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TRITA-BKN. MASTER THESIS 458, 2015 ISSN 1103-4297,

ISRN KTH/BKN/EX–458–SE

Verification of the response

of a concrete arch dam

subjected to seasonal

temperature variations

OSKAR ANDERSSON MAX SEPPÄLÄ

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Verification of the response of a concrete

arch dam subjected to seasonal temperature

variations

Oskar Andersson & Max Seppälä

June 2015

TRITA-BKN. Master Thesis 458, 2015

ISSN 1103-4297,

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Department of Civil and Architectural Engineering Division of Concrete Structures

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Abstract

Many dams existing today were constructed around fifty years ago. Condition mon-itoring is essential for maintaining high safety and determining the current level of safety and stability for these dams. There is a need for new monitoring tech-niques and finite element coupled monitoring could be one of these techtech-niques. A concrete arch dam located in Sweden is modelled and calibrated with respect to concrete temperature measurements. The temperature distribution is then defined as a prescribed strain in a structural mechanical model in which a parametric study is performed. The results from the parametric study are compared to measurements of the crest deformation and a combination of parameters is found giving the lowest difference between measurements and model results for the mid-section.

The results show that the finite element model can be used to predict the behavior of the dam with acceptable deviation. The parametric study indicates that the refer-ence temperature of the concrete has little effect on the amplitude of the deformation and that the governing factor is the coefficient of thermal expansion.

Keywords: Concrete arch dams, thermal effects, ice load, condition monitoring, finite element analysis

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Sammanfattning

Många av de dammar som finns idag byggdes för omkring femtio år sedan. Till-ståndsövervakning är avgörande för att kunna bestämma nivån av säkerhet och stabilitet för dessa dammar. Det finns ett behov av ny övervakningsteknik och finita element-kopplad övervakning kan vara en av dessa tekniker. En betongvalv-damm modelleras och kalibreras med avseende på uppmätt betongtemperatur. Den beräknade temperaturfördelningen definieras sedan som en föreskriven töjning en strukturmekanisk modell i vilken en parametrisk studie utförs. Resultaten från pa-rameterstudien jämförs med mätningar av kröndeformation och en kombination av parametrar identifieras som ger lägsta skillnad mellan mätningar och modellresultat för mittsektionen.

Resultaten visar att modellen kan användas för att förutsäga dammens beteende med acceptabel avvikelse. Parameterstudien indikerar att referenstemperaturen för betongen har liten inverkan på amplituden för deformationen och att den styrande faktorn är längdutvidgningskoefficienten.

Nyckelord: Betongvalvdammar, termiska effekter, tillståndsövervakning, islast, finita element analys

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Preface

This is a Master of Science thesis performed at the Division of Concrete Structures, Department of Civil and Architectural Engineering at KTH Royal Institute of Tech-nology in collaboration with WSP and Fortum during the period January-June 2015. The thesis subject was initiated by Rikard Hellgren, WSP, and supervised by Dr. Richard Malm, KTH.

We would like to thank Dr. Richard Malm and Rikard Hellgren for the invaluable guidance and support they have given us during this period. We would also like to thank Sezar Moustafa at Fortum for providing us with material for the case study and for believing in the subject and sharing useful knowledge with us. We would further like to thank Dr. Fredrik Johansson at the Division of Soil and Rock Mechanics for taking the time to discuss the subject and give us useful comments. We would finally like to thank Scanscot for providing us with necessary software licenses.

Stockholm, June 2015

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Contents

Abstract iii Sammanfattning v Preface vii 1 Introduction 1 1.1 Background . . . 1

1.2 Aim and scope . . . 2

1.3 Structure of the thesis . . . 2

2 Concrete arch dams 3 2.1 Loads . . . 4

2.1.1 Static loads . . . 4

2.1.2 Seasonally varying loads . . . 5

2.1.3 Dynamic and other loads . . . 6

2.2 Thermal effects on concrete arch dams . . . 7

2.2.1 Temperature distribution . . . 7

2.2.2 Deflection . . . 7

2.2.3 Thermal properties of concrete . . . 8

2.3 Finite element modelling in Abaqus/Standard . . . 10

2.3.1 Geometry . . . 11

2.3.2 Heat transfer analyses of concrete arch dams . . . 12

2.3.3 Convective heat transfer coefficients . . . 13

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3.2 Properties of the concrete arch . . . 19

3.3 Material properties . . . 20

3.3.1 Heat transfer analyses . . . 20

3.3.2 Structural mechanical analyses . . . 20

3.4 Measurements . . . 20

3.4.1 Temperature measurements with PT100 sensors . . . 21

3.4.2 Water level measurements . . . 23

3.4.3 Deformation measurements . . . 23

3.5 Finite element model . . . 25

3.5.1 The arch system . . . 25

3.5.2 The rock foundation . . . 26

3.5.3 The slab . . . 27

3.6 Establishing the mesh . . . 28

3.7 Heat transfer model . . . 31

3.7.1 Temperature application . . . 32

3.8 Structural mechanical model . . . 34

3.8.1 Boundary conditions . . . 34 3.8.2 Loads . . . 35 3.8.3 Output-data . . . 35 3.8.4 Parametric study . . . 36 3.8.5 Stress distribution . . . 37 3.8.6 Contact conditions . . . 37 3.9 Verification . . . 37 3.9.1 Error evaluation . . . 37 3.9.2 2D model . . . 38

4 Results and discussion 39 4.1 Heat transfer analyses . . . 39

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4.2 Structural mechanical analyses . . . 45

4.2.1 Standard deviation from the parameter study . . . 45

4.2.2 Results from best fit - Case 2 . . . 46

4.2.3 Comparison of the principal behavior during summer and winter 49

4.2.4 2D-model . . . 52 4.2.5 Comparison of cases . . . 53 4.2.6 Possible deflection due to ice load . . . 54

5 Conclusions 55 5.1 Limitations . . . 55 5.2 Future research . . . 56 Bibliography 57 A Element distribution 59 B Numerical results 61

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Chapter 1

Introduction

1.1

Background

Many of the dams existing today were constructed around fifty years ago (DSIG, 2015). Determining the level of safety for existing dams can be a difficult task as knowledge, in terms of drawings and material as well as the level of degradation, can be insufficient. There is a need for new diagnostic monitoring tools and tech-niques in order to evaluate the stability and level of safety of existing dams (Nilsson A, 2014). Condition monitoring is an important part of the dam safety work and can provide early warnings of dam failure or incidents (Nilsson A, 2014). Condition monitoring is a collective term that includes activities such as surveillance, mea-surements, inspections and evaluation of data. Programs for condition monitoring must be thoroughly planned and executed. Data must be evaluated with respect to the dam safety and events that should be acted upon needs to be determined (Nils-son A, 2014). The recent development in numerical analysis techniques, and finite element analysis (FEA) in particular, provides the opportunity to integrate numer-ical models with the condition monitoring program. There are several possibilities and advantages of integrating finite element analyses with a condition monitoring program. When the model is calibrated to a satisfactory degree, it may be used to predict the behavior of the dam. New measurements can be compared to model predictions to determine whether they indicate a change of behavior or if they corre-spond to the dam’s natural variations. In other words, a finite element model offers the opportunity to separate natural variations from actual events and thus facilitate the risk assessment. These behavior predictions can further be used for extreme loading situations regarding temperature, water levels and ice loads. A finite ele-ment model can also be used as a tool for identifying areas where monitoring needs to be intensified, such as areas of high tensile stresses that may result in cracking of the concrete. As many of the existing dams are subject to reconstruction work due to degradation, a finite element model can also be used to predict the effect of such planned reconstruction work. This thesis deals with the construction and calibration of a finite element model for a concrete arch dam located in Sweden. The work presented here should be viewed as a feasibility study for a general approach to the long-term aim of finite element coupled condition monitoring of dams.

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1.2

Aim and scope

The aim of this thesis is to develop a finite element model of a concrete arch dam and calibrate the model with respect to data of measured concrete temperatures and crest deformation provided by the dam owner. The studied dam is a combined concrete arch dam and embankment dam with a section of spillways in between, but only the arch dam is considered in the thesis. Furthermore, the concrete is in this study assumed to be un-cracked why the analysis is limited to the linear-elastic range. The rock foundation is simplified into one solid mass and shear zones, crush zones and fractures are neglected. The foundation is an important aspect regarding the stability and safety of the dam, but the non-linearities of the rock are assumed to not affect the seasonal movements in a significant manner.

1.3

Structure of the thesis

This thesis begins with a theory chapter, Chapter 2, aimed at providing some back-ground to the behavior of concrete arch dams, loading conditions, modelling con-siderations and the finite element method. Special attention is given to thermal effects on concrete arch dams and factors that influence the thermal behavior of such structures.

Chapter 3 constitutes the method section of this thesis and describes the geom-etry and properties of the dam under study, the measurements and input-data used in the analysis, the construction of the finite element models, the analysis procedure and methods of verification.

The modelling results are presented in Chapter 4.

Conclusions drawn from the results and suggestions for future research are pre-sented in Chapter 5.

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Chapter 2

Concrete arch dams

A dam is constructed with the purpose to raise the water level and regulate the water flow (Bergh H, 2014). There are two aspects that separate dams from most other types of structures:

• The dam is exposed to a large horizontal force due to the hydrostatic pressure but also ice and sediment loads as well as seismic forces.

• The expected life time is large: from over a hundred years to a thousand years for tailing dams.

Arch dams primarily transfer the horizontal hydrostatic pressure to the abutments by arch action. Depending on the loading situation and type of arch dam, some of the load is also transferred vertically to the foundation by the monoliths. An arch dam foundation requires rock of good quality as the forces acting on the foundation are large but in return requires less concrete than a gravity dam. Arch dams can be designed with a constant or a varying radius. With a varying radius, the pressure is reduced as the distance between abutments is smaller close to the bed and thus requires less thickness (Bergh H, 2014). Figure 2.1 shows some conceptual terms regarding arch dams.

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Figure 2.1: Conceptual terms for an arch dam

2.1

Loads

The loads acting on an arch dam can be categorized into static and dynamic. Static loads are loads that do not change with time, or change very slow compared to the dam’s periods of natural vibration. The response to static loads is governed by the stiffness of the structure. Dynamic loads change rapidly with time why also the inertial and damping characteristics of the dam affect the behavior (FERC, 1999). There are primarily eight different types of loads acting on a dam, the importance and magnitude of which differs depending on the location of the dam. As the dam studied in Chapter 3 is located in northern Sweden, and since the effect of seasonal temperature variations is analyzed, special attention is given to the ice and thermal loads in this section.

2.1.1

Static loads

• Outer water pressure – This is a hydrostatic water pressure that acts on the upstream and downstream faces of the dam as a result from the reservoir and tail water levels during operation.

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2.1. LOADS

in the rock, the hydrostatic pressure will push the dam upwards. The uplift pressure varies with geological conditions of the rock foundation and the size and distribution of cracks in the concrete. When analyses or measurements indicate cracking of the concrete, full uplift should be assumed to exist over cracks exposed to the reservoir (FERC, 1999).

• Pore pressure – The porosity and permeability of concrete and rock results in uplift pressures.

• Dead load – The dead load is the total weight of the concrete of the arch dam and additional structures such as spillway gates, bridges and so on. The weight of these appurtenant structures is typically negligible compared to the weight of the arch dam and can therefore be excluded from a static analysis (FERC, 1999). The concrete density is based on laboratory tests and is usually in the range of 2300-2450 kg/m3. The weight of the steel should be considered in reinforced structures.

• Silt load – It is normally assumed that sediment settles close to the dam and exerts an active earth pressure. However, sediment contents of the rivers in Sweden are small and is therefore usually neglected (Bergh H, 2014).

2.1.2

Seasonally varying loads

• Ice load – In regions where the reservoir is expected to freeze, the ice load has to be considered. As with loads in general, ice loads can be of static or dynamic nature. The static ice load acts on the dam when the reservoir is completely frozen and the dynamic ice load can occur when floating sheets of ice collide with the dam (FERC, 1999). The thickness of the ice cover governs the magnitude of the ice load. The maximum ice load is assumed to be reached at an ice cover thickness of 1 m and a design force of 50-200 kN/m is used in south-north Sweden (RIDAS, 2011). The knowledge of this magnitude is however beset with uncertainties. In a recent study, the magnitude of the ice load was estimated by analyzing the difference in dam crest deformations between measured values and results from numerical analyses in which the ice load was neglected (Johansson F, Malm R, Fransson L, 2014). The study concludes that, albeit difficult, it is theoretically possible to use the outlined procedure for ice load estimation, that it requires data for several years and that crest deformation using direct pendulums is likely the best technique to use for this purpose. It is also stated in the report that the largest difference in model results and measurements seem to occur in April-May.

• Temperature load - Arise as a result of the difference between the temper-ature when the construction joints between the monoliths are grouted, called the reference temperature, and the temperatures in the concrete during opera-tion of the dam. The reference temperature is a design parameter that affects the thermally induced tensile stresses in the dam. If the reference

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tempera-ture is unknown it can be assumed to be equal to the mean annual concrete temperature or mean annual ambient temperature. (FERC, 1999)

2.1.3

Dynamic and other loads

• Dynamic loads – Earthquakes generate accelerations in both vertical and horizontal direction. The effect of seismic load is normally not considered in Sweden where the probability of large magnitude earthquakes is small. Other dynamic loads can occur as a result of the impact of debris, boats or the aforementioned floating ice sheets, blast-induced forces and dynamic effects from movable gates and traffic.

• Other loads – Loads can also occur as the result of uneven settlements in different parts of the foundation as well as from the interaction between mono-liths.

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2.2. THERMAL EFFECTS ON CONCRETE ARCH DAMS

2.2

Thermal effects on concrete arch dams

Thermal effects, such as freezing and thawing and temperature variations, play a significant role in the deterioration of concrete dams. In cold regions, the tempera-ture difference between summer and winter reach magnitudes of 45 °C (Daoud M, Galanis N, Ballivy G, 1997). Dams located in cold regions are especially exposed to freeze-thaw effects as the upstream face generally is saturated whereas the down-stream face is exposed to seasonal temperatures (FERC, 1999). These variations can induce stresses that exceed the tensile strength of the concrete, resulting in cracks. Two important factors that affect the temperature distribution on the downstream face of arch dams are solar radiation and convection heat transfer (Jin F, Chen Z, Wang J, Yang J, 2010).

Solar radiation increases the temperature of the structure and the effect depends on the slope and orientation of the exposed surface as well as the latitude. The increase in temperature reduces the temperature loads during winter and increases the loads during summer. The effect of solar radiation should be accounted for on faces of the dam not covered by reservoir water (FERC, 1999).

2.2.1

Temperature distribution

The thickness of an arch dam and the duration of the temperature governs the nature of the temperature distribution. A linear temperature distribution from the upstream face to the downstream face is a reasonable approximation for relatively thin arch dams. In arch dams with thicker sections, the temperature distribution is non-linear as the temperatures close to the faces respond quickly to fluctuations in the air and water temperatures whereas the temperature in the middle remains near the reference temperature, defined in Section 2.1.2. The temperature distribution in thick arch dams can preferably be determined using the finite element method described in Section 2.3.

2.2.2

Deflection

The dam is pushed in the downstream direction by the hydrostatic pressure. During winter, when temperature drops below the reference temperature, the concrete con-tracts resulting in deflections in the downstream direction. During summer, when the temperature rises above the reference temperature, the deflection is in the up-stream direction. The reference temperature thereby governs the movement of the arch dam (FERC, 1999).

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2.2.3

Thermal properties of concrete

Thermal conductivity

Governing factors for the thermal conductivity is aggregate type, temperature, mois-ture content and porosity. For normal conditions, the thermal conductivity is be-tween 1.2-3.0 W/mK (ICOLD, 2009).

Specific heat

The specific heat is a measure of the amount of energy required to raise the tem-perature in a material. For concrete this value is typically between 0.85-1.15 kJ/kg. The most influencing factor regarding this coefficient is the water content due to the high specific heat for the water (ICOLD, 2009).

Coefficient of thermal expansion

Concrete expands with heating and contracts with cooling. The strain that devel-ops with a change in temperature depends on the coefficient of thermal expansion, α, and the temperature difference (FERC, 1999), see Equation 2.2. The relation between linear expansion with temperature change and α is defined in Equation 2.1 (Burström P.G, 2007):

∆L = α · L · ∆T → α = ∆L

L · ∆T (2.1)

Re-writing Equation 2.1, the relation between the temperature change, coefficient of thermal expansion and strain, ε, becomes evident:

∆L

L = ε = α · ∆T (2.2)

As concrete is a composite material, the coefficient of thermal expansion is dependent on the constitutive parts, as described in Equation 2.3 (Ljungkrantz C, Möller G, Petersons N , 1994):

αc= αm(Vm− 0.23) + αg(Vg+ 0.23) (2.3)

where

αc=the coefficient of thermal expansion for the concrete

αm=the coefficient of thermal expansion for the mortar, including aggregates <4mm Vm=the volumetric share of the mortar

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2.2. THERMAL EFFECTS ON CONCRETE ARCH DAMS

Vg=the volumetric share of the stones

The coefficient of thermal expansion of the aggregate, αg, has a large impact as

aggregates occupy about 70% of the concrete. The magnitude of αg depends on the

mineralogical composition and ranges from 6-12·10−6 1/K for standard concretes

(Ljungkrantz C, Möller G, Petersons N , 1994). Generally, the thermal expansion

for concrete varies between 4-14·10−6 1/K (ICOLD, 2009).

Estimated thermal properties for some concrete dams in the USA are listed in Table 2.1 (Ljungkrantz C, Möller G, Petersons N , 1994):

Table 2.1: Thermal concrete properties for a few concrete dams (Ljungkrantz C, Möller G, Petersons N , 1994). Object Temperature ° C Coefficient of thermal expansion αc 10−6·1/K Specific heat cc kJ/(kg·K) Conductivity λc W/(mK) Hoover 38 9.5 0.94 2.88 Kortes 38 9.4 0.93 2.77 Hungry horse 38 11.2 0.97 2.95 Angostura 38 7.2 0.99 2.56 Libby 36 11.7 0.92 3.86

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2.3

Finite element modelling in Abaqus/Standard

The finite element method, FEM, is a method for numerical solution of field prob-lems. A field problem consists of the spatial distribution of one or more dependent variables (Cook R.D, Malkus D.S, Plesha M.E, Witt R.J, 2002). The FEM is ad-vantageous for many reasons:

• applicable to any field problem such as stress analysis, heat transfer analysis etc.

• the structure to be analyzed may have any shape, no restriction in geometry • no restriction in boundary conditions or loading

• material properties can change between elements or even within elements • a finite element model may contain several different types of elements

• the level of approximation can be improved by gradually refining the mesh where appropriate

There are several element types available when constructing a finite element model as shown in Figure 2.2. Most of these elements are displacement-based (Cook R.D, Malkus D.S, Plesha M.E, Witt R.J, 2002). Isoparametric solid elements are suitable for the description of an arch dam. This is because the isoparametric formula-tion permits quadrilateral elements to deform in non-rectangular ways (Ghanaat Y, 1993).

The most commonly used elements are shown in Figure 2.2.

Figure 2.2: Available elements in Abaqus. From (Hibbitt et al., 2006)

All degrees of freedom, such as displacements, are calculated at the element nodes. In any other point within the element, the element data is interpolated from the nodal data. The interpolation order is determined by the number of nodes in each element. Three types of elements are displayed in Figure 2.3 below:

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2.3. FINITE ELEMENT MODELLING IN ABAQUS/STANDARD

• Quadratic element

• Modified second-order element

The names refer to the interpolation order. Linear elements only have nodes in each corner and use linear interpolation in each direction whereas the other two types use quadratic interpolation between nodes. As a result, the linear brick elements have 8 nodes whereas quadratic brick elements have 20 nodes as shown in Figure 2.3.

Figure 2.3: Available elements in Abaqus. From (Hibbitt et al., 2006)

For heat transfer analyses, Abaqus/Standard provides diffusive elements allowing heat storage and heat conduction. Results from such an analysis, for instance a temperature distribution within a structure, can be used as input in a structural mechanical model.

The element formulation used for stress/displacement elements in Abaqus/Standard is normally based on the Lagrangian description of behavior, meaning that the el-ement deforms with the assigned material. For convective heat transfer analyses, Abaqus/Standard uses the alternative Eulerian description in which elements are fixed in space as material flows through them. (Hibbitt et al., 2006)

2.3.1

Geometry

There are numerous considerations to be made when analyzing dams with numer-ical modelling. The size and distribution of the finite element grid, or the mesh, should be constructed so that the elements are smaller where higher gradients are expected. Higher gradients occur both in relation to the geometry and the loading condition. Another important aspect when modelling dams is the definition of the domain dimension surrounding the dam. The domain definition is significant when dynamic or heat transfer analyses are carried out, but is also important when a static analysis is performed.

For static loads and linear analyses, the extensions of the vertically and horizon-tally surrounding foundation should be at least equal to one of the characteristic dimensions of the structure such as height or width at the base for a two-dimensional model.

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Some types of dams and problems can be modelled in 2D whereas other prob-lems require a 3D-model. 3D-models may be required for concrete gravity dams and embankment dams built in narrow valleys. (ICOLD, 2005)

2.3.2

Heat transfer analyses of concrete arch dams

The mesh should be constructed so that at least three elements are included through the thickness of the dam. The amount of elements depends on the element type. Elements with quadratic interpolation order require fewer elements than elements

with linear interpolation order. Water and air temperatures are applied to the

respective boundaries of the model whereas the foundation can be assumed adiabatic or given a mean annual air temperature, see Figure 2.4. When determining the heat distribution using computer analyses, it is important to let the analysis run long enough for the temperature cycles to stabilize. Preferably, the analysis should be performed in both steady-state and transient steps setting the initial temperature equal to the mean annual air temperature or the closure temperature (FERC, 1999).

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2.3. FINITE ELEMENT MODELLING IN ABAQUS/STANDARD

2.3.3

Convective heat transfer coefficients

This section is based on Engineering thermodynamics and fluid mechanics (Chalmers, 2012).

When a fluid flows along a solid body or inside a pipe and there is a temperature difference between the fluid and body, the heat transfer is affected by the movement of the fluid. This phenomenon is known as convection. Convection is divided into two groups depending on the way the fluid was set in motion:

• Forced convection: The fluid is forced in motion by fans, wind, pumps etc. • Natural convection: The convection arises due to a difference in density

gen-erated by a temperature difference.

The convective heat transfer is calculated using a number of empirically established equations containing several dimensionless numbers briefly described in the following sections.

Reynolds number

The forced convection is divided into laminar and turbulent flow. The division is determined by the Reynolds number, Re:

Re = ul

ν (2.4)

where

u = The average speed in the flowing medium [m/s] l = The characteristic length [m]

ν = The kinematic viscosity [m2/s]

The Nusselt number

Another number used in calculating the convective heat transfer is the Nusselt num-ber, Nu, which describes the temperature field in the flowing medium and is defined as:

N u = αl

λ (2.5)

where

α = The heat transfer coefficient [W/m2K]

λ = The thermal conductivity of the flowing medium [W/mK] l = The characteristic length [m]

The Nusselt number is a function of the Prandtl number and the Reynolds number in case of forced convection and the Prandtl number and the Grashof number in the case of natural convection.

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The Prandtl number

The Prandtl number, Pr, characterizes the flowing medium and is defined as: P r = νρcp

λ (2.6)

where

ν = The kinematic viscosity of the flowing medium [m2/s] ρ = The density of the flowing medium [kg/m3]

cp = The specific heat capacity of the flowing medium [J/kgK] λ = The thermal conductivity of the flowing medium [W/mK]

The Grashof number

The Grashof number, Gr, describes the flow field in the case of natural convection and is defined as:

Gr = gβl

3∆t

ν2 (2.7)

where

g = the gravity of Earth, 9.81 [m/s2]

β = The volume expansion coefficient [K−1]

l = The characteristic length in the observed structure [m]

∆t = The temperature difference between the flowing medium and the observed structure [K]

ν = The kinematic viscosity of the flowing medium [m2/s]

Natural convection

In situations of natural convection, when the Prandtl number lies between 0.5 and 200, the Nusselt number can be expressed as:

N u = A(GrP r)B (2.8)

where B = 14 in cases of laminar flow and 13 in cases of turbulent flow. A is a

constant depending on the geometry of the structure. For a vertical, plane surface, A is 0.59 for laminar flow and 0.13 for turbulent flow. The transition from laminar to turbulent flow occurs when the factor GrP r > 109.

Forced convection

For turbulent flow across a flat surface, assuming a constant surface temperature, the Nusselt number can be expressed as:

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2.3. FINITE ELEMENT MODELLING IN ABAQUS/STANDARD N u = 0.037 · Re 0.8· P r 1 + 2.443 · Re−0.1(P r2/3− 1) = αl λ (2.9) where

l = The characteristic length=the surface length in the direction of the flow [m]. The equation is valid for:

0.6 < P r < 2000 5 · 105 < Re < 107

Table 2.2 shows the magnitude of the convective heat transfer coefficient for different conditions:

Table 2.2: Magnitudes of the convective heat transfer coefficient (Chalmers, 2012)

Convection type α [W/(m2°C)]

Forced turbulent tubular flow

Water, u=0.5-5m/s 1500-20000

Air, u=2-10m/s 10-40

Forced laminar tubular flow

Water 50-500

Air 2-20

Forced air flow across plates

u=1-10m/s 10-100

Natural convection

Water 200-1000

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2.3.4

FE-coupled monitoring

All structures undergo an ageing process during long-term operations. Monitoring must therefore confirm the structural adequacy and detect detrimental changes in the behavior of the structure. Data from the monitoring activity can, among other things, be used to confirm the validity of design assumptions or calibrate math-ematical models used for design or back analyses in which parameters of certain interest can be estimated using the difference obtained from numerical results and measurements (ICOLD, 2005).

The parameters supplied by monitoring systems to the mathematical model can be categorized as follows:

• Actions of the environment • Physical-mechanical quantities

Actions of the environment includes reservoir water level, air and water tempera-tures at different depths, ice thickness and so on whereas the physical-mechanical properties for concrete dams include the dam-reservoir-foundation response in terms of absolute displacements between dam and foundation, relative displacements be-tween blocks, temperature distribution in the structure, stress and strain state in the dam-foundation system, uplift pressures and seepage.

By iteratively modifying the physical-mechanical properties of the model and thereby reducing the error between measured and computed features, a better understanding of the state of the real structure is obtained. This process is referred to as structural identification and can be based on static and dynamic monitoring data.

The following actions are pointed out regarding the safety reassessment and reha-bilitation of existing dams, (ICOLD, 2005):

• a detailed study of the structural history and behavior of the dam and its foundation

• constructing a suitable numerical model for the simulation and explanation of the observed behavior

• reassessing the failure mechanism and safety margins, and thereby the struc-tural stability, using numerical models simulating both usual and unusual loads • if necessary, designing remedial works to improve durability, serviceability and

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Chapter 3

Case study of a concrete arch dam

The dam power station under study was constructed between the years 1947-1952 and is located in the river of Ljusnan in Sweden. The dam consists of three parts: an embankment dam across the southern part of the river, a section of spillways in the middle followed by an arch dam on the north river side, see Figure 3.1. The spillway acts as one of the abutments to the arch dam. The northern abutment of the arch dam is connected to rock. The highest part of the dam is 45 meters. The arch dam is divided into 18 monoliths and has a radius of 100 meters and a length of 160 meters.

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To decrease the effect from solar radiation and the temperature variations in the dam, an insulating wall was installed in 1952. Weak rock and crush zones under the arch led to the construction of a concrete slab at the downstream side. This slab was built in 2003 to increase the pressure on these weak zones. Figure 3.2 shows the slab and the arch dam. During 2003 - 2009 several sensors were installed to monitor the behavior of the arch dam. These sensors measure temperatures, leakage, deformation, pore pressure and water level.

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3.1. ROCK MASS PROPERTIES

Figure 3.3: The arch dam seen from above and from the upstream side

3.1

Rock mass properties

In a previous study of this dam, (Johansson F, 2005), the modulus of elasticity for the rock mass was back calculated. The results from these calculations resulted in a modulus of elasticity of 0.5 - 3.2 GPa. The lower values were found in the weaker shear zones and the higher values at greater depths. In the same study, the coefficient of thermal expansion was set to 8.3·10−6 1/°C.

3.2

Properties of the concrete arch

When considering long term loading, the modulus of elasticity should be reduced to account for concrete creep. The hydrostatic pressure is considered a long term load while the seasonal variation in temperature is not. Given that the temperature variation is the load that gives the greatest deformation variations over the year, the elastic modulus is not reduced. In (Johansson F, 2005), a modulus of elasticity of 30 GPa was used for the concrete when analyzing the temperature load.

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3.3

Material properties

3.3.1

Heat transfer analyses

Table 3.1: Material parameters used in the heat transfer analyses

Parameter Concrete Rock

Conductivity, λ W/(mK) 1.7 * 1.7 **

Specific heat, c kJ/(kg·K) 1000 * 1000 *

Density, ρ, kg/m3 2700 *** 2600 *

* - (Johannesson G., 2012)

** - (Eppelbaum L., Kutasov I., Pilchin A., 2014) *** - assumed value

3.3.2

Structural mechanical analyses

Table 3.2: Material parameters used in the structural mechanical analyses

Parameter Concrete Rock

Young’s modulus, GPa 30 * 3.2 **

Expansion coefficient, α 12 · 10−6, 10 · 10−6, 8 · 10−6 8.3 · 10−6 ** *** Density, ρ, kg/m3 2700 2600 Reference temperature, °C 15, 10 , 5 *** 0 *** * - (Johannesson P, Vretblad B, 2011) ** - (Johansson F, 2005) *** - (assumed value)

3.4

Measurements

The dam owner provided hourly measurements ranging from 2009-11-06 to 2015-02-12. Measurements included deflection of the crest for monoliths 1, 4, 10, 14 & 18, see figure 3.3, water and concrete temperatures for monolith 10, pore pressures, temperature in the foundation and ambient air temperatures. As some of the data had different starting dates, only measurements from 2010-01-01 to 2015-02-12 were included in the analysis. Due to the limitations in this thesis, only temperatures and deflections were used to calibrate the model. Furthermore, since neither water or air temperatures nor deflections change noticeably every hour, the weekly average

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3.4. MEASUREMENTS

3.4.1

Temperature measurements with PT100 sensors

Figure 3.4 shows sensors embedded in monolith 10 measuring the temperature in

concrete and the reservoir. The water temperature sensors are labeled TV201,

TV202 and TV203. The concrete temperature sensors are labeled TV101, TV102, TV103, TV107, TV108, TV109, TV113, TV114 and TV115.

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Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015 0 5 10 15 20 25 Temperature [ ° C]

Measured water temperatures

TV201 - 3m depth TV202 - 20m depth TV203 - 38m depth

Figure 3.5: Measured water temperatures at TV201, TV202 and TV203

Figure 3.5 shows the measured water temperatures at monolith 10. It is clear that there is little difference between the three sensors which indicates that there is no sieving of the water.

Jan 2008 Jan 2009 Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015 −30 −20 −10 0 10 20 30 40 Temperature [ ° C]

Measured ambient temperatures

Hourly measurements, dam owner Weekly average

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3.4. MEASUREMENTS

Figure 3.6 shows the measured temperatures and the weekly average used in the analyses. Ambient air temperatures for the years 2008 and 2009 (marked with red) were retrieved from the Swedish meteorological and hydrological institute, SMHI (SMHI, 2015), and used for model stabilization as described in Section 3.7. The temperatures measured by SMHI correspond well to measurements carried out by the dam owner.

3.4.2

Water level measurements

The water level is measured every 15 minutes with ultrasonic sensors Milltronics The Probe PL -511 4-20mA. The average water level over one year is +338.8 m and the annual variation is about ±0.2 m.

3.4.3

Deformation measurements

A hanging, or direct, pendulum is used to measure crest displacements. It is an-chored at the crest and provides information on displacements relative to the crest. The hanging pendulums installed at the dam studied are Plumb line GL50. In Figure 3.7 FP denotes the fixed reference point, G weight of the pendulum, LTD pendulum wire, VDD2 damping vessel and KK84 the coordiscope used for manual measuring.

Figure 3.7: Principle sketch of a hanging pendulum (Huggenberger AG, 2015)

Deformations of the concrete arch crest are registered by hanging pendulums in-stalled at sections A-E in Figure 3.3. The measuring device has a system accuracy

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of ±0.1mm and measures the movement in both radial and tangential directions. The recordings from the pendulum are made on a monthly basis with manual read-ings. Records of deformation were provided for the period 2010-01-01 to 2015-01-12, with an average interval of one month between readings.

Deformations of the rock mass are measured with inverted pendulums type Huggen-berger SL30 attached to the bottom of drill holes varying from 31 - 42 m depth on the downstream side of the dam. Movements of the inverted pendulums are measured automatically, meaning that data from the sensors are transmitted and stored centrally once every hour. The total crest deformation is gained by adding the readings from the hanging and inverted pendulums, see figure 3.8.

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3.5. FINITE ELEMENT MODEL

3.5

Finite element model

The numerical FE model was defined with Abaqus/CAE and both heat transfer and mechanical analyses were performed in Abaqus/Standard. As described in Section 2.3, different element types and procedures are used for heat transfer analyses and structural mechanical analyses. Therefore, two models were created having iden-tical mesh-distribution and geometric features: one heat transfer-model and one structural mechanical model. These models are described further in Sections 3.7 & 3.8.

3.5.1

The arch system

The arch was first constructed as a 2D model in Autocad and then revolved to create a circular 3D shape. In order to facilitate the modelling, the monolith was modelled with a curved shape on the downstream side in the 2D model. The geometry model was then imported to Abaqus as a solid. This solid had to be cut due to the natural slope of the banks. Figure 3.9 shows the simplified geometry. The next step was to divide the arch into 18 monoliths. This was done by first defining a vertical datum axis 100 meters towards the arch center corresponding to the radius of the dam. By creating reference points along the dam crest, a cutting plane was defined and the monoliths split into separate parts. The whole dam was thereafter defined with concrete material properties. To account for the weight of the reinforcement, the density of the concrete was given a higher value.

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Figure 3.9: Monolith 10. Simplified (left) and actual geometry (right). Distances in meters

3.5.2

The rock foundation

After defining the arch dam, a square 3D solid was created to represent the founda-tion, see Figure 3.10. The geometry of the arch served as a template for defining the foundation. By using this approach, all surfaces between the arch and foundation system were in direct contact, and the force from the arch dam to the abutments was defined perpendicular to the face of the abutments. The foundation was thereafter defined with material properties corresponding to intact, linear elastic, rock. The right abutment, seen from the upstream side, is in reality attached to a concrete spillway. The reservoir south of the arch was excluded from the analysis and mod-elled as rock. The height of the foundation was 100 m, width 300 m and the depth 150 m.

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3.5. FINITE ELEMENT MODEL

Figure 3.10: The foundation system used in the FE-analysis

3.5.3

The slab

Monoliths 7 to 12 are in reality fixed in the downstream direction by the rock foundation. Above this foundation the concrete slab is cast. In the finite element model, the rock foundation and slab were simplified into one solid consisting of concrete, as seen in Figures 3.11 & 3.12. The reason for doing this simplification was uncertainties regarding the distribution between rock, fill material and concrete. As for the arch, the density of the concrete was given a higher value to account for the weight of the reinforcement.

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Figure 3.12: The modelled slab (left) and the real slab (right)

3.6

Establishing the mesh

The different parts were meshed differently depending on the geometry. The rock foundation and some of the monoliths were best suited for a tetrahedral mesh type, while the other monoliths and slab were meshed using hexagonal elements. Depend-ing on the accuracy required, the element size varied along the model. This was done by seeding all edges with unique element size or element numbering. Seeding is a marking technique used to illustrate the mesh density. To achieve sufficient accuracy regarding both temperature and deformations, at least four elements throughout the thickness of each monolith was used. The number and type of elements used for the different parts are shown in Figures 3.13-3.16 and Table A.1.

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3.6. ESTABLISHING THE MESH

Figure 3.13: The foundation and slab mesh

Figure 3.14: Monolith 10 and the arch

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Figure 3.16: The assembled model seen from the downstream direction

The model was divided into approximately 61200 linear tetrahedron and hexagonal elements. See appendix A for details.

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3.7. HEAT TRANSFER MODEL

3.7

Heat transfer model

The heat transfer analysis was performed in two steps: • Steady state analysis

• Transient analysis

The steady state analysis was performed in order to get a reasonable initial tem-perature distribution in the model prior to the transient analysis. Data for the air temperature for the year 2009 was retrieved from the Swedish meteorological and hydrological institute, SMHI (SMHI, 2015), and the annual average was computed and used as input. The water temperature was given the same value.

A transient heat transfer analysis was thereafter performed to evaluate the temper-ature distribution throughout the structure. The total time for the analysis was set to 370 weeks, corresponding to the time period January 2008 to February 2015. The analysis was performed in 370 increments, thus each increment corresponded to one week. The temperature data was provided by the dam owner, except for the air temperature data for the first two years, 2008 & 2009, which was retrieved from SMHI as in the case of the steady state analysis, see section 3.4.1. The water temperature for the years 2008-2009 were given the same values as for the years 2010-2011. The results from the steady state analysis and the transient analysis for the years 2008 & 2009 were used only to get the model behavior to stabilize. The water temperature measurements provided by the dam owner showed little difference from point TV201 to TV203, see Figure 3.5. These data were at first assumed to be wrong, being that the vertical distance between TV201 and TV203 is 35 m. However, when introducing these values in the model, the modelled and measured concrete temperatures corresponded well. The temperature model was calibrated by comparing the computed temperatures and measured temperatures corresponding to measurement points TV101-TV115 as previously shown in Figure 3.4. Solid elements were embedded in the model at points corresponding to the lo-cations of the temperature sensors in the concrete. The results were then extracted from these elements. The connections between the different materials were described with interactions. Heat transfer coefficients for the concrete/air and concrete/water surfaces were calculated using the method and equations described in Section 2.3.3. These coefficients were then used as surface film coefficients in Abaqus to get a more realistic heat distribution. The values for the insulated and top part of the arch were first given the same values as an uninsulated part and then iteratively changed until the results corresponded well with the measured concrete temperatures, see Figure 3.18 in section 3.7.1. The top part was iterated due to uncertainties regarding the geometry and the thickness of the surface layer. The connection between the arch and rock foundation was modelled as a tie constraint. Being that the downstream face was insulated to minimize the effect of solar radiation, only convection was considered in the heat transfer analysis.

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3.7.1

Temperature application

The upstream surface was divided into three segments: 17, 18 and 9 m in height respectively, as seen in Figure 3.17. The measured water temperatures for monolith 10 (TV201-TV203), shown in Figure 3.5, were then applied to these surfaces.

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3.7. HEAT TRANSFER MODEL

Figure 3.18: Surface division for air temperature application

The convective heat transfer coefficients used in the heat transfer analysis are pre-sented in Table 3.3. The letters a,b,c & d refer to the surfaces displayed in Figure 3.18.

Table 3.3: Convective heat transfer coefficients used in the heat transfer analysis

Surface

Convective heat transfer coefficient α [W/(m2K)]

Initial, calculated value

Convective heat transfer coefficient α [W/(m2K)]

Final, iterated value

Upstream face 1000 Not iterated

Downstream face (a) 13 5

Top surface (d) 13 9

Rock surfaces (b) 13 Not iterated

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3.8

Structural mechanical model

3.8.1

Boundary conditions

Two cases regarding the boundary conditions were considered: case 1 and case 2. In case 1, interactions between the outermost monoliths and the abutments were modelled with a tie connection. The slab and monoliths were also modelled with a tie connection against the rock foundation i.e. all translational d.o.f. (degrees-of-freedom) are constrained. As mentioned in Section 3.5.3, the slab is cast on the bedrock and the monoliths are supported in the downstream direction by both the bedrock and the slab. By defining a tie connection between the slab and monoliths in contact with the rock foundation, the model was assumed to behave in a more realistic manner. Regarding the monoliths, they were free to move relative to each other with a friction coefficient of 0.8. The same friction condition applied for the interaction between the monoliths and the slab. In case 2, all interactions were modelled with a tie constraint. The final model was locked for translations in all directions except the positive z-direction, i.e. the vertical direction, see figures 3.19 and 3.20. These conditions were applied perpendicular to the outer rock faces.

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3.8. STRUCTURAL MECHANICAL MODEL

Figure 3.20: Global boundary conditions, seen from the downstream side Table 3.4: Interactions used in the deformation analysis for the two cases

Interaction Case 1 Case 2

Monoliths-Rock (bottom) Tie Tie

Slab-Rock Tie Tie

Monoliths-Monoliths Friction=0.8 Tie

Monoliths-Slab Friction=0.8 Tie

Monoliths-Rock (abutments) Tie Tie

3.8.2

Loads

All loads were applied with Abaqus ramp-function, meaning that the loads were increased linearly, and divided into several smaller increments. The gravity was applied first. In this step, only the arch and slab were subjected to gravity. The next step was to apply the load from the hydrostatic pressure. The hydrostatic pressure was applied on all arch surfaces in contact with water and also on the surrounding rock. The water level was set to a constant value of +338,8m due to the small annual variations. As the arch was modelled with a tie constraint against the rock foundation, the uplift pressure was neglected. The ice load was intentionally ignored in the FE-model. As mentioned in Section 2.1.2, the ice load could in this way be detectable in the comparison between modelled and measured deformations.

3.8.3

Output-data

As only the seasonal variations were to be considered, the initial deformations from the static loads were removed before analyzing the results from the FE-model. This was also done for the measured deformations.

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3.8.4

Parametric study

A parametric study has been performed where the thermal expansion coefficient, α, and the concrete reference temperature have been varied. These two parameters were varied only for case 1. Case 2 was then modelled with the best parameter combination of case 1. The best combination of parameters is here defined as the combination resulting in the lowest standard deviation, σ, defined in Section 3.9.1.

Table 3.5: Parameter combinations used in the parametric study

Job Case Expansion Reference temperature

coefficient, α in the concrete, °C

1-a 1 12 · 10−6 15 1-b 1 12 · 10−6 10 1-c 1 12 · 10−6 5 1-d 1 10 · 10−6 15 1-e 1 10 · 10−6 10 1-f 1 10 · 10−6 5 1-g 1 8 · 10−6 15 1-h 1 8 · 10−6 10 1-i 1 8 · 10−6 5

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3.9. VERIFICATION

3.8.5

Stress distribution

The stress distribution during summer and winter was evaluated to see if there were any tensile stresses in the dam. Tensile stresses could indicate that the dam is cracked at these locations and non-linear effects could be needed in order to get adequate results.

3.8.6

Contact conditions

The connection between monoliths and rock foundation was modelled as a tie con-straint. By doing so, all effects from the uplift pressure had to be neglected. As a check to see if there is a possibility that the monoliths will separate from the rock, a case with interaction between the monoliths and rock was modelled and the possible gap calculated.

3.9

Verification

To implement the FE-model in a condition monitoring program, the measured value at a given time is considered expected if the difference between this value and the value predicted from the model falls within a given deviation interval. This range, denoted d, is commonly assumed to be 2 · σ, where σ is the standard deviation of the mean error, defined in Equation 3.3 (SCOD, 2003) corresponding to a 95% confidence interval.

3.9.1

Error evaluation

Results from the parametric study were compared to measurements for 40 available observations, n, for monolith 10. The comparison was evaluated with respect to the standard deviation as defined in Equation 3.3.

The error for each observation is defined as:

xierror = ximodel − ximeasurement (3.1)

The mean error then becomes: ¯ xerror = 1 n n X i=1 xierror (3.2)

The standard deviation is defined as:

σ = v u u t 1 n − 1 n X i=1 (xierror− ¯xerror)2 (3.3)

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The maximum allowable deviation is defined here as:

d = 2 · σ (3.4)

3.9.2

2D model

As an additional verification of the 3D-model, a 2D-model of monolith 10 was devel-oped. The model was analyzed with the parameter combination giving the lowest standard deviation as described in Section 3.9.1. The monolith was restrained for all translations at the bottom edge.

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Chapter 4

Results and discussion

The results from the numerical analyses are presented in this chapter, starting with the results from the heat transfer analysis in Section 4.1. The heat transfer results are only validated visually. The parameter combination of the structural mechanical analyses giving the lowest standard deviation, σ, is plotted for monolith 10 in Sec-tion 4.2.2 showing the correlaSec-tion between measurements and model results. These results are followed by a comparison of the residual in relation to the allowable devi-ation range, d. The principal behavior of the dam during winter and summer as well as the stress distribution in the arch and contact conditions between the arch and rock are presented and briefly discussed in Section 4.2.3. Results from the 2D-model are presented in Section 4.2.4. A comparison between the best fitted result from the parameter study and the 2D-model for monolith 10 are presented in Section 4.2.5. The magnitude of the crest deflection due to an ice load is estimated in Section 4.2.6. Additional results are also found in Appendix B.

4.1

Heat transfer analyses

The results from the heat transfer analyses are presented in Figures 4.1, 4.2 & 4.3 below. It is clear that the results show an overall correlation with the measurements. It can further be concluded that the reference points located close to the middle or upstream surface of the dam shows better agreement than reference points located closer to the downstream side. This is possibly due to the fact that the uncertainties regarding the heat transfer conditions at the downstream side are greater due to the insulated wall, which was not included geometrically in the model.

The bottom three points, TV113-TV115, show a greater deviance, especially be-tween 2010-2011 where two consecutive peaks are seen in the measurements. The measurement device for reference point TV115 is assumed to be malfunctioning as there is no variation in temperatures, see Figure 4.3. Unfortunately, no registra-tions of water temperatures were available for the years before 2010, but as the air temperatures for the years 2008 & 2009, seen in Figure 3.6 in Section 3.4.1, have a similar variation as the years 2010-2015 used in the model, it is possible that the

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bottom water temperatures were slightly higher the years before 2010. The width of monolith 10 at the position of these reference points spans 6.5-9.4 meters, as previously shown in Figure 3.9 in Section 3.5.1. One assumption could therefore be that if the water temperatures were higher the years before 2010, that is 2008-2009, the thermal inertia of the concrete could result in the two peaks. The remaining years, 2013-2015 show a better agreement between measurements and results from the numerical model indicates that during this time, the concrete temperatures at the bottom have stabilized. The remaining years still show a greater deviation than the middle and top sections. This is likely due to the fact that the modelled mono-lith is around one meter wider than the actual monomono-lith at the sections of these reference points whereas the difference is significantly less for the middle and top reference points.

Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015 −10 0 10 20 30 Temperature [ ° C] Concrete temperatures FEA TV101 Measured TV101

Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015 −10 0 10 20 30 Temperature [ ° C] FEA TV102 Measured TV102

Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015 −10 0 10 20 30 Temperature [ ° C] FEA TV103 Measured TV103

Figure 4.1: Results from the model compared to measured temperatures in monolith 10, top.

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4.1. HEAT TRANSFER ANALYSES

Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015 −10 0 10 20 30 Temperature [ ° C] Concrete temperatures FEA TV107 Measured TV107

Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015 −10 0 10 20 30 Temperature [ ° C] FEA TV108 Measured TV108

Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015 −10 0 10 20 30 Temperature [ ° C] FEA TV109 Measured TV109

Figure 4.2: Results from the model compared to measured temperatures in monolith 10, middle.

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Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015 −10 0 10 20 30 Temperature [ ° C] Concrete temperatures FEA TV113 Measured TV113

Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015 −10 0 10 20 30 Temperature [ ° C] FEA TV114 Measured TV114

Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015 −20 −10 0 10 20 30 Temperature [ ° C] FEA TV115 Measured TV115

Figure 4.3: Results from the model compared to measured temperatures in monolith 10, bottom.

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4.1. HEAT TRANSFER ANALYSES

Jan 2010−8 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015

−6 −4 −2 0 2 4 6 8 10 Temperature [ ° C]

Difference between measurements and model results

TV 101 TV 102 TV 103 TV 107 TV 108 TV 109 TV 113 TV 114

Figure 4.4: Difference between measured temperatures and results from the numer-ical modelling.

Figures 4.4 show the difference between measured temperatures and model results. The malfunctioning sensor at TV 115 was excluded. It can be concluded that the largest difference occurs at TV 114 and that the difference is larger for the first two years, 2010 & 2011 for which the magnitude of the difference is around 6 °C. Figure 4.5 and 4.6 shows the temperature distribution for summer and winter. It can be seen that the core temperatures are kept at an almost constant value while the surface temperature varies more over the year.

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Figure 4.5: Temperature distribution in monolith 10 during summer. Units in Kelvin

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4.2. STRUCTURAL MECHANICAL ANALYSES

4.2

Structural mechanical analyses

Results from the structural mechanical analyses are presented in this section. Section 4.2.1 contains a summary of the error for each of the 9 parameter combinations of case 1 as well as the error for case 2 and the combination with the best fit is determined. In section 4.2.2 results from the best fit are plotted together with the measured deformation for monolith 10 and the allowable deviation range for the error of the best fit, plotted together with the residual. Section 4.2.3 is intended to give a graphical representation of the principal behavior of the arch during winter, spring and summer. Section 4.2.4 Shows the results from the 2D-model together with the measurements for monolith 10. A plot of the best fit and the 2D-model for monolith 10 together with the measurements are shown in Section 4.2.5.

4.2.1

Standard deviation from the parameter study

The parameter study resulted in a lowest standard deviation for job 1-i. The param-eter combination of this job was input to case 2 resulting in case 2 giving the lowest standard deviation of all 10 jobs, see Table 4.1. In order to make out any difference between the results, five significant figures are given for the standard deviation even though the accuracy of the model does not correspond to this level of certainty.

Table 4.1: Results for monolith 10 from the parameter study

Job Case Expansion Reference temperature Standard

coefficient, α

1 K

in the concrete, °C deviation, s,

mm 1-a 1 12 · 10−6 15 5.7929 1-b 1 12 · 10−6 10 5.7929 1-c 1 12 · 10−6 5 5.7929 1-d 1 10 · 10−6 15 4.1504 1-e 1 10 · 10−6 10 4.1443 1-f 1 10 · 10−6 5 4.1504 1-g 1 8 · 10−6 15 2.6904 1-h 1 8 · 10−6 10 2.6112 1-i 1 8 · 10−6 5 2.6104 2-a 2 8 · 10−6 10 1.9794

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4.2.2

Results from best fit - Case 2

The analyses showed that the best correlation was given for case 2. As mentioned in Section 3.8.1, all interactions in case 2 were modelled with a tie constraint. The best agreement between calculated and measured crest displacement was obtained for monolith 10. This result was expected given that the temperature calibration was performed for this monolith. As mentioned in Section 3.5.2, the foundation and the abutments were modelled as solid rock. This simplification is reflected in the results and is most noticeable for the outermost monoliths, 1 and 18. Furthermore, the total annual measured deformations for monolith 1 and 18 are in the range of 1-2 mm. Such small deformations are difficult to capture for a structure of this size. The results are also affected by the decreasing intensity of measurements during the last two years.

Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015 −15 −10 −5 0 5 10 15 20 25 Deformation [mm]

Radial crest deformations at monolith 10 − Case 2 α=8⋅10−6 °C−1 Reference temperature=5 °C

FEA Measured

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4.2. STRUCTURAL MECHANICAL ANALYSES

Acceptable deviation

Figure 4.8 shows the difference between measurements and model results for mono-lith 10 and the acceptable deviation here defined as two standard deviations. It is clear that the residual lies within the acceptable range for most of the period studied but with two peaks at which the residual is too large. The first peak occur in december, whereas the largest, second peak, occurs in october. These two peaks, however, only occur once. It is not possible by means of the graph, 4.8, to deduce any effect from the ice load. This could partly be explained by the limited amount of measurement data available.

Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015

−10 −8 −6 −4 −2 0 2 4 6 8 10 Deformation [mm]

Allowable deviation for monolith 10 − Case 2

α=8⋅10−6°C−1 Reference temperature=5 °C

Allowable deviation, 2σ Residual

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Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015 −15 −10 −5 0 5 10 15 20 25 Deformation [mm]

Allowable deviation for monolith 10 − Case 2

α=8⋅10−6°C−1 Reference temperature=5 °C

FEA ± allowable deviation 2·σ Measurements

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4.2. STRUCTURAL MECHANICAL ANALYSES

4.2.3

Comparison of the principal behavior during summer

and winter

Figures 4.10 & 4.11 show the deformation of the arch during summer and winter. The magnitude of the displacement is greater in the upstream direction (during

summer) than in the downstream direction (during winter). This is reasonable

since, as mentioned in Section 2.2.3, concrete expands with heating and contracts with cooling. This displacement is counteracted only by the hydrostatic pressure and the friction against the bedrock and abutments. During winter, when the concrete contracts, the behavior is the opposite and there is a straightening of the arch which is counteracted by the compressive resultant from the abutments and the arching effect.

Figure 4.10: The deflection of the concrete arch during winter seen from above. Units in meters

Figure 4.11: The deflection of the concrete arch during summer seen from above. Units in meters

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Stress distribution

The figures below show the stress distribution in the arch during summer and winter. During the summer, when the arch is expanding, the highest stress levels are found at mid height. During winter the stress level is greatest at the top and bottom. As seen in the figures below, there are no existing tensile stresses in the arch.

Figure 4.12: Stress distribution during summer seen from the upstream side. Units in Pascal

Figure 4.13: Stress distribution during winter seen from the upstream side. Units in Pascal

Figure 4.14: Stress distribution during summer seen from the downstream side. Units in Pascal

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4.2. STRUCTURAL MECHANICAL ANALYSES

Figure 4.15: Stress distribution during winter seen from the downstream side. Units in Pascal

Contact conditions

The contact between the arch system and the underlying rock is shown in figures 4.16 and 4.17. It can be seen in the figures that the biggest gap is close to the left abutment and is present during the whole year. Also, an opening occur during the winter and is located close to the middle of the arch.

Figure 4.16: Contact conditions for the arch system in contact with the underlying rock during summer. Distances in meters

Figure 4.17: Contact conditions for the arch system in contact with the underlying rock during winter. Distances in meters

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4.2.4

2D-model

Figure 4.18 shows the results for monolith 10 from the 2D-model compared with the measurements. The 2D-model captures the behavior but has a more rapid response to temperature fluctuations. This is likely due to the fact that the 2D-model can not reflect the arching effect in the dam.

Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015

−20 −15 −10 −5 0 5 10 15 Deformation [mm]

Radial crest deformations at monolith 10 − 2D model α=8⋅10−6°C−1 Reference temperature=5 °C

FEA 2D Measured

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4.2. STRUCTURAL MECHANICAL ANALYSES

4.2.5

Comparison of cases

Figure 4.19 summarizes the results from case 2 and the 2D-model for monolith 10. The arching effect is naturally absent in the 2D-model which significantly affects the behavior. The computational time for the 2D-model is however only around 3% of that of the 3D-model. If the 2D-model was refined to better capture the amplitude of the seasonal movements, it could be a useful tool in FE-coupled condition mon-itoring, even if the error would be greater than the error given by the 3D-model. It should be noted that in a future FE-coupled monitoring program, where only one step needs to be added to the model at a time, the computational time for the 3D-model is likely acceptable for implementation in the program.

Jan 2010 Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015

−20 −15 −10 −5 0 5 10 15 20 25 Deformation [mm]

Radial crest deformations at monolith 10 α=8⋅10−6°C−1 Reference temperature=5 °C

FEA 3D - Case 2 FEA 2D Measured

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4.2.6

Possible deflection due to ice load

The model response to ice load is shown in figure 4.20. The RIDAS guideline values are 50-200 kN/m which corresponds to a crest deflection in the range of 1-9 mm. No distinction of an ice load could be made from the model results. However, since the standard deviation for the model is close to 2 mm, it is reasonable to assume that with intensified measurements and an extended monitoring period the ice load effect could be detected.

50 100 150 200 250 300 350 400 450 500 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Ice load [kN/m] Deformation [mm]

Deformation due to ice load

Ice load

RIDAS guideline values

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Chapter 5

Conclusions

It can be concluded from the parameter study that the expansion coefficient, α, governs the result, while the reference temperature has little influence on the stan-dard deviation. However, the reference temperature determines the zero level for the movement of the arch. This means that when evaluating the effects of the ice load with the method used in this thesis, the reference temperature is a significant factor that needs careful consideration. Reducing α results in a lower standard deviation, refer to Table 4.1 in Section 4.2.1. The standard deviation also decreased when all contact surfaces were modelled with a tie constraint, as in case 2. The analyses were focused on finding a parameter combination giving the smallest standard deviation for the error since this parameter was believed to be the best estimate for evaluating the behaviour of the concrete arch dam. The parameter-study resulted in a best fit with α=8·10−6 1/K, reference temperature 5 °C and case 2 interactions, as pre-viously shown in Figure 4.7. The allowed deviation plotted in Section 4.2.2 shows that the residual usually falls within the acceptable range. There was no noticeable effect of an ice load.

5.1

Limitations

The model behavior was only evaluated with respect to the mid-section. Capturing the behavior at the abutments is more difficult due to the complex conditions and small deformations at those sections. Typically, arch dams are connected to rock on both sides. The outer most monoliths, 1 & 18, at the dam studied, are connected to concrete spillways on one side and a combination of concrete and rock on the other. Capturing these conditions would require an extended model and a deeper analysis of these parts.

FE-coupled monitoring is beneficial, but it is important to note that there are large uncertainties connected to the deviation range. Values that fall outside the range could therefore detect points of interest but may not necessarily indicate a serious event. Another drawback with FE-coupled monitoring is that it requires a unique model for each new dam under study.

References

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