MAXIMUM EFFECTIVE PRESSURE DURING CONTINUOUS BRITTLE CRUSHING OF ICE

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1 MAXIMUM EFFECTIVE PRESSURE DURING CONTINUOUS BRITTLE CRUSHING OF ICE Devinder S. Sodhi 1 1 Retired from CRREL, Hanover, NH, USA ABSTRACT After presenting a brief review of the ductile and brittle modes of ice crushing against a vertical structure, the discussion is focused on estimation of maximum forces (or effective pressure) during continuous brittle crushing by the use of a correlation length parameter and statistical arguments. The maximum effective pressure is found to depend on the average effective pressure on a wide structure, the aspect ratio of structure width to ice thickness and the ratio of ice thickness to correlation length parameter. The plots of maximum effective pressure versus nominal contact area (structure width x ice thickness) show a large scatter, as reported in the literature on ice mechanics. However, the plots of maximum effective pressure versus aspect ratio fall on the same curve without any scatter, indicating its importance in plotting data on effective pressure resulting from continuous brittle crushing. KEY WORDS Ice forces; Effective Pressure; Ice crushing process; Continuous Brittle Crushing. INTRODUCTION To estimate ice forces on a vertical structure, one assumes that shear stresses imparted by the wind and the water on the top and the bottom of an ice cover are large enough to cause crushing failure at the ice structure interface. For the case of a free floating ice floe impacting a structure, it is also assumed that the momentum or the kinetic energy of the floe is large enough to cause full penetration of the structure into the floe. The resulting ice force on a structure depends on the mode in which ice crushing takes place. It is convenient to express the ice force in terms of an effective pressure p e, which is defined as the global ice force divided by the nominal contact area. In the case of edge indentation of an ice sheet against a vertical structure, the nominal contact area is defined as the product of the structure width -235-

2 and the ice thickness. If the actual contact area during an ice crushing process is less than or equal to the nominal contact area, the local pressure in the contact area is, respectively, more than or equal to the effective pressure. The local contact pressure and the area over which it acts are needed to design the external plating and the supporting frames of a structure, and the effective pressure is needed to estimate the global ice force for designing the foundation of a structure. Ice crushing takes place in either ductile or brittle manner depending on the rate of indentation. Ductile mode of ice crushing comprises creep and extensive micro cracking of ice at low rates of indentation, resulting in gradually increasing, or full, contact within the nominal contact area. In contrast, the continuous brittle crushing of ice is dominated by fracturing of ice at high rates of indentation, resulting in random, non-simultaneous, partial, short duration contacts within the nominal contact area. The rate of indentation of a structure into an edge of an ice floe depends on the speed of the moving ice feature as well as the compliance of the structure. It should be noted that the local or the global compliance of a structure should be considered in relation to the magnitude of the force that a moving ice sheet can exert on the structure. At intermediate speeds, an ice sheet moving against a compliant structure can cause alternating ductile brittle crushing, resulting in a saw tooth form of ice force records and severe structural vibrations. The effective pressure also depends on the aspect ratio, which is defined as the ratio of structure width to ice thickness. For ice moving at very low speed against a wide structure, an ice sheet undergoes creep buckling instead of ductile crushing, resulting in very low effective pressure. However, a fast moving ice sheet fails in continuous brittle crushing against narrow and wide structures. It has been observed during small scale and medium scale indentation tests that the ice deforms in a ductile manner at indentation speeds less than 1 3 mm s -1 (Sodhi et al. 1998, Masterson et al. 1999). Observations made during full-scale measurements of ice forces on wide structures in the southern Beaufort Sea have revealed that continuous brittle crushing takes place at indentation speeds higher than 100 mm s -1 (Blanchet 1998). For ice crushing at intermediate speeds, there is a possibility of alternating ductile brittle mode of crushing. It is difficult to specify the demarcations in ice speed for various modes of ice crushing, because these depend on the compliance of a structure, the ice properties and the ice thickness. To summarize the effects of relative indentation speed and the width of a structure, Table 1 lists possible modes of ice crushing against a structure. Table 1. Modes of ice crushing with respect to indentation speed and structure width Indentation Speed Width of Structure Narrow Wide Low Intermediate High Creep Indentation Creep Buckling Alternating Ductile Brittle Crushing Continuous Brittle Crushing -236-

3 CRUSHING ICE FORCES (a) Ductile Crushing During an interaction involving ductile deformation of ice against a narrow structure, the force between an ice sheet and a structure gradually increases, attains a peak value, and then gradually reduces to a steady-state value at 50 60% of the peak force without any structural vibration (Michel and Toussaint 1977, Sodhi 1991). During the time ice force increases gradually, the contact area between the ice edge and the structure also increases gradually to full contact of the nominal contact area. The range of effective and contact pressure for ductile crushing against narrow structures has been measured to be MPa in small-scale indentation tests. In the literature, there are two methods to estimate ice forces resulting from ductile crushing of ice: (1) Empirical method proposed by Michel and Toussaint (1977), and (2) Reference stress method proposed by Ponter, et al. (1983). Both these methods define an effective strain rate based on the ice speed and the structure width, and obtain the effective pressure from the uniaxial strength of ice at an effective strain rate. It should be noted that comparisons of the effective pressure from these two methods have only been made with the effective pressure measured during small scale and medium scale indentation tests. Because of buckling and flexural failures of ice sheet against wide structures at low ice speeds, it is not useful to compare theoretical values of effective pressure for ductile indentation with measured ice forces against wide strutures at low ice speed. (2) Brittle Crushing Continuous brittle crushing takes place when an ice sheet moves against a narrow or a wide structure at high speed (>100 mm s -1 ). The salient feature of brittle crushing is that the non simultaneous, short duration contacts at the ice structure interface are over a fraction of the nominal contact area (Kry 1978, Ashby et al. 1986). A series of such contacts were first described as line like contact in the nominal contact area (Joensuu and Riska 1989, Gagnon 1998). As an ice sheet advances towards a structure, the pressure at an individual contact area builds up to create a bulb of highly micro cracked ice and to propagate a macro crack. As each asperity is loaded, the local force increases until the ice sheet fails locally by some fracturing process, creating irregular surface and more asperities. Because of local loading and unloading, there is a large variation of local ice pressure. The process, as described above, is supported by measurements of contact pressure at the ice structure interface by tactile sensors (Sodhi et al. 1998, Sodhi 2001a). At an instant in time, summation of all local pressure over the thickness of an ice sheet and an infinitesimal width gives the local force f(t) per unit width of a structure, and summation of all local force across the structure width gives the global force g(t) on a structure. Because of random formation of asperities, there is no correlation of local forces, except in the vicinity of a point. Dunwoody (1991) proposed a formulation to relate the average (μ f ) and the standard deviation (σ f ) of the local ice force per unit width to the average (μ g ) and the standard deviation (σ g ) of the global ice force by assuming a negative exponential expression, exp( x /L), for the correlation function of the local ice force f(t), where -237-

4 x is the distance between two points along the width of a structure and L the correlation length parameter. The average global force μ g = wμ f, and the standard deviation of the global force σ g = σ f [2L{w-L(1-exp(-w/L))}] 1/2, where w is the structure width (Dunwoody 1991). A similar derivation of the average and the standard deviation of the global forces based on the model proposed by Kry is given in the Appendix. Assuming the global ice force during continuous brittle crushing has a normal distribution, an estimate of the maximum global force F max is (Sodhi 2001a): Fmax = Ar pwwh, (1) where w and h are, respectively, the structure width and the ice thickness, p w is the effective pressure for very wide structures (w/h>>1), and w h σ f L h L h = + h A L r e. (2) μ f h w h w Rewriting equation 1, an estimate of the maximum effective pressure p e is given by: p e = F max /(wh) = A r p w (3) The factor A r is to account for the effect of aspect ratio w/h, which was first reported by Afanasyev et al. (1972). The factor A r approaches [1+3(σ f /μ f )] for w/h<<1, and it approaches 1 for w/h>>1. Because A r approaches 1 for very wide structures, the corresponding effective pressure p w is independent of the structure width. Its dependence on the ice thickness, p w h α, is currently being debated, where α ranges from 0 to (Hardy et al. 1998, Sodhi 2001b, Timco and Johnston 2004). Full scale measurements of ice forces on wide caisson structures in the Canadian Beaufort Sea indicate that the value of p w is in the range of MPa for continuous brittle crushing, whereas medium-scale indentation tests with sea ice near its melting temperature give a value of p w in the range of MPa (Sodhi et al. 2001). The measured values of effective pressure indicate a dependence of p w on temperature. To estimate ice forces on bridge piers, Canadian Standard Association (CSA 2000) recommends four different values of effective pressure (1.5, 1.1, 0.7 and 0.4 MPa) to incorporate the effect of temperature and the ice condition. Recently, Timco and Johnston (2004) compiled data on measured ice forces on wide caisson structures in the southern Beaufort Sea, and found the average and the standard deviation of effective pressure p w for ice crushing to be, respectively, 1.09 MPa and 0.22 MPa, and they recommend these to be independent of the ice thickness. The results of small-scale and medium-scale tests indicate that the correlation length parameter L is of the order of one tenth, or smaller, of the ice thickness for continuous brittle crushing (Sodhi et al. 1998, Sodhi 2001a). However, more indentation tests with tactile sensors need to be conducted to ascertain the magnitude of L in terms of the ice thickness. Assuming a value of p w to be 1.5 MPa (which is approximately equal to the average plus two times the standard deviation of the measured effective pressure mentioned above), Figures 1 and 2 show plots of effective pressure, obtained from equation 3, with respect to nominal contact area wh and aspect ratio w/h for various ice thickness and the following values of the parameters: σ f /μ f =2 and h/l=10. For comparison purposes, Figures 1 also shows plots of effective pressure as proposed by Masterson and Frederking (1993), because their recommendations form the basis for estimation of ice forces in CSA (1992) and API RP 2N (1995). While the -238-

5 pressure area plots in Figures 1 have a large scatter for small nominal contact area wh, the plots of the same effective pressure with respect to the corresponding values of aspect ratio w/h fall along the same curve, as shown in Figures 2. These plots show that the pressure area plots of measured effective pressure are another manifestation of the aspect ratio effect inherent in continuous brittle crushing for various ice thicknesses. The expression for effective pressure, as given eq. 3, is also valid for small segments of a wide structure, giving an estimate of maximum local effective pressure over a small area (Fig. 1) for designing of external plates and supporting frames. For a narrow structure being acted upon by ice sheets, equation 3 gives lower estimates of the ice forces than those estimated from the pressure-area equation proposed by Masterson and Frederking (1993). Figure 1. Effective pressure versus nominal contact area wh for p w = 1.5 MPa, σ f /μ f =2 and h/l=

6 Figure 2. Effective pressure versus aspect ratio w/h for p w = 1.5 MPa, σ f /μ f =2 and h/l=10 CONCLUSIONS To estimate ice forces, it is necessary to consider the modes of ice crushing that are likely to take place at a particular location of a structure. As mentioned earlier, the ductile mode of ice crushing does not take place against wide structures because of creep buckling or flexure at low ice speed. For narrow structures, such as bridge piers in a river, ice fails in brittle mode because of high ice speed. In case there is a possibility of ductile crushing of ice against a narrow pier, a sloping face of a pier will eliminate such possibility. It should also be noted that, when ice moves against a narrow, sloping pier at high speed, it is possible for an ice sheet to fail in continuous brittle crushing. The above comments are for either low or high ice speeds. For ice moving at intermediate ice speed against a compliant structure, there is a real possibility of high ice forces to develop because of alternating ductile brittle crushing, as was experienced when a multi-year ice floe slowed down after impacting the Molikpaq structure in 1986 (Jefferies and Wright 1989). The expression for effective pressure, as given eq. 3, is also valid for small segments of a wide structure, giving an estimate of maximum local effective pressure over a small area (Fig. 1) for designing of external plates and supporting frames

7 REFERENCES Afanasyev, V.P., Dolgopolov, Iu.V., and Shvashteyn, Z.I. (1972) Ice pressure on separate supporting structures in the sea. Draft Translation 346, U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, New Hampshire. API RP 2N (1995) Recommended Practice for Planning, Designing, and Constructuing Structures and Pipelines for Arctic Conditions, 2nd Edition, American Petroleum Institute, API Publications, 1220 L Street N.W., Washington, DC Ashby, M.F., Palmer, A.C., Trouless, M., Goodman, D.J., Howard, M.W., Hallam, S.D., Murrell, S.A.F., Jones, N., Sanderson, J.O., and Ponter, A.R.S. (1986) Non-simultaneous failure and ice loads on structures. In Proceedings, Offshore Technology Conference, Houston, Texas, p Blanchet, D. (1998) Ice loads from first-year ice ridges and rubble fields. Canadian Journal of Civil Engineering, 25: CSA-S471 (1992) General requirements, design criteria, the environment and loads. Codes for the Design, Construction and Installation of Fixed Offshore Structures, Canadian Standards Association, Rexdale, Ontario, Canada. CSA (2000) Design of Highway Bridges, A National Standard of Canada, CAN/CSA-S6-88, Rexdale, Ontario, Canada. Dunwoody, A.B. (1991) Non-simultaneous ice failure. A report to Amoco Production Company, Tulsa, Oklahoma. Gagnon, R.E. (1998) Analysis of visual data from medium scale indentation experiments at Hobson s Choice Ice Island. Cold Regions Science and Technology, 28: Hardy, M.D., Jefferies, M.G., Rogers, B.T., and Wright, B.D. (1998) DynaMAC: Molikpaq ice loading experience. PERD/CHC Report 14-62, Klohn-Crippen, Calgary, Alberta, Canada. Jefferies, M.G., and Wright, W.H. (1988) Dynamic response of Molikpaq to ice structure interaction. In Proceedings, 7th International Conference on Offshore Mechanics and Arctic Engineering (OMAE), Houston, Texas, vol. IV, p Joensuu, A., and Riska, K. (1989) Contact between ice and structure (in Finnish). Laboratory of Naval Architecture and Marine Engineering, Helsinki University of Technology, Espoo, Finland, Report M-88. Kry, P.R. (1978) A statistical prediction of effective ice crushing stress on wide structure. In Proceedings, 4th IAHR Symposium on Ice Problems, Lulea, Sweden, p Masterson, D.M., and Frederking, R.M.W. (1993) Local contact pressures in ship/ice and structure/ice interaction. Cold Regions Science and Technology, 3(4): Masterson, D.M., Spencer, P.A., Nevel, D.E., and Nordgren, R.P. (1999) Velocity effects from multi-year ice tests. In Proceeding, 18th International Offshore Mechanics and Arctic Engineering Conference, St. John s, Newfoundland, Canada, OMAE99/P&A1127. Michel, B., and Toussaint, N. (1977) Mechanisms and theory of indentation of ice plates. Journal of Glaciology, 19(81): Ponter, A.R.S., Palmer, A.C., Goodman, D.J., Ashby, M.F., Evans, A.G., and Hutchinson, J.W. (1983) The force exerted by a moving ice sheet on an offshore structure. Cold Regions Science and Technology, 8: Sodhi, D.S. (1991) Ice-structure interaction during indentation tests. In Ice-Structure Interaction: Proceedings of IUTAM-IAHR Symposium, edited by S. Jones et al., Springer-Verlag, Berlin, p

8 Sodhi, D.S., Takeuchi, T., Nakazawa, N., Akagawa, S., and Saeki, S. (1998) Medium-scale indentation tests on sea ice at various speeds. Cold Regions Science and Technology, 28: Sodhi, D. S. (2001a) Crushing failure during ice-structure interaction. Journal of Engineering Fracture Mechanics, 68: Sodhi, D. S. (2001b) Absence of size effect in brittle crushing and breakthrough loads of floating ice sheets. IUTAM Symposium on Scaling Laws in Ice Mechanics and Ice Dynamics (Editor: J. P. Dempsey and H. H. Shen), Kluwer Academic Publishers, pp Sodhi, D.S., Takeuchi, T., Kawamura, M., Nakazawa, N., and Akagawa, S. (2001) Measurement of ice forces and interfacial pressure during medium-scale indentation tests. Proceedings, 16th International Conference on Port and Ocean Engineering under Arctic Conditions, Ottawa, Canada, Vol. 2, pp Timco, G. W. and M. Johnston (2004) Ice loads on the caisson structures in the Canadian Beaufort Sea. Cold Region Science and Technology, 38: Wright, B.D. and Timco, G.W. (1994) A review of ice forces and failure modes on the Molikpaq. In Proceedings, 12th IAHR Symposium on Ice, Trondheim, Norway, vol. 2, p APPENDIX To investigate the effect of non-simultaneous ice failure during brittle crushing, Kry (1978) assumed the structure width w to be divided into n segments of width Δ (i.e., w=nδ). The local ice forces are assumed to be perfectly correlated within each segment, but unrelated to the local forces in other segments. The average (μ p ) and the standard deviation (σ p ) of the global effective pressure are μ p =μ and σ p =σ/(n) 1/2, where μ and σ are, respectively, the average and the standard deviation of effective pressure in each segment (Kry 1978). Following the derivation given by Dunwoody (1991), the average of the global force μ g(t) =wμ f(t), which is the same result as given in the previous paragraph in terms of effective pressure. Assuming the correlation of local forces at any point on the structure to be perfectly correlated to local force acting within a distance of (+/-)Δ/2, the following expression for the standard deviation of global force can be derived: σ g(t) =σ f(t) (wδ) 1/2, which is the same result as given in the previous paragraph in terms of effective pressure. Assuming the global ice force to have a normal distribution, an estimate of the maximum global force during continuous brittle crushing is given by F max = μ g(t) + 3 σ g(t). Dividing both sides by the nominal contact area (wh), an estimate of the maximum effective pressure is given by: p e max = (μ f(t) /h)[1 + 3 (σ f(t) /μ f(t) ) {Δ/w} 1/2 ] = (μ f(t) /h)[1 + 3 (σ f(t) /μ f(t) ) {1/n} 1/2 ] As Δ/w approach approaches zero for a wide structure, the maximum effective pressure approaches μ f(t) /h