A series of full scale trials on the vessel Canmar Kigoriak and Robert Lemeur ...... Carstens, T., Loset, S., Lovas, S.M., "Ice Drift Modelling for the Barents Sea" ...
Claude Daley Kaj Riska
Review of Ship-ice Interaction Mechanics Report from Finnish-Canadian Joint Research Project No. 5 "Ship Interaction With Actual Ice Conditions" Interim Report on Task 1A
Report M-102 – Ship Lab, Helsinki University of Technology
Approved:
Petri Varsta Professor Otaniemi 1990-09-27 ISBN 951-22-0387-1 ISSN 0784-5650
ii
ABSTRACT A review of the technical literature concerning ship/ice interaction is presented. The context of the review is a joint Canada-Finland research project (JRPA No.5) to study ship interaction with actual ice. The constituents of ship/ice interaction are described. The literature concerning six ship/ice interaction scenarios is examined in detail. The six scenarios are; ramming a thick ice edge, impact with a single floe, contact with a thin ice edge, level icebreaking, ridge transit, and pack ice transit. Application of the technology to actual conditions is discussed, including ice load measurements and transit and risk simulations. The source and nature of randomness in ice loads is discussed. Directions for research are proposed.
ACKNOWLEDGEMENTS This work forms part a joint Finnish/Canadian research project. On the Canadian side funding and support for the work is provided by the National Research Council and the Canadian Coast Guard (Northern). In Finland, funding is provided by the Technology Development Centre (TEKES), the Academy of Finland and the Ministry of Trade and Industry. This funding support is gratefully acknowledged.
iii CONTENTS
1 INTRODUCTION .......................................................................................1 1.1 Overview ......................................................................................1 1.2 Ship-ice Interaction Scenarios .....................................................2 2 COMPONENTS OF ANALYSIS ................................................................4 2.1 Ice Mechanics and Failure ...........................................................4 2.2 Response of the Vessel ...............................................................10 2.2.1 Rigid Body Mechanics ....................................................10 2.2.2 Elastic Motions of the Ship .............................................12 2.3.3 Consideration of Vessel Failure ......................................15 2.3 Analysis of the Interaction ............................................................15 2.4 Hierarchy of Ice Load Analysis .....................................................17 3 METHODS OF ANALYSIS ........................................................................19 3.1 Ramming a Very Thick Ice Edge ..................................................19 3.1.1 Popov and Nogid ............................................................19 3.1.2 Vaughan's Ramming Models ..........................................21 3.1.3 Popov's and Vaughan's Methods Discussed ..................24 3.1.4 Riska's Investigations and Ramming Model ..................27 3.1.5 Other Ramming Investigations ......................................32 3.1.6 Discussion .....................................................................34 3.2 Impact With a Single Floe ...........................................................36 3.3 Contact with a Thin Ice Edge .......................................................36 3.3.1 Quasi-static Bending ......................................................36 3.3.2 Dynamic Bending ...........................................................39 3.4 Transiting Level Ice ......................................................................47 3.4.1 Background ....................................................................48 3.4.2 Ice Resistance Equations ...............................................49 3.4.3 Ice Resistance Calculation Methodologies .....................53 3.4.4 Experimental Methods ....................................................56 3.5 Transiting Ridges .........................................................................56 3.5.1 Field Measurements of Ridges ......................................57 3.5.2 Ridge Mechanics ...........................................................58 3.5.3 Failure Criteria ...............................................................60 3.5.4 Ship/Ridge Interaction Mechanics .................................63 3.5.5 Ship Transit in Ridged Ice ..............................................65 3.5.6 Field Measurements of Ships in Ridges .........................66 3.6 Transiting Pack Ice.......................................................................68
iv 3.6.1 General...........................................................................68 3.6.2 Ship/Pack Ice Interaction ................................................69 4 APPLICATION TO ACTUAL ICE CONDITIONS .......................................72 4.1 Ice load Measurements ................................................................72 4.1.1 Baltic Ice Loads ..............................................................72 4.1.2 High Arctic Ice Load Measurements ...............................75 4.1.3 Additional Ice Load Measurements on Ships .................77 4.1.4 Discussion of Measured Ice Loads .................................77 4.2 Transit and Risk Simulations ........................................................80 4.3 Discussion....................................................................................83 4.4 Directions for Research................................................................86 4.4.1 Ship Transit Research ....................................................86 4.4.2 Ice Cover Research ........................................................86 4.4.3 Ship/Ice Interaction Mechanics Research ......................86 5 CONCLUDING REMARKS ........................................................................87 6 REFERENCES ..........................................................................................87 6 REFERENCES
100
1 1 INTRODUCTION 1.1 Overview This report is a review of available methods to solve the mechanics of ship-ice interaction. The term "mechanics" is defined as "the branch of physics which seeks to formulate general rules for predicting the behavior of a physical system under the influence of any type of interaction with its environment". It is important to note that the interaction of the structure with its environment is a key part of all mechanics. The difficulty of creating mechanical models is directly dependent on the complexity of the environment. This review is intended to establish a starting point to develop methods suitable for tackling questions related to actual ice conditions. To date, ship-ice interaction mechanics has focused on simple ice geometries. Cases involving impact with an infinitely thick ice edge or a thin ice sheet of infinite extent have been analyzed. Cases involving single floes or single idealized ridges have also been analyzed, but with less success. Cases involving actual conditions have not been attempted. Even the description of ice conditions on seas is inadequate for ice load analysis. This review is the first task of a larger project titled "Ship Interaction with Actual Ice Conditions". A discussion of the aims and directions of the project is given in (JRPA No.5, 1989). The scope of this review was broadly set in this research plan. Since the publication of the plan the scope of this review has been further refined. During ice transiting, many different processes take place on a wide variety of length and time scales. The mechanics of ice crystal fracture and the ice cover formation and movement are examples of processes from small and large scales. Both these are present to a certain extent in ship/ice interaction. However, for this review, the term "ship-ice interaction" will be mainly confined to events on the scale of the vessel. The main focus will be on single events (single impacts or rams). The ice mechanics will be viewed in a general way. This means that an ice edge will normally be assumed to have certain indentation (or interaction) characteristics. The study of the origin of those characteristics is only lightly addressed here. A review of the ice indentation mechanics is given in (Det norske Veritas 1986). Goldsmith (1960) discusses the mechanics of colliding bodies.
2 1.2 Ship-ice Interaction Scenarios Arctic sea ice comes in many different forms as illustrated in Figure 1.1. The smallest ice features are much lighter than the mass of a ship, while the largest features are much larger than a ship. Contact with the ice may take place symmetrically on the bow, on the shoulders, amidships or on the stern. The interaction may continue with ice blocks deep below the waterline as in the case of transiting level ice or an ice ridge. The ship may be moving only straight forward (surge direction) during the interaction. It may be moving in a combination of pitch, roll, yaw, sway and heave as well as surge. The number of possible ship/ice contact scenarios is large. To limit this to a manageable and appropriate number, the following scenarios will be considered in this review (illustrated in Figure 1.2); - ramming a very thick ice edge (head on) - impact with a single floe (head-on or shoulder-on) - contact with a thin ice edge (head on) - level icebreaking (straight ahead) - transiting ridges or rubble (straight ahead) - transiting pack ice (straight ahead).
3
Figure 1.1 Ice Conditions Considered
Figure 1.2 Ship-Ice Interaction Scenarios
4 2 COMPONENTS OF ANALYSIS
2.1 Ice Mechanics and Failure When a ship strikes ice, forces and pressures are transmitted through the contact area. The forces developed during the contact depend on the mechanical response of the ice and ship. The mechanical response of the ice may be described using the mechanical properties of the ice which express the contact forces as some function of the nature of the contact surfaces. The analysis can be separated into three components as shown in Figure 2.1. This separation of components is useful because the interface can be characterized by relating the contact force to the average quantities of the contact area. Figure 2.2 illustrates the idealization for a ball contacting a semi-infinite solid. The nature of the contact is quite complex and as yet no comprehensive theories exist. A general idealization relates the contact force and the average pressure to the projected area. Local ice deformation is ignored in this idealization. A similar approach has been taken in analyzing the case of an ice edge as shown in Figure 2.3. The overall bending at the floe edge is considered, but the local contact is again idealized by some average pressure over the apparent area.
Figure 2.1 Components of Ship-Ice Interaction Analysis It was stated earlier that this review is primarily concerned with ship/ice interaction mechanics, rather than ice failure and contact zone mechanics. From this point of view it is the contact force related to some average description of contact area which is of interest. The contact force can be fully defined by the apparent area and the average pressure. The average pressure may be a function of the apparent area (and/or of time and ice properties). In this way the distribution of local pressures within the contact will be disregarded.
5
Figure 2.2 Idealization of Contact Process.
Figure 2.3 Idealization of Contact on an Ice Sheet Edge.
The simplest way of getting a value for the average pressure is to use the ice strength, Sc, as defined by a uniaxial compressive test. Unfortunately the contact process is quite different from uniaxial compression. Average values of impact pressure can be much higher or lower than, Sc, depending on the shape and size of the contact and on the time involved. Fig. 2.4 (Johnson and Benoit 1987) shows ice pressure on a spherical face being indented into iceberg ice. The average pressures drop to approximately Sc/4 for contact areas above 1.5 m2.
6
Figure 2.4 Pressure-Area Relationship for Mobil's Spherical Indentor Tests (Johnson and Benoit 1987). Normalized Pressure is Pressure divided by Sc, Amax is 3 m2 Another suggestion to obtain average ice pressures is to use the "specific breakup energy" (Kheysin et.al., 1973). The specific breakup energy is the crushing energy per unit volume that is expended during indentation. The equivalence of specific energy to pressure is seen as follows;
E Fx d = =P V Axd
(2.1)
where;
= specific breakup energy (MJ/m3) (crushing energy per unit volume) E = total crushing energy V = volume of crushed ice P = average ice pressure (MPa) (note: MJ/m3 = MN-m/m3 = MN/m2 = MPa)
Figure 2.5 (Jordaan and McKenna, 1988) shows a compilation of ice crushing energy data. Lines of equal specific energy are also shown (these being lines of equal pressure). It can be seen that the average pressures range over more than three orders of magnitude. For the Kheysin and Likhomanov (1973) data (small steel casings dropped onto ice) the average pressures were 2x to 6x the uniaxial strength.
7
Figure 2.5 Crushed Ice Volume vs Crushing Energy. The dashed lines represent constant specific energies. (Jordaan and McKenna, 1988) The concept of a "nominal pressure" Pnom was developed (Riska 1987) to reflect the ice strength in describing the contact force. It is defined to be the uniform pressure required to initiate failure of intact ice in the contact geometry under investigation. The nominal ice pressure corresponds thus to the maximum average pressure ice can sustain in each penetration geometry. Its value can be determined experimentally for each contact case, or by calculations using, for example, the TsaiWu failure criterion. To calculate the nominal ice pressure, it is necessary to have a complete failure envelope defined (Tsai-Wu for instance). For the calculations made, data collected from triaxial strength tests on multiyear ice was used (Riska and Frederking, 1987). Figure 2.6 illustrates a finite element model and results used to calculate the nominal ice pressure for a wedge indenter in multiyear ice. Failure numbers () were calculated by applying a unit pressure (1 MPa) to the contact face and then calculating the fraction of failure at every point in the ice. (The fraction of failure is the ratio of the length of a stress vector from the origin to the state of stress, to the length of a collinear vector from the origin to the failure surface). Figure 2.6 shows that the maximum was
8 .088. Since 1 MPa causes .088 of failure, the nominal pressure is 1/.088 = 11.3 MPa. Figure 2.7 a shows data from two types of indenter tests (one full-scale and one laboratory). In Figure 2.7 b all data points were divided by the nominal pressure (11.3 MPa in the case of the wedge (a ship's bow) and 4.1 in the case of an inclined plane). The calculated values of pnom agree well with the upper limits of experimentally measured pressures. The unexplained aspects of this approach is the decrease in pressure above some critical area.
Figure 2.6 Development of Nominal Ice Pressure.(Riska, 1987) A finite element model and the calculated failure numbers () are shown. ( represents the fraction of failure stress using the Tsai-Wu criteria). Three methods of obtaining a description of ice pressure for use in a ship/ice interaction analysis, have been presented. Unfortunately all are insufficient for the formulation of a general ship/ice interaction model, because they describe ice pressures and forces at best only partially. Also these methods are strongly dependent on empirical results. In fact each theory stems from a limited set of specific measurements. Other theories to explain contact ice forces include those based on ice damage (Jordaan and McKenna 1990, Santoaja 1990), finite crushing depth
9 associated with viscous flow of crushed ice and non-simultaneous ice failure in failure zones (Kry 1980). A comprehensive contact force theory based on physical principles is still lacking. Much work is currently underway on this topic. Some ideas in this direction are presented later in this review.
Figure 2.7a Measured Average Ice Pressure Values from Two Penetration Tests (Riska, 1987).
Figure 2.7b Average Contact Pressure from two test cases divided by the nominal ice pressure, plotted vs. the contact area. The lines drawn are least squares regressions. (Riska, 1987)
10 2.2 Response of the Vessel The response of the vessel includes elastic and plastic deformation and rigid body motions. These are quite straight forward to determine once the force is known. The reason for this is that methods available in the field of mechanics of elastic bodies, which can be employed when studying ship/ice interaction. Despite powerful tools, the mechanics of elastic bodies can be quite complex. Application of the complete theory of elasticity to real problems will present numerous computational difficulties. This results in a need to simplify the problem at hand. The question then becomes: how appropriate are the assumptions leading to the simplification and how accurate is the computed solution? In the next paragraphs the ship response is divided into two parts; rigid body motions and elastic deformations.
2.2.1 Rigid Body Mechanics The complete equations of motion for a general rigid body are;
F x = Mx F y = My F z = Mz T x = I x x z y y z T y = I y y x z x z T z = I z z x y x y where: Fx, Fy, Fz : force acting in x, y, z, directions Tx, Ty, Tz : torque acting about x, y, z axes (through center of mass of the body) M
: mass of the body
Ix,Iy, Iz : mass moments of inertia about principle (x, y, z) axes
x,y,z : acceleration of body in x, y, z directions
(2.2)
11
x , y , z : angular accelerations about x, y, z axes = angular velocity These complete equations are coupled for general rotations.This set of equations is also "body-fixed" meaning that the x, y and z axes are moving with the body. This means that the applied loads must be expressed in the moving frame of reference. In the above formulation all effects from water surrounding the ship which classically are treated as added mass, damping, and buoyancy terms are contained within the force term. For cases of ship-ice interaction it is usual to assume that the angular velocities () are small and therefore the above equations can be written without these coupling terms. The difficulty in the above complete set of equations (2.2) lies in treating the force. The ice force is a function of the ship motions, creating an interaction. If this function is nonlinear the solution of the equations requires numerical methods, even if the coupling of angular velocities is ignored. So far, the complete set of the equations has only been solved using strong assumptions. A rigorous set of equations is still to be developed and the influence of various assumptions demonstrated. In cases where the ice impact does not move significantly along the shell of the ship a further simplification can be made [Popov, 1968]. The ships motion and properties can be reduced to the motions of the point of contact along the shell normal. A reduced mass is defined as the force per unit acceleration of the point. Let us consider first the equations of motion for a point force on the ship shell. The ship response to an ice impact is usually a transient of short duration. In this case the damping and buoyancy terms may be neglected. Then once the x, y, z masses are defined separately so that added mass terms can be included if desired. The equations of motion are;
l F = M xx m F = M yy n F = M zz F = I x x F = I y y F = I z z
(2.3)
12 where; l, m, n = direction cosines of point of impact = lever arms from point of impact to principle axes (which together with F define the applied torques) The equations are uncoupled and can be solved explicitly for linear and angular accelerations due to F. These accelerations can be used to calculate the (normal) acceleration of the impact point. From this the reduced mass can be shown to be;
MR =
1 2
2
2
2
2
2
l +m + n + + + Mx My Mz I x I y I z
(2.4)
this allows the problem to be formulated as a single degree of freedom system, as;
F = MR
(2.5)
measuring the indentation into ice from the initial point of contact along the normal of the shell. The advantage of using the coordinate is that it describes the indentation into ice. The major drawback with this formulation is the neglect of the buoyancy terms. However this does not result in a large discrepancy compared with the solution including these (Niemela, 1990). When using equation(2.5), the nonlinearity of the contact force does not preclude an explicit solution. 2.2.2 Elastic Motions of the Ship During ship/ice interaction, the ship will deform. These deformations contribute to the interaction in that the resulting displacement in the contact area influences the ice force. The extent of the contribution varies from case to case. If the ice load is local and of relatively small magnitude, then only the deformation of the shell structure is important. Here a static analysis of stiffened plates is adequate. At small areas the deformation of the shell effects the ice pressure distribution. If the force is larger then the deflection of the whole hull beam must be considered. These deformations may be
13 analyzed separately but cases occur where they are coupled to the ice load. This demands numerical methods. The most detailed analysis of ship motions is performed using the finite element technique. The ship is idealized as an assemblage of structural elements which usually are shell and beam elements. The level of refinement is at the discretion of the analyst, as set by the target of the analysis and limited by the time and resources available to create the model. Numerous finite element routines are available for use. In certain cases the surrounding fluid can also be modelled. Hakala(1985) describes how fluid/structure and structure/structure models can be coupled to produce more general solutions. In the most general case the ship/ice interaction problem may be analyzed with; - Fluid Finite Element Model - Ice Finite Element Model - Global Ship Finite Element Model - One or more Local Ship Finite Element Models - Coupling Routine of all of above. In this way both local and global ship motions (and stresses) could be considered, as well as fluid and ice forces. Such a model represents the limit of what can presently be contemplated. Figure 2.8 illustrates this approach to the problem. Variations could include use of boundary elements to describe the fluid and/or distinct elements or discrete elements to describe the ice behavior. The ice behavior could also be described with a simple force algorithm. Concerning dynamic analysis of structures, it is usual to employ the concept of natural modes. Mathematically, modes are eigen functions which permit the structural behavior to be described by a small set of uncoupled equations. The complete solutions can be found by modal superposition (addition of modal responses). This approach works very well, but is based on the following assumptions; - uncoupled structural damping - small amplitude motions - linear elastic behavior - lowest few modes are dominant
14
Figure 2.8 Aspects of Ship-Ice-Water Interaction Modelling (Composite based on Hakala(1985) and Riska(1987)) The number of modes employed is at the discretion of the analyst. The decision is based on the level of accuracy of both force and response that is required. Normally only a few modes are required to accurately predict the ice force. Many more may be required to predict the stress response, particularly if local or semi-local effects (double bottoms or deck openings) are considered. The modal superposition method works well if the force is only time dependent. The interaction force, if a linear function of displacements, may be solved using eigen functions. However, a general nonlinear case requires numerical methods such as a direct time-step solution.
15
2.3.3 Consideration of Vessel Failure The ice force may be high enough to cause stresses above the yield limit, resulting in permanent deflections and ultimately failure of the ship. Some design philosophies consider localized yielding as acceptable in rare cases. Thus the response in the materially or geometrically nonlinear region should be considered in the interaction analysis. The difficulty here is that in ship/ice interaction it is usual to conduct the analysis without regard for failure. The results are then checked to see if failure would have occurred. Certain numerically intensive calculation methods will handle structural failure. The methods employ a direct time step integration procedure, rather than model superposition. The details of these approaches (some of which are found in commercially available programs) are beyond the present scope. The value of these methods is currently limited to global failure. This is because global ice penetration models exist which are suitable for analysis of ship/ice interaction including global (hull girder) failure. Local ice interaction models (i.e. models which accurately predict local pressures) are not as advanced. The question of pressure redistribution during deformation is insufficiently well understood at present to allow for consideration of local failure (ie shell rupture).
2.3 Analysis of the Interaction The analyses of both the ice indentation and the ship response include the contact ice force F. The contact force is the integral of the ice pressure p and surface traction p over the contact area. The understanding of the development of the contact force while ice is breaking is still lacking (c.f. chapter 2.1). Thus the first assumption in treating ice force is to assume the traction opposes motion and is px p where is the coefficient of friction. The second major assumption is to use only average quantities, either for the whole contact area or, if local structures are considered, for the structural area. The interaction scenario considered in this review are such that the local force distribution does not influence the interaction and thus the latter alternatives are not treated here. The calculation of the normal ice force is the solution of the interaction. Once the time history of the force is known, the ship and ice responses may be calculated in a
16 straight forward manner. The problem here lies in the fact that many tests have shown the ice force to be a function of the indentation depth of the structure into ice either directly or indirectly through the contact area. Thus the contact force is;
F = f((t),tt
(2.6)
where the dependence on indentation speed may come from potential strainrate, inertial or viscous effects. The indentation can be obtained from the relative displacement between the ice and ship using geometry. The dependence of contact force on the ship and ice response creates a true interaction between these. The solution of interaction equations may be obtained by dividing the indentation components, such as crushing squeezing and elastic indentation, and dividing the ship and ice motions into components such as rigid body and elastic displacements. Each of the components have an equation of motion. These can numerically be solved relatively easily by tying the components together with a kinematic condition (Riska 1989b). Rather than discussing the solution technique, some general aspects of analyzing the interaction are given here. The most important of the early analytical ship/ice interaction model is that of Popov (1968). In that case the problem was treated as a simplified two body collision. The masses of ship and ice were reduced for treatment as a single degree of freedom system where the buoyancy forces are neglected. The contact force is described as a simple function of the penetration depth. The resulting equation is (see Fig. 2.2);
2
d
MR
dt
2
a
= c c = -F (2.7)
where a is a constant depending on the ice edge geometry and c is a constant depending on ice and ship geometry. The first integral of this equation is directly obtained by the substitution v = . The result at the moment of maximum penetration max (v=0) is a+1
2 c c max v0 = M R(a+1) 2
(2.8)
17 where vo is the initial impact speed at zero penetration. Solving this for
max
we
get the maximum force as;
a+1
F max = c c max c c
a (a+1)
v 0 M R (a+1) 2 c c
(2.9)
This direct integration solution is possible because of the simplicity of the ice force equation and the omission of the damping and buoyancy. A more ambitious example of a direct solution is given by Vaughan (1986). Vaughan considered the rigid body and first flexural modes of the ship hull in a head-on ramming situation. As the buoyancy terms are included in the ship description some simplifications had to be made. Here the simplification is to omit ice crushing and require that the ship bow velocity changes direction at the moment of impact from forward to along the stem. This is to say that the ice edge is like a roller on which the bow does not bounce. Based on this kinematic assumption the equations of motion are integrated. The results are not fully satisfying, however, because the first impulse may cause high bending moments and yet when ice crushing is omitted, the impulse is not treated in a reliable manner. The limits of direct analytical solutions have not been yet probed. Especially there is a need to gain an understanding of which ship parameters influence the maximum hull response. Numerical solutions to all ramming geometries do not give that understanding because the direct influence of ship parameters is lost in lengthy calculations.
2.4 Hierarchy of Ice Load Analysis Analysis of ice loads may be divided into a hierarchy of levels. The main difference between levels is scale, i.e. how large a system is considered. The ice and ship interact on all levels, although the type of interaction varies greatly. At the level of transportation system design, there is interaction between the ice environment and the vessel design. Design parameters are optimized to produce the greatest return on investment. For icebreaking ships the return is determined only in a general way. The return may be measured by safety, efficiency, speed or money. The importance of proper ice load analysis varies depending on the measure-of-merit used. On this level the interaction is between design ice conditions and ship design parameters. Figure 2.10 illustrates this scale.
18 At the level of vessel navigation there is again an interaction between the ship and ice. The vessel is continually reacting to the ice situation and modifying the route. In this way the ship's specific ice environment is changed. Available ice information forms the critical issue at this level. When visual observations are inadequate, remote sensing systems are used. Interpretation of the remote sensed images becomes a crucial safety issue. On the scale of the whole ship, single impacts constitute ship/ice interaction. This will be discussed in detail in the following sections. This is the level on which ship/ice interaction is treated in the review. There are levels present at smaller scales in which structural parts or subsystems of ships are considered. For example on the scale of a panel of the shell plating the ship and the ice interact to determine the distribution of ice pressure.
Figure 2.9 Ship-Ice Interaction Viewed Economically.
Figure 2.10 Ship-Ice Interaction during Navigation. Ship and ice Interact in Routing Decisions
19
3 METHODS OF ANALYSIS This chapter will discuss the analysis of six types of ship/ice interaction. These cases are the basic constituents of broader analysis of ship-ice interaction in actual ice conditions. The emphasis in the following discussion is to pinpoint the deficiencies in present analyses, so that further research may be best directed at filling the gaps and avoiding pitfalls. 3.1 Ramming a Very Thick Ice Edge The case of a ship colliding onto a single thick ice edge has been of practical interest for several years. It is normally considered the limiting scenario for ships in very heavy ice. Total ice force, accelerations and hull girder bending will all reach maximum values in this scenario (see again Fig. 2.1). The case of impact with an ice wall,.ie a wall-sided iceberg, is usually disregarded as too unlikely. The thick ice edge may be a multiyear ice floe or an ice island. 3.1.1 Popov and Nogid The earliest comprehensive treatment of ship/ice interaction mechanics is found in (Popov et.al, 1967), which references the earlier papers (Nogid, 1959) and (Shimansky, 1938). The ship/ice impact was viewed as a case of two body collision, as illustrated in the Figure 3.1. Both ship and ice have, in general, six rigid body motions. The ice will crush at the point of contact and may bend locally as well. The problem was simplified by considering only motions on the line of impact i.e. in the direction normal to the ship's hull at the point of impact. In this way the problem was reduced to a single degree of freedom system as shown schematically in Figure 3.2. The kinetic energy in the direction of impact is consumed by the penetration into the ice normal to the impact. Popov's stated assumptions are; 1) ship is treated as a rigid body 2) ice floe is a thin plate 3) hydrodynamic aspects of impact may be treated only with added mass 4) vessel motions other than along the line of impact are ignored 5) friction can be ignored 6) ice crushing requires a constant energy per unit volume.
20 The reduced equations of motion which Popov derived are those discussed earlier (equations 2.5 and 2.6). The solution for the head-on ramming case can be found as follows; for a "wedge" bow (see Figure 3.3), force and penetration are related by; 2
FN =
N tan c 2
cos sin
(3.1)
where; FN = force normal to stem N = penetration normal to stem c = specific energy of crushing = stem angle (from vert.) = waterline angle (from fwd.) Substituting equation (3.1) into (2.7) and (2.9) the maximum normal force FNmax is;
max FN =
tan c 2
cos sin
R v R (3.2)
which is, not surprisingly, exactly the same as the equation derived by Popov for impact against an angular edge (page 37, *Popov, 1967). There is a problem in applying Popov's approach to the ramming situation. Assumptions 3, 4 and 5 are not valid in this case. The ramming case involves substantial motions tangential to the contact.
Figure 3.1 General Ship-Ice Impact Problem [from Popov et.al. 1967]
21
Figure 3.2 Ice Impact Problem in Reduced Coordinates
Figure 3.3 Geometry of Ice Notch for Head-On Ram by Wedge Bow. 3.1.2 Vaughan's Ramming Models In a series of two papers (Vaughan 1983, Vaughan 1986), an analytical model for ships ramming heavy ice was developed. In the earlier paper the ship was assumed to be a rigid body. In the later paper the flexural motions of the ship were included. Vaughan's approach is as follows (refer to Figure 3.4); - the ship strikes the ice edge with initial velocity vo - the ice edge is idealized as a point contact upon which bow will not bounce i.e. a rigid roller support with a coefficient of restitution of zero) - the size of the impact impulse is solved based on the angular and linear momentum balance using a kinematic condition requiring the bow to slide up the ice edge after the initial impulse - once the initial impact is complete, the ship slides up the ice until all kinetic energy is lost
22 In the case of the rigid vessel, Vaughan calculates the impact impulse which changes the movement of the bow as;
I T = M(1 + a h) v o
tan v cos v
(3.3)
where 2
(1 + a h) (1 + a h) s K=1+ tan v tan v + 2 (1 + a v) (1 + c m) k
(3.4)
Vaughan continues to show that much of the kinetic energy (90 % to 99 %) remains after the impact phase. The lost energy is an upper bound on the energy available for flexural motions, (see Fig. 3.5).
Figure 3.4 Ship Ramming Models as Idealized by Vaughan.The model assumes a very quick initial impact which serves to re-direct the bow motion from horizontal to a slide-up direction. This is followed by a beaching phase.
23
Figure 3.5 Percentage of Kinetic Energy Lost During First Impact. (from Vaughan, 1986)
Figure 3.6 Flexural Response of Ship During Ice Impact. (from Vaughan, 1986) When considering flexural motions Vaughan considers the first flexural mode of a free-pinned beam (Fig. 3.6). By including the flexural motion in the momentum balance equations, Vaughan shows that the impact impulse is significantly reduced compared to a rigid ship. The reduced impulse is of the order of 65 % of the rigid ship impulse. The flexural response to the impact is based on the application of a step charge in vertical velocity to the bow of the ship. This constrains the ship to behave as a pinned beam in free vibration. The maximum bending moment during the impact phase is given by the formula; max
M1
= .228 m L f 1 v o tan v
(3.5)
24 Vaughan's treatment of the beaching phase seems much less rigorous. The beached moment calculation includes an assumption about the trim at the end of the ram, xiz. the trim is sufficient to bring the fore foot to the water surface. This is an extremely conservative assumption.
3.1.3 Popov's and Vaughan's Methods Discussed Popov's method contains assumptions which make it inappropriate for the beaching phase of the ramming problem. It is interesting to note that, while Vaughan and Popov approach the problem of the impact some what differently, their results are almost identical. In Popov's case he calculates the normal impulse as;
I = MR v R
(3.6)
while Vaughan gives the equation;
I T = m(1 + a h) v o
tan v cos v
(3.3)
If friction is ignored (as Popov did), it can easily be shown (using equations 2.5 and 3.4) that;
m(1 + a h) cos v
(3.7)
v R = v o tan v
(3.8)
MR =
and
meaning that equations 3.6 and 3.3 are identical. The additional benefit of Vaughan's approach is the inclusion friction. Are Vaughan's assumptions regarding the ramming problem correct? His major assumption is the kinematic assumption that the bow motion changes direction immediately and then follows the ice edge. This contains the tacit assumption that the force during impact is greater than the beaching force, i.e. no further ice crushing occurs after impact, only sliding. The validity of this assumption can be investigated using the equations for impact and beaching.
25 Vaughan(1983) suggests an approximate design formula for the maximum normal beaching force; .9
1.2
F BEACH = .63 m v o
(3.9)
The general form of the impact force (neglecting friction) is; 2
v M (a + 1) F max = c R R 2c
a a+1
(3.10)
where;
vR = vo l MR = m C' 2
2
2
l n 3n C' = + + (1 + a s) (1 + a h) (1 + c ) m
(head-on impact)
l, n = direction cosines = cos, sin (head-on impact) 1.88 R c, a = , 1.5 1.5 l n (spoon bow) tan = ,2 2 ln (wedge bow) = stem angle (from vert.) = waterline angle (from fwd.) R = Radius of spoon bow (metres)
For the purpose of calculation assume as =.1, ah = .9, cm = .6, = .69 (i.e. k =.24L) and = 60. This results in l = .5, n = .866 and C' = 2.66. Further, assume that = 45(for wedge bow) and the R = 39 m1/3(for spoon bow). This leaves only the mass m and ice strength undefined. For the assumption of a roller support to be correct the following inequality must be true;
F max > F BEACH
(3.11)
Using the above assumptions and equation 3.9, 3.10 this inequality can be restated as; Wedge Bow
Spoon Bow
26
> crit
> crit .7
-.4
crit m v o
crit m
.58
(3.12)
The values of the critical ice strength which is the minimum ice strength necessary to fulfill the "roller" condition are given in Table 3.1 below(assuming vo = 6 m/s for a wedge bow).
crit
crit
MPa, Wedge
MPa, Spoon
7
5.2
6.3
40
18
18
150
45
38
mass 1000's Tonnes
Table 3.1 Values of critical ice strength necessary to fulfill the "roller"condition during ram. The ice strength as defined here is not the uniaxial crushing strength, but rather the average ice pressure over the entire imprint. The value for this p av will be in the range of 2 MPa, based on the MV Arctic ramming trials as discussed by (Riska, 1987). This suggests that even smaller spoon-bow vessels like the CANMAR Kigoriak will not fit Vaughan's assumptions. The presence of an old notch from a previous ram tend to increase the likelihood that the impact force will exceed the beaching force. To examine this possibility equation 3.10 can be restated including the size of the indentation; If EQ.(2.8) is integrated after the substitution v from vo to zero and from 1 to max where 1 is the indentations made by previous ram(s) on an intact ice edge. The first integral gives max(vo) and thus the maximum force is;
F max = c p av
(
2 +1)MR v o
2 c p av
+1
+1
+ 1
(3.13)
27 Assuming pav = 2 MPa and vo = 6 m/s, the minimum required indentations for equation 3.11 to hold are presented in Table 3.2;
mass 1000's Tonnes
min
Wedge min/1
min
Spoon min/1
7
1.5
.96
1.6
1.07
40
6.0
2.1
4.3
1.6
150
17
4.0
8.6
2.1
Table 3.2 Minimum Indentations Needed to Achieve the "Roller" Condition during a Ram (pav = 2 MPa). Table 3.2 indicates that for a small ship using the stated assumptions, a second ram may result in the conditions necessary to make Vaughan's "roller" assumption valid and thus making equation 3.5 for calculating moment valid. However, for larger vessels, and especially for larger vessels with wedge shaped bows, the validity of the "roller" assumption can be questioned. It is more likely that crushing and sliding motions happen together in large ships. These developments show that explicit solutions can shed light on the interaction between parameters and the validity of assumptions. 3.1.4 Riska's Investigations and Ramming Model A doctoral thesis published in 1987 (Riska 1987) addressed the topic of a vessel ramming very thick ice. The thesis is a description of the mechanics involved. In a number of subsequent papers (Riska 88, 89a, 89b) the discussion of the process is extended. The problem into the following components; - description of mechanical properties of multi-year ice leading to the presentation of a general failure criteria of Tsai-Wu type presented in Figure 3.7 - formulation of the ramming interaction to include ship rigid body and elastic motion, ice elastic motion and crushing penetration - discussion of the nature of the contact/penetration process - description of the concept and calculation of a nominal ice pressure based on the failure criteria and the nominal contact geometry presented in Figure 2.6 - comparison with experimental data and selection of empirical constants of pressure-area relationship for ice as presented in Figure 2.7 b - determination of linear hydrodynamic properties of the ship
28 - numerical solution of ramming equations - comparison of results with experimental measurements as shown in Figure 3.8
Figure 3.7 Tsai-Wu Failure Envelope (Riska, 1987)
29 Figure 3.8 M.V. Arctic Ramming Results, Comparing Experimental Values with Calculations. (Riska, 1987) Riska presented many results about the nature of the ramming process. Figure 3.9 shows the influence ice strength on the force time history. The ice strength varied from 75 kPa to 608 kPa. In weaker ice there is a only one peak, corresponding to the beached condition. In stronger ice the first peak is very pronounced and leads to dynamic magnification of the first mode flexural response of the ship.
Figure 3.9 The Influence of Model Ice Compressive Strength on Ice Impact Force. (Riska, 1987) Riska also presents the result that Froude scaling of ice strength leads to incorrect ice penetration at model scale. The nominal model-scale uniaxial ice strength should have been 125 kPa using Froude Scaling of 5 MPa full-scale crushing strength, and a scale factor of 1/40. As can be seen in Figure 3.10, the required strength (in F.G.Ice) was approx. 20 kPa, or 1/250th rather than 1/40th of the full-scale value. Figure 3.11 plots the major results of Riska's experiments i.e.; vertical ice force, hull bending moment, ice penetration vs velocity, showing data obtained in various ice materials. The influence of ice strength is visible.
30
Figure 3.10 Ramming Penetrations vs. Ice Strength in Model Scale for Two Ice Types. The arrows indicate the target values obtained from full-scale tests. (Riska, 1987)
Figure 3.11 Maximum Values of Vertical Force, Bending Moment, Ice Penetration vs Ramming Speed for the M.V. Arctic Model. (Riska, 1987)
31 Riska's model permits the calculation of the relative energy components as a function of time as shown in Figure 3.12 for one model scale ram of the M.V. Arctic. The figure shows how little energy is involved in the flexural motions (terms V i , i = 3 to 12 in the figure). The rigid body motions and the crushing/friction losses represent almost all of the energy. Nevertheless, it must be remembered that the flexural motions are of major importance for the safety of the vessel because they provide the critical hull stresses.
Figure 3.12 Variation of the Energy Components during a Ram of the M.V.Arctic Model. (Riska, 1987) In (Riska 89a) the influences of the bow angles (stem and water line), nominal ice pressure, pressure/area parameter and coefficient of friction are investigated for the M.V. Arctic and the SA-15 vessels. For the normal range of these variables they are shown to have very little influence on either total force or bending moment. In (Riska 89b) the influences of ice thickness and floe size are investigated. Figure 3.13 illustrates the influence of thickness and diameter on the vertical force and bending moment for the M.V. Arctic. Until the floe is quite large, i.e.10 times the vessel mass, there is a strong influence of floe size on the force and response. In concluding remarks the contact process is identified as the key remaining problem in a full
32 description of the single impact process. Cases involving multiple interactions, as in level ice breaking, are discussed as areas for further research.
Figure 3.13 Influence of Ice Floe Thickness on Maximum Vertical Ice Force and Hull Bending Moment. (Riska, 1989b) 3.1.5 Other Ramming Investigations Many other investigations has made valuable contributions to the understanding of the ramming interaction. A series of full scale trials on the vessel Canmar Kigoriak and Robert Lemeur was reported in (Ghoneim et.al. 1984). Full scale data and the results of a ramming model are described. Another set of full scale data is presented in (Tunik et.al. 1988). Data from some of the 39 impacts on the USCG Polar Sea measured in 1985 are presented. As well, a design oriented force equation is presented. The equation is based on an analytical model developed by Tunik and based on the work of the Soviet scientists Kurdymov and Kheisin. The work of Riska was based on a joint Canada-Finland research project (JRPA #3) to investigate the ramming process using a set of physical models. The writers were the initial principle investigators on the Canadian and Finnish sides. The first joint report of those tests is given in (Riska and Daley 1986). A second phase of testing involving dynamic models of the Kigoriak (in Finland) and a 150.000 tonne tanker (in Canada) has been completed. A report of the Canadian tests has been completed (Howard,Menon and Daley 1989). Figure 3.14 illustrates the behavior of the 150.000
33 tonne vessel during a 4 kt ram into very thick ice. Figure 3.15 shows a ram into an old imprint (the imprint left from the ram in Figure 3.14) It can be seen that the force is higher and rises faster in this second ram. If the process were quasi-static, then the moment would be a constant multiple of the force, i.e. Mstatic = k x Fstatic. This is the same as saying that the normalized
Force (kN) Mnor/Fnor
1.2 1 0.8 0.6 0.4 0.2 0 4.5
Mnorm/Fnorm (dynam. mag.)
Note Scale Difference 5
5.5
6
6.5
7
7.5
Moment (kNm) 1.2 1 0.8 0.6 0.4 0.2 0 8
Time (s) Force
Mnor/Fnor
Moment
Figure 3.14 Vertical Force and Bending Moment during a 4 kt. Ram of a 150,000 Tonne Tanker Model. (from Howard et. Al., 1989) The process is nearly quasi-static, with bending moment following the force quite closely. 2 Force (kN)
1.5
1 Bending Moment (kNm) 0.5 0 4
5
6
7
8
9
10
Time (s) Force
Moment
Mnor/Fnor
Figure 3.15 Force and Bending Moment during a 4 kt. Ram of a 150,000 Tonne Tanker Model into a old Notch (of Ram in Fig. 3.14). (Howard et. al., 1989) The
34 bending moment during the first peak is considerably amplified, while the later beaching is nearly quasi-static. moment devided by the normalized force (both divided by their values when the force is maximum and shown on the plots as Mnor/Fnor) would be 1 at all times. However the bending moments at the first peak in the second ram are clearly amplified compared to the initial ram. The report of the tests in Finland and a joint report summarizing the series will be produced during 1990.
3.1.6 Discussion Ramming very thick ice is a design and operational concern for vessels operating in Polar regions. A number of full scale trials, model tests and mathematical model studies have been performed. The important mechanics of the phenomena are quite well understood. This is to say that workable physical and mathematical methods exists which can determine ice loads and vessel response with quite good accuracy. However it is not yet possible to point to an analytical statement of the mechanics which can adequately predict loads and response for all ship types and ice conditions. Vaughan's treatment is partly correct but fails to treat, for instance, large ships adequately. Consider Figure 3.16 which attempts to illustrate the range of behaviors that can take place during repeated ramming on the spot. The tests have indicated that the type of behavior changes (qualitatively and quantitatively) dependant on a parameter which may be called the "hardness" of the ice. A qualitative description of "soft", "medium" and "hard" rams is as follows; Soft Rams - There is one overall peak, far both the ice force and bending moment. A small amplification of the moment during the penetration phase (a bulge in the leading edge of the force and moment curves) is seen. This is a case of nearly "quasi-static" beaching. Medium Rams - An initial peak in the force-time history appears, but is lower than the later beached peak. The bending moment experienced sufficient dynamic magnification during the first peak to approximately equal the later peak. Hard Rams - In this case first peak contains the maximum force and bending and ice followed by a loss of contact (force is zero for a time). The bending moment experiences considerable magnification.
35 The factors which contribute to the "hardness" of the ram are; basic variables:
which change:
- ship displacement
- pitch/surge period
- ship stiffness
- flexural period
- ice penetration strength
- rate of rise of ice force
- bow shape
- rate of rise of ice force
- size of old imprint
- rate of rise of ice force
A short small vessel with a full bow striking an old notch in hard ice will tend to experience a "Hard" impact (as in the case of the Kigoriak). A long, large vessel with a fine bow striking a square ice edge will tend to experience a "soft" impact. Z
force moment
"SOFT"
1st Ram
X
time Z
force moment
2ndRam
"MEDIUM"
X time
force moment
"HARD"
Z 3rd Ram
Bounce
Bounce
X
time Force & Moment Time History
Bow Position Plot
Figure 3.16 Typical Force & Moment and Penetration Records for "Soft", "Medium" and "Hard" Rams.
36 3.2 Impact With a Single Floe A ship striking a single finite ice floe forms on of the basic ship/ice interaction scenarios. It is the constituent of many, geometrically more complicated situations. Popov (1967) treated this problem by reducing the properties of the ship and the ice floe to a single impact impulse value:
I=
Mship x C
vo l 1+
Mship C
x
C Mice
(3.14)
where both the ship mass (Mship/C') and the ice mass (Mice/C") are accounted for. The calculation of the ice force is performed by equating the impulse (available momentum) to the integral of the force-time history, as described in chapter 3.1. This calculation will not show any qualitative differences between rams on finite or practically infinite floes. These exist as Figure 3.12 shows for head-on rams. There are also some recent results (Niemella, 1990) indicating that in the case of a glancing impact the mass of the ice floe has a large influence on the ramming process. Despite the fundamental importance of the case of striking a finite floe on the shoulder, it has not been actively investigated. 3.3 Contact with a Thin Ice Edge Contact with a thin ice edge arises in cases of level ice breaking and in maneuvering in pack ice or broken channels. This section will be limited to the case of infinite extent of ice. 3.3.1 Quasi-static Bending If the inertial forces in the ice plate or the water beneath the ice plate are ignored, the solution of the plate bending equations are quite straight forward. Popov (1967) used this assumption in examining the case of a ship's side striking the edge of a floe as illustrated in Figure 3.17. Popov presents a solution for this impact by first assuming that the ice sheet does not fail. This solution is similar in nature to the case of contact with very thick ice presented earlier, but is not of interest here. When considering the case of ice edge failure Popov employs the semi-empirical formulation for the force to break the ice;
37
.52 St h P max = sin
2
(3.15)
h is ice thickness. The empirical constant which was based on an empirical formula from Kashteljan (1960), and the assumption that in-plane and inertial forces are not significant. The problem of a quasi-statically loaded ice edge was the subject of a detailed review by Kerr (1976). Kerr discusses the approaches of several investigators, notably Nevel (1965). The fact that the ice sheet is floating on a fluid foundation and that it has a non-uniform through-thickness elastic modulus due to temperature gradient presents analytical difficulties. Kerr is skeptical of the simplicity of the Kashteljan equation, noting particularly the need to consider the wedging-in moments between the wedges formed by the radical cracks (Fig. 3.18).
n ••• de llisio i e n id io lS o l S oll is se •••C e s ss C Ve rin g Ve fo re Du Be Ic e Floe Before Collision
P cos Xu
X 1
Ic e Floe During Collision P
P sin
Figure 3.17 Ship Striking an Ice Edge (Popov 1968).
38
X
Y Radial Cracks
P Circumferential Cracks
b
Wedging Moments
Figure 3.18 Edge Loaded Ice Sheet (Kerr 1976). Kerr suggests an approximate equation for the failure load for each of the wedges considered; 1 2
b 3(1 - .9 E
Pf =
4
tan(
4
h
5
+
.9
2 h2 S f (3.16)
where is poissons ratio, is the specific weight of water and Sf is the flexural strength of ice. The equation can be further simplified to the form;
P
=A 1 + A 2 b lc Sf h 2
(3.17)
where A1, A2 are constants, lc is the characteristic length (D/)1/4 and D is the plate modulus;
D=
Eh
3 2
12(1-
(3.18)
Based on upper-bound test results, Kerr selects the constant to be;
P max
=.58 + .27 b 2 lc Sf h
(3.19)
39 It should be noted that this formulation drops the dependence on the angle The case of an edge loaded wedge with an inclined load (see Fig.3.19) was investigated by Riska (1980). Riska shows results which indicate that only a small error ( < 5%) arises when ignoring the horizontal component of an inclined load, for loads inclined less than 75° from the vertical. At 30° inclinations the errors are negligible.
P
Pv
Figure 3.19 Ice Wedge with Inclined Load (Riska 1980)
3.3.2 Dynamic Bending The question of dynamic bending of an ice edge by ship contact was investigated by Varsta (1983) in his doctoral thesis. The first part of the thesis deals with the development of the contact pressure. The contact was investigated empirically by crushing the rectangular edge of ice pieces. Figure 3.21 shows data from those tests. The pressures indicated are arrived at by dividing the total force by the apparent contact area calculated from the penetration. Varsta discusses the nature of the contact process in terms of "wet" and "dry" contact. Varsta states that it is assumed that up to c = 10 mm only solid "dry" contact occurs". If this were true then one would expect, in situations where everything else is similar, the "wet" and "dry" pressures for c < 10 mm to be equal. Figure 3.21 shows that this was not the case. The question then remains why the tests termed "wet" produce higher pressures. Many questions such as this remain to be answered.
40
Figure 3.21 Ice Pressures vs. Crushing Depth (from Varsta 1983). Regardless of the details of the contact, one can investigate the dynamic bending of an ice edge during impact. Varsta presents a detailed discussion of this phenomenon. Figure 3.22 shows the edge of an ice wedge as studied by Varsta. To examine the behavior of the ice wedge Varsta began by calculating the natural modes of the floating ice edge using a finite element model. He then solved the transient load cases by the piece-wise exact solution method, in a time step process. Figure 3.23 shows the contribution of the first 50 modes to the failure stress. Modes up to 40 are involved. Figure 3.24 shows the evolution of stress with time on a line perpendicular to the ice edge. Varsta converted the stress data to a failure reference number based on the Tsai-Wu failure criteria, such that failure occurred at the point where = 1. Parametric results of Varsta's numerical model are shown in Figures 3.25 and 3.26 The angle is the direction of the load (from horizontal). Full scale data is also shown in Figure 3.26. The source of the scatter was not explained, but may be due to natural variability in the ice cover.
41
Figure 3.22 Ice Wedge Geometry (Varsta 1983)
Figure 3.23 Participation of Ice Edge Flexural Modes in the Failure Stress. (Varsta 1983)
42
Figure 3.24 Development of Ice Edge Stress Field During Impact. (Failure Stress is Exceeded at 20.5 ms) (Varsta 1983)
Figure 3.25 Ice Edge Failure Load vs. Frame angle and Ship Speed. (Varsta 1983)
43
Figure 3.26 Ice Edge Maximum (Normal) Load vs Speed Compared with Measured Data (Varsta 1983) The main difficulties in Varsta's calculations are the treatment of the underlying water and calculations of the plate deflection. The problems in treating the hydrodynamics arise from the transient nature of the bending and the open water boundary condition. The first was solved assuming a steady state harmonic solution and the second by linearized high frequency boundary condition. A problem in treating the plate arises if the plate is assumed to be semi-infinite. The horizontal deflection due to the in-plane force has a singularity (is infinite) and the vertical deflection has a continuous spectrum. Both problems were avoided by summing the plate to be finite and fixed at the far end. The validity of these assumptions is difficult to assess but comparison with full-scale values shows reasonable agreement. The formulation above about ice load centers on ice pressure. To make the calculations applicable for practical designers an ice force force formulation may be obtained by fitting equations to Varsta's numerical results (Varsta 1984). Starting with the normal force;
F n = h c L p av
(3.20)
44 where hc and L are load height and length. Load height is given as;
hc =
7.1 1 + 1.5(v sin ) 0.4 h 1.7 c - 8.7
m (3.21)
and the ice pressure as;
p av = 2.3 L s
-.27
+4
MPa (3.22)
where s is the frame spacing and is the waterline angle. The pressure equation above corresponds to the estimated annual maximum in Baltic ice. A similar expression was found by (Daley 1985), resulting in the equation; 2
1.4 ST h Fn = sin + cos
v gh
.25
(3.23)
These two formulations are only slightly different because they are based on the same data set. A further corroboration of Varsta's results can be found from (St. John et.al. 1986), in which ice loads in level ice in the Antarctic are presented. Figure 3.29 shows normal ice force data for a range of ice thickness (.9 m to 1.8 m). Unfortunately, ice thickness for specific events are not known. It is known that there were a wide range of velocities for each ice thickness. The lines on Figure 3.27 are from equation 3.23 where the nominal ice strength of 400 kPa was used.. The equation clearly brackets most of the measured results. The reason for the few very high values (up to 2,5 MN) is not known, but may be due to variations in contact geometry. The variations may also be due to effects discussed by (Valanto 1989).
45
Figure 3.27 Normal Force vs. Level Ice Impacts in the Antarctic. The ice thickness ranged from .9 to 1.8 m. (from St.John et al. 1986) Lines represent Equation 3.23. Ice flexural strength was 400 kPa. Another group of researchers have employed finite element analysis of the problem posed by Varsta (Jebaraj et.al. 1988), and found very similar results. They employed 3-dimensional solid elements for the ice model and obtained slightly lower values for the load at failure. A recent publication investigates the ice load development during a twodimensional impact with an ice edge (Valanto 1989). Valanto focuses on the horizontal component of the ice force (resistance component) but presents as well values of maximum vertical force. The experimental arrangements is shown in Figure 3.28. Valanto extended earlier efforts by considering a number of new components in his investigations. His major extension of Varsta's work was to consider the mechanics of the broken ice slab (see Figure 3.30). This included modelling of the fluid above and below the slab, the contact with the ship, including consideration of the gap at the contact point, and the "hinge" between the broken slab and the intact ice. Valanto's investigations employed numerical modelling and physical model experiments. The lack of analytical modelling limits the generality of Valanto's work, but regardless, the work has shed light on a number of phenomena. Valanto gives a table of maximum values for horizontal (Fx) and vertical (Fy) forces. Assuming that these two forces occur simultaneously, values for maximum normal force (Fn) were calculated by simple vector addition. The bow angle was 15°, and the mean friction factor was about .05. Therefore the value for Fn were only a few percent lower than the Fy values. These
46 normal forces are plotted vs. velocity on Figure 3.29. Also in Figure 3.29 are Nondimensional values of normal force. By applying equation 3.16 (with = 0°) one can derive the dimensionless formula for quasi-static cantilever beam strength;
P f lc 2
ST h b
= .785 (3.24)
The left hand scale and the open diamonds ◊ show the data in non-dimensional form, with the quasi-static value indicated as a horizontal line. The data seems to indicate an approximately linear relationship between maximum normal force and velocity. The reason why this differs from the cusp situation described earlier which has a 1/4 power on velocity, is not clear. The explanation seems to lie in the fact that in Valanto's model, the higher speed impacts result in peak forces after the cusps are formed. The critical speed in model scale is about .2 m/s. Figure 3.30 compares Valanto's data with Eqn. 3.23. The values diverge at about .2 m/s, at which point Valanto says the peak values begin to occur after the cusp is formed. Valanto's work is therefore a logical extension of Varsta's investigations. It should be recognized that Valanto ignored one aspect of the problem which might be critical. He did not consider the effect of previously broken cusps. A submerged cusp puts an additional upward (and bracing) force on the cusps considered by Valanto. This might have a substantial effect on the peak loads. This must be left to later investigations.
Figure 3.28 Experimental Arrangement for Ice Sheet Impact Tests (Valanto 1989)
47 4.00
60.00
F l Fd = n c h2 w
3.50
50.00
3.00 Dimensionless Force Values F
d
2.50 2.00 1.50
40.00
Normal Force (N)
30.00
Fn
20.00
1.00 10.00
0.50
Quasi-Static Beam Load
0.00
0.00 0
0.1
0.2
0.3
0.4
0.5
Velocity (m/s)
Figure 3.29 Results from Impact between a Ramp Bow and a Cantilever Beam (Valanto 1989).
60 Valanto's Data
50
Vertical Force (N)
40 30 20 10
Range of Varsta's Data
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Velocity (m/s) Figure 3.30 Valanto's and Varsta's Results Compared. Peak Vertical Force During Ice Edge Breaking.
3.4 Transiting Level Ice Transiting level ice forms the classical case of icebreaking. The majority of ice navigation studies have concentrated on this. The reason for this is that conceptually,
48 level ice is easy to define, even though usually only a small fraction of a natural ice cover is level ice. This section will discuss icebreaking in level ice with particular emphasis on resistance. This field is still largely empirical and experimental. After a review of the early work on ice resistance, a list of ice resistance equations will be tabulated and discussed. Then the more complex and recently developed ice resistance calculation methodologies will be discussed. A short discussion on experimental approaches will conclude the section. 3.4.1 Background Runeberg (1889) is credited with being the first to study ships in ice in a rational fashion and develop a design formula for resistance. He viewed the problem as being a matter of breaking the ice with the bow of a ship. Thus he estimated that all the ice resistance arises in breaking the ice. It is notable that he explicitly considered friction in his formulation while leaving ice strength as an implicit variable. The next major development was made by Shimansky (1938), who proposed a semi-empirical method for determining the ice resistance qualities of ship. Shimansky realized that the ice resistance arises from forces all along the hull. He developed parameters which depended on the hull waterline and frame angles up to the maximum bean. Vinogradov (1958) is credited with developing the first theoretical method for analyzing ice resistance. He considered the problem of level icebreaking by repeated rams, and employed an energy balance formulation to compute resistance. Based on many years of work by several investigators in the Soviet Union, Kashteljan (1968) produced the first comprehensive discussion of ice resistance. He presented a semiempirical resistance equation which included components for breaking, submergence & turning of ice blocks and speed. His equation was based on extensive model scale tests of the Icebreaker YERMAK performed in 1958. The tests were performed in the world's first ice model basin constructed in 1955 at the Arctic and Antarctic Research Institute in Leningrad. Kashteljan's work began the modern efforts to quantify and understand the icebreaking process. Enkvist (1972) presented a detailed discussion of icebreaking and the issues involved in model testing. Lewis and Edwards (1970, 1972, 1976) developed methods based on extensive model testing, dimensional analysis and multiple regression. Recently these approaches have given way to resistance
49 calculations based on more detailed calculation of the components of the mechanics of resistance (Milano 72, Naegle 80, Kotras et.al. 83, Ishibashi 84, Luk 88). 3.4.2 Ice Resistance Equations Since 1889, investigations have been publishing formulas for ice resistance. The equations, tabulated in Table 3.3, have been getting more complex as time goes by. More terms have been used to describe the ship. It may be surprising that the ice description continues to be contained in ice thickness, strength, friction factor and density, all of which but thickness are normally assumed fixed. Ice resistance equations are often preliminary design tools and thus reflect the designers experience. These equations can not be mathematically derived from physical arguments. Table 3.3 shows the resulting scatter in terms included. These equations contain terms not defined here. The reader is referred to the original reference for the definitions. The Table is meant primarily to reflect the evolution of ice resistance equations Figure 3.31 (Bacher 1983) shows a comparison of most of these formulations for a particular ship. The range of resistance at 4 m/s among the equations is from about 670 to 1540 kN.
50
Figure 3.31 Comparison of Ice Resistance Equations (Bacher 1983)
51
Table 3.3 Ice Resistance Equations
52
Table 3.3 Ice Resistance Equations (Continued)
53
Table 3.3 Ice Resistance Equations (Continued)
3.4.3 Ice Resistance Calculation Methodologies The complexity of the icebreaking process has lead to the development of methodologies which require computer programs to solve. Table 3.4 gives a summary of four such methods, although several others are known to exist. Milano (1972) was the first to attempt an analytical calculation of resistance. His method required the determination of a number of energy components during the icebreaking cycle. Another approach was developed by Naegle (1980) and developed into the ship resistance program, SPLICE by Kotras et.al. (1983). The model is still based on the force component breakdown by Kashteljan, but contain much more detailed and complex calculation of the force values. Notably SPLICE was the first program to include a velocity dependent ice cusp size. SPLICE, for all it's complexity, is not a rigorous formulation/solution of a mechanics problem. SPLICE contains empirical coefficients which are optimized to fit with full scale measurements.
54 Ishibashi (1984) presents a quite similar methodology to SPLICE, but it is a time step solution of the equations of motion base on an assumed ice failure pattern. Luk (1988) presents an ice resistance model which only examines the icebreaking process to determine ice piece sizes and not the forces. The ice resistance calculation is then based on the momentum balance during the turning and submergence of the ice pieces. Carter (1983, 1987) describes an ice resistance model based entirely on the mechanics of icebreaking in which all effects after breaking are ignored. Carter's model examines the solid-ice interaction on the complete forebody waterline of the ship and determines resistance from the integral of the longitudinal components of the ice forces. Carter's and Luk's model explicitly simplify the icebreaking mechanics in exactly opposite ways, but both then rigorously solve the remaining mechanics problem and both claim good agreement with full scale data.
55
Table 3.4 Ice Resistance Calculation Methodologies
56 3.4.4 Experimental Methods Many investigators will still argue, as did Kasteljan (1968) and Enkvist (1972), that the icebreaking process is too complex to permit an analytical solution. Usually the quantities which influence resistance are selected based on simple physics, but with numerous empirical constants. In response physical modelling has been extensively used to examine ice resistance. Enkvist (1972) added greatly to the development of physical modelling methods. Edwards et.al. (1976) describe an extensive experimental program leading to an empirical ice resistance formulation. Problems of scaling have resulted in the on-going search for better ice models. The most usual model ice material is ice, although other materials have been tried. After saline ice was developed at Arctic and Antarctic Institute, the next development was the use of urea as a dopant (Timco 1981) and later the development of EGADS ice (Timco 1986). Saline, urea and EGADS ice are all seeded ice covers which grow as vertical crystals into the water column. The model ice is warmed after growing to weaken it down to a correct strength. Enkvist (1983) describes a new model ice developed at the Wartsila Arctic Research Centre in Helsinki. It is called FG (fine grained) ice and is produced by a spay of doped water onto the surface. A layer of wet snow (or slush) is formed which is brought to correct strength by cooling. A number of recent experimental investigation has attempted to better define the mechanics of the icebreaking process. Varsta (1983) examined the dynamic failure of an ice edge. Valanto (1989) examined the dynamic failure, turning and submergence of a ice edge in two dimensions. Ettema, Stern and Lazaro (1988a,b) have examined the dynamics of ship motion during level icebreaking. The subject of level ice resistance deserves much wider treatment than is possible here. The reader is referred to Jones (1988) and to the annual ITTC (International Towing Tank Conference) reports. 3.5 Transiting Ridges Ridged ice is one of the most common ice features, particularly in dynamic ice conditions. Ridges have been and continue to be the focus of much study. There does not exist any comprehensive theory of ridge formation or appearance. The published work on ridges can be divided into the following sub-topics; * Field measurements of ridge characteristics * Studies ridge-building and mechanical properties * Mechanics based ship/ridge interaction models * Studies of ship operations in ridge fields
57 * Field measurements of ships in ridges. The first two items are somewhat outside the scope of this review, so only a few recent references will be given. For the last three items the published work is relatively scarce and easy to cover. An attempt will be made to identify the state of development of ship/ridge interaction mechanics. 3.5.1 Field Measurements of Ridges There have been numerous published descriptions of the frequency and gross geometric characteristics of ridges. Much unpublished work is also known to exist. Two recent references will serve to illustrate the available data and lead to other references. Lepparanta and Hakala (1989) describe the structure of a number of firstyear pressure ridges in the Baltic. Figure 3.32 and 3.33 taken from the paper show both an overall cross section and details of the internal structure of a first-year ridge.
Figure 3.32 Cross-Section of an Ice Ridge in the Baltic (Lepparanta and Hakala 1989)
Figure 3.33 Detailed Cross-Section of an Ice Ridge Sail (Lepparanta and Hakala 1989)
58 Sayed and Frederking (1989) examined ridges in the southern Beaufort Sea. They present statistical description of ridge characteristics (Figure 3.34) and plots of ridge blocks sizes (Figure 3.35). Sayed and Frederking conclude the "Ice ridging is a complex phenomenon that remains poorly understood".
Figure 3.34 Cumulative Distribution of Ice Ridge Sail Heights (Sayed and Frederking 1989).
Figure 3.35 Histogram of Ice Block Thicknesses in Ridges (Sayed and Frederking 1989). 3.5.2 Ridge Mechanics Recently there has been considerable effort given to the study of basic ridge mechanics. This has mainly involved the study of the mechanics properties of ice rubble but has also included physical and numerical analyses of ridge formation. Table 3.5 summarizes most of the published material on the shear behavior of ice rubble. It is noteworthy that most investigations report ice as being a cohesive Mohr-Coulomb material with low but non-zero valves of cohesion and friction angles up
59 to 65°. Lepparanta, reporting on the only in-situ measurement, reports a 0° friction angle. This then results in the simpler Tresca failure criterion which is normally used in solids but not in granular materials. Wong et.al. dismiss the Mohr-Coulomb model as only a rough approximation of the true behavior. Source
Ice Rubble Characteristics
Test Results
Material
Temperatu Normal
Cohesion Friction AngleComments
re (°C)
(kPa)
Porosity
Pressure
(deg)
(kPa) Keinonen,
Blocks:
Nyman 1978
Saline Ice 0.37
Prodonovic 1979Blocks:
0.35 to
/
/
0.5 to 1.5 0.11
47
/
0 to 2.7
47 to 53
Saline Ice
0.26 to 0.58
(8:1) Weise et al. 1981 Blocks:
0.19 to
-4 to -20 0 to 28
1.7 to 3.4 11 to 34
5 to 25 mm/s
Ice chips /
0
0 to 4
5.8
54
described two
Commercia/
0
0 to 4
3.0
43
successive
Saline Ice 0.50 (4:1) Hellman 1984
l Ice
shear failure
Milled Urea/
0
0 to 4
+2
0 to 8
0
65
modes
Ice
Fransson and
Blocks:
0.2
0.8 to 1.0 12 to 21
Sandvist 1985 Fresh and
10 mm/s high scatter two
Saline (well
successive
graded)
failure modes
Wong et. al.
Angular
0.41 to
-0.4 to -1.6 51 to 140 No Clear Failure -
.01 to 0.46
1987
pieces of 0.51
Hyperbolic Stress/Strainmm/s (slow
Commer.
Relationship
speed)
1.7 to 2.4 0
0.5 mm/s 100-
Ice Lepparanta,
In-situ ridge .23 to .32 -1
Hakala 1989
keel
2.8
200 mm blocks
Table 3.5 Shear Properties of Ice Rubble
60 3.5.3 Failure Criteria Before continuing it is worthwhile to define three common failure criteria, those of Tresca, Von-Mises and Mohr-Coulomb. These criteria all are used to define a limiting state of stress, beyond which the material fails or yields. These criteria do not describe the post elastic behavior, which may be ideal fracture, ideal plasticity or some more complex type of deformation. The Tresca(1864) criteria is a pressure-independent failure criteria which is based on the assumption that failure results when the maximum shear stress within a material exceeds a critical value. In terms of principle stresses this can be expressed as;
MAX
1
2
3
y
or
y
y
(3.25)
where; 1, 2, 3
= principal stresses
y
= uniaxial yield stress
y
= maximum shear stress
Plotted on the principal stress axes (Figure 3.36) equation 3.25 represents a hexagonal surface, extending infinitely along the hydrostatic axis (1 = 2 = 3). On a Mohr's circle diagram the failure region is a horizontal line equal to the half the axial yield stress. This criteria was developed from the study of ductile metals. The discussion about failure criteria that are independent of hydrostatic stress leads to dividing the stress into two components; hydrostatic stress and stress perpendicular to it called deviatoric stress. Based on this division Von Mises (1913) developed a failure criterion on the assumption that deviatoric stress determines the failure. It ca be written as; 1 1 1 y
(3.26)
61 This equation can be seen (Fig. 3.37) to be a circular prism, extending infinitely along the hydrostatic axis. Physically, the Von-Mises criteria is defined by the maximum energy of distortion rather than a limiting stress. Energy of distortion is the elastic strain energy due to shear distortion, neglecting the strain energy due to hydrostatic pressure.
2
Fa ilure Regio n
3
2
Hydrostatic Axis
Fa ilure Surfa ce
1
1
3 =0 Failure Envelope
a) Mohr's Circle
3
b) Principal Stress Axes
Figure 3.36 Tresca Failure Criteria Plotted on a) Mohr's Circle and b) Principle Stress Axes.
Fa ilure Envelo pe Can not Be Dra wn on Mo hr's Circl e
2 Fa ilure Surfa ce
Both are Fa ilure States
3
2
1
Hydrostatic Axis
1
3 =0 Failure Envelope
a) Mohr's Circle
3
b) Pri ncipal Stress Axes
Figure 3.37 Von Mises Failure Criteria Plotted on a) Mohr's Circle and b) Principle Stress Axes.
62 Von-Mises' criteria is also used to describe the behavior of metals. Actual failure data usually lies between these two criteria. For general materials, the Mohr-Coulomb (Coulomb 1773, Mohr 1900) criteria is normally used. Coulomb, considering the strength of stone constructions, developed the criteria;
cp
(3.27)
which states the limiting shear stress is equal to a linear function of the normal stress. Mohr generalized this to the equation;
f (p)
(3.28)
Figure 3.38 illustrates the linear form of the Mohr-Coulomb failure criteria (really the Coulomb criteria). The failure envelop is not a prism but a six-sided pyramid, closed on the tension side, and aligned with the hydrostatic axis. Shown on a Mohr's circle, the failure envelope is a sloping curve. A comprehensive discussion of these and many other macroscopic failure criteria can be found in (Paul, 1968)
Figure 3.38 Linear Mohr-Coulomb Failure Criteria Plotted on a) Mohr's Circle and b) Principle Stress Axes.
63 3.5.4 Ship/Ridge Interaction Mechanics There is only one published ship/ridge interaction model (Keinonen, 1979) based on ridge mechanics. However, Mellor published a description of ship resistance in thick brash ice(Mellor, 1980) which is a very similar problem to the unconsolidated ridge fields which Keinonen considered. Keinonen approaches the ship/ridge interaction by making the following assumptions: - The ridge is a floating sheet of granular material, uniform in thickness and material properties (i.e. a uniform ridge field). - Ridge material follows the Coulomb failure model with
= .011 + tan 47° p
(3.29)
where and p are the average shear stress and average normal stress on the failure plane - Ice rubble density is constant (before and after transit of ship) - The ship is idealized with either a landing craft bow or a simple inclined wedge. Aft of the bow the body lines are rectangular. - Only low velocities are considered. - No inertial forces are considered. - The ice is assumed to continually form a failure plane at the point of contact between the top layer of the ridge and the ship's bow. Keinonen paid particular attention to the development of the ice rubble profile around the vessel. Figure 3.39 shows the development assumed by Keinonen for a landing craft bow. The actual profile is needed so that the normal stresses on the failure plane can be determined, and so the frictional forces on the bow, sides and bottom can be determined. Keinonen referred to his method as giving "pure ridge resistance". He presumed that either the entire ridge was unconsolidated (which is usually only true when the ridge or rubble field is active), or that the resistance due to the consolidated layer could be analyzed. He stated that the pure ridge resistance is the largest part of the resistance. His equation for pure ridge resistance has the components; RR = R + R + R+ RpB + RpS
(3.30)
64 where; RR
= total pure ridge resistance
R
= upper shear plane force
R
= lower shear plane force
R
= end shear plane force
RpB
= bottom frictional resistance
RpS
= side frictional resistance
The equations for each of the terms would be too long to present here. Keinonen's comparisons with full scale data are particularly disappointing. He concludes that the theoretical values need to be multiplied by .51. A number of Keinonen's assumptions must consequently be questioned. The most significant are these: - Separation of "pure ridge resistance". There seems little justification for the assumption that pure ridge resistance and resistance due to the consolidated layer do not interact. The failure plane patterns would surely be affected by the thick layer of solid ice. The assumed smooth flow of granular material would be affected by large solid blocks. The breakup of the consolidated would likely be quite different from the breakup of a floating ice sheet. - The continuous formation of shear failure planes. Would not these failure planes form intermittently, between which some other type of material distortion would take place (as would happen in sand)? - Coulomb behavior. While this was probably the only reasonable assumption, one must question its validity, or at least hope for field verification measurements. The relatively large block size in rubble leads one to question the soil analogy. Mellor, 1980, was concerned with the problem of Great Lakes bulk carriers in brash ice. He also employed the linear Coulomb failure criteria. Mellor's analysis dealt primarily with vertical or near vertical at the waterline, even at the bow. He thus determined total resistance from the normal and frictional force on the bow and the frictional force aft of the bow. Mellor employed somewhat more advanced soil mechanics principles that did Keinonen. He considered both active and passive pressures, earth pressure coefficients, and dealt with the buildup of rubble above the
65 waterline and ahead of the vessel. Mellor presented no full scale or model observations, but his work is nevertheless thought provoking, as it follows sound physical reasoning. Ship/ridge interaction mechanics is an area still demanding a great deal of work, which must begin with some form of accurate field observations of ridge characteristics and ship/ridge interaction tests.
Figure 3.39 Development of Pure Ridge Resistance (Keinonen 1979) 3.5.5 Ship Transit in Ridged Ice A number of investigators have studied vessel operations in ridged ice (e.g. Tunik, 1984, Lee and Wang, 1987). The situation is illustrated in Figure 3.40. The problem is idealized as a series of simple ridges in an otherwise level ice cover. Two simple resistance equations are determined for the level ice and ridges. With a thrust equation and the vessel mass, it becomes possible to determine if the vessel will be able to move continuously or need to back and ram. Speed made good can then be calculated as a function of the few ship and ridge parameters. These models are useful for transit analysis, but lack a rigorous foundation. The major uncertainties lie in the true ridge geometry, frequency and lack of knowledge of ship/ridge interaction mechanics.
66
Figure 3.40 Ridge Transit Position During Continuous and Back-and-Ram Operations 3.5.6 Field Measurements of Ships in Ridges There are few published records of ship/ridge interaction. Figure 3.41, from Makinen et al (1975) (reproduced in Keinonen, 1983), shows a detailed record of the progress of an icebreaker through a large Baltic ridge. Keinonen, 1983, refers to a number of unpublished reports by private companies of studies of ships in ridges. These may be expected to become public in time.
67
Figure 3.41 Ship Progress in a Baltic Ridge. (Makinen et al. 1975) A recent report of trials of the icebreaker ODEN (Johansson and Liljestrom, 1989) discusses the performance of various icebreakers in ridges. Based on 25 tests of the ODEN in ridges of thicknesses from 7 to 14 m and for ship velocities of 1 to 5 m/s, the authors present a ridge resistance equation for the ODEN (based on linear regression of the points) as:
R = Rcons + Rridge
(3.31)
where Rcons is the resistance in consolidated ice (= c1+c2 V"), Rridge is the pure ridge resistance ( = (.71 - .075V) Hridge ), c1, c2 are constants, V is velocity (m/s) and Hridge is the average unconsolidated ridge rubble thickness The intriguing aspect of this data is the decreasing ridge resistance with increasing velocity and that the thickness of the consolidated part of the ridge does not influence the resistance. The authors report a further decrease of 40% in resistance when the heeling system is employed (this only at one speed, 2.5 m/s). No explanation of this phenomenon is presented. One can only speculate that the velocity and heeling tend to disturb the ice rubble and reduce its shear strength. This phenomenon is found in some types of soils, particularly saturated clays with high
68 porosity (Sowers and Sowers, 1970). Catastrophic loss of strength can result in these soils as a result of shock or vibration. These phenomena are, unfortunately, quite difficult to reproduce in the laboratory as they are dependant on the detailed internal structure of the material. This is an avenue of further research.
3.6 Transiting Pack Ice 3.6.1 General The term "pack ice" represents a broad range of ice conditions, from a very open distribution of free floes to a compact mass of separate floes under pressure. It is therefore quite difficult to state a particular ship/ice interaction scenario for pack ice. Investigators have considered the following types of scenarios: - repeated single finite floe impacts - wedging impact between two floes - multi-floe impact including floe-floe interaction - treatment of pack as granular material in contact with the ship. The case of single floe impact is discussed in an earlier section. Wedging is normally treated by the same methods as single floe impact, but with lateral motion restricted. This section will review methods for treating floe groups. The constitutive properties of pack ice have been the focus of much work which is directed at understanding the large scale behavior of ice covers. For instance (Coon 1972) examined the behavior of compacted ice floes, and proposed a thickness dependent failure envelope for pack ice. The failure envelope consists of a Coulomb yield criteria, together with a circular compressive stress limit. The stress limit was determined from considerations of the overturning stability of small ice floes at stresses much less than crushing failure. A more recent treatment of ice field mechanics can be found in OstojaStarzewski et al, 1986. In that work the ice field was considered as an ensemble of interacting ice floes with random physical and geometric properties. A visco-elasticplastic model is proposed to treat floe interactions. The paper proposes an approach to sea ice dynamics but does not present solutions.
69 3.6.2 Ship/Pack Ice Interaction Analytical investigations of ship/pack ice interaction have only recently been begun. Scale model investigations have been performed (Nogid, 1959; Chu, 1974), which have resulted in empirical formulations. Unfortunately these have lacked any but the simplest basis in mechanics. Vinogradov (1986) discusses a simulation methodology for ship/pack ice interaction. The methodology deals with contact between a ship and a cluster of floes of arbitrary (random) size and shape, illustrated in Figure 3.42. The floe cluster geometry is represented by a tree of vectors. In this way the equations of motion for the whole can be written in a compact form for relatively easy solution. A number of rather restrictive assumptions are made: - ice floes are not damaged i.e. are not crushed, nor split - ice floes stay in plane i.e. are not tipped nor rafted - vessels are wall sided. Vinogradov, at the end of the paper, states that the methodology was then being implemented. New finite element techniques referred to as discrete and distinct element methods have been developed to analyze multi-body interaction problems, including consideration of the breakup of one element into many. These methods appear to have some application to ice mechanics. Hamza (1989) gives an overview of the use of a two-dimensional distinct element program (DEM2D) to calculate the interaction between a ship and pack ice. The three-dimensional problem was solved by the use of three two-dimensional partial models. Many of the details of the method are omitted, with reference made to an unpublished report (Hamza, 1988). Aboulazm and Muggeridge (1989) present an analytical model for ship resistance in pack ice. The model (Figure 3.43) makes the following assumptions: - pack ice is modelled as a set of discrete ice floes - resistance arises from energy loss due to collision with individual floes - floes do not interact (all are initially free floating and stationary) - ice/ship collisions can be modelled as having a fixed coefficient of restitution - ship is wall sided (classic impact mechanics) - impact is within waterplane (no tipping of floes)
70 Based on these assumptions the authors present a resistance equation: 2
2
R i = C ihV sin (B + d) 2
2
2
(1 +k) x 2 2
2 r csc - (cot - (1 + e)) + e (1 + e) R
(3.3
2) where; C is ice concentration (0...1), d = 2r is floe diameter, k is ice floe shape factor = a/d2, a is floe area, e is coefficient of restitution, R is floe radius of gyration
71
Figure 3.42 Pack Ice Floe Cluster Geometry (Vinogradov 1986)
Figure 3.43 Pack Ice Resistance Model (Aboulazm and Muggeridge 1989) Using a friction coefficient of .2, a coefficient of restitution of .2 and an added mass coefficient of 1, a floe diameter of 2 m and a thickness of 1 m, the authors calculated resistance values which compared favorably to published values for the USCGC KATMAI BAY in brash ice, showing the classical quadratic dependence on speed. Since the assumptions limit the model to application in very open pack ice, full scale verification is still lacking.
72 4 APPLICATION TO ACTUAL ICE CONDITIONS The goal of this study is to develop methods of analysis suitable for actual and not idealized ice conditions. The analysis methods discussed in the earlier sections refer to idealized ice conditions. This section will discuss aspects of current knowledge which are closest to actual ice. Ice load measurements are seldom in ideal ice conditions. In some cases measurements have been made during an actual transit i.e. not during a test. The second topic will be transit and risk simulation models which attempt to treat a full range of actual conditions. A discussion of the random nature of actual ice and directions for research will complete this section. 4.1 Ice load Measurements Attempts to make direct measurements of ice loads began in 1963 in the Soviet Union (Likhomanov in Kheysin and Popov, 1973). These early efforts were surprisingly ambitious, even by present standards. Likhomanov describes a system of 40 strain gauge channels applied to the motorship "Olenegorsk". Tests were conducted in both first year and multiyear sea ice. Shell plating stresses due to local ice loads were found to follow an exponential distribution in all cases. In North America similar efforts began in the late 1960's with the instrumentation of the MANHATTAN during its 1969 voyage. During the 1970's a number of Canadian and United States Coast Guard icebreakers, and one cargo vessel, the MV ARCTIC, were fitted with strain gauges in an attempt to determine ice loads (for instance, Levine et al, 1973; German and Milne,1973; Edwards et al, 1981; German et al, 1981). All of these systems were similar in that they were primarily stress monitoring systems. Their ability to measure true ice loads was quite questionable. Only after assuming most, and sometimes all, of the ice load's geometric properties could the magnitude of the load be found. Even then, many anomalies arose during analysis of the strain gauge data. Later systems were capable of measuring actual ice loads. 4.1.1 Baltic Ice Loads The first direct measurements of ice pressures were accomplished onboard the MT IGRIM in March 1978 (Korri and Varsta, 1979). The ice pressures were measured on four specially constructed ice pressure gauges which were later often referred to as Varsta gauges. These gauges were 180 mm diameter diaphragm type pressure sensors, were the deflection of the diaphragm is directly and linearly related to the ice pressure. Figure 4.1 shows the distribution of ice pressure maxima for three of the gauges.
73 In 1978 measurements of ice pressures were made on the Icebreaker SISU (Vuorio et al, 1978). The instrumentation consisted of three channels to measure stress on the inside of the shell plating and two channels for two of the ice pressure gauges mentioned above. The ice pressure gauges were 200 mm in diameter and capable of accurately measuring ice pressures over the sensing area. The system was operated for two and a half months on an almost continuous basis. Results for one of the ice pressure gauges, the distribution of daily maxima, are shown in Figure 4.2. The data was clearly quite statistical, however it did not easily fit any of the usual extreme value distributions. The measurement program on the SISU was later extended to include loads measured on frames (Riska and Varsta, 1983; Kujala and Vuorio,1986). Both short term and long term statistics were presented. The 1986 report concluded that "the stochastic nature of the ice induced loads is clear. The measured peak amplitudes follow the exponential statistical distribution correspond fairly accurately to the calculated estimates with the Gumbel asymptotic distribution" In a follow-on project to the SISU work, the chemical tanker M/S KEMIRA was instrumented in 1985 to measure ice loads. Measurements have been conducted for five successive winters (Kujala, 1986; Gylden and Riska, 1989), on the regular commercial voyages of the KEMIRA. Figure 4.3 shows the ice load on one of the frames (12 hr maxima) plotted versus return period. The database of ice loads data in the Baltic is quite extensive. Kujala, 1989a, has performed extensive extreme value analysis of much of this data to examine the influence of different environmental parameters such as ice thickness, geographic region on the ice pressure and frame ice load statistics.
74
Figure 4.1 Ice Pressure Maximum Values from M.T.IGRIM (Korri and Varsta 1979)
Figure 4.2 Ice Pressure Daily Maximum Values from I.B.SISU (Vuorio et.al. 1978)
75
Figure 4.3 Ice Frame Load, 12 Hour Maximum Values from M.S.KEMIRA (Kujala 1989b) 4.1.2 High Arctic Ice Load Measurements The first direct measurements of ice pressures in the high arctic were accomplished with the use of an array of small diameter pressure sensors (7.8 mm diameter) on the LOUIS S. ST. LAURENT in 1980 (Glen et al 1981). Individual pressure peaks as high as 53.3 MPa were recorded on the ST. LAURENT. The measurements on the ST. LAURENT were of little immediate use to designers. The pressures were measured over areas too small to be of significance to the shell plating. Furthermore, there was no theoretical model available then which would account for the very high local pressures. The next step was the instrumentation of the Canmar KIGORIAK in 1981, in a manner very similar to the SISU (Dome, 1982 - unpublished report). Like the SISU, the KIGORIAK system derived ice load from shear differences in the webs of a number of adjacent frames. This system was capable of determining the shape and intensity of the ice load over an area of several square meters. This data has not yet been made public, although some data is given in Ghoneim and Keinonen, 1983 (Figure 4.4). In 1982, a program to measure ice loads on the USCGC POLAR SEA was begun. An array of 80 channels of strain gauges installed on 10 adjacent frames
76 allowed the contiguous measurement of ice pressures over an area of 10 m 2 with a spatial resolution of .15 m2. The system was deployed during the years 1982, 1983 and 1984 in the Beaufort Sea, Bering and Chukchi Seas, as well as in McMurdo Sound in the Antarctic. The system employed solely digital data collection, which permitted the immediate conversion of strain readings to ice pressures (Daley et al, 1984; Daley et al, 1986). A total of 3680 impact events were recorded in a variety of ice conditions. Figure 4.5 shows distributions of ice pressures for three different ice type groups.
Figure 4.4 Ice Load Measurements on Kigoriak (Ghoneim and Keinonen 1983)
77
Figure 4.5 Ice Pressure Measurements from USCDG Polar Sea (Daley et. al 1986) 4.1.3 Additional Ice Load Measurements on Ships Two other projects deserve a short description. In 1984 the German vessel POLARSTERN was fitted with two ice load measuring devices (Hoffman, 1985). The ice load cells were fitted into pockets constructed in the ship's bow and shoulder. The unique feature of these load cells was their capability to measure the tangential (frictional) as well as the normal component of the ice load. The system was deployed off the Canadian Labrador Coast in mixed first year ice conditions, and later (Muller and Payer, 1987) around Spitzbergen in multiyear ice. The friction coefficient was seen to be inversely dependent on the normal load, and ranged from .05 to .25. The low friction coating was said to have steadily eroded during the Labrador tests. Concern for the ice worthiness of small vessels led to installation of an ice loads measuring system on the Canadian fishing trawler CAPE BRIER (Daley et al, 1988). An array of 35 strain gauges was used to measure the shear in the webs of 5 adjacent main frames in the bow of the vessel. All measurements were analyzed statistically, and are reported. 4.1.4 Discussion of Measured Ice Loads All field measurements of ice loads show considerable scatter. Normally this scatter is dealt with by fitting data to known statistical distributions. This approach will
78 continue until the ship/ice interaction mechanics are better understood. Unfortunately, the curve fitting is not producing any physical insight into the ice loading process, and at best produces some general properties of the statistical distributions. Consider the three statistical distributions given in Figure 4.5. Each of the three distributions represent one of the three types of extreme value distributions (Gumbel, 1954). The three types can be seen as special cases of the Jenkinson exceedence distribution (Jenkinson, 1955), which has the form: -1
F(p) = e
c - 1- (p - A 1) A2
c
(4.1)
where A1 and A2 are constants and c is a constant which indicates the "curvature" of the distribution. The Gumbel extreme value distributions are special cases of this distribution. For the cases of c=0, c0 the distribution corresponds to Gumbel I, II and III distributions, respectively. When plotted on extreme value paper (as in Figure 4.5), the Type I(C=0) equation appears as a straight line, the Type II (C0) are curved upwards and downwards respectively. For the data given in Figure 4.5 the values of parameters are: Ice Type
A1(
c MPa)
A2(
No.of Events
MPa)
Known Multiyear
0
2.89
1.52
266
Known Firstyear
.042
1.71
.63
723
Mixed MY & FY
-
2.37
.92
1017
.128
The disturbing question is why the mixed conditions should lead to a distribution predicting higher values than either first year or multiyear ice. Is this just a property of this quite large sample or is it an indication of some complex mechanical process, or a result of some problem with the analysis? It appears that the latter is, unfortunately, the case. Figure 4.6 illustrates the problem. On Figure 4.6 the two extreme value equations representing the Known Multiyear and Known Firstyear values (as in Figure 4.5) are shown. As well, the simple combination of the two is given, by combining the
79 data sets and re-ranking the combination. The combined data set contains 989 data points (266 +723). When a Jenkinson distribution is "fitted" to the combination the terms C, A1, A2 become -.128, 1.95 and .92 respectively. This curve is plotted on Figure 4.6 and is of Type II (C
