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Numerical modelling of ice ridge keel action on subsea structures
Cold Regions Science and Technology 67 (2011) 107–119
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Numerical modelling of ice ridge keel action on subsea structures Nicolas Serré ⁎ Norwegian University of Science and Technology (NTNU), Department of Civil and Transport Engineering, Høgskoleringen 7a, 7034 Trondheim, Norway Barlindhaug Consult AS, Sjølundveien 2, Postboks 6154, 9291 Tromsø, Norway
a r t i c l e
i n f o
Article history: Received 22 July 2010 Accepted 24 February 2011 Keywords: Ice keels Rubble ice Finite element model Sensitivity analysis
a b s t r a c t The present paper describes a numerical simulation of the keel–structure interaction where the keel geometry, ice thickness and interaction speed present a scaling ratio of 1:20. The rubble is represented by a Drucker–Prager material with cohesive softening and no dilatation. The partial differential equations are solved with the non-linear Eulerian finite element method of Abaqus Explicit V6.8.2. The results are compared to physical experiments conducted in the ice basin of the Hamburg ship model basin (HSVA). Two identical cubic subsea structures were impacted into the unconsolidated keel portion of two ice ridges with different thermal properties. The results indicate that only slight dilatation occurs, and the models are able to estimate the rubble action and deformed shape. A progressive failure of the rubble occurs. The rubble action is influenced more by the friction angle than it is in punch tests, due to higher confinement at the failure surface. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In a number of Arctic regions, the design load on a structure is determined by the action of first-year sea ice ridges. The keel action can be the main contributor to the load; however, little is known about its mechanical behaviour and how it should be modelled mathematically. The mechanical properties of full-scale ice ridges have been investigated in only a few full-scale experiments. The two main types of experiments that have been performed are: • The shear test (Bruneau et al., 1998; Croasdale et al., 2001), in which a vertical wall is introduced into the keel of an ice ridge and pushed horizontally. Loads and wall displacements are recorded. • The punch test (Leppäranta and Hakala, 1992; Heinonen and Määttänen, 2000; Heinonen, 2004), in which a horizontal plate is penetrated into the keel while loads and plate displacements are recorded. However, a lack of control over the boundary conditions and a limited amount of recorded data on the rubble behaviour and its physical properties make the interpretation of these tests challenging. Some offshore structures have been instrumented with load panels, video cameras and underwater sonars to relate keel loads to keel shapes. Data are available in the literature for: • The Molikpaq structure when it was in operation in the Beaufort Sea (Timco et al., 1999; Wright and Timco, 2000; Timco and Johnston, 2003). ⁎ Norwegian University of Science and Technology (NTNU), Department of Civil and Transport Engineering, Høgskoleringen 7a, 7034 Trondheim, Norway. Tel.: +47 77622679; fax: +47 77622699. E-mail address: [email protected]. 0165-232X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2011.02.011
• The Nörströmsgrund lighthouse in the Gulf of Bothnia (Kärnä and Jochmann, 2003; Bonnemaire and Bjerkås, 2004). • Two piers of the Confederation Bridge (Lemee and Brown, 2004). The pier loads were compared to the loads on the Kemi I lighthouse in the Baltic Sea in Brown and Määttänen (2009). These measurements provide useful information on the load levels and failure modes that are likely to occur during an interaction with an ice ridge. Unfortunately, none of them provide details on the specific action from the keel and the underwater deformation of the rubble. Data on the failure mode are of primary importance because it is necessary to verify that the theoretical models are able to correctly predict it and because numerous assumptions of the analytical keel load models are based on the type of failure occurring in the rubble. The Dolgopolov model (Dolgopolov et al., 1975) assumes a local failure of the keel. In some cases, for example, with small ice ridges, the keel can fail globally, and the deformations are similar to those of a plug of rubble pushed in front of the structure. Croasdale and Cammaert (1993) proposed a model for such a scenario. In a more recent approach, the keel load is given as the minimum load computed by the two previous models, for different penetrations of the structure into the keel (Brown et al., 1996). This idea was extended in general passive failure models proposed by Weaver in Croasdale, Associates Ltd. (1996) and by Kärnä et al. (2001). The load on the structure is computed all along the structure penetration into the ridge. For each position, different straight failure planes with varying inclinations are assumed, and the keel load is computed as the minimum load necessary to reach failure on one of these planes. This last generation of analytical models removes the need for the assumption of global or local failure but still requires assumptions with regard to the failure shape.
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Serré (2011). The numerical model uses a continuum description of rubble based on the Drucker–Prager constitutive model (Heinonen, 2004; Serré, 2011). It is equivalent to the Mohr–Coulomb material law, which has been suggested by the ISO/FDIS 19906 (2010), but its mathematical formulation renders it more suitable for use with an explicit finite element method. This paper studies how the numerical model simulates experiments and investigates the validity of the material law. The model helps in understanding the rubble failure mode, and a parametric analysis is performed, indicating which mechanical tests are the most suitable for determining the mechanical properties. 2. Experimental set-up 2.1. Ice ridges
Fig. 1. Interaction of physical, numerical and analytical modelling in design (Randolph and House, 2001).
The ridge failure experimental study that has most closely approached full-scale conditions was performed by Liferov and Høyland (2004). They built a medium-scale ice ridge in the Van-Mijen Fjord, Svalbard, and moved it against the seabed. Failure indicators were placed into the keel and showed that the failure was not simultaneous (global keel failure, deformation propagating throughout the entire keel) but progressive (local keel failure, deformations only in the vicinity of the structure). Due to the extensive resources required for a proper investigation of the failure of full-scale ridges, numerical simulations are an attractive approach for the study of ridge behaviour. They remove the need for failure shape assumptions and can account for irregular boundary conditions and better constitutive models than analytical methods (Timco et al., 2000). Nevertheless, numerical models that include the main complexities of rubble deformations require long computation times, which prevent their use in probabilistic analyses or in specific time-domain algorithms. Therefore, the major part of the design of an arctic offshore oil field is generally undertaken through simple analytical models. Because full-scale experiments are not available for their calibration, their development requires numerical modelling of the full-scale boundary value problem. The complementary role of physical and computational modelling is well explained in Randolph and House (2001) and can be represented as in Fig. 1. The present paper falls mostly on the left hand side of Fig. 1, i.e., the validation of modelling techniques by comparison between physical and numerical modelling. However, it also highlights some mechanisms of rubble deformation that should be considered in the development of conceptual models. The physical model used herein is presented in Serré et al. (2009a), and the rubble material parameters required for input of the numerical model are derived in Serré et al. (2009b) and
The ice tank in the Hamburg ship model basin (HSVA) is 78 m long, 10 m wide and 2.5 m deep. A 5-m-deep section of 10 m by 12 m is located at the end of the tank. Four ice sheets (corresponding to test series 1000, 2000, 3000 and 4000) were produced with one ice ridge per ice sheet. The ridges were wider than what would be expected from a typical keel depth to width relationship but were appropriate for the purpose of investigating ice rubble failure loads. With a scaling ratio of 1:20, they corresponded to a target full-scale size ridge with a 10-m-deep and 110-m-wide keel. A thermodynamic scaling of the rubble was not attempted. The geometries and velocities in the boundary value problem were scaled according to the Froude scaling law (ratio 1:20) to ensure conservation of the inertial to gravity forces ratio as given by: Vc ffi Fr = pffiffiffiffiffiffi gLc
ð1Þ
where Vc and Lc are the characteristic speed and length, respectively, and Fr is the Froude number. If the characteristic length is scaled by λ and the ice density is kept constant, then the masses scale as λ3 and the velocities and time as λ1/2. The ice ridges were confined on the sides by the tank walls. Ridges 2000 and 4000 were built from colder ice and tested soon after their production, with very little consolidation (degradation) time allowed. The rubble density (ρr) was computed from: ρr = η ⋅ ρw + ð1−ηÞρi
ð2Þ
where ρi is the ice density, η is the ridge porosity and the water density ρw is 1006 kg/m3. A layer of level ice was pushed over the rubble to model the consolidated layer. A visual inspection of the ridges after the interaction showed that some blocks of ice were frozen to the level ice. The consolidated layer did not interact with the subsea structures and was therefore not modelled in the numerical analysis. The principal characteristics of the ice ridges are represented in Fig. 2 and
Fig. 2. Model ice ridge characteristics.
N. Serré / Cold Regions Science and Technology 67 (2011) 107–119
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Table 1 Ice ridge characteristics. Ridge #
1000 2000 3000 4000
(warm) (cold) (warm) (cold)
hk (m)
kw (m)
kbw (m)
η (%)
ρi (kg/m3)
ρr (kg/m3)
Consolidation Time (h)
Tair (°C)
0.5 0.45 0.45 0.55
5 6 6 5.5
3 4.5 4 4.5
32 36 41 43
795 868 798 844
863 917 883 914
18 5 18 5
4 −0.5 3.6 − 0.2
given in Table 1, where hk is the keel depth measured from the waterline, kw is the keel width and kbw is the keel bottom width. More details on the physical properties of the ridges and their measurement methods are given in Repetto-Llamazares et al. (2009a). The water had a salinity of 7 ppt and a freezing temperature of − 0.5 °C. Its temperature during the interaction test was −0.3 ± 0.2 °C. 2.2. Interaction test To model shallow water conditions, bottom elements (false bottom) were inserted into the tank to reduce the water depth. The structures were fixed on the false bottom, which was displaced along the tank by a motor-driven carriage able to provide a maximum towing load of 50 kN. Two types of structures were studied: a cubical shape for subsurface keel interactions and a simple conical structure for surface ridge interactions. An analysis of the keel–cone interaction is given in Serré and Liferov (2010), and an analysis of the subsea structure interaction is presented in this paper. A bird view of the testing procedure is shown in Fig. 3, and an artistic 3D view of the ridge interaction with one subsea structure is given in Fig. 4. The principal dimensions of the structures are given in Fig. 5. The friction coefficient of the ice against the painted plywood of the subsea structures was measured to be 0.11. For the measurement, a 0.15 × 0.15 m block of ice was extracted from the ice sheet and placed in dry conditions on a surface built from the same material as the subsea structure. Different confinement weights were applied on the ice, which was then displaced over the material. The applied shear force was measured, and its ratio over the applied weight gave the friction coefficient value. The ice was extracted from the ice sheet before the ridge testing and tested the next day. The ice was kept overnight in a water basin in a room with an air temperature of −1.6 °C.
Fig. 4. 3D illustration view of the sub-surface interaction.
with Abaqus/Explicit V6.8.2. A description of the model is given in the following section. 2.4. Geometry The numerical model of the ice ridge keel is shown in Fig. 6, and the keel geometries are provided in Table 1 together with the rubble density. A 2-m-thick slice of the model ice ridge keel was considered. During the model development, the results from the simulations were compared to a model considering the full length of the ridge to verify that the slice was thick enough to ensure that the simulated load was independent of its thickness. A 0.4-m-thick elastic layer referred to as an “air layer” was placed above the keel to balance the buoyancy forces. One numerical model was developed for ridges 1000 and 2000. Due to technical problems with the load cells during the test series 3000 and 4000, the data from these two tests were not analysed. The dimensions of the cubic subsea structure and its position relative to the water line and false bottom are given in Fig. 5 and in the experimental set-up description.
2.3. Numerical method
2.5. Loads and boundary conditions
The keel–structure interaction was analysed with the finite element method using a coupled Eulerian–Lagrangian formulation to allow for large displacements. The computations were performed
The friction coefficient between the structure and the rubble ice was measured to be 0.1. In the first step, only buoyancy forces were
Fig. 3. Test set-up, bird view.
Fig. 5. Dimensions of the cubic structure.
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Rubble Void Other material
Elastic forces
Subsea structure
Buoyancy Eulerian forces mesh
Motion Symmetry
2m
Fig. 6. Cut view of the numerical model of the rubble/structure interaction.
applied to the rubble. They were modelled by an upward gravity. The gravity coefficient was computed such that, when multiplied by the rubble density, the result was equal to the buoyancy forces of the rubble. It was given by: g′ = g
ρw −1 ρr
ð3Þ
where g′ is the “upward gravity” (buoyancy) coefficient and g is the gravitational acceleration (9.81 m/s2). In the same step, a vertical pressure was applied on the bottom surface of the air contacting the rubble to achieve static equilibrium. Its value was given by: p = hk ρr g ′
ð4Þ
where hk is the keel depth. In the second step, the structure was displaced with a velocity of 0.125 m/s until it had crossed the entire rubble. This velocity was larger than the experimental velocity (0.045 m/s) to save computation time. A comparison with a numerical model featuring the experimental velocity did not show velocity effects on the computed rubble action. A symmetry boundary condition was applied such that only one half of the structure and the rubble must be modelled (the 2-m slice becomes equivalent to 4 m). To represent the effect of the remaining part of the rubble on the 2-m slice considered, a fixed frictionless wall boundary condition was applied on the other side of the rubble slice. The consolidated layer effect on the rubble is considered by preventing any horizontal motion of the nodes located at the rubble top. The vertical motion is left free because some vertical motion of the ridge sail was observed during the interaction (Serré et al., 2009a). 2.6. Material model The structure was modelled as a rigid body. The air layer was purely elastic, with a Young modulus of 3 kPa. Its compression produced the same vertical stress range on the rubble as the gravity load from the ice lifted out from the water. The ice rubble was modelled with the linear Drucker–Prager plasticity model. Similarly to the Mohr–Coulomb material law, it is a hydrostatic-pressuredependent type of model characterised mainly by the Drucker–Prager friction angle β and cohesion d. A detailed description of the Drucker– Prager model applied to the rubble ice is given below.
2.6.1. Yield surface The linear Drucker–Prager yield surface is: t = d + p tanðβÞ;
ð5Þ
where p is the hydrostatic pressure. The deviatoric stress measure t is defined in Abaqus as: t=
3 1 1 1 r ; q 1 + − 1− 2 K K q
ð6Þ
where K is the ratio of the yield stress in triaxial tension to the yield stress in triaxial compression, r is the third stress invariant and q is the Von Mises stress. Abaqus allows one to select a K value between 0.778 and 1 (above 0.778 to ensure convexity of the yield surface). No triaxial tests were performed on the rubble, so the yield stress ratio was selected to be equal to 1 such that the yield surface corresponded to the original Drucker and Prager (1952) model. Consequently, the influence of the third stress invariant was neglected. 2.6.2. Hardening law The failure of the freeze bonds inside the rubble is represented by a softening cohesive behaviour: when the stress state lies on the yield surface, the material yields in shear, and the cohesion d decreases linearly from its original value di to 0 as the first 0.02 of equivalent plastic strain (εp) is applied. The cohesion and equivalent plastic strain are then related through: 8 > < > :
i
d = di − d=0
d ⋅ εp if εp ≤ 0:02 ; 0:02 if εp N 0:02
ð7Þ
pffiffiffi where εp is equal toγpl = 3 and γpl is the engineering shear strain. The cohesive softening in Abaqus is implemented by a hardening law, where the yield shear stress at 0 Pa of hydrostatic pressure (i.e., the cohesion) is tabular specified as a function of the equivalent plastic strain. 2.6.3. Plastic flow potential When the stress state lies on the yield surface, plastic deformations occur in the material. The plastic flow is defined by a non-associated flow potential G given as: G = t−p tanðψÞ;
ð8Þ
where ψ is the dilatation angle. A dilatation angle equal to 0° corresponds to the case of incompressible inelastic deformations,
N. Serré / Cold Regions Science and Technology 67 (2011) 107–119
while a positive dilation angle indicates that the material dilates during shear deformations. The strain is computed by multiplying the strain rate by the length of the time increment. An additive strain rate decomposition is assumed: el
pl
dε = dε + dε ;
ð9Þ
where dε is the total strain rate, dεel is the elastic strain rate and dεpl is the plastic strain rate computed as: dγpl ∂G pl ; dε = 2 pffiffiffi 1 ∂σ 3 1+ K
ð10Þ
with the rubble through the Lagrangian/Eulerian contact formulation of Abaqus. The mesh in the interaction zone was composed of cubic Eulerian elements with a length of 0.05 m. An analysis of the result dependency on the mesh size was performed and showed that the computed load became stable for an element length below 0.075 m (Figs. 7 and 8). The numerical model used for the mesh sensitivity analysis was based on the geometry and material properties from ridge 2000 but with a uniform cohesion of d = 735 Pa. 2.8. Parametric analysis The parameters selected for analysis were:
where σ is the stress tensor. 2.6.4. Mechanical parameters The parameters necessary to define this model are d, β and ψ. The elastic domain is defined by the standard Hooke's law, requiring the Young modulus E and the Poisson's ratio ν, which was chosen to be 0.3, a typical value for granular materials (Harr, 1977). The mechanical properties were derived from different mechanical tests performed on the four ice ridges (Serré et al., 2009b; Serré, 2011) and are given in Table 2. A different cohesion value di was attributed to ridges 1000 and 2000 due to their different ages and temperatures. Ice ridge 2000 was colder and younger than ridge 1000, as indicated in the keel temperature profile measurements of Repetto-Llamazares et al. (2009b). The temperature of ridge 1000 was above the solidification temperature of the water in the ice tank in contrast to that of ridge 2000. The upper 10 cm of keel 2000 were also significantly colder than the rest of the keel. The short degradation time in the basin prevented the heat from the water from reaching the upper part of the keel and degrading the freeze bonds located there. The freeze bond strength distribution was therefore represented by a linear distribution of the cohesion in the keel of ridge i at the top of the rubble to 2000, with an initial value varying from dtop i d bottom at the bottom. For this purpose, each horizontal layer of nodes in the keel was assigned a cohesion value in relation to its vertical position. The cohesion value in the material was then interpolated by Abaqus. In the present model, the dilatation angle ψ was set to 0°, and the elastic domain was not limited along the hydrostatic pressure axis, that is, the rubble did not deform plastically under hydrostatic compressive forces.
• The ice density, varied according to Evers and Jochmann (1993) from 750 to 930 kg/m3. • The keel depth, varied from 0.3 to 0.5 m. • The Drucker–Prager friction angle, varied according to the pile test experiment in Serré (2011) from 40 to 50°. • The Drucker–Prager cohesion, varied from 0 to 2.5 kPa (arbitrary values).
2.7. Mesh The meshed zone was divided into three areas. The upper one was 10 cm high and above the water line. It was composed of void elements. The second one was the ridge keel, whose elements were composed of rubble. The third area was under the keel and was composed of void elements. Void elements could be filled with rubble upon deformation. A representation of the mesh is given in Fig. 6. The air layer shown as the white layer above the keel was modelled with Lagrangian elements. It overlaps the upper void layer and can interact Table 2 Ice rubble mechanical properties for ridges 1000 and 2000 (Serré, 2011). Parameter E [MPa] ν β [°] i [Pa] dtop i [Pa] dbottom ψ [°]
111
Ridge 1000
Ridge 2000 0.9 0.3 50
612 612
1470 0 0
Fig. 7. Estimated load for a mesh size of 0.075 m (a) and 0.0375 m (b).
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a
Structure
False bottom
Motion (125 mm/s)
b Fig. 8. Estimated load as a function of the mesh size.
Although the cohesive softening behaviour of the freeze bonds and rubble ice has been highlighted by numerous authors, full-scale rubble cohesion values considering softening are reported only in Heinonen (2004), while only Serré (2011) reports model-scale values. There is therefore no cohesion range from which input values could be selected for a parametric analysis. However, several authors such as Ettema and Urroz (1989) have reported that rubble ice submitted to continuous deformation behaves as a cohesionless material. The Drucker–Prager cohesion range was thus selected to span from 0 Pa to approximately two times the base case cohesion value (1.47 kPa). For an easier comparison with the rubble description given by the ISO/FDIS 19906 (2010), the cohesive and frictional parameters of the rubble can be represented by the Mohr–Coulomb parameters c (cohesion) and ϕ (friction angle). The relation between the Drucker– Prager and Mohr–Coulomb parameters can be extracted from Chen
Fig. 9. Experimental (black) and numerical (red) rubble load on the subsea structure, test 1000.
Motion Fig. 10. View of the numerical estimation of the rubble accumulation on the front of the subsea structure (a) and corresponding situation in the experimental model (b), test 1000.
Fig. 11. Experimental (black) and numerical (red) rubble loads on the subsea structure, test 2000.
N. Serré / Cold Regions Science and Technology 67 (2011) 107–119
113
Structure Pressure (Pa) 2500 False bottom Motion (125 mm/s)
0
Fig. 12. Hydrostatic pressure in the rubble during test 2000.
and Saleeb (1982) for a plane strain approximation. The corresponding equations are: tan β =
pffiffiffi 3 sin ϕ
Fig. 14. Rubble accumulation on the front of the subsea structure; position: 61.5 m, test 2000.
ð11Þ
pffiffiffi d = 3c cos ϕ:
ð12Þ
3. Results
measured during the experiments (Tables 1 and 2) over-estimates the rubble action from test 1000 by 250%. Fig. 10 shows that the rubble accumulation on the structure's front is 20 cm deeper in the numerical model than in the experiment.
3.1. Test 1000, rubble load and deformation Fig. 9 compares the simulated and measured rubble loads on the subsea structure. The numerical model based on the parameters
a
Structure Horizontal velocity (mm/s) 240
Structure
False bottom
125 False bottom
Motion (125 mm/s)
0
Motion
b
b
Structure Vertical velocity (mm/s) 200
0
Horizontal velocity (mm/s) 240
False bottom 125 Motion
0
Structure False bottom Motion (125 mm/s)
-100 Fig. 13. Test 2000, rubble velocity, horizontal (a) and vertical (b); structure position: 61.5 m.
Fig. 15. Rubble accumulation on the side of the subsea structure, experimental on the starboard of the structure (a) and numerical on the portside (b); position: 61.5 m, test 2000.
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3.2. Test 2000, rubble load
the numerical model because they are caused by a standing wave with a period unique to the HSVA basin (Serré et al., 2009a).
The numerical estimation of the rubble load on the subsea structure is plotted together with the experimental rubble load for test 2000 in Fig. 11. The numerical model predicts the load well except when the structure exits the rubble at the end of the interaction, where the model appears slightly conservative. Oscillations with a 30-s (1.35-m) period can be observed for all of the experimental load time series. They should not be considered in the comparison with
a
3.3. Test 2000, stress levels Fig. 12 shows a section of the rubble along the vertical symmetry plane of the structure and shows the hydrostatic pressure during the interaction. The hydrostatic pressure is below 2500 Pa in the main part of the rubble, but in the area in the upper corners of the structure, it can reach 5000 Pa. The size of this area does not exceed the size of one element and is a result of stress concentrations at the corners. 3.4. Test 2000, rubble deformation A plot of the rubble shape and ice velocity is given in Fig. 13. It shows that deformations occur only in the vicinity of the structure. They correspond to a progressive (local) failure of the rubble. When the structure travels into the rubble, ice blocks accumulate on the front and are pushed on the sides. The rubble accumulation rapidly
a Structure
Motion
False bottom
b Structure
False bottom Motion
b
Structure
Horizontal velocity (mm/s) 240 125
False bottom Motion
Fig. 16. Rubble accumulation seen from behind the subsea structure, experimental (a) and numerical (b); position: 61.5 m, test 2000.
Structure
False bottom
0 Motion (125 mm/s) Fig. 17. Rubble accumulation on the front of the subsea structure when exiting the keel, experimental (a) and numerical (b); position: 62.9 m, test 2000.
N. Serré / Cold Regions Science and Technology 67 (2011) 107–119
covers the whole front of the structure and then slightly increases in size until the structures exit the keel. The extent of the rubble accumulation on the front of the subsea structure compares well with the underwater image (Fig. 14). Numerical and experimental results also agree with respect to the rubble coverage on the side of the structure (Fig. 15) and as seen from the back (Fig. 16), where a wake is created behind the structure. The wake is first freed from the rubble up to the roof of the structure, and then the two rubble mounds on its side collapse in it due to the action of the buoyancy force until their slopes reach stability. As for the rubble load, a comparison between the numerical and experimental rubble shapes at different stages of the interaction also shows a good match for most of the test but a difference at the end (Fig. 17). Experimentally, a deep portion of the rubble is accumulated on the front of the structure and is almost entirely separated from the rest of the keel; only very little rubble covers the side of the structure. However, the numerical model presents a shallower but wider accumulation, also covering the sides of the structure and in contact with the keel. This observation is in agreement with the conservatism of the numerical model when the structure exits the keel.
a
115
3.5. Test 2000, parametric analyses The effects of the ice density, keel depth, friction angle, cohesion and porosity on the computed rubble load were studied through parametric analyses. Only one parameter was changed at a time along a range of possible values. The corresponding numerical simulations are given in Figs. 18 and 19. Few authors have derived the rubble cohesion while considering cohesion softening. Therefore, the literature does not propose a cohesion range from which input values can be selected for a proper parametric analysis. Fig. 19 shows the effect of maximum cohesion (at the top of the rubble) from 0 to 2.5 kPa on the rubble load, which increases from 750 to 1200 N. For an extreme cohesion of 9 kPa, the load reaches 1600 N. The highest 1-m average load for each simulation is given in Table 3. It corresponds to the portion of the load curve that presents the highest average load over a structure displacement of 1 m. The results of the parametric study given in Table 3 are plotted in Fig. 20. The variation of each parameter as compared to their base case value (parameter values given previously for test 2000) is given in percent along the horizontal axis, and the resulting variation of the
b 1600 1600 1400
1400 i = 750
kg/m
3 1200
1200
Load [N]
Load [N]
1000 1000 800
800 600
600 400
hk = 0.5 m i=
3
400
930 kg/m
200
200 0 56
57
58
59
60
61
62
63
0 56
64
hk = 0.3 m 57
58
59
Distance [m]
60
61
62
63
64
62
63
64
Distance [m]
c
d 1400 1600
= 0.3 1400
o
= 45 , = 50
1200
o
1000
1000
Load [N]
Load [N]
1200
800
= 0.45
800
600
600 400 400 o
200 0 56
200
o
= 30 , = 40 57
58
59
60
61
Distance [m]
62
63
64
0 56
57
58
59
60
61
Distance [m]
Fig. 18. Effect of the ice density (a), keel depth (b), friction angle (c) and porosity (d) on the simulated rubble action, test 2000.
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Fig. 19. The numerical rubble load on the subsea structure for different levels of initial cohesion; the base case corresponds to the cohesion value for the simulation of test 2000.
estimated load is given along the vertical axis. A slope is computed between the extremities of each curve. A slope close to the horizontal (slope = 0) indicates that the parameter in question has no influence on the load. The closer the curve is to the vertical, the more important the role of the parameter is in the rubble load computation. The ice density curve is the closest to the vertical, showing that this parameter has the largest influence on the load. The base case corresponds to a numerical simulation of test 2000. 4. Discussion 4.1. Measurement inaccuracies The poor concordance between the experimental and numerical results for test 1000 is not necessary linked to a weakness in the numerical model. Actually, this test was the first to be performed, and the research team was not used yet to the different measurement methods. Therefore, a higher uncertainty exists for this test with respect to the ice density, the keel depth and the rubble porosity. Furthermore, the rubble load measurement also appears questionable because the measured load on the structure was as low as at least half the rubble action measured during tests 2000, 3000 and 4000 (Serré et al., 2009a). It should, however, be remembered that the measurements from the 3000 and 4000 series were approximate due to damaged load cells. To estimate the influence of these uncertainties, the rubble action from ridge 1000 was simulated based on a set of “plausible” parameters selected among the parameters that were measured for the four different ice ridges. The modified parameters are given in Table 4. The friction angle is given by the lower bound value derived from a rubble pile test analysis performed on ridge 2000 and described in Serré (2011). In this test, the Mohr–Coulomb friction angle ranged between 30 and 45°. Table 3 Parametric analyses for the interaction test, showing the effect of selected parameters on the average peak load. Parameter 3
ρi [kg/m ] hk [m] β (ϕ) [°] η d (c) [kPa]
Parameter range
Average peak load [N]
[750; 930] [0.3; 0.5] [40 (30); 50 (45)] [0.3; 0.45] [0 (0); 2.5 (2)]
[1550; 700] [475; 1200] [530; 1050] [907; 1100] [700; 1200]
Fig. 20. Influence of different parameters on the rubble load on the subsea structure. The ice density curve is the closest to the vertical, showing that this parameter has the largest influence on the load. The base case corresponds to numerical simulation of test 2000.
It is not known why the rubble friction angle in ridge 1000 is different from that of ridge 2000. However, the ice blocks composing the rubble from these two ridges did not present the same aspect, as shown in Fig. 21, where pictures extracted from the underwater videos revealed that the ice block size was larger in test 2000 than in test 1000. Either this aspect variation or the cause of this variation might have affected the value of the rubble friction angle. The measured and simulated rubble action according to the modified parameters in Table 4 is given in Fig. 22a. The numerical model still produces conservative results, but the numerical and the experimental load curves agree with an error of 20% if the keel depth is reduced to 45 cm instead of the original 50 cm (Fig. 22b). The deformed rubble shape computed by the 45-cm-deep keel model (Fig. 23) is closer to the experimental observation (Fig. 10b) than the original numerical model (Fig. 10a). Because the person in charge of the keel depth measurement differed from ridge to ridge and because this measurement is rather subjective (Repetto-Llamazares et al., 2009a), a 5-cm error in the keel depth is possible. Therefore, by considering the different uncertainties linked to the measurements, in spite of the poor agreement between the experimental and the original numerical load curve, the test 1000 results do not invalidate the numerical model.
4.2. Dilatation and volumetric hardening The determination of the deformed shape of the rubble is dependent on a proper selection of the dilatation angle. In the numerical model, no dilation occurs because ψ is equal to 0°. It is difficult to accurately estimate the experimental amount of dilatation. However, the good comparison between the numerical and experimental deformed rubble shapes in test 2000 (see Figs. 13–16) reveals that only slight dilatation occurred in the experiment. The small amount of dilation can be explained by the high porosity of the rubble ice, which then behaves as a loose granular material (Chen and Saleeb, 1982). Table 4 Modified values of the parameters used in the simulation of test 1000. η [%]
ρi [kg/m3]
ρr [kg/m3]
g′ [m/s2]
β [°]
43
868
927
0.84
40
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Fig. 21. Images of the maximum block size for test 2000 (a) and test 1000 (b).
The low hydrostatic pressure levels estimated numerically confirm the possibility of neglecting the volumetric plastic deformations linked to hydrostatic compaction. Serré (2011) performed oedometer tests on model rubble ice and presented a curve relating the hydrostatic yield strength of the rubble to its volumetric strain. It was found that a hydrostatic pressure of 2.5 kPa (Fig. 12) corresponded to a plastic volumetric strain below 7%. Therefore, only slight volumetric plastic deformation occurs during the interaction process, and it is thus not necessary to consider this failure mechanism in the present numerical model.
Fig. 22. Experimental (black) and numerical (red) rubble load on the subsea structure, test 1000 with modified ice density, rubble porosity and friction angle. Original keel depth in (a) and a 5-cm-smaller keel depth in (b).
4.3. Parametric analysis Fig. 20 indicates that the ice density most influences the rubble load on the subsea structure, followed by the keel depth, the friction angle, the porosity and the cohesion. The same ranking was observed in the punch test sensitivity analysis reported by Serré (2011), but two differences are worth noting:
Horizontal velocity (mm/s)
Structure
290 False bottom
• The effect of the friction angle on the rubble load is significantly larger in the interaction test than in the punch test. • The cohesion plays a role throughout the entire length of the load curve in the interaction test (Fig. 19), while it acts only on the peak of the punch test load curve and leaves the post-peak portion unaffected.
125 0
Motion (125 mm/s)
Fig. 23. Computed rubble accumulation on the front of the subsea structure for a 45-cmdeep keel, for comparison with the experimental rubble accumulation in test 1000.
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4.4. Frictional resistance The friction angle plays a more important role in the rubble resistance for the interaction tests than for the punch test due to the different rubble failure modes that occur. In punch tests, the failure has the shape of an upward cone at the beginning of the interaction. The conical rubble portion extends to the keel bottom and is pushed downward by the punch plate (Heinonen and Määttänen, 2000; Jensen et al., 2001). The same behaviour was observed in the punch test numerical simulations of Liferov et al. (2003) and Serré (2011). During this failure process, the conical portion of the rubble is pushed downward by the punch plate and is separated from the rest of the keel. Therefore, the confinement at the failure plane decreases during the failure process, as shown by Serré (2011). In the interaction case, however, the buoyancy pushes the failed portion of the rubble upward against the rest of the keel, even after failure has occurred, and the motion of the structure compresses this portion against the keel, causing downward rubble extrusion (see the velocity plots of Fig. 13). Both of these effects increase the confinement at the failure plane as the structure penetrates into the rubble, as shown in the pressure curve of Fig. 24. Due to the different failure behaviours for these two tests, the confinement at the failure zone increases for the interaction test, while it decreases for the punch test. This explains why the friction angle is a more important parameter in the interaction test than in the punch test. Therefore, it is unwise to estimate the friction angle with the punch test when this value is to be used for the simulation of a different boundary value problem that is more sensitive to this parameter. This is one of the reasons why punch tests should not be the sole characterisation tool for rubble mechanical properties. Alternative approaches include the pile test (Keinonen and Nyman, 1978; Serré, 2011), the shear box test (Ettema and Urroz, 1991; Liferov and Bonnemaire, 2005), the shear test as mentioned in the introduction and the retaining wall test (Keinonen and Nyman, 1978; Serré et al., 2009b). 4.5. Cohesive resistance The parametric analysis shows that, in spite of cohesive softening, the cohesion affects the entire length of the simulated load curve. It
represents the process well in which the structure has to break freeze bonds all along the interaction due to the failure zone which reaches undisturbed rubble areas as long as the structure has not reached the end of the keel (progressive failure, as shown in the velocity plots of Fig. 13). The progressive failure of the ridge keel impacted by the subsea structure is in good agreement with the medium-scale observations from the ridge shear-off test of Liferov and Høyland (2004). It also shows that the concept of primary and secondary failure of the rubble given in Liferov and Bonnemaire (2005) can occur locally without being necessarily initiated throughout the entire rubble. The primary failure mode is controlled by the initial strength of the rubble skeleton (the frictional resistance of the rubble is not mobilised), while in the secondary failure mode, substantial deformations have already occurred and the rubble strength is dominated by the frictional resistance.
5. Conclusion Numerical modelling of the ice ridge keel–subsea structure interaction was performed and compared to model-scale experiments. It was found that only slight dilatation and compaction occurred in the rubble. The numerical model used herein is based on the Drucker–Prager material law and can compute the rubble action and deformation. However, the precision of the results depends on the measurement quality of the physical parameters involved in the rubble resistance. The effects of four parameters were studied. The ice density is the parameter with the largest effect on the rubble action, followed by the keel depth, the friction angle and the cohesion. The frictional resistance plays a more important role in the rubble action in situations where an ice ridge impacts a structure than in a punch test. The failure zones of these two scenarios present two very different stress states. These results show that the mechanical properties of the rubble cannot be characterised by one test alone and that the punch test is poorly suited for an accurate estimation of the friction angle. A progressive failure occurs in the rubble (local failure), and therefore cohesive resistance is present all along the interaction, even if the failed portion of rubble becomes cohesionless.
Fig. 24. Average hydrostatic pressure at a selected failure zone as the structure travels through the keel, test 2000.
N. Serré / Cold Regions Science and Technology 67 (2011) 107–119
Acknowledgments The help of Dr. Pavel Liferov and Pr. Knut Høyland in reviewing this paper is appreciated. Many thanks are also due to Simulia Scandinavia for advice regarding numerical modelling with Abaqus. The work was supported by the PetroArctic project, part of the Petromaks project of the Research Council of Norway and by the European Community's Sixth Framework Programme through a grant to the budget of the Integrated Infrastructure Initiative HYDRALAB III within the Transnational Access Activities, Contract no. 022441. References Bonnemaire, B., Bjerkås, M., 2004. Ice ridge–structure interaction, part I: geometry and failure modes of ice ridges. Proceedings of the 17th International Symposium on Ice, IAHR, Saint Petersburg, Russian Federation, vol. 1, pp. 123–130. Brown, T.G., Croasdale, K.R., Wright, B., 1996. Ice loads on the Northumberland Strait bridge piers — an approach. Proc. of 6th ISOPE Conference, Los Angeles, USA. Brown, T.G., Määttänen, M., 2009. Comparison of Kemi-I and Confederation Bridge cone ice load measurement results. Cold Reg. Sci. Technol. 55 (1), 3–13. Bruneau, S., Crocker, G., McKenna, R., Croasdale, K., Metge, M., Ritch, R., Weaver, J., 1998. Development of techniques for measuring in situ ice rubble shear strength. Ice in Surface Waters, Proc. of the 14th International Symposium on Ice, IAHR, vol. 2. Potsdam, New York, USA, pp. 1001–1007. Chen, W.-F., Saleeb, A.F., 1982. Constitutive equations for engineering materials. Volume 1: Elasticity and Modelling. John Wiley & Sons, Ltd, Chichester, England, p. 455. Croasdale, Associates Ltd., 1996. Ice Load Models for First Year Pressure Ridges and Rubble Fields. A Joint Industry–Government Project, Partly sponsored by PERD through the National Energy Board, Calgary, and supported by Mobil and Exxon. Croasdale, K.R., Bruneau, S., Christian, D., Crocker, G., English, J., Metge, M., Ritch, R., 2001. In-situ measurements of the strength of first-year ice ridge keels. Proceedings of the 16th International Conference on Port and Ocean Engineering Under Arctic Conditions, POAC'01, vol. 3, pp. 1445–1454. Ottawa, Ontario, Canada. Croasdale, K.R., Cammaert, A.B., 1993. An improved method for the calculation of ice loads on sloping structures in first-year ice. Proceedings of the First International Conference on Exploration of Russian Arctic Offshore, St. Petersburg, Russia, pp. 161–168. Dolgopolov, Y.V., Afanasiev, V.P., Korenkov, V.A., Panfilov, D.F., 1975. Effect of hummocked ice on the piers of marine hydraulic structures. Proceedings of the 3rd International Symposium On ice Problems, Hanover, New Hampshire, pp. 469–477. Drucker, D.C., Prager, W., 1952. Soil mechanics and plastic analysis or limit design. Q. Appl. Math. 10, 157–165. Ettema, R., Urroz, G.E., 1989. On internal friction and cohesion in unconsolidated ice rubble. Cold Reg. Sci. Technol. 16, 237–247. Ettema, R., Urroz, G.E., 1991. Friction and cohesion in ice rubble reviewed. Proceedings of the 6th International Speciality Conference Cold Regions Engineering, Hanover, USA, pp. 316–325. Evers, K.U., Jochmann, P., 1993. Advanced techniques to improve the mechanical properties of model ice developed at the HSVA ice tank. 12th International Conference on Port and Ocean Engineering under Arctic Conditions, POAC, pp. 877–888. Harr, M.E., 1977. Mechanics of Particulate Media. Mc-Graw-Hill Inc., USA, 540p. Heinonen, J. (2004) Constitutive modelling of ice rubble in first-year ridge keel. Doctoral Thesis, TKK. VTT Publications 536. Espoo, Finland, 2004, 142 p, ISSN 12350621. Heinonen, J., Määttänen, M., 2000. Ridge loading experiments, field experiments in winter 2000. LOLEIF Progress Report No. 10, TKK. 40 pp.
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ISO/FDIS 19906, 2010. Petroleum and natural gas industries — Arctic offshore structures, ISO TC 67/SC 7. Final Draft International Standard. International Standardization organization, Geneva, Switzerland. 434 p. Jensen, A., Løset, S., Høyland, K.V., Liferov, P., Heinonen, J., Evers, K.U., Maattanen, M., 2001. Physical modelling of first-year sea ice ridges — part II: mechanical properties. The 16th International Conference on port and Ocean Engineering under Arctic Conditions, POAC, Ottawa, Canada, vol. 3, pp. 1493–1502. Kärnä, T., Rim, C.W., Shkhinek, K.N., 2001. Global loads due to first year ice ridges. Proceedings of the 16th International Conference on Port and Ocean Engineering under Arctic Conditions, POAC'01, pp. 627–638. Ottawa, Canada. Kärnä, T., Jochmann, P., 2003. Field observations on ice failure modes. Proceedings of the 17th International Conference on Port and Ocean Engineering under Arctic Conditions, POAC'03, vol. 2, pp. 839–848. Trondheim, Norway. Keinonen, A., Nyman, T., 1978. An experimental model-scale study on compressible, frictional and cohesive behaviour of broken ice masses. Proc. of the Int. Symp. on Ice. Lulea, Sweden, vol. 2, pp. 335–353. Lemee, E., Brown, T., 2004. Review of ridge failure against the confederation bridge. Cold Reg. Sci. Technol. 42 (1), 1–15. Leppäranta, M., Hakala, R., 1992. The structure and strength of first-year ice ridges in the Baltic sea. Cold Reg. Sci. Technol. 20 (3), 295–311. Liferov, P., Bonnemaire, B., 2005. Ice rubble behaviour and strength: part I. Review of testing and interpretation of results. Cold Reg. Sci. Technol. 41 (2), 135–151. Liferov, P., Høyland, K.V., 2004. In-situ ridge scour tests: experimental set up and basic results. Cold Reg. Sci. Technol. 40, 97–110. Liferov, P., Jensen, A., Høyland, K., 2003. 3D finite element analysis of laboratory punch tests on ice rubble. Proceedings of the 17th International Conference on Port and Ocean Engineering under Arctic Conditions, POAC, Trondheim, Norway, vol. 2, pp. 611–621. Randolph, M.F., House, A.R., 2001. The complementary roles of physical and computational modelling. IJPMG Int. J. Phys. modelling Geotech. 1, 01–08. Repetto-Llamazares, A., Serré, N., Evers, K.-U., Jochmann, P., Høyland, K., 2009a. Model testing of ridge keel loads on structures part II: ridge building and physical properties. Proceedings of the 20th International Conference on Port and Ocean Engineering under Arctic Conditions. POAC, Luleå, Sweden. Repetto-Llamazares, A., Jochmann, P., Evers, K.-U., Høyland, K., 2009b. Model testing of ridge keel loads on structures part IV: preliminary results of freeze bond shear strength experiments. Proceedings of the 20th International Conference on Port and Ocean Engineering under Arctic Conditions. POAC, Luleå, Sweden. Serré, N., Liferov, P., Evers, K.U., 2009a. Model testing of ridge keel loads on structures part I: test setup and main results. Proceedings of the 20th International Conference on Port and Ocean Engineering under Arctic Conditions. POAC, Luleå, Sweden. Serré, N., Liferov, P., Jochmann, P., 2009b. Model testing of ridge keel loads on structures part III: investigation of model ice rubble mechanical properties. Proceedings of the 20th International Conference on Port and Ocean Engineering under Arctic Conditions. POAC, Luleå, Sweden. Serré, N., 2011. Mechanical properties of model ice ridge keels. Revised version submitted to Cold Regions Science and Technology. Serré, N., Liferov, P., 2010. Loads from ice ridge keels — experimental vs. numerical vs analytical. 20th IAHR International Symposium on Ice, June 14 to 18, Lahti, Finland. Timco, G.W., Johnston, M., 2003. Ice loads on the Molikpaq in the Canadian Beaufort Sea. Cold Reg. Sci. Technol. 37, 51–68. Timco, G.W., Wright, B., Johnston, M., Frederking, R., 1999. First-year ice ridge loads on the Molikpaq. Proceedings of the 4th International Conference on Development of the Russian Arctic Offshore, RAO99, Part II, pp. 172–179. St. Petersburg, Russia. Timco, G.W., Croasdale, K., Wright, B., 2000. An overview of first year sea ice ridges. Technical report HYD-TR-047, PERD/CHC reports 5-112. Wright, B.D., Timco, G.W., 2000. First-year ridge interaction with the Molikpaq in the Beaufort Sea. Cold Reg. Sci. Technol. 32, 27–44.
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Cold Regions Science and Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o l d r e g i o n s
Numerical modelling of ice ridge keel action on subsea structures Nicolas Serré ⁎ Norwegian University of Science and Technology (NTNU), Department of Civil and Transport Engineering, Høgskoleringen 7a, 7034 Trondheim, Norway Barlindhaug Consult AS, Sjølundveien 2, Postboks 6154, 9291 Tromsø, Norway
a r t i c l e
i n f o
Article history: Received 22 July 2010 Accepted 24 February 2011 Keywords: Ice keels Rubble ice Finite element model Sensitivity analysis
a b s t r a c t The present paper describes a numerical simulation of the keel–structure interaction where the keel geometry, ice thickness and interaction speed present a scaling ratio of 1:20. The rubble is represented by a Drucker–Prager material with cohesive softening and no dilatation. The partial differential equations are solved with the non-linear Eulerian finite element method of Abaqus Explicit V6.8.2. The results are compared to physical experiments conducted in the ice basin of the Hamburg ship model basin (HSVA). Two identical cubic subsea structures were impacted into the unconsolidated keel portion of two ice ridges with different thermal properties. The results indicate that only slight dilatation occurs, and the models are able to estimate the rubble action and deformed shape. A progressive failure of the rubble occurs. The rubble action is influenced more by the friction angle than it is in punch tests, due to higher confinement at the failure surface. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In a number of Arctic regions, the design load on a structure is determined by the action of first-year sea ice ridges. The keel action can be the main contributor to the load; however, little is known about its mechanical behaviour and how it should be modelled mathematically. The mechanical properties of full-scale ice ridges have been investigated in only a few full-scale experiments. The two main types of experiments that have been performed are: • The shear test (Bruneau et al., 1998; Croasdale et al., 2001), in which a vertical wall is introduced into the keel of an ice ridge and pushed horizontally. Loads and wall displacements are recorded. • The punch test (Leppäranta and Hakala, 1992; Heinonen and Määttänen, 2000; Heinonen, 2004), in which a horizontal plate is penetrated into the keel while loads and plate displacements are recorded. However, a lack of control over the boundary conditions and a limited amount of recorded data on the rubble behaviour and its physical properties make the interpretation of these tests challenging. Some offshore structures have been instrumented with load panels, video cameras and underwater sonars to relate keel loads to keel shapes. Data are available in the literature for: • The Molikpaq structure when it was in operation in the Beaufort Sea (Timco et al., 1999; Wright and Timco, 2000; Timco and Johnston, 2003). ⁎ Norwegian University of Science and Technology (NTNU), Department of Civil and Transport Engineering, Høgskoleringen 7a, 7034 Trondheim, Norway. Tel.: +47 77622679; fax: +47 77622699. E-mail address: [email protected]. 0165-232X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2011.02.011
• The Nörströmsgrund lighthouse in the Gulf of Bothnia (Kärnä and Jochmann, 2003; Bonnemaire and Bjerkås, 2004). • Two piers of the Confederation Bridge (Lemee and Brown, 2004). The pier loads were compared to the loads on the Kemi I lighthouse in the Baltic Sea in Brown and Määttänen (2009). These measurements provide useful information on the load levels and failure modes that are likely to occur during an interaction with an ice ridge. Unfortunately, none of them provide details on the specific action from the keel and the underwater deformation of the rubble. Data on the failure mode are of primary importance because it is necessary to verify that the theoretical models are able to correctly predict it and because numerous assumptions of the analytical keel load models are based on the type of failure occurring in the rubble. The Dolgopolov model (Dolgopolov et al., 1975) assumes a local failure of the keel. In some cases, for example, with small ice ridges, the keel can fail globally, and the deformations are similar to those of a plug of rubble pushed in front of the structure. Croasdale and Cammaert (1993) proposed a model for such a scenario. In a more recent approach, the keel load is given as the minimum load computed by the two previous models, for different penetrations of the structure into the keel (Brown et al., 1996). This idea was extended in general passive failure models proposed by Weaver in Croasdale, Associates Ltd. (1996) and by Kärnä et al. (2001). The load on the structure is computed all along the structure penetration into the ridge. For each position, different straight failure planes with varying inclinations are assumed, and the keel load is computed as the minimum load necessary to reach failure on one of these planes. This last generation of analytical models removes the need for the assumption of global or local failure but still requires assumptions with regard to the failure shape.
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Serré (2011). The numerical model uses a continuum description of rubble based on the Drucker–Prager constitutive model (Heinonen, 2004; Serré, 2011). It is equivalent to the Mohr–Coulomb material law, which has been suggested by the ISO/FDIS 19906 (2010), but its mathematical formulation renders it more suitable for use with an explicit finite element method. This paper studies how the numerical model simulates experiments and investigates the validity of the material law. The model helps in understanding the rubble failure mode, and a parametric analysis is performed, indicating which mechanical tests are the most suitable for determining the mechanical properties. 2. Experimental set-up 2.1. Ice ridges
Fig. 1. Interaction of physical, numerical and analytical modelling in design (Randolph and House, 2001).
The ridge failure experimental study that has most closely approached full-scale conditions was performed by Liferov and Høyland (2004). They built a medium-scale ice ridge in the Van-Mijen Fjord, Svalbard, and moved it against the seabed. Failure indicators were placed into the keel and showed that the failure was not simultaneous (global keel failure, deformation propagating throughout the entire keel) but progressive (local keel failure, deformations only in the vicinity of the structure). Due to the extensive resources required for a proper investigation of the failure of full-scale ridges, numerical simulations are an attractive approach for the study of ridge behaviour. They remove the need for failure shape assumptions and can account for irregular boundary conditions and better constitutive models than analytical methods (Timco et al., 2000). Nevertheless, numerical models that include the main complexities of rubble deformations require long computation times, which prevent their use in probabilistic analyses or in specific time-domain algorithms. Therefore, the major part of the design of an arctic offshore oil field is generally undertaken through simple analytical models. Because full-scale experiments are not available for their calibration, their development requires numerical modelling of the full-scale boundary value problem. The complementary role of physical and computational modelling is well explained in Randolph and House (2001) and can be represented as in Fig. 1. The present paper falls mostly on the left hand side of Fig. 1, i.e., the validation of modelling techniques by comparison between physical and numerical modelling. However, it also highlights some mechanisms of rubble deformation that should be considered in the development of conceptual models. The physical model used herein is presented in Serré et al. (2009a), and the rubble material parameters required for input of the numerical model are derived in Serré et al. (2009b) and
The ice tank in the Hamburg ship model basin (HSVA) is 78 m long, 10 m wide and 2.5 m deep. A 5-m-deep section of 10 m by 12 m is located at the end of the tank. Four ice sheets (corresponding to test series 1000, 2000, 3000 and 4000) were produced with one ice ridge per ice sheet. The ridges were wider than what would be expected from a typical keel depth to width relationship but were appropriate for the purpose of investigating ice rubble failure loads. With a scaling ratio of 1:20, they corresponded to a target full-scale size ridge with a 10-m-deep and 110-m-wide keel. A thermodynamic scaling of the rubble was not attempted. The geometries and velocities in the boundary value problem were scaled according to the Froude scaling law (ratio 1:20) to ensure conservation of the inertial to gravity forces ratio as given by: Vc ffi Fr = pffiffiffiffiffiffi gLc
ð1Þ
where Vc and Lc are the characteristic speed and length, respectively, and Fr is the Froude number. If the characteristic length is scaled by λ and the ice density is kept constant, then the masses scale as λ3 and the velocities and time as λ1/2. The ice ridges were confined on the sides by the tank walls. Ridges 2000 and 4000 were built from colder ice and tested soon after their production, with very little consolidation (degradation) time allowed. The rubble density (ρr) was computed from: ρr = η ⋅ ρw + ð1−ηÞρi
ð2Þ
where ρi is the ice density, η is the ridge porosity and the water density ρw is 1006 kg/m3. A layer of level ice was pushed over the rubble to model the consolidated layer. A visual inspection of the ridges after the interaction showed that some blocks of ice were frozen to the level ice. The consolidated layer did not interact with the subsea structures and was therefore not modelled in the numerical analysis. The principal characteristics of the ice ridges are represented in Fig. 2 and
Fig. 2. Model ice ridge characteristics.
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Table 1 Ice ridge characteristics. Ridge #
1000 2000 3000 4000
(warm) (cold) (warm) (cold)
hk (m)
kw (m)
kbw (m)
η (%)
ρi (kg/m3)
ρr (kg/m3)
Consolidation Time (h)
Tair (°C)
0.5 0.45 0.45 0.55
5 6 6 5.5
3 4.5 4 4.5
32 36 41 43
795 868 798 844
863 917 883 914
18 5 18 5
4 −0.5 3.6 − 0.2
given in Table 1, where hk is the keel depth measured from the waterline, kw is the keel width and kbw is the keel bottom width. More details on the physical properties of the ridges and their measurement methods are given in Repetto-Llamazares et al. (2009a). The water had a salinity of 7 ppt and a freezing temperature of − 0.5 °C. Its temperature during the interaction test was −0.3 ± 0.2 °C. 2.2. Interaction test To model shallow water conditions, bottom elements (false bottom) were inserted into the tank to reduce the water depth. The structures were fixed on the false bottom, which was displaced along the tank by a motor-driven carriage able to provide a maximum towing load of 50 kN. Two types of structures were studied: a cubical shape for subsurface keel interactions and a simple conical structure for surface ridge interactions. An analysis of the keel–cone interaction is given in Serré and Liferov (2010), and an analysis of the subsea structure interaction is presented in this paper. A bird view of the testing procedure is shown in Fig. 3, and an artistic 3D view of the ridge interaction with one subsea structure is given in Fig. 4. The principal dimensions of the structures are given in Fig. 5. The friction coefficient of the ice against the painted plywood of the subsea structures was measured to be 0.11. For the measurement, a 0.15 × 0.15 m block of ice was extracted from the ice sheet and placed in dry conditions on a surface built from the same material as the subsea structure. Different confinement weights were applied on the ice, which was then displaced over the material. The applied shear force was measured, and its ratio over the applied weight gave the friction coefficient value. The ice was extracted from the ice sheet before the ridge testing and tested the next day. The ice was kept overnight in a water basin in a room with an air temperature of −1.6 °C.
Fig. 4. 3D illustration view of the sub-surface interaction.
with Abaqus/Explicit V6.8.2. A description of the model is given in the following section. 2.4. Geometry The numerical model of the ice ridge keel is shown in Fig. 6, and the keel geometries are provided in Table 1 together with the rubble density. A 2-m-thick slice of the model ice ridge keel was considered. During the model development, the results from the simulations were compared to a model considering the full length of the ridge to verify that the slice was thick enough to ensure that the simulated load was independent of its thickness. A 0.4-m-thick elastic layer referred to as an “air layer” was placed above the keel to balance the buoyancy forces. One numerical model was developed for ridges 1000 and 2000. Due to technical problems with the load cells during the test series 3000 and 4000, the data from these two tests were not analysed. The dimensions of the cubic subsea structure and its position relative to the water line and false bottom are given in Fig. 5 and in the experimental set-up description.
2.3. Numerical method
2.5. Loads and boundary conditions
The keel–structure interaction was analysed with the finite element method using a coupled Eulerian–Lagrangian formulation to allow for large displacements. The computations were performed
The friction coefficient between the structure and the rubble ice was measured to be 0.1. In the first step, only buoyancy forces were
Fig. 3. Test set-up, bird view.
Fig. 5. Dimensions of the cubic structure.
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Rubble Void Other material
Elastic forces
Subsea structure
Buoyancy Eulerian forces mesh
Motion Symmetry
2m
Fig. 6. Cut view of the numerical model of the rubble/structure interaction.
applied to the rubble. They were modelled by an upward gravity. The gravity coefficient was computed such that, when multiplied by the rubble density, the result was equal to the buoyancy forces of the rubble. It was given by: g′ = g
ρw −1 ρr
ð3Þ
where g′ is the “upward gravity” (buoyancy) coefficient and g is the gravitational acceleration (9.81 m/s2). In the same step, a vertical pressure was applied on the bottom surface of the air contacting the rubble to achieve static equilibrium. Its value was given by: p = hk ρr g ′
ð4Þ
where hk is the keel depth. In the second step, the structure was displaced with a velocity of 0.125 m/s until it had crossed the entire rubble. This velocity was larger than the experimental velocity (0.045 m/s) to save computation time. A comparison with a numerical model featuring the experimental velocity did not show velocity effects on the computed rubble action. A symmetry boundary condition was applied such that only one half of the structure and the rubble must be modelled (the 2-m slice becomes equivalent to 4 m). To represent the effect of the remaining part of the rubble on the 2-m slice considered, a fixed frictionless wall boundary condition was applied on the other side of the rubble slice. The consolidated layer effect on the rubble is considered by preventing any horizontal motion of the nodes located at the rubble top. The vertical motion is left free because some vertical motion of the ridge sail was observed during the interaction (Serré et al., 2009a). 2.6. Material model The structure was modelled as a rigid body. The air layer was purely elastic, with a Young modulus of 3 kPa. Its compression produced the same vertical stress range on the rubble as the gravity load from the ice lifted out from the water. The ice rubble was modelled with the linear Drucker–Prager plasticity model. Similarly to the Mohr–Coulomb material law, it is a hydrostatic-pressuredependent type of model characterised mainly by the Drucker–Prager friction angle β and cohesion d. A detailed description of the Drucker– Prager model applied to the rubble ice is given below.
2.6.1. Yield surface The linear Drucker–Prager yield surface is: t = d + p tanðβÞ;
ð5Þ
where p is the hydrostatic pressure. The deviatoric stress measure t is defined in Abaqus as: t=
3 1 1 1 r ; q 1 + − 1− 2 K K q
ð6Þ
where K is the ratio of the yield stress in triaxial tension to the yield stress in triaxial compression, r is the third stress invariant and q is the Von Mises stress. Abaqus allows one to select a K value between 0.778 and 1 (above 0.778 to ensure convexity of the yield surface). No triaxial tests were performed on the rubble, so the yield stress ratio was selected to be equal to 1 such that the yield surface corresponded to the original Drucker and Prager (1952) model. Consequently, the influence of the third stress invariant was neglected. 2.6.2. Hardening law The failure of the freeze bonds inside the rubble is represented by a softening cohesive behaviour: when the stress state lies on the yield surface, the material yields in shear, and the cohesion d decreases linearly from its original value di to 0 as the first 0.02 of equivalent plastic strain (εp) is applied. The cohesion and equivalent plastic strain are then related through: 8 > < > :
i
d = di − d=0
d ⋅ εp if εp ≤ 0:02 ; 0:02 if εp N 0:02
ð7Þ
pffiffiffi where εp is equal toγpl = 3 and γpl is the engineering shear strain. The cohesive softening in Abaqus is implemented by a hardening law, where the yield shear stress at 0 Pa of hydrostatic pressure (i.e., the cohesion) is tabular specified as a function of the equivalent plastic strain. 2.6.3. Plastic flow potential When the stress state lies on the yield surface, plastic deformations occur in the material. The plastic flow is defined by a non-associated flow potential G given as: G = t−p tanðψÞ;
ð8Þ
where ψ is the dilatation angle. A dilatation angle equal to 0° corresponds to the case of incompressible inelastic deformations,
N. Serré / Cold Regions Science and Technology 67 (2011) 107–119
while a positive dilation angle indicates that the material dilates during shear deformations. The strain is computed by multiplying the strain rate by the length of the time increment. An additive strain rate decomposition is assumed: el
pl
dε = dε + dε ;
ð9Þ
where dε is the total strain rate, dεel is the elastic strain rate and dεpl is the plastic strain rate computed as: dγpl ∂G pl ; dε = 2 pffiffiffi 1 ∂σ 3 1+ K
ð10Þ
with the rubble through the Lagrangian/Eulerian contact formulation of Abaqus. The mesh in the interaction zone was composed of cubic Eulerian elements with a length of 0.05 m. An analysis of the result dependency on the mesh size was performed and showed that the computed load became stable for an element length below 0.075 m (Figs. 7 and 8). The numerical model used for the mesh sensitivity analysis was based on the geometry and material properties from ridge 2000 but with a uniform cohesion of d = 735 Pa. 2.8. Parametric analysis The parameters selected for analysis were:
where σ is the stress tensor. 2.6.4. Mechanical parameters The parameters necessary to define this model are d, β and ψ. The elastic domain is defined by the standard Hooke's law, requiring the Young modulus E and the Poisson's ratio ν, which was chosen to be 0.3, a typical value for granular materials (Harr, 1977). The mechanical properties were derived from different mechanical tests performed on the four ice ridges (Serré et al., 2009b; Serré, 2011) and are given in Table 2. A different cohesion value di was attributed to ridges 1000 and 2000 due to their different ages and temperatures. Ice ridge 2000 was colder and younger than ridge 1000, as indicated in the keel temperature profile measurements of Repetto-Llamazares et al. (2009b). The temperature of ridge 1000 was above the solidification temperature of the water in the ice tank in contrast to that of ridge 2000. The upper 10 cm of keel 2000 were also significantly colder than the rest of the keel. The short degradation time in the basin prevented the heat from the water from reaching the upper part of the keel and degrading the freeze bonds located there. The freeze bond strength distribution was therefore represented by a linear distribution of the cohesion in the keel of ridge i at the top of the rubble to 2000, with an initial value varying from dtop i d bottom at the bottom. For this purpose, each horizontal layer of nodes in the keel was assigned a cohesion value in relation to its vertical position. The cohesion value in the material was then interpolated by Abaqus. In the present model, the dilatation angle ψ was set to 0°, and the elastic domain was not limited along the hydrostatic pressure axis, that is, the rubble did not deform plastically under hydrostatic compressive forces.
• The ice density, varied according to Evers and Jochmann (1993) from 750 to 930 kg/m3. • The keel depth, varied from 0.3 to 0.5 m. • The Drucker–Prager friction angle, varied according to the pile test experiment in Serré (2011) from 40 to 50°. • The Drucker–Prager cohesion, varied from 0 to 2.5 kPa (arbitrary values).
2.7. Mesh The meshed zone was divided into three areas. The upper one was 10 cm high and above the water line. It was composed of void elements. The second one was the ridge keel, whose elements were composed of rubble. The third area was under the keel and was composed of void elements. Void elements could be filled with rubble upon deformation. A representation of the mesh is given in Fig. 6. The air layer shown as the white layer above the keel was modelled with Lagrangian elements. It overlaps the upper void layer and can interact Table 2 Ice rubble mechanical properties for ridges 1000 and 2000 (Serré, 2011). Parameter E [MPa] ν β [°] i [Pa] dtop i [Pa] dbottom ψ [°]
111
Ridge 1000
Ridge 2000 0.9 0.3 50
612 612
1470 0 0
Fig. 7. Estimated load for a mesh size of 0.075 m (a) and 0.0375 m (b).
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a
Structure
False bottom
Motion (125 mm/s)
b Fig. 8. Estimated load as a function of the mesh size.
Although the cohesive softening behaviour of the freeze bonds and rubble ice has been highlighted by numerous authors, full-scale rubble cohesion values considering softening are reported only in Heinonen (2004), while only Serré (2011) reports model-scale values. There is therefore no cohesion range from which input values could be selected for a parametric analysis. However, several authors such as Ettema and Urroz (1989) have reported that rubble ice submitted to continuous deformation behaves as a cohesionless material. The Drucker–Prager cohesion range was thus selected to span from 0 Pa to approximately two times the base case cohesion value (1.47 kPa). For an easier comparison with the rubble description given by the ISO/FDIS 19906 (2010), the cohesive and frictional parameters of the rubble can be represented by the Mohr–Coulomb parameters c (cohesion) and ϕ (friction angle). The relation between the Drucker– Prager and Mohr–Coulomb parameters can be extracted from Chen
Fig. 9. Experimental (black) and numerical (red) rubble load on the subsea structure, test 1000.
Motion Fig. 10. View of the numerical estimation of the rubble accumulation on the front of the subsea structure (a) and corresponding situation in the experimental model (b), test 1000.
Fig. 11. Experimental (black) and numerical (red) rubble loads on the subsea structure, test 2000.
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113
Structure Pressure (Pa) 2500 False bottom Motion (125 mm/s)
0
Fig. 12. Hydrostatic pressure in the rubble during test 2000.
and Saleeb (1982) for a plane strain approximation. The corresponding equations are: tan β =
pffiffiffi 3 sin ϕ
Fig. 14. Rubble accumulation on the front of the subsea structure; position: 61.5 m, test 2000.
ð11Þ
pffiffiffi d = 3c cos ϕ:
ð12Þ
3. Results
measured during the experiments (Tables 1 and 2) over-estimates the rubble action from test 1000 by 250%. Fig. 10 shows that the rubble accumulation on the structure's front is 20 cm deeper in the numerical model than in the experiment.
3.1. Test 1000, rubble load and deformation Fig. 9 compares the simulated and measured rubble loads on the subsea structure. The numerical model based on the parameters
a
Structure Horizontal velocity (mm/s) 240
Structure
False bottom
125 False bottom
Motion (125 mm/s)
0
Motion
b
b
Structure Vertical velocity (mm/s) 200
0
Horizontal velocity (mm/s) 240
False bottom 125 Motion
0
Structure False bottom Motion (125 mm/s)
-100 Fig. 13. Test 2000, rubble velocity, horizontal (a) and vertical (b); structure position: 61.5 m.
Fig. 15. Rubble accumulation on the side of the subsea structure, experimental on the starboard of the structure (a) and numerical on the portside (b); position: 61.5 m, test 2000.
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3.2. Test 2000, rubble load
the numerical model because they are caused by a standing wave with a period unique to the HSVA basin (Serré et al., 2009a).
The numerical estimation of the rubble load on the subsea structure is plotted together with the experimental rubble load for test 2000 in Fig. 11. The numerical model predicts the load well except when the structure exits the rubble at the end of the interaction, where the model appears slightly conservative. Oscillations with a 30-s (1.35-m) period can be observed for all of the experimental load time series. They should not be considered in the comparison with
a
3.3. Test 2000, stress levels Fig. 12 shows a section of the rubble along the vertical symmetry plane of the structure and shows the hydrostatic pressure during the interaction. The hydrostatic pressure is below 2500 Pa in the main part of the rubble, but in the area in the upper corners of the structure, it can reach 5000 Pa. The size of this area does not exceed the size of one element and is a result of stress concentrations at the corners. 3.4. Test 2000, rubble deformation A plot of the rubble shape and ice velocity is given in Fig. 13. It shows that deformations occur only in the vicinity of the structure. They correspond to a progressive (local) failure of the rubble. When the structure travels into the rubble, ice blocks accumulate on the front and are pushed on the sides. The rubble accumulation rapidly
a Structure
Motion
False bottom
b Structure
False bottom Motion
b
Structure
Horizontal velocity (mm/s) 240 125
False bottom Motion
Fig. 16. Rubble accumulation seen from behind the subsea structure, experimental (a) and numerical (b); position: 61.5 m, test 2000.
Structure
False bottom
0 Motion (125 mm/s) Fig. 17. Rubble accumulation on the front of the subsea structure when exiting the keel, experimental (a) and numerical (b); position: 62.9 m, test 2000.
N. Serré / Cold Regions Science and Technology 67 (2011) 107–119
covers the whole front of the structure and then slightly increases in size until the structures exit the keel. The extent of the rubble accumulation on the front of the subsea structure compares well with the underwater image (Fig. 14). Numerical and experimental results also agree with respect to the rubble coverage on the side of the structure (Fig. 15) and as seen from the back (Fig. 16), where a wake is created behind the structure. The wake is first freed from the rubble up to the roof of the structure, and then the two rubble mounds on its side collapse in it due to the action of the buoyancy force until their slopes reach stability. As for the rubble load, a comparison between the numerical and experimental rubble shapes at different stages of the interaction also shows a good match for most of the test but a difference at the end (Fig. 17). Experimentally, a deep portion of the rubble is accumulated on the front of the structure and is almost entirely separated from the rest of the keel; only very little rubble covers the side of the structure. However, the numerical model presents a shallower but wider accumulation, also covering the sides of the structure and in contact with the keel. This observation is in agreement with the conservatism of the numerical model when the structure exits the keel.
a
115
3.5. Test 2000, parametric analyses The effects of the ice density, keel depth, friction angle, cohesion and porosity on the computed rubble load were studied through parametric analyses. Only one parameter was changed at a time along a range of possible values. The corresponding numerical simulations are given in Figs. 18 and 19. Few authors have derived the rubble cohesion while considering cohesion softening. Therefore, the literature does not propose a cohesion range from which input values can be selected for a proper parametric analysis. Fig. 19 shows the effect of maximum cohesion (at the top of the rubble) from 0 to 2.5 kPa on the rubble load, which increases from 750 to 1200 N. For an extreme cohesion of 9 kPa, the load reaches 1600 N. The highest 1-m average load for each simulation is given in Table 3. It corresponds to the portion of the load curve that presents the highest average load over a structure displacement of 1 m. The results of the parametric study given in Table 3 are plotted in Fig. 20. The variation of each parameter as compared to their base case value (parameter values given previously for test 2000) is given in percent along the horizontal axis, and the resulting variation of the
b 1600 1600 1400
1400 i = 750
kg/m
3 1200
1200
Load [N]
Load [N]
1000 1000 800
800 600
600 400
hk = 0.5 m i=
3
400
930 kg/m
200
200 0 56
57
58
59
60
61
62
63
0 56
64
hk = 0.3 m 57
58
59
Distance [m]
60
61
62
63
64
62
63
64
Distance [m]
c
d 1400 1600
= 0.3 1400
o
= 45 , = 50
1200
o
1000
1000
Load [N]
Load [N]
1200
800
= 0.45
800
600
600 400 400 o
200 0 56
200
o
= 30 , = 40 57
58
59
60
61
Distance [m]
62
63
64
0 56
57
58
59
60
61
Distance [m]
Fig. 18. Effect of the ice density (a), keel depth (b), friction angle (c) and porosity (d) on the simulated rubble action, test 2000.
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Fig. 19. The numerical rubble load on the subsea structure for different levels of initial cohesion; the base case corresponds to the cohesion value for the simulation of test 2000.
estimated load is given along the vertical axis. A slope is computed between the extremities of each curve. A slope close to the horizontal (slope = 0) indicates that the parameter in question has no influence on the load. The closer the curve is to the vertical, the more important the role of the parameter is in the rubble load computation. The ice density curve is the closest to the vertical, showing that this parameter has the largest influence on the load. The base case corresponds to a numerical simulation of test 2000. 4. Discussion 4.1. Measurement inaccuracies The poor concordance between the experimental and numerical results for test 1000 is not necessary linked to a weakness in the numerical model. Actually, this test was the first to be performed, and the research team was not used yet to the different measurement methods. Therefore, a higher uncertainty exists for this test with respect to the ice density, the keel depth and the rubble porosity. Furthermore, the rubble load measurement also appears questionable because the measured load on the structure was as low as at least half the rubble action measured during tests 2000, 3000 and 4000 (Serré et al., 2009a). It should, however, be remembered that the measurements from the 3000 and 4000 series were approximate due to damaged load cells. To estimate the influence of these uncertainties, the rubble action from ridge 1000 was simulated based on a set of “plausible” parameters selected among the parameters that were measured for the four different ice ridges. The modified parameters are given in Table 4. The friction angle is given by the lower bound value derived from a rubble pile test analysis performed on ridge 2000 and described in Serré (2011). In this test, the Mohr–Coulomb friction angle ranged between 30 and 45°. Table 3 Parametric analyses for the interaction test, showing the effect of selected parameters on the average peak load. Parameter 3
ρi [kg/m ] hk [m] β (ϕ) [°] η d (c) [kPa]
Parameter range
Average peak load [N]
[750; 930] [0.3; 0.5] [40 (30); 50 (45)] [0.3; 0.45] [0 (0); 2.5 (2)]
[1550; 700] [475; 1200] [530; 1050] [907; 1100] [700; 1200]
Fig. 20. Influence of different parameters on the rubble load on the subsea structure. The ice density curve is the closest to the vertical, showing that this parameter has the largest influence on the load. The base case corresponds to numerical simulation of test 2000.
It is not known why the rubble friction angle in ridge 1000 is different from that of ridge 2000. However, the ice blocks composing the rubble from these two ridges did not present the same aspect, as shown in Fig. 21, where pictures extracted from the underwater videos revealed that the ice block size was larger in test 2000 than in test 1000. Either this aspect variation or the cause of this variation might have affected the value of the rubble friction angle. The measured and simulated rubble action according to the modified parameters in Table 4 is given in Fig. 22a. The numerical model still produces conservative results, but the numerical and the experimental load curves agree with an error of 20% if the keel depth is reduced to 45 cm instead of the original 50 cm (Fig. 22b). The deformed rubble shape computed by the 45-cm-deep keel model (Fig. 23) is closer to the experimental observation (Fig. 10b) than the original numerical model (Fig. 10a). Because the person in charge of the keel depth measurement differed from ridge to ridge and because this measurement is rather subjective (Repetto-Llamazares et al., 2009a), a 5-cm error in the keel depth is possible. Therefore, by considering the different uncertainties linked to the measurements, in spite of the poor agreement between the experimental and the original numerical load curve, the test 1000 results do not invalidate the numerical model.
4.2. Dilatation and volumetric hardening The determination of the deformed shape of the rubble is dependent on a proper selection of the dilatation angle. In the numerical model, no dilation occurs because ψ is equal to 0°. It is difficult to accurately estimate the experimental amount of dilatation. However, the good comparison between the numerical and experimental deformed rubble shapes in test 2000 (see Figs. 13–16) reveals that only slight dilatation occurred in the experiment. The small amount of dilation can be explained by the high porosity of the rubble ice, which then behaves as a loose granular material (Chen and Saleeb, 1982). Table 4 Modified values of the parameters used in the simulation of test 1000. η [%]
ρi [kg/m3]
ρr [kg/m3]
g′ [m/s2]
β [°]
43
868
927
0.84
40
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117
Fig. 21. Images of the maximum block size for test 2000 (a) and test 1000 (b).
The low hydrostatic pressure levels estimated numerically confirm the possibility of neglecting the volumetric plastic deformations linked to hydrostatic compaction. Serré (2011) performed oedometer tests on model rubble ice and presented a curve relating the hydrostatic yield strength of the rubble to its volumetric strain. It was found that a hydrostatic pressure of 2.5 kPa (Fig. 12) corresponded to a plastic volumetric strain below 7%. Therefore, only slight volumetric plastic deformation occurs during the interaction process, and it is thus not necessary to consider this failure mechanism in the present numerical model.
Fig. 22. Experimental (black) and numerical (red) rubble load on the subsea structure, test 1000 with modified ice density, rubble porosity and friction angle. Original keel depth in (a) and a 5-cm-smaller keel depth in (b).
4.3. Parametric analysis Fig. 20 indicates that the ice density most influences the rubble load on the subsea structure, followed by the keel depth, the friction angle, the porosity and the cohesion. The same ranking was observed in the punch test sensitivity analysis reported by Serré (2011), but two differences are worth noting:
Horizontal velocity (mm/s)
Structure
290 False bottom
• The effect of the friction angle on the rubble load is significantly larger in the interaction test than in the punch test. • The cohesion plays a role throughout the entire length of the load curve in the interaction test (Fig. 19), while it acts only on the peak of the punch test load curve and leaves the post-peak portion unaffected.
125 0
Motion (125 mm/s)
Fig. 23. Computed rubble accumulation on the front of the subsea structure for a 45-cmdeep keel, for comparison with the experimental rubble accumulation in test 1000.
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4.4. Frictional resistance The friction angle plays a more important role in the rubble resistance for the interaction tests than for the punch test due to the different rubble failure modes that occur. In punch tests, the failure has the shape of an upward cone at the beginning of the interaction. The conical rubble portion extends to the keel bottom and is pushed downward by the punch plate (Heinonen and Määttänen, 2000; Jensen et al., 2001). The same behaviour was observed in the punch test numerical simulations of Liferov et al. (2003) and Serré (2011). During this failure process, the conical portion of the rubble is pushed downward by the punch plate and is separated from the rest of the keel. Therefore, the confinement at the failure plane decreases during the failure process, as shown by Serré (2011). In the interaction case, however, the buoyancy pushes the failed portion of the rubble upward against the rest of the keel, even after failure has occurred, and the motion of the structure compresses this portion against the keel, causing downward rubble extrusion (see the velocity plots of Fig. 13). Both of these effects increase the confinement at the failure plane as the structure penetrates into the rubble, as shown in the pressure curve of Fig. 24. Due to the different failure behaviours for these two tests, the confinement at the failure zone increases for the interaction test, while it decreases for the punch test. This explains why the friction angle is a more important parameter in the interaction test than in the punch test. Therefore, it is unwise to estimate the friction angle with the punch test when this value is to be used for the simulation of a different boundary value problem that is more sensitive to this parameter. This is one of the reasons why punch tests should not be the sole characterisation tool for rubble mechanical properties. Alternative approaches include the pile test (Keinonen and Nyman, 1978; Serré, 2011), the shear box test (Ettema and Urroz, 1991; Liferov and Bonnemaire, 2005), the shear test as mentioned in the introduction and the retaining wall test (Keinonen and Nyman, 1978; Serré et al., 2009b). 4.5. Cohesive resistance The parametric analysis shows that, in spite of cohesive softening, the cohesion affects the entire length of the simulated load curve. It
represents the process well in which the structure has to break freeze bonds all along the interaction due to the failure zone which reaches undisturbed rubble areas as long as the structure has not reached the end of the keel (progressive failure, as shown in the velocity plots of Fig. 13). The progressive failure of the ridge keel impacted by the subsea structure is in good agreement with the medium-scale observations from the ridge shear-off test of Liferov and Høyland (2004). It also shows that the concept of primary and secondary failure of the rubble given in Liferov and Bonnemaire (2005) can occur locally without being necessarily initiated throughout the entire rubble. The primary failure mode is controlled by the initial strength of the rubble skeleton (the frictional resistance of the rubble is not mobilised), while in the secondary failure mode, substantial deformations have already occurred and the rubble strength is dominated by the frictional resistance.
5. Conclusion Numerical modelling of the ice ridge keel–subsea structure interaction was performed and compared to model-scale experiments. It was found that only slight dilatation and compaction occurred in the rubble. The numerical model used herein is based on the Drucker–Prager material law and can compute the rubble action and deformation. However, the precision of the results depends on the measurement quality of the physical parameters involved in the rubble resistance. The effects of four parameters were studied. The ice density is the parameter with the largest effect on the rubble action, followed by the keel depth, the friction angle and the cohesion. The frictional resistance plays a more important role in the rubble action in situations where an ice ridge impacts a structure than in a punch test. The failure zones of these two scenarios present two very different stress states. These results show that the mechanical properties of the rubble cannot be characterised by one test alone and that the punch test is poorly suited for an accurate estimation of the friction angle. A progressive failure occurs in the rubble (local failure), and therefore cohesive resistance is present all along the interaction, even if the failed portion of rubble becomes cohesionless.
Fig. 24. Average hydrostatic pressure at a selected failure zone as the structure travels through the keel, test 2000.
N. Serré / Cold Regions Science and Technology 67 (2011) 107–119
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