MASTER'S THESIS
Numerical Modeling of Deformations and
Stresses in the Face Slab for High
Concrete Face Rockfill Dams
Sebastian Andersson
Fredrik Eriksson
2015
Master of Science in Engineering Technology Civil Engineering
Luleå University of Technology
Master of science program in civil engineering Luleå University of Technology
Department of Civil, Environmental and Natural resources Engineering Division of Mining and Geotechnical Engineering
Numerical modeling of deformations and
stresses in the face slab for high Concrete
Face Rockfill Dams
Sebastian Andersson Fredrik Eriksson
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Preface
The work that resulted in this master thesis is the fulfillment of our M.Sc. degree in Civil Engineering with focus on soil and rock construction at Luleå University of Technology. This master thesis covers 30 ECTS and was carried out during the period August-December for the China Institute of Water Resources and Hydropower Research, which will be referred to as IWHR in this report, at their research center in Beijing, China. As we have written our thesis we have become to understand that China is in the frontline of developing hydropower dams and we are grateful to have gotten the chance to be a part of their research. We would like to thank Professor Zeping Xu who have welcomed us and been very supportive. He has together with Song Xianhui, Liang Jianhui, Zhang Yanyi and Wu Junming made our stay in Beijing to such a great experience.
We would also like to acknowledge Professor James Yang who presented us with the opportunity to go to Beijing and do this master thesis, which has been a great experience both personally and professionally. We would additionally like to thank Energiforsk who have founded our work and therefore made this possible. The project is funded by Energiforsk AB within the frame of dam safety, with Mr. Cristian Andersson and Ms. Sara Sandberg as program directors,
www.energiforsk.se
. Some funding is even obtained from Luleå University of Technology, which facilitates the accomplishment of the project.Furthermore we would also like to thank Hans Mattsson for his support, great input and critics on our master thesis, this have made us work harder.
Last but not least we would like to thank our classmates and friends for our time at Luleå University of Technology and we wish you all the luck in the future.
Beijing, November 2015 Sebastian Andersson Fredrik Eriksson
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Abstract
Chinas economy is growing rapidly and this creates an enormous demand for energy. China’s government has over the years constructed many hydropower stations to meet some of this demand. A crucial part of a hydropower station is the reservoir, in order to create one a dam is needed. With the reservoir the hydropower station can store and produce electricity on demand.
The concrete face rockfill dam (CFRD) is a dam type that consists of a concrete slab that lies on the upstream slope of an embankment. The Concrete face works as an impervious layer. The slab is supported by the rockfill embankment, which is constructed in different zones with different purposes and features. CFRD has been accepted in China because of the safety and also for economic reasons. The CFRD has been constructed at numerous places in China, the dam type is under continuous development at the China Institute of Water resources and Hydropower Research (IWHR). As the development of CFRD continues the height of these dams increases. There is one CFRD in China that is in the planning stage right now which will have a total height of 257 meters and in the future China plans to build dams with heights over 300 meter.
There have been several high CFRDs that have had failure due to rupture in the face slab. Movement in the embankment causes larger stresses in the face slab. If these stresses increase enough it will lead to spalling failure in the face slab. The rupture will then significantly escalate the seepage through the dam body. These problems have come as a surprise as it were not foreseen by either numerical modeling or empiric design methods.
IWHR conducts research on the problem with rupture in the slab of high CFRD, this master thesis is a small part of that work. In this study two numerical analyses with the finite element method (FEM) have been conducted. The first analysis tries to answer how stresses and deformations change with an increased height of the dam. The second analysis models the performance of the dam when stiffness is decreased in one of the rockfill zones in the dam.
When the dam height is increased the deformations in the cross-valley direction magnifies, these will cause larger compression stresses in the center of the slab. Since the stresses in the cross-valley direction increase faster than the confining water pressure there is an enhanced risk of rupture in the face slab. When the stiffness modulus of the downstream rockfill zone, referred to as zone 3C, is decreased the result will be an increased deflection of the concrete slabs upper part. The stresses in the cross-valley direction do not enlarge with lower stiffness of zone 3C. Therefore the risk of rupture in the slab will not increase.
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Sammanfattning
Kinas ekonomi växer snabbt och detta har skapat en ökad efterfrågan på elektricitet. Kinas regering har de senaste åren byggt många vattenkraftverk för att delvis möta denna efterfrågade ökning. En vital del av en vattenkraftstation är reservoaren, för att kunna skapa en sådan behöver man bygga en damm. Med en reservoar kan vattenkraftstationen lagra energi och producera elektricitet när det krävs.
Concrete face rockfill dam (CFRD) är en dammtyp som består av en betongskiva
som vilar på uppströmssidan av en bank som består av krossat bergmaterial. Betongskivan fungerar som ett impermeabelt lager. Betongplattan stöds av sprängstensfyllnadsbanken, som är uppbyggd av flera zoner vilka har olika syften och egenskaper. CFRD har accepterats i Kina på grund av säkerheten men också av ekonomiska skäl.
CFRD har uppförts på ett flertal platser i Kina, dammtypen är under konstant utveckling av China Institute of Water resources and Hydropower Research (IWHR). Allt eftersom utvecklingen av denna dammtyp pågår, ökar också höjden på dessa dammar. En CFRD som planeras här i Kina kommer att ha en total höjd på 257 meter men i framtiden vill de bygga dessa dammar över 300 meter.
Det är flera höga CFRD som har haft problem med vida vertikalsprickor i betongskivan. Rörelser i banken orsakar stora spänningar i betongskivan. Om dessa spänningar ökar tillräckligt kommer det leda till ett spjälkningsbrott i betongskivan. Den spricka som då uppstår kommer att öka läckaget genom dammen väsentligt. Detta problem har kommit som en överraskning då det inte gick att förutse med varken numerisk modellering eller empiriska metoder. IWHR genomför forskning på detta problem med sprickor i betongskivan hos höga CFRD, detta examensarbete är en liten del i denna forskning. Arbetet har två numeriska analyser med finita element metoden (FEM) genomförts. Den första analysen försöker svara på hur spänningar och deformationer i betongskivan påverkas av dammhöjden. Den andra analysen studerar hur dammen beter sig när styvheten i en av stenfyllnadszonerna minskas.
När dammhöjden ökar kommer deformationerna vinkelrätt mot dalens riktning förstärkas, detta kommer att ge större kompressionsspänningar i de centrala delarna av betongskivan. På grund av att spänningarna vinkelrätt mot dalens riktning ökar i en högre takt än det mothållande vattentrycket finns det en utökad risk för en vertikal spricka i betongskivan. När styvhetsmodulen i nedströms sprängstensfyllnadszon, zon 3C, minskas resulterar det i en ökad deflektion i betongskivans övre del. Spänningar vinkelrätt mot dammens riktning visar ingen tendens till att ökas vid sänkt styvhet i zon 3C. Därav finns det ingen ökad risk för vinkelräta sprickor i betongskivan.
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Table of contents
1 Introduction ... 1
1.1 Background ... 1
1.2 Purpose and Objective ... 2
1.3 Limitations ... 3
2 Theoretical background ... 5
2.1 Concrete Face Rockfill Dams ... 5
2.1.1 Design ... 5
2.1.2 Performance ... 7
2.2 Material ... 10
2.3 Case study ... 11
2.3.1 Failures in dams ... 11
2.3.2 Deformation and strain data from the rupture at Mohale dam ... 12
2.3.3 Lesson learned ... 15
2.4 Finite Element Analysis... 15
2.4.1 Computer software ... 16
2.5 Duncan’s E-B Constitutive model ... 16
2.5.1 Nonlinearity ... 17
2.5.2 Stress dependency ... 19
2.5.3 Inelasticity... 21
2.5.4 Nonlinear volume change accounted for by using constant bulk modulus ... 22
2.5.5 Variation of bulk modulus with confining pressure ... 23
2.5.6 Restriction on the range of the bulk modulus parameters ... 24
2.5.7 Parameters of the hyperbolic stress-strain relationship ... 24
2.5.8 The limitations of Duncan’s E-B model ... 25
3 Numerical Simulation ... 26
3.1 Presentation of the problem and design of the analysis ... 26
3.2 Geometrical model and material properties ... 27
4 Results and Interpretation ... 30
4.1 Analysis with varied height ... 30
4.2 Analysis with varied material parameters of zone 3C ... 33
5 Conclusion and Discussion ... 38
6 References ... 40
Appendix A: Distributions of stresses and deformations in the face slab for the height 300m from the analysis with varied height. ... 42
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List of figures
Figure 2.1 Cross-section of a CFRD with zoning. (1A) Impervious layer, (1B) Random fill, (2) Transition zone, (3A) Fine rockfill, (3B) Rockfill, (3C) Coarse rockfill (Fell, et al., 2005) ___________________________________________ 5
Figure 2.2 Upstream face of a CFRD with different types of joints (Fell, et al., 2005) ____________________________ 6
Figure 2.3 Sketch of deformations during construction and impoundment (Fell, et al., 2005) __________________ 7
Figure 2.4 Construction sequence of CFRD (Fell, et al., 2005) _____________________________________________________ 8
Figure 2.5 Movement directions and stress zones in the upstream face (Fell, et al., 2005). _____________________ 8
Figure 2.6 Directions of movements and how they effect the perimetric joint (Fell, et al., 2005). ______________ 9
Figure 2.7 Direction of the different modulus, during construction and filling (Fell, et al., 2005) _____________ 10
Figure 2.8 Rupture of compressive joint at Mohale dam (Johannesson & Tohlang, 2007). ____________________ 12
Figure 2.9 Horizontal deformations in the downstream direction at Mohale dam (Johannesson & Tohlang, 2007). _______________________________________________________________________________________________________________ 13
Figure 2.10 Settlement at Mohale dam (Johannesson & Tohlang, 2007). _______________________________________ 13
Figure 2.11 Horizontal deformation in the cross-valley direction at Mohale Dam (Johannesson & Tohlang, 2007) _______________________________________________________________________________________________________________ 14
Figure 2.12 Strain and water level at Mohale dam (Johannesson & Tohlang, 2007). __________________________ 14
Figure 2.13 Face of Mohale dam (Johannesson & Tohlang, 2007). ______________________________________________ 15
Figure 2.14 The hyperbolic stress-strain curve (Duncan, et al., 1980) __________________________________________ 18
Figure 2.15 The transformed hyperbolic stress-strain curve (Duncan, et al., 1980) ___________________________ 19
Figure 2.16 Initial tangent modulus with confining pressure (Duncan, et al., 1980) __________________________ 20
Figure 2.17 Unloading-reloading modulus (Duncan, et al., 1980) ______________________________________________ 22
Figure 2.18 Nonlinear and dependent strain and volume change curves, point A on the stress-strain curve can be compared with point A’ that is related to the volume change (Duncan, et al., 1980) ____ 23
Figure 2.19 Variation of bulk modulus with confining pressure (Duncan, et al., 1980) ________________________ 24
Figure 3.1 The face slab and the different zones of the model ___________________________________________________ 28
Figure 3.2 Shows zone 3B and 3C _________________________________________________________________________________ 28
Figure 4.1 Horizontal deformation in the cross-valley direction across the dam, at the height with the
maximum deformations for each dam. ___________________________________________________________________________ 31
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Figure 4.3 Maximum deflection of face between the different dam heights ____________________________________ 32
Figure 4.4 Sigma y in the center of the face slab for the different dam heights also the water pressure is plotted _____________________________________________________________________________________________________________ 33
Figure 4.5 Maximum deflection in cross-valley direction ________________________________________________________ 34
Figure 4.6 Settelment across the dam crest for the different cases _____________________________________________ 34
Figure 4.7 Maximum deflection of face slab with decreased stiffness ___________________________________________ 35
Figure 4.8 Sigma y with the change of material properties _____________________________________________________ 36
Figure A.1 Distribution of horizontal deformations in the face slab in cross-valley direction, the axis units are meters [m] the deflection is also in meters [m] __________________________________________________________________ 42
Figure A.2 Distribution of deflection in the face slab, the axis units are meters [m] the deflection is also in meters [m] _________________________________________________________________________________________________________ 42
Figure A.3 Distribution of sigma-y in the face slab, the axis units are meters [m] the units of sigma y is in [MPa] _______________________________________________________________________________________________________________ 42
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List of tables
Table 2.1 Shape-factor for the four dams that have had ruptures (parameters taken from Freitas & T. Cruz, 2007). _______________________________________________________________________________________________________________ 11
Table 2.2 Summary of the hyperbolic parameters (Duncan, et al., 1980). ______________________________________ 25
Table 3.1 Model data of the 5 different models __________________________________________________________________ 27
Table 3.2 Material parameters for the analysis with varied height ____________________________________________ 29
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Symbols
Latin characters
A Area of the face slab
B Bulk modulus
c Cohesion
E Young’s modulus
𝐸𝑖 Initial tangent modulus
𝐸𝑡 Tangent modulus
𝐸𝑢𝑟 Unloading- reloading modulus number
𝐸𝑟𝑐 Modulus during construction
𝐸𝑟𝑓 Modulus during first filling
H Height of the dam
K Modulus number
𝐾𝑢𝑟 Unloading-reloading modulus number
𝐾𝑏 Bulk modulus number
m Bulk modulus exponent
n Modulus number
𝑝𝑎 Atmospheric pressure
Sigma y The stress in the cross-valley direction
𝑣𝑡 Possions ratio
Greek characters
∆𝜎𝑋 Normal stress increment
∆𝜎𝑌 Normal stress increment
∆𝜏𝑋𝑌 Shear stress increment
∆𝜀𝑋 Normal strain increment
∆𝜀𝑌 Normal strain increment
∆𝑦𝑋𝑌 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
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1 Introduction
1.1 Background
China is a fast growing country, with it an increasing demand for energy. To meet this demands the government of China to construct; coal-plants, nuclear power stations and hydropower stations (Hammar & Lennartsson, 2013). Hydropower is relatively ecofriendly in the perspective that is does not pollute the air and it only affects the nearby surrounding environment. When producing electricity from hydropower plants dams are essential to create a water reservoir so that the water stream can be controlled. China has a landscape with many rivers suitable for hydropower. The annual generation of hydropower in China was 1066 MW in the year of 2014, this makes China the biggest producer of hydropower in the world (Xu, 2015).
Embankment dams was the first dam type to be used in the world, they were constructed with earth materials. It is also one of the most widely accepted and developed dam types in the world. The two main types of dams that are constructed of rockfill; earth core rockfill dams (ECRD) and concrete faced rockfill dams (CFRD) (Xu, 2015). The idea of the CFRD is that the concrete slab at the surface of the dam will work as an impervious face. The concept of dams with an impervious face is over 150 years old, the first dam of this type where constructed in California in 1856. With the advancement of geotechnics and construction equipment during the 1950s-1960s the concept of CFRD developed and with it the height of the dams (Cruz, et al., 2009).
The reasons that CFRDs have been widely accepted in China is mainly because of three reasons; security, material at site and construction time. It is a safe dam since even if the face slab cracks the stability of the dam is not jeopardized, because of a semi-impermeable layer that can handle leakage to a certain level. Furthermore the downstream slope is not likely to erode since it is constructed of rockfill in large fractions. As mentioned the main material for construction can be taken at site, which is of economic benefits. The dam is constructed of rock material which is little to non-affected by climate, so the construction can be performed all year, which saves time and money (Xu, 2015).
The concept of building concrete faced rockfill dams has from experience been both economical and safe. CFRD can be built at a low cost in comparison with rockfill dams with impervious core and concrete dams. The reason that the CFRD is efficient seen to economy is that the rockfill can be taken from site (Cruz, et al., 2009). Another advantage is that grouting for the CFRD can be done parallel with the embankment construction, which can shorten the construction time. CFRD can be profitable to other embankment dams if there is no suitable soil for the core nearby. The slopes of a CFRD can be made steeper than a dam with earth core, which will decrease the fill quantities and save construction costs (Fell, et al., 2005).
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China has constructed the highest CFRD Shuibuya with a total height of 233 m, now China is planning a CFRD with a height of 257 m (Xu, 2015). With increasing of the dam height the dam will be subjected to larger stresses. The concerns with these high dams are their performance in settlements and face deflection during reservoir filling and operation. To large deflection of the concrete face will make it crack and this will increase the seepage from the dam. With Numerical analyses the crucial section of the rockfill, which is subjected to large stresses can be found in the design stage (Cruz, et al., 2009).
In the last two decades there has been recorded several high CFRD having problems with vertical cracks in the center part of the face slab. Some of the dams facing this problem; Tianshengqiao, is referred to as TSQ1 in this report, (178 m) in China. Barra Grande (185 m) and Campos Novos (202 m) both in Brazil, Mohale (145 m) in kingdom of Lesotho. All these dams have experienced high compressive stresses in the center part of the face slab and this has caused the vertical cracks. When designing high CFRD in the future, the high strain in the center part of the slab needs to be taken into consideration (Freitas & Cruz, 2007).
1.2 Purpose and Objective
As previously mentioned there are four high concrete face rockfill dams that have had large vertical cracks, called ruptures, in the concrete face slabs. The broad purpose of the research project is to investigate if these ruptures will become more common as concrete face rockfill dams are built higher. This master thesis is a small part of this investigation.
The focus of this work was to examine how an increasing height might change stresses and deformations in the concrete face slab for high CFRDs. In order to do so numerical analyses were performed with five different heights, ranging from 200m to 300m.
Furthermore the aim was to examine if an increasing difference in stiffness of the rockfill zones would enhance the risk of rupture. Numerical analysis were conducted with varying stiffness of the downstream rockfill zone, the other rockfill zones had a constant stiffness, for a concrete face rockfill dam with the height of 300m.
Questions to investigate:
Ruptures are a troubling problem for high concrete face rockfill dams. Ruptures are caused by deformations in the rockfill material, which leads to deflections and greater stresses in the face slab and if these stresses become large enough they could trigger a rupture. The research questions of this thesis;
Could an increasing height give larger stresses-, deformations in the cross-valley direction and deflection in the face slab for high CFRDs?
Could an increasing difference in modulus of compression between the rockfill
zones give larger stresses-, deformations in the cross-valley direction and deflection in the face slab for high CFRDs?
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1.3 Limitations
To be able to focus on the main task, which is the structural stability of the dam embankment delimitations were made. The limitations in this master thesis are:
No problems associated with the foundation were taken into account.
The deformations and stresses will not be compared to any failure criteria for
the face slab.
No problems associated with seepage were taken into account.
No consideration to how time affects the deformation was investigated only
immediate deformation was analyzed.
Material parameters were obtained from the China institute of water resources
and hydropower research, which is abbreviated to IWHR, who have done extensive research in this field.
The sections, proportions and zoning of the dam were not optimized to reduce
deformation during the numerical modeling.
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2 Theoretical background
2.1 Concrete Face Rockfill Dams
2.1.1 Design
As CFRD are developed from conventional embankment dams these have similar design features. The CFRD design is under continuous development in order to build higher dams, which has to withstand larger deformations and stresses. However the zonings presented by (Sherard & Cooke, 1987) are basically applicable for the main part of the CFRD’s built (Freitas & Cruz, 2007). These zones have been named differently by (Fell, et al., 2005) and (Cruz, et al., 2009); in this thesis the names from the latter are used. Figure 2.1 shows the general design of a modern CFRD with the different zones.
Figure 2.1 Cross-section of a CFRD with zoning. (1A) Impervious layer, (1B) Random fill, (2) Transition zone, (3A) Fine rockfill, (3B) Rockfill, (3C) Coarse rockfill (Fell, et al., 2005)
The upstream face of the dam is constructed of concrete, which is grouted in slabs. The thickness of these slabs is usually between 0.25 and 0.6 meters (Fell, et al., 2005). At the toe of the dam these slabs are connected to a plinth, which is cast on the ground to unite the foundation with the dam. The plinth also decreases the hydraulic gradient and seepage through the foundation. Between slabs as well as between slabs and the plinth there are joints. These can be divided into horizontal-, vertical and perimetric joints. The joints are designed to allow some movement of the concrete slabs, both contraction and extension. There are different joint types that can be chosen depending on if there is compression or tensional stresses in the area in which the joint is located (Freitas & Cruz, 2007). In figure 2.2 is the upstream concrete face with the different types of joints illustrated.
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Figure 2.2 Upstream face of a CFRD with different types of joints (Fell, et al., 2005)
The embankment is composed of four main zones with different purposes and properties as can be seen in figure 2.1. Zone 1 is a safety measure, which covers the perimeter joint and lower part of the dam and ensures that if a joint opens it will not lead to leakage. Closest to the concrete is an impermeable layer called Zone 1A, this can be supported by a less expensive supporting layer named Zone 1B (Cruz, et al., 2009).
Zone 2 is located directly behind the concrete face and can be compared with the filter in a conventional earth and rockfill dam. This filters, or as it is also called transition zone, have several features. It consists of finer material than the rockfill, which makes a smoother surface on which the concrete can be cast on. Furthermore this layer is a semi-impervious layer preventing unrestrained seepage if the reservoir impounds during construction or if a failure in the slab or joint would occur. (Cruz, et al., 2009).
Zone 3 is rockfill, which main purpose is to support the face slab. Because of this the rockfill should have a low compressibility and high shear strength to reduce the deformation and creeping to acceptable levels. Furthermore the rockfill should be free draining to prevent high pore pressures during construction and to drain seepage water from upstream layers. This material should also be easily accessible because of the large volume needed. Zone 3 consists of three variations, which are:
Zone 3A, fine rockfill.
Zone 3B, rockfill.
Zone 3C, coarse rockfill.
Zone 3A is a transition zone between zone 2 and 3B and works as a filter to zone 2 in case of seepage. Because the water load mainly passes into the foundation through Zone 3B this zone is the most compacted during construction. This zone is usually wetted and compacted in layers of 1 meter by using large vibrating steel drum rollers and 4-8 passes. Because zone 3C is not as subjected to the water load the compaction modulus do not need to be as high as for zone 3B. This means that zone 3C can be
7 compacted in layers with thickness of 1.5-2 m in order to save funds. (Fell, et al., 2005). The downstream face of the dam is often covered with a riprap of large boulders. According to (Sherard & Cooke, 1987) this is because it creates a stable and aesthetically pleasing surface but also make use of large boulders that otherwise would need to be disposed of.
2.1.2 Performance
The concrete face is designed as an impermeable membrane with no or very reduced ability to withstand any deformation without the support of the embankment. The deflection of the concrete face will because of this be a complete reflection of the displacement in the embankment and primarily the rockfill in zone 3B. The displacement in dams is three-dimensional. The vertical settlements as well as the horizontal deformations in the upstream-downstream and cross-valley directions are all affecting the concrete face (Cruz, et al., 2009).
The deformations in the dam are developed during construction of the embankment, during impoundment and while the dam is in operation. The deformations throughout construction and reservoir filling are schematically illustrated in figure
2.3.
Figure 2.3 Sketch of deformations during construction and impoundment (Fell, et al., 2005)
According to (Xu, 2015) are high CFRDs constructed in phases and could even partially be taken into operation before construction of the whole dam is completed. Accordingly the concrete face is also cast in phases. Figure 2.4 shows an example how a construction sequence could look like. As the first section (1) of the embankment is built and the second section (2) is due, the concrete face, which is marked with red could be cast. As seen in figure 2.4 the concrete slab is not grouted to the crest of section (1), the casting is stopped around 20m from the crest. This is to reduce creep deformation. Because the top of section (1) has no overburden and will be compressed as section (3) is built. When section (3) is finished the rest of the concrete face is cast. According to (Cruz, et al., 2009) the construction sequence should be further investigated for high CFRD as the face slab deflection behaves differently from a dam, which has been built and impounded in one stage.
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Figure 2.4 Construction sequence of CFRD (Fell, et al., 2005).
During impoundment the water load will cause a deformation in the rockfill normal to the upstream face. This creates horizontal movement both in the valley direction and in the cross-valley direction. Furthermore the embankment will settle from the self-weight and the water load. As the concrete follows these displacements of the rockfill and bends there will be movements in the joints. The tendency of movements in the joints is that tensional joints near the abutment will open and the compressive joints in the central compressed area will close. Figure 2.5 illustrate the direction of movements and stress zones on the face slab. In narrow valleys the rockfill will have a movement toward the bottom of the valley, this will cause high stresses in the slab (Cruz, et al., 2009). The valley shape causes this, for a V-shape valley the embankment will experience uneven settlements, with the largest in the middle. This is one of the reasons that the slab will move toward the middle of the valley (Xu, 2015). The deflections are also going to cause shear and tensile movements in the perimetric joints. Figure 2.6 shows how the directions of the movements will affect the perimetric joint (Fell, et al., 2005).
9 Figure 2.6 Directions of movements and how they effect the perimetric joint (Fell, et al., 2005).
The stress-strain characteristic for rockfill is that when it is subjected to stresses larger than the pre-consolidations pressure, large deformations will occur. This is the case for the embankment as it is built, as the reservoir is filled-up for the first time and as the reservoir height is increased during operation. If there during operation are fluctuations in the reservoir level, beneath its highest level, it will be as loading and unloading beneath the pre-consolidation pressure. The deformations will therefore be limited (Fell, et al., 2005). However the movements, which start as the reservoir is filled up, continue during operation as creep deformations but this displacement rate is reduced with time (Cruz, et al., 2009).
These creep deformations can if they become large enough lead to problems for the concrete face. The arching effect in narrow valleys is reduced with time and as a consequence the settlements developed by creep may be larger than in open valleys, which have less arching effects (Cruz, et al., 2009). Sufficiently large deformations can cause tensional cracks to open or trigger cracking in compression joints or the face slab itself, resulting in leakage through the dam. This seepage causes efficiency loss and can if it becomes severe enough endanger the stability of the whole dam.
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2.2 Material
The modulus of compressibility of the rockfill controls how the rockfill will deform when subjected to pressure. It is an important parameter of the dam’s total performance. The rockfills compressibility modulus is determined by several factors; rock type, strength, shape and gradation of the rock. Furthermore it depends on the layer thickness and the rollers weight and number of passes, as well as if water has been added during compaction (Fell, et al., 2005).
The void ratio is a key parameter to recognize, since it tells how well compacted the rockfill is, most dams have a void ratio less than 0.25 (Fell, et al., 2005).
The rockfill should have a medium to high rock strength, the uniaxial compressive strength is between 30 to 80 Mpa. To obtain high compaction density it is important to have a well-graded material. In the effort of reducing deformation, a high compression density is always sought (Xu, 2009). If two rockfills is compacted under the same conditions and one is well graded and the other is uniformly graded. The well-graded rockfill is less likely to undergo internal breakage since a less deformable soil-skeleton has been created (Cruz, et al., 2009).
During the first reservoir filling the water load will act in the normal direction of the concrete face, this will cause a rotation of the principal stresses acting inside the rockfill. The shear strength will decrease in some areas of the upstream zone. This will generate deformations, rockfill will perform as a pre-consolidated soil (Cruz, et al., 2009). Since the principal stresses will rotate during the first reservoir filling, the rockfill will have two modulus; one modulus during construction 𝐸𝑟𝑐 and another
modulus during first filling 𝐸𝑟𝑓see figure 2.7 (Fell, et al., 2005). The modulus during
first filling 𝐸𝑟𝑓 will always have a higher value than the modulus during contruction
𝐸𝑟𝑐. It is common that the ratio between 𝐸𝑟𝑓/𝐸𝑟𝑐 is between 2-4 (Cruz, et al., 2009). As
Fell et al. (2005) points out the modulus of the first filling is not a true modulus of the rockfill, it is used to get an understanding of why settlements occure and how to predict them during first reserviour filling.
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2.3 Case study
2.3.1 Failures in dams
In CFRDs there are three types of general failures in the concrete face:
Fissures (≤3mm) are thermal induced cracks and these can appear as the
concrete cure. These cracks are usually horizontal and have a width of less than 3mm. Fissures are not seen as a problem to the stability of the dam and are therefore not usually repaired (Cruz, et al., 2009).
Cracks (>3mm) could be caused by bulging of the lower part of the face during construction of the embankment, which is reversed during impoundment. These can appear during construction, the first filling or after several years and are often horizontal (Cruz, et al., 2009).
Ruptures are caused by large stresses in the central part of the face and could
appear in a compressive joint or in the slab. For well-packed dams, rupture in combination with large leakage (>1000liter/s) is a relatively new problem and could not be foreseen by either Finite Element Analysis or empirical design principle (Cruz, et al., 2009).
Ruptures are triggered by the large stresses, which are induced by displacement in the embankment after the concrete face is cast. For CFRDs in general the post-construction deformation is correlated with the factors: width and shape of the valley, height of the dam and the rockfill modulus (Guocheng & Zeping, 2010).
For Tianshengqiao (TSQ1), Barra Grande, Campos Novos and Mohale which all have had problems with ruptures there are several mutual factors. For example, these dams are constructed in relatively narrow valleys. To describe the shape of the valley (Pinto & Marques Filho, 1998) have suggested a shape-factor
𝐴
𝐻2 (1)
where A is the face slab area and H is the height of the dam (Cruz, et al., 2009). In table
2.1 the shape-factor is presented for these dams.
Table 2.1 Shape-factor for the four dams that have had ruptures (parameters taken from Freitas & T. Cruz, 2007).
These dams are also located in V-shaped valleys. The problem especially for a narrow V-shaped valley is the arching effect the valley creates; this will decrease with time. Deformations will because of this postpone from construction to the impoundment and operation (Guocheng & Zeping, 2010). The result of constructing
Dam Height [m] Face slab area [m^2] A/H ^2
Barra Grande 185 108000 3.15
Campos Novos 202 106000 2.59
Mohalé 145 87000 3.8
12
high CFRDs under these conditions have in these cases lead to rupture in the face slab.
2.3.2 Deformation and strain data from the rupture at Mohale dam
As described earlier Mohale dam, with a height of 145m, in the kingdom of Lesotho is one of the dams that have had rupture in the face slab. The rupture in this dam was reported on February 14, 2006 during a fast impoundment up to spilling level caused by heavy rain. As a new highest level was registered large settlement and horizontal movement, both in the valleys direction and perpendicular to the valley, were detected. This increased the already high compressive stresses in the central part of the face slab and led to spalling in a compressive joint in the middle of the dam, from the crest of the dam and about 40m downward. Moreover observations were made of increased widths for the vertical and perimetric joint at the flanks. As a result of this the leakage increased drastically. Figure 2.8 shows the rupture, from the crest of the dam looking downward, after loose concrete have been removed (Johannesson & Tohlang, 2007).
Figure 2.8 Rupture of compressive joint at Mohale dam (Johannesson & Tohlang, 2007).
As seen in figure 2.8 one slab has lifted up, which was caused by the low confining water pressure near the top of the dam. This relief in stress triggered shear failure in the lower part of the dam, due to larger vertical strains, in two stages during 15-16 of March. Figures 2.9-2.11 shows how the deformation developed during the time that led up to the concrete failure and after. Also how strain in the center of the face increased as these deformations advanced. Figure 2.9 shows the horizontal movement of the parpett wall, located on the dam crest, in the direction up-stream to down-stream between the slopes of the valley. Between the 25 of January and the 16 of February there is a relatively large deformation, this period overlap the time of rupture (Johannesson & Tohlang, 2007).
13 Figure 2.9 Horizontal deformations in the downstream direction at Mohale dam (Johannesson & Tohlang, 2007). Figure 2.10 shows the vertical movement of the parpett wall across the valley. There
is also a relative large settlement during the same period, 25 of January to 16 of February.
Figure 2.10 Settlement at Mohale dam (Johannesson & Tohlang, 2007).
Figure 2.11 shows the horizontal cross-valley movement of the parpett wall, between
the abutments, and between 25 of February and 9 of March there are large displacement. Which is just after that the rupture occurred and before the shear failures.
14
Figure 2.11 Horizontal deformation in the cross-valley direction at Mohale Dam (Johannesson & Tohlang, 2007) Figure 2.12 shows the plot of strain, measured by the strain cells in the joint where the
rupture occurred, and water level. In this figure the strain cells are named SG-1, 3, 5, 7 and the location of them is shown in figure 2.13. The line marked with red in figure
2.12 is the water level in the reservoir.
15 Figure 2.13 Face of Mohale dam (Johannesson & Tohlang, 2007).
As seen in figure 2.12 the strains near the top of the dam is the largest and decreases for the gauges further down. Also there is a sudden release in strain as the water level is rising to a new highest level near the maximum capacity of the dam. The rupture in the concrete face causes this release. Moreover as seen in figure 2.12 this release is largest in the strain gauge that is situated nearest the dam crest. Then the strain release is decreasing further downward. At SG1 there is no strain release but in contrary an increase in strain after the rupture.
2.3.3 Lesson learned
To avoid similar problem, to what the Mohale dam had, for other high CFRDs in the future, safety-measures should be implemented. Both to decrease the post-construction deformation and to improve the face slabs ability to manage larger stresses (Guocheng & Zeping, 2010). These measures could for example be: increasing compaction and consequently the modulus of the rockfill, postpone the start of casting the concrete as much as possible, increase the thickness of the upper part of the face slab and design the face slab with anti-spalling reinforcement (Guocheng & Zeping, 2010).
As mentioned before there have also been failures in the concrete face slab in three other dams. As a result of these four concrete face ruptures, the vertical joints near the middle of the dam have been redesigned, for high CFRDs, with a compressive filler of wood or material with similar properties (Cruz, et al., 2009). Furthermore in China research is initiated to try to acquire a deeper understanding of the phenomenon rupture and how the risk of rupture could be limited as high CFRDs are becoming higher (Xu, 2015).
2.4 Finite Element Analysis
The Finite element method (FEM) is a numerical method that calculates an approximated solution of a statically indeterminate problem. This is done by solving partial differential equations. FEM is a good method to use when the problem has a complex geometry and the materials are isotropic. Since these problem often have several unknown parameters that FEM can solve. In the finite element theory the
16
body of the problem is divided in to elements, these elements is linked to one another in points called node points. The system that consists of elements and nodes is called the mesh (Liu & Quek, 2003).
The field parameters (e.g., stresses and displacements) inside the continuum are unknown, the assumption is made that inside the elements the change in field parameters can be obtained with simple approximations. These are called interpolation models, they are defined as changes of the parameters at the node points (Rao, 2004).
The nodes have a certain degree of freedom, which correlates to an unknown parameter. In the computations these unknown parameters are obtained. To calculate a displacement of an element an interpolation is done between the nodes in the element. An increase of the number of nodes will therefore give more accurate results but also increase the amount of data and slow down the computation speed (Rao, 2004).
The finite element method can for example be used to analyze stresses and movements in earth masses. If the results are to be accurate and trustworthy, it is important that the stress-strain behavior of the soil in the analysis is handled in a proper way. A constitutive model describes the relationship between stress-strain in a soil in the finite element method (Duncan, et al., 1980).
2.4.1 Computer software
The analysis is carried out with a FEM-software called SDAP3D, this software calculates stresses and strains in a three dimensional body. The software is DOS-based and the input and output data is text DOS-based. Zeping Xu developed SDAP3D at IWHR, the software uses Duncan’s E-B constitutive model. The software GID 7.2 is a graphical interface program that is used to visualize; the dam body, the zones, the mesh, the element and nodes.
2.5 Duncan’s E-B Constitutive model
Duncan el al. (1980) has developed a non-linear elastic model with the intent to simulate soil behavior. Duncan’s E-B model is an updated model; the original model from 1970 was developed by Duncan and Cheng from data obtained by triaxial tests. In the original model the Poisson’s ratio is assumed to be constant. In the Duncan’s E-B model the Poisson’s ratio varies with a stress dependent bulk modulus (Duncan, et al., 1980).
In Duncan’s E-B model the volume change behavior is modeled with a bulk modulus that varies with confining pressure and is independent of the strength mobilized. This is a suitable model for modeling soil behavior when large differential stresses are analyzed. The model is good at determine soil movement within a stable earth mass (Duncan, et al., 1980). This is the case for large rockfill dams, where the stress levels vary depending on were in the cross section the analysis is performed. This means that the strength of the rockfill will vary even though it is the same material.
17 The hyperbolic stress-strain relationship is a nonlinear incremental model that can be utilized in analyses of soil deformation. In each increment the stress-strain relationship is linear elastic ruled by Hooke’s law. For a plane strain condition it can be expressed as { ∆𝜎𝑋 ∆𝜎𝑌 ∆𝜏𝑋𝑌 } = 3𝐵 9𝐵−𝐸[ (3𝐵 + 𝐸) (3𝐵 − 𝐸) 0 (3𝐵 − 𝐸) (3𝐵 + 𝐸) 0 0 0 0 ] { ∆𝜀𝑋 ∆𝜀𝑌 ∆𝛾𝑋𝑌 } (2)
where ∆𝜎𝑥 and ∆𝜎𝑌 are the normal stress increments, ∆𝜏𝑋𝑌 is the shear stress
increment, 𝐵 is the bulk modulus, 𝐸 is the young’s modulus, ∆𝜀𝑥 and ∆𝜀𝑌 are the
normal strain increments and ∆𝛾𝑋𝑌 is the shear strain increment (Duncan, et al.,
1980).
With equation (2) it is possible to model three features of the stress-strain behavior of soils; nonlinearity, stress dependency and inelasticity. This is conducted by varying the values of Young’s modulus and bulk modulus as the stress in the soil changes (Duncan, et al., 1980).
2.5.1 Nonlinearity
It has been proven that stress-strain curves for numerous soils can be approximated with a hyperbola, as illustrated in figure 2.14 this hyperbola can be expressed by the following equation (𝜎1− 𝜎3) = 1 𝜀 𝐸𝑖+ 𝜀 (𝜎1−𝜎3)𝑢𝑙𝑡 (3)
where 𝜎1 is the major principal stress, 𝜎3 is the minor principal stress, 𝜀 is the strain,
𝐸𝑖 is the initial tangent modulus and (𝜎1− 𝜎3)𝑢𝑙𝑡 is the ultimate stress difference
between major- and minor principal stress which is connected to the soil strength. The value of (𝜎1− 𝜎3)𝑢𝑙𝑡 is always greater than the stress difference at failure, see
18
Figure 2.14 The hyperbolic stress-strain curve (Duncan, et al., 1980)
The hyperbolic equation (3) can be transformed to equation (4) and will have linear appearance 𝜀 (𝜎1−𝜎3)
=
1 𝐸𝑖+
𝜀 (𝜎1−𝜎3)𝑢𝑙𝑡 (4)𝜎1, 𝜎3and 𝜀 can be obtained from triaxial tests, these values can be plotted against the line in figure 2.15, the values will differ from that line. Through numerous tests of different soils a method which gives satisfying results have been obtained. The method is performed by drawing a line between the points where 70- and 95% of the strength is mobilized (Duncan, et al., 1980).
19 Figure 2.15 The transformed hyperbolic stress-strain curve (Duncan, et al., 1980)
2.5.2 Stress dependency
The stress dependent stress-strain ratio is achieved by varying 𝐸𝑖, (𝜎1− 𝜎3) and the
confining pressure 𝜎3, which is the total pressure. The variation of the initial tangent
modulus and the confining pressure can be presented by
𝐸𝑖 = 𝐾𝑃𝑎(𝜎3
𝑃𝑎)
𝑛 (5)
where 𝐸𝑖 is the initial tangent modulus, K is a dimensionless modulus, number n is a
dimensionless exponent and 𝑃𝑎 is the atmospheric pressure. The expression is
20
Figure 2.16 Initial tangent modulus with confining pressure (Duncan, et al., 1980)
A relationship between the ultimate stress difference and the stress difference at failure can be expressed
(𝜎1− 𝜎3)𝑓 = 𝑅𝑓(𝜎1− 𝜎3)𝑢𝑙𝑡 (6)
where (𝜎1− 𝜎3)𝑓 is the stress difference at failure, 𝑅𝑓 is the failure ratio and
(𝜎1− 𝜎3)𝑢𝑙𝑡 is the ultimate stress difference.
As mentioned earlier the value of the ultimate stress difference is always higher than the stress difference at failure see figure 2.14, the value of 𝑅𝑓 is smaller than one and
varies from 0.5 to 0.9 for most soils (Duncan, et al., 1980).
The stress difference at failure (𝜎1− 𝜎3)𝑓 can vary and the relation between 𝜎3 and
(𝜎1− 𝜎3)𝑓 can be expressed with the Mohr-Coulomb strength relationship, as follows
(𝜎1− 𝜎3)𝑓 =
2𝑐 𝑐𝑜𝑠∅+2𝜎3 𝑠𝑖𝑛∅
1−sin ∅ (7)
21 The tangent of the hyperbolic curve at a certain stress level is the tangent modulus, 𝐸𝑡. If equation (3) is differentiated with respect to 𝜀, and equations (5), (6) and (7) is inserted in to that equation, the equation for the tangent modulus will be
𝐸𝑡 = [1 −𝑅𝑓(1−𝑠𝑖𝑛∅)(𝜎1−𝜎3) 2𝑐 𝑐𝑜𝑠∅+2𝜎3 𝑠𝑖𝑛∅ ] 2 𝐾𝑃𝑎(𝜎3 𝑃𝑎) 𝑛 (8)
With this equation the tangent modulus can be calculated for any stress condition (Duncan, et al., 1980).
2.5.3 Inelasticity
The inelastic behavior of soils is defined in this model by using different modulus for initial loading and unloading-reloading of the soil. When unloading a sample in the triaxial test the stress-strain curve is steeper than the initial loading curve, if the sample is loaded again it will have almost the same curve as during unloading. The unloading- and reloading tangents is not the same, but it is seen as a good
approximation to combine them together to a unloading-reloading modulus 𝐸𝑢𝑟.
The mentioned loading steps and the elasto-plastic behavior can be seen in figure
2.17. Theexpression for unloading-reloading modulus is
𝐸𝑢𝑟 = 𝐾𝑢𝑟𝑃𝑎( 𝜎3
𝑃𝑎)
𝑛
(9)
where 𝐸𝑢𝑟 is the unloading-reloading tangent modulus and 𝐾𝑢𝑟 is the dimensionless
loading-reloading number. Equation (9) has the same mathematical structure as the equation for the initial tangent 𝐸𝑖 equation (5), the difference is that 𝐾𝑢𝑟 always is
greater or equal to K. The exponent, n is the same for the two equations in this model (Duncan, et al., 1980).
With the unloading-reloading modulus the Duncan’s E-B model can account for the impact of stress history of the soil (Cruz, et al., 2009).
22
Figure 2.17 Unloading-reloading modulus (Duncan, et al., 1980)
2.5.4 Nonlinear volume change accounted for by using constant bulk modulus
The bulk modulus is according to the theory of elasticity expressed by
𝐵 =∆𝜎1+∆𝜎2+∆𝜎3
3𝜀𝑉 (10)
where ∆𝜎1, ∆𝜎2, ∆𝜎3 are the changes in the principal stresses and 𝜀𝑉 is the related
change in volumetric strain. In the conventional triaxial test the confining stress is equal to 𝜎2 = 𝜎3 and constant and the deviatoric stress is increased, the following
expression is valid
𝐵 =(𝜎1−𝜎3)
3𝜀𝑉 (11)
For a conventional triaxial test the bulk modulus can be obtained, using any value of deviatoric stress (𝜎1− 𝜎3) on the stress-strain curve and compare that to the volume
23 Figure 2.18 Nonlinear and dependent strain and volume change curves, point A on the stress-strain curve can be compared with point A’ that is related to the volume change (Duncan, et al., 1980)
It is not self-evident on how to decide the bulk modulus, B, since soil experience volume change both under shear stress and normal stress. Large studies have been conducted on deciding how to determine the bulk modulus, where numerous soils have been tested. The studies show that if a horizontal volume change tangent is reached prior to that 70% of the strength is mobilized, use that point. If not a horizontal volume change tangent is reached prior to 70 % of the strength is mobilized use the point that corresponds to a stress level of 70% (Duncan, et al., 1980).
2.5.5 Variation of bulk modulus with confining pressure
When the bulk modulus is calculated for the same soil type with an increasing confining pressure the value of the bulk modulus generally grows as shown in figure
2.19. The variation of the bulk modulus in relation to confining pressure can be expressed as
𝐵 = 𝐾𝑏 𝑃𝑎(𝜎3
𝑃𝑎)
𝑚
24
where 𝐾𝑏 is the bulk modulus number and m is the bulk modulus exponent, both are
dimensionless. 𝑃𝑎 is the atmospheric pressure. The value of the bulk modulus
exponent depends on the soil type (Duncan, et al., 1980).
Figure 2.19 Variation of bulk modulus with confining pressure (Duncan, et al., 1980)
2.5.6 Restriction on the range of the bulk modulus parameters
When applying Duncan’s E-B model in finite element software the Poisson’s ratio, 𝑣𝑡
may be restricted to positive values and equal or less than 0.49. These restrictions are needed for the calculations in the software to run more smoothly, since the underlying equations can give extremely low values of Poisson’s ratio and unrealistically high values (Duncan, et al., 1980).
2.5.7 Parameters of the hyperbolic stress-strain relationship
There are 9 parameters in total that have to be determined to be able to use Duncan’s E-B model the parameters are presented in table 2.2. These parameters can be determined with conventional triaxial tests, from either drained- or unconsolidated-undrained tests. It is important that the test conditions represent the reality. (Duncan, et al., 1980). How to evaluate the parameters is described by Duncan et al. (1980).
25 Table 2.2 Summary of the parameters in the hyperbolic stress-strain relationship (Duncan, et al., 1980).
Parameters Name Function
𝐾, 𝐾𝑢𝑟 Modulus number
Relate 𝐸𝑖 and 𝐸𝑢𝑟 to 𝜎3
n Modulus exponent
c Cohesion intercept
Relate (𝜎1− 𝜎3)𝑓 to 𝜎3
∅, ∆∅ Friction angle parameters
𝑅𝑓 Failure ratio Relate (𝜎1− 𝜎3)𝑢𝑙𝑡 to 𝜎3= 𝑃𝑎
𝐾𝑏 Bulk modulus number Value of 𝐵/𝑃𝑎 at 𝜎3 = 𝑃𝑎
𝑚 Bulk modulus exponent Change in 𝐵/𝑃increases in 𝜎𝑎 for ten-fold
3
2.5.8 The limitations of Duncan’s E-B model
The Hyperbolic stress-strain relationships has some significant limitations that are essential to know and understand when using the model in finite element software. The hyperbolic stress-strain relationships is built on the generalized Hooke’s Law, see equation (2), therefore the model is more suitable to analyze the stresses and strains before failure. The model can simulate stresses and correlated movements to a stage where there is local failure in some elements. Nonetheless the model fails when it tries to analyze earth mass that is controlled by elements, which have already failed. Therefore the model is valid for stable earth masses (Duncan, et al., 1980). The hyperbolic relationships do not consider volume change related to changes in shear stress. Therefore the model is not suitable for soil types that can show dilatancy, such as dense sand (Duncan, et al., 1980).
It is important that the parameters in the model, as in all soil simulations, describe the reality. Test conditions are crucial for good results, for example the density-, water content- and stress conditions of the soil in the test have to relate to the conditions in the field (Duncan, et al., 1980).
26
3 Numerical Simulation
3.1 Presentation of the problem and design of the analysis
As briefly described in the introduction the purpose was to investigate if the height and material parameters affects the risk of spalling failure in the concrete face for high CFRDs. The focus was to study if there is any trend in how deformations and stresses were affected with an increasing height or changed material parameters in the downstream Rockfill zone, zone 3C, for a high CFRD with a specific cross-section. Finite element analyses together with Duncan’s E-B material model described previously were used for these computations. The FE model is three-dimensional to be able to study the deformations in three dimensions and therefore get a more comprehensive view of what transpires in the dam body.
The study had to be divided into two sub analyses, one in which the height varied, the other one where the material parameters were varied. For the analysis with variable height, five cases were investigated, with a proportionally scaled cross-section. For the analysis with varied material parameters for zone 3C five cases were also studied. As described in chapter 2.1.1 the zone with coarse rockfill, zone 3C in
figure 3.2, is preferably constructed in thicker layers than the zone with rockfill, zone
3B, in order to reduce costs. When the layer thickness is increased the compaction during construction will decrease, if all other compaction parameters are kept constant. As the coarse rockfill zone is less compressed the result will be a reduced modulus of compressibility. For the second analysis the material parameters of the rockfill layer 3B were constant the material parameters of the coarse rockfill layers 3C were decreased in five stages. The focus of these analyses was to investigate how the differences in these parameters affects the deformations-, stresses in the cross-valley direction and the deflection in the face slab. These differences affects the risks of rupture in the face slab.
The deformations in the cross-valley direction and the deflection perpendicular to the face slab were analyzed. The stresses in the cross-valley direction, referred to as sigma y, were analyzed. The numerical computations was composed of two parts, the first represents the completion of the dam and the second part after the impoundment of the reservoir. However as the largest deformations and stresses were sought after, only the result from the second part were used.
27
3.2 Geometrical model and material properties
The geometrical model is constructed in different zones with the face slab and the plinth made of concrete. Zone 2A, 3A, 3B and 3C consists of granular material. The model is constructed to imitate a dam in a narrow V-shaped valley see figure 3.1. In the first analysis the total dam height was increased, five models were generated with the height of 200-, 225-, 250-, 275- and 300 meters. The models have the same proportions the difference is the scale between them. The models were constructed in different steps like the construction steps in the field. The analysis was conducted in several impoundment steps, to attempt realistic conditions see table 3.1.
In the second analysis the effect of the material parameters of the rockfill was investigated, and the height was fixed to 300 m, the same model as in the previous analysis was used. The material parameters were varied in zone 3C see figure 3.2. The
material parameters that control the stiffness; modulus numbers K, 𝐾𝑢𝑟 and the bulk
modulus number 𝐾𝑏 were varied in the analysis. These parameters were varied in
relation to the same parameters of zone 3B. The differences between the analyses were the ratio between the stiffness parameters of Zone 3B and 3C see table 3.3.The other material properties for: slab, plint, 2A, 3A and 3B were the same as for the analysis with the varied height see table 3.2. All material parameters, both the varied and the constant, were obtained from IWHR.
Table 3.1 Data of the 5 different models
Dam height Number of
elements Number of nodal points Construction steps Impoundment steps Iteration steps
200 m 4044 4614 15 21 13824
225 m 4658 5290 16 23 15870
250 m 6018 6786 26 26 20358
275 m 7554 8474 29 29 25422
28
Figure 3.1 The face slab and the different zones of the model
29 Table 3.2 Material parameter values for the analysis with varied height
d kN/m3 o () Δ () K Kur n Rf Kb m Face slab 2.45 40 0.00 220000 220000 0.00 0.00 180000 0.00 2A 2.25 58.0 9.0 1500 3000 0.35 0.85 900 0.20 3A 2.23 55.0 9.0 1400 2800 0.35 0.85 800 0.20 3B 2.20 52.0 9.0 1200 2400 0.35 0.85 600 0.20 3C 2.15 50.0 9.0 900 1800 0.35 0.85 500 0.20 Plinth 2.40 40 0.00 200000 200000 0.00 -1.00 175000 0.00
Table 3.3 Material properties of zone 3B and 3C in the analyses were the material parameters were varied
Analysis number Zone K 𝑲𝒖𝒓 𝑲𝒃 𝐑𝐚𝐭𝐢𝐨 𝐛𝐞𝐭𝐰𝐞𝐞𝐧 𝐳𝐨𝐧𝐞 𝟑𝐂𝒏 𝒂𝒏𝒅 𝟑𝑩 * 3B 1200 2400 600 * 1 3C1 1200 2400 600 1 2 3C2 960 1920 480 0.8 3 3C3 720 1440 360 0.6 4 3C4 600 1200 300 0.5 5 3C5 480 960 240 0.4
30
4 Results and Interpretation
The deformations and stresses presented in this chapter are all obtained after the impoundment, as mentioned in chapter 3.1, since it is the maximum values that are interesting for determining the risk of rupture. The elements and nodes that are presented in this chapter are all from the face slab. In order to make the amount of data reasonable the most significant data will be presented in this chapter. Plots of the stress- and deformation distribution in the slab will be presented in appendix A. These plots are supposed to help the reader get a deeper understanding of the performance of the dams during impoundment.
4.1 Analysis with varied height
The maximum deformation in the cross-valley direction increases with height see
figure 4.1. This trend is expected since one of the driving forces of this phenomenon is
the uneven settlement in the rockfill, with greater heights the irregular settlements increases. The slab can be seen as two half’s that is moving towards the middle. There is no deformation in the middle of the slab and almost none near the edges. The maximum deformation is found in the middle of each half. The movement of these two half’s makes the center part of the slab compressed.
The node with the maximum deformation in the cross-valley direction is found on the upper part of the dam. For the dam with the height of 200 m the node is found approximately 3/4 from the bottom. And for the highest dam the node is found approximately 8/9 from the bottom. Note that the models are symmetric and origo is found at -30 meters on the x-axis.
31 Figure 4.1 Horizontal deformation across the dam, at the height with the maximum deformations for each dam The settlement of the crest can be seen in figure 4.2. The trend that the settlement of the crest increases with height is clear. The settlement of the crest increases with 49% when the dam height is increased from 200- to 300 meters. The distribution of the settlements along the crest see figure 4.2 can be explained by the V-shaped valley mentioned earlier.
Figure 4.2 Settlements for different dam heights
-0,30 -0,25 -0,20 -0,15 -0,10 -0,05 0,00 0,05 0,10 0,15 0,20 0,25 0,30 -330 -230 -130 -30 70 170 270 Deformation [m]
Left abutment Distance between abutments [m] right abutment
Maximum deformation in the cross-valley direction
Dam 200 m (at height of 151m)
Dam 225 m (at height of 186m)
Dam 250 m (at height of 196 m)
Dam 275 m (at height of 226m)
Dam 300 m (at height of 261m) -0,12 -0,1 -0,08 -0,06 -0,04 -0,02 0 -360 -300 -240 -180 -120 -60 0 60 120 180 240 300 Deformations [m]
Left abutment Distance between abutments [m] Right abutment
Settlement in the slab across the dam crest
200m 225m 250m 275m 300m
32
With an increase of height the deflection of the face slab will increase see figure 4.3. However there is no clear tendency in the rate in which this deflection grows, to determine if there is a change of rate further analyses has to be conducted. The maximum deflection is located at the upper center part of the slab. The deflection is caused by deformations in the underlying rockfill, the biggest settlement of the rockfill is found in the center due to the v-shaped valley as presented in chapter 2.1.2. This will as mentioned before cause the rockfill near the abutments to move towards the center. The node of maximum deflection is at the same height as the node of maximum deformation in the cross-valley direction.
Figure 4.3 Maximum deflection of face for the different dam heights
The direction of sigma y is in the axis of the dam. The maximum value of sigma y is increased with the total height of the dam see figure 4.4, where compression is positive and tension is negative. The four dams, with the heights of; 200-, 225-, 250- and 275 m, are moved vertically so that the water pressure can be compared with sigma y for each dam. It is a trend that the value of maximum sigma-y is increasing with the height of the dam and the node with that value is located further to the top of the dam when the height is increased. The movement that is presented in figure 4.1 drives this increase, and as it develops so should sigma y. In percent the maximum sigma y is increased with 70 % from the height of 200m to 300m and in the same way the water pressure is increased with 30%. This means that sigma y is increasing more rapidly than the confining pressure. The water pressure is assumed to go up to the crest that is not realistic. It is a simplified calculation of the water pressure, with the purpose not to show the absolute value but the change of the confining pressure.
-1,2 -1,1 -1 -0,9 -0,8 -0,7 -0,6 -0,5 200 225 250 275 300 D efl ect io n [m]
The different dam heights [m]
