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Equations for local ice energy dissipations during ship ramming
Cold Regions Science and Technology, 18 (1990) 101-115
101
Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
Equations for local ice energy dissipations during ship ramming D. Blanchet ~, H.R. Kivisild 2 and J. Grinstead 3 1Amoco Canada Petroleum Company, Ltd., Box 200, Calgary, Alia. T2P 2H8 (Canada) 2HRK Consultants Ltd., Calgary, Alta. (Canada) 3Central Region, Canadian Coast Guard, Toronto, Ont. (Canada)
(Received April 10, 1989; revised and accepted December 19, 1989)
ABSTRACT The principle of energy conservation is used to represent the behaviour of ice during ship ramming. The summation of all ship and ice, local and global energydissipations throughout a ram was done using a time-step digital simulation model. Theoretical derivations of six local ice energies are described: bending, cracking, crushing, flaking,ship/ice frictional and removal. The most important local ice and global ship energiesare calculatedas percentagesof the total input energy (ship kinetic plus propulsion energies) during a ram.
INTRODUCTION The prediction of forces during the ramming of a thick multi-year floe by a ship is complicated by the dynamic effects and the interdependence of the ship and ice. It is possible, however, to estimate global forces on the bow of a ship from ice breaking and deformation energy dissipations. Early publications by Kheysin et al. (1976), Popov et al. ( 1969, 1973 ), Likhomanov and Kheysin (1974) and Kurdyumov (1975) have demonstrated the applicability of the energy conservation concept in modelling ship/ice interactions. More recently, Kivisild et al. (1987 ) presented a discussion on the variation of all energies during ship ramming. Breaking and deformation energies depend on the failure modes of the ice. Failure mechanisms change due to the influence of ice and scale parameters, interaction geometries, impact speed and the duration of load. They are influenced also by the ship and water responses, and the global response of the ice floe. To date, only limited data have been collected on the local and global behaviours of ice floes during ramming. This paper presents mathematical models of the
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phenomena involved in the transfer of the ship kinetic and propulsion energies to the individual local ice energies. Quantities for each of these local ice energies are determined. The sum of all the ice energies to determine the energy balance along with the most important ship energies for each phase of the interaction is presented. Special geometries were adopted in order to verify the measurements made on board the Canmar M.V. Kigoriak ice breaker with its spoon bow hull form.
L O C A L ICE F A I L U R E Z O N E S The total local ice energy throughout the ram, including the bow impact, slide up and forefoot impact phases (Kivisild et al., 1987 ) at a given time is the sum of the six local ice energies, i.e., bending, cracking, flaking, crushing, ship/ice friction and ice removal energies (see Fig. 1. ). In the past, the local behaviour of the ice under impact was derived from observations and measurements during small- and medium-scale indentation tests (Michel and Toussaint, 1977; Kry, 1981; Taylor, 1981; Michel and Blanchet, 1983; Timco, 1986). Today, a number of full-scale data sets are available from ship trials
© 1990 ElsevierScience Publishers B.V.
102
D. BLANCHETET AL
NOMENCLATURE a A A1 A~ b B
edge distance of vertical load for bending energy coefficient for bending energy intermediate contact area area of crushed ice width of contact constant for bending energy scale effect CA coefficient for dynamic effects on bending energy CD coefficient for bending energy C~ contact factor (0.3) CFA strain rate factor Cs. diameter of the indenter ( 3.6 m ) d dm increment of mass of crushed ice ds increment of displacement along horizontal plane increment of time dt dx increment of displacement along sliding surface plate modulus o f inertia = Eh 3/ 1 2 ( 1 - v 2) D E Young's modulus of ice local bending energy EBL & crushing energy E~ local cracking energy E~ elastic energy ship/ice friction energy local ice energy EL reference Young's modulus (at Ao)=0.01 E0 m2=4500 MPa) plastic energy EpL ice removal energy ER F momentum force of escaping ice F~ crushing force FN normal bow force on ice edge friction bow force Fvo-r F~ vertical ship load on ice (24.5 MN ram KR 426 ) function of ( ) f( ) g gravity acceleration G deformation modulus of ice (G~, Gt) H~,/4~ geometrical horizontal forces for ice removal energy thickness of ice h h1 height of loaded area k subgrade modulus or unit weight of water K bulk modulus of ice l characteristic length of ice sheet m[, m 2 masses of broken ice P elastic radius of ice sheet e~,e~ geometrical vertical forces for ice removal energy p total elastic, plastic and dynamic stress crushing energy per unit volume - a class of avpc erage pressure p. total elastic and dynamic stress Q plastic deformation modules of ice Q,,Q~ volumes of broken ice Q, tangent plastic deformation modulus of ice r shape factor in dynamic loading shape factor for dynamic loading including /'con confinement dynamic impedance of ice governing forces which resist high speed indentations
Ri,Rs R~ R2
R3 s u v v~ V Wi Wp x
geometrical resultant forces for ice removal energy subscript refers to rods and elongated bodies subscript refers to plane stress conditions subscript refers to half infinite space horizontal displacement under the indenter speed of ejecting dislodged ice initial impact speed sliding velocity of bow on ice volume of ice broken ice mass for removal energy work required to dislodge the crushed ice geometrical distance for ice removal energy
For cracking energy a¢ crack length CR constant dependent upon reference grain size = 9 E Young's modulus of ice m, constant in crack f o r m u l a e - 1.7 × 10-6 k-1 MI constant in crack formulae - 1.7 × 10-3 N crack density at E$ and X¢ Nc critical crack density in the following conditions: = 1.04× 10 -4 xC=0.47 mm N¢=0.055 t = time of crack formation ig= 1.33 X 107m - l d l = l mm s=l K= 1 b=0.34 a - r ( T = 2 6 3 K ) = 2 . 5 × 10-4s -I Qtc crack activation energy QcL local crack propagating energy R~ universal gas constant T, T~,7"2 temperature for crack formation Tk crack forming temperatures (degrees K = Tf¢) ye surface energy=0.09 J m -s ami, crack initiation stress Greek symbols A displacement before flaking= 0.06 h strain strain rate (3.3× 10 - 2 ) E~ elastic static strain % plastic strain # friction coefficient u Poisson ratio of ice (0.3) P(Pl,P2) density of ice (910 kg m -3) O" stress trb break limit of indenter pressure trbs break limit of indenter pressure for static conditions ( 18.2 MPa) ad dynamic stress ae elastic static stress O'min minimum stress for first crack ap plastic stress au unconfined compressive strength angle of repose ( 30 ° ) ~/ internal fraction angle (8.5 ° )
103
LOCAL ICE ENERGY DISSIPATIONS DURING SHIP RAMMING
Local Ice Energies EL
I
.aln
I--~'~
Energy I S en'
componentsJE' gY [
Loc., Ilcr° hing Crackingl J EnI ;gy Energy I
Ei"
,I
I-.lls- ce Energy EFL
#
Friction Energy EFR
T
Energy Sub-Components
Fig. 1. Flow chart for local ice energy dissipation.
ICE ENERGIES LOCAL
Zone1 I
J
Zone2 ~ Ice Removal
~ Flaking\
Zone3 Zone4
Ice Removal ~
(':("i(':':i"! ~,aking
t'7:"-':'):'."
ELEVATION VIEW
(A) Figs. 2A, B. Identification of local ice indentation zones dnnng ship ramming tests.
PLAN VIEW
(R)
104
(Ghoneim and Keinonen, 1983; Kigoriak and Robert LeMeur, 1983; Daley et al., 1984). In addition, high-speed indentation tests and drop ball tests in ice (Nawwar, 1979; EI-Tahan et al., 1983) and indentation tests in rock (Ladanyi, 1966) have also provided useful data on the behavior of ice under impact loads. Local phenomena observed during these tests are those occurring in the vicinity of the applied indenter force. Typically, the formation o f four zones can occur (Figs. 2A and B). Zone 1 represents the zone of complete removal of the ice where the bow of the ship was located at the end of the impact. Zone 2 represents the zone of a thin layer of crushed ice and water including flakes. Zone 3 is the zone of damaged ice resulting from micro and macro-cracking, and radial and circumferential cracks and, zone 4 is the zone of elastic deformations where the ice remains intact but deforms elastically under static and dynamic stresses. The extension of those zones will depend on the ice type, temperature, impact speed, ice thickness, ship and ice stiffnesses, interaction geometries and the duration of the load (Ladanyi, 1966; Michel, 1978; Sodhi and Morris, 1986). The local ice energy dissipations shown in Fig. 1 are limited to zones 2 and 3. For zones 2 and 3, average stresses are first derived for each identified failure, deformation and friction mechanism. These stresses are then converted into total energy dissipations.
MODEL PARAMETERS Many parameters affect the local ice energy dissipations. They are, namely, confinement, interaction geometries, unevenness of the top and bottom ice surfaces, presence of cracks or flaws, presence of water (viscous layer), impact velocity, ship's added mass, load duration, scale parameters and finally the contact factor. For the purposes of this study, many parameters were simplified. For example: - The bow geometry is fiat with a 25 ° stem angle. - The aspect ratio (contact width/contact depth) is constant and equal to eight in the case of the Kigoriak. - The ice floe is fiat with a uniform square edge.
D. BLANCHET ET AL.
- The ice type is defined as being multi-year ice with random crystal sizes and orientation mostly made of granular grains. - Flawed ice is naturally present in any ice at micro scales. This effect is included in the ice properties. - Only first rams are considered since cracks will result from a first ram. Loads from subsequent rams would be affected by the damaged ice zone. - For the ship/ice interaction model, the stress response of the ship is not considered. - The horizontal added mass was included in the total of the input energies from the ship.
L O C A L ICE E N E R G Y D I S S I P A T I O N MODELS A model for each local ice energy is presented. Crushing energy dissipation model
In order to derive crushing ice energy dissipation equations, stress equations must be developed. Stresses in ice, during indentation arise from strain and stress waves. A generalized stress-strain relationship in static and dynamic loading is shown in Fig. 3. The general formulation for the failure criterion for high-speed impacts has been presented by Kivisild et al. (1987). The detailed derivation of the static and dynamic stresses in ice follows. All coefficients and symbols are defined in the Nomenclature. Elastic static stress
A simplified expression can be written for the surface stress in a half infinite space of ice under a circular static load on the plane surface. For an elastic medium, the ratio between the displacement and the diameter of the indenter is expressed by:
S/d= ( n / 4 ) ( I - v)ae/E=0.550 adE
(I )
By inverting Eq. 1, the elastic static stress is given by:
a~=(4/n)(E/(1-u))s/d=e~E/O.550f(~)
(2)
The static surface stress depends also on the strain rate as shown above. By using the bulk modulus K
LOCAL ICE ENERGY DISSIPATIONS DURING SHIP RAMMING
105
20
15
Legend: (7d = dynamic stress O ' p = plastic stress O"e -- elastic stress "~
10
Residual
I Energy to Failure
Op
t ,,0.0006
g5 i ~ ¢0 ,-nc
~ / 0.0004
/
0 0.010
Unconfined Compression
°el
0.05
0.10
0.15
Displacement Related to Diameter of Load
Fig. 3. Definitionsof local elastic,plasticand dynamicstresses. and the deformation modulus G of the ice to replace E and u, Eq. 2 can be written in a form which is directly usable in plastic ice and in remolded states of ice:
~ = ( 4/~)6G/ ( I + ( 4 /3 ) G / K ) ~ j ( ~) ~ = (41g)
Q~f(~)
(3) (4)
where:
Q= 6G/(1 + ( 4 / 3 ) G / K )
(5)
The limits of these parameters can be calculated for different conditions such as for rigid non-deforming and moldable materials. The ratio G/K in the elastic range is expressed by:
G/K=~[ ( 1 - 2 v ) / 1 + vl
(6)
For ice, u=0.3 which gives G/3K--O.154, Q=3.71 G and Eq. 4 becomes: ae =4.72 Ge~f(~)
(7)
Elastic static and dynamic stress During high-speed impacts, static and dynamic stresses add uo. Using the rod analogy of dynamic stresses, the total elastic and dynamic stress is expressed by:
Pe=ad +ae
(8)
or:
Pe= ( rpR ) ~/2v+ 4.72G ~ef( ~)
(9)
The dynamic component develops almost instantaneously as the ship hits the face of the ice. Further load is added during the continuation of the penetration. The dynamic impedance R depends on the state of the stress caused by the indentation. This depends on the geometry of the target ice. The factor r has been introduced to consider the influence of the geometry of the load. Contact areas during the impact are idealized as a patch load on a half infinite body of ice. In this confined condition, the dynamic impedance R3 of ice can be shown to have two components. Part of the response arises from the action of the remolded ice as a fluid (viscous fluid or granular material). Another part of the response arises from the deformation resistance of the material itself:
R3=K+(4/3)G
(10)
The bulk modulus is unchanged with the change of material from the elastic range to remolded range. It is also valid for fluids:
106
D. BLANCHETET AL.
K=E/3(1-2v)
(11)
The second term is a function of the deformation modulus for ice. In the fully plastic and in the remolded states, G becomes small compared to K. Fluid effects start to dominate. The insertion of these values gives the following expression for impact pressure by an indenter. Equation 9 becomes:
pe=[pr(K+(4/3)G~)]l/213+4.72~eGf(~)
(12)
The subscript "s" represents static conditions. In the elastic region, the impedance R is equal to 135E (R3), 0.91E (R2) and E(Rl) for, respectively, a half infinite space, plane stress conditions (ice sheet ), and an elongated body where E = 4195 MPa (static) or E=6991 MPa (dynamic) and tr~= 26% of trb~ (Wang and Ralston, 1983 ).
Plastic static stress The remaining of the stress shown in Fig. 3 is the plastic stress or stress during damaging of the ice. This stress can be simply approximated by subtracting the total stress (at the breaking limit) and the static stress at the elastic limit during penetration:
ap=abs--tre=4/n(1--v)f(~) f Q d e
(13)
or: ~p =
1/0.55f(~) at (ep)
(14)
where ~p= (ebb-- ~)- The subscript "t" represents tangent. Secant modulus may be used to relate stress to strain.
Total elastic plastic static and dynamic stress
ity, whereas the plastic and elastic portions of the total stress will decrease due to their strain rate or velocity rate dependency. This equation will be converted into a crushing energy equation which is presented in the following section.
Conversion of stress into crushing energy dissipation Crushing is defined as the complete pulverization of the ice from initial micro-fissuring which typically results in uniform ice particles of very small dimensions (less than l cm of linear dimension). The crushing energy is given by the following general equation:
Ec/V=pc = fFc ds/fAcds
(16)
where Pc represents the amount of energy to failure per unit volume of ice crushed. Essentially, three methods can be used to calculate the crushing energy per unit volume of ice crushed. Method 1: Actual ice pressure data measured during icebreaker interactions with ice (such as the M.V. Canmar Kigoriak impact tests) can be used and corrected for different ice and impact conditions. Method 2: The uniaxial crushing strength of ice at small scale can be used and corrected for different ice and impact conditions taking the size effect into account. Method 3: Method 2 can be further developed by breaking down the crushing strength of ice into two parts; the elastic and the plastic limits during the deformation of the ice. This method is used in this paper. The energy per unit volume of ice crushed is given by: Pc = CSR CA CFA (EEL + EpL ) / V+ ( rconpRavg:e,p) 1/213
The total stress at failure will be given by:
(17)
/
p = p e -I-6p
(15)
where subscript "avg:e,p" represents the average elastic impedance of the ice for elastic/plastic ranges.
Equation 15 represents the total stress during indentation which is the summation of the dynamic, elastic and plastic stresses. The dynamic portion of the total stress will increase with the impact veloc-
This energy dissipation is the result of the initial deformation of the ice under the initial low stresses. Wang and Ralston (1983) calculated the elastic limit of the stress to be about 26% of the yield/plas-
p= [pr(K+ (4/3)Gs) ],/z13 +4.72 ~eGf(~) + 4.72¢pGt f([)
Local elastic ice energy dissipation EEL
LOCALICEENERGYDISSIPATIONSDURINGSHIPRAMMING
tic strength. This is close to the uniaxial compressive strength of ice. Elastic deformations during high-velocity impacts are very small. The amount of energy involved, before fracture or plastification occurs, is given by: EEL V= 0.5 O'ere
(18)
EEL/V=O. 106 (a~/G) f( ~)
(19)
for: 0 < v < 0 . 9 1 abJ (pE) 1/2
(20)
which gives v<3.3 m s -1, using E = 0 . 5 4 Eo and ab~= 0.3 trob~,EEL/V= 0 otherwise. Local plastic ice energy dissipation EpL As the stresses increase, fracture and permanent deformations of the ice occurs. Plastification is then initiated until complete failure of the ice occurs. The energy dissipation associated with plastic deformations is given by:
EpL / V--- ( (O'bs- O'b)/2)Epf(d )
(21)
107
A theoretical coefficient of friction of/t = 0.15 was used. The effects of impact velocity and type of ice were not included.
Ice removal energy dissipation model Ice removal, in this paper, represents two phenomena: (a) the expulsion of the crushed ice out of the contact zone, and (b) the remolding of the crushed ice which accumulates on top of the ice at the contact zone during the impact. The ice removal energy dissipation is calculated assuming the broken ice is a friction material with negligible cohesion. The following equations were developed for a load per unit width. Forces occurring along the front face (Fig. 4a) are disregarded. From field observations, it is suggested to assume 50% voids in the broken ice mass. A value for the escape speed u can be obtained by comparing ice volumes Q2 and Q1 (Fig. 4a). The mass of ice which is accumulated on top of the floe is the same as the mass broken off the floe. From geometry, we obtain the following relationships:
EpL/V=O.55( (Crbs-- re)/2)(ab --ae) f(d)/Q, (22) for: 0 < v < 0 . 6 7 crbs/(pE) 1/2
(23)
which gives v< 2.5 m s- i. EpL/V= 0 otherwise. For ship velocities greater than 0.91 abs/(pE) 1/2, which gives v=3.3 m s -l, EpL and EpL=0. Therefore, the work for indenting fractured ice is derived from the requirements of dislodging the material:
Wp=4.5 pv2A(d)h I
.,,
x
/
f.-
Front Face
I I
b" d x
0.3e4
(24) H$
Ship/ice friction energy dissipation model The friction between the ship's hull at the bow and the ice contributes significantly to the total energy dissipation process. Only dynamic friction is considered. The ship/ice friction energy is given by: EFR = f FToTdX
(25)
where: f T o T = fNfl
(26)
dx=vsdt
(27)
Pi
Fig. 4. a. Description of parameters included in the derivation of the removal energydissipation. b. System of forces for the remolding of crushed ice on top of the ice sheet.
108
D. BLANCHETETAL.
m, = mz = O.085bs2pl = Q l p l
(28)
= Q2pz = O. 129x2pzb which gives u = 1.15 v. Escape m o m e n t u m at constant speed, and a width of indentation b in the direction parallel to the bow hull are as follows: F=bu (dm/dt)
(29)
By inserting the geometrical parameters given in Fig. 5a, the following result is obtained: F = 0 . 2 0 v 2 bs p
(30)
where Fr~=0.2 v 2 bs p cos 12.5 °. As found above, escaping ice moves past the ship's hull at a low differential speed. This is still sufficient to engage the friction of the ship. A substantial resistance is obtained from the ice surface. Resulting forces may be split into vertical (P) and horizontal ( H ) components to derive equations for escaping ice (see Fig. 4b). The relationship o f vertical forces is: Ps + Wi=Pi
(31)
Ps = - Wi + e i = Q l p b g + e i
(32)
where/t = tan g/= 0.15 and g/= 8.5 °. It should be noted that the derivation of friction forces is obtained as the small difference of two large values. Therefore, there could be occasional jamming with ensuing sudden release and large impulses. There could also be periods o f easy flow when friction against the ship becomes inactive or when the ship moves faster than the ice and would actually drag dislodged material with it. In addition to these effects there are silo effects in the indenter zone which will cause further impediments to the flow of ice. The horizontal force can be evaluated by introducing the dynamic impulse which has been defined above ( H + FH = 0): Ps = - 0 . 3 2 vZpbs-O.O8pgbs 2
(37)
The horizontal force Hs is given by: H~ = - P ~ t a n (25 ° - ~u)
(38)
= - 0 . 3 ( - 0 . 3 2 p v Z b s ) - 0 . 3 ( -O.08pgbs z) = 1.0 v2pbs+ 0.02 pgbs 2 The energy equation may be written as follows:
where, from geometry, we obtain the following equations:
ER = 1.0 p f vZbs ds+ 0.02 pgf bs2ds
P~ =R~cos(25 ° - ~u),
Ice bending energy dissipation model
(39)
H~ = - P, tan (25 o _ ~), Pi = Ri cosq~, Hi = Pitanq~
(33)
The repose angle ~ is assumed to be equal to 30 °. To find Pi, we need to s u m the horizontal forces. The total horizontal force may be written as follows: n = Hs + Hi + Hactive
(34)
where nactive is defined as the horizontal force generated by the friction, if the ship is moved past the mast. Hpassive would apply if squeezing o f the broken ice mass occurs. Hac,ive =P~( 1 - s i n 0 / 1 + sin~) = (1/3)Ps
(35)
Using Hi=P~ tan q~and Eq. 31, Eq. 35 and H s = - P ~ tan (25 ° - ~), Eq. 34 becomes: H = - 0 . 3 Ps +0.049 pg bs2 +0.333 Ps +0.577 Ps (36)
Local bending of the edge o f a semi-infinite ice sheet occurs during the transfer of the bow vertical load to the ice during ramming, if the ice is sufficiently thin or weak. Under the impact of the ship, the edge of the ice sheet will be subjected to a combination of bending and compressive stress waves. It is assumed that the ice sheet is separately responding to pure bending and compressive waves. This leads to a small under-estimation o f the total bending strain energy dissipations. A quasi-static analysis of the pure bending o f the edge o f a semi-infinite and isotropic ice plate is proposed. However, the energy equation can be corrected to account for the following factors: (a) The parameter a representing the distance between the concentrated load F~ and the undeformed ice edge estimated to be equal to 3l. (b) The ratio between a semi-infinite to an infinite ice sheet can be generalized by using the coef-
LOCAL ICE ENERGY DISSIPATIONS DURING SHIP RAMMING
ficient CE(2 < CE < 5 ) (Westergaard, 1925 ). (c) The dynamic amplification factor representing the effect of the added mass of water increasing with the increase of the rate of loading of the ice sheet is generalized by using the coefficient CD(4
Em=4B2Drta 2 1 + v) +z~K [ ( 1/2)AZa2+ (1/2)ABa4+ (1/6)BZa61 -FyA (40) where:
A+Ba2[(2/3)+16D (l+v)/kaZ]=O
(41)
and:
A+ ( l/2)BaZ=Fy/(n ka z)
(42)
109
level was estimated to be around 0.6 MPa (Gold, 1972). Parameswaran and Jones (1975) using a brittle model derived by Petch (1968), calculated the fracture stress at 0.3 MPa. Their results, however, gave a fracture stress varying between 0.1 and 0.5 MPa. The crack initiation stress is given by (Sinha, 1984): O'min
=E(M1 - ml T)l"s/CR
Assuming that the stress Crrninis an average value of the stress for creating the first crack over the entire bow print area, the activation energy for cracks forming at times T~c and T2c and temperatures of TI and T2, is given by:
Qtc={RJ[ ( 1 / T , ) - ( 1 / T z ) ]}Xln
{(aT21ar,)x (In[1 - [ ( M 1 m~T~)/CR](E/a)*]/ (48)
The general Eqs. 41 and 42 given above, become: A = 0.0772 Fy/(k/2)
(43)
B= - 0.0093 Fy/(kl 4)
(44)
When inserted in Eq. 40, the above equations give the following bending energy dissipation equation:
EBL = -- 0.038CEF2/(kl 2)
(45)
or in more details: EBL=0.13 CEF2/(k2/ZE~/2) (CDh) 3/z
(46)
where: hdynami c = CDhstatic For this study, only the results for thick ice are relevant to compare to measurements. The above equations can also be used as a component in calculations for thin ice. Large bending energies from thin ice sheets will not be discussed since verification measurements were lacking.
Local ice cracking dissipation model Local cracking is a contributor to other phenomena such as crushing and flaking. However, when crushing and flaking are ongoing processes in zone 2 during the interaction, the remainder of the cracking energy is spent creating and propagating cracks in zone 3. The work to initiate a crack at the first micro-scale
(47)
ln[l - [ M - m , T2) / CR] (E/a)*] ),/b} For ice at - 10 ° to - 5 ° C, typically, Qtc is equal to 80-100 KJ mo1-1 at a=0.6 MPa (or tr/ E>5×10-5). The other parameter to be studied is the crack density or the number of cracks created under a given stress per area. Sinha (1984) suggested the following equation: N=Ncexp [~( CRdl/K) (a/E) s {1 - e x p [ -
(avt)b]}-xO]
(49)
Assuming a given strain level in zone 3, the number of cracks can be calculated or estimated using Gold (1972). The strain energy in zone 3 will be equal to the cracking energy when the threshold average strain is reached. In addition to this cracking initiation energy, the local cracking propagation energy is present in the damaged area (zone 3). It is given by: QCL= (rttr2a2/E-4ac?~b)h
(50)
Ice flaking energy dissipation model Flaking is the mode of ice failure which ultimately dislodges pieces of ice. Flaking can be
110
D. BLANCHETETAL. h~'~~ ~ F l n l t e
L ~~.lce
Block
PARTIAL CONFINEMENT
NOT CONFINED Difficult for Extrusion
Infinite Block of Ice
t ~ ~ I'~ ~.
Indenlor~ _ Effectof
"~oped
Difflcul for Extrusion
j v. "~ Sides O~ SHIP/ICE INTERACTION (PARTIAL CONFINEMENT)
]
FULL CONFINEMENT
at 25 ° angle to one of the sides. The total load reaches a value which is sufficient for pieces of ice to be broken-out by the flaking process. The condition at which it occurs has been expressed deterministically using the Mohr-Coulomb failure theories, combined with various results on the strength of ice in different confinements. Using partial confinement conditions (Fig. 5 ), the maximum centre-located load area is given by (Kivisild et al., 1987 ): A1=0.76 (hi) 2
The load applied on this area will be applied at full confinement stresses which gives a maximum force value of: F = 3 . 1 4 (hi)2tru
Fig. 5. Different confinement scenarios for flaking.
TABLE 1 Average stress ratios related to unconfined compression with partial geometric confinement Slope angle interface
Collision
Molded
Line formations 90 ° ridge 65 ° ridge 45 ° ridge 25 ° ridge 0 ° ridge - 90 ° cavity (25 ° +65 ° ridge)
0.57 1.21 1.86 2.70 4.14 10 1.67
1.00 1.87 2.57 3.27 4.14 10 1.87
Cylinderformation 90 ° walls
0.77
1.00
through failure planes such as in unconfined compression or there could be shear faces curving up to the surface as in normal bearing type loading. Figure 5 shows flaking failures which could occur at different confinement scenarios described above. As shown in Fig. 5 and Table 1, various types of loading result in various flaking stresses, starting with uniaxial stress and reaching higher stresses o f several times this load at full confinement (Kivisild et al., 1987). A solid, flat, unlimited indenter is assumed to collide with a 90 ° corner of a quarter infinite block
(51)
(52)
This analysis leads to the result that the face of the ice will have areas covering 0.40 (contact factor) of the total aggregate ice area under crushing loads which are equal to bearing capacity. The colder the ice and the faster the interaction, the lower the contact factor is to be expected. At high impact speeds, larger than 2 m s- t, the contact factor measured during the MN Canmar Kigoriak Impact Tests was about 0.3 to 0.4. Work by a contact force occurring at a maximum flake may b e expressed as follows: EEL = trbAA 1= 0.758trb (h t ) 2A
(53)
EXAMPLES AND COMPARISON WITH F I E L D DATA The purposes of the above models is to predict the average global load on the bow of a ship during the bow impact, slid-up and forefoot impact phases. The models also help to identify the most important failure processes during each phase of the interaction. The dissipation of the total input energy (ship kinetic and propulsion energies) into different ice and ship failure, deformation or friction mechanisms can also be followed throughout the
ram. A comparison with some of the measurements made during the 1981 Kigoriak Impact Tests was conducted and some of the results are presented.
LOCAL ICE ENERGY
DISSIPATIONS
DURING
lll
SHIP RAMMING
30
Ram KR426 (Kigoriak 1983)
~6 22
18 14
v 6. "O
g-~lpfias~s:'
F
' .i
o Up r
. . . .
e
;~
. . . .
f
o
~
o
. . . .
;~
t
'
sE~.'
|
Prior
-10
To
-14
Impact
-18 .I
B.B2
BOW
Fig. 6. Bow force time history KR426.
D u r i n g the period from 5-13 October, 1983, 202 rams were completed by the Kigoriak in h e a v y multi-year ice. During this r a m m i n g period, the highest global load on the bow o f the Kigoriak was m e a s u r e d during r a m K R 4 2 6 (Fig. 6). The initial i m p a c t speeds was 4.6 m s - l . Borehole jack and uniaxial and confined compressive strengths o f ice were collected during the ship trials in 1981 and 1983. They were extrapolated to r a m m i n g conditions o f crushing velocity varying f r o m 2 m s - l to zero which gives an average strain rate o f 3.3 × l0 -2 s e c - i. Table 2 together with Fig. 7 a n d the following section summarizes the energy balance during r a m
TABLE 2 Kigoriak Ram KR426 ( 1983 ) (calculated energy dissipation) Energy (Mg) Input energies Ship kinetic Ship propulsion Total input energy Distributed ice energies Crushing Flaking Friction Removal Bending Cracking Forefoot crushing Forefoot friction Total ice energies Potential energy Total distributed ice plus ship Ship kinetic left Total distributed ice energy (KE+ice+ ship) Unaccounted energy (ice + ship) Time from initial impact (s) Max. meas. force (ran)
Before bow impact
End of bow impact phase
End of slide-up phase
End of ram (ship stopped )
103.9
103.9
103.9
103.9
103.9 0.8 104.7
103.9 1.4 105.3
0.0 0.0 103.9 103.9 0 0 0
8-10* 0.2-0.5* 8.1 0.8-0.9 0.3-0.6 1.5-2.0"(o) 20.6 (avg) 0.9 21.5
9-11 0.2-0.5 15.3 0.8-0.9 0.1 1.5-2.0 28.5 (avg) 19.2 47.7
9-11 0.2-0.5 19.7 1.3-1.45(F) 0.1 1.5-2.0 31.4"* 2.8 67.5 (avg) 33.6 101.1
81.7 103.2 ( avg )
53.7 101.4 ( avg )
0.3 101.4 ( avg )
- 0.7
- 3.3
- 3.9
1.8 14.0
3.0 10.0
0.62 24.5
(o) Estimated cracking energy from strain measurements (Table 4) =0.08 MJ. *Crushing, flaking and cracking measured energy- 12.7 MJ from force/penetration diagram. Method 1 was used. (F) Forefoot impact included. **Includes cracking and flaking,
1 12
D. BLANCHETET AL.
T
Bow-Ice Friction
Crushing
80-
1) 2) 3) 4)
Remaining Distributed Energies Propulsion Energy Fore Foot Friction Input Energy
60. I,IJ
._= hi
m
c
in
i
40-
20.
00
2-0
1-0
Start
End
~ Initial
Initial
Impact
~ Impact
3-0
First
Time S
~ Fore Foot Contact
Ship ~ Stopped
Fig. 7. Breakdown of input and distributed energies for RAM KR 426.
KR426. All parameters required to calculate each energy are given in the Nomenclature. The input energy is made up of the initial kinetic and propulsion energies which gives the total input energy at any instant. The distributed energies are made up o f all ice and ship energies and the remaining kinetic energy which summed to equal the input energy. During the bow impact, slide-up and forefoot impact phases of ram KR426, these energies can be summarized as follows. Prior to bow impact
(A) Input: Initial kinetic energy plus propulsion energy.
(B) Distributed: No energy is distributed before the impact.
After bow impact before slide up
(A) Input: The propulsion energy contributes an additional input. It is a small addition as duration is short but the value depends on power setting. (B)Distributed: The distributed energy at the end of the bow impact phase can be identified as follows: - Ice crushing energy; about 10% of initial KE. - Ice friction energy; about 7-8% of initial KE. - Ice removal energy; about 3% of ice crushing. - Ice flaking energy; about 3-5% of ice crushing and depends on the initial velocity. - Potential energy; as there is relatively little vertical ship motion, the potential energy at this stage is only about 3% o f the total potential energy present in the ship when it finally stops. The kinetic energy consumed is approximately 20% o f initial KE.
LOCALICEENERGYDISSIPATIONSDURINGSHIPRAMMING - Ice bending, local cracking, ship global and local strain energies; collectively consume less than 2% of initial kinetic energy. Local strain is negligible throughout the ram. Global strain is a function of the bow force. During the initial impact it rises to a m a x i m u m of less than 2% of initial KE. Prior to forefoot c o n t a c t after s l i d e up
(A) Input: Propulsion energy; about 50% o f propulsion system total increment has been added. (B) Distributed: The distributed energies at the end of the slide-up phase can be identified as follows: - Friction energy; has continued to rise as ship slides up and represents about 15% of initial kinetic energy. - Crushing, flaking, ice removal, and cracking energies; little additional ice is destroyed during this phase. Only about 11% is added to values at end of the bow impact. - Bending energy; more than 50% has been recovered. - Ship global and local strain energies; together much less than 1% o f initial KE. They are recoverable. - Kinetic energy; about 40-50% of initial K E has been consumed. The remaining K E has been transformed into rotational and translational KE. - Potential energy; the vertical displacement o f the bow of the ship has increased significantly during this phase. The potential energy represents one of the largest distributed energies at this stage and can have risen to about 50% of the PE end value or 20% o f the initial KE.
113 TABLE 3 Crushing energy calculations during bow impact phase ram A7 Crushing energy (a) Crushing energy= (contact area) (pressure) (penetration) (sin 24 °). (b) Friction energy= (average pressure) (average contact area) (length of contact) (coefficient). (c) Pressure= (average pressure on contact area) (d) Penetration= (penetration rate) (time) Time (s)
A
P
PEN
Ec
Contact area (m 2)
Pressure (MPa)
Penetration (m)
Crushing energy (MJ)
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
4.5 5.7 5.7 4.6 6.2 7.1 8.5 5.7 6.3 6.5 3.5 1.2
1.8 1.5 0.9 1.4 1.1 1.2 1.8 2.4 2.4 2.1 1.5 1.1
0.28 0.12 0.11 0.10 0.09 0.07 0.06 0.05 0.09 0.02 0.01 0.00
0.92 0.42 0.23 0.26 0.25 0.24 0.37 0.28 0.25 0.11 0.03 0.00
End of bow impact Average contact area= 5.5 m 2 Average pressure= 1.6 MPa. Total penetration = 0.95 m. Total crushing energy = 3.36 MJ, calculated = 5-6 MJ. Contact length = 3.0 m (from movie data). Coefficient of friction = 0.15. Friction energy (impact phase)=(5.5) (1.6) (3.0) (. 15 ) = 3.96 MJ, calculated 5 MJ.
S h i p s t o p p e d after forefoot i m p a c t
(A) Input: Propulsion energy; in rams considered, only adds about 1-2% to initial kinetic energy. (B) Distributed: Forefoot crushing, friction, and removal energies; collectively represents the dominant combined ice energy. The total value is greater than the combined values o f crushing and friction due to bow. This energy accounts for approximately 40% of the initial KE. The distributed energies at the end o f this phase can be identified as follows: - Bow friction energy; 20% of initial KE. - Ice bow crushing, flaking, ice removal, and cracking energies; another 11% added in values given in the previous ram phase.
- Kinetic energy; has been fully expended. - Potential energy; represents about 35% o f initial KE. The total unaccounted energy given in Table 2, shows the possible errors due to both measurements and theoretical calculations, Ram KR-A7 (Kigoriak 1981)
The measurements of local ice loads and the global movement of the Canmar Kigoriak in ramming were conducted in August and October 1981. The instrumentation of the Kigoriak and the data collected
D. BLANCHETETAL.
114 TABLE 4 Uniaxial strain energy in ice Ram
B 227 B 230 B 231 B 234 B 235 B 236
ES
ESA
(J m -3 )
(MJ)
0.009 0.26 0.082 0.121 0.067 0.121
0.003 0.078 0.025 0.036 0.020 0.036
Impact velocity
Average strain
(ms -1)
(X10 -6)
1.39 1.03 1.54 3.09 3.09 3.60
2.99 5.12 9.0 11.0 8.2 11.0
Peak global strain energy/volume = ( 1/2) E e 2 Apparent modulus of elasticity= 2 × l03 MPa Averageglobal strain in ice Approximate volume = 3 X 10s m 3 E S A Average global strain energy in ice.
Es
E e V
were presented by G h o n e i m and Keinonen (1983). In addition to the ship information, strain measurements on the ice surface were conducted by B.P. ( G o o d m a n and Duckworth, 1981 ) in October 1981. Six wire strain meters were fixed to the ice using ice screws. Crushing, friction and strain energies were estimated, respectively, for R a m s KR-A7 and KRB227, B230, B231, and B234-B236. The results are given in Tables 3 and 4. The "crushing" energy in these calculations include cracking and flaking energies. In Table 3, the critical crushing and friction energy values for r a m A7 are also given. They compare well with the calculated values if the crushing energy includes flaking and cracking energies.
CONCLUSIONS ( 1 ) Six local ice p h e n o m e n a were observed during full-scale impact tests. They are: crushing, friction, removal, flaking, bending, and cracking. Energy dissipation equations have been derived for each phenomenon. (2) The crushing energy dissipation has been derived based on an analysis of dynamic, static elastic and plastic stresses during indentation. (3) Friction, bending, and cracking energy dissipations were derived from original or classical the-
ories. New theoretical solutions for ice removal and ice flaking energy dissipations were presented. (4) For the Kigoriak impact tests, the most important distributed energy dissipations which account for more than 95% of all the distributed energy are: - Ice knife crushing; - Ice friction (including ice knife); Ice Crushing (Bow); and - Ship potential energy. (5) The impact velocity is the most important parameter affecting all but the flaking energy dissipation. It also considerably affects the ice pressures and forces. (6) There are two important ranges of ice behaviour during ship ramming: - At low ramming speeds (0.5-2.5 m s - I ), the ice behaves in a brittle manner and the ice loads are low due to static conditions. - At high ramming speeds ( > 2.5 m s - ~) ice loads become large amplitude impulses. Elastic stress waves propagate in remolded ice and increase the resistance. (7) The combination of the ship/ice friction and the ice knife/ice friction represents about 20% of the total distributed energy.
ACKNOWLEDGEMENTS The authors would like to thank Canadian Marine Drilling Ltd., Canadian Coast G u a r d Northern and Lavalin Offshore Inc. for the opportunity to work on this most interesting project and for their permission to publish this paper. We greatly acknowledge A. Churcher (simulation model) of Canadian Marine Drilling Ltd., Dr. A. Keinonen and Mr. Colin Revill for their contribution to the project, " S h i p / I c e Interaction Energies". Appreciation is also due to the Panel of Energy Research and Development, who under Task 6.1, financed the original project. REFERENCES
Daley, C.G., St. John, J.W. and Seibold, F., 1984. Analysis of extreme ice loads measured on USCGC polar sea. The So-
LOCALICE ENERGYDISSIPATIONSDURINGSHIPRAMMING ciety of Naval Architects and Marine Engineers, Annu. Meet., New York, N.Y., Pap. 8. EI-Tahan, H., Swamidas, A.S.J., Arockiasamy, M. and Reddy, D.V., 1984. Behaviour of iceberg and artificial snow ice under high strain rates and impact loads. Proc. 3rd OMAE Symp., Dallas, Tex., Vol. IV, pp. 158-165. Ghoneim, G.A.M. and Keinonen, A., 1983. Full scale impact tests of Canmar M.V. Kigoriak in thick ice. 7th POAC Conf., Helsinki, Vol. III: 329-346. Gold, W.W., 1972. The failure process in columnar-grained ice. NRCC Rep. 12637, Div. Building Res., Canada; and in Philos. Mag., 8th Ser., 26(2): 311-328. Goodman, D.J., Duckworth, R., 1981. A study of ice deformation by the Dome Petroleum ice breaking vessel Kigoriak during impact on multi-year sea ice floes and ridges. B.P. Petroleum, Rep. proprietary to Dome and B.P. Kheysin, D., Ye., Likhomanov, V.A. and Kurdyumov, V., 1976. Determination of specific breakup energy and contact pressures produced by the impact of a solid against ice. CRREL, Translation 539. Kigoriak and Robert LeMeur, 1983. Full Scale Ice Impact Tests. Rep. TP6028E Canadian Marine Drilling Ltd. for Transport Canada, Canada Coast Guard Northern, 2 Vol. Kivisild, H.R., Blanchet, D. and Revill, C., 1987. Ship/ice interaction pressures and energies during ship ramming. 9th Int. OMAE Conf., Houston, Tex. Kry, R., 198 I. Scale effects in continuous crushing of ice. Proc. IAHR Int. Syrup. on Ice problems, Quebec, Voi. II: 665579 (From APOA Projects 93, 66, and 106). Kurdyumov, V., 1975. Determination of loads due to the ice impacts on the hull of an icebreaker. M.Sc. thesis, The Shipbuilding Inst. Leningrad, Leningrad, USSR (in Russian ). Ladanyi, B., 1966. Failure mechanism of rock under a plate load. Proc. First Congress Int. Soc. Rock Mechanics, Lisbon, Vol. 1. Likhomanov, V.A. and Kheysin, D., Ye., 1974. Experimental investigation of solid body impact on ice.Problem Arktiki I Antarktiki, 38. Masterson, D.M., 1983. Interpretation ofin-situ borehold ice strength measurement tests. Symp. Mechanics of Ice, Rocks and Soils, ASME Applied Mechanics, Bio-Engineering and Fluids Engineering Conf., Houston, Tex. Michel, B., 1978. Ice Mechanics. First Edition, Les Presses de L'Universite Laval, Que. Michel, B. and Blancher, D., 1983. Indentation of a $2 floating ice sheet in the brittle range. Ann. Glaciol. 1: 180-187. Michel, B. and Toussaint, N., 1977. Mechanics and theory of
115 indentation of ice plates. J. Glaciol. 19(81 ): 285-300. Nawwar, A.M., 1979. Experimental investigation of ice impact pressure under various conditions. Rep. DFR 536C by Arctec Canada Ltd., submitted to F.G. Bercha and Associates Ltd. Parameswaran, V.R. and Jones, S.J., 1975. Brittle fracture of ice at 77K, J. Glaciol., 2154(71 ): 305-315. Petsch, M.J., 1968. Metallographic aspects of fracture. In: H. Liebowritz (Editor), Fracture and Advanced Treatise. Vol. I. Academic Press, New York, N.Y., pp. 351-393. Popov, Yw.N., Feddeev, O.V., Kheysin, D.Ye. and Yakovlev, A.A., 1969. Strength of ships sailing in ice. U.S. Army Foreign Sci. Technol. Centre, Washington, D.C., Translation No. FSTC-HT-23-96-68. Popov, Yw.N., Tegkhayeva, T.Kh. and Feddeev, O.V., 1973. Determination of the ice loads on an icebreaker's hull with consideration of reflected impact. Ice Navigation Qualities of Ships. Sinha, N.K., 1984. Delayed elastic model for initiation and accumulation of creep cavitation at high temperatures. Adv. Fract. Res., Proc. 6th Int. Conf. on Fracture, New Delhi, India, pp. 2295-2302. Sodhi, D.S. and Morris, C.E., 1986. Characteristic frequency of force variations in continuous crushing of sheet ice against rigid cylindrical structures. Cold Reg. Sci. Technol., 12(1): 1-12. Taylor, T.P., 1981. An experimental investigation of the crushing strength of ice. Proc. 6th POAC Conf., Quebec, Vol. 1:332-345 (From APOA Project 106). Timco, G.W., 1986. Indentation and penetration of edgeloaded fresh water ice sheets in the brittle range. Proc. 5th Int. Symp. OMAE, Japan, Vol. IV: 444-452. Timoshenko, S. and Woinowsky-Kreiger, S., 1959. Theory of Plates and Shells. Second Ed. Engineering Societies Monographs. McGraw-Hill, New York, N.Y. Tunik, A.L, 1984. Dynamic ice loads on a ship. IAHR Ice Symp., Hamburg, Vol. III: 247-313. Wang, Y.S, and Ralston, T.D., 1983. Elastic-plastic stress and strain distributions in an ice sheet moving against a circular structure. Proc. 7th POAC Conf., Helsinki, Finland, Vol. III. Westergaard, H.M., 1925. Stresses in concrete pavements computed by theoretical analysis. Paper presented at the Annual Meeting of the Highway Research Board, Washington, D.C., Vol. 7(2 ) Public Roads, pp. 25-35. Yu, Yee-Tak, 1945. Virtual masses of rectangular plates and parallelepipeds in water. J. Appl. Phys., 16, (Nov.): 724729.
101
Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
Equations for local ice energy dissipations during ship ramming D. Blanchet ~, H.R. Kivisild 2 and J. Grinstead 3 1Amoco Canada Petroleum Company, Ltd., Box 200, Calgary, Alia. T2P 2H8 (Canada) 2HRK Consultants Ltd., Calgary, Alta. (Canada) 3Central Region, Canadian Coast Guard, Toronto, Ont. (Canada)
(Received April 10, 1989; revised and accepted December 19, 1989)
ABSTRACT The principle of energy conservation is used to represent the behaviour of ice during ship ramming. The summation of all ship and ice, local and global energydissipations throughout a ram was done using a time-step digital simulation model. Theoretical derivations of six local ice energies are described: bending, cracking, crushing, flaking,ship/ice frictional and removal. The most important local ice and global ship energiesare calculatedas percentagesof the total input energy (ship kinetic plus propulsion energies) during a ram.
INTRODUCTION The prediction of forces during the ramming of a thick multi-year floe by a ship is complicated by the dynamic effects and the interdependence of the ship and ice. It is possible, however, to estimate global forces on the bow of a ship from ice breaking and deformation energy dissipations. Early publications by Kheysin et al. (1976), Popov et al. ( 1969, 1973 ), Likhomanov and Kheysin (1974) and Kurdyumov (1975) have demonstrated the applicability of the energy conservation concept in modelling ship/ice interactions. More recently, Kivisild et al. (1987 ) presented a discussion on the variation of all energies during ship ramming. Breaking and deformation energies depend on the failure modes of the ice. Failure mechanisms change due to the influence of ice and scale parameters, interaction geometries, impact speed and the duration of load. They are influenced also by the ship and water responses, and the global response of the ice floe. To date, only limited data have been collected on the local and global behaviours of ice floes during ramming. This paper presents mathematical models of the
0165-232X/90/$03.50
phenomena involved in the transfer of the ship kinetic and propulsion energies to the individual local ice energies. Quantities for each of these local ice energies are determined. The sum of all the ice energies to determine the energy balance along with the most important ship energies for each phase of the interaction is presented. Special geometries were adopted in order to verify the measurements made on board the Canmar M.V. Kigoriak ice breaker with its spoon bow hull form.
L O C A L ICE F A I L U R E Z O N E S The total local ice energy throughout the ram, including the bow impact, slide up and forefoot impact phases (Kivisild et al., 1987 ) at a given time is the sum of the six local ice energies, i.e., bending, cracking, flaking, crushing, ship/ice friction and ice removal energies (see Fig. 1. ). In the past, the local behaviour of the ice under impact was derived from observations and measurements during small- and medium-scale indentation tests (Michel and Toussaint, 1977; Kry, 1981; Taylor, 1981; Michel and Blanchet, 1983; Timco, 1986). Today, a number of full-scale data sets are available from ship trials
© 1990 ElsevierScience Publishers B.V.
102
D. BLANCHETET AL
NOMENCLATURE a A A1 A~ b B
edge distance of vertical load for bending energy coefficient for bending energy intermediate contact area area of crushed ice width of contact constant for bending energy scale effect CA coefficient for dynamic effects on bending energy CD coefficient for bending energy C~ contact factor (0.3) CFA strain rate factor Cs. diameter of the indenter ( 3.6 m ) d dm increment of mass of crushed ice ds increment of displacement along horizontal plane increment of time dt dx increment of displacement along sliding surface plate modulus o f inertia = Eh 3/ 1 2 ( 1 - v 2) D E Young's modulus of ice local bending energy EBL & crushing energy E~ local cracking energy E~ elastic energy ship/ice friction energy local ice energy EL reference Young's modulus (at Ao)=0.01 E0 m2=4500 MPa) plastic energy EpL ice removal energy ER F momentum force of escaping ice F~ crushing force FN normal bow force on ice edge friction bow force Fvo-r F~ vertical ship load on ice (24.5 MN ram KR 426 ) function of ( ) f( ) g gravity acceleration G deformation modulus of ice (G~, Gt) H~,/4~ geometrical horizontal forces for ice removal energy thickness of ice h h1 height of loaded area k subgrade modulus or unit weight of water K bulk modulus of ice l characteristic length of ice sheet m[, m 2 masses of broken ice P elastic radius of ice sheet e~,e~ geometrical vertical forces for ice removal energy p total elastic, plastic and dynamic stress crushing energy per unit volume - a class of avpc erage pressure p. total elastic and dynamic stress Q plastic deformation modules of ice Q,,Q~ volumes of broken ice Q, tangent plastic deformation modulus of ice r shape factor in dynamic loading shape factor for dynamic loading including /'con confinement dynamic impedance of ice governing forces which resist high speed indentations
Ri,Rs R~ R2
R3 s u v v~ V Wi Wp x
geometrical resultant forces for ice removal energy subscript refers to rods and elongated bodies subscript refers to plane stress conditions subscript refers to half infinite space horizontal displacement under the indenter speed of ejecting dislodged ice initial impact speed sliding velocity of bow on ice volume of ice broken ice mass for removal energy work required to dislodge the crushed ice geometrical distance for ice removal energy
For cracking energy a¢ crack length CR constant dependent upon reference grain size = 9 E Young's modulus of ice m, constant in crack f o r m u l a e - 1.7 × 10-6 k-1 MI constant in crack formulae - 1.7 × 10-3 N crack density at E$ and X¢ Nc critical crack density in the following conditions: = 1.04× 10 -4 xC=0.47 mm N¢=0.055 t = time of crack formation ig= 1.33 X 107m - l d l = l mm s=l K= 1 b=0.34 a - r ( T = 2 6 3 K ) = 2 . 5 × 10-4s -I Qtc crack activation energy QcL local crack propagating energy R~ universal gas constant T, T~,7"2 temperature for crack formation Tk crack forming temperatures (degrees K = Tf¢) ye surface energy=0.09 J m -s ami, crack initiation stress Greek symbols A displacement before flaking= 0.06 h strain strain rate (3.3× 10 - 2 ) E~ elastic static strain % plastic strain # friction coefficient u Poisson ratio of ice (0.3) P(Pl,P2) density of ice (910 kg m -3) O" stress trb break limit of indenter pressure trbs break limit of indenter pressure for static conditions ( 18.2 MPa) ad dynamic stress ae elastic static stress O'min minimum stress for first crack ap plastic stress au unconfined compressive strength angle of repose ( 30 ° ) ~/ internal fraction angle (8.5 ° )
103
LOCAL ICE ENERGY DISSIPATIONS DURING SHIP RAMMING
Local Ice Energies EL
I
.aln
I--~'~
Energy I S en'
componentsJE' gY [
Loc., Ilcr° hing Crackingl J EnI ;gy Energy I
Ei"
,I
I-.lls- ce Energy EFL
#
Friction Energy EFR
T
Energy Sub-Components
Fig. 1. Flow chart for local ice energy dissipation.
ICE ENERGIES LOCAL
Zone1 I
J
Zone2 ~ Ice Removal
~ Flaking\
Zone3 Zone4
Ice Removal ~
(':("i(':':i"! ~,aking
t'7:"-':'):'."
ELEVATION VIEW
(A) Figs. 2A, B. Identification of local ice indentation zones dnnng ship ramming tests.
PLAN VIEW
(R)
104
(Ghoneim and Keinonen, 1983; Kigoriak and Robert LeMeur, 1983; Daley et al., 1984). In addition, high-speed indentation tests and drop ball tests in ice (Nawwar, 1979; EI-Tahan et al., 1983) and indentation tests in rock (Ladanyi, 1966) have also provided useful data on the behavior of ice under impact loads. Local phenomena observed during these tests are those occurring in the vicinity of the applied indenter force. Typically, the formation o f four zones can occur (Figs. 2A and B). Zone 1 represents the zone of complete removal of the ice where the bow of the ship was located at the end of the impact. Zone 2 represents the zone of a thin layer of crushed ice and water including flakes. Zone 3 is the zone of damaged ice resulting from micro and macro-cracking, and radial and circumferential cracks and, zone 4 is the zone of elastic deformations where the ice remains intact but deforms elastically under static and dynamic stresses. The extension of those zones will depend on the ice type, temperature, impact speed, ice thickness, ship and ice stiffnesses, interaction geometries and the duration of the load (Ladanyi, 1966; Michel, 1978; Sodhi and Morris, 1986). The local ice energy dissipations shown in Fig. 1 are limited to zones 2 and 3. For zones 2 and 3, average stresses are first derived for each identified failure, deformation and friction mechanism. These stresses are then converted into total energy dissipations.
MODEL PARAMETERS Many parameters affect the local ice energy dissipations. They are, namely, confinement, interaction geometries, unevenness of the top and bottom ice surfaces, presence of cracks or flaws, presence of water (viscous layer), impact velocity, ship's added mass, load duration, scale parameters and finally the contact factor. For the purposes of this study, many parameters were simplified. For example: - The bow geometry is fiat with a 25 ° stem angle. - The aspect ratio (contact width/contact depth) is constant and equal to eight in the case of the Kigoriak. - The ice floe is fiat with a uniform square edge.
D. BLANCHET ET AL.
- The ice type is defined as being multi-year ice with random crystal sizes and orientation mostly made of granular grains. - Flawed ice is naturally present in any ice at micro scales. This effect is included in the ice properties. - Only first rams are considered since cracks will result from a first ram. Loads from subsequent rams would be affected by the damaged ice zone. - For the ship/ice interaction model, the stress response of the ship is not considered. - The horizontal added mass was included in the total of the input energies from the ship.
L O C A L ICE E N E R G Y D I S S I P A T I O N MODELS A model for each local ice energy is presented. Crushing energy dissipation model
In order to derive crushing ice energy dissipation equations, stress equations must be developed. Stresses in ice, during indentation arise from strain and stress waves. A generalized stress-strain relationship in static and dynamic loading is shown in Fig. 3. The general formulation for the failure criterion for high-speed impacts has been presented by Kivisild et al. (1987). The detailed derivation of the static and dynamic stresses in ice follows. All coefficients and symbols are defined in the Nomenclature. Elastic static stress
A simplified expression can be written for the surface stress in a half infinite space of ice under a circular static load on the plane surface. For an elastic medium, the ratio between the displacement and the diameter of the indenter is expressed by:
S/d= ( n / 4 ) ( I - v)ae/E=0.550 adE
(I )
By inverting Eq. 1, the elastic static stress is given by:
a~=(4/n)(E/(1-u))s/d=e~E/O.550f(~)
(2)
The static surface stress depends also on the strain rate as shown above. By using the bulk modulus K
LOCAL ICE ENERGY DISSIPATIONS DURING SHIP RAMMING
105
20
15
Legend: (7d = dynamic stress O ' p = plastic stress O"e -- elastic stress "~
10
Residual
I Energy to Failure
Op
t ,,0.0006
g5 i ~ ¢0 ,-nc
~ / 0.0004
/
0 0.010
Unconfined Compression
°el
0.05
0.10
0.15
Displacement Related to Diameter of Load
Fig. 3. Definitionsof local elastic,plasticand dynamicstresses. and the deformation modulus G of the ice to replace E and u, Eq. 2 can be written in a form which is directly usable in plastic ice and in remolded states of ice:
~ = ( 4/~)6G/ ( I + ( 4 /3 ) G / K ) ~ j ( ~) ~ = (41g)
Q~f(~)
(3) (4)
where:
Q= 6G/(1 + ( 4 / 3 ) G / K )
(5)
The limits of these parameters can be calculated for different conditions such as for rigid non-deforming and moldable materials. The ratio G/K in the elastic range is expressed by:
G/K=~[ ( 1 - 2 v ) / 1 + vl
(6)
For ice, u=0.3 which gives G/3K--O.154, Q=3.71 G and Eq. 4 becomes: ae =4.72 Ge~f(~)
(7)
Elastic static and dynamic stress During high-speed impacts, static and dynamic stresses add uo. Using the rod analogy of dynamic stresses, the total elastic and dynamic stress is expressed by:
Pe=ad +ae
(8)
or:
Pe= ( rpR ) ~/2v+ 4.72G ~ef( ~)
(9)
The dynamic component develops almost instantaneously as the ship hits the face of the ice. Further load is added during the continuation of the penetration. The dynamic impedance R depends on the state of the stress caused by the indentation. This depends on the geometry of the target ice. The factor r has been introduced to consider the influence of the geometry of the load. Contact areas during the impact are idealized as a patch load on a half infinite body of ice. In this confined condition, the dynamic impedance R3 of ice can be shown to have two components. Part of the response arises from the action of the remolded ice as a fluid (viscous fluid or granular material). Another part of the response arises from the deformation resistance of the material itself:
R3=K+(4/3)G
(10)
The bulk modulus is unchanged with the change of material from the elastic range to remolded range. It is also valid for fluids:
106
D. BLANCHETET AL.
K=E/3(1-2v)
(11)
The second term is a function of the deformation modulus for ice. In the fully plastic and in the remolded states, G becomes small compared to K. Fluid effects start to dominate. The insertion of these values gives the following expression for impact pressure by an indenter. Equation 9 becomes:
pe=[pr(K+(4/3)G~)]l/213+4.72~eGf(~)
(12)
The subscript "s" represents static conditions. In the elastic region, the impedance R is equal to 135E (R3), 0.91E (R2) and E(Rl) for, respectively, a half infinite space, plane stress conditions (ice sheet ), and an elongated body where E = 4195 MPa (static) or E=6991 MPa (dynamic) and tr~= 26% of trb~ (Wang and Ralston, 1983 ).
Plastic static stress The remaining of the stress shown in Fig. 3 is the plastic stress or stress during damaging of the ice. This stress can be simply approximated by subtracting the total stress (at the breaking limit) and the static stress at the elastic limit during penetration:
ap=abs--tre=4/n(1--v)f(~) f Q d e
(13)
or: ~p =
1/0.55f(~) at (ep)
(14)
where ~p= (ebb-- ~)- The subscript "t" represents tangent. Secant modulus may be used to relate stress to strain.
Total elastic plastic static and dynamic stress
ity, whereas the plastic and elastic portions of the total stress will decrease due to their strain rate or velocity rate dependency. This equation will be converted into a crushing energy equation which is presented in the following section.
Conversion of stress into crushing energy dissipation Crushing is defined as the complete pulverization of the ice from initial micro-fissuring which typically results in uniform ice particles of very small dimensions (less than l cm of linear dimension). The crushing energy is given by the following general equation:
Ec/V=pc = fFc ds/fAcds
(16)
where Pc represents the amount of energy to failure per unit volume of ice crushed. Essentially, three methods can be used to calculate the crushing energy per unit volume of ice crushed. Method 1: Actual ice pressure data measured during icebreaker interactions with ice (such as the M.V. Canmar Kigoriak impact tests) can be used and corrected for different ice and impact conditions. Method 2: The uniaxial crushing strength of ice at small scale can be used and corrected for different ice and impact conditions taking the size effect into account. Method 3: Method 2 can be further developed by breaking down the crushing strength of ice into two parts; the elastic and the plastic limits during the deformation of the ice. This method is used in this paper. The energy per unit volume of ice crushed is given by: Pc = CSR CA CFA (EEL + EpL ) / V+ ( rconpRavg:e,p) 1/213
The total stress at failure will be given by:
(17)
/
p = p e -I-6p
(15)
where subscript "avg:e,p" represents the average elastic impedance of the ice for elastic/plastic ranges.
Equation 15 represents the total stress during indentation which is the summation of the dynamic, elastic and plastic stresses. The dynamic portion of the total stress will increase with the impact veloc-
This energy dissipation is the result of the initial deformation of the ice under the initial low stresses. Wang and Ralston (1983) calculated the elastic limit of the stress to be about 26% of the yield/plas-
p= [pr(K+ (4/3)Gs) ],/z13 +4.72 ~eGf(~) + 4.72¢pGt f([)
Local elastic ice energy dissipation EEL
LOCALICEENERGYDISSIPATIONSDURINGSHIPRAMMING
tic strength. This is close to the uniaxial compressive strength of ice. Elastic deformations during high-velocity impacts are very small. The amount of energy involved, before fracture or plastification occurs, is given by: EEL V= 0.5 O'ere
(18)
EEL/V=O. 106 (a~/G) f( ~)
(19)
for: 0 < v < 0 . 9 1 abJ (pE) 1/2
(20)
which gives v<3.3 m s -1, using E = 0 . 5 4 Eo and ab~= 0.3 trob~,EEL/V= 0 otherwise. Local plastic ice energy dissipation EpL As the stresses increase, fracture and permanent deformations of the ice occurs. Plastification is then initiated until complete failure of the ice occurs. The energy dissipation associated with plastic deformations is given by:
EpL / V--- ( (O'bs- O'b)/2)Epf(d )
(21)
107
A theoretical coefficient of friction of/t = 0.15 was used. The effects of impact velocity and type of ice were not included.
Ice removal energy dissipation model Ice removal, in this paper, represents two phenomena: (a) the expulsion of the crushed ice out of the contact zone, and (b) the remolding of the crushed ice which accumulates on top of the ice at the contact zone during the impact. The ice removal energy dissipation is calculated assuming the broken ice is a friction material with negligible cohesion. The following equations were developed for a load per unit width. Forces occurring along the front face (Fig. 4a) are disregarded. From field observations, it is suggested to assume 50% voids in the broken ice mass. A value for the escape speed u can be obtained by comparing ice volumes Q2 and Q1 (Fig. 4a). The mass of ice which is accumulated on top of the floe is the same as the mass broken off the floe. From geometry, we obtain the following relationships:
EpL/V=O.55( (Crbs-- re)/2)(ab --ae) f(d)/Q, (22) for: 0 < v < 0 . 6 7 crbs/(pE) 1/2
(23)
which gives v< 2.5 m s- i. EpL/V= 0 otherwise. For ship velocities greater than 0.91 abs/(pE) 1/2, which gives v=3.3 m s -l, EpL and EpL=0. Therefore, the work for indenting fractured ice is derived from the requirements of dislodging the material:
Wp=4.5 pv2A(d)h I
.,,
x
/
f.-
Front Face
I I
b" d x
0.3e4
(24) H$
Ship/ice friction energy dissipation model The friction between the ship's hull at the bow and the ice contributes significantly to the total energy dissipation process. Only dynamic friction is considered. The ship/ice friction energy is given by: EFR = f FToTdX
(25)
where: f T o T = fNfl
(26)
dx=vsdt
(27)
Pi
Fig. 4. a. Description of parameters included in the derivation of the removal energydissipation. b. System of forces for the remolding of crushed ice on top of the ice sheet.
108
D. BLANCHETETAL.
m, = mz = O.085bs2pl = Q l p l
(28)
= Q2pz = O. 129x2pzb which gives u = 1.15 v. Escape m o m e n t u m at constant speed, and a width of indentation b in the direction parallel to the bow hull are as follows: F=bu (dm/dt)
(29)
By inserting the geometrical parameters given in Fig. 5a, the following result is obtained: F = 0 . 2 0 v 2 bs p
(30)
where Fr~=0.2 v 2 bs p cos 12.5 °. As found above, escaping ice moves past the ship's hull at a low differential speed. This is still sufficient to engage the friction of the ship. A substantial resistance is obtained from the ice surface. Resulting forces may be split into vertical (P) and horizontal ( H ) components to derive equations for escaping ice (see Fig. 4b). The relationship o f vertical forces is: Ps + Wi=Pi
(31)
Ps = - Wi + e i = Q l p b g + e i
(32)
where/t = tan g/= 0.15 and g/= 8.5 °. It should be noted that the derivation of friction forces is obtained as the small difference of two large values. Therefore, there could be occasional jamming with ensuing sudden release and large impulses. There could also be periods o f easy flow when friction against the ship becomes inactive or when the ship moves faster than the ice and would actually drag dislodged material with it. In addition to these effects there are silo effects in the indenter zone which will cause further impediments to the flow of ice. The horizontal force can be evaluated by introducing the dynamic impulse which has been defined above ( H + FH = 0): Ps = - 0 . 3 2 vZpbs-O.O8pgbs 2
(37)
The horizontal force Hs is given by: H~ = - P ~ t a n (25 ° - ~u)
(38)
= - 0 . 3 ( - 0 . 3 2 p v Z b s ) - 0 . 3 ( -O.08pgbs z) = 1.0 v2pbs+ 0.02 pgbs 2 The energy equation may be written as follows:
where, from geometry, we obtain the following equations:
ER = 1.0 p f vZbs ds+ 0.02 pgf bs2ds
P~ =R~cos(25 ° - ~u),
Ice bending energy dissipation model
(39)
H~ = - P, tan (25 o _ ~), Pi = Ri cosq~, Hi = Pitanq~
(33)
The repose angle ~ is assumed to be equal to 30 °. To find Pi, we need to s u m the horizontal forces. The total horizontal force may be written as follows: n = Hs + Hi + Hactive
(34)
where nactive is defined as the horizontal force generated by the friction, if the ship is moved past the mast. Hpassive would apply if squeezing o f the broken ice mass occurs. Hac,ive =P~( 1 - s i n 0 / 1 + sin~) = (1/3)Ps
(35)
Using Hi=P~ tan q~and Eq. 31, Eq. 35 and H s = - P ~ tan (25 ° - ~), Eq. 34 becomes: H = - 0 . 3 Ps +0.049 pg bs2 +0.333 Ps +0.577 Ps (36)
Local bending of the edge o f a semi-infinite ice sheet occurs during the transfer of the bow vertical load to the ice during ramming, if the ice is sufficiently thin or weak. Under the impact of the ship, the edge of the ice sheet will be subjected to a combination of bending and compressive stress waves. It is assumed that the ice sheet is separately responding to pure bending and compressive waves. This leads to a small under-estimation o f the total bending strain energy dissipations. A quasi-static analysis of the pure bending o f the edge o f a semi-infinite and isotropic ice plate is proposed. However, the energy equation can be corrected to account for the following factors: (a) The parameter a representing the distance between the concentrated load F~ and the undeformed ice edge estimated to be equal to 3l. (b) The ratio between a semi-infinite to an infinite ice sheet can be generalized by using the coef-
LOCAL ICE ENERGY DISSIPATIONS DURING SHIP RAMMING
ficient CE(2 < CE < 5 ) (Westergaard, 1925 ). (c) The dynamic amplification factor representing the effect of the added mass of water increasing with the increase of the rate of loading of the ice sheet is generalized by using the coefficient CD(4
Em=4B2Drta 2 1 + v) +z~K [ ( 1/2)AZa2+ (1/2)ABa4+ (1/6)BZa61 -FyA (40) where:
A+Ba2[(2/3)+16D (l+v)/kaZ]=O
(41)
and:
A+ ( l/2)BaZ=Fy/(n ka z)
(42)
109
level was estimated to be around 0.6 MPa (Gold, 1972). Parameswaran and Jones (1975) using a brittle model derived by Petch (1968), calculated the fracture stress at 0.3 MPa. Their results, however, gave a fracture stress varying between 0.1 and 0.5 MPa. The crack initiation stress is given by (Sinha, 1984): O'min
=E(M1 - ml T)l"s/CR
Assuming that the stress Crrninis an average value of the stress for creating the first crack over the entire bow print area, the activation energy for cracks forming at times T~c and T2c and temperatures of TI and T2, is given by:
Qtc={RJ[ ( 1 / T , ) - ( 1 / T z ) ]}Xln
{(aT21ar,)x (In[1 - [ ( M 1 m~T~)/CR](E/a)*]/ (48)
The general Eqs. 41 and 42 given above, become: A = 0.0772 Fy/(k/2)
(43)
B= - 0.0093 Fy/(kl 4)
(44)
When inserted in Eq. 40, the above equations give the following bending energy dissipation equation:
EBL = -- 0.038CEF2/(kl 2)
(45)
or in more details: EBL=0.13 CEF2/(k2/ZE~/2) (CDh) 3/z
(46)
where: hdynami c = CDhstatic For this study, only the results for thick ice are relevant to compare to measurements. The above equations can also be used as a component in calculations for thin ice. Large bending energies from thin ice sheets will not be discussed since verification measurements were lacking.
Local ice cracking dissipation model Local cracking is a contributor to other phenomena such as crushing and flaking. However, when crushing and flaking are ongoing processes in zone 2 during the interaction, the remainder of the cracking energy is spent creating and propagating cracks in zone 3. The work to initiate a crack at the first micro-scale
(47)
ln[l - [ M - m , T2) / CR] (E/a)*] ),/b} For ice at - 10 ° to - 5 ° C, typically, Qtc is equal to 80-100 KJ mo1-1 at a=0.6 MPa (or tr/ E>5×10-5). The other parameter to be studied is the crack density or the number of cracks created under a given stress per area. Sinha (1984) suggested the following equation: N=Ncexp [~( CRdl/K) (a/E) s {1 - e x p [ -
(avt)b]}-xO]
(49)
Assuming a given strain level in zone 3, the number of cracks can be calculated or estimated using Gold (1972). The strain energy in zone 3 will be equal to the cracking energy when the threshold average strain is reached. In addition to this cracking initiation energy, the local cracking propagation energy is present in the damaged area (zone 3). It is given by: QCL= (rttr2a2/E-4ac?~b)h
(50)
Ice flaking energy dissipation model Flaking is the mode of ice failure which ultimately dislodges pieces of ice. Flaking can be
110
D. BLANCHETETAL. h~'~~ ~ F l n l t e
L ~~.lce
Block
PARTIAL CONFINEMENT
NOT CONFINED Difficult for Extrusion
Infinite Block of Ice
t ~ ~ I'~ ~.
Indenlor~ _ Effectof
"~oped
Difflcul for Extrusion
j v. "~ Sides O~ SHIP/ICE INTERACTION (PARTIAL CONFINEMENT)
]
FULL CONFINEMENT
at 25 ° angle to one of the sides. The total load reaches a value which is sufficient for pieces of ice to be broken-out by the flaking process. The condition at which it occurs has been expressed deterministically using the Mohr-Coulomb failure theories, combined with various results on the strength of ice in different confinements. Using partial confinement conditions (Fig. 5 ), the maximum centre-located load area is given by (Kivisild et al., 1987 ): A1=0.76 (hi) 2
The load applied on this area will be applied at full confinement stresses which gives a maximum force value of: F = 3 . 1 4 (hi)2tru
Fig. 5. Different confinement scenarios for flaking.
TABLE 1 Average stress ratios related to unconfined compression with partial geometric confinement Slope angle interface
Collision
Molded
Line formations 90 ° ridge 65 ° ridge 45 ° ridge 25 ° ridge 0 ° ridge - 90 ° cavity (25 ° +65 ° ridge)
0.57 1.21 1.86 2.70 4.14 10 1.67
1.00 1.87 2.57 3.27 4.14 10 1.87
Cylinderformation 90 ° walls
0.77
1.00
through failure planes such as in unconfined compression or there could be shear faces curving up to the surface as in normal bearing type loading. Figure 5 shows flaking failures which could occur at different confinement scenarios described above. As shown in Fig. 5 and Table 1, various types of loading result in various flaking stresses, starting with uniaxial stress and reaching higher stresses o f several times this load at full confinement (Kivisild et al., 1987). A solid, flat, unlimited indenter is assumed to collide with a 90 ° corner of a quarter infinite block
(51)
(52)
This analysis leads to the result that the face of the ice will have areas covering 0.40 (contact factor) of the total aggregate ice area under crushing loads which are equal to bearing capacity. The colder the ice and the faster the interaction, the lower the contact factor is to be expected. At high impact speeds, larger than 2 m s- t, the contact factor measured during the MN Canmar Kigoriak Impact Tests was about 0.3 to 0.4. Work by a contact force occurring at a maximum flake may b e expressed as follows: EEL = trbAA 1= 0.758trb (h t ) 2A
(53)
EXAMPLES AND COMPARISON WITH F I E L D DATA The purposes of the above models is to predict the average global load on the bow of a ship during the bow impact, slid-up and forefoot impact phases. The models also help to identify the most important failure processes during each phase of the interaction. The dissipation of the total input energy (ship kinetic and propulsion energies) into different ice and ship failure, deformation or friction mechanisms can also be followed throughout the
ram. A comparison with some of the measurements made during the 1981 Kigoriak Impact Tests was conducted and some of the results are presented.
LOCAL ICE ENERGY
DISSIPATIONS
DURING
lll
SHIP RAMMING
30
Ram KR426 (Kigoriak 1983)
~6 22
18 14
v 6. "O
g-~lpfias~s:'
F
' .i
o Up r
. . . .
e
;~
. . . .
f
o
~
o
. . . .
;~
t
'
sE~.'
|
Prior
-10
To
-14
Impact
-18 .I
B.B2
BOW
Fig. 6. Bow force time history KR426.
D u r i n g the period from 5-13 October, 1983, 202 rams were completed by the Kigoriak in h e a v y multi-year ice. During this r a m m i n g period, the highest global load on the bow o f the Kigoriak was m e a s u r e d during r a m K R 4 2 6 (Fig. 6). The initial i m p a c t speeds was 4.6 m s - l . Borehole jack and uniaxial and confined compressive strengths o f ice were collected during the ship trials in 1981 and 1983. They were extrapolated to r a m m i n g conditions o f crushing velocity varying f r o m 2 m s - l to zero which gives an average strain rate o f 3.3 × l0 -2 s e c - i. Table 2 together with Fig. 7 a n d the following section summarizes the energy balance during r a m
TABLE 2 Kigoriak Ram KR426 ( 1983 ) (calculated energy dissipation) Energy (Mg) Input energies Ship kinetic Ship propulsion Total input energy Distributed ice energies Crushing Flaking Friction Removal Bending Cracking Forefoot crushing Forefoot friction Total ice energies Potential energy Total distributed ice plus ship Ship kinetic left Total distributed ice energy (KE+ice+ ship) Unaccounted energy (ice + ship) Time from initial impact (s) Max. meas. force (ran)
Before bow impact
End of bow impact phase
End of slide-up phase
End of ram (ship stopped )
103.9
103.9
103.9
103.9
103.9 0.8 104.7
103.9 1.4 105.3
0.0 0.0 103.9 103.9 0 0 0
8-10* 0.2-0.5* 8.1 0.8-0.9 0.3-0.6 1.5-2.0"(o) 20.6 (avg) 0.9 21.5
9-11 0.2-0.5 15.3 0.8-0.9 0.1 1.5-2.0 28.5 (avg) 19.2 47.7
9-11 0.2-0.5 19.7 1.3-1.45(F) 0.1 1.5-2.0 31.4"* 2.8 67.5 (avg) 33.6 101.1
81.7 103.2 ( avg )
53.7 101.4 ( avg )
0.3 101.4 ( avg )
- 0.7
- 3.3
- 3.9
1.8 14.0
3.0 10.0
0.62 24.5
(o) Estimated cracking energy from strain measurements (Table 4) =0.08 MJ. *Crushing, flaking and cracking measured energy- 12.7 MJ from force/penetration diagram. Method 1 was used. (F) Forefoot impact included. **Includes cracking and flaking,
1 12
D. BLANCHETET AL.
T
Bow-Ice Friction
Crushing
80-
1) 2) 3) 4)
Remaining Distributed Energies Propulsion Energy Fore Foot Friction Input Energy
60. I,IJ
._= hi
m
c
in
i
40-
20.
00
2-0
1-0
Start
End
~ Initial
Initial
Impact
~ Impact
3-0
First
Time S
~ Fore Foot Contact
Ship ~ Stopped
Fig. 7. Breakdown of input and distributed energies for RAM KR 426.
KR426. All parameters required to calculate each energy are given in the Nomenclature. The input energy is made up of the initial kinetic and propulsion energies which gives the total input energy at any instant. The distributed energies are made up o f all ice and ship energies and the remaining kinetic energy which summed to equal the input energy. During the bow impact, slide-up and forefoot impact phases of ram KR426, these energies can be summarized as follows. Prior to bow impact
(A) Input: Initial kinetic energy plus propulsion energy.
(B) Distributed: No energy is distributed before the impact.
After bow impact before slide up
(A) Input: The propulsion energy contributes an additional input. It is a small addition as duration is short but the value depends on power setting. (B)Distributed: The distributed energy at the end of the bow impact phase can be identified as follows: - Ice crushing energy; about 10% of initial KE. - Ice friction energy; about 7-8% of initial KE. - Ice removal energy; about 3% of ice crushing. - Ice flaking energy; about 3-5% of ice crushing and depends on the initial velocity. - Potential energy; as there is relatively little vertical ship motion, the potential energy at this stage is only about 3% o f the total potential energy present in the ship when it finally stops. The kinetic energy consumed is approximately 20% o f initial KE.
LOCALICEENERGYDISSIPATIONSDURINGSHIPRAMMING - Ice bending, local cracking, ship global and local strain energies; collectively consume less than 2% of initial kinetic energy. Local strain is negligible throughout the ram. Global strain is a function of the bow force. During the initial impact it rises to a m a x i m u m of less than 2% of initial KE. Prior to forefoot c o n t a c t after s l i d e up
(A) Input: Propulsion energy; about 50% o f propulsion system total increment has been added. (B) Distributed: The distributed energies at the end of the slide-up phase can be identified as follows: - Friction energy; has continued to rise as ship slides up and represents about 15% of initial kinetic energy. - Crushing, flaking, ice removal, and cracking energies; little additional ice is destroyed during this phase. Only about 11% is added to values at end of the bow impact. - Bending energy; more than 50% has been recovered. - Ship global and local strain energies; together much less than 1% o f initial KE. They are recoverable. - Kinetic energy; about 40-50% of initial K E has been consumed. The remaining K E has been transformed into rotational and translational KE. - Potential energy; the vertical displacement o f the bow of the ship has increased significantly during this phase. The potential energy represents one of the largest distributed energies at this stage and can have risen to about 50% of the PE end value or 20% o f the initial KE.
113 TABLE 3 Crushing energy calculations during bow impact phase ram A7 Crushing energy (a) Crushing energy= (contact area) (pressure) (penetration) (sin 24 °). (b) Friction energy= (average pressure) (average contact area) (length of contact) (coefficient). (c) Pressure= (average pressure on contact area) (d) Penetration= (penetration rate) (time) Time (s)
A
P
PEN
Ec
Contact area (m 2)
Pressure (MPa)
Penetration (m)
Crushing energy (MJ)
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
4.5 5.7 5.7 4.6 6.2 7.1 8.5 5.7 6.3 6.5 3.5 1.2
1.8 1.5 0.9 1.4 1.1 1.2 1.8 2.4 2.4 2.1 1.5 1.1
0.28 0.12 0.11 0.10 0.09 0.07 0.06 0.05 0.09 0.02 0.01 0.00
0.92 0.42 0.23 0.26 0.25 0.24 0.37 0.28 0.25 0.11 0.03 0.00
End of bow impact Average contact area= 5.5 m 2 Average pressure= 1.6 MPa. Total penetration = 0.95 m. Total crushing energy = 3.36 MJ, calculated = 5-6 MJ. Contact length = 3.0 m (from movie data). Coefficient of friction = 0.15. Friction energy (impact phase)=(5.5) (1.6) (3.0) (. 15 ) = 3.96 MJ, calculated 5 MJ.
S h i p s t o p p e d after forefoot i m p a c t
(A) Input: Propulsion energy; in rams considered, only adds about 1-2% to initial kinetic energy. (B) Distributed: Forefoot crushing, friction, and removal energies; collectively represents the dominant combined ice energy. The total value is greater than the combined values o f crushing and friction due to bow. This energy accounts for approximately 40% of the initial KE. The distributed energies at the end o f this phase can be identified as follows: - Bow friction energy; 20% of initial KE. - Ice bow crushing, flaking, ice removal, and cracking energies; another 11% added in values given in the previous ram phase.
- Kinetic energy; has been fully expended. - Potential energy; represents about 35% o f initial KE. The total unaccounted energy given in Table 2, shows the possible errors due to both measurements and theoretical calculations, Ram KR-A7 (Kigoriak 1981)
The measurements of local ice loads and the global movement of the Canmar Kigoriak in ramming were conducted in August and October 1981. The instrumentation of the Kigoriak and the data collected
D. BLANCHETETAL.
114 TABLE 4 Uniaxial strain energy in ice Ram
B 227 B 230 B 231 B 234 B 235 B 236
ES
ESA
(J m -3 )
(MJ)
0.009 0.26 0.082 0.121 0.067 0.121
0.003 0.078 0.025 0.036 0.020 0.036
Impact velocity
Average strain
(ms -1)
(X10 -6)
1.39 1.03 1.54 3.09 3.09 3.60
2.99 5.12 9.0 11.0 8.2 11.0
Peak global strain energy/volume = ( 1/2) E e 2 Apparent modulus of elasticity= 2 × l03 MPa Averageglobal strain in ice Approximate volume = 3 X 10s m 3 E S A Average global strain energy in ice.
Es
E e V
were presented by G h o n e i m and Keinonen (1983). In addition to the ship information, strain measurements on the ice surface were conducted by B.P. ( G o o d m a n and Duckworth, 1981 ) in October 1981. Six wire strain meters were fixed to the ice using ice screws. Crushing, friction and strain energies were estimated, respectively, for R a m s KR-A7 and KRB227, B230, B231, and B234-B236. The results are given in Tables 3 and 4. The "crushing" energy in these calculations include cracking and flaking energies. In Table 3, the critical crushing and friction energy values for r a m A7 are also given. They compare well with the calculated values if the crushing energy includes flaking and cracking energies.
CONCLUSIONS ( 1 ) Six local ice p h e n o m e n a were observed during full-scale impact tests. They are: crushing, friction, removal, flaking, bending, and cracking. Energy dissipation equations have been derived for each phenomenon. (2) The crushing energy dissipation has been derived based on an analysis of dynamic, static elastic and plastic stresses during indentation. (3) Friction, bending, and cracking energy dissipations were derived from original or classical the-
ories. New theoretical solutions for ice removal and ice flaking energy dissipations were presented. (4) For the Kigoriak impact tests, the most important distributed energy dissipations which account for more than 95% of all the distributed energy are: - Ice knife crushing; - Ice friction (including ice knife); Ice Crushing (Bow); and - Ship potential energy. (5) The impact velocity is the most important parameter affecting all but the flaking energy dissipation. It also considerably affects the ice pressures and forces. (6) There are two important ranges of ice behaviour during ship ramming: - At low ramming speeds (0.5-2.5 m s - I ), the ice behaves in a brittle manner and the ice loads are low due to static conditions. - At high ramming speeds ( > 2.5 m s - ~) ice loads become large amplitude impulses. Elastic stress waves propagate in remolded ice and increase the resistance. (7) The combination of the ship/ice friction and the ice knife/ice friction represents about 20% of the total distributed energy.
ACKNOWLEDGEMENTS The authors would like to thank Canadian Marine Drilling Ltd., Canadian Coast G u a r d Northern and Lavalin Offshore Inc. for the opportunity to work on this most interesting project and for their permission to publish this paper. We greatly acknowledge A. Churcher (simulation model) of Canadian Marine Drilling Ltd., Dr. A. Keinonen and Mr. Colin Revill for their contribution to the project, " S h i p / I c e Interaction Energies". Appreciation is also due to the Panel of Energy Research and Development, who under Task 6.1, financed the original project. REFERENCES
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