Stability Analysis of Non-overflow Section

of Concrete Gravity Dams

A Longtan Dam case study

Dan Valtersson
Lukas Johansson

Civil Engineering, master's level

2018

Luleå University of Technology

Department of Civil, Environmental and Natural Resources Engineering
Dan Valtersson
Lukas Johansson

Stability Analysis of Non-overflow Section

of Concrete Gravity Dams
A Longtan Dam case study

Master Thesis
X7009B
Civil Engineering

Specialization in Mining and Geotechnical Engineering

2018-09-01

Master of Science, Civil Engineering

Luleå University of Technology

Department of Civil, Environmental and Natural resources engineering

Division of Mining and Geotechnical Engineering

PREFACE
This master thesis has been carried out as the fulfilling part of our M.Sc. degree in Civil
Engineering with specialization towards Mining and Geotechnical Engineering at Luleå Uni-
versity of Technology, Luleå, Sweden. The thesis covers 30 ECTS and was conducted at
Hohai University, Nanjing, China during the period April – July 2018. We want to thank
Energiforsk and Vattenfall for financing the project and particularly Professor James Yang for
creating this opportunity for us.

We want to express our gratitude to Professor Wenhong Dai at the college of Water Con-
servancy and Hydropower Engineering, Hohai University for supervising this project. We
would also like to thank Mengjiao Ding and Tao Hu for helping us on the project and making
our stay in China pleasant.

We also want to thank our examiner Associate Professor Hans Mattsson at Luleå University
of Technology for his support during our stay in China and the time he spent taking on this
thesis.

Nanjing, July 2018

Dan Valtersson
Lukas Johansson

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Stability Analysis of Non-overflow Section of Concrete Gravity Dams

ii
ABSTRACT
The rapid growth of China’s economy results in drastic increase of energy demand amongst
its 1.4 billion inhabitants. To meet the demands, the government of China has over the years
invested in hydropower structures. The climate of China is suitable for hydropower dams
where the stations can store and produce electricity from the high-water reservoirs.

Roller-compacted concrete (RCC) dams is a type of concrete gravity dams where the stability
of the dam relies on the self-weight of the structure. Compared to other types of gravity dams,
RCC dams are more time and cost efficient. Such a dam is Longtan Dam, located in Tian’e
County on the Hongshui river, China. It is one of the tallest of its kind with the height 216.5
m and length 849 m. During this study, Longtan Dam will be used as reference.

The objective of this thesis is to investigate the safety of the non-overflow section of a con-
crete gravity dam. As consequences of dam failure are devastating, the failure modes and
seepage of the dam needs to be analyzed. This is conducted by determining the limit states
and seepage of the dam. Three scenarios are considered – normal water condition, critical
water condition and an earthquake hitting the normal water scenario.

The results show marginal difference between the water cases when considering the limit
states and seepage. In fact, the numerical simulation show that the stress distribution and values
are almost identical in the normal and critical case. However, the increase of water level does
result in a higher displacement, where the dam tilts 0.5 cm more in the critical case compared
to the normal case with 7.0 cm tilt.

The numerical results indicate that the dam is going to fail due to tensile stress at the dam
heel, with the stress ranging from -6.0 MPa to -2.0 MPa. The induced compressive stress
shows stable results, being below the compressive strength of concrete. The seepage analysis
shows no significant risk for the stability of the dam, where high concentrations of seepage
velocity concentrates near the dam heel and grout curtains. The low seepage velocity implies
no risk of erosion or inducement of high stresses.

As a conclusion, the dam is deemed stable against compressive stress and anti-sliding. How-
ever, the risk for tensile failure is high and therefore reinforcements in the dam heel should
be applied. The study implies that the dam is safe in some regards, but this thesis is heavily
based on several simplifications and assumptions. Further study such as using the analysis type
QUAKE/W should be conducted to compare the normal earthquake scenario with the hand
calculations.

Keywords: RCC, Longtan Dam, limit states, seepage, FEM

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Stability Analysis of Non-overflow Section of Concrete Gravity Dams

iv
SUMMARY IN SWEDISH
Kinas ekonomi har utvecklats starkt under de senaste åren, vilket har resulterat i en drastisk
ökning av energikonsumtion bland landets 1,4 miljarder invånare. För att bemöta efterfrågan
har Kinas regering över åren investerat i vattenkraft. Kinas klimat visar sig vara passande för
vattenkraft där energi produceras från höga vattenreservoarer.

Roller-compacted Concrete (RCC) dammar är en klassifikation av betongdammar där kon-

struktionens stabilitet förlitar sig på dammens egenvikt. Jämfört med andra typer av betong-
dammar så är RCC dammar mer tids- och kostnadseffektiva. Ett exempel på en sådan damm
är Longtan dammen som är belägen i Tian’e County på Hongshui floden, Kina. Dammen är
en av de högsta i dess klass med höjden 216,5 m och längden 849 m. Longtan dammen
kommer under denna studie användas som referens.

Målet med detta examensarbete är att undersöka stabiliteten hos en betongdamm. Konse-
kvenserna vid ett dammbrott är förödande, därmed måste dammens stabilitet och läckage
analyseras. Denna studie tar hänsyn till tre olika fall, nämligen normalt vattenstånd, kritiskt
vattenstånd och även det normala vattenståndet samtidigt som en jordbävning tar plats.

Resultaten visar att det är marginell skillnad mellan vattenståndsfallen med hänsyn till brott-
kriterier och läckage. Faktum är att den numeriska simuleringen visar att spänningsfördel-
ningen och dess värden är nära identiska i det normala och kritiska vattenståndet. Däremot
leder en ökning av vattennivåer till ökade deformationer, där dammen lutar 0,5 cm mer i det
kritiska tillståndet jämfört med 7,0 cm i det normala.

Resultaten från de numeriska analyserna antyder att dammen kommer att gå till brott på grund
av dragspänningar på dammhälen, spänningarna kretsar kring -6,0 MPa till -2,0 MPa. Den
inducerade tryckspänningen visar stabila resultat med marginal under tryckhållfastheten för
dammens betong. Läckageanalysen visar ingen signifikant risk för dammens stabilitet, högre
flöden kretsar kring hälen och vid injekteringsskärmarna. Det låga läckaget antyder att ingen
risk för erosion eller inducerade spänningar förkommer.

Sammanfattningsvis bedöms dammen att vara säker mot tryckspänningar och stabilitetsbrott.
Däremot är risken för dragbrott hög och därmed bör dammhälen förstärkas. Denna studie
antyder att dammen är stabil och säker i viss mån, detta examensarbete är baserat på flertalet
antaganden och förenklingar. Vidare studier som exempelvis en jordbävningsanalys i
QUAKE/W bör beaktas för att jämföra numeriska resultat med handberäkningar i jordbäv-
ningstillståndet.

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Stability Analysis of Non-overflow Section of Concrete Gravity Dams

vi
TABLE OF CONTENTS
1 INTRODUCTION ..................................................................................................1
1.1 Background..........................................................................................................2
1.1.1 Longtan Dam ................................................................................................2
1.2 Objective and Purpose .........................................................................................3
1.3 Limitations ...........................................................................................................3
2 THEORETICAL BACKGROUND .........................................................................5
2.1 Concrete Dams ....................................................................................................5
2.1.1 Roller-compacted Concrete Dams ................................................................6
2.2 Layout Design ......................................................................................................6
2.2.1 Slope Turning Point and Top Elevation ........................................................6
2.3 Induced Forces .....................................................................................................7
2.3.1 Gravitational Force ........................................................................................7
2.3.2 Hydrostatic & Uplift Pressure ........................................................................8
2.3.3 Wave & Sediment Pressure ...........................................................................9
2.3.4 Earthquake Inertia Force ...............................................................................9
2.4 Partial Safety Factor Method .............................................................................. 10
2.4.1 Limit Equilibrium Method .......................................................................... 11
2.5 Stress and Strain ................................................................................................. 11
2.5.1 Linear Elastic ............................................................................................... 11
2.5.2 Hooke’s Law ............................................................................................... 12
2.5.3 Plane stress .................................................................................................. 12
2.5.4 Plane strain.................................................................................................. 13
2.6 Slope Stability .................................................................................................... 13
2.6.1 Mohr Coulomb’s Failure Criterion.............................................................. 13
2.6.2 Morgenstern-Price Method ......................................................................... 14
2.7 Darcy’s law & Seepage ....................................................................................... 15
2.8 Finite Element Method ...................................................................................... 15
2.8.1 Basic Principle ............................................................................................. 16
3 METHODOLOGY ................................................................................................. 17
3.1 Layout................................................................................................................ 17
3.2 Acting Forces ..................................................................................................... 18
3.3 Limit States ........................................................................................................ 19
3.4 Modeling in GeoStudio...................................................................................... 20
3.4.1 Model setup ................................................................................................ 20
3.4.2 SIGMA/W ................................................................................................. 21
3.4.3 SEEP/W ..................................................................................................... 22
3.4.4 SLOPE/W .................................................................................................. 22

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Stability Analysis of Non-overflow Section of Concrete Gravity Dams

4 RESULTS ............................................................................................................... 25
4.1 Dam Dimensions ............................................................................................... 25
4.2 Stability Analysis ................................................................................................ 26
4.2.1 Forces and Moments ................................................................................... 26
4.2.2 Limit States ................................................................................................. 27
4.3 Numerical Analysis ............................................................................................ 27
4.3.1 SIGMA/W ................................................................................................. 28
4.3.2 SEEP/W .................................................................................................... 32
4.3.3 SLOPE/W ................................................................................................. 36
4.4 Comparing Hand- and Numerical Results ......................................................... 36
5 ANALYSIS AND DISCUSSION ............................................................................ 39
5.1 Model Validation ............................................................................................... 39
5.2 Limit States ........................................................................................................ 39
5.3 Seepage ............................................................................................................. 40
5.4 Future Study ...................................................................................................... 40
6 CONCLUSIONS .................................................................................................... 43
7 REFERENCES ....................................................................................................... 45
APPENDIX I – FEM LAYOUT ....................................................................................... a
APPENDIX II – BOUNDARY CONDITIONS ............................................................. c
APPENDIX III – OVERVIEW LAYOUT ....................................................................... e

viii
LIST OF FIGURES
Figure 1.1. China’s installed energy capacity in 2016 (IEA, 2017). ......................................1
Figure 1.2. Location of Longtan Dam, China (Google, n.d.). ..............................................2
Figure 2.1. Illustration of gravity, arch and buttress dams (Australian Geographic, 2011). ....5
Figure 2.2. Uplift distribution with and without drain. .......................................................8
Figure 2.3. Relation between stress and strain in linear elastic conditions (GEO-SLOPE
International Ltd, 2013). ................................................................................................... 11
Figure 2.4. Mohr Coulomb’s failure criterion (GEO-SLOPE, 2013) ................................. 14
Figure 2.5. Illustration of the slice method (GEO-SLOPE International Ltd, 2012b). ........ 14
Figure 2.6. Structure discretized to a finite element model, showing different element types
(FEM Expert, n.d.). .......................................................................................................... 16
Figure 3.1. Layout parameters of the non-overflow section. .............................................. 17
Figure 3.2. Illustration of the forces acting on the non-overflow section of Longtan dam. . 18
Figure 3.3. Limit states of the non-overflow section. ........................................................ 19
Figure 3.4. Model layout in GeoStudio. ........................................................................... 20
Figure 4.1. Design for non-overflow section of the dam. .................................................. 25
Figure 4.2. Max. Shear Stress, normal water case. ............................................................. 28
Figure 4.3. Max. Shear stress in relation to boundary distance, heel and toe. ..................... 28
Figure 4.4. Max. Total Stress, normal water case............................................................... 29
Figure 4.5. Max. Total Stress in relation to boundary distance, toe.................................... 29
Figure 4.6. Min. Total Stress, normal water case. .............................................................. 30
Figure 4.7. Min. Total Stress in relation to boundary distance, heel. ................................. 30
Figure 4.8. XY-displacements, normal water case. ............................................................ 31
Figure 4.9. Deformed mesh, normal water case................................................................. 31
Figure 4.10. Water Total Head, normal water case. .......................................................... 32
Figure 4.11. Water Total Head, critical water case. ........................................................... 32
Figure 4.12. Pore-water Pressure, normal water case......................................................... 33
Figure 4.13. Pore-water Pressure, critical water level. ....................................................... 33
Figure 4.14. Water Flux, normal water case. ..................................................................... 34
Figure 4.15. Subdomain elements chosen for water rate plot. ............................................ 34
Figure 4.16. Water rate through the base layer. ................................................................. 35
Figure 4.17. Factor of safety against sliding, normal water case. ......................................... 36
Figure 8.1. Imported CAD layout for FEM analysis. ........................................................... a
Figure 9.1. Visualization of uplift pressures for respective section. .......................................d
Figure 10.1. Elevation arrangement of the dam................................................................... e
Figure 10.2. Horizontal arrangement of the dam................................................................. f

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Stability Analysis of Non-overflow Section of Concrete Gravity Dams

x
LIST OF TABLES
Table 3.1. Input parameters for layout design (Dai, 2018). ................................................ 17
Table 3.2. Simulated water level scenarios in GeoStudio. .................................................. 21
Table 3.3. Material input data for SIGMA/W analysis (Dai, 2018). ................................... 21
Table 3.4. Material input data for SEEP/W analysis (Dai, 2018). ...................................... 22
Table 3.5. Material input data for SLOPE/W analysis (Dai, 2018). ................................... 23
Table 4.1. Top water elevation for design and critical water levels. ................................... 25
Table 4.2. Summarized horizontal and vertical design loads. ............................................. 26
Table 4.3. Summarized positive and negative moments..................................................... 26
Table 4.4. Center of mass for the arm of force in X- and Y-direction. .............................. 26
Table 4.5. Factory of safety against sliding......................................................................... 27
Table 4.6. Factor of safety against compressive stress. ........................................................ 27
Table 4.7. Design tensile stress at dam heel. ...................................................................... 27
Table 4.8. Factor of safety against sliding, comparing hand- and numerical results. ............ 37
Table 4.9. Factor of safety against compressive failure, comparing hand- and numerical results.
........................................................................................................................................ 37
Table 4.10. Tensile stresses at dam heel, comparing hand- and numerical results. .............. 37
Table 9.1. Uplift pressure forces for respective Section (Figure 9.1). Values in kN. ............. c

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Stability Analysis of Non-overflow Section of Concrete Gravity Dams

xii
SYMBOLS
Roman uppercase

E Young’s modulus
F Force
FEM Finite element method
G Material shear modulus
GK Standard value of permanent effect
H Section height
L Arm of force
Lm Wave length
M Moment
P Pressure
Q Boundary flux
QK Standard value of variable effect
R Structural resistance function
RCC Roller-compacted concrete
S Action effect function
SF Factor of safety
Sp Height of steel penstock
U Uplift pressure
W Self-weight

Roman lowercase

a Drain pressure attenuation coefficient

ah Horizontal earthquake acceleration
aK Characteristic value of geometry parameter
c Cohesion
f Friction coefficient
fK Standard value of material properties
h Water height
i Hydraulic head gradient
k Hydraulic conductivity
m Downstream slope inclination
n Upstream slope inclination
q Seepage discharge
t Time
w Section width

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Stability Analysis of Non-overflow Section of Concrete Gravity Dams

Greek uppercase

Γ Shear material strain

∇ Elevation
Σ Resultant
𝛹𝛹 Design condition coefficient

Greek lowercase

α Dynamic distribution coefficient

γ Unit weight
𝛿𝛿 Limit states coefficient
ε Strain
η Reduction factor
θ Volumetric water content
ν Poisson’s ratio
ξ Comprehensive influencing coefficient
ρ Density
σ Stress
τ Shear
φ Friction angle

xiv
 Introduction

1 INTRODUCTION
China’s economy has grown rapidly the last years and with a population of 1.4 billion people,
the demand for energy is drastically increasing. As a result, China has become the largest
energy consumer and producer in the world, where in 2016 63% of China’s energy was
produced through coal (International Energy Agency [IEA], 2017), see Figure 1.1. Nearly
three-quarter of the global growth in carbon emission between 2010-2012 occurred in China.
With the current growth rate, is it predicted that Chinese emissions could rise by more than
50% in the next 15 years (Liu, 2016).

Large emission of CO2 and pollution has become a big environmental issue for China. The
Chinese government has stated a goal that by 2020, 15% of all energy consumed should have
renewable energy source background. In 2013, China’s carbon emission was 9.2 Giga ton
CO2 which corresponds to 25% of the global carbon emissions that year (Energy Information
Administration [EIA], 2015).

Figure 1.1. China’s installed energy capacity in 2016 (IEA, 2017).

An environmental friendly way to produce energy is through hydropower. Luckily, China’s

landscape is suitable for hydropower and is the second largest energy source (Figure 1.1). In
2016 the installed capacity was 330 GW, which equals around 20% of China’s total energy
capacity. Chinese hydropower capacity represents more than 25% of the global capacity, and
with China’s government 2020 goal, it is aimed to increase the capacity to 480 GW (Inter-
national Hydropower Association [IHA], 2016). To achieve this goal, more dams and hydro-
power dams is planned to be constructed.

1
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

1.1 Background

Hydropower dams regulates the water flow in rivers and uses it to extract energy, this is done
by building large constructions. The safety is a critical part in these constructions, as the con-
sequences of a failed dam is catastrophic. The growing energy demand creates a need to build
more and higher hydropower dams which results in higher stresses and larger deformations.
This creates a need to understand the behavior of the structures, one way to analyze this is to
conduct a case study on an already constructed dam. This thesis covers the design procedure
of a concrete gravity dam using existing Longtan Dam in China as case study.

1.1.1 Longtan Dam

The Longtan Dam is a roller-compacted concrete (RCC) gravity dam located in Tian’e
County on the Hongshui River, China, see Figure 1.2. The purpose of the dam is hydroe-
lectric power production, flood control and navigation. With the dimensions of the height
216.5 m and body length 849 m, it is one of the tallest of its kind. Formal construction began
in 2001 and in 2009 all generators became operational, making the dam reach full capacity
with the annual generation of 18.7 TWh and capacity 6426 MW (Electric Light & Power
[ELP], 2013).

Figure 1.2. Location of Longtan Dam, China (Google, n.d.).

2
 Introduction

1.2 Objective and Purpose

The aim of this thesis is to investigate the safety of the non-overflow section of a concrete
gravity dam. The consequences of dam failure are devastating and therefore the behavior and
stability of the dam must be analyzed with time.

This thesis covers a small part of a wider investigation where the focus in this report is the
non-overflow section of the dam. The task is to design the layout of the non-overflow section
and determine if the design is stable enough against considered failure modes.

The following questions are aimed to be answered in this thesis:

• Is the chosen design layout stable against the limit states of anti-sliding, compressive
stress and tensile stress?
• What are the critical stability areas in the dam concerning material failure?
• Is there any risk of failure due to seepage in the dam?
• What influences has natural events such as earthquake and floods on the stability and
deformations of the dam?

1.3 Limitations

The design methodology followed in this thesis only covers the non-overflow section in a
gravity dam, with the Longtan Dam as reference. In addition, the following limitations are
present in this design:

• The shape, slope inclinations and bottom elevation were based on reference data due
to time limitations.
• Occurring loads such as ice, wind and sub-atmospheric are not considered in this
thesis.
• The numerical analysis was conducted on a 2D cross-section of the non-overflow
section. To analyze the effects of widespread stresses, a 3D model should be established
and analyzed.
• Analysis in GeoStudio© is conducted using a linear elastic material model, thus assum-
ing the material never deforms plastically.
• No earthquake analysis is computed in GeoStudio© due to time limitations.
• Numerical computations were conducted on a private laptop, limiting the coarseness
of the mesh in the numerical analysis.

3
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

4
 Theoretical Background

2 THEORETICAL BACKGROUND

2.1 Concrete Dams

Dams are generally divided into two groups: embankment dams and concrete dams, where
Longtan Dam is classified as a concrete dam. Concrete dams are commonly built using con-
crete, but older constructions are made of masonry and classifies as concrete dams. Compared
to embankment dams, concrete dams require less material due to concrete’s strength which
results in a slenderer construction. However, concrete dams requires to be built upon a hard
and resistant foundation compared to embankment dams. Concrete dams are categorized in
three types: gravity, arch and buttress dams (Association of State Dam Safety Officials
[ASDSO], n.d.), see Figure 2.1.

Figure 2.1. Illustration of gravity, arch and buttress dams (Australian Geographic, 2011).

Gravity dam is the most common concrete dam, where the design utilizes the weight of the
dam to transfer the induced reservoir water load downward. The self-weight resists the hor-
izontal water pressure against the dam. Each section of a gravity dam is independently stable,
being divided into overflow and non-overflow sections (Tata & Howard, 2016).

Arch dams are designed and built curved in a planar view with the convexity facing the
upstream reservoir. This form makes it possible to transfer the induced loads to the adjacent
rock formations, compared to gravity dams where the force is relocated downwards. There-
fore, the design is only optimal in canyons with solid rock that can resist the pressure and bear
the induced bulking. When compared to other concrete dams, arch dams require less materials
(ASDSO, n.d.).

Buttress dams has similar design as a gravity dam but has a reduced mass of concrete where
vertical and sloping buttresses resists the induced loads and transfer it downwards. The but-
tresses act like support and are constructed along the dam at certain intervals on the down-
stream side (Tata & Howard, 2016).

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Stability Analysis of Non-overflow Section of Concrete Gravity Dams

2.1.1 Roller-compacted Concrete Dams

RCC is a blend of concrete that has the same composition as conventional concrete, but with
other ratios such as occasional fly ash. The use of RCC for gravity dam construction has
rapidly increased as it proves to be more time and cost efficient compared to conventional
construction methods. As the mix has lower cement content and use fly ash, the generated
heat declines during curing and therefore a higher rate of concrete placement is achieved
(Abdo, 2008).

During dam construction, RCC sections are built layer-by-layer in successive horizontal
manner, resulting in a slope facing the downstream side. After a layer is built, it immediately
supports the continuation of the next layer. The method is like paving, where the material is
delivered by conveyers or dump trucks, spread by small dozers and finally compacted by
vibratory rollers (Abdo, 2008).

2.2 Layout Design

The purpose when designing a gravity dam section is to select a section design that satisfies
the required stability and strength, minimizes the volume to reduce the environmental im-
pact, costs and be constructed and operated easily. To start with, the main load of the dam is
considered – which defines the basic section and dimensions according to safety and economic
requirements. The basic design is analyzed depending on stress, strain and stability, where it’s
modified to a practical design until the most reasonable profile regarding stability and costs is
achieved (United States Bureau of Reclamation [USBR], 1976).

2.2.1 Slope Turning Point and Top Elevation

Two essential design conditions of a concrete gravity dam are the slope turning point and top
elevation. If using a design with an upstream slope, the slope turning point determines the
height of the slope which stabilizes the dam. According to Changsong (2007), the elevation
of slope turning point is defined after an assumption based on two criteria;
1� ℎ < 𝐻𝐻 < 2� ℎ (2.1)
3 𝑠𝑠 3
𝛻𝛻𝐻𝐻𝑠𝑠 < 𝛻𝛻𝛻𝛻 (2.2)

where h = reservoir height [m]; Hs = height from bottom elevation to slope turning point
[m]; ∇ Hs = slope turning point elevation [m] and 𝛻𝛻S = bottom elevation of the steel penstock
[m].

6
 Theoretical Background

The top elevation is the peak of the dam, which is designed after the water elevation and any
additional reservoir height due to accidental situations (Sheng-Hong, 2015). This is done by
∇𝐻𝐻 = ∇ℎ + ∆ℎ (2.3)

where ∇h = water elevation [m] and Δh = additional reservoir height [m], which is defined
by
∆ℎ = ℎ1% + ℎ𝑧𝑧 + ℎ𝑐𝑐 (2.4)

where h1% = wave height with a frequency of 1% [m], hz = height difference between the
central line of the wave and the flood storage level [m] and the height hc = freeboard height
[m].

2.3 Induced Forces

It’s essential to determine the induced forces acting on the dam structure. To start with, the
structure must be stable to withstand certain loads that varies with time and events such as
earthquakes. Therefore, the loads are classified according to their time dependence and spatial
variation, effect property to the structure, and occurring frequency (US Army Corps of En-
gineers [USACE], 1995). The loads are classified into:

• Permanent loads - Loads that under the design reference period do not change value
with time, or such small change is overlooked. Permanent loads include self-weight,
earth pressure, etc.
• Variable loads - Loads that under the design reference period change value with time.
Variable loads include water pressure, uplift, and wave pressure.
• Occasional loads - Loads that under the design reference period has a very low prob-
ability of occurring, e.g. loads with high impact but short duration. Occasional loads
include water pressure under the critical water level and seismic inertia.

2.3.1 Gravitational Force

The gravitational force is the self-weight of the dam and acts as the major resisting force. In
a 2D analysis, the force is calculated by the heaviness of the concrete and the area (USACE,
1995)
𝑊𝑊 = 𝛾𝛾𝑐𝑐 × 𝐻𝐻 × 𝑤𝑤 (2.5)

7
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

where W = cross-section self-weight of the dam [kN]; γc = unit weight of concrete [kN/m3];
H = height of the section [m] and w = width of force [m].

2.3.2 Hydrostatic & Uplift Pressure

Hydrostatic pressure has the largest impact on the dam. The pressure is induced due to water
acting against the dam on both the upstream and downstream side (USACE, 1995). The
pressures are calculated by
𝑃𝑃𝑤𝑤 = 𝛾𝛾𝑤𝑤 × ℎ × 𝑤𝑤 (2.6)

where Pw = cross-section hydrostatic pressure [kPa] and γw = is the unit weight of water
[kN/m3].

Uplift pressure is a result of water penetrating the foundation, creating a pressure that acts
upwards on the dam base. The uplift is evaluated through a pressure diagram from the hy-
draulic head on the upstream and downstream sides, see Figure 2.2. The uplift can be reduced
by installing drains in the bedrock, if no drains is installed the pressure are assumed to be linear
between the down and upstream side (Shen-Hong, 2015).

Figure 2.2. Uplift distribution with and without drain.

Drains create seepage pressure which is calculated using

𝑈𝑈 = 𝑎𝑎 × 𝛾𝛾𝑤𝑤 × ℎ (2.7)

where U = cross-section uplift pressure [kPa] and a = pressure attenuation coefficient from
drains.

8
 Theoretical Background

2.3.3 Wave & Sediment Pressure

Wave pressure is induced due to wind blowing over the reservoir, where the top surface is
pulled along the wind’s direction and form waves (Sheng-Hong, 2015). This is defined as
1
𝑃𝑃𝑊𝑊𝑊𝑊 = × 𝛾𝛾𝑤𝑤 × 𝐿𝐿𝑚𝑚 × (ℎ1% + ℎ𝑧𝑧 ) (2.8)
4

where Pwk = cross-section wave pressure [kPa] and Lm = wave length [m].

The incoming stream brings sediment such as silt which deposits on the reservoir bed. The
sediment will be carried to the dam face and the elevation of the deposit ascends with time.
This creates horizontal pressure defined as active earth pressure (USACE, 1995)
1 1 − sin 𝜑𝜑𝑠𝑠
𝑃𝑃𝑠𝑠𝑠𝑠 = × 𝛾𝛾𝑠𝑠 × � � × ℎ𝑠𝑠2 (2.9)
2 1 + sin 𝜑𝜑𝑠𝑠

where Psk = cross-section sediment pressure [kPa]; γs [kN/m3] is the unit weight of the sedi-
ment; φs [°] the friction angle and hs [m] the height of the sediment.

2.3.4 Earthquake Inertia Force

If a dam is situated in a region with moderate risk of earthquake shakes, it is appropriate to

take the acceleration into account. The acceleration is either considered horizontally (up-
stream or/and downstream) or vertically (upward or/and downward), based on the impact of
damage (Sheng-Hong, 2015).

The force act like self-weight, where the earthquake inertia applies to the centroid of mass
element i. The inertia force in horizontal direction is determined using
𝑎𝑎ℎ × 𝜉𝜉 × 𝑊𝑊𝑖𝑖 × 𝛼𝛼𝑖𝑖
𝐹𝐹𝑖𝑖 = (2.10)
𝑔𝑔

where Fi = horizontal earthquake inertia force of the mass element i [kN]; Wi = respective
self-weight for mass element i [kN]; ah = horizontal earthquake acceleration [m/s2]; ξ = com-
prehensive influencing coefficient and αi = dynamic distribution coefficient for mass element
i. For gravity dams, αi is defined by

𝐻𝐻 4
1 + 4 � 𝐻𝐻𝑖𝑖 �
𝛼𝛼𝑖𝑖 = 1,4 × (2.11)
𝑛𝑛 𝑊𝑊𝑗𝑗 𝐻𝐻𝑗𝑗 4
1+ 4 ∑𝑗𝑗=1 × � 𝐻𝐻 �
𝑊𝑊

9
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

where H = total height of the dam [m]; Hi = height for mass element i [m]; W = total self-
weight of the dam [kN]; Wj = accumulated self-weight for the mass elements [kN] and Hj =
accumulated height for the mass elements [m].

When a dam undergoes earthquake motion, the body of the water facing its front tends to
remain in place and induce seismic dynamic water pressure on the dam body. Neglecting the
vertical motion of water, the horizontal pressure can be simplified and calculated by (Ministry
of Water Resources [MWR], 1997)
𝐹𝐹0 = 0,65 × 𝑎𝑎ℎ × 𝜉𝜉 × 𝑝𝑝𝑤𝑤 × ℎ2 × 𝜂𝜂𝑐𝑐 (2.12)

where F0 = resultant hydrodynamic force [kN]; ρw = density of water [kg/m3] and ηc = re-
duction factor determined by the inclination of the upstream dam face.

2.4 Partial Safety Factor Method

The general design requirement for partial safety factor method for hydraulic structures ac-
cording to Shen-Hong (2015) considers three situations during the structure’s lifespan:

• Permanent situations: Long-term duration, usually corresponds to the design reference

period. The normal operation situation is an example of a permanent situation.
• Temporary situations: Mainly corresponding to a construction-, service- or other
temporary situation during the operation of the dam.
• Accidental situations: Lower probability situations, but once occurred will result in
serious consequences. Such situations can be critical flood, earthquake or ineffective
drainage.

The probabilities and situations are considered by determining the collapse and serviceability
limit states. For a gravity concrete dam, such collapse limit states are stability against sliding
and compressive strength, while tensile stress is a serviceability limit state.

10
 Theoretical Background

2.4.1 Limit Equilibrium Method

The limit equilibrium method defines the factor of safety where the resisting function are
compared to the action effect function. The method consists of both basic and contingency
principles of force combination - dependent on the load situation (Shen-Hong, 2015)
1 𝑓𝑓𝐾𝐾
𝛿𝛿0 𝛹𝛹𝛹𝛹�𝛿𝛿𝐺𝐺 𝐺𝐺𝐾𝐾 , 𝛿𝛿𝑄𝑄 𝑄𝑄𝐾𝐾 , 𝑎𝑎𝐾𝐾 � ≤ 𝑅𝑅 � , 𝑎𝑎𝐾𝐾 � (2.13)
𝛿𝛿𝑑𝑑1 𝛿𝛿𝑚𝑚
1 𝑓𝑓𝐾𝐾
𝛿𝛿0 𝛹𝛹𝛹𝛹�𝛿𝛿𝐺𝐺 𝐺𝐺𝐾𝐾 , 𝛿𝛿𝑄𝑄 𝑄𝑄𝐾𝐾 , 𝐴𝐴𝐾𝐾 , 𝑎𝑎𝐾𝐾 � ≤ 𝑅𝑅 � , 𝑎𝑎𝐾𝐾 � (2.14)
𝛿𝛿𝑑𝑑2 𝛿𝛿𝑚𝑚

where γ0 = structural importance coefficient; 𝛹𝛹 = design condition coefficient; S = action

effect function; R = structural resistance function; 𝛿𝛿G = sub coefficient of permanent effect;
GK = standard value of permanent effect; 𝛿𝛿Q = sub coefficient of variable effect; QK = standard
value of variable effect; aK = characteristic value of geometry parameter [m]; AK = contin-
gency representative value; 𝛿𝛿d1 = basic composite structure coefficient; 𝛿𝛿d2 = accidental com-
bined structure coefficient; 𝛿𝛿m = material properties coefficient and fK = standard value of
material properties.

2.5 Stress and Strain

Subjecting a material to a planar analysis, the material can be induced by normal- and shear
stresses. While the normal stress act perpendicular to the plane, the shear stress acts along it.
The result of induced stress on a material body is deformations, determined by strain. The
relationship between stress and strain can be described by the linear elastic material model and
Hooke’s law, which is described in this chapter.

2.5.1 Linear Elastic

Linear elastic analysis indicates the proportional behavior of strain when induced with a stress,
see Figure 2.3.

Figure 2.3. Relation between stress and strain in linear elastic conditions (GEO-SLOPE International Ltd,
2013).

11
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

Many stress and strain problems can be analyzed in a two-dimensional plane in elastic condi-
tion for a first estimate. The theory of elasticity consists of two general types, namely plane
stress and plane strain. These analyses can be done with the assumption that the material is
homogenous and isotropic (GEO-SLOPE International Ltd, 2013).

2.5.2 Hooke’s Law

Hooke’s law is a linear elastic behavior where the stresses are expressed in relation to the
strains and is calculated using (GEO-SLOPE International Ltd, 2013)
𝜎𝜎 = 𝐸𝐸𝐸𝐸 (2.15)

where σ = normal stress [Pa]; E = Young’s modulus of the material [Pa] and ε = material
normal strain [%]. Hooke’s law also applies to a two-dimensional shear stress analysis, where
the shear stress and strain are calculated with
𝜏𝜏 = 𝐺𝐺𝐺𝐺 (2.16)

where τ = planar shear stress [Pa]; G = material shear modulus [Pa] and Γ = shear material
strain [%]. The shear modulus is determined by
𝐸𝐸
𝐺𝐺 = (2.17)
2(1 + 𝑣𝑣)

where ν = Poisson’s ratio of the material.

2.5.3 Plane stress

Plane stress can be visualized as stress action along a thin plate where the z-direction are
considered to be zero, therefore σzz = σyz = σxz = 0 (Efunda, n.d.). The plane stress condition
leads to the compliance matrix

𝜀𝜀𝑥𝑥𝑥𝑥 𝜎𝜎𝑥𝑥𝑥𝑥
1 1 −𝑣𝑣 0
�𝜀𝜀𝑦𝑦𝑦𝑦 � = �−𝑣𝑣 1 0 � �𝜎𝜎𝑦𝑦𝑦𝑦 � (2.18)
𝐸𝐸
𝜀𝜀𝑥𝑥𝑥𝑥 0 0 1 + 𝑣𝑣 𝜎𝜎𝑥𝑥𝑥𝑥

By inverting the compliance matrix, the stiffness matrix for plane stress is found, which is
given by
𝜎𝜎𝑥𝑥𝑥𝑥 1 𝑣𝑣 0 𝜀𝜀𝑥𝑥𝑥𝑥
𝐸𝐸
𝜎𝜎
� 𝑦𝑦𝑦𝑦 � = �𝑣𝑣 1 𝜀𝜀
0 � � 𝑦𝑦𝑦𝑦 � (2.19)
𝜎𝜎𝑥𝑥𝑥𝑥 1 − 𝑣𝑣 2
0 0 1 − 𝑣𝑣 𝜀𝜀𝑥𝑥𝑥𝑥

12
 Theoretical Background

2.5.4 Plane strain

Plane strain can be visualized in an object where one dimension is much larger than the other,
for example a long wire with stresses acting perpendicular to its length. The strains in the z-
direction are neglected which gives ε zz = ε yz = ε xz = 0 (Efunda, n.d.). The strain-stress
stiffness for an isotropic material gives the stiffness matrix
𝜎𝜎𝑥𝑥𝑥𝑥 1 − 𝑣𝑣 𝑣𝑣 0 𝜀𝜀𝑥𝑥𝑥𝑥
𝐸𝐸
𝜎𝜎
� 𝑦𝑦𝑦𝑦 � = � 𝑣𝑣 1 − 𝑣𝑣 𝜀𝜀
0 � � 𝑦𝑦𝑦𝑦 � (2.20)
𝜎𝜎𝑥𝑥𝑥𝑥 (1 + 𝑣𝑣)(1 − 2𝑣𝑣)
0 0 1 − 2𝑣𝑣 𝜀𝜀𝑥𝑥𝑥𝑥

Inverting the stiffness matrix leads to the plane strain’s compliance matrix
𝜀𝜀𝑥𝑥𝑥𝑥 0 𝜎𝜎𝑥𝑥𝑥𝑥
1 + 𝑣𝑣 1 − 𝑣𝑣 −𝑣𝑣
�𝜀𝜀𝑦𝑦𝑦𝑦 � = � −𝑣𝑣 1 − 𝑣𝑣 0� �𝜎𝜎𝑦𝑦𝑦𝑦 � (2.21)
𝐸𝐸
𝜀𝜀𝑥𝑥𝑥𝑥 0 0 1 𝜎𝜎𝑥𝑥𝑥𝑥

2.6 Slope Stability

The identification of potential failure surfaces and slip locations are crucial when determining
the stability of a concrete gravity dam. The analysis of slope stability can be conducted using
several different methods. In this study, the interslice method Morgenstern-Price is computed
using the Mohr Coulomb failure criterion.

2.6.1 Mohr Coulomb’s Failure Criterion

Mohr Coulomb is a failure criterion which describes the shear strength of geotechnical ma-
terials according to a set of linear equations of the surrounding principle stresses. (GEO-
SLOPE, 2013) The relation is expressed as

𝜏𝜏 = 𝑐𝑐 + 𝜎𝜎𝑛𝑛 tan 𝜑𝜑 (2.22)

where τ = shear stress on the failure plane [Pa]; c = apparent cohesion [Pa]; σn = normal stress
on the failure plane [Pa] and φ = internal friction angle [°]. The relation is visually represented
in Figure 2.4, where the half circles represent the Mohr circles.

13
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

Figure 2.4. Mohr Coulomb’s failure criterion (GEO-SLOPE, 2013)

2.6.2 Morgenstern-Price Method

The Morgenstern-Price method is a limit equilibrium method which is used in computational

analysis of slope stability. The basic theory behind limit equilibrium is that forces, moments
or stresses that resists the unstable motion are compared to those who causes it. This method
is applied by considering a specific slip surface where the failure surface is divided into vertical
slices, see Figure 2.5.

Figure 2.5. Illustration of the slice method (GEO-SLOPE International Ltd, 2012b).

In Morgenstern-Price method, the interslice forces and shear forces are assumed to act be-
tween the slices by considering the half-sine function. The half-sine function concentrates
the shear forces of respective interslice to the middle of the sliding mass and decreases area
around the edges, crest and toe (D. G. Fredlund and J. Krahn, 1977).

14
 Theoretical Background

2.7 Darcy’s law & Seepage

Darcy’s law is a water flow formula for both saturated and unsaturated soil and is defined by
(GEO-SLOPE International Ltd, 2012a)
𝑞𝑞 = 𝑘𝑘𝑘𝑘 (2.23)
where q = the specific discharge [m3/s]; k = the hydraulic conductivity [m/s] and i = the
gradient of total hydraulic head.

The specific discharge in Darcy’s law are often defined as Darcian velocity, where the actual
average velocity at which water moves through the soil is linear. In GeoStudio©, SEEP/W
computes and presents only the Darcian velocity.

The governing differential equation used in SEEP/W finite element formulation is defined as
(GEO-SLOPE International Ltd, 2012a)
𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
�𝑘𝑘𝑥𝑥 �+ �𝑘𝑘𝑦𝑦 � + 𝑄𝑄 = (2.24)
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

where H = the total head [kPa]; kx = hydraulic conductivity in the X-direction [m/s]; ky =
hydraulic conductivity in the Y-direction [m/s]; Q = applied boundary flux [m3/s/m2]; θ =
volumetric water content [%] and t = time [d].

The formula states that the difference in water flow (flux) through an element volume at some
point in time is equal to the difference in water storage in the soil systems. Under steady-state
conditions, the flux in and out through an elemental volume remains constant, which applies
to Steady-State analyses in SEEP/W
𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕
�𝑘𝑘𝑥𝑥 �+ �𝑘𝑘𝑦𝑦 � + 𝑄𝑄 = 0 (2.25)
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

2.8 Finite Element Method

The application of the laws of physics for space- and time-dependent issues are often expressed
in terms of partial differential equations. In practices like civil engineering, complicated ge-
ometry, boundary conditions and material properties can prove analytical methods to be dif-
ficult, if not impossible. Thus, the application of numerical methods can approximate the
partial differential equations in a way making it possible for the engineer to validate the results.
This procedure is called discretization which approximates the partial differential equations
with numerical model equations. The finite element method (FEM) is one such method that
computes such approximations (COMSOL, n.d.).

15
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

2.8.1 Basic Principle

The basic principle behind the finite element method is the division of a structural body into
elements, called finite elements. The division makes it possible to instead of solving a contin-
uous problem for a whole structure, the finite elements are solved independently and thus a
solution of the entire structure can be obtained (COMSOL, n.d.). The discretization of a
structure is divided into elements and nodes, creating a finite element model (Figure 2.6).

Figure 2.6. Structure discretized to a finite element model, showing different element types (FEM Expert,
n.d.).

The choice of the most suitable element type depends on the type of structure that is analyzed.
Triangular elements and quadrilateral elements, which can be seen in Figure 2.6, are the most
common in civil engineering and are used for structural issues such as plate bending and planar
stress and strain (Malm, 2016).

16
 Methodology

3 METHODOLOGY

3.1 Layout

In this thesis, the layout for two cases are considered, namely design and critical water level.
The design water level is the water reservoir based on normal conditions, where the critical
water level considers accidental situations. Using some of Longtan Dam’s parameters and di-
mensions as reference (Table 3.1), the slope turning point and top elevation for both cases are
calculated (Figure 3.1).

Figure 3.1. Layout parameters of the non-overflow section.

Table 3.1. Input parameters for layout design (Dai, 2018).

∇b [m. a.s.l.] n m
195 0.15 0.75

Table 3.1 shows ∇b = bottom elevation [m]; m = inclination of the downstream slope and n
= inclination of the upstream slope.

Using the top elevation, the total width is determined by applying 8 ~ 10 % of the top ele-
vation (Dai, 2018), which gives the width 17 m. For overview layout of the dam, see AP-
PENDIX III – OVERVIEW LAYOUT.

17
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

3.2 Acting Forces

The forces covered in chapter 2.3 are calculated in this thesis and are illustrated in Figure 3.2.

Figure 3.2. Illustration of the forces acting on the non-overflow section of Longtan dam.

The loads in Figure 3.2 are the following:

1. Seismic dynamic water pressure

2. Horizontal sediment pressure
3. Vertical sediment pressure
4. Wave pressure
5. Horizontal hydrostatic pressure, upstream
6. Vertical hydrostatic pressure, upstream
7. Gravitational force, self-weight
8. Horizontal earthquake force
9. Uplift pressure
10. Vertical hydrostatic pressure, downstream
11. Horizontal hydrostatic pressure, downstream.

In reality, the dam consists of void spaces containing turbines, utility tunnels and storage.
However, the additional weight is neglected as it is assumed that the weight of voids filled
with concrete cancel out the weight of machineries. Therefore, the dam is considered to
consist of 100% concrete for simplicity (Federal Energy Regulatory Commission, 2016).

18
 Methodology

The loads are determined for both the normal and critical water level. However, the earth-
quake forces are not calculated for the critical water level as it’s assumed that the possibility
of a critical flood and earthquake occurring at the same time is unlikely (Dai, 2018). The
forces are summarized for both conditions in horizontal and vertical directions.

When determining the moments and moment arm, the bottom of the dam is used as reference
point. Therefore, the moments acting in counter clockwise direction is considered positive
and clockwise negative. The moments are divided into restoring- and overturning moments.

3.3 Limit States

The limit states considered in this thesis is the anti-sliding stability of concrete interface be-
tween dam body and bedrock, compressive stress at the dam toe and tensile stress of the dam
heel. Acting forces and moments for respective limit state can be seen in Figure 3.3.

Figure 3.3. Limit states of the non-overflow section.

The stability for respective limit state is calculated for both basic and contingency load com-
binations. For the basic combination the normal water level is analyzed and for contingency
the critical water level and earthquake situation are considered. The unreduced compressive
strength of the concrete is 25.4 MPa (Dai, 2018).

19
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

3.4 Modeling in GeoStudio

The numerical analyses in this thesis are conducted using the software GeoStudio©. By draw-
ing the models and simulating the results, conclusions can be made based on the software’s
computation. GeoStudio© is a modeling software which is mainly used by geo-engineers and
earth scientists.

In this thesis the analysis types of SIGMA/W, SEEP/W and SLOPE/W are used to analyze
the non-overflow section of the concrete gravity dam and conclude the objective of this
study.

3.4.1 Model setup

The dimensions of the non-overflow section of the Longtan Dam are directly imported into
GeoStudio© using AutoCAD©. Regions and respective material layers are created based on
given layout from the project background (Dai, 2018). See APPENDIX I – FEM LAYOUT
for imported CAD layout. Mesh properties are set to triangles only with 5 m approximate
global element size. See Figure 3. 4 for model layout.

R1

R2 Curtains

Base1

Base2

Figure 3.4. Model layout in GeoStudio.

20
 Methodology

Two scenarios were simulated in GeoStudio©. The scenarios are dependent on the upstream
and downstream water levels, where the first scenario reflects a normal case and the second
scenario replicates flooded water levels – referred as critical case. See Table 3.2 for applied
water levels.
Table 3.2. Simulated water level scenarios in GeoStudio.

Scenario Upstream [m] Downstream [m]

Normal 182.1 30.5
Critical 185.8 65.2

3.4.2 SIGMA/W

Analysis in SIGMA/W simulates the resulting stresses and deformations that occurs in the
dam dependent on the water level scenario. The analysis is set up with a Load/Deformation
case and an in-situ analysis as its source. This is to establish in-situ conditions on the dam
body and bases, which allows the Load/Deformation analysis to compute the end condition
based on the in-situ case (GEO-SLOPE International Ltd, 2013). The time duration applied
to the Load/Deformation analysis is only one day with the time increment one day.

The material category used is Total Stress Parameters with the Linear Elastic (Total) model
explained in Chapter 2.5. Applied material properties are shown in Table 3.3.
Table 3.3. Material input data for SIGMA/W analysis (Dai, 2018).

Material E [GPa] γ [kN/m3] ν [-]

R1 20 24 0.163
R2 20 24 0.163
Curtain 20 26.5 0.25
Base1 16 25 0.3
Base2 16 25 0.3

Boundary conditions are defined with a zero value X-displacement along the edge boundary
walls and a zero X/Y-displacement at the boundary base. The water levels from Table 3.2
are defined as hydrostatic pressures. The boundary conditions of the induced uplift pressures
can be found in APPENDIX II – BOUNDARY CONDITIONS.

21
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

3.4.3 SEEP/W

Seepage analysis in SEEP/W are computed to determine the pore water pressure, hydraulic
head and water flux distribution and penetration through the dam body and base layers. The
analysis of seepage velocity and direction is crucial for a stable and safe dam (GEO-SLOPE
International Ltd, 2012a). SEEP/W is formulated on the theory of Darcy’s law of water flow
through both saturated and unsaturated soil, see Chapter 2.7.

The seepage through the dam is analyzed using the Steady-State Seepage analysis. The mate-
rial model used is Saturated/Unsaturated where the water volume content is undefined, mak-
ing the hydraulic conductivity and its anisotropy the only input data. See Table 3.4 for ma-
terial seepage properties.
Table 3.4. Material input data for SEEP/W analysis (Dai, 2018).

Material Hyd. Conductivity, k [-] Anisotropy, ky’/kx’ [-]

R1 2.5E-9 1
R2 2.5E -11
1
Curtain 2.5E -9
1
Base1 2.5E -7
1
Base2 - 1

The upstream and downstream water pressures are defined as Water Total Head boundary
conditions with the constant normal/critical water levels for each boundary. On the down-
stream slope a Water Rate boundary with the constant rate of 0 m3/s is drawn to simulate the
impermeable concrete.

3.4.4 SLOPE/W

Stability analysis of earth structures in SLOPE/W is simulated to determine a key issue in

geotechnical engineering, namely if the structure will remain stable or not. SLOPE/W applies
the theory of limit equilibrium formulations explained in Chapter 2.4.1, making it possible
to deal with complex stratigraphy, various shear strength models and many kinds of slip surface
shapes (GEO-SLOPE International Ltd, 2012b). The factor of safety method applied in this
study is Morgenstern-Price which is covered in Chapter 2.6.2.

The material models used differs on the material type. To start with, the drainage curtain itself
is assumed to have no impact on the slope stability, thus it is not defined in the model. The
concrete of the dam is defined as a high strength material with no cohesion or friction angle.
The upper base layer has cohesion and friction angle defined but the lower has none due to
it being bedrock, see Table 3.5.

22
 Methodology

Table 3.5. Material input data for SLOPE/W analysis (Dai, 2018).

Material γ [kN/m3] c [MPa] φ [°]

Concrete 24 - -
Base1 23 1.09 54.1
Base2 - - -

Induced water pressure is defined by a piezometric line spanning from the upstream water
level along the dam slope to the downstream water level. In addition, sediment, uplift and
wave pressures are simulated by point loads in the model.

23
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

24
 Results

4 RESULTS

4.1 Dam Dimensions

According to the procedure in Chapter 2.2, the slope turning point is assumed to be 90 m
based on the criteria from Chapter 2.2.1. The calculated additional water levels and reservoir
elevation for both design and critical case is presented in Table 4.1.
Table 4.1. Top water elevation for design and critical water levels.

Design [m] Critical [m]

Δh 4.35 1.82
∇Top 381.05 380.80

The table shows that the top elevation for the dam is the design elevation, which has a higher
elevation than the critical case. The top elevation is therefore 381 m a.s.l. and the width or
crest 17 m, see Figure 4.1.

Figure 4.1. Design for non-overflow section of the dam.

25
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

4.2 Stability Analysis

The induced forces and moments acting on the dam have been calculated according to Chap-
ter 2.3. The results are in turn applied in the procedure of calculating stability according to
limit states presented in Chapter 2.4.

4.2.1 Forces and Moments

The acting forces and moments are summarized in tables below. See Table 4.2 for horizontal
and vertical loads and Table 4.3 for positive and negative moments acting on the dam.
Table 4.2. Summarized horizontal and vertical design loads.

ΣFH [MN] ΣFV [MN]

Normal 200.6 290.8
Critical 174.5 287.5

Table 4.3. Summarized positive and negative moments.

ΣM+ [MNm] ΣM- [MNm]

Normal 8348.8 -13789.7
Critical 9224.1 -13922.3

The arm of force in both X- and Y-direction from dam center (Figure 3.3) are calculated and
presented in Table 4.4.
Table 4.4. Center of mass for the arm of force in X- and Y-direction.

X [m] Y [m]
Normal 23.1 56.6
Critical 18.7 54.3

26
 Results

4.2.2 Limit States

The factor of safety against sliding is presented in Table 4.5.

Table 4.5. Factory of safety against sliding.

Calculation Condition 𝛿𝛿0 𝛹𝛹 S [MN] 1/ 𝛿𝛿0 R [MN] SF

Normal 202.6 256.0 1.26
Critical 163.1 253.7 1.56
Normal + Earthquake 187.6 256.0 1.37

The induced compressive stress compared to the concrete’s compressive strength is listed in
Table 4.6.

Table 4.6. Factor of safety against compressive stress.

Calculation Condition 𝛿𝛿0 𝛹𝛹 S [MPa] 1/ 𝛿𝛿0 R [MPa] SF

Normal 5.2 14.1 2.73
Critical 4.5 14.1 3.13
Normal + Earthquake 4.8 14.1 2.93

Table 4.7 shows the tensile stress induced on the dam heel, where values < 0 indicates ten-
sile failure.

Table 4.7. Design tensile stress at dam heel.

Calculation Condition 𝛿𝛿0 𝛹𝛹 S [MPa]

Normal 0.9
Critical 0.6
Normal + Earthquake 0.5

4.3 Numerical Analysis

The computations and results presented in this chapter are obtained using the setup according
to Chapter 3.4. Results of interest from the SIGMA/W analysis are the max. Shear stress,
max. & min. total stress and XY-displacements, while from SEEP/W the interesting results
are the water total head, pore-water pressure and the water flux. As for SLOPE/W the safety
factor against sliding is of interest. Numerical computation for both normal and critical cases
has been conducted and the differences are commented.

27
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

4.3.1 SIGMA/W

The maximum shear stress acting on the dam for normal water level show high concentrations
of stress at the dam heel and dam toe, see Figure 4.2. At the range of 2 m inwards from the
dam heel the maximum shear stress spans from 1.4 MPa to 3.2 MPa.

Figure 4.2. Max. Shear Stress, normal water case.

Compared to the normal water level, the critical maximum shear stress shows the same pattern
where high concentrations occur in the dam heel and toe. In addition, similar stresses are
induced, with only a slight difference of 0.2 MPa near the boundaries, see Figure 4.3 for
horizontal boundary plots.

Max. Shear Stress, heel Max. Shear Stress, toe

4 3,5
Normal Normal
3,5 3
Critical Critical
Stress [MPa]

Stress [MPa]

3
2,5
2,5
2
2

1,5 1,5

1 1
0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5
Distance from boundary [m] Distance from boundary [m]

Figure 4.3. Max. Shear stress in relation to boundary distance, heel and toe.

28
 Results

The max. Total stresses induced on the dam show high concentration of compressive stress
at the dam toe, see Figure 4.4. According to the contour the amount of stress rapidly increases
at the dam toe’s boundary. The stress spans from 4.0 MPa to 6.5 MPa.

Figure 4.4. Max. Total Stress, normal water case.

The critical analysis of the max. Total stress shows the same behavior and concentration of
stresses. However, the amount of compressive stress is declining when applying the critical
water levels. The main difference in stress is due to the increase of the downstream level,
which is double the normal case. See Figure 4.5 for stress plot.

Max. Total Stress, toe.

7
Normal
6,5
Critical
Stress [MPa]

5,5

4,5

4
5 4 3 2 1 0
Distance from boundary [m]

Figure 4.5. Max. Total Stress in relation to boundary distance, toe.

29
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

The min. total stress result show tensile stresses acting on the dam heel, see Figure 4.6. The
range from the boundary and inwards span from -6 MPa to -2 MPa, indicating that there are
tensile stresses present which could lead to failure due to the tensile strength of concrete being
0 MPa according to Chinese standards.

Figure 4.6. Min. Total Stress, normal water case.

Computing the critical water case, the difference of tensile stress is noted to be approximately
0.3 MPa higher compared to the normal case. This indicates higher amount of tensile failure
in the dam heel in the critical case compared to normal case, but only marginally so according
to Figure 4.7.

Min. Total Stress, heel.

-7
Normal
-6
Critical
-5
Stress [MPa]

-4

-3

-2

-1
0 1 2 3 4 5
Distance from boundary [m]
Figure 4.7. Min. Total Stress in relation to boundary distance, heel.

30
 Results

The displacements in XY-direction for normal water level show displacements up to 7 cm at

the top, aiming to tilt the dam towards the downstream side, see Figure 4.8. The displacement
is a combination of both X- and Y-displacements, where the X-displacement is around 6.5
cm at the top and Y-displacement 1.2 cm on the dam upstream face. For the deformed mesh,
see Figure 4.9.

Figure 4.8. XY-displacements, normal water case.

Figure 4.9. Deformed mesh, normal water case.

When comparing the normal case to the critical case, it is observed that the XY-displacements
in the critical case has a 0.5 cm higher displacement. This is due to the X-displacement being
1 cm higher, which is a result of the increase of elevation in the critical case.

31
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

4.3.2 SEEP/W

The water total head for normal water level show the head pressure interval after the defined
upstream and downstream levels, namely 182.1 m and 30.5 m, see Figure 4.10. The equipo-
tential lines of the flow net are adjusted after the structure of the dam, where the influence of
the drainage curtain can be observed in the figure.

Figure 4.10. Water Total Head, normal water case.

When comparing to the critical case, similar flow net and equipotential lines in the base layer
can be observed. However, the increase in both upstream and downstream levels result in
higher water total head pressure in the dam body. Specifically, on the downstream side where
the water level is doubled in the critical scenario, see Figure 4.11.

Figure 4.11. Water Total Head, critical water case.

32
 Results

The pore-water pressure acting on the dam show an increase in pore-water pressure in rela-
tion to depth. In the dam structure the water pressure increases based on the difference in
water levels of the upstream and downstream sides, see Figure 4.12.

Figure 4.12. Pore-water Pressure, normal water case.

For the critical case, the pore-water pressure declines with depth as expected. But the pore-
water pressure also follows the downstream slope due to the higher water level, see Figure
4.13.

Figure 4.13. Pore-water Pressure, critical water level.

33
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

The water flow for the normal water condition show a flow chart with high concentrations
of flux at the dam heel and the grout curtains, see Figure 4.14. Due to the impermeable
conditions of concrete and rock, no water flux is present in the dam and the lower base layer.

Figure 4.14. Water Flux, normal water case.

Like the water flow in the normal case, the flux concentrates at the dam heel and around the
grout curtains. The main difference between the cases is the amount of water flux, where the
flow is higher in the normal case. This is visualized by plotting the water rate through the
base layer in relation to the dam curtains, see Figure 4.15 for subdomains and Figure 4.16 for
plot.

Figure 4.15. Subdomain elements chosen for water rate plot.

34
 Results

Water Rate
-1,5E-06
Curtain 1 Normal
-1,0E-06
Critical
Water Rate [m3/s]

-5,0E-07 Heel

0,0E+00
Toe
5,0E-07

1,0E-06
Curtain 2
1,5E-06
0 25 50 75 100 125 150 175 200 225

Distance [m]

Figure 4.16. Water rate through the base layer.

The graph shows that at the point of grout curtains, the water rate is higher in the normal
scenario than the critical scenario. However, due to the higher downstream water level the
water rate at the dam toe is higher in the critical case than in the normal case.

35
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

4.3.3 SLOPE/W

The numerical computation of slope stability of the dam shows 55 potential slip surfaces
where all show stable results. In Figure 4.17 are the most critical slip surfaces visualized. The
results show that the lowest safety factor is 2.26. It is observed that all potential slip surfaces
are very shallow in the base layer.

Figure 4.17. Factor of safety against sliding, normal water case.

Applying the critical water case, a similar result is observed. Specifically, the safety factor of
the most critical surface increases from 2.26 to 2.36. The same potential slip surfaces apply to
the critical case as the normal, although slightly higher safety factor.

4.4 Comparing Hand- and Numerical Results

In this chapter, the hand calculated limit states are compared with its respective numerical
analysis. The comparisons apply to the normal and critical scenarios, while the normal +
earthquake scenario are not computed in GeoStudio©.

The hand calculated anti-sliding is compared to the SLOPE/W analysis using Morgenstern-
Price method according to Chapter 3.4.4. The safety factor retrieved from the numerical
analysis is the worst-case slip surface. See Table 4.8 for comparison.

36
 Results

Table 4.8. Factor of safety against sliding, comparing hand- and numerical results.

Calculation Condition Force, hand [MN] Resistance, hand [MN] SF SF, numerical
Normal 202.6 256.0 1.26 2.26
Critical 163.1 253.7 1.56 2.36
Normal + Earthquake 187.6 256.0 1.37 -

The results show stable safety factor for both methods. However, the hand calculations imply
a lower factor of safety against sliding when compared to the numerical results.

When comparing the factor of safety against compressive failure, numerical data from the
boundary and 2 meters inwards from the dam toe was used. This results in a stress interval
and thus a factor of safety interval, see Table 4.9.
Table 4.9. Factor of safety against compressive failure, comparing hand- and numerical results.

Calculation Condition Stress, hand [MPa] SF Stress, numerical [MPa] SF, numerical
Normal 5.2 2.73 5.8 – 6.6 2.13 – 2.43
Critical 4.5 3.13 5.4 – 5.9 2.39 – 2.66
Normal + Earthquake 4.8 2.93 - -

The table show that the factor of safety in the numerical analysis is lower than the hand
calculations, but show similar patterns comparing the normal and the critical case. The factor
of safety are above 2.0 for both methods.

Tensile stress data from the boundary and 2 meters inwards are collected for the comparison
with the hand calculations, see Table 4.10. The tensile strength of concrete in this analysis is
set to 0 MPa.
Table 4.10. Tensile stresses at dam heel, comparing hand- and numerical results.

Calculation Condition Stress, hand [MPa] Stress, numerical [MPa]

Normal 0.9 (-5.8) – (-4.2)
Critical 0.6 (-6.2) – (-4.5)
Normal + Earthquake 0.5 -

The comparison shows that the hand calculations indicate that the concrete is stable against
tensile failure while the numerical results show the complete opposite with a huge marginal.

37
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

38
 Analysis and Discussion

5 ANALYSIS AND DISCUSSION

5.1 Model Validation

One limitation mentioned in Chapter 1.3 is the hardware used for the numerical modeling,
which limited the number of elements and nodal points which determines the accuracy of
the computation. Arguably more elements give better results, but it affects the computation
speed and due to the time limitation, the amount of computation and mesh size had to be
limited. More accurate numerical computation could be a potential subject for further study
in this matter.

However, it was noted that when computing a finer mesh, the result itself did marginally
differ from a more courser mesh. Most notably was the stress at the dam heel and toe bound-
aries which increased rapidly when using a finer mesh. This is due to stress singularity where
the defined uplift pressure acts like a point load, causing the stress to increase rapidly as the
element area the force is distributed lowers. Therefore, it is argued that the true values were
located a few meters inwards of the dam and not at the boundary. This was later confirmed
when returning to Sweden where computations on finer meshes show the same stresses as the
courser mesh in China a few meter inwards from the dam body’s boundary.

5.2 Limit States

The results show no big difference between the normal and critical water level cases. The
stress distribution and values are almost identical in both cases, where a small increase in
stresses and displacement can be observed in the critical case. The increase of water levels does
result in a higher displacement, where the dam tilts 0.5 cm more than the normal case of 7.0
cm.

Points of interest lies mainly on occurring tensile stress at the dam heel and compressive stress
at the dam toe. Using the tensile strength of 0 MPa for concrete, the numerical results indi-
cates that the dam is going to fail at the dam heel. The tensile stresses range from -6.0 MPa
to -2.0 MPa, but the highest value is present on the boundary which rapidly decreases further
inwards. This is due to stress singularities in FEM, where the value increases the finer the
mesh is due to the uplift pressure being defined as a point load.

When compared to hand calculations, no tensile stresses are shown at the heel which implies
that the dam is safe. But the hand calculations are very simplified compared to GeoStudio©
as the computation takes a more precise approach with material parameters, stress distribution
and dam geometry into account.

39
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

Thus, GeoStudio© show more reliable results than the hand calculations. But in a real scenario
the tensile strength of concrete is arguably higher than 0 MPa. But as the area is one of the
cores in dam stability, one should reinforce it to hinder any tensile failure.

When comparing the compressive stresses in GeoStudio© and hand calculations, both show
a stable result with stress below the compressive strength of 14.1 MPa. In addition, the stress
values from FEM and hand calculation show similar results according to Table 4.9. Therefore,
it can be concluded that the dam is safe from compressive failure as two different approaches
show stable results and similar stresses.

The anti-sliding results show that the dam is safe against sliding. However, it is noted that in
the hand calculations the safety factor for the normal case is close to 1.0. Arguably the safety
factor should be over the limit with reasonable marginal to ensure the stability of the dam.
The numerical results show higher safety factors when compared to hand calculations. As a
result, it is concluded that the dam is safe against sliding.

5.3 Seepage

Judging from the seepage results there are no significant risk for the stability of the dam due
to seepage. The results show high concentrations of seepage velocity near the heel and the
grout curtains, while underneath the dam lower velocity is noted. This implies that any risk
of erosion and high stresses are induced at the heel and grout curtains, but with the maximum
velocity of 3.4E-7 m3/s/m2 it is deemed unlikely. This applies to both the normal and critical
case, where notably the critical case is more stable due to the phreatic surface being higher.

5.4 Future Study

In this analysis a linear elastic material model was used for the concrete. This means in the
computation the strain will keep increasing constantly with the stress even though it may have
surpassed the failure limit. Since our computation show tensile stresses present in the dam
heel and concrete being a brittle material, it is of interest to do a plastic analysis to analyze
how the stability of the dam behaves when exposed to tensile failure. But in a linear elastic
analysis this is not taken to account, where the concrete is assumed to keep deforming with
increased stress. For example, cracks will most likely occur in the dam judging from the re-
sults, which causes water to seep into the cracks and create uplifting pore pressure. The impact
of such events on the stability of the dam is worth investigating.

40
 Analysis and Discussion

In addition, the numerical simulation only covered the normal and critical water scenarios.
Further study using the analysis type QUAKE/W should be conducted to compare the nor-
mal earthquake scenario with the hand calculations. Also, the SLOPE/W analysis should be
complemented as the material properties and model are based on assumptions by the modelers
and supervisor. Due to time limitations only one SLOPE/W analysis was computed, more
methods should be investigated where the results are compared to reach a conclusion.

Lastly, this report only covers the non-overflow section of the dam. A stability study of the
overflow section should be conducted and with the remaining sections a detailed 3D analysis
of the whole dam should be investigated.

41
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

42
 Conclusion

6 CONCLUSIONS
The results covered in this thesis conclude that the non-overflow section of the concrete
gravity dam is safe against compressive stress and anti-sliding. However, the risk for tensile
failure is high and therefore reinforcements in the dam heel should be applied. Even though
the simulations and the procedure conducted in this study implies that the dam is stable in
some regards, the study is based on several simplifications and assumptions. While the dam
design has the potential to be recommended for practical use, further studies discussed in
Chapter 5.4 should be considered. In addition, the design methodology and stability limits
are based on Chinese standards, which should be converted or considered when applying the
design internationally.

43
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

44
 References

7 REFERENCES
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Hydro Review. Retrieved from http://cement.org/water/RCC%20Design&Construc-
tion.pdf

Association of State Dam Safety Officials (n.d). What Are The Different Types of Dams and
How Do They Work?. Retrieved 2018-04-24 from https://damsafety.org/different-types-
dams

Australian Geographic. (2011). To dam or not to dam?. Retrieved 2018-04-24 from

http://www.australiangeographic.com.au/topics/science-environment/2011/01/to-dam-
or-not-to-dam

Changsong, S. (2007). Hydraulic Structure (China Water & Power Press). Nanjing: Hohai
University.

COMSOL. (n.d.). The Finite Element Method (FEM). Retrieved 2018-06-13 from
https://www.comsol.com/multiphysics/finite-element-method

D. G. Fredlund and J. Krahn. (1977). Comparison of slope stability methods of analysis. Canada:
University of Saskatchewan.

Dai, W. (2018). Design tasks and background about the gravity dam. Nanjing: Hohai University.

Efunda. (n.d.). Mechanics of materials – Hooke’s Law. Retrieved 2018-09-01 from

https://www.efunda.com/formulae/solid_mechanics/mat_mechanics/hooke.cfm

Electric Light & Power. (2013). The world’s top ten power generation assets. Retrieved 2018-
05-02 from https://www.elp.com/articles/slideshow/2013/07/the-world-s-top-ten-
power-generation-assets/pg003.html

Energy Information Administration. (2015). China. Retrieved from

https://www.eia.gov/beta/international/analysis_includes/countries_long/China/china.pdf

Federal Energy Regulatory Commission. (2016). Gravity Dams. Retrieved 2018-05-07 from
https://www.ferc.gov/industries/hydropower/safety/guidelines/eng-guide/chap3.pdf

FEM Export. (n.d.). FEM Theory, Classification. Retrieved 2018-06-13 from

http://femexpert.com/tutorials/fem-theory/classification/

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Stability Analysis of Non-overflow Section of Concrete Gravity Dams

GEO-SLOPE International Ltd. (2012a). Seepage Modeling with SEEP/W. Retrieved 2018-
06-11 from http://downloads.geo-slope.com/geostudiore-
sources/8/0/6/books/seep%20modeling.pdf?v=8.0.7.6129

GEO-SLOPE International Ltd. (2012b). Stability Modeling with SLOPE/W. Retrieved

2018-06-22 from http://downloads.geo-slope.com/geostudiore-
sources/8/0/6/books/slope%20modeling.pdf?v=8.0.7.6129

GEO-SLOPE International Ltd. (2013). Stress-Deformation Modeling with SIGMA/W. Canada:

Calgary, Alberta.

Google. (n.d.). [Google Maps location for Longtan Dam, Tian’e County, China]. Retrieved
2018-04-25 from https://goo.gl/maps/PuFhYCFXfTN2

International Energy Agency. (2017). World Energy Outlook 2017: China. Retrieved 2018-
04-24, from https://www.iea.org/weo/china/

International Hydropower Association. (2017). Hydropower status report 2017. Retrieved

from https://www.hydropower.org/sites/default/files/publications-docs/2017%20Hydro-
power%20Status%20Report.pdf

Liu, Z. (2016). China’s Carbon Emissions Report 2016. Boston: Harvard Kennedy School.
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tion/China%20Carbon%20Emissions%202016%20final%20web.pdf

Malm, R. (2016). Guideline for FE Analyses of Concrete Dams. Sweden: Energiforsk AB.

Ministry of Water Resources. (1997). Specifications for seismic design of hydraulic structures
(SL203-97). China: Beijing.

Sheng-Hong, C. (2015). Hydraulic Structures (1. ed). China: Wuhan, Wuhan University.

Tata & Howard. (2016). Types of dams. Retrieved 2018-04-24 from https://tataandhow-
ard.com/tag/gravity-dam/

United States Bureau of Reclamation. (1976). Design of Gravity Dams. Retrieved from
https://www.usbr.gov/tsc/techreferences/mands/mands-pdfs/GravityDams.pdf

US Army Corps of Engineers. (1995). Gravity Dam Design. Washington, DC: Department
of the army.

46
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

APPENDIX I – FEM LAYOUT

Figure 8.1. Imported CAD layout for FEM analysis.

a
 Appendix

b
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

APPENDIX II – BOUNDARY CONDITIONS

Uplift Pressure

Table 9.1. Uplift pressure forces for respective Section (Figure 9.1). Values in kN.

Section 1 Section 2 Section 3

Case Normal Critical Normal Critical Normal Critical
P1 8037 8199 2529 2629 520 621
P1,5 10714 10932
P2 4443 4549 2137 2222 400 475
P2,1 3412 3610 1540 1611
P2,2 3252 3575 1290 1343
P2,3 3092 3541 1030 1074
P2,4 2933 3506 770 806
P2,5 2773 3472 520 537
P2,6 2613 3438 250 269
P2,7 2454 3403
P2,8 2294 3369
P2,9 2134 3334
P2,10 1974 3300
P2,11 1815 3266
P2,12 1655 3231
P3 150 320 0 0 0 0
P4 1489 4156

c
 Appendix

Figure 9.2. Visualization of uplift pressures for respective section.

d
Stability Analysis of Non-overflow Section of Concrete Gravity Dams

APPENDIX III – OVERVIEW LAYOUT

Figure 10.3. Elevation arrangement of the dam.

e
 Appendix

Figure 10.4. Horizontal arrangement of the dam.