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Effects of uplift on a gravity arch dam by linear elastic finite element analysis

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Title:
Effects of uplift on a gravity arch dam by linear elastic finite element analysis
Alternate title:
Effects of uplift
Creator:
Lund, Guy Stephen
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Language:
English
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xv, 228 leaves : illustrations ; 28 cm

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Subjects / Keywords:
Dams -- Design and construction ( lcsh )
Dam safety ( lcsh )
Dam safety ( fast )
Dams -- Design and construction ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaves 227-228).
General Note:
Spine title: Effects of uplift.
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Civil Engineering.
Statement of Responsibility:
by Guy Stephen Lund.

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University of Colorado Denver
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Auraria Library
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All applicable rights reserved by the source institution and holding location.
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ocm23603022
Classification:
LD1190.E53 1990m .L86 ( lcc )

Full Text
EFFECTS OF UPLIFT ON A GRAVITY ARCH DAM
BY LINEAR ELASTIC FINITE ELEMENT ANALYSIS
by
Guy Stephen Lund
B.S., Colorado State University, 1982

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirement for the degree of
Master of Science
Department of Civil Engineering
1990


(7) 1990 by Guy Stephen Lund
All right reserved.


This thesis for the Master of Science
degree by
Guy Stephen Lund
had been approved for the
Department of
Civil Engineering
by
Date
Andreas S. Vlahinos


Lund, Guy Stephen (M.S., Civil Engineering)
Effects of Uplift on a Gravity Arch Dam by Linear Elastic
Finite Element Analysis
Thesis directed by Professor John R. Mays
Present dam safety procedures require that many
older dams undergo structural analysis to determine their
safety before obtaining re-licensing. Unfortunately,
present criteria for the dam analysis is more stringent
then criteria used at the time of design, and may include
loads, such as the internal hydrostatic pressure due to
uplift, that were not considered during design.
The uplift criteria for design and analysis differ
between concrete gravity dams and arch dams. Conflicts
have arisen during the evaluation of older dams trying to
determine which criteria should govern the analysis. The
criteria for gravity dams respects uplift as an important
factor in the determination of the dam's structural
safety. The criteria for arch dams have normally ignored
uplift loads in structural analysis since these uplift
loads are believed to have negligible effects on the
stresses within the dam, and because arch dams do not
depend on shear resistance for stability as do gravity
dams. However, many older dams under evaluation are
classified as thick, or gravity arch dams and fall
between gravity and arch dams' criteria. To assure
adequate analysis, governing agencies have required that


gravity arch dams be subjected to the gravity dam
criteria, which include uplift.
This thesis used the linear elastic finite element
method to model the behavior of a gravity arch dam due to
normal static loads and uplift. Two models were
constructed to study the dam's behavior. The first model
compared the effectiveness of a softened foundation
modulus (simulating the inability of the foundation to
develop tension) a with cracked base along the cross-
canyon base of the dam. This model applied the uplift
force, which was determined based on present criteria for
evaluation of existing dams, to the cross-canyon base of
the dam and not the abutments. The second model studied
the cracked base with uplift along the entire
dam/foundation contact.
The results show that the application of uplift
forces to the dam decreases the dam's internal cantilever
action and increases the arch action. This indicates
that the dam behaves more favorably to uplift forces than
its gravity dam counterpart.
The form and content of this abstract are approved. I
recommend it publication.
Signed
v


CONTENTS
Figures........................................ ix
Tables........................................ xiii
Acknowledgements ............................... xv
CHAPTER
1. INTRODUCTION .................................... 1
2. LITERATURE REVIEW ............................... 5
2.1 General................................ 5
2.2 Uplift Begins:
Hypothetical Crack Philosophies ... 9
2.3 Seepage Philosophies .................... 18
2.4 Present Criteria . .................. 21
2.4.1 Gravity Dams............................. 21
2.4.2 Arch Dams................................ 25
2.5 Cracked Base Analysis.................... 25
2.6 Dam Safety and Analysis.................. 27
3. PRELIMINARY STUDIES ......................... 30
3.1 General.................................. 30
3.2 Two-dimensional Analysis on a Gravity
Dam...................................... 31
3.2.1 Model Description ....................... 31
3.2.2 Material Properties ..................... 34
3.2.3 Loads.................................... 35


3.2.4 Load Combination.................... 3 6
3.2.5 Analysis................................ 37
3.2.6 Results................................. 38
3.3 Three-Dimensional Block Model ... 42
3.3.1 Model Description ...................... 42
3.3.2 Material Properties.................... 42
3.3.3 Load.................................... 44
3.3.4 Load Combinations....................... 45
3.3.5 Analyses................................ 45
3.3.6 Results................................. 46
3.4 Summary................................. 47
4. SITE DESCRIPTION AND MATERIAL PROPERTIES . . 48
4.1 General................................. 48
4.2 Description............................. 48
4.3 Material Properties .................... 51
5. MODEL I..................................... 54
5.1 General................................. 54
5.2 Model Development ...................... 55
5.3 Material Properties .................... 59
5.4 Loads................................... 59
5.5 Load Combinations....................... 61
5.6 Analyses................................ 62
5.7 Results................................. 67
5.8 Computation of Safety Factor .... 103
5.9 Summary................................ 110
vii


6. MODEL II........................................ 114
6.1 General................................ 114
6.2 Model Development ..................... 115
6.3 Material Properties ................... 119
6.4 Loads.................................. 119
6.5 Load Combinations...................... 120
6.6 Analyses............................... 121
6.7 Results................................ 125
6.8 Computation of Safety Factor .... 146
6.9 Summary................................ 149
7. CONCLUSIONS..................................... 151
APPENDIX
A. GRAVITY DAM CRACKED BASE STUDY USING
GRAVITY METHOD OF ANALYSIS .................... 156
B. COMPUTATION OF NODAL FORCES FOR THE
ISOPARAMETRIC THICK SHELL ELEMENT ............. 162
C. PRE-PROCESSING FILE PREDAM...................... 167
D. POST-PROCESSING FILES .......................... 218
LIST OF REFERENCES................................. 227
viii


FIGURES
Figure
2.1 Limits of the resultant force based on the
middle-third principle.......................... 10
2.2 Levy's hypothetical crack theory............... 13
2.3 Lieckfeldt's hypothetical crack theory. . 15
2.4 Link's hypothetical crack theory............... 17
2.5 Solid material with voids allowing water
molecules to penetrate.......................... 19
2.6 Present uplift theory shown on a gravity dam. 22
3.1 Two-dimensional plane strain finite element
model of gravity dam............................ 33
3.2 Two-dimensional plane strain finite element
model of gravity dam and foundation............. 33
3.3 Stress distribution of loads due to gravity,
reservoir, and uplift loads assuming a rigid
uncracked base.................................. 40
3.4 Stress distribution of loads due to gravity,
reservoir, and uplift assuming rigid cracked
base............................................ 40
3.5 Stress distribution of loads due to gravity,
reservoir, and uplift with an elastic
foundation and assuming an uncracked base. . 41
3.6 Stress distribution of loads due to gravity,
reservoir, and uplift with an elastic
foundation and assuming a cracked base. . 41
3.7 Three-dimensional finite element model using
isoparametric thick shell elements ............ 43
4.1 Cutler Dam specification drawing............ 49


57
58
72
73
74
75
79
80
81
82
85
86
87
88
92
93
Four isometric views showing the finite element
model of Cutler Dam using stepped foundation.
Element numbers for finite element model: (a)
Upstream elements looking downstream; (b)
Downstream elements looking downstream. .
Arch and cantilever stresses on the upstream
face for LC-1...............................
Arch and cantilever stresses on the downstream
face for LC-1...............................
Principal stresses on the upstream face for LC-
1...........................................
Principal stresses on the downstream face for
LC-1........................................
Arch and cantilever stresses on the upstream
face for LC-2...............................
Arch and cantilever stresses oh the downstream
face for LC-2...............................
Principal stresses on the upstream face for LC-
2...........................................
Principal stresses on the downstream face
for LC-2....................................
Arch and cantilever stresses on the
upstream face for LC-3......................
Arch and cantilever stresses on the
downstream face for LC-3....................
Principal stresses on the upstream face for
LC-3........................................
Principal stresses on the downstream face
for LC-3....................................
Arch and cantilever stresses on the
upstream face for LC-4......................
Arch and cantilever stresses on the
downstream face for LC-4....................
x


5.17 Principal stresses on the upstream face for
LC-4......................................... 94
5.18 Principal stresses on the downstream face
for LC-4..................................... 95
5.19 Arch and cantilever stresses on the
upstream face for LC-5....................... 99
5.20 Arch and cantilever stresses on the
downstream face for LC-5.................... 100
5.21 Principal stresses on the upstream face for
LC-5........................................ 101
5.22 Principal stresses on the downstream face
for LC-5.................................... 102
5.23 Plan and profile of dam showing elements
shear friction factors of safety for LC-1. . 107
5.24 Plan and profile of dam showing elements
shear friction factors of safety for LC-5. . 108
5.25 Radial crest deflections for Cutler Dam due
to the load combinations LC-1 through LC-5. 112
6.1 Four isometric views of Cutler Dam finite
element model using the radial foundation. . 116
6.2 Cross-section of Cutler Dam finite element
model taken at the dam's centerline showing the
transition element between dam and foundation. 118
6.3 Crest deflections of Cutler Dam for load
combinations LC-1 and LC-6.................. 122
6.4 Arch and cantilever stresses on the upstream
face for LC-6............................... 127
6.5 Arch and cantilever stresses on the downstream
face for LC-6............................... 128
6.6 Principal stresses on the upstream face for LC-
6........................................... 129
6.7 Principal stresses on the downstream face for
LC-6........................................ 130
xi


133
134
135
136
139
140
141
142
145
147
148
Arch and cantilever stresses on the upstream
face for LC-7...............................
Arch and cantilever stresses on the downstream
face for LC-7...............................
Principal stresses on the upstream face for
LC-7. ...........................
Principal stresses on the downstream face
for LC-7....................................
Arch and cantilever stresses on the
upstream face for LC-8......................
Arch and cantilever stresses on the
downstream face for LC-8....................
Principal stresses on the upstream face for
LC-8........................................
Principal stresses on the downstream face
for LC-8....................................
Radial crest deflections of the dam crest
for load combinations LC-5, LC-6, LC-7, and
LC-8........................................
Plan and profile of dam showing elements
shear friction factor of safety for LC-6.
Plan and profile of dam showing elements
shear friction factors of safety for LC-8. .
xii


34
45
46
53
53
59
62
71
71
78
78
84
84
91
TABLES
Summary of material properties used in finite
element studies.............................
Summary of the isoparametric thick shell
element model applied loads.................
Summary of isoparametric thick shell element
results.....................................
Summary of material properties..............
Summary of the allowable stresses and safety
factors.....................................
Summary of the static loads applied to the
finite element model. ......................
Summary of load combinations................
Summary of maximum arch and cantilever stresses
for load combination LC-1...................
Summary of average arch and cantilever stresses
and deflections for load combination LC-1. .
Summary of maximum arch and cantilever stresses
for load combination LC-2...................
Summary of average arch and cantilever stresses
and deflections for load combination LC-2. .
Summary of maximum arch and cantilever stresses
for load combination LC-3...................
Summary of average arch and cantilever stresses
and deflections for load combination LC-3. .
Summary of maximum arch and cantilever stresses
for load combination LC-4...................


5.10 Summary of average arch and cantilever
stresses and deflections for load
combination LC-4................................ 91
5.11 Summary of maximum arch and cantilever
stresses for load combination LC-5.............. 98
5.12 Summary of average arch and cantilever
stresses and deflections for load
combination LC-5................................ 98
5.13 Summary of maximum stress safety factors. . 104
5.14 Summary of the average element safety
factor for LC-1 and LC-5....................... 105
6.1 Summary of load combinations................... 120
6.2 Summary of maximum arch and cantilever stresses
for load combination LC-6...................... 126
6.3 Summary of average arch and cantilever stresses
and deflections for load combination LC-6. . 126
6.4 Summary of maximum arch and cantilever stresses
for load combination LC-7...................... 132
6.5 Summary of average arch and cantilever stresses
and deflections for load combination LC-7. . 132
6.6 Summary of maximum arch and cantilever stresses
for load combination LC-8...................... 138
6.7 Summary of average arch and cantilever stresses
and deflections for load combination LC-8. . 138
6.8 Summary of maximum stress safety factors. . 146
6.9 Summary of the average element safety factor
for LC-1 and LC-5.............................. 149
xiv


ACKNOWLEDGEMENTS
I would like to give special thanks to Professor
John Mays for his academic guidance and efforts directing
this research. I would also like to thank Assistant
Professors Judith Stalnaker and Andreas Vlahinos for
their academic guidance throughout my graduate study.
A special appreciation is given to Mr. Howard
Boggs, who gave me technical guidance throughout this
research project.
A special thanks goes to ATC Engineering
Consultants, Inc., and Charles C. Hutton, who gave me
financial support towards my graduate degree and offered
me the opportunity to pursue my engineering goals towards
three-dimensional finite element analyses of concrete
dams.
Finally, to my lovely wife Nancy who has endured
the strain of living with a graduate student and, as a
result, sacrificed many things. Without her support and
patience over the last two-and-one-half years, this
graduate degree would only be a dream, not a reality.
xv


CHAPTER 1
INTRODUCTION
Dam engineers have used many different techniques to
analyze three-dimensional concrete dams over the past
century. One technique used frequently is the finite
element method (FEM) of analysis. Since the FEM gives
the engineer flexibility in applying load combinations
and analyzing structures with complicated geometrical
configurations, it has become a powerful tool in the
design and analysis of concrete dams.
Although the demand for the design and analysis of
new dams still exists, the emphasis has shifted to the
analysis of existing dams over the last couple of
decades. Recently the governing agencies that oversee
the inspection of most of the United States' non-
federally operated dams have required that structural
analyses be performed in order to obtain re-licensing.
The present criteria used in structural analyses include
some additional loads that were, in many cases, not
included in the design of the dam. One of these loads is
the internal hydrostatic pressure, or uplift.
Uplift has only been considered in the design of
two-dimensional gravity dams from the early part of the


twentieth century. Prior to that time the importance of
uplift forces was not fully understood; therefore, uplift
was not considered in the design of a gravity dam's
factor of safety. Presently the uplift is analyzed for
both foundation stability and structural soundness of
gravity dams.
Similarly, the design and analysis of arch dams has
only addressed the uplift forces since the first part of
this century. However, the design of arch dams has been
more concerned with the effects of uplift in regards to
the stability of the foundation, and not so much with the
structural analysis of the dam. Uplift has been
neglected in the structural analysis of arch dams because
it is believed to have a negligible effect on the
stresses within the dam.
Recently the Federal Energy Regulatory Commission
(FERC) has required that the uplift forces acting on some
thick arch dams be included in the structural analysis.
This requires structural engineers to develop analytical
models which include the uplift forces on the dam.
However, this problem is very non-linear in its behavior,
and at the present time the use of non-linear assumptions
for the structural analysis of arch dams is not
economically feasible. Therefore, the engineers must
develop techniques to model this non-linear behavior with
2


linear techniques.
This thesis is an attempt to develop a model using a
general linear elastic finite element analysis computer
program that will accurately predict the behavior of an
arch dam under load combinations that include uplift.
The first phase of this study was to develop a two-
dimensional finite element model of a concrete gravity
dam and subject it to the normal loads due to gravity,
reservoir, and uplift. The results of this study were
compared to the results of hand computations using the
gravity, or rigid body, method of analysis. By comparing
these two different approaches, the accuracy of the
finite element model could be verified.
The second phase of this study was to practice the
techniques used in the two-dimensional study on a simple
three-dimensional finite element model.
The third phase of this study applies the techniques
and programs developed in the first two phases to a thick
arch dam. The arch dam chosen for this study is Cutler
Dam, located on the Bear River about 13 miles west of
Logan, Utah. Four approaches were used to analyze the
dam behavior due to uplift. First, the dam was studied
without uplift and an uncracked base. Second, the dam
was studied with uplift and an uncracked base. The third
and fourth studies looked at the dam with a cracked base,
3


without and with uplift respectively.
All of the studies for this analysis have been done
on microcomputers. With the advances in microcomputers
over the last few years, it is now possible to conduct
finite element analysis without expensive computer time.
Therefore, this study used the programs available on the
microcomputer to carry out the analysis.
4


CHAPTER 2
LITERATURE REVIEW
2.1 General
Attempts to develop the earth's water resources are
as present today as they have been in the past. Dams for
water supply have been documented throughout history and
were probably one of the earliest structures devised by
mankind [1]. The ancient Egyptians, Persians, Romans,
and Indians are just a few who have left evidence of dam
construction dating as early as 3000 B.C. These civil
structures have played roles in the rise and decline of
many civilizations dependent on water for irrigation,
transportation, and municipal uses.
The dam builders of the past used materials that
were readily available in construction. Many of these
early dams were built with soils and gravel and a few
were constructed of cut masonry. The engineering was
haphazard at best, and builders relied heavily on
previous designers' experience and trial and error
methods [2] since they only had a slight understanding of
mechanics of materials.
Presently, engineers have a much better
understanding of the mechanics of construction, in situ


materials, and the loads acting on dams. This better
preparedness results in a minimizing of the risk in dam
design and construction. Due to human needs these water
resources will continue to be developed in the future and
the engineering principles used to design and construct
these dams will also progress.
Just as the understanding of mechanics of materials
has evolved throughout history so has the understanding
of loads and load combinations. It is probably true that
the primary load, perhaps the only load, analyzed for
many of the early dams was the water load on the upstream
surface [3]. Some of these early structures have
survived the test of time, but many have not been so
lucky. Statistically, for every one-thousand dams built,
ten have failed [1].
The new methods of design have progressed a long way
from those considering only the primary reservoir loads.
Today concrete dams are designed for a multitude of loads
which range through gravity, hydrostatic pressures due to
reservoir water and silt surfaces, volumetric changes in
concrete due to the maximum/minimum ambient air and water
changes, ice, internal hydrostatic pressures (uplift),
and seismic loads.
The crude methods used in the ancient design of dams
have also evolved into a complex form of analysis. The
6


ancient dam builders had very little knowledge on which
to base their design, and perhaps used very simple
experiments to help verify their structure. The
techniques used by the early pioneers who designed dams
such as Kurit, 13th century Iran, or Elche, 17th century
Spain, relied on rules for building arches for bridges
[3]. Today the engineer relies heavily on numerical
analysis to design dams. Techniques such as the finite
element method (FEM) have allowed the engineer to analyze
complex structures for an array of loads that only one-
half century ago were considered impossible. With the
help of the FEM the dam engineer is pushing the knowledge
of dam design to new limits and eliminating the over-
designed empirical methods of the past.
Consideration of the loads due to uplift has
developed in dam design over the last 150 years. Prior
to that time the existence of uplift was unknown.
Theories, such as the hypothetical crack and seepage
philosophies, have evolved into the present required
uplift criteria applicable to the analysis of dams. The
understanding of uplift is still being studied today at
many research laboratories, including the University of
Colorado, Boulder.
The effects of uplift and cracking have been
primarily studied in the behavior of gravity dams. The
7


uplift effects on arch dams, although not neglected, have
been ruled relatively small [4]; therefore, the uplift
loads have been normally excluded in arch dam design and
analysis.
Today, the emphasis has shifted from design and
construction to dam safety. Many older arch dams are in
the process of re-licensing and must undergo structural
analyses to satisfy dam safety requirements. However,
many of these arch dams are classified as gravity arch
dams, or thick arch dams, because they have a base
thickness to height ratio greater than 0.3 [5]. These
types of arch dams fall in the gray area between gravity
and arch dams criteria. Therefore, they are usually
subjected to the more stringent gravity dam criteria on
uplift, a load not included in the design of most of the
older dams.
The design criteria, as published by the USBR [4],
for both gravity and arch dams are very similar.
However, the uplift loads are excluded in the structural
analyses of arch dams. Since uplift has been excluded in
the analysis of arch dams, its inclusion with the present
numerical analyses technique, such as FEM, has not been
well defined or documented. This thesis investigates a
technique used to apply uplift pressures to a gravity
arch dam in a three-dimensional linear elastic finite
8


element analysis.
2.2 Uplift Begins:
Hypothetical Crack Philosophies
The history of uplift theory really begins in the
late 19th century. Prior to that time uplift was not a
concern in dam design. The criteria used by most
practicing dam designers was the middle-third principle,
or sometimes called the Rankine method. This principle
assumed a linear variation of stress across the base and
stated the following [1]:
On any horizontal plane within a gravity dam the
resultant of the forces should act within the middle
third. (Jansen 1980, 126)
Dams designed so that the resultant of all forces was
located within the middle third of the base assured that
the entire base would be in compression. Figure 2.1
shows the allowable envelope for a gravity dam's
resultant forces, based on the middle third principle and
assuming the reservoir is at no time empty. It was
believed that keeping the base in compression would
prevent cracks from forming on the upstream face, and
thus no internal pressures could develop within the dam.
However, some dam engineers began to realize the
consequential effects that uplift loads could have on the
dams factor of safety. The initial uplift theories were
developed based on the hypothetical crack philosophy.
9


FIGURE 2.1 Limits of the resultant force based on the
middle-third principle.
10


This philosophy assumes the dam is an impermeable barrier
in which a horizontal crack forms. This crack has the
potential of developing an uplift pressure which would
decrease the dams effective weight, thus reducing the
factor of safety.
Alfelt Dam, 1889, in Germany was one of the first
dams to incorporate uplift loads into the computation of
the dams factor of safety. Even though some projects
such as Alfelt Dam were beginning to look at uplift in
the design it was still an unrecognized load as far as
most dam engineers were concerned.
Probably the most influential event which started the
inclusion of uplift forces on gravity dams was the
failure of Bouzey Dam, 1895 [2]. The Bouzey dam was
located on L'Avier, a tributary of the Moselle River in
France. The dam was a straight masonry gravity structure
with a height of 72 feet and a length of 1732 feet. The
crest was 13 feet thick and the base was 37 feet thick.
The dam's construction was completed in 1881 and the
reservoir began filling at that time. Apparently the
builders excluded the middle third principle in the
design of the dam. This resulted in leakage totaling 900
gallons per minute and observed cracks on the upstream
face soon after the reservoir filled [1]. Although the
maximum reservoir was lowered about 9 feet due to leakage
11


and observed cracks, the dam's operational regime
remained relatively unchanged. The dam failed in April
1895 causing the deaths of over 100 people [1].
Investigations discovered that the water pressure in and
around the dam played an important role in contributing
to the failure [1].
Unfortunate as the failure was, it created an
unprecedented state of nervous tension among dam
designers, and this resulted in many discussions on the
uplift controversies. It has been said that the Bouzey
Dam failure is a milestone in the development of uplift
in dam design [2].
Following the Bouzey Dam failure in 1895 a French
Engineer Maurice Levy published a revolutionary paper on
uplift [2]. Levy's paper expanded on Rankine's
philosophy of the middle third principle. The middle-
third principle only requires that zero stress exist on
the upstream face on the dam. Levy argued that in order
to achieve an adequate factor of safety the stress on the
upstream face should be greater than the hydrostatic
pressure from the reservoir. This was illustrated with a
solid dam of impermeable material, as shown in figure
2.2. Assume that a hypothetical crack forms in the dam
at depth, h. The pressure from the reservoir will act on
the crack and try to pry it open. It was, therefore,
12


FIGURE 2.2 Levy's hypothetical crack theory.
13


obvious that the compressive stress at the upstream tip
of the crack should be greater than the water pressure
trying to open it. If this criteria was satisfied then
the uplift would not be allowed to enter the dam and
effect the stability. Levy's method became widely
accepted and was used extensively by the dam designers
until the middle 1920's.
The next influential development in uplift theory was
by a German designer, Lieckfeldt, in 1898 [2].
Lieckfeldt's argument was also an extension of the
hypothetical crack philosophy; however, it was one of the
first applications which allowed water pressure to enter
the crack. Similarly to Levy's illustration, shown in
figure 2.3, a dam is assumed built of impermeable
material in which a hypothetical crack forms. Lieckfeldt
divided the base into two parts, cracked and uncracked.
The cracked portion would be subject to pressure
equivalent to the reservoir head, and the uncracked
portion would have the compressive stress from the dam
and would not have any water pressure acting on it.
Lieckfeldt's criteria allowed the dam to be designed with
the crack such that the crack depth would not exceed
certain predetermined limits of safety.
14


FIGURE 2.3 Lieckfeldt's hypothetical crack theory.
15


Further improvements on the hypothetical crack theory
came in 1910, when Link represented the uplift force as a
linear varying pressure across the entire base of the dam
as shown in figure 2.4. Link assumed that the crack
existed across the base and that the water pressure
flowed into the crack. The pressure at the upstream face
was equivalent to the product of the reservoir head and a
factor, where the factor was dependent on the different
dam conditions and/or authors. This pressure diagram is
in agreement with the assumptions that the water is
moving, which was similar to the newly discovered seepage
phenomenon.
16


FIGURE 2.4 Link's hypothetical crack theory.
17


2.3 Seepage Philosophies
The seepage theory of uplift began to show up in the
early 1900's. As reported by S. Leliavsky, Water Power.
1959, the theory of seepage effect on dams was presented
by Oscar Hoffman in 1928 [2].
This principle is based on the fact that any
material, no matter how solid, contains pores which are
capable of absorbing a certain amount of water when
subjected to enough pressure. Figure 2.5 shows the
micropores of a solid and the potential path that
molecules of water may follow. This theory allows the
water to filter throughout the pores in the dam and
foundation. Therefore, the impermeable barrier, which
the hypothetical crack principle relied heavily, does not
exist. Instead, the pores in the dam and foundation
allow the water pressure to penetrate and develop uplift
pressures. Laboratory test performed by Leliavsky, and
reported in Water Power. 1959 [2], and Design Textbooks
in Civil Engineering: Volume IV. 1981 [6], showed that
for high hydrostatic pressure the seepage could develop
tensile stresses within the concrete.
18


FIGURE 2.5 Solid material with voids allowing water
molecules to penetrate.
Using the seepage theory for uplift within the intact
concrete in the dam was challenged in 1977. An engineer
named Bazant showed that based on the porosity and
subsequent permeability of the concrete via analogy with
temperature, that uplift would take over a century to
develop within a concrete dam. None-the-less, because of
the numerous cracks, seams, and joints within the
19


foundations, the seepage theory is still the basis for
developing uplift criteria on the base of the dam.
Today the uplift criteria uses both the seepage
theory and the hypothetical crack theory. The
hypothetical crack theory is used for the concrete and
the seepage theory is used for the foundation. Also, the
organizations that have developed present day uplift
criteria have used recorded measurements from hundreds of
dams to assist in determining the criteria for uplift.
Although there are some minor differences between dam
designing agencies, virtually all agree on the following
criteria:
a) uplift should be included in the design of dams,
b) the uplift is applied to the entire base of the
dam,
c) the upstream pressure is equivalent to the
reservoir hydrostatic head unless field data supports
otherwise,
d) the pressure varies linearly from the upstream
face to the downstream face, and
e) if a crack develops in the dam or at the contact
the uplift within the crack shall be equivalent to
the reservoir head.
These rules may vary slightly for dam with drainage
systems, grout curtains, or seepage cutoffs. However the
20


basic principle is the same. Figure 2.6 shows the
present accepted uplift criteria for a gravity dam.
2.4 Present Criteria
Presently there are many different criteria regarding
the application of uplift loads towards dam design.
Agencies such as United States Bureau of Reclamation
(USBR), United States Army Corps of Engineers (COE), and
the Federal Energy Regulatory Commission (FERC) have all
published methods in which dam engineers may use as
guidelines in the design of a new dam.
2.4.1 Gravity Dams
The USBR assumes that the internal hydrostatic
pressure exists throughout the dam and foundation pores,
cracks, and seams [7]. The pressure along a horizontal
section is assumed to vary linearly from full reservoir
head at the upstream face to zero or tailwater head at
the downstream face for dams which do not contain drains.
For dams which have effective drains the pressure is less
than that of dams without drainage systems. The uplift
pressure is assumed to vary linearly from full reservoir
head at the heel, to the tailwater pressure plus one-
third the difference between the reservoir and tailwater
pressure at the drains, to tailwater pressure at the
downstream toe.
21


FIGURE 2.6 Present uplift theory shown on a gravity dam.
22


The COE criteria for uplift at the base of the dam is
similar to the USBR criteria; however, the uplift at an
effective drain is assumed to be reduced by 25 to 50
percent of the difference between the reservoir and
tailwater heads [8]. However, the COE does differ from
the USBR in determining uplift forces within the concrete
of the dam. The uplift within the concrete is assumed to
be 50 percent of the hydrostatic head, and is based on
the maximum reservoir duration and the time required to
develop internal pore pressure in the concrete due to
porosity.
The uplift criteria which FERC published for gravity
dams also assumes the uplift pressure on a dam without
drainage varies from full head at the upstream face to
tailwater head at the downstream face [9]. However, FERC
as gone into considerable detail to outline the uplift
criteria for dams which include drainage, grouting, and
cutoff collar systems.
It must be noted that the USBR and COE have produced
their uplift criteria based on the design of new dams.
The result is a stringent criteria for dam analyses.
However, society's increased consciousness towards the
environment and increasing economic costs have
diminished, even eliminated, the demand for new dams. In
fact, the majority of the dam engineers work is now in
23


dam safety and requires investigation of the stability
and adequacy of existing dams. Most of these existing
dams were designed without regards to the present uplift
criteria and many did not include uplift in the design.
To apply the strict rules governing the design of new
dams may lead to costly repairs which do not result in an
additional factor of safety.
The following is an excerpt from an article in Hydro
Review. April 1990, which addressed the most applicable
criteria available for the evaluation of existing dams
[10].
FERC's experience indicated that the Bureau of
Reclamation's and the Corps of Engineers criteria
doesn't give guidance in several key areas which
commonly occur at existing non-federal projects. ...
FERC has adopted the other federal agencies' uplift
criteria, with modifications to handle situations
found at existing dams that available criteria do not
cover. (Foster 1990, 70)
It appears that the most relevant criteria are probably
FERC's because they address the approaches to be used in
the analysis of existing dams. All federal agencies
agree on the basic concepts of uplift pressure acting on
the dams. However, FERC's additional criteria on the
uplift loads for existing dams with various drainage
systems is refreshing for the dam engineers working on
older existing dams.
24


2.4.2
Arch Dams
At the present time the uplift criteria for arch dams
is not as strongly documented as it is for gravity dams.
The USBR addresses the issue of uplift, as applied to
arch dams, in their engineering monograph No. 19, Design
Criteria for Concrete Arch and Gravity Dams [4]:
Internal hydrostatic pressures reduce the
compressive stresses acting within the concrete,
thereby lowering the frictional shear resistances.
Unlike gravity dams, which depend on shear resistance
for stability, arch dams resist much of the applied
load by transferring it horizontally to the abutments
by arch action.
The effects of any internal hydrostatic
pressures in arch dams, therefore, will be
distributed between both vertical and horizontal
elements. A recent analysis of these effects on
an arch dam of moderate height showed a stress
change of approximately 5 percent of the
allowable stress. The capability of analyzing
the effects of internal hydrostatic pressure has
not been incorporated as a regular part of the
analysis because of the minor change in stress.
(USBR 1977, 10)
Neither the COE nor FERC have design criteria for arch
dams at the present time.
2.5 Cracked Base Analysis
Historically whether or not a dam's base has cracked
was determined based on linear elastic beam formula
[11]. Although this equation is limited to shallow beams
this method has been widely accepted by dam engineers.
Presently there is research underway at the University of
Colorado, Boulder, which is attempting to prove the
25


validity of the shallow beam theory. However, the present
criteria is still influenced by the linear elastic beam
formula.
The primary influence uplift has on a dam is to
decrease the vertical compressive stress acting across
the horizontal plane of the structure. This lower
compressive stress results in a reduction of the shear-
friction safety factor of the dam [4], If the uplift
pressure acting on a horizontal section, in conjunction
with the other loads on the dam, is enough to develop
tensile stresses at the dam heel then the section is
assumed to crack. All agencies agree that the uplift is
equivalent to full reservoir head within any portion of
the base that has cracked. However, there are different
criteria on how the tensile stress at the heel is
computed.
The USBR computes the stress across the base in two
steps [12]. First, the stress is computed due to all
normal loads on the dam excluding uplift pressure.
Second, the stress distribution across the base is
computed for uplift. These two stress diagrams are
compared and a crack is assumed to develop when the
compressive stress at the heel, due to the normal loads,
is less than[the potential tensile stress. The potential
i
tensile stress is equivalent to the difference between
26


the product of reservoir head, a reduction factor for
drains, and the allowable tensile strength of the
material. Should a crack form then the crack is assumed
to propagate to the point where the compressive stress
and internal hydrostatic pressure are equivalent.
The COE computes the stress across the base with all
the loads acting on the dam, including uplift [7]. The
crack is assumed to form anywhere a tensile stress
exists.
FERC states that the base stresses should be computed
excluding uplift and compared to the stresses due to
uplift [10]. A cracked section exists where the
compressive stress is less than the uplift stress. The
cracked portion will be subjected to full reservoir head
and the uncracked portion will be a varying pressure
based on the type of drainage system present and the
length of crack.
2.6 Dam Safety and Analysis
As stated previously, there is not a strong demand
for the design of new dams. Instead, the dam engineering
emphasis has shifted to dam safety; the analyses of
existing dams for potential hazards.
It is important to note that dam engineering is not
an exact science. Therefore, to keep up with the new dam
safety policies established by organizations such as
27


FERC, the engineers are pressured to develop state-of-
the-art techniques to more accurately analyze existing
dams for repair. Numerical analyses techniques, such as
FEM, are used frequently today to assist the engineers in
the analyses of concrete arch dams.
The finite element method of analysis has progressed
from the advent of the computer and now is one of the
most powerful tools available to the dam designer.
Because of the flexibility in finite element analysis it
is easy to analyze special situations due to loading
conditions and for all type of variations in geometry.
The FEM also allows the engineer to model the effects
of a dam due to an elastic foundation. This is important
since the yielding of the foundation has great effects on
the stresses within the dams.
The advancement of today's microcomputers has allowed
the engineer to develop desk top computing power that a
decade ago was only available to large engineering firms
with mainframe computers. Similarly, the graphic
software available has helped the engineer produce superb
graphic presentations of results which aid the engineers
in the analysis of stresses. Finally, the microcomputer
helps the engineer keep costs down, which is very
important in today's economy.
It has become necessary, therefore, to develop a
28


procedure in which the uplift loads can be applied to a
three-dimensional finite element analysis of an arch dam.
Furthermore, it would be advantageous to develop the
process on the microcomputer so that the cost of computer
time can be kept to a minimum and the savings be passed
along to the clients.
The remainder of this paper discusses the procedures
used to develop the uplift pressures on the base of a
gravity arch dam using a linear elastic finite element
analysis. All of the analyses were conducted on the
microcomputer, thus demonstrating the ability to keep
cost to a minimum. Similarly, all pre- and post-
processing were done on the microcomputer.
29


CHAPTER 3
PRELIMINARY STUDIES
3.1 General
Before the .uplift analysis was performed on the
gravity dam, preliminary studies were conducted to better
understand the cracked base analysis and the application
of uplift using the finite element method.
Two studies were conducted using simplified models
to verify the proposed finite element techniques to be
used on the gravity dam's uplift analysis.
The first analysis was performed on a two-
dimensional gravity dam. The dam was subjected to normal
static and uplift loads and was analyzed using the
criteria described in Chapter 2. The dam was studied
assuming a rigid base and with a significant portion of
the foundation included in the model.
The second analysis was performed on a simple three-
dimensional block structure using the isoparametric,
variable-noded, thick shell elements. This analysis
investigated the effectiveness of uplift loads using both
pressures and nodal forces.
Both of the studies are described in this chapter.


3.2 Two-dimensional Analysis on a Gravity Dam
A finite element analysis was performed on a simple
two-dimensional gravity dam model to determine the
effectiveness of internal forces, simulating uplift
pressure, acting at nodes located near the base of the
dam.
A parallel study was also performed on the two-
dimensional gravity dam using the gravity method of
analysis, as described in USBR's Design of Gravity Dams
[12]. This analysis was used to assist in verification
of the finite element model.
3.2.1 Model Description
The two-dimensional gravity dam used has a vertical
upstream face, a downstream slope of 0.75 vertical: 1.0
horizontal, and a crest thickness of 10 feet. The base
thickness is 75 feet and the dam height is 100 feet. The
finite element model used to analyze the dam is shown in
figure 3.1.
The model was constructed with 260 four-noded, plane
strain elements, with 10 elements across the base of the
dam. The dam was analyzed using a roller base boundary
condition. This boundary condition restrained the base
nodes against vertical displacement only, except for the
downstream face node, which was also restrained against
31


horizontal displacement.
Additional studies were performed that showed the
fixed base restraint against both horizontal and vertical
translation was incorrect, and produced poor results.
The dam was also analyzed with an elastic
foundation. The finite element model was modified to
contain a significant portion of the foundation around
the dam. The foundation modeled extended 100 feet
upstream, downstream, and below the limits of the dam and
used 100 four-noded, plane strain elements. The
foundation boundary condition was the roller type,
similar to the boundary condition used in the previous
study. The finite element model used to study the dam
with the foundation is shown in figure 3.2.
32


100
y
10 -o-
FIGURE 3.1 Two-dimensional plane strain finite element
model of gravity dam.
10 -0"
FIGURE 3.2 Two-dimensional plane strain finite element
model of gravity dam and foundation.
33


3.2.2 Material Properties
The material properties used in this study were
chosen to correspond.to the material properties
identified for Cutler Dam and described in detail in
Chapter 4. The summary of the material properties are
shown in table 3.1.
Description Value
Concrete: Young1s Modulus, E 3,500,000 lbs/in2
Shear Modulus, G 1,458,333 lbs/in2
Unit Weight 155.0 lbs/ft3
Poisson's Ratio 0.2
Foundation: Young's Modulus, E 2,400,000 lbs/in2
Shear Modulus, G 1,000,000 lbs/in2
Unit Weight 0.0 lbs/ft3
Poisson's Ratio 0.2
Reservoir: Water Unit Weight 62.5 lbs/ft3
TABLE 3.1 Summary of material properties used in finite
element studies.
34


3.2.3
Loads
The following loads were applied to the structure:
o Gravity
o Reservoir hydrostatic pressure
o Internal hydrostatic pressure (uplift)
The gravity was applied as a monolithic load using
the unit weight of concrete and a gravitational
acceleration of 386.4 in/sec2.
The hydrostatic pressure due to the reservoir was
applied as a stepped pressure distribution on the
upstream face of the dam. The pressure for each element
was equivalent to the hydrostatic head at the centroid of
the element.
The internal hydrostatic pressure due to uplift was
simulated using nodal forces. An upward force (positive
Y-axis), equivalent to the product of the element's
horizontal length and the internal hydrostatic pressure
at the element's centroid, was applied to the row of
nodes located about 2.75 feet above the base of the dam.
The uplift pressure was assumed to vary linearly from
full reservoir head at the upstream heel to zero head at
the downstream toe for an uncracked section. For a
cracked section the uplift was set equivalent to the full
reservoir pressure within the limits of the crack, and
assumed to vary linearly from full head at the crack tip
35


to zero head at the downstream toe. This application of
the uplift was in accordance to the manner similar
described in FERC's Hydropower guidelines [9].
To keep the analysis simple the effects due to
drainage systems were neglected for three reasons.
First, the study focused on the accuracy of the method to
apply uplift, and the methods used to apply uplift to the
finite element model do not change significantly for a
dam with drains. Second, by leaving the drains out of
the study made verification of the results more straight
forward. Third, Cutler Dam, the concrete gravity arch
dam under consideration in the final phase of this
thesis, does not contain drains. Therefore, these
studies were more in direct correlation with the actual
field conditions of Cutler Dam.
3.2.4 Load Combination
Each of the loads were applied independently to the
model to study their effects. All the loads were
superimposed on the model to determine the effects
cracking would have on the base.
The load combination included the gravity, reservoir
hydrostatic pressure, and the internal pressure due to
uplift.
36


3.2.5 Analysis
The general linear elastic finite element program
SAP386 was used for this study [13]. This program is the
microcomputer equivalent to the well known linear elastic
finite element code SAPIV [14].
The first analyses were performed for a rigid base.
The initial study assumed an uncracked base; therefore,
uplift pressure varied linearly from full reservoir head
at the upstream face to zero head at the downstream face.
The stresses, normal to the base of the dam, at the
upstream heel were analyzed to determine if the base
would form a crack. If cracking occurred the uplift
load was adjusted for the cracked length and the model
was re-analyzed. This procedure was repeated until the
crack length converged.
The crack was modeled by relaxing the modulus of
elasticity of selected elements, located at the base of
the dam. The relaxation was accomplished by reducing the
elements modulus to 10 percent of the original modulus,
or 350,000 lbs/in2. Relaxing the modulus of these
selected elements simulated the crack's inability to
develop tensile stresses across the base.
The uplift load was adjusted to be equivalent to the
full reservoir head within the limits of the crack, and
varied linearly from full head at the crack tip to zero
37


head at the downstream toe of the dam. The nodal forces
simulating uplift were adjusted to reflect this
assumption.
A rigid cracked base analysis using the criteria
discussed in FERC's guidelines was also performed on the
two-dimensional gravity dam to assist in the verification
of the finite element analysis [9]. This method is also
referred to as the gravity method in the USBR
publications [12]. These computations are shown in
Appendix A.
Studies were also conducted to determine the dam's
behavior when the foundation was included in the finite
element model.
The application of uplift, with the foundation
included, is similar to the rigid base model. However,
additional downward forces (negative Y-axis) must be
applied to the nodes at the dam base, on the top of the
foundation. These forces are equal and opposite to the
nodal forces applied to the dam.
The analyses for the cracked base follows the same
procedure as described above.
3.2.6 Results
The gravity method of analysis indicated that a
crack would form and extend about 32 feet downstream.
The results for the finite element model with the
38


roller base boundary conditions were similar to the
gravity method results. The stresses due to gravity,
reservoir, and uplift for the uncracked and cracked base
are shown in figures 3.3 and 3.4. The finite element
analyses indicated that.a crack would form and would
extend between 30 and 45 feet downstream. Since the
finite elements mesh used in the model was coarse, 7.5
foot element lengths, the crack length could only be
approximated. However, the model did show stress
distribution across the base similar to the gravity
method computations; therefore, the model was concluded
to be an accurate approximation of the dam.
The finite element model which included the
foundation showed a reduction of the crack length. The
stresses due to gravity, reservoir, and uplift for the
uncracked and cracked base are shown in figures 3.5 and
3.6. The reduction in crack length is expected since the
foundation, having a modulus that was less than the
concrete modulus, would allow the dam to rotate. This
rotation causes the stresses across the base to
redistribute and prevents the crack from propagating as
far downstream as shown in the rigid base analysis.
39


Reservoir
FIGURE 3.3 Stress distribution of loads due to gravity,
reservoir, and uplift loads assuming a rigid
uncracked base.
o
CV
FIGURE 3.4 Stress distribution of loads due to gravity,
reservoir, and uplift assuming rigid cracked
base.


FIGURE 3.5 Stress distribution of loads due to gravity,
reservoir, and uplift with an elastic
foundation and assuming an uncracked base.
FIGURE 3.6 Stress distribution of loads due to gravity,
reservoir, and uplift with an elastic
foundation and assuming a cracked base.
41


3.3 Three-Dimensional Block Model
Preliminary investigations were conducted on a
three-dimensional solid finite element model. The
isoparametric, variable noded, thick shell element,
called the type-8 element in SAP386, was used to model a
simple structure to study the behavior of the solid
finite elements when subjected to internal loads
simulating uplift pressure. Both pressure loads and
nodal forces were studied in these analyses.
3.3.1 Model Description
Figure 3.7 illustrates the model used to analyze the
three-dimensional elements.
Sixteen elements were used to model the 24 x 24 x 36
inch solid cube. Each element was a 12 x 12 x 12 inch
cube. The base of the cube was fixed against translation
in both horizontal and the vertical direction and the
internal loads were applied to the top and bottom edges
of the middle row of elements.
3.3.2 Material Properties
The material properties used in the finite element
model were identical to the concrete materials in Cutler
Dam and are summarized in the table 3.1.
42


Y
FIGURE
3.7 Three-dimensional finite element model using
isoparametric thick shell elements; (a) model
geometry and element 1-4 location, (b)
uniform pressure, (c) linearly varying
pressure, (d) nodal forces.
43


3.3.3
Load
The loads were applied to the model simulated an
internal hydrostatic pressure, uplift. Two approaches
were studied to determine the most effective way to apply
the uplift load to the thick shell element. First nodal
forces were applied to the top and bottom nodes of the
middle row of elements to simulate the uplift force. The
forces due to internal hydrostatic loads were determined
based on an uniform pressure. The computation of the
nodal forces was more difficult then the straight forward
method used in the plane strain analyses of the gravity
dam [15]. The calculations of the nodal forces are
described in detail in Appendix B.
The second approach applied both uniform and
linearly varying pressures to the top and bottom faces of
the middle row of elements to simulate uplift.
44


3.3.4
Load Combinations
The summary of the load combinations is shown in
table 3.2.
Model
TEST1
TEST2
TEST3
______Description______
Uniform pressure applied equal and
opposite along Y-direction to top and
bottom faces of middle row of elements.
Summation of pressure force equals 80
lbs.
Linear varying pressure applied equal and
opposite along Y-direction to top and
bottom faces of middle row of elements.
Summation of varying pressure force
equals 80 lbs.
Nodal forces applied equal and opposite
along Y-direction to top and bottom nodes
of middle row of elements. Summation of
forces equals 80 lbs.
TABLE 3.2 Summary of the isoparametric thick shell
element model applied loads.
3.3.5 Analyses
The same general linear elastic finite element
program SAP386 that was used in the two-dimensional
analysis was also used for this three-dimensional study.
45


3.3.6
Results
The centroid stresses for the middle row of elements
were compared to estimated, hand computation, stress
values to indicate the accuracy of the model. The
estimated stress was equivalent to the total normal load
divided by the cross-sectional area of the model. A
summary of the computed and FEM stresses and the percent
difference are shown in table 3.3.
Element
Load Number
Computed
Stress
(centroid)
F.E.M.
Stress
(centroid)
Percent
Error
TEST1* 1,2,3,4
0.1389 0.1416
1.9 %
TEST2* 1,2
3,4
0.2084 0.2094 0.5
0.0695 0.0737 6.0
TEST3* 1,2,3,4
0.1389 0.1415
1.9
Stresses are computed at the centriod of the
middle row of elements for total normal load of
80 lbs.
TABLE 3.3 Summary of isoparametric thick shell element
results.
46
o\o cN>


3.4 Summary
The results from both the two-dimensional gravity
dam and the three-dimensional thick shell studies showed
that the uplift load must be simulated using equal and
opposite forces, or pressures, of the dam and foundation.
This assures that the foundation behaves properly in the
analysis.
The two-dimensional analysis showed that the
relaxing of selected elements modulus of elasticity to
simulate the crack at the base of the dam was an
appropriate method.
The results of the three-dimensional studies showed
that both the nodal forces and the pressure loads can be
successfully used to simulate the internal uplift loads.
However, due to the nature of the thick shell element,
exceptional care must be taken to calculate the nodal
forces. Therefore, the pressure load seems to be more
appropriate for simulating the uplift load on a thick
shell element.
47


CHAPTER 4
SITE DESCRIPTION AND MATERIAL PROPERTIES
4.1 General
The analyses of uplift pressures on a thick arch dam
were performed on a model of Cutler Dam, Utah.
4.2 Description
Cutler Dam is located on the Bear River about 13
miles northwest of Logan, Utah. The dam was designed and
constructed during the period of 1924-1927. The
construction specification drawing is shown in figure
4.1.
The structure is located in a flat bottomed valley,
has a crest length of 465 feet, and a structural height
of 114 feet. The crest is 7 feet thick at elevation
4409.0 and the base is about 53 thick at elevation
4295.0. The upstream face of the dam is vertical except
for a corbel across the spillway section above elevation
4369.5. The downstream face is vertical from the crest
down to elevation 4397.9, then slopes at 0.45 horizontal
to 1.0 vertical.


FIGURE 4.1 Cutler Dam specification drawing.
49
CUTLER DAM
PLAN.ELEVATION a SECTIONS


The overall length of the crest is 545 feet which
includes two canal intakes, one located on each abutment.
The irrigation canal intakes discharge water into two
canals excavated into the canyon walls. The canals are
approximately 20 feet wide and 12.5 feet deep. Each
canal intake is equipped with two 8-foot by 8-foot
vertical lift gates operated by electric hoists.
The gated, four-bay, overflow spillway is located
near the center of the dam. It has four 30-foot wide
openings equipped with 14-foot high tainter gates. The
ogee crest is at elevation 4391.5. The stilling basin is
located at the base of the overflow section and consists
of a 90-foot long by 150-foot wide concrete apron
bordered by concrete training walls. The stilling basin
invert is at elevation 4300.0.
Five concrete piers divide the spillway bays and
support the tainter gates. The piers are about 8 feet
wide, support the bridge deck, and extend upward to
elevation 4410.25.
A power water intake is located in the reservoir
about 60 feet upstream from the dam. The intake is
equipped with a trashrack and a cylindrical gate operated
with an electric screw hoist. The intake is at elevation
4376.0. A bridge deck allows access to the intake tower
from the dam crest. The power water intake connects to
50


an 18-foot diameter steel lined conduit which passes
through the base of the dam and connects to an 18-foot
diameter steel penstock. The penstock continues from the
dam to the powerhouse which is located about 1,100 feet
downstream from the dam.
The foundation of the dam consists of hard durable
dark limestone of the Cutler formation. There is no
evidence of faulting at the site; however there is
faulting further downstream. The rock is massive with a
near vertical jointing system and a strike of 45 west
from the project north-south base line.
4.3 Material Properties
A core drilling program was conducted to determine
the concrete material properties of Cutler Dam in 1989.
Three 4-inch diameter drill holes were bored into the
dam, two of which extended from the dam crest into the
rock foundation. A total of 28 core samples were
selected from the concrete drill cores and subjected to
compressive, tensile, and modulus of elasticity testing.
The average compressive stress was determined to be 5,600
lbs/in2 and the average tensile stress 835 lbs/in2. The
average instantaneous modulus of elasticity was 5,330,000
lbs/in2 with a corresponding compressive strength of 5988
lbs/in2. These test results were used to estimate the
sustained modulus of elasticity of 3,500,000 lbs/in2
51


based on criteria that the sustained modulus should be
60% to 70% of the instantaneous laboratory modulus [4].
The unit weight of the concrete was assumed to be 155
lbs/ft3 based on results from previous investigations
[16].
The Poisson's ratio of the concrete was assumed to
be 0.2 based on published data by the U.S. Bureau of
Reelamation [4].
The rock core sample extracted from beneath the dam
had a Rock Quality Designation (RQD) ranging from 18 to
38 percent which is indicative of a low quality rock.
Limestone similar to the formation at Cutler Dam has been
tested to have a laboratory foundation modulus of
elasticity of about 6,000,000 lbs/in2 [17], Geologic
references suggest that the in situ foundation modulus
rages from 20 to 60 percent of the laboratory modulus
[18]. Therefore, the confined in situ foundation modulus
was assumed to be 2,400,000 lbs/in2 base on an average 40
percent reduction. The material properties for the
concrete and foundation used in the analysis are
summarized in Table 4.1.
The allowable stresses were computed based on USBR
[4] and FERC [9] criteria for high hazard dams and are
summarized is Table 4.2.
52


Material
Property
5,600
835
560
Value
Concrete: Compressive Strength
Tensile Strength
Shear Strength
lbs/in2
lbs/in2
lbs/in2
Modulus of
Elasticity
Poisson's Ratio
Unit Weight
3,500,000 lbs/in2
0.20
155 lbs/ft3
Foundation: Modulus of
Elasticity
Poisson's Ratio
Unit Weight
2,400,000 lbs/in2
0.20
0 lbs/ft3
Reservoir: Unit Weight
62.5 lbs/ft3
Silt Vertical Unit Weight 120.0 lbs/ft3
Horizontal Unit Weight 85.0 lbs/ft3
TABLE 4.1 Summary of material properties.
Concrete Compressive Stress:
Usual Load Combination 1,500 lbs/in'
Concrete Tensile Stress: Usual Load Combination 150 lbs/in'
Concrete Factors of Safety: Usual Load Combination 3.0
Foundation Factors of Safety: Usual Load Combination 4.0
TABLE 4.2 Summary of the allowable stresses and safety
factors.
53


CHAPTER 5
MODEL I
5.1 General
This analysis used general linear elastic finite
element computer program to set up a finite element mesh,
called Model I, of Cutler Dam with a stepped foundation.
The objective of this analysis was to study the effects
of uplift when applied only to the flat cross-canyon base
of the dam.
Five separate load combinations were analyzed; each
load combination included the normal static loads due to
gravity, reservoir, and silt and used Model I. Load
combinations LC-1 through LC-3 investigated the behavior
of the dam without uplift pressures. Load combination
LC-4 and LC-5 combined the normal static loads with the
uplift pressures for an uncracked and cracked base,
respectively.
Two techniques were used to model the behavior of
the dam at the dam/foundation contact. The first
approach relaxed (by lowering the modulus of elasticity)
selected foundation elements, adjacent to the contact to
relieve any tensile stresses in the concrete at the base
of the dam on the upstream face. This approach has been


used in arch dam analyses to model the inability of the
contact to transmit tensile stresses [19].
The second approach assumed a crack formed along the
cross-canyon base at the dam/foundation contact due to
tensile stresses on the upstream face. The crack
relieved the tensile stresses on the upstream face of the
dam at the contact.
Both of theses analyses are discussed in this
chapter.
5.2 Model Development
The same general linear elastic finite element
computer program SAP386, as described in Chapter 3, was
used in these studies.
The computer program PREDAM was developed to assist
in the generation of the finite element model. PREDAM
allows the engineer to develop the necessary nodal
coordinates, element connectivity, loads, and load
combinations to perform a finite element analysis of any
gravity, thick or thin arch, or double- or three-centered
curvature arch dam. The program has dynamic storage
capabilities allowing the engineer to model a dam with
virtually an unrestricted number of nodes and elements.
PREDAM is described in more detail in Appendix C.
The dam was modeled using 242 variable node,
isoparametric thick shell elements with two elements
55


through the thickness of the dam. The model's geometry
includes the spillway corbel on the upstream face of the
dam and the thickened penstock block located to the right
of the spillway (looking downstream). Due to the
geometric complexity of the spillway piers they were not
included in the model. Similarly, due to the piers load
complexity, and because computations of the additional
load on the piers and gates showed it to be a negligible
percentage on the total load on the dam, the loads of the
piers and the gates were also excluded from the model.
The dam and the foundation finite element model is shown
in four isometric views in figure 5.1 and the element
numbers in the dam are shown in figure 5.2.
The foundation was modeled using 528, eight node,
isoparametric thick shell elements with six elements,
from upstream to downstream edge of the foundation,
through the thickness The foundation used a stepped
shape to follow the dam/foundation contact and was
extended out from the dam at least one half the dam
height on both the right and left abutments and one dam
height upstream, downstream, and below the dam. The
nodes located at vertical boundaries of the foundation
were fixed against translation in the horizontal
direction. The nodes located at the base or on a
56


-c'-rrrtJB.E

^e finite
showing Y1 stepP
. --.otri0 vievTvier Dam uSl g
umir isomehr_ cutler
57


(a)
(b)
FIGURE 5.2 Element numbers for finite element model: (a)
Upstream elements looking downstream; (b)
Downstream elements looking downstream.
58


horizontal boundary were fixed against translation in the
vertical direction.
5.3 Material Properties
The material properties of the dam and the
foundation rock used in these analyses were obtained from
concrete and rock core sample tests and are discussed in
more detail in Chapter 4.
The allowable stresses in the concrete used in these
analyses were also discussed in detail in Chapter 4.
5.4 Loads
The loads included in these analyses are summarized
in Table 5.1.
Loads
Gravity
Reservoir
Silt
Uplift
________Description_________
Dead weight of dam.
Hydrostatic water pressure
applied to upstream face of dam
due to the Normal Water Surface
El. 4404.0.
Additional pressure due to
equivalent horizontal unit
weight of 85 lbs/ft3 applied as
a hydrostatic load on the
upstream face of the dam.
Internal hydrostatic water
pressure applied to the base of
the dam.
TABLE 5.1 Summary of the static loads applied to the
finite element model.
59


The gravity loads were applied as a homogeneous
monolithic load to the dam using the average unit weight
of concrete of 155 lbs/ft3 [20].
The reservoir water pressure acts hydrostatically on
the upstream face of the dam. This was simulated using
the hydrostatically varying pressure load available in
the program SAP386 [13,14]. A linearly varying pressure
was applied to the upstream face of the upstream elements
in the dam based on the reservoir water surface elevation
4404.0 and an unit weight of 62.5 lbs/ft3.
The silt level in the reservoir has been reported to
be at approximately elevation 4389.0. There was no data
available to determine the density or unit weight of the
silt so the general properties for silt as outlined by
the USBR were used in this study [4]. The equivalent
saturated horizontal pressure of 85 lbs/ft3 acts as a
hydrostatic pressure on the upstream face of the dam.
The pressure was simulated by applying 22.5 lbs/ft3, the
difference between the silt's equivalent horizontal
pressure (85 lbs/ft3) and the reservoir unit weight (62.5
lbs/ft3) as a hydrostatic linearly varying pressure to
the upstream face of the upstream elements in the dam
based on the silt surface elevation 4389.0.
There were no data available on the effective head
of the uplift at the upstream face of the base of the
60


dam. Therefore, in accordance with the FERC guidelines
[9] the uplift at the upstream face was assumed to be
equivalent to the full reservoir pressure.
The uplift load was applied as an equal and opposite
pressure to the top and bottom faces of the row of thin
elements at the base of the dam. For an uncracked base
the uplift pressure was assumed to vary linearly from
full reservoir head at the upstream face to zero head at
the downstream face. For the cracked base analysis, a
uniform pressure equivalent to full reservoir head was
applied across the entire crack length, and varied
linearly in the uncracked portion from full reservoir
head at the crack tip to zero head at the downstream toe
of the dam.
5.5 Load Combinations
The load combinations are summarized in table 5.2.
Each of the individual loads were applied independently
to the model to study their effects. Gaining an
understanding of the dam's behavior due to each
individual load is useful in determining which load
significantly influences the dam during a given load
combination. The final stresses and deflections for each
load combination were obtained by summing the
corresponding stresses and deflections of the individual
loads, based on the laws superposition.
61


Name
Load Description
LC-1
*
LC-2**
LC-3
***
Gravity load.
Reservoir load at El. 4404.0.
Silt load at El. 4389.0.
Gravity load.
Reservoir load El. 4404.0.
Silt load El. 4389.0.
Gravity load.
Reservoir load El. 4404.0.
Silt load El. 4389.0.
LC-4 - LC-2.
Uplift pressure for uncracked
base.
LC-5*** - LC-3.
Uplift pressure for cracked
base.
uniform foundation modulus.
with relaxed foundation modulus.
with uniform foundation modulus and cracked
base.
TABLE 5.2 Summary of load combinations.
5.6 Analyses
Load combination LC-1 includes the normal static
loads due to gravity, reservoir and silt, and used Model
I with a uniform dam and foundation modulus.
The results produced by LC-1 show tensile stresses
on the upstream face of the dam near the dam/foundation
contact, as expected. The wide base of the thick arch
dam across the U-shaped canyon relies heavily on the
internal cantilever action to support the loads. This
results in vertical tensile stresses on the upstream face
62


caused by the cantilever bending moment. The tensile
stresses, produced by the reservoir and silt loads acting
on the dam, are the result of the linear elastic
assumptions used for the foundation and concrete material
properties and are not a true indication of the dam's
behavior. Visual observation of the abutment rock showed
closely spaced joint sets and the rock core recovered
from the base of the dam had a Rock Quality Designation
(RQD) of 18 to 38 percent [21]. This suggests that the
dam/foundation contact cannot develop tensile stresses as
predicted by the model.
To analyze the behavior of the dam at the
dam/foundation contact some minor modifications were
performed on Model I. The high tensile stress at the
upstream face are an indication that the model is too
stiff at the contact. To alleviate these tensile
stresses selected elements in the foundation upstream and
directly underneath of the contact were relaxed. The
relaxation was accomplished by reducing the foundation
deformation modulus by 90 percent to 240,000 lbs/in2
upstream of the dam and by 99 percent to 24,000 lbs/in2
underneath the dam below elevation 4315.0, which
corresponded to the location of the higher tensile
stresses.
Load combination LC-2 used the modified Model I with
63


the relaxed foundation and was analyzed under the normal
static loads. The tensile stresses on the upstream face
near the dam base, evident in LC-1, were reduced, under
LC-2. Detailed analysis of the individual loads showed
that the relaxed foundation deformation modulus allowed
the dam cantilevers to deflect downstream without
developing the higher tensile stresses shown in LC-1.
Allowing the dam cantilevers to deflect further
downstream resulted in an increase in the arch
compressive stress, thus transferring the load to the
abutments.
Load combination LC-3 used a second approach to
model the behavior of the dam/foundation contact. Since
the stresses located at the base on the upstream face of
the dam in LC-1 were less than the hydrostatic head at
the same location it can be assumed that the base would
form a crack [9]. This approach was demonstrated in
Chapter 3 with the two-dimensional gravity dam analyses.
Two assumptions were made in analysis LC-3 so that
the existing Model I could be used without requiring
major modifications. First, the crack would be located
in the thin row of elements at the base of the dam.
Second, the crack would be modeled within the cross-
canyon base of the U-shaped canyon and would extend to
64


half the thickness of the base. Third, since the crack
would relieve the tensile stress, evident in LC-1, the
foundation deformation modulus was assumed to be uniform.
Simulating the crack to half the thickness of the base
was done by relaxing the upstream elements 223, 225, 227,
... 241. The relaxation was accomplished by reducing the
crack elements modulus of elasticity to 10 lbs./in2.
Assuming the crack to exist in the thin row of elements
at the base of the dam simplified the application of
uplift pressure in latter load combinations. The uplift
pressure could easily be applied to the top and bottom of
the thin row of elements at the base of the dam (It was
preferred to use the dam elements because the computation
of uplift pressure was relatively simple for the square
bottom elements of the dam, as compared to the
complicated computations that would be required for the
foundation elements).
Assuming the crack to extend to half the thickness
of the base was believed to be a conservative initial
assumption since the actual condition was expected to
crack less than half.
The results from load combination LC-3 with the
cracked base, showed that the tensile stresses on the
upstream face have been eliminated. The dam behaves
similarly to LC-2, allowing the cantilevers to deflect
65


further downstream and increase the compressive stress in
the arches.
Load combinations LC-4 and LC-5 used the models and
loads constructed for LC-2 and LC-3, respectively, but
included the uplift pressures on the cross-canyon base of
the dam. The uplift pressures acting on the dam
abutments were not included in this study. Since the
hydrostatic pressure reduces as the elevation increases,
the uplift pressure acting on the dam will decrease from
the dam base towards the dam crest. Computation of the
uplift forces acting on the dam abutments indicated that
it would be much less than the uplift force at the base.
Therefore, the uplift pressure acting on the dam at the
abutments was assumed to have a negligible effect on the
dams behavior.
The uplift in LC-4 follows the criteria for an
uncracked base and varies linearly from full reservoir
head at the upstream face to zero head at the downstream
face. The uplift load was applied as an equal and
opposite pressure to the thin row of elements at the base
of the dam. The base was assumed uncracked because the
compressive stresses on the upstream face at the
dam/foundation contact, resulting from LC-2, were close
in magnitude to the hydrostatic pressure; thus, the dam
would not crack significantly. Modifying the model to
66


simulate a crack to half the thickness of the base would
not be an accurate representation of the conditions
portrayed in LC-2. Therefore, LC-4 was modeled with
uplift for an uncracked base.
Similarly, the uplift load in LC-5 was applied as an
equal and opposite pressure to the top and bottom faces
of the thin row of elements at the base of the dam.
However, the model used in LC-3 assumed that the base of
the dam was cracked, therefore, the uplift load had to
follow the rules for the same cracked base. The uplift
pressure is equivalent to full reservoir head within the
crack and varies linearly from full reservoir head to
zero head in the uncracked portion of the base.
5.7 Results
The stresses from the finite element program SAP386
are output in the global X-, Y-, and Z-axis coordinate
system for the isoparametric thick shell element.
The global stresses for each load combination were
output for the centroid of the upstream and downstream
face of the dam elements and the centroid of the
foundation elements. The program POSTDAM was developed
to convert the dam elements global stresses into a local
stresses, tangent to both the dam axis and the
upstream/downstream face of the element. POSTDAM also
67


computes the dam elements tangential, radial, and
vertical deflection at the center of the dam element's
upstream and downstream face. POSTDAM is discussed in
more detail in Appendix D.
To assist in the analyses of the load combinations
the resulting stresses and deflections have been
displayed in two tables: maximum arch and cantilever
tensile and compressive stresses, and the average arch
and cantilever stresses along with the average radial
deflections.
The maximum arch and cantilever tensile and
compressive stresses are the normal means of displaying
results in arch dam analysis and give the engineer a
quick means to compare the maximum stresses to the
allowable stresses of the material.
The average arch stresses and radial deflections
were developed to give a general indication of the
behavior of the dam. Two areas of the dam were used to
compute the average stress and deflection. The area
entitled "Spillway" includes the elements within the
middle 37.4 degrees of the dam (this area includes the
elements underneath the spillway). The area entitled
"Right/Left" includes the elements located outside the
middle 37.4 degrees of the dam. The average arch stress
and average radial deflections were computed using the
68


top row of elements in the given area. The average
cantilever stresses were computed using the elements
adjacent to the foundation within the given area. The
deflection sign convention is positive for radial
downstream, and negative for radial upstream.
The dam elements stresses are displayed in vectorial
plots of arch and cantilever stresses and principal
stresses. The stress vectors are plotted to scale,
meaning that the vector length represents the stress
magnitude and the angle shows the orientation. The
stress sign convention is positive for tension and
negative for compression. Stresses are displayed in
pounds per square inch (lbs./in2) and plotted on the
profile of the dam, looking downstream, at the centroid
of the element.
The summary of the maximum tensile and compressive
stresses for load combination LC-1 are shown in table
5.3. The average arch and cantilever stresses and
average radial deflections for LC-1 are shown in table
5.4. The plots of arch and cantilever stresses are shown
in figures 5.3 and 5.4, the plots of principal stresses
are shown in figures 5.5 and 5.6.
The maximum arch tensile stress is 97 lbs/in2 and
the maximum cantilever tensile stress is 121 lbs/in2.
Both peak tensile stresses occur on the upstream face in
69


elements adjacent to the foundation. The maximum
compressive arch stress is 134 lbs/in2 and is located in
the upper central portion of the dam. The maximum
cantilever compressive stress is 419 lbs/in2 and is
located at the downstream toe of the dam. All of these
maximum stresses are will within the allowable stress
limits of the concrete; however, the tensile stresses in
70


Upstream Downstream
Description Stress Stress
(lbs/in2) (lbs/in2)
Arch Tension 97 0
Compression 128 134
Cantilever Tension 121 1
Compression 27 419
TABLE 5.3 Summary of maximum arch and cantilever
stresses for load combination LC-1.
Upstream Downstream
Description Stress Stress
(lbs/in2) (lbs/in2)
Average Upper Arch Stress
Right/Left Area -30 -45
Spillway Area -89 -82
Average Base Cantilever Stress
Right/Left Area +61 -254
Spillway Area +92 -380
Average Upper Arch Radial Deflection (inch.)
Right/Left Area +0.0457
Spillway Area +0.1229
TABLE 5.4 Summary of average arch and cantilever
stresses and deflections for load combination
LC-1.
71


FIGURE 5.3 Arch and cantilever stresses on the upstream
face for LC-1.
NOTES:
1. Stresses are computed at
elements surface centroid.
2. Stresses are in psi.
3. + = Tension
= Compression.
4. = Tension.
----- = Compression.
the


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CUTLER DAM
LOAD COMBINATION NO.1
UPSTREAM FACE
ARCH AND CANTILEVER STRESSES


NOTES:
1. Stresses are computed at the
elements surface centroid.
2. Stresses are in psi.
FIGURE 5
CO Q)
e K
I"
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II II
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4 Arch and cantilever stresses on the downstream
face for LC-1.
73


FIGURE 5.5 Principal stresses on the upstream face for
LC-1.
NOTES:
1. Stresses are computed at the
elements surface centroid.
2. Stresses are in psi.
J. + = Tension
= Compression.
4. = Tension.
------ = Compression.
o
Scale of Stress
(pst)
600
Jj
CUTLER DAM
LOAD COMBINATION NO.1
UPSTREAM FACE
PRINCIPAL STRESSES


FIGURE 5.6 Principal stresses on the downstream face for
LC-1.
75


the elements adjacent to the foundation are greater than
the dam/foundation capability.
The tensile stresses from LC-1 at the dam
foundation contact were expected. These tensions are the
result of the assumed linear elastic properties of the
material in the finite element analysis. Because of this
assumption the material property can computationally
develop tensile stresses. However, as discussed
previously, the foundation rock is unable to develop
tension at the dam/foundation contact. Therefore, these
tensile stresses should not exist and are not a true
indication of the dam's behavior.
The first approach to address the tensile stresses
used relaxed foundation elements. The deformation
modulus of the foundation elements adjacent to the
upstream face of the dam were relaxed to simulate the
inability of the foundation rock to develop tensile
stresses. Model I was modified for load combination LC-2
based on these assumptions.
The summary of the maximum tensile and compressive
stresses for load combination LC-2 are shown in table
5.5. The average arch and cantilever stresses and
average radial deflections for LC-1 are shown in table
5.6. The plots of arch and cantilever stresses are shown
in figures 5.7 and 5.8 and plot of the principal stresses
76


are shown in figures 5.9 and 5.10.
The maximum arch tensile stress is 48 lbs/in2 and
the maximum cantilever tensile stress is 82 lbs/in2.
Both peak tensile stresses occur on the upstream face in
elements adjacent to the dam/foundation contact. The
maximum compressive arch stress is 125 lbs/in2 and is
located on the upstream face in the upper central portion
of the dam. The maximum cantilever compressive stress is
459 lbs/in2 and is located at the downstream toe of the
dam. All of these maximum stresses are will within the
allowable stress limits of the concrete. However, the
tensile stresses at the base of the dam on the upstream
face are greater than the dam/foundation contact
capability.
The relaxed foundation deformation modulus, used
in LC-2, results in a decrease of cantilever tensile
stresses on the upstream face of the dam near the base.
The relaxed foundation has also allowed the dam's
cantilevers and arches, located within the right and left
area of the dam, to deflect further downstream. This is
shown in table 5.6 where the average right/left radial
deflection has increased from LC-1. The increased radial
deflection has resulted in an increased cantilever
compressive stress on the downstream toe and an increase
in the arch compressive stresses.
77


TABLE
TABLE
Descriotion Upstream Stress Downstrea Stress
Arch Tension (lbs/in2) 48 (lbs/in2) 0
Compression 125 59
Cantilever Tension 82 2
Compression 59 459
5.5 Summary of maximum arch and cantilever
stresses for load combination LC-2.
Upstream Downstream
Descriotion Stress Stress
(lbs/in2) (lbs/in2)
Average Upper Arch Stress
Right/Left Area -31 -46
Spillway Area -84 -78
Average Base Cantilever Stress
Right/Left Area +34 -309
Spillway Area -23 -419
Average Upper Arch Radial Deflection (inch.)
Right/Left Area +0.0462
Spillway Area +0.1189
5.6 Summary of average arch and cantilever
stresses and deflections for load combination
LC-2.
78


FIGURE 5.7 Arch and cantilever stresses on the upstream
face for LC-2.
NOTES:
1. Stresses are computed at
elements surface centroid.
2. Stresses are in psi.
3. + = Tension
= Compression.
4. - = Tension.
----- = Compression.
the
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Im of
(pit)
CUTLER DAM
LOAD COMBINATION N0.2
UPSTREAM FACE
ARCH AND CANTILEVER STRESSES


I
FIGURE 5.8 Arch and cantilever stresses on the downstream
face for LC-2.
80


FIGURE 5.9 Principal stresses on the upstream face for
LC-2.


FIGURE 5.10 Principal stresses on the downstream face
for LC-2.


Although the tensile stresses evident in LC-1 were
significantly reduced in LC-2, they were not eliminated
because the stiffness in the adjacent foundation elements
were not eliminated. The deformation modulus in the
selected foundation elements adjacent to the dam were
only reduced, as discussed previously; therefore, the
stiffness of the dam's internal cantilevers were not
eliminated. This allowed tensile stresses on the
upstream face of the dam near the contact to develop.
The second approach to address the tensile
stresses, located on the upstream face of the dam near
the contact in LC-1 assumed the cross-canyon base of the
dam to form a crack. The modulus of elasticity of the
crack elements were relaxed. This simulated the
inability of the crack to develop tensile stresses.
Model I was modified for load combination LC-3 based on
the these assumptions.
The summary of the maximum tensile and compressive
stresses for load combination LC-3 are shown in table
5.7. The average arch and cantilever stresses and
average radial deflections for LC-3 are shown in table
5.8. The plots of arch and cantilever stresses are shown
in figures 5.11 and 5.12 and plot of the principal
stresses are shown in figures 5.13 and 5.14.
83


Description
Upstream
Stress
(lbs/in2)
Downstream
Stress
(lbs/in2)
Arch
Tension 51
Compression 125
0
160
Cantilever Tension
Compression
15
34
2
520
TABLE 5.7 Summary of maximum arch and cantilever
stresses for load combination LC-3.
Description Upstream Stress Downstrec Stress
(lbs/in2) (lbs/in2)
Average Upper Arch Stress
Right/Left Area -34 -49
Spillway Area -87 -80
Average Base Cantilever Stress
Right/Left Area 0 -337
Spillway Area 0 -471
Average Upper Arch Radial Deflection (inch.)
Right/Left Area +0.0447
Spillway Area +0.1234
TABLE 5.8 Summary of average arch and cantilever
stresses and deflections for load combination
LC-3.
84


FIGURE 5.11 Arch and cantilever stresses on the
upstream face for LC-3.
NOTES:
1. Stresses are computed at the
elements surface centroid.
2. Stresses are in psi.
J. + = Tension
- = Compression.
4. ----- = Tension.
------ = Compression.
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