Numerical simulations of thermal loads exerted on
dams by laterally confined ice covers
Thèse
Ekaterina Kharik
Doctorat en génie civil
Philosophiae Doctor (Ph.D.)
Québec, Canada
Numerical simulations of thermal loads exerted on
dams by laterally confined ice covers
Thèse
Ekaterina Kharik
Sous la direction de :
Brian Morse, directeur de recherche
Mario Fafard, codirecteur de recherche
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RÉSUMÉ
Les barrages hydro-électriques construits dans les climats froids subissent des poussées des glaces importantes. Ces charges doivent être prises en compte pour l’évaluation de la sécurité de ces structures. Bien que les barrages aient été construits et exploités dans différents pays pendant de nombreuses années, il n’y a pas de consensus sur les critères de charge de glace. Les valeurs de conception varient considérablement – plus de 3 fois d’une autorité internationale à l’autre. La nature complexe des poussées des glaces implique de nombreux facteurs et incertitudes qui empêchent encore la pleine compréhension des processus en jeu, malgré les avancées récentes. Cette recherche doctorale vise à donner une nouvelle approche aux calculs des poussées des glaces sur les barrages et, ainsi, faire progresser l’objectif d’harmonisation des critères de conception pour les poussées des glaces. L’approche méthodologique de cette recherche consiste à compiler les modèles existants de comportement rhéologique de la glace et les inclure dans la simulation numérique des principaux événements historiques de poussées des glaces mesurés au Canada.
La construction d’un modèle numérique commence par un choix de processus et d’hypothèses clés, ainsi que par des modèles rhéologiques à inclure. Par conséquent, une revue compréhensive de la littérature sur le comportement rhéologique de la glace a été réalisée. Cette thèse décrit un modèle de fluage 3D du comportement rhéologique pour deux types de glace (glace columnaire et glace de neige). Le modèle construit peut être implémenté dans un logiciel d’éléments finis pour simuler le comportement de la glace sous les charges statiques. Au cours du projet, le logiciel ANSYS a été utilisé pour construire un modèle d’éléments finis pour des simulations d’événements mesurés. Le modèle d’éléments finis a été validé et étalonné par rapport à des expériences en laboratoire provenant de la littérature. Par la suite, il a été utilisé dans la simulation d’événements de poussées des glaces sur des barrages au Canada. Seule expansion thermique de la glace a été simulée par le modèle d’éléments finis. Les fluctuations de niveau d’eau ont été indirectement incluses dans le modèle par la simulation de la glace confinée latéralement. L’influence de différents facteurs sur la poussée des glaces simulées a été étudiée. Les
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conseils et les lignes directrices pour le choix et l’implémentation du modèle rhéologique de fluage dans un logiciel d’éléments finis commercial sont également fournis.
La thèse est basée sur des résultats d’études qui ont été publiés dans deux articles de colloques et dans un article de journal. La thèse inclut l’introduction des poussées des glaces sur les barrages, les objectifs du projet, les revues de littérature sur les processus influents des poussées des glaces, sur les propriétés rhéologiques des deux types de glace fréquents sur un réservoir et la modélisation du comportement de la glace, ainsi que la présentation des données de terrain, la description du modèle, les études non encore publiées, la conclusion et les recommandations. La thèse discute du lien entre les expériences en laboratoire sur la glace et les mesures in situ des poussées des glaces sur les barrages. L’étude documente le fait que le fluage élastique retardé de la glace a un effet limité sur les poussées des glaces résultantes pour les grandeurs de grains de 3 mm à 20 mm, et son calcul peut être omis dans des évaluations approximatives des poussées des glaces avec cettes grandeurs de grains moyennes. Il est également montré que la variabilité du coefficient de dilatation thermique de la glace avec la température n’est pas importante pour les températures étudiées. L’importance des conditions initiales et du comportement de la glace en tension pour les poussées des glaces résultantes est également montrée. La thèse met l’accent sur l’importance de la considération correcte du confinement latéral et du type de glace pour les calculs de poussées des glaces sur les barrages. Il a été montré que la hausse du niveau d’eau et le confinement latéral de la glace peuvent produire très grandes forces de poussées, si la glace est principalement columnaire. De ce fait, il devient important d’évaluer la possibilité d’avoir des hivers secs ayant peu d’accumulation de neige sur la glace (les conditions favorables à la formation de la glace columnaire) pour la sélection de la charge de conception.
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ABSTRACT
Hydroelectric dams are subjected to ice loads in cold climates. Withstanding such loads is an important safety requirement for the structures. Although dams have been built and operated by different countries for many years, there is no consensus on a design ice load criteria. Design values vary considerably – by more than 3 times from one international authority to another. The complex nature of ice loads involves many factors and uncertainties that still prevent the full understanding of ice field processes in spite of much progress done in this direction. This doctoral research aims to give new insights into reservoir ice loads, and, thus, to move forward the objective of harmonization of ice load design criteria. The main approach of this research is a compilation of existing models of ice material behaviour and their inclusion in the numerical simulation of key historical ice load events measured in Canada.
A construction of a numerical model begins with a choice of key processes and key assumptions as well as material models to include. Therefore, a comprehensive literature review on the topic is required. During the project, diverse literature on ice material behaviour was reviewed. This thesis describes a 3D creep material model for rheological behaviour of two types of ice, namely columnar (S2) ice and snow (T1) ice. The constructed model can be implemented into software to simulate behaviour of ice under the static loads. In this project, the commercial software ANSYS was used for this purpose to build a finite element model (FEM) for simulations of field events. After the FEM was built, it was validated and calibrated against laboratory experiments taken from the literature. Subsequently, the resulting FEM was used for simulation of key Canadian field events. Only thermal expansion of ice was simulated by the FEM with indirect accounting of the impact of water level fluctuations through the modelling of ice cover under lateral confinement. The influence of different factors on the simulated ice load was studied, namely, the influence (i) of selected material constants, (ii) of lateral confinement, (iii) of grain size and of delayed elastic creep, (iv) of variable or constant coefficient of thermal dilatation, (v) of ice type, (vi) of initial stresses as well as (vii) the influence of ice behaviour in tension. The advice and guidelines for the choice and implementation of a creep material model into commercial finite element software are also given.
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The thesis is based on studies conducted for two conference papers and one journal paper. The thesis includes introduction to the ice loads on dams, objectives of the project, literature reviews on governing ice loading processes, on material properties of two types of ice of frequent occurrence on a reservoir and on modelling of ice behaviour as well as field data presentation, model description, yet unpublished studies, conclusion and recommendations. The thesis discusses the linkage between laboratory experiments on ice and in situ measurements of reservoir ice loads on dams through the numerical modelling of field events. It documents the fact that delayed elastic creep has a limited effect on the resulting ice load for grain sizes of 3 mm – 20 mm, and its calculation can be omitted in approximate assessments of field ice loads for ice cover with these average grain sizes. It is also shown that variability of coefficient of thermal dilatation of ice with temperature is not important for the temperature range studied. The importance of initial conditions and ice behaviour in tension for the resulting ice load is also shown. The thesis emphasizes the importance of correct accounting of lateral confinement and ice type for ice load calculations. It is shown that when water level rises and the ice is confined laterally, if the reservoir consists primarily of columnar ice, the loads can be very substantial. As such, it is recommended that engineers pay particular attention to the possibility of having cold dry winters with little snow accumulations on an ice cover (favorable conditions for columnar ice formation) when choosing their design criteria.
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TABLE OF CONTENT
RÉSUMÉ ... iii
ABSTRACT ... v
TABLE OF CONTENT ... vii
LIST OF FIGURES ... x
LIST OF TABLES ... xiii
LIST OF ABBREVIATIONS ... xiv
LIST OF SYMBOLS ... xv
THESIS ACKNOWLEDGEMENTS ... xviii
1. Introduction ... 1
2. Literature review on governing ice loading processes ... 6
2.1. Field studies on reservoir ice ... 6
2.2. Contributing factors ... 7
2.2.1. Thermal expansion and ice loads... 7
2.2.2. Water level fluctuations and ice loads ... 8
2.3. Spatial-temporal variability in static ice forces ... 11
3. Field data presentation ... 12
3.1. Multi-day event (Seven Sisters, MB) ... 12
3.2. Diurnal events (Barrett Chute, ON) ... 15
4. Literature review on material properties of two most common reservoir ice types and on their modelling ... 27
4.1. Material properties of two most common reservoir ice types ... 27
4.1.1. Deformation process of an ice sample ... 28
4.1.2. S2 columnar ice ... 29
4.1.2.1. Physical properties ... 29
4.1.2.2. Laboratory studies on ice behaviour under uniaxial tension ... 29
4.1.2.3. Laboratory studies on ice behaviour under uniaxial compression ... 29
4.1.2.4. Laboratory studies on ice behaviour under compressive loading with lateral confinement ... 29
4.1.3. T1 snow ice ... 31
4.1.3.1. Physical properties ... 31
4.1.3.2. Laboratory studies on ice behaviour under uniaxial tension ... 32
4.1.3.3. Laboratory studies on ice behaviour under uniaxial compression ... 35
4.1.3.4. Laboratory studies on ice behaviour under compressive loading with lateral confinement ... 36
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4.2. Modelling of ice behaviour ... 37
4.2.1. Rheological models for ice material behaviour ... 37
4.2.2. Scale effects ... 38
4.2.3. Modelling of ice covers ... 38
5. Finite element model ... 41
5.1. Structural behaviour of ice ... 41
5.1.1. Governing assumptions ... 41
5.1.2. Base material model and its generalization into 3D ... 42
5.1.3. Model validation ... 47
5.1.4. Model calibration ... 50
5.2. Thermal behaviour of ice ... 56
5.2.1. Governing assumptions ... 56
5.2.2. Thermal strain ... 56
5.2.3. Validation of thermal strain calculation ... 56
5.3. Thermal-structural problem statement for simulations of field events ... 58
6. Numerical applications ... 61
6.1. Choice of mesh and time step ... 61
6.1.1. 02/23 Barrett Chute event summary ... 61
6.1.2. Simulation ... 62
6.2. The importance of the selected material constants and of confinement ... 66
6.2.1. Seven Sisters event simulation ... 66
6.2.2. 02/22 – 02/24 Barrett Chute events ... 68
6.2.3. Barrett Chute events modelling ... 69
6.2.4. Discussion ... 70
6.3. Importance of ice type ... 72
6.3.1. Seven Sisters event modelling ... 72
6.3.2. 02/22 – 02/23 Barrett Chute events modelling ... 73
6.3.3. Discussion ... 74
6.4. The importance of the grain size and of the delayed elastic term in the material model ... 77
6.4.1. Seven Sisters event modelling ... 77
6.4.2. 02/22 Barrett Chute event modelling... 78
6.4.3. Discussion ... 79
6.5. Necessity of including the variable coefficient of thermal dilatation ... 81
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6.5.2. 02/22 Barrett Chute event modelling... 81
6.5.3. Discussion ... 82
6.6. Importance of initial conditions ... 83
6.6.1. 02/23 Barrett Chute event modelling... 83
6.6.2. 02/22 Barrett Chute event modelling... 86
6.6.3. Seven Sisters event modelling ... 89
6.6.4. Discussion ... 90
6.7. Importance of tensile cracks ... 91
6.7.1. Details on simulations of the whole 2011 winter at Barrett Chute ... 91
6.7.2. Results of simulations ... 93
6.7.3. Discussion ... 99
7. Key findings, conclusion and recommendations... 102
References ... 109
Appendices ... 117
Appendix A. Principle of calculation of Sinha’s model (1983) ... 117
Appendix B. Stress history of delayed elastic strain in ANSYS ... 121
Appendix C. Initial temperature distribution in ANSYS ... 125
Appendix D. Usermat3d ... 129
Appendix E. Lists of parameters for numerical simulations ... 151
Appendix F. Example of an auxiliary program for calculation of ice load ... 216
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LIST OF FIGURES
Figure 2.1. Typical ice temperature profiles during a warming event. ... 7
Figure 2.2. Types of cracks produced by water level fluctuations on a dam reservoir (photos courtesy of George Comfort, Comfort et al. 2016). ... 9
Figure 2.3. Circumferential cracks on a dam reservoir (inspired by Lupien 2013). ... 10
Figure 2.4. An illustration of the mechanism by which water level fluctuations increase ice loads (inspired by Comfort et al. 2003). ... 10
Figure 3.1. Location of Seven Sisters dam on the map. ... 13
Figure 3.2. Layout of BP sensor groups at Seven Sisters dam (photo of BP sensors from Taras et al. 2011). ... 13
Figure 3.3. Field data (Seven Sisters, data from Comfort and Gong 1999). ... 14
Figure 3.4. Location of Barrett Chute dam on the map. ... 15
Figure 3.5. Layout of BP sensor groups, Carter panels and biaxial gauges at Barrett Chute dam (Figure 1 from Taras et al. 2011). ... 16
Figure 3.6. Field data (Barrett Chute, data from the field campaign reported in Morse et al. 2011b, Taras et al. 2011, Comfort et al. 2016). ... 17
Figure 3.7. Average Near Field stresses (in kPa) interpolated over the depths. ... 19
Figure 3.8. Variation of stress measurements at the wall with depth. ... 22
Figure 3.9. Some stresses measured by panels at different depths. ... 23
Figure 3.10. Ice growth in the crack near the dam during the whole winter. ... 24
Figure 3.11. Horizontal motion of crack gage away from dam along with water level rise or drop for the study period 2011/02/17 – 2011/03/01. ... 25
Figure 3.12. Vertical motion of crack gage and water level rise or drop for the study period 2011/02/17 – 2011/03/01. ... 26
Figure 4.1. A scheme of a reservoir ice cover consisted of T1 ice and S2 ice and the structure of these two types of ice (inspired by Lupien 2013). ... 27
Figure 4.2. Scheme of a biaxial experiment conducted by Frederking (1977) on S2 columnar ice (inspired by Sanderson 1988). ... 30
Figure 4.3. Plane strain strength to uniaxial strength ratio as a function of strain rate – S2 ice (deduced from Frederking 1977). ... 31
Figure 4.4. Dependence of tensile strength of T1 ice on strain rate (inspired by Hawkes and Mellor 1972; Michel 1978b)... 34
Figure 4.5. Dependence of tensile strength of T1 ice on temperature (inspired by Michel 1978b; Haynes 1979). ... 34
Figure 4.6. Dependence of tensile strength of T1 ice on grain size (inspired by Michel 1978b; Currier and Schulson 1982). ... 35
Figure 4.7. Scheme of a biaxial experiment conducted by Frederking (1977) on T1 snow ice (inspired by Sanderson 1988). ... 36
Figure 4.8. Scheme of Burger’s model. ... 37
Figure 5.1. Coordinate system for columnar ice assumed in the FEM (S2 ice (blue) superimposed by T1 ice (transparent), p – symmetry plane for S2 columnar ice). ... 47
Figure 5.2. Schemes for numerical tests used for the FE model validation (z-axis is directed towards the observer)... 48
Figure 5.3. Comparison of numerical results of uniaxial constant load tests (100 kPa, 500 kPa, 900 kPa) with analytical solutions. ... 49
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Figure 5.4. Comparison of FE model predictions with results of constant strain rate
experiments (5·10-7 s-1, 1.7·10-6 s-1, 1·10-5 s-1, 3·10-5 s-1; Sinha 1982). ... 50
Figure 5.5. Data of uniaxial and biaxial experiments on T1 snow ice (Frederking 1977). .. 51
Figure 5.6. Data of uniaxial and biaxial experiments on S2 columnar ice (Frederking 1977). ... 51
Figure 5.7. The front view for a 3D solid (or the structural scheme for a 2D solid), z-axis is directed towards the observer (plane strain for the 2D solid is in z-axis). ... 52
Figure 5.8. Calibration of the model using uniaxial and biaxial experiments on T1 snow ice. ... 53
Figure 5.9. Calibration of the model using uniaxial and biaxial experiments on S2 columnar ice. ... 54
Figure 5.10. The x-component of thermal strain in strains (ANSYS). ... 57
Figure 5.11. The front view for a 3D solid (or the structural scheme for a 2D solid), z-axis is directed towards the observer (plane strain for the 2D solid is in z-axis). ... 59
Figure 6.1. 02/23 Barrett Chute event summary. ... 61
Figure 6.2. Dependence of the numerical solution on the mesh size. ... 63
Figure 6.3. Dependence of the numerical solution on the time step size. ... 64
Figure 6.4. Figure 6.2 with reference case with 4040 mesh and time step of 20 s. ... 65
Figure 6.5. Figure 6.3 with reference case with 4040 mesh and time step of 20 s. ... 65
Figure 6.6. Observed and calculated line loads (Seven Sisters multi-day event). ... 67
Figure 6.7. 02/22 – 02/24 Barrett Chute events (zoomed in from Figure 3.6a and Figure 3.6d). ... 69
Figure 6.8. Observed and calculated line loads (Barrett Chute diurnal events). ... 70
Figure 6.9. Observed average and calculated loads for Seven Sisters multi-day event (different ratios between T1 and S2 ice types: 0%/100%, 50%/50%, 90%/10% of the ice cover). ... 73
Figure 6.10. Observed average and calculated loads for 02/22 and 02/23 Barrett Chute diurnal events (different ratios between T1 and S2 ice types: 0%/100%, 50%/50%, 90%/10% of the ice cover). ... 74
Figure 6.11. Observed and calculated loads (F-FC-b model with different grain sizes for Seven Sisters event). ... 78
Figure 6.12. Observed and calculated loads (FC-b model with different grain sizes and F-FC-b-EVO model for Seven Sisters event). ... 78
Figure 6.13. Observed and calculated loads (F-FC-b model with different grain sizes for 02/22 Barrett Chute event). ... 79
Figure 6.14. Observed and calculated loads (FC-b model with different grain sizes and F-FC-b-EVO model for 02/22 Barrett Chute diurnal event). ... 79
Figure 6.15. Calculated loads with variable and constant coefficients of thermal dilatation (multi-day Seven Sisters event). ... 81
Figure 6.16. Calculated loads with variable and constant coefficients of thermal dilatation (02/22 diurnal Barrett Chute event). ... 82
Figure 6.17. Calculated ice loads with 0 kPa, +63 kPa, and +114 kPa initial stresses for 02/23 Barrett Chute event. ... 84
Figure 6.18. Calculated ice loads with 0 kPa, +63 kPa, +114 kPa, and -114 kPa initial stresses for 02/23 Barrett Chute event. ... 84
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Figure 6.19. Calculated ice loads with different initial stresses for 02/23 Barrett Chute event (t = 0 at 07:15). ... 85 Figure 6.20. Calculated ice loads with different initial stresses for 02/23 Barrett Chute event (t = 0 at 09:00). ... 86 Figure 6.21. Diurnal event: a) observed and calculated loads with different initial stresses; b) calculated stress profiles within the ice cover (initial conditions and time of maximum calculated load); c) calculated stresses and temperatures (ice surface and ice bottom). ... 88 Figure 6.22. Multi-day event (Seven Sisters). ... 89 Figure 6.23. Far Field ice temperatures (in °C) for full winter calculations. ... 92 Figure 6.24. Creep factors (CF) for tension calculations: a) Case 3, b) Case 4, Case 5/Case 7. (Note the difference in y-scales for a) and b).) ... 93 Figure 6.25. Measured and calculated ice loads (cases 1 – 5) along with measured ice surface temperature... 94 Figure 6.26. Calculated stresses at different depths (from the ice surface): a) Case 1, b) Case 2, c) Case 3, d) Case 4, e) Case 5... 96 Figure 6.27. The structural scheme for 2D solid (plane strain in z-axis directed towards the observer). ... 97 Figure 6.28. Measured and calculated ice loads (cases 1, 6, and 7) along with measured ice surface temperature... 98 Figure 6.29. Calculated stresses at different depths in the middle of the ice solid a) for case 6; and b) for case 7 (ice detachment at -25 kN/m in both cases, but tensile behaviours of two cases are different – shown by dashed red ovals). ... 99
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LIST OF TABLES
Table 1.1. Objectives of the thesis. ... 5
Table 3.1. Observed peak line loads and their confinement at Barrett Chute. ... 18
Table 4.1. Summary of experiments on tensile strength of T1 ice. ... 33
Table 5.1. Summary of the base material model (* parameter value at the reference temperature of 263 K or -10°C). ... 46
Table 5.2. Elastic constants (Sanderson 1988; Sinha 1989b) used in the finite element model. ... 47
Table 5.3. Coefficients for equivalent stress formulation (Zhan 1993; Naumenko 2006 – von Mises) used in the finite element model. ... 47
Table 5.4. Parameters of numerical tests and boundary conditions for validation. ... 49
Table 5.5. Parameters of the solids for calibration. ... 52
Table 5.6. Parameters for S-UC, F-UC, and F-FC models (* parameter value at the reference temperature of 263 K; refer to Sinha (1979)). ... 53
Table 5.7. Field mechanisms included and not included into the FEM. ... 60
Table 6.1. Norms of Figure 6.2 cases relative to reference case norm. ... 63
Table 6.2. Norms of Figure 6.3 cases relative to case reference norm. ... 64
Table 6.3. Norms of Figure 6.4 and Figure 6.5 cases relative to one reference case norm (T1 – S2 (4040, dt = 20 s)). ... 65
Table 6.4. Observed (Obs.) and Calculated (Calc.) peak line loads and their respective ratios. (*Estimated maximum near field line load at Seven Sisters is 3.6 times measured maximum far field line load based on the ratio between average maximum near field and average far field loads at Barrett Chute. All other observed line loads in the table are truly observed.) ... 68
Table 6.5. Calculated peak loads. ... 86
Table 6.6. Summary of seven numerical scenarios. ... 93
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LIST OF ABBREVIATIONS
Abbreviation Definition
CF Creep Factor
FEM Finite Element Model
FF Far Field
F-FC Frederking – Fully Confined (biaxial constants)
F-FC-b Frederking – Fully Confined (biaxial constants) – biaxial confinement (confined in x- and z-directions)
F-FC-b-EVO Frederking – Fully Confined (biaxial constants) – biaxial confinement (confined in x- and z-directions) – Elastic and Viscous terms Only
F-UC Frederking – UnConfined (Uniaxial Constants)
F-UC-u Frederking – UnConfined (Uniaxial Constants) – uniaxial confinement (confined in x-direction only)
F-UC-b Frederking – UnConfined (Uniaxial Constants) – biaxial confinement (confined in x- and z-directions)
IREQ Hydro-Quebec’s Research Institute
NF Near Field
S-UC Sinha – UnConfined (Uniaxial Constants)
S-UC-u Sinha – UnConfined (Uniaxial Constants) – uniaxial confinement (confined in x-direction only)
S-UC-b Sinha – UnConfined (Uniaxial Constants) – biaxial confinement (confined in
x- and z-directions)
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LIST OF SYMBOLS
Symbol Definition (Units)
Coefficient of thermal dilatation (K-1)
Δ Increment of a value
Total strain (-)
Viscous strain (-)
1
Viscous strain rate corresponding to 1 (s-1) cr Creep strain (d + ) (-)d Delayed-elastic strain (-)
e Elastic strain (-)
m Mechanical (or structural) strain
th Thermal strain (-)
Poisson ratio
p Poisson ratio of S2 ice at plane of isotropy xz (-) T1 Poisson ratio of T1 ice (-)
yp Poisson ratio in the direction of columns of S2 ice (-)
Density of ice (kg/m3)
p Density of pure ice (917 kg/m3)
Stress (kPa)
u Unit stress (kPa)
dev Stress deviator (kPa)
σeq Equivalent stress (kPa)
FJ1, FJ2, FJ3, FJ4 Stress measured by FJ1, FJ2, FJ3, FJ4 (kPa) parallel Stress parallel to a dam face (kPa)
perpendicular Stress perpendicular to a dam face (kPa)
σxx, σyy, σzz Normal stress components of creep strain in engineering notation (kPa)
σxy, σyz, σxz Shear stress components of creep strain in engineering notation (kPa)
aT Inverse relaxation time at temperature T (s-1)
b Time exponent (-)
c1 A constant (-)
d Grain size (mm)
d1 Unit of grain size (mm)
di Diameter (cm)
e Porosity (-)
e0 Reference porosity (0.285)
E Young’s modulus (kPa)
E0 Young’s modulus of ice (kPa)
ET1 Young’s modulus of T1 ice (kPa)
Ep Young’s modulus of S2 ice (plane of isotropy xz) (kPa)
Ey Young’s modulus in the direction of columns of S2 ice (kPa)
FJ1, FJ2, FJ3, FJ4 Flatjack 1, flatjack 2, flatjack 3, flatjack 4 at depth 0.1 m, 0.3 m, 0.5 m, 0.7 m from the ice surface, respectively
Gyp Shear modulus in y-direction for S2 ice (kPa)
h Ice thickness (m)
h1, h2, h3, h4 Ice thicknesses attributed to FJ1, FJ2, FJ3, FJ4 (m)
jj xx, yy, or zz (-)
jl xy, yz, or zx (-)
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l Original length of a specimen (m)
L Final length of a specimen (m)
m1, m2, m4, m5 Coefficients for equivalent stress calculation (-)
n Stress exponent (-)
p xz plane – plane of isotropy for S2 ice
P Constant load (kPa)
P1, P2, P3 P4, P6, P8
Panel 1, panel 2, panel 3, panel 4, panel 6, panel 8
P11(9), P13(11) Panel 11(9), panel 13(11)
Q Activation energy (67000 J/mol)
R Gas constant (8.31 J mol-1 K-1) [S] Compliance matrix for Hooke’s law
S1,2 Shift function (-)
S2 Columnar ice
t Time (s)
T, T1, T2 Temperature (K)
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THESIS ACKNOWLEDGEMENTS
I would like to acknowledge the contribution of the following people and agencies:
Brian Morse, my research director, for his continuous support since the start of my studies in Canada, for his constant availability (in person, by email, by phone and by Skype), for his help, guidance and advice during the whole project, for reading and correcting my work, for connecting me with world cold regions experts and for the financial support during the last phase of the project.
Varvara Roubtsova, my supervisor in Hydro-Quebec’s Research Institute (IREQ), for her support before my coming to Canada, for introducing me to Canadian reality, for organizing my workplace in IREQ, for introducing me to previous work on the topic in IREQ, for her constant availability and for her help and advice in theoretical and programming aspects.
Alain Côté, chief of the project in IREQ, for finding funding for the first phase of the project, for providing me administrative support, continuous access to my workplace in IREQ and to field measurements done by IREQ and BMP Fleet Technologies, for reading, correcting and commenting my work, for advice and for help and critical discussions.
Mario Fafard, my research co-director, for his support particularly for the theoretical aspects, for being available for joint Skype discussions, for correcting my work, for financial support during one of the last terms of the project, for critical review of my research and for his advice.
André Taras, researcher in IREQ, for providing field data files of excellent quality, for introducing me to his analyses of field data, for reading, correcting and commenting my work, for discussions, for critical review of my research and for help and advice.
George Comfort, Ice Engineering Specialist in G. Comfort Ice Engineering Ltd., for providing results of his model for some distinct field events and explanations on calculations, for providing field data measured by BMP Fleet Technologies and for his scientific support during my research.
Robert Frederking, Research Council Officer in National Research Council Canada, for helping interpret his laboratory data, for his availability when I had questions about his measurements, for providing high quality scans of his paper figures, for accepting to be an external examiner at my oral exam.
Roger Lupien, researcher in IREQ, for introducing me into his work on the topic, for providing me his literature review on the topic and for discussions particularly at the early stages of the project.
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Edward Stander, professor at SUNY Cobleskill, for his scientific and technical help in my research and for reviewing some important parts of my work.
Luca Sorelli, professor at Université Laval, for his continuous availability to examine my progress during studies and for his course that enriched my experience in presentation.
Brian Morse, Mario Fafard, Robert Frederking, Luca Sorelli, Jean-Loup Robert, my jury, for accepting to examine my thesis.
Jean Côté, Director of Graduate Program (Civil Engineering), for his help in organising the final defence and for accepting to be a president of the jury.
Dany Crépault, Martin Lapointe, Denis Jobin, René Malo, technicians at Université Laval, for their support in the cold room laboratory and in the field.
Mathieu Dubé, Master student at Université Laval, for his help in preparation of thin sections of snow ice in laboratory and for scan samples of snow ice for study of their porosity.
Marlyne Fergusson, Kathy Champagne, Lorraine Malouin, Julia Lebreux, Isabelle Boucher for their administrative support at Université Laval, for their help with documents for scholarships, contracts and expenses.
Fonds de recherché du Québec – Nature et technologies (FRQNT), Natural Science and Engineering Research Council of Canada (NSERC) and IREQ for their financial support.
I would like to thank the following people for personal reasons:
My beloved husband Evgeny for always being there for me, for his continuous encouragement, support and belief in me even during the moments of my disbelief, for his enormous support in all aspects of life, for his patience and for being the first listener of all my presentations.
My father Konstantin, my mother Olga, my grandparents Mariya, Ivan and Zoya and my family-in-law (Oleg, Alfiya, Ludmila, Igor) for their huge patience, comprehensiveness and for not asking some inconvenient questions too frequently. My friends (Yevgen, Yuliya, Isabelle, Andrey, Yuliya, Nikita, Ekaterina, Yuliya,
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1. Introduction
Hydroelectric dams in northern climates experience numerous ice loads during a winter period. Determining the potential maximum ice loads on a dam is an important factor in the design of a safe structure to protect people living along the rivers downstream of dams in Canada and in other countries with cold climates (United States, Northern Europe, etc.).
Reviewing reports of field research programs conducted in Canada, there are two types of extraordinary events measured in situ at Seven Sisters dam (multi-day) and Barrett Chute dam (diurnal) (see Chapter 3; Comfort et al. 2003; Taras et al. 2011). Both of them were the result of a number of unfavorable conditions, such as absence of snow cover on ice, intermediate water level fluctuations, ice crack configurations that greatly resist ice sheet movements, and a warming period that led to large ice temperature change. These loads are of concern because they demonstrate that load values can be very significant when unfavorable conditions occur at the same time.
Engineers must ensure the safety of the dams. The problem is that these events suggest that loads can be substantial, and yet it is not known how substantial the design load should be. One method to find out it is to measure loads at a large number of dams over a large number of years. This was tried by Hydro-Quebec and others, but it is difficult to get a sample size big enough to cover all scenarios. Another approach is to empirically model the driving forces using relationships developed from in situ measurements (Comfort et al. 2004). A third approach (proposed by Sodhi and Carter 1998) is to examine the failure mechanism of ice cracks near a dam face and to base the design on the capacity of the ice to pass on environmental loads to the dam. A fourth approach (proposed by Sanderson 1988) is to introduce an empirically-based contact area factor to account for indentation.
All of these approaches have their advantages and should be used where appropriate. Knowing that field studies are costly, time consuming, and difficult, knowing that empirical relationships may not cover the most crucial circumstances, knowing that some ice covers (Barrett Chute and Seven Sisters) seem to be stronger than the failure mechanisms predicted by Carter et al. (1998), knowing that generating mechanisms still need to be better understood, this thesis attempts to model some key processes and fill in some of the
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data and knowledge gaps through the construction of a 3D creep material model for ice based on the relevant literature. The effort follows the numerical modelling initiative of thermal ice loads undertaken by Azarnejad and Hrudey (1996) and Côté et al. (2016) with particular emphasis on the influence of lateral confinement on ice material behaviour.
Comfort et al. (2003) emphasized that two main mechanisms generating ice loads are ice temperature changes and water level fluctuations. Ice properties depend significantly on its temperature. During cold weather, once equilibrium is reached, there is a temperature gradient within an ice cover (upper ice layers are colder than lower layers). When non uniform temperature distribution exists within a reservoir ice cover it has a complex stress state. Thermal ice loads are quite well understood theoretically and were modelled numerically (for example, Donald Carter & Associés 1996; Azarnejad and Hrudey 1996). However, the effect of lateral confinement of a reservoir ice cover has almost never been correctly taken into account.
Indeed, the ice cover in numerical simulations is often assumed to be isotropic and to consist of one type of ice, although several ice types can be present (often, S2 transversely isotropic columnar ice1 superimposed by T1 isotropic snow ice2, Michel and Ramseier’s classification 1971). Normally material (rheological) models based on uniaxial small-scale laboratory experiments are used (e.g. Lupien et al. 2013; Côté et al. 2016) because there are no constitutive equations for ductile behaviour of ice deduced from medium-scale / large-scale experiments at low strain rates. Sometimes the resulting uniaxial load is increased by 15% – 20% to account for plane strain conditions (e.g. Lupien et al. 2013). In case of T1 ice, this seems to be sufficient to represent ductile behaviour of ice cover under lateral confinement. But for S2 ice, it is not (see Section 4.1.2.4). Therefore, it is essential to take correctly into account the effect of lateral confinement on material behaviour of reservoir ice cover.
1 The formation of S2 ice occurs when reservoir water freezes gradually from the top of primary ice cover
(with a random crystal orientation) towards a reservoir bed.
2 The formation of T1 snow ice occurs because of flooding of snow resting on an ice cover. The flooding can
occur when the weight of the snow exceeds the buoyancy of the ice cover. The flooded snow layer (usually called slush) forms the snow ice layer after freezing (Ashton 2011).
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Intermediate water level fluctuations cause the highest ice loads because these water level changes produce vertical or near vertical cracks with new ice growth in them (Comfort et al. 2003). One possible theoretical explanation is presented by Stander (2006): new ice forming in open crack along the dam walls makes ice cover to become larger with respect to its enclosing reservoir whose water level varies cyclically about a mean value. As water level rises and crack closes, the cover adjusts to the basin dimension by compressing the adjacent ice sheet (Morse et al. 2011). Another possible explanation is that lateral confinement of ice cover (and, therefore, ice cover strength) could be increased by water level fluctuations when ice cover goes through its most confining position (Section 4.1.2.4 and Section 6.2).
A load exerted by ice on a hydraulic structure is not uniform: there is a certain spatial-temporal variability in static ice forces within an ice cover that has been discussed by many researchers (Sanderson 1988; Prat 2010; Morse et al. 2011; Taras et al. 2011 – Section 2.3). There are different theories for explanation of this fact. The pressure variation along the dam face may be explained by imperfect contact between crack walls (faces) in through-ice cracks. Prat (2010) showed that stresses parallel to the dam were greater near the dam reaching their maximum at 5 m from the dam and diminished quasi-exponentially as a function of distance from the dam face. Possibly, the further ice is located from the interaction area, the less its confinement is.
Université Laval in collaboration with Hydro-Quebec’s Research Institute (IREQ) established a research program for studying ice loads on dams. Several research projects were completed to date both in field and in numerical simulation. The main objective of this thesis is to push forward numerical simulations of ice loads for the research and to give new insights into reservoir ice loads and, therefore, to advance to the harmonization of ice load design criteria. This main objective is divided into several sub-objectives presented in Table 1.1.
This chapter provides the introduction to the main problem as well as thesis objectives. Diverse literature is synthesized for three main topics: (i) governing ice loading processes, (ii) material properties of two most common reservoir ice types, and (iii) modelling of ice behaviour (Chapters 2 and Chapter 4). Chapter 3 presents the field data used for numerical
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simulations during the project. It also gives additional analysis of the data before their comparison with simulations. Chapter 5 describes the 3D creep material model for ice, the finite element model, and its validation and calibration. Only thermal expansion is modeled by the FEM but the impact of water level fluctuations is indirectly quantified through the inclusion of lateral confinement. Chapter 6 presents numerical applications of the model to study the influence (i) of selected material constants, (ii) of confinement, (iii) of grain size and of delayed elastic creep, (iv) of variable coefficient of thermal dilatation, (v) of ice type, (vi) of initial stresses, and (vii) the influence of tensile cracks on the resulting ice load. Chapter 7 summarizes numerical studies and key findings of the project and provides recommendations for following research directions. Appendix A – Appendix F provide some precisions on the principle of calculation of delayed elastic strain, additional information for avoiding difficulties encountered during a material model implementation into commercial finite element software such as ANSYS, algorithm of user material subroutine, lists of parameters for numerical simulations, and an example of auxiliary program for an event calculation. Appendix G provides some additional parametric studies on the model. The studies are reported at two conferences and in one journal paper (Kharik et al. 2015; 2016; 2018).
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Table 1.1. Objectives of the thesis.
Main objective Chapter(s) /
Section push forward numerical simulations of ice loads to improve current
knowledge on it and on field processes during ice loading on a dam
This thesis
Sub-objectives Chapter(s) /
Section compile up-to-date literature on governing ice loading processes Chapter 2 present field data and conduct additional analyses suitable for further
numerical modelling comparison
Chapter 3 compile up-to-date literature on material properties of S2 and T1 ice and
on modelling of ice material behaviour
Chapter 4 compile existing models and methods into a 3D rheological law for
modelling of ice material behaviour under static loads
Chapter 5 implement correctly ice material behaviour into commercial finite
element software ANSYS
Chapter 5, Appendix B – Appendix D identify material constants appropriate for numerical modelling Section 6.2 study the importance of lateral confinement for ice loading Section 6.2 study the importance of ice cover structure for ice loading Section 6.3 study the importance of grain size and of delayed elastic creep for ice
loading
Section 6.4 study the importance of variable coefficient of thermal dilatation for ice
loading
Section 6.5 study the importance of initial stress states for ice loading Section 6.6 study the influence of ice tensile behaviour on ice loading Section 6.7
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2. Literature review on governing ice loading processes
2.1. Field studies on reservoir ice
The main field studies available in the literature can be classified by the authors who reported them. The most important studies are described below, and their findings are summarized in Sections 2.2 and 2.3.
Carter et al. (2001) undertook a four-year program to measure static ice forces in five reservoirs located in Quebec. Having suggested that an ice cover cannot transmit a force to a structure larger than its own internal resistance, Carter et al. (2001) derived an upper bound for static forces by determining the in-plane compression force at which a fragmented ice cover collapses. Empirical formulae were presented also for walls, gates, and piers. The formulae correlated well with the field data, and it was suggested that the maximum ice load may be defined as a function of ice thickness, contact geometry, and lateral extent (indentation effect) only.
Comfort et al. (2003; 2004) reported an 11-year investigation that was conducted to study static ice loads on dams. Field measurements were done at nine sites over nine years. Parallel work was conducted to develop predictors for ice loads on the basis of the field studies. An environmental model was created to predict ice thickness, snow depth, slush, and ice temperature. Algorithms were developed to predict ice loads on long dam faces, on stop-logs, and on gates. Thermal ice loads, ice loads due to water level fluctuations, residual ice loads, and contingency loads were taken into consideration.
Stander (2006) reported the results of two field programs (winters of 1992 and 1993) during which ice stress, temperature, and water level data had been collected on a hydro-electric reservoir in Quebec. The collected data suggested that large compressive stresses can be developed by the repetitive opening and freezing of circumferential cracks in ice covers. Such cracks developed in reservoirs whose water level varies cyclically about a mean value.
Université Laval in collaboration with Hydro-Quebec (2008 – 2012): A research project was undertaken to measure ice loads using different sensors (Morse et al. 2009; 2011; Taras
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et al. 2011). Three types of instruments were compared, i.e., Carter panels, BP gauges, and Geocon biaxial gauges (Taras et al. 2011). Ice load measurements were carried out at four dams located in Quebec and in Ontario. Analytical and numerical studies were conducted in parallel. On the basis of the work a formula for ice load calculations was suggested by Lupien et al. (2013).
2.2. Contributing factors
Comfort et al. (2003; 2004) have collected a database of 155 events at nine dams during a nine-year field program. They have found that the main mechanisms generating ice loads are:
1) ice temperature and
2) water level fluctuations.
2.2.1. Thermal expansion and ice loads
The air temperature influences the temperature of bare ice surface, while the bottom of ice cover is at 0°C due to its contact with water. The high thermal loads are generally produced after an extended warming period (Comfort et al. 2004). Thermal load is characterized by ice temperature profile area (area lying between the ice temperature distributions at the start of an event and at the moment of peak loading). Typical ice temperature profiles during a thermal loading event are shown in Figure 2.1.
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It can be concluded from the figure that ice within an ice cover has a complex stress state: upper layers (subjected to temperature increase) have a tendency to expand while bottom layers (not yet subjected to a temperature change) may resist this expansion. Ice material properties depend significantly on the temperature. For example, the ice stiffness reduces with depth as the temperature of the ice increases with the depth (Comfort and Gong 1999). Ice pressure due to thermal expansion acts on a dam. The dam must provide sufficient strength to induce the strain rate into the ice equal to the rate of thermal expansion (Donald Carter & Associés 1990).
2.2.2. Water level fluctuations and ice loads
Comfort et al. (2003; 2004) reported that intermediate water level fluctuations (of about an amount equal to the ice thickness) increase ice load due to producing greatly resistant ice crack and ice cover conditions, i.e., vertical or near vertical cracks “with a great deal of new ice growth in them”. Different crack configurations are shown in Figure 2.2 (Comfort et al. 2016). Vertical or near vertical cracks (the most resistant to ice rotation) will cause the highest loads due to intermediate water level fluctuations. S-shaped cracks are less resistant to ice rotatation than vertical or near vertical cracks. They will produce lower loads due to intermediate water level fluctuations. Hinge-shaped cracks are not resistant to ice rotation. They will cause insignificant loads due to water level fluctuations. Ice cracks usually run parallel to dam wall and reservoir shores (Figure 2.3). To study crack configuration, ice block with crack is usually cut in front of a dam face. The region of cut is schematically shown by the black-boundary rectangle in Figure 2.3.
Stander (2006) reported that stresses associated with rising of a water level had been of the order of 4 – 5 kPa per cm rise and had been superimposed on thermal stresses, giving a maximum ice load. Morse et al (2011b) reported similar values: 3.5 – 5.5 kPa per cm rise. Morse et al. (2011b) also informed that the greatest stress rise (related to water level rise) had occurred nearest the dam. It could be explained by better confinement of ice cover near the dam as stresses parallel to dam face are greater near the dam (Prat 2010).
An illustration of the mechanism by which water level fluctuations increase ice loads is given in Figure 2.4. At a mean water level, an ice cover has certain dimensions and
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circumferential cracks are closed (Figure 2.4a). When water level goes up or down, cracks open (Figure 2.4b and Figure 2.4c). Open cracks allow the ice cover to expand quite freely. Also, new ice growth can appear in open cracks. Both new ice growth and free thermal expansion of ice increase the dimensions of the ice cover. When water level returns to its mean value, the ice cover has to match quickly its previous dimensions. This process produces lateral confinement, effectiveness of which depends on crack configurations (Figure 2.2). Effective lateral confinement can considerably increase ice load in case of S2 columnar ice. Indeed, lateral confinement of S2 ice changes its material behaviour by making S2 ice stronger because of its structure (Frederking 1977; Section 4.1.2.4).
Figure 2.2. Types of cracks produced by water level fluctuations on a dam reservoir (photos courtesy of George Comfort, Comfort et al. 2016).
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Figure 2.3. Circumferential cracks on a dam reservoir (inspired by Lupien 2013).
Figure 2.4. An illustration of the mechanism by which water level fluctuations increase ice loads (inspired by Comfort et al. 2003).
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2.3. Spatial-temporal variability in static ice forces
Morse et al. (2009) reported a spatial heterogeneity of ice forces on a dam wall. They placed twelve Carter panels (consisted of four flatjacks each) against a dam. It was noticed that some flatjacks registered stresses, while others did not. Taras et al. (2011) reported the same behaviour. Additional Barrett Chute data will be presented in Section 3.2 in support of the observations of stress variation along a dam face. The spatial heterogeneity of ice forces may be explained by imperfect contact between faces of through-ice circumferential cracks (Sanderson 1988).
Prat (2010) did a thorough analysis of the spatial-temporal strain evolution in the ice cover on a dam reservoir for winter 2008-2009. He showed that stresses parallel and perpendicular to the dam were greater near the dam and diminished as a function of distance from the dam face, although stress fluctuations at different distances had the similar pattern. The possible explanation of it is that the further ice is located from the interaction area, the less it is laterally confined.
Also, ice movement within an ice cover was analyzed by Prat (2010). It was found that new ice formation in cracks near shore provoked an ice movement towards a center of a reservoir. The largest ice movement was near the dam face and near shore lines. The movement attenuated at the center of a reservoir. The calculated strain rate near the dam fluctuated and could be approximated by a sinusoidal curve with limits [-10-6 s-1; 10-6 s-1]. Morse et al. (2011a) reported that most of strain rates inside an ice cover on a reservoir are typically of the order of 10-9 s-1 – 10-7 s-1. The ice loads on a dam are slow and can last up to several days (Comfort et al. 2003).
Stresses within an ice sheet are redistributed due to creep of the ice which is dependent upon many factors including the ice type, the initial temperature gradient of the ice, and rate of temperature rise (Comfort and Gong 1999). To model ice loads on dams, it is necessary to know ice types in a case ice cover and their physical and material properties. Field data for numerical case studies are presented in the next chapter. And literature review on material properties of two most common reservoir ice types is presented in Chapter 4.
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3. Field data presentation
Ice loading events measured against a dam face can be divided into two major types: multi-day and diurnal events. This chapter presents both types. Section 3.1 presents a multi-multi-day event, and Section 3.2 presents diurnal events. These observed events will be simulated in Chapter 6.
3.1. Multi-day event (Seven Sisters, MB)3
One of the largest Canadian multi-day ice load events (5 days 17 hours) was registered at Seven Sisters dam, MB (Figure 3.1), mid-January 1999. At that time, the ice cover was 0.51 m thick and consisted of S2 ice only (Comfort and Gong 1999). Pressures in ice cover were measured with two BP sensor groups (Duckworth and Westermann 1989) at about 24 m from the dam face (considered to be “far field” – FF, Figure 3.2). Line loads were calculated “by integrating the measured stresses over their respective areas” (Comfort and Gong 1999). The average FF line load was then calculated. The peak FF line load reached 356 kN/m during the event. Such a load value was unexpectedly high for 0.51 m ice thickness. This and other similar events measured during nine-year field studies contributed to conclusion that intermediate water level fluctuations amplify thermal ice loads on dams (Comfort et al. 2003). Indeed, the load resulted from superposition of: (i) large ice temperature changes produced by very cold winter on a poorly-insulated cover; and (ii) intermediate water level fluctuations. The thermal strain rate at the time of peak load was ~10-9 s-1. Figure 3.3 presents a summary of the event (sign convention: compression is positive, tension is negative). The event started at January 13 at 06:45 (green line in Figure 3.3). Ice loads peaked after 61 hours (≈ 2.5 days) on January 15 at 20:00 when the ice surface temperature was -5.7°C (ΔT = 18.7°C; 84% of maximum ΔT during the event = 22.3C). The temperature at the ice surface rose from -24.4°C to -2.1°C during 86.5 hours approximately (≈ 3.6 days; average rate of temperature rise – 0.26°C/h). The time lag between the time of the peak load and time of the maximum temperature at the ice surface was 25.5 hours. The air temperature rose from -28C to -1.1C during the event. The water level fluctuations were ±0.12 m about mean January water level. There
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were no instruments deployed to measure lateral stress confinement or to measure the loads generated at the dam face (“near field loads”).
Figure 3.1. Location of Seven Sisters dam on the map.
Figure 3.2. Layout of BP sensor groups at Seven Sisters dam (photo of BP sensors from Taras et al. 2011).
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3.2. Diurnal events (Barrett Chute, ON)4
Diurnal events recorded at Barrett Chute dam, ON (Figure 3.4), in February 2011 (Taras et al. 2011) are among the largest loads of this type ever measured against a dam face in Canada. Barrett Chute dam was instrumented with BP sensor groups, Carter panels (Carter et al. 1998) and biaxial gauges (Morse et al. 2011b) – Figure 3.5. Two BP sensor groups were deployed at 27 m from the wall (FF). Line load from these two sensor groups was calculated by the same method as in previous section.
Figure 3.4. Location of Barrett Chute dam on the map.
Pressures in ice cover were also measured with 11 Carter panels against the dam wall (near field – NF) separated by 2.5 m from each other. Each panel had four flatjacks at different depths (0.1 m, 0.3 m, 0.5 m, and 0.7 m from the ice surface). The flatjacks registered stresses every 5 minutes. Line load for each panel was calculated by multiplying registered stresses at each depth and their attributed ice thicknesses – Equation (3.1) – list of symbols is given at page xv. The average line load was then calculated from line loads of individual panels (Taras et al. 2011).
Panel Line Load = FJ1·h1 + FJ2·h2 + FJ3·h3 + FJ4·h4 (3.1)
Biaxial gauges were also deployed in the reservoir ice cover. They were placed in four rows, “all within 10 m of the dam face” (Morse et al. 2011b). The gauges registered
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magnitude and direction of principal stresses. Stresses perpendicular and parallel to the dam were then retrieved from the measurements by Ed Stander (2016, personal communication).
Figure 3.5. Layout of BP sensor groups, Carter panels and biaxial gauges at Barrett Chute dam (Figure 1 from Taras et al. 2011).
Seven diurnal events were measured during the winter (Figure 3.6). Among them, three events were of significant importance, namely events measured on February 22, 23, and 24. The maximum FF line loads for all seven events are given in the second column of Table 3.1. During the winter 2011, reservoir ice at Barrett Chute had through-ice circumferential crack, the faces of which may have frozen together for short periods of time. Given that the contact of the ice cover with dam and reservoir shoreline is quite erratic (due to the presence of cracks in the ice sheet parallel to the boundaries) and given that the ice sheet can contain many material imperfections and some lateral instability, the line load in the FF has less time-dependent variability and less amplitude variability than maximum loads recorded at (and near) the dam face (near field – NF). Also, the line load in the FF increases slower during events than the maximum NF load.
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Figure 3.6. Field data (Barrett Chute, data from the field campaign reported in Morse et al. 2011b, Taras et al. 2011, Comfort et al. 2016).
The line load recorded at the dam face depends on the width of over which pressures are averaged (Taras et al. 2011) because the degree of contact between the ice sheet and the dam face depends on highly variable contacts through the ice cracks. This phenomenon of
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lower average load as a function of integrated width is often referred to as “indentation” and is present in all ice measurements done throughout the world. The peak NF line loads averaged over the 11 panels (spanning 25 m) are given in the third column of Table 3.1. These average NF values correspond to about 1.9 times the FF values. The local maximum NF values are given in the fourth column of Table 3.1 and are, on average, 3.6 times maximum FF values (Max local NF
FF)5, 6.Table 3.1. Observed peak line loads and their confinement at Barrett Chute. Event date (mm/dd) Measured peak line load (kN/m) parallel / perpendicular
FF Ave. NF Max. (local NF)
02/17 21 60 181 0.37 02/20 27 101 176 0.33 02/22 50 117 208 0.50 02/23 76 145 295 0.54 02/24 81 128 199 0.70 02/26 61 94 174 0.61 02/27 82 110 203 0.73
Such high loads were recorded during a normal winter with 0.45 m thick reservoir ice (0.225 m of S2 ice covered by 0.225 m of T1 ice). Most of the loads resulted from: (i) a thin or no snow cover; (ii) a cold spell followed by a warm period with rain (among all events, maximum ΔT = 8.4C); and (iii) intermediate water level fluctuations. The water level fluctuation pattern was quite irregular with sharp rises and drops (Figure 3.6e). The relationship between water level rise and pressure rise was reported by Stander (2006) and Morse et al. (2011b). Pressure increases by 3.5 – 5.5 kPa per cm rise of water level. The stresses parallel to the dam face (parallel) were 0.33 – 0.73 times stresses perpendicular to
the dam face (perpendicular) for peak loads of reported events (fifth column in Table 3.1).
These stresses parallel to the dam face show lateral confinement and possible stresses generated in this direction. Thermal strain rates at time of the reported peak loads were of order 10-10 s-1 to 10-8 s-1.
5 Similar analysis of three events only (February 22, 23, and 24) gives value of 3.4.
6 This value is important because it will be used for estimation of local maximum NF load for multi-day
Seven Sisters event (in Chapter 6) as there were no NF measurements at Seven Sisters. This estimation is used because the model described in Chapter 5 is expected to model local maximum NF load (Section 6.2).
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Figure 3.6d shows the average NF load (full black line) but also the minimum and maximum recorded load at any one of the 11 Carter panels (local load). Note that there is a significant difference between the minimum and maximum values. The maximum value of local line load was not attributed to a certain panel always measuring maximum stresses. If it was the case, the stress and load variation could be attributed to an offset in measurements of that panel. Instead, the maximum local load switched from one panel to another in a chaotic fashion as events developed (Taras et al. 2011). The variability of the loads from one panel to the next may be a result of the changing contact along the crack in the ice in front of the dam as well as the change in confinement due to water level fluctuations. Figure 3.6d presents loads that are a combination of many contributing factors: ice thickness, ice type, ice temperature, rate of temperatures rise, water level fluctuations, ice confinement, cracks, bending, buckling, rain, snow, wind etc.
NF stresses are presented in Figure 3.7. Although, the presented stress values were first averaged over all eleven panels for each depth, a significant variability is still visible in the figure. This shows that, sometimes, the ice is pushing against the dam near the top while, at other times, it is mostly pushing at a mid-level (for example, at the end of February 17th or at the end of February 27th – beginning February 28th).
Figure 3.7. Average Near Field stresses (in kPa) interpolated over the depths7.
7 Although, tensile (negative) stresses are shown in the figure, flatjacks cannot measure tension. Negative
values are probably due to slight problems with temperature corrections (Stander 2018, personal communication). No effort was undertaken by the author of this thesis to correct the data as the author had not been involved into the data processing of panels.
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This variability is further explored in Figure 3.8 where measurements of 3 different pressure sensors (flatjacks) are shown for each of four depth. Note that flatjacks of different panels (panel 1, panel 2 etc. – P1; P2 etc.) are selected for each depth (FJ1 – flatjack at depth 0.1 m, FJ2 – flatjack at depth 0.3 m, FJ3 – flatjack at depth 0.5 m, and FJ4 – flatjack at depth 0.7 m). Flatjacks of each distinct panel are shown by one color. Figure 3.8 shows that stresses vary significantly at all depths (at 0.1 m, 0.3 m, 0.5 m and 0.7 m), whereas Figure 3.6a shows significant temperature changes at depths no lower than 0.3 m from the ice surface. If the stresses were of thermal nature only, flatjacks at 0.1 m depth had measured significant stresses, flatjacks at depths of 0.3 m had shown some less important stresses and flatjacks at 0.5 m and 0.7 m depths had not registered important stresses. Thus, quite important measured stresses shown at depths 0.3 m, 0.5 m, and 0.7 m (Figure 3.8b – Figure 3.8d) originate not only from ice temperature changes but from its superposition with other non-thermal field processes or just from non-thermal field processes alone. Non-thermal processes can include water level fluctuations, cracks, bending, buckling, lateral confinement, rain, snow, wind etc.
Stresses measured at different panels and/or at different depths can, in turn, dominate the others during an event. To illustrate this statement, Figure 3.9 shows stress measurements at different depth for three panels: Panel 1 (P1), Panel 3 (P3), and Panel “13” (i.e., 11th
panel along the wall) in Figure 3.9a-c, respectively. As can be seen from Figure 3.6, the 02/17 event was primarily generated by water level fluctuations (±0.1 m) as ice was warm (minimum temperature was -2C at ice surface). In Figure 3.9, flatjacks FJ1 (0.1 m depth) and FJ3 (0.5 m depth) of P1 and P13(11) responded similarly to the fluctuations: they increased and decreased simultaneously, and, then, FJ1 dominated at the end of the event. FJ2 (0.3 m depth) of P1 registered no stress at the time, while FJ2 of P13(11) registered some stress during the event. FJ1 – FJ3 of P3 registered higher stresses at all depths during the event than those of P1 and P13(11). Moreover, FJ3 stress of P3 was much higher than FJ1 and FJ2 stresses during the first half of the event. At the middle of the event, FJ3 stress suddenly dropped down and let stresses of FJ1 and FJ2 dominate for the second part of the event. This example shows that ice stresses are highly influenced not only by their primary generators (ice temperatures and water level fluctuations) but also by other chaotic
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processes in the field (cracks, bending, buckling, lateral confinement, rain, snow, wind etc.).
One of such processes was quantified by placing a set of four displacement transducers on both sides of the crack in front of the dam (Morse et al. 2011b). The measurements were conducted from 2011/01/16 to 2011/03/13. It was shown that new ice growth in the crack was about 0.2 m for the period of measurements (Figure 3.10a). Figure 3.10b shows that vertical gages descended by 0.06 m, approximately. It means that new ice grew at the surface of the ice cover that was confirmed by measurements (Taras et al. 2011). For the studied period, 2011/02/17 – 2011/03/01, there was 0.05 m of new ice growth in horizontal direction, while vertical gages descended by 0.015 m, approximately (Figure 3.11 – Figure 3.12).
These and other chaotic processes overlap each other at each moment of time in different combinations. The chaotic overlapping makes measured stresses difficult to analyze, unless they are averaged enough. After additional stress data analysis, it was decided to concentrate on comparing the average near field line load to numerical simulations. An accurate building of a numerical model for ice load calculation requires the knowledge of material properties of ice. Next chapter presents the literature review on material properties of two ice types that usually form on a dam reservoir. Next chapter also presents the literature review on modelling of behaviour of these two types of ice and reservoir ice covers.
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Figure 3.11. Horizontal motion of crack gage away from dam along with water level rise or drop for the study period 2011/02/17 – 2011/03/01.
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Figure 3.12. Vertical motion of crack gage and water level rise or drop for the study period 2011/02/17 – 2011/03/01.
