.."iSBH

VERTICAL VIBRATION ANALYSIS OF RIGID FOOTINGS

ON A SOIL LAYER WITH A RIGID BASE

by

MEHMET S. ASIK, B.S., M.S.

A DISSERTATION

IN

CIVIL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Chairperson df the Committee

Accepted

Dean of the Graduate School

December, 1993

- ifSjl

^^^^ d ACKNOWLEDGEMENTS /UD, //O

I am deeply indebted to Dr. C.V.Girija Vallabhan, my

dissertation committee chairman, for his suggestion of the

topic and for his excellent guidance throughout this work

and Y.C Das for the valuable discussions at the beginning of

my thesis work. Also, I am grateful to the members of

my committee, Dr. Atila Ertas, Dr. William P. Vann,

Dr. Priyantha W. Jayawickrama, Dr. Partha P. Sarkar for

serving as committee members.

I would like to extend my special thanks to Dr. Jose M.

Roesset for the valuable discussion that we had in the Civil

Engineering Department at University of Texas at Austin.

I am grateful to my wife Serap Turut Asik for her

editorial comments, and her patience throughout this work.

My special thanks go to Mustafa Ulutas for allowing me

to modify his Effective Simple Numerical Integration Code

for my research.

1 1

m^M

CONTENTS

ACKNOWLEDGEMENTS ii

ABSTRACT v

LIST OF FIGURES vi

NOMENCLATURE viii

CHAPTER

I. INTRODUCTION 1

General 1

History of Studies 3

General 3

Dynamic Winkler Model 5

Wave Propagation Models: Analytical, Semi-Analytical and Numerical 11

Scope of Present Study 18

II. STRIP FOOTING ON LAYERED SOIL 21

Introduction 21

Strip Footing on a Layer 21

Formulation 21

Solution of Field Equations Given in Eq. (2.7) 26

Displacement (Compliance) Functions /j and /j 28

Forced Response of Rigid Footings 34

Numerical Results 37

An Illustrative Example of a Vibrating Strip Footing 44

Solution 46

m

^a^m-}

^ f l l T * r"'"" '"FMHlHiiiii

Strip Footing on a Non-Homogeneous Soil Medium

with Linearly Varying Modulus 4 7

Formulation and Numerical Results 4 7

III. CIRCULAR FOOTING ON LAYERED SOIL 54

Introduction 54

Circular Footing on a Layer 54

Formulation 54

Solution of Field Equations for a Circular Footing 59

Displacement (Compliance) Functions /j and /j 61

Forced Response of Rigid Circular Footings 63

Numerical Results 64

An Illustrative Example of a Vibrating

Circular Footing 70

Solution 70

Circular Footing on a Non-Homogeneous Soil

Medium with a Linearly Varying Modulus 72

IV. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 74

Summary 74

Conclusions 75

Recommendations 77

REFERENCES 7 9

IV

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ABSTRACT

A simple semi-analytical method is developed to compute

the response of a rigid footing subjected to a harmonic

excitation and resting on a layered soil deposit with a non-

compliant rock or rock-like material at the base. The method

is based on variational principles and minimization of

energy using Hamilton's principle. Nondimensional equations

are developed for a rigid strip and circular footings

resting on a layered soil media with constant or variable

modulus lying on a rigid rock at the base. The method is

relatively simple to use and has the ability to provide

variable elastic modulus of the soil with depth. Dynamic

response characteristics are plotted using nondimensional

parameters.

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LIST OF FIGURES

1.1 Winkler Model 6

1.2 Winkler-Voigt Model 9

2.1 Soil-Footing System With Constant Shear

Modulus 22

2.2 Physical Interpretation of Foundation Impedance 2 9

2.3 Soil Reactions and Spring-Dashpot

Equivalence 31

2.4 Soil-Footing System and Soil Reactions 33

2.5 Forces and Reactions on the Soil Footing

System 36 2.6 Comparison of Results for Different Mass

Ratios 38

2.7 Vertical Decay Function (j) for Different y Values 4 0

2.8 Effect of Damping Ratio on the Response of Strip Footing 42

2.9 Effect of H/B Ratio on the Response of Strip Footing 43

2.10 Effect of Poisson's Ratio on the Response of Strip Footing 45

2.11 Soil-Footing System with Variable Shear Modulus 4 9

2.12 Vertical Decay Function (J) for Varying Shear Modulus 51

2.13 Effect of Variable Modulus on the Response of

Strip Footing 53

3.1 Soil-Circular Footing System 55

3.2 Comparison of Resonant Frequencies 65

V I

•MJiMI

3.3 Comparison of Results for Different Mass Ratios 67

3.4 Effect of Damping Ratio on the Response of Circular Footing 68

3.5 Effect of Poisson's Ratio on the Response of Circular Footing 69

3.6 Effect of Variable Modulus on the Response of Circular Footing 73

V I 1

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s^^-'^mmm^

NOMENCLATURE

dr.

a Qm

B

b

= Nondimensional frequency

= Nondimensional resonant frequency

= Half length of the footing in the x direction

= Length of the footing in the y direction (=1)

= Nondimensional mass ratio

= M](GU)Y'^^±'

H ^(t) = ^i^)^'^^ dz

e

/l,/2

F

Giz)

'H

H

h

H ji^^dz u

= Modulus of elasticity of footing

= Radius of the eccentricity

= Reissner's displacement functions

= W" 1-1

= Shear modulus of elasticity of soil

Go H

= Shear modulus of elasticity of soil at the surface

= Shear modulus of elasticity of soil at the bottom

= Depth of the soil

= Height of the footing

Vlll

ii^l

-- v^ [• mmm >/W-«V..v^V>

^b

K

K.

= Moment of inertia of footing about y-axis

= Impedance of the soil

= Static stiffness of the soil

M

H 2 ( l - v ) p /jf(|) ]Gi^fdz

i dz (l-2v)J

= Dimensionless mass r a t i o

m 2(1-V)

(l-2v)_J JGW^dx

m

H

= Jp^^^z

m.

N

= Unbalanced mass (on machine)

= Shear force at the edge

n

D O

G(^f^ ax

Pit)

Po

q(x,t)

q(x,t)

R(t)

Ro

R

T

2t

V

= Harmonic force on the footing

= Amplitude of harmonic force on the footing

= Vertical traction on the soil surface

= External applied force

= Total soil reaction

= Amplitude of total soil reaction

= Radius of the circular footing

= Kinetic energy in the soil-footing system

H

= JG<l)'rfz

= Potential energy in the soil-footing system

I X

MOUHMli?]

"l^-^^^^'^flKm^ •^m.

w

w w

a

= Shear wave velocity of soil

= Vertical displacement at a point in the soil

= Surface disturbance

= Amplitude of the displacement under footing

'2(1-v)c, 1 - f l n

{l-2v) c,{H/Bf '

Material damping

= Variational notation

= Normal strain in the x direction

= Normal strain in the z direction

= Vertical decay function

1-2VA 2(1-V) B

a l + 2a

•fln

Y^ = Shear strain

V = Poisson's ratio

p^ = Mass density of footing

p = Mass density of soil

o = Normal stress in the x direction

a = Normal stress in the z direction

T = Shear stress

Q = Operational frequency of machine

X

iMMMHliit^

" ' • " • '•*•-*

CHAPTER I

INTRODUCTION

General

The design of foundations to resist dynamic loading,

either from supporting machinery or external sources, has

been a major topic of research and study for the last thirty

years. In particular, the dynamic analysis of machine

foundations is made to study the vibration of a soil-

foundation system subjected to some frequencies and

amplitudes such that operating machines supported by the

foundation or machines in the immediate vicinity of the

foundation are not subjected to severe amplitudes of

vibrations during their normal operations and, besides,

people around these machines are not disturbed. Then, the

determination of amplitudes and frequencies for performance

criteria related to performance of foundations is an

important step. A design engineer can only accomplish this

by choosing an appropriate analysis technique to be applied,

if it is available. Even though numerous analysis techniques

have been developed for such machine foundation systems, the

design engineer is still confronted with a complicated

design procedure, where he has to deal with oversimplified

soil parameters.

Sij-j^a,i^tmmititm ^M^uidMidiHBMiii»;^^^^^^^^^^MBd^^^BS

smiMKJi.i .- i-i-iwi'mw,'"?i^^

If examined, one can easily understand what a complex

phenomenon the vibration of a footing on a soil medium is.

The dynamic analysis of the system needs wide knowledge of

soil and structural dynamics, wave propagation and soil

properties, which are separate disciplines in engineering

that need to be combined.

Soil-foundation systems can be classified according to

the material and geometric characteristics of the footing

and the soil underneath. Therefore, a soil-foundation system

can be identified by its shape such as strip, circular,

rectangular or arbitrary; amount of embedment such as on the

surface, embedded or placed deep into the soil; the type of

soil profile such as layered medium on rock, layered medium

on a half space or a deep uniform medium; and the footing

flexural rigidity such as flexible or rigid footing. To

analyze the vibration of the footing, an analysis method

based on material and geometric characteristics of the soil-

foundation system should be selected.

The most important problem in the analysis of footings

is to predict the frequency of the footing by determining

impedance and inertia characteristics of the overall medium

and the amplitude of the footing by determining damping

characteristics (geometric and material damping) of the soil

medium. The Raleigh waves carry away 67% of the energy

(Fung, 1965). Then these types of surface waves cause the

radiation damping which plays an important role in the

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amplitude of the footing, especially with regard to layered

soil on rock.

History of Studies

General

Before any method of analysis was developed, machine

foundations were designed by the rule of thumb. For example,

to be sure about the safe design, one had to build a massive

foundation. In this case, the total weight of foundation was

equal to three to five times the weight of the supported

machine. By doing this, the resonant frequency of the system

was being decreased due to the increase in mass of the

foundation.

In the 1930's, as a result of tests done by German

DEGEBO (Richart et al. , 1970), valuable information was

obtained to develop some empirical analysis procedures.

These procedures were used until the 1950's. The important

aspect in these methods was the attempt of the researchers

to find the resonant frequency of the system by defining

"in-phase mass" and "reduced natural frequency" concepts.

In the "in-phase mass" concept, some empirical soil mass

was being added to the footing mass to more closely get the

resonant frequency of the system. Actually, this added "in-

phase mass" represents the inertia of the soil, since the

inertia forces developed in the soil have an effect on the

ttmrnm

mm lM^'^f:^i

resonant frequency through dynamic interaction. Of course,

this was not what the researchers had in mind.

The other method, which is called the "reduced natural

frequency" method was developed for the same purpose as the

"in-phase mass" method. Because of the inertia of the soil,

the natural frequency of the footing decreases. The early

researchers thought that the natural frequency of the system

was a function of or was affected by the contact area,

contact pressure and the properties of the underlying

medium. They tried to incorporate the above effects by

multiplying the natural frequency of the system with the

square-root of the average vertical contact pressure. This

method is also known as the Tschebotarioff's "reduced

natural frequency" method (Gazetas, 1983).

The early methods, stated above, were only concerned

with the resonant frequency of the system. They were not

able to predict the amplitude of vibration at the operating

or resonant frequency.

If one looks at the historical development of the

methods of analysis for footing vibrations, it can be

observed that there are two main schools of thought which

were developed parallel to, but at the same time almost

independent of each other until recent years. One replaces

the effect of the underlying medium or soil with a spring or

a bed of springs without considering dynamic interaction and

this concept originated from the classical Winkler model

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used for the analysis of elastic foundations. The Winkler

method developed in 18 67 is also known as "elastic subgrade

hypothesis" (see Fig. 1.1). The other is the wave

propagation modeling that inherently considers a dynamic

interaction between the footing and the soil, and originated

from the study of the vibration of an elastic semi-infinite

medium (half-space) by Lamb (1904) for vertical harmonic

line loading. Each school of thought will be discussed under

a separate heading.

Dynamic Winkler Model

The Winkler Model is a very popular method to model

footings resting on a semi-infinite medium. Winkler first

introduced this method in 18 67 in the analysis of rail-road

tracks resting on soil as a static problem. He was very

successful because of the physical appropriateness of the

problem. His model replaces the soil medium by vertical

independent springs which only represent elastic behavior of

soil or stiffness of the soil as a more specific term. In

the equilibrium of forces in the vertical direction, inertia

forces can be easily introduced to solve the problem

dynamically. This is known as the Dynamic Winkler Model. The

most important simulation failures of this model are: lack

of the representation of continuity of the medium, ignoring

the damping of the medium because of geometry, and lack of

B -m • i t i i ia i i i ia i l l l l i i i i iMl^

Bf •PPPmiiPPVMi>«">"*""'^>*>wK>*«iaM

P(t)

k< k< k < k<: k< k<: k< k < k

Fig. 1.1 Winkler Model

irli^iMliiilMiiii*»r

ff^ffl

consideration of dynamic interaction between the soil and

the footing. Despite its inherent inadequacies, it is still

in use among many engineers for the analysis of both static

and dynamic problems. Its attractiveness arises from the

very simple form of the equations. Also, the value of the

spring constant, k, is being determined more accurately

because of the availability of reliable data from field

experiments done with the sophisticated instruments

available at present.

During the 1960s, Barkan (1962) conducted some plate

bearing tests in the field to get the spring constant k.

From these tests, he has prepared tables and empirical

formulae to easily estimate design values of the subgrade

reaction for several types of soil for each possible mode of

vibration. The Winkler model with k from plate bearing tests

can give reasonable information about the response of a

foundation in the low-frequency range. By ignoring geometric

damping, the results obtained by this method around the

resonant frequency may not be accurate. But this model was

the one which was used for the design of machine foundations

in India and it was incorporated as a design procedure into

the 197 0 "Indian Standard Code of Practice for Design of

Machine Foundations" (Prakash, 1981). There is no doubt that

results are conservative and provide reasonable estimation

of the resonant frequency. But the designer has to notice

that in case of high damping, the resonant frequency of the

aaaiifadta mmgmmtfiig^

•gi—ggBW . • ..jj^^^gwawjtf*^ '"^'^'^^^^^''"""^""^'"^^^ff^^^gB^^^S ??r<rrt*s^' -•;—

8

system will be importantly affected and amplitudes will

drastically decrease. Therefore, for some modes of

vibration, the designer may end up with highly conservative

results. For the rotational modes of vibration, the designer

can even end up with unsafe designs (Gazetas, 1983).

Amplitudes of vibrations are directly related to the

dissipation of the energy from the system. To simulate this

energy dissipation, viscous dampers can be added to the

system parallel to the elastic springs (see Fig. 1.2). This

procedure is called the "Winkler-Voigt" model and the 1971

USSR machine foundation code is based on this model (Barkan

and Ilyichev, 1977). Dynamic tests are made to determine the

dashpot constant. From these tests, it is observed that

there is a discrepancy between the spring constants obtained

from dynamic and static repeated loading tests. Barkan

introduced an in-phase soil mass to polish this discrepancy

regarding the size of foundation, properties of the medium,

mode of vibration, etc.

The Winkler and Winkler-Voigt models are both completely

empirical. In static analysis, the Winkler model is known as

a one-parameter model regarding the spring constant, k.

Developments have been made by many researchers to improve

the Winkler model by adding one more parameter to describe

the soil. But these developments were limited to static

analysis only. In 1954, Pasternak developed a two-parameter

model regarding a parameter, s, for the shear layer in

mm

•pi^^Hm^^^^—".—™^^"^^liW»^«llBijJi«i«iil I i.,.~,v.

P(t)

7777

cLiJ k CL±i

7777

Fig. 1.2 Winkler-Voigt Model

ifliiiiiMMilfl

fS""^

10

addition to the spring constant, k. Vlasov and Leont'ev

(1966) derived formulae which relate Pasternak parameters to

the subgrade displacement profile by recoursing to a virtual

work principle. Vallabhan and Das (1991) developed a new

model that is mathematically consistent in determining the

so-called "y" parameter and named their model the "Modified

Vlasov Model." All of their studies are about static

analysis (Vallabhan and Das, 1988a, 1988b, 1991). For

dynamic analysis, Baranov (1967) developed a model by

utilizing the wave propagation concept to define the

behavior of the system. He utilized wave equation solutions

for simplified conditions. His improvement was the

integration of an internal coupling mechanism by connecting

the vertical thin soil strips with horizontal springs which

work horizontally and vertically because of the displacement

differences in adjacent strips. Nogami (198 9) utilized the

two-parameter model concept. He improved the Pasternak model

by adding more shear layers which are connected to each

other by vertical springs. In his derivation of related

equations, he used the energy principle. In another study,

Nogami and Leung, (1990) improved Baranov's approach, which

is based on a Winkler approximation.

IMHUMlilPI

11

Wave Propagation Models: Analytical. Semi-Analytical and Numerical

The reason for calling the following models "wave

propagation models" is that they consider the propagation of

stress waves in a semi-infinite or even finite soil medium,

and the dependence of the stress wave propagation on the

solution of the dynamic behavior of the footing. The

origination of this model should be credited to the

classical solution by Lamb (1904).

Since Lamb's work is the cornerstone for the development

of these wave propagation models, any student studying the

theory should start from his solution. Lamb first studied

the response of the elastic half space subjected to

oscillating vertical forces acting along a line. Thus, he

solved the two-dimensional wave propagation problem. He

repeated his work for horizontal forces and forces inside or

on the surface of the medium. Later he solved the three-

dimensional case in which a single concentrated dynamic load

is acting on the surface of a body, and the problem was

called the "Dynamic Boussinesq Problem." This solution

formed the basis for the study of oscillation of footings

resting on a surface of half-space (Reissner, 1936; Sung,

1953; Quinlan, 1953; Richart et al., 1970).

In 1936, Reissner (1936) developed the first analytical

solution for the vertically loaded cylindrical disk on an

elastic half-space. His solution is considered to be the

first engineering model. He reached a solution by

^^^^^^n •••"'• ' ^ " ' ^

12

integrating Lamb's solution over a circular area for the

center displacement, ZQ;

^ Pr,Exp(i(£i) , ^

Gi 0

where,

Po= amplitude of the vertical force,

G= shear modulus of the medium,

ro= radius of the circular disk,

f\Si~ Reissner's displacement functions.

His solution was approximate because he assumed a state

of uniform stress under the footing which provided

mathematical simplicity in the solution of the problem. His

important contribution is that his work shed light on

radiation damping that had not been realized until then.

During the dynamic interaction, stress waves originating

from the contact surface propagate through the medium as

body and surface waves. These waves carry energy away from

the footing and this causes a significant drop in the

amplitude of vibration as a result of energy dissipation.

This reduction was realized as "geometric damping" in later

years.

The assumption of a uniform stress distribution is

unrealistic for many foundations. During the mid-1950s,

Quinlan (1953) derived equations for parabolic, uniform and

rigid base stress distributions. He introduced the solution

for only rigid base stress distribution. In the same year

^^^^^m\

wm VHP ii^gH^^S^.

13

Sung (1953) also set up equations for three different stress

distributions and presented solutions for all cases. The

studies of Quinlan and Sung were an extension of Reissner's

solution. The approximated stress distributions used by Sung

(1953) under the footing are as follows:

(1) Uniform approximation:

a. = P^Expiiayt)

7cr'

0 = 0

for r <r^

for /')ro;

(2) Parabolic approximation:

G. = PoExpiioyt)

nr^

a =0

for r<r.

for r)rQ;

(3) Rigid base approximation:

a, = P^ExpiiCiyt)

2nro4ro ' r'

a.=0

for r<j'r

for r>/o

In these solutions, the displacement is described at the

center of the circular disk. The parabolic and uniform

stress distributions produce larger displacements at the

center than at the edge, which correspond to the

displacement pattern of the flexible footings. The rigid

base stress distribution produces a uniform displacement

pattern under the footing. Quinlan's and Sung's results are

valid both for circular and rectangular footings lying on an

elastic half space.

^SSiSSS^mBaaM^

' fl^fTPsn^l mmmk^'^^-^F^

14

Other approximate solutions were found by Arnold,

Bycroft and Warburton who assumed a static stress

distribution under the footing. Arnold (Arnold et al., 1955)

studied the vibration of circular footings for four modes of

vibration, which are horizontal, vertical, rocking, and

rotational. Bycroft (1956) and Warburton (1957) presented

similar studies. Bycroft was the first who presented the

solution for vibration of a footing on a layered medium.

Most of the researchers who deal with this problem today

know that the stress distribution under the footing changes

with the frequency of vibration. With a static stress

distribution the resonant frequency obtained becomes higher

than the actual resonant frequency.

The aforementioned solutions depend on the assumption of

the stress distribution under the footing. In reality, the

problem is a mixed boundary-value problem, which means the

described pattern of the displacement is imposed under the

footing and zero stresses on the rest of the area. By

considering the above boundary conditions and ignoring

secondary contact stresses (relaxed boundary) under the

footing for the sake of simplicity, researchers started to

solve the footing vibration problems in a more realistic

way. Awojobi and Grootenhuis (1965) presented a solution for

strip and circular footings on an elastic half space. In

1965, in his doctoral thesis, Lysmer (1965) studied the

vertical vibration of circular footings by discretizing the

•MM m^.

circular area into concentric rings, each having a frequency

dependent uniform stress distribution. His model was

consistent with the boundary conditions. By modifying

Reissner's solution, Hsieh (1962) was the first who showed

that a footing lying on an elastic medium can be represented

as a single degree of freedom system. Later, Lysmer

introduced a frequency independent stiffness coefficient,

K^, and dashpot coefficient, C^, for the medium vibrating in

the low frequency range. For low frequencies, the footing

resting on a semi-infinite medium is treated as a single

degree of freedom mass-spring-dashpot system, which is known

as Lysmer's analog. As a result of his studies, K^ and C^

are defined for low frequencies as follows:

K,= AGR l-\)

and C„ = 3AR^ 1-0)

V ^ where G is the shear modulus of soil, R is the radius of the

circular footing, V is Poisson's ratio of the half-space,

and p is the soil density.

After Lysmer, similar studies were done by Hall (1967)

and Richart and Withman (1967) for other modes of vibration.

In their studies, they could not find good agreement between

the resonant frequency of the actual system and the simple

Lysmer mass-spring-dashpot system. Thus they suggested that

a fictitious soil mass be added to the foundation mass for

each mode as shown in Table 1.1

•..:AI UUdftdiyi l i^

m am q m ^^^s^^^sm

16 Table 1.1 Equivalent lumped parameters for analysis of

circular foundations on elastic half space

S t i f f n e s s K

Mass r a t i o m

Damping r a t i o

F i c t i t i o u s a d d e d mass

Mode V e r t i c a l

4GR

1-1)

m( l - 'u )

4pR'

0.425

0.27m

m

H o r i z o n t a l

8G/?

2-1)

m(2-D)

8p/?'

0.29

0.095m

m

R o c k i n g

8Gi?'

3(1-D)

3 / , ( l -D)

8pi?'

0.15

(l + m)m'^^

0.247,

Tn

T o r s i o n

16GR'

3

7.

0.5

H-2m

0.247,

m

1^,1, = mass moment of inertia around a horizontal and vertical axis,

respectively; Damping ratio =C/C^ where C^=2(/s:w)'" or C^=2{Kiy" for

transiational or rotational modes of vibration, with / = / or 7 for

rocking or torsion, respectively.

In addition to the aforementioned studies, Luco and

Westman (1971), Karasudhi et al. 1968 and Veletsos and Wei

(1971) spent considerable effort to obtain impedance

functions for strip, circular and rectangular footings on a

half space by considering the problem as a mixed boundary

value problem. During the mid seventies, Luco (1974) and

Gazetas (1975) developed analytical solutions for the same

type of footings on the surface of a layered medium with

rigid rock as the last layer or a layered medium lying on a

half space. Further, they continued studies on the vibration

of footings (Gazetas and Roesset, 1976, 1979). These results

were not only about extreme cases of a layer on rigid rock

m

ne ^^mamm^^--

17

and half space, but also about a layer on rock with a finite

rigidity. Gazetas and Dobry (Gazetas et al., 1991a, 1987a,

1987b, 1985, and Dobry et al., 1984a, 1984b, 1985) have done

more work on the stiffness and damping of arbitrarily shaped

foundation. They have published many useful charts, tables

and graphs that assist engineers to design practical

footings subjected to vibration.

As has been happening in all areas of engineering, the

use of digital computers with large storage memory has

boosted different approaches in footing vibration research.

The derivation of dynamic finite element formulations and

the introduction of viscous and consisting boundaries made

it possible to simulate infinite media reasonably and to

analyze any kind of foundation (on the surface, embedded or

having arbitrary shape, etc.). Some problems, which formerly

could not possibly to be solved analytically because of

complex geometry, have been handled easily by this method.

It helped to verify and calibrate analytical solutions. In

spite of its versatility, the finite element method is too

complex and requires too much memory and computation time on

the computer in solving three-dimensional problems.

Researchers have used the finite element method effectively

to solve two-dimensional problems in plane strain and in

axisymmetric cases even with nonlinear properties of the

medium.

— — ' " " • ' " " miM

lfE5!! TSSTf-nTS

During the last two decades, a new method called the

Boundary Element Method (BEM) has been employed for a wide

class of engineering problems. The most important advantage

of this method is that it is possible to reduce the

dimensionality of the problem by one, and it requires

discretization only on the boundary instead of the whole

domain. This technique allows a considerable saving of

memory on the computer. First, Dominguez and Roesset (1978)

solved the dynamic response of rigid foundation by the BEM

in the frequency domain. After that, the BEM was used by

Spyrakos and Beskos (198 6a, 198 6b) to solve the dynamic

field equations for two-dimensional rigid and flexible

foundations and to find the compliance functions for a rigid

strip footing on a viscoelastic soil, etc. In the paper

written by Ismail and Ahmad (1989), the dynamic response of

rigid strip foundations on a viscoelastic soil subjected to

vertical excitation was investigated in detail for three

types of soil profiles, namely, a homogeneous half space, a

stratum-over-half space and a stratum-over-bed rock.

Scope of the Present Study

Previous studies had done little on the vibration of the

surface strip and circular footings on layered media. In

this study, a very simple new model is introduced to solve

the problem of vibration of footings on layered media. The

Medium can be a single soil layer with constant modulus

^ ^ iBSB^i^tiiy^l

'J.._iJlU-!«

iim-nwiM -as i:i.iti±fir3L---w..5»*.-J5fca*t-»'

19

along the depth or a layer with a variable modulus along the

depth as an approximation to a multi-layered medium.

Chapter II is devoted to strip footings resting on an

elastic soil layer, with constant or variable modulus, which

in turn rests on the rigid rock or rock-like material. The

soil is infinite (i.e., there is an open boundary) in the

horizontal direction. The governing differential equations

are derived using variational principles and Hamilton's

approach, and these equations are in coupled form. An

iterative numerical procedure is applied to solve these

coupled equations. In the variable modulus case, due to the

complexity of the differential equation for a decay

function, (j), in the vertical direction, the finite

difference method is employed to solve the related

differential equation. Also, a tri-diagonal matrix solver is

employed to take advantage of lesser storage. Even very

small PC's can be used to solve the above problem by the

computer program written for the new model. Results are

compared with the available published data to verify the

present results for rigid strip footings resting on a layer

underlain by rigid rock.

Equations for a circular footing resting on a layer with

constant modulus and on a layer with a variable modulus are

derived in Chapter III. The procedure used for the strip

footing is followed for the circular footing, too. The

solution to the differential equation of a surface

^a ms^^mtim

•II f l - - ^

20

disturbance for circular footing is a Modified Bessel

Function and numerical values of the Bessel Function are

included in the program. An iterative procedure is applied

to determine the parameters which depend on the

nondimensional frequency, a^. In order to achieve

verification of the model, a comparison is done with the

available published data for a rigid circular footing

resting on a layer underlain by rigid rock.

-•aH^BBBBBi tf^l

^S^^^^^^^V* - .a w?aW5Kfv7^*«jrB»p g^UJfflMUIiWWWIillil Ill I I iiiiiiiillli IIIMIIilll

CHAPTER II

STRIP FOOTING ON LAYERED SOIL

Introduction

As it is known, footings are generally placed on a semi-

infinite soil medium which is layered, and they are usually

embedded and resting on a rigid base. But, in this study

footings placed on the surface are studied. For small

deformations, the soil can be assumed as elastic and linear.

If the soil is stratified, the modulus of elasticity of the

soil is variable. Such a problem is too complicated and

engineers use an approximation for the continuum using an

average modulus of elasticity for the soil. An idealization

of this type of continuum can be represented as shown in

Fig. 2.1. In the model developed here, it is possible to

incorporate a variable modulus of elasticity of the soil

with respect to depth. The modification required for this

analysis will be discussed later in this chapter.

Strip Footing on a Layer

Formulation

To derive the governing differential equations for the

present model, Hamilton's principle is the perfect

mathematical tool. The principle is

b\'\T-V)dt = 0 (2.1)

21

^

ip^^ m^^mmn^mmK

22

_ ^

Surface

2B \1 \

V S o i l

Medium Ps . G , V

//////// v/////////////////////////////y Rigid Bedrock

>x

77777

F i g . 2 . 1 S o i l - F o o t i n g System wi th Constant Shear Modulus

mgm» i|u .^L^iim^itygggEggs.

23

where, T is the kinetic energy of the footing and soil, and

V is the potential energy of the footing and soil. By

assuming that the footing and soil experience only vertical

vibration (i.e., M(x,j,z,r) = 0 in the soil), the kinetic energy

in the system can be written as follows;

=irj:M<?» -ii>j> (?) ^ ' (2.2)

where.

W{x,z,t)= vertical displacement of a point {x,z) in the soil,

W{x,0,t)-W{x,t)= vertical displacement of the footing,

p = density of the footing,

p = density of the soil,

jb= width of the footing (=1 in plain strain case) ,

B= half length of the footing (in the x direction),

h= height of the footing (in the z direction),

H= depth of the soil layer.

The total potential energy, V, in the soil and footing is

written as follows;

V = V -\-V . ^ ^bending ~ ^soil

av. H

- e e O

= \\\E,z\—Yfdzdx^^\\{a^z,+QX,+i:^Udzdx -BO ^^

B

-bjqix,t)W{x,t)dx

( 2 . 3 )

-B

where, E^= the modulus of e l a s t i c i t y of the footing.

ttlUMi^

mmfi'^f^ -JS-iSi

24

q{x,t)= external applied force,

^x^^:^'^xz - the horizontal, vertical and shear stresses,

respectively,

;t' 2'Y« ~ "th® horizontal, vertical and shear strains,

respectively.

Since the displacement in the x direction is zero and

defining W{x,z,t) = W{x,t)(^{z) with (|)(0) = 1 and (1)(77) = 0, it follows

that e^=0, ^ = ^ = w^ y =il^=(t)^. Constitutive relations

give the following stress-strain relations (Hooke's law) for

the soil assuming plane strain case:

' (l + v)(l-2v) dz (l-2v) dz

^ ^ Eil-v) ^^,#_^2(l-v)^^,#

' (l + v)(l-2v) dz (l-2v) dz (2.4)

(j)-_ = G(j)— '" 2(1 +V)' dx

Then, the potential energy takes the following form for the

plane strain case;

V = dw

B

- jq{x,t)Wdx -B

where.

(2.5)

1= the moment of inertia of the footing about the

y-axis,

-.-^*lii >.v.iM.w.:..V^ ^ - .11. 1 ilj J.l*MUiU^a

25

G= the modulus of elasticity of the soil,

v= Poisson's ratio of the soil.

Equation (2.5) is applicable to a footing which can have

some flexibility, such as bending in x-z plane. But in this

study, the footing is assumed to be rigid and thus, the

first term in equation is equal to zero. And the derivatives

of W under the footing with respect to x are zero.

After applying variational principle in Eqs. (2.1), (2.2)

and (2.5) and taking the variation in W and (}), the

following field equations and boundary conditions are

obtained:

1. For -B<x<B (soil surface under the footing) with bW^O

{pj.h + m)W+kW = q

with boundary conditions,

2t-z—^W = 0 ; dx -^

2. For x<-B&x>B (soil surface outside the footing)

mW-2t^ + kW=0 dx^

with boundary conditions,

2,^61^-^=0 and 2r^6l^|>0; a : - dx *

3. For 0<z<H (inside the soil) and ^^0

dz^ H

with boundary conditions.

(2.6)

(2.7)

(2.8)

%~s.Mi-

f^WBPKW- l-U IJW.' 4JUI1 I. .!•-;• g x j ^ j ^ . l

S B •J'.'V.«JIH,<.(..VS. .VMUHU '-tvr^^v^t"'i~

m—6(j) dz

H

= 0 ;

where.

//

m = jp>^^z = p 77c,

(l-2v)J dz (l-2v)77

w 2r = jG(l)'^z = G77c,

where c^ and c, a r e r ede f ined as

and

c.=//}(J)Vz , c,=-ifj<t,^& 0 J

26

m = 2(1-V)

(l-2v)

OO

^GW^dx

n=]G{^fdx •' dx

H m

Solution of Field Equations Given in Eq.(2.7)

Equation (2.7) is a homogeneous equation. If the system

is linear and the footing is subjected to a harmonic force

with a frequency Q, the response of the system is also

^ ^ • ^ -ffWS^ffWBWI^Hroffrtifriioriiiniiii • IIIr • I. Min' n

27

harmonic with the same frequency in a steady state

condition. It is assumed that,

W{x,t) = wix)e'''' and 1 =-QV(A-)e'"'.

Then equation (2.7) takes the following form:

aV k-mQ.^^ —i--( 1 )w=0. dx' 2t

(2.9)

The solution to this differential equation is a

X w{x) = C,e * +C2^^

.a.2 k-mQ^ where, (—) = . Applying the boundary condition at .x = oo,

B 2t

for a finite value of displacement, C2 = 0 and at x = B, for

the rigid body displacement of the footing Vr = Ho (at the

surface of the soil under the footing), C^^W^e^^. Then, for the vibration of the surface of the soil outside the

footing, the solution is as follows:

B<X<oo

-oo<X<-B .

--(x-B) .„

W{x,t) = W.e^ e'""

-{x+B) ._,,

W{x,t) = W^e^ e'"' (2.10)

Similarly, the differential equation (2.8) has the

following solution after application of the boundary

conditions (t)(z) = l at z = 0 and <j)(z) = 0 at z = 77;

(j>(z) = 1

l-e 2Y

1-z -^z (2.11)

^11 iipi^if 11 II' ^ssssam^^i

•J.S."KI.V..'.ai5;V Mfajmy.ijtrT

28

Displacement (Compliance) Functions / and /.

Although the topic of this section is displacement

(compliance) functions, it is better to start with the word

"impedance" as the physical interpretation of impedance is

easy, and the displacement function (compliance) and

impedance (stiffness) function are reciprocal to each other.

Figure 2.2 introduces the physical interpretation of the

impedance function. Impedance is defined as the ratio

between the force R{t) and the steady-state displacement W

at the centroid of the base of the massless foundation and

is a function of frequency (Gazetas, 1987b) . Then the

vertical impedance is defined by

Wit)

and, therefore the compliance function is

(2.12)

(2.13)

As may be seen in Equations (2.12) and (2.13), the impedance

and compliance functions are both complex functions since

the energy given into the system is not conserved. Part of

the energy dissipates through the relative motion between

soil particles and pore fluid (viscous nature) or the

irreversible sliding between soil particles (hysteretic

nature) or both, which is called material damping (Kausel,

1974). Energy also dissipates through the infinite medium as

the elastic stress waves propagate to infinity. This loss of

abyfiiiMiiiiiMBiliHaHp

.-™-iAaaaancis::^- i9r<3.v.«w.w.:.V\ I' MK^MJi' \.ifm.'.i JJKAsmim

29

t R(t)

Rigid and massless

W=-R(t)

Fig. 2.2 Physical Interpretation of Foundation Impedance

BiBifiiMi&

Eli...,J*.^J.UWI •j^sm^B^Biam f " t,™- T ! I ' ^ | W»MttaK'H*v.

30

energy is called geometric damping or radiation damping

(Gazetas, 1987b). In Equation (2.13), /j represents energy

conserved in the system (or the recoverable component of the

deformation) and f^ represents energy dissipated in the

system.

For a massless rigid footing, the displacement under the

footing at time t is W = WQ which is a constant. Thus, Eq.

(2.6) takes the following form:

mW+kW = q (2.14)

where, W stands for the second derivative of W with

respect to time and q is the vertical traction on the soil

surface. Figure 2.3 is a mathematical model in which the

soil medium is replaced by forces and edge reactions. Using

Fig. 2.3, the equilibrium of forces can be written as:

2Bq + 2N = Rit) (2.15)

where R = RQe'^ is a line harmonic force per unit length of

the strip footing and R^ is the amplitude of this force. In

Eq. (2.15), N represents the effect of the continuous

medium on both edges of the footing. As a result of the

variational formulation,

N = 2t dw dx

( 2 . 1 6 ) t=±fi

dw 1 dw and in wave p r o p a g a t i o n , we know t h a t ~^~ = ~~^ (Lysmer and

ox V^ ot

Kuhlemeyer, 1969; Gazetas, 1987b), where V^ is the shear

ma

mtmmtmmmi

31

y \ A A A A A A A A A A A A A A A A

N

R(t)

N

Above figure is equivalent to following

R(t)

/TTZ /Tzr

Fig.2.3 Soil Reactions and Spring-Dashpot Equivalence

ill,..^u.^..>, ..-^'-^ ^ ^ s ^ ^ ^ ^

wave velocity. Substituting dw ~d^

32

and 2t in equation (2.16), A'

takes the following form:

N = iG{H IB)c,a,W,e"" = N.e'"" (2.17)

I f ? 2 where c, = — J ^ dz and flp = V.

is the nondimensional

frequency. In Fig. 2.4, N^ is the lumped form of all the

shear stresses along the depth at x=B at time t. It is

assumed that these stresses are taking energy away from the

system. Therefore, A represents a shear force which will be

propagated out to infinity because of the geometry of the

problem which causes geometric damping or radiation, as

mentioned before. In Fig. 2.3, dashpots at the edges

represent geometric damping mechanisms in the system. By

combining Equations (2.14), (2.15) and (2.17), the following

nondimensional force-displacement relationship for a

massless rigid footing without material damping in the soil

is obtained:

2(l-v) c, (l-2v)(H/B)

-2(H/B)c,al+i2(H/B)c,a, \ W.G

0^ _

R. = 1 (2.18)

H # ,

where c,=H\(—)^dz. i dz 0

Material damping is not included in Eq. (2.18). There

are two ways to introduce material damping: one is to use a

complex modulus; the other is to resort to the

iSmMSA

5ij

33

/ / / / / / / / / / / / / / / / / / / / / / / / / / / /

\

\

\

\

[3

Above figure is equivalent to following:

/ r"7nfr7FirT~?rT~?

N

±.

R(t)

N

Fig. 2.4 Soil-Footing System and Soil Reactions

jommtmM

34

correspondence principle of viscoelasticity (Dobry and

Gazetas, 1985) . Both ways have been reported to give the

same results. The author also obtained almost the same

results using from the two ways. Here, for the

correspondence principle, the following equations are

written. If P is defined as material damping ratio and, k

and c are defined as frequency independent dynamic

stiffness and damping coefficients, respectively, then the

impedance can be written as follows (Gazetas, 1983) :

K = K^{k + ica^){l-¥2i^) (2.19)

where K^ is the static stiffness coefficient with <2o=0. The

displacement functions are

F = [KY and /i=Re[F], /2 = Im[F] (2.20)

where both /, and f^ are functions of the dimensionless

frequency, a^, the soil depth to footing half-length ratio,

H/B, and Poisson's ratio, v. Then, to incorporate the

effect of material damping, Eq. (2.18) is multiplied by

(H-2/p) (Gazetas, 1983; Kausel, 1974) .

Forced Response of Rigid Footings

Since rigid foundations are often subjected to harmonic

forces which are produced by vibratory machines, the force

acting on the footing can be assumed to be a harmonic force

as follows:

P = Poe iQt (2.21)

HMM

sSa^X ;?:' ""-' '' ' Miffltiiirn

35

where Q. is the operational frequency of the machine and PQ

is the amplitude of the force. In general, P^ can be assumed

to be a constant or equal to m^eQ.^ which is created by the

vibratory machine, where m^ is the unbalanced mass and e is

the radius of eccentricity.

For a harmonic excitation, the reaction R(t) from the

soil as shown in Fig. 2.5 will also be harmonic, i.e.,

R(t) = RQe' '. The dynamic equilibrium equation for the footing

can be written as follows:

MW+R{t) = Pit) (2.22)

where M is the total mass of the foundation and the

machine(s) on this foundation. Following Reissner(1936),

Quinlan(1953) or Sung(1953), the uniform harmonic

displacement under the footing can now be written in terms

of the soil reaction and displacement functions, / and f^:

JClt W = ^if,+if,)e'-. (2.23)

The physical meaning of the functions /j and f^ is

discussed later. By combining Equations (2.21), (2.22) and

(2.23), the following nondimensional amplitude of vibration

is obtained:

W,G_ / > / . ^ V^

2 /• x2

a-bXAr+{bXf2) j ( 2 . 2 4 )

rmrt •• hi iM « t t f n u25S .. - -.--r , i>Atea;

R ( t )

S o i l Layer

Rigid bedrock

36

R(t) W(t)

±.

Fig. 2.5 Forces and Reactions on the Soil-Footing System

"'""- "".'• •' •'- '— "T.Tr-^KMJW.'JWiuuwj iiii iHiiw iiJMft . •' Ti THr n n r rnMWMti i rMHTin ^"—"*" eB8SaSB B HtSi ^ = =

37

where \ is a dimensionless mass ratio, br,= and <3n is a

,. . , ^ 05 05 dimensionless frequency, cL-—r—=- .

It is convenient to define a nondimensional displacement

parameter as

W G W ^ o = - ^ - (2.25)

Po

This parameter is a function of c^, c,, H/B, v and a^.

Numerical Results

After setting up the mathematical model and related

equations, the first thing to be done is to check or verify

the model by comparing it with the available solutions. This

comparison is shown in Fig. 2.6. A series of curves is

plotted to compare the results from the new model with those

obtained from the wave propagation model of Gazetas (1983) .

This figure shows the nondimensional amplitude of vibration

WQ versus the nondimensional frequency a^ for different mass

ratios b^ with fixed parameters p = 5%, v = 0.4 and 77/5 = 2. It

is observed that, for mass ratios greater than five, the

resonant frequencies of the two solutions are very close and

the absolute error percentage changes from 0 to 8. But the

error percentage for the amplitude is high: it changes from

19 to 35- The reasons for the occurrence of this important

jt^M»>».»«»r'.'..^iini,ii • I I.I'•'•»•--..'^if

TMrriiiiiiiiiiiiiiiiiii ii ^iiiiiiiibflUMiiTNiTr

38

1.40 -1

1.20 -

§

•g 1.00 3

• Q .

E jO 0.80 CO c o

"co S 0.60

c o 0.40 -

0.20 -

0.00

0.00

Dashed Lines --> Gazetas Model

Solid Lines - > New Model

P = 5%

v = 0.4

H/B = 2

0.50 1.00 1.50 2.00

Non-dimensional frequency, ^Q

2.50

Fig. 2.6 Comparison of Results for Different Mass Ratios

isssaas

39

difference are that different assumptions are made in the

two models. In the current model, the horizontal

displacement u is assumed to be zero in the entire soil

medium. However, in Gazetas' solution, the horizontal

displacement is assumed to be zero only along the interface,

just under the footing. Assumption of the u displacement

equal to be zero in the entire soil region makes the system

stiffer than the wave propagation model of Gazetas (1983).

Also, the equations for the current model are derived by

using variational and Hamilton's concepts. It is known that

variational formulation with an assumed displacement

function makes the system stiffer, too. If a system is

stiffer, it gives higher resonant frequencies and smaller

amplitudes than the actual system.

The current model creates a shape of the vertical decay

function (|) (Fig. 2.7) which is similar to the shape of the

vertical decay function created by Raleigh waves. It is

interesting to observe that the behavior of the function ^

along depth for different y values depends on the

nondimensional frequency, a^.

The assumption of zero displacement in the horizontal

direction causes more geometric damping for the vertical

vibration. It is assumed that only shear stresses at the

edges of the footing along the depth are taking away energy

Ml Hill H ^ W ^ " ^ *

-J^ •*-*• ""ilffn^ffTr?f#IF'^g?™^«^ffl>'ffag»Ba

40

0.8 -

c o "o 0.6 c

^

> . CO o 0

• D

1 0.4 CD

>

0.2 -

0.2 0.4 0.6

Non-dimensional soil depth, Z/H

0.8

Fig. 2.7 Vertical Decay Function (j) for Different y Values

dtf

^af i iBBSaBfi fanf i^ais

41

which affects the amplitude of the vertical vibration.

However, part of the energy is actually being taken away by

the stresses in the horizontal direction, which affects the

amplitudes of the horizontal motion. This causes a decrease

in the energy which creates vertical vibration. This can

only be accomplished by introducing displacements in the

horizontal direction in addition to the displacements in the

vertical direction in the current model.

Figure 2.8 is a classical plot in vibration problems. It

shows the effect of the material damping ratio p. As is

known, the amplitude of vibration decreases when the

material damping ratio increases. The resonant frequency of

the system is not affected very much by the damping ratio.

The geometry of the system has an important effect on

the resonant frequency of the system. Figure 2.9 shows the

effect of the ratio of the soil depth to the half-length of

the footing, H/B on the behavior of the system. This ratio

has an especially strong effect on the resonant frequency.

As the H/B ratio increases, the resonant frecjuency of the

system decreases. The amplitude of the vibration is not

affected very much by the H/B ratio in a shallow layer. For

high H/B ratios, (deep soils), such as H/B equal to 10 or

more, the nondimensional amplitude versus the nondimensional

frecjuency curves show the behavior of half space response

curves as expected.

^ ^ ^ ^

iMiiiyiiywiiiiBfiifiiin n irmrTniitiTTTTinrifiiiiiiir

42

1.2 n

'5^ 1 -

0 • D

J 0.8 Q. E CO

CO c .g

'co c 0 E I

C

o

0.6 -

0.4 -

0.2 -

1 r

0.5 1 1.5 2

Non-dimensional frequency, aQ

2.5

Fig. 2.8 Effect of Damping Ratio on the Response of Strip Footing

HBM x\K\%.w.Kr.: ,.v-*svv / Y^.^i.S^~'A}^ -^'JTv^

w«l^ ' 4^

43 -x:

1 -1

0.2 -

I §°0.8

0" T3 D

• ^

Q. E 0.6 CO

nal

0 "(0

1 0.4

-di

c 0 Z

X

/ ^ s

» / /

/ ,y

/ ^

..'-'

* \

\ \ / \ « A

H/B=2

H/B=3

H/B=4

H/B=5

H/B=10

P = 5%

v = 0.4

bo=4

\

\

—1 1 1 r

0.5 1 1.5 2

Non-dimensional frequency, a^

2.5

Fig. 2.9 Effect of H/B Ratio on the Response of Strip Footing

i^i^iii±jii^i±i^&sisees&^

3 3 j B i ^ — P W ^ Biim iiT^TinniTfirnir I - IVi ' . ' . ' -

44

Figure 2.10 shows the effect of Poisson's ratio of the

soil on the behavior of the system. It is observed from the

curves that Poisson's ratio affects both the amplitude and

the resonant frequency of vibration. The amplitude of

vibration decreases as Poisson's ratio increases, but the

resonant frecjuency increases as Poisson's ratio increases.

An Illustrative Example of

a Vibrating Strip Footing

The strip footing in Fig. 2.1 has a width, 2B=3.6

meters, a thickness, h=2.25 meters and unit mass,

p, = 2403 kg/m^. An engine operating with frequency of 865 cpm

has a total weight of 90 kN and an amplitude, P = 45 kN. The

soil depth H is 3.6 meters. The soil layer is lying on rigid

rock. The properties of the soil are as follows:

ps = ll50 kg fm^; G = 20100 kN Im^; and V = 0.4. The Material

damping ratio, p = 5%. Here the problem is to determine:

(a) the amplitude of the vertical displacement at the

operating frecjuency.

(b) the resonant frequency for the footing-soil system.

_^^

^•^^mSBSSSB^^^SS^SI^SSiM

45

1.2 -1

1 -'sr

0

J 0.8 Q. E CO

"co

.2 °- "co c 0 E

c o z

0.4 -

0.2 -

/ \

/

/ • '

/ W • ; w \ • .

y / / \ \ A \

- - v=0.25

•-v=0.3

-•v=0.35

• • v = 0 . 4

— v=0.45

P = 5%

H/B = 2

bo=5

—I 1 1 r

0.5 1 1.5 2

Non-dimensional frequency,aQ

2.5

Fig. 2.10 Effect of Poisson's Ratio on the Response of Strip Footing

II I Hill IHI i i ig i i i i i i

46

Solution

(a) First, the mass ratio has to be determined as

follows:

6>o = IT, where M is the total mass which is equal to the

mass of the footing plus the mass of the machinery operating

on it. Then,

M=(3.6) (2.25) (1.0) (2403)+90000/9.81=28638.61 kg.

M bo =—-^=5

9sB

and

a^ = _ Q B _ 60

2nf^ 1 ^ ^ 1 . 8 60

" ^[GJVS K 108.76 = 1.4992 = 1.5

From F i g . 2 . 6 f o r H /B=3 .6 /1 .8=2

WnG H Q =—^i— = 0.0933 from the curve for the new model

^0

WnG VFQ = — ^ ^ = 0.1333 from the curve for the Gazetas

^0

solution.

Then,

W Q =0.0203 cm (obtained from the new model)

W^0= 0-028 cm (obtained from the Gazetas solution)

•-nmz^f

47

(b) From the same figure, the nondimensional resonant

frequencies are

^0 = 0.978 for the new model

ao=0.91 for the Gazetas solution.

Then,

^^ r. ^oK (0.978)(108.76) ,^ ^^ <3o =-|T-=^ i = — — = = 59.09 rev Is (from the new model)

^^ r^ ^o^s (0.91)(108.76) , , ^„ flo =-r—=> ^ =-^^-^ = ^ - = 5A.9%rev s ( f rom t h e G a z e t a s

V, B 1.8

solution).

Strip Footing on a Non-Homogeneous Soil

Medium with Linearly Varying Modulus

Formulation and Numerical Results

In practical problems, engineers are mostly faced with

layered soils with material properties varying with depth.

Even with one type of soil, such as a sandy soil, the

elastic modulus can increase with depth; for example, the

effect of confinement due to an overburden pressure will

easily increase the stiffness of the soil. Realistic

modeling of this kind of medium can only be accomplished by

considering a variable modulus along the depth. For

practical purposes, to simulate the elastic modulus

realistically, empirical formulas have been developed which

relate the modulus or shear wave velocity to the z-

^^'mmrmmnrrrrwmiirr Wimm UTiMTBfifi ijeBH^S£is&;±&dBS

48

coordinate along the depth for different soil types. For

heterogeneous deposits, the wave velocity changes with depth

in proportion to (l + azY (Gazetas, 1982), where a is a

positive constant and represents the rate of heterogeneity,

and s depends on the soil category. Then, a more general

form of this relation in soil mechanics can be written as:

G(z) = G,(l+azy. (2.26)

In the current model, it is assumed that the shear modulus

of the soil varies linearly with depth. This is similar to

_(G^/Go-l) assuming a= H

and s=\f then a shear modulus that is

linearly varying with depth is:

G{z) = Go 1 + (^_1)± Go 77

(2.27)

where G^ is the shear modulus at the soil surface (z=0) and

Gu is the shear modulus at z = 77 (Fig. 2.11). By recoursing

to a variational formulation with a variable shear modulus,

ec^uations are derived for the inhomogeneous soil. Most of

the equations are the same as the ones which are derived for

the homogeneous layer. The following ecjuations are the ones

which differ from the previous ones. Eq. 2.8 with the same

boundary conditions for 0<z<77 and h^^O becomes

± G{z)^ _(X)2G(z)(j) = 0 az\ dz J 77

(2.28)

.x»,i. M..^^.uu^.,.^y»x....^^^...«>^

49 I

//////A Rigid Bedrock

Fig. 2.11 Soil-Footing System with Variable Shear Modulus

•rrwy-* ' i i w • ny - ^ - - T T / ^ W E . F V W T ^ ^ ' " ' ^ ' " 'JVfrfT iiimwiiiii iwKwu^'im .. uimi..LUiui.i...^u.*tAHr. 35T"

_ f where G{z) = l + (^-l)

77 It is to be noted that G{z)

50

IS a

nondimensional function. Coefficients k and 2t in Eqs. (2.6)

and (2.7) take the following form;

* = §1^}G„G(Z)4)^..^„2(1-V)C, (l-2v)i dz Vl-2v^77 (l-2v)77

and

H 2t = JG,G{z)(^'dz = G,Hc^

w h e r e q a n d c, a r e r e d e f i n e d at

H - , d(^ c,=HJG{z)i^)'dz , c , = - i fG(z)(^

rs dz H\i 'dz

E q . ( 2 . 1 7 ) a l s o c h a n g e s i n t h e f o l l o w i n g way

N = N.e'"^ = /G(771 B)c,a,W,e''" (2.29)

where c,=^^U[G{z)f (Sf'dz

Equation (2.28) is a hypergeometric type differential

equation. An analytical solution of this ecjuation is quite

cumbersome. Here, Eq. (2.28) is solved numerically by using

the central finite difference method and integrations are

carried out numerically, too.

Figure 2.12 presents the results of the numerical

solution to Eq. (2.28) for several values of y. If Eq. (2.8)

and Eq. (2.28) are compared, the <j) function correspondence

^LjuiJUUUUiJUiLuuum^^iiiaMwgg;

51

0.8 -

0.6 -

c .g "o c >> CO O 0

• D

O 0.4

0 >

0.2 -

0 0.2 0.4 0.6 0.8 1

Non-dimensional soil depth, Z/H

Fig. 2.12 Vertical Decay Function ^ for Varying Shear Modulus

•y^_l ^IL ^^rr^Jl a',tfr "«-V»W«ivv.»,., r f^ft.aa&.i>ti»;i;,^vvt- ^^•^'^u'-ffm

52

to the linearly increasing shear modulus has a higher decay

rate, which makes the system stiffer and gives smaller

amplitudes than that of the soil layer with constant shear

modulus along the depth.

Figure 2.13 shows the response of the soil deposit which

has a linearly varying modulus. The shear modulus at the

bottom is higher than shear modulus at the surface of the

soil for most soil deposits. This means smaller

displacements and higher resonant frec^uencies for the

footing. In Fig. 2.13, this phenomenon is observed. From a

practical point of view, an analysis with the modulus

varying with depth is moreappropriate than an analysis with

constant modulus.

In the above equations, GQ is assumed to be different

from zero. For most of the soil types, this assumption is

valid. But, for the Gibson soil, which is a very special

case, GQ is equal to zero. In order to use the same computer

program to find the response of a footing resting on a

Gibson soil, some of the equations need simple

modifications.

Al iM MMttMiliDd^

iii^4

0.7 -1

0.6 -

eg'^o.s 0"

• D 13

Q. E CO

"co

g CO

c 0 E

T5 I

C

o z

0.4 -

0.3 -

0.2 -

G /Go = 1 GH/GO = 2

GH/GO = 3

Gj^/Go = 5

G H / G O = 1 0

0.1 -

P = 5% v = 0.4

bo=5

1 r 0.5 1 1.5 2

Non-dimensional frequency, aQ

2.5

Fig. 2.13 Effect of Variable Modulus on the Response of Strip Footing

tfHi

r:F»J«-.--.'*BL-^. - - ^•^.•^•i.^ rvi'iTi'rwgjtMM^ lij^'itiftiib^JB — ^ • ^ > * i ^ ^ j » *

CHAPTER III

CIRCULAR FOOTING ON LAYERED SOIL

Introduction

Most of the footing vibration problems faced in the

field by engineers are three-dimensional. A plain strain

approximation is rarely found to be applicable. In this

chapter, a new technicjue is proposed to analyze the vertical

vibration of a circular footing resting on a layered medium

with a rigid rock boundary at the bottom and with an open

boundary (semi-infinite medium) in the r direction, as shown

in Fig. 3.1. The circular footing and rotating machinery are

assumed to be axisymmetric. In order to verify the results

obtained from the current model, a comparison is made with

the results obtained from the approximate model developed by

Warburton (1957).

The current model makes it possible to introduce a shear

modulus varying with depth. Vibration of a circular footing

resting on a soil layer with a varying modulus will be

discussed later in this chapter.

Circular Footing on a Layer

Formulation

To derive the governing differential ecjuations of motion

for the circular footing, Hamilton's principle is used

54

f^-y^^^' • — I I H I T '

55

Surface

S o i l Medium

R

ft / G , V

//////// f//////////////////////////////y/ \ \ ///// Rig id Bedrock

F ig . 3.1 So i l -C i rcu la r Footing System

•.!.«|...k«-.»aiBa.-aecoc» ,-„

-^^'^'^"'•''ffff^MfflTiTflwwyfrtriiii-i i' iiiiiKifli;'- -'—t'^''"^v^ ^

56

(Eq.(2.1)). By assuming that the footing and soil experience

only vertical vibration, the kinetic energy in the

systemcan be written as follows:

• dW^-, , _ , f// r^n poo 3 ^ /.27C fR fj\y . / / -271 r« AW ,

ar dt (3 .1)

w h e r e .

W(r,z,t)= vertical displacement of a point (r.z) in the soil,

W{r,0,t) = W{r,t)= vertical displacement of the footing,

p^= density of the footing,

p = density of the soil,

R= radius of the footing (in the r direction),

h= height of the footing (in the z direction),

H= depth of the soil layer.

The total potential energy, V, in the soil and footing is

written as follows:

bending soil

2jt rR

Jo Jo

2Tr. 2 2( l -V)aW^aV (V^W)

dr dr rdrdQ +

(3 .2) 27C rR -1 rH pin i-oo r2.n pK _

w h e r e .

D = Eh'

12(1-v^J = flexural rigidity of the circular footing.

Ep= modulus of elasticity of the circular footing,

V = Poisson's ratio of the circular footing.

11 m I I ..«Pi»«»"Ta.'5aWssri.\X'oi.-v^».-,nsv»:t-:~-^3^':ac.-.l

PBli.LJ ._.!....

57

q{r,t)= external applied force,

^r^^-.^^fz - the radial, vertical and shear stresses,

respectively,

r' 2'Y;7 ~ the radial, vertical and shear strains,

respectively.

Since the displacement in the r direction is zero (u = 0),

and defining W{r,z,t) = W{r,t)^{z) with (|)(0) = 1 and <Sf{H) = 0, it

dw dis> dw dw f o l l o w s t h a t £^=0, e =—— = V^—!-,y^ = = ^——• C o n s t i t u t i v e az dz dr dr

relations give the following stress-strain relations

(Hooke's law) for the soil in the axisymmetric case:

(l + v)(l-2v) dz (l-2v) dz

^ z =

£(l-v) ^^,^^^^2(l-v)^^,#

(l-hv)(l-2v) dz (l-2v) dz (3.3)

'C. = E ^dW ^^ dw

( | ) ^ ^ = G ( t ) - — dr 2(1+ v) ar

Then, t h e p o t e n t i a l e n e r g y t a k e s t h e f o l l o w i n g form;

J 'ln fR

0 Jo

. , 2(i-v)ai a'w^ (V'l^)

ar dr'

dw

rdrdQ +

i f r fa 2 ( i z Z ) ( W . ^ ) ^ + ( ^ ^ f rdrd6dz-rf-qir,,)Wrdrd6 2 Jo Jo Jo ( l - 2 v dz dr •'° •'

( 3 . 4 )

_(1

w h e r e ,

G= modulus of elasticity of the soil

v= Poisson's ratio of the soil.

iosaXSSiilr:

^«" FfTff^tiftfiiirfffffrs; M.j...«iai>iir.w&«—gga -t * • sss

58

Equation (3.4) is applicable to a footing which can have

some flexibility, such as bending in r-z plane. But in this

study, footing is assumed to be rigid and thus, the first

term in the ecjuation is equal to zero. And the derivatives

of W under the footing with respect r are zero, too.

After applying Hamilton's principle and taking the

variations in W and (|), the following field ec^uations and

boundary conditions are obtained:

1. For 0<x<R (soil surface under the footing) with bW^O

(p.h + m)W+kW = q (3.5)

with boundary conditions,

2t^bw'=0; dr

2. Fo r R<r <oo ( s o i l s u r f a c e o u t s i d e t h e f o o t i n g )

mW-2tV'W + kW =0

w i t h b o u n d a r y c o n d i t i o n s ,

2t---6W = 0 ; ar ^

3 . Fo r 0<z<77 ( i n s i d e t h e s o i l ) and b^^O

w i t h b o u n d a r y c o n d i t i o n s .

(3 .6)

( 3 . 7 )

m—offl dz

H

= 0 ;

'fillftid^fi if I

where.

H

m = |pycfz = p 77c,

59

^=SJ<)^-=-^^ -v)c. (l-2v)J Vz (l-2v)77

H

2t = JG(^'dz = GHc,

where q and c, are redefined as

- 4 > • -^fi* 'dz Vo J

and

m = 2(l-v) (l-2v)

JGW'rdr

"-j«?> r^r

77 w

The above ec^uations are similar to the ones derived for

vibration of a strip footing except that in the last three

equations, there is rdr instead of dx.

Solution of Field Equations for

a Circular Footing

By assuming that

Wir,t) = w(r)e"^ and W =-QV(r)^'"',

...-i-JCiri.\a-ivv>.«M'Hs'si:w .

""•-fffffiffiiffgimrnimiaiiiii iiHiiii KI 'II n aB f l t e j t f aa tg^ iMB^r i rrn'n

60

e c j u a t i o n ( 3 . 6 ) t a k e s t h e f o l l o w i n g f o r m :

W-( 2r~ )w = 0. (3.8)

Ecjuation (3.8) is a modified Bessel ecjuation and leads to

the following Bessel solution:

w(r) = C/o(^r) + C A ( ^ r ) 7v R

.a,2 k-mQ' where (—) = , and 7 and A„ are the modified Bessel

R 2t ' "*

functions of the first kind and the second kind of order

zero, respectively. Cj=0, because the Bessel function 7 at

r = oo becomes infinity. The boundary condition at r = R that

W W = Wr, gives C. = — . Then, the solution to the field

' K,{a)

ecjuation outside the footing is as follows:

W(r,t) = ^K,(^r)e^^ Koia) R

R<r<oo ( 3 . 9 )

Similarly, the differential Ecjuation (3.7) has the

following solution after the application of the boundary

conditions, which are assumed to be (|)(z) = l at z = 0 and (j)(z) = 0

at z = 77 :

-X. i^(z) = -^{e"'-e''e"').

i-e 2Y

(3-10)

The above ecjuation is the same as given in Eq. (2.11).

•'^^-"~-ra~rm-m,.m..:

r^isg?«iHBaawHffl?Tn^n^-r^T^

61

Displacement (Compliance) Functions /, and f.

Most of this part is similar to that of the plane strain

case. Ecjuations are rearranged for the axisymmetric case.

For a massless rigid footing, the displacement under the

footing at time t is \ = Wo which is a constant, and VW = 0,

Ecjuation (3.5) takes the following form:

rnW+ kW = q (3.11)

where W stands for the second derivative of W with respect

to time. Then, the ecjuilibrium ecjuation can be written for

the axisymmetric case from the same Fig. 2.3 in Chapter II,

but this time N is a circumferential reaction and the

footing is a circular one :

KR'q + 2KRN = R{t). (3.12)

In Equation (3.12), N represents the effect of the

continuous medium along the edge (outer circle) of the

footing and R(t) = R,e"" is the concentrated force at the

center of the footing. As a result of the variational

formulation.

N = 2t dW

dr (3.13)

c=R

dw 1 dw and from wave propagation we know that - ^ - y ^ (Lysmer

dw and Kuhlemeyer, 1969, Gaze tas , 1987b). S u b s t i t u t i n g — and

2t i n Eq. ( 3 . 1 3 ) , N t a k e s t h e fo l lowing form:

N = iG{H I R)c,aQW/'' = N/ jat ( 3 . 1 4 )

la^imaBgiWfft-ih'MiSfflr-

62

1 f ? ^ where c, = — I (|) z . N^ is the lumped form of all the

circumferential shear stresses along the depth at r=R at

time t, which is the same as in Fig. 2.4.

By combining Ecjuations (3.11) , (3.12) and (3.14), the

following nondimensional force-displacement relationship for

a massless rigid circular footing without material damping

in the soil is obtained:

71-2(1-V) c, (l-2v)(77/7?)

- 7C(77 / R)c,al + i2K{H/R)c,aQ \ Wr^GR

o' -" _

Ro = 1 (3.15)

H d(^

where q = 77J(—)^^z . dz

As in the plane strain case, by resorting to the

correspondence principle of viscoelasticity (Dobry and

Gazetas, 1985) and by defining p as the material damping

ratio, and k and C as frec^uency-dependent dynamic stiffness

and damping coefficients, respectively, the impedance can be

written as follows (Gazetas, 1983):

K = K^{k + ica,)a + 2if^) (3.16)

where K^ is the static stiffness coefficient with ^ = 0 . The

displacement functions are

F = [K]-' and /, = Re[F], f, = H^'] ^''^

where /j and f^ are both functions of the dimensionless

frec^uency a^, the soil depth to footing half-length ratio,

H/R, and Poisson's ratio V.

mmMimk

tr^^ss^ss^^^^^^^ -•.-^,.,^^-^jjffifff^o^v.rTrniriHniim

63

Forced Response of Rigid Circular Footings

In this case, ecjuations can be derived as in the plain

strain case. A concentrated force acting at the center of

the footing is assumed to be a harmonic force as follows;

Pit) = P,e iCit (3.18)

where, Q. is the operational frecjuency of the machine and PQ

is the amplitude of force which can be constant or ecjual to

m^eQ.' which is created by the machine on the footing (where,

m is the unbalanced mass and e is the radius of €

eccentricity).

Let the harmonic reaction of the soil be Rit) as shown

in Fig. 2.5 in Chapter II. Then the dynamic equilibrium

equation can be written as follows

MW-^Rit) = Pit) (3.19)

where M is the total mass of the circular footing and

machine (s) on this foundation and /?(f) = T e'"'. The uniform

harmonic displacement under the circular footing can be

written in terms of soil reaction and the nondimensional

displacement functions,/j and /jt

Ro iOt

KjrK

(3.20)

By combining Equations (3.18), (3.19) , (3.20), the following

nondimensional amplitude of vibration for the axisymmetric

case is obtained:

WQGR_ f'+fi v^ (i-Vo7i) +(Vo72) J

(3 .21 )

&fii&£Bi

t-i-nTOrnnrirriHiTi-n' inmiiii' i •',' ' AITIIInt'ifrf ^vai^iitxi.

M

PsR'

64

where ^ _ — — is a dimensionless mass ratio and

^"-~;== — is a dimensionless frecjuency. " VoT^'v,

A nondimensional displacement parameter for the

axisymmetric footing is defined as

W.o = ^ ^ ( 3 . 2 2 )

and this parameter is used later for the related figures

Numerical Results

As a more realistic approach to real problems, the

circular foundation model is studied by plotting some of the

results. At first, it is better to compare the results from

the new model with the results from Warburton's model

(Warburton, 1957). Results are presented in Fig. 3.2. There

are differences between the two solutions due to different

assumptions made in formulating these two models. As is

known, Warburton assumed a stress distribution under the

footing as for a static case, whereas the stress

distribution under the footing depends on the frecjuency in

the new model. For H/R=l, the difference is around 5% and

for H/R=2, it is around 10%. Also, Warburton used a relaxed

boundary condition, which is the assumption of zero

secondary (horizontal) stress between the footing and the

^'^9 T^ii^'LiisiJ MiMinmrnrMrnffinftiriii imriiMiM

35 -1

65

30 -

.2 25 CO k_ CO CO CO

E 20 "co c ,g 'co ^ 15

-a c o 10 -

5 -

'. Dashed Lines -- Warburton Results

New Model

0.2 0.4 0.6 0.8 1 1.2 1.4

Non-dimensional frequency, aQm

1.6

- y

• 1 ^

F i g . 3.2 Comparison of Resonant Frec^uencies

c ^ - 1

3

inirTnnr-ii-<lr •a hVrffirm-il'iirr

66

soil, whereas, in the present case, there is secondary

stress under the footing.

Figure 3.3 is a plot of the response of the circular

footing for different mass ratios, b.= -. A result like

this is not presented in the literature for the circular

footings according to the knowledge of the author. General

trends in the behavior, such as amplitude increases and

frequency decreases as the mass ratio increases, are

observed. But the amplitudes are just about half of the

corresponding amplitudes for the plane strain case and the

resonant frecjuencies are slightly higher than those of the

plane strain case (see Fig. 2.6) . This means that the

axisymmetric system is stiffer than the plane strain system.

But, from a practical point of view, the more realistic

model is the axisymmetric one because most problems are

three-dimensional.

Figure 3.4 shows the effect of the material damping

ratio, p, on the response of the circular footing. The

resonant frec5uency range for the circular footing is wider

than that for the strip footing in the plane strain case.

Figure 3.5 shows the effect of Poisson's ratio on the

response of the circular footing. Poisson's ratio affects

both the amplitude and the resonant frecjuency of vibration.

The amplitude of vibration decreases as Poisson's ratio

ii

'"•"'^"'IliiriifliVi TlmllTTi

67

0.5 -1

i gPo.4

0 • D D

• Q .

E 0.3 CO

"co c o "co c 0 E I

C o

0.2 -

0.1

0.00

P = 5%

v = 0.4

H/R = 2

T 1 1 r

0.50 1.00 1.50 2.00

Non-dimensional frequency, aQ

2.50

Fig. 3.3 Comparison of Results for Different Mass Ratios

•:»..;n---,-T-..-4S3W3i: . V

I'lTimr-' •" ii>*Mai I « • riraiVin^^*!

68

0.4 -,

0.35 -

5 0 "D D

0.3 -

I" 0.25 CO

CO c g CO

c 0 E I

C o Z

0.2 -

E 0.15 -

0.1 -ZT^"

0.05 -

0.5 1 1.5 2

Non-dimensional frequency, aQ

2.5

Fig. 3.4 Effect of Damping Ratio on the Response of Circular Footing

> i r t - - . i l l 111 1 • ill 1 t i-fe*i*.iii...^ ^3^^

rj-jc- .-iLiv--.rj jaaBBBaesBsansss^

69

0.4 -1

^S: 0.3 -

0 • D U

"Q. E CO

"co

'co c 0 E I c o

v=0.25

v=0.3

v=0.35

v=0.4

v=0.45

P = 5% H/R = 2

0.1 - • -

T 1 r

0.5 1 1.5

Non-dimensional frequency,aQ

Fig. 3.5 Effect of Poisson's Ratio on the Response of Circular Footing

70

increases. But the resonant frecjuency increases as Poisson's

ratio increases. Both effects are substantial.

An Illustrative Example of a Vibrating

Circular Footing

The circular footing in Fig. 3.1 has a radius, 7? = 1.7

meters, a thickness, h = i.5 meters, and a density of mass,

p. = 2403 kg/m^. An engine operating with a frecjuency of 850

cpm has a total weight of 97 kN and an amplitude, 7*0=45 kN.

The soil depth, H, is 3.4 meters. The soil layer is lying on

a rigid rock. The properties of the soil are as follows:

p^ = ll50 kg fm^r G = 20100 kN Im^, and v = 0.25. The material

damping ratio, p = 5%. Here the problem is to determine:

(a) the resonant frequency for the footing-soil system,

and

(b) the amplitude of vertical displacement at the

operating frecjuency for V = 0.4 .

Solution

(a) First, the mass ratio has to be determined as

follows:

M h - ^ where M is the total mass which is equal to the

p.-R

mass o f the footing plus the mass of the machinery operating

on it:

M=(3.14) (1.7) (1.7) (1.5) (2403)+97000/9.81=42597.27 kg,

Si--,fir'r-^**i S3SS3SSSES ngit^^"^ >

71

^0 = M

9sB =5.

Then, from F i g . 3 .2 t h e n o n d i m e n s i o n a l r e s o n a n t f r ec juenc ies

a r e

aQ=0.951 from the curve for the Warburton solution

^0=0.829 from the curve for the new model.

OB «o =

V. Q =

a^, (0.957)(108.76)

B L7 = 65.22 rev I s ( o b t a i n e d from t h e

•tM

Warbur ton s o l u t i o n )

^ 0 =

OB

V. ^ floK (0.83)(108.76) ^, , / . w^ • ^ ^ Q = — - = - -— ^ = 53.1r^v/5 ( o b t a i n e d from t h e new B 1.7

model) .

(b)

OB 2nf ^ 2 71850^^

an = 60 60

' ^GT^S n 108.76 = L39

From F i g . 3 . 3 f o r H/R=3 .4 /1 .5=2

w = ^ ! ^ = 0.183. :2

Then, t h e a m p l i t u d e of t h e v e r t i c a l d i s p l a c e m e n t i s t : .^

Wo =0.0234 cm.

i i m - i i I- rTuMiniMrTtiTiliniimniiniii'm '•""fit'i.i'.i'i^Yi kLUIUti

BWTBri-r-T"-•^'^—M-.ft%iw^in V I •V.-a-.ii^MM

72

Circular Footing on a Non-Homogeneous Soil Medium with a Linearly Varying Modulus

The formulas which are necessary to be modified for a

variable modulus in the axisymmetric case are the same as

the corresponding formulas in the plane strain case, except

for the shear force, which is

jiit N = N^e"" = /G(77 IR)c,aQWQe iat ( 3 . 2 3 )

where c. = 1 (?fc

H J{G(Z)Y'i,'d2

Obviously, parameters are taking different numerical values.

Figure 3.6 shows the response for the soil deposit when

it has linearly varying modulus. The shear modulus at the

bottom is higher than the shear modulus at the surface of

the soil for most soil deposits. This leads to smaller

displacements and higher resonant frecjuencies for the

footing. If Fig. 2.13 and Fig. 3.6 are compared, the

variable modulus condition has more effect on the response

of the circular footing than that of the strip footing. The

increase of the modulus with depth makes the resonant

frequency of the system increase substantially. And also,

the amplitude of the response is decreasing while the

modulus of the medium is increasing.

~3

iMMMiiiaii

73

0.25 -i

G H / G O = 1

G H / G O = 2

GH/GO = 3

G^/Go= 5 G^/Go=10

P = 5%

v = 0.4

1 1 r 1 1.5 2 2.5 3

Non-dimensional frequency, aQ

Fig. 3.6 Effect of Variable Modulus on the Response of Circular Footing

mM m • - ^ I v - I s v ' v ^ V . "

rili»iiiilHIHIIiilllllllll Iillll .IWLlL'^g'^' |l^^l^^"^*it^J<.< "Tl'm

CHAPTER IV

SUMMARY, CONCLUSIONS AND

RECOMMENDATIONS

Summary

A new mathematical model for the response of strip and

axisymmetric footings resting on a layer with constant or

variable modulus with depth is developed by using Hamilton's

principle combined with a variational approach. This method

is semi-analytical and uses an improved form of the

classical Vlasov Model for elastic foundations to solve

steady-state dynamic problems. Using variational principles,

the governing differential ecjuations are developed for the

surface disturbance (displacement) Wix.t) and vertical decay

function (j) which are coupled through parameters k, 2t, m,

and n that describe the characteristics of the soil medium.

Two additional parameters a and y are introduced that are

dependent on the properties of the overall system and

operating frecjuency ^. An iterative procedure is applied to

find these a and y parameters.

To make the model more appealing for practical

applications, a linearly varying modulus of the soil is

incorporated into the governing differential equations. By

the mathematical procedure adopted here, it is very

convenient to incorporate this characteristic of the soil.

74

U^SSi ^k^s^

r r ^ j T / ' ^ - — j : ^ " iiHiMiiiiHiiiiii I Ki'iiii'iii'ii'ii i i T - r - ' - i r " " ^ " " ^ ""'^' - n - ^ " — ^

75

Responses of strip footings are studied in Chapter II.

Related equations and formulas for the recjuired parameters

are derived. Results from the new model are compared for

verification with previously published data. The effects of

some important ratios on the overall behavior of the system

are investigated.

Responses of circular footings are studied in Chapter

III. The same calculations are carried out for circular

footings as for the strip footings.

Geometric damping forces which are important in open

systems experiencing the propagation of waves are

represented by edge shear forces in the plane strain case

and circumferential shear forces in the axisymmetric case.

Edge and circumferential shear forces are calculated from

the elastic wave theory making them inversely proportional

to the particle velocity. Therefore, edge shear forces in

the plane strain case and circumferential shear forces in

the axisymmetric case are considered as the viscous energy

absorbing mechanisms.

Conclusions

Following are the important conclusions reached as a

result of the present study:

1. A very simple model is developed to determine the

response of strip and axisymmetric footings restings

r::^*u^MMnmtmtyrnimmHfut,.jj^mMt,nufw^^^^^^mgfBaKmB^mm^^ ^

76

on a soil layer having a constant or a linearly

varying modulus.

Non-dimensional dynamic stiffness and geometric

damping are dependent on the frecjuency, H/B ratio in

the plane strain case and H/R ratio in the

axisymmetric case, and Poisson's ratio of the soil.

The model is developed using Hamilton's principle

with the assumption of zero displacements in the

secondary direction. This assumption makes the

system stiffer.

The errors in resonant frecjuencies and in amplitudes

are in the acceptable range for practical design

conditions, which demand accurate values of the soil

modulii. The errors in soil modulii are often higher

than the errors in the frecjuency and amplitude.

A variable modulus is an important characteristic of

the soil. The effect of this phenomenon on the

response of the circular footing is more than on

that of the strip footing. It should be considered

in the design of footings.

Compared to large sophisticated computer codes

developed for this type of problems the code

developed here is simple and easy to understand and

recjuires very little memory in the computer.

- * iiimiiHiyuiiiLiii, ' ^ •WWIHB ^BS^

77

3.

4.

Recommg^ndations

The following recommendations are made to further expand

the potentiality of the model.

1. Research should be directed to eliminate some of the

assumptions made in the present study, such as u=0,

to improve the model and to obtain better results.

2. It is possible to expand this modeling concept to

the other types of loadings and footings.

For further research, displacement in the horizontal

direction shall be introduced to make it more

realistic.

The energy dissipated into the continuum by the edge

shear forces can be predicted from experiments.

Therefore, it is desirable to compare the results

that are obtained from the current model and

experimental results. This comparison can give

important insight to the geometric damping and it

would be very useful to designers.

5. This research can be extended to analyze the dynamic

response of flexible footings.

6. This study can be extended to dynamic analysis of

rectangular plates. As is known, it is very

difficult to solve the dynamic behavior of

rectangular plates resting on a soil medium by the

finite element or any other numerical method.

78

7. It is recommended to verify the results obtained

from this model by controlled experiments in the

laboratory.

' """""VAWWmi '" T . ^ . ^ 1 ^ . - I .11 - U

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