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Dynamic response of foundations resting on layered soil by cone model
P.K. Pradhan*, D.K. Baidya, D.P. Ghosh
Department of Civil Engineering, Indian Institute of Technology, Kharagpur 721302, India
Received 20 February 2004
Abstract
Impedance functions for a rigid massless circular foundation resting on a soil layer underlain by a rigid base subjected to vertical harmonic
excitation are found using one-dimensional wave propagation in cones. The static stiffness predicted by the model for different depths of
layer agrees well with published results. The frequency –amplitude response of a foundation is then found using impedance functions.
The predicted resonant frequency is compared with published results based on both analytical and experimental investigation, which shows a
good engineering accuracy (deviation of ^12%) with a vast variation of parameters. The predicted frequency –amplitude response and
resonant amplitudes are also compared with reported experimental results, which shows a good agreement. Also the influence of key
parameters such as mass ratio, Poisson’s ratio, depth of the layer and hysteretic material damping ratio on the vertical response of the
foundation is investigated. The results are presented in the form of simple and versatile dimensionless graphs, which may prove to be useful
in understanding the harmonic response of foundations resting on layered soil under vertical excitation.
q2004 Elsevier Ltd. All rights reserved.
Keywords: Impedance functions; Circular foundation; Wave propagation; Resonant frequency; Resonant amplitude; Layered soil and cone model
1. Introduction
The determination of resonant frequency and resonant
amplitude of foundations has been a subject of considerable
interest in the recent years, in relation to the design of
machine foundations, as well as the seismic design of
important massive structures such as nuclear power plants.
One of the key steps 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 (spring and
dashpot coefficients) of an ‘associated’ rigid but massless
foundation, using a suitable method of dynamic analysis.
With the help of these functions the amplitude of vibration
can be calculated using the equations of motion of a single
degree of freedom oscillator. Over the years a number of
methods have been developed for foundation vibration
analysis, such as: (1) Single degree of freedom mass –spring
dashpot model; (2) elastic half-space theory; (3) cone
model using the physical concept of wave propagation;
(4) analytical solutions based on integral transform
techniques; (5) semi-analytical and boundary element
formulations requiring discretization of top surface only;
(6) dynamic finite element methods using special wave
transmitting boundaries; and (7) hybrid methods combining
analytical and finite element techniques.
The solution of the ‘dynamic Boussinesq’ problem of
Lamb [21] formed the basis for the study of oscillation of
footings resting on a half-space. Reissner [31] first
developed the analytical solution for a vertically loaded
cylindrical disk on elastic half-space assuming uniform
stress distribution under the footing. Later, extending
Reissner’s solution, many investigators (Sung [34], Quinlan
[30], Bycroft [8], Richart et al. [32], Luco and Westman
[23], Nagendra and Sridharan [28], to name a few) studied
different modes of vibrations with different contact stress
distributions. Also, Gazetas [13] presented simple formulas
and charts for impedance functions of both surface and
embedded foundations for various modes of vibration,
which can be readily used by the practicing engineers.
In most of the studies the soil medium below the
foundation was assumed to be a homogeneous elastic
half-space. In reality, however, soils are rarely homo-
geneous. The presence of a hard rock at shallow depth is one
of the common features of soil in natural state. The vertical
vibration response of a circular footing on the surface of
an elastic layer underlain by a rigid base was evaluated by
Warburton [38]. He studied the effect of an elastic layer on
0267-7261/$ - see front matter q2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.soildyn.2004.03.001
Soil Dynamics and Earthquake Engineering 24 (2004) 425–434
www.elsevier.com/locate/soildyn
*Corresponding author. Tel.: þ91-3222-220010; fax: þ91-3222-
282254.
E-mail address: pkpradhan1@yahoo.co.in (P.K. Pradhan).
the resonant frequency of the footing for two values
of Poisson’s ratio. Gazetas and Rosset [15] have developed
a solution for the vertical vibration response of a strip
footing on the surface of an elastic soil layer overlying rock.
They showed that the presence of a thin layer tends to
increase the resonant frequency and amplitude compared to
the half-space values. Luco [22], Kausel et al. [20], Hadjian
and Luco [17], Kagawa and Kraft [19], Tassoulas and
Kausel [35], Gazetas [11], Wong and Luco [44], Apsel
and Luco [2], Wolf [39], Baidya and Sridharan [6], Ahmad
and Rupani [1], Asik and Vallabhan [3] to name a few, also
considered the effect of layering or nonhomogenity in their
analyses. Most of these works were confined to analytical or
semi-analytical type.
Gazetas and Stokoe [16] discussed different types of
experimental investigation related to vibrating foundations
and also discussed the relative merits and limitations of each
method. They also indicated that the case histories and field
experiments are the best, since the propagation of elastic
waves is not interrupted by the presence of artificial lateral
boundaries as in laboratory tests. Studies by Sridharan et al.
[33], Baidya and Muralikrishna [4,5] and Baidya and
Sridharan [7] can be mentioned as some of the important
experimental works on layered soil.
The cone model was originally developed by Ehlers [10]
to represent a surface disk under translational motions and
later for rotational motion [24,36].Bycomparisonto
rigorous solutions, the cone models originally appeared to
be such an oversimplification of reality that they were used
primarily to obtain qualitative 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 have
encouraged a number of researchers to match discrete
element representation of exact solutions in frequency
domain by curve fitting [9,37,43]. Proceeding in another
direction, Gazetas [12] and Gazetas and Dobry [14]
employed wedges and cones to elucidate the phenomenon
of radiation damping in two and three dimensions. Later
Meek and Wolf [25] presented a simplified methodology 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 [26] and on flexible rock, Wolf and
Meek [41]. Meek and Wolf [27] performed dynamic analysis
of embedded footings by idealizing the soil as a translated
cone instead of elastic half-space. Wolf and Meek [42] 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 [18] 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 [42] is that
the damping coefficient can become negative at lower
frequency, which is physically impossible. Pradhan et al.
[29] have computed dynamic impedance of circular
foundation 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 summarized in Wolf [40].
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,
Nomenclature
a0dimensionless frequency ð
v
r0=csÞ
b0nondimensional mass ratio
Bnondimensional modified mass ratio
cappropriate wave velocity
csshear wave velocity
cpdilatational wave velocity
cða0Þnormalized damping coefficient
ddepth of the soil layer
Gshear modulus of soil
kða0Þnormalized stiffness coefficient
Kstatic stiffness coefficient on homogeneous half
space
Kða0Þdynamic impedance
mmass of the foundation or total vibrating mass
(mass of foundation plus machine) in case of
machine foundation
meunbalanced mass (on machine)
P0harmonic interaction force
Qharmonic force on foundation
lQlforce amplitude on the foundation
r0radius of circular foundation or radius of
equivalent circle for noncircular foundation
u0harmonic surface displacement for the layer
u0harmonic surface displacement for homo-
geneous half space
lu0ldisplacement amplitude for the layer
u0Gr0
Qnondimensional amplitude
Greek symbol
n
Poisson’s ratio of the soil
v
circular frequency of excitation
r
mass density of soil
u
angle for setting eccentricity in the oscillator
j
hysteretic material damping ratio
P.K. Pradhan et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 425–434426
the cone models which provide conceptual clarity with
physical insight and is easier for the practicing engineers to
follow. Most of the published results using cone model are
confined to the determination of the dynamic response of the
foundation in the form of impedance functions and their
comparison with rigorous elastodynamic solutions based on
finite element or boundary element methods. To the best of
authors’ knowledge no literature is available with regard to
the study of frequency– amplitude response of the
foundation using cone model, though resonant frequency
and resonant amplitude are the two main design
parameters. Hence, in the present investigation the dynamic
response (frequency –amplitude) of the foundation resting
on a soil layer underlain by a rigid base under vertical
harmonic excitation is found using wave propagation in
cones. The validity of the model is checked by comparing
the predicted response with reported analytical and
experimental results. The foundation response is also
studied varying widely the parameters like mass ratio,
Poisson’s ratio, depth of the layer and hysteretic material
damping ratio.
2. Dynamic response of the foundation
To study the dynamic response of a machine foundation
resting on the surface of a soil layer underlain by a rigid
base, a rigid massless ‘associated’ foundation of radius r0is
addressed for vertical degree of freedom (Fig. 1). The layer
with depth dhas the shear modulus, G;Poisson’s ratio,
n
;
mass density,
r
;hysteretic damping ratio,
j
:The interaction
force P0and the corresponding displacement u0are
assumed to be harmonic. The dynamic impedance of the
massless foundation (disk) is given by
Kða0Þ¼ P0
u0¼K½kða0Þþia0cða0Þ ð1Þ
where
Kða0Þdynamic impedance
kða0Þspring coefficient
cða0Þdamping coefficient
a0¼
v
r0
cs
;dimensionless frequency
cs¼ffiffiffiffiffi
G=
r
p;shear wave velocity of the layer
K¼4Gr0
12
n
;static stiffness coefficient of homogeneous
half-space with material properties of the
layer
The effects of hysteretic material damping is isolated
using an alternate expression to Eq. (1) for dynamic
impedance
Kða0Þ¼K½kða0Þþia0cða0Þð1þ2i
j
Þð2Þ
Using the equations of dynamic equilibrium, the dynamic
displacement amplitude of the foundation with mass mand
subjected to a vertical harmonic force Qis expressed as
lu0l¼Q
K½kða0Þþia0cða0Þ2Ba2
0
ð3Þ
where
lu0ldynamic displacement amplitude under the
foundation resting on the layer
lQlforce amplitude
B¼12
n
4b0;with b0¼m
r
r3
0
;the mass ratio
In general, lQlcan be assumed to be constant or equal to
mee
v
2which is generated by the eccentric rotating part in
machine, where meis the eccentric mass, eis the
eccentricity and
v
is the circular frequency.
Dynamic displacement amplitude given in Eq. (3) can be
expressed in the nondimensional form as given below
u0Gr0
Q
¼12
n
4l½kða0Þþia0cða0Þ2Ba2
0l21ð4Þ
3. Wave propagation in cones
Fig. 2(a) shows wave propagation in cones beneath the
disk of radius r0resting on a layer underlain by a rigid base
under vertical harmonic excitation, P0:The dilatational
waves emanate beneath the disk and propagate at velocity c
equal to the dilatational wave velocity cpfor
n
#1=3 and
twice the shear wave velocity, csfor 1=3,
n
$1=2:
These waves reflect back and forth at the rigid base and
free surface, spreading and decreasing in amplitude. Let
the displacement of the (truncated semi-infinite) cone be
denoted as
uwith the value
u0under the disk (Fig. 2(b)),
modeling a disk with same load P0on a homogeneous half-
space with the material properties of the layer. The
parameters of cone model shown in Fig. 2(b) are given
in Table 1. This displacement
u0is used to generate the
displacement of the layer uwith its value at surface, u0:
Fig. 1. ‘Associated’ massless foundation soil system under vertical
interaction force.
P.K. Pradhan et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 425–434 427
Thus,
u0can also be called as the generating function.
The first downward wave propagating in a cone with apex 1
(height z0and 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 know if at a specific depth a rigid interface is
encountered or not. Thus, the aspect ratio defined by the
ratio of the height of cone from its apex to the disk is made
equal for cone of the half-space and first cone of the layer.
Since the incident wave and subsequent reflected waves
propagate in the same medium (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. 2(a). The displacement amplitude of the incident
wave propagating 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
uðz;
v
Þ¼ z0
z0þze2i
v
ðz=cÞ
u0ð
v
Þð5Þ
The displacement of the incident wave at rigid base
equals
uðd;
v
Þ¼ z0
z0þde2i
v
ðd=cÞ
u0ð
v
Þð6Þ
Enforcing the boundary condition that the displacement
at rigid base vanishes, the displacement of the first reflected
upward wave propagating in a cone with apex 2
(vide Fig. 2(a)) equals
2z0
z0þ2d2ze2i
v
ðð2d2zÞ=cÞ
u0ð
v
Þð7Þ
At the free surface the displacement of the upward wave
derived by substituting z¼0 in Eq. (7) equals
2z0
z0þ2de2i
v
ð2d=cÞ
u0ð
v
Þð8Þ
Enforcing compatibility of the amplitude and of elapsed
time of the reflected wave’s displacement at the free surface,
the displacement of the downward wave propagating in a
cone with apex 3 is obtained as
2z0
z0þ2dþze2i
v
ðð2dþzÞ=cÞ
u0ð
v
Þð9Þ
In this pattern the waves propagate in their own cones
and their corresponding displacements are found out.
Thus, after jth impingement at rigid base, the displacements
of upward and downward waves propagating in cones with
Fig. 2. (a) Wave propagation in cones for the layer, (b) cone model for the
half space.
Table 1
The parameters of semi-infinite cone modeling a disk on homogeneous
half-space under vertical motion [40]
Cone parameters Parameter expressions
Aspect ratio, z0
r0
p
4ð12
n
Þc
cs
2
Static stiffness coefficient, K
r
c2A0
z0
Normalized spring
coefficient, kða0Þ
12
m
p
z0
r0
c2
s
c2a2
0
Normalized damping
coefficient, cða0Þ
z0
r0
cs
c
Dimensionless frequency, a0
v
r0
cs
Coefficient
m
for trapped
mass contribution
m
¼0 for
n
#1=3;
m
¼2:4pð
n
21
3Þ
for 1=3#
n
#1=2
Appropriate wave
velocity, c
c¼cpfor
n
#1=3;c¼2cs
for 1=3#
n
#1=2 where
cp¼csffiffiffiffiffiffiffiffiffiffiffiffi
2ð12
n
Þ
122
n
r
P.K. Pradhan et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 425–434428
apex 2jand 2jþ1 are expressed as
ð21Þjz0
z0þ2jd2ze2i
v
ðð2jd2zÞ=cÞ
u0ð
v
Þð10Þ
ð21Þjz0
z0þ2jdþze2i
v
ðð2jdþzÞ=cÞ
u0ð
v
Þð11Þ
The resulting displacement in the layer is obtained by
superposing all the down and up waves and is expressed in
the following form
uðz;tÞ¼z0e2i
v
ðz=cÞ
z0þz
u0ð
v
ÞþX
1
j¼1ð21Þj
z0e2i
v
ðð2jd2zÞ=cÞ
z0þ2jd 2zþz0e2i
v
ðð2jdþzÞ=cÞ
z0þ2jd þz
"#
u0ð
v
Þð12Þ
At the rigid base udð
v
Þ¼uðd;
v
Þvanishes as required by
the rigid base boundary condition and at the free surface the
displacement of the foundation is obtained by setting z¼0
in Eq. (12)
u0ð
v
Þ¼uðz¼0;
v
Þ
¼
u0ð
v
Þþ2X
1
j¼1
ð21Þj
1þ2jd
z0
e2i
v
ð2jd=cÞ
u0ð
v
Þð13Þ
u0ð
v
Þ¼X
1
j¼0
EF
je2i
v
ð2jd=cÞ
u0ð
v
Þð14Þ
with
EF
0¼1ð15aÞ
and for j$1;
EF
j¼2ð21Þj
1þ2jd
z0
ð15bÞ
EF
jcan be called as echo constant, the inverse of sum of
which gives the static stiffness of the layer normalized by
the static stiffness of the homogeneous half-space with
material properties of the layer.
4. Dynamic impedance
The interaction force displacement relationship for a
massless disk resting on homogeneous half-space using the
cone model can be written as
P0ð
v
Þ¼ðK2DM
v
2þi
v
CÞ
u0ð
v
Þð16Þ
where K2DM
v
2is the spring coefficient and C is the
dashpot coefficient
DMis the trapped mass and is given by
DM¼
mr
r3
0ð17Þ
with trapped mass coefficient
m
;the values of which
recommended by Wolf are given in Table 1. The trapped
mass DMis introduced in order to match the stiffness
coefficient of the cone model with rigorous solutions for
incompressible soil i.e. 1=3.
n
$1=2;Wolf [40]. After
simplification Eq. (16) reduces to the form
P0ð
v
Þ¼K12
m
p
z0r0
c2
v
2þi
v
z0
c
u0ð
v
Þð18Þ
Using Eq. (14) in Eq. (18), the interaction force
displacement relationship for the layer-rigid base system
reduces to
P0ð
v
Þ¼K
12
m
p
z0r0
c2
v
2þi
v
z0
c
X
1
j¼0
EF
je2i
v
ð2jd=cÞ
u0ð
v
Þð19Þ
Substituting echo constant given by Eq. (15) in Eq. (19),
the dynamic impedance equals
Kð
v
Þ¼ P0ð
v
Þ
u0ð
v
Þ¼K
12
m
p
z0r0
c2
v
2þi
v
z0
c
1þ2X
1
j¼1
EF
je2i
v
ð2jd=cÞð20Þ
In the expression of the dynamic impedance
Kð
v
Þgiven
by Eq. (20), 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 impingement. Numerically j
is terminated at a value, such that lEF
jþ12EF
jl#0:01:
Table 2
Normalized static stiffness of a circular foundation resting on a soil layer underlain by rigid base
d=r0Normalized static stiffness of the layer, KL=K(Cone model) 1þ1:28 r0
d;
for all
n
[11]
n
¼0:0
n
¼0:2
n
¼0:3
n
¼0:4
n
¼0:49 Avg
2 1.550 1.640 1.677 1.663 1.550 1.606 1.640
4 1.272 1.320 1.334 1.327 1.272 1.299 1.320
6 1.180 1.213 1.221 1.217 1.180 1.198 1.213
8 1.135 1.160 1.165 1.162 1.135 1.148 1.160
10 1.107 1.128 1.132 1.129 1.107 1.118 1.128
12 1.089 1.107 1.110 1.107 1.089 1.098 1.107
Note: KL;static stiffness of layer-rigid base system.
P.K. Pradhan et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 425–434 429
5. Results and discussion
In the static case, the stiffness of the layer normalized by
the stiffness of the homogeneous half-space with material
properties of the layer using the model KL=K¼ðPEF
jÞ21;
which is dependent on
n
and d=r0:The normalized stiffness
of the layer is found out for five different values of
n
and
d=r0varying from 2 to 12 (Table 2). As the variation of
normalized stiffness of the layer with
n
is found to be very
less, the average values are used for comparison with
Gazetas’s [11] values, which shows a very good agreement
Fig. 6. Comparison of predicted resonant amplitude with experimental
results.
Fig. 5. Comparison of predicted resonant frequency with experimental
results.
Table 3
Shear modulus values for sand and sawdust [4]
Static weight
(kN)
Eccentric angle,
u
(8)
Shear modulus, G
(kN/m
2
)
Sand Sawdust
8.0 8 19,473 –
10 19,346 –
12 19,028 –
14 18,491 –
8.9 8 19,664 –
10 19,219 –
12 19,155 –
14 18,837 –
4.0 4 – 1354
8 – 1288
12 – 1260
4.9 4 – 1578
8 – 1465
12 – 1381
Note:
g
sand ¼17 kN/m
3
,
g
sawdust ¼2:6 kN/m
3
,
n
sand ¼0:3 (assumed),
n
sawdust ¼0:0 (assumed).
Fig. 4. Comparison of predicted resonant frequency with Warburton’s results.
Fig. 3. Comparison of normalized static stiffness.
P.K. Pradhan et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 425–434430
(Fig. 3). This figure indicates that the value of KL=K
decreases as the d=r0increases and reduces to nearly unity at
d=r0¼6:Thus, from this static analysis it is observed that
the layer-rigid base system behaves as a homogeneous
half-space when d=r0$6:
The validity of the model is checked by comparing the
predicted resonant frequencies with the published analytical
results. Fig. 4 presents the comparison between the results
obtained by the cone model and the one proposed by
Warburton [38]. It is seen from Fig. 4 that for
n
¼1=4;the
variation of nondimensional resonant frequency with mass
ratio is in good agreement with the results of Warburton [38]
for all the five values of d=r0:
Also to verify the validity of the model a more detailed
comparison of frequency– amplitude response and in
particular the resonant frequencies and resonant amplitudes,
computed by the model is made with the experimental
results of Baidya and Muralikrishna [4]. They studied the
dynamic response of a foundation resting on a layer
underlain by a rigid base by conducting vertical vibration
tests using a model concrete footing of size
400 £400 £100 mm
3
and a mechanical oscillator (Lazan
type) with different static weights and different force levels.
Sand and sawdust were used as material of the layer. For
each material six different depths of layer were used
(d=r0¼1:77;2.66, 3.55, 4.43, 5.32 and 5.98). Static weights
of 8.0 and 8.9 kN for sand layer and 4 and 4.9 kN for
sawdust layer were used. Tests on sand layer were
conducted at four different force levels, i.e. eccentricity
u
¼8;10, 12 and 148. For sawdust layer three force levels
(
u
¼4;8 and 128) were applied. Thus, a total of 84 tests
were conducted. Experimentally evaluated shear modulus
Fig. 7. Comparison of frequency–amplitude response curves for various depths of sand layer under static weight 8.0 kN and
u
¼88:
P.K. Pradhan et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 425–434 431
values reported by Baidya and Muralikrishna [4] are shown
in Table 3. Resonant frequencies and resonant amplitudes
are predicted using the proposed model for the above 84
tests (assuming material damping ratio
j
¼5% for sand and
2% for sawdust) and compared with experimental results of
Baidya and Muralikrishna [4] in Figs. 5 and 6, respectively.
Fig. 5 indicates that the predicted resonant frequencies
match well with observed values as the maximum deviation
is found to be 12% for the stiffer layer (sand) and at the
lowest value of the depth of layer considered ðd=r0¼1:77Þ:
Comparison of predicted resonant amplitudes with their
respective observed values (Fig. 6) shows a very good
agreement in case of stiffer (sand) layer. But in case of softer
(sawdust) layer a little more deviation is observed at the
larger depth of the layer, i.e. when the layered soil
approaches homogeneous half-space. Thus, it indicates
that the model predicts a little higher damping in case of
softer homogeneous half-space.
As the most significant parameter in the present study is
the depth of the soil layer, the computed frequency–ampli-
tude response are compared with the experimental response
curves of Baidya and Muralikrishna [4] for six different
depths of layer under a given static weight (8 kN for sand
and 4 kN for sawdust) and a given dynamic force level
(
u
¼88for sand and 48for sawdust), which are presented in
Figs. 7 and 8. From these figures it is observed that the
predicted response match well with the experimental
response curves.
Fig. 9 presents a plot of the response of the foundation for
different mass ratio, b0by cone model. An increase in the
amplitude and decrease in resonant frequency is observed
with increase in mass ratio.
Fig. 8. Comparison of frequency –amplitude response curves for various depths of sawdust layer under static weight 4.0 kN and
u
¼48:
P.K. Pradhan et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 425–434432
For six different values of Poisson’s ratio the foundation
response is obtained using cone model and plotted in Fig. 10.
It is observed that the amplitude of vibration decreases and
resonant frequency increases with increase in Poisson’s ratio.
Fig. 11 shows the effect of material damping ratio on the
response of the foundation. A remarkable decrease in the
amplitude is observed with increase in material damping
ratio. But no distinct change in the resonant frequency is
observed with increase in material damping ratio.
The effect of the depth of the layer on the dynamic
response is presented in Fig. 12. It is observed that both
amplitude and resonant frequency decrease with increase in
the depth of the layer. Also it is observed that when
d=r0¼6;the resonant frequency for the layer-rigid base
system is very close to the half-space value though the
corresponding amplitude is higher compared to half-space
value. Thus, for d=r0$6;half-space theory may be applied.
6. Conclusions
In contrast to rigorous methods, which address the very
complicated wave pattern consisting of body waves and
generalized surface waves working in wave number domain,
the procedure based on wave propagation in cones with
reflections at layer-rigid base interface and free surface
considers only one type of body wave for the vertical degree
of freedom. The sectional property of the cones increases in
the direction of wave propagation downwards as well as
upwards. Physical insight with conceptual clarity thus results.
Also, the model gives accurate results when compared against
reported analytical and experimental results with vast
variation of parameters. The method based on wave
propagation in cones is well suited for foundation vibration
analysis, as it provides physical insight—which is often
obscured by the complexity of rigorous numerical solutions,
exhibit adequate accuracy, easier to use and offer a cost-
effective tool for the design foundations under dynamic loads.
Fig. 9. Response curves for different mass ratios.
Fig. 10. Response curves for different values of Poisson’s ratio.
Fig. 11. Effect of material damping ratio on the response of the foundation.
Fig. 12. Effect of d=r0ratio on the response of the foundation.
P.K. Pradhan et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 425–434 433
Based on the parametric studies, the following con-
clusions can be drawn.
1. With increase in the depth of the layer the static stiffness
decreases and reaches a value of static stiffness of homo-
geneous half-space at d=r0¼6:Also the resonant fre-
quency and resonant amplitude decrease with increase in
the depth of the layer and at d=r0¼6;the resonant
frequency approaches the half-space value though the
resonant amplitude observed is slightly higher than the
half-space value.
2. The resonant amplitude decreases and resonant
frequency increases with increase in Poisson’s ratio.
3. With increase in mass ratio the resonant frequency
decreases and resonant amplitude increases.
4. With increase in material damping ratio the resonant
amplitude decreases, but the resonant frequency remains
unchanged.
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