DISCRETE ELEMENT SIMULATION OF ICE PILE-UP AGAINST AN INCLINED STRUCTURE Jani Paavilainen, Jukka Tuhkuri and Arttu Polojärvi Helsinki University of Technology, Laboratory for Mechanics of Materials, Finland ABSTRACT A two dimensional discrete element method has been used to study the ice pile-up process against an inclined plate. In a simulation, an intact ice sheet, driven against an inclined plate, failed into discrete ice blocks which then accumulated into a pile. The pile-up process, forces acting on the plate and the shape of the rubble pile were studied. Simulation results were compared to laboratory experiments. KEY WORDS: pile-up, inclined structure, simulation INTRODUCTION The failure of an ice cover against an inclined structure is a process where an ice sheet breaks into discrete ice blocks which then accumulate in the ice-structure interface and form a rubble pile. The first models for calculating the ice load on sloping structures separated the failure process into failure of the ice sheet through bending, and ride-up of the ice along the structure (for a review, see Sanderson, 1988). The main variables in this kind of models are the slope angle, thickness and other properties of the ice sheet, as well as the friction coefficient. Later, also other failure modes and effects of weight and buoyancy of the rubble have been included in the calculation models (e.g. Määttänen, 1986). While solutions for different events can be obtained through simple analyses, calculation of the total ice load is not as easy. This is because the maximum value of each load component does not necessarily occur at the same time, and also because the different ice failure events can interact. As an example, the bending failure of the ice sheet is affected by the rubble formation. This kind of uncertainties can be avoided if the ice-structure interaction is studied as a process and simulated. One method to simulate the ice failure processes is the discrete element method (DEM), which has already been used in analyses of ice pile-up on an inclined ramp (Hopkins, 1997) and ice sheet interaction with a rubble pile adjacent to a structure (Evgin et al., 1992). In addition, it is important to note, that the pile-up process is closely related to the ice ridging process, which has been extensively simulated with DEM (e.g. Hopkins, 1998; Hopkins et al. 1999). -177-
In this paper, a two dimensional DEM is used to simulate laboratory scale experiments of ice sheet failure and rubble formation against an inclined structure. The simulations provide a method to study the rubble formation process and the forces related to it, while the experimental data is used to benchmark the simulation method. DISCRETE ELEMENT METHOD Discrete element method (DEM) is a technique for modelling media consisting of discrete particles. In DEM, the interaction between the particles within the system occurs through pairwise collisions from which the forces acting on each particle are derived according to the chosen interaction laws. On each time step of a DEM simulation, the contacting particles are identified, the contact geometries are solved, the contact forces are determined, and the particles are moved to new locations. The method for searching contacting particles was adopted from Munjiza and Andrews (1998) and the method for calculating contact forces between interacting ice blocks was adopted from Hopkins (1992). The contact forces between particles were calculated with an elastic-viscous-plastic material model combined with an incremental Mohr-Coulomb tangential force model. Inelasticity was modelled by using a plastic limit for the material. Determination of the elastic contact forces was based on the overlap area and the rate of change of the overlap area of contacting particles. The normal component of the elastic contact force is F ne = k ne (A e + A)n, (1) where k ne is the normal stiffness of the contact, A e is the elastic area of the overlap from the previous time step, A is the change in the overlap area between time steps and n is the unit vector normal to the line of contact defined for the contact geometry. The viscous component of the normal force is given by F nv = k nv ( A t ) n, where k nv is the viscous damping coefficient and t is the length of time step. The plastic limit is defined as (2) F p = σ p l n, (3) where σ p is the plastic limit of the material and l is the length of the line of contact. Using F p as the upper limit for the normal contact force, the total normal contact force is given by F n = min {F ne + F nv,f p }. (4) If the plastic limit is exceeded, the elastic area for the next time step is updated by solving the equation A e t = 1 k nv ( F p k ne A e ). (5) The tangential contact force F t,i on timestep i is calculated using its value from time step i 1, tangential contact stiffness k te and relative tangential velocity of interacting particles v t. The -178-
upper limit for F t,i, using Mohr-Coulomb model, is achieved by using friction coefficient µ and F n, hence F t,i = min {F t,i 1 k te v t t, µf n }. (6) The moment acting on each particle due to contacts is given by M = r F, (7) where r is the vector pointing from the centroid of a particle to the centroid of the overlap area and F is achieved as a sum of the tangential and normal contact forces. When simulating an ice sheet failure against a structure, an intact ice sheet and the failure of the ice sheet need to be modelled, in addition to the discrete particles. This intact 2D sheet was modelled with discrete particles that were combined together with beam elements. For the beam elements, a non-linear Timoshenko plane beam model was used (Paavilainen, 2006). From the positions and orientations of two adjacent particles in the intact sheet, the strainsh T = [ε γ κ] of the beam element combining the particles are calculated. The stress resultants z T = [N V M] are then defined from the strains, and the internal forces p of the beam element are calculated from the stress resultants as follows p = B T z dx. (8) L 0 The matrix B is obtained from the strain-displacement relation and the integral is taken over the initial length L 0 of the beam element. Internal forces of the beam element are then added to the corresponding forces acting on the particle. A fracture criterion, σ x σ b, where σ x is the stress in the surface of the beam and σ b is the bending strength of the beam, was used for the ice sheet. On each time step, the fracture criterion was examined on both the upper and the lower surface of the beam. If σ x σ b, a crack was assumed to initiate and propagate with constant velocity through the beam during the following time steps. The crack velocity in many brittle materials is of the order of 0.3 E/ρ, where E is Young s modulus and ρ is density (Broek, 1991). Parsons et al. (1987) measured a much lower value 0.01 E/ρ for sea ice. Crack velocity in the model ice simulated here is not known, and a value 0.05 E/ρ was used in the simulations. SET-UP OF THE SIMULATIONS AND THE LABORATORY EXPERIMENTS The numerical DE simulations performed were simulations of the model scale laboratory experiments conducted by Saarinen (2000). In both experiments and simulations, an ice sheet was pushed against an inclined plate with a constant velocity. Figure 1 shows the geometry of the set-up and Table 1 gives the parameters used in the simulations and either measured during the laboratory experiments or known for the model scale laboratory ice. In the experiments, a 4.6 m wide ice sheet was used, but the simulations were two dimensional. All the individual particles in the simulation were square shaped. At the left boundary, the ice sheet rotation was constrained, vertical displacement was allowed, and a constant horizontal velocity was given as an input value. Distance from the inclined plate to the left boundary was 217.5h, where h is the ice thickness. The number of ice particles in the beginning of the simulation was 217 and at the end 1006. Both ends of the inclined plate were rigidly supported. -179-
Table 1: Parameters used in the simulations and either measured during the laboratory experiments or known for the model scale laboratory ice. Parameter Simulations Experiments Ice particles thickness [m], h 0.05 0.045 0.052 width [m] 0.05 - Ice properties Young s modulus [MPa], E i 30.0 14 90 Poisson ratio, ν i 0.3 - density [kg m 3 ], ρ i 930 930 friction coefficient, µ i 0.35 0.32 0.36 friction coefficient (under water), µ is 0.3 - normal stiffness [MPa], k ne 30.0 14 90 tangential stiffness [MPa], k te 20.0 - viscous damping [MPa s], k nv 1.0 - plastic limit [kpa], σ p 50.0 15 40 bending strength [kpa], σ b 30.0 30 56 internal damping [kpa s], c i,int 1.0 - crack propagation speed [m s 1 ] 10.0 - ice sheet velocity, v i 0.1, 0.05 0.014 Water density [kg m 3 ], ρ w 1000 1000 Wall properties inclination angle [ ], α 50 50, 65, 80 Young s modulus [GPa], E W 10.0 - Poisson ratio, ν W 0.3 - internal damping [kpa s], c W,int 100.0 - Ice-wall contact friction coefficient, µ iw 0.35 0.07, 0.35 friction coefficient (under water), µ iws 0.3 - Simulation time step [s], t 1.0 10 5 - SIMULATION RESULTS Figure 1 shows snapshots of the simulation and Figure 2 gives the total force acting on the inclined plate in the horizontal direction as a function of the length of pushed ice L. Figure 2 shows also the force obtained in the laboratory experiments. For comparison of the forces and the stage of simulation, the subfigures of Figure 1 are pointed out in the F(L)-record in Figure 2 with arrows. The growth of the cross sectional area of the rubble pile was also studied. Cross sections from the simulation are presented in Figure 3 at L = 10, 20, 30 and 40 m. Initially, the ice sheet failed against the inclined plate by bending (Figure 1a) and started to form a pile (Figure 1b) which grew both vertically and horizontally. After a pile was formed, the ice sheet failed against the pile to form a sail (Figures 1d and 1g) or slided on top of the pile to the inclined plate and rode-up along the plate (Figures 1c and 1e). Figures 1e and 1f illustrate a collapse of the pile, when a failure of the ice sheet resulted in decrease of the horizontal force, followed by movement of the pile down and left. The cycle, growth of the pile against the plate and collapse of the pile, occurred several times during the simulation. During the initial stages (0 L < 20 m) of the simulation these collapses of the rubble pile caused the force to drop to zero as the contact between the sheet -180-
(a) (b) (c) (d) (e) (f) (g) (h) Figure 1. Snapshots of a simulation. Ice sheet (white) is pushed with a constant velocity of 0.05 m/s against an inclined plate (black). The water layer is coloured gray. The length of the pushed ice L in subfigures (a)-(h) can be found from Figure 2. and the plate was lost. This is evident from the F(L)-record. Another phenomenon observed in the simulations was a clockwise rotational motion of the rubble pile. In later stages, the location of the active failure prosess moved away from the plate (Figures 1g and 1h). Two velocities, 0.1 m/s and 0.05 m/s, were used in the simulations. In this range, the velocity did not affect the calculated F(L)-record or the final shape of the rubble pile profile. -181-
1800 1600 1400 simulation experiment (e) F [N/m] 1200 1000 800 600 400 200 (a) (c) (g) 0 (f) (h) (b) (d) 200 0 5 10 15 20 25 30 35 40 L [m] Figure 2: Horizontal force F acting on the plate as a function of the length of the pushed ice L as obtained from the simulation and experiments. The letters refer to Figure 1. DISCUSSION AND CONCLUSIONS Visual observations of the simulations and laboratory experiments were similar. In both cases cycles of rubble pile growth and collapse were observed. More quantitative data can be obtained through comparison of the rubble pile profiles and measured forces. Figure 3 shows rubble pile profiles during the simulation and Figure 4 compares the rubble profiles from the simulations and experiments. The shapes of the rubble piles from the experiments are similar with those achieved from the simulations. However, it appears that in the simulations the rubble piles were more loosely packed. The porosity ν p of the piles was in the range of 0.07...0.18 in the experiments and 0.31...0.35 in the simulations. Here porosity ν p is defined as (1 ν p )A = hl, where A is the cross sectional area of the pile as measured from 0.5 L=30 m L =40 m L =20 m L =10 m 0.0 0.5 1.0 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 Figure 3: Rubble pile profiles from the simulation with different lengths of the pushed ice L. -182-
0.5 0.5 0.0 0.0 0.5 0.5 1.0 1.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 (a) 2.5 2.0 1.5 1.0 0.5 0.0 0.5 (b) Figure 4. Comparisons of rubble pile profiles from laboratory experiments and simulations. (a) L = 14.6 m with an angle of inclination of 65 in experiments, and (b) L = 16.6 m with an angle of inclination of 50 in experiments. Data from simulations is presented with bold lines and experimental data with dashed lines. the cross sectional profiles. One reason for this difference in porosity may be edge crushing. During the experiments, edge crushing leads to formation of very small ice particles which may fill the pores in the rubble. In the simulations, the ice blocks failed only through bending, although an upper limit for the normal contact force is defined by Equation 4. The F(L)-records in Figure 2 show that the average force obtained from the simulation and experiments reached the same level of about 400 N/m. However, during the initial stages the F(L)-records look rather different. This is due to the loss of contact between the pile and the inclined plate during initial stages of the simulation. When the contact is lost, the simulated force drops to zero. This is caused by the 2D nature of the simulations and is thus not observed during the experiments. In the later stages of the simulation, when the contact between ice and the structure was not lost, the simulated F(L)-record is similar with the experimental data. The ice force obtained from the simulations can further be compared with results from analytical methods. A textbook solution to the horizontal ice force F h on a sloping-sided structure (Sanderson, 1988) divides the force into an ice breaking force F b and an ice ride-up F r force ( ) ρw gh 5 1/4 ( F h = F b + F r = σ b 0.68 sin α + µ ) iw cos α E cosα µ iw sin α ( sin α + µiw cosα + zhρ i g (sin α + µ iw cosα) cos α µ iw sin α + cosα ) sin α, (9) where z is the height of ride-up and the other symbols are as explained in Table 1. At the simulation event shown in Figure 1e, z = 0.75 m and the simulated force was about 1400 N/m. By using the values given in Table 1 and z = 0.75 m, a force F h = 1353 N/m is obtained by Equation 9. -183-
ACKNOWLEDGEMENTS The financial support from the Academy of Finland, the Finnish Cultural Foundation, and Heikki and Hilma Honkanen Foundation is gratefully acknowledged. REFERENCES Broek, D. (1991), "Elementary engineering fracture mechanics", Kluwer, 516 p. Evgin, E., Zhan, C. and Timco, G.W. (1992), "Distinct element modelling of load transmission through grounded ice rubble," Proc. of the 11th OMAE Symp., Calgary, Canada, Vol IV, pp. 273-279. Hopkins, M.A. (1992), "Numerical simulation of systems of multitudinous polygonal blocks", Cold Regions Research and Engineering Laboratory, CRREL, Report 92-22, 69 p. Hopkins, M.A. (1997), "Onshore ice pile-up: a comparison between experiments and simulations," Cold Regions Science and Technology, 26: 205-214. Hopkins, M.A. (1998), "Four stages of pressure ridging," Journal of Geophysical Research, 103, pp. 21,883-21,891. Hopkins, M.A., Tuhkuri, J. and Lensu, M. (1999), "Rafting and ridging of thin ice sheets," Journal of Geophysical Research, 104, pp. 13605-13613. Munjiza, A., Andrews K. F. R. (1998), "NBS Contact detection algorithm for bodies of similar size", Int. J. Numer. Meth. Engng., 43, pp. 131-149 Määttänen, M. (1986), "Ice sheet failure against an inclined wall", Proc. of the IAHR Ice Symposium 1986, Iowa City, USA, Vol I, pp. 149-158. Paavilainen, J. (2006), "Modelling of beams in discrete element simulations", To appear in the Proceedings of IX Finnish Mechanics Days. Parsons, B.L., Snellen, J.B. and Hill, B., (1987), "Preliminary measurements of terminal crack velocity in ice", Cold Regions Science and Technology, 13: 233-238. Saarinen, S. (2000), "Description of the pile-up process of an ice sheet against an inclined plate", M.Sc. thesis, Helsinki University of Technology, Dept. of Mechanical Engineering, 78 p. Sanderson, T.J.O. (1988). "Ice Mechanics, Risks to Offshore Structures". Graham & Trotman. 253 p. -184-