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 . . . . . . . . ... Accepted by...... Chairman, Departmental Committee dents of the Department of Civil Thesis Supervisor on Graduate Stu- Engineering ARCHIVES APR 10 1975 1BRARIES
Transcript
DYNAMIC STIFFNESS FUNCTIONS OF STRIP AND
RECTANGULAR FOOTINGS ON LAYERED MEDIA
by
GEORGE CONSTANTINE GAZETAS
Diploma of Civil EngineeringNational 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 . . . . . . . . ...
Accepted by......Chairman, Departmental Committeedents of the Department of Civil
Thesis Supervisor
on Graduate Stu-Engineering
ARCHIVES
APR 10 19751BRARIES
j_morris
Typewritten Text
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 1975in partial fulfillment of the requirements for the degree ofMaster of Science in Civil Engineering
The dynamic response of a rigid strip or rectangularfooting perfectly bonded to an elastic layered halfspaceand excited by horizontal and/or vertical forces and byrocking and/or twisting moments is studied.
The solution is derived using a Fast Fourier Trans-form for a unit load under the footing and then integrat-ing over the width and imposing the condition of rigid bodymotion for the footing.
The results for the halfspace compared with knownanalytical solutions show very good agreement. The effectof the rigidity of the rock on which the soil layer(s) restsis primarily investigated. The solution converges to thehalfspace one if the rock has the same properties as thesoil layer.
Thesis Supervisor Jose' M. Roesset
Professor of Civil EngineeringTitle
Acknowledgements
The work presented in this document constitutes the Master'sThesis of Mr. George Gazetas, submitted to the M.I.T. Departmentof 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 SeismicEffects published under this research grant.
Professor Jose M. Roesset's guidance and assistance through-out all stages of this research is gratefully acknowledged. Thanksare extended to Mrs. Malinofsky for the typing of the thesis.
Table of Contents
PageTitle Page
Abstract
Acknowledgements
Table of Contents
List of Figures
List of Symbols
Chapter 1 - Introduction
1.1 Early Approximate Solutions
1.2 Scope of this Work
1.3 Soil Properties
Chapter 2 - Strip Footing on a Layered Soil - Formulation
2.1 Derivation and Solution of Basic DifferentialEquation
2.2 Layered System2.3 Boundary Conditions
Chapter 3 - Parametric Studies
3.1 Halfspace
3.la Effect of Number of Points and of theirDistance on the Stiffness Functions
3.1b Comments on the Curves
3.2 Layer on Rock
3.2a Effect of Number of Points and of theirSpacing on the Stiffness Function
Table of Contents
3.2b
3.2c
Chapter 4
4.1
4.2
Continued
Layer on Rigid Rock
Layer on Elastic Rock
- Rectangular Footing
Formulation
Results
Summary and Conclusions
References
Page
67
75
104
106
List of FiguresPage
1-1 Evolution of Solution for Dynamic Motion of Rigid 16Loaded Area
1-2 Definition of Equivalent Modulus & Damping Ratio for 17a Hysteretic Material
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 27Resulting from a P-wave Incident in a Layered System
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 42Fourier Transform
3-3 Explanation Why the Actual Width of the Footing 42Should be Taken between B and B'
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
3-10 Correlation between Wavelength of AX 52
List of Figures Continued Page
3-11 k' vs. a0 53
3-12 Imaginary k vs. a0 54
3-13 k' vs. ao 55
3-14 k' vs. ao 56
3-15 k vs. a0 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 Theory of 1-D Amplification: Natural Modes of 68Vibration
3-23 F vs. a0 (Smooth and Rough) 69
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 4) 84
3-32 Fz vs. a0 (C r/Cs = 4) 85
3-33 k vs. a0 (Cr s = 4) 86
List of Figures Continued Page
3-34 k vs. a0 (Cr/Cs = 4) 87
3-35 k vs. a0 (C r/Cs = 4) 88
3-36 kz vs. a0 (C r/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 (C r/Cs = 2) 93
4-1 System of Forces and Moments 95
4-2 Grid Used for the Evaluation of the Fourier Trans- 96form and the Flexibility Coefficients for Pointsunder the Footing
1 1
4-3 k and k vs. a 102
4-4 kt and k; vs. ao 103
List of Symbols
a - dimensionless frequency (with respect to footingCs half-width)
B = halfwidth of strip footing
Cp = dilatational (P) wave velocity
Cs = shear (S) wave velocity
p = soil density = y/g
= normal stresses (a, Iy, az)
T = shear stresses (T ,T T xz zy
H = thickness of the soil layer
v = Poisson's ratio
n n th natural cyclic frequency (rad/sec)
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
Ft
F
= vertical flexibility function
= torsional flexibility function
= real part of F
F = imaginary part of F
[K] = stiffness matrix = [F]IV
K = swaying stiffness functionxx
K = rocking stiffness function
K = cross-coupling stiffness function
k = vertical stiffness function
Kt = torsional stiffness function
t = time
= percentage of critical
= rotation (rocking)
0 = rotation (torsion)
internal damping of the soil
List of Symbols Continued
w = vertical displacement
E = strain (c , ' y, ez)
y = shear strain (y , yyz' zx)
Wj = rotation with respect to i,j
E = V change in unit volume (= e + + 6)V x y z
2 2 2V2 = Laplace operator = + Dy +
m,k,n = directional cosines of the wave front
h,k = dilatational (P) and shear (S) wave numbers
T,B = top and bottom matrices
* * * * * *
P, PY, Pz , M , 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
= Fourier transform of u(x) at z = 0,
= Fourier transform of w(x) at z = 0,
= Fourier transform of G(x) at z = 0,
= frequency of excitation (Hz)
= { u(x)e- x
= w x -ix~00
= faa
(x)e x dx
= natural frequency of soil layer (Hz)
= 2M+l = total number of points representing the freesurface
= 2m+l = total number of points under the footing.
U ()
W()
S ()
13
CHAPTER I - INTRODUCTION
The machine foundation problem has recently received very much
attention due to the new trend towards larger machines and the detri-
mental 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 supporting the sourceof dynamic energy;
(2) the response of the nearby structures due to the transmis-sion of energy through the soil; and
(3) the response of the structure supporting the machinery dueto 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 struc-
tures.
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 stress-
strain behavior of the soil. The simplest and most widely used is the
linear viscoelastic model, with the hypothesis of a homogeneous, iso-
tropic semi-infinite elastic solid (halfspace). Use of this model does
not imply that soil is actually thought to be a fundamentally visco-
elastic material. Rather, this model is used because it can be easily
handled mathematically, and, by suitable choice of parameters, its re-
sponse can be made to fit the key features of the response of a hys-
teretic 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 per-
fectly 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 con-
centrated 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 stres-
ses changes and the accuracy of the solutions decreases. Lysmer and
Richart derived solutions by taking into account the frequency depen-
dence 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 vibra-
tions 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, homogen-
eous and isotropic layers resting on top of an elastic or rigid rock.
ASsumeastress Ad§G+ri((A +1iOV
(ylC19 2 )
QaUj- \ICxv\ 0 qS D)
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oef v+.v
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OP SOLUTIONS FCoK -A(NArAIC
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Locu~s ol +fps ck S e-Es loco? S
S h ecLrstrain
LooPS FKOM C'/CL1'Z LOATONGx IN
SIMPLE SHEAR.
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cxdsoy-bea I0{ (00?(bv'j)
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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 half-
space, a single layer of soil on top of rigid rock, and a layer on
top of elastic rock of variable stiffness. Comparisons are made be-
tween 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 visco-
elastic 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 ef-
fects 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 hystere-
sis 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 = X l(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
loss due to hysteretic damping. The ratio G2/G = tand is called
the loss tangent. 6 is the loss coefficient: that is, the phase lag
between force and displacement during cyclic loading.
In general the damping capacity $ equals:
$ = 27r tan6 2TrdT (for small a).
Defining as damping ratio the ratio of the viscous damping coeffi-
cient to the value of this coefficient necessary to suppress the peri-
odic free vibrations, the relationship
P = 4Trr is true.
Therefore 6 = 2a and the shear modulus can be written:
G = 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
medium and plane strain conditions are:
x+ xz _ 2
D xz + =
and isotropic
2w
at
The strain-displacement relations are:
x aua x E awz az au + aw
xz z ax
and the constitutive relations are:
a = xE + 2 G x
az = xE + 2 G z
xz = G yxz
whereE u + aw -ax a z + z
S, p G are the Lame constants
_2v G (v = Poisson's ratio).and
-SS o -oodv
s zero
2,W51> S -
fl ) Iv ) C , I Vn
Pr) CS V
>-T E NDS To INFINITY
G-R TRiP FooTING ON A.LA'fEE.ED SOILFicoonsF 2. 1.
Calling wxz - and after some manipulation,
easily get two independent differential equations:
(+ 2G) ( 2)(3x2
2
G z3x(
or alternatively
with
V2E
oxz
+ -y3z)
2= Pcp
Cs
one can
2xz
at
a2 E
2aw
c = X + 2Gp p
cs =
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:
(1)
(2)
(3)
(4)
32 W+ xz
z2 )
F ( x + nz + t) with k 2 + n2C
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 de-
scribed by F remains unchanged as the wave propagates.
In this report only harmonic excitations of the footing are con-
sidered. 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 = P0e iwt, the response is:
E = E(x,z) eiot
= W (x,z) eittxz xz
Substitution of the above expressions in the wave equations leads
to solutions of the form:
E =A eiwt e±iw(x + nz)/cp , k2 + n2 _ 1
Wxz =B eiwt etiw( x + n z)/cs, k, + n'2= 1
or = A e ±lot t h(px + nz)]E eit
= B ei[wt ± k(t x + n z)]toxz
2 2h 2 , and k 2 are the dispersion relations,
p s
h & k are the wave numbers of the dilational and shearwaves respectively.
h= h ,5 hn =hz'
k =
k9, =k kn k
h = h2 + hx z and k2 = k + k .x z
That is, k , kz (or h , h z) 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 + k z /cs, provided that k , k<.
Significance of a complex wave number
If either k of kz is greater than k, say k > k, then
k = k2 -k
= lax
= ± i k -k 2
a = real, positive.
W x = B e-az ai(ot ±k X Xxz a
This equation represents a Rayleigh wave propagating in the o eaie xdrcinwt eoiyc=-
postiv (+)Rpositive or negative x direction with velocity c - k <and with amplitude decaying exponentially with depth.
where:
and
Setting
Then
- wave ront
X+ z = C+ot)
WAVE FRONT AND WAVE NUMBE..
~U-)
XC -k
Z jw
Fi(GuR E 2.3. S I GN IFICANCE
(gAY LA G I
OF CoMPLE)X WAVE NUMBER.
WAVE )
kz
Figure 2.2..
)
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 inter-
face between two layers, there will be a system of P-waves (longi-
tudinal 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, LOxxz
E = E eihnz + El e-ihnz ei(wt - hkx)
=xz (w eikn z + " e-iknz) ei(wt - kk x)
I IWhere the terms E , 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:
U 1 DE 2 awxzu = - zh k (6)
W = + 2 xz
h z k 2 DX
and the components of the stress
PSI
[c1oL~-iA-
Feuoga 2.4. COMPLICATaTB SYSTEM OF REFLECTED
AND REFRACTED WAVES RESULTING
FRoM A P-WAVE INcN1)ENT' iN A
LAN(EReD $NCTEF.
lQa(eK i-
I
Sysdcm of co-cotinacxe- cL-Ae5
-P-I-z
'pk
-TM
Fi~ure 2.G. RE'i' PROBLEM To' PJQ3it F00TING
FiND TRE ]'ISPLACEMEWNTS A7 -ttE
(OF -THE FK?.E- SUR-FAC-C-) FoR.- A uI-T
S1 EAP MUI ATr T-HE op.4GrI
FORMIULATioW.
W~OKMAL ANDJ
FtIwe2. 5.
i I I a
m
S= E + 2G
T G (U +3)
Using equations (5) into (6) and (7):
u = + i k eihnz E' + i 9 e-ihnz E)
+ (ikn z + 2 i n- e - ikn+ (-2i ne o k )
w - + -ine ihnz El + ine- ihnz Ell(_h h
ei(wt - hzx)
z e\ i(wt - k9 x)
ei(wt - h9,x)
- (2 j-eikn zw' + 2i 4 e-ikn z w.e i(wt - kk
a = + (x + 2Gn 2 eiknz E + e- ihnz E ei(wt - h9x)
+(4Gn 1 eikn z ' _ e-ikn z " ei(wt - k9 x)
T = (2 Gn. eihnz E - eihnz E ) ei(wt - hgx)
- +(2G(n 12
(8)
9 12) e ikn z W' + e-ikn z ) e (t - k9 x)
2.3 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
(7)
or in matrix notation,
u. (H.) = u
w. (H.) = W
a (H.) = a
T (H.) = T
U. (H.) = U
(0)
(0)
(0)
(0)u
(0), where U = w
In order to satisfy these equations for any x,
I I
H.k. = h. = k s .. = k. z,. j = 1, 2, ... , 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(WtJ 3 33J
- hkx)
= B A. f(xt)
and the right-hand side as
Ugg (0) = T. A f(x,t)
U BOT = BA f(x,t),
E ,A = ,
E"li
UTOP = TA f(x,t), where
mairi cesand B, T the "bottom" and "top" respec-tively are as given below:
(9)
-2ij n
-21
-2i
4Gn
2G(n 12 2)
-2i n q
-2i k q
4Gn 9. q
X +2Gn 2
2Gn P.
20
-2i
-4Gn k
2G(n'2_ 2
h
(42Gn2)
-2Gn .
S9
*n 9-1i g
(A+2Gn2 -1g
2 n -12i 1' q
-2i . q~1
I I I-4Gn p, q
-2Gn Pg
where g = eihnH
2G(n'2_ '2 )q 2Gnkg 1 2G(n'2_ '2 -1
q eikn H
With the above notation the boundary conditions (9) can be writ-
ten for the successive interfaces starting from 1-2 and ending with
n-rock. Eliminating f(x,t), which is a common factor, we get:
B An n -T Ar r
i h g
Cg+2Gn 2)g
= T2
= T3
We can write thus
A (B1 T 2 B 1 T3 ... B T) A
-B_1 T B_1 T B_1 T B_1 U (H)1 2 2 3 ... n-i n n n n
or
U(0) =T, B 1T2 B 2... 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
) R 21 R22
and therefore
= R1 (ubottomr
= R21 (U)
W botton
+ R12 Gbottom
+ R22 (a)T bottom
where top refers now to the free surface of the soil deposit, and
bottom to the soil-rock interface.
In particular, if (u)wbottom
= 0
(u) axw top =R 1 2 (T)
top bottom = R12 R22 T top
This expression relates then displacements at the free surface
to forces (stresses) applied at the same level.
(U)w top
T top
(11 )
For the case of elastic rock
U(O) = R Un(Hn) = R Tr Ar = Q Ar
" 11 Q12 Ar
L ~i(Q) Q21 Q22 Ar
Since there are no incoming waves from the rock for a surface
excitation Ar = 0
and therefore
w top
T top
uotop
= Q12 Ar
= Q22 Ar
Q12 Q2 (G)top
(12)
Notice that expressions (11) and (12) can be considered equiv-
alent. Only equation (12) need be used if one defines
R = T B 1
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) = 3(E) ei x dE
with +00 -~S() = {a(x) e~" dx
and similarly for T (x), T (E).
One can thus solve equation (12) for any particular E, by set-
ting for each layer h . = k . = - , leading thus to
U(3) 3 S(E
{w()}= 01 022 )W(E) Q2E)Q2 T(E)
and the surface displacements are then
u(x) = 1 { U() eiEx dE
w(x) = 2 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 out-
side 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 dis-
placements for a unit normal and shear stress pulses centered at the
origin.
It is possible to solve then for each one of these unit rec-
tangular pulses a simple boundary value problem as before, obtaining
for any point i on the surface
PE d1. d1 P0w 0 d2 d2]2 {P}u iZ ol ol Z{iJ [oi42P 0 d21 012
z~~0 0 i 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,
D11 D12 D11 D1200 00 ol 0 1
D21 D22 D21 D 2200 00 ol ol
l D12 D11 D12
ol ol o o
Diom D12om
D21 D22om om
Diiol
. 11 D12om om
. 21 D 22om om
D11 D)12.. D(m-l)
D 12ol D00 D12oo oo0
D21 D22 D21 D22ol ol oo oo
C2(m+1) x 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 trans-
lation V, and rotation q), which are related to the u. , w i displace-
ments of the (m+l) points under the half footing by the following
relations:
u0
w
u 1
u2
w2
u r
w J
p0x
g0z
pPx
pz
pmx
Pmz
u. = v
w = w + i xi
i = 0, 1, ..., 2m
and in matrix notation
1 0
0 x0=0
1 0
xm I
2(m+1) x 1 = 2(m+1) x 3
The resultants of the applied point forces (stress distribution
under the footing) are
m
-m
m
-m
m
-m
Px.
Pz.1
Pz xi +i
or alternatively
ut
Vw
V
=[T] 4)
w
S3 x 1 )
+ +
V
w
0 1 0 l . .. 1 0
0 0 x1 0 ... 0 x-m
1 0 1 0 ... a 1
Relation (18) is solved
PD]x [D*]I
Px
[T]TPz
for x= [D*]~ [T]
and due to (20)
* N
Pz
[T]T [D ] [T]
V
W
[K*I = [T]T [D*]I [T] is the stiffnessmatrix of the system.
By inversion, the flexibility matrix can be obtained:
[F] = [T]~- [D* I
and the force-displacement relation can be written as:
*
Px
Pz
(20)
where
[K*V
W
-l1[T]T
39
V F F 0 Pxx$ x
*- = F F 0 M$x $$-4
*1 0 0 F Pzz Z
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 soil-
structure 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 gen-
eral 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 re-
sponse of a massless rigid footing was primarily investigated. The
results of this investigation are presented in plots of either dimen-
sionless flexibility, or dimensionless stiffness functions versus
dimensionless frequency a .
Another significant contribution of the above method is the pos-
sibility 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 continu-
ous 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-
RiGiT.) f:oo-rING
FIGL)PE 3A.
R cz u rt cm 3 .'2. !S-TR.F-SC DIS-rRMUTIGN L)t4DE:R. -r4r FOOTIM(jr
VERTiCAL VISKATiON) Durc To Ti4E FouPIGF -rFA-NSFOPt-e\ -
o ~r C1) of eo6 AiC the {c.. l.vcdjs Y-,rLo-o is ouQreg(rnrajtecLt
N ~ocI(IINO
ICA~
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 ex-
tend from -w to + 0. As a result, a first question that must be
investigated is the total number of points in the discrete trans-
form 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 distribu-
tion under the foundation.
3.1 Halfspace
3.la. Effect of number of points and of their distance on the stiff-
ness 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 , x = S.
M = L1024) m 9 x}
2.S [-
II- /
N
Li-
(I)I-st-
u~ 'U
//
kOCK-I N G
\9
'28+/
o.4
Qo f .0. 6
Figure 3.4. Rocking Stiffness vs. a
sW
V17111171=17
H ALS PACE
2.o0-
.7 \
1.51- \.
x
0
..-
\ /
(~1=
256 G
I024,
G4)
U,
yy= <, x=.
1) I.xmvi'=% Ax=I0.m'= 9 = 6x o.
| (04O
Figure 3.5. Swaying Stiffness vs. ao
1.0I-
V/
0.00.6O.O ^
AY(N G
~;>
M
~p,.4
j s5\at}I1S
'0
VI)
/LO C
)(N-j
E-
II Ii
0C\j
Y-'A
U
A
L
y,(-a
l
versus dimensionless frequency ao B/cs, having as parameters the
numbers of points M', n'.
The curves with Ax = 10 and 9 points under the footing, hav-
ing different total number of points, fluctuate around the curve
having Ax = 10, m' = 9, and maximum total number of points M' = 1024.
The curves with Ax = 5 and m' = 17 points under the footing fluctuate
around the curve having Ax = 5, m' = 17 and M = 1024, which is almost
parallel, but below the curve with (Ax = 10, m' = 9, M = 1024). This
must be expected, since the "total" width in the second case is 2B =
80 + 5 = 85 < 2B' = 80 + 10 = 90, of the first case, while we still
normalize with respect to 2B = 80. (With the kxx/G this did not occur,
since the normalization is done with respect to G only).
The curves obtained by dividing by the "total" width B = B + 2,
instead of B, showed the reverse trend, since the actual stress distri-
bution (Fig. 3.3) is (more likely) the 1 and not the 2 , which is
tacitly assumed when normalizing with respect to B'. So k /GB'2 was
overestimated by the moment of the areas A between 1 and 2 with
respect to centerline of the footing, which is larger when Ax is lar-
ger (1st family of curves).
An intermediate value between B and B was considered the most
appropriate in this case. So a Bequiv = B-B' = Ax yi(m + 0.5)
was used.
The results are shown in Figures 3.7, 3.8, 3.9. The kg/GB2
curves for the above two cases (AX = 10, m = 9, M' = 1024, and
,St
fne
.5
'~i~cn
Re C
kq)/G
B2
0 Q,
C/1
,
In.'
07 -~
ii0
~Ln
-. ...
- - ~ - -- ~ - -- M 102.4 M'= ,
o.' 0.4
Figure 3.8.
GE70
vs. a0
R A L FS PA CE
x= 0.
Z2x l- +
-0.5
Io
U
-o.4
-0.3
-0.2
-0.1 I I
0.6
Corrected k
SWAY I N G
vi'= 02.4m g - IAL.FS PA C F.
2.0 --
'I)
1-0J
II
c,
0.5
CLO
7t
Figure 3-9. Corrected k vs. a0
Ax = 5, m' = 17, M' = 1024) differ only slightly, and for very low
frequencies. For dimensionless frequency ao ~ 0.47t to 0.5Tr they
almost coincide. The k /GB vs. a curves (Fig. 3.8) show a very
similar trend. So the new "definition" of B was considered appro-
priate.
Comparing on the other hand the family of curves with M = 256
and variables m and Ax for low frequencies, it is evident that the
larger the number of points under the footing (and hence the smaller
the Ax), the larger is the fluctuation of the resulting curve. Thus,
the (m' = 9, x = 10) curve is worse than the (m' = 5, Ax = 20). This
can be explained as follows: for low frequencies, the wavelengths are
large, according to the dispersion relation:
X = C/f.
Thus the resulting Rayleigh waves attenuate only after a long dis-
tance (due to internal damping of the soil).
Since the distance in the x direction covered by the Ax = 20
case is twice as much as the one considered by the Ax = 10, and the
points are taken close enough to reproduce the large wavelength, the
(m' = 5, Ax = 20) curve is more accurate than the (m' = 9, Ax = 10)
one. For example, for Cs= 1600 ft/sec and frequency f = 4 cps,
x = 1600 - 400' = 20 x Ax
Ax = 20 can reproduce the motion very well
//
~1
L arcle wauieyx 4 +ke ra4i~oK e ecfi I4 SMOAIO
ca~mof 6e repr6(LcecdNY{ or- a. jcA64a bcUiumf~oer of
-x
4ke c4ress JiiS-i~tmno und e r 4K e- A iis noi~ well reoLL-c WiAh Ire L.'x
FI(3uR&E 3.10 CO RKELA-TION
AN~D 6 X .13TVJG GN WA~VE Lr A QT H
ver LuaetQO +& .
U-TTIUIV.1v. n A, . . . n A. .
3 21012.3.
2.O)
M' = 10 24
- -- M' = 2.56 r
S-M'= 256
A M o24
o M' 2.6
SANG
H-AL FSPA C
m 17 AX =M S.
'= 1 1, A X-= S.
m'= 5 , x 20.
rn =33 ,x =2.5
i nc reas r-n r' u3i tn croSu-s
-e 9 ue-AnC riodcifcc~Ateovi)
Lo Q0
Figure 3.11. k vs. ao
0.0
x 10 -- 01
H ALF -SPACE
0
C/) 7
1-0
Figure 3.12. Imaginary kxx vs. ao
0 M =256
M" =10z4,
-- - - M'1= 2 5 6
mv~ 17m'=l7 ,
, Y '-: )
1.5 D/)I
AX 5.
Ax =2= .
20
21-
\.j -
I I I.O)0
25
2D
0
77 W
M _ -i102.4 ) M _ 1-7 Ax = 5.
-- -- - -- M ' V56 , m'=( ,- 6 - =6.
m~ 256 , 'Z A S x=20.
A M W=toa4 m 33 , AX=2.5
o lncreastn rw) Loit incyeas(
-freqLtenc
n q
(neu>O YVodLicd-onL )
M-AL~$PAC~
0.51-
-II ^- - 1. 5 2. .U. tj
Figure 3.13. k v
0.0 5
vs . ao
oC.RI
xIU-)
*1L
O0
0-
-x
UL-
CI o M 2:5 6o ,m' 17 ,AAx S.
1 02.4, m' Z 1'7 s, S.-~M '. '= 5L , yyl x = 20
A M 10'-4,
o.S
m'= 33 , b'>( = 2.S
1.0
Dimensov\est regue-ncJ
Figure 3.15. k
- I __
7UJ- QS
vs. ao-. 1
2.o
x
f.0
01 x W 'eA'i i N G
G F G -Re (xx '^ RALF S PA C E
-- - -.- G FxI G -Im= I.xx)c roo
0/o -G---- G F, Loco AN1> waS-MN O'
6>4TRAPOLATE-D 9- coaRCeseoNCo cePRINc(RLE- .
o.4 -
ot
0O 0.2.-A O.G .a,
Figure 3.16. Comparisons of Fxx with Known Solutions
KOc.K cG
Gz FGS-Re CF ,9)2
G~R
LuCo AND VESTMANN
FYATRAPOLATeDe coKESePo-OGNc 0
?R1 NcI PLE
0.2 0.4.0
FA.iFigure 3.17. Comparisons of F #with Known Solution
LL0
(D
.4
U
E0
o.G
=G i?2- .1 cF99
60
However, for higher frequencies the (Ax = 10, n = 9) curve
becomes smooth and coincides with the (Ax = 24, rn' = 5). For fre-
quencies higher than 12 cps the (Ax = 20, m = 5) deviates signifi-
cantly from the (M' = 1024, Ax = 5, m = 17) curve, which covers the
same x distance, but with closer-spaced points, and thus is more
accurate. However, the (M' = 256, Ax = 5, m = 17) curve very slightly
differs from the M = 1024 one. This indicates that for high frequen-
cies (20 - 40 cps) the important parameter is the spacing between the
points and not the total covered length along the free surface. The
above concepts are illustrated in Figure 3.10.
The need to change the number of points under the footing, and
hence their spacing Ax, as the frequency increases, led to imposing a
criterion of "good reproduction" of the motion; as it was disclosed
from the previous discussion 8 points per wavelength are sufficient
for this purpose. That is,
Ax< - 5-88f
or, using dimensionless frequency,
a _B 2rfB .. m = 4
or total number of points under the footing
m =8 + 1.7IT
Some typical values are:
a 0
0.5
1.0
1.5
2.0
m
5
9
13
17
The computer program was implemented so as to automatically
increase the number of points under the footing as the frequency
under consideration surpassed the above limit. Figures 3.11 and
3.13 show the points obtained by the modified program. The total
number of points, M , is of relatively secondary importance, except
for small frequencies. In this research it has been, almost always,
taken as 1024 points. 512 or even 256 points would give almost as
good results.
3.lb Comments on the curves
Figures 3.16 and 3.17 show dimensionless flexibility functions
vs. a0, as well as a comparison with solutions by Luco and Westman.
Figures 3.11 to 3.15 show stiffnesses versus a .
The horizontal flexibility starts from an infinite value at
zero frequency (static solution), decreases very rapidly with frequency
up to a = 0.47T - 0.57, and continues to decrease very slowly there-
after. The Luco and Westman solution, which was extrapolated by use
of the correspondence principle, gives very similar results.
The rocking stiffness starts with a value 2.35, which is in
good agreement with the one computed from the formula:
k
=2i 2 x (--.3 = 2.25 (static solution)GB2 -2lv 2x(10)
Then it decreases almost linearly up to a frequency a0 ~ r, and there-
after it has a constant value. The rocking compliance agrees well
also with the Luco and Westman solution.
3.2 Layer on Rock
3.2a Effect of number of points and of their spacing on the stiffness
functions
A very important parameter in the dynamic response of the rigid
footing on a layer of soil is the H/B ratio, as will be further illus-
trated in this chapter. Due to the increase of the actual width of
the footing by Ax (or of the "equivalent" width by (,fn(m+O.5) - m) Ax),
the H/B ratio is different for different Ax. This is clear from a
comparison of two cases: 1st, (M' = 1024, Ax = 2.5) and 2nd, (M - 256,
Ax = 10). The results (Figs. 3.18, 3.20) are significantly different
even for very low frequencies, despite the fact that the total length
covered in the x direction is the same in both cases.
In order to maintain the same geometry for a given ax, the thick-
ness H of the stratum should be modified so that
3.0 -0
+ 2.0 --f-rY
x/
s'OC
C+-
/ ( :/ 1
V////////
-440+ 7x
ao /JtFigure 3.10. Layer: kxx vs. ao
0 -
0---
u.0-
-
10
LL-
m=2s6 m I=S C-Bl (G
024m 12(C4 l. )
K26 '= 5 (g = .)
~5CKI NG
I/iL / L11 I
7n 17 7 711RIG-
R /RG
kf /GB
0.0 O.Z
Figure 3.19.
040.4
a.JC
Layer: kog vs. ao
4.O-
/G
Z .0-
1.01-
3.ol-
c
F-iC).5 - 1.oo 5
LL
.4 .o
.31-
.21- o so
=2sG) m I la (A -
M 1024 17O/n .%
Y// /////\77 -
I myR0G1R OC K(tis K .it
- - -~
CLO
.0Figure 3.20.
O.4-Layer: k vs. ao
a2
0-
<2
c~.L
C-1-
U
LL
V
-1-i
(J)
.010
-R 9 (: -x 9) /ci P,
H _ H _ constant for all m'
B 1+ B and Ax.
After the modification, the (256, 10) curve was only very slightly
different from the 1024, 2.5) one.
3.2b Layer on a rigid rock
In terms of stiffness and flexibility functions, the halfspace
and the elastic layer resting on rigid rock differ in two ways:
- First, the static stiffness increases due to the presence of
the rigid rock, so that at frequencies near zero the displace-
ment f or fzz has a-ffiite value instead of being infinite.
- Second, in the case of the halfspace, the vertical radiation is
large, especially at low frequencies, and slightly decreases
with increasing frequency. This leads to smooth response curves,
plotted against frequency. In other words, there are no reson-
ance phenomena in the case of a halfspace. In the case of a
stratum over a rigid medium, however, there is no radiation damp-
ing at low frequencies, since the generated body waves reflect
on the rigid rock, propagate upward, reflect at the free surface
or the footing, and go downward and so on, until they decay due
to the internal damping of the soil. As a result, there will be
certain frequencies of vibration at which resonance occurs. At
these resonant frequencies, the motion tends to infinity for zero
internal damping, because no energy is required to sustain the
motion and the vibrating footing continuously transfers energy
to the soil.
These frequencies can be predicted approximately by the theory
of one-dimensional amplification, according to which the natural fre-
Fi are . L
68RIGID Foo-TlNG
\OO. ?e t
V 0. .40
R ICT I D Rk cK C t<\\'4
A' ER o F SO(L oN (2.ICrID
2= Mode
FiT -.7 . T E -b AMPL(FcA~T ON-
Ro c K
r- Mode
\l(BRA-TloN,NArURAL MObES OF
Yd I .- =-Y 0
142.2
-r - o o . = 0 5
c__ _ GsooG .osx~
o.4 -
RIGID
ROCK
G F G Re(F
- -- -G F '/ = G Imn (Ex)
swoo
ct 0
Figure 3.23. Fxx vs. a0 (Smooth and Rough)
I\RocKIN C
V)=o
Q4 YOC+' o+A
GB2 F ('rou'" {ooing )Qoo~ 4(oo4c) "s&v'Aool"
S
\D
LLL
0-
oJ
o..
/1'
/4,
0.2.
Figure 3.24.
0.4 aOSF
F vs. ao(Smooth and Rough)
I or-
24
GB'
-
0.2--
0. r 0.
0.91i
.4t-
i
2-Boo -
o)= I y 00
s--= oo
ID
eQF rouL
cr b F "Soo"
srmooW
0.4
a.OSL
Figure 3.25. F vs. a (Smooth and
.U ~
.0 ~
0o -
z7
-. 05
Rough)
4d
- .28 -
- -C=.oo. KO- . v=o.4
RIGID ROCK
smootk'oo.4-
K
x
c.o (S.
Figure 3.26. Fz vs. ao (Smooth and Rough)
-j
+ .
U
d
0
-. I-G Fi
//
GFa
iMLI YYLR=2
73
quencies of a stratum coincide with the natural frequencies of vibra-
tion of a shear beam of soil having length equal to the depth of
the layer, fixed at the bottom and free at the top.
For shear waves the natural frequencies are:
Cf = s (2n - 1), n = 1, 2, 3,
with Cs = A/p = shear wave velocity.
For longitudinal (dilatational) waves they are:
C= (2n - 1), n = 1, 2, 3,
S+ 2G 2(1-v)with C = = CS 1 -2V
Figures 3.23, 3.24, 3.25, and 3.26 show the compliance func-
tions fx, f , f0, fzz for the case of a layer with C5 = 800 ft/sec,
y = 100 pcf, v = 0.4, and H/B = 2. An internal damping 6 = 5% of
critical was taken for the soil layer in all the cases studied. f
has three peaks within the studied range of frequencies, 0 < ao = Tr
The maximum peak occurs at a frequency a /Tr = 0.24, or
a /Tr - C _ 0.24 x 800 =r 2B 40
which is almost the same as the 1st natural frequency of the stratum
fs _ 4800 = 5 cps.
At frequencies lower than 4.8 cps, the imaginary part of the
fxx has very small values, corresponding to the internal damping
of the soil, since there is no radiation in this range of frequencies.
Just above the resonant frequency the radiation damping significantly
increases due to Rayleigh waves which carry away most of the energy
transmitted to the soil.
A second resonant frequency at a0/Iz 0.58 corresponds to the
propagation of dilatational waves
a ls 21 )=0.24)( 2 0.4) ~ 0.588
'_'"Av1 - 2 x 0.4
The third peak at a/r = 0.72 corresponds to the second natural
mode of shear vibration of the soil
= 3 x 0.24 = 0.72Tr
Similarly, f has a peak at a frequency somewhat less than the
natural vertical frequency a1 , since rocking is influenced primarily
by dilatational waves and secondarily by shear waves. This leads to
a less narrow peak compared with the f . For higher frequencies both
real and imaginary parts have values which are almost constant (i.e.,
they are independent of frequency).
The cross-compliance function f is negligible for a 0.4r.
At a = 0.48Tr and a = 0.571T , it has two peaks, one positive and
the second negative.
The vertical displacement function fzz shows three peaks at
a0/Tr = 0.44, 0.58, and 0.65. The second resonance apparently corre-
sponds to the 1st vertical mode of vibration. The first, which
reaches a much higher peak, is due to an unknown combination of S-
and P-waves, and cannot be predicted by the one-dimensional theory.
The same can be said for the 3rd peak which is negative.
Generally speaking, the motion is a complex combination of
waves and cannot be completely predicted by the one-dimensional theory,
which only predicts some of the resonant frequencies.
3.2c Layer on elastic rock
It is interesting to examine the effect of the rigidity of the
rock on the dynamic response of the foundation. The case of an in-
finitely rigid rock (Csrock = oo) has been examined in section 3.2b.
Two other cases were run with Cs = 3200 ft/sec (stiff rock) and C5 =
1600 ft/sec (medium stiff rock). The results are presented in Figures
3.27 to 3.40.
As expected, the peaks of the flexibility curves are lower and
wider than those of the rigid rock case, due to the radiation of energy
from the soil stratum into the rock. The resonant frequencies, however,
change only very slightly.
Figures 3.27 and 3.28 show the real parts of the f xx and fog func-
tions for the cases of halfspace (1), layer on rock with Cs = 1600 (2),
layer on rock with Cs = 3200 (3) and layer on rigid rock (4).
C
Cr
0.2.
H 2
(hafspace)
0.4.
Figure 3.27. Influence of Rock
x
0s
CLO:rt
Flexibility on F x VS. ao
L.0j-
Ll
Cr .4-z - -oEi=
C;s
C',S
Wo.6
U-S
Q)
o .
0.0o.2 0.4 0 .6 o
Figure 3.28. Influence of Rock Flexibility on FQvs. a0
The "static" part of the swaying flexibility (f ) veryxx
slightly increases as the Cs of the rock decreases from co to 3200 ft/
sec. For smaller C s(1600), however, the flexibility tends to infin-
ity, being a little smaller than the halfspace one.
The peak at the 1st resonant frequency of the curve (3) (Cs k
= 3200) is 33% lower than that of the curve (4) rigid rock), this
being the most important effect of the elastic over the rigid rock.
In the case of Cs = 1600 (2), this peak has been very much suppressed
and the subsequent valley flattened.
The second resonant frequency has been decreased in the case
(3) to the value of a = 0.5r (contrasted to the 0.58r), which implies0
less participation of the dilatational mode of vibration. This was
expected since the P-waves propagating downward are partially refrac-
ted in the elastic rock, and therefore they do not contribute to the
vibration of the foundation. The third peak almost disappears, and
at higher frequencies the flexibility is practically zero. The curve
(2) does not exhibit even the 2nd peak and is generally very flat,
being more like the halfspace curve (1) than the rigid rock one (4).
The rocking flexibility is much less influenced by the rigidity
of the rock. Curves (3) and (4) are almost identical to (2) except
for the peak, which has very slightly shifted to the left. Curve (2)
has a very smooth peak at resonant frequency a0 = 0.4r (< 0.477r of the
rigid rock or 0.45r of the stiff elastic rock), which means even less
participation of the dilatational mode in the vibration of the footing.
The flexibility to vertical vibration changes greatly as the
stiffness of the rock decreases. The static compliance tends to
infinity instead of having a finite value (0.165). The second peak,
corresponding to the vertical resonant frequency, completely disap-
pears, and the third negative peak is very much suppressed. The
first peak shifts towards the left but decreases much less than the
second and third ones. The explanation is again the transmission of
vertical P-waves in the rock and the lesser participation of the
vertical mode of vibration in the resonance. Again, the curve for
not-stiff rock (C = 1600) is very similar to the halfspace one, which
justifies, to an extent, the continuing use of halfspace solutions to
predict the motion.
"Internal Damping"
The effect of decreasing the internal damping of the rock is
shown in Figures 3.29 and 3.30. Rocking is almost independent of S,
while swaying shows some sensitivity to it. But since the value of
8 = 0.005 of critical, it is unlikely to be so low; it can be conclu-
ded that the material damping of the rock is unimportant. The impor-
tance of the soil internal damping has been extensively examined by
Victor Chang Liang and was not considered necessary to be reinvesti-
gated in this work.
"Smooth" versus "rough" footing
In the case of a rigid disk perfectly bonded to an elastic lay-
ered halfspace, stresses and displacements are continuous at the inter-
face between disk and soil. This problem is commonly referred to
as the complete mixed boundary value problem ("rough" footing).
The solutions presented so far are solutions to the complete problem.
If it is assumed that at least one of the components of sur-
face traction at the interface is zero, then a relaxed boundary value
problem ("smooth" footing) results. The relaxed problem, extensively
studied so far, assumes that for vertical and rocking vibrations the
contact surface is free of shear stresses, while for horizontal vibra-
tions the contact surface is free of normal tractions. Consequently,
the horizontal displacements under the disk are unconstrained for
vertical and rocking vibrations, and the vertical displacements are
unrestrained for horizontal vibrations.
Veletsos and Wei and Luco and Westmann have obtained numerical
results for this relaxed problem ("smooth" footing). With the program
developed based on the above (chapter 2) formulation, both the "smooth"
and the "rough" footing cases can be studied.
The flexibilities of a "smooth" foundation on a layer of soil
resting (1) on a rigid rock and (2) on an elastic rock with Cs = 1600,
are compared with the ones of a rough footing(Figures 3.37, 3.38, and
3.23 to 3.26).
In the case of a layer of soil on a medium-soft rock, the differ-
ence between "smooth" and "rough" is very small. Only the imaginary
part of the rocking shows a little higher peak (at dimensionless fre-
quency a0/7r = 0.5), while the swaying (real + imaginary) curves are
almost identical.
In the case of the soil on the rigid rock, however, the rock-
ing flexibility of the "smooth" footing showed much higher peaks
in both real and imaginary parts. The sliding flexibility, however,
is virtually the same.
Large depths
For large depths of the soil stratum and very small Ax (which
is required for a good reproduction of the motion at high frequencies),
the factor g = e ihnH, encountered in the "Bottom" matrix, becomes
very large. Indeed, since
h = -/ n =\1- 12
g = exp -i 1 - 12 H)- exp (i ({)2
and as H/Ax surpasses some certain limit, g becomes very large, lead-
ing to an overflow.
The explanation of this is that for high frequencies the wave-
length is small and decays at very shallow depths. Thus, the exist-
ance at a large depth rock does not influence the motion of the footing.
This explanation is the basis of the correction made. After a
certain H of the soil stratum, such that the H/Ax ratio exceeds a cer-
tain value, the soil profile is modified by considering (elastic)
rock below this depth.
GRe CFJ') r .s
G I C FCKX,G Fx procI - C
/
\0.4 \
0.2 / \
0.00.0 0.2 0.4.
Figure 3.29. Fxx vs. ao
N
CLo3t
(Cr/Cs) = 4)
0.6
IY'Ay -=
I / / / /
(,%
/ I
N-
Gz. Re(Fq)G -I P CF9)
-=o.o
r,= 0.05
/
//
/
- C.005
u.q.
Cr o
F4 vs. a 0 (Cr/Cs
o*G
Figure 3.30. = 4)
lo-O
0.4 d-
6
0.0
O.GF- k
0.2.t-
-f =z283
Roc4K IG - SWAY[MG
.15
.t-s --- GT-CFC c- 00.L ROC..K
.05 -
CL7-
.00
-. 05
.10o / /F-o
Figure 3.31. F vs. a0 (Cr/Cs =4
0.12
.4
G F =CG Re (FzQJB
d .0-i- .12
0
00(
Figure 3.32.
// - I00 -
- ' =040-ELASTIC ROCK
c= azoo.
G Fz"= G I CFz)
0.4 0. GCLr
F7z vs. a0 (Cr/C = 4)
=- 2.
IL-z
~~iKJ)/G
CS Soo. =0-05SoV .40.
-LASTIC ROCK
cs Szoo. f -=-0,V o.30
U)
0.4Figure 3.33. k vs. a0 (Cr C = 4)
LO-o(I3t
0-I riY77---O
R.0QkrdG
'O
10.4
ELASTIC ROCK
30.
cs~3zoo. p 0oosv = 0.30
-S7
A-
~0
clJ
0
U
LV
(I)F.
-7
04Figure 3.34. k vs. a (C r/Cs = 4)
3. -
Ir~4 (k~/G~Z~. I-
0.0
Re (k 9) /cdE
-=,2
ar o OO/J
- e ~ OO.~- -
ELASTic Rock
c5=zoo. bO
0 o. 50
C32I0 0 os0.5t-
I-T c) IC8sico
V)K
CA)
""0.40.4
Clo
k 0vs.
.S r-
I4Z5
700
ReGI
0.0
-0.51-
- 1.0 1--
0.2-6 (
Figure 3.35. ao (Cr/Cs =4
-:' =2
Re (kz')/G
-T Ik ) /G
a. I JFigure 3.36.
6
~zN
.4
+~J 4
ii
N
-2
-4
-61
(Cr/Cs =4kz vs. ao
Io.
t,X
0.- CTGF G-ReCFJ
II----- G F 4 V G xF ~'xx
0.- GF for o oo5- 0.6 --- G MC,Y,
02
-9j
Li-
U
07WAN7
F77nm77/?l=- 28B .
II . c,='3Oo. ~=0.0S
ELASTIc ROCKtN lao
c 130 =0.0
0 .. --- t
s moo"
040
0.0 0.2. 0.4 0 -. o<
Figure 3.37. F vs. ao (Cr/Cs = 2)
GB ?- e -FGf)- -e 2Im (F )
G z. Re CF,4)G B2 - _I v (99)
C " s w o o H f O L \ q )
Q" svok& {oUA')
'I
0~
-S 0-
-
U
4N
LLG~
0.2.0.0Clo
Figure 3.38. F vs. a0 (Cr/Cs = 2)
4- 2
Yt 0
,ZB.Y 100. -
LAST~C ROCK
0Ioo. ( j SGoo.0-0
o ~ c--- o
//
/7/7
7
ROCKING -s AAY ING
V7~/wtAYo
RThe (1F4- GB.10 - :
. -) --
.051
0
.00
-. 051
-cy Koo.ELASTIC ROCK
-V . o
7
7
0.0 0.2. 0.4 0.6 O.S
Figure 3.39. F vs. a (Cr/Cs = 2)
li
0-R
Figure 3.40. Fz vs/ ao(Cr/Cs = 2)
U-
+
U-
0(I
NU-
0
C
U
U-
VU
0
/0.2
0.0
1-7GFz
GFz/7-
y=O.40.
0LASTIcRloc <
cs= Meoo. =o.os,v = 0.30
CHAPTER 4 - RECTANGULAR FOOTING
4.1 Formulation
The differential equations of motion
(+2G) ( 2E + 2E _ + 2E 32E(x+G) 2 + 2+--- -p2
3x ay zat(32 32 2 32G 2 + 2 + 2 ~ at2
3x 3y 32 / t1
can be directly solved for u,v,w for any layer of soil.
u = (L [A'eihnz+ A"e- ihnz + [B'eikn'z+B"e-ikn'z ei(wt-hx-hmy)
v = (-[A'eihnz+ Ae-ihnz + [Cleikn'z+ Ceikn'z) ei(wt-hkx-hmy)
W= (n -A'eihnz+A"e- ihnz + [D'eikn'z+ D"e ikn'zi) ei(wt-hx-hmy)
with 92 + m2 + n2 = 1
A'2 + M'2 + n'2 _
h m k =2 _
p s
B'k' + C'm' - D'n' = 0
B" + C"m' - D"n' = 0
FiGrEu 4.j. SYS-TEM OF FORCES , MoMENTS
CORRESPONDING TO THE MoDas
or- VIBRATION CONsITDRep .
u D
-VL kL-o~
TOTAL NUM5P,
_ F _ _ _ _ __
4
F~cxuRtC 4.2.. GRID UsrGD FOP, -THE E VA LU)AT toN OF 7HE Voutlg~~
'TPANSF-o?JA AND ThIe FL6)q~lLlT'Y COGEFF'ICICeNrS foe PotiJTs
UND-R TikE FooTItACr
x
m vy
~I
W
-J
14:
YTAY-1
P%
The boundary conditions at the interface between any two suc-
cessive layers leads to the conditions
h = k 2' =h +9+ =k 9,pp p p p+1 p+1 p+l p+l
hm =k m' =h m =k m'pp p p P p+l p+l p+1 p+l
and after elimination of the intermediate layer matrices we get:
u C
v} =Q2 Q x (2)w Ty
in a quite similar way as in the two-dimensional case.
For a unit pulse (normal or shear) applied at the origin of
the coordi tes on the free surface, it is sufficient to write
a(x,y) = I f' fS(§,C) e1 e'iy dE dC4Tr CO 00
with S(EC) = J 0 c(x,y) e~ e - y dx dy-00 -00
and similarly to T (x,y), T y(x,y).
For a particular set of E and C, one can solve equation (2)
after setting for each layer:
h k = k 1 = -
h.m. = k.m. = -C33 33 m
Thus
u(5C s(,C)
v=
or
U= Ql2 ' 2 Q2 ( 9 S (C
and by the Fourier Transformation
{U(x,y)} = (2 ,C e ey d dc47T -0
the displacements at the free surface are obtained.
The formulation for the rigid footing, thereafter, is quite simi-
lar to the one for the strip footing. The only difference is that
at any given normal or shear stress on a point there correspond three
displacements and thus the flexibility coefficients are 3 x 3 and not
2 x 2. Thus, finally:
u F F 0 0 0 0 P
1 F F 0 0 0 0 M
v = 0 0 F F 0 0 Pyy yc , y,
$ 0 0 F F 0 0 M22*2 2 y
w 0 0 0 0 Fz 0 Pz
e L 0 0 0 0 0 Ft Mt
where F, 9 , F = swaying & rocking flexibilities in theS 1 xz plane
F , Fyp Fq p = swaying & rocking flexibilities in the'' 'y2 2 2 yz plane
Fz = flexibility in vertical translation
Ft = flexibility in torsion.
4.2 Results
Effect of number of points
Figures 4.3 and 4.4 show the dimensionless stiffness functions
k , kz, kt versus dimensionless frequency a0 = wB/Cs in case
of a square footing resting on an elastic halfspace. The total number
of points as well as the number of points under the footing are taken
much smaller than those of a plane footing (strip), because otherwise
the capacity of the IBM 360 is surpassed.
Two kinds of curves are shown. The total number of points is 64
in each direction (64 x 64). The number of points under the footing
is taken as 5 (solid curves) or 3 (dashed curves) with the correspond-
ing x equal to 10 or 20, respectively. As expected, according to the
theory presented in Chapter 3, the dashed curves are less wavy than
the solid ones and, apparently, more accurate.
Static spring constants
According to the so-called correspondence principle, it is always
possible to write the stiffness functions as
K = K (k + i a C) (1 + 2i B)
100
where Ko is the real part of the stiffness function in the static
case (static "spring constants") and k, c are the stiffness and vis-
cous damping coefficients, functions of the frequency a That is,
K0 =Re [K(w = 0)]
o eK( = 0
a 0 m k(w = 0)
The halfspace solutions are
Kxo 2 8GR v swaying
K = 3 R rocking
K 4GR
Kto = 3
vertical translation
torsion
In case of a rectangular footing, the equivalent radius is given
R = 2B for translation
_ 2B for rocking and torsion
and thus dimensionless static spring constants are:
by:
101
K X/GB = 5.15
K /GB3 = 5.26
K zo/GB = 6
K to/GB 3 = 7.9
The corresponding values which were found are: 5.60, 6.50, 6.50
and 11.0 respectively. The difference is rather small for such a
small number of points except for the torsion.
2B)s
Ka Spckce.
Re(ecc /Gai
N f =G4 Yfl 5 L~. VJ*
M,'= G-4, my = -5
D ime5jones5 F requ.e.n~ 7
=' 20.
.1 0.20 .- Z 7 .
Figure 4.3. k and k vs. a0
10. t-
8)4-. ay
VI
Li..
0.0 0.3
E tr'lZ ,
m )((D
5.--B
L / 1 --
10.
0 5
I I
0.0 0.1 .2 0.3
imension less Freq u e -Acy ao /x = t v Vr
Figure 4.4. kt and kz vs. a0
104
SUMMARY AND CONCLUSIONS
The response of a rigid strip or rectangular footing resting
on a layered soil stratum was studied. Results were given in terms
of dimensionless compliance (flexibility) or stiffness functions
versus dimensionless frequency. Each layer of soil was assumed to be
homogeneous, isotropic and linearly viscoelastic (theory of "equiva-
lence"). The layers were considered to be welded to each other, and
the footing to be welded to the soil surface. Thus tensile stresses
between footing and soil could be developed.
The solution presented was based on direct integration of the
differential equations of motion while satisfying the boundary condi-
tions at the interfaces and at the surface. The complexity of the
latter was overcome by using a fast Fourier transform for a unit load
pulse under the footing and then integrating over the width of the
footing while imposing the conditions of rigid body motion. All the
possible modes of vibration can be handled with this method; horizontal
(both directions) or vertical translation and rocking (both directions)
or twisting were studied.
The effect of the number of isolated points by which the free
surface was represented and of their distance was studied first. It
was shown that the required number of points for a good solution, as
well as their distance, are functions of the shortest (shear) wave-
length. A number of 8 points per shear wavelength was found to be suf-
ficient for a good solution.
105
The results for a halfspace are compared with known analytical
solutions. The agreement found was very good.
The effect of the rigidity of the rock on which a layer of soil
rests was primarily investigated. There is a considerable change in
the response curves as the rigidity of the underlying rock decreases
from o to some value of 20 times the rigidity of the soil above it.
The peaks at the resonant frequencies decrease or even disappear
(higher modes.). The solution converges to the halfspace as the shear
wave velocity of the rock approaches the one of the soil.
The effect of the "smoothness" of the footing was then studied.
"Smooth" footing is one in which the secondary stresses in the contact
area between footing and soil are neglected (relaxed boundary condi-
tions). Only in the case of a layer of soil on a rigid rock is this
effect important, and only for the rocking vibration. For an elastic
rock this effect becomes less and less important as the stiffness of
the rock decreases.
There is only one case of change in the internal damping of the
elastic rock which was studied. The effect was not important, except
for the swaying at the first resonant frequency (C s/4H).
106
References
1. Agabein, M.E., Parmelee, R.A., and Lee, S.L., "A Model for theStudy of Soil-Structure Interaction," Proc. Eighth Congress ofthe Intl. Assoc. for Bridge and Structural Engng., pp. 1-12, NewYork, 1968.
2. Ang, A.H.-S, and Harper, G.N., "Analysis of Contained PlasticFlow in Plane Solids," Journ. Engineering Mechanics Div., ASCE,Vol. 90, No. EM5, pp. 397-418, 1964.
3. Arnold, R.N., Bycroft, G.N., and Warburton, G.B., "Forced Vibra-tions of a Body on an Infinite Elastic Solid," Journ. AppliedMechanics, Trans. ASME, Vol. 22, No. 3, pp. 391-400, 1955.
4. Awojobi, A.O., "Approximate Solution of High-Frequency-FactorVibrations of Rigid Bodies on Elastic Media," Journ. AppliedMechanics, Trans. ASME, Vol. 38, Ser. E, No. 1, pp. 111-117, 1971.
5. Awojobi, A.O., and Grootenhuis, P., "Vibration of Rigid Bodieson Semi-infinite Elastic Media," Proc. Royal Soc. London, Ser. A.,Vol. 287, pp. 27-63, 1965.
6. Baranov, V.A., "On the Calculation of Excited Vibrations of anEmbedded Foundation," (in Russian) Voprosy Dynamiki i Prochnocti,No. 14, Polytechnical Inst. of Riga, pp. 195-209, 1967.
7. Beredugo, Y.O., and Novak, M., "Coupled Horizontal and RockingVibration of Embedded Footings," Canadian Geotechnical Journal,Vol. 9, No. 4, pp. 477-497, 1972.
8. Bland, D.R., The Theory of Linear Viscoelasticity, Pergamon Press,1960.
9. Bycroft, G.N., "Forced Vibration of a Rigid Circular Plate on aSemi-infinite Elastic Space and an Elastic Stratum," Phil. Trans.Royal Soc. London, Ser. A., Vol. 248, pp. 327-386, 1956.
10. Chakravorty, M.K., Nelson, M.F., and Whitman, R.V., ApproximateAnalysis of 3-DOF Model for Soil Structure Interaction, ResearchReport 71-11, Dept. of Civil Engrg., MIT, Cambridge, Mass., June1971.
11. Chang-Liang, Victor, "Dynamic Response of Structures in LayeredSoils," Ph.D. thesis, MIT, 1974.
107
12. Dimaggio, F.L., and Bleich, K.H., "An Application of a DynamicReciprocal Theorem," Journ. Applied Mechanics, Trans. ASME,Vol. 29, pp. 678-679, 1959.
13. Elorduy, J., Nieto, J.A., and Szekely, E.M., "Dynamic Responseof Bases of Arbitrary Shape Subjected to Periodical VerticalLoading," Proc. Intl. Symp. on Wave Propagation and DynamicProperties of Earth Materials, Univ. of New Mexico, Albuquerque,pp. 105-121, 1967.
14. Fleming, J.F., Screwvala, F.N., and Kondner, R.L., "FoundationSuperstructure Interaction under Earthquake Motion," Proc. 3rdWCEE, pp. 1-22, to 1-30, New Zealand, 1965.
15. Fung, Y.C., Foundations of Solid Mechanics, Prentice-Hall, 1965.
16. Gladwell, G.M., "Forced Tangential and Rotatory Vibration of aRigid Circular Disc on a Semi-infinite Solid," Intl. Journ. Eng.ineering Science, Vol. 6, No.10, pp. 591-607, 1968.
17. Gupta, D.C., Parmelee, R.A., and Krizek, R.J., "Coupled Slidingand Rocking Vibrations of a Rigid Foundation on an Elastic Medium,"Tech. Report, Dept. of Civil Engng., Northwestern Univ., Evanston,Ill., 1972.
18. Hall, J.R., Jr., "Coupled Rocking and Sliding Oscillations ofRigid Circular Footings," Proc. Intl. Symp. on Wave Propagationand Dynamic Properties of Earth Materials, Univ. of New Mexico,Albuquerque, pp. 139-148, 1967.
19. Hsieh, T.K., "Foundation Vibrations," Proc. Institution of CivilEngineers, Vol. 22, pp. 211-226, 1962.
20. Iljitchov, V.A., "Towards the Soil Transmission of Vibrationsfrom One Foundation to Another," Proc. Intl. Symp. on Wave Propa-gation and Dynamic Properties of Earth Materials, Univ. of NewMexico, Albuquerque, pp. 641-653, 1967.
21. Jones, T.J., and Roesset, J.M., Soil Amplification of SV and PWaves, Research Report R70-3, Dept. Civil Engng, MIT, Cambridge,Mass., Jan. 1970.
22. Karasudhi, P., Keer, L.M., and Lee, S.L., "Vibratory Motion of aBody on an Elastic Half Plane," Journ. Applied Mechanics, Trans.ASME, Vol. 35, Ser. E, No. 4, pp. 697-705, 1968.
108
23. Kausel, E., "Forced Vibrations of Circular Foundations onLayered Media," Ph.D. thesis, MIT, 1974.
24. Kobori, T., Minai, R., and Suzuki, T., "The Dynamical GroundCompliance of a Rectangular Foundation on a Viscoelastic Stratum,"Bull. of the Disaster Prevention Research Institute, Kyoto Univ.,Vol. 20, pp. 289-329, Mar. 1971.
25. Kobori, T., Minai, R., Suzuki, T., and Kusakabe, K., "DynamicalGround Compliance of Rectangular Foundations," Proc. SixteenthNatl. Congress for Applied Mechanics, 1966.
26. Kobori, T., Minai., R., Suzuki, T., and Kusakabe, K., "DynamicGround Compliance of Rectangular Foundation on a Semi-infiniteViscoelastic Medium," Annual Report, Disaster Prevention ResearchInstitute of Kyoto Univ., No. llA, pp. 349-367, 1968.
27. Kobori, T., and Suzuki, T., "Foundation Vibrations on a Visco-elastic Multilayered Medium," Proc. Third Japan Earthquake Engng.,Symp., pp. 493-499, Tokyo, 1970.
28. Krizek, R.J., Gupta, D.C., and Parmelee, R.A., "Coupled Slidingand Rocking of Embedded Foundations," Journ. Soil Mech. and Foun-dations Div., ASCE, Vol. 98, No. SM12, pp. 1347-1358, 1972.
29. Kuhlemeyer, R., Vertical Vibrations of Footings Embedded in Lay-ered Media, Ph. D. thesis, Univ. of California, Berkeley, 1969.
31. Luco, J.E., and Westmann, R.A., "Dynamic Response of a RigidFooting Bonded to an Elastic Half Space," Journ. Applied Mechanics,Trans. ASME, Vol. 39, Ser. E., No. 2, pp. 527-534, 1972.
32. Lycan, D.L., and Newmark, N.M., "Effect of Structure and FoundationInteraction," Journ. Engng. Mechanics Div., ASCE, Vol. 87, No. EM5
33. Lysmer, J., Vertical Motion of Rigid Footings, Dept. of Civil Engng.,Univ. of Michigan, Report to WES Contract Report No. 3-115, 1965..
34. Lysmer, J., and Kuhlemeyer, R.L., "Finite Dynamic Model for Infin-ite Media," Journ. Engineering Mechanics Div., ASCE, Vol. 95,No. EM4, pp. 859-877, 1969.
109
35. Lysmer, J., and Richart, F.E., Jr., "Dynamic Response of Footingsto Vertical Loading," Journ. Soil Mech. and Foundations Div.,ASCE, Vol. 92, No. SMl, pp. 65-91, 1966.
36. MacCalden, P.B., and Matthiesen, R.B., "Coupled Response of TwoFoundations," Paper 238, Proc. 5th WCEE, Rome, 1973 (in publica-tion).
37. Meek, J.W., and Veletsos, A.S., "Simple Models for Foundations inLateral and Rocking Motion," Paper 331, Proc. 5th WCEE, Rome,1973 (in publication).
38. Merritt, R.G., and Housner, G.W., "Effects of Foundation Compli-ance on Earthquake Stresses in Multi-story Buildings," Bull Seism.Soc. Am., Vol. 44, No. 4, pp. 551-570, 1954.
39. Novak, M., "The Effect of Embedment on Vibration of Footings andStructures," Paper 337, Proc. 5th WCEE, Rome, 1973 (in publica-tion).
40. Novak, M., and Beredugo, Y.0., "Vertical Vibrations of EmbeddedFootings," Journ. Soil Mech. and Foundations Div., ASCE, Vol. 98,No. SM12, pp. 1291-1310, 1972.
41. Oien, M.A., "Steady Motion of a Rigid Strip Bonded to an ElasticHalf Space," Journ. Applied Mechanics, Trans. ASME, Vol. 38,Ser. E, No. 2, pp. 328-334, 1971.
42. Ratay, R.T., "Sliding-Rocking Vibration of Body on Elastic Medi-um," Journ. Soil Mech. and Foundations Div., ASCE, Vol. 97, No.SMl, pp. 177-192, 1971.
43. Reissner, E., "Stationsre, axialsymmetrische, durch eine schtt-telnde Masse erregte Schwingungen eines homogenen elastischenHalbraumes," Ingenieur-Archiv, Vol. 7, pp. 381-396, 1936.
44. Richardson, J.D., Forced Vibrations of Rigid Bodies on a Semi-infinite Elastic Medium, Ph.D. thesis, Univ. of Nottingham,England, May 1969.
45. Richardson, J.D., Webster, J.J., and Warburton, G.B., "The Re-sponse on the Surface of an Elastic Half-Space near to a Harmonic-ally Excited Mass," Journ. Sound and Vibration, 14 (3), pp.307-316, 1971.
46. Robertson, I.A., "Forced Vertical Vibration of a Rigid CircularDisk on a Semi-infinite Solid," Proc. Cambridge Philosophical Soc.,Vol. 62A, pp. 547-553, 1966 .
110
47. Roesset, J.M., Whitman, R.V., Dobry, R.: "Modal Analysisfor Structures with Foundation Interaction," ASCE Journal of Struc-Div., Vol.99, March, 1973.
48. Roesset, J.M., and Whitman, R.V., Theoretical Background for Ampli-fication Studies, Report No. 5 on "Effect of Local Soil Conditionsupon Earthquake Damage," Research Report R69-15, Dept. of CivilEngng., MIT, Cambridge, Mass., March 1969.
49. Sakurai, J., and Minami, J.K., "Some Effects of Nearby STructureson the Seismic Response of Buildings," Paper 274, Proc. 5th WCEERome, 1973 (in publication).
50. Sarrazin, M.A., Soil-Structure Interaction in Earthquake ResistantDesign, Research Report R70=59, Dept. Civil Engng., MIT, Cambridge,Mass., Sept. 1970.
51. Sinitsyn, A.P., "The Interaction between a Surface Wave and a Struc-ture," Proc. Intl. Symp. on Wave Propagation and Dynamic Propertiesof Earth Materials, Univ. of New Mexico, Albuquerque, pp. 521-527,1967.
52. Thomson, W.T., "A Survey of the Coupled Ground-Building Vibration,"Proc. 2nd WCEE, Tokyo, 1960.
53. Urlich, C.M., and Kuhlemeyer, R.L., "Coupled Rocking and LateralVibrations of Embedded Footings," Canadian Geotechnical Journal,Vol. 10, No. 2, pp. 145-160, 1973.
54. Veletsos, A.S., and Verbic, B., Vibration of Viscoelastic Founda-tions, Report No. 18, Structural Research at Rice, Dept. of CivilEngng., Rice Univ., Houston, Jan. 1971.
55. Veletsos, A.S., and Wei, Y.T., Lateral and Rocking Vibration ofFootings, Report No. 8, Structural Research at Rice, Dept. of CivilEngng., Rice Univ., Houston, Jan. 1971.
56. Waas, G., Earth Vibration Effects and Abatement for Military Facili-ties, Report 3, "Analysis Method for Footing Vibrations throughLayered Media," Tech. Report S-71-14, U.S.A.E.W.E.S., Sept. 1972;also Ph.D. thesis, Univ. of California, Berkeley, June 1972.
57. Warburton, G.B., "Forced Vibrations of a Body on an Elastic Stratum,"Journ. Applied Mechanics, Trans. ASME, Vol. 24, No. 1, pp. 55-58,1957.
58. Warburton, G.B., Richardson, J.D., and Webster, J.J., "Forced Vibra-tions of Two Masses on an Elastic Half Space," Journ. AppliedMechanics, Trans. ASME, Vol. 38, Ser. E., No. 1, 1971.
111
59. Warburton, G.B., Richardson, J.D., and Webster, J.J., "HarmonicResponse of Masses on an Elastic Half Space," Journ. of Engng.for Industry, Trans. ASME, Paper No. 71-Vibr-59, 1971.
60. Wei, Y.T., "Steady State Response of Certain Foundation Systems,"Ph.D. thesis, Rice Univ., Houston, 1971.
61. Whitman, R.V., and Richart, F.E., Jr., "Design Procedures forDynamically Loaded Foundations," Journ. Soil Mech. and FoundationsDiv., ASCE, Vol. 93, No. SM6, pp. 169-193, 1967.
62. Zienkiewicz, 0.C., The Finite Element Method in Engineering Science,McGraw-Hill, 1971.
