The Yang Qu dam: Optimization of Zones by Numerical Modelling on this New Type of Dam

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MASTER'S THESIS

The Yang Qu dam

Optimization of Zones by Numerical Modelling on this New Type of Dam

Elias Hammar Daniel Lennartsson

2014

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Master of Science Program

The Yang Qu dam

Optimization of zones by numerical modelling on this new type of dam

Elias Hammar Daniel Lennartsson

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Preface

This thesis is the final 30 ECTS in our Master of Science degree in Civil Engineering with specialization in Mining and Geotechnical Engineering at Luleå University of Technology (LTU) in Luleå, Sweden. One year ago we get in contact with Professor James Yang at Vattenfall. He provided us with the opportunity to write the master thesis in Beijing, China, among people in the frontline of dam construction. We are very grateful for this and want to thank him and Elforsk, which have been involved in the financing of our work.

The thesis was written on the research facilities of the Chinese Institute of Water resources and Hydropower Research (IWHR) in Beijing. Topic was chosen based on our interest and with the guidance of Professor Xu Zeping. Throughout the work we have learned how IWHR conducts their top of the line research within hydropower dams.

We would like to express our gratitude to Professor James Yang at Vattenfall, Elforsk and Professor Xu Zeping at IWHR for giving us this opportunity to be a part in the development of the Chinese hydropower work. We are very grateful for the warm welcome we experienced at IWHR and would beside our supervisor Professor Xu Zeping like to thank Song Xianhui, Liang Jianhui and Wu Junming for their commitment to help us with all our personal matters during our entire stay in China.

Especially Song Xianhui for her patience and hard teaching of the Chinese language.

Beijing, November 2013

Daniel Lennartsson Elias Hammar

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Summary in Swedish

Kina har ett enormt energibehov och för att klara av detta investerar staten i vattenkraft.

Stora vattenkraftsanläggningar har både byggts och planeras. Dammarna byggs ständigt högre. En dammtyp som påvisat god förmåga, både säkerhetsmässigt och kostnadseffektivt är Concrete Face Rockfill Dam (CFRD). När dessa byggs väldig höga, upp mot 300 meter, bildas höga spänningar i betongplattan som kan leda till läckage. På grund av det stora vattendjupet kan reparationsarbete vara svårt och kostsamt. För att komma närmare en lösning på detta har China Institute of Water Resources and Hydropower Research (IWHR) börjat utveckla en helt ny dammtyp som är en kombination av en betongdamm och en CFRD. Denna typ av damm minskar betongplattans längd och således spänningarna samt underlättar reparationsarbete genom en tunnel som konstrueras i betongdammen.

En prototyp med höjden 150 meter, Yang Qu dammen, kommer byggas i Gula floden.

Denna kommer, förutom att producera energi, fungera som ett pilotprojekt inför att i framtiden bygga denna dammtyp upp emot 300 meter hög.

Det kritiska området är övergångszonen mellan de två dammtyperna. För att minimera förskjutningarna i denna zon kommer hårdfyllning användas i närheten av det kritiska området. Detta material utgörs av bergkross och sand som blandas med cement vilket minskar kompressibiliteten jämfört med bergkross.

Yang Qu analyseras i tre typfall med olika fördelning på hårdfyllningen. Detta utförs med hjälp av numerisk modellering baserat på finita elementmetoden. Varje typfall varieras med tre olika värden på E-modulen hos hårdfyllningen. Den maximala tillåtna förskjutningen vinkelrätt mot betongplattans lutning i övergångszonen är 5 cm. Utan hårdfyllningen uppgår denna till 7,7 cm. Alla tre undersökta typfall ger en förskjutning under 5 cm. Ett fall påvisar bäst resultat med minst åtgång av hårdfyllning. Det utgörs av hårdfyllning placerat i samma lutning som betongdammen med bredden 50 m (i tvärsektion).

Maximala totala sättningar i dammkroppen undersöks också. Gränsvärdet för sättningarna är 1,5 % av totala dammhöjden. Alla typfall inklusive det utan hårdfyllning ger totala sättningar långt under detta gränsvärde.

Nyckelord: Yang Qu dammen; CFRD; Hårdfyllning; Duncan´s E-B modell;

Dammanalys; FEM

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Abstract

China has a huge need of energy and to meet this demand the government invest in hydropower. Large hydropower plants are both constructed and planned. The dams will continuously be constructed larger and higher. A dam type that has demonstrated decent performance is the Concrete Face Rockfill Dam (CFRD), both in terms of safety and costs. Very high CFRD´s, up to 300 meters, are subjected to large stresses in the face slab, which can cause leakage. It can be hard and costly to repair due to large water depths. To approach a solution, China Institute of Water Resources and Hydropower Research (IWHR) have start developing a new type of dam. It is a combination of a CFRD and a Concrete Gravity Dam (CGD). This dam type will reduce the length of the face slab, and thus decrease the stresses, and enable maintenance work through a gallery in the CGD.

A dam prototype with a height of 150 meters, Yang Qu dam, will be built in the Yellow River. The goal with this prototype is, besides energy production, to develop the technique and in the future build this new type of dam to the height of 300 meters or more.

The critical area is the transition zone between the CFRD and CGD. To minimize the displacements in this zone hardfill will be used. This material consists of gravel and sand mixed with cement, which reduces the compressibility compared to ordinary rockfill.

Yang Qu is analysed in three cases with different distributions of the hardfill material.

This is performed by numerical modelling based on the finite element method. Each case is varied with three different values of the Young´s modulus of the hardfill. The maximum allowable displacement perpendicular to the slope of the face slab in the transition zone is 5 cm. Without hardfill the displacement reaches 7.7 cm. All three investigated cases give a displacement under 5 cm. One case shows the best results with the least consumption of cemented fill. It consists of hardfill placed in the same slope as the concrete dam with a width of 50 m (in cross section).

Maximum total settlement of the dam body is also being investigated. The maximum allowed settlement is 1.5% of the dam height. All cases including the case without hardfill provides a maximum settlement far below this allowed value.

Keywords: Yang Qu dam; CFRD; Hardfill; Duncan´s E-B model, Dam Analysis, FEM

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List of Figures

Figure 1-1: Cross section of Yang Qu dam ________________________________________________ 2 Figure 1-2: Cross section of the new Yesa dam (modified from Euroestudios, 2013) ________________ 3

Figure 2-1: Zoning of a modern CFRD. (1A) Impervious soil, (1B) Random fill, (2) Processed small rock transition, (3A) Selected small rock, (3B) Quarry run rockfill, about 1 m layers, (3C) Quarry run rockfill, about 1.5 to 2.0 m layers, (T) Central zone. (Modified from Cruz et al., 2009) _____________________ 6 Figure 2-2: Detail of plinth area (modified from Fell et al., 2005) ______________________________ 7

Figure 2-3: Perimeter joint in detail. (1) Face slab, (2) Plinth, (3) Corrugated waterstop, (4) GB-filler, (5) Copper waterstop, (6) GB sealant (modified from Zeping, 2013) ____________________________ 8 Figure 2-4: Stress-strain path of the linear elastic material model _____________________________ 10 Figure 2-5: Stress-strain curve represented as a hyperbola by eq. (3) (Duncan et al., 1980) _________ 12 Figure 2-6: Transformed hyperbolic stress-strain curve (Duncan et al., 1980) ___________________ 13 Figure 2-7: Initial tangent modulus as a function of confining pressure (Duncan et al., 1980) _______ 14 Figure 2-8: Mohr-Coulomb strength relationship (Duncan et al., 1980) ________________________ 15 Figure 2-9: Unloading-reloading in the stress-strain plane (Duncan et al., 1980) _________________ 16

Figure 2-10: Nonlinear and stress-dependent stress-strain and volume change curves (Duncan et al., 1980) _____________________________________________________________________________ 18 Figure 2-11 Variation of bulk modulus with confining pressure (Duncan et al., 1980) _____________ 19

Figure 2-12. Mohr circles for triaxial tests on a shell soil-material of Oroville dam (Duncan et al., 1980) _________________________________________________________________________________ 21 Figure 2-13. Logarithmic variation of friction angle with confining pressure (Duncan et al., 1980) ___ 22

Figure 2-14: Example of initial tangent modulus variation with confining pressure (Duncan et al., 1980) _________________________________________________________________________________ 23 Figure 3-1: Mesh of the finite element model ______________________________________________ 27 Figure 3-2: Element and node points of the joint in the transition zone _________________________ 28

Figure 3-3: Zoning of case 0; (1) Concrete, (2A) Filter, (3A) Transition zone, (3B) Main rockfill, (3C) Less compacted rockfill, (4) Joint _______________________________________________________ 30 Figure 3-4: Zoning of case 1; (1) Concrete, (2A) Filter, (3A) Transition zone, (3B) Main rockfill, (3C) Less compacted rockfill, (4) Joint, (5) Hardfill ____________________________________________ 30

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Figure 3-5: Zoning of case 2; (1) Concrete, (2A) Filter, (3A) Transition zone, (3B) Main rockfill, (3C) Less compacted rockfill, (4) Joint, (5) Hardfill _____________________________________________ 31

Figure 3-6: Zoning of case 3; (1) Concrete, (2A) Filter, (3A) Transition zone, (3B) Main rockfill, (3C) Less compacted rockfill, (4) Joint, (5) Hardfill _____________________________________________ 31 Figure 4-1: Displacement as a function of modulus number for each case _______________________ 37 Figure 4-2: Maximum settlement as a function of modulus number for each case _________________ 38 Figure 4-3: Location of maximum settlement ______________________________________________ 38

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Symbols

Latin characters

B Bulk modulus

c Cohesion

E Young´s modulus

Ei Initial tangent modulus Et Tangent modulus

Eur Unloading – reloading modulus

K Modulus number

Kur Unloading-reloading modulus number

n Modulus exponent

pa Atmospheric pressure Rf Failure ratio

S Displacement

Greek characters

Δσx Normal stress increment Δσy Normal stress increment Δτxy Shear stress increment Δεx Normal strain increment Δεy Normal strain increment Δγxy Shear strain increment

ε Strain

(σ1 – σ3) Stress difference

(σ1 – σ3)ult Ultimate stress difference σ1 Major principal stress σ3 Minor principal stress (σ1 – σ3)f Stress difference at failure

ϕ Friction angle

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1 Introduction

1.1 General background

China is one of the most growing economies in the world and the demand for energy is increasing rapidly. To meet this need, the government of China is establishing nuclear power plants, coal plants and hydropower plants. Hydropower plants are an effective and in a perspective of pollutions, an environmentally friendly way to produce electricity.

The total hydropower capacity in China was 2012 approximately 860 TWh. This makes China to the largest hydropower producer in the world (REVE, 2013). This stands for a large part of the country’s total energy production. Many hydropower dams are also in a planning stage and under construction (Zeping, 2013).

This progress requires continuous development, which occurs constantly in China.

Higher hydropower dams need to be constructed to meet this energy demand. Higher dams results in higher stresses and larger deformations, hence increased need for more investigations and more advanced technology. Two of these high dams in design phase are Cihaxia dam with a dam height of 253 meters and Rumei dam, 340 meters (Cruz et al., 2009). The chosen dam type for Cihaxia are Concrete Face Rockfill Dam (CFRD) and the dam type of Rumei is not decided yet (Zeping, 2013).

CFRD´s have proven to be an economical, safe and effective option. Shuibuya dam with a height of 233 meters was completed 2009. This dam has proved that a CFRD of this height has a satisfactory performance. In an economical perspective, some of the main construction material, i.e. rockfill or earthfill, can be taken from the site when excavating the dam. A CFRD can be a good alternative to an ordinary embankment dam in sites where there is a lack of soil material for an impervious core. CFRD’s can be founded on alluvium soils with a good result. This is an advantage compared to the dam types that need to be founded on bedrock, i.e. when a complete removal of the soil overburden is necessary. The slopes can be constructed relatively steep, which saves material and operational costs. (Cruz et al., 2009).

Very high CFRD´s are subjected to high stresses and that will lead to deformations in the dam body. The crucial characteristic of this type of dam is the ability of the concrete face to withstand water penetration. To maintain this water resistance of the concrete it has to be as few cracks in the concrete as possible. Deflections in the face slab can produce cracks with leakage as a result. (Cruz et al., 2009).

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When the cracks are developed, it can be hard to repair. The water depth is too large for cheap and easy repair actions, so the reservoir level needs to be lowered, with a high cost as a result. There will also be high axial stresses along the concrete face slab because of the high pressure from its own weight. Lowering the slopes of the dam, which will decrease the pressure on the face slab and counteract spalling failure, can solve this problem. With lower inclination of the slopes there will however be increased material costs of the rockfill and the concrete. (Zeping, 2013).

To deal with these problems, a new type of dam is under development by China Institute of Water Resources and Hydropower Research (IWHR). It is a combined dam construction of a Concrete Gravity Dam (CGD) and a CFRD; the proposed layout is presented in Figure 1-1. This type of construction will reduce the length of the face slab and therefore reduce the axial stresses. It will also be possible to install a gallery in the CGD, which will lead to more effective surveillance and maintenance of the dam construction. (Zeping, 2013).

Figure 1-1: Cross section of Yang Qu dam

The idea of this type of combined dam origins in one earlier dam named Yesa dam, located in the Aragón River in north part of Spain. Yesa dam was first a CGD with a dam height of 48 meters, impounded for the first time 1959. It was 1983 decided to increase the dam height, due to increased need for drinking water. The construction phase begun 2001 and the heightening were performed by an added concrete face slab and rockfill to increase the reservoir capacity. The height of the new combined dam is 117 meters. Figure 1-2 shows the new Yesa dam. (Garcia et al., 2006).

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Figure 1-2: Cross section of the new Yesa dam (modified from Euroestudios, 2013)

It has been decided to build a prototype of this new type of dam in Yellow river, China.

This dam will have a height of 150 meters and it is named Yang Qu. The goal with this prototype is, beside energy production, to develop the technique. It is planned to apply this dam design to build even higher dams (300 meters or more) in a sustainable and effective way, both economically and structurally. (Zeping, 2013).

1.2 Objective and purpose

The general purpose with this work is to gain more options and possibilities to construct large dams and by that produce more energy to the fast growing country. Through investigation of this new design of dam, with focus on a detail in the design phase; this work is a minor part on the long way for China to solve the issue of energy.

This work is focused to investigate displacements of the concrete face slab on the CFRD near the transition zone, where the construction goes over from concrete in the CGD to rockfill in the CFRD.

The objective of this work is to show on what grounds further design analyse procedures should be conducted. The design of the 150 meters high Yang Qu dam is investigated with different zoning of dam materials. The expected displacement in the transition zone between the crest of the concrete gravity dam and the face slab of the CFRD need to be kept below an accepted maximum. Furthermore these results can be used for development of this new type of dam with heights over 300 meters.

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Question to investigate:

• How should the cross section of this new type of dam be designed to limit the displacement in the transition zone between the crest of the CGD and the face slab of the CFRD to a tolerated level?

Or:

• How should Hardfill material be divided with different Young’s modulus into the dam body of the new 150 meters high Yang Qu dam to achieve tolerated displacements in the critical zone?

1.3 Limitations

This work is written for IWHR´s department of hydraulic engineering, Beijing, China. It only treats structural stability issues of this new type of dam. The foundation of this type of dam is not taken into consideration. When optimizing the structure there is two major parts in this case, the CGD construction and the CFRD construction. This thesis focuses on the CFRD part. The work is based on data obtained from analyses with the FEM and focuses on the displacements along the upstream side of the face slab in the area for the transition zone between the CGD and CFRD. This thesis does not present studies on the design of the concrete or seepage prevention.

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2 Theoretical background

2.1 What is a CFRD?

A Concrete Face Rockfill Dam (CFRD) contains of different zones of materials with specific functions in the dam construction. Cooke and Sherard developed the basic designs and presented the result in papers 1987 and they are now standard for CFRD´s (Cruz et al., 2009). Figure 2-1 shows the zoning. The main zones and parts in a CFRD is listed here:

• The concrete face slab is situated on the upstream side and serves as the impervious element. It consists of 0.25 – 0.6 m thick reinforced concrete (Fell et al., 2005).

• Zone 1 – Impervious

Consists of compacted soil, silt is preferable, which is impervious (Zone 1A). It will serve as a cover for the perimetric joints in the lower part of the face slab. It will thus seal potential cracks in the concrete face. Zone 1A can be supported by a less costly material (Zone 1B). CFRD’s without Zone 1 have been successfully constructed, therefore it is not always necessary, but it is useful if problems develop. It is an economical matter but it is recommended to have this zone in high CFRD´s.

• Zone 2 – Filter/transition zone

This layer is a filter and transition zone between the face slab and the rockfill. It consists of fine graded soil material, in the range from sand-sized particles to fines. It has two purposes; to make a stable and even surface for the face slab and a semi-impervious layer with a filtering function. Since the material is well- graded, it will prevent any large leaking.

• Zone 3 – The main rockfill

Zone 3A is a transition zone between Zone 2 and 3B and consists of finer rockfill. Zone 3B will transfer most of the water load into the foundation. It consists of well-compacted rockfill and need to have low compressibility. Large settlements of Zone 3B should be avoided because it can make the face slab deflect with cracks and leakage as a result. Zone 3C is not subjected to the same amount of load as Zone 3B. To save compaction and material costs, this zone can be allowed to have higher compressibility.

• Zone T – Central zone

This “dead zone” is situated between Zone 3B and 3C. To save costs, it can be constructed with poorer rockfill quality since the area is not subjected to large loads compared to the other zones.

(Cruz et al., 2009)

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Figure 2-1: Zoning of a modern CFRD. (1A) Impervious soil, (1B) Random fill, (2) Processed small rock transition, (3A) Selected small rock, (3B) Quarry run rockfill, about 1 m layers, (3C) Quarry run rockfill, about 1.5 to 2.0 m layers, (T) Central zone. (Modified from Cruz et al., 2009)

The maximum tolerated settlement in a CFRD is 1.5 % of the total dam height based on practical experience of high CFRD´s. (Zeping, 2013)

2.1.1 Plinth and perimetric joint

The toe of the dam on the upstream side is a critical and important area of a CFRD. A plinth (also called toe slab) is constructed in modern CFRD´s. Upstream of the plinth the water reservoir is situated and downstream the rockfill. The purpose of the plinth is to hold the face slab in position. (Fell et al., 2005).

To make the area between the face slab and plinth watertight and sustainable during settlement and movement of the face slab, especially during water impoundment, a perimetric joint is constructed (Cruz et al., 2009). A simple figure of the plinth, the perimetric joint and the face slab is presented in Figure 2-2.

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Figure 2-2: Detail of plinth area (modified from Fell et al., 2005)

The perimetric joint is created and constructed between the plinth and the face slab with a material that allows small movements between this two construction elements.

Without the joint it would have been cracks and thereby leakage in this zone. To satisfy the requirements of waterproofness, the type of perimetric joint for this type of dam has been chosen to be the one showed in Figure 2-3. (Zeping, 2013)

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Figure 2-3: Perimeter joint in detail. (1) Face slab, (2) Plinth, (3) Corrugated waterstop, (4) GB-filler, (5) Copper waterstop, (6) GB sealant (modified from Zeping, 2013)

This type of joint is filled with GB-filler that is an Ethylene-propylene-diene monomer (EPDM), a high-density rubber that is very durable. It is sealed on top with a corrugated waterstop, (EPDM). In the bottom a combined copper waterstop and GB sealant prevents the water to flow in to the dam body. (Zeping, 2013).

This construction type of joint between plinth and the face slab is relatively waterproof.

Like any other type of connection between these two elements in the CFRD, it is a sensibility of movement in the joint construction. To have this type of joint functioning in a good way without leaking too much, displacements perpendicular to the face slab need to be kept beneath five centimetres. (Zeping, 2013).

2.2 Hardfill

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The hardfill can be considered as a cohesive-frictional material. It is complex to obtain a perfect constitutive model for this material, and there is a need for further studies.

(Zeping, 2013).

Hardfill can be described with a linear elastic model from a concrete viewpoint and with a nonlinear stress-strain relationship from a geotechnical viewpoint. A combined model with both rockfill- and cementation characteristics has been developed out from uniaxial compression tests and triaxial tests. It has been discovered that the stress-strain behaviour is very dependent of the age of the material. (MengXi et al., 2011).

The compressibility of hardfill is much lower than for regular rockfill, due to the cement added. This will lead to smaller strains as well. The strains in the hardfill material do not lead to failure, this because that the material is defined as a linear elastic material.

(MengXi et al., 2011).

In this thesis, the CSG material is assumed to behave as a linear elastic material. The design value of Young’s modulus number is according to tests performed, reached after 25 days (Cai et al., 2012).

2.3 Linear-elastic material model

The concrete- and hardfill material in the finite element model of the dam are defined by the linear elastic material behaviour. This model is based on Hooke´s law. The Young’s modulus is defined as

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where 𝜎 is the axial stress and 𝜀 the axial strain. The linear elastic material model has no failure criterion, which means that the Hardfill only going to deform elastically and never goes to failure. The behaviour of the model is illustrated in Figure 2-4.

E =σ ε

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Figure 2-4: Stress-strain path of the linear elastic material model

2.4 Duncan’s E-B model

The Duncan - Cheng material model is a method to simulate soil behaviour. It assumes a hyperbolic stress-strain relation and was developed based on data from triaxial testing in the decade of 1970 (Duncan and Cheng, 1970).

The original material model from 1970 developed by Duncan and Cheng assumes a constant Poisson’s ratio, but in this case the revised model is used, Duncan’s E-B model. This updated model contains a variation of Poisson’s ratio by means of a stress- dependent bulk modulus. (Duncan et al., 1980)

The model is very suitable for modelling soil material with large differential stresses. In very high rockfill dams, the stresses in the bottom are much larger then stresses near the slopes. This leads to very different strength parameters for the same material depending on where in the dam construction it is situated. (Duncan et al., 1980)

The hyperbolic stress-strain relationship is a nonlinear incremental material model to make analyses of soil behaviour in terms of deformation. Each increment in deformation is assumed to be linear elastic with Hook´s law applied. This can, for plane strain condition, be expressed by

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where Δ𝜎_! and Δ𝜎_! are the normal stress increments, Δ𝜏_!" is the shear stress increment, B is the Bulk modulus, E is the Young´s modulus, Δ𝜀_! and Δ𝜀_! are the normal strain increments and Δ𝛾_!" is the shear strain increment. (Duncan et al., 1980)

With (2) it is possible to model three characteristics of stress-strain behaviour;

nonlinearity, stress dependency and inelasticity, which will be described here. This is performed by varying the values of bulk- and Young´s modulus respectively, as the stresses in the soil mass will vary. (Duncan et al., 1980).

2.4.1 Nonlinearity

It has been discovered that stress-strain curves can be approximated with acceptable accuracy with hyperbolas. This hyperbola is gained by

(3)

where 𝜎_! is the major principal stress, 𝜎_! is the minor principal stress, 𝜀 is the strain, 𝐸_! is the initial tangent modulus and the initial slope in the stress-strain hyperbola curve and (σ₁−σ₃)_ult is the ultimate stress difference, and it is closely related to soil strength.

The hyperbola stress-strain curve is showed in Figure 2-5. (Duncan et al., 1980).

σ₁−σ₃

( )

⁼ ₁ ^ε

E_i + ε (σ1−σ3)_ult

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Figure 2-5: Stress-strain curve represented as a hyperbola by eq. (3) (Duncan et al., 1980)

To evaluate 𝐸_! and (σ₁−σ₃)_ult, the hyperbolic expression in (3) can be transformed to a linear relationship with

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From test data, where ε^, σ₁and σ₃ are evaluated, 𝐸_! and (σ₁−σ₃)_ult can be determined. When test data are plotted in the transformed graph, they will diverge and not appear in a perfect straight line. Through investigation of many soil samples one method has showed to give a good match. This method is conducted by drawing a straight line between the points where 70 % and 90 % of the strength is mobilized. The hyperbolic expression transformed to a linear relationship is showed in Figure 2-6.

(Duncan et al., 1980).

ε

(σ₁−σ₃)= 1

E_i + ε (σ₁−σ₃)_ult

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Figure 2-6: Transformed hyperbolic stress-strain curve (Duncan et al., 1980)

2.4.2 Stress dependency

The stress dependent stress-strain behaviour is characterized by varying 𝐸_! and (σ₁−σ₃)_ultwith different confining pressure, σ₃. The variation of initial tangent modulus and confining pressure can be evaluated by

(5)

where 𝐸_! is the initial tangent modulus, K is the dimensionless modulus number, n is the dimensionless modulus exponent and pa is the atmospheric pressure. The atmospheric pressure pa can be of different units, and by that it is possible to convert the system of units to another. Figure 2-7 shows the corresponding plot of (5) with logarithmic axes, which forms a straight line. (Duncan et al., 1980).

E_i= Kpa

σ3

p_a

!

"

# $

%&

n

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Figure 2-7: Initial tangent modulus as a function of confining pressure (Duncan et al., 1980)

The relation between the ultimate stress difference and stress difference at failure can be defined by

(6) where is the ultimate stress difference, is the stress difference at failure and Rf is the failure ratio. The failure ratio is always less than 1 because the stress difference at failure is always smaller than the ultimate. The value differs between soils; Rf is for most of them in the interval between 0.5 and 0.9. (Duncan et al., 1980).

When the stress difference at failure can be obtained, the classic Mohr-Coulomb strength relationship can be used to relate σ₃ to , with the following expression

(σ₁−σ₃)_ult (σ₁−σ₃)_f

(σ₁−σ3)_f σ₁−σ₃

( )

f = R_f

(

σ₁−σ₃

)

ult

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where c is the cohesion and

φ

is the friction angle. Figure 2-8 shows the Mohr- Coulomb strength relationship. (Duncan et al., 1980).

Figure 2-8: Mohr-Coulomb strength relationship (Duncan et al., 1980)

The slope, i.e. the derivative with respect to ε in (3), at a certain point of the stress-strain hyperbola curve is the tangent modulus, E_t. The derivative function with (5), (6) and (7) inserted will give the following expression for the tangent modulus

(8)

This expression enables calculation of the tangent modulus value for any stress conditions with different values of σ₃ and (σ₁−σ₃) if the factors K, n, c,

φ

and R_f are determined, see section 2.4.7. (Duncan et al., 1980).

E_t = 1−R_f(1− sinφ)(σ1−σ3) 2 ccosφ+ 2σ₃sinφ

"

#$ %

&

'

2

Kp_a σ3

p_a ( )* +

,-

n

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2.4.3 Inelasticity

The inelastic behaviour is characterized by use of different modulus values in terms of loading and unloading states. If a triaxial soil sample is subjected to first loading and then unloading, the stress-strain curve will be steeper when unloading, see Figure 2-9. If the sample thereafter is loaded again, it will almost follow the earlier unloading curve.

The resulting strain from the primary loading is only to some extent recovered during unloading. (Duncan et al., 1980).

This is inelastic behaviour. The curves of unloading and reloading will not be linear and has a small difference between each other but it is considered sufficient enough to approximate them together to a line, as in Figure 2-9. The expression for the unloading- reloading modulus is

(9)

where 𝐸_!" is the unloading-reloading modulus and 𝐾_!" is the dimensionless unloading- reloading modulus number. Equation (9) has the same form as the initial tangent modulus (5) and is in relation to the confining pressure, 𝜎_!. 𝐾_!" is always greater or equal to K (5), the initial modulus number of primary loading. The value of the exponent n is similar to the one of initial loading, they are in this model assumed to be equivalent. (Duncan et al., 1980).

E_ur = Kurp_a σ3

p_a

!

"

# $

%&

n

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2.4.4 Nonlinear volume change accounted for using constant bulk modulus

The theory of elasticity is defining the value of bulk modulus by

B =Δσ1+ Δσ2+ Δσ3

3Δεv

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where B is the bulk modulus; Δ𝜎_!, Δ𝜎_! and Δ𝜎_! are the changes in principal stresses and Δ𝜀_! is the corresponding change in volumetric strain. When the deviatoric stresses increases and the confining pressure is held constant, like in conventional triaxial testing, the bulk modulus may be expressed as

B = Δσ1

3Δεv

=(σ1−σ3) 3εv

(11)

For a conventional triaxial test, the bulk modulus can be calculated using the value of the deviatoric stress corresponding to any point on the stress-strain curve in Figure 2-10, such as point (A), and the corresponding point (A’) on the volume change curve. (Duncan et al., 1980).

(σ₁−σ₃)

(σ₁−σ3)

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Figure 2-10: Nonlinear and stress-dependent stress-strain and volume change curves (Duncan et al., 1980)

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70 % of the strength should be mobilized and in that point the corresponding point on the volume change curve will be selected. If the volume change curve does not reach a horizontal tangent prior to the point of 70 % strength on the corresponding stress-strain curve, the point where the volume change curve becomes horizontal will be chosen.

2.4.5 Variation of bulk modulus with confining pressure

When the bulk modulus is examined for a soil as a function of the confining pressure σ3, the modulus will usually increase with increasing confining pressure. In Figure 2-11 this is shown and can be approximated by

(12)

where 𝐾_! is the bulk modulus number and m the bulk modulus exponent, both are dimensionless. The atmospheric pressure 𝑃_! is expressed in the same units as the bulk modulus (B) and the minor principal stress (𝜎_!). (Duncan et al., 1980).

Figure 2-11 Variation of bulk modulus with confining pressure (Duncan et al., 1980)

B = KbP_a σ3

P_a

!

"

# $

%&

m

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2.4.6 Restrictions on the range of values of bulk modulus

In finite element programs the values of νt (tangent Poisson’s ratio), may be restricted to positive values. The value of νt may also be restricted to values equal or less then 0.49.

In the case when the value of B approaches 𝐸_! 3 in the equation υ_t = 1 2 − E_t 6B there will be a problem. Using B = E_t 3 in cases where (12) indicates lower values can solve this problem. Similarly where (12) indicates higher values can B =17E_t be used.

2.4.7 Evaluation of strength parameters

In the hyperbolic stress-strain relationship, nine parameters need to be determined. They can all be evaluated from a conventional triaxial test (both unconsolidated undrained and drained tests) and the progress is described in this chapter.

Before performing the test it is important to select suitable test conditions. If the test should be performed to cohesionless material such as sand, gravel and rockfill it is important that the tested material will have the same density, water content and drainage conditions as in reality. The tests should also be performed with corresponding stress conditions, compared to the field. (Duncan et al., 1980).

2.4.7.1 Friction angle for cohesionless soils

For all soils, especially cohesionless soils such as sand, gravel and rockfill the Mohr- Coulomb envelope will not be perfectly linear; some curvature will be expected when the soil is subjected to different levels of stress. It can be some difficulties to evaluate one friction angle (ϕ) to represent the whole soil mass of a dam. For example, in the centre near the bottom of the dam the soil suffers from much higher stresses compared to areas near the surface. The friction angle will be higher in the zone near the surface and less where the stresses are higher in the bottom. Figure 2-12 presents three triaxial tests on a shell material from Oroville dam plotted as Mohr-circles in a τ-σ plane, note that the friction angle is decreased with increased stresses.

One method is to let the friction angle of the material vary with the confining pressure (𝜎_!), which can be evaluated from each triaxial test. Since it is a cohesionless soil, the envelope for each circle can be assumed to intersect the origin of stress. The friction angle can be obtained either by the envelope or by

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Figure 2-12. Mohr circles for triaxial tests on a shell soil-material of Oroville dam (Duncan et al., 1980)

It has been discovered that the value of ϕ decreases in relation to the logarithmic value of 𝜎_!. The relationship is expressed

(14)

Where φ₀ is defined as the value of φ when σ₃ is equal to the atmospheric pressure p_a and Δφ is the reduction in φ with a ten-fold increase of σ₃. Figure 2-13 shows the relation in a graph with the same example as before, the Oroville dam. (Duncan et al., 1980). The values of φ₀ and Δφ are tabulated for many different cohesionless soils and can be found in Duncan et. al (1980).

φ=φ₀− Δφlog₁₀ σ₃ p_a

#

$% &

'(

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Figure 2-13. Logarithmic variation of friction angle with confining pressure (Duncan et al., 1980)

2.4.7.2 Modulus parameters, K and n

To calculate the modulus parameters, K and n, there is two steps to follow. First of all the value of Ei has to be calculated (4). The second step is to plot the value of Ei/Pa

against the corresponding value of σ3/Pa on a logarithmic scale, see Figure 2-7. The parameter n can be determined from the slope of this curve or numerically by

n = (15)

Δ log E_i P_a

"

#$ %

&

' Δ log"σ3

$ %

'

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Figure 2-14: Example of initial tangent modulus variation with confining pressure (Duncan et al., 1980)

2.4.7.3 Unloading-Reloading modulus number, Kur

In most cases, where sufficient data has been available to check, it has been found an accurate assumption for Kur. It can thus be determined that the unloading-reloading exponent, n is the same as for the modulus exponent for primary loading. This simplifies the evaluation of Kur. (Duncan et al., 1980).

With help of the unloading – reloading curve in Figure 2-9, the Modulus number of unloading – reloading can be determined. This is made by choosing the most appropriate inclination of this line in the diagram. When both the modulus exponent, n and the unloading – reloading modulus, Eur is known, the unloading – reloading modulus number, Kur can be calculated by

(16) K_ur = E_ur

P !σ₃

# $

&

n

(39)

where σ3 is the confining pressure during unloading. There is not so common that data for unloading is available. In these cases it is necessary to assume the value of Kur. It has been shown on available data that Kur always is greater than the value of K. The ratio between this two has shown to vary with . The lower value is for dense soils and the higher for softer soils. (Duncan et al., 1980).

2.4.7.4 Bulk modulus parameters, Kb and m

To determine the two bulk modulus parameters, Kb and m, there is also here two steps with similarities to the Young’s modulus parameter evaluation. First the value of B has to be calculated according to section 2.4.4. The second step is to plot the values of B against σ3 on a logarithmic scale, see Figure 2-11. On the same way as in section 2.4.7.2, the slope of the trend line can be evaluated to get m and check where the trend line passes the confining stress value to get Kb. (Duncan et al., 1980).

2.4.8 Summary of hyperbolic parameters

Totally, the hyperbolic stress-strain relationship is dependent of nine parameters. The parameters and their function are presented in Table 1.

Table 1: Summary of parameters (Modified from Duncan et al., 1980)

Parameter Name Function

K Modulus number These are related to initial

tangent modulus (Ei) and unloading-reloading modulus (Eur) by σ3

Kur Unloading-reloading modulus number

n Modulus exponent

c Cohesion These are related to (σ1 – σ3)f

with Mohr-Coulomb failure criterion by σ3

ϕ Friction angle

Δϕ Reduction in friction angle for a ten-fold increase in σ3

Rf Failure ratio Relates (σ1 – σ3)f by (σ1 – σ3)ult

Kb Bulk modulus number These are related to the bulk 1.2 < K_ur/ K < 3

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the material. The model is efficient to foresee nonlinear relationships between loads and movements in stable earth masses. It is possible to analyse the behaviour to local failure states in some elements but not if the earth mass is controlled by elements that already have been failed. The result is then not longer reliable. (Duncan et al., 1980).

The hyperbolic relationships do not account for volume changes due to changes in shear stress. It can therefore be limited in the accuracy to model deformations in dilatant soils, such as dense sand with low confining pressure. (Duncan et al., 1980).

The parameters included in the model are only describing the soil under special conditions, decided by the type of tests performed. The parameter values are dependent on the density, water content, stress conditions and drainage conditions. It is therefore important to get parameter values from tests that are closely related to the conditions in the field. (Duncan et al., 1980).

2.5 Numerical modelling with FEM

2.5.1 General about FEM

The finite element method (FEM) is a mathematical method that was developed to find numerical solutions to complicated differential equations. This method is very effective to describe complex geometries and boundary conditions; it is often used for problems with strength character.

The Finite element theory is dividing the continuum that is representing the geometry of the model, into smaller volume elements. Three- and four sided elements are used for the two-dimensional analyse, these elements are linked with nodes, called node points.

Every node has a certain number of degrees of freedom that corresponds to discrete values of the unknown parameters, which will be solved. The degree of freedom can, for example, describe the displacement of the node. To calculate the displacement of the elements, an interpolation is done between the nodes in the element. For complex models, there is a need for more nodes in each element. This is though slowing down the computation because of the large amount of data. (Ottosen and Petersson, 1992).

To describe how soil is acting when the stress state is changing, a soil model need to be used. The soil model is a mathematical description of how the soil behaves. When boundary conditions are defined, a given value for the degrees of freedom in the nodes that represent the volume elements on the boundary are set. By help of the known solutions of the boundary equations, the rest of the differential equations can be solved.

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For example, displacements can be computed. For a complete mathematical description see (Ottosen and Petersson, 1992).

2.5.2 Computer software

The numerical analysis is performed by a FE-computer software named SDAP2D that is developed by Professor Xu Zeping and his colleagues at IWHR. It is a DOS-based program with only text in the output file and no graphic presentation. The program is based on FEADAM software. The program has been developed for determination of the stress-strain and volume change parameters using least square procedures for fitting the curves illustrated in Figure 2-13 and Figure 2-14 (Duncan et al., 1980).

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3 Methodology

The construction method of the new type of dam is evaluated with the aim to find a first design idea with help of the FEM theory mentioned. This is done, by studying the joint and the movement of the face slab connected to it. All material parameters used in the FE model have been obtained from previous laboratory works by IWHR, Beijing, China.

The Duncan’s E-B material model and the linear elastic material model presented in chapter 2 are applied to the 2-dimensional FEM software SDAP2D. The mesh consists of 916 three- and four sided elements linked with 964 node points. It is optimized by IWHR to have good enough accuracy for this type of first analyse of the dam design.

The mesh for the model is shown in Figure 3-1.

Figure 3-1: Mesh of the finite element model

Essentially, the modelling aims to find where in the dam body it is necessary to have the hardfill, to avoid large displacements in the face slab near the plinth and the transition zone. Settlements, i.e. total vertical deformation in the dam body are also taken into consideration. The maximum tolerated vertical deformation in the Yang Qu dam is set to the same as for CFRD´s i.e. 1.5 % of the dam height.

Four different cases are investigated to find the most appropriate design. One case serves as a reference case where no hardfill is used. In the three cases with hardfill, the Young´s modulus of the hardfill is varied in three ways for each case.

To evaluate the displacements in the transition zone in a satisfying way, an element for the joint has to be introduced in the FE model. This joint is, as earlier mentioned, placed

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set in a way that allows the concrete face slab to move free from the plinth in all ways except compression (Zeping, 2013).

This is done by giving material parameters to the joint similar to concrete in compression. However, the failure ratio of the joint is set to be negative. This means that the joint has very low strength in tension and shearing, the face slab will then be allowed to move in harmony with the soil material’s movement.

To evaluate which case that will be most appropriate for this type of construction, the joint-element is investigated. This element is connected with node points 52, 53, 901 and 902 in the bottom of the concrete face slab, Figure 3-2: Element and node points of the joint . Node point number 52 and 53 are points on the concrete plinth and will not have any significant effect because they are stable on the CGD.

Figure 3-2: Element and node points of the joint in the transition zone

From the model, horizontal (x-axis) and vertical displacements (y-axis) are gathered.

These displacements are converted to a direction perpendicular to the upstream slope angle, S901, S902 and are then compared for the different cases. A more detailed calculation procedure is presented in Appendix A. The calculations are done in two

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3.1 Material parameters

The material parameters for the rockfill are, as earlier mentioned, evaluated on the laboratory department of the IWHR, Beijing, China. For the hardfill material, the parameters are based on old research by IWHR. For a complete list of the material parameter values used in the finite element programme, see Appendix B. Every material except the hardfill are set to have constant parameters through out the cases. Hardfill is displayed as number 5 in Table 2.

In Table 2, the different modulus numbers and Young’s modulus are shown and for which zone it is applied on. For zone 5, Young’s modulus is varied in three different ways. The location of the zones are found in Figure 3-3 (case 0), Figure 3-4 (case 1), Figure 3-5 (case 2) and Figure 3-6 (case 3) respectively.

Table 2: Modulus numbers for the different zones

Modulus number

Zone K Kur Kb

2A 1023.3 2046.6 500.0

3A 1438.6 2877.2 7915

3B 1412.5 2825.0 772.0

3C 571.0 1142.0 229.0

Young’s modulus, E (GPa)

1 28

12 3 4 5 4

5*

*For the hardfill material the Young’s modulus is varied in three different ways 3.2 Case 0

Case 0 is a case where no hardfill is applied. It is used as a reference case, to see how the dam will behave without hardfill. The zoning is showed in Figure 3-3.

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Figure 3-3: Zoning of case 0; (1) Concrete, (2A) Filter, (3A) Transition zone, (3B) Main rockfill, (3C) Less compacted rockfill, (4) Joint

3.3 Case 1

This case consists of hardfill material placed horizontally over the whole dam section.

The material is placed in a 30-meter thick layer in the bottom of the dam, Figure 3-4.

Figure 3-4: Zoning of case 1; (1) Concrete, (2A) Filter, (3A) Transition zone, (3B) Main rockfill, (3C) Less compacted rockfill, (4) Joint, (5) Hardfill

3.4 Case 2

Case 2 consists of hardfill material placed in a 40 meters thick layer over the half of the dam section, through the dam centre axis, illustrated in Figure 3-5.

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3.5 Case 3

To reduce the amount of hardfill, the material can be placed in smaller zones near the sensitive area around the joint. The zones will have the hardfill thickness of 40 meters and distributed in the same inclination as the downstream slope of the CGD. Case 3 is divided into two parts, a and b, where the amount of hardfill is increased in two steps as Figure 3-6 shows. The width of the hardfill in Case 3a is 32 meters and in Case 3b 50 meters.

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4 Result and analysis

The modelling cases described in section 3 produce large amount of data, which will not be fully presented here due to the vast amount of space it will require. Instead, the most significant data will be presented. In the last section of this chapter, a diagram shows a comparison between the different cases and material properties.

The following subsections present the deformations caused by the hydrostatic loads from the initiation of the reservoir impoundment i.e. the dam is filled. All displacements presented in this chapter are the displacements of the node points in the transition zone, which is earlier described in section 3.

The maximum settlement in the dam body is also investigated for each case. Figure 4-3 is a schematic figure of where these settlements are located in the dam. The maximum estimated settlements are far from reaching the maximum tolerated, which is 2.25 meter (1.5 % of the dam height) in any case.

A complete table with stresses (in both the CGD and CFRD), horizontal displacements and settlements in the after construction- and after reservoir impoundment state can be found in Appendix C.

4.1 Case 0

The displacement of Case 0 is in node point 901 7.72 cm and in point 902 7.73 cm. The amount of displacements is exceeding the tolerated value of 5 cm, which is an expected result due to the dam design. According to this behaviour, hardfill need to be introduced in the model.

Maximum settlement is 86.2 cm and is located in node point 668. Case 0 will naturally give the largest settlement, but it can still be considered as relatively small.

4.2 Case 1

The displacements in the transition zone of Case 1 are presented in Table 3. The Young´s modulus of the hardfill has a great impact on the displacement. No significant difference occurs between modulus of 3 and 4 GPa but when it is increased to 5 GPa, the displacement is reduced with around 56 %. The maximum allowed displacement of 5 cm is not reached.

This case will give the lowest level of settlements, around 70 cm for all three values of Young´s modulus, as Table 3 shows.

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Table 3: Displacements and settlements of Case 1

Young´s Modulus*

(GPa)

Node Point Displacement (cm) Maximum settlement (cm)

3

901 4.60 -

902 4.60 -

716 - 70.6

4

901 4.60 -

902 4.59 -

716 - 70.5

5

901 2.06 -

902 2.06 -

716 - 70.4

*Young´s modulus of hardfill 4.3 Case 2

Case 2 indicate a very small amount of displacement, only around 2 cm as Table 4 shows. The amount of settlement is low, i.e. some optimization could be done with still adequate results with less amount of hardfill.

The maximum settlements are presented in Table 4, which shows a slight increase with 10 cm compared to case 1.

Table 4: Displacements and settlements of Case 2

Young´s Modulus*

(GPa)

3

901 2.03 -

902 2.03 -

668 - 80.8

4

901 1.95 -

902 1.95 -

668 - 80.8

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4.4 Case 3

The displacements and maximum settlements of the to subcases, Case 3a and 3b, are presented in Table 5 and Table 6 respectively. Between this two cases large differences are detected. Case 3a exhibits relatively large displacements, it almost reaches the maximum allowed amount. When the zone of hardfill is extended as in Case 3b, it will reduce the displacements significantly.

Table 5: Displacements and settlements of Case 3a

Young´s modulus*

(GPa)

3

901 4.94 -

902 4.94 -

668 - 85.8

4

901 4.86 -

902 4.85 -

668 - 85.8

5

901 4.80 -

902 4.80 -

668 - 85.8

*Young´s modulus of hardfill

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Table 6: Displacements and settlements of Case 3b

Young´s modulus*

(GPa)

3

901 2.33 -

902 2.33 -

668 - 85.4

4

901 2.24 -

902 2.24 -

668 - 85.4

5

901 2.22 -

902 2.21 -

668 - 85.4

*Young´s modulus of hardfill

4.5 Comparison

The material models for rockfill and hardfill are as earlier mentioned different. And the compressibility parameters are expressed in Young’s modulus for the hardfill and in modulus number K for the rockfill. In this section the two comparison figures has only the Young’s modulus on the x-axis. This means that the point of the reference value will not be exact in the given position in Figure 4-1 and Figure 4-2. The reference value symbolizes the displacements in the transition zone without hardfill material. That depends on the earlier mentioned variation in strength parameters because of the big stress differences of the rockfill in the dam body. The purpose of this section is though to show how the displacements and the settlements vary.

The displacements of Node Point 901 and 902 will not differ significantly in any Case.

The largest variation is only 1 mm. The comparison diagram, Figure 4-1, consists of mean values of S901 and S902 for each case and modulus.

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Figure 4-1: Displacement as a function of modulus number for each case

Figure 4-2 is a comparison diagram of the settlements for each case and different Young´s modulus. Case 1 gives the lowest settlements. This is because the hardfill is applied over the whole dam section; in the other cases there is no hardfill on the downstream side of the dam axis. It is then less rockfill and more hardfill beneath the location of maximum settlements in case 1. The rockfill will settle more than hardfill due to higher Young´s modulus.

The maximum settlements will be located in the dam body as Figure 4-3 shows. In every case it will occur in the rockfill, Zone 3C. Maximum settlement of case 1 is more upstream and upwards compared to the other cases. This is also due to the hardfill distribution.

0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09

0 1 2 3 4 5

S (m)

Young's modulus, E (GPa)

Displacement vs Young's modulus

Case 1 Case 2 Case 3a Case 3b Max value

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Figure 4-2: Maximum settlement as a function of modulus number for each case

Figure 4-3: Location of maximum settlement 60

65

70

75

80

85

90

0 1 2 3 4 5

Se;lement (cm)

Young´s modulus, E (GPa)

Se;lement vs Young´s modulus

Case 1 Case 2 Case 3a Case 3b

The Yang Qu dam: Optimization of Zones by Numerical Modelling on this New Type of Dam

MASTER'S THESIS

The Yang Qu dam

Optimization of Zones by Numerical Modelling on this New Type of Dam

Elias Hammar Daniel Lennartsson

2014

The Yang Qu dam

Preface

Summary in Swedish

Abstract

Table of contents

List of Figures

Symbols

1 Introduction

2 Theoretical background

( )

( )

(

)

φ

φ

3 Methodology

4 Result and analysis

Displacement vs Young's modulus

Se;lement vs Young´s modulus