
41 m
c
cs
2
Static stiffness coefficient Kqc2pr2
0
z0
Normalized spring coefficient k(a0) 1 l
p
z0
r0
c2sc2
a20
Normalized damping coefficient c(a0)z0
r0
cs
c
Dimensionless frequency a0xr0
csCoefficient l for trapped mass
contribution
l = 0 for m B 1/3
l 2:4p m 13
for
1/3\ m B 1/2
Appropriate wave velocity c c = cp for m B 1/3
c = 2cs for
1/3\ m B 1/2
where, cp cs
ffiffiffiffiffiffiffiffiffiffiffi21m
12m
q
Half-space
(c, )
g
h
Free surface
Layer (c, )Interface
f
1
2
3
d
z0
z0+d
z0
Fig. 3 Incident, reflected and refracted waves at layer interface
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impingement. Numerically j is terminated at a value,
such that EFj1 EF
j
0:01 .4 Experimental Program
In the present study the effect of layering on the
dynamic response of foundation soil system is pro-
posed to investigate experimentally. Vertical vibration
tests using mechanical oscillator (Lazan Type) on
various depths of top layer with different static
weights, W and different dynamic force level (eccen-
tric settings in oscillator, h) are conducted. Detailed
program of thestudy is presented in Table 2 and Fig. 4.
Table 2 presents the various depths of top layer and the
dynamic force level considered in the investigation
whereas Fig. 4 shows two different test conditions.
4.1 Test Pit
To simulate the condition of proposed soil layering in
the investigation only choice is to conduct the test in a
tank or a pit of finite dimension. In the laboratory
tests, an optimization is needed between tank and
footing size to minimize the effects caused by
restricting lateral boundary. In spite of this, it is very
difficult to simulate the field conditions in the
laboratory. In order to overcome the limitations of
laboratory tests, the authors are inspired to conduct the
field tests. Present investigation is carried out in a pit,
excavated at the adjoining area of S.R. Sengupta
Foundation Engineering Laboratory, Indian Institute
of Technology, Kharagpur which is sufficiently larger
(width is 5 times the width of the footing) than that
required for the static condition. The density of in situ
soil is approximately equal to 18.0 kN/m3. Suitability
of the dimensions of the pit with respect to the size of
the footing for possible boundary effects is consid-
ered. The side of the pit is made of local soil of density
18.0 kN/m3 and moisture content is around 11% and
is expected to be extending up to infinite distance.
4.2 Material Properties
The density of the gravel used in this test is 17.2 kN/
m3 and frictional angle from direct shear test is 49.
The relative density of the gravel achieved in this
experiment was 85%. The study of grain sizedistribution of the soil at the pit site indicated sand
(30%), silt (61%) and clay (9%). Liquid limit, plastic
limit, and shrinkage limit of the site soil were 36%,
23%, and 12%, respectively. Experimental values of
dynamic shear modulus of both gavel and the in situ
soil at different static and dynamic loading conditions
are given in Table 3.
4.3 Preparation of Layers
4.3.1 Series I
The in situ soil is excavated from the top in steps of
200 mm. The excavated surface of the soil is then
leveled. Each time the total depth of pit is replaced by
locally available gravel. Thus, six different depths of
top gravel layer (400 mm, 600 mm, 800 mm,
1,000 mm, 1,200 mm, and 1,400 mm) are prepared.
Table 2 Details of field tests
Depth of top gravel
layer (d) in mm
Total number of tests considering all
variables
400 For each depth of top gravel layer tests
are conducted at two static weights,
8.0 kN and 10.0 kN and three
eccentric settings, 12, 16, 20 for
each static weight). Hence, total
number of tests is 72 being 36 on
each series
600
800
1000
1200
1400
d Gravel
Natural soil
Series I
Gravel
Rigid baseSeries II
d
Fig. 4 Different layered-soil systems
Table 3 Shear modulus values for gravel and in situ soil
Static
weights (kN)
Eccentric
setting (h)
Shear modulus (G) MN/m2
Gravel In situ soil
8.0 12 21.36 17.26
16 20.87 16.41
20 20.25 16.26
10.0 12 25.84 19.07
16 22.98 18.56
20 21.11 17.96
Note: msoil = 0.3 and csoil = 18.0 kN/m3
; mgravel = 0.25 and
cgravel = 17.2 kN/m3
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To maintain a uniform condition throughout the test
program, the pit is filled in steps of 200 mm thick
layer of gravel and each layer is compacted using a
plate vibrator (250 N static weights and vibrating at a
frequency of 3,000 rpm) by constant compactive
effort to achieve a density of approximately 17.2 kN/
m3. Calculated amount of dry gravel for 200 mmdepth maintaining uniform density (17.2 kN/m3) is
poured and compacted to bring it to 200 mm. Thus,
gravel layers of six different thicknesses are prepared
over in situ soil according to the experimental
program given in Table 2.
4.3.2 Series II
The test pit is excavated up to 1,700 mm depth. At
the base a 300 mm PCC concrete slab is cast to
represent rigid base. After casting and curing ofconcrete slab the gravel layer is placed. The different
depths of gravel layers are prepared over rigid base as
per experimental program. Necessary steps have been
taken to maintain the uniform density through out the
test. The tests are conducted on the level surface of
each layer.
4.4 Experimental Procedure
A model concrete footing of size 400 9 400
9 100 mm and a Lazan type mechanical oscillatorare used to conduct model block vibration test in
vertical mode. The concrete footing is first placed
centrally over the prepared gravel layer. A rigid mild
steel plate is tightly fixed on the concrete footing to
facilitate load-fixing arrangement. Oscillator is then
placed over the plate and a number of mild steel
ingots are placed on the top of the oscillator to
provide required static weight. Sufficient rubber
packing between two ingots is given for tight fixing.
The whole set-up is then tightened to act as a singleunit during vibration. Proper care is taken to maintain
the center of gravity of whole system and the footing
to lie in the same vertical line. In this investigation,
8.0 and 10.0 kN static weights are used to simulate
two different foundation weights and under each
static weight three different eccentric settings
(h = 12, 16, and 20) are used to simulate three
different dynamic force level. The frequency
dependent dynamic force amplitude in N was
expressed by
meex2
Wee
gx2
0:9sinh=2
gx2 24
The oscillator is connected through a flexible shaft
to a variable DC motor (3 H.P. frequency range up to
3000 rpm). A B&K piezoelectric-type vibration
pickup (type 4370) is placed on top of the footing to
measure the displacement amplitude with the B&K
vibration meter (type 2511). Figure 5 shows the
schematic diagram of the experimental set-up. The
oscillator is then run slowly through a motor using
speed control unit to avoid sudden application of highmagnitude dynamic load. Thus the foundation is
subjected to vibration in the vertical direction. Photo
tachometer and vibration meter recorded frequency
Motor
Vibration
meter
Static
weight
Mechanical
oscillator
Speed
control unit
Shaft
Rigid base to
simulate bedrockTopsoil layer:
varying thickness
1.7m
0.3m
Fig. 5 Experimental set-up
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Table 4 Comparison of
resonant frequencies and
resonant amplitudes for
gravel layer over rigid
basestatic weight =
8.0 kN
Depth
(mm)
Depth ratio
(d/r0)
h
(degree)
Resonant
frequency (Hz)
Diff.
(%)
Resonant
amplitude (mm)
Diff.
(%)
Expt. Pred. Expt Pred.
400 1.77 12 31.40 36.25 15.44 0.16 0.20 25.00
16 30.53 35.83 17.35 0.20 0.27 35.00
20 29.83 35.41 18.71 0.23 0.33 43.47
600 2.66 12 29.38 33.33 13.44 0.21 0.12 -42.85
16 28.81 32.91 14.22 0.21 0.16 -23.80
20 28.60 32.50 13.63 0.23 0.20 -13.04
800 3.54 12 29.08 30.41 4.58 0.25 0.14 -44.00
16 28.33 30.00 5.88 0.30 0.20 -33.33
20 27.93 29.58 5.90 0.34 0.23 -32.35
1,000 4.43 12 28.40 29.58 4.16 0.19 0.11 -42.10
16 28.02 29.58 5.59 0.28 0.15 -46.42
20 27.78 29.16 4.97 0.30 0.24 -20.00
1,200 5.32 12 28.21 27.91 -1.06 0.25 0.12 -52.00
16 27.90 27.91 0.06 0.24 0.16 -33.33
20 27.48 27.50 0.06 0.32 0.26 -18.75
1,400 6.20 12 28.10 26.66 -5.10 0.21 0.11 -47.61
16 27.73 26.25 -5.34 0.23 0.15 -34.78
20 27.30 26.25 -3.84 0.32 0.22 -31.25
Table 5 Comparison of
resonant frequencies and
resonant amplitudes forgravel layer over rigid
basestatic
weight = 10.0 kN
Depth
(mm)
Depth ratio
(d/r0)
h
(degree)
Resonant
frequency (Hz)
Diff.
(%)
Resonant
amplitude (mm)
Diff.
(%)
Expt. Pred. Expt Pred.
400 1.77 12 31.18 35.00 12.23 0.08 0.10 25.00
16 30.21 32.92 8.93 0.12 0.13 8.33
20 29.15 31.67 8.63 0.16 0.16 0.00
600 2.66 12 28.51 32.50 13.96 0.10 0.14 40.00
16 27.68 30.83 11.37 0.12 0.17 41.66
20 26.81 29.58 10.31 0.14 0.20 42.85
800 3.54 12 28.15 30.00 6.57 0.16 0.10 -37.50
16 27.50 28.33 3.03 0.19 0.14 -26.31
20 26.55 27.08 2.01 0.24 0.17 -29.16
1,000 4.43 12 27.91 29.58 5.97 0.13 0.11 -15.38
16 26.36 27.92 5.87 0.15 0.15 0.00
20 26.10 26.67 2.17 0.22 0.18 -18.18
1,200 5.32 12 27.76 28.33 2.04 0.12 0.10 -16.67
16 26.18 26.67 1.84 0.15 0.14 -6.67
20 25.10 25.41 1.26 0.22 0.18 -18.18
1,400 6.20 12 27.63 27.08 -1.99 0.11 0.10 -9.09
16 26.03 25.41 -2.36 0.15 0.14 -6.67
20 24.95 25.41 1.87 0.21 0.17 -19.05
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Table 6 Comparison of
resonant frequencies and
resonant amplitudes for
gravel layer over in situ
soilstatic
weight = 8.0 kN
Depth
(mm)
Depth ratio
(d/r0)
h
(degree)
Resonant
frequency (Hz)
Diff.
(%)
Resonant
amplitude (mm)
Diff.
(%)
Expt. Pred. Expt Pred.
400 1.77 12 27.52 29.50 7.20 0.073 0.050 -31.50
16 27.03 29.16 7.89 0.083 0.060 -27.71
20 26.81 29.16 8.76 0.093 0.080 -13.97
600 2.66 12 28.28 29.83 5.48 0.077 0.052 -32.46
16 27.98 29.50 5.41 0.087 0.068 -21.83
20 27.25 29.16 7.03 0.100 0.084 -16.00
800 3.54 12 27.78 30.00 7.97 0.077 0.054 -29.87
16 27.21 29.66 9.00 0.093 0.070 -24.73
20 26.85 29.33 9.24 0.077 0.086 11.68
1,000 4.43 12 28.01 30.00 7.07 0.080 0.054 -32.50
16 27.30 29.66 8.66 0.093 0.072 -22.58
20 27.05 29.33 8.44 0.107 0.090 -15.88
1,200 5.32 12 28.25 29.83 5.60 0.083 0.056 -32.53
16 27.80 29.66 6.71 0.100 0.074 -26.00
20 27.08 29.16 7.69 0.107 0.092 -14.01
1,400 6.20 12 28.16 29.66 5.32 0.080 0.054 -32.50
16 27.61 29.50 6.81 0.097 0.072 -25.77
20 27.18 29.00 6.68 0.100 0.088 -12.00
Table 7 Comparison of
resonant frequencies and
resonant amplitudes for
gravel layer over in situsoilstatic
weight = 10.0 kN
Depth
(mm)
Depth ratio
(d/r0)
h
(degree)
Resonant
frequency (Hz)
Diff.
(%)
Resonant
amplitude (mm)
Diff.
(%)
Expt. Pred. Expt Pred.
400 1.77 12 27.05 28.50 5.36 0.050 0.040 -20.00
16 26.85 27.00 0.56 0.067 0.056 -16.42
20 26.42 25.83 -2.21 0.080 0.072 -10.00
600 2.66 12 27.38 29.00 5.90 0.053 0.042 -20.75
16 26.68 27.33 2.43 0.063 0.060 -4.76
20 25.58 26.16 2.28 0.070 0.074 5.71
800 3.54 12 27.22 29.33 7.78 0.050 0.044 -12.00
16 27.00 27.50 1.85 0.060 0.060 0.00
20 26.08 26.33 0.96 0.073 0.076 4.11
1,000 4.43 12 27.18 29.50 8.52 0.057 0.046 -19.3016 27.08 27.67 2.15 0.067 0.062 -7.46
20 26.42 26.33 -0.31 0.080 0.078 -2.50
1,200 5.32 12 27.28 29.33 7.51 0.060 0.048 -20.00
16 27.03 27.50 1.73 0.070 0.064 -8.57
20 25.72 26.33 2.39 0.080 0.080 0.00
1,400 6.20 12 27.17 29.33 7.97 0.057 0.048 -15.79
16 26.87 27.33 1.74 0.067 0.060 -10.45
20 25.85 26.17 1.23 0.077 0.074 -3.89
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and corresponding displacement amplitude of vibra-
tion respectively. To obtain a foundation response and
locate the resonant peak correctly, the displacement
amplitudes are noted at a frequency interval approx-
imately of 25 to 50 rpm.
A sufficient time between two successive mea-
surements has been given to reach equilibrium, which
facilitates accurate measurement of frequency and the
corresponding displacement amplitude. The displace-
ment amplitude corresponding to each frequency isrecorded and the response curves are plotted for
different layered systems under various static and
dynamic loading conditions.
5 Cone Model versus Experiment
The frequency-amplitude response for all the cases
mentioned in Table 2 are computed using the
solutions of cone model. The experimental values
of dynamic shear modulus given in Table 3 are used
in the above computation. Material damping ratio 2%
and 1% was assumed for top gravel layer and bottom
in situ soil respectively. The predicted resonant
frequencies and resonant amplitudes are compared
quantitatively with respective experimental values,
which are presented in Tables 47 and Figs. 6 and 7.
The comparison of resonant frequencies for gravel
layer over concrete rigid base (series II) shows adifference of-5% to 19% under static weight 8.0 kN
and -2% to 14% under static weight 10.0 kN
(Tables 4 and 5). The maximum difference is
observed at lower depth and at higher force level.
But the predicted amplitudes for the above case are
found to deviate from corresponding experimental
values in the range -52% to 43% and -37% to 42%
under static weight 8.0 kN and 10.0 kN respectively.
For the case of gravel layer over in situ soil (series I)
20 25 30 35 4020
25
30
35
40(a)
Data Points
45 LineResonantFrequencyPredicted(Hz)
Resonant Frequency Observed (Hz)
0.0 0.1 0.2 0.3 0.4 0.5 0.60.0
0.1
0.2
0.3
0.4
0.5
0.6(b)
ResonantAmplitu
dePredicted(Hz)
Resonant Amplitude Observed (Hz)
Data Points
45 Line
Fig. 6 Comparison of (a)
resonant frequencies and
(b) resonant amplitudes for
gravel layer over rigid base
20 25 30 3520
25
30
35(a)
Data Points
45 LineResonantFrequencyPredicted(Hz
)
Resonant Frequency Observed (Hz)
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.200.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20(b)
ResonantAmplitudePredicted(Hz
)
Resonant Amplitude Observed (Hz)
Data Points
45 Line
Fig. 7 Comparison of (a)
resonant frequencies and
(b) resonant amplitudes forgravel layer over in situ soil
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the deviation of predicted resonant frequencies are in
the range 5% to 9% and -2% to 8% under static
weight 8 kN and 10 kN respectively. The predicted
resonant amplitudes for the above case shows a
negative difference in majority of cases when com-
pared against their respective experimental values,
maximum being -32% and -20% under staticweight 8 kN and 10 kN respectively. In general
considering the comparison of all the test results it is
observed that the predicted resonant frequencies
are very close to their experimental values (max.
deviation 19%), thus showing a good engineering
accuracy (Figs. 6 and 7). But in case of amplitudes
the deviation of predicted values are negative in most
of the cases indicating that the model predicts a
higher damping, giving rise to lower values of
amplitudes. Also the authors feel that this may be
due to poor selection of material damping. The
material damping (hysteretic) considered in themodel is strain dependent and hence it should vary
with the variation of force level. But it is not taken
into consideration, rather irrespective of the force
level a constant material damping ratio 2% for gravel
and 1% for in situ soil is considered. This may be the
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
PredictedExperimental
d/r0 =1.77
DisplacementAmplitu
de(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
PredictedExperimental
d/r0=2.66
DisplacementAmplitu
de(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
PredictedExperimental
d/r0=3.54
DisplacementAmplitude(
mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
PredictedExperimental
d/r0=4.43
DisplacementAmplitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
Predicted
Experimental
d/r0=5.32
DisplacementAmplitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
Predicted
Experimental
d/r0=6.20
DisplacementAmplitude(mm)
Frequency (Hz)
Fig. 8 Comparison of
frequency-amplitude
response curves for gravel
layer over rigid base (static
weight = 8.0 kN andh = 16)
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reason for which the predicted amplitudes are lower
compared to experimental ones. In spite of such
deviations, it is observed that the predicted ampli-
tudes match well with experimental values (Figs. 6
and 7).
In case of layered soil the dynamic response of
foundation is greatly influenced by the depth of thetop layer and relative rigidity of layers. In the present
investigation two different cases of relative rigidity
(series I and series II) and six different depths of top
layer are considered. The nature of variation of
frequencies and amplitudes due to variation of
above two parameters are observed to be same in
both experimental and model predicted results
(Tables 47).
In case of gravel layer over in situ soil, change in
the resonant frequencies and resonant amplitudes
with variation of the depth of top layer are negligible(Tables 6 and 7) as the relative rigidity is very close
to one. Thus, this case may be considered closer to a
homogeneous half-space. Hence for comparison of
frequency-amplitude response for layered soil only
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Predicted
Experimental
d/r0=1.77
DisplacementAm
plitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Predicted
Experimental
d/r0=2.66
DisplacementAm
plitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20Predicted
Experimental
d/r0=3.54
DisplacementAmplitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Predicted
Experimental
d/r0=4.43
DisplacementAmplitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Predicted
Experimental
d/r0=5.32
DisplacementAmplitud
e(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Predicted
Experimental
d/r0=6.20
DisplacementAmplitud
e(mm)
Frequency (Hz)
Fig. 9 Comparison of frequency-amplitude response curves for gravel layer over rigid base (static weight = 10.0 kN and h = 16)
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the case of gravel layer over rigid base with variation
of depth of top layer is presented for a given force
level (h = 16) under two values of static weights
(Figs. 8 and 9). From Figs. 8 and 9, it is observed
that the predicted and experimental resonant frequen-
cies and amplitudes are closer at higher static weight.
With increase in the depth of top layer decrease ofresonant frequency is also observed in both experi-
mental and predicted response curves.
The effect of damping ratio on the behaviour
between the displacement amplitude and frequency
has been studied for different depths of gravel layer
over rigid base under a given dynamic force level and
two different static weights. The damping ratio varied
from 0.00 to 0.03, and results obtained with different
damping ratios are presented in Figs. 10 and 11. Ingeneral a decrease in the resonant amplitude and
negligible change in the resonant frequency is
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
d/r0=1.77
Displaceme
ntAmplitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
d/r0=2.66
Displaceme
ntAmplitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
d/r0=3.54
DisplacementA
mplitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
d/r0=4.43
DisplacementA
mplitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
=0.00=0.01
=0.02
=0.03
d/r0=5.32
DisplacementAmp
litude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
d/r0=6.20
DisplacementAmp
litude(mm)
Frequency (Hz)
=0.00
=0.01
=0.02=0.03
=0.00
=0.01=0.02
=0.03
=0.00
=0.01=0.02
=0.03
=0.00
=0.01=0.02
=0.03
=0.00=0.01
=0.02
=0.03
Fig. 10 Effect of damping ratio on frequency-amplitude response for gravel layer over rigid base (static weight = 8.0 kN and
h = 16)
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observed from Figs. 10 and 11 with the increase of
damping from 0.0 to 0.03. The average decrease in
the resonant amplitude from that corresponding to
zero damping is observed to be 25%, 40% and 50%
when the damping is increased to 0.01, 0.02 and 0.03
respectively under 10 kN static weight. The order of
decrease was observed to be 30%, 45% and 55% for8 kN static weight. Also, it is observed from Figs. 8
11 that the response curve at damping ratio 0.02 is
closer to the experimental response curve indicating a
good assumption of damping ratio.
6 Conclusions
Compared to available rigorous analytical methods
for foundation vibration analysis on layered soil cone
model is found to be very simple as it considers only
one type of body waves for the mode of vibration
considered and the analysis is performed using abasic strength of material approach. Though the
model predicts a little higher damping, a good
engineering accuracy is achieved when compared
against 72 field test results. Thus, it may be used as a
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
d/r0=1.77
DisplacementAmplitude(mm)
Frequency (Hz)
=0.00=0.01=0.02=0.03
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
d/r0=2.66
DisplacementAmplitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
d/r0=3.54
DisplacementAm
plitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
d/r0 =4.43
DisplacementAm
plitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
d/r0=5.32
DisplacementAmplitude(mm)
Frequency (Hz)
16 18 20 22 24 26 28 30 32 34 36 38 400.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
d/r0=6.20
DisplacementAmplitude(mm)
Frequency (Hz)
=0.00=0.01=0.02=0.03
=0.00=0.01=0.02=0.03
=0.00=0.01=0.02=0.03
=0.00=0.01=0.02=0.03
=0.00=0.01=0.02=0.03
Fig. 11 Effect of damping ratio on frequency-amplitude response for gravel layer over rigid base (static weight = 10.0 kN and
h = 16)
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cost effective tool for the analysis of machine
foundations on layered soil with due reliability.
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