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:

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elements surface centroid.

2. Stresses are in psi.

FIGURE 5

CO Q)

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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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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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CUTLER DAM

LOAD COMBINATION N0.3

UPSTREAM FACE

ARCH AND CANTILEVER STRESSES