THE KINEMATIC INTERACTION PROBLEM
OF EMBEDDED CIRCULAR FOUNDATIONS
by
JOSEPH PARKER MORRAY, JR.
BS, University of California
(1973)
Submitted in partial fulfillment of
the requirements for the degree of
Master of Science in Civil Engineering
at the
Massachusetts Institute of Technology
(June, 1975)
Signature of Author.
Depar ient'of Civil
4Eineeriig, Ray 9,
.
Certified by..................
.
.
1975
.
Thesis Supervisor
Accepted by ....
-
-
..
..
..
Chairman, Department1 Committee on Graduate
Students of the Department of Civil Engineering
ARCHIVES
JUN 201975
ABSTRACT
THE KINEMATIC INTERACTION PROBLEM
OF EMBEDDED CIRCULAR FOUNDATIONS
by
JOSEPH PARKER MORPAY,
JR.
Submitted to the Department of Civil Engineering on May 9, 1975
in partial fulfillment of the requirements for the degree of
Master of Science in Civil Engineering
The applicability of 1-dimensional wave theory for
the determination of support motion of a circular
embedded foundation is studied. Rules are derived
by means of a series of parametric studies to account for embedment of the foundation and thickness of the soil stratum.
Using the derived rules, a comparison is made with
a full 3-dimensional finite element model to determine the accuracy and applicability of the rules.
The comparisons are in the form of transfer functions, frequency spectra, time histories of acceleration, and response spectra. It is found that
the motions obtained from the derived rules correlate very well with that of the 3-D model, suggesting that it can be a useful tool for design
purposes.
Finally, the effect of varying the soil and structural properties on the components of support motion is studied.
Thesis Supervisor:
Title:
R.V. Whitman
Professor of Civil Engineering
ACKNOWLEDGMENTS
The author is deeply indebted to Professor R.V. Whitman and
Dr.
Eduardo Kausel for their extensive aide and guidance in
researching and writing the thesis.
In addition, the author
would like to thank Professor J.M. Roesset for suggesting
the topic of study, as well as making numerous helpful suggestions.
The final draft was meticulously typed by the au-
thors wife, Claudia, despite at times almost cryptic references.
Finally, the author would like to thank Mr. Chuck Reeves and
Stone & Webster Engineering Corporation for their participation in the Engineering Residency Program, which made this
study possible.
To Claudia,
without whom it would never
have been possible.
TABLE OF CONTENTS
Page
I.
TITLE PAGE
1
II.
ABSTRACT
2
III. ACKNOWLEDGMENTS
3
IV.
TABLE OF CONTENTS
4
V.
LIST OF FIGURES
6
VI.
CHAPTER 1
INTRODUCTION
10
1.1 GENERAL CONSIDERATIONS
10
1.2 SCOPE OF WORK
12
1.3 METHODS OF ANALYSIS OF SOIL-STRUCTURE
INTERACTION EFFECT
14
1.4 PROGRAM USED FOR ANALYSIS
23
1.5 PAST WORK
24
CHAPTER 2
PARAMETRIC STUDIES
26
2.1 MODEL
26
2.2 COMPARISON OF TRANSLATIONAL MOTION
30
2.3 COMPARISON OF ROTATIONAL MOTION
39
2.4 CORRELATION FOR A LAYERED MEDIA
50
2.5 EFFECT OF FLEXIBLE SIDEWALLS
53
CHAPTER 3
STUDY OF FULL SCALE MODEL
62
3.1 MODEL
62
3.2 MASSLESS FOUNDATION MOTION
65
3.3 STRUCTURE'S RESPONSE
78
5
3.4 EFFECT OF NEGLECTING ROTATIONAL COMPONENT
OF FOUNDATION MOTION MOTION
92
3.5 EFFECT OF MAT FLEXIBILITY
97
CHAPTER 4
VII.
SUMMARY AND RECOMMENDATIONS
REFERENCES
103
106
VIII. APPENDIX
A. INPUT EARTHQUAKE
107
LIST OF FIGURES
Page
Title
Figure
1-1
Schematic Representation of Spring Method
17
1-2
Coordinate System for Free Field
20
1-3
Layered Stratum
22
2-1
Discretized Foundation Model
27
2-2
Points of Measurement on Foundation and
Free Field
29
Comparison of Translation at Massless
Foundation Using 1-D and 3-D Models For:
31
2-3
H/R= 2.0,
E/R= 1.5
2-4
H/R = 2.5, E/R = 1.0
32
2-5
H/R= 2.0, E/R= 1.0
33
2-6
H/R= 1.5, E/R = 1.0
34
2-7
H/R = 2.5, E/R = 0.5
35
2-8
H/R = 2.0, E/R = 0.5
36
2-9
H/R = 1.5, E/R= 0.5
37
2-10
Scale Factor for Rotation Vs.
E/R
40
2-11
Scale Factor for Rotation Vs. H/R
41
Comparison of Rotation at Massless
Foundation Using 1-D and 3-D Models For:
=
=
2-12
H/R
0.5
43
2-13
H/R= 2.0, E/R= 0.5
44
2-14
H/R= 2.0, E/R
1.0
45
2-15
H/R= 2.5, E/R= 0.5
46
2-16
H/R
=
1.5,
1.5,
E/R
E/R
=
=
1.0
47
2-17
H/R = 2.5, E/R = 1.0
48
2-18
H/R = 2.0, E/R= 1.5
49
2-19
Stratum with Variable Shear Wave Velocity
50
2-20
Comparison of Translation at Foundation with
Layered Media for 1-D and 3-D Models
51
2-21
Comparison of Rotation at Foundation with
Layered Media for 1-D and 3-D Models
52
2-22
Comparison of Translation at Foundation for
Models with Rigid and Elastic Sidewalls
55
2-23
Comparison of Rotation at Foundation for
Models with Rigid and Elastic Sidewalls
56
2-24
Rotation of Foundation with Rigid and
Flexible Sid ewalls
57
2-25
Phase Angle for Rotation of a Foundation
without Sidewalls
59
2-26
Phase Angle for Rotation of a Foundation
with Rigid Sidewalls
60
2-27a
Rotation of Foundation with Rigid Sidewalls
61
2-27b
Rotation of Foundation with No Sidewalls
61
3-1
Full Scale Model
63
3-2
Comparison of Translational Motion at
Foundation for 1-D and 3-D Models
66
3-3
Comparison of Rotational Motion at
Foundation for 1-D and 3-D Models
67
3-4
Frequency Spectrum of Absolute Value of
Translational Acceleration at Foundation
Using Finite Element
69
3-5
Frequency Spectrum of Absolute Value of
Translational Acceleration at Foundation
Using 1-D Approximation
70
3-6
Frequency Spectrum of Absolute Value of
Rotational Acceleration at Foundation
Using Finite Elements
71
3-7
Frequency Spectrum of Absolute Value of
Rotational Acceleration at Foundation
Using 1-D
72
3-8
Time History of Translational Acceleration
at Foundation Using Finite Elements
73
3-9
Time History of Translational Acceleration
at Foundation Using 1-D Approximation
74
3-10
Amplified Response Spectra of Translational
Acceleration at Foundation Using Finite
Elements
75
3-11
Amplified Response Spectra of Translational
Acceleration at Foundation Using 1-D
Approximation
76
3-12
Amplified Response Spectra of Translational
Acceleration at Foundation Using 1-D
77
3-13
Lumped Mass Model
79
3-14
Acceleration X at Level 8 of Structure 1
From Finite Elements
81
3-15
Acceleration X at Level 8 of Structure 1
From 1-D Approximation
82
3-16
ARS Acceleration X at Level 8 of Structure
1 From Finite Elements
83
3-17
ARS Acceleration X at Level 8 of Structure
1 From 1-D Approximation
84
3-18
Acceleration X at Level 1 of Structure 1
From Finite Elements
85
3-19
Acceleration X at Level 1 of Structure 1
From 1-D Approximation
86
3-20
ARS Acceleration X at Level 1 of Structure
1 From Finite Elements
87
3-21
ARS Acceleration X at Level 1 of Structure
1 From 1-D Approximation
88
3-22
ARS Acceleration X at Level 1 of Structure
2 From Finite Elements
89
3-23
ARS Acceleration X at Level 1 of Structure
2 From 1-D Approximation
90
3-24
ARS Acceleration X at Level 1 of Structure
1 From Finite Elements - Translation Only
93
3-25
ARS Acceleration X at Level 1 of Structure
2 From Finite Elements - Translation Only
94
3-26
ARS Acceleration X at Level 1 of Structure
1 From 1-D Approximation - Translation Only
96
3-27
ARS for Horizontal Motion at Top of Containment for Flexible Mat
99
3-28
ARS for Horizontal Motion at Top of Containment for Rigid Mat
100
3-29
ARS for Horizontal Motion at Top of Mat
for Flexible Mat
101
3-30
ARS for Horizontal Motion at Top of Mat
for Rigid Mat
102
1- Introduction
1.1- General Considerations:
The seismic design of commercial structures is based on the
premise that they be able to survive, without damage, moderately strong ground motions, but that some damage may be tolerated in the event of large and relatively infrequent
earthquakes.
In the case of nuclear containment structures,
however, the basic requirements are that under the severest
earthquake, the plant must be able to safely shut down without leakage of the radioactive material contained within.
Thus the requirements are considerably more demanding than
had been previously encountered in structural engineering.
To this end much attention has been given recently to the topic
of soil-structure interaction as it relates to the dynamic analysis of nuclear containment structures.
Of particular
importance is the effect embedment has on the dynamic response
of the structure.
Both experimental evidence and theoretical
considerations have shown that the embedded structure will
respond quite differently from the corresponding structure
resting on the surface, both in terms of the foundation motion and the stiffness of the soil underlying it.
It will be shown later that the soil-structure interaction
problem can be divided into three steps, which, for a linearly elastic solution, can be studied independently and com-
11
bined using the principle of superposition.
1.
These steps are:
Definition of the "support" motion which excites the
structure.
This is called the kinematic interaction
problem.
2.
Determination of soil stiffness coefficients of the
underlying stratum.
3.
The response of the structure given the support motion from (1) and the stiffness coefficients from
(2).
This is referred to as the inertial interaction
problem.
Most of the past research work has been addressed to the last
two steps, while the first has been systematically ignored.
This is partly due to the fact that in a complete solution,
where soil and structure are solved together in a single
step - for instance, in a finite element model - step 1 is
automatically incorporated.
Since the complete solution is generally expensive in terms of
computation time, it is desirable to establish simple rules to
predict the "support" motion (to be defined later), and thus
obtain a substantial saving in the cost of the analysis.
This
frees the engineer to study a wider range of soil parameters,
enabling him to make a more reliable prediction of the maximum structural response.
1.2- Scope of this Work:
The study will address itself to the first of the three steps
of the interaction problem.
The results will be evaluated by
comparing with the results obtained from the direct solution.
The comparisons will be in the form of transfer functions,
time histories, and response spectra.
Much controversy currently exists as to the correct criteria
for defining the "control" motion for a seismic analysis.
The "control" motion is the motion that would occur at some
point in the soil in the free field, before any structure had
been built.
This controversy stems primarily from a disagree-
ment over the reliability of the models currently used to reproduce the motion of a soil stratum.
In nature, earthquake
stress waves propogate in all directions; thus, any arbitrary
point in the soil will displace in a random three-dimensional
fashion.
There are, however, preferred directions of motion
if the site is not immediately adjacent to the epicenter.
This justifies the use of more simplified two-dimensional and
one-dimensional models to study the propogation of earthquake
waves in the immediate neighborhood of a soil site.
In addi-
tion, amplification studies using 2-D models show amplification patterns very similar to those predicted by the 1-D
model, suggesting that the simpler 1-D model can be used.
The validity of the use of one-dimensional wave propogation
13
theory, which expounds that the earthquake motion consists of
vertically propagating stress-waves, will not be discussed
here.
While there is still disagreement among the experts,
there exists considerable theoretical and experimental basis
for the use of this theory.
Thus, 1-D theory will be used as
the basis for this study.
Two methods for the definition of the control motion will be
discussed here.
In the first, the control motion is defined
at the free surface, and then by means of 1-dimensional wave
theory the motion is resolved at the foundation of the structure.
The second method to be discussed, which was proposed
by Whitman (io), recommends that the control motion be applied
directly under the foundation.
In this method, the support
and control motions become equivalent.
Though this method is
conservative, it is attractive in its simplicity and economy
of computation time.
Chapter 1 will review past work conducted and summarize methods for analyzing foundation motion of embedded structures.
Chapter 2 will investigate by means of parametric studies the
applicability of 1-dimensional wave propagation theory to the
analysis of foundation motion, and simple rules will be derived for its application to embedded structures.
Of partic-
ular interest in this chapter will be the motion of a massless
foundation.
Then using realistic models, chapter 3 will in-
14
vestigate the accuracy of the approximate rules derived from
chapter 2, and secondly, investigate the effect of the flexibility of the structure's foundation on the structural response.
This will permit an assesment of the applicability of
the 3-step solution (the spring method) which is based on the
assumption of a perfectly rigid foundation.
Finally, chapter
4 will discuss conclusions and recommendations from the study.
1.3- Methods of Analysis of Soil-Structure Interaction Effect:
Two equivalent methods to predict the dynamic soil-structure
effect are:
a.
A complete solution in which soil and structure are
modeled together (usually with finite elements and
linear members) and solved for the prescribed motion.
As mentioned earlier, this is a very expensive method
of solution since the dynamic equations for the soil
and the structure are solved in one step.
b.
A 3-step solution (spring method) (5) in which the
analysis is conducted on the three phases mentioned
in the introductory considerations.
The three steps
are as follows:
1.
The motion that would occur at the level of the
foundation without the structure having mass
is first determined.
In the case of surface
structures, the control and support motions are
15
equivalent, and therefore this step may be bypassed.
The motion is in the form of a hori-
zontal translation (provided, of course, that
1-D wave propagation theory is being used).
When the structure is embedded there will be a
rotational as well as a translational motion.
The determination of these two motion components is as costly and. complicated as the direct solution for soil and structure.
It is
here that we see the impetus for chapter 2 of
this study, in which simple rules will be derived to estimate the motion at the massless
foundation, from the motion occurring in the
free field.
2.
Frequency dependent compliances or stiffnesses
for the massless foundation are computed.
Ex-
tensive theoretical and numerical solutions
are available for several cases.
3.
A dynamic analysis of the structure is performed in the frequency domain, with the stiffnesses from step 2 modeling the soil, and the
foundation motion from step 1 representing the
seismic excitation.
This was referred to in
the introduction as the
"support" motion.
The mathematical justification for this solution scheme can
16
be seen from the general equation of motion for the entire
soil-structure system.
The equation can be described in ma-
trix form by:
o
0
(1)
rMU+CY+K-Y=
Y= Vector of relative displacements
where:
U= Vector of absolute accelerations
W*
*
.of
Y=U-Ug
Ug
=
Generalized ground acceleration vector.
Now, imposing the fundamental superposition theorem, (linear
solution implied) equation (1) can be expressed as the sum of
two other equations:
where:
MU+CY,+KY=0
(2)
MY+CY+KY=-MU
(3)
U=Y+Ug
U=U+Y
Y=Y+Y
M= Mass of system excluding mass of structure
M= Mass of structure
Equation (2) describes the motion of a massless structure sub-
jected to a prescribed ground motion.
step 1 of the analysis.
very rigid
This corresponds to
If one assumes that the structure is
(a good assumption in the case of a nuclear con-
tainment), it will move essentially as a rigid body, and its
17
motion can be described in terms of two components.
the translation and rotation of the foundation.
These are
Thus, with
the assumption of rigidity of the containment, no structure
needs to be incorporated in this step of the analysis, and a
massless rigid foundation is used.
Equation (3) describes the
motion of the structure due to nodal inertial loads applied to
it.
To further clarify the method of solution a schematic representation can be made:
-pl
-A
Schematic Representation of "Spring Method"
Figure 1-1
It is important to note that by performing the analysis to determine the motion at the massless foundation, given the control motion at some other locale in the soil stratum, the
spring method implicitly assumes that the structural foundation is sufficiently rigid that the two components of motion
18
(translation and rotation) suffice to describe the motion.
In
chapter 3 this assumption will be tested.
Since in this study 1-dimensional wave theory will be investigated as a means for calculating the foundation motion for an
embedded structure, a description of 1-D theory would be beneficial.
One-dimensional wave theory is well understood (6),
only a brief description is needed.
and thus
1-D motion is thought to
occur in the free field, far away from the structure, where it
is not affected by the structure's motion.
tion of the soil without the structure.
Thus it is the mo-
The assumption is
made that all seismic waves propagate vertically through the
soil strata in the form of shear and pressure waves.
This
assumption is made due to the following:
1.
Wave velocity generally decreases from the earth's
interior toward its surface, and thus, due to succesive refraction, the stress waves will be almost
vertical when they reach the surface.
2.
Waves propagating with a shallow inclination in the
firm ground, upon hitting the soil stratum will be
refracted to a nearly vertical path by Snell's law
of refraction.
The differential equation of motion for a one-layered system
19
4)
is:
where:
f
=
Mass density per unit volume
G = Shear modulus
?= Coefficient of viscosity
f(t)= External force per unit volume
If one defines the forcing function as a displacement at bedrock such that %/t
z
is harmonic, say Csinat, the solution
will have two types of terms:
The first is a series of terms with frequency ., the natural frequency of the soil layer, and with coefficients
depending on the initial conditions.
This is known as
the free vibration portion.
The second term of the solution containsJ2 , the frequency of the imposed base displacement, and is known as
the forced vibration portion of the solution.
If the system has damping, the free vibration portion decays
exponentially with time and is referred to as transient.
The
forced vibration has constant coefficients and is referred to
as the steady-state solution.
20
C'1'
Coordinate System for Free Field
Figure 1-2
The steady-state response for equation 4 will be of the form:
y=u(x)e'
Substituting into equation 4, the following results are obtained :
ciLe1
UCe
-ce
or,
Calling:
P'
I
and imposing the boundary conditions:
U(H)=O , U(H)=0
one obtains the ratio between the absolute acceleration at the
top of the layer to the absolute acceleration of the base.
This ratio, termed the amplification of the soil stratum is:
where p is a complex number for the general case of a damped
system.
Alternatively calling:
(9)
The amplification can be expressed as:
I
(10)
For small values of 22L:
- cos
I
+o 5in?(3
(11)
If there is no viscosity:
IA(nl
cosf
(12)
and thus for cos)=0 the amplification will be infinite.
This
corresponds to a frequency 2 of:
2 = 1)1--
(13)
which are the natural frequencies of the layer.
If one extends the theory to multiple layers, each with different properties, such that:
22
i
5i +
2(14)
and writing the displacements as:
uj
Fe
e
(15)
one can express the amplification (or deamplification) between
any two layers.
0
X
Layered Stratum
Figure 1-3
The solution is obtained by imposing continuity of displacement and stresses at the layer interfaces, as well as the conditions of zero stress at the free surface and prescribed displacement at bedrock.
This procedure results in expressions
of the form:
E + =aE
(16)
F.++ =bF
and therefore, the amplification between bedrock and the free
surface can be expressed as:
A(6a)
L-
Q ni-I
t
-
En t nF,,
...;
a +,
( 17))
23
Though the continuous solution outlined above is straight for-
ward, a discrete technique is required with the use of finite
elements in order to satisfy consistency requirements.
This
insures that an earthquake motion filtered through a stratum
modeled with finite elements and appropriate boundary conditions will produce the same motions at the various layer interfaces, and in particular at the free surfaces as the 1-dimensional discrete theory.
The technique is straight forward
and will be omitted here.
1.4- Program Used for Analysis:
The finite element program used for this study was developed
by Kausel (4), and will be referred to as TRIAX.
It is a 3
dimensional finite element program for axisymmetric structures
where both the soil and the structure are modeled with isoparametric toroidal quadrilateral finite elements having 3 degrees of freedom per nodal ring.
The dynamic equations are
solved in the frequency domain, using Fourier transformation
techniques.
A fundamental feature of this program is the exact representation of the model boundary, which separates the finite element
region from the semi-infinite continuum (free field).
This
energy transmitting boundary, which was developed for the
plane strain case by Waas and Lysmer (8), and extended to the
3-D case by Kausel, is equivalent to a virtual extension of
the finite element mesh to infinity.
Thus a significant re-
duction in the number of elements is achieved, since the exact
boundary conditions are solved, allowing for the boundary to
be placed very close to the structure.
In addition, an accu-
rate model of the radiation damping is achieved, avoiding the
so-called "box effect".
1.5- Past Work:
Extensive work has been done on the use of 1-dimensional wave
theory for the analysis of layered media.
Whitman and Boesset
(6) have presented a complete and thorough treatment of 1-D
amplification theory.
Good agreement between measured and
predicted motions by means of 1-D amplification were obtained
by Seed and Idriss (7).
In addition there exist numerous cases
of measured soil amplification such as Duke (2), Donovan
and Mathesen (3),
and Wiggins (9), among others.
In all of
these studies, the purpose for the use of 1-D theory is to derive the support motion for structures with little or no embedment.
Very little literature exists, however, for the use of 1-D
theory for predicting the support motion of embedded structures.
The presence of an embedded footing alters fundamen-
tally the motions that occur in the vicinity of the foundation
(even if this foundation has no mass), and thus the support
motion can no longer be described by 1-D theory alone.
Docu-
25
mentation of this exists by Kausel (4) using a massless foundation.
In this study, he found good agreement between the
3-D and 1-D motions up to the first natural frequency of the
embedment region, but a marked divergence in the higher frequency range.
In addition, Chang-Liang (1) found that 1-D mo-
tion overestimates the response in the low frequency range,
until past the fundamental frequency, and under estimates the
response in the high frequency range.
It will be shown, never-the-less, that useful approximations
can indeed be made on the basis of simplified 1-dimensional
models.
2- Parametric Studies
In this section, the relationship between the motion at a
buried massless, rigid foundation and the motion in the free
field at the foundation level will be studied.
Rules will be
derived for predicting the foundation motion (to be used as
support motion in the 3-step procedure) for different embedment ratios, given the control motion in the free field.
2.1- Model:
Since the primary concern of this study is seismic motion as
it relates to nuclear containment structures, the foundation
is modeled as rigid circular plate, with rigid sidewalls.
To
achieve the desired properties, the plate and walls are formulated by a series of massless finite elements, with stiffness
1,000 times greater than that of the underlying soil.
The ra-
dius of the plate is taken as unity and a plate thickness of
0.1 is employed.
The underlying homogeneous stratum is modeled by a series of
finite elements, with the mass-density and shear wave velocity
taken as unity and internal damping taken as 5% throughout the
stratum.
Obviously, this is a highly idealistic state, since
soils are generally nonhomogeneous and layered.
The solution of the 3-D foundation motion is obtained using
27
TRIAX, which employs a consistent energy transmitting boundary
to model the far field.
The motion in the free field is pre-
dicted by means of 1-dimensional wave propagation theory, as
discussed earlier.
(See figure 2-1).
_-
___FREE
FIELD
-I-D
THEORY
_-
H
ENERGY ABSORBING
BOUNDARY
FINITE ELEMENT
REGION
Discretized
Foundation Model
Figure 2-1
During a seismic excitation, an embedded massless structure
will experience two plane symmetric displacement modes.
The
first is a translational motion which contributes the major
portion of the excitation to the structure.
Under a transla-
tional motion, a structure will experience both a swaying response and a rocking motion caused by the inertial forces on
28
the structure.
The second type of motion experience by the
foundation will be a rotational motion, caused by the shearing
stresses developed along the wall-soil interface.
These
shearing stresses result from the differential horizontal
translation of the soil in the embedment region, producing a
"pseudo" rotation of the soil.
The structure on the other
hand being rigid, cannot deform in this manner, and thus will
rotate.
This rotation is mainly resisted by the subgrade
stiffness acting on the mat, and to a smaller extent by the
sidewalls.
Seven different embedment ratios were studied, covering a
range of values encountered in nuclear reactors:
= 1.5
1)
E/R=0.5,
2)
=0.5,
=2.0
3)
=0.5,
=2.5
4)
=1.0,
=1.5
5)
=1.0,
=2.0
6)
=1.0,
=2.5
7)
=1.5,
=2.0
H/R
A unit harmonic displacement is placed at the free surface and
deconvolved to bedrock, modeling the seismic excitation.
Using TRIAX, frequency dependent transfer functions for the
displacements are determined at the mat-soil interface (see
figure 2-2, points A & B) and at the top of the sidewalls
29
(point C).
Similarly, frequency dependent transfer functions
are determined in the free field at the foundation level for
a unit displacement at the free surface (points F & G, respectively), using 1-D theory.
/
/
/
Points of Measurement on Foundation and Free Field
Figure 2-2
Since the mat and the sidewalls are very rigid compared to
the soil, and the structure is axisymmetric, the rotation of
the foundation is defined as:
V
___-Ul
(18)
and the horizontal motion u is defined at point A.
In the
free field, the horizontal motion is determined at the foundation level (u.), and then compared with the horizontal motion at the foundation.
The free field rotation or "pseudo"
rotation can be defined in two possible manners.
The first
is simply proportional to the horizontal displacement of the
soil at the free surface minus the soil displacement at the
foundation level:
30
0__
(19)
__
E
A second method is to take the weighted mean of the "pseudo"
rotations of the individual soil layers throughout the embedment region.
Using this method the rotation takes the form
of:
E
Where:
+
N = Number of layer interfaces in embedment region
n = Number of layers in embedment region
ui= Horizontal translation of i'th layer interface
'= Average horizontal translation of embedment region
E= Depth of embedment region
Preliminary studies showed that the first option gave better
results, and in addition is much simpler to use.
Thus it will
be used as a basis for comparison with the "actual" 3-D rotation of the foundation.
2.2- Comparison of Translational Motions:
Figures 2-3 through 2-9 compare the absolute values of the
transfer functions for the translational motion of the foundation (3-D model) with the horizontal motion of the soil in
I-D MOTON
1.oo00
o.0
016
0.24
0-o2
0.40
0o.4
0o:.8
o'e4
FREQUENCY
0.72
COMPARISON OF TRRNSLATION AT FOUNDATION USING 1-D AND 3-0 MODELS
STUDY OF FOUNDATION MOTION WITH A MASSLESS FOUNDATION H/R=2.0,E/R=1.5
I-D MOTION
-o
'0
o
'.ie o'24
o'.z
0'.40
0.4
0.56
0'.e4
0.72
FREQUENCY
COMPARISON OF TRANSLATION AT FOUNDATION USING 1-D AND 3-D MODELS
STUDY OF FOUNDATION MOTION HITH A MASSLESS FOUNDATION H/R=2.5.E/R=1.O
FIGURE 2-4
I-0 MPRONX
I-P APPROX.
~
'0.00
o'.0
o'.1e
0'.24
o'.sZ
0'.40
0:4
0
o'.5
o'.4
0o'.72
'.o
o'.e
0.s
1.04
1'.12
1.zo
FREQUENCY
COMPARISON OF TRANSLATION AT FOUNDATION USING 1-D AND 3-0 MODELS
STUDY OF FOUNDATION MOTION WITH A MASSLESS FOUNDATION H/R=2.0,E/R=1.O
FIGURE 2-5
1-D MOTION
1.00
0.;0
0-o18
O.24
0.32
0.40
0;.4
056s
0.64
C.72
FREQUENCY
COMPRRISON OF TRANSLARTION AT FOUNDATION USING 1-0 AND 3-0 MODELS
STUDY OF FOUNDATION MOTION WITH A MASSLESS FOUNDATION H/R=1.5.E/R=1.0
FIGURE 2-1p
I
MOTION
..........
...........
......
3-0 MOTIOt
'0:00
o 6os
o0:1
0.24
0.32
0.:4
0'.49
o'.5
0o'.4
0.72
'.80
0.88
o'.98
1'.04
1'.12
1'.20
FREQUENCY
COMPARISON OF TRANSLATION AT FOUNDATION OF 1-D AND 3-D MODELS
PARAMETRIC STUDY WITH MASSLESS FOUNDATION. H/R=2.5.E/R=0.5: CASE 9
FIGURE 2-7
i-D MOTION
i"c
IO
tr,
O
cV
o.08
0.16
0.24
0.32
0040
0.56
0'.64
FREQUENCY
O'.72
COMPARISON OF TRANSLATION AT FOUNDATION USING 1-0 AND 3-0 MODELS
STUDY OF FOUNDATION MOTION WITH A MASSLESS FOUNDATION H/R=2.0.E/R=0.5
FIGURE 2-8
I-D MOTION
3-
MOTION
I-D APPROX.
l.a
'0.00
0'.06
0'.18
0.24
0.82
0.40
0.4
0o'.58
0"84
0'.72
0.680
0o88
0.96
1.04
1. 12
1.20
FREQUENCY
COMPARISON OF TRANSLATION AT FOUNDATION OF 1-D AND 3-D MODELS
PARAMETRIC STUDY WITH MASSLESS FOUNDATION. H/R=1.5,E/R=0.5: CASE 8
FIGURE
2-9
the free field at the foundation level (1-D model).
The free
field motion depicts the 1-D transfer functions with minima
occuring at the natural frequencies of the soil column above
the foundation level:
S
4E
C
and maxima at
The 3-D foundation motion follows very closely the 1-D motion
up to roughly .75 of the first natural frequency of the embedment region.
After that point the 3-D solution oscillates
with moderately small amplitudes, while the 1-D motion displays a sinusoidal nature.
Because of this, the 1-D motion
severely underestimates the response in the region of the natural frequencies of the embedment region (f.), while severely overestimating the response in the intervals between the
natural frequencies.
Notice, however, if one follows the 1-D
motion to .7 of the first natural frequency of the embedment
region and then keeps the transfer functions constant after
that point, one arrives at what appears to be a good estimate
of the actual 3-D motion.
While the procedure is arbitrary
and there is no theoretical justification for the 0.7 factor,
the effect is to partially consider the deamplification due to
embedment but not to the extreme predicted by the 1-D curve.
Notice that the classic method in which support motion and
free field surface motion are equated is a particular case of
39
the procedure described above.
In this case the arbitrary
factor is taken as 0.0, instead of 0.7, and the transfer functions have a constant value given by the ordinate at zero frequency, that is, 1.
In addition, the above criteria satisfies the particular case
of no embedment, where the natural frequency of the embedment
region is infinite.
In this case the transfer functions again
have a constant value of 1.
In chapter 3 the comparison will be made in terms of time histories and amplified response spectra, in addition to transfer
functions, which will enable a better assesment of the proposed method.
2.3- Comparison of Rotational Motions:
The rotational motion induced by the embedment of the structure shows a very good correlation between the 1-D and 3-D
motions up to a multiplicative constant.
This suggests the
use of an empiric correlation between the two types of motion
of the form:
where:
= Rotation of foundation (3-D) (Equation 18)
O
1.
=
"Pseudo" rotation in the free field (l-D)
(Equation 19)
40
o(= Scale factor dependent on E/R ratio
3
= Scale factor dependent of H/R ratio
Plotting the multiplicative constants for each of the seven
cases vs. E/R and H/R, an equation correlating the two types
of motion can be derived.
1.5"
R
R
= 1.5
C.R
0.5+
__
I
0.I
__
02.
0.5
Scale Factor for Rotation vs. E/R
Figure 2-10
0.+
H
R
S0.5
E.O
I
I
I
o.1
o..Z
o.s
Scale Factor for Rotation vs. H/R
.-
0.4
Figure 2-11
As can be seen from figures 2-10 and 2-11, the correlation is
much more sensitive to variations in embedment depth than
change in stratum thickness.
This agrees with theory since
the rotational motion is induced by the shearing stresses along the sidewalls, and thus directly proportional to the contact area.
The stratum thickness will have a secondary role,
affecting the foundation stiffnesses (rocking and swaying
"springs") and thus, indirectly, the rotational motion.
The numerical correlation between the rotational motions for
a rigid structure embedded in a homogeneous stratum may then
be expressed as follows:
0 (f)=((.257*(E/E))*(.037*(H/R)+.926))*,
(P)
42
where 0, and A,are frequency dependent transfer functions.
For most practical embedment depths, however, the factor
which depends on H/R can be set equal to 1 without loss of
accuracy (equation 21).
This expression has the advantage
that it can be used for cases where the stratum approaches
the half-space condition, as well as for shallow stratums.
,,(f )=0.257*(E/R ) *-,(D(O)
(21)
Figures 2-12 through 2-18 compare the free field "pseudo" rotational motion with the 3-D foundation rotational motion for
different embedments after the application of the scale factor given by equation 21.
The agreement in the low frequency
range in all cases is very good, with only minor variations
between the two
transfer functions.
In the higher frequency
range, as was the case with the translational motion there
is some discrepancy in the two types of motion, while the
trends remain similar.
Since this rotational component contributes little to the total motion of the structure (most of the rocking motion in the
structure is induced by the translational component and the
inertia in the structure), a divergence of 20% between the
transfer functions under discussion will produce only minor
variations in the response spectra.
expected to be good.
Thus the correlation is
CORRECTED I-D ROTATION
ON
'.08
0.18
0.24
0.32
0.40
0.48
06.5
0.684
0.72
0.80
0.88
FREQUENCY
COMPARISON OF ROTATION AT FOUNDATION OF 3-D AND 1-D (WITH SCALE FACTOR) MDODELS
PARAMETRIC STUDY WITH MRSSLESS FOUNDATION. H/R=1.5.E/R=0.5: CASE 8
FIGURE 2-12
I-P ROTATION
3-D ROrATION
r
I:'.0
o.e
o0.24
o'0~
o'40o
o0.4
1.72
O.0
0o.e
COMPARISON OF ROTATION AT FOUNDATION OF 3-0 AND 1-0 (WITH SCALE FACTOR) MODELS
STUDY OF FOUNDATION MOTION WITH A MASSLESS FOUNDATION H/R=2.0.E/R=0.5
FIGURE 2-13
CORRECTED
1-D ROTATION
r
'.os
o'.Se
o'.4
04o2
o'.
04
o'.4
o'.se
0.64
0.72
o.so
3-D ROTATION
046e
FREQUENCY
COMPARISON OF ROTATION AT FOUNDATION OF 3-D AND 1-0 (WITH SCALE FACTOR) MODELS
STUDY OF FOUNDATION MOTION WITH A MASSLESS FOUNDATION H/R=2.0.E/R=1.0
FIGURE 2-14
coiREcrTED I-0 ROTATION
~1~
1= I
I
I.08
0'.18
0'.24
0.32
0.40
0.48
0'.58
0.84
0.72
0.80
Ifni
~X~\\\
0.88
FREQUENCY
COMPARISON OF ROTATION AT FOUNDATION OF 3-0 AND 1-0 (WITH SCALE FACTOR) MOODELS
PARAMETRIC STUDY WITH MASSLESS FOUNDATION. H/R=2.5.E/R=0.5: CASE 9
FIGURE 2-15
D ROTATION
-0 ROT'ATION
.00
0.08
.1
0.4040
02484
0.32
0.40
0.46
0.6e
0-84
0'72
FREQUENCY
COMPARISON OF ROTATION AT FOUNDATION OF 3-0 AND 1-0 (NITH SCALE FACTOR) MODELS
STUDY OF FOUNDATION MOTION WITH A MASSLESS FOUNDATION H/R=1.5.E/R=1.O
FIGURE 2-16
CORRECTED I-D ROTATION
r 3-D ROTATION
7.00
0.08
0.16
0.24
0.32
0.40
0.48
0566
0-.64
0.72
0'60
O'.0
FREQUENCY
COMPARISON OF ROTATION AT FOUNDATION OF 3-D AND 1-D (WITH SCALE FACTOR) MODELS
STUDY OF FOUNDATION MOTION WITH A MASSLESS FOUNDATION H/R=2.5.E/R=1.0
FIGURE 2-17
CORECTEP I-D ROTATION
IROTATION
1L20
Ios
o0:16
0.24
0.s32
0.40
0.48
0o.e
0o:64
0~.72
000o
o.e
FREQUENCY
COMPARISON OF ROTATION AT FOUNDATION OF 3-D AND 1-D (WITH SCALE FACTOR) MODELS
STUDY OF FOUNDATION MOTION WITH A MASSLESS FOUNDATION H/R=2.0,E/R=1.5
FIGURE 2-18
50
2.4- Correlation for a Layered Media:
Up to this point, the results obtained are for a uniform stratum, where it is assumed that the shear wave velocity remains
constant with depth.
Since in a realistic case the soil be-
comes stiffer with depth, it is of interest to investigate
what effect this has on the previous discussion.
The model chosen represents a realistic distribution of shear
wave velocity.
All other parameters, however, remain the same
as before.
0.5
Stratum With Variable Shear Wave Velocity
Figure 2-19
The correlation between the horizontal translation at the
foundation and in the free field
(Figure 2-20) shows very sim-
ilar trends to those obtained for the uniform stratum case.
Again we see a very good agreement between the frequency content of the transfer functions up to close to the first na-
I-0 MOTION
3-0 MOTION
I-D APPIROX.
,1
'0.00
o'.10o
.
0.20
o
030
0.40
o'so
0o.60
0.70
O'80
0.90
1.00
1.10
zo20
1.30
1.40
1'.50
FREQUENCY
COMPARISON OF TRANSLATION AT FOUNDATION WITH LAYERED MEDIA FOR 1-D AND 3-0 MDLS.
STUDY OF FOUNDATION MOTION WITH A MASSLESS FOUNDATION H/R=2.0.E/R=1.0
FIGURE 2-20
l-D ROTATION (CORRECTED)
r-3-D ROTATION
1.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
FREQUENCY
0.90
1'.00
'.10
COMPARISON OF ROTATION AT FOUNDATION WITH LAYERED MEDIA FOR 1-0 AND 3-0 MODELS
STUDY OF FOUNDATION MOTION WITH A MASSLESS FOUNDATION H/R=2.0.E/R=1.0
FIGURE 2-21
tural frequency of the embedment region.
After that point the
motions diverge, but again the .7 factor gives a good estimate
of the motion.
The rotational motion, as can be seen from figure 2-21, also
agrees well with the previous correlations, although it is
somewhat more conservative.
This is due in large part to the
fact that the embedment region is weaker than in the previous
cases, and thus, as will be shown later, the foundation rotates less.
As in previous cases the 1-D motion has been
scaled by the value computed from equation 21.
Thus it can be
concluded that layering does not affect the results of this
study significantly.
2.5- Effect of Flexible Sidewalls
In this section the effect of introducing flexible sidewalls
into the model will be studied.
The properties chosen for the
sidewalls are proportional to the actual concrete properties,
and are as follows:
Concrete:
Cs = 8,000 ft/sec.
X= 150.00 lb/ft.
Soil:
Cs = 1,000 ft/sec.
(=
120.00 lb/ft.
This yields the nondimensional shear modulus for the concrete
of 80.00 for a unit shear modulus in the soil.
The mat was
assumed rigid as before, in order to be able to define a
translation and rotation.
Figure 2-22 compares the translational motion at the foundation due to a unit sinusoidal displacement at the free surface
for two identical 3-D models, except for the rigidity of the
sidewalls.
The comparison bears out the intuitive prediction
that the rigidity of the sidewalls has practically no effect
on the translational motion at the foundation.
The rotational motion, however, is considerably affected by
the flexibility of the sidewalls.
Since the rotational motion
of the foundation results from the differential lateral soil
displacement of the embedment region acting on the structure,
we would expect the rotational motion to decrease for decreasing rigidity of the sidewalls.
For a structure with perfectly
rigid sidewalls, the differential motion will be totally taken
up by the rotation of the foundation.
On the other hand, a
structure with flexible sidewalls will respond to the differential lateral motion partially in flexure and shear distortion of the sidewalls, and partially by rotation of the foundation.
RIGID SIDEWALL
ELASTIC SIDEWALL
.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
FREQUENCY
0.90
1.00
1.10
COMPARISON OF TRANSLATION AT FOUND. FOR MODELS WITH RIGID AND ELASTIC SIDEWALLS
STUDY OF FOUNDATION MOTION WITH A MASSLESS FOUNDATION H/R=2.0,E/R=1.0
FIGURE 2-22
-RIGID SIDEWALL
-ELASTIC 5IDEW/ALL
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10t
FREQUENCY
COMPARISON OF ROTATION AT FOUND. FOR MODELS WITH RIGID AND ELASTIC SIDEWALLS
STUDY OF FOUNDATION MOTION WITH A MASSLESS FOUNDATION H/R=2.0.E/R=1.0
FlGURE
2-23
Rigid Sidewalls
Flexible Sidewalls
Figure 2-24
As can be seen from figure 2-23, the rotation decreases 23%
with the introduction of flexible sidewalls.
Thus an adjust-
ment to equation 21 must be made to account for the flexibility.
Further research is needed, beyond the scope of this
study, to fully be able to asses this factor.
The effect of
reducing the stiffness of the sidewalls on the rotation of the
foundation should be similar to the effect of decreasing the
soil properties of the embedment region (disturbance of the
backfill).
Both will decrease the rotation of the foundation.
Notice that for the extreme case of no sidewalls at all, the
phase angle is almost 180 degrees out of phase from the case
of perfectly rigid sidewalls (figures 2-25 and 2-26).
The
reason for this, though there are a variety of counteracting
stress fields in the neighborhood of the foundation, appears
to be the following:
58
In figure 2-27a the case of the rigid sidewalls can be divided into two steps:
In the first, assume there are hinges
at the mat-wall connection and thus the mat does not rotate.
In the second step moments are applied to the sidewalls and
mat in order to satisfy the condition of rigidity of the
connection, resulting in a clockwise rotation of the foundation.
In figure 2-27b, the case of no sidewalls can be divided into
three steps.
In the first step, the plate with soil on top
of it is subjected to a rock motion, producing only a translational motion.
In step 2 the fill is removed and replaced
by equivalent stresses to satisfy equilibrium.
plate does not rotate.
Again the
Finally, in step 3, to simulate no la-
teral walls the fictitious forces in step 2 must be reduced
to zero.
This is achieved by applying equal and opposite for-
ces to those in step 2.
These new forces produce a counter-
clockwise rotation of the plate.
Thus, superimposing steps
2 and 3 the following results are obtained:
i) Zero lateral wall stresses.
ii) A counter-clockwise rotation of the plate.
The case of flexible sidewalls represents an intermediate
state between these two extreme conditions, and therefore the
rotation should be small.
,--4O
0 a0
.1
a
'.n e
TMNFER FlNCTIN FO
IPARAETRIC 8TUDY
'.1
S'.4
. 5
U
ROTATION OF TNE FU IrTIs
NOTION OF A NASSLE8s
o
PW MLE
FOUWODATIsN
ITHOUT SIDEWFLL.,
CASE 6
FIGURE 2-25
UI
0
Or
°ON
S
e. e
o'.
o.a
o'.
e.
e'.m
e.w
e'.- e'
FEIWENCY
'*.e .to
I'a
TROINFER FUNCTION FOR ITITIN OF THE FOUNDTIgON PWME OiWLE
STUDY OF FOUNDATION NOTION WITH A HR8SLE33 FOUDTIOM WO H/R=20.E/R=1.0sCRE 2
'.
C-0
17.0
FIGURE 22
o
SOIL EL
SMENT
-'17\
f /f
\/
'4,
LA W.
ve~
Rotation of Foundation with Rigid Sidewalls
Figure 2-27a
I
I
--,'I
r
I/
III
I
I
e -eVA--\ -NYO-----A I I I \-
------------ -- -
Rotation of Foundation with No Sidewalls
Figure 2-27b
3-
Study of Full Scale Model
Using the approximate methods derived
in chapter 2, this chap-
ter will determine their accuracy by studying the response of
a realistic structure subjected to support motions derived
from 1-D and 3-D models.
In addition, an assesment will be
made on the importance of the rotational component of foundation motion by studying the response of a structure subjected
only to the translational component of foundation motion.
Finally, the assumption that the mat of a nuclear containment
can be modeled as an infinitely rigid plate, as required for
the 3-step solution, will be evaluated.
Throughout this chapter the 1-D foundation motion shall refer
to the 1-D motion which has been modified by the rules derived
in chapter 2.
3.1- Model:
The physical properties and dimensions used for the full scale
model are described in figure 3-1.
E/R
=
0.815
H/R
=
1.704
The relevant ratios are:
63
1*
Full Scale Model
Figure 3-1
64
An 8% linear histeretic damping is assumed throughout the soil
stratum.
The soil and foundation are discretized by layers and finite
elements (linear expansion), while the structure is modeled
by a series of lumped masses.
Since the response of the
structure is to be used only as a method of comparison of the
two definitions of support motion, the loss of accuracy incurred by modeling the structure as a series of lumped masses
does not affect the results.
The steps in the analysis are as follows:
1.
The response of the 3-D massless foundation is computed
with the control motion at the free surface in the free
field (kinematic step).
The resulting motion is in the
form of a rotation and a translation.
In addition, the
1-D motion in the free field is computed and modified
as described in chapter 2.
The translation of the
foundation is defined at the center line of the mat and
at the mat-soil interface.
Since the foundation is as-
sumed very rigid, the rotation is defined as the vertical displacement of the end of the mat divided by the
mat radius.
As in the parametric studies of chapter 2,
the 1-D free field translation is defined at the foundation level and the pseudo rotation proportional to
the free surface translation minus the foundation level
65
translation (equation 19).
Frequency-dependent stiffnesses for the massless
2.
foundation are computed using a 3-D finite element
model.
3.
Using the computed foundation motions and stiffnesses,
the response of the structure is computed (dynamic
step).
3.2- Massless Foundation Motion:
The comparison of the 1-D and 3-D foundation motions are in
the form of transfer functions, frequency spectrum, time histories and amplified response spectra.
Figure 3-2 compares the transfer functions for translation at
the massless foundation for the 3-D and 1-D motions.
The two
curves follow very closely up to approximately 0.7 of the
first natural frequency of the embedment region.
After that
point, the 3-D motion oscillates about the constant line defined for the 1-D motion with fairly small amplitude suggesting that the 1-D approximation as defined in chapter 2 will
be good.
In the case of the rotational motion at the massless
foundation, we see from figure 3-3 that the 1-D motion predicted by equation 21 is considerably more conservative than the
actual 3-D motion.
The reason for this lies in the weaker
backfill of the embedment region, which cannot be easily modeled into the 1-D equation.
However, the trends of the two
TRAN5LAION
1:oo
t'.oo
s'.oo
400oo
'.oo
s'.oo
"oo
'oo
FREQUENCY
.0oo
COMPARISON OF TRANSLATIONAL MOTION AT FOUNDATION FOR 1-0 AND 3-D MODELS
FOUNDATION MOTION OF FULL SCALE MODEL WITH MASSLESS FOUNDATIONt CASE 1
FIGURE 3-2
rr--*" ~c~r
n
~ r
I-D ROTATION
oo
z'.oo
3'.oo
400
5s.00
.00oo
7.0oo
'.00
9 00
FREQUENCY
COMPARISON OF ROTATIONAL MOTION AT FOUNDATION FOR 1-0 AND 3-0 MODELS
FOUNDATION MOTION OF FULL SCALE MODEL WITH MASSLESS FOUNDATION: CASE 1
FIGURE 3-3
68
curves are very similar, suggesting that with further research
an additional multiplicative factor could be introduced to
account for this effect.
The good correlation between the 1-D and 3-D translation is
further demonstrated if one compares the frequency spectrum
of the two motions (figures 3-4 and 3-5).
In the low fre-
quency range the 1-D motion is slightly more conservative.
However, notice that the maxima for the two types of motion
are almost identical, occuring at the same frequency.
In the
high frequency range the frequency content of the 1-D and 3-D
motions are for all intents and purposes identical, suggesting
that the response of the structure using the approximate 1-D
support motion will be in good agreement with the response of
the structure using the true 3-D motion.
In the case of the rotational component for the massless foundation, the conservatism predicted by the transfer functions
is again reflected in the frequency spectrum of the motion
(figures 3-6 and 3-7).
Even though the actual values are some
50% of the 1-D motion, the relative content between each of
the frequencies is almost identical for the two motions, again
suggesting that with the addition of a multiplicative constant
to correct for weak lateral soil, the correlation should be
good.
.o00
2.00
3.o00
7.00
0800
FREQUENCY
00
Ib.oc
11.00
FREQUENCY SPECTRUM OF ABS. VAL. OF TRRNS. ACCEL. AT FOUND. USING FINITE ELEM.
FOUNDATION MOTION OF FULL SCALE MODEL WITH MASSLESS FOUNDATION: CASE 1
.00
MAX
AT
0.06811
0.781
1.00o
14.00
1.00
- --
-TZ I"""
t'.oo
9
2.00o
3.00
4a00
s.00
-
-
.00oo 7.oo0
oo00
FREQUENCY
00
i'00
1b.oo
FREQUENCY SPECTRUM OF RBS. VARL. OF TRRNS. ACCEL. RT FOUND. USING 1-0 APPROX
FOUNDATION MOTION OF FULL SCALE MODEL OF MASSLESS FOUNDATION' CASE I
.. oo
HRX
RT
0.06908
0.781
ib.oo
14.00
th.oo
MAX
0.00013
AT
5.029
m
co
a t.
a
C
'.0
o
1'.00
2.0o
3'.00
4'o00
6.00
o'.00
7.00
8o00
'.o00
ioo
11-00
tk-oo
.o00
14.00
tb.00
FREQUENCY
FREQUENCY SPECTRUM OF ABS.
VAL. OF ROTAT.
ACCEL. AT FOUND. USING FINITE ELEM.
FOUNDATION MOTION OF FULL SCALE MODEL WITH MASSLESS FOUNDATION: CASE 1
FIGURE 3-Zo
f
MAX
AT
.oo
'00
oo
'0oo
'o
4'.0
00
'.0oo
e.oo
7.oo
'.o0
900
FREQUENCY
FREQUENCY SPECTRUM OF RBS. VAL. OF ROTAT. ACCEL. AT FOUND. USING 1-D
FOUNDATION MOTION OF FULL SCALE MODEL WITH MASSLESS FOUNDATION: CASE 1
0.00028
5.029
MAX
0.07531
AT
5.540
kAN
'.0oo
'.oo0
'.oo
4.00
5'.00
'.00
700o
8.o00
9.00
lb.oo
TIME (SEC)
TIME HISTORY OF TRANSLATIONAL ACCEL. AT FOUNDATION USING FINITE ELEMENTS
FOUNDATION MOTION OF FULL SCALE MODEL WITH MASSLESS FOUNDATION: CASE 1
MAX
AT
-0.07205
5.820
IAf\JAA
1.00
2.00
:.00
4'00
5.00
8.00
7-00
8.00
TIME (8EC)
e.oo
TIME HISTORY OF TRANSLATIONAL ACCEL. AT FOUNDATION USING 1-0 APPROX
FOUNDATION MOTION OF FULL SCALE MODEL OF MASSLESS FOUNDRTION: CASE 1
FIGURE 3-9
MAX
0.4148
AT
0.584
DAMP 0.010
0.65
0.64
PERIOD (SEC)
AMPLIFIED RESPONSE SPECTRA OF TRANS. ACCEL. AT FOUND. USING FINITE ELEMENTS
FOUNDATION MOTION OF FULL SCALE MODEL WITH MASSLESS FOUNDATION: CASE 1
MAX
AT
0.4744
0.596
DAMP 0.010
PERIOD (SEC)
AMPLIFIED RESPONSE SPECTRA OF TRANS. ACCEL. AT FOUND. USING 1-0
APPROX
FOUNDATION MOTION OF FULL SCALE MODEL OF MASSLESS FOUNDATION: CASE I
MAX
AT
0.4702
0.596
DAMP 0.010
3'08
0:.1
02244
0'.2
0.40
0.48
0.58
0.84
PERIOD (SEC)
0.72
0:80
0o88
AMPLIFIED RESPONSE SPECTRA OF TRANS. ACCEL. AT FOUND. USING 1-0
FOUNDATION MOTION OF FULL SCALE MODEL WITH MASSLESS FOUNDATION: CASE 1
78
The comments pertaining to the transfer functions and frequency spectrum are further reinforced by the time histories
and response spectra of the two motions (figures 3-8 through
3-11).
Notice in particular the response spectra for trans-
lation of the massless foundation for the 1-D and 3-D models.
For the 1-D translation, the values in the range of the lower
periods follow very closely that of the 3-D motion, producing
a marked improvement over the pure 1-D response spectrum (no
straight line approximation beyond the .7f, factor - figure
3-12), where there is a considerable deamplification in the
vicinity of period 0.3 seconds.
(The natural period of the
soil column in the free field, above the foundation level.)
It is of interest to note the effect that embedment has on the
deamplification of motion.
A comparison of figure A-1 (appen-
dix) and figure 3-8 shows a deamplification for the translational motion from a maximum acceleration of 0.125 g to 0.075
g's, representing a reduction of 40%.
Thus, if one neglects
the deamplification due to embedment, i.e. making the control
motion and support motion equivalent, we would expect a conconservatism in the response of the structure in the order of
40%.
3.3- Structure's Response:
Though the comparison of the foundation motions is a good indication of the applicability of the 1-D model, the true test
lies in the comparison of the response of the structure subjected to the two support motions.
The lumped mass model used for this study consists of two
spring-connected lumped mass systems.
The first, modeling
the external containment structure, consists of eight masses,
while the second "lollypop", modeling the internal structure
(reactor pressure vessel, pedestal and crane wall), consists
of five masses.
There is one common mass, which is the mat.
The total weight of the structure is approximately 3500 KipSlugs.
4
5TRUCTURE
(EXTERNAL)
I -~
'
3
STRUCTURE 2
(INTERNAL)
7
8
Lumped Mass Model
Figure 3-13
To test the accuracy of the rules derived in chapter 2, the
two motion components obtained from the massless foundation
are applied as support motion to the lumped mass model.
Fre-
quency dependent soil stiffnesses derived from the 3-D case
are used for all studies.
Three points of comparison are tak-
en: at the top of the mat, at the top of the containment
structure, and at the top of the internal structure.
The com-
parisons are in the form of time histories of acceleration and
response spectra.
Figures 3-14 through 3-17 depict the time histories and response spectra at the top of the mat for the 1-D and
port motions.
3-D sup-
For both the time history and response spectrum
the structure subjected to the 1-D support motion has a response some 10% higher than the 3-D counterpart.
In the case
of the response spectra, the two responses follow very closely
up to a period of roughly 0.4 sec.,
after which point the 1-D
diverges slightly to the conservative side.
Notice, however,
that the trends remain very similar for the two motions past
the 0.4 sec. period point.
The same trends are encountered in
the comparison of the response at the top of the containment
and at the top of the internal structure
3-23).
(figures 3-18 through
In both cases the comparison of the response of the
structure subjected to the 1-D and 3-D support motions are
good up to a period
of roughly 0.4 sec., after which point the
motions diverge somewhat, but with similar trends.
MAX
AT
-0.08538
11.170
PhrwI
11.oo
ACCELERATION
X
k.oo0
AT LEVEL 8 OF STRUCTURE 1
STRUCTURE WITH FOUNDATION MOTION GENERATED FROM MASSLESS FOUND. AND FIN. ELEM.
FIGURE 3-14
HRX
RT
-0.08819
5.860
-A N,
ACCELERRTION
X
RT LEVEL 8 OF STRUCTURE 1
STRUCTURE WITH FOUNDATION tOTION GEER RT 0 FROM PASSLESS FOUND. AND 1-D APPROX.
FIGURE 3-15
MAX
0.5219
AT
0.596
DAMP 0.010
)
008
0;18
ARS ACCELERATION
0.24
X
0";2
0;40
0'48
0566
0;84
0.72
0490
06688
06
1;.04
1.12
1.20
PERIOD (SEC)
RT LEVEL 8 OF STRUCTURE I
STRUCTURE WITH FOUNDATION MOTION GENERATED FROM MASSLESS FOUND. AND FIN. ELEM.
FIGURE 3-11
MAX
AT
DAMP
ARS ACCELERATION
X
AT LEVEL 8 OF STRUCTURE 1
STRUCTURE WITH FOUNDATION MOTION GENERATED FROM MASSLESS FOUND. AND 1-D APPROX.
0.6010
0.596
0.010
FIGURE 3-17
MRX
AT
ACCELERATION
X
-0.18876
4.210
AT LEVEL 1 OF STRUCTURE 1
STRUCTURE WITH FOUNDATION MOTION GENERATED FROM MASSLESS FOUND. AND FIN. ELEM.-
FIGURE 3 18
HRX
AT
0.21937
4.460
.A
ACCELERATION
A
X AT LEVEL 1 OF STRUCTURE 1
STRUCTURE WITH FOUNDATION MOTION GENERATED FROM MASSLESS FOUND. AND 1-0 APPROX.1
FIGURE 3-19
lAX
AT
1.4878
0.417
DAMP 0.010
PERIOD (SEC)
ARS ACCELERATION
X
AT LEVEL I OF STRUCTURE 1
STRUCTURE WITH FOUNDATION MOTION GENERATED FROM MASSLESS FOUND. AND FIN. ELEM.
FIGURE 3-20
MAX
AT
1.7849
0.463
DAMP 0.010
S
0o'.o
o'.1e
RS RCCELERATION
0'.4
X
'.32
0'.40
0'.40
0.56
0.04
0.72
0.0
070
0.98
1.04
1-.1
PERIOD (SEC)
AT LEVEL 1 OF STRUCTURE I
STRUCTURE WITH FOUNDATION MOTION GENERATED FROM MASSLESS FOUND. AND 1-D APPROX.1
FIGURE 3-21
MAX
AT
1.6168
0.417
DAMP 0.010
PERIOD (
RRS ACCELERATION
X
AT LEVEL 1 OF STRUCTURE 2
STRUCTURE WITH FOUNDATION MOTION GENERATED FROM MASSLESS FOUND. AND FIN. ELEM.
FIGURE 3-22
MAX
AT
1.9342
0.463
DAMP 0.010
0:C8
0- .6
IRS ACCELERATION
024
0'-2
0.40
0.49
0.66
0.84
PERIOD (8EC)
0-.72
080
0.88
0.96
X AT LEVEL 1 OF STRUCTURE 2
STRUCTURE WITH FOUNDATION MOTION GENERATED FROM MASSLESS FOUND. AND 1-D APPROX.
1.04
1.12
1.20
FIGURE 3-23
It will be shown later that the primary reason why the 1-D
support motion produces a somewhat higher response in the
structure after period 0.4 seconds is the conservatism introduced by the rotational component of support motion.
This
conservatism, which affects the response in the vicinity of
2-3 c.p.s. contributes roughly 50% of the conservatism.
Thus,
if the rotational component of support motion could be improved, the improvement in the total response of the structure
would be significant.
The remaining portion of the conservatism in the response of
the structure results from the translational component of support motion.
As stated earlier, the conservatism is confined
to the lower frequency range, predominately in the vicinity of
2 c.p.s.
The question may be asked:
Is 10 or 15% too conservative a
value to be used in the design of a nuclear structure? Clearly, since the trends of the two motions are very similar, and
given the uncertainties of the local soil conditions, the nature of the earthquake motion and the countless other variables
which enter into a seismic analysis, a conservatism of 10 or
15% is very reasonable.
This is further reinforced by the
fact that up to only recently, the deamplification effect due
to embedment in the kinematic phase was totally neglected in
92
connection with the half space spring method, and thus derived
responses were at times 100% more conservative than those predicted by the 3-D embedded model.
The method proposed here,
while not requiring finite element techniques for its application, predicts results which are very close to those of the
complete solution, and thus a useful tool for design purposes.
Therefore, it can be concluded from this study that the 1-D
foundation motion will produce a somewhat more conservative
response in the structure, with the conservatism generally lying in the low frequency (high period) range, while the higher
frequencies are reproduced very well.
In the entire frequency
range, the trends of the two motions are very similar, as evidenced by the response spectra.
3.3- Effect of Neglecting Rotational Component of Foundation
Motion:
Using the model previously discussed, it is of interest to investigate the importance of the rotational component of foundation support motion on the total response of the structure.
To this end, the lumped mass model is excited by only the
translational 3-D component of foundation motion, with the response computed at the top of the containment and at the top
of the internal structure.
A comparison of figures 3-20 and 3-24 reveals that at the top
tIRX
1.2439
MAX
1.2439
AT
0.463
DAMP 0.010
o
Lu
C-)0
a
01
o
c.00
o'.0o
o0.1
0.24
o.82
0.40
0o.4
o.ss
0o.64
0.72
oeo
o'.
oe.98
.o4
1z.12
PERIOD (SEC)
/ AT LEVEL I OF STRUCTURE 1
ARS ACCELERATION
STRUCTURE WITH CONSISTENT MOTION FROM FINITE ELEMENT-TRANSLRTION ONLY
FIGURE
3-24
'.20
MqX
RT
1.3775
0.454
DRMP 0.010
o'.o
S
o'.6
RRS RCCELERRTION
o'.24
X
032
0.40
o.se
5
0"40
-64
0o72
0.o0
o;nb
PERIOD (SEC)
RT LEVEL I OF STRUCTURE 2
STRUCTURE WITH CONSISTENT MOTION FROM FINITE ELEMENT-TRANSLATION ONLY
95
of the containment the peak of the response spectra is reduced some 16% by neglecting the rotational component of support motion.
As can be observed, the reduction occurs primar-
ily over the frequency range of 2 to 4 c.p.s., with the very
high and low frequencies generally unaffected.
Notice, how-
ever, that throughout the entire frequency range, the trenes
of the two curves are very similar.
Much the same trends are exhibited in the comparison of the
internal structure (figures 3-22 and 3-25).
The peak response
is reduced some 15% from the response of the structure subjected to both motion components.
However, the range of rel-
ative significance is reduced to a band of roughly 2-3 c.p.s.,
as opposed to 2-4 c.p.s. for the containment structure.
In order to evaluate the contribution of each of the 1-D components of support motion to the total response of the structure, the lumped mass model is now excited by the 1-D translational component of support motion.
Figure 3-26 depicts the response of the top of the containment
subjected to the 1-D translational support motion.
When com-
pared to the response of the containment subjected to the 3-D
translational support motion (figure 3-24) the correlation between the responses is considerably better than the response
with both support motion components.
Thus, even though con-
ARX
AT
1.4227
0.463
DRMP 0.010
aIid
CW
C
C
C:
w
c
OI
lb'.oo
o'.0o
o.16
RRS RCCELERRTION
o'.24
o.32
.oo's 04e
0.40
0i5S0i6
or4
PERIOD (SEC)
0.72
'
0.10
0.11
-
0Do
-
1.e04
- 112
-
--
20
X AT LEVEL 1 OF STRUCTURE I
STRUCTURE WITH CONSISTENT MOTION FROM 1-0 RPPROX.-TRANSLRTION ONLY
FIGURE 3-2o
97
tributing only 15% to the total motion, the rotational component contributes greatly to the discrepancy in the response
of the structure subjected to the 1-D and 3-D support motions.
It has been suggested that the rotational component of support
motion be totally neglected in the analysis of a containment.
In terms of absolute numbers, the resulting reduction in the
response of the structure seems significant, especially since
the reduction produces an unconservatism.
However, in terms
of the uncertainties introduced when performing a seismic analysis, the reduction may be insignificant.
Therefore, one
must weigh the 16% reduction in response with the relative
certainty of the other imput parameters in order to make an
accurate assesment of its importance.
3.4- Effect of Mat Flexibility:
As stated earlier, an essential assumption for the use of the
3-step solution method is that the mat of the containment can
be modeled as an infinitely rigid plate.
Thus it is of inter-
est to investigate the difference in the response of a structure modeled with an infinitely rigid mat, and a structure
with a mat flexibility proportional to concrete.
To make this comparison, the structure discussed in this chapter is modeled with both a flexible and a rigid mat.
The con-
trol motion is applied in the form of inertial forces on the
98
structure modeled with finite
elements
(Whitman's Method),
by-
passing the kinematic step as discussed in chapter 1.
Two points of comparison are used: at the top of the mat, and
at the top of the containment structure.
As is evidenced by
figures 3-26 through 3-29, the difference in the response of
the two structures is neglegible.
At both the top of the con-
tainment and top of the mat, the maximum response for the flexible and rigid mat structure are almost equal, occuring at the
same frequency.
Throughout the entire frequency range, with
one notable exception, the values of the responses are almost
identical for the two structures.
The one exception occurs in
the vicinity of 5 c.p.s., which is roughly the natural frequency of the containment structure, where the response of the
structure with a rigid mat is noticeably greater than that of
a flexible mat.
However, this difference in response is over
a very small band of frequencies and thus should not be significant.
It can therefore be concluded that the mat of the containment
structure can be modeled as an infinitely rigid plate without
great loss of accuracy.
_
_q
MAX
2.4103
AT
0.338
DAMP 0.010
o'.0
S
o'.16
0o.24
o'02
0.40
O.4
o.5s
PERIOD
0.64
(SEC)
0.72
0.0eo
o;0.
0o.
AMPLIFIED RESPONSE SPECTRA FOR HORIZONTAL MOTION AT TOP OF CONTAINMENT
FINITE ELEMENT ANALYSIS WITH CONTROL MOTION AT FOUNDATION AND FLEXIBLE MAT
MAX
AT
0.9241
0.463
DAMP 0.010
0.5o
0o.4
PERIOD (SEC)
AMPLIFIED RESPONSE SPECTRA FOR HORIZONTAL MOTION AT TOP OF MAT
FINITE ELEMENT ANALYSIS WITH CONTROL MOTION AT FOUNDATION AND FLEXIBLE MAT
lAX
AT
0.9062
0.463
DRMP 0.010
oy
a
o
.0
o'oe
o'.00
o'.2
o'4
o'.
od.S
d40
4
o'"
PERIOD (8EC)
RPLIFIED RESPONSE SPECTRA FOR HORIIONTAL NOTION
RT
't72
e'.e
o'.$@
'o4
t.1
t'.to
TOP OF NAT
FINITE ELEMENT ANALYSIS NITH CONTROL MOTION RT FOUNDATION AND RIGID MRT
FIGURE 3- 30
103
4- Summary and Recommendations:
Rules for the use of 1-dimensional wave theory to determine
the motion that occurs at the foundation of a circular embedded foundation are presented here.
Factors are derived to
account for the embedment depth and stratum thickness which
modify both the translational and rotational components of
support motion.
The recommended steps to determine the motion that occurs at
the foundation of a circular embedded foundation (support motion) given the control motion at the free surface in the free
field is as follows: Using 1-dimensional wave theory, determine the translational motion that occurs at the level of the
foundation in the free field.
From this the translational
component of support motion can be computed by using the
transfer functions of the motion in the free field at the
foundation level up to a frequency of .7 of the first natural
frequency of the embedment region.
After that point the value
of the transfer functions remain constant, with a value corresponding to that occuring at .7f.
The rotational component
of support motion is determined by computing the "pseudo" rotational motion in the free field (equation 19), and then applying the corrective factor to account for embedment depth
(equation 21).
104
Using a realistic model, a comparison is made of the response
of a structure using the above recommended procedure and the
response of the same structure using finite elements to determine the support motion.
The agreement in the responses is
shown to be very good, especially in the high frequency range
where the motion is reproduced almost exactly.
In the low
frequency range, the suggested procedure produces a somewhat
higher response than the 3-D finite element response, primarily due to an overestimation of the rotational component of
support motion.
The trends of the two responses, however, are
still very similar.
It is shown that the properties of the backfill and sidewalls
have little effect on the translational component of support
motion, but greatly influence the rotational component.
A
decrease in the strength of either the backfill or sidewalls
will result in a decrease of the rotational motion.
For the particular model studied, the rotational component of
support motion contributes roughly 15% to the total response
at the top of the containment structure.
The zone of primary
influence is in the vicinity of 2 to 4 c.p.s., with the very
high and very low frequencies depending almost exclusively on
the translational component.
105
Finally, the assumption that the mat of the containment can
be modeled as an infinitely rigid plate, an essential assumption for the 3-step solution, is shown to be accurate enough
for design purposes.
It is recommended that further tests be conducted for a variety of physical and geometric conditions, to be able to
fully assess the accuracy of the recommended procedure.
In
particular, a study into the relationship between backfill
properties and rotational component of support motion is needed for a more general and accurate reproduction of that component.
106
REFERENCES
1)
Chang-Liang, V.: "Dynamic Response of Structures in
Layered Soils," ScD. Thesis, M.I.T., 1974.
2)
Duke, C.M.: "Effects of Ground on Destructiveness of
Large Earthquakes," Proc. ASCE, Vol. 84, No. SM3,
August, 1958.
3)
Donovon, N.C. and R.B. Matthiesen: "Effects of Site
Conditions on Ground Motions During Earthquakes,"
State-of-the-Art Symposium, Earthquake Engineering
of Buildings, San Francisco, California, February,
1968.
4)
Kausel, E.: "Forced Vibrations of Circular Foundations
on Layered Media," ScD. Thesis, M.I.T., 1974.
5)
Kausel, E. and J.M. Roesset: "Soil-Structure Interaction Problems for Nuclear Containment Structures,"
ASCE Power Division Specialty Conf., Denver,
August, 1974.
6)
Roesset, J.M. and R.V. Whitman: "Theoretical Background for Amplification Studies," M.I.T. Dept. of
Civil Engineering Report, 1969.
7)
Seed, H.B. and I.M. Idriss: "The Influence of Soil
Conditions on Ground Motions During Earthquakes,"
Proc. ASCE, Vol. 95, No. SM1, Jan., 1969.
8)
Waas, G.: "Linear Two-Dimensional Analysis of Soil
Dynamics Problems in Semi-Infinite Layer Media,"
Ph.D. Thesis, U. of Cal., Berkeley, 1972.
9)
Wiggins, J.H.: "Effect of Site Conditions on Earthquake
Intensity," Proc. ASCE, Vol. 90, ST2, April, 1964.
10) Whitman, R.V.: Unpublished paper.