Gazetas, Dynamic Stiffness Functions of Strip and Rectangular Footings on Layered Media 1975

DYNAMIC STIFFNESS FUNCTIONS OF STRIP AND RECTANGULAR FOOTINGS ON LAYERED MEDIA by GEORGE CONSTANTINE GAZETAS

Diploma of Civil Engineering National Technical University of Athens (July 1973)

Submitted in partial fulfillment of the requirements for the degree of Master of Science in Civil Engineering at the Massachusetts Institute of Technology February, 1975

. . . Signature of Author Department of Givil Engineering, Novemoer 5, 1974

Certified by

. . . . . . . . ... Thesis Supervisor

Accepted by...... Chairman, Departmental Committee on Graduate Students of the Department of Civil Engineering ARCHIVES

APR 10 1975 1BRARIES

Page 66 is missing from the original.

ABSTRACT DYNAMIC STIFFNESS FUNCTIONS FOR STRIP AND

RECTANGULAR FOOTINGS ON LAYERED MEDIA by

GEORGE GAZETAS Submitted to the Department of Civil Engineering in February 1975 in partial fulfillment of the requirements for the degree of Master of Science in Civil Engineering The dynamic response of a rigid strip or rectangular footing perfectly bonded to an elastic layered halfspace and excited by horizontal and/or vertical forces and by rocking and/or twisting moments is studied. The solution is derived using a Fast Fourier Transform for a unit load under the footing and then integrating over the width and imposing the condition of rigid body motion for the footing. The results for the halfspace compared with known analytical solutions show very good agreement. The effect of the rigidity of the rock on which the soil layer(s) rests is primarily investigated. The solution converges to the halfspace one if the rock has the same properties as the soil layer.

Thesis Supervisor Title

Jose' M. Roesset Professor of Civil Engineering

Acknowledgements The work presented in this document constitutes the Master's Thesis of Mr. George Gazetas, submitted to the M.I.T. Department of Civil Engineering. It was made possible through a Research Grant, No. GI-35139, by the National Science Foundation. It is the fifth of a series of reports on Nonlinear and Coupled Seismic Effects published under this research grant. Professor Jose M. Roesset's guidance and assistance throughout all stages of this research is gratefully acknowledged. Thanks are extended to Mrs. Malinofsky for the typing of the thesis.

Table of Contents Page Title Page Abstract Acknowledgements Table of Contents List of Figures List of Symbols Chapter 1

-

Early Approximate Solutions Scope of this Work Soil Properties

1.1 1.2 1.3 Chapter 2

-

2.1

3.1

3.2

Strip Footing on a Layered Soil - Formulation

Derivation and Solution of Basic Differential Equation Layered System Boundary Conditions

2.2 2.3 Chapter 3

Introduction

-

Parametric Studies

Halfspace 3.la

Effect of Number of Points and of their Distance on the Stiffness Functions

3.1b

Comments on the Curves

Layer on Rock 3.2a Effect of Number of Points and of their Spacing on the Stiffness Function

Table of Contents Continued

Page

3.2b

Layer on Rigid Rock

67

3.2c

Layer on Elastic Rock

75

Chapter 4 - Rectangular Footing 4.1 Formulation 4.2 Results Summary and Conclusions

104

References

106

List of Figures Page 16

1-1

Evolution of Solution for Dynamic Motion of Rigid Loaded Area

1-2

Definition of Equivalent Modulus & Damping Ratio for a Hysteretic Material

17

2-1

Strip Footing on a Layered Soil

21

2-2

Wave Front and Wave Number

25

2-3

Significance of Complex Wave Number (Rayleigh Wave)

25

2-4

(Complicated) System of Reflected and Refracted Waves Resulting from a P-wave Incident in a Layered System

27

2-5

System of Co-ordinate Axes

28

2-6

Key Problem to Rigid Footing Formulation

28

3-1

Typical Cross-section

41

3-2

Stress Distribution under the Footing Due to the Fourier Transform

42

3-3

Explanation Why the Actual Width of the Footing Should be Taken between B and B'

42

3-4

Rocking Stiffness vs. A

44

3-5

Swaying Stiffness vs. a0

45

3-6

Imaginary Stiffnesses vs. a0

46

3-7

Corrected kp

vs. a0

48

3-8

Corrected k> , vs. a0

49

3-9

Corrected kxx vs. a0

50

Correlation between Wavelength of AX

52

3-10

List of Figures Continued

Page

3-11

k' vs. a0

3-12

Imaginary k

3-13

k'

3-14

k' vs. ao

56

3-15

k

57

3-16

Comparisons of F

with known solution

58

3-17

Comparisons of F

with known solution

59

3-18

Layer:

k

vs. a0

63

3-19

Layer:

k

vs. a0

64

3-20

Layer:

k

vs. ao

65

3-21

Layer of Soil on Rigid Rock

68

3-22

68

3-23

Theory of 1-D Amplification: Natural Modes of Vibration F vs. a0 (Smooth and Rough)

3-24

F

vs. a0 (Smooth and Rough)

70

3-25

F

vs. a0 (Smooth and Rough)

71

3-26

Fz

vs. a0 (Smooth and Rough)

72

3-27

Influence of Rock Flexibility on F

vs. a0

76

3-28

Influence of Rock Flexibilityon F'

vs. a0

77

3-29

F

vs. a0 (Cr s = 4)

82

3-30

F

vs. a0 (Cr

s = 4)

83

3-31

Fxr vs. a0 (Cr/Cs

3-32 3-33

53 vs. a0

54

vs. ao

55

vs. a0

69

4)

84

Fz

vs. a0 (Cr/Cs = 4)

85

k

vs. a0 (Cr

86

s = 4)

List of Figures Continued

Page

3-34

k

vs. a0 (Cr/Cs = 4)

87

3-35

k

vs. a0 (Cr/Cs = 4)

88

3-36

kz

vs. a0 (Cr/Cs = 4)

89

3-37

F

vs. a0 (Cr/Cs = 2)

90

3-38

F

vs. ao (Cr/Cs = 2)

91

3-39

F

vs. a0 (Cr/Cs = 2)

92

3-40

Fz

vs. a0 (Cr/Cs = 2)

93

4-1

System of Forces and Moments

95

4-2

Grid Used for the Evaluation of the Fourier Transform and the Flexibility Coefficients for Points under the Footing

96

4-3

k

and k

vs. a

102

4-4

kt and k;

vs. ao

103

1

1

List of Symbols

a

Cs

- dimensionless frequency (with respect to footing half-width)

B

=

halfwidth of strip footing

Cp

=

dilatational (P) wave velocity

Cs

=

shear (S) wave velocity

p

=

soil density

=

normal stresses (a,

T

=

shear stresses (T

H

= thickness of the soil layer

v

=

y/g

Iy, az) ,T

xz T

zy

Poisson's ratio

nth natural cyclic frequency (rad/sec)

n

o

=

= cyclic frequency of excitation (rad/sec) =

Lame constant

G

=

shear modulus

7T

=3.14159 ...

u

= horizontal displacement

List of Symbols Continued [F]

=

compliance matrix

zz = vertical flexibility function Ft

=

torsional flexibility function

F

=

real part of F

F

=

imaginary part of F

[K] = stiffness matrix = [F]IV Kxx = swaying stiffness function K

=

rocking stiffness function

K

=

cross-coupling stiffness function

k

=

vertical stiffness function

Kt

=

torsional stiffness function

t

=

time

= percentage of critical internal damping of the soil = rotation (rocking)

0

= rotation (torsion)

List of Symbols Continued w

= vertical displacement

E

=

y

= shear strain

strain (c ,

y,' ez)

(y

,yyz'

zx)

= rotation with respect to i,j

Wj

VV

=

E

change in unit volume (=

2

V2

= Laplace operator =

2 + Dy

e +

x

y + 6) z

2 +

= directional cosines of the wave front

m,k,n

= dilatational (P)and shear (S)wave numbers

h,k

T,B = top and bottom matrices *

*

*

*

PY, Pz , M

P,

*

,

*

M2, M

=

forces and moments acting on the footing

F

= swaying flexibility in the x direction

F

=

swaying flexibility in the y direction

F

=

rocking flexibility function

= cross-compliance (flexibility) function

List of Symbols Continued U()

=

Fourier transform of u(x) at z = 0,

W()

=

Fourier transform of w(x) at z = 0,

=

=

=

S()

{ u(x)ew x ~00

x

- ix

faa

(x)e

=

Fourier transform of G(x) at z = 0,

=

frequency of excitation (Hz)

=

natural frequency of soil layer (Hz)

=

2M+l = total number of points representing the free

surface =

2m+l = total number of points under the footing.

x dx

13 CHAPTER I - INTRODUCTION

The machine foundation problem has recently received very much attention due to the new trend towards larger machines and the detrimental effects of the resulting vibrations of the ground on nearby structures.

The whole problem can be divided into a number of sub-

problems: (1) the dynamic response of the footing of dynamic energy;

supporting the source

(2) the response of the nearby structures due to the transmission of energy through the soil; and (3) the response of the structure supporting the machinery due to the vibrations of the machine and the footing. The objective of machine foundation design is to keep, for a given frequency, the amplitudes and velocities or accelerations of the footing of the structure it supports, or a nearby structure below certain critical values which depend on the function of these structures. The parameters on which the response of the footing depend for a given frequency and applied force of the machinery are: (1) the geometry of the footing (shape and dimensions, embedment, mass and mass moment of inertia); and (2) the soil properties (layers and their dynamic properties). The latter parameter is very difficult to determine. Various models have been suggested to simulate the dynamic stressstrain behavior of the soil.

The simplest and most widely used is the

linear viscoelastic model, with the hypothesis of a homogeneous, isotropic semi-infinite elastic solid (halfspace). Use of this model does not imply that soil is actually thought to be a fundamentally viscoelastic material.

Rather, this model is used because it can be easily

handled mathematically, and, by suitable choice of parameters, its response can be made to fit the key features of the response of a hysteretic material. The whole machine foundation problem is a very complicated one. It is a wave-propagation problem with mixed boundary conditions: that is,force and displacement compatibilities.

In other words, it re-

quires matching the displacements of the soil and the structure under the footing while leaving the free surface without normal or shear stresses. Most of the studies and research done on this subject assume perfectly elastic halfspace.

Very recent solutions based on the finite

element method consider the soil as a series of layers resting on rigid rock. In this work, both a rigid strip footing and a rectangular footing are considered, resting on a more realistic soil profile-that is, a series of layers resting on an elastic rock, through which waves can be transmitted. The rather unusual but simpler case of a flexible footing (simple boundary value problem) can also be treated with the developed computer program. The solution was derived using a fast Fourier transform for a concentrated load under the footing and integrating across the width while

imposing the condition of rigid body motion in this area.

1.1

Early Approximate Solutions Figure 1.1 shows some of the early approximate solutions in

historical sequence, indicating the assumption made concerning the distribution of stresses on the contact area.

If the distribution

of stresses in the contact area is predetermined, the displacements will generally not be uniform, and hence the solution will not be completely accurate.

Sung and Bycroft used the static stress distri-

bution. Thus the solutions arrived at are probably good for very low frequencies, but at higher frequencies the distribution of stresses changes and the accuracy of the solutions decreases.

Lysmer and

Richart derived solutions by taking into account the frequency dependence of the stress distribution under the footing. Use of the Finite Element Method with energy-absorbing boundaries gave great impetus to the whole field, and thereafter a vast number of solutions have been obtained by various researchers.

1.2 Scope of this Work The problems considered here are the steady-state harmonic vibrations of a massless rigid strip or rectangular footing resting on the surface of a layered system and being excited by forces applied on it. The soil is considered as a series of linearly viscoelastic, homogeneous and isotropic layers resting on top of an elastic or rigid rock.

ASsumea

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'

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In Chapter 2 the formulation is presented for the case of the rigid strip footing.

In Chapter 3 compliance (flexibility) and

stiffness functions are presented for the case of an elastic halfspace, a single layer of soil on top of rigid rock, and a layer on top of elastic rock of variable stiffness. Comparisons are made between the responses of the above cases.

Flexibility functions for

a so-called smooth footing (relaxed boundary conditions) are then compared with those above.

In Chapter 4 the formulation of a rec-

tangular footing is summarized and results are presented for a few cases.

Conclusions and recommendations are finally given in Chapter 5.

1.3 Soil Properties At strains less than 10-5, soil is nearly elastic, with viscoelastic action present to a very small degree. At somewhat larger strains, however (of the order of 10-3) nonlinear effects begin to show.

Figure 1-2 presents the stress-strain relationship of most soils

subjected to symmetric cyclic loading conditions.

Each cycle of load-

ing results in energy loss due to hysteresis, and time-dependent effects are secondary in importance compared with nonlinear effects. For strains higher than 10-3, time-dependent effects may become important, while still secondary to nonlinear effects. According to the theory of "equivalence," the shear modulus ("equivalent") of the linear system is taken equal to the slope of the line connecting the tips of the hysteresis loop in the T-y axes, and the damping of the linear system is taken so that the area of the sys-

19 tem's hysteresis loop equals the area of the real material hysteresis loop. The key in the theory of equivalence lies in picking parameters consistent with the expected level of strain. The above model can be mathematically expressed by considering the soil moduli in complex form: G = G1 (w)+ i G2 (w) ~G + i G2 X = Xl(w) + i "2(w)

X1 + i X2

where, according to the above-mentioned, G , Xi (i = 1,2) are almost frequency independent.

Poisson's ratio is taken as a real quantity.

The imaginary part of the moduli is associated with the energy The ratio

loss due to hysteretic damping.

G2/G

= tand

is called

6 is the loss coefficient: that is,the phase lag

the loss tangent.

between force and displacement during cyclic loading. In general the damping capacity $ equals: $ = 27r tan6

Defining as damping ratio

2

TrdT

(for small a).

the ratio of the viscous damping coeffi-

cient to the value of this coefficient necessary to suppress the periodic free vibrations, the relationship P = 4Trr

6 = 2a

Therefore G

=

is true.

and the shear modulus can be written:

G(l + 21M).

This formula was used in this work.

20 CHAPTER 2 - STRIP FOOTING ON A LAYERED SOIL - FORMULATION

2.1

Derivation and Solution of Basic Differential Equations The equations of motion for a linear, elastic and isotropic

medium and plane strain conditions are:

xz _

x+

2

2

Dxz +

=

w

at

The strain-displacement relations are:

x

aw

E

au

ax

z

az

xz

au + aw

z

and the constitutive relations are: a

= xE + 2 G x

az = xE + 2 G z = G

xz

yxz

where E

S, p

axu

+ aw

az

-

+ z

G are the Lame constants

and _2v G (v = Poisson's ratio).

ax

-SSo

-oodv

s zero

2,W 51> S

fl

>-T E NDS

1.

Iv) C, I Vn

)

Pr)

FicoonsF 2.

-

CS To

V

INFINITY

G-R TRiP FooTING

ON

A.LA'fEE.ED

SOIL

Calling wxz

-

and after some manipulation, one can

easily get two independent differential equations:

(+ 2G)

(

2 + )(3x2

2 G

3x(

z +

-y

(1)

3z)

2

32 W xz z2 )

xz

at

(2)

or alternatively V2E

a2E

= 2 P cp

(3)

2 aw (4)

oxz Cs

with c= p

cs

X + 2G p

=

V=

32 + 2

ax2

32 3z

These are the classical uncoupled wave equations of the dilatational (3)and shear (4)waves. The general solution of the wave equation can be written in the form:

with k 2 + n2

F ( x + nz + t) C

where k and n are the projections on the xz axes of a unit vector normal to the wave front, i.e., normal to planes of constant phase. The F function describes a disturbance which is propagated through the medium with the velocity c. The form of the waves which is described by F remains unchanged as the wave propagates. In this report only harmonic excitations of the footing are considered.

The response consists of two parts: the free vibration and

the forced vibrations (steady-state response) with the frequency of excitation. So for an excitation of the form P = P0eiwt, the response is:

E = E(x,z) eiot

xz

= W

(x,z) eitt

xz

Substitution of the above expressions in the wave equations leads to solutions of the form: E =A eiwt Wxz =B eiwt

or

E toxz

=

A e

eit

e±iw(x + nz)/cp , etiw(

x + n z)/cs,

±loth(px + nz)] t

= B ei[wt ± k(t x + n z)]

k2 + n2

_ 1

k, + n'2= 1

where:

h

2 2 , and k

2 2 are the dispersion relations,

p

s

and

h & k are the wave numbers of the dilational and shear waves respectively.

Setting

h= h ,5 k

hn =hz'

=

k9,

=k

h

=

kn

h2 + h

x

k k2 = k + k . z x

and

z

That is, k , kz (or h , hz) can be interpreted as the x,z components of a vector k (or

I)perpendicular

to the wave front and having magni-

2 2 2 = 2 2 Wdta tude k= k2 + kz /cs, provided that k , k k, then

k

=

k 2 -k

=

±

k -k

i

2

=lax a

=

real, positive.

Wx = B e-az ai(ot ±k XX

Then

xz

a

This equation o eaie represents xdrcinwt a Rayleigh wave propagating eoiyc=in the postiv positive

or negative (+)R x direction with velocity c - k <

and with amplitude decaying exponentially with depth.

- wave

X+

kz

Figure

2.2..

ront

WAVE

FRONT

AND

WAVE

z = C+ot)

NUMBE..

)

~U-)

XC

-k

Z jw

Fi(GuR E 2.3.

S I GN IFICANCE (gAY

LA I G

OF

CoMPLE)X

WAVE )

WAVE

NUMBER.

2.2

Layered System Since every incident shear or dilatational wave produces two

reflected (S and P) and two refracted waves (S and P) at the interface between two layers, there will be a system of P-waves (longitudinal waves) and S-waves (shear waves) propagating in the positive and negative x and z directions. A treatment based on the principles of reflection and refraction is possible but mathematically complicated. A more straightforward approach is to consider for each layer a local system of coordinates and expressions for E, LOx xz

E =

ei(wt - hkx)

E eihnz + El e-ihnz eikn z+

=xz (w

I

Where the terms E

ei(wt - kk x)

" e-iknz)

I

w represent waves travelling in the negative

,

z direction (upwards) and the terms E

,

,

waves travelling downwards.

The components of the displacements in the layer can be obtained by the simple relations: 2 uU = - 1 DE h k +

W =

h

z

awxz z (6)

2

xz

k2

DX

and the components of the stress

lQa(eK iPSI

[c1oL~-iA-

I Feuoga 2.4.

COMPLICATaTB AND

SYSTEM

REFRACTED

A FRoM LAN(EReD

P-WAVE $NCTEF.

WAVES

OF

REFLECTED RESULTING

INcN1)ENT'

iN

A

FtIwe2. 5.

Sysdcm of

co-cotinacxe- cL-Ae5

-P-I-z 'pk i

m

I

I

a

-TM

Fi~ure 2.G.

RE'i'

PROBLEM

FiND TRE (OF S1

-THE

To' PJQ3it

]'ISPLACEMEWNTS

FK?.E- SUR-FAC-C-) EAP MUI

ATr

F00TING A7 -ttE

FoR.- A

T-HE

FORMIULATioW.

op.4GrI

uI-T

W~OKMAL

ANDJ

S=

E + 2G

(7) G (U

T

+3)

Using equations (5) into (6)and (7): u = +

i k

e\ i(wt - k9 x)

+ 2 i n- e - ikn z k )

(ikn z

+ (-2i +ne

w - +

ei(wt - hzx)

eihnz E' + i 9 e-ihnz E)

o

ei(wt - h9,x)

-ine ihnz El + ine-ihnz Ell h (_h

- (2 j-eikn zw' + 2i

4

i(wt - kk

e-ikn z w.e

(8) a = + (x

+(4Gn 1

eikn z

'

_ e-ikn z

"

ei(wt

-

h9x)

ei(wt - k9 x)

=(2 Gn. eihnz E - eihnz E ) ei(wt - hgx)

T

-

2.3

eiknz E + e- ihnz E

+ 2Gn 2

+(2G(n 12

912)

eikn z W' + e-ikn z

)

e (t - k9

x)

Boundary Conditions Relations (8)hold for every layer. Since the layers are con-

sidered as welded at the interfaces, the boundary conditions at the interface between jth and j + 1)th layer are

u. (H.)

=

u

(0)

w. (H.)

=

W

(0) (9)

a

(H.)

=

a

(0)

T

(H.) =

T

(0)

U. (H.)

or in matrix notation,

=

U

u

(0), where U =

w

In order to satisfy these equations for any x, I

H.k. = h.

s .. =

= k

I

k.

j = 1, 2, ...

z,.

,

y

(10)

(Snell's law of refraction) Due to (10) the left-hand side of equations (9)can be written in matrix form as U. (H ) = B. A. ei(Wt J 3 33J

- hkx)

= B A. f(xt) and the right-hand side as Ugg (0)= T.

UBOT = BA f(x,t), E, A =

, E"

li

A

f(x,t)

UTOP = TA f(x,t), where mairi ces

and B, T the "bottom" and "top" respectively are as given below:

-2ijn

20

-21 h

(42Gn2) -2Gn

.

S9

-2i

-2i

X +2Gn

4Gn 2)

2G(n 12

k

q

i

Cg+2Gn 2)g

4Gn 9. q

-2Gn Pg

2G(n'2_ ' 2 )q

n q-1 2i2 1'

*n 9-1 g

(A+2Gn2 2Gnkg

q

2

2G(n'2_

ih g

g = eihnH

where

-4Gn k

2Gn P.

-2i n q -2i

2

-1g 1

-2i

.

q~1

-4Gn I p,I qI

2G(n'2_ '2 -1

eikn H

With the above notation the boundary conditions (9)can be written for the successive interfaces starting from 1-2 and ending with n-rock.

Eliminating f(x,t), which is a common factor, we get:

=T2 =T3

Bn An -T r Ar

We can write thus A

(B1 T 2 B

1

-B_1 T B_1 T 1 2 2 3

T 3 ...

B

T)

A

B_1

T B_1 U (H) n-i n n n n

...

or U(0) =T, B1T2 B2... Tn B~nU (H )=

RUn (Hn

For the case of rigid rock, where the displacements (u) are specified at the nth interface

=

U'(0) = )

R

R

R21

R22

and therefore (U)

w top

= R 1 (ubottomr + R12 Gbottom W botton

T top

= R2 1

(U)

+ R2 2 (a)

T bottom

where top refers now to the free surface of the soil deposit, and bottom to the soil-rock interface. =0 In particular, if (u) wbottom (u)

ax

w top =R 1 2 (T) = R12 R22 bottom top

T

top

(11 )

This expression relates then displacements at the free surface to forces (stresses) applied at the same level.

For the case of elastic rock U(O) = R Un(Hn) = R Tr Ar = Q Ar

"

11

(Q)

Q2 1

L

~i

Q12

Ar

Q2 2

Ar

Since there are no incoming waves from the rock for a surface excitation

Ar = 0

w top T

top

=

Q1 2 Ar

=

Q2 2 Ar

and therefore uo top

Q12

Q2

(G)

(12) top

Notice that expressions (11) and (12) can be considered equivalent. Only equation (12) need be used if one defines R =

T B1

Q = R for rigid rock

Q = R Tr for elastic rock.

That is to say, by performing an additional post multiplication of the matrix R by Tr in the case of elastic rock.

For the case of a

half space, the matrix R is an identity matrix and the matrix Q is simply Tr' Boundary conditions at the free surface Equation (12) relates forces and displacements at the free surface of the soil deposit for any layered stratum. If stresses o(x),

T(x)

are specified at the free surface (simple boundary value

problem), it is then sufficient to write

cy(x)

=

S()

=

3(E) ei +00

with

x dE

-~

dx

{a(x) e~"

and similarly for

T (x),

T (E).

One can thus solve equation (12) for any particular E, by setting for each layer

h.

=

k

U(3)

.

= -

,

3

{w()}= W(E)

022 01 Q2E)Q2

S(E

)

T(E)

and the surface displacements are then u(x) = 1

w(x)

=

2

{

U()

leading thus to

eiEx dE

W(E) eiEx dE

Rigid footing formulation For the case of a rigid footing, we have a mixed boundary value problem, where stresses are specified at the free surface outside the footing, but displacements are imposed under the footing. To solve this situation, we consider a set of 2M+l equally spaced points on the free surface, and determine first their displacements for a unit normal and shear stress pulses centered at the origin. It is possible to solve then for each one of these unit rectangular pulses a simple boundary value problem as before, obtaining for any point i on the surface

PE d. d 1

1

P0

{iJ [oi42P 0d21 u iZ

w0

d2ol

z~~0

i

d2]2

ol

{P} Z

012 0 oi.

The terms d0o are flexibility coefficients, or displacements under unit loads. Noticing in particular that, as the load moves to any other point, the displacements at all points would just be shifted by the amount the load has moved, it is possible to write for the set of points under the footing,

u

D11 00

0

w

D21

u1

l

00

ol

D12

D11 ol

00

D21

D22

00

ol

D12 01

D22 ol

12 D12 D1 1 D

ol

o

o

. 11 om

D12 om

p0x

. 21 om

D22 om

g0

D11 ..

D)12 D(m-l)

p Px

z

pz u

2

w2

Di om D12 om

ur

12 Dii ol Dol

J C2(m+1) w

D2om1 D22 om

x

D21 ol

D22 ol

D00

oo

D12

oo0

D22 D21 oo oo

pm x Pm z

1 = 2(m+1) x 2(m+1) x 2(m+1) xl

) = [D*1

(k')

(18)

Due to the rigid body motion of the footing, it has three degrees of freedom, namely: vertical translation W, horizontal translation V, and rotation q),which are related to the u., wi displacements of the (m+l) points under the half footing by the following relations:

u.

= v

w

=

w

+

i

xi

, 2m i = 0, 1, ... and in matrix notation

1

0

0

x0=0

1

0 V

V

ut

=[T]

Vw

w

xm

4) w

I

2(m+1) x 1 = 2(m+1) x 3

S3 x 1 )

The resultants of the applied point forces (stress distribution under the footing) are m Px.

+

+

-m m -m

Pz.

1

m -m or alternatively

Pz xi i

+

*

Px

Pz

0 1 0

l .

0

0

x1

0 ...

0

x-m

1

0

1

0 ...

a

1

..

1

0

Px

[T]T for x

(20)

Pz

Relation (18) is solved

PD] x

=

[D*]~

[T]

[D*]I

and due to (20) * N

V

V [T]T [D ]

Pz where

[K*

[T]

W

W

[K*I = [T]T [D*]I

[T]

is the stiffness matrix of the system.

By inversion, the flexibility matrix can be obtained: -l1 [F] = [T]~- [D* I

[T]T

and the force-displacement relation can be written as:

39

V -

1

F

=

F xx$

0

P x

F $x

F 0 $$-4

M

0

0

P Z

Fzz

*

*

since, clearly, only swaying and rocking are coupled, while vertical translation is independent of the other two.

CHAPTER 3 - PARAMETRIC STUDIES

In most of the analytical studies in the area of dynamic soilstructure interaction, the "soil" has been treated as a homogeneous, isotropic and elastic halfspace.

Only recently, the "soil" has been

considered as a series of layers resting on a "rigid" base. With the method described in detail in Chapter 2, the more general case of a system of layers resting on "elastic" rock can be solved as well. Throughout this chapter the influence of the "soil" properties (halfspace vs. layers on rigid or elastic rock) on the dynamic response of a massless rigid footing was primarily investigated.

The

results of this investigation are presented in plots of either dimensionless flexibility, or dimensionless stiffness functions versus dimensionless frequency a . Another significant contribution of the above method is the possibility of examining the case of a "smooth" footing with the same computer program, since this case (relaxed boundary conditions) is the one that has been normally solved in previous studies. The influence of the geometry (mainly the H/B ratio) has been investigated in the previous work of Victor Chang Liang for continuous strip footing and of Eduardo Kausel for a circular footing.

So

it was not given particular emphasis in this research. The solution scheme described in Chapter 2 is based on the use of the Fourier transform.

From a practical point of view, it is con-

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e)

SH*OULD

&N iD bf.

of

cdjs

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ACT7UAL LWI1TR IBE TAKGN

venient to use the Fast Fourier transform algorithms, which are extremely efficient.

It must be noticed, however, that in this

case we have really a discrete transform, rather than the actual continuous transform, and therefore the integrals do not truly extend from -w to + 0. As a result, a first question that must be

investigated is the total number of points in the discrete transform needed to get a good accuracy. A second point of concern is the number of points under the footing needed to reproduce accurately the unknown stress distribution under the foundation.

3.1

Halfspace

3.la.

Effect of number of points and of their distance on the stiffness functions for a halfspace. Because of the discontinuity of the applied load at the edge

of the footing (Pm at x < m x (Ax),

0 at x > m x(Ax)), the Fourier

transform does not converge at the exact value, but at the f(x+0 2+ f(x-0) .

So the assumed stress distribution may be like the

one shown in Figure 3.2 (for vertical vibration).

This distribution

corresponds to an increase of the width of the footing by a fraction of Ax. This becomes clear by running cases with different number of points under the footing (and therefore different ,x) and plotting dimensionless stiffnesses, k ,/GB,

k /GB2 .

Figure 3-4 shows k /GB2

x14, ' = 17 , M = L1024) m

N

2.S

9

x

= S. kOCK-I

x}

[-

\9 '28+/

II- /

Li-

/

I-st-

/

(I)

u~'U

0. 6

o.4

Qo f Figure 3.4.

.

Rocking Stiffness vs. a

N G

sW

AY(N G

V17111171=17 H ALS PACE

2.o0-

x .7

1.51-

\

\.

0

U,

..-

1.0I\

V/

/

yy=