INOM
EXAMENSARBETE SAMHÄLLSBYGGNAD,
AVANCERAD NIVÅ, 30 HP ,
STOCKHOLM SVERIGE 2020
Numerical study of the effect of
thermal ice loads on concrete dams
WASSEEM KORDOGHLY
IMAL ZUWAK
Numerical study of the effect
of thermal ice loads on
concrete dams
Imal Zuwak, Wasseem Kordoghly
Master of Science Thesis
Stockholm, Sweden 2020
TRITA-ABE-MBT-20318 ISBN: 978-91-7873-579-2
KTH School of ABE SE-100 44 Stockholm SWEDEN © IMAL ZUWAK, WASSEEM KORDOGHLY 2020
Royal Institute of Technology (KTH)
Department of Civil and Architectural Engineering Division of Concrete Structures
Abstract
It is essential to understand the mechanics of ice load and how it affects concrete dams located in a cold climate, such as Sweden, where the temperature becomes sufficiently cold to freeze the surface of the reservoir. The purpose of this thesis is to study ice load distribution along concrete dams, and its response during the application of an ice load. Two types of concrete dams were analysed, an arch dam and a buttress dam. For these dams, the influence from different parameters on the ice load distribution along the dams is studied. In addition to this, a study on how the ice load affects dam stability had also been performed.
Stability analyses based on the finite element method were performed using both linear and nonlinear formulation of the interaction behaviour between the base of the dam and the underlying rock. A parametric study of ice sheet expansion on different dam types and geometries were performed. The expansion of the ice sheet was assumed to either be caused by a constant temperature 15 ˚C uniformly distributed over the ice thickness, or by a temperature gradient from 15 ˚C at the top surface of the ice sheet and 0 ˚C at the bottom. The parametric study also includes an investigation about influence of the shape of the reservoir beaches, where it either had a perpendicular shape towards the surface of the dam, or it had an angle of 30˚ with the dam surface.
In the linear stability analysis, the structure continued to deform with increasing of the resultant pressure until it reached nonlinearity. The dam deflection had a linear relation with the applied ice load force until it reached the point when structure behaviour was nonlinear. The structure failed due to sliding, overturning or combination of both sliding and overturning. A material failure can also occur if the nonlinear material behaviour is considered, however this was not considered in this study.
The parametric study showed that the ice load distribution was less near the beaches, and the distribution of the load on the concrete dam was higher near the top surface of the ice sheet. It was also shown in the study that the distribution of the ice load along the dam was as a cosine function where it had the maximum value at the buttress and the minimum at monolith connections. The result also showed that the load distribution over the thickness of the ice sheet was the same along the dam, regardless of the shape of the beaches or the length of the ice sheet.
Sammanfattning
Det är viktigt att förstå hur islasten beter sig och hur den påverkar betongdammar som är belägna i kallt klimat, som t.ex. Sverige, där temperaturen blir tillräckligt låg för att frysa ytvattnet i en flod.
Syftet med detta examensarbete är att studera isbelastningsfördelningen längs en betongdamm och dess respons under en belastningen. Två olika typer av betongdammar har analyserats, vilka är valvdamm samt lamelldamm. För dessa, studerades det hur olika parametrar påverkar lastfördelningen från istrycket längsmed dessa dammar. Slutligen har det studerats hur islasten påverkar dammsäkerheten och risken för dammbrott.
Inverkan från interaktionen mellan dammen och det underliggande berget som linjär eller olinjärt har studerats i stabilitetsanalyser baserade på finita elementmetoden. En parameterstudie har också genomförts för olika dammtyper och geometrier där islasten orsakades av en expansion av isytan. Denna expansion antogs vara orsakad av antingen av en jämn fördelad temperatur över istjockleken på 15 ˚C, eller en temperaturgradient över istjockleken med +15 ˚C på den övre ytan och 0 °C vid isens bottenyta. Den parametriska studien beaktar även inverkan från utformningen av stränderna, där den har definierats som antingen vinkelrät mot dammen eller med en lutande vinkel på 30 grader.
I fallet med linjära stabilitetsanalyser kommer konstruktionen att fortsätta att deformeras som ett resultat av ökande resulterande tryckkraft. Dammens deformation har ett linjärt förhållande med den applicerade islasten till dess att den når en punkt då strukturens beteende övergå till olinjärt. Strukturens brottmod kan uppstå på grund av glidning, stjälpning eller i en kombination av både glidning och stjälpning. materialbrott kan uppstå om icke-lineariteterna beaktas. Dammen gick till brott på grund av glidning, vältning eller i kombinationen av dessa då. Materialbrott kan uppstå om icke-linjära materialmodeller inkluderas, men detta beaktades dock ej i denna studie.
Den parametriska studien visar att isbelastningen är mindre nära stränderna och att belastningen på betongdammen är högre vid isens ovanyta. Studien visar att
islastfördelningen längsmed dammen liknar en cosinusfunktion som når sitt maximum vid stödskivan och sitt minimum vid monolitanslutningen. Resultatet visar även att islastfördelningen genom islastens tjocklek har samma form längsmed dammen oavsett utformningen av stränder eller istäckets längd.
Preface
This master thesis has been performed at the division of concrete structures and department of Civil and Architectural Engineering at KTH, the Royal Institute of Technology.
The authors are thankful to Docent Richard Malm for his supervision, support and guidance throughout this thesis. During the thesis, Docent Richard Malm has shared knowledge with authors and authors has learned much from that. In addition to that, the authors also want to show an appreciation for Assoc. Prof. Bert Norlin, who has taken his time and helped us when needed.
Stockholm, June 2020
Imal Zuwak
Table of content
Abstract ... i Sammanfattning ... iii Preface ... v 1 Introduction ... 1 1.1 Background ... 11.1 Aim and purpose ... 2
1.2 Limitations ... 3 1.3 Thesis structure... 3 2 Concrete dams ... 5 2.1 Gravity dams ... 5 2.1.1 Mass dams ... 5 2.1.2 Buttress dams ... 6 2.2 Arch dams ... 7 2.3 Loads ... 11 2.3.1 Gravity ... 12 2.3.2 Uplift pressure ... 12 2.3.3 Hydrostatic pressure ... 12 2.3.4 Ice load ... 13 2.3.5 Thermal loads ... 13 2.4 Material properties ... 14 2.4.1 Concrete ... 14
2.5 Ice load measurements... 15
2.5.1 Thermal effects on ice load ... 15
2.5.2 Reservoir water level effect on ice load ... 17
3.1 Geometrical model ... 19
3.2 Meshing ... 21
3.2.1 Shape function ... 22
3.2.2 The discretisation of the elements ... 23
3.3 Boundary conditions and interactions ... 23
3.3.1 Interactions between the surfaces ... 24
3.3.2 Modelling of the rock ... 25
4 Case studies ... 27
4.1 Dam geometries ... 27
4.1.1 Concrete buttress dams ... 28
4.1.2 Concrete arch dam ... 30
4.2 Material properties ... 31
4.3 Mesh ... 32
4.4 Loads and loading procedure ... 36
4.4.1 Case 1 ... 36
4.4.2 Case 2 ... 38
4.5 Boundary conditions and interactions ... 38
4.5.1 Case 1 ... 38
4.5.2 Case 2 ... 40
5 Results ... 43
5.1 Ice load force and deformation relation on dams ... 43
5.1.1 Linear behaviour of the concrete dam ... 43
5.1.2 Nonlinear behaviour of the concrete dam ... 45
5.2 Variation in ice load due to dam geometry ... 47
5.2.1 Arch Dam ... 47
5.2.2 Buttress Dams ... 49
5.3 Influence of reservoir length ... 55
5.4 The inclination of the beaches ... 57
5.5 Pressure distribution over the ice thickness ... 59
6 Discussion ... 61
6.4 Pressure distribution over the ice thickness ... 64
7 Conclusions and further research ... 67
7.1 Conclusions ... 67
7.2 Further research ... 69
Bibliography ... 71
1.1.BACKGROUND
1
Introduction
1.1
Background
Following the increased demand for electrical energy and water supply, which are essential for any growing society. Societies have used dams constructed from different materials as a solution to ensure a steady supply of water and for flood control. At the beginning of the 20th century, the use of generating electricity was introduced to the dam’s function. There are 10 000 dams of all types in Sweden, and 2 000 of those are used for energy production. The purpose of dams is to store water, and then a powerhouse is built to produce energy from the kinetic energy of the water through turbines. (Malm, 2020)
In Sweden, when it comes to hydropower concrete structures design, the engineers should perform the design in accordance with the Swedish dam safety guideline, RIDAS (Swedish hydropower companies guidelines for dam safety). RIDAS refers to the Eurocodes for the design of concrete structures with some modifications to increase the safety requirements. In addition, RIDAS is to a large extent based on the international guidelines developed by ICOLD (International Commission of Large Dams). RIDAS, thereby define how to evaluate hydraulic structures including, the type of loads to consider and their magnitude, type of design scenarios to consider and design requirements for dam stability and cross-sectional design. Yet during the lifetime of the structure, the loads may not reach to the ultimate state load, but it will vary and cause deflections and deformations in the structure. The deformation may lead to cracks, and these cracks will affect the strength capability of the structure. This, together with inadequate assessment and maintenance, may lead to the total failure of the structure. In cold regions, such as Sweden, the temperature variation is significant between summer and winter and can reach up to 70 °C. The winter is usually sufficiently cold that the surface of the river or a lake freezes and any movement of the ice sheet applies pressure to the dams. It is an essential factor for the safety of the dams’ structure to have a better understanding of the ice loads effect on dams and use accurate descriptions of the ice load. (Malm et al.,2018)
The designing ice loads were considered as a unit load, and the value for the load varies between 50-200 kN/m, according to RIDAS (2017). The value for the design load depends on the geographical location (Gebre et al., 2013), water depth and inclination of the waterside (Comfort et al., 2003). Adolfi och Eriksson (2013) has collected the previous ice load data and derived a lognormal distribution of the maximum annual ice load with an average value of 81 kN/m, and a standard deviation of 86 kN/m.
In Norway, the ice load is assumed as a line load as well, but the values varies between 100-150 kN/m according to the Norwegian water resources and energy directorate, (NVE, 2003). The seasonal temperature variation during a year is fairly similar from one year to another, which means that the seasonal ice load on a concrete dam should be quite similar from one year to another. The background is unknown for the theoretical values of the design codes (Jeppsson, 2003).
As mentioned previously, the assumptions about the ice load in the design codes introduces uncertainties to the calculations and it is unclear if the ice load can be considered as a constant line load along a dam. This assumption differs from the real distribution along the dam that has been investigated in this thesis. There have been previous researches conducted to measure the ice load on concrete dams, for example, Malm et al. (2017), Comfort et al. (2003), Carter et al. (1998), Petrich et al. (2015). These studies aimed to determine the actual size of the ice load and the parameters that impact its magnitude. However, there is a lack of studies on the load distribution along different dam geometries and the structural response due to this ice load.
1.1
Aim and purpose
The purpose of this thesis is to study the ice load distribution along a concrete dam and its response during an ice load. The main focus is on the influence of thermal ice load caused by the expansion of the ice sheet due to an uneven or constant temperature profile over the ice thickness. The ice load distribution will be investigated for different dam geometries, to study the influence of the dam stiffness. These investigations will be performed with the finite element method.
The research questions for this project are presented below.
• Is it possible to perform structural analyses that can capture the distribution in thermal ice loads acting on a concrete dam?
• How is the ice load distribution along the dam influenced by different dam types and geometries?
• What is the effect of the beaches shape on the ice load distribution along arch and buttress dams?
• Will the length of the ice sheet affect the results produced in the parametric study?
• Can a simplified finite element study also be used to capture the variation in ice pressure over the ice thickness?
1.2.LIMITATIONS
1.2
Limitations
The performed studies in the thesis are limited to only two types of dams, gravity buttress dams and arch dam. The properties of freshwater ice have been taken into consideration and not saline ice.
The properties of concrete and its behaviour are considered regarding Eurocodes. In the studies of the pressure distribution over the ice thickness, only three monoliths section of the dam will be considered.
Many parameters may influence the ice load effect on dams. However, this thesis focuses on the parameters about the dam geometry, thermal expansion of the ice, temperature variation, ice sheet length and influence from the beaches (different angles of the reservoir with respect to the upstream of the dam).
1.3
Thesis structure
In Chapter 2, the theoretical background of the subjects of this study is summarised. Different types of dams will be described, and typical mechanical loads that act on dams together with a description of ice load and the parameters that influence its size and spatial distribution against a concrete dam.
The theory behind the finite element (FE)-method regarding modelling and analysing concrete structures is explained in Chapter 3. The FE concepts about modelling concrete dams and identifying the boundary conditions and the contact between the dam structure and the foundations are described.
Chapter 4 is focused on describing the study case for this thesis, where the studied models are described including their geometries, properties, loads and the underlying assumptions and approximations that have been considered in this study.
Chapter 5 presents the results from the FE-analysis, in the first part, the results from analyses based on linear and nonlinear interaction between the dam and rock is presented. In the rest of this chapter, the results of the parametric study are presented consisting of displacements, stresses and moments at the contact between the ice sheet and the dam surface.
In Chapter 6, a discussion of the results that were presented in chapter 5 is given. This chapter also contains reflections about the results of this thesis.
Chapter 8 contains the procedure and the outcome of this work and offer some recommendations for further research.
2.1.GRAVITY DAMS
2
Concrete dams
2.1
Gravity dams
Concrete gravity dams depend on the material weight of the structure to counter the forces that applied to it. It is thereby designed so that its self-weight provide sufficient stability and prevent the risk of sliding or overturning failure caused by external loads, i.e. hydrostatic pressure, ice load, uplift pressure. The main characteristic of the gravity dams is that they are made of individually stable sections (monoliths). Each of the monoliths can be considered as a fixed cantilever at the foundation, the horizontal forces acting on the dam cause a moment which generates tensile stresses on the upstream surface of the dam and compressive stresses on the downstream (Chen, 2015). The monoliths are joined with vertical joints, and these monoliths can be constructed of either filled triangular uniform cross-section along the dam, or to save the use of materials the monoliths compose from parts which are the front plate and the buttress. The dams built of the previously mentioned two types of monoliths are called massive dams and buttress dams, respectively (Bergh, 2014).
2.1.1
Mass dams
Mass dams are defined by its characteristically filled triangular cross-section along a dam with a wide base that distributes the loads to the foundation ground and works on stabilising the structure using the structure weight (Bergh, 2014).
The dam is constructed by casting vertical independent parts called monoliths, with a uniform cross-section see Figure 2.1. Each of the monoliths is stable individually and joined together by vertical sealed joints. These contraction joints counteract the thermal dilation effect, i.e. the contraction and expansion of the concrete dam monoliths due to temperature variations after the construction phase is done (Bergh, 2014).
One of the disadvantages of this type of dams appears in the case of uneven foundations. Which leads to uneven settlements of the monoliths, therefore leakage occurs in the contracting joints and cracks form. Leakage and cracking results in increased moisture transport in the concrete and deterioration of concrete and reinforcement through processes such as leaching, frost-damage, carbonisation, corrosion, etc. In addition to this, the bulk of concrete yields low dissipation of the heat generated by the hydration process, which makes the concrete more likely to crack, which may result in deterioration of the dam strength. Another disadvantage of this type of dams is that large amounts of concrete are required to satisfy the stability conditions, which makes it a costly choice economically and environmentally because of high carbon emissions (Chen, 2015).
Figure 2.1 Cross-section for a massive dam
2.1.2
Buttress dams
Concrete buttress dams are a type of gravity dams that have the advantage of saving a significant amount of concrete compared to massive dams. Buttress dams also have reduced footprint compared to massive dams, which reduce the area subjected to uplift pressure. Finally, to increase the stability, an inclined upstream surface is often used in buttress dams (Chen, 2015).
A buttress dam is classified as a gravity dam since it relies on its self-weight for stability, and is constructed of monoliths, similar to massive dams. The monoliths consist of two parts, an upstream plate which is supported by a triangular buttress see Figure 2.2. These two parts have the following properties;
• The front plate is a relatively thin plate and could be constructed as a vertical or inclined plate. A vertical plate resists the water pressure and transfers the loads to the buttress, whereas an inclined plate is not only relying on the self-weight for stabilization but also take advantage of the vertical resultant of the hydrostatic pressure to help stabilize the structure.
• The buttress (or the support), is placed downstream the front-plate. It transfers the loads from the front plate and distributes them on the foundations.
2.2.ARCH DAMS
foundations due to the higher concentration of forces that the buttress transfers to the foundation, which requires a higher quality of the rock.
Buttress dams can be distinguished according to the monolith shapes into three types; massive head, flat slab and multi-arch buttress dams (Chen, 2015).
Figure 2.2 Buttress dams: (a) Massive head buttress type, (b) flat slab buttress type, (c) multi-arch buttress type (Akıntuğ, 2012).
2.2
Arch dams
Arch dams are curved on the upstream side, and partially transfers the hydrostatic pressure by arch action in the horizontal direction to the abutments. Arch dams can be classified into;
• Single curvature arch dams are generally constructed from thick parts joined with a single curve; i.e. horizontal curve see Figure 2.3a.
• Double curvature arch dams are in general slenderer and more constructed as vertical parts joined to form a shell or dome construction. The dam then is curved vertically and horizontally, which improve the dam efficiency of transferring the loads, see Figure 2.3b.
(a) (b)
Figure 2.3 (a) Cross-section of a single curvature arch dam, (b) Cross-section of a double curvature arch dam.
Arch dams are most suitable to be built in narrow canyons and are in general built with a small ratio of span/height up to five. This value may be exceeded, but for higher values, other types of dams are in general, more suitable. Arch dams have been constructed in different geographies, hydrology and climate conditions, which mean that these are not conditional factors (Pedro, 1999).
Other factors to be considered in the choice of arch dams- like any other type of dams- are economical and reliability factors. Arch dams are durable, reliable and have a significantly smaller volume, especially in comparison to mass gravity dams, yet cast on situ arch dams require a more expensive framework and usage of high-quality concrete in addition to the layout challenges. To overcome the economical high cost and maintain the durability for arch dams (Pedro, 1999) concludes that roller-compacted arch dams optimise the economy/reliability balance.
Arch dams transfer loads horizontally to the abutments and vertically to the foundation, hence a large pressure obtained in the foundation, this arises the awareness of the water tightness and foundation stability problems, which were the cause of the dramatic accidents like Malpasset dam (1958), and the Vajont reservoir (1962) (Pedro, 1999) see Figures 2.4 to 2.7. Pedro (1999) concluded that arch dams require rock foundation of high quality with special characteristic regarding drainage, consolidation and water tightness. In addition to this, other essential aspects are making the interface between the dam and the rock as smooth as possible and extend the dam into the foundations,
2.2.ARCH DAMS
Figure 2.4 Malpasset dam after the end of construction (Marinos, 1958).
Figure 2.6 Vajont reservoir dam after the end of construction (Casagrande, 2014).
2.3.LOADS
2.3
Loads
Loads acting on a dam could be divided into static and dynamic loads and permanent and time-dependent loads. The loads that a dam is subjected to are:
• The gravity load, i.e. the dead weight of the dam structure and facilitating parts.
• Hydrostatic pressure, i.e. static water pressure from the headwater acting on the upstream and the tailwater is acting on the downstream part of the dam.
• Uplift, i.e. pore water pressure acting on the surface of contact between the dam foot and the foundation ground.
• Thermal loads, i.e. temperature-related loads from ambient temperature and chemical reactions in concrete during the hydration.
• Seismic loads, i.e. earthquakes.
• Ice loads, i.e. load acting on the contact surface between the ice sheet and dam upstream at winter.
• Wind pressure is a mechanical load due to wind pressure on the dam structure. • Wave pressure, i.e. the pressure loads from streams in the water acting on the
dam upstream.
• Forces in the concrete due to internal reactions.
In this section, a brief description of the static loads will be presented. Figure 2.8 is an illustration for some of the loads mentioned above.
2.3.1
Gravity
Gravity load is the equivalent downward gravity force due to the total weight of the concrete dam and additional appurtenances such as gates, bridges and the rest of the dam components. The weight of the accessories such as gates is negligible compared to the dam weight; therefore, the gravity load can, in general, be calculated using the following equation.
𝐺𝐺 = 𝛾𝛾c∗ V (1)
where,
𝐺𝐺 is the gravity load [kg]
𝛾𝛾𝑐𝑐 is the density of concrete [kg/m3] V is the volume of the concrete dam [m3]
2.3.2
Uplift pressure
Stored water behind the upstream face penetrates the concrete dam through the pores. The water can also be transported through cracks of the foundation material, cracks in the concrete structure and joints between the dam toe and the foundation. The penetrated water results in an upward hydrostatic pressure underneath the dam, which is denoted uplift pressure (Ghanaat, 1993).
The uplift pressure counteracts the vertical downward force, thus lower the resistance to sliding failure. The uplift has a considerable effect on the sliding stability of gravity dams and thick arch dams but can usually be neglected in thin arch dams (Ghanaat, 1993).
2.3.3
Hydrostatic pressure
The hydrostatic pressure is a triangular distributed pressure load that acts perpendicular to the upstream face, where the pressure intensity depends on the water elevation. The pressure force increases linearly from the top to the bottom of the reservoir. In some cases, the presence of water on the downstream face of the dam results in a pressure that works as stabilising force. For safety reasons, the hydrostatic downstream pressure is usually neglected. However, downstream water pressure results in increased uplift pressure. Therefore it is always considered in the uplift load.
2.3.LOADS
The hydrostatic pressure is calculated using the following equation:
𝑝𝑝 = 𝜌𝜌w. 𝑔𝑔. ℎ (2) where,
𝑝𝑝 is the hydrostatic pressure [N/m2] 𝜌𝜌𝑤𝑤 is the water density [kg/m3]
𝑔𝑔 is the gravitational constant [m/s2] ℎ is the height of the hydraulic head [m]
2.3.4
Ice load
The influence of ice load should be considered in cold climates regions. According to RIDAS Sweden is divided into three regions;
• South of Sweden with a horizontal ice pressure of magnitude 50 kN/m and an ice sheet thickness of 0.6 m.
• The middle part of Sweden with a horizontal ice pressure of 100 kN/m and an ice sheet thickness of 0.6 m.
• The north of Sweden with a horizontal ice pressure of 200 kN/m and an ice sheet thickness of 1.0 m.
The ice pressure is assumed to act horizontally on the contact surface of the structure. The ice pressure is caused by several mechanisms, acting separately or combined. The most important causes for the ice load pressure are; temperature variations which lead to thermal expansion and contraction, water level variations and drag forces from wind and currents (Johansson et al.,2013).
The horizontal ice pressure caused by a thermal variation is represented by triangular distributed load degraded from the maximum at the top surface to zero at the bottom surface. The resultant force located one-third of the ice sheet thickness from the top surface (Malm, 2020).
2.3.5
Thermal loads
Temperature variations cause the concrete dams to expand or contract, and if this volume change is restrained, stresses are introduced. The expansion and contraction may vary depending on the type and amount of aggregates and varies typically between ±50 mm, Contraction joints or dilation joints are placed between monoliths or between a dam sections to help the concrete expand and contract without the risk of cracking (Chen, 2015).
If the resulting tensile stresses are equal to the concrete tensile strength, cracks will occur. The restrained volume change may induce cracks in hydropower plants, especially in members with thinner thickness subjected to a temperature gradient from possible inner heat from producing electricity and the outdoor climate. In this dams, since the reservoir water not used in the hydration of concrete, hence the Concrete dries out faster, and as a result, shrinkage occurs much faster. If the design of joints and reinforcement is not sufficient enough to limit the width of the cracks caused by the shrinkage, then deterioration of the concrete dam strength occurs (Chen, 2015).
2.4
Material properties
2.4.1
Concrete
Three parameters can describe the linear behaviour of the concrete in mechanical analyses; elastic modulus poisons ratio and the density of the concrete. The density is only required for calculating the gravity loads in static analyses.
The mean value for concrete that is 28 days old can be calculated by Eq. (2), according to Eurocode 2 (2008).
𝐸𝐸𝑐𝑐𝑐𝑐 = 22 ∗ (𝑓𝑓10 )𝑐𝑐𝑐𝑐 0.3 (3) where,
𝐸𝐸𝑐𝑐𝑐𝑐 is the mean value of elastic modulus of concrete
𝑓𝑓𝑐𝑐𝑐𝑐 is the mean value of cylindrical compressive strength of concrete
The elastic modulus, i.e. the stiffness of concrete, increases with hardening while creep or cracking cause a reduction in the elastic modulus.
According to Eurocode 1 (2011), the density of the concrete is assumed to 24 kN/m3. In the case of the concrete dams, it is used 23 kN/m3 RIDAS (2011), the reason for this is gravity load is beneficial in the stability analyses.
According to Eurocode 2 poisons ratio is 0.2 for uncracked concrete and 0 for cracked concrete.
FE-analysis also requires the thermal expansion coefficient a which varies between 0.74 ∙ 10−5 ≤ 𝛼𝛼 ≤ 1.3 ∙ 10−5 [𝐾𝐾−1] but according to Eurocode 2 the assumption of alfa can be 𝛼𝛼 = 1.0 ∙ 10−5 [𝐾𝐾−1].
2.5.ICE LOAD MEASUREMENTS
2.5
Ice load measurements
The ice load varies depending on external variables and may be caused by different factors. According to Johansson et al. (2013), the ice load can be influenced by the following mechanisms:
• Temperature variations. • Water level variations. • Water currents and winds.
These mechanisms can affect the ice load separately or as a combination of two or more mechanisms at the same time, which makes it hard to study the influence of one factor on the ice load without enough data. That data needs to be extracted during a long period of time in different locations of the dam, to describe the variation of one or more factors and record their effect on the ice load. Therefore, the complexity of the experiments that have been made varied depending on the resources that were available for researchers. When considering combined factors for the ice load, the complexity of data analysis increases significantly. Therefore, a higher risk of interpretations and decrease of precision will occur, which will make it more difficult to draw conclusions about the ice load, according to Comfort et al. (2003).
The size of ice loads has been found in Previous measurements were between 100 kN/m to 460 kN/m (Adolfi and Eriksson, 2013; Saether, 2012). These measurements show a significant difference compared to the design ice line-loads between 50 kN/m and 200 kN/m that RIDAS (2017) recommends.
2.5.1
Thermal effects on ice load
The influence of thermal effects on the ice load is governed by the thermal expansion of the ice. The temperature of the bottom of the ice sheet remains constant 0˚C, while the top surface varies due to the ambient air temperature see Figure 2.9 (Johansson et al.,2013).
Figure 2.9 Thermal loads (Malm et al., 2017)
According to Hellgren et al. (2019), the ice load due to temperature gradient along the ice thickness has an inverse relation with the ambient air temperature. The daily measurement of ice load and air temperature shows that the ice load peaks occur when the temperature drops. This relation is inverted compared to the normal expectation that ice load increases due to the expansion from increasing temperature. Hellgren et al. (2019) presents a figure showing how the ice load differs daily with the change of the temperature during one spring see Figure 2.10.
Figure 2.10 Ice loads and temperature variation during one spring (Hellgren et al., 2019)
2.5.ICE LOAD MEASUREMENTS
and expansion of the ice sheet. Due to the expansion of the ice sheet, the ice load pressure on the concrete dam increase (Johansson et al., 2013).
When the top surface of the ice sheet expands and contracts, a bending moment occur over the ice thickness. This bending moment cannot be carried by the ice sheet since the ice sheet is floating on the water surface. Hence, the ice sheet cannot bend upwards nor downwards, since the ice is restrained against bending by its self-weight. Considering that the ice sheet is restrained from bending while a low resistance to the tensile forces caused by bending moments, the ice sheet will crack(Bergdahl, 1977).
Another cause of the ice sheet to crack iswhen the temperature changes unevenly along the ice sheet. For instance, if the ambient temperature rises, causing the heated surface to expand. The expanding part of the ice sheet will face resistance from the surrounding boundaries, and this resistance leads to rising in tension and the ice sheet cracks to relieve this tension (Johansson et al., 2013)
There are more parameters that affect the thermal effect on an ice sheet. Some examples are; the reservoir surface shape and its boundary, if an ice sheet is free to grow in one or more directions, is an ice sheet constrained by the surrounding geometry, and the snow cover over an ice sheet (Malm et al., 2018 ).
The stress distribution within an ice cover is quite nonlinear; however, for simplicity, the stress distribution is often considered linear in previous researches. The expected stresses at the top of the sheet are to be -200 kPa and +200 kPa at the bottom for a typically cold winter. For severe cold peaks, the tension may go up to -600 kPa at the top surface, and the compressions in the bottom surface also reach +600 kPa (Kharik et al., 2017).
2.5.2
Reservoir water level effect on ice load
Water level variation results in tensile and bending stresses near the bond in the interface between an ice sheet and a dam due to sag. The reservoir water level effect varies depending on some factors :
• The difference in water level variation. • The sequence of the water level variation. • The duration takes for the water level to change.
In an eight-year research program, Comfort et al. (2000) derived the effect of these factors into three main classes:
• Negligible effect: when the water height change is small (10 - 15 cm ) with a low rate for changing (0 - 0,5 times/day).
• Medium effect: the water height change between (10 - 30 cm) with relatively frequent changes (1 - 2 times/day).
• Large effect: with water height change between (35 - 45 cm ) and frequent. The results from seven measured points on different dams show that an ice load reaches as high as 300 kN/m due to the variation in reservoir water level.
A study by Stander (2006) supports the assumption of an ice sheet bonded to the dam upstream face would crack when the reservoir water level is lowered. These cracks will be filled with water and freeze, which will lead to the expansion of the ice sheet. This expansion will cause pressure on the dam face under some condition that Stander (2006). To mention one of the conditions and not limited to, an ice sheet should be limited from expanding to the free water direction. Otherwise, the expansion will cause negligible compressive stresses on the dam surface, to read more see Stander (2006). The previous condition was one of the assumptions that had been considered while modelling the ice sheet in this thesis.
2.5.3
Other factors that effects the ice load
The effect of dam geometries will be studied in this thesis since it was encouraged to investigate ice load distribution along with dams due to the lack of studies in this regard. An example of previous studies there is the studies of Morse et al. (2011) and Taras et al. (2011), where the stress distribution studied with sensors placed three meters apart along the dam. The outcomes of these two studies show that the ice load acting on the dam varies significantly along a few meters of the dam and has high and the variation is described as chaotic. In addition, when at one sensor high load is measured, the magnitude measured in that sensor is always higher than the average measured stress over two or more sensors.
Another property that influences ice load on a dam which has not been investigated to any larger extent is the influence of reservoir beaches. Therefore in this thesis, the influence of the angle of the reservoir beaches with respect to the dam surface was studied, i.e. the shape of the reservoir. As mentioned in Hellgren ( 2019 ), the only reference is found that studied the influence of reservoir beaches is a study by Ko et al. (1994). Ko et al. ( 1994 ) study the influence of beach slope not beaches angle, the study divides thermal ice load into three categories according to the slope of the beaches. Steep beaches which have the slope smaller than 45° the design load is increased by 1.5 times compared to flatter beaches with a slope higher than 20°.
3.1.GEOMETRICAL MODEL
3
Finite element analyses of concrete
dams
When it comes to using FEM software like Abaqus, the general procedure consists of three steps, which are; processing, analysis, and post-processing. The first step, pre-processing, is defining the required parameters such as geometry, material properties, boundary condition, and loads. The next stage is the analysis where, for instance, the defined geometry or model in the software can be analysed depending on the purpose of the study. The last step is post-processing, which is evaluating the obtained results (Malm, 2016).
3.1
Geometrical model
For continuous gravity dams, it is better to utilise the plane strain theory and use 2D analyses. If material nonlinearity is included in the analyses, 2D analyses prevent the out of plane cracking in the model. Figures 3.1 and 3.2 below are shown as an example of a situation where plane and strain theory can be preferred to utilise (Malm, 2016).
Figure 3.31 Gravity dam with a fracture in the rock at the upstream toe (Ruggeri, 2004).
For arch dams, where the cross-section various along the width of the dam, or when the load is carried in the lateral direction of the dam, i.e. through arch action, then Malm (2016) recommends performing the analyses with 3D solid elements. Concrete buttress dams can be analysed by using either 2D or 3D solid elements. If the 2D analysis is used, then the lateral bending of the front plates is disregarded. Therefore, 3D solid or shell elements are a better option for even the buttress dams. Figure 3.3 below shows the example of where 3D solid elements are used for the analyses (Malm, 2016).
3.2.MESHING
problem, the shell elements may have to be defined with an offset to represent the real geometry, and its real stiffness can be represented (Malm, 2016).
3.2
Meshing
There are several element types to choose from to mesh their model, as an engineer, it is important to know about them so that these can be used where appropriate. Different element types have a different degree of freedom; below is a list of the common element types used in the FEM (Malm, 2016).
The simplification of a real structure is the hardest part of the numerical analysis for an inexperienced engineer, due to the lack of understanding the fundamental behaviour. For instance, to translate the real structural behaviour into boundary conditions, loads, interaction etc.
Figure 3.4 is showing the most common element types in structural analysis. Most common types of elements and their degrees of freedom in FEM are :
• Truss elements (three degrees of freedom, translations, at the nodes) • Solid elements (three degrees of freedom, translations, at the nodes)
• Beam elements (six degrees of freedom, translations and rotations, at the nodes) • Shell elements (six degrees of freedom, translations and rotations, at the nodes)
Figure 3.4 Common elements for structural analyses (Malm, 2016).
There are few more differences than only in degrees of freedom in elements. For the design of the beam, the cross-sectional forces are interesting. In the Euler Bernoulli theory, the shear deformations are neglected, but the theory of Timoshenko considers the shear deformation. Similarly, for shell elements, the option is between Kirchhoff that neglect the shear and Mindlin theory includes it (Malm, 2016).
When connecting a shell or a beam to a solid element, only the translation can be restrained. This means that the rotations in the beam or shell are unrestrained and acts as a hinge. In order to solve this problem, another vertical shell element has to be created
in the connection between the elements almost like a plate. This solution gives a distributed pressure over the face of a solid element, see Figure 6.2 for an illustration.
3.2.1
Shape function
FE-theory is based on nodes which are discrete values that describe the behaviour of a structure. Between these nodes are the solution which is interpolated either by linear shape function or quadratic. Interpolation is made by shape function, and an illustration is presented in Figure 3.5 below (Malm, 2016). The line that is straight between the nodes in the linear and the line that is bent is the quadratic shape function.
Figure 3.5 Shape function (Malm, 2016).
To describe the bending of the structure, a linear shape function is a poor choice. A better option for bending is to choose a quadratic shape function. If the serendipity elements are excluded, the quadratic shapes have twice as many nodes as the linear shape functions; this doubles the total degree of freedom for the structure and leads to expensive calculations. An example can be seen in Figure 3.6, where it presents a 2D plane stress or plane strain element. The left one is four noded elements, and the right one is eight noded elements (Malm, 2016).
Figure 3.6 2D plane stress elements, left one is four noded, and the right one is eight noded elements (Malm, 2016).
3.3.BOUNDARY CONDITIONS AND INTERACTIONS
3.2.2
The discretisation of the elements
The definition of the mesh should be on a way where skew elements are avoided. Figure 3.7 below shows the error in percentage when a point load is applied to a cantilever beam. One important thing to have in mind is to perform convergence test, means that analysis should be done with both higher and lower grade elements and it is preferred to have lower grade linear elements in case material nonlinearity includes which requires smaller elements. Smaller element size obtains a better representation of the crack pattern. To increase the accuracy of the FE analysis, either the linear upgrade elements to quadratic elements or increase the total number of the elements (Malm, 2016).
Figure 3.7 Error of elements (Eriksson 2002 and Malm 2016).
3.3
Boundary conditions and interactions
In the definition of boundary conditions and interactions, one of the many factors that influence the accuracy of the numerical model is the boundaries, and how large part of the actual dam is included in the analyses. The models must capture the mechanical modes of action for the part that is analysed. If the simplifications of the model are made correctly, then there is practically no significant difference between the model and the real structure. This means that engineering judgment is an important factor when evaluating the results.
In some cases, it is necessary to model the whole structure, for instance, in the case of arch dams, according to Malm (2016). When analysing a whole dam, special consideration may be needed at the contraction joints. A valid assumption could be in this case to includes the joints and not allow force transmission over the joints. Those joints often consist of PVC, steel, or bitumen.
Another important factor in FE modelling is the interaction between the rock and the dam. Interaction can be handled by three approaches, including the large part of the rock, using spring elements, defining the boundary condition at the bottom of the dams (Malm, 2016).
Special consideration also has to be taken into account when combining solid elements with shell elements. A direct connection between a shell and solid elements results in a
high displacement and stress at the connection node, which works like a hinge and no bending moment can be transmitted from solid to shell. To transfer the bending moment from solid to shell, it should be connected through a rigid link. Figure 3.8 illustrates both types of connections on how to couple solid to shell elements.
Figure 3.8 To the left it is a direct connection between shell and solid element. To the left a common displacement restraint between the nodes is shown. To the right, a rigid
link connection is shown which also ensures moment restraint between the solid and the shell strutures.
3.3.1
Interactions between the surfaces
One way to simplify the contact between the rock and concrete when analysing dams is by to assume complete bond. This can be a fair assumption in some cases if the concrete has some bond strength to the foundation, for instance, when the rock surface is blasted and cleaned from loose particles. However, it must be ensured to not subject the dam body that is closest to the rock for high tensile stresses since the bond and shear strength is lower than the tensile shear strength (Malm, 2016).
In order to make the assumption of the FE-model more realistic, a contact formulation between the rock and the dam can be used. This contact formulation should include the frictional resistance of the horizontal plane and high stiffness for compressive forces in the normal direction. It is important that the contact formulation also allows separation between the rock and the dam when subjected to tensile forces, which may occur in case of an overturning failure. According to RIDAS (2011), the friction coefficient can be set as μ= 1.0 for concrete dams built on rock (Malm, 2016).
In contact formulations, it is important to consider which nodes that are defined as master and slave. The surfaces with coarser mesh are recommended to be defined as a master surface, according to Malm (2016). The reason for this is that the mid-nodes on the slave surface can be adjusted to remove penetration based on a linear interpolation between the nodes of the master surface. This increases the convergency and the convergency rate (Malm, 2016).
3.3.BOUNDARY CONDITIONS AND INTERACTIONS
separation and the normal pressure is decides the pressure-overclosure relationship (Dassault Systèmes, 2014). Figure 3.9 and Figure 3.10 below shows the normal pressure when using these two types of contact.
Figure 3.9 Normal pressure when using soft contact (Dassault Systèmes, 2014).
Figure 3.10 Normal pressure when using hard contact (Dassault Systèmes, 2014).
3.3.2
Modelling of the rock
An option for the boundary condition is to include the rock mass into the numerical model, but then there are some recommendations for this type of boundary condition that needs to be considered in the FE-analysis (Malm, 2016).
The boundary condition should be in the outer part of the rock mass, so that the displacements of the rock is restricted in the direction perpendicular to each external surface.
4.1.DAM GEOMETRIES
4
Case studies
In this chapter, the two different case studies used to analyse the influence of ice load on concrete dams are described.
Case 1 involved stability analyses of two different concrete dams to assess the influence of an ice load on the dam behaviour. The first dam was an arch dam and the second dam was a monolith from a buttress dam. These geometries had been modelled in the FEM-software Abaqus. The stability analyses were performed using two different approaches for the concrete-rock interface, where both complete bond and a contact formulation had been studied.
Case 2 was consisting of a parametric study regarding the loads acting on the two different concrete dams caused by the expansion of an ice sheet. The parametric study also included investigations of the shape of the reservoir beaches, where it either had a perpendicular shape towards the surface of the dam, or it had an angle on 30 degrees. In both cases that are mentioned above, the analyses are performed with linear elastic material behaviour. In the analyses with linear elastic material behaviour, the reinforcement had a minor influence on the results. The reason for this is that the stresses of embedded reinforcement are small in uncracked concrete. Therefore, in instability analyses, the reinforcement had a minor effect of the behaviour of the dam, and only influence the gravity load. This can, however, easily be compensated by defining a density of the concrete that includes the weight of reinforcement (Malm, 2020). For this reason, the modelling of reinforcement had been not taken into consideration since it is not going to affect the stability of the dam.
4.1
Dam geometries
Five different geometries of the concrete buttress dam and one arch dam had been studied. The buttress dam consists of a concrete buttress dam built out of 3 and 9 monoliths as seen in Figure 4.2 – 4.6 in the geometry consisting of 9 monoliths, three of the monoliths on each side were placed with an angle of 30 degrees, see Figure 4.6. The last one is the same arch dam that was studied previously. It is connected to a spillway and abutment on sides, see Figure 4.7. All the different geometries studied in case 2 are summarised in Table 4.1.
4.1.1
Concrete buttress dams
The concrete dam consists of one single monolith of a small typical Swedish concrete buttress dam. Figure 4.1 illustrates the model that had been analysed in Case 1.
Figure 4.1 FE-Model consist of one monolith of a buttress dam considered the foundation to rock.
All the geometries that had been studied in case 2 are included in table 4.1. Table 4.1 Dams geometries that are considered in the study of case 2
Geometries that had been analysed in case 2 The numbers of figures that are presenting the model Three monoliths dam with perpendicular ice sheet Figure 4.2
Three monoliths dam with inclined ice sheet Figure 4.3 Nine monoliths dam with perpendicular ice sheet Figure 4.4
Nine monoliths dam with inclined ice sheet Figure 4. 5 Rotated Monoliths dam Figure 4.6 Arch dam with perpendicular ice sheet Figure 4.8
Arch dam with inclined ice sheet Figure 4.9 Monolith
4.1.DAM GEOMETRIES
Figure 4.2 Three monoliths dam with the perpendicular ice sheet.
Figure 4.3 Three monoliths dam with the inclined ice sheet.
Figure 4.4 Nine monoliths dam with the perpendicular ice sheet.
Figure 4.6 Rotated Monoliths dam.
4.1.2
Concrete arch dam
The dam is a typical Swedish arch dam previously used in the 14th ICOLD International Benchmark Workshop on Numerical analysis of Dams, see Malm et al. (2018). In Figure 4.7, the arch dam is illustrated. It is supported by abutment and spillway on each side, from underneath there is a rock support foundation for the arch dam.
Figure 4.7 FE-Model for an existing arch dam in Sweden.
Arch Dam
4.2.MATERIAL PROPERTIES
Figure 4.8 Arch dam with the perpendicular ice sheet.
Figure 4.9 Arch dam with inclined ice sheet.
4.2
Material properties
The material properties that were used for concrete in this thesis are presented in Table 4.2, and Table 4.3 presents the ice sheet properties that were used in the models. The values specified for the arch dam can be referred to the 14th ICOLD Benchmark proceedings (Malm et al. 2018)
Table 4.2 Material properties for concrete. Material properties Unit Concrete Rock
Quality C30/37
Density (Monolith) kg / m3 2300 2700
Density (Arch dam) kg / m3 2400 2700
Poisson's ratio 0.2 0.15 Young's modulus GPa 33 40
Table 4.3 Material properties for the ice sheet. Coefficient Value Youngs modulus 8.7*10^9 Expansion Coeff. 1*10^-5 Poissons ratio 0.31 Mass density 918.9 Conductivity 2.3
All the models were analysed with the fixed time steps using 10 increments. The only load that was applied to dams’ models was from the ice sheet; the rest of the loads were deactivated. The water level was assumed to 1.5 below the crest. In order to give analyse the stresses at the edges, an inclination on 30 degrees were given to the edges.
4.3
Mesh
A mesh in Abaqus was created to analyse the monolith model. The used elements are four noded tetrahedral(C3D4). The details for the mesh are presented in Table 4.4 and shown in Figure 4.10.
4.3.MESH
Table 4.4 Mesh properties for one monolith model. Part Element
type Element length (m) Number of nodes Number of elements
Ground C3D4 1.2 1707 7535
Monolith C3D4 0.35 9481 45514
Figure 4.10 The mesh that is used for one monolith model.
Mesh details for three monoliths and the ice sheet is presented below in Table 4.5. Element types are linear four noded tetrahedral that are used for monoliths, and four noded quadrilateral elements are used for the ice sheet.
Table 4.5 Mesh details for three monoliths and the ice sheet. Part Element
type Element length (m) Number of nodes Number of elements Ice sheet,
perpendicular S4R 0.75 1089 1024 Ice sheet, inclined S4R, S3 0.75 2566 2484 Monoliths C3D4 0.75 3870 15012
Mesh details for nine monoliths and the ice sheet is presented below in Table 4.6. Element types are linear four noded tetrahedral that are used for all monoliths, and four noded quadrilateral elements are used for the ice sheet.
Table 4.6 Mesh details for nine monoliths and the ice sheet. Part Element
type Element length (m) Number of nodes Number of elements Ice sheet,
perpendicular S4R 0.75 3201 3072 Ice sheet, inclined S3,S4R 0.75 22420 22405
All Monoliths C3D4 0.75 54450 45036
Mesh details for the arch and the ice sheet is presented below in Table 4.7. Element types are linear four noded tetrahedral that are used for all monoliths, and four noded quadrilateral elements are used for the ice sheet.
4.3.MESH
Table 4.7 Mesh details for the arch dam and the ice sheet. Part Element
type Element length (m) Number of nodes Number of elements Ice sheet,
perpendicular S4R 0.75 3201 3072 Ice sheet, inclined S4R 0.75 22420 22405
Arch C3D4 0.75 66697 330384 Foundation C3D4 0.75 50551 259038 Spillway C3D4 0.75 33493 174343 Abutment C3D4 0.75 4441 21832 Small Rock C3D4 3 8165 36945 Large rock C3D4 3 58517 306085
Mesh details for rotated monoliths and the ice sheet is presented below in Table 4.8. Element types are linear four noded tetrahedral that are used for all monoliths, and four noded quadrilateral elements are used for the ice sheet.
Table 4.8 Mesh details for the model with rotated monoliths. Part Element
type Element length (m) Number of nodes Number of elements The ice sheet,
straight S4R 0.75 3201 3072 Monolith C3D4 0.75 53250 53036
4.4
Loads and loading procedure
4.4.1
Case 1
The loads that had been taken into account in the analyses are; gravity load, hydrostatic pressure, ice load and uplift load from the hydrostatic press. The models were defined with three load steps, where the gravity load was applied as a first step. In the second step, the effect from the gravity load is propagated, and the hydrostatic pressure is applied with the default hydrostatic distribution in Abaqus. In the last step, the ice load is applied as pressure with a triangular distribution. The ice load is defined as a height of one meter, starting from the point that is just below the crest and applied along the whole dam length. An illustration of the ice load distribution is seen in Figure 4.11. The maximum ice load pressure was defined as 400 kN/m2, which results in a pressure of 200 kN/m along the dam crest for both dams. With this distribution, the resultant force is applied 1/3 from the top surface in accordance with the specifications in RIDAS.
Figure 4.11 The ice load distribution on the point just below the crest.
How monolith is loaded for in analysis model is illustrated in Figure 4.12. The ice load is acting at the top of the monolith; the surface it is acting on is also marked in Figure 4.12 with a rectangle. The rest of the surface is subjected to hydrostatic pressure, and an arrow that is pointing downwards is the gravity load.
4.4.LOADS AND LOADING PROCEDURE
Figure 4.12 Loads illustration for monolith in case 1.
Illustration for loads acting on the arch dam is represented in Figure 4.13. The gravity load in the figure is pointing downwards and at the top of the dam is the ice load, and the rest of the surface is subject to hydrostatic pressure.
4.4.2
Case 2
The load that is acting on the concrete dams in case 2 was caused by thermal expansion of the ice sheet through the thickness. This expansion had been defined as a predefined field function in Abaqus. It had either been set as a constant temperature on the ice sheet or as a temperature gradient where the top surface and the bottom surface had two different temperatures. One of the expansions had been set at a time to analyse all models in case 2. The top surface temperature is set to + 15 ˚C and the bottom to 0 ˚C for the gradient. The constant temperature on the ice sheet was set to +15 ˚C. The pressure in the models had been extracted as a nodal force. The unit for these nodal forces are kN/m, so it had been divided by the height of the contact surface, i.e. the ice thickness of 1 m, to convert it to pressure.
4.5
Boundary conditions and interactions
4.5.1
Case 1
The interaction between the monolith and rock was defined with a surface to surface contact using nonlinear interaction. The behaviour of the contact in the normal direction, was set as a type of hard contact, see Section 3.3.1. In the tangential behaviour, the friction value in the buttress dam was defined as 1.0 according to RIDAS, and the contact thereby allows for sliding behaviour. The arch dam friction value was defined as a 40 to represent the reality, where the dam was going through the foundation, and therefore, the sliding of the dam was restricted. See figure 4.14 for illustration how the bottom surface of the monolith is connected to the foundation below, where the highlighted surfaces had the hard contact formulation.
4.5.BOUNDARY CONDITIONS AND INTERACTIONS
The boundary condition was set for monolith in case 1 is represented below by Figure 4.15. The arrows are showing the restricted directions for translations in the model.
Figure 4.15 Boundary condition for monolith in case 1.
The boundary condition was set for the arch dam in case 1 is represented below by Figure 4.16. The arrows are showing the restricted directions for translations in the model.
4.5.2
Case 2
In case 2, the interaction between the rock and the concrete dams surfaces had been considered as fixed boundary conditions see Figure 4.17. This means that the rock was neglected in the FE analysis, and only the concrete dams had been considered. Figure 4.18 shows an example of the connection between the monoliths in studied buttress dams. The connection between the monoliths surfaces had been set to tie connection. This means that all the monoliths were considered as one structure.
Figure 4.17 Boundary condition for the arch dam in case 2.
Figure 4.18 Tie connection between the monoliths
The connection between the concrete dams and the ice sheet had been defined as a tie connection in Abaqus for all the models that were included in case 2, see Figure 4.19 for
4.5.BOUNDARY CONDITIONS AND INTERACTIONS
the ice sheet as a rigid link see Figure 4.20. A tie connection between an ice sheet and a concrete dam is presented in Figure 4.19.
Figure 4.19 Direct tie connection between the monoliths and ice sheet.
Figure 4.20 Rigid link connection between the monoliths and ice sheet.
Figure.4.21 represents how the boundary conditions for perpendicular ice sheet and buttress dam that are set in the models with constant temperature on the ice sheet. The ice sheet has been restricted from bending upwards or downwards by setting a constrain.
Figure 4.21 Tie connection between the monoliths and perpendicular ice sheet, including all the boundary conditions.
Figure 4.22 shows the boundary condition for the buttress dam and inclined ice sheet with a temperature gradient; the bending in the ice sheet had been allowed.
Figure 4.22 Tie connection between the monoliths and ice sheet, including all the boundary conditions.
5.1.ICE LOAD FORCE AND DEFORMATION RELATION ON DAMS
5
Results
In this chapter, the most important results that had been generated from the numerical analyses are presented. Two main aims of interest for this project are presented in the results; one is to model the ice load as a simple triangular load and study the relation between ice load and deflection of the dam, second is to model the ice sheet and study the variation in ice load due to different dam geometries. Data of the deflection was taken at the dam crest since the maximum deflection occurred at the crest, for more results see Appendix A.
5.1
Ice load force and deformation relation on
dams
To establish an idea about the force and displacement relation for the effect of ice load on a concrete dam, models mentioned in chapter 4.1.1 had been used (one monolith dam and arch dam). Depending on the dam foundation contact condition, the deflection-force relation had either linear or nonlinear behaviour.
5.1.1
Linear behaviour of the concrete dam
When the interaction between rock and studied dams assumed as fixed, i.e., the translations and rotations were prevented in all directions. As a result, Linear relationship between the ice load pressure applied and the deflection was produced. The structure would continue to deform with the increase of the resultant pressure force caused by the triangular ice load until it reached the maximum applied load, As shown in Figures 5.1 and 5.2 for buttress and arch dams respectively. Figure 5.3 is presenting the deformed shape of the arch dam with a linear relation between displacement and ice load force.
Figure 5.1 Linear displacement-force graph in one monolith of a buttress dam.
Figure 5.2 Linear displacement-force graph in the arch dam.
0 50 100 150 200 250 300 350 400 1,19 1,26 1,33 1,39 1,46 1,53 1,60 1,67 1,74 1,81 1,88 Fo rc e (k N) Displacement (mm) 0 50 100 150 200 250 300 350 400 17,20 17,54 17,89 18,24 18,59 18,94 19,29 19,64 19,99 20,34 20,69 Fo rc e (k N) Displacement (mm)
5.1.ICE LOAD FORCE AND DEFORMATION RELATION ON DAMS
Figure 5.3 Deformed shape of the arch dam with linear displacement-force relation.
5.1.2
Nonlinear behaviour of the concrete dam
When the interaction between rock and dam was considered more complex than the first case, i.e. there was hard contact between the dam and the rock with friction coefficient. The dam deflection had a linear relation with the applied ice load force until it reached the nonlinear relation state. The structure either slide where the deformation started to increase significantly even for a constant load, as presented in Figure 5.4 and Figure 5.6 for the monolith and arch dam respectively. To reach the nonlinear behaviour, the ice pressure load had to be applied significantly higher than the maximum design load and cannot be reached in real-life situations. In Figure 5.5, the deformed shape of one monolith buttress dam after sliding failure is shown.
Figure 5.4 Nonlinear displacement-force graph in one monolith of a buttress dam.
Figure 5.5 Deformed shape of one monolith of a buttress dam after sliding failure.
0 150 300 450 600 750 1 1,5 2 2,5 3 3,5 4 Fo rc e (k N) Displacement (mm)
5.2.VARIATION IN ICE LOAD DUE TO DAM GEOMETRY
Figure 5.6 Nonlinear displacement-force graph in the arch dam.
5.2
Variation in ice load due to dam geometry
This parametric study was performed to analyse how the thermal expansion of an ice sheet caused displacements and normal forces along the dam for different dam geometries. The studied geometries are presented in the following for two different dam types; arch dams and buttress dams. The normal forces acting on the concrete dam were caused by thermal expansion of the ice sheet. This ice sheet was subjected to either constant temperature or a temperature gradient between the top and the bottom surface of the ice sheet.
5.2.1
Arch Dam
• Constant temperature
The displacement along the crest of the arch dam and the variation in normal forces along the crest is presented in Figure 5.7 for a constant temperature increase of +15 ˚C in the ice sheet (the positive displacement sign is towards the downstream direction, the origin was placed at the spillway). The maximum displacement, and subsequently the lowest force, appeared almost in the middle of the arch. The displacement was zero at both ends, and the normal forces at the ends of the arch were maximum due to the boundary conditions that were set in the FEM model. Figure 5.8 is presenting the deformed shape of the arch dam due to constant temperature expansion of the ice sheet.
0 150 300 450 600 16,5 17 17,5 18 18,5 19 19,5 20 20,5 Fo rc e (k N) Displacement (mm)
Figure 5.7 Normal force and displacement along the arch dam due to constant temperature increase in the ice sheet.
Figure 5.8 Deformed shape of the arch dam due to constant temperature expansion of the ice sheet.
• Thermal gradient
The displacement and normal force in the arch dam due to the expansion of the ice sheet subjected from a thermal gradient is shown in Figure 5.9. The thermal gradient in the ice sheet was defined to be + 15 ˚C at the top surface and 0 ˚C at the bottom (the positive displacement sign is towards the downstream direction, the origin placed at the spillway). The maximum displacement in the arch dam occurred in the connection between the spillway and arch then gradually decreased along the arch. Figure 5.10 is
Spillway Aboutment 0,0055 0,011 0,0165 0,022 0,0275 0,033 0,0385 0,044 0,0495 0,055 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0 17 34 51 68 85 102 119 136 153 170 Di sp la cemen t [ m] No rma l fo rc e (N /Nm ax )% Arch length (m)
5.2.VARIATION IN ICE LOAD DUE TO DAM GEOMETRY
Figure 5.9 Normal force and displacement along the arch dam due to thermal gradient.
Figure 5.10 Deformed shape of the arch dam due to a thermal gradient.
5.2.2
Buttress Dams
• Constant temperature
The displacements and forces when the thermal expansion occurred due to constant applied temperature in the ice sheet are presented in Figures 5.11, 5.15 and 5.16 (the positive displacement sign is towards the downstream direction). In Figure 5.11 displacement for models -three and nine monoliths- is compared, in these models, the ice sheet was with straight edges. The displacement for both dams was zero at the dam boundaries and increased to follow a harmonic shaped curve with the maximum displacement between two monoliths and the lowest at the buttresses. Figures 5.12 to 5.14 are presenting deformed shape of three and nine monoliths buttress dams due to constant temperature expansion of the ice sheet.
Spillway -0,00005 -0,00004 -0,00003 -0,00002 -0,00001 0 0,00001 0,00002 0,00003 0,00004 -40% -20% 0% 20% 40% 60% 80% 100% 0 17 34 51 68 85 102 119 136 153 170 Di sp la cemen t [ m] Nor m al for ce (N /Nm ax ) % Arch lenghth (m)
Figure 5.11 Displacement for three and nine monoliths buttress dam due to the constant temperature.
Figure 5.12 Section view for the deformed shape of three monoliths buttress dam due to constant temperature. 0 0,002 0,004 0,006 0,008 0 10 20 30 40 50 60 70 Di sp la cemen t ( m) Dam length (m) 9 Monoliths 3 Monoliths
