foundation 8

8/4/2019 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


D. K. Baidya


D. P. Ghosh


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



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.

(%)


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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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.

(%)


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




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.

(%)


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

462 Geotech Geol Eng (2008) 26:453468

123


11/16

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)


(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)


(b) resonant amplitudes forgravel layer over in situ soil

Geotech Geol Eng (2008) 26:453468 463

123


12/16

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


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


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


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


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


Frequency (Hz)

Fig. 8 Comparison of

frequency-amplitude

response curves for gravel

layer over rigid base (static

weight = 8.0 kN andh = 16)

464 Geotech Geol Eng (2008) 26:453468

123


13/16

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


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


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)

Geotech Geol Eng (2008) 26:453468 465

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14/16

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)

466 Geotech Geol Eng (2008) 26:453468

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15/16

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


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


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


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


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)

Geotech Geol Eng (2008) 26:453468 467

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16/16

cost effective tool for the analysis of machine

foundations on layered soil with due reliability.

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foundation 8

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