U.S.C.G.C MACKINAW General Dimensions L.0.A. 290' L.B.P. 280' Beam, extreme, molded 74' -4" Depth, molded, to maindeck amidships at side 28'-0-5/8" Draft 19' -0 Displacement 5125 L.T.F.W. Number of screws 1 forward 2 aft Total maximum BHP 10,000 Diesel-Electric Drive Frame spacing 16" Plating forward near waterline 1-5/8"1

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR INVESTIGATION OF STRUCTURAL STRESSES IN ICE-BREAKING VESSELS EDITED BY JESSE ORMONDROYD Professor of Engineering Mechanics Project M720-6 U. S. NAVY DEPARTMENT, OFFICE OF NAVAL RESEARCH CONTRACT NO. N6-onr-232, TASK ORDER VI June, 1950

INTRODUCTION A series of tests was carried out on the U.S.C.G.C. Mackinaw, the purpose of which was to study the forces which act upon a vessel engaged in breaking ice and the reactions occurring in the hull girder of such a vessel. The project was broken down into three principal divisions: (1) Theoretical investigations of the forces required to break ice and the stresses produced in the ice due to the ice-breaking operation; (2) experimental investigation of the motion of the icebreaker including the rigid body motion and elastic deformation. The theoretical work was done at the University of Michigan, Ann Arbor. Experimental studies of the physical properties of ice were made both at the University of Michigan, using artificial ice, and aboard the Mackinaw in January, 1948, using lake ice. A special testing machine was built to determine the ultimate strength, modulus of elasticity, shearing strength, Poisson's ratio, and the coefficient of friction between ice and steel. Experimental investigations of the rigid body motion and elastic deformation of the vessel were conducted in September, 1947, and January, 1948. The September tests were planned to find the normal mode shape factors in deep and shallow water and to test an accelerometer, to find the order of magnitude of the ship beam stress, and to lay out a final testing procedure. A vibration generator was mounted in the eyes of the ship and run at various speeds and eccentricities. Records were made of the motion of the ship by means of a portable pallograph. Strain gages were located in six positions on the ship's beam structure, and records were made during vibration tests by means of a recording oscilloscope. The tests conducted in January, 1948, were designed to find the stresses in the hull and vibration of the hull during ice-breaking operations, the manner in which ice is broken and the forces exerted by the vessel on the ice. Speed and power measurements were made while the vessel was running through the ice. Simultaneously, records were made of about twenty-four strain gages on the hull girder and on the plating. Vibration was recorded by a pallograph and three accelerometers. Moving picture cameras were mounted at the bow and amidships to photograph the ice-breaking process. All of the electronic equipment and most of the mechanical equipment used in these tests was supplied by the David Taylor Model Basin. The personnel operating this equipment belonged to the staff of the Model Basin. The tests were planned by members of the staff of the U. S. Coast Guard and by members of the staff of the Engineering College of the University of Michigan. The tests were executed by the personnel listed in the attached passenger lists and by members of the crew of the U.S.C.G.C. Mackinaw. The analyses of the test data were carried out by the staff of the University of Michigan. iii

TABLE OF CONTENTS Page INTRODUCTION iii SUMMARY OF IMPORTANT RESULTS AND TEE MOST IMPORTANT vii RECOMMENDATIONS CHAPTER I A. HISTORY OF THE PROJECT I-1 B. OUTLINE OF TESTS MADE IN 1947 AND 1948 I-29 1. Test Party, September, 1947 I-29 2. Test Party, January, 1948 I-30 35. U.S.C.G.C. Mackinaw Ice-Breaking Tests - Schedule of Tests I-32 4. U.S.C.G.C. Mackinaw Ice-Breaking Tests - Station Bill I-36 C. INSTRUMENTATION FOR ICE-BREAKiNG TESTS OF U.S.C.G.C. MACKINAW I-40 CHAPTER II DISCUSSION OF TEST RESULTS II-1 CHAPTER III RECOMMENDATIONS FOR FUTURE TESTS III-1 APPENDIX A MOTIONS OF ICE-BREAKER HULL SECTION a RIGID BODY MOTIONS OF THE HULL Part I The Velocity of the Icebreaker Through the Ice AaI-l v

Page Part II Trim of the U.S.C.G.C. Mackinaw While Moving Through Thin Ice AaII-1 Part III Squat Test Analysis AaIII-1 SECTION b ELASTIC DEFORMATIONS OF THE HULL Part I Vibration Tests of Hull Girder to Determine the Natural Frequencies of the Three Lowest Normal Modes of Vibration in the Vertical Plane AbI-l Part II Analysis of Longitudinal Girder Strain Gage Records and Pallograph Records Made During Ice-Breaking Runs in January, 1948 AbII-l Part III Analysis of the Hog-Sag Test Data AbIII-1 Part IV Accelerometer Records AbIV-1 Part V Long Base Strain Gage in 301-A AbV-1 APPENDIX B FORCES ACTING ON THE ICEBBEAEER HULL SECTION a ICE THRUST MEASUREMENTS Ba-l SECTION b STEADY STATE GIRDER STRESS ANALYSIS Bb-l SECTION c PLATE STRESSES CAUSED BY EXTERNAL IMPACTS OF ICE Bc-1 APPENDIX C PROPERTIES OF ICE AND STRESSES IN ICE S EETS SECTION a PHYSICAL PROPERTIES OF LAKE ICE WITH PARTICULAR REFERENCE TO STRENGTH Ca-l SECTION b BIBLIOGRAPHY ON LAKE ICE Cb-l SECTION c STRESS IN A PLANE SHEET OF ICE LOADING IN THE PLANE OF THE ICE SHEET Cc-i

Page SECTICI d Part I Bending Stresses in a Continuous SemiInfinite Ice Field CdI-l Part II Bending Stresses under a Concentrated Load on the Edge of a Semi-Infinite Ice Sheet CdII-l SECTION e STRESS ANALYSIS OF A CRACKm ICE SHEET Ce-l SECTION f REPORT ON ICE CRACK PHOTOGRAPHY, U.S.C.G. CUTTER MACKINAW Cf-1 vii

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I SUMMARY OF IMPORTANT RESULTS AND THE MOST IMPORTANT RECOMMENDATIONS 1. MECHANISM OF BREAKING ICE - THE FORCES INVOLVED As the ice-breaking vessel advances through thin sheet ice, the ice is split in cracks radiating forward from the forefoot of the ship. Bending cracks, roughly at right angles to the split cracks, form continuously several feet forward of the bow. The cracked ice is "ploughed under" the hull. The ice ultimately breaks into large pieces 5 to 6 feet wide and 10 to 15 feet long along the side of the ship. Contacts between these pieces and any given point on the hull near the ice level is intermittent and irregular. This intermittent contact excites the ship into continuous vertical vibrations in the lowest mode of vertical vibration motion. On the Mackinaw, this vibration is 2-noded and has a natural frequency of 3.45 cycles per second. The amplitude of this vibration varies, in a beat-like fashion, with variable beat frequency. During the icebreaking tests of January, 1948, the largest double amplitude of vertical vibration at Frame 10 was 1/3-inch. This motion led to the largest dynamic stress in bending recorded at gage No. 5 + 1875 lb/in2. This vibration, in combination with the forward velocity of the ship, probably determines the average width of the broken ice cakes.

ENGINEERING RESEARCH INSTITUTE X | UNIVERSITY OF MICHIGAN The contact forces between the hull and the ice are of five kinds: splitting forces at the bow, probably small; bending forces which break the ice, relatively large; inertia forces which change the velocity of ice from zero to the "ploughing under" velocities, undoubtedly the largest force of all, especially at high speeds; flotation forces from the broken ice beneath the ship, about five pounds per cubic foot of submerged ice; and friction forces which are not more than 10 percent of the magnitude of the other forces. A very important part of the friction forces arises from the "machining" action of the raised weld beads on the ice. These forces give rise to resistance to the forward motion of the ship; they lift the forward part of the ship, changing the trim angle; and they squeeze the ship. The breaking forces vary as the thickness of ice raised to the 5/4 power (h5/4) and the inertia forces vary as the weight of ice, the thickness of the ice to the first power and the velocity squared (t h v2). This indicates that the resisting, lifting and squeezing forces all vary in general in the following form: F = aD h5/4 + b D h v2 b where a and b are constants to be determined by test D = the mean draft of the ship h = the ice sheet thickness v = the velocity of the ship. A general picture of this relationship at one value of D is given below:

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN xi MAXIMUM STALLING THRUST FORCE'2 _OF THE SHIP FORCE F h3 HIGHEST SPEED OF THE /SHIP IN AN ICE SHEET OF THICKNESS h2 STALLING THRUST h2 IN ICE OF THICKNESS h3 hi ~ \ C/V ZPN VELOCITY OF SHIP —V At resistances less than the maximum stalling thrust the ship proceeds through the ice at a constant speed with trims by the stern increasing with ice thickness. Above the maximum stalling thrust of the ship the ship can advance through the ice only by "charging" and heeling and by periodic changes of trim as the ship climbs onto the ice and breaks it down. During the January, 1948, tests the calculated ice resistance varied between 86 and 99 percent of the combined sum of ice resistance plus water resistance. Maximum operating ship velocities ranged from one-third to onehalf open water velocities with the same propeller rotational speeds. 2. PHYSICAL PROPERTIES OF FRESH WATER ICE Weight density A = 56 lbs/ft3 YounIg's modulus E = 0.5 x 106 to 1.0 x 106 lbs/in2 Poisson's ratio = 0.35

ENGINEERING RESEARCH INSTITUTE xii UNIVERSITY OF MICHIGAN Ultimate stress, tension Ta. = 200 lbs/in2 to 250 lbs/in2 Ultimate stress, shear 7max = 100 lbs/in2 3. SHELL PLATING STREESSES CAUSED BY ICE IMPACTS ON THE U.S.C.G.C. MACKINAW The low magnitude of shear ultimate strength probably acts as a safety.valve and limits the normal surface loading between ice and hull to magnitudes around 200 lbs/in2. The spacing between frames and the width of contact area on the ice cakes in combination with the limiting normal loading probably limit the maximum plating stress in sheet ice of any thickness to be encountered in the Great Lakes to below 20,000 lbs/in2. This is well below the yield point of high tensile steel. The maximum stresses encountered in the ice impacts in January, 1948, (thickness between 7 inches and 18 inches) were as follows: Run 4AIIIb, Rosette Strain Gage No. 29: = - oo6400 lbs/in2; y = - 3000 lbs/in2; y = - 1120 lbs/in2 From these data the maximum stress in the shell plating midway between two frames was - 6735 lbs/in2. From this it can be inferred that the maximum stress at the juncture of the frame and the plating was of the order of 13,000 lbs/in2. 4. MAXIMUM BENDING STRESSES IN THE HULL GIRDER OF TEE U.S.C.G.C. MACKINAW CAUSED BY ICEBREAKING The maximum bending stresses measured in January, 1948, and calculated under assumed conditions are given below for the positions of longitudinal strain gages No. 4, 5, and 6.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN xiii Dynamic Ship Ship on Run Run Hog- Stress Run Hung Up Standar 5 A 5 B Sag 2 Noded 4BIc on Ice TroGage 15' 20' Test Mode 7"-18" Ledge at choida Windrow Windrow 7'"-18" Ice Frame 11 Wave No. Meas- Meas- Meas- Ice Calcu- Calcu- Calcuured ured ured Meas- lated lated late lbs/in2 lbs/in2 lbs/in2 ured lbs/in2 lbs/in2 lbs/in2 lbs/in2 4 - 593 -695 -645 --— 163 - 2890 - 3400 5 - 732 - 778 -990 +1875 - 194 - 1920 - 4175 6 - 760 - 8o5 870 ---- - 147 - 1590 - 4715 For a maximum ice thickness of 40 inches the dynamic stress will probably not exceed +1 4000 lbs/in2 at gage No. 5, and the steady, unidirectional stress at the same gage will probably not exceed - 2000 lbs/in2, and it might need a windrow to bring it that high. An important limiting factor on these stresses is the low velocities of advance which will be possible in thick ice. The two automatic "safety valves" on ice loadings are the low value of ultimate strength of ice in shear and the low velocities of advance which thic ice will enforce. 5. MAXIMUM VERTICAL LOAD ON THE FOREFOOT OF THE U.S.C.G.C. MACKINAW The maximum calculated vertical load which can be exerted by the forefoot of the Mackinaw is about 10 percent of the displacement under static conditions and about 20 percent of the displacement under dynamic (charging) conditions. At about 5000 tons displacement this load would give rise to about - 3000 lbs/in2 tr at gageas No. 4 and the dynamic (charging) condition would give rise to about - 6ooo lbs/in2 stress at the same gage.

ENGINEERING RESEARCH INSTITUTE xiv UNIVERSITY OF MICHIGAN Under static conditions a solid sheet of ice about 12 feet thick could be cracked on its underside Just beneath the load. Under dynamic conditions a solid sheet of ice about 17 feet thick could be cracked on its underside Just beneath the load. 6. THE SHIP HULL AS A. DYNAMOMETER TO MEASURE ICE LOADS Perhaps the most interesting possibility developed during the analysis of the ice-breaking tests was that of converting the whole ship girder into a dynamometer capable of measuring bending moments and compression continuously. This can be done in the following manner. Longitudinal strain gages can be attached to the ship girders in pairs, one above the neutral axis and one below the neutral axis, both in the same plane, with 6 to 10 pairs distributed along the ship. These will give signals which can be analyzed into direct compressive strains and simultaneous bending strains. If the stern propellers are operating, with the bow propeller idling, the direct compressive strains will be proportional to the combined wind, water and ice resistance forces. The bending strains will give bending moment distribution curves which, when differentiated graphically twice, will give vertical ice loadings. This assumes that the location of the neutral axis is known with some accuracy at each cross section. These measurements can be checked against the power and thrust measurements. This proposal might even suffice alone to give direct evidence of ice loadings. If the scheme really worked out well, it could be made part of the standard equipment of an icebreaker to gather continuous information on ice conditions encountered.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Xv 7. MOST IMPORTANT RECOMMENDATIONS In order to benefit from the experience and information gained to date, repeat tests should be run at times when the ice has attained its maximum thickness. A program of testing lasting over two or three ice seasons could probably be run for much less than 5 percent of the cost of design and construction of a vessel of the same dimensions as the Mackinaw. Such tests might indicate that thinner plating and wider frame spacings could be used, compatible with safe operation. If this should be the case, the cost of testing would be regained in the cost of construction of a single vessel. The repeat tests could be run with simplified instrumentation, accurate and effective planning, and a small number of test personnel. A reliable wheel ice log should be designed and built, capable of being run by one man. The general specifications for such a wheel are contained in the body of the report. The wheel ice log is the simplest means of getting ship velocities. The reduction of data from a wheel ice log is not expensive nor time-consuming. This is in sharp contrast to any other method which can be used for this purpose. Ice thickness must be measured at regular intervals over each test course after each run is completed. The test runs should be run at pre-set operating conditions, such as 1) Power necessary to start from standstill 2) Half maximum available power 3) Full available power.

ENGINEERING RESEARCH INSTITUTE XVi UNIVERSITY OF MICHIGAN These power conditions should be repeated for each draft, trim and propeller combination used. The wind resistance should be minimized in all test runs either by running tests only when the wind velocity is less than 20 miles per hour or by taking the wind on the beam in all runs. One set of test operating conditions used over and over again until consistent data are obtained would be far more valuable than any number of hastily executed tests under widely varying conditions. The twenty-one shell plating strain gages should be replaced by one or, at most, two vertical columns containing three to four rosette strain gages distributed between the 14-foot water level and the 19-foot water level. These should be placed in a convenient space such as the dry cargo hold. The January 1948 tests indicated that all the gages received roughly the same number of ice impacts, so that one or two columns of four gages each would supply the same information just as well as twenty-one gages. Only rosettes should be used. Finally, all records should be developed as taken and complete calculations on carefully prepared forms should be made on the ship as the tests progress. This will detect faulty tests and lead to their replacement by test runs which are satisfactory. This procedure will also lead to the speedy production of the report on the tests.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN CHAPTER I A. ISTORY OCF THE PROJECT In March, 1946, Coast Guard personnel made observations on the U.S.C.G.C. Mackinaw during ice-breaking operations. A study was made of the variables involved in icebreaking. These are listed in Table I. It was also learned that the ship's inclinometer, draft gages, and tank gages were not accurate enough for scientific work. From the information obtained, suggestions were made for instrumentation and procedure for future tests (see the Agenda). In the following year an attempt was made to run a series of tests on the Mackinaw and on a 180-foot tenders according to the Agenda. The duty assigned these vessels, however, precluded a complete series of tests. Instruments were put aboard the Mackinaw, and some tests were run between assignments. Data were obtained on power, drift, trim and ice conditions (Table II). Further knowledge of instrumentation was obtained. With this background of exploration tests, Rear Admiral Ellis ReedHill wrote to Professor J. A. Baier of the Department of Naval Architecture and Mearine Engineering, University of Michigan, in May, 1947, suggesting a coamprehensive series of tests aboard the Mackinaw to be conducted by University of Michigan personnel. The project was undertaken by the Department of Engineering Mechanics with the Department of Naval Architecture. Acting as

I-2 | ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN consultant, Admiral Reed-Hill outlined the study which the Coast Guard anticipated in a letter dated June 11: (a) The development of methods and instruments to obtain the following physical characteristics of ice: (1) The thickness of natural ice in a floating field (2) Tensile strength (3) Yield strength (4) Shear strength (5) Impact strength (6) Elongation (7) Density (b) Measurement of strains in selected locations in the hull of a vessel actually working in ice (c) Determination of the deflections experienced by a vessel working in ice (d) Determination of the vibration forces acting upon a vessel working in ice (e) Determination of the horsepower required during the ice-breaking operation From the results it was hoped to evaluate: (a) The optimum hull form for breaking ice (b) The optimum distribution of structural material in an ice-breaking vessel (c) The minimum horsepower required to break ice of a given thickness.

I-3 TAB3LE I COAST GUARD INSTRLENTATION EVALUATION OF FACTORS IINVOLVED IN ICEDEEAK SIGN EQUIPIE:NT AMD MAICEI1AW FACT OR 'VARIABLE UNITS EQUIPMENT METHODS MA( INSTRLWENTS PROBA]: PN~ 1. Ice Strength (a) Compression Pounds per 8q.in. (b) Tension " it it t (c) Shear it * it it (d) Viscosity 1012 Poises (e) Elasticity Poissonls Ratio Observed Velocity of Waves (f) Brittleness (g) Hardness Moh's Scale Resistance to Abrasion 1.5 (h) Rigidity Modulus of Rigidity Twisting Cylinders of Ice (i) Salinity Parts NaCl per 1000 Salinity Indicator Melt Taste or Analyse O (j) Crystals Size, inches Magnifying Glass Observe Flowers of Ice 1 to surface Cohesion (k) Size Thickness, inches Water lance, Ruler Scale Drilled Hole or 25" x 1 sqI mi. Field Area, sq.ft. Chart Broken Cake (1) Discontinuity Cracks Eye Grooves Ruler Conglomerate Water lance % Adhesion " q% water ti it (m) Temperature Degrees F HQ. Thermometer Bury in ice 32~F (n) Pressure Lbs. per sq.in. Strain gauges (o) Time Seconds Sweep Sec. Clocks A.C. Tracing Records 2. Breaking Force(a) Displacement Tons Draft Marks Read Dead in Water 18' - 20' (b) Form WL - ButtockL8 Scale Protractor.Fram Lines 200 - 30~ WL - Center of " " " From Curves of Form 140' - 200' flotation (c) Trim Trim Zs Draft Marks Stalled on Ice O - 3' total Sensitive I[Inclinometer Intersal Reading Transit) Rods Rod, Horizon Sights Ice Wheel Photographic (d) Squat Trim, Elevations Transit, Level Rod Level Sights while in open water tow- on Buoy towing; float dising at breaking speeds tances below bulwark (e) Contact Points Contact-Center of Curves of Form 0- - 140' from FP Flotation Lines Distaste in feet Vibration Meter Strain Gauges (f) Heel List Sensitive Incl:Lno Interval Rieadings 0 - 5~ meter 3. Total Work (a) Thrust Pollrds Strain Gauge Hold Thrust Block with O - 250,000 lbs. Hydraulijc Hydraulic Gauge Piston EquiLpment (b) Torg-ue Pound - Feet Torque Gauge (c) Horsepower Kilorwa~tts Volt, Ammeters and Use Model Basin Data O - 10,000 1g? Movie Cameras Recording IW meter (d) Distribution No. of Propeller Volt., Ammeters anal Use Model Basin Date O - 3 Propellers Movie Camneras (e) E~hetic Energy Speed Chante Ice Wheel Tachometer o - 16 E (f ) Angle of Thrust 4. Power Loss (a) Cavitation (b) Restrictive Water Depths Sound~ing Lead Mlodel:3asin Data Wakre Ice Thickness Ruler, Water Lance Model 33asin Data (c) Submerging Ice Ice Thickness Rulers Water Lance Calcu:late (d) Moving Ice Ice Thickness Ruler} Water lance (e) Ice Friction Coefficient of Protractor L of Repose f =.36 0 20~ (f) Water Lubri- Temperature Thermcmeter fire hose along water cat ion line; heat on shell 5. Hull Strenlgth(a) Shell Stress pounds per Strain Gauges or Stress Ma~terialJ sg. in. Coats Dimensions (b) Frames" Jo t1 t t Spac ing ( c) Girders t t t t t Joints (d) Decks,, t t Wt Design (e) 23eas" " Vibration (f ) Bulkheads " t tt of (g) Stem;" t *Ad *t (h) Rudder* *,,,,t ( i) BOSS >t t t t ttt fj) Propeller) Shafts" "" "

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-5 University of Michigan personnel experts in the various phases of the work, were assigned to parts of the test. Proposals were submitted by these men and sent to the Coast Guard August 13, giving the scope of work to be done, and the methods of obtaining the necessary data (see Outline from Letter of August 13). These suggestions were reviewed by the Coast Guard in a letter dated September 12. The parts approved were: theoretical investigation of forces required to break ice, experimental determination of physical properties of ice, experimental determination of the rigid body, elastic motion of the icebreaker, and electrical measurements of power. Dock tests to determine the ship's vibration characteristics had already been carried out from September 2-6. The analysis of propeller thrust was not approved, as it was felt that sufficient work had been done on this by Taylor Model Basin. The model study of the hydraulic effects in icebreaking was not considered feasible in view of scale effect and limited intrumentation. Development of an instrument for measuring ice thickness was not approved since a similar instrument was believed to be already under developrment Measurement of trim was found to be one of the critical measurements in ice-breaking tests requiring accurate measurement. Various methods were tried in the preliminary tests of March, 1946, and March, 1947. The ship's mercury-alcohol differential inclinometer was found to be not sensitive enough. Liquid oscillation, inertia, and temperature effects provided sources of error. Level rods and level determinations were also found to be unsatisfactory, owing to the time required to obtain a set of readings, and to the necessity for clear weather and a true horizon. The Coast Guard suggested the use of a gyro stabilizer, such as is used to stabilize fire control radar antennas. Two methods of indication were investigated: mechanically,

1-6 JI ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN by means of pens secured to the rotors of the control transformers; and electrically using the output of the control transformers. The mechanical method was discarded since the pens would load the control transformers, introducing error. The speed of the ship through the ice was another important quantity which was hard to measure accurately. In March, 1947 tests, a bicycle wheel fitted with a tachometer was to be run on the ice ahead of the ship by means of an outrigger. This equipment could not be used because the operations were in windrowed ice. A new device was built for the January, 1948 tests, consisting of a bicycle wheel rotated by a wire anchored in the ice. This is described in detail in other parts of this report.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-7 0utl'ne from Letters- August 13, 1942 Chief, Naval Engineering Division United States Coast Guard Washington 25, D. C. Dear Sir: We have received your telegram dated August 6 with regard to your conference August 5 at the Taylor Model Basin about the experimental investigation on icebreaking. The following method of handling pay and travel expenses for Model Basin personnel working on the project is proposed. The Department of Engineering Research, University of Michigan, can hire these men on the project and pay their travel expenses. It is suggested that the Taylor Model Basin personnel working on the project take leave without pay for the period of the tests and enroll as staff members of the Department of Engineering Research. The personnel of the University of Michigan working on this project will be available during the periods indicated below. (1) Professor Jesse Ormondroyd, Mr. George K. Hess Jr., and Mr. Robert L. Hess will be at the Taylor Model Basin from August 18 to August 29, 1947, and will be available for detailed consultation on the plans for the project. (2) The entire group on the project at the University of Michigan will be available for dock tests from September 1 to September 17, 1947. (3) The entire group will also be available for ice-breaking tests from December 22, 1947, to January 3, 1948, and from January 19 to February 4, 1948. The ship should be scheduled for dock tests and ice-breaking runs during these periods. The following paragraphs outline the proposed investigations to be made in connection with the study of icebreaking on the Great Lakes. This letter is not an official proposal from the Department of Engineering Research of the University of Michigan. It is an outline on which we can construct a proposal with as much detail as desired after the conference in Washington between August 18 and August 29, 1947. The investigation has been divided into several phases and assigned to the personnel indicated below.

ENGINEERING RESEARCH INSTITUTE I-8 UNIVERSITY OF MICHIGAN I. Theoretical investigation of the forces required to break ice and the stresses produced in the ice during the icebreaking operation. (I. A. Wojtaszak, Professor of Engineering Mechanics, University of Michigan. ) 1. OBJECT: To determine the stresses and displacement in an infinite and semi-infinite ice field due to loads similar to those produced by an icebreaker. 2. PERSONnEL: The analysis and calculation required for this study will require the services of two experts in the theory of elasticity (for about 100 hours) and at least two computers (100 hours). 3. EQUIP1,ENT: The equipment required, including the computing machines, is available at the University of Michigan. II. Experimental determination of the physical properties of ice, the coefficient of friction between ice and steel plates, and the development of equipment to determine the physical properties of ice in the field. (J. T. Wilson, Professor of Geology, University of Michigan.) 1. OBJECT: To determine the physical properties of ice and the coefficient of friction in appropiate kinds of ice under varying conditions of temperature and history and to develop equipment to determine these quantities in the field by simple tests. 2. PEESONNEL: The supervision and implementation of this program will require at least two graduate students in addition to the services of Professor Wilson. 3. EQUIPMEN-: -The equipment required for the tests to be conducted at the University of Michigan is either available or will be constructed at the University of Michigan. III. Experimental investigation of the motion of the icebreaker including rigid body motion and elastic deformation. (Jesse Ormondroyd, Professor of Engineering Mechanics, University of Michigan. ) 1. OBJECT: To determine the rigid body motion and the elastic deformation of the ship during icebreaking and to determine the natural frequencies of the ship.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN 2. PEESONNEL: The test program for this phase of the project will require the services of the electronics technician and Mr. R. T. McGoldrick to be loaned by the Taylor Model Basin, Coast Guard personnel, and about ten persons provided by the University of Michigan to operate the accelerometers, the vibration generator, the gyro stable vertical, the strain recording system, and accessory equipment. 3. EQUIPMENT: The following equipment will be supplied by the Taylor Model Basin: electrical time signal apparatus to synchronize all recording equipment, six vertical recording accelerometers, the vibration generator, gyro stable vertical, SR 4 strain recording system, and accessories. The University of Michigan will provide velocity recording equipment in the form of an ice borne log to record the main forward motion of the ship. IV. Effect of thrust developed by propellers on icebreaker progress through ice. (L. A. Baier, Professor of Naval Architecture and Marine Engineering, University of Michigant Chairman of the Department of Naval Architecture and Marine Engineering. ) 1. OBJECT: To determine tow-rope pull on ship operating under icebreaking conditions. This must be done by a combination of test data and calculations. In order to obtain all the data necessary for the calculations we still need: (a) Copies of the displacement and form curves. (b) Overload tests on a powered model. Professor Baier proposes that the Taylor Model Basin run these tests on a University of Michigan requisition (paid for by the project). Professor Baier will go over details of these tests with the Model Basin. 2. PERSONNEL: The calculations will be made by Professor Baier and one assistant when the test data is available. 3. EQUIPMEST: No equipment is needed at the University of Michigan. The model at the University of Michigan is not large enough for these tests and there are no facilities here for making powered model tests to at these data.

ENGINEERING RESEARCH INSTITUTE I-10 UNIVERSITY OF MICHIGAN V. Electrical measurements on the test run. (M. B. Stout, Professor of Electrical Engineering, University of Michigan). 1. OBJECT: To get time records of current, voltage, power, and motor speed on recording electrical meters. 2. PERSONNIEL: Professor Stout and assistance from the ship's crew. 3. EQUIPMENT: Recording ammeter, voltmeter, wattmeter, and a voltmeter to measure motor speed. We hope to borrow this equipment from the local power companies or provide it from the instrument storeroom of the University of Michigan. VI. Model study of the hydraulic effects in the icebreaking operation. (R. A. Dodge, Professor of Engineering Mechanics, University of Michigan). 1. OBJECT: To make measurements of the pressure forces on the hull of the model and on the simulated ice sheet with and without forward advance of the model, to evaluate the effect of the forward propeller, and to investigate the use of powered models on thin ice as an aide to design and the evaluation of the icebreaking characteristics of a ship by this means. 2. PERSONNEL: Professor Dodge and two assistants. 3. EQUIPMENJT A powered model with forward and stern propellers is needed. It may be obtained from the Taylor Model Basin or constructed at the University of Michigan. VII. Development of an instrument for measuring the thickness of ice. (J. R. Frederick, Research Physicist, Department of Engineering ResearchUniversity of Michigan) 1. OBJECT: To develop a portable instrument to measure ice thickness using ultrasonic vibrations. 2. PERSONNEL: Mr. J. R. Frederick

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-1l 3. EQUIPMENT: Electrical apparatus to be purchased. (Mr. Frederick has estimated the cost for the first model to be $10,000 and for each additional model $1,500. Mr. Frederick will be on the staff of Brown University after September 20, 1947. It is evident in view of the estimated cost that this development should be a separate project. This project could be given to Frederick at Brown University or another physicist may be located at the University of Michigan to carry on the development. ) VIII. Dock tests to be made in the period Septemter 1 to 15, 1947. 1. OBJECT: Vibration generator tests to find vertical modes of hull vibration; installation of strain gages to measure strain in ship plating and frame girders in the ice-'breaking operation, and bending of ship during vibration generator tests; hogging and sagging ship to calibrate strain gages on keel and deck plating for bending moment. 2. PERSONNEL: The services of Mr. R. T. McGoldrick and the electronics technician to be loaned by the Taylor Model Basin, Coast Guard personnel, and personnel from the University of Michigan will be needed for this phase. 3. EQUIPMENT: The equipment needed for this phase of the project consists of the vibration generator, SR 4 strain gage system, accelerometers, and accessories. These are to be borrowed from the Model Basin. The above paragraphs outline the investigations to be carried out on this project and are submitted for approval. Final plans can be arranged when we are at the Model Basin in Washington, D. C., next week. The attached enclosures are copies of the original suggestions made by the several persons working on this project here at the University of Michigan. Sincerely yours, Jesse Ormondroyd Professor of Engineering Mechani c s Enclosures A, B, C, D, and E.

I|1 2ENGINEERING RESEARCH INSTITUTE I-12 | UNIVERSITY OF MICHIGAN Enclosure A THEORETICAL INVESTIGATION OF FORCES REQUIRED TO BREAK ICE AND TBE STRESSES IN ICE DUE, TO ICE( BREAKING (Submitted by Professor I. A. Wojtaszak and P. F. Chenea) The object of the analytical study of the forces and stresses involved in icebreaking is to determine the stress- distribution due to a given load on an ice field as well as the deflection in the ice sheet. For the purposes of this investigation the ice sheet will be assumed to be an elastic plate on an elastic foundation. The force acting on the ice will be taken at least initially as a concentrated load.. The boundaries of the ice field will be assumed to be at infinity with the exception of an open area similar to the path of an icebreaker. The shape of the open area wil be assumed such that the actual conditions are approximated and still permit an expedient analysis. The problem of a semi-infinite sheet of ice will be a special case of the above analysis. The results of this analysis will be presented so as to show the stress distribution, deflections, and potential crack lines in the ice field. If the analysis permits, other distributions of the load will be investigated which more nearly represent the actual forces exerted by the icebreaker. Physical constants used in this analysis will be based upon the experimental studies described elsewhere in this report. Personnel required are estimated as two stress analysts and two computers. Equipment required including computing machines is available at the University of Michigan.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-13 Enclosure B Department of Geology July 19, 1947 Professor Jesse Ormondroyd 411-A West Engineering Campus Dear Professor Ormondroyd: Enclosed is a stmmary of the various ice tests that I think we might carry out. So far I have been unable to find a local source of lake ice, but I anticipate no difficulties in getting some in the northern part of the state. I do not have my graduate student help lined up as yet, as they are all out of town, but I should have some answers on that very shortly. One of them is supposed to be available August 7, and as he is a trained meteorologist I will start him to work on the bibliography and the job of going through a lot of the papers that may give is some help. I think we should be able to get a great deal of the preliminary work in the cold chamber done during the latter part of August and the early part of September, and long before the tests on the Mackinac in February, we should be able t;o get some ice from some of the local lakes or some brought down by the Coast Guard. I am a little afraid of basing too much on ice that has been stored through the summer, even though it was lake ice to begin with. I have been unable to find any references that deal at all directly with the problem of the loaded ice edge, but I still have a few places to look, and I will let you know if I find anything. Sincerely yours, James T. Wilson JTW:rlw Encl.

ENGINEERING RESEARCH INSTITUTE I-14 UNIVERSITY OF MICHIGAN PROPOSED ICE TESTS Object and Scope The main object of the tests is to determine various properties of lake ice that are of interest in the general problem of icebreaking. The breaking strength (in the plane of the ice sheet) and the coefficient of friction for steel on ice are thought to be two of the more important properties. Both of these will depend on temperature and crystal orientation. The crystal orientation may in turn depend upon the freezing rate and wind conditions at the time of freezing. The effects of temperature and orientation will be studied. As it is desired to design breaking strength tests that may be performed quickly and with portable equipment, a study will be made of the expected variation for individual tests and of the effects of specimen size. Young's modulus and the rigidity will be measured as they may be more representative cf the ice than the breaking strength. A considerable amount of information on ice is available. A comprehensive bibliography will be compiled and abstracts made of articles pertinent to the tests. Methods The breaking strength will be measured by loading at one or more points bar-shaped specimens constrained at the ends. The experiments will be conducted in a cold chamber so that tests may be carried out over a range of temperature greater than is expected under natural conditions. Specimens of various sizes will be tested in order to determine the smallest size that can be expected to give representative results. The Young's modulus of ice cannot be determined accurately by bend tests, but representative values may be obtainable and of use. The results of bend tests may depend upon the rate of loading and this will be investigated. It is planned to make dynamic tests for Young's modulus and the rigidity. It is possible that these tests, while not so simple, may give results that are more consistent and representative. Particular attention will be paid to the effects of crystal orientation and to the effects of orientation and temperature history of the ice. The friction measurements will be made by sliding blocks of ice over steel plates. Particular attention will be paid to temperature effects. An attempt will be made to evaluate the effect of pressure.

ENGINEERINGC RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN -15 Artificial ice will probably be used to set up the tests, but it is hoped that lake ice can be obtained for most of the work. Some of the tests can be repeated in the refrigerator compartment of the Mackinaw this winter using ice cut from the lakes at that time. This will give an indication of the practicability of the tests and will also serve as a check on the work done with cut and stored ice. Personnel At least two graduate students will be employed to help in the work. It is hoped that one of them can start on the compilation of a bibliography and abstracts in the near future. Schedule It is planned to begin the assembly of equipment immediately and to conduct some preliminary tests during the latter part of August. James T. Wilson Department of Geology University of Michigan July 19, 1947

ir~-16 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Enclosure C EXPERIMENTAL INVESTIGATION OF TIE LOADING AND MOTION CHAEACTERISTICS OF ICEBREAKERS USED ON THE GREAT IAKES (Submitted by George K. Hess, Jr.,) The object of the experimental investigation is to obtain information which will be of use in the design of icebreakers. It is expected that situations will arise where icebreakers must be designed for conditions other than those occurring on the Great Lakes, and that is will be necessary to know the relationships between such quantities as ice strength, ship strength, ship tonnage., power requirements, etc. The investigation has been divided into a study of the following topics: a. Physical properties of ice, and development of field test equipment. b. The way in which an ice sheet breaks under the action of loads which may be applied by an icebreaker. c. The hydraulic effects due to propeller action on the loads applied to the ice. d. The motion of the icebreaker in an icebreaking operation. e. Propulsive efficiency of the ship at several conditions. f. A measurement of the power needed to break ice. g. Measurement of ice thickness. The problem of determining the motion of an icebreaker can be divided into two parts: measurement of the rigid body motion and measurement of the elastic motion. The information obtained in these measurements can be used to determine the loadings put on the ice, the effectiveness of the ship as an icebreaker, the structural strength required in a vessel to be used for an icebreaking operation, and the elastic characteristics of such a vessel. These phases of the problem are discussed in more detail in the outline below.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-17 OUTLINE OF TBE XPERIMENTAL INVESTIGATION OF TEE MOTION OF AN ICEBREAKER IN TEE GREAT LATS I. MEASUREMEN32 OF TME RIGID BODY MOTION. The rigid body motion of the ship is to be determined for the purpose of estimating the forces acting on the ice and for use in connection with the measurement of the propulsive characteristics of the ship. Of the six degrees of freedom of the ship as a rigid body, only three are of interest in this problem. A. FORWARD MOTION. The measurement of the forward motion is needed-to find the relation between the rate of advance through the ice field and the power consumption. It is to be recognized that there are two cases of interest: steady advance through the ice and unsteady advance. We assume that the ice conditions to be met will be such that steady advance through the ice field will be possible. Then for the purposes for which this information is to be used it will be sufficient to measure the average rate of paying out of a light line over the side of the ship, the end being anchored in the ice. B. VERRICAL MCI2ION. The measurement of the vertical motion of the ship is needed to find the force acting on the ice. It is expected that the inertia of the ship will affect the forces produced on the ice so that it will be necessary to measure the vertical acceleration of the center of mass of the ship. The vertical motion of the ship vill be accompanied by a change in the buoyancy force which will in turn be reflected in the forces acting on the ice. The vertical motion of the ship can be deduced from the records of vertical recording accelerometers. C. TRIM. Measurement of the angle of trim is necessary for use in connection with the evaluation of the effectiveness of the ship's propulsion equipment. This measurement can be made by recording the output of the standard gyro stable vertical used as an element of fire control equipment. Dynamic action due to Ii. II. MEASURENEME F T~E ELASTIC MOTION. The elastic motion of the ship is to be measured for use in the evaluation of the elastic characteristics of the ship, the strength requirements for the ice-breaking operation, and the loading to which the ship is subjected. It is expected that the vertical beamlike flexure of the ship will be the most important elastic deformation. The differential equation of motion for the vertical vibration of the ship is SI' S ^ =f

ENGINEERING RESEARCH INSTITUTE I-18 UNIVERSITY OF MICHIGAN We are interested in finding the loading W as a function of time and distance along the ship. This differential equation suggests that, if the acceleration and bending moment can be measured, knowing the mass distribution of the ship, we can calculate the loading. The measurement of the elastic deformation falls into two parts. A. MEASREENT OF STRAIN. The ship may be fitted with strain gages of the electrical recording type at places that would indicate the actual strains corresponding to flexure of the ship. It is to be expected that the keel would be a satisfactory place to attach these gages. We recognize that there will be strain concentrations which should be elifminated from the measurements. If the gages are calibrated in terms of known bending moments by means of hog-sag loadings, then this calibration will include the effect of strain concentration and the gages may be read directly in terms of bending moment. This information may be plotted to obtain a time record of the bending moment as a function of distance along the ship from which we can calculate the first term in the above differential equation. B. MEASUREMENT CF VCERICAL ACCETLRATION. The vertical acceleration of the ship may be measured at the same stations at which the strain measurements are made. The usual recording type accelerometer may be used. It is to be noted that the information from both the strain gages and the accelerometers must be time synchronized. III. MEASUBEI OF TEE ELASTIC CHARACTERISTICS OF THE SHIP. The elastic characteristics of the ship can be determined by a still water test with a vibration generator. This will give information on the flexural characteristics of the ship and natural frequencies to be used in the evaluation of the dynamic action of the elastic ship on the ice. The information will also be of interest to the Bureau of Ships and the Taylor Model Basin in connection with certain researches on the dynamics of ship's structures.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-19 MAJOR EQUIPMElNT EEQURE4ENTS FOR TEE EXPERIMENTAL FIASTURJEMEI OF THE MOTION OF THE SHIP The following items of equipment will form the list of the major items needed for the measurement of the elastic and rigid body motion of the ship as described in the previous paragraphs. 1. Rotating drum, light line, and revolution counter for the purpose of determining forward motion of the ship. 2. Electrical time signal apparatus to be used to synchronize automatic recording apparatus. 3. Vertical recording accelerometers (six) to be used to measure the vertical motion of the ship at six stations. 4. Strain recording apparatus (SR 4 - 24 element system) to be used to measure the elastic deformation of the ship In vertical beamlike flexure. 5. Vibration generator to be used to determine the natural frequencies of oscillation of the ship and the other dynamical properties associated with elastic motion. 6. Gyroscopic stable vertical equipment to be used to measure the angle of trim of the ship. There should also be prepared recording apparatus to go with this equipment so that a time record of the trim angle may be obtained. PEBSONNEL REQUIRME1NTS FOR THE EXPERIMENAIAL MEASUREMENTS OF TEE MOTION OF THE SHIP The above equipment requires that personnel be available who are familiar with the operation of the same. It is estimated that the services of the electronics technician and Mr. R. T. McGoldrick to be loaned by the Taylor Model Basin, Coast Guard personnel, and personnel to be provided by the University of Michigan will be sufficient to operate this equipment.

ENGINEERING RESEARCH INSTITUTE I-20 UNIVERSITY OF MICHIGAN Enclosure D August 8, 1947 PROPOSAL FOR THE DEVELOPMENT OF AN INSTRUMENT FOR MEASURING THE TRICKCIS OF ICE The method proposed is that of measuring the time for an ultrasonic vibration to travel through the thickness of ice and back to the surface. The apparatus required would consist of the following units: 1. A device to produce the waves, such as a hammer, a pulsed magnetostriction oscillator or a pulsed piezo-electric crystal. 2. A means of detecting the echo which could be either a magnetostrictive, electrodynamic, or piezo-electric crystal type pickup. 3. An amplifier and cathode-ray oscilloscope to observe the received pulses. 4. Timing and synchronizing circuits. If variations of the elastic properties of the ice were sufficient to produce significant and unknown changes in the velocity of the ultrasonic vibrations as the equipment was moved from one location to another, the method would become somewhat more involved. Either sacmne means of calibrating the wave velocity in the interior of the ice in terms of a measurable wave on the surface, or a method of simultaneous observation at two or three points would be necessary. The apparatus could be constructed in two parts. One would be a probe unit to be taken onto the ice and weighing 10 or 15 pounds. This would be connected by an electric cable to the other unit of about 50 pounds on shipboard. The latter unit would contain the cathode-ray tube and most of the electrical circuits. The cost of the development of such equipment I would estimate to be about $10,000.00. Duplication of the final design would amount to $1500.00, depending, of course, on the simplicity achieved in the end product of the development.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-21 Whether or not I personally would be able to undertake or aid in the program of developing this instrument would depend upon the policy and present program of research of the Physics Department of Brown University, since I shall be on the staff there after September 20, 1947. Respectfully submitted, Julian R. Frederick JRF: ljp

ENGINEERING RESEARCH INSTITUTE 1-22 I UNIVERSITY OF MICHIGAN Enclosure E I. The principal object of the hydraulic tests is to determine the effectiveness of the forward propeller. The general method would be to determine the load on the ice and the change in this load produced by three different methods of operating this propeller, that is,forward, reversed, and static. There are three ways of proceedings, amely: A. To determine pressure distribution under the ice and on the hull. B. To determine the resultant force exerted on the ice by the hull. C. To conduct field tests of the powered model on controlled thin ice. II. To Determine Pressure Distribution (See A above). A supported wooden platform would be used to simulate the ice. It would be held in the ice position and would surround the hull as ice surrounds the proto-type. The model would be held in the stall position. Numerous pressure taps would be made (1) in the platform near the bow of the model and (2) in the hull of the model. These taps would be connected to a manometer bank so the readings can easily be recorded by photograph. The tests would be made over a range of thrusts. By integrating the pressure distribution on the hull and the ice and by considering the known forces such as weight, it will be possible to determine the effect of the forward propeller on the -forces which break the ice. The equipment necessary for Part II is: 1. A powered model. 2. A platform to surround the model. 3. A manometer bank equipped with a suction arrangement. 4. Photographic apparatus for recording manometer readings. III. To Determine Vertical Force and Horizontal Thrust on Ice. In this test the powered model and the platform described in Part II will be used, The model will be supported at a point on the bow, at the ice line, by wires leading to balances. These balances will determine the forces

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-23 produced under various operation conditions of the forward propeller. The vertical force found by this method should agree with the one found by the test described in Part II. The balance is the only equipment required for Part III in addition to the powered model and platform used in Part II. IV. Field Tests A powered model or a towed model could be tested and observed while breaking ice of controlled thickness. This could be done on a lake in cold weather. A channel would be cleared of ice and then allowed to freeze to the required thickness. The model could be powered or it could be towed by a dynamometer. Motion pictures taken by a camera mounted on the model or on a towing carriage would record the forces and the breaking pattern simultaneously. The equipment necessary for Part IV is: 1. A powered or towed model 2. Thrust meters or a dynamometer car. 3. Ice cutting equipment. 4. Motion picture camera. Respectfully submitted, R. A. Dodge AD:ljp

ENGINEERING RESEARCH INSTITUTE I-24 UNIVERSITY OF MICHIGAN PROPOSED METEIRING ARRANGEENTS PROPULSION MOTORS, U. S. C. G. CGUTIER 1'4ACI:NAW." Meters The recording meters will be Leeds and Northrup "Speedomax G." Each set will consist of three recorders, one each for motor armature, current, armature power input, and motor RPM. There will be two sets, one connected to the bow motor, and the other arranged to be switched to either port or starboard stern motor. These meters are self-balancing potentiometers, in which the voltage to be measured is balanced against a voltage on a slide-wire (standardized by means of a standard cell). In this way, no current is drawn from the measured circuit, making the readings independent of the resistance and length of the connecting leads. The meters have 10-inch, 100-division charts with a chart speed of three inches per minute. The pen of the recorder is positioned mechanically by a motor, controlled by an electronic balancing circuit. Pen position is thus independent of Jolting or vibration. The balancing process is rapid, being capable of pen traverse from zero to full-scale in less than two seconds. Current The current recorder is a 50 millivolt instrument, which gives the following full-scale ranges when used with the switchboard shunts on the ship,

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I I-25 Stern motor, 0-6000 amperes (forward direction only) Bow motor, 0-4000 amperes (forward direction only) The current recorder has an auxiliary transmitting slide-wire for pavower measurements. Power The auxiliary slide-wire in the current recorder has impressed upon it the motor armature voltage, through a suitable series resistor. The output voltage of this slide-wire is thus a function of the position of the slider (i.e.,, current) and the voltage of the motor, and hence is a measure of the power. This is an accurate method for the measurement of DC power, and should give values accurate to within 1%. A switch is provided on the control box to give multipliers of 1, 2 and 4, with corresponding ranges of 1500, 3000 and 6000 KW on a stern motor and 1000, 2000 and 4000 EW on the bow motor. Speed Speed is measured by the voltage of the tachometer generator. A series resistor is needed to adapt the range of the Speedomax (300 mv) to the tachometer voltage. A variable resistor is provided for calibration purposes. A filter is also provided to avoid possible trouble from commutator ripple at the lower speeds. Coordination of Records Provision is made for starting all chart drives simultaneously as a means of providing a common time origin for all the recording devices used on the project. The asswumption is made on the sketch that this will

ENGINEERING RESEARCH INSTITUTE I-26 | UNIVERSITY OF MICHIGAN be a 115V AC circuit; however, a different voltage can be used if desired by changing the coil of the operating relay. Protective Arrangements The high voltage (90OV) of this system requires special precautions to protect operating personnel and equipment. The Speedomax instruments, moreover, are not generally operated with more than 200 volts from the circuit to the case. In this application, there would be anything from zero to 900 volts (depending on ground conditions on the circuit), from the live parts to the case if the case were grounded. The current and power recorders will be mounted on a wooden panel, and surrounded by a wooden cabinet with wire-mesh or glass doors. Door switches will be provided so that the high voltage connections will be broken as soon as a door is opened. In this way the meters will be completely safe for servicing, changing of charts, etc. The 115V AC power for the amplifiers and the chart drives will be fed to the current and power recorders through l-to-l insulating transformers, to prevent breakdown of the electrical equipment in the recorders. M. B. Stout Dec. 26, 1947

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-27 AGENikA FOR TEST ON U.S.C.G.C. MACKIItAW 1. The following procedures for testing are to be followed in determining the ice-breaking characteristics of the MACKINAW. A kilowatt recording meter will be installed in port motor room for obtaining the horsepower on the port propeller. A recording volt meter will be installed in the starboard motor room and a 35 mm motion picture camera set to record the switchboard volt meter and ammeter instruments for the horsepower on the starboard propeller. A sweep second clock will be located in view of this camera. Readings of voltage, amperage and time in the bow motor room will be recorded by an observer on signal or by 35 mm movie camera. Far the majority of these tests the bow motor will idle to obtain no thrust either forward or aft. 2. Strain gauges will be located on the shell and frames forward in points assumed to have maximum stress and shall be wired up to the oscilloscope. Time will be impressed upon the oscnlscope manually by breaking and making one of the circuits with a switch. 3. For obtaining the squat of the vessel under power at no speed, an ice anchor imbedded on an ice flow and connected to the towing reel hawser, the towing engine hawser shall be rove throughout a snatch block centered in the towing notch. Another snatch block located approximately frame 183 shall be attached to a line with a weight of approximately 2000 pounds suspended over the starboard side and the deflection of the towing hawser shall be indicated by means of a surveyor rod on deck. This will permit a determination of the thrust relative to the horsepower being developed. The squat will be measured by means of board floats suspended over the bow and port and starboard amidship stations whereby change and trim relative to no power conditions may be measured in undisturbed waters. In addition to the draft readings by mean of steel tapes and floats, an inclinometer will be installed abaft the anchor windlass for a check. 4. For measuring the change and trim in ice, bicycle wheels will be fitted on outriggers forward and port and starboard amidships to give changes in distances between ice and bulwark top. Scales will be fitted on the upper end of these outriggers for obtaining measurements and a remote reading tachometer will be fitted on the forward wheel. A 35 mm camera set will be set up on the bow station to record the changes and elevation of the forward outrigger and the tachometer reading on the ice. As another check on trim, a surveyor's transit or level shall be set up on the bridge wing with surveyor's rods fitted to the jack staff forward and crane tower aft so that a reading of the horizon line on the surveyor's rod will give a rod reading indicative of the change and trim. These readin g shall be taken periodically on signal. 5. A Cox vibration recorder will be set up on the bow to record vlbratory amplitudes and periods. Observers located on the ice will measure ice thiclkness from blocks broken or with a water lance, also, giving general length and width dimensions of individual cakes. Markers thrown over

ENGINEERING RESEARCH INSTITUTE I-28 UNIVERSITY OF MICHIGAN from the starboard midship station will indicate locations of various time readings recorded on shipboard. Distances between markers will be recorded by stadimeter or tape. 6. Attempts will be made to obtain the coefficient of friction between snaw coated ice blocks and steel plates by means of a reading the maximum angle of repose on steel plate incline. This test will be made on blocks coated with a cold dry show and with snow flushed with water. 7. With the ship at rest, check readings will be taken of the drafts and the bicycle wheel outrigger elevation readings. With the vessel stalled in an ice windrow, check the maximum trim obtained from a charging run. Temperatures of ice and water to be recorded periodically throughout the test by the ice field station. 8. The docking circuit lJV will be used for controlling the test giving starts and stops of run in interval times of approximately 15 seconds.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I I-29 B. OUTLINE OF TESTS MADE IN 1947 AND 1948 1. Test Party, September, 1947 - U.S.C.G.C. Mackinaw Vibration Tests Taylor Model Basin 1. Mr. R. T. McGoldrick 2. Mr. V. S. Hardy 3. Mr. L. D. Anderson 4. Mr. J. P. Hendrican University of Michigan 5. Professor J. Ormondroyd 6. Professor E. L. Erickson 7. Professor W. W. Hagerty 8. Professor J. T. Wilson 9. Mr. R. L. Hess 10. Professor M. B. Stout U.S. Coast Guard 11. Capt. D. R. Simonson 12. Mr. S. W. Lank 13. Mr. R. C. Browning 14. Mr. P. G. Tomalin 15. Comdr. Doebler, C. 0., U.S.C.G.C. Mackinaw

ENGINEERING RESEARCH INSTITUTE I-30 UNIVERSITY OF MICHIGAN 2. Test PaLrty, January, 1948 - U.S.C.G.C. Mackinaw Ice-breaking Tests Bureau of Ships 1. Capt. L. V. Honsinger, U.S.N. Chiefs Design Section 2. Mr. Johannsen 3. Mr. E. C. Lloyd, Machinery Section U. S. Maritime Ccmimiissicn 4o. Mr. John Vasta, Research Section, Technical Division Gibbs and Cox 5. Mr. J. P. Doyle, Machinery Division 6. Mr. A. B. Gray, Hull Division American Bureau of Shipping 7. Ir. L. D. Weston, Chicago Office Fairbanks Morse 8. Mr. Walter Fischer, Research Division U. S. Coast Guard 9. Capt. D. R. Simonson 10. Comdr. R. T. Alexander 11. Comdr. R. D. Schidtman 12. Lieut. Comdr. S. F. Schumacher 13. Lieut. J. P. Latimer 14. Mr. P. G. Tomn1in 15. Mr. S. W. Lank 16. Mr. R. C. Browning 17. Lieut. Comdr. Case, Executive Officer, U.S.C.G.C. Mackinaw, in command

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-31 Taylor Model Basin 18. Mr. L. D. Andewrson 19. Mr. J. P. Hendrican, Electronic Section 20. Mr. V. T. Almasy 21. Mr. V. S. Hardy University of Michigan 22. Professor C. W. Spooner 23. Professor J. T. Wilson 24. Professor W. W. Hagerty 25. Professor E. L. Ericksen 26. Professor J. Ormondroyd 27. Mr. R. L. Hess 28. Mr. J. R. Packard 29. Mr. J. M. Horeth, Assistant to Professor Wilson Westinghouse Electric and Manufacturing Carpany 30. Mr. Tirk 31. Mr. J. A. Wasmund Detroit Edison Co. 32. Mr. E. D. Kane 33. Mr. H. G. Hammond Kingsbury Machine Works Inc. 34. Mr. W. E. Colley U.S. Coast Guard Photographic Unit 35. C. W. Wicks, PhoMlc 36. P. A. Biscuti, PhoMSc 37. G. H. Watson, PhoM3c

ENGINEERING RESEARCH INSTITUTE I-32 UNIVERSITY OF MICHIGAN 3. U.S.C.G.C. Machinaw Icebreaking Tests - Schedules of Tests 1. 26, January 1948 Calibration of gyro stable vertical and differential level gage. At dock or anchor. (Inspect after peak for ice.) Shift water from after peak to fore peak in increments (as ordered) taking draft readings, gyro oscillograph, and differential gage. 2. 30, January 1948 Squat Test A. Astern, both wheels full —then full ahead with both stern wheels to dead in water and ahead to 6 knots. B. Astern, both stern wheels full —then full ahead on all 3 wheels to dead in water and ahead to 6 knots. C. Same with bow wheel astern and ahead to 6 knots. Data to be recorded each condition: TMB Oscillograph Stable vertical Thrust Bicycle wheel interruption at dead in water Statham Accelerometers Westinghouse Oscillograph All power channels Detroit Edison Meters All power and RPM Float-tape record Inclinations from zero speed to 6 knots —speed to be measured by ship log. Material Required: Chip log for bicycle wheel Float tapes and floats (Differential level gage to be installed in Wardroom Pantry just aft of Windlass Room) 3. 27, January 1948 Strain Gage Exploration Test A. Two runs with even gages (I) Trim to 15' fw'd. (II) Trim to 17' fw'd. B. Two runs with odd gages (I) Trim to 15' fw'd. (II) Trim to 17' fw'd.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-33 Extra Run 3B16 Conditions Stern wheels only Speed (4 generators) (2 per wheel) Data to be recorded as follows: Ice thickness Wind direction and velocity Draft gages Gyro Stable Vertical Ice log speed 2 Thrust meters Power and RPM Westinghouse Leeds and Northrup Ship's Instruments Strain gages Pallograph 3 Accelerometers Long base strain gage transverse 3 Cameras Bow platform Sternhead Quarterdeck 4. Record Tests A. 28, January 1948 Trim to 15' at bow. I. Two stern wheels ahead No bow wheel (a) Barely moving thru ice (b) Intermediate power (c) Full power (6 generators) II. Two stern wheels ahead Bow wheel ahead (a) Barely moving through ice (b) Intermediate power (c) Full power (4 generators aft) (2 generators fwd) III. Two stern wheels ahead Bow wheel astern (a) Barely moving through ice (b) Intermnediate through ice (c) Full power (4 generators aft) (2 generators fr 'd)

I-34 ENGINEERING RESEARCH INSTITUTE - UNIVERSITY OF MICHIGAN B. Trim 17' at bow displacement as at 15'. I. Two stern wheels ahead 28, January 1948 No bow wheel (a) Barely moving through ice (b) Intermediate power (c) Full power (6 generators) II. Two stern wheels ahead 29, January 1948 Bow wheel ahead (a) Barely moving through ice (b) Intermediate power (c) Full power (4 generators aft) (2 generators Nw'd) C. 29, January 1948 Trim even keel 19' draft. I. Two stern wheels ahead No bow wheel (a) Barely moving through ice (b) Intermediate power (c) Full power (6 generators) II. Two stern wheels ahead Bow wheel ahead (a) Barely moving through ice (b) Intermediate power (c) Full power (4 generators aft) (2 generators Nw'd) III. Two stern wheels ahead Bow wheel astern (a) Barely moving through ice (b) Intermediate power (c) Full power (4 generators aft) (2 generators Nw'd) Data to be collected: Ice thickness Wind direction and velocity Draft gages Gyro stable Vertical Trim Ice Log speed 2 Thrust meters Power and RPM Westinghouse Leeds and Northrup Ship' s instruments Strain gages Pallograph 3 Accelerometers

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-35 Long Base strain gage transverse 3 Cameras Bow platform Sternhead Quarterdeck 5. 29, January 1948 Windrowed Ice. A. Full power two after screws (6 generators) B. Full power three screws (ahead) Condition: 17' draft at bow or as indicated. Ship to charge windrow from a distance of 3 ship lengths. Data to be collected as follows: Depth to windrow (ice lance) Strain gages (selected shell) Strain gages (3 hull girders selected) 3 Accelerometers 2 thrust Stable vertical Power and RPM Westinghouse Leeds and Northrup Pallograph Long base strain gages (transverse) Camera aimed at ice edge quarter deck Camera aimed at bow from crow's nest 6. 29, 30, January 1948 Hog sag tests (Night Test) A. Peak tanks full B. Peak tanks empty C. Peak tanks full Data: Draft and Tank conditions Hull girder strain gages Long Base strain gages 7. 30, January 1948 Calibration Run for Westinghouse 32 spots - 50-100-150 RPM (approximately) in open water.

ENGINEERING RESEARCH INSTITUTE i-36 I UNIVERSITY OF MICHIGAN 4. U.S.C.G.C. Mackinaw Ice-breakin Tests - Station Bill Station 1 - Bridge Master Control of all Tests Data: Wind velocity and direction, drafts, soundings, navigation. Personnel: Lt. Comdr. Cass, Acting C.O., U.S.C.G.C. Mackinaw Prof. J. Ormondroyd, Test Engineer Capt. D. R. Simonson, U.S.C.G. Liason Officer Comdr. R. D. Schidtman, Chief, Test Staff, Bridge Talker, Quartermaster, Recorder Station 2 - Weather Deck Aft Speed of Ship through Ice and Ice Samples Data: Ice log revolutions counted mechanically and by observation, photographs of ice, ice samples. Personnel: Prof. E. L. Eriksen, Head of Ship Speed Party Mr. S. W. Lank, Coordinator and Talker, PhoM c, U.S.C.G. Photographer Prof. J. T. Wilson, Head of Ice Party Mr. J. M. Horeth, Assistant 2 Seamen, Crane Operators Station 3 - Weather Deck Forward Ice Crack Photography Data: Photographs fram two positions of the ice being broken. Personnel: Prof. W. W. Hagerty, Head of Ice Crack Photography and Talker Mr. J. R. Packard, Photographer Station 4 - Boatswain's Locker Recording Room Data: 3 oscillographs, pallograph, Leeds and Northrup instruments, gyro stable vertical. Personnel: Mr. R. C. Browning, Coordinator and Talker Mr. L. D. Anderson, Chief of TMB Oscillograph Party Mr. J. P. Hendrican, TMB Electronic Engineer Mr. V. T. Almasy, TMB Electronic Engineer Mr. Tirk, Westinghouse Representative and Oscillograph Operator. Mr. J. A. Wasmund, Westinghouse Representative and Recorder

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN 1 I-37 Mr. H. G. Hammnond, Detroit Edison Representative and Recorder Mr. E. D. Kane, Detroit Edison Representative and Operator Mr. Johannsen, Bureau of Ships Representative, Gyro Operator Mr. V. H. Hardy, TMB Representative, Pallograph and Mechanical Strain Gage Operator Station 5 - Damage Control Office Computing Room Recording of all Test Data Personnel: Mr. R. L Hess, Chief Recorder Mr. P. G. Tomalin, Assistant Recorder and Talker 3 Messengers from ship's force. Station 6 - Forward Motor Room Data: Ship's instrument board Personnel: Prof. C. W. Spooner, Roving Assignment in all Motor Rooms and Shaft Alleys Lt. J. P. Latimer, Recorder and Talker Station 7 - Starboard Aft Motor Room Data: Ship's instrument board Personnel: Comdr. S. F. Schumacher, Recorder and Talker Station 8 - Port Aft Motor Room Data: Ship's instrument board Personnel: Comdr. S. F. Schumacher, Recorder and Talker Station 9 - Forward Shaft Alley Data: Thrust bearing oil pressures Personnel: Mr. W. E. Colley, Roving Assignment in all Motor Rooms and Shaft Alleys Mr. J. P. Doyle, Recorder, Talker, and Thrust Meter Adjuster Capt. L. V. Honsinger, U.S.N., Alternate

I-38 UNERSITY OF MICHIGANll Station 10 - Starboard Aft Shaft Alley Data: Thrust bearing oil pressures Personnel: Mr. L. D. Weston, Recorder, Talker, and Thrust Meter Adjuster Mr. E. C. Lloyd Station 11 - Dairy Room Physical Properties of Ice Data: Compressive and shearing strength of ice, modulus of elasticity, Poisson's ratio, and coefficient of friction against steel. Personnel: Prof. J. T. Wilson, Chief of Ice Tests Mr. J. M. Horeth, Assistant on Ice Tests 2 Seamen, Crane operator (same as on Station 2) Station 12 - Dark Room Development of oscillograph records as taken, and development of samples of moving picture film. Personnel: Photographer's Mate, U.S.C.G. Station 13 - Compartment A-301-A Long Base Mechanical Strain Gage Personnel: Mr. V. HE. Hardy, Mechanical Strain Gage Recorder and Talker Mr. J. Vasta, Assistant Station 14 - Squat Test Stations Data: Float tape readings Station 2 - port, main deck amidships Personnel: Prof. J. T. Wilson, Tape Reader and Recorder Amidships Station 2 - starboard main deck amidships Personnel: Mr. J. M. Horeth, Tape Reader and Recorder Amidships Station 3 - Bow, weather deck Personnel: Lt. J. P. Latimer, Tape Reader and Recorder Prof. W. W. Hagerty, Talker Forward

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-39 Station Bill: - Hog-Sag Tests Data: Strain gages 1-6, long base strain gages, drafts by gage, trim by gage and gyro, air and water temperature, tank soundings. Personnel: Mr. L. D. Anderson, Strain gages 1-6 Mr. J. P. Hendrican, Strain gages 1-6 Mr. V. T. Almasy, Strain gages 1-6 Mr. J. Vasta, Long Base Strain Gages - Transverse Mr. V. H. Hardy, Long Base Strain Gages - Longitudinal Prof. J. Ormondroyd, Drafts by gage - Bridge Mr. S. W. Lank, Tank Soundings Comdr. R. D. Schidtman, Trim by differential level gage Lt. J. P. Latimer, Air and water temperature.

ENGINEERING RESEARCH INSTITUTE I-40 i UNIVERSITY OF MICHIGAN C. lINSTRUMENTATION FOR ICE-B.EAtING TESTS OF U.S. C, G.,C. MACINEW Six sweep second Telechron clocks, synchronized, were provided for use at test stations. In addition, a public address system was used to start and stop test readings at all stations simultaneously. The tests were controlled from the bridge. Ice thickness was estimated by measuring samples of ice broken by the ship. The samples were hoisted aboard by the ship's crane fitted with special ice tongs. Wind direction relative to the ship was estimated by means of a pennant on the bridge. Wind velocity was measured by the ship's indicating anemometer, and was read on the bridge. Drafts forward and aft were read from the indicating draft gage on the bridge. Trim was measured by means of a standard Navy gyro stable vertical element Mark 8, mod 2, 4, as is used to stabilize fire control radar aerials. (See Figures 1 and 2.) The signal was recorded by a TB recording oscillograph No. OS140. The gyro was installed in the bosun's locker by Mr. Johannsen of the New York Navy Yard between 24 and 26 January 1948 at Manitowoc, Wisconsin. It was calibrated on 26 January with the ship at the dock. Water was shifted from the after peak tank to the fore peak tank in increments, while drafts forward and aft were read from a small boat, and by the gage on the bridge. The gyro calibration factor, using a 56-volt per inch transducer, was: 1 inch on chart = 36.9 inches trim, or tan 0.0124 in 248 feet, between draft marks (Figure 3).

I-41 I1.?. Figure la Gyroscope Stable Vertical Assembled with cover on

1-42 Figure lb Gyro8cope Stable Vertical Port side view- cover off

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5000.. HELICAL POTENTIOMETER TO ONE PHASE OF STABLE VERT. TO RECORDING GALV. OR OSCILLOGRAPH OUTPUT 36 /DEGREE ROLL STEP DOWN TRANSFORMER 50:1, 100:1, 26.7:1, 33'/3:1 The circuit used with the stable vertical gyro trim meter is8 shown above. The helical potentiometer was set to give a deflection of about 1 inch on the chart for the greatest trim expected. I NSTRUMENTATI ON GYRO FIGURE 2

1-45 GYRO STABLE VERTICAL CONVERSION CURVES U.S.C.G.C. MACKINAW ICE-BREAKING TESTS JAN.1948 40 _.016 351.014 30 _ _.012 FlU~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_ w w OD ~~~~~~~~~~~~z X 25.010 < Lz, ~~2O.008i20 G~~~~~~~~~~~ z z IlI0.006 5.002 0 0 0 20 40 60 80 100 120 MAGNITUDE OF GYRO TRIM SIGNAL HUNDRETHS OF INCHES I FIG. 3

i-46 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Speed through the ice was measured in two ways: by paying out a fine wire anchored in the ice, and by moving pictures. A bicycle wheel 5.76 feet in circumference and a spool (Figure 4) containing 15000 feet of 0.018-inch diameter wire were mounted in a wooden base which was fastened to the rail with "C" clamps. From the spool, the wire was led 1-1/2 times around the wheel and then extended aft. A 30-foot length of cord connected the end of the wire to several short lengths of truck tire chain which served as an anchor. A metal peg welded to the hub of the bicycle wheel contacted a small cantilever spring at every revolution, thereby closing an electrical circuit. The revolutions were recorded by three methods: visually by an observer, mechanically by means of a bicycle cyclometer, and electrically by means of oscillograph No. 0S140 located in the bosun's locker. The electrical impulses were produced by a contact soldered to one cog of the cyclometer wheel which contacted a fixed spring every fifth revolution, thereby closing an electrical circuit. During the squat tests, which were run in open water, a chip log replaced the concrete block in the bicycle wheel speed-measuring apparatus. The photographic record of speed was obtained by a 35 mm moving picture camera hand held on the weather deck aft aimed at the ice athwartships. The procedure in analyzing the films was to identify points on the ice and measure their displacements in successive frames. A Kingsbury 27-inch horizontal-type thrust meter was installed on the forward shaft to measure thrust in either the ahead or astern direction. (Figure 5) The starboard shaft aft was fitted with a 34-inch horizontal-type Kingsbury thrust meter to measure thrust in the ahead direction only. (Figure 6) The United States Coast Guard furnished the thrust meters and

I-47 Figure 4 a, b Bicycle Wheel for Ice Log The apparatus is shown in place on the starboard rail. The spindle for the wire reel is at the right. The cyclometer and electrical apparatus can also be seen. Wire from the reel was led under the wheel and 1-1/2 times around it before it was extended aft.

Figure 4 a Bicycle Wheel

Figure 4 b Contact Indicator

1-50 Figure 5 a, b Forward Thrust Bearing This bearing is equipped with a Kingsbury Thrust Meter to measure thrust in both the astern and ahead directions. Duplicate sets of piping and gages can be seen. The TMB elastic tube pressure gages are mounted near the bottom of the panel in which the large Bourdon tube gages are located.

NOWreI51 Figure 5 a

I-.52 Figure 5 b~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN 1 -53 installed them in January 1948. Mr. E. C. Colley of Kingsbury Machine Works Inc. made the final adjustments in the installation. Hydraulic pumps for the meters were provided by the Taylor Model Basin. Both meters were equipped with TNB Elastic Tube pressure gages of 5000 psi. capacity made by the U, S. Coast Guard as well as Bourdon tube gages to measure the oil pressure in the bearing. The Bourdon tube gages were read visually in the shaft alleys while the elastic tube gages were recorded by oscillograph No. OS140 in the bosun's locker. TMB type 1-A strain indicators provided the signal for the oscillograph. Elastic tube gage numbers and locations are as follows: Instrument No. Gage Station Gage No. Position TMB 119 P-1 CG-4(ET-4) Forward face, bow TM1B 159 P-2 CG-3(ET-3) After face, bow TMB 112 P-3 CG-2(ET-2) Forward face, starboard aft CG-1(ET-1) Spare The elastic tJube gages were zeroed after every run by dropping the oil pressure in the thrust meter to zero. Calibration was done in the laboratory at the Taylor Model Basin. Calibration Factors in Micro-Inches per Inch per 1000 psi Gage No. 20~F 80~F 120~F 0-l40OF Temp. range, Max. zero shift CG-l 452 453 449 40 Micro in. /in. in 1400 CG-2 448 452 449 35 CG-3 518 519 519 15 CG-4 455 457 456 300 The oil temperature in the bearings varied from about.100~ to 115m0F for the starboard motor aft, and about 850 to 950F. for the forward motor,

Figure 6 Starboard After Thrust Bearing This shows the piping and equipment used in connection with the Kingsbury Thrust Meters. The Bourdon tube gage is at the left. The T.M.B. elastic tube gage can be seen mounted on the board just to the right of the bourdon tube gage.

9 amn~.{F

Thrust is obtained by multiplying the measured pressure by the piston area of the thrust meter. This was 50 square inches for the 27-inh meter on the bow motor, and 75 square inches for the 34-inch meter on the starboard aft motor. Power and RPM of propulsion motors were recorded by Leeds and Northrup Speedomax meters, Figure 7, and a Westinghouse Oscillograph. Figure 8, all located in the bosun's locker. Check-readings of the instru ments on the ships switch board were taken during each test run Two sets of Leeds and Northrup instruments were provided: one by the Detroit Edison Company, and the other by the University of Milchigan. Each set consisted of two Speedomax Type G recorders No. 60101-489 range 0-500 millivolts, and one Speedomax Type G recorder No. 60101l489S range 0-50 millivolts0 One set was used for the bow motor, the other set was arranged to be switched to either the port or starboard motor aft, In practice it was used exclusively for the starboard motor. The 50-millivolt instrument of each set was arranged to read motor arture current in amperee one of the 300-millivolt instruments read motor power input in KW. These instruments were calibrated in the laboratory prior to installation on the ship. Calibration factors were as follows: Stern set: Amperes: 6o000o amps full scale, or 60 amps per divisiont Kilowatts: (Three ranges) 1500, 3000 6000 m full scae or 15, 30, 60 KW per Division T:n!. ace

I-57 Figure 7 a, b Leeds and Northrup Speedomax Type G Recording Meters Six of these instruments were used: Three on the forward motor and three on the starboard motor aft, Four No. 60101 - 489 instruments were used to measure power input in kilowatts and RPM. Two No. 60101 - 489S instruments were used to measure armature current.

i-58 Figure 7 a Instrument Closed

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I-60 Figure 8 a, b Westinghouse Oscillograph This is a nine-element oscillograph which was used to record armature current, voltage, and RPM for the three motors.

I-41 - - iiiiiB " a I D i~.6i: iibt r -- 4 js~ jiie a BAB ijdiaspnaassrsss4nssnaaa~-is : i i j J- " i $ -i i-: -? ;:;; ~:;; ic i: 3 r -1 It_ —8,,:,,; _.~i Iii: I- i i;- -Eal;i - " r; u Fiure 8 a.

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ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-63 The second 500-millivolt instrument in each set was arranged in conJunction with a tachometer-generator to read RPM. These instruments were calibrated by comparison with a hand revolution counter. Calibration factors were: Stern set: RPM - chart reading x 2.024 Bow set: RPM - chart reading x 2.568 The Leeds and Northrup meters were installed by Professor M. B, Stout of the University of Michigan, E. D. Kane and H. G. ammond of Detroit Edison Company at Manitawac, Wisconsin, 24-26 January, 1948. A Westinghouse nine-element oscillogSraph was used to obtain additional readings. It recorded motor voltage, armature current, and RPM for each of the three motors simultaneously. This was calibrated on the ship by Westinghouse representatives in a special test run. Westinghouse representatives Tirk and J. A. Wasmund installed the oscillograph during 24-26 January 1948 at Manitowac, Wisconsin. The ship's standard indicating ammeter, voltmeters, wattmeter, and tachometer for each motor were read during each run by an observer in each motor room. The six type A-i, SR-4 electric strain gages installed on the ship's longitudinal structure for the vibration tests were used also in the icebreaking tests. See Appendix A, Section b, Part I. Their locations are repeated here for convenience: 1. On inner bottom plating in the forward shaft alley 26 inches forward of bulkhead 57 on centerline, 6 feet 0 inches above the baseline.

I-64 l UNIVERSITY OF MICHIGAN 2. In generator room No. 2 on web of 10-inch channel over 12-inch I-beam 13 feet 10 inches abaft bulkhead 93 on centerline, 7 feet 0 inches above the baseline. 3. On inner bottom plating 13 inches abaft bulkhead 153 on centerline, 4 feet 8 inches above the baseline. 4. Centered on lover flange of upper deck girder 7 feet 11-3/4 inches off centerline to starboard, 6 inches abaft frame 43, 37 feet 6 inches above the baseline. 5. On longitudinal bulk~head 7 feet 7 inches off centerline to port, 2 inches below upper deck plating, midway between frames 113 and 114, 35 feet 6 inches above the baseline. 6. On underside of flange of main deck girder 8 feet off centerline to port, 9 inches forward of bulkhead 148, 27 feet 6 inches above the baseline. In addition to the six gages installed for the vibration tests, twenty-one rosette gages, Type AR-1, SR-4, were installed for the ice-breaking tests. All of these gages were located on shell plating of 1-5/8-inch thickness. The gages were placed midway between the frames. All elements perpendicular to the frames were wired up and numbered for the preceding frame. At the four stations where the entire rosette was wired, the suffix letters A, B, and C were added. The longitudinal elements were suffixed A, the elements at 45 degrees to the frames were suffixed B, and those parallel to the frames were suffixed C.

SHELL PLATING STRAIN GAGE LOCATIONS I-65 Frame and Gage No. Compartment No. Height Above Baseline Frame Spacing 8 A1W 14'-6" 16.0" 11 l 16.0222" 16A *l 16B 16C t n 20 14 -0 " 22 A301A 14'-6" 16.0033" 25 It f 29A t 29B 29C 33 t t 37 ft ft It 43 A7W " 16 inches 47 51 14 ' - " 55A 14'-61" " 55B t tr n 55C t 59t 63t.. 67 It f.t 71 B7F 75 ft It t 79A n ft n 79B 79C " t 83 " 86 " " "t

I-66 Figure 9 Rosette strain gages were attached to the inside of the shell plating midway between two frames. The water-proof covering is shown over one of these rosettes.

1-67 ~~:::_:; -I I~;- " | || Figure 9

ENGINEERING RESEARCH INSTITUTE I-68 UNIVERSITY OF MICHIGAN Strain gages outside the fuel oil tanks were moisturesproofed by tackwelding 1-3/4-inch lengths of 6-inch pipe around the gage; the gages were covered with Petricine A wax and the pipe was filled wt1h Ozite-B and sealed with a metal plate. (See Figure 9.) The shell was heated to the melting temperature of the Petricine wax by means of a low heat acetylene welding torch. Strain gages in the fuel oil tanks were moisture-proofed by covering with Petricine A wax, placing oatmeal box bottoms over the gages, and filling with Flintcote. Flintcote was used when a shortage of Ozite-B developed. It appeared to be an equivalent product. The Coast Guard prepared for the installation of strain gages by cleaning the tanks, grinding the plating smooth, and sanding with 000 sandpaper, atnd by marking the location and orientation of the gages. The installation of the gages was done by Messrs. L. D. Anderson and V. T. Almasy of Taylor Model Basin, and Messrs. Ts E. Quinsey, C. Lee and G. Geisendorfer of the University of Michigan. The installation was made during 9-12 October 1947 while the vessel was in dry dock at Manitowac Shipbuilding Company,, Manitowac, Wisconsin, A long base mechanical strain gage, gage length 10 feet, was located in the forward cargo hold, compartment A-301-A (Figure 10). It was placed athwartships across the centerline at frame 30 on the first platform A second similar lIonlg base strain gage was located on the upper deck running fore-and-aft between frames 104 and 112. (See Appendix A, Section b, Part I.) It was used only during the hog-sag tests, They were read visually by observers at the gage stations, Indication was by means of dial gages ibration of the vessel was recorded by four Stats accelereters and one pallograph (Figure 11).

I-69 Figure 10 Long base strain gage, transverse, located in Ctment A-301-A at frame 30. The gage length is 10 feet, Reading was done visually by means of a dial gage0

I-70 II'7 Figure U1 Type B Pallograph This was located on the main deck at frame lO1 The instrument records on a waxed paper strip, had a natural frequency of 1/4 cps and a magification or ~o

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I-71 STATHAM ACCELEROMETERS Station Serial No. Range Model Nat. Fre._c.p.s. TMB No. A-1 735 lOg R-10-240 none A-2 736 2g S-2-120 102 A-3 737 2g S-2-120 101 A-4 635 5g R-5-120 185 Accelerometer A-1 was located on the main deck at frame 10 on the centerline, compartment AlO3A. A-2 was located on the main deck at frame 51 on the centerline, compartment A-105. A-3 was on the main deck at frame 106 on centerline, in the engineer's office. A-4 was on the second deck at frame 210 on the centerline in the steering gear room. The accelerometers were equipped with Baldwin indicators: Accelerometer No. Strain Indicator Type Indicator No. 735 1-B 12 G (no No.) 736 1-B 102 737 1-B 102 635 1-B 101 Recording was through oscillograph No. OS140. One portable pallograph-type accelerometer, Type B, was used. (See Figure 11.) For these tests it was not moved, but was stationed at frame 10 on the main deck. This was a self-recording instrument, with a natural frequency of 1/4 cps and magnification of 2. Two Consolidated Type 5-10lB fourteen-channel recording scillographs Nos. OS134 and 0S140 were provided to record strain gage readings, thrust, trim, ice speed, and vibration. (Figure 12.) They were located in

I-72 Figure 12 Comsoiated Type 5101O1B, FourteensChanel s cillographs Strain gage readings,accelercmeterse elastic tube pressure gages, gyro stable vertic al ice log, and time ere recorded by these instr ntas

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ENGINEERING RESEARCH INSTITUTE I-o,74 )UNIVERSITY OF MICHIGAN the bosun's locker. Twenty-five TIB Type 1-A strain indicators were used with the electric strain gages. (See Figures 13, 14, 15, and 16.) Serial numbers were: 1062 109, 110, 111, 112, 113,, 118, 119, 120, 148, 152, 155, 156, 157, 159, 170, 176, 185, 186, 188, 189, 193, 194, 195, 198, and 200. Two TMB Type 1-B strain indicators, numbers 101 and 102, and one TMB Type 1-B, 12G strain indicator were used with the accelerometers as noted above. Type 1-B is a modification of Type 1-A to accommodate the Statham accelerometers. Twelve TMB Type 10-A power supplies were used. Serial numbers were: 103, 106, 110, 123, 132, 133, 135, 139, 140, 3076, and 2895, and one had none. Three regulators were used: TMB 2597 500 VA Sola TMB 2448 500 VA Sola TIMB TrO9 1000 VA Sorenson A Hathaway power supply was provided for each oscillograph: TMB OS1254 (OS134) TMB OS124A (os140) The accelerometers, oscillographsa strain indicators and acccmpanying electronic equipment were installed during the period 21-25 January 1948 by Mr. Anderson, Mr. Eardy, Mr bendrican, and Mr, Almasy of the Taylor Model Basin, assisted by Mr. Martin and Mr. Geisendorfer of the University of Michigan. Photographs of the ice being broken by the ship were taken from two locations. A hinged framework of angle bars was constructed extending about 12 feet ahead of the bow of the ship at about the 21-foot 6-inch waterline,

I-75 Figure 13 General View of Strain Indicator Bank and. Oscillographs Located in the bosun's locker

176 Figure 14 Strain Indicators and Power Supplies used with the strain gages and accelerometers. Front view located in the bosun' s locker.

I-77 Figure 14 MW~Si ~~i~~s~~~a~~~i ~ ~~~ ~W

I-78 Figure 15 General view of the rear of the Strain Indicator Bank and Switchboard. Located in the bosun's locker.

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I-80 Figure 16 Switchboard Connections between the strain indicators and the oscillographs were made here. By means of this switchboard, changes could be made in the gages, etc., recorded. The numbers on the board correspond to the strain gages. Located in the bosun's locker.

Figure 16

ENGINEERING RESEARCH INSTITUTE I-82 UNIVERSITY OF MICHIGAN A 16 mm Eastman Cine-Kodak Special moving picture camera was mounted on a Pivoted Support at the extremity of the framework, pointing toward the bow of the ship. A 3/16-inch aluminum box served as a protective case and mount for the camera. To prevent flying snow and ice particles from impinging on the lens, a Lucite disk was placed in front of the lens and rotated by a small electric motor. A string was led to the deck to operate the shutter. To permit rapid elevation of the camera as windrows were approached, a safety chain was fastened to the top of the camera box. A firm pull on the chain would release the box from its supports. Exposures from this position showed close-up action of the ice-breaking process at the bow. (See Appendix C, Sections c and d.) During some of the tests, this camera was used to take pictures from the weather deck forward. From this position the ice-breaking action over a wider area could be recorded, and the total area affected observed. Bending and shear tests on ice samples were conducted on the main deck of the ship and at Ann Arbor by means of a special testing machine. Two I4-inch I-beams about three feet in length were held one above the other by four adjustable rods. A hydraulic Jack secured to the lower I-beam fitted with a pressure gage supplied the breaking force. Various blocks were fitted to the Jack and the upper I-beams to bend the specimens, or to break them in shear. (See Appendix C, Section a.) (Figures 17, 18, 19, 20 and 21 show the locations of instruments in the ship.)

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN CHAPTER II DISCUSSION OF TEST RESULTS Objectives The testing program recorded in this report had for its objectives: a) The determination of structural stresses in the U.S.C.G.C. Mackinaw while breaking ice. The stresses of interest were those arising from contact between ice and the hull of the vessel. b) The determination of the ice forces acting on the hull of the U.S.C. G.C. Mackinaw while breaking ice. Tests Conducted In order to achieve the objectives several auxiliary tests and an extensive series of ice-breaking tests were carried out. Auxiliary Tests: a) Vibration tests of the hull of the U.S.C.G.C. Mackinaw by means of the Taylor Model Basin medium vibration generator. This gave the dynamic characteristics of the hull. b) Calibration of the stable vertical gyro trim meter by trim tank loadings and ship draft marks. c) Hog-sag tests by means of peak tank loadings. d) Squat test to find effects of ice on the trim of the vessel e) Tests on the physical properties of ice - tensile strength, shear strength, Young's modulus of elasticity, Poisson's ratio, and friction.

UNIVERSITY OF MICHIGAN Main Tests: a) Ice-breaking tests in sheet ice of thicknesses varying between 7 inches and 18 inches. b) Ice-breaking tests in windrowed ice with windrows 15 feet and 20 feet thick. The quantities measured and the instruments used during the main tests were: a) Drafts, forward and aft, floating at standstill Bridge draft gages and ship draft marks b) Trim changes during the test Stable vertical gyro trim meter recorded on the oscillograph paper c) Speed of vessel through the ice Bicycle wheel ice log recording on the oscillograph paper; moving pictures d) Ice thickness - spot checks Direct measurement. e) Wind velocity Ship indicating anemometer f) Wind direction Ship's flag g) Motor voltage, amperes and RPM Indicated on ship's meters, recorded on an electromagnetic oscillograph, and on Speedomax meters h) Thrust bearing pressures on starboard motor and bow motor Kingsbury thrust meters and Taylor Model Basin thrust meter recording on oscillograph paper i) Vertical accelerations Statham resistance type accelerometers located along the ship on the main deck at Frames 50 and 52 and on the second deck at Frame 208, recorded on oscillograph paper

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN II-3 j) Vertical motions at Frame 10 on the main deck Recorded on a Taylor Model Basin Pallograph k) Bending and compressive strains Six metalectric, single leg, strain gages, three above the neutral axis, three below the neutral axis, recording on the oscillograph paper 1) Bending strains in the shell plating Twenty-one metalectric strain gages, rosettes and single leg, located on the inner surface of the shell plating on the starboard side at the 4I-ft. 6-in. water line, spaced more or less evenly from Frame 8 to Frame 86; they recorded on the oscillograph paper m) Lateral contraction of the ship on the first platform at Frame 51 Indicated on a 10-ft. long strain gage, dial indicator, visual observations n) Ice-breaking process Photographed at the bow by a moving picture camera. No method of direct measurement of contact forces between the ice and the exterior of the hull was devised or used. Any statements made about such forces are the results of theoretical calculations or inferences from strain gage readings. Test Results The results of the auxiliary tests and the ice-breaking tests can be summarized as follows: Vibration Tests: The hull vibration tests are completely described in Appendix A, Section b, Part I. The results can be summarized as follows: Force Amplitude No. Amplitude Frequency Frequency Node Locations Vertical Mode of ~tFae cycles cycles rm E0i a Nodes at Fram second minute Frame Mi a 3, lbs. Frame 10 1 2 6,520 3.45 207 68 and 149.040 inch 2 3 20,800 6.15 370 40 110 end 171.025 inch 3 4 11,100 9.80 588 34,83,l16 and 178.015 inch

ii-4 ENGINEERING RESEARCH INSTITUTE II-4 | UNIVERSITY OF MICHIGAN This test was successful and does not need to be repeated. The normal elastic curve for the second mode is not accurate because the heavy load on the vibration generator (20,800 lbs. amplitude) made accurate speed control impossible. This is not important for the interpretation of the icebreaking data. Thebreaking ice excites the first mode because most of the violent impacts between the hull and the ice occur forward of Frame 68, where the forward node is located. The breaking ice does not excite the second and third modes, since as much ice breaks forward of the forward nodes as aft of the forward nodes in these modes. The second mode is strongly excited by the propellers of the ship. Calibration of the Stable Vertical Gyro Trim Meter: The trim meter puts out signals in the form of a 60-cycle carrier wave. The width of the envelope of the carrier wave is directly proportional to the angle of trim. The calibration showed that.01235 radians gave 1-inch width of carrier band. The draft marks on the ship are 248 feet apart. The difference in elevation of the draft marks is 3.06 feet for 1-inch width of carrier band. If this device is used for future tests, it should be recalibrated and a tell-tale circuit should be devised to indicate + and - trim angles. Hog-Sag Tests: These tests are discussed in Appendix A, Section b, Part III. In the hog tests, space A-l-w contained 29,780 gallons of fresh water, space C-4-w contained 26,621 gallons of fresh water, and space C-5-w contained 12,200 gallons of fresh water. In the sag condition A-l-w contain 150 gallons, C-4-w contained 1750 gallons, and C-5-w contained 40 gallons. The longitudinal strain gages indicated differences of strain between these conditions which could be compared with calculated strains based on the changed loading conditions and treating the ship hull as a beam of known

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN, dimensions and neutral axis location. The calculated strains were about 10 percent higher than the measured.strains. This discrepancy is related to the fact that the calculated cross-sectional moments of inertia contain only the fully active material in the ship and that the location of the neutral axis is not precise. These tests indicate that the ship can be treated as a beam for purposes of checking test results. This test does not have to be repeated. Squat Tests: These tests are discussed in detail in Appendix A, Section a, Part III. These tests were made to provide a correction to the trim angle to be applied in the ice-breaking runs. They were not suitable for this purpose. For future ice-breaking tests, this purpose could be achieved by model tests of trim and, sinkage at different speeds and initial drafts and trim in open water. Further model tests should be attempted in which the effect of the suppression of the bow wave on trim angle could be measured. Tests on the Physical Properties of Fresh Water Ice: These investigations, experimental and bibliographic, are described in detail in Appendix C, Section a and Appendix C, Section b. For general purposes the results can be summed up as follows: Young's modulus, at high rates of loading E = 1.0 x 106 lbs/in2 Young's modulus, at low rates of loading E = 0.5 x 106 lbs/in2 Poisson's Ratio $ = 0.35 Ultimate stress, tension - = 200 lbs/in2 Ultimate stress, shear t = 100 lbs/in2 Coefficient of friction, dry H = 0.10 Coefficient of friction, wet H = 0.01 (or less)

ENGINEERING RESEARCH INSTITUTE II-6 l UNIVERSITY OF MICHIGAN The low values of ultimate strength in bending and shear keep the loadings on the icebreaker hull within reasonable limits. The low value of shear ultimate strength is especially important in this matter. For ice that is free to bend or float (without serious jambing) the ultimate strength of 100 lbs/in2 in shear probably limits the maximum unit loading normal to the exterior surface of the ice-breaking vessel to values around 200 lbs/in2 at contact points and surfaces. The friction forces which arise from welding beads "ploughing" or "machining" the ice is determined by the ultimate strength of the ice in shear. If h is the height of a welding bead, the "friction" force of "machining" is approximately F = 2 h ' = 200 h lbs/in. contact length. Ice-Breaking Tests: In the large table which records the main test data the ice-breaking runs are designated by a symbol of the form 4AIIc. The number 4 indicates that the run was on sheet ice. The symbols A, B and C indicate draft conditions at standstill in open water. Symbol Draft Forward Draft Aft A 15'4" 18'10" B 17'0" 17'2" C 18,8" 19'l" The symbols I, II and III indicate propeller combinations. Symbol Propeller Combination I 2 stern propellers, bow propeller idle. II 2 stern propellers, bow propeller, ahead. III 2 stern propellers, bow propeller, backing. The a, b and c indicate relative speeds. Symbol Relative Speed a low b intermediate c high

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN II-7 These symbols did not always coincide with reality: in a few cases ice conditions made speed b higher than speed c. Changes in Trim: Throughout the ice-breaking tests the changes in trim were always small. Sometimes, if the zero readings are correct, these changes are negative. Only in the windrowed ice runs 5A and 5B were large positive changes of trim recorded. These will be discussed later. The stable vertical gyro worked in a satisfactory manner except that the carrier wave envelope gave no indication as to + or - in the trim angle; this should be corrected in future tests. Speed of Ship Through the Ice: This is very important information. Perhaps the most damaging failure in the whole program was the partial failure of both methods used to gain speed information. The photographic method was hard to reduce, it was not consistent, and it failed largely from the lack of a good length scale on the ice and from lack of fixity in position. The ice log broke its driving wire in about one-half of the tests. What success it did have arose from the skill of the operating crew in speeding up the bicycle wheel by hand so that there was no sudden jerk when the wire became taut. On January 29, 1948, nearly all the speed data was lost for the reason that members of the party which operated the wheel with comparative success on January 28, 1948, were shifted to other duties. The new crew did not begin to function properly until the day was over. The extreme importance of these wheel failures became painfully apparent when the d.ata were needed for computation.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN In Appendix A, Section a, Part I the necessary conditions for a successful ice log are treated in detail. The analysis given there should have - and could have - been made before the ice-breaking tests were run. The large moment of inertia of the bicycle wheel and the spool of driving wire broke the recording wire after the ice anchor was thrown overboard and became anchored on the ice. The only things which could have saved the wire were large wire length streamed overboard before the wire became tight, stronger wire, or hard cranking of the wheel up to ship speed before the driving wire became taut. Only the latter device was used, with only partial success. This method can be made reliable and it is the only method which permits easy reduction of data, is insensitive to irregular ice surface elevations, and permits use with the "chip lag" in open water. At the high speeds (18 ft/sec) over l400 feet of wire should have been paid out before permitting it to become taut, to give sufficient flexibility to prevent breakage. This was never done. The following general specifications will produce a successful wheel ice log. 1) Use a small diameter, light wheel. 2) Use a small diameter wire spool. 3) Use larger diameter wire. 4) With same diameter wire use a 90-foot nylon lead between the ice anchor and the drive wire. 5) Use a sturdy flush-type contact. 6) Provide wire guides where the wire enters and leaves the wheel. Proper design can make this device reliable and operatable by one man.

ENGINEERING RESEARCH INSTITUTE II-9 UNIVERSITY OF MICHIGAN I Ice Thickness: The other glaring inadequacy in the test procedure was in the measurement of ice thicknesses. Only haphazard thickness samples were measured. These important data must be known with some accuracy over the whole length of the course for each test run. The ice thickness at several points along the test run course should be measured and recorded. The longest run (60 seconds at 18 ft/sec) covered a course less than 1100 feet long. This is only three ship's lengths. An ice party with ordinary yardsticks could measure the thickness of floating ice blocks or up-ended ice blocks at stations 100 feet apart over this course in about 20 minutes. The shortest run was less than 300 feet long. On this run 5 to 10 minutes would suffice to obtain thickness readings at stations 100 feet apart. These data are just as vital as accurate speed measurements. The method that was used was totally inadequate for the purposes it was supposed to serve. Wind Velocities: The wind reached velocities around 50 MPH on January 28, 1948. It was a mistake to test on such a day, since the wind resistance was as large as the ice resistance in some runs. If tests must be run on windy days, the relative wind should be taken across the beam of the ship. The tests to discover the forces arising from ice contact should be measured without the complicating presence of the wind. Wind resistance can be calculated with only fair accuracy. As a calculated correction factor it might not be so bad, but as an element entering the study of ice forces acting on the icebreaker it should be eliminated. The indicating anemometer was a satisfactory device to use, provided the calibration was good. It was assumed to be 100 percent correct in the resistance calculations.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Wind Direction: Wind direction relative to the ship was measured very approximately by observing the ship's flag. This direction was indicated on the special bridge data sheet by a pencil line in the same direction. This must have been highly inaccurate; a wind vane with an indicating rosette would have been much more satisfactory. Wind resistance is calculated by the formula 2 Rwind = CD A 2 where CD is a function of the angle between the wind direction and the foreand-aft centerline of the ship. The data used for CD was gotten from Admiral Taylor's "Speed and Power of Ships" and was based on tests in water with inverted models. Much more modern data gotten in wind tunnels on ship models is now available at the David Taylor Model Basin. In the next tests these new data should be used. This, in combination with eliminating the wind resistance as much as possible, will make the ice resistance determination more accurate. Motor Voltage, Amperage, Power, Thrust and RPM: These measurements are all necessary and can be made better during future tests than in these tests. This can be done in two ways: 1) By using either an oscillograph alone for recording or Speedomax meters alone. The oscillograph is probably the more accurate of the two means of recording. 2) By using in all tests a very few set predetermined electrical conditions or motor RPM conditions, whichever is more practical from an operating viewpoint.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN II-11 The results of all these measurements are shown on the master data sheet, and the calculations of wind resistance, water resistance and ice resis tance are given in detail in Appendix B, Section a. The absence of complete and reliable data on ship speed made these calculations incomplete. Incomplete as these calculations are, they indicate that a large percentage of the resistance offered to the motion of a ship through ice is due to the ice. In ice the ship loses between 7 and 11 knots in speed compared to open water operation under the same conditions of power and propeller RPM. With the bow propeller backing, the loss in speed in ice ran between 6 and 9 knots. The resisting force of the ice is related to the speed of the ship in the following way: Ri = a + b v2 where Ri = resisting force in lbs v = velocity of ship in ft/sec a = a function of ice thickness and draft b = a function of ice thickness and draft. One of the purposes of the tests was to discover whether this relationship is correct and to find the functional relationship of the constants a and b to ice thickness and draft. It would be expected that a and b would increase when the ice thickness increases and that they would also increase as the mean draft increases. The data gotten are not good enough to note any functional relationship except the overall form. A "shotgun" diagram of the calculated ice resistance for all runs versus the measured velocities could be represented by the

II-12 I ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN equation Ri = 30,000 + 500 v The data poured into this include three different draft conditions and ice thicknesses varying from 7 inches to 18 inches. An infinity of equations of this form could also represent the shotgun data just as readily. The constant a could be taken anywhere from 15,000 lbs. to 50,000 lbs. The constant b could be taken over an equally wide range of values. The values of a and b given in the above equation are not to be taken too seriously; but the form of the equation is to be taken seriously. I believe that the present data warrant this conclusion. This form is physically reasonable. As the speed of the ship approaches zero only the strength properties of the ice resist the motion. As the ship speeds up the inertia resistance of the ice is added to the elastic resistance. The inertia forces are proportional to the square of the velocity. The equation given above is the rough average of two families of curves in which ice resistance to ship motion is plotted against variations in draft and in ice thickness. Therefore, it has no physical significance except as to form. Accelerations: The accelerometers attached rigidly to the decks picked up the unbalance vibration of the gyro exciter, which ran at 3600 RPM. The resultant 60-cycle disturbance was picked up by the accelerometers and magnified until it obscured the accelerations wanted in a forest of "hash" A study of spring mounting these accelerometers indicates that any spring which gives reasonable records of the vertical acceleration would still contain very large amplitude 60-cycle "hash". A low pass electrical filter which would pass everything under 30 cycles/second and nothing above that limit would be necessary in the circuit of the Statham accelerometers to make them

ENGINEERING RESEARCH INSTITUTE II-13 UNIVERSITY OF MICHIGAN work in this application. However, it was a mistake from the very beginning to introduce these accelerometers into this problem. First of all, there were not enough used to be of significant use. Second, the reduction of data from even successful accelerometer records is a tedious process. If enough accelerometers give perfect, well synchronized records and the dead weight load distribution of the ship is known, along with the distribution of virtual mass, then the inertia loading per foot is known as a function of position and the bending moments at that position can be calculated. From these the bending strains can be calculated,and the bending stresses. However, we never know the virtual mass distribution, we never synchronize the timing on accelerometers closely enough for accurate matching of records, and we didn't have enough accelerometers. The final, and really conclusive, reason that the accelerometers were not needed is the fact that we could and did measure bending strains directly. On this Job ten bending strain gages are many times as valuable as ten vertical motion accelerometers could be. For this reason, on future tests no accelerometers should be used, and the number of bending strain gages should be increased as much as possible. Pallograph at Frame 10 and Bending Strain Record: The pallograph at Frame 10 was a valuable instrument, as it gave a direct tie-in between the vibration generator tests and the ice-breaking tests. This connection between the two tests is discussed in detail in Appendix A, Section b, Part II. Perhaps the most noticeable and outstanding effect of breaking ice on the icebreaking vessel itself is the vertical vibrations created by that regularly "irregular" process. The breaking ice sets the hull into violent, intermittent irregularly repeatable vertical vibrations in the 2 noded mode of vertical

ENGINEERING RESEARCH INSTITUTE II-14 UNIVERSITY OF MICHIGAN motion. The breaking ice takes the place of the vibration generator in creating and maintaining this motion. The relationship between the strains measured by metalectric gage No. 5 and the 10-foot long mechanical strain gage near to gage No. 5 and the amplitude of motion at Frame 10 in the vibration tests was Xo 2700 in. where A = strain double amplitude at Frame 113, gage No. 5, in in/in. xo = double amplitude of vertical vibration in 2 noded mode at Frame 10, in in. This same relationship held, at least statistically, on the average, during the ice-breaking tests. Therefore, we can evaluate the amplitude of equivalent vertical disturbing force at Frame 3 in the ice-breaking tests. During the vibration generator tests an unbalanced force with an amplitude of 6520 lbs. acting at Frame 3 caused a double amplitude strain in the long base strain gage of 28.4 x 10-6 in/in. with the hull in resonance in the 2 noded mode. The maximum double amplitude strain in gage No. 5 in the 2 noded mode during the ice-breaking tests was 125 x 10-6 in/in. Therefore, at that instant the equivalent amplitude of disturbing force from the breaking ice referred to Frame 3 is 125 6520 lbs. = 28,700 lbs. This occurred in Run 4AIIb, for which the speed is 11.56 ft/sec., and the ice thickness was probably between-12 and 15 inches. However, this reading is so much out of line with the other readings that it is certain that

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN iI-15 either the speed is wrong or the ice thickness is wrong or both are wrong. However, it is the worst that happened to the ship dynamically in bending throughout the whole week of ice-breaking tests. The strain of 125 x 10-6 in/in. corresponds to a double amplitude of stress of 3750 lbs/in2, or a single amplitude of 1875 lbs/in2, at the position of gage No. 5, Frame 113, 35.5 feet above the baseline near the longitudinal centerline of the ship. The worst thing which happened to the hull girder in steady state bending occurred during the windrow ice runs 5A and 5B. In run 5A the ship went through a windrow 15 feet deep at a speed of 15 ft/sec. and registered a maximum compressive strain of 24.4 x 10-6 in/in. on gage No. 5. This corresponds to a stress of 732 lbs/in2. This should be compared with the single amplitude of dynamic stress mentioned above, 1875 lbs/in2. The steady state compressive stress was 39 percent as large as the largest dynamic strain amplitude. In run 5B the ship went through a 20-foot deep windrow at a speed of 17 ft/sec and registered a maximum compressive stress at gage No. 5 of 778 lbs/in2. This is 41.5 percent of the maximum dynamic stress amplitude experienced during the ice-breaking tests. It is interesting to compare these test values with the computed values of stresses calculated in Appendix B, Section b, under the assumption that the ship is hung up on an unbroken ledge of ice at a point 16 feet abaft the forward perpendicular. At gage No. 5 the compressive stress is 1920 lbs/ in2, a stress roughly equal to the worst amplitude of dynamic stress experienced during the ice-breaking runs. The ship on a standard trochoidal wave has a stress at gage No. 5 of - 4175 lbs/in2, a value beyond any steady stress possible in ice.

I 1116 I (ENGINEERING RESEARCH INSTITUTE II-16 l UNIVERSITY OF MICHIGAN There are physical reasons for believing that the amplitude of dynamic bending stress at every point in the ship is of the form = a + b 2 in which a and b are functions of ice thickness, draft, and position of the stressed point, both increasing as the thickness and draft increase. Our speed and ice thickness data do not permit evaluation of a and b, but the overall information does check the general form of the relationship. If a prediction is justifiable with present information, the worst dynamic stress amplitude at gage No. 5 probably would not exceed 4000 lbs/in2 in the thickest ice the Great Lakes could produce. The worst steady state stress at gage No. 5 could not come near 1920 lbs/in2 unless the ship runs aground on a ledge of rock, since no sheet of ice will ever occur in the lakes which can resist breakage under the bow of the Mackinaw. Strains in the Shell Plating: These are discussed in Appendix B, Section c. These 21 gages were near the level of the neutral axis and in a perfect beam would not show effects from bending. However, the ship is not a perfect beam, the plating can "pant" in a direction normal to its own plane, and many of the gages show distinct traces of the normal mode bending frequencies. These are not significant. The significant readings on these gages record the effects of impacts between the external surface of the hull and the ice. The gages were on the 14-foot, 6-inch waterline. They were on the level with the ice sheet only in draft condition A. In all other draft conditions they were below the ice sheet. The effect of submergence of the gages is shown below:

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN II-17 Maximum Stress Number of Impacts Run?dr Afts Recorded on Registered Rosette Gage No. 29 on All Gages A 15'4" 18'10" - 6735 655 B 17'0" 17'2" - 3700 363 C 18'8" 19'1" - 2500 80 The maximum stress condition does not last more than a small fraction of a second. A very striking condition was the almost total absence of registered impacts during the high speed runs. During these runs the bow propeller was backing, driving forward, and idling. Therefore, the bow propeller could not cause this, but the initial impact on each cake of ice may have given it enough downward velocity to separate it from the hull. In any case, an insulating layer of water could be created between the ploughed-under ice and the ship to prevent subsequent impacts. This point is so strange that it must be checked in later tests. The distribution of impacts along the ship in all runs is very irregular with gage 8 Just aft of Frame 8 receiving 130 impacts, and gage 55 Just aft of Frame 55 receiving 159 impacts. Gage 86 just aft of Frame 86 received the minimum number of impacts of all the gages: 55 impacts throughout all the runs. However, if enough records had been taken, probably all the gage stations would have registered roughly equal numbers of impacts. This suggests that a single station covered vertically from the 14-foot 6-inch waterline up to the 19-foot waterline by four rosette strain gages would give the maximum

ENGINEERING RESEARCH INSTITUTE II-18 I UNIVERSITY OF MICHIGAN amount of useful information in future tests. The maximum stress registered happened to be in the only rosette gage used consistently throughout the ice-breaking runs - gage No. 29. This maximum stress was 6735 lbs/in2 in compression. It seems probable that the actual maximum stress which existed near gage No. 29 was no more than 13,500 lbs/in2. It is also possible that the maximum impact stresses which the plating would ever encounter even in the thickest ice possible in the Great Lakes would never go over 20,000 lbs/in2. This tentative conclusion must be checked in future tests. One very strong recommendation which must be given here is that nothing but rosette gages be used in future tests on the ship plating; a few rosettes will give much more real information than a large number of single leg strain gages. The Lateral Contraction of the Ship: The transverse long base strain gage indicating on a dial gage showed that the ship was squeezed by the ice with stresses of the order of 10,000 lbs/in2 on the first platform. In future tests this gage should be replaced by metalectric strain gages which can record on an oscillograph. Motion Pictures of the Bow Action of an Icebreaker: These motion pictures are submitted separately; they are discussed in detail in Appendix C, Section f. These pictures reveal the details of the ice-breaking process. The forefoot of the icebreaker splits the ice along radial lines which branch out from the bow as a center. Cracks at right angles to the splitting cracks are also produced simultaneously several feet forward of the bow. The cracked ice is ploughed under the hull. Along the side of the ship the ice is broken into slabs several feet wide parallel to the ship and around 10 to 12 feet

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN II-19 long at right angles to the ship. The process is an irregular one which could be called statistically continuous. The widths of the slabs alongside the ship are connected with the speed of the ship and the natural frequency of vibration of the hull in the 2 noded mode. This frequency is 3.45 cycles/second. At a ship speed of 18 ft/ sec this should result in slabs about 5 feet wide. The bending strains vary in amplitude in a beat-like fashion, but with no constant beat frequency, indicating that the breaking ice and the hull react together in the 2 noded cycle in an irregular way. Changes in the ice thickness or stiffness put the process out of synchronism. This is very fortunate for the ship. A true synchronism, long continued, might easily build up vibration amplitudes large enough to fatigue the ship at the cross section of maximum bending moment in the first mode. On the upswing of the ship's vibration it advances over ice as yet unbroken. On the dawnswing the ice is broken off. The amplitude of vibration increases with each swing until the process gets out of phase because of local changes in the ice properties or thickness. The amplitude dies down to zero and starts to build up again. There are five types of forces acting between the icebreaker and the ice: 1) Splitting forces at the forefoot 2) Bending forces at the forefoot and along the sides of the vessel aft to the widest section 3) Inertia forces caused by changing the velocity of the ice from zero to the "ploughing-under" velocities at all points of contact

UNIVERSITY OF MICHIGAN 4) Flotation forces of the ice under the hull of the ship at all points of contact 5) Friction forces at all points of contact. The magnitude of the splitting force is totally unknown either by measurement or by theory. Theory indicates that it is small. At the sharp edges of the forefoot the least load will cause infinite plane stresses. Those edges should always be kept reasonably sharp and square. It might possibly happen that the splitting resistance to forward motion of the ship could be minimized by giving the forefoot a single central sharp edge. This could be tried by welding a "false" sharp edge on the present flat forefoot. The bending forces are relatively large at all speeds. The inertia forces increase in general with the square of the speed and they are relatively large. The flotation forces amount to about 6 pounds per cubic foot of submerged ice, and this is reduced by the inertia forces induced by following the ship's lines under water. Probably an average force of 5 pounds per cubic foot of ice is exerted on the ship bottom wherever submerged ice is in contact with the hull. The friction forces are of the order of 10 percent of the breaking and inertia forces where the ice is dry, and they are negligible where the ice is wet. However, fairly large friction forces exist where the weld beads "machine" the ice in contact with the ship. These forces are limited by the ultimate strength of ice in shear and may be of the order of 200 h pounds per inch of contact length, where h is the height of the welding bead. The main matchnineg contact is at the ice sheet level and there the length of contact is determined by the thickness of the ice sheet. For ice 20 inches thick and a

ENGINEERING RESEARCH INSTITUTE II-21 UNIVERSITY OF MICHIGAN weld bead 1/2 inch high, a retarding force of about 2000 pounds might be created for each bead in machining contact. The total resisting force caused by this might readily be of the same order of magnitude as the resistance forces arising from the water. An increase in economy might be affected by grinding off all the welding beads if this would have no adverse effects on the welded joints. A given point below the water plane on the hull of an ice-breaking vessel probably spends most of its life with only water forces acting on it. Intermittently and repeatedly the point is loaded with large bending, inertia, flotation, and friction forces when contact is made with the cakes of ice. Some ideas as to time-averages of these forces can be gained by theoretical analysis. This is done by considering the ice as a cheese-like substance which has the bending and shear stiffness of true ice and the weight properties of true ice. This cheese-like substance will plough under the hull continuously in contact with the hull. In Appendix C, Section c, it is shown that the vertical loading per unit length on the edge of a floating beam of ice of constant cross section and of infinite length, with E = 1 x 106 lbs/in2, will be approximately l = 2.5 h5/4 where ql = vertical edge loading per inch length, lbs/in. h = thickness of ice, in. The inertia reaction forces per inch to speed the ice from zero velocity to the velocity with which it is ploughed under the ship is of the order of

ENGINEERING RESEARCH INSTITUTE II-22 UNIVERSITY OF MICHIGAN where k = a number which depends on the angle between the ice plane and the hull of the vessel, about 1.1 = weight per cubic inch for ice, lbs/in3 g = the acceleration of gravity, 386 in/sec2 h = thickness of ice, in v = velocity of the ship, in/sec. But q2 is the effective force without considering the impact energy lost in shattering the edges of the ice. The true inertia reaction loading q3 could easily be q3 X 4 2 The number 4 is a frank guess, suggested by the ice resistance calculations of the actual tests. A good analogy to this is a blow on the end of a spring: most of the force is used to accelerate the spring. At the other end of the spring only the effect of spring deflection is felt. The coefficient of restitution, e, between impacting ice and steel is very smll, meaning that the efficiency of speeding up ice by impact is very low. This, too, would suggest inertia reaction forces much higher than those effective in speeding up the ice. The extra force is scattered (if we can speak of scattering force) in a plane parallel to the ship, and is used in shattering the ice. These ideas find numerical expression in the table below. h, 92 - lbs/in 9. - lbs/in ql q3 v=7 ' v=144 v=216 v=72 v=.41 - v=216 inch lbs/in in/sec in/sec in /sec in/ sec in/sec 12 56 6 24 54 24 96 216 24 133 12 48 108 48 192 432 36 233 18 72 162 72 288 648

ENGINEERING RESEARCH INSTITUTE II-23 UNIVERSITY OF MICHIGAN These values are all calculated from theoretical considerations and free speculation. They represent time and space averages and are to be considered only orders of magnitude. The ship receives its forces intermittently and over small areas. From consideration of the low ultimate strength of ice in shear (100 lbs/in2) there is some reason to believe that the local normal pressure on the hull of the vessels never goes much beyond 200 lbs/in2 on the local areas of contact between ice and the ship. Such loading would lead to maximum stresses of the order of magnitude of 20,000 lbs/in2 in the ship plating. This must be verified by tests in thick ice before it is believed. Summary of Test Results The ice during the test runs never exceeded 18 inches in thickness, except in the case of loosely layered ice in windrows, and the maximum measured stresses were low. Maximum steady stress, gage No. 5, windrow ice - 778 lbs/in2 Maximum dynamic stress, gage No. 5, sheet ice + 1875 lbs/in2 Maximum dynamic stress, gage No. 29 on plating, - 6755 lbs/in2 sheet ice Since the loadings go up roughly with the ice thickness, these maximum stresses will probably not exceed twice as much as shown above in any ice which can be met in the Great Lakes. The steady stresses calculated on the standard trochoidal wave are five times as high as the maximum measured steady stress in bending, and perhaps twice as high as any which will be met in any possible ice in the Great Lakes.

I|-2 4ENGINEERING RESEARCH INSTITUTE II-24 | UNIVERSITY OF MICHIGAN These conclusions are not to be taken as final until further testing in ice verifies them. Calculations: Some of the theoretical developments are of interest to summarize. Analysis of Forces in the Wire of a Wheel Ice Log: In Appendix A, Section a, Part I, the bicycle wheel ice log is analyzed to find out why the wire broke. Inertia loading of the wire had been foreseen, but it was felt that speeding up the wheel by hand would be sufficient to save the wire. This turned out to be a most disastrous conclusion, since at least half the speed measurements were actually lost. The relationship between the maximum tensile force in the wire and the other features of the ice log design, including the possibility of hand cranking and nylon leads between wire and ice anchor, is (Fman ~ Ff)2 r%1o 2 EwAw 1 w 2 - [ v J r2 Aw 1 + EwAwnJ/EnAnL g when no nylon lead is used (On = 0) this becomes r2 "w g (Fmax; F = [1 _ ok 2 Eww w v2l when, in addition, no hand cranking is used (4o = 0) it becomes (Fmax Ff )2 = EwA w v2 r2 U g where F = breaking strength of wire max Ff = brake friction force on rim of wheel r = radius of the wheel to the wire

ENGINEERING RESEARCH INSTITUTE II-25 UNIVERSITY OF MICHIGAN p = radius of gyration of the wheel Ew = modulus of elasticity of the~ wire Awt = cross-sectional area of the wire = length of wire streamed over the side at the time of the starting jerk w = weight of wheel plus weight of wire spool g = acceleration of gravity v = velocity of the vessel at time of Jerk. The last equation solved for O, shows that at least 1400 feet of wire should have been streamed over the side before the jerk at the highest speed. And 350 feet of wire was needed for safety at halt the highest speed. These long lengths of wire were never used, so the wire broke. The analysis shows how a successful wheel ice log can be designed. A Method for Calculating Vertical Load Distribution on An Icebreaker in Action: Appendix A, Section b, Part III outlines a method for arriving at an approximate idea of the vertical load distribution under the hull while breaking ice. The scheme calls for several bending strain gages distributed along the ship hull both above and below the neutral axis. Two strain gages are placed in the same cross section one above the neutral axis and one below it; and area A, moment of inertia I, the distance above the neutral axis c1 of the upper gage and the distance below the neutral axis c2 of the lower gage are known. The simultaneous strains E in the upper gage and & in the lower gage are measured. Then the compressive force P on the section is P = EA 2 11kE C1 + C2

II-26 I ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN and the bending moment M on the section is M = E I C1 + C2 The bending moment in several planes is measured simultaneously and these moments are plotted against location along the ship. The first derivative of the M versus. x curve gives the shear at location x dM dx S The derivative of the S versus x curve gives the load per unit length at location x dS dx qx This will be the combined effect of changed buoyancy and ice loading if the gages have been zeroed with the vessel floating freely in calm water. This scheme, while it depends on graphical differentiation, which is notoriously inaccurate, is still worth trying on future tests. Incidentally, the force P found at each section gives a direct measurement of the wind, water and ice thrusts acting on the ship. This method, in effect, makes the ship a dynamometer capable of measuring directly bending and resistance forces Steady State Hull Girder Stress Analysis: Appendix B, Section b, gives an alternative means of measuring the same quantities as were treated in Appendix A, Section b, Part III. In this method it is assumed that initial drafts are known with the ship at standstill and that power readings have made it possible to calculate propeller thrusts, wind resistance, water resistance and ice resistance. It is also assumed that the locations of the centers of

ENGINEERING RESEARCH INSTITUTE II-27 UNIVERSITY OF MICHIGAN these forces are known. It is also assumed that the change in trim angle caused by the ice can be measured. Finally, the ship's lines are necessary for the calculation of certain integrals. When all this is known, various assumed load distributions can be evaluated as follows: If the ice load is assumed concentrated at one point at the bow, the point x1,where the old and new water planes intersect,and the ice force P at the bow are: 1 = M + By - 2C and adP = 2'i 2AC - B2 A M B B If a triangular distribution of ice loading is assumed with the largest load per unit length ql at the bow, then xi 1 /Im_ BA.2 - 2C -) "1 1 - Al/6B [21B B B 2AC - B A and 4 2 " B B Other assumed loadings can be given their proper scales to fit the power measurements. If the loading per unit length is calculated by the previous method, it can be checked against this method. In these expressions A = f p(x) dx ft2

II-28 I ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN B J J p(x) dxdx ft3 0 0 C = f p(x) dx x dx ft4 where p(x) is half the width of the water plane at station x, measured forward from the after perpendicular, 2t = length of the ship, ft = the change in trim angle caused by the upward thrust of the ice, radians = weight of water per unit volume, lbs/ft3 XM = Rwa hwa + Rwi hwi + Rice hice - Tp hp Rwa = water resistance, lbs Rwi = wind resistance, lbs Rice= ice resistance, lbs Tp = thrust of after propellers, lbs hwa = distance of center of water resistance above baseline, ft hwi = distance of center of wind resistance above baseline, ft hice= distance of center of ice resistance above baseline, ft hp = distance of propeller center above baseline, ft. The buoyant resistance to change of pitch angle (trim) is EK = 2[ [(R2 - xl)A + (21 - xj)B - 2C 0 Similar single, double and triple integrals of the water plane could be used for calculating buoyant resistance to lateral inclination.

ENGINEERING RESEARCH INSTITUTE -29 UNIVERSITY OF MICHIGAN II-29 It is interesting to note that in the evaluation of xl, P or ql, the term IM does not enter in a very sensitive way. For most conditions the terms consisting wholly of the integrals A, B and C are the large terms, and these can be evaluated quite accurately. The Meaning of "Thin" Ice: In Appendix C, Section d, Part II, a formula taken from "The Theory of Plates and Shells" by S. P. Timoshenko is given to find the load necessary to crack the bottom of the edge of a semi-infinite thick sheet of ice floating on water. The breaking load P is P = rx max h 0.529(1 + 0.54) [logl(Eh /Kb ) - 0.7 where x max = ultimate strength of ice in tension = 200 lbs/in2 h = thickness of ice, in. = Poisson's ratio for ice E = Young's modulus for ice (taken as.5 x 106 lbs/in2 in this calculation) K = weight of water per cubic inch b = 0.325 h, considering P as a concentrated load. The maximum static vertical force P which the icebreaker can exert at the base of the forefoot just forward of the bow propeller skeg is about 10 percent of the displacement of the vessel. This is well over 1,000,000 pounds at maximum displacement, and this load will crack an ice sheet 12 feet thick. Dynamically, if the vessel is run up onto the ice sheet the breaking load will almost double and will defeat 16 or 17 feet of ice. Naturally, if this latter method were used the bow propeller would have to go in "fighting" - rotating in the ahead direction in order to chop

ENGINEERING RESEARCH INSTITUTE II-30 I UNIVERSITY OF MICHIGAN out a hole for itself in the ice. Otherwise it would be bent or broken. From this it can be seen that any ice that the Mackinaw can encounter in the Great Lakes will be "thin" for it. Stresses and Deflections of a Semi-Infinite Sheet of Floating Ice: Appendix C, Section d, Part I is an attempt to use accurate mathematical theory to accomplish what was accomplished approximately in the previously discussed section. The differential equation for the situation is easy to construct, but no known analytical methods are available for its solution. The difference method was attempted, as shown in this section. The difference method failed at the load - failed in accuracy by a very large percentage. It indicates breaking loads obviously too small. This section is a monument to a glorious failure. It is included in this report because some record should be kept of the attempt. In what turned out to be poor judgment, more money was spent on this problem at the University of Michigan than on any other single part of the analysis of the test. This money could have been used to better advantage on investigations which were not made. This analysis, the failure of the bicycle wheel ice log, and the failure to measure ice thickness over each test course are the three major blunders made in this investigation. These three blunders remind the writer of Benjamin Franklin's famous statement that "Experience is a hard school; but fools will learn in no other." The utter failure to handle ice sheets mathematically and numerically indicates the desirability of studying the deflection and breaking of model ice sheets in a cold box. This would be a worthwhile auxiliary test to go along with any future ice-breaking tests which may be made.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN II-31 stress Analysis of a Cracked Ice Sheet: Appendix C, Section e shows an analysis of edge loads on the end of an infinitely long floating beam of ice of constant cross section. Such a beam is bent in standing waves which die away in amplitude as infinity is approached. That means that there are an infinite number of stations along the ice beam where the bending moments have maximum values. The first maximum is of direct interest, for there is where the ice beam will break off. If x is the distance from the loaded end of the beam to the first maximum, tan x = 1 and Xx = (/4 x = /4X 2 2 - Xh =h x max bb/ hx max max 6 e sin Rx where = the total vertical breaking load on the free end of the ice beam h = the thickness of the ice b = the width of the beam 6 ultimate strength of ice in tension - 200 lbs/in2 max X = Xs 23(1 2) / = Poisson's ratio =.35 = weight of water per unit volume = 62.4/1728 lbs/in3 E = Young's modulus for ice = 500,000 lbs/in2. Then (Q) (~x manx.3(1 - ~2) i/E h/4 max 6e sin (i/4)

II-32 UNIVERSITY OF MICHIGAN Inserting values of the constants (~) = 2.16 h5/4 lbs/in This holds for a floating beam of ice of constant cross section. (Qo/b)max is the breaking edge loading. The attempt to carry this accurate solution for a constant cross section beam over into an approximate solution for beams of ice with variable cross section is interesting, may not be accurate, and has not been used to draw any conclusions in this report. Plating Stresses Caused by Normal Loads: In Appendix B, Section c some expressions are given for stresses caused in infinitely long flat plates simply supported and fixed along the two parallel edges. Point and line loads are discussed. Another interesting case is for an infinitely long plate with fixed edges loaded uniformly over an area equal in length to its width. The maximum stress at the center of the loaded area is x max = 0.25 qa2/h2 The maximum stress at the built-in edge is x max = 0.50 qa2/h2 To apply this to the Mackinaw, let q = 200 lbs/in2, a = 16 in. (frame spacing) and h = 1.625 in. (shell plating thickness). Then max (center of load) = 4850 lbs/in2 rx max (fixed edge) = 9700 lbs/in2

ENGINEERING RESEARCH INSTITUTE I II-33 UNIVERSITY OF MICHIGAN, There will be stresses in the y direction (parallel to the plate edges) and shear stresses which when added to the stresses across the plate will give larger values than shown here for the center of the plate. But the maximum stress at the built-in edge will be 9700 lbs/in2 and the maximum stress at the center will not reach this amount. If the idea that the ultimate strength of ice in shear (100 lbs/in2) limits the normal loading on the hull to 200 lbs/in2 is accepted, this calculation shows that the maximum stresses in the plate will be of the order of 10,000 lbs/in2 with frame spacings of 16 inches and plate thickness of 1.625 inches. Suppose this to be true - although further testing is necessary to prove it - and that the steel in the shell plating can stand a maximum stress of 40,000 to 50,000 lbs/in2. Then the frame spacing could be doubled, keeping; the plate thickness constant, or the plate thickness could be halved, keeping the frame spacing constant. This possibility is so important that its proof or disproof alone would make further testing desirable. It should be remembered that the maximum bending stresses which can be reached in ice are probably less than those calculated on the standard trochoidal wave. This, too, indicates the possibility of thinner shell plating in a ship with the general dimensions of the Mackinaw. These possibilities are very important and warrant further testing to prove or disprove their feasibility. Distribution of Vertical, Horizontal and Longitudinal Forces along the Ice Plane of the Hull This is an investigation planned but never carried out because of lack of time and personnel. The effect of the ice on the hull is threefold. At every point of the hull at the ice sheet level there are forces with

ENGINEERING RESEARCH INSTITUTE I~I-34. UNIVERSITY OF MICHIGAN components which retard the velocity of the vessel, which tend to lift the vessel and which tend to squeeze the vessel. These components depend on the shape of the vessel. The shape is given in the faired lines of the ship taken from mold loft offsets. Suppose a rectangular coordinate system with x positive outward to the port side, y positive upward, and z positive forward had an origin which could travel along the edge of the water plane. The angles made by the plating relative to these axes can be scaled from the faired lines with some accuracy at as many points as desired. The direction of the normal to the ship plating at each point could then be determined. The direction of flow of the ice relative to the ship plating could also be determined at each point. The variation of time averages of breaking loads, inertia loads, and friction loads could then be plotted at each point in three mutually rectangular directions for as many water planes as were of interest. This is an enormous job of routine nature which was considered but avoided not only for lack of time, personnel, and-money, but also because the raw data on bending loads and inertia load time averages are not considered too good. If further testing is considered, at least the geometrical part of this calculation should be carried out for several water planes.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN CHAPTER III RECOMMENDATIONS FOR FUTURE TESTS Based on the experience gained during the ice-breaking tests, the following recommendations are made: 1) Future tests should be run when the ice thickness is at its maximum value - at the end of February or the beginning of March. 2) The future test program can be streamlined by eliminating the following tests: a) Eliminate the squat test These tests as run give rise to complicated and uncontrollable conditions. They did not give the information sought. Replace these tests by model tests for: 1) Trim versus speed in open water at various drafts and initial trims. 2) Trim versus speed with bow wave suppressor attached to the front of the model. b) Eliminate the hog-sag tests The one test already made gives adequate information. It indicated that the usual calculations, considering the ship as a beam, are adequate and accurate enough.

ENGINEERING RESEARCH INSTITUTE III-2 I UNIVERSITY OF MICHIGAN c) Eliminate the long-base strain gages The electric strain gages are better for all purposes. d) Eliminate the ice strength tests on board vessel Present information as good as we can obtain by further tests of this kihd. e) Eliminate the vibration generator tests Present information adequate. f) Eliminate the accelerometers Bending strain gages give the same information in an easier form for interpretation. g) Eliminate duplicating electrical power, RPM measurements Concentrate on oscillographs alone or Speedomax meters alone. h) Eliminate most of the shell plating strain gages Replace with 3 (or at most 4) rosette strain gages arranged either. 1) Equally spaced at one water line along the hull. Or 2) Arranged at one frame in a vertical distribution to get adequate records at all drafts to be tested. i) Eliminate moving pictures of ice 1) The ice in late February or early March may be covered with snow. In this case, the moving pictures are useless. 2) The ice may be badly clouded by that time so that the pictures would not be useful. 3) The present pictures indicate the main features as well as we need to know them.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN III-3 i) Moving pictures of ice (continued) If the moving pictures are used again, Hagerty's report, Appendix C, Section f, gives detailed recommendations. j) Eliminate the low speed operation Concentrate on high speed conditions, since the maximum forces will be developed there. Replace low speed runs by measurement of power required to start moving. 1) With stern propellers + bow propeller ahead 2) With stern propellers + bow propeller backing and possibly 3) With stern propellers alone. Fix for all tests 1) Propeller speed or 2) Electrical conditions in all motors. k) Eliminate all but one good method of measuring the velocity of the ship. 3. Certain other changes in test procedure are also recommended. General Recommendations: The propeller combinations - two stern propellers, two stern propellers plus bow propeller forward, and two stern propellers plus bow propeller aft - must be tested separately to obtain complete information. But before doing this, it would be far better to make many repeated tests at one propeller combination, one draft and trim, two set, predetermined electrical conditions (or propeller speeds) with the relative wind on the beam, measuring vessel speed and ice thickness as they occur. The ice thickness might be controlled roughly by varying the distance of the vessel from the open water edge of the ice field.

~~III-4 | ENGINEERING- RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Detailed recommendations are made in the following paragraphs. a) Wind velocity Rut all tests on days when the wind speed is less than 20 MPH. Run all tests with the relative wind on the beam. This will eliminate all resistance except that of water and ice. b) Trim records Provide stable vertical gyro with a tell-tale circuit to indicate + or - trim. c) Ship velocity Use a well designed ice-log which can be operated by one man. A small wheel and small spools of wire will eliminate most of the inertia which broke the wire. Stronger wire and nylon "leaders" will protect the operation from shock failure. Lengths of chain linkage for ice anchors should be used. Wire guides, flush-type contacts and a wheel speed-up crank will make the device fit for one-man operation. The great advantage of the wheel ice log is that it puts the speed data on the oscillograph film where it is needed. No data reduction difficulties. d) Ice thickness Ice thickness should be measured along the test course by an ice party. Measurements at five intervals would be enough. With test runs one minute in duration, the longest run will not be much over 1100 feet in length.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN e) Longitudinal (bending) strain gages Find good location near the keel around the center of the ship and attach two or three longitudinal strain gages. These should be exactly below strain gages above the neutral axis. Increase the number of longitudinal strain gages above the neutral axis from three to ten, spaced conveniently along the ship's centerline as far above the neutral axis as possible with good connection with the bending action of the ship. f) Pallograph Use this instrument in the same location. Reduce the paper speed to one-half inch per second and have timing signals from a common source with the oscillographs. g) Power, RPM and thrust measurements Thrust meters should be installed on all shafts. Use only one method of recording, either oscillograph alone or Speedomax meters alone. Provide accurate and reliable ship speed measuring device. Operate at one or two predetermined propeller speeds or at one or two predetermined electrical conditions in all tests. h) Wind direction A wind direction vane with protractor dial should be installed and used. i) Model tests Adequate model basin tests should be run for exactly the same conditions under which the ship will run on the actual ice-breaking runs.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN 1) Same displacements and trims 2) Same (model) speed of advance of model under varying powers with self-propulsion. j) Running check on results The results of each day's tests should be calculated and plotted on board the vessel each day on forms prepared in advance. This involves complete development on board of all films taken for that day. Faulty data should be replaced by repeated runs. k) Oscillograph procedure The oscillographs were operated with emphasis completely on dynamic conditions to the detriment of steady state conditions. Zeros were shifted to space the record traces. This procedure should be changed as follows: Set zeros with the ship at rest in open water. Set sensitivity scale (to lower sensitivity than was used if necessary). Make the short circuited galvanometer trace coincide with the actual freely floating zero. Leave the galvanometer traces evenly spaced. When the galvanometer is shorted, balance the bridge to zero output current; then the two zeros must coincide. Provide a calibration pulse at each end of the run. Put this on with the ship floating freely at the end of the run after the ship has been backed off the broken ice. Run the oscillograph slowly after the zero procedure until the actual test starts, then speed it up to one-half inch per second. At the end of the test, run the paper slowly again until the ship is backed off the ice and apply the final calibration pulse. Reduce the sensitivity

ENGINEERING RESEARCH INSTITUTE -7 UNIVERSITY OF MICHIGAN I until the traces are not too mixed up. Zero drift was not a serious problem over the few minutes which could include the test run and backing the vessel off the ice. Speed and velocity of wind at time of zero setting should be noted. Also the bearing, altitude and brightness of the sun (light meter reading) should be noted. We must see the change which takes place between the ice borne and the completely water borne states. 1) Ice properties More experimental work should be done on the "machining" of ice. Some of the friction acting on the ship comes from the shear of ice by the raised weld seams. This friction can be several thousand pounds in magnitude. Model tests on bending and splitting ice sheets in a cold room test are badly needed since the theory of bending and splitting floating ice sheets turned out to be 80so difficult and unrewarding. 4. Computations The effect of the ship's form on the distribution of average forces between the ship and the ice was outlined, but is not included in this report. It required an enormous amount of routine calculation for which time, personnel and money were lacking. If new tests are undertaken, these calculations might be worth doing. Everything not recommended for elimination or change should be done as in the January, 1948, ice breaking tests. The test party should be kept to the minimum necessary, and the numerous unattached observers should be omitted.

ENGINEERING RESEARCH INSTITUTE III-8 UNIVERSITY OF MICHIGAN The most important recommendation that can be made is that these tests be repeated for two or three years during the last week in February and the first week in March (or at whatever time the operating log of the icebreakers show maximum thickness of ice). Each year the tests will be performed in a better and more efficient manner. At the end of such a series of tests, the forces acting on an icebreaker will be known with some accuracy - enough accuracy to affect the actual design of future ships. Only in this way can the large expenditures made for this firs test be Justified. Experience is the only sound teacher in testing icebreakers as in all other forms of engineering activity.

APPENDIX A Section a - Part I THE VELOCITY OF THE ICEBREAKER THROUGH THE ICE BY JESSE ORMONDROYD June, 1950

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX A Section a - Part I THE VELOCITY OF THE ICEBEAER THROUGH THE ICE Two methods of measuring the velocity of the U.S.C.G.C. Mackinaw through the ice were used during the tests in January, 1948. (1) By taking motion pictures of the passing ice field from the stern of the vessel. (2) By measuring a length of wire which was anchored at one end on the ice and paid out from the ship by a length metering device. This device could be called an "ice log". PHOTOGRAPHIC METHOD In the photographic method the camera was held by a Coast Guard photographer. During each run the photographer attempted to hold the camera exactly in the same position and at the same orientation. To translate the motion picture into velocity the film speed was needed and a scale on the ice was needed. This method of measuring speed turned out to be very laborious (measuring the film frame by frame) and very erratic. The camera holder did not succeed in holding the camera in the same position and orientation - nor could he resist completely the urge to move relative to the ship to follow a particular object in the ice field.

ENGINEERING RESEARCH INSTITUTE AaI-2 ~ UNIVERSITY OF MICHIGAN This method might work if the camera were fastened firmly to the bow of the ship, oriented directly down with its optical axis perpendicular to the surface of the ice. A longitudinal scale could be put on the ice in the field of the camera to give accurate length calibration at the end of each run. But the labor of working over the film, frame by frame, makes this method uneconomical. The results of the motion picture method of measuring speed are given in Table III, of Chapter I. The results are very erratic and untrustworthy. BICYCLE WHEEL SPEEDOMETER (ICE LOG) The second method of metering wire anchored at one end in the ice shows far more promise of being a practical, accurate and economical method of measuring the speed of the icebreaker relative to the ice. This can be stated even though the method worked only 50% of the time during the tests in January, 1948. A bicycle wheel was mounted on a wooden base which was attached to the rail of the ship by C-clamps, a spool with a few thousand feet of.018-inch diameter steel wire wound on it was also set on this wooden base. From this spool the wire was wrapped around the bicycle wheel one and one-half times and then extended aft from the top of the wheel. At the beginning of the test runs a small cord of 30-foot length was fastened to the wire and on the end of this cord was attached a block of concrete with spikes in all directions (porcupine). This porcupine was thrown overboard on the ice and as the ship moved forward it very quickly stretched cord and wire into a straight line along the side of the ship. See Figure 4, Chapter I, b 5.

ENGINEERING RESEARCH INSTITUTE AaI-3 UNIVERSITY OF MICHIGAN AaI-3 A small peg was welded to the hub of the bicycle wheel, and this peg swiped a five-tooth Veeder Counter which had a small cantilever leaf spring on one tooth. This contact closed an electric circuit every five revolutions of the wheel. The opening and closing of the electric. circuit was recorded on an oscillograph in the bosun locker forward. The apparatus was quite difficult to manage, mostly due to the fact that it was not easy to give the wheel and spool the initial starts when the wire began to straighten out alongside the ship, and if that initial start was not close to the speed of the ship the wire would snap and the result was no record for that run. Wire breakage was avoided in about 50 percent of the runs, and in those cases the measurements of the speed are quite exact. In order to soften this starting snap in the wire small chain lengths were used instead of the porcupine. These chains were bought in Cheboygan and were nothing more than cross-links of a regular truck tire chain. The advantages of these links were that they would slide on the ice or snow and very quickly come into line with the side of the ship. During this sliding it was relatively easy to increase the speed of the wheel and the spool so that no breaking of the wire occurred. When the chain came into line along the side of the ship it soon would hook itself behind a piece of floating ice.

ENGINEERING RESEARCH INSTITUTE AaI-4 UNIVERSITY OF MICHIGAN ANALYSIS OF OPERATION OF BICYCLE WHEEL SPEEDOMETER The bicycle wheel speedometer failed because the measuring wire broke when the wire became taut and started the free-wheeling bicycle wheel into rotation. The speed was measured successfully whenever the operators could speed up the wheel by hand fast enough to cause the slack in the wire to be taken up slowly instead of suddenly. It was evident that the inertia resistance of the bicycle wheel and the wire spool to angular acceleration was loading the wire beyond its ultimate strength. ANALYSIS OF FORCE ACTING ON WIRE A certain length of wire is paid out when the anchor is thrown overboard. When the wire becomes taut and starts to rotate the wheel it will have a spring constant which is K = AE/A where A is the cross-sectional area of the wire, E is the modulus of elasticity of the wire and A is the length of the wire when it becomes taut. The wire is stretched by an amount s = vt - rG where v is the velocity of the ship, t is the time measured from the instant the wire becomes taut, r is the radius of the bicycle wheel and G is the angle the wheel has rotated. Stretch in wire (1) s = vt - rG Force in wire (2) F = Kvt - KrG Torque on the wheel (3) T = Krvt - Kr29 Then (4) I d = Krvt - Kr2 dt Or r (5) 2 m2 2 d~+2 m 20= ~2v

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AaI-5 m = mass of wheel p = radius of gyration of wheel If 2 K r2 p 2 2 then (6) d + vt dt2 when t = 0 and = eG 00 = angular velocity of wheel at instant the wire becomes taut. This initial angular velocity is created by spinning the wheel in the right direction by hand. Then (7) 9 = A sin ct + B cos aCt + v t r (8) 4 = Aa cos At - Ba sin at + v (9) B = o (10) A = at r a and (11) 0 r (at [ -] sin ait) The force in the wire is (12) F 1= kJ l- sin at The first maximum force in the wire is the force which breaks the wire. This force is (13) Fvau =a the =i el] =8 V If a friction force is used to keep the wheel fraoa overrunning, then its equiva lent value at the rim of the wheel is Ff.

ENGINEERING RESEARCH INSTITUTE AaI-6 UNIVERSITY OF MICHIGAN (14) Fma = r j + Ff (15) I f F2 = [ i EA W 72 Lax A L vJ 2 A g The length of the wire must be such that Fmax will be equal to (or less than) the breaking strength of the wire. Solving for I we get (16) = FrlAW v22 LrJ 2 [F_ ], 2 g For the conditions which existed in the January, 1948, tests at the maximum velocity of 12 mi/hr or 18 ft/sec: p = radius of gyration of bicycle wheel 0 11 in. r = radius to wire - 11.04 in. E = modulus of elasticity of wire = 30 x 106 lbs/in2. d = diameter of wire -.018 in. W = weight of bicycle wheel plus onehalf the weight of small wire spool (4-1/4 + 1-3/4) - 6 lbs. F = ultimate strength of wire - 20 lbs. Ff = braking friction at rim = 2 lbs. v = velocity of ship = 18 ft/sec. = 216 in/sec g = acceleration of gravity = 87 in/sec2 With these data the safe length of steel wire would be (17) e = - _ x 1400 ft.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN I AaI-7 If no hand cranking of the wheel was done (o = 0) in the high speed run over 1400 feet of wire would have to be paid out before it was permitted to become taut if the wire is to be saved from breaking. This amount of wire was never paid out before the wheel was being turned by the wire pull. Even at half the top speed at least 350 feet of wire waq necessary for safety in the first Jerk. In most of the runs the wire did break so that it is evident that not enough wire was paid out to give the safe flexibility. Those runs which were successful were due to sufficient hand cranking of the wheel to give 0o a high value which would reduce the factor [1- rQ~ to a safe value. EFFECT OF ADDED FLEXIBILITY The thought arises that the addition of a very flexible spring between the end of the wire and the ice anchor would make it possible to achieve safety with less paid-out wire. Such a very flexible spring could be made of nylon cord. Nylon cord has a modulus of elasticity En = 70,000 lbs/in2. It has a yield point of about 3200 lbs/in. A 3/32-inch diameter nylon cord could carry a load of 22 pounds before it would reach the yield point. For.018-inch diameter steel wire and 3/32-inch diameter nylon cord the ratio (18) Ew A = 15.82 En An If EnAn is put in Equation (16) instead of EwAw the safe length becomes 2 I=~l r'o I~ 88.6 ft. 15.82

ENGINEERING RESEARCH INSTITUTE AaI-8 UNIVERSITY OF MICHIGAN Equation (14) can be written (19) F = [- ja v + Ff (20) F -mazw r rg (20) Fmv ]= [-1 - 4 W 2 But wire and nylon in series give a combined spring constant Kn v 1 E.v w A (21) nK 1. Kn+K g:, A, _ 4_ En En An 4w so that (22) [w-7 F- 1 En An w (23) roe EW A W [ v]r2 [F Ff] - En An. v,r2 Fmax-Ff] 2 - Using the same values as before for the highest speed we get (24) XLw + 15.82 n [1 - 2 1400o ft Considering the case in which there is no hand cranking of the wheel (0o = 0) the safe length of steel wire is (25) w + 15 82 = 1400 ft. (26),w - 1400 - 15.82 n (ft.)

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AaI-9 From Equation (26) the following table is prepared. n w 0 ft. 1400 ft. 20 ft. i084 ft. 40 ft. 767 ft. 60 ft. 451 ft. 80 ft. 134 ft. 88.5 ft. O ft. Obviously, a lead of 3/32-inch diameter nylon cord 90 feet long would make all starts certain and safe even if a bicycle wheel is used with no hand cranking. These calculations indicate that it was surprising that any successful runs were made. This analysis should have been made before the test runs since the ship velocity is an indispensible measurement for most of the calculations made with the test data. RECOMMENDATIONS FOR FUTURE DESIGNS OF WHEEL SPEEDOMETERS This apparatus could be made to work 100 percent of the time by the following changes: (1) Make a smaller wheel of aluminum like a disc with a relatively deep groove; guide the wire as it enters the wheel as well as when it leaves the wheel through small pipes mounted rigidly on the wooden base. This will prevent the wire from jumping the groove and the smaller wheel will have less inertia than the original bicycle wheel and thereby diminish the snap in the wire.

AaI-10 ENGINEERING RESEARCH INSTITUTE AaI-l10 UNIVERSITY OF MICHIGAN (2) Make a number of spools with enough wire on each spool for one run, about 1200 to 1500 feet on each spool. This will diminish the inertia of the spool and also help to minimize the snap in the wire. (3) Redesign the arrangement of the electric contact, as the one used in the original design is not sturdy enough for this kind of work. Flush surface, side contact slip ring, and contact would make the best design. (4) Use nylon cord leads between the wire and the ice anchor. (5) Provide a ratchet hand crank to give initial pay-out velocity to the wheel. (6) Use a section of chain as an ice anchor. This anchor was very successful.

APPENDIX A Section a - Part II TRIM OF TBE U.S.C.G.C. MACKINAW WHILE MOVING TROUGH THIN ICE BY JESSE ORMONDROYD June, 1950

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX A Section a - Part II TRIM OF THE U.S.C.G.C. MACKINAW WHILE MOVING TlROUGH THIN ICE The gyro stable vertical which measured the angle of inclination (trim) of the ship gave a signal in the form of a carrier wave of 60-cycle/ second frequency. The width of the envelope of the carrier band was directly proportional to the trim angle. Calibration against draft marks established the following relationship between the trim angle in radians and the width of the carrier band in inches: Trim angle.01235 radians Trim angle 0 inch width of carrier band With the magnitudes of the trim angles encountered in the icebreaking runs and even in the calibration measurements, the angle 0 in radians was quite accurately equal to the tangent of 0, since the maximum angle 0 was never greater than 1 degree. The circuit used had one disadvantage. There was no way of telling whether the trim angle was positive or negative. If the band fluctuated in width around zero, there was no way of telling which side of zero the angle was varying. This could-have been remedied by using a simple tell-tale signal for + and - angles. For the purposes of power calculations average values of the trim angle tangent were measured from the records.

ENGINEERING RESEARCH INSTITUTE AaII-2 UNIVERSITY OF MICHIGAN Table I shows the drafts forward and aft read dead on the water by the bridge draft gages and the average trim used in each run for the power calculations. This table can be summarized as follows: Average Draft Gage Readings Average Gyro Stable Trim Angle Vertical Zero Reading Runs Fwd Aft [ Radians in Radians Trim Angle 4A 15t4" 18' 10".O140(Bow up).009* 4B 17' 0" 17'2".0005 (Bow up).000 to.0087** 4c 18'8" 19' 1".0017(Bow up) to.002 *The average zero readings of the stable vertical do not check the average draft gage readings. **This reading was evidently taken with the ship hung up on the ice at standstill. The gyro stable vertical trim readings fluctuated continuously with the pitching of the ship. The pitching period was about five seconds. The amplitude of pitching in still water was about.0003 radians and during the ice breaking operation the pitching amplitude was as much as.006 radians. During the squat tests when the ship was subjected to violently changing hydrodynamic conditions the pitching amplitude was.0005 radians in open water. The magnitude of the pitching can be better understood by the statement that an amplitude of.0005 radians is equivalent to difference in elevation between the bow and the stern of 1.8 inches. This is extremely small.

AaII-3 TABLE I DRAFTS DEAD IN WATER AND AVERAGE GYRO TRIMS Run Draft Fwd Draft Aft Tan Gyro Trim Angle Average 4A Ia 15'4" 18'9".0105 4A Ib 15'35" 18'9".0110 4A Ic 14 ' 10" * 18' 10".0110 4A IIa 15't4l 18'10".0004* 4A IIb 15' 3" 18' 9".0140 4A IIc 15'2" 18' 10".0153 4A IIIa 15'3" 18 9".0096 4A IIIb 15'7" 18'8" -.0089 4A IIIc 15'4" 19'6" *.00914 4B Ia 15'11" * 17'2".00093 4B Ib 17'1" 17'2".00060 4B Ic 16'4" * 17'2".00302 4B IIa 17'0" 17'2".00102 4B IIb 17'1" 17'2".00075 4B IIc 17' 0" 17'2".0005 4B IIIa 17' 0" 17' 1".0005 4B IIIb 17'1" 17'1".00145 4B IIIc 16'11" 17'2".00075 4C Ia 18'8-1/2" 19' o".00047 4C Ib 18' 8" 19' 0".00120 4C Ic 18'8" 19''l".00072 4C IIa 18'7" 19'2".0007 4c IIb 18' 7-3/4" 19'1".00218 4C IIc 18'6" 19'3".00212 4C IIIa 18' 8" 19' 1".0015 4C IIIb 18'7" 19'1 ".00202 4C IIIc 18'8" 19' l".00185 5A 18'11" 18'9".00275 5B 18'7" 18'9".00385 *These readings are evidently out of line.

ENGINEERING RESEARCH INSTITUTE AaII-4 UNIVERSITY OF MIbIIGAN EFFECT OF ICEBBEAING ON TRIM It seems reasonable to suppose that an icebreaker advancing through ice will have its trim by the stern increased, since all the breaking forces and most of the upward inertia forces of the broken ice being ploughed under are concentrated at the forward end of the vessel. However, there are many factors which change trim. Some of these which may affect the trim of the U.S.C.G.C. Mackinaw may be listed: (1) The windage (2) The water resistance (3) The bow wave (which is suppressed in the ice) (4) The direction of rotation and the speed of the bow propeller (5) The upward thrust of unbroken ice (6) The upward thrust of ice being ploughed under (7) The distributed flotation upward thrust of the broken ice under the ship We have no adequate information on any of these factors. Some general information concerning the trim and sinkage of vessels totally dissimilar to the U.S.C.G.C. Mackinaw is contained in two reports issued by the Taylor Model Basin, Navy Department, Washington 7, D. C. These reports are: (1) "Shallow Water Effect on the Performance of Single Screw Vessels of U.S. Maritme Commission Design Determined From Resistance, Propulsion and Sinkage Tests of Models", By W. H. Norley, Report 640, February, 1948. Unrestricted Report. (2) "The Change of Trim of Destroyers Running at Various Speeds" By Captain H. E. Sanders, U.S.N., and Lt. W. J. Alfriend, U.S.N.R., Report

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AaII-5 R-236, June,1944. Confidential Report. The merchant vessels trim slightly by the bow up to full speed. The destroyers trim very slightly by the stern up to speed ratios V = 1.1 The maximum speed ratio of the U.S.C.G.C. Mackinaw during the ice-breaking runs was v = o.6 Under steady running conditions it might be assumed that the trim of the Mackinaw is very slightly influenced by speed through open water with a fully developed bow wave. When the bow wave is suppressed by the ice sheet being broken the ship, may even trim slightly by the bow. We do not know that, however. The squat test which was made to give information on this score raises more questions than it answers. In all the squat tests the trim by the stern increased as the ship gained forward speed through the water. At the end of all tests the trim angle was levelling off around.007 radians (bow up). Unfortunately, the readings were not continued long enough to see if this was a transient condition. It probably was. The windrow ice tests showed a positive increase in trim angle (bow up). In run 5A the ship went through a windrow 15 feet deep (hot water lance measurement) and the trim angle increased from 0 radians to.0037 radians (maximum), i.e., the bow rose relative to the stern some 13-1/2 inches. In run 5B, the ship went through a windrow 20 feet deep andtie to the trim changed from.00123 radians to.00556 radians - the bow rising relative to the stern some 15-1/2 inches.

ENGINEERING RESEARCH INSTITUTE AaII-6 UNIVERSITY OF MICHIGAN The effect of the bow propeller on trim is not known. However, when rotating in such a direction as to drive the ship forward, it must increase the speed of the water relative to the hull, thereby lowering the pressure under the bow and increasing the trim by the bow. In rotating the opposite way it must lower the velocity of the water relative to the bow, increase the pressure, and increase the trim by the stern. This is pure speculation. An attempt to correlate the changes in trim angle between standstill (zero readings) and readings during each run showed the usual inconsistencies. The changes were always very small and sometimes they were negative, sometimes positive. In future tests the change in trim should be measured in a different way. The oscillograph paper should be run at very slow speed before the run starts, the zeroes should not be redistributed, and after the run is over the oscillograph should be run slowly until the ship is backed off the ice into open water. In this way we can get a direct measure of any changes in trim. The ice encountered in these test runs was too thin to give rise to large forces and consequent large changes in trim. Only the windrow tests were run in such a way as to show distinctly a change in trim. It was in the direction expected, but was of such a size as to indicate that a thin sheet of ice would have very little effect.

APPENDIX A Section a - Part III SQUAT TEST ANALYSIS BY A. C. McCLURE August, 1949

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX A Section a - Part III SQUAT TEST ANALYSIS A series of three tests, 2A, 2B, 2C, were run on the ship to determine the change in trim and squat at various speeds compared to that with the ship at rest. The purpose of the tests was to find a correction to be applied to the total trim when breaking ice. The test procedure was to run the vessel full astern, then switch the motors to full power ahead. The test records were started when the ahead signal was given. Three practice runs were made to familiarize the test personnel with the measurements to be made. Freeboard forward and amidships port and starboard were measured by means of floats and tapes. Tape readings were taken before the test with no power on, during the test going astern under full power, and at the moment when the ship was dead in the water. Trim was measured during the run by means of the gyro stable vertical. Thrust, power, speed, vibration, and plating strain gages were recorded simultaneously by the oscillographs. The tests were run under three different power conditions: Run 2Atwo stern motors from full astern to full ahead; Run 2B - all three motors full astern to full ahead; and Run 2C - the bow motor maintained astern while the stern motors run from full astern to full ahead.

ENGINEERING RESEARCH INSTITUTE AaIII-2 UNIVERSITY OF MICHIGAN Data taken during the runs is given in Table I. The bicycle wheel ice log was equipped with a float in place of the drag used in the ice-breaking runs. To check the accuracy of the measured speed, an estimate was made using the results of TMB model tests in the following manner: BHP, thrust, and shaft speed in revolutions per second were obtained from the squat test data. The thrust coefficient, CT, was calculated according to the formula: CT T T n2p2D2 where: T = Thrust on one propeller in pounds. n = Shaft revolutions per second. P = Propeller pitch in feet. D = Propeller diameter in feet. Propeller characteristic curves provided by TMB were entered with CT to obtain real slip ratio, SR. A second set of curves of real slip ratio plotted against brake horsepower for several speeds was.entered with BHP and SR to obtain speed. Speeds were calculated in this manner for all of the runs in the ice-breaking tests for which accurate speed measurements had been made. Correction curves of measured versus estimated speed were drawn. The results of the above calculation were corrected accordingly. The points on the curves (Figures 2,3, and 4) which were calculated as above are circled, the speeds taken from the ice log are marked by X's. There is no indication that the vessel had reached a steady speed in the ahead direction before the conclusion of the run. Gyro trim plotted against time shows a gradual increase from negative trim (down by the head) to positive trim after the motors were reversed. Toward the conclusion of

AaIII-3 SQUAT TEST TABLE I Appendix A Section a - Part III Run: 2A 2B 2C Time: Start 10:22:00 10:42:00 10:53:00 Dead in Water 10:22:29 10:42:35 10:53:58 End of Run 10:22:59 10:43:06 10:54:28 Drafts Before Test Fwd. 17 L2-1/2" 17'-2" 17'-6" Aft. 17 ' -7" 17' - 171' -2" Freeboard by Tapes And Floats: Main Deck Amidships, Port No Power 13' - 3-1/4" 13' - 3'1/4" 13'-4-3/4" Full Power, Ship Dead 13' - 4-3/4" 13' - 2-3/4" 13'-4" Main Deck Amidships, Star. No Power 13' - 4-3/4" 13' - 4-3/4" 13' - 3-1/4" Full Power, Ship Dead 13' - 4-1/4" 13' - 3-1/2" 13'-1" Mean Change in Freeboard Amidships +1/2" - 1-1/4" - 1-1/2" Upper Deck at Bow No Power 25'-10" 25'-10" 25'-10" Full Power, Backing 25'-8" 24'-5" 25'-8" Full Power, Dead in Water 26'-0" 25'-11" 26'-0" Change in Trim From No Power to Full Power, Dead in Water: Float-tapes 3" 3-3/4" 7" Gyro 8" 3" 14-1/2" Date of Test - January 30, 1948

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ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AaIII- 9 the run the trim leveled off. It is likely that if the gyro record had been continued longer, it would have shown a decrease in trim from the maximum reached during the period of acceleration. There was at least a qualitative agreement between the trim obtained from float-tape readings, and the gyro record. Comparing the trim when the ship was dead in the water with full power ahead to the trim at rest, the float-tape readings for Run 2B showed a change of about 3-3/4 inches. The gyro registered a 3-inch change in trim. In Run 2A the float-tape readings gave a change in trim of about 3 inches, while the gyro indicated a change of 8 inches. In Run 2C the float-tape readings gave a change in trim of 7 inches, while the gyro showed a 14-1/2 inch change. The float-tape readings for Run 2A taken at the instant the vessel was dead in the water with full power ahead indicate a decrease amidships of 1/2 inch as compared to the draft in still water. (See Figure L) This represents a loss in displacement of about 16 tons. It is hard to conceive a decrease in displacement of this magnitude, since the vertical component of propeller thrust at this instant was only about 3 tons. It is more likely that the vessel would be lowered bodily with respect to the undisturbed surface of the water due to reduction in pressure under the hull in the vicinity of the propellers. The actual change in draft was undoubtedly small, making it hard to measure accurately. Turbulence caused by the rapid change in direction of motion of the vessel, waves, rolling, and pitching, however slight, would all contribute to inaccuracy. The float-tape readings for Runs 2B and 2C indicate increases in displacement of 28.8 and 49.5 tons, respectively. In these runs the changes

ENGINEERING RESEARCH INSTITUTE AaIII- 10 UNIVERSITY OF MICHIGAN are in the direction expected, but their magnitude is subject to the same errors as outlined above. Curves of thrust, trim, and RPM (Figures 2,3,4) were plotted against time to show the variation of these quantities throughout the run. The thrust curves show wide fluctuations between the time the motors were reversed and the arresting of the ship's sternward motion. This is to be expected owing to the turbulence of the water suddenly being reversed in direction. About the time the ship lost way, the thrust steadied down and only the normal higher frequency fluctuations were apparent. The propeller RPM increased rapidly until the thrust fluctuations began to steady down. At this point the RPM dropped sharply, then began to rise again as the thrust leveled off.

APPENDIX A Section b - Part I VIBRATION TESTS OF FULL GIRDER TO DETERMINE THE NATURAL FREQUENCIES OF THE THREE LOWEST NORMAL MODES OF VIBRATION IN THE VERTICAL PLANE BY ROBERT L. HESS JESSE ORMONDROYD September, 1947

DEPARTMENT OF ENGINEERING RESEARCH I I UNIVERSITY OF MICHIGAN PURPOSE OF TEST The test run was planned to find the normal mode shape factors in deep and shallow water, to test an accelerometer, to find the order of magnitude of ship beam stress and to lay out a final testing procedure. It was also planned to calibrate six strain gages in terms of bending moment by means of a hog-sag test.

ENGINEERING 'RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX A Section b - Part I VIBRATION TESTS OF HULL GIRDER TO DETERMINE TEE NATURAL FREQUENCIES OF THE THREE LOWEST NORMAL MODES OF VIBRATION IN THE VERTICAL PLANE Instrumentation for Vibration Tests of U.S.C.G.C. Mackinaw, September 2-6, 1947 (1) A Taylor Model Basin medium vibration generator No. 100052, capable of delivering a force amplitude of 20,000 pounds, was installed on the centerline of the ship on the upper deck, centered over Cant Frame No. 3. It was used to generate vertical vibrations. The force amplitude generated with this machine is: PO = 0.154 sin (a/2) N2 where a = the eccentric setting in degrees N = the RPM of the generator. The vibration generator was operated at various speeds and eccentricity in order to find the vertical natural frequencies of the ship hull and the normal elastic curves of the ship hull at the natural frequencies. The vibration generator and its controls are completely described in Taylor Model Basin Report No. 524, April, 1944. Figure 1 shows the vibration generation generator and its controls installed on the upper deck, forward, of the U.S.C.G.C. Mackinaw in September, 1947.

AbI.~~~~2 F IGUIE 1. Appendix A, Section b, Part 1. Taylor Model Basin medium vibration generator and controls installed on the centerline of the upper deck, centered over Can~t Frame 5, U.S.C.O.C. Mackinaw, September 2-6, l9ky7 The vibration generator was adjusted to give vertical forces.,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~aE~~~~~~~~~~~~~~~~~~~~~00;00S;00000:000000000000000000000000:X~t0000000000000000000000000000000000000:0000000000000000000000000000000 llllllllllllarr~~~: I >1, _IIII~llillljl 11111 * Xiiiiiiiiiiiii~iiq _ _ I 010 i~~~~~~~~~~~Q~l 11 i Ap~peuix A, Section b, Prat l Taybsnr MideJ:1asin mediu nribra-tion geneart~2;rb ~d conlrolo inetwlled on the center11ne 09 the upzper dgck, cenlered OureP Cart; Frae 3, {J*S.C.CC, Mtackrinaw, Septembe~r 2-d, 15>47. The vibpation gEsnera-oa: tJaa tajusted Lo give vefftlesl folpeee,

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AbI-3 TEST DATA Vibra- Condition Mode Resonant Natural Amplitude tion RPM of Frequency of Exciting Test Vibration Cycles Force at Ship Motion Number Generator sec. Resonance 1 Underway, 104 Variable Variable Variable Variable RPM, 15 mi/hr on Lake Huron 2 At anchor, 80 2 nodes 207 3.45 6,588 lbs ft. of water under hull 3 At dock in Che- 2 nodes 153 2.55 3,588 lbs boygan, Mich., 2 ft. of water under hull 4 At anchor, 80 3 nodes 368 6.15 20,860 lbs ft. of water under keel 5 At dock in Che- 3 nodes 267 4.45 8,700 lbs boygan, Mich., 2 ft. of water under keel 6 At anchor, 80 4 nodes 587 9.80 11,040 lbs ft. of water under keel The amplitude of the rigid, body motion of the ship was negligible in size in comparison with the resonant vibration amplitudes. Table I shows the forces developed at various speeds and eccentric settings of the vibration generator.

AbI-4 TABLE I FORCES DEVELOPED AT VARIOUS SPEED AND ECCENTRIC SETTINGS Angle Speed for Speed for Angle Speed for Speed for of Eccen- 5000-lb. 20,000-lb. of Eccen- 5000-lb. 20,000-lb. Setting tricity Driving Driving Setting tricity Driving Driving ax e - Force Force a e Force Force Degrees Inches RPM RPM Degrees Inches RPM RPM 3 0.105 1114 2230 93 2.90 211 423 6 0.209 788 1578 96 2.97 209 418 9 0.314 643 1287 99 3.04 207 413 12 0.418 557 1115 102 3.11 205 409 15 0.522 499 997 105 3.17 203 405 18 0.626 455 911 108 3.24 201 401 21 0.729 422 844 111 3.30 198 397 24 0,.832 395 790 114 3.36 197 393 27 0o.938 372 744 117 3.41 195 390 30 1.035 354 708 120 3.46 193 387 33 1.136 338 676 123 3.52 192 384 36 1.236 324 649 126 3.56 191 381 39 1.335 312 623 129 3.61 190 379 42 1.434 301 602 132 3.65 189 377 45 1.531 291 583 135 3.70 187 375 48 1.627 282 565 138 3.74 186 373 51 1.722 274 549 141 3.77 186 371 54 1.816 267 535 144 3.80 185 369 57 1.909 261 521 147 3.84 184 368 60 2.000 255 510 150 3.86 183 367 63 2.09 249 499 153 3.89 183 366 66 2.18 244 488 156 3.91 182 365 69 2.27 239 478 159 3.93 182 364 72 2.35 235 470 162 3.95 181 363 75 2.44 231 462 165 3.96 181 362 78 2.52 227 454 168 3.98 181 361 81 2.60 224 447 171 3.99 181 361 84 2.68 220 440 174 3.99 181 361 87 2.75 217 434 177 4.00 180 360 90go 2.83 214 428 180 4.00 180 360

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AbI-5 (2) A portable pallograph, TMB Type V, No. 100131, was employed to measure the amplitude of the vertical vibrations at sixteen stations along the main deck. The instrument had a natural frequency of 20 cycles per minute, a magnification factor of 6, and it was self-recording on wax-surfaced paper. The amplitude measurements were made at the following stations on the centerline of the main deck: Frame 10 Frame 21 Frame 32 Frame 46 Frame 57 in the head Frame 74 in the pantry Frame 86 in the galley Frame 109 in the Engineer's office Frame 113 in the motor generator room Frame 128 Frame 145 Frame - at the crane centerline Frame - forward of Hatch 173 Frame - aft of Hatch 173 Frame - at sounding C5W Frame - in the fan tail. Figure 2 shows the TMB Type V pallograph. (3) A long base mechanical strain gage was installed on the port side of the upper deck between Frames 104 and 112 in the fore-and-aft direction. It served as a check on the electrical strain gages. The gage length of this strain gage was 10 feet. It indicated the double amplitude of the strains on a dial gage heading in units of.0001 inch. Figure 3 shows the long base strain gage installed on the upper deck of the U.S.C.G.C. Mackinaw, on September 2-6, 1947. The motor-driven scratch target shown in the figure was not used for the test readings taken. The damping is so small in the ship that very sharp tuning exists at resonance. Only those readings of.030 inch and over were taken near the actual 2 noded mode resonance. All readings below this were taken when the

AbI-6 FIGURE 2 Appendix A, Section b, Part 1 Taylor Model Basin Type V Pallograph. Shown installed at Frame 10 on the main deck of the U.S.C.G.C. Mackinaw, September 2-6, 1947. The instrument was adjusted to measure vertical vibrations.

AbI-7 FIGURE 3 Appendix A, Section b, Part I Taylor Model Basin long base strain gage - gage length, 10 feet. Installed on the port side of the upper deck between Frames 104 and 112 in the fore-and-aft direction. The longitudinal bulkhead of the deckhouse is to the left of the strain gage. U.S.C.G.C. Mackinaw, September 2-6, 1947.

ENGINEERING RESEARCH INSTITUTE AbI-8 l IUNIVERSITY OF MICHIGAN TABLE OF READINGS ON LONG BASE STRAIN GAGE 2 Noded Mode 3 Noded Mode 4 Noded Mode Double Amplitude Double Amplitude Double Amplitude Inches Inches Inches.032.027.017.030.00001.00050.032.027.019.029.00010.00012.033.028.022.030.00005.00005.034.028.024.030.030.029.025.029.030.030.021.029.026.030.025.029.027.008.027.029.026.021.019.029.028.016.019.029 vibration generator was operating above or below resonance. These readings must be divided by 120 inches (gage length of the strain gage) to find unit strains in the neighborhood of the gage. It is interesting to compare these strains with those obtained by the metalectric strain gage No. 5. Strain gage No. 5 is mounted at Frame 113-1/2, 22.5 feet above the neutral axis. The long base strain gage extends longitudinally between Frames 104 and 112. It is 23.5 feet above the neutral axis. The point of maximum bending stress (the anti-node) in the 2 noded mode of motion is located at Frame 117. The bending moment is near its maximum value around this frame, so80 that we would expect the strain on the upper deck to be somewhat greater than the strain in gage No. 5.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AbI-9 Strain in the long base strain gage at 2 noded resonance is.034/120 = 28.3 x 10-6 in/in. This corresponds to a stress range of 850 lbs/in2 (+ 425 lbs/in2). The maximum stress range recorded on gage No. 5 in the 2 noded mode motion was 586 lbs/in2 (+ 293 lbs/in2). (4) Six Type A-1, SR-4, single element electric strain gages were installed to read fore and aft strains on the ship's longitudinal beam structure. Figures 4, 5, 6, 7, 8, and 9 show these strain gages as installed and waterproofed. (5) An electromagnetic oscillograph No. TMB-2008 was used to record the SR-4 strain gage signals. Figure 10 shows this oscillograph and related electronic equipment. The SR-4 strain gages and the electronic equipment were installed on September 2 and 3, 1947, by L. D. Anderson and J. P. Hendrican of the Taylor Model Basin. (6) A Statham accelerometer, Model R-5-120, No. 635, was located on the main deck abaft bulkhead 10. This instrument had a natural frequency of 185 cycles per second, and could record accelerations of + 5 g (+ 1930 in/sec2). Strain indicator, TMB Type l-B, serial no. 101, provided the signal for the osc illograph. This instrument was merely used to get advanced information for the accelerometer installation on the ice-breaking tests to be run in January, 1948.

AbI-10 FIGURE 4 Appendix A, Section b, Part I Strain gage No. 1 (SR-4, Type A-l, single element) installed on inner bottom plating in the forward shaft alley, 26 inches forward of bulkhead 57, on the centerline of the ship, 6 feet above the base line. The figure shows the metal cover over the waterproofing which covers the strain gage element. This gage measures longitudinal strains. U.S.C.G.C. Mackinaw, September 2-6, 1947.

AbI-11 FIGURE 5 Appendix A, Section b, Part I Strain gage No. 2 (SR-4, Type A-l, single element) installed in generator room No. 2 on the upper flange of 10-inch channel over 12-inch I-beam. 15 feet, l0 inches, abaft bulkhead 93 on the centerline of the ship, 7 feet above the base line. The gage measured longitudinal strains. This gage never responded to the bending of the ship. U.S.C.G.C. Mackinaw, Sep.tember 2-6, 1947.

AbI-12 FIGURE 6 Appendix A, Section b, Part I Strain gage No. 3 (SR-4, Type A-i, single element) installed on inner bottom plating 13 inches abaft bulkhead 153 on the centerline of the ship, 4 feet 8 inches above the base line. This gage measured longitudinal strains. This gage developed a leak to ground and never gave good records. Uo.SC.G.C. Mackinaw, September 2-6, 1947.

AbI-13 FIGURE 7 Appendix A, Section b, Part I Strain gage No. 4 (SR-4, Type A-1, single element) installed on the center of lower flange of upper deck girder, 7 feet 11-3/4 inches aft the centerline to starboard, 6 inches abaft Frame 43, 37 feet 6 inches above the base line. This gage measured longitudinal strains. UJ.S.C.GvC. Mackinaw, September

AbI-14 FIGURE 8 Appendix A, Section b, Part I Strain gage No. 5 (SR-41 Type A-l, single element) installed on longitudinal bulkhead 7 feet 7 inches off the centerline to port, 2 inches below upper deck plating, midway between Frames 113 and 114, 35 feet 6 inches above base line. This gage measured longitudinal strains. U.S.C.G.C. Mackinaw, September 2-6, 1947.

AbI-15 FIGURE 9 Appendix A, Section b, Part I Strain gage No. 6 (SR-4, Type A-l single element) installed on underside of flange on main deck girder 8 feet off centerline to port, 9 inches forward of bulkhead l48, 27 feet 6 inches above base line, This gage measured longitudinal strains. U.S.C.G.C* Mackinaw, September 2-6, 1947.

AbI-16 FIGURE 10 Appendix A, Section b, Part I Taylor Model Basin amplifier, power supplier, and oscillograph installed on port side of bosun's locker, main deck forward, U.S.C.G.C. Mackinaw, September 2-6, 1947.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AbI-17 Test Party Passenger List for Vibration Survey, U.S.C.G.C. Mackinaw, September 2-6, 1947 U.S. Coast Guard 1) Capt. D. R. Simonson 2) Comdr. Doebler, C. O. U.S.C.G.C. Mackinaw 3) Mr. R. C. Browning 4) Mr. S. W. Lank 5) Mr. P. G. Tomalin David W. Taylor Model Basin 6) Mr. L. D. Anderson 7) Mr. V. H. Hardy 8) Mr. J. P. Hend.rican 9) Mr. R. T. McGoldrick University of Michigan 10) Prof. E. L. Eriksen 11) Prof. W. W. Hagerty 12) Mr. R. L. Hess 13) Prof. J. Ormondroyd 14) Prof. M. B. Stout 15) Prof. J. T. Wilson The test party, with instruments, boarded the ship at 1200 in Detroit, Michigan, on Tuesday, September 2, 1947. Installation of the TMB medium vibration generator and its controls was finished during the morning of September 3, 1947, and a general survey of the vibrational characteristics of the ship hull was made on September 3, while the ship was underway toward Cheboygan, Michigan. The normal mode vibration tests were made at the dock at Cheboygan, Michigan, bottom 2 feet below the keel and in deep water (bottom 80 feet below the keel) off Cheboygan on September 4 and 5. Strain gages were installed and tested during the period September 2-5. The test party, with instruments, left the ship at Detroit, Michigan, at 2100 on Saturday, September 6, 1947.

DEPARTMENT OF ENGINEERING RESEARCH AbI-18 UNIVERSITY OF MICHIGAN SUMMARY OF REPORT Tests on board the Mackinaw were begun 3 September 1947. A vibration generator capable of delivering a 20,000-pound force was mounted in the eyes of the ship and run at various speeds and eccentricities. Records were made of the motion of the ship by means of a portable pallograph. Strain gages were located in six positions of the ship's beam structure and records made during vibration tests by means of a recording scill scope. An accelerometer was tested and found to deliver a good signal. Hog-sag tests were made invalid by action of the sun and were discontinued. Pallograph records showing (a) Amplitude versus Frequency at Frame 10 (b) First Mode Shape in deep water* (c) First Mode Shape in shallow water** (d) Second Mode Shape in deep water (e) Second Mode Shape in shallow water (f) Third Mode Shape in deep water have been taken, measured, recorded, and plotted. Frequencies of these modes were (a) First Mode -- 3.4 - 3.5 c.p.s. in deep water 2.5 - 2.6 c.p.s. in shallow water (b) Second Mode -- 6.1 - 6.2 c.p.s. in deep water 4.4 - 4.5 c.p.s. in shallow water * Deep water - Ship was anchored in eighty feet of water. ** Shallow water - Ship was moored to wooden' dock at Cheboygan, port-side to, with 2 feet of water below keel.

DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN AbI-19 (c) Third Mode -- 9.8 c.p.s. Oscilloscope records were measured which were used to calculate stresses. The order of magnitude of these stresses in the first mode is 500 psi. The shapes of the first three modes of vibration were calculated from the uniform beam theory. (See reference on page 18.)

DEPARTMENT OF ENGINEERING RESEARCH AbI-20 UNIVERSITY OF MICHIGAN STUDIES OF FORCES WHICH ACT UPONI A VESSEL ENGAGED IN BREAKING ICE AND REACTIONS OCCURRING IN THE HULL GIRDER OF SUCH A VESSEL STRAIN GAGE LOCATIONS Six strain gages of the single element SR-4 type were located about the ship. These gages were waterproofed and leads run to the recording room (the bosun's locker). Gage No. 1 - 26 inches forward of Bulkhead 57, on center line of ship, on floor plating of forward shaft alley. Gage No. 2 - 13 feet 10 inches aft of Bulkhead 93 on center line of ship, on 10-inch channel over 12-inch I-beam, in Engine Room No. 2. Gage No. 3 - 13 inches aft of Bulkhead 153, on center line of ship, on plating on top of inner bottom tank. Gage No. 4 - 6 inches aft of Frame 43, 8 feet starboard of center line of ship, lower flange of beam, underside of upper deck. Gage No. 5 - Halfway between Frames 113 and 114, 7 feet 7 inches port of center line of ship, 2 inches below deck plating of upper deck, on longitudinal bulkhead.

DEPARTMENT OF ENGINEERING RESEARCH AbI UNIVERSITY OF MICHIGAN AbI-2. Gage No. 6 - 9 inches forward of Bulkhead 149, 8 feet port of center line of ship, on lower flange of I-beam, on underside of main deck. Gage No. 1 - 6 feet above M.B.L. Gage No. 2 - 7feet above M.B.L. Gage No. 3 - 4 feet 8 inches above M.B.L. Gage No. 4 - 37 feet 6 inches above M.B.L. Gage No. 5 - 35 feet 6 inches above M.B.L. Gage No. 6 - 27 feet 6 inches above M.B.L. M.B.L. - moulded base line.

DEPARTMENT OF ENGINEERING RESEARCH AbI-22 UNIVERSITY OF MICHIGAN HULL DIMENSIONS AND TANK LOADINGS LBP 280' - 0" LOA 290' - 0" Breadth extreme (Mld) 74' - 4" Frame spacing, aft of bulkhead 10 to stern cants = 16" Top of bridge at center line of frame 68 (Mld) (Wheel house top) above base line 56' - 2-3/8" Upper deck at center line of frame 66 (old) above base line 37' - 6-1/8" Main deck at stern (mild) above base line 28' - 3-7/8" Drafts in feet and inches Shallow Water Test Deep W. T. Forward 15' Aft 16 ' 11" Tank Soundings in Tons A1W Fore Peak 123 Tons B901W Fresh Water 4800 gal. B920W Ballast Tank 13 Tons C901W Ballast Tank 6 Tons C-4W Trim Tank 100 Tons C-SW After Peak 55 Tone B10-1/2 480 480 312-1/2 11560 11020 B7-1/2 Stbd Diesel Oil Service Tank 11800 gal 11590 gal B8F Port Diesel Oil Wing Tank 14600 14600 B91 Stbd Diesel Oil Wing Tank 20200 20200 Bl0F Port Diesel Oil Wing Tank 7000 7000 BllF Stbd Diesel Oil Wing Tank 33200 33200 B12F Port Diesel Oil Wing Tank 11625 11625 B906F Port Diesel Oil Bottom Tank 14875 14875 B909 Port Diesel Oil Bottom Tank 575 575 B91OF Port Diesel Oil Bottom Tank 12475 12475 B911F Stbd Diesel Oil Bottom Tank 13600 13000 B912F Port Diesel Oil Bilge Tank 12900 12900 B9137 Stbd Diesel Oil Bilge Tank 4275 4275 19147 Port Diesel Oil Bottom Tank 3725 3725 B915F Stbd Diesel Oil Bottom Tank 1500 1500 3916 11610 11610 B917 13680 13680 All Machinery, and piping including heeling and trimming piping, full.

DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICT IGAN VIBRATION TEST NO. 1 3 September 1947 Amplitude versus Frequency at Frame 10 Ship underway at 104 RPM (15 MPH). Vibration generator in eyes of ship running at 36~ eccentricity. Pallograph at Frame 10 on main deck on center line of ship. Magnification of motion is six. Each record shown covers a 1-second interval of time.

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DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN AbI-25 VIBRATION TEST NO. 2 5 September 1947 First Mode Ship at anchor with 80 feet of water below keel. Vibration generator located in eyes of ship running at 171~ eccentricity. Pallograph magnifies motion six times.

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DEPARTMENT OF ENGINEERING RESEARCH b27 UNIVERSITY OF MICHIGAN AbI-27 VIBRATION TEST NO. 3 3 September 1947 First Mode Ship made fast to dock at Cheboygan, Michigan, with two feet of water under keel. Vibration generator located in eyes of ship running at 1710 eccentricity. Pallograph magnifies motion six times.

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DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN AbI-29 VIBRATION TEST NO. 4 5 September 1947 Second Mode Ship at anchor with 80 feet of water below keel. Vibration generator located in eyes of ship running at 171~ eccentricity. Pallograph magnifies motion six times.

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DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN AbI-31 VIBRATION TEST NO. 5 4- September 1947 Second Mode Ship made fast to dock at Cheboygan, Michigan, with 2 feet water under keel. Vibration generator located in eyes $f ship running at 1050 eccentricity. Pallograph magnifies motion six times.

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DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN IAbI-3 VIBRATION TEST NO. 6 5 September 1947 Third Mode Ship at anchor with 80 feet of water below keel. Vibration generator located in eyes of ship running at 24~ eccentricity. Pallograph magnifies motion six times.

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DEPARTMENT OF ENGINEERING RESEARCH 1i UNIVERSITY OF MICHIGAN AbI-35 CALCULATED NORMAL MODE SHAPES Reference: "Theorectical Research on the Dynamics of a Ship's Structure", University of Michigan, Department of Engineering Research, Project M670-4, Contract N 50ri-116, 15 September 1947. Deflections for forced vibration are given by y(x, t, xo) L Xn 2 nL nrl n V.J. where X An(xo)C+(anx) + B'n(xo)S+(anxjf Can is tabulated in Table 12 and P is maximum force of vibration generator )'(is mass per foot of ship L is length of ship 0 is tan-1 2fR R is srzn w is frequency of force rn2 EIA (anL)4 f= a rn c is damping per foot of ship

DEPARTMENT OF ENGINEERING RESEARCH AbI-36 UNIVERSITY OF MICHIGAN E is Young's Modulus IA is average moment of inertia (%nL) are roots of cos(anL)cosh(anL)= 1. We choose the following numbers as approximating an equivalent uniform beam: L = 280 feet E = 42 x 108 lb.(ft.)-2 IA = 1040 ft.4 t - 800 slugs (ft.)'l P = 5000 lbs. We calculate alL = 4.73 a2L = 7.85 a3L - 10.99 rl= 3.3 r2 = 9.1 r3 = 15.6 for c = 200 for c = 100 f, 0.038 fl = 0.019 =2 0.0138 f2 = o.oo69 f3 =.oo008 f3 = 0.004 First Mode Y(c-200) = ~ Y(c=10 ) 220

DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN - AbI-37 Second Mode Y(c=200) ' 1230 Y(c1 ' Third Mode Y(c=200) 210

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DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN AbI-43 NORMAL MODE FUNCTIONS TABLE NO. XII vaX Xl(x,O) X2(x,O) X3(x,0) 0.0 -4.oo00 -4,00 -.oo 0.5 -2.04 -2.01 -2.01 1.0 -0.21 -o.16 -0.14 1.5 +1.29 +1.41 +1.40 2.0 +2.22 +2.39 +2.39 2.5 +2.40 +2.65 +2.64 3.0 +1.8o +2.17 +2.16 3.5 +0.55 +1.14 +1.11 4.o -1.27 -0.20 -0.25 4.5 -3.10 -1.49 -1.56 5.0 -5.07 -2.38 -2.51 -2.64 -2.84

DEPARTMENT OF ENGINEERING RESEARCH AbI-44 UNIVERSITY OF MICHIGAN RESULTS OF OSCIILOSCOPE RECORD TABLE NO. XIII |age No. 1 2 3 4 5 6 Stress (psi) eront 171' ecc. 104 --- 95 154 656 586 Mode Stress Seconi 105 aecc. 109 --- 94 --- 57 -'Mode Stress 420 ecc. 70 --- 107 --- 296 _ | (psi) Third 240 ecc. 60 --- 80 --- 148 The strains indicated by the records were multiplied by Young's Modulus [ 30 x 106p41 to give the stresses in the Table above.

APPENDIX A Section b - Part II ANALYSIS OF LONGITUDINAL GIRDER STRAIN GAGE RECORDS AND PALLOGRAPH RECORDS MADE DURING ICE-BREAKING RUNS IN JANUARY, 1948 BY JOSEPH F. SHEA JESSE ORMONDROYD June, 1950

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX A Section b - Part II ANALYSIS OF LONGTMUDINAL GIRDER STRAIN GAGE RECORDS AND PALLOGRAPH RECORDS MADE DURING ICE-BREAKING RUNS IN JANUARY, 1948 Table I shows the location of the six longitudinal strain gages. The hog-sag test showed that the ship could be treated as a beam for longitudinal strain caused by bending moments in the vertical plane. The pallograph was located at Frame 10 on the main deck throughout the tests. Figure 1 shows the results of the normal elastic curves gotten on the vibration tests of 1947. There is good reason to suspect that the 3 noded mode normal elastic curve is not accurate. For one thing, the vibration generator was operating at over load capacity with 20,800 pounds amplitude, which made speed control difficult. Secondly, this curve indicates that the maximum bending strain in the second mode should occur at Gage No. 6. Actually, it does occur at Gage No. 4. Finally, the amplitude at Frame 0 should have been about.0o60 inch instead of.020 inch. This ratio is suggested by the resonance curve run with ship underway in Lake Huron. However, the curves as given permit us to predict the modes which wilL show on each gage.

TABLE IH Gage 1 2 3 4 5 6 Frame 55 103 154 43 113 148 Height above Base Line 6 ft. 7 ft. 4.65 ft. 37.5 ft. 35.5 ft. 27.5 ft. Height of Neutral Axis above Base Line 14.95 ft. 13.1 ft. 12 ft. 16.18 ft. 13.1 ft. 11.63 ft. Distance of Gage above or -8.95 ft. -6.1 ft. -7-35 ft. +21.32 ft. +22.40 ft. +15.87 ft. below Neutral Axis C -107.2 in. -73.1 in. -88.1 in. +256 in. +269 in. +190.2 in. Cross section 11.85x200ft4 12.8x200ft4 5.6x200ft4 10.1x200ft4 ll.9x200ft4 66x200ft4 Moment of 4 Inertia I 48.75xlO6in4 52.6x106in4 23.1xlO6in4 41.6xlO6in4 49xlO6in4 27.2xlO6in4 Section Modulus I/C 45.5xlO4in3 71.8xl04in3 26.2x104in3 16.27xl04in3 18.2x104in3 14.3x104in3 Note: The data in this table were taken from Figure 4, Ab2.

AbII-3 4 1 2 5 6 3.050 ' g r i, 2 NODED MODE.040 -_ RELATIVE BENDING MOMENT 4 ' DISTRIBUTION IN THE BENT LONG BASE STRAIN GAGE W c0.030 3 0 -r.O3O MODE OF VIBRATION.. -.020 -2 CD a. rz w 0010.020 -3.45 CYCLES/SECOND 6520* FORCE AMPLITUDE FRAME 3.030 1 1 1 1 1 I I i I I ' I z 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 FRAME NUMBER.050 ~ 3 NODED MODE.040 — cn.03-.020 6.15 CYCLES/SECOND 20800 FORCE AMPLITUDE FRAME 3.0 30I I I I I I I 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 FRAME NUMBER.030 z.010 j.020 9.815 CYCLES/SECOND 211100 FOR CE AMPLITUDE FRAME 3.030 I I I I I I I I I I 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 FRAME NUMBER A.b.2 NORMAL ELASTIC CURVE OF US.C.GC. MACKINAW FIG. IDE

Ab. IsI^ a,^ ^> -4,UNIVERSITY OF MICHIGANII-4 AbUII. I~ E UNIVERSITY OF MICHIGANS Gage Nos. 2 and 3 were never of any use. Gage No. 2 was placed on a girder which was not tied in completely with the hull and. Gage No. 3 was water soaked so that it had low resistance to ground. Gage No. 1 should show oscillations of the 2 noded, 3 noded, and 4 noded modes. Gage No. 4 should show all the modes of vibration. Gage No. 5 should show large 2 noded mode vibrations, No. 3 noded mode vibrations and fairly large 4 noded mode vibrations. Gage No. 6 should show all the modes. Gage No. 5 is the most important gage since it is located near the frame of the ship where the maximum bending moment in the 2 noded mode occurs. Figure 2 shows a typical record of the longitudinal strain gage readings. It is located in a pocket at the end of this report. Figure 3 shows the corresponding pallograph record taken on the same AUn* The records show irregular strains and vertical motions in which the most prominent component is that caused by the 2 noded mode of vibration. Measuring all the records at the largest oscillations of the 2 noded mode it is found, on the average, that the amplitudes of strain in the working gages are in the following ratio: Gage No. 1.161 Gage No. 4.375 Gage No. 5 1.000 Gage No. 6.815 Since the ship acts as a beam on bending,the bending moment at each gage is M = -. E. E C

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ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AbII-7 where M = bending moment in lb-in. I = section moment of inertia in4 C = distance between gage and the neutral axis, in E = modulus of elasticity lbs/in2 - = strain in/in The data in Table I and the ratios of strain given above gave the following relative bending moments at the four useful gages in the 2 noded mode of vibration Gage No. 1 1.2 Gage No. 4 1.0 Gage No. 5 3.0 Gage No. 6 1.9 These four points form the basis for the relative bending moment distribution curve shown in Figure 1 with the 2 noded. mode normal elastic curve. Table II shows the maximum double amplitude of strain on Gage No. 5 and the maximum double amplitude of vertical motion at Frame 10, both for the 2 noded mode of motion. The pallograph recording paper was run too fast, making it difficult to find the maximum amplitude accurately. No signal was used to tie the oscillograph records to the pallograph records. For this reason column 4 of Table II shows the ratio of the two double amplitudes given in Columns 2 and 3. These ratios should be constant if the two records could be measured at the same oscillation. They actually vary quite widely. However, if the three ratios 500, 502 and 637 are rejected, the

AbII-8 TABLE II Strain Gage No. 5 Pallograph Frame No. 10 Micro inches/inches Double Amp. Double Amp. Run (micro inches) (inches) inches inches 4A Ia 17.7.060 295 b 55.0.190 290 c ----.100 --- 4A IIa 17.1.05 342 b 125.0*.250 500* c 116.0.300 387 4A IIIa 35.0.100 350 b 35.4.130 273 c 43.0.100 430 4B Ia 18.3.050 366 b 49.4.125 396 c 59.3.210 282 4B IIa 34.2.050 687* b 57.8.120 482 c 80.0.225 356 4B IIIa 20.3.050 406 b 50.5.100 505* c 58.6.130 453 4C Ia 20.0.055 364 b 64.5.150 430 c 90.5.250 362 4C IIa 28.6.065 440 b 71.3.165 431 c 114.5.300 382 4C IIIa 24.8.065 382 b 66.8.200 334 c 59.0.200 296 372 Avg. Vibration Tests, 1st. mode Long Base 25.0 Avg..074 338 Avg. Strain Gage 28.2 Max..074 382 Max. * Ratios omitted in taking average Note: ABC Drafts. Fdw. Aft IIIIII propeller Combination Ship Speeds A 15' 3" 18'9" I Bow idle 2 stern a low B 17' 17' II Bow Fwd 2 stern b medium C 18'8" 19' III Bow Aft 2 stern c high

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AbII-9 rest give an average ratio of 372. This average ratio compares very favorably with the ratio gotten from the long base strain gage in the September vibration tests. This ratio was 382. Using a round number 370 as the average ratio between the double amplitude of C, the strain shown on Gage No. 5 in the 2 noded mode of motion and the double amplitude IX0, the vertical motion at Frame l0,we have the relationship E Xo inches 2700 inches The maximum strain shown in Table II is 125 x 10-6 in/in which corresponds to a stress range of 3750 No./in2 or an amplitude of stress of 1875 No./in about the average stress. These strain variations undoubtedly are functions of draft, ice thickness and speed. Unfortunately, only spot checks were made on the ice thickness, which varied between 7 inches and 18 inches throughout the tests. The speed records are not reliable enough to get a good correlation between strain amplitude and speed. A glance at Table II indicates that the strain amplitudes and vibration amplitudes increase with speed. There is good reason to believe that even at the speeds approaching zero, there would be a finite amplitude of motion when the ice sheet broke. There is also good reason to suppose that - as in many dynamic situations - the forces acting between the breaking ice and the ship's hull are proportional to the square of the speed. By taking very great liberties with the admittedly poor data the following relationships hold between the drafts, speeds of the ship in ft/sec, and strain amplitudes:

| ENGINEERING RESEARCH INSTITUTE AbII-10 UNIVERSITY OF MICHIGAN Runs 6Double Amplitude versus Velocity A 106E = 15.3 +.34v2 B 106f = 17.4 +.21V2 C 106 = 20.3 +.44V2 These equations should not be taken too seriously in view of the poor data used to derive them, but they do indicate trends. The first constant on the right-hand side increases with draft - as, perhaps, it should. The deeper the ship is in water the farther forward the breaking ice on the bow - which is favorable to producing larger amplitudes of motion in the 2 noded mode. The coefficients of V2 are probably wholly unreliable. No attempt was made to take ice thickness into account. The coefficient of V2 will probably vary in proportion to the first power of ice thickness. Perhaps the constant term would increase with ice thickness also. The zeroes of all the strain gages were adjusted (after the ship hit the ice in each run) in such a way that the record traces were evenly distributed and separated on the record paper. This eliminated the average strain from the record,leaving only the dynamic strain. This defect does not exist in the two windrow ice runs 5A and 5B. During these runs the strain gage traces were evenly distributed on the oscillograph paper and no adjustment was made after the run started. In these records we have both average and dynamic strain recorded. These results are given in the following chart.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AbII-11 Max.Double Amp. Max. Double Amp. Max.Increase in Max. Increase in Strain Stress Run Gage Average Strain Average Stress 2 Noded Mode 2 Noded Mode 5A No.4 19.76x10O6in/in -592 lbs/in2 No.5 24.4 xlO-6in/in -730 lbs/in2 73.2xlO-6in/in 2200 lbs/in2 No.6. 25.3 xl0-6in/in -760 lbs/in2 58.3xl0-6in/in 1750 lbs/in2 The fact that the trace moves downward toward the time signals indicates that the average stress is compression. Gage No. 4 moves away from the time signals, but the trace for this gage is reversed throughout the whole series of runs. This can be seen by the fact that the vibratory strains in the 2 noded mode motions are out of phase with the strains on gages No. 5 and No. 6. Physically, the vibratory strains in gages Nos. 4, 5, and 6 must be in phase since the bending moment is either plus or minus along the whole ship at any given instant. The equation I indicates the following bending moments at the three gages; Gage No. 4 96 x 106 lb-in moment Gage No. 5 133 x 106 lb-in moment Gage No. 2 108.5 x 106 lb-in moment The ship is in the sag condition, the bow hung up on the windrowed ice, the stern receiving increased water buoyancy forces and the center of the ship losing water buoyancy with an apparent increase of dead.load downward. During Run 5A the trim of the ship increased from 0 radians to.0037 radians - that is, the bow rose relative to the stern some 13-1/3

ENGINEERING RESEARCH INSTITUTE AbII-12 | UNIVERSITY OF MICHIGAN inches as the bow passed over the center of the windrow. Similar data can be given for Run 5B: Max. Increase Max. Increase Max. Double Max.Double in Average in Average Amp. Strain Amp.Stress Run Gage Strain Stress 2 Noded Mode 2 Noded Mode in/in lbs/in2 in/in lbs/in2 5B No.4 23.2 x 10-6 - 696 29.6 x 10-6 888 No.5 25.9 x 10-6 - 777 97.6 x 10-6 2928 No.6 26.8 x 10-6 804 48. x 10-6 1440 The bending moments at each gage are: Gage No. 4 113 x 106 lb-in. moment Gage No. 5 141.5 x 106 lb-in. moment Gage No. 6 111 x 106 lb-in. moment The trim in Run 5B increased from.001235 radian to.00556, the bow rising 15-1/2 inches. It should be noted that the maximum dynamic stress range is three to four times the average stress change. It should also be noted that the change of bending stress caused by a windrow deep enough to go below the keel of the vessel is very small - less than 1000 lbs/in2 average and less than 3000 lbs/in2 range dynamic. In breaking sheet ice up to 18 inches in thickness the average strain in the ship would change a negligible amount. Most of the ice-breaking runs show very little change in trim during the run, which is another way of saying that the ship as a beam is not being subjected to large changes in average bendirng stress.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AbII-13 The squat tests, run in open water, indicate clearly that the appearance of 2 noded mode vibrations is due entirely to breaking ice and that it is not caused by propeller excitation. In all the squat test records no 3.45-cycle/second vibrations appear. Strain frequencies of 6-1/4 to 7 cycles/second, 15 to 16-1/2 cycles/second, and 18 to 21-1/2 cycles/second show up. The pallograph recorded only 6-1/4 to 7-cycle/second vibrations with a maximum double amplitude of.075 inch. The second and third mode of vibration appear in the longitudinal girder strain gage records at several places. They are not large enough to be useful in estimating the forces acting on the ship. The second mode is excited largely by the propeller exciting forces. Since the brunt of the icebreaking occurs between Frames 0 and 70, the second and third modes get almost equal amounts of positive and negative excitation from the ice forces, which keeps the motions small.

NO. 6 GAGE NO. 5 GAGE NO.4 GAGE 27 '/ ABOVE [ 35 /2 ABOVE 3 7 3/ ABOVE _ /.- _ -20 14 / / / / / - l:8 / /. / n 1 / _/ -1 6 -',/ - I FT.12 II~~~ Z z J!ce p 10 ZNO. 2 GAGE 7 ~~~~~~~~~~~NO. I GAGE ~~~~~~NO. 3I GAGE /NO. 3 GAGE 4 UNIT: MACKINAW TITLE: HULL GIRDER CHARACTERISTICS o_ UNITED STATES 200 180 160 140 120 100 80 60 40 20 o COAST GUARD FRAME NO. A. b. 2 FIG.4

APPEINDIX A Section b - Part III ANALYSIS OF THE HOG-SAG TEST DATA BY JOSEPH F. SHEA August, 1948

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX A Section b - Part III ANALYSIS OF THE HOG-SAG TEST DATA In order to determine the stress in the hull due to beam-like flexure under known bending moments, a hog-sag test was run on the ship. Strain sizes on the hull girder could then be calculated in terms of bending moment free of the effect of stress concentration. These tests were conducted at night between 11 P.M. and 2 A.M. to eliminate the effect of changing temperature due to the sun. Heating in the ship was maintained constant. Three sets of readings were taken: hog condition, then sag condition, and back to hog. The sag condition was with the ship "as it was" with peak tanks empty. The hog condition was with peak tanks filled (see Table 1), other conditions unchanged. In the first hog condition readings were taken at three different times to check for drift. The first time interval was six minutes, and the second was thirty minutes. Drift was slight, ranging from zero to five microinches per inch, and showed no general trends. Readings were taken at one time in each of the other two conditions. The differences in strain gage readings between the first hog condition and sag, and sag and second hog, were averaged to obtain the figures in Table 4. This section includes (1) an analysis of the hog-sag test data, and (2) a proposed method whereby such data can be used to obtain the upward thrust of the ice upon the ship during operation.

ENGINEERING RESEARCH INSTITUTE AbIII-2 UNIVERSITY OF MICHIGAN TABLE 1 - HOG-SAG TEST DATA HOG SAG Drafts ' F-16'-ll" A-18'-7" F-16'-6" A-17'-11" Trim 1 ' -8" AFT 1 '-5" AFT SPACE A-l-W Tape Reading 23'-11.5" 1 -6" Capacity in Gal. 29, 780 150 Est. c.g. (long.) Frame No. 16.5 SPACE C-4-W Tape Reading 20'-8" 3'-4" Capacity in Gal. 26,621 1750 Est. c.g. (long.) Frame No. 180 SPACE C-5-W Tape Reading 15'-9" 0'-2" Capacity in Gal. 12,200 40 Est. c.g. (long.) Frame No. 197 Total Gal. in Tanks 68,601 1940 *Total Wt. in Tanks 255.5 tons 7.22 tons AW = 248.3 tons * Factor used: 3.72 x 10-3 long tons/gal.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AbIII-3 1. Analysis of the Hog-Sag Test Data We were interested in evaluating only the change in bending moment in the ship due to the difference in loading between the hog and sag conditions. It is an established fact that when change in bending moment due to additional loading is sought, the initial loading need not be known minutely. It is only necessary that the load and supporting system be assumed to coincide. The calculation was done on this basis. Strain gages were placed upon the hull of the ship at various stations and at varying distances from the neutral axis. Stresses at these points were calculated directly from the strains by the formula S = E e where S = stress in lbs/in2 E = modulus of elasticity in lbs/in2 = 30 x 106 lbs/in2 e = strain in inches/inch. In order to get an approximate check on the accuracy of the measured strains, the ship was also analyzed as a beam. Stresses at the strain gage locations were computed from the formula M C I where S = stress in lbs/in2 M = bending moment in lb-in C = distance from gage to neutral axis in inches I = cross section moment of inertia in in4. The bending moment was calculated from the ship's lines and from tank soundings taken during the test.

ENGINEERING RESEARCH INSTITUTE AbIII-4 UNIVERSITY OF MICHIGAN The Bon-Jean's curves for the ship were drawn from molded offsets provided by the U. S. C. G. The drafts in the sag condition were known, and from the Bon-Jean's curves the buoyancy curve was drawn (Table 2 and Figure 1). This curve was assumed to be the load curve in the sag condition. From the drafts in the hog condition the buoyancy curve was calculated (Table 3 and Figure 2*). From the tank soundings the weight of water added in each of the tanks was obtained and the longitudinal centers of gravity were estimated by observation of the profile drawing as shown on Table 1. These weights were given an arbitrary distribution and added to the assumed weight curve for the sag condition (Figure 3). This was used as the load curve in the hog condition. The weight curve and the buoyancy curve for the hog condition were checked with a planimeter and the areas and c.g. checked to within one percent. The two curves were then superimposed (Figure 4) and their difference considered the load on the beam. By integrating in the usual manner the shear curve (Figure 5) and the bending moment curve (Figure 6) were obtained. Table 4 gives the results of the stress calculations. It will be noted that the stresses calculated from the beam formula, S = MC/I, are, in general, higher than those calculated from strain measurements. This is to be expected, owing to the fact that only fully effective material is considered in the calculation of the moment of inertia. Actually there is considerable partially effective material which reduces the stress in the rest of the structure. * Note: The radius of curvature was estimated from the simple beam formula p = C/e = 21,935,800 inches, and the deflection with respect to the ends was approximately.1 inch. Therefore the waterline was assumed to be straight

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AbIII-5 TABLE 2 - CALCULATIONS FOR SAG CONDITION Drafts - F = 16'-6" A = 17'-11" ORD SM Area f(Vol.) Lvr. f(LM) 0 1/2 0 0 10 0 1/2 2 1 2 9-1/2 19 1 1 21 21 9 189 1-1/2 2 65 130 8-1/2 1105 2 3/2 190 285 8 2280 3 4 350 1400 7 9800 4 2 500 1000 6 600ooo 5 4 635 2540 5 12700 6 2 750 1500 4 6000 7 4 831 3324 3 9972 8 2 900 1800 2 3600 9 4 945 3780 1 3780 10 2 958 1916 0 55445 (f) 11 4 955 3820 1 3820 12 2 920 1840 2 3680 13 4 862 3448 3 10344 14 2 777 1554 4 6216 15 4 650 2600 5 13000 16 2 493 986 6 5916 17 4 320 1280 7 8960 18 3/2 158 237 8 1896 18-1/2 2 87 174 8-1/2 1479 19 1 29 29 9 261 19-1/2 2 16 32 9-1/2 304 20 1/2 0 0 10 0 Totals: f(V) 33698 f(LM) 55876 (a)

AIii i-ENGINEERING RESEARCH INSTITUTE AbIII-6 I UNIVERSITY OF MICHIGAN TABLE 2 - CONTINUED 33698 Mean Area = X = 561.63 Vol. = 561.63 x 280 = 157,256.4 ft3 Displ. = 4370 tons (F.w.) L.C.B. = 0.18 t. - -33698

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AbIII-7 TABLE 3 - CALCULATIONS FOR HOG CONDITION Drafts - F = 16'-"11 A = 18'-7" ORD SM Area f (Vol.) Lvr. f (LM) 0 1/2 0 0 10 0 1/2 2 4 8 9-1/2 76 1 1 27 27 9 243 1-1/2 2 75 150 8-1/2 1275 2 3/2 204 306 8 2448 3 4 369 1476 7 10332 4 2 526 1052 6 6312 5 4 666 2664 5 13320 6 2 785 1570 4 6280 7 4 869 3476 3 10428 8 2 942 1884 2 3768 9 4 986 3944 1 3944 10 2 1005 2010 0 58435 (a) 11 4 999 3996 1 3996 12 2 970 1940 2 3880 13 4 912 3648 3 10944 14 2 823 1646 4 6584 15 4 694 2776 5 13880 16 2 539 1078 6 6468 17 4 356 1424 7 9968 18 3/2 184 276 8 2208 18-1/2 2 105 210 8-1/2 1785 19 1 34 34 9 306 19-1/2 2 21 42 9-1/2 399 20 1/2 0 0 10 0 Totals: f(V) 35637 f(LM) 60418 (f)

ENGINEERING RESEARCH INSTITUTE AbIII-8 UNIVERSITY OF MICHIGAN TABLE 3 - CONTINUED Mean Area = 3563 594 Vol. = 594 x 280 = 166,320 ft3 Vol. Displ. = V- = 4620 tons (F.W.) Diff. in Displ. = 250 tons Hog vs. Sag 0.8% Diff. in Weight = 248 tons Hog vs. Sag L.C.B = 27762 =. 766 ft. AnFT 'Hog 35637 Curves were checked by planimeter: Hog - Computed Area - 166,320 ft3 Planimeter - 166,960 ft3 Error =.38 % Sag - Computed Area - 157,256 ft3 Planimeter - 157,800 ft3 Error =.35 %

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AbIII-9 TABLE 4 - STRAIN GAGE DATA Strain Gage 1 2 3 4 5 6 Location - Aft of Frame No.55 No.103 No.43 No.113 No.147 Distance of Gage Above B.L. 6' 7' 37.5' 35.5' 27.5' Distance of Gage from Neutral Axis (c) -8.95' -6.11' 21.35' 22.4' 15.73' Moment of Ipertia = I - ft4 2342 2560 2020 2380 1320 Strain (Hog to Sag) 1 x 10- in. 9 4.5 21.5 33 29 Bending Moment - ft-tons (From Figure 6) 4780 7390 3560 7330 5270 Stress - from S = MC/I lbs/in2 297 274 585 1073 977 Stress - from S = Ee lbs/in2 270 135 645 990 870 Percent Deviation + 10 +103 - 9.3 + 8.4 +12.3

AbIII-10 U.S.C.G. CUTTER MACKINAW DISPLACEMENT CURVE SAG CONDITION U. DRAFTS- FORE: 16'- 6" AFT: 17 1'-1 " 1200. 0 0 N 1000 m ~800 | / I SQ IN.- 8000 C.TOTAL DISPLACEMENT 157,800 CU. FT. 10 880 8 120 90 0 4 380 TONS w 4,3F0 I SQ. IN= 8000 CU. FT DISPLACEMENT o (36 CU. FT. I TON) LL U. lo: 4150 120 90 60 FRAMES FRAME SPACING: 16" I": 40 FT.

AbIII-l1 U.S.C.G. CUTTER MACKINAW DISPLACEMENT CURVE HOG CONDITION LiU ~~DAT- FORE: 16'-8, 0200 0 21 000 150 10 90 60 30 w FR: TOTAL DISPLACEMENT 166,960 CUFIGUREFT. Z0 4,630 TONS U U) o- 600 I SQ. IN.= 0L 80000U. FT. DISPLACEMENT o200 210 180 150 120 90 60 30 0 FRAMES I1 =40' A. b.3 FI GUR E 2

AbIII-12 U.S.C.G. CUTTER MACKINAW LOAD CURVE HOG CONDITION ORDINATE EVALUATED IN 1200 l d TERMS OF DISPLACEMENT — 1200 I. -1000 0 oF I" ILl ~L 4 21" 18" Il" 12" FRAMES A.b.3. FIGURE 3

AbIII-13 U.S.C.G. CUTTER MACKINAW DISPLACEMENT CURVE VS. LOAD CURVE 1200- l HOG CONDITION I000 --- I - DISPLACEMENT od CURVE o -LOAD CURVE N 800 ~~- 600.I u.I 210 180 150 120 90 60 30 0 FRAMES 1"- 40' A.b.3 FIGURE 4

AbIII-14 U.S.C.G. CUTTER MACKINAW SHEAR CURVE FOR CHANGE OF LOAD BETWEEN HOG 8i SAG CONDITION INTEGRATED FROM FIG. 4 -— 4000 I SQ. IN. =80,000 CU. FT. (MOMENT) LL 5 210 180 150 120 90 60 30 w FRA 0I'" I 40 'U A b4000 A. b.3 FIGURE 5

AbIII-15 U.S.C.G.C. MACKINAW BENDING MOMENT CURVE INTEGRATED FROM FIG.5 SHOWING POSITIONS OF STRAIN GAGES LL 0 280 u-WI 1 4-2400 a U %P0 = - 120 210a180 150 120 90 630 a zW~3 3 210 I 100 I40 -, A.b.,3 FIG. 6

ENGINEERING RESEARCH INSTITUTE AbIII-16 UNIVERSITY OF MICHIGAN 2. Proposed Method for Measuring Vertical Compression Ice Force The foregoing analysis suggests a possible method of evaluating the upward thrust of the ice during breaking operations. Strain gages similar to the ones used should be placed at approximately 15 frame intervals and a series of hog-sag tests run to calibrate the gages in terms of bending moment as was done in Section 1. The gages should be attached to the hull when the drafts are accurately known; call this condition (1). The flotation curve at those drafts can be used as the original load curve, so that we consider the bending moment zero at all points. After the hog-sag tests the ship should be returned to a known load condition at known drafts*. If this condition is the same as (1) we can assume zero initial shear or bending moment. When the ship is run through ice, the gages will indicate a certain bending, moment. This must be due to a change -in the vertical load condition, i.e. the upward thrust of the ice. From the data obtained from the gages the bending moment curve for this condition can be drawn. By a suitable method of graphical differentiation, a shear curve due to the ice load may be derived. Upon differentiating this curve, the actual load of the ice on the ship would be obtained. In the event that the conditions at the start of the run differ from (1), the bending moment curve can be calculated from the change of load and the new drafts. This can be subtracted from the bending moment curve obtained from the strain gages during the run, the result being the bending moment curve due to the ice load. This can then be handled as before. The accuracy oftheSe *Gages should be zeroed during a run in open water so as to eliminate any errors due to change of draft underway and the axial thrust of the propellers.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN AbIII-1 data will depend upon (1) accuracy of calibration of the strain gages, and (2) accuracy if methods of graphical differentiation. Graphical Differentiation*: In general, the use of graphical differentiation places a limit on the accuracy of the resulting curves. The best method of differentiation is essentially the inverse of the integration method used for computing the shear and bending moment in Section 1. If the integral curve y = f(x) is given, we may construct the derivative curve y = y/x by using the principle that the ordinate of the derivative curve at any point P(x,y) is equal to the slope of the integral curve or of the tangent line to the curve at the point P(x,y). The practical construction of the derivative curve is shown in Figure 7. The bending moment curve derived in Section 1 was differentiated and the resulting shear curve can be compared with Figure 5. It is expected that the upward ice force will vary more uniformly than this load curve does, and if so, accuracy will be even better than shown by this example. The method should provide a good indication of the load pattern and can be expected to produce values of ice force within ten percent of the true value. Graphical and Mechanical Computation, Lipka, John Wiley and Sons, 1918.

AbIII-18 U.S.C.G. CUTTER MACK I NAW SHEAR CURVE BY GRAPHICAL DIFF. 280 280- _ G IBENDING MOMENT CURVE DRAWN AS A SERIES OF Z- L l l |/ __ | X wTANGENTS. SLOPE.OF:Ded l / EACH LINE IS AVERAGE SHEAR ORDINATE FOR O 20 THE INTERVAL 0 aD 160 120, ~ IL cci ~180 150 120 90 60 3.0 FRAMES 4000,,2000 180 / 150 120. 90 60 -30 / o 2000, 0 SHEAR I j! A~~b.33 -- CURVE COMPARES WITH A.b.3 FIGURE 7

APPENDIX A Section b - Part IV ACCELEROMETER RECORDS BY JESSE ORMONDROYD June, 1950

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX A Section b - Part IV ACCELEROMETER RECORDS The accelerometer records were a total loss because they became a blur of high frequency oscillation which completely blanked out the lower frequency information which was being sought. The frequency of the "hash" is 60 cycles per second. It undoubtedly comes from the exciter generator for the gyro stable vertical which ran at 3600 RPM. The accelerometers were mounted rigidly on the deck plating. This caused the 60-cycle motion to be transmitted to the accelerometers directly. Accelerometers are sensitive to high frequencies and relatively insensitive to lower frequencies. The accelerometers could have been spring mounted with the natural frequency of the accelerometer case and its base on a spring to the deck about twice as high as the highest frequency which was expected to be of interest. A disagreeable amount of 60-cycle "hash" would still come through this mounting. The real solution for this problem would have been a low pass filter in the electronic circuit which would pass everything up to 40 cycles/ second and nothing above 59 cycles/second. Thi.s filter circuit should pass zero frequency components without distortion.

APPENDIX A Section b - Part V LONG BASE STRAIN GAGE IN 301-A BY JESSE ORMONDROYD June, 1950

APPENDIX A Section b - Part V LONG BASE STRAIN GAGE IN 301-A Run Maximum Average Number Inches Inches Notes 3AI 3BI 3AII.022.007 3BIb.018.004 3BII.018.003 4Ala.0005 Steady 4Alb.012.002 4Alc.011.002 The gage was ten feet long and 4A2a Steady - No Motion 4A2b.018.002 had a dial indicator. 4A2c.030.003 4A3a Missing These readings indicate steady 4A3b.012.003 4A3c.011.002 unidirectional compressions with 4Bla Steady at Zero fairly large surges of compression 4Blb.012.001 4Blc.028.002 superimposed. The readings are not 4B2a.003 Steady 4B2b.012.002 reliable enough to warrant analysis. 4B2c.032.004 4B3a Steady? If the readings are taken at their 4B3b.012.002 4B3c.035.004 face value they indicate compressive 4Cla Steady stresses as high as 10,000 lbs/in2 4Clb.009.002 4Clc.030.004 during Run 5B (windrowed ice run) 4C2a Missing 4C2b Missing with an average compressive stress 4C2c.018.00oo6 4C3a Missing of 1250 lbs/in2 in the same run. 4C3b Missing 4C3c.033.004 5B.040.005 5A.024.003 2A A 2B | Not Recorded

APPEnDIX B, SECTION a ICE THRUST MEASUREMEIS BY L. A. BAIER C. W. SPOO00R A. C. McCLUIRE August 1948

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX B, SECTION a TESTS JANUARY, 1948, ON U.S.C.G.C. MACKINAW ICE THRUST MEASUREMENTS This portion of the report deals with the measurement of the thrust of the vessel against the ice. Since a direct thrust measurement was impractical, the ice thrust was deduced from other measurements. The outline of the procedure is as follows: 1. Thrust measurements were made on two of the shafts and estimated on the third and corrected to water thrust on the propellers and for "thrust deduction" which is caused by interaction of propellers and hull. 2. From model tests at the David Taylor Model Basin the open water resistance of the vessel was estimated. 3. The difference between the algebraic sum of the corrected thrusts and the open water resistance was presumed to be the fce resistance. Propeller shaft thrust meters had been developed by the Kingsbury Machine Works (1), (2) to a high degree of perfection and two of these were installed, one in the starboard stern propeller thrust block and one in the bow propeller thrust block; this latter, being a double meter, was able to measure when the shaft was turning either ahead or astern. Thus, the thrusts on the two shafts equipped with meters could be calculated. The port stern shaft thrust was estimated from readings taken on the starboard one. Rather large errors may have been introduced by this procedure, and a meter should be installed in future tests of this sort because of the difficulty of synchronizing the two stern shafts as to speed and power and because the two propellers

ENGINEERING RESEARCH INSTITUTE Ba-2 UNIVERSITY OF MICHIGAN were undoubtedly not identical (they were of the same design but of opposite rotation, being outboard turning). Further influencing factors may have been helm angle and amount of ice going through the propeller discs. The thrust meters were read by observers in the shaft alleys and als by recording oscillograph$ in the bosun's locker. In most cases the oscillograph readings were used, although because of the small deflection used, the degree of error in these readings may have been 2 percent of full scale. The direct readings made in the shaft alleys proved invaluable in checking for gross errors on the other readings. The thrust meter reading for the port stern shaft was estimated from the starboard side by plotting measured thrust in pounds versus motor Brake Horsepower divided by shaft revolutions per minute. BHP/RPM is proportional to torque, and it was presumed before the test that for two supposedly identical propellers in the thrust-torque relationship would be similar enough to permit an accurate estimate of the thrust on the shaft without the meter. Data were, however, quite variable and the estimate introduces another unknown error into the test results. Corrections to the thrust meter readings were made for the fore and aft components of the weights of all rotating members on the shafts. Shaft inclinations from the base line of the ship were obtained from hull drawings and trim of the ship was measured by the stable vertical gyro. Weights of motor rotors, turning gears, thrust shafts, tail shafts, propellers, fairwaters, and nuts were obtained from the Coast Guard records (3). It is believed that the weights furnished were also estimates, because port and starboard weights were identical and the weights of the propellers (from the ship's file) were not in agreement by 5400 pounds in the case of the bow propeller and by lesser amounts in the after propellers. However, the error, if any, is

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ba-3 about 1 percent of the smallest measured thrust and less in other test runs. The ship's file did not include weights for other parts than propellers. Further corrections were made for static water pressure, tending in all cases to push the shafts into the hull. The static heads of water over the shafts where they entered the ship were measured from static draft data and the forces calculated. The last correction to the thrust was made on the stern shaft because of dynamic pressure forces in the oil film of the thrust bearing on the slack side (as opposed to the side taking the thrust). A curve (4) for making this correction was furnished by the Kingsbury Machine Works. The correction was automatically made for the bow shaft meter by reading the forward and after sides simultaneously. Sample Thrust Calculation RUN NO. 4A II a - - Propeller Turning Ahead BOW Fwd Face Press.fRdg. 710 psi (Oscill. Rdg.) Aft " " " 237 psi Differential Press. 437 psi Piston Area (total 1 side) 50 sq. in. Meter Thrust 23,650 lb. fwd. Tangent of Ship's Trim Angle = 0.0094 Aft " "l Shaft Angle to t = 0.0739 Fwd Horiz. = o.o645 Fwd Wt. Rotating Members = 80,887 lb. Wt. Component Fwd. = 0.0645 x 80,887 = 5220 lb. Fwd. Shaft Sleeve Area at Stuffing Box: x 141.375 = 162.3 sq.in. Avg. Draft Fwd. for "4A" Runs = 15'3 Propeller A -ABV t, approx. = 6'0" Shaft Below W.L. 9'3" Static Pressure on Shaft 4.01o psi Static Force of Water on Shaft 65o lb. Act

ENGINEERING RESEARCH INSTITUTE Ba-4 UNIVERSITY OF MICHIGAN Meter Thrust (+) 23,650 lb. Fwd. Shaft Wt. Component (-) 5,220 lb. Fwd. Static Water Force (+) 650 lb. Aft Net Propeller Thrust (+) 19,080 lb. Fwd. STBD STERN Fwd. Face Pressure Rdg. 203 psi (Oscill.Rcdg.) Assume Brg. Oil Pressure 0 psi Differential Pressure 203 psi Piston Area (Total) 75 sq. in. Meter Thrust (+) 15,220 lb. Fwd. Slack Side Thrust, Est. from (-) 840 lb. Aft Kingsbury Curve K.O. 3104 Tangent of Ship's Trim Angle = 0.0094 " "Shaft Angle to I = 0.0523 "? "T "? "? "T Horiz. = O.0617 Wt. of Rotating Members = 119,000 lb. Wt. Component Aft (+) 7,340 lb. Aft Shaft Sleeve Area at Stuffing Box: x 12 = 254.5 sq.in. Avg. Draft Aft for "4A" Runs = 18'10" Propeller Above I = 7' 6" Shaft Below W.L. lit 4?t Static Water Force (-) 1200 lb. Fwd. Meter Thrust (+) 15,220 lb. Fwd. Slack Side (-) 840 lb. Aft Shaft Wt. Component (+) 7,340 lb. Aft Static Water Force (-) 1,200 lb. Fwd. Net Propeller Thrust (+) 20,520 lb. Fwd. PORT STERN Values estimated from curve of BHP/RPM vs. thrust from Stbd. Rdgs.

ENGINEERING RESEARCH INSTITUTE Ba UNIVERSITY OF MICHIGAN BRAKE HOSEPOWER MEASUREENTS Electrical power input to the propulsion motors was measured on all three motors by representatives of the Westinghouse Electric Company by recording oscillographs and on the two shafts with thrust meters by representatives of the Detroit Edison Company using high speed recording wattmeters. The power readings were not in good agreement and no constant meter factor was found to rectify the readings. To eliminate some uncertainty, the corrected thrusts were plotted against their respective brake horsepowers divided by revolutions per minute and the horsepower readings faired. (This also permitted an estimate of the thrust on the port stern shaft as previously mentioned. ) The revolutions per minute of all shafts were also recorded by both the above representatives and were in substantial agreement. Propulsion motor data were furnished by the Westinghouse Electric Company (5) and the motor brake horsepowers were calculated as follows: Sample BlIP Calculations RUN NO. 4A II a BOW MOTOR Readings: Armature Current, Ia, 1240 Amps Armature Voltage, Ea, 290 Volts Electric Power Input, 360 Kilowatts Speed, N, 80.4 RPM Armature Ia2R loss incl. series field 14.91 Kw Brush Loss, (2 volts x Ia)/1000 2.48 Core Loss, (N/Nfl)2 x (F.L. Loss = 2352 KW) 3.75 Brush Friction, (N/Nfl)2 x(F.L. Loss = 2.7 KW) 0.44 Bearing Friction and Windage, (N/Nfl)2 x (F.L.Loss = 2.71 KW) 0.44 22.02 KW

ENGINEERING RESEARCH INSTITUTE Ba-6 UNIVERSITY OF MICHIGAN Load Loss: 1% of Output Brake KW = (360 - 22) 0.99 = 335 KMT Brake H.P. = 335/0.746 449 STERN MOTORS SIMILAR Shaft horsepowers (measured on the propeller side of the thrust blocks) were not calculated as the thrust bearing losses are extremely small. The corrected propeller thrusts and brake horsepowers of the motors are shown in Table I. ESTIMATE OF WATER RESISTANCE The resistance of a ship's hull moving through water may be accurately estimated from model tests. Such tests were performed by the David Taylor Model Basin as their model number 3771, the results being plotted as effective horsepower (EHP) versus ship speed in knots. This model test was performed at a displacement of 5138 long tons, and these results were used by correcting the horsepower in direct proportion to the displacement. This correction is not exact, but nearly so in most cases. It should be pointed out that a relatively small proportion of the ERP goes to creating waves at these low ship speeds, about 22 percent at 9 knots and less at reduced speeds. Under ice-breaking conditions there was little or no wave formation, but nearly the same energy must have been expended in moving the water out of the way of the hull. The unknown error in the wavemaking part of the hull resistance applied to 22 percent of the hull resistance, which in the case of the fastest test run, 47A IIc, was 19,830 pounds. This in turn is a small fraction of the total resistance of 174,160 pounds. The resulting error must, then, have been a

TABLE I CORRECTED THRUSTS AND BRAKE HORSEPOWERS BOW STBD. AFT PORT AFT TOTAL Run MEASURED FAIRED MEASURED FAIRED ESTIMATED MEASURED NET No. THRUST RPM BEP THRUST RPM BHP BHP/RPM THRUST RPM BHP BHP/RPM THRUST BHP 4A I a 24,100 57.1 574 10.05 19,700 57.3 471 8.22 43,800 1,04 b 64,o60 106.8 2,863 26.81 52,200 103.5 2,260 21.83 116,260 5,12 c 80,070 150.8 5,043 33.44 74,500 146.3 4,558 31.14 154,570 9,601 4A II a 19,080 80.4 438 20,520 61.6 527 8.55 25,800 69.0 744 10.78 65,400 1,709 b 48,750 156.6 2,136 41,380 97.0 1,678 17.30 40,200 95.9 1,608 16.78 130,330 5,422 c 56,680 190.6 3,016 68,630 131.8 3,779 28.67 64,000 131.1 3,509 26.75 189,300 10,304 P 4AIII a -31,450 114.0 1,358 68,930 97.3 2,561 26.32 36,000 77.3 1,064 13.77 73,390 4,983 | b -32,210 130.7 1,600 73,600 109.9 3,089 28.11 66,000 107.6 2,714 25.22 107,390 7,40l c -43,440 157.1 2,587 82,290 122.8 3,860 31.43 45,600 102.8 1,792 17.43 84,450 8,239 4B I a 21,720 52.8 469 8.88 19,800 55.9 454 8.12 41,520 923 Do b 50,790 105.5 2,198 20.83 48,600 109.0 2,170 19.91 99,390 4,368 c 101,070 132.7 5,496 41.42 96,800 131.8 5,226 39.65 197,870 10,722 q 4B II a 12,390 78.7 280 19,560 58.9 470 7.98 21,600 60.3 532 8.82 53,550 1,282 b 29,440 117.5 974 49,370 99.7 2,017 20.23 43,300 96.6 1,713 17.73 122,110 4,704 | c 58,o40 175.4 2,844 69,850 125.2 3,583 28.62 70,500 124.9 3,611 28.90 198,390 10,038 4BIII a -20,100 79.2 601 38,860 72.9 1,064 14.60 35,900 80.7 1,088 13.48 54,660 2,753 | b -34,240 102.9 1,348 47,520 99.7 1,811 18.16 51,300 98.7 1,906 19.31 64,580 5,065 c -54,290 169.6 3,500 84,050 122.9 3,855 31.61 78,400 125.6 3,706 29.50 108,160 11,091 4c I a 12,030 47.2 222 4.71 21,300 56.6 471 8.32 33,330 693 b 64,150 109.4 2,749 25.13 54,900 103.5 2,230 21.55 119,050 4,979 c 105,610 138.7 5,875 42.36 88,800 136.6 4,732 34.63 194,410 10,607 4c II a 11,530 67.6 223 27,470 58.0 625 10.77 23,600 52.4 484 9.24 62,600 1,332 b 32,740 138.9 1,278 49,010 99.8 1,918 19.22 43,600 98.0 1,676 17.10 125,350 4,872 c 55,910 187.9 2,933 74,530 129.4 3,782 29.23 70,800 125.6 3,490 27.79 201,240 10,205. 4cIII a -10,120 68.1 257 22, 110 58.4 474 8.12 24,600 64.9 588 9.06 36,590 1,319 ' b -26,630 114.0 1,153 57,280 96.6 2,046 21.18 48,600 95.2 1,707 17.93 79,250 4,906 c -56,130 154.3 3,294 85,440 123.6 3,897 31.53 78,100 121.4 3,500 28.83 107,410 10,691

ENGINEERING RESEARCH INSTITUTE Ba-8 UNIVERSITY OF MICHIGAN small part of the total and could hardly exceed 1 percent. The speed of the ship was measured by a drag on soft steel wire. The drag was heaved overside onto the ice after the vessel was underway. The wire ran from a spool around a bicycle wheel and thence to the drag.. The drag which was most successful was a three-pound length of 1/8-inch chain. However, the wire broke frequently because of too much inertia of the moving parts. The rotation of the bicycle wheel was counted by an observer and electrically. Also, the ice alongside th ship was photographed by a movie camera during each run. The camera speed was calibrated by photographing the second hand of a clock; however, there were no targets of known size or distance on the ice, and results by this method were negligible. ESTIMATE OF WIND RESISTANCE The wind resistance is a function of the apparent wind velocity, the frontal area of the ship, and the angle of the apparent wind. The formula used was taken from Taylor's "Speed and Power of Ships," 1943 edition, page 160 (6) and is as accurate as is usually warranted in ship trials. The frontal area of the ship was estimated at 2450 square feet and a constant of 0.0044 assumed as being representative of a ship of this nature. R wind K x C x A x(V )apparent lb apparent where K = factor for wind angle relative to the ship's centerline C = a constant (range 0.0025 to 0.0055) A = frontal area of the ship, sq. ft. V = apparent wind velocity, knots. The factor K becomes a maximum for an apparent wind angle of 30 degrees off

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ba-9 the centerline, unity with wind dead ahead or astern, and zero with wind abeam. This angle was estimated by an observer near the pilot house using a small wind pennant on a stay overhead. The wind velocity was obtained from the ship's indicating anemometer in the pilot house. It is extremely unfortunate that the record trials were partly run off on a day when the wind was blowing 40 to 50 miles per hour, thus giving wind forces approaching or even exceeding the ice forces. ICE RESISTANCE The corrected total water thrust on the propellers is subject to a further correction to arrive at the force which would be obtained by means of a towrope because of the interaction of the hull with the propellers. The forward end of the vessel abaft the bow propeller was subjected to the propeller wash, which must have exerted a sternward force on that part of the ship. This force partially offsets the thrust created by the propeller, the total of which was measured by thrust meter. Similarly, there was a region of low pressure ahead of the stern propellers, which was created by them and exerted a sternward force on the after end of the vessel in the region. On a model test it is possible to self-propel the model to obtain the thrusts and also to tow the model at the same speeds and obtain the relation between the two forces. ~Let:~ (l-t) = Total Towrope pull eTotal propeller shaft thrust then, t is known as the thrust deduction. Values of thrust deduction were taken from model test data for both the Mackinaw (7) and Northwind Class (8), which data were incomplete in that the model of the Mackinaw was not tested at conditions comparable to these

ENGINEERING RESEARCH INSTITUTE Ba-lO UNIVERSITY OF MICHIGAN trials, and data from the similar ships were used instead. Table II shows a tabular compilation of the ice resistance. In ten of the twenty-seven runs, the vessel speed was unknown and the calculations, therefore, could not be completed. The results of Table II are given in Figure 1. ICE THRUST Transient conditions of ice thrust during tests on windrowed ice were not analyzed, in view of the lack of dependable speed measurement and the-difficulties and inaccuracies encountered in the above analysis. SPEED ESTIMATION An attempt was made to estimate the speeds for the runs on which the speed measuring device failed. Three methods were used: 1. Use of propeller characteristic curves 2. Comparison of speed through ice with speed in open water 5. Use of curves of towrope pull versus power from model tests (1) It is known that for each thrust and propeller RPM there is a corresponding speed. Using the measured thrust and RPM, an estimate of speed was made. Two sets of curves were used: Characteristic Curves for Propeller No. 2412-13, U.S.C.G. Drawing No. 121CR4400-2, Exp. Model Basin, Navy Yard, Washington, D. C. Shaft Horsepower and Towrope Pull, Model 3771, Tests Nos. 2a, 3a, D. W. Taylor Model Basin. The propeller curves contain a plot of thrust coefficient, CT, against real slip ratio. Thrust coefficient was calculated from the available test data according to the formula:

Ba-ll 200 150 oB ID -J 0 0 0 w A 00 50 An U.S.C.G0 MACKI-NAW FIG B.a. FIG, I

ENGINEERING RESEARCH INSTITUTE Ba-12 UNIVERSITY OF MICHIGAN CT T n2 p2 D2 where CT = thrust coefficient T = thrust on one propeller in pounds n = shaft speed in revolutions per second P = propeller pitch in feet D = propeller diameter in feet. By means of the curves, real slip was obtained. The second set of curves contains a plot of real slip ratio against brake horsepower for several speeds, from which speeds were estimated. Speeds were thus calculated for every test run. For the runs in which speed measurement was obtained, this calculated figure was plotted against the measured speed to obtain a correction curve, Figure 2. Table III gives the calculated speeds and the estimates. (2) The speed the vessel would have attained had she been in open water was estimated for each test run. Model test curves furnished by the D. W. Taylor Model Basin were used. These were self-propelled tests on Model No. 3771 using propellers nos. 2412-13-14. Test No. 3, file no. QS13(21), 20 February, 1943, in which the stern propellers were driving with the bow propeller idling, most closely approached the conditions of the ice-breaking tests. Speed-power curves were not made for conditions of the bow propeller driving ahead or astern. Since the model tests were run at a draft corresponding to a displacement of 5138 tons and the ice-breaking tests were run at 4495 and 5125 tons displacement, the horsepower had to be corrected. The ratio of displacements was considered sufficiently accurate for this correction. Curves of shaft horsepower versus speed in knots were entered, and speed read off. Table IV gives the results.

Ba-13 U.S.C.G.C. MACKINAW FIGURE 2 SPEED FROM CURVES OF PROPELLER AND HULL CHARACTERISTICS KEY:O I BOW PROPELLER IDLING l: '" AHEAD a TfI, I, ASTERN A- 17.29 FT. MEAN DRAFT 3.5 FT TRIM AFT A = 4495 L.T.F.W. B- 1708 FT., II 0 TRIM 1=4495 L.T.FW. 0- 18.86 FT. " ii 0.5 FT TRIM AFT A = 5125 L.T. F.W. 12 I 0 w.l I0 6 I A,L w 4 E 1J O 2 4 6 8 10 12 B,.a. COMPUTED SPEED-KNOTS FIG.2

ENGINEERING RESEARCH INSTITUTE Ba-14 UNIVERSITY OF MICHIGAN TABLE III SPEED FROM PROPELLER CHARACTERISTIC CURVES 2 _ C 1/2 T p2D = 26150 CT 2 2 2 D = 14 D2 = 196 Run Thrust - 2 n CT SR Speed Measured Corrected No. Stern Props. r.p.s. Knots Speed-VK Estimate 4A I a 43,800 0.953.922 1+ 0- 1.21 b 116,260 1.755.721.728 6.9 5.96 c 154,570 2.475.481.418 14. 11.3 4A II a 42,750 1.090.689.685 4.5 4.9 b 77,110 1.608.570.535 7.5 6.85 c 128,150 2.190.510.457 10.5 9.71 4AIII a 100,290 1.455.905 1+ 0- 5.85 b 132,190 1.815.768.795 3 6.81 c 123,300 1.880.667.657 6 8.52 4B I a 41,520 0.905.971 1+ 0- 1.69 b 99,390 1.790.594.566 10.5 8.07 c 197, 870 2.200.781.815 5.5 7.46 4B II a 37,250 0.994.722.730 3 3.8 b 89,460 1.635.640.620 6 6.1 c 114,690 2.090.501.445 11 10 4BIII a 61,130 1.280.714.720 4 7.2 b 93,730 1.660.651.638 6 8 c 157,260 2.060.709.715 5.5 7.8 4C I a 33,330 o.865.850.925 3 3.8 b 119,050 1.775.723.736 7 7.29 c 194,410 2.295.705.71 7.4 6.54 4C II a 46,000 0.920 1.037 1+ O- 1.5 b 87,700 1.650.616.593 7 7.04 c 140,410 2.125.595.567 8 8.17 4CIII a 41,040 1.027.745.765 4 1.87 b 100,950 1.600.803.850 2 6.27 c 158,81o 2.040.730.740 5 5. 30

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ba-15 TABLE IV SPEED ESTIMATION DIFFERENCE BETWEEN ESTIMATED OPEN WATER SPEED AND ICE SPEED Speed Speed Run BHP at BHP at Knots Knots Differ- Estimated No. Trial A A=5138 Open Ice ence Speeds Water 4A I a 1,045 1,190 10.15 1.21 8.94 b 5,123 5,850 15.6 5.96 9.64 c 9,601 10,950 17.5 10 7.5 4A II a 1,709 1,950 11.8 9 2.8 b 5,422 6,200 15.75 6.85 8.90 c 10,304 11,880 17.9 9.71 8.19 4AIII a 2,267 2,595 12.75 5.85 6.90 b 4,203 4,800 14.95 6.81 8.14 c 3,065 3,505 13.80 8.52 5.28 4B I a 923 1,052 9.8 1.69 8.11 b 4,368 5,000 15.1 8.07 7.03 c 10,722 12,200 18.0 7.46 10.54 4B II a 1,282 1,465 10.85 9 1.85 b 4,704 5,370 15.3 10 5.3 c 10,038 11,450 17.7 10 7.7 4BIII a 1,551 1,773 11.45 7 4.45 b 2,369 2,710 12.90 7 5.9 c 4,061 4,640 14.80 7 7.8 4C I a 693 693 8.5 7.5 1.0 b 4,979 4,979 15.1 7.29 7.81 c 10,607 10,607 17.5 6.54 10.96 4C II a 1,332 1, 332 10.55 9 1.55 b 4,872 4,872 15.0 7.04 7.96 c 10,205 10,205 17.33 8.17 9.16 4CIIT a 805 805 8.90 1.87 7.03 b 2,600 2,600 12.75 6.27 6.48 c 4,100 4,o100 14.35 5.30 9.05

ENGINEERING RESEARCH INSTITUTE Ba-16 UNIVERSITY OF MICHIGAN The difference between this estimated open water speed and the measured speed was fo-~und to be nearly constant, varying from 7 to 11 knots, mostly between 8 and 10. In general the difference was greater at the higher speeds than at the lower speeds. For the tests in which the bow propeller was backing, the differences were smaller, ranging from about 6 to 9 knots. (3) Two sets of curves of towrope pull for several speeds plotted against shaft horsepower were used. One set was of tests on Model 3705, a model of the same type as the Mackinaw, but of somewhat different dimensions; and the other set was of tests on Model 3771. Both sets were drawn for selfpropelled tests performed at the D. W. Taylor Model Basin. The curves of Model 3705 were used because the tests on Model 3771 did not cover the power range required. The towrope pull was taken to be the sum of the ice and wind resistances, or the total thrust developed minus the water resistance. Since water resistance had to be estimated, a preliminary estimate of speed was needed for the runs in which speed was not measured. This was taken from the first method of estimating speed. Negligible error was introduced since the water resistance was only a small part of the total thrust, and because the model test curves are not sensitive. Except for the very low speeds (1 to 2 knots), reasonable agreement was obtained between the speeds thus estimated and the speeds measured during the ice-breaking tests (Table V). An effort was made to correlate the thickness of the ice with speed and ice resistance. Ice resistance per inch of thickness was found for the runs in which sufficient data had been collected. This was plotted against speed (Figure 3). Nearly all the points are contained within two lines fanning apart from the origin at an angle of about 22 degrees. This variation

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ba-17 TABLE V SPEED ESTIMATION SPEED FROM CURVES OF TOWROPE PUIL TMB MODELS 3705, 3771 Run BHP Towrope Speed, Knots Speed,Knots Test No. Uncorrected Pull 3705 3771 Speed 4A I a 1,045 36,880 6 1.21 b 5,123 87,800 8 5.96 c 9,601 116,020 11 12 4A II a 1,709 56,370 6 b 5,422 109,060 7 6 6.85 c 10,304 154,330 8 8 9.71 4AIII a 4,983 63,900 7 5.85 b 7,403 94,520 6 1 6.81 c 8,239 66,940 10 4 8.52 4B I a 923 34,600 7 1.69 b 4,368 70,800 8 8.07 c 10,722 157,150 10.5 10 7.46 4B II a 1,282 45,88o 7 b 4,704 101,210 7 7 c 10,038 168,300 8 7 4BIII a 2,753 49,770 1.5 b 5,065 52,990 7 4 c 11,091 91,800 11 2.5 4C I a 693 27,800 7.5 b 4,979 88,290 8.5 9 7.29 c 10,607 155,220 9.5 10 6.54 4C II a 1,332 53,250 6.5 b 4,872 102,360 7 6.5 7.04 c 10,205 168,310 8.5 7.5 8.17 4CTII a 1,319 34,470 1 2 1.87 b 4,906 67,170 4 o 6.27 c 10,691 97,700 10 3 5.30

U.S.C.G.C. MACKINAW 13- FIGURE 3 12~ KEY:O I BOW PROPELLER IDLING o 12 TI,,, AHEAD / x I,I ASTERN 0( II CIRCLED POINTS ARE THOSE FOR - m WHICH SPEED WAS ESTIMATED - l FROM OPEN WATER SPEED. A- 17.29 FT. MEAN DRAFT 3.5 FT TRIM AFT A = 4495 L.T.F.W. C 10 B- 17.08 FT., ' 0 TRIM A= 4495 L.T.F.W. W g l C- 18.86 FT. " " 0.5 FT TRIM AFT = 5125 L.T.F.W. w OC 0 OB C') W: V) S( O 1 2 3 4 5 6 7 8 9 10 B.a. SPEED IN KNOTS FIG 3 "'"~~~~~~~lo B.a. SPEED IN KNOTS FG

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ba-19 is probably the result of inconsistent properties or conditions of ice, and the snow covering which increases friction. No definite trends are apparent regarding the other variables of the test: draft, trim, and propeller combinations. Considering that the ice thickness is only known from a few spot checks near the end of a few runs and that the information on speed is very incomplete, the data shown in Figure 3 should be considered as representing trends only. It is probable that the resistance of the ice has a positive value as zero speed is approached. It is also probable that the chief effect of speed on resistance will be caused by the inertia forces of the breaking and broken ice. Therefore, the resistance per inch thickness could be represented by the relationship R = a + bV2 where R = resistance/inch thickness of ice, lbs./in. V = speed, knots. a and b = constants determined by ice-breaking experiments. The data in Figure 3 could be interpreted roughly on this basis in the following equations: Draft Condition Ice Resistance Equation A R = 2000 + 100 V2 B R = 3000 + 100 V2 C R = 2740 + 160 V2 The constants are not accurate. The information is so rough that either the straight line function or the quadratic function can be abstracted from it.

ENGINEERING RESEARCH INSTITUTE Ba-20 UNIVERSITY OF MICHIGAN In order to show the effect of different propeller combinations, a curve of towrope pull (total thrust minus water resistance, or the sum of ice and wind resistance) was plotted against brake horsepower (Figure 4). Since thrust and water resistance are functions of the vessel itself, independent of external forces, this eliminated the effect of wind resistance which caused scattering of the points in the curve of ice resistance against brake horsepower. Definite trends were noted which show the effect of using the bow propeller ahead, astern, or idling. It is to be expected that the towrope pull was lower with the bow propeller astern than with it ahead or idling, assuming the same total brake horsepower. The curve also shows that the towrope pull is higher when power ahead is put on all three screws than when the bow propeller idles and all the power is delivered to the stern propellers. The points for which the speed was estimated by the three methods above checked very well with those for which complete data had been obtained. With the data available, the effects of trim and draft were not apparent. RECOMMENDATIONS 1. One condition of draft and one propeller arrangement should be tested over and over again until the data obtained are known to be reliable. 2. Thrust meters should be installed on all shafts. 3. Results from each day's data should be calculated and plotted on board the ship each day on forms made ahead of time so that faulty data may be discovered and replaced before too late. This is especially true of data from oscillographs, where the films may be spoiled or zeros, etc., unchocked. 4. A wind direction vane with protractor dial should be used instead of a

U.S.C.G.C. MACKINAW FIGURE 4 KEY: 0 I BOW MOTOR IDLING 0 II, ", AHEAD A mEE " ' ASTERN CIRCLED POINTS ARE THOSE FOR 170 WHICH SPEED WAS ESTIMATED FROM OPEN WATER SPEED. 160 A-17.29 FT. MEAN DRAFT 3.5 FT TRIM AFT _/ B-17.08 FT. MEAN DRAFT 0 TRIM 0 -C-18.86 FT. MEAN DRAFT 0.5 FT TRIM AFT AOC TOWROPE PULL = THRUST- WATER RESIST., / = ICE THRUST+ WIND RESIST. 140 120 ~/ ~~~o, I I I =AO 1 00 B w 8 0 0 I a-. /o B o I 60 40 20 0 2 4 6 8 10 12 14 B.a. BHP X I0-3 FIG. 4

ENGINEERING RESEARCH INSTITUTE Ba-22 UNIVERSITY OF MICHIGAN rough estimate by means of a pennant. 5. Trials should not be run in a high wind, say, anything over 20 miles per hour. 6. Adequate model basin tests should be run for exactly the same conditions under which the ship trial is run, namely: A. Displacements and trims. B. Speed of advance of model under varying powers with self-propulsion. To obtain correct thrust deduction factors and water resistances, some attempt should be made to install bow-wave suppressers on the model to find out the effect on trim. 7. Ideal ice conditions should be sought, particularly as to constant thickness. 8. An adequate and reliable method of vessel speed measurement should be found, since this is of prime importance in this type of calculation. A survey party with instruments on the ice would seem to be a possible procedure and not too arduous if wind velocities are not high. An improved and reliable bicycle wheel ice log is also possible.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ba-23 REFERENCES 1. Kingsbury Machine Works booklet on thrust meters, Ref. No. 3104 and 3105. 2. Transactions of the Society of Naval Architects and Marine Engineers, 42, 151, 1934. 3. Letter from Chief, Naval Engineering Section, U.S.C.G., Washington, D. C., File CG-31, dated 3 March 1948. 4. Curve sheet from Kingsbury Machine Works, K.0. 3104. 5. Propulsion Motor Data from Westinghouse Electric Company. 6. "Speed and Power of Ships", D. W. Taylor, published by U. S. Maritime Commission, 1943. 7. Model test curves for U.S.C.G.C. Mackinaw, David W. Taylor Model Basin, Model No. 3771, Correspondence File QS 13(21). 8. Model test curves for U.S.C.G. Icebreakers, David W. Taylor Model Basin, Model No. 3705, Correspondence File QS 13(20). 9. Principal data procured during the tests of January, 1948; see Table VI. * This data sheet is located in a pocket at the end of this report.

ENGINEERING RESEARCH INSTITUTE Ba-24 UNIVERSITY OF MICHIGAN Ref. 3. UNITED STATES COAST GUARD Address Reply to THE COMMANDANT (ENE-16) Washington 25, D.C. Refer to File: CG-31 3 March 1948 Professor Charles W. Spooner 326 West Engineering Building University of Michigan Ann Arbor, Michigan Dear Professor Spooner: Your letter of February 10, 1948 has been forwarded to this office for reply by the Commander, Ninth Coast Guard District, Cleveland. Below are listed the weight items and shaft bearing sizes as requested in your letter. Forward shaft: Weights: Propeller with fairwater cap 15,773 lbs. Motor rotor and shaft 28,345 Shafting and bearing sleeves 36 364 Attached turning gear 405 Total 80,887 lbs. Shaft sleeve diameter: 14-3/8" After shafts each: Weights: Propeller with fairwater cap 21,184 lbs. Motor rotor and shaft 45,870 Shafting and bearing sleeves 51,474 Attached turning gear 485 Total 119,013 lbs. Shaft sleeve diameter: 18" By direction of the Commandant. Very truly yours, G. F. Hicks Captain, USCG Chief, Naval Engineering Division CC: Comdr. 9 CGD

-0 - -.......__...__ - 160 1800 150 140 1600 ~~~~~~~1400~~130,4001 I I 120 1200 oi 11 AL1L:<1:X~10 a3 W 1000 800 90,- | \ | > 50 l |DOUBLE SIX SHOE BEARING 34" _ 600 N END PLAY-.044" ct ___ OIL - USN 2190 o4 7~40 o ASSUMPTIONS: _400_ | _ _ _ RATE OF OIL SUPPLY TO BOTH BEARINGS-20 TO 30 GPM. TEMPERATURE OF INCOMING OIL - 87 TO 90 OF. 200 0 20 40 60 80 100 120 MEASURED LOAD ON AHEAD SIDE. LBS X 103

ENGINEERING RESEARCH INSTITUTE Ba-26 UNIVERSITY OF MICHIGAN Ref. 5 Information furnished by Mr. J. A. Wasmund: U. S. COAST GUARD CUTTER MACKINAW PROPULS ION MOTORS Armature Stern R =.00347 at 750C. Bow R =.00467 at 75~C. Commutating and Stern R =.003345 at 750C. Compensating Bow R =.00503 at 750C. Brush Loss - (2xIa) F.L.Ia - Stern = 4400 F.L.Loss = 8.8 K.W. aI F.L.Ia - Bow = 2890 F.L.Loss = 5.9 K.W. Load Loss = (1% Output) Stern F.L.Loss = 37.3 K.W. aI2 x Speed Bow F.L.Loss = 24.6 K.W. Core Loss Stern F.L.Loss = 24.9 K.W. a (Speed)2 Bow F.L.Loss = 23.2 K.W. Brush Friction Stern F.L.Loss = 3.9 K.W. a (Speed) Bow F.L.Loss = 2.7 K.W. Bearing Friction and Windage Stern F.L.Loss = 2.10K.W. a (Speed)2 Bow F.L.Loss = 2.71K.W. *Shunt Field F.L.If = 62.6 Stern R = 3.22 at 75~C. F.L.If = 38.5 Bow R = 4.90 at 750C. * Motor Separately Excited Stern Motors 5000 HP, 900 Volts, 4400 Amps. 136/170 RPM. Bow Motor 3300 HP, 900 Volts, 2890 Amps. 175/200 RPM. WESTINGROUSE ELECTRIC COMPANY

APPENDIX B, SECTION b STEADY STATE GIRDER STRESS ANALYSIS BY J. ORMONDROYD A. C. McCLURE August, 1949

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX B, SECTION b STEADY STATE HULL GIRDER STRESS ANALYSIS The stresses in the ship's girder were calculated analytically for the condition in which the vessel advanced through the ice field with uniform velocity. The ship was acted upon by various forces due to water, ice, wind and propeller thrust. Some of these were known, and others were calculated. From the forces, loading, shear and bending moment curves were drawn. Curves of compressive forces were drawn and the moment of these forces added to the bending moment above. Stresses were calculated by the formulas: Mc for bending stress, and P 6- = for compressive stress when 6 = stress M = bending moment c = distance of point in question from neutral axis I = moment of inertia of cross section P = compressive force A = cross-sectional area of ship's structure. Figure 1 shows the various steady forces acting on the ship as she advances through the ice field with uniform velocity. The ship rotates about a point on the original waterline at anunknown distance, xlfronm the after perpendicular. The unknown vertical component of the ice force, Q, is assumed

FIG. I U.S.C.G.C. MACKINAW FREE BODY DIAGRAM -'Xe I RWi hwi. [D.W.L. EPROP SHAFT hwi. A.. B( b.

ENGINEERING RESEARCH INSTITUTE Bb-3 UNIVERSITY OF MICHIGAN l for the first approximation to act at the forward perpendicular. The three equations of static equilibrium were written. %.H = 0 was satisfied in the thrust calculations in order to determine the ice thrust. The other two equations, 2VT = 0 and 1M = O, were solved simultaneously to obtain Q and xl. Symbols: x = distance from after perpendicular to any point p(x) = half breadth at waterline = change in angle of trim Thus, xl0 = depression of stern (xl-x)0 = height of "wedge" at any point x ja = length of ship q(x) = buoyant force of water per unit length at each point, x, positive upward. = density of water = 62.4 lbs. per cu. ft. q(x) = 2X p(x) (xl-x)0 Summation of vertical forces: 270 '9 p(x) (xl-x) dx + Q = 0 2 ' j1, p(x) dx - f x p(x) dx + o 2 El fo p(x) dx - () dx + 1p (x) dx dx + Q = O (1) 2 x Lxl-1) p(x) dx + jdxx Q 0 (-_ =_0)

Bb-14 | ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Summation of moments: the intersection of the keel and the after perpendicular was taken as the moment center. The moment equation is: (2) Tp.hp - Ti.hi - Rwi.hwi - w.ha - Rwa - hK -- 0 (~M= O) in which K0 is the moment of the change in buoyancy: (3) KO = 2:0 x(xl-x) p(x) dx = 2 0[xl I x p(x) dx - x2 p(x) dx 2=. L xll-12) p(x)dx + (21-x1) Tp(x)dxdx - p2 J p X (x)dxdxji where Tp = propeller thrust Ti = ice thrust Rwi = wind resistance Rwa = water resistance h = lever arm, according to subscript Figure 1 shows the locations of the various forces. The integration was done graphically. The change in shape of the waterplane as the vessel trimmed was neglected. Since the trim in open water was measured only approximately by means of the draft gages, trim before the run was assumed zero. The trim indicated by the gyro was accordingly taken to be the change in trim. This necessitated the omission of data from test series 4A since the ship was run at a considerable trim. Also, it was assumed that the entire change in trim was caused by ice forces, neglecting the effect of water forces.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Sample Calculation: Run 4BIc, 17.08' mean draft, trim angle.00302 radians Quantities from integration: Let A - p(x) dx = 6,680 ft. 0 B - p(x) dx d = 913,000 ft.3 x x c -i P(x) dx dx dx 76,780,000 ft. The evaluation of these integrals is shown in Figure 2. Figure 3 shows the same evaluation for 18.86-foot draft. Equation (1) can be written (la) 2 A xl + Q = 2 (A - B) and Equation (2) can be written (2a) 2 /i (B -At) - x - Q = M - 2 (A2 2B + 2C) where =M = Tihi + Rwi hwi + Rwa-hwa - Tp.hp The solutions for these equations are: Xl _ iB -2C + Z M X1 B 2 r,.B Q = 2t > 2AC - B2 A For Run 4BIc:

7 I__ ___ _____ _sf__tINTEGRAL I" = 1000 FT2 WATER PLANE INTEGRATION 17' WATER LINE 2nd INTEGRAL 6" 5I" =lo 20 18 12 14 12 10 8 6 4 2 0 B.b. STATIONS FIG. 2 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~, 20 1 8 1 6 1 4 1 2 1 0 8 6/ 4

7" I t INTEGRAL I" = I000 FT 2 WATER PLANE INTEGRATION 2"d INTEGRAL 18.86' WATER LINE _ I"= 163,333 FT3 -6" /I 3rd INTEGRAL I" 15 106 FT4 5 2___ 20 18 16 14 12 10 8 6 4 2 0 B.b. STATIONS FIG.3

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN A = 6,680 ft2; Tp = 169,770 lbs; hp = 7.6 ft; Tp'hp = 1.29 x 106 lb-ft. B = 913,000 ft3; Ti = 142,770 lbs; hi = 17.0 ft; Ti~hi = 2.43 x 106 lb-ft. | = 76,780,000 ft4; Rwi = 14,380 lbs; hwi = 19.0 ft; Rwi.hwi = 0.27 x 106 lb-ft. = 0.00302 radian; Rwa = 12,620 lbs; hwa = 10.5 ft; Rwa hwa = 0.13 x 106 lb-t. 1'= 62.4 lb/ft3 _M = 1.54 x 106 lb-ft. I= 280 ft For these conditions x I = 116.4 ft. Q = 67,672 lbs. For a triangular distribution of ice load with the load per foot equal to q.l at the bow and equal to 0 at the midship section, Equations (1) and (2) become (lb) 2 9 A xl' + ~ ~ = 2 t (A2 - B) (2b) 2 (B - A) xl' - = M - 2 (At2 - 2B+ 2C) The solutions for these equations are: X- ' -A: /~ L + _ M A - A -- 1 A= B B~ 6 11-AA/66B 4 2 2AC- B 4 Q 4l 1 - A2/6B A! 1-Ae/6B Z For the data of Run 4BIc: x1' = 102.2 ft. ~q a= 1460 lb/ft.

ENGINEERING RESEARCH INSTITUTE Bb-9 UNIVERSITY OF MICHIGAN The vertical ice load per foot is assumed to decrease at a constant rate from the bow to the midship section, where it is assumed to be zero. At the forward quarter point, the ice loading per foot would be 730 lb/ft. The force, Q, is the resultant of four forces: the breaking force of the ice sheet, acceleration of the broken pieces of ice, buoyant effect of the ice under the hull, and the vertical component of friction of the ice moving around the hull. These were approximated by various means. Buoyant force was estimated by assuming a distribution under the hull: that 50 percent of the solid ice sheet flowed under the hull at the bow, and no ice was under the hull at the stern. Distribution per foot of width varied linearly from bow to stern. As the ice flowed around the curved surface of the hull, it experienced an acceleration. The value of the acceleration was obtained by plotting several flow lines, differentiating graphically twice to obtain velocity and acceleration curves. The expression for acceleration is: a = v2 d2y/dx2 J7 (dy/dx) 2c = 2 d2y/dx2 * (dy/dx) 2 where a = acceleration v = forward velocity of ship x,y = coordinates of flow lines Q = angle of flow line to vertical at any point x,y. Figures 4, 5 and 6 show how the acceleration a was calculated. Since the flow lines are convex at all points, the acceleration was upward, causing a reduction in buoyant force. The average acceleration was.0028 v2. Taking the

7 FLOW LINE NO. I 6" F 1UNBROKEN SHEET OF ICE 6d d2y ACCELLERATION = 2 dy 2 dx dx FLOW LINE 1"= 4' 22,,4 =-.o0 IFT-2 2"..,.d 1"=.2 FT/FT. B~b.STAIONS"flG.4 20 18 16 14 12 I0 8 6 4 2 B. b. STATIONS

FLOW LINE NO.2 UNBROKEN SHEET OF ICE 6!l d2y d X2 ACCELLERATION c _ ddy 2 2 I y1".005 FT.~ dX2 3* FLOW LINE 2" ~Y. I"=.I FT/FT dx 20 Is 16 14 12 10 8 6 4 2 0 B.b. STATIONS FIG.5

FLOW LINE NO.3 to -UNBROKEN SHEET OF ICE _ d x - ACCELLERATION =,_ 1+ (dy 2 y 1".I FT/FT 5": 4'_" 5" I" 2, I".0025 FT1 20 18 16 14 12 10 8 6 4 2 0 B.b. STATIONS FiG.6

ENGINEERING RESEARCH INSTITUTE Bb-13 UNIVERSITY OF MICHIGAN density of ice to be 56.4 lbs. per cu. ft., the flotation force is 6 lbs. per cu. ft. submerged. The velocity during Run 4BIc was 12.60 ft. per sec., resulting in an acceleration force of about 1/2 lb. and a buoyant force of about 5-1/2 lbs. per cu. ft. With the ice thickness averaging 16-1/2 in., the estimate of total buoyant force during Run 4BIc was 25,000 lbs. The center of this force was found to be 28.3 ft. forward of amidships. At the moment of impact with the bow of the ship the ice was accelerated rapidly downward. (This is not to be confused with the upward acceleration of the ice moving around the curved surface of the hull.) The well-known principle of dynamics, change of momentum equals force times time, was used to evaluate this effect: Ft = M2 V2 - M1 V1 where F = force t = time interval M = mass of ice accelerated V = velocity of ice 1,2 = refer to conditions before and after impact. Since the mass can be considered constant for a given piece of ice, and the initial velocity is zero, the equation can be simplified to: Ft = MV2 M or F =- V2 where M/t can be interpreted as the mass of ice accelerated in a unit time, V2 equals the vertical component of the velocity imparted to the ice. Calling the angle between the ship's plating at the waterline and the waterplane i,

Bb-14 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN the velocity of the ship V ft. per sec., the density of ice f and the thickness of the ice sheet h, the equation can be written: F = h tV2 tan Q for unit width. g Tan G was found at four points along the bow of the ship, and the total force was assumed to be the sum of the forces at these points times the width of a strip centered about each point. A considerable frictional force arises out of the pressure of the ice against the hull. The coefficient of friction was assumed to be 0.10 (see Bibliography on Lake Ice by Richard Strong, p. 61). The normal force of the ice against the hull was calculated from the ice thrust, which was deduced from the power calculations. This thrust includes friction, so the horizontal component of friction must be subtracted from it. If 0 is the angle of the buttocks to the waterline plane, T is the total ice thrust, and 0 is the angle of friction (tan-lh4), the frictional force can be written: F = T - F cos 9 sin =,,T _ F cos e' sin 9 sin 0 _ T sin 9 (1 +t cot G) The vertical component of the frictional force is: Fv = F sin 9 1 + cot 9 Substituting 0.1 for 4, and 30~ for 9, the expression becomes:

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Bb-'i Fv =.0852 T Distribution was assumed to be uniform across the beam of the ship. Thus the force per foot of length is proportional to the tangent of the angle between the waterline and the centerline at any point. The breaking force of the ice was estimated by considering the ice sheet a homogeneous elastic plate of infinite extent on an elastic support with a load distributed uniformly along a straight edge. The final equation for the load per foot along the edge, Q0, is: = rX h2 e /4 W Q0 = e 6F where 6' = tensile strength of ice, 200 psi h = thickness of ice sheet k = density of supporting medium (water) = 62.4 lbs. per cu. ft. E = modulus of elasticity of ice, - 5,000,000 psi V = Poisson's Ratio for ice, 1/3. Distribution was assumed to be uniform along a 70-foot strip on each bow, the force being Qo/2.70 lbs. per foot along each side of the hull. The moments of the ice force components were computed and adjusted to equal the moment of Q, the total ice force. From the adjusted moments, corrected ice forces were obtained. The sum of these forces equalled a new "Q" acting at the centroid of the component forces. The distance from the after perpendicular to the center of rotation, x, was then recalculated. The change in displacement in lbs. per foot of length was calculated at several stations: AB = 2 0 ' p(x) (xl - x)

Bb-16 I ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN where AB = change in displacement per foot of length. The difference between the change in buoyancy and the ice forces at any point is the net vertical load at that point. This was plotted and integrated twice to obtain shear and bending moment curves (Figure 7). A curve of compression loading due to propeller and ice thrust, wind and water resistance was drawn (Figure 8). These forces were considered to act at the neutral axis by addition of a couple. Adding the moment at each point along the hull due to compression to the bending moment resulting from the vertical loading, the total bending moment was found. The stresses in the region of each of the six hull girder strain gages were computed using the expressions: = MI for bending stress, and r = P for compression stress Their sum was assumed to be the total stress at each location (see Table I). The order of magnitude was found to be extremely low: about 200 psi on the compression side, and 50 psi on the tension side. Stresses were computed only for Run 4BIc for several reasons: The trim was greater than that in any other run; consequently the total ice force was greater. Also, there was reasonably good agreement between theoretical computations and measured quantities. Strain gage readings are not available to check the computed stresses. However, stresses similarly calculated from the hog-sag test data showed an average excess of 10 percent over stresses calculated from strain gage readings. This difference is to be expected, owing to the manner in which the moment of inertia and sectional area of the hull girder were calculated. Only

ICE LOADING 3"F H SHIP STEADILY BREAKING ICE SPEED = 7.45 KNOTS URVE ICE THICKNESS = 15 -18I / RUN 4BITC TOTAL BENDING 2 "_ MOMENT MOMENT IS' ____________~ __ _I" 800,000 FT # SHEAR DIAGRAM /BENING MOMENT DOE TO COMPRESSION L LOAD CURVE I"- 400 _ _ 2u 20 18 16 14 12 10 8 6 4 2 0 B.b. STATIONS FIG. 7

COMPRESSION LOADING 3" SHIP STEADILY BREAKING'ICE SPEED = 7.45 KNOTS ' ICE THICKNESS =15"- 18" RUN 4BIC COMPRESSION MOMENT: I - 00,000 FT*& 2 OMPRESSION LOAD 0 20 18 16 14 12 10 8 6 4 2 0 B.b. STATIONS FIG.8

TABLE I STEADY STATE HULL GIRDER STRESS ANALYSIS CALCULATED STRESSES AT THE LOCATIONS OF STRAIN GAGES 1-6 15-INCH TO 18-INCH ICE Run 4BIc A = 4495 LTFW DIST. GAGE OMMOM. COMP. I SECT. MS P/A SESS NAFO FT-LBS LBS F4 AREA P.S.I D.... P.S.I. P.SI. P.S. 1 + 9.0 2,310,000 146,000 2370 2610 + 61 -56 + 5 2 + 6.1 2,440,000 155,000 2560 2830 + 40 -5 - 15 3 + 7.33 820,000 163,000 1130 2260 + 37 -72 - 35 4 -21.4 1,530,000 120,000 2020 2400 -113 -50 -163 5 -22.4 2,070,000 157,000 2380 2770 -137 -57 -194 6 -15.8 650,000 164,000 1320 2370 - 78 -69 -147 STRESS AT POINT OF MAXIMUM BENDING MOMENT STA. 7-1/2 -22.2 2,875,000 150,000 2726 2850 -16 -53 -216

|bI2 0ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN fully effective material is taken into account, whereas there is considerable material that is partially effective in resisting loads. In view of the fact that the estimated stresses in the hull girder under steady ice-breaking conditions were extremely low, it was considered desirable to investigate the most severe static condition that could be encountere&-. This is the case in which the ship is unable to break the ice, and runs out on the ice as far as the forward skeg will allow. For the purpose of determining the hull girder stresses, the load can be assumed concentrated just forward of the skeg as shown in Figure 1. For the first approximation to the load, the change in shape of the waterplane as the vessel trimmed was neglected. Calling: F = the concentrated load h = change in draft at the point of application of the load t = sinkage, negative; or decrease in draft if the load is considered to act at the center of flotation r = distance of the load ahead of the center of flotation MTl" = moment to trim 1" TP1 = tons per inch 2 = length of ship between perpendiculars. Two expressions for the load can be written: (1) F = 12(h - t) (2/r)MTl" r (2) F = t ~ TP1 in which F and t are unknown. Expressing t as a function of F: t = F rn1

ENGINEERING RESEARCH INSTITUTE Bb-21 UNIVERSITY OF MICHIGAN Expression (1) can be rewritten: F = 12(h - Solving for F we have: F = 12 h 2 MTI" TP1 TP1 r2 + i MlTl" The point of application of the load is 16 feet abaft the forward perpendicular and the center of flotation is 141 feet abaft, making r 125 feet. At the 19 -foot draft, the moment to trim one inch is 500 foot-tons, and the tons per inch is 33. The change in draft at the load point is taken to be the distance of the keel below the original waterline; that is, the keel at that point is forced up to the water level. Inserting these values, we have: F = 575 tons. The change in buoyancy at each station was then calculated, plotted, and integrated twice to obtain shear and bending moment curves. An error introduced by neglecting the change in shape of the waterplane showed up in the bending moment curve. This was corrected by drawing another set of curves erring in the opposite direction, and interpolating. A final set of curves, Figure 9, was then drawn, yielding a value of: F = 516 tons maximum bending moment of 53,900,000 ft.-lbs. (at station 7-1/2), and maximum stress of 3050 psi in compression (see Table II). The only direct measurements of steady state strains available in the ice-breaking test data occur in the Windrowed Ice Runs 5A and SB. In these two runs the gages were not zeroed to achieve even distribution on the oscillo

SHIP HUNG UP ON EDGE OF ICE SHEET ___ 16'- O" ABAFT FR P DISPLACEMENT 5,125 L.T. EW. BENDING MOMENT I" 161800000 FT# "'INS CHANGE IN BO YANCY 1: 4000 / FT 20 18' 16 14 12 10 8 6 4 2 0 B.b. STATIONS FIG.9

Bb-23 TABLE II STEADY STATE HULL GIRDER STRESS ANALYSIS Stress In Hull Girder for Condition in Which Ship Is Hung Up on Ice 16'-0"Abaft Forward Perpendicular Concentrated Load = 1,157,000 lbs = 516- T A = 5125 LTFW DIST. GAGE BELOW MOMENT I4 PS= Mc/I N.A.. N.A. FT-LBS FT P.S.I. 1 + 9.0 48,800,000 2370 1287 2 + 6.1 47,050,000 2560 779 3 + 7.33 21,750,000 1130 979 4 -21.4 39,300,000 2020 -2890 5 -22.4 41,600,000 2380 -1920 6 -15.8 19,150,000 1320 -1590 POINT OF MAXIMUM BENDING MOMENT STA. 7-1/2 -22.22 53,900,000 2726 -3050

Bb-24 | ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN graph record. Also, the ship started to gain speed in open water before hitting the windrow. For this reason, it is possible to measure the difference in bending strains in the water and in the ice. Depth Change in Calibration Maximum of Zero Line Factor Strain Change Run Wind- Gage of the of Gage Stress in Trim row Record microinches inches lbs/in Angle feet inches inch inch radians No.4 -.4 49.4 19.76 x 10-6 - 593 5A 15 No.5 -.5 48.8 24.40 x 10-6 -732 +.03705 No.6 -.5 50.6 25.30 x 10-6 -760 No.4 -.47 49.4 23.2 x 10-6 -695 5B 20 No.5 -.53 48.8 25.9 x 10-6 -778 +.043225 No.6 -.53 50.6 26.8 x 10-6 -805 The stresses in these gages are compressive stresses, indicating that the ship was sagging with upward ice forces from windrow distributed along the ship forward of the midship section and increased upward flotation forces acting aft of the midship section. The stresses in Runs 5A and 5B are about one-quarter to one-half of those calculated in Table II, where the ship was assumed to be hung up on a ledge of ice. They are about four times the stresses calculated for Run 4BIc. In the stress calculations for Run 4BIc, a fairly large percentage of the total stress was caused by direct compression. Run 4BIc Gage Gage Percent Direct Percent Compression Stress Bending Stress 4 31 69 5 29 71 6 47 53 If these same proportions are assumed to hold in Runs 5A and 5B, the stress and

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN B moment distribution is as follows: Assumed Bending Moment Measured Direct Assumed Corresponding Run Gage Stress Compression Bend ing to Assumed Number lbs/in2 Stress Bending Stress lbslbs/ lb-/in lb-in 4 593 184 409 66.5 x 106 5A 5 732 209 523 95.2 x 106 6 760 357 403 57.7 x 106 4 695 216 479 78.0 x 106 5B 5 777 225 552 100.4 x 106 6 805 378 427 61.1 x 106 In order to determine the comparative severity of the stresses during ice-breaking operations, a standard strength calculation was made. Only the sagging condition was considered, since that is the condition when breaking ice. In standard sagging condition, the ship is considered as poised on a trochoidal wave equal in length to the ship's length, and one-twentieth as high as it is long, with the trough amidships. Assuming for comparative purposes that the stress in the hull in still water was zero, the loading was then the difference between the still water load line and the wave profile (Figure 10). Stresses calculated from this (Table III) are of the order of 20 percent greater than the stresses estimated when the ship was run up on the ice (Table II) and of the order of 15 times as great as when the ship was steadily breaking ice. It can be concluded from the foregoing that the stresses in the hull girder of a vessel breaking ice are less than the stresses due to waves of "standard" proportions, and that a vessel built for icebreaking need not have greater hull girder strength than ordinary vessels. It is obvious that hogging in a "standard" wave and hogging on an ice ledge could not occur simultaneously.

AREA (LOAD) VOLUME SHEAR MOMENT 3 _ 100 FT2_ I"= 6000 FT3 =400,000 FT4 1"-1 I0 FT 2AEHX STEADY STATE 2" SHIP POISED ON STANDARD -2" TROCHOIDAL WAVE 280' LONG _ 14' HIGH CRESTS AT ENDS OF SHIP. DISPLACEMENT OF SHIP 4495 L.T EW. MEAN DRAFT 17'-0". DRAFT AMIDSHIPS 12.2' AT ENDS 26.2' 20 18 16 14 12 10 8 6 4 2 0 B.b. STATIONS FIG.1O

Bb-27 TABLE III STEADY STATE HULL GIRDER STRESS ANALYSIS Increment of Stress Due to Sag on a Standard Trochoidal Wave 280 Ft. Long, 14 Ft. High, Crests at Ends of Ship. Displacement = 4495 LTFW DIST. GAGE BELOW 4 GAGE BELOW MOMENT I ~ = Mc/I N.A. FT-LBS FT P.S.I. 1 + 9.0 13,100,000 2370 346 2 + 6.1 58,400,000 2560 965 3 + 7.533 59,900,000 1130 2700 4 -21.4 46,200,000 2020 -3400 5 -22.4 63,800,000 2380 -4175 6 -15.8 56,800,000 1320 -4715 POINT OF MAXIMUM BENDING MOMENT STA. 10 -22.22 66,100,000 2726 -3745

APPENDIX B, SECTION c PLATE STRESSES CAUSED BY EXTERNAL IMPACTS OF ICE BY P. F. CHENEA P. NAGBDI J. SHEA J. ORMONDROYD August, 1949

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX B, SECTION c PLATE STRESSES CAUSED BY EXTERNAL IMPACTS OF ICE Twenty-one SR-4 mitalectric strain gages were attached to the inner surface of 1-5/8-inch thick shell plating midway between frames at the 14-foot 6-inch waterline. This is nearly the same level as the first platform. It is also very nearly the level of the neutral plane (where bending stresses are zero) of the ship forward of the midship section. The variation of strain signalled from these gages can be caused by several distinct actions as follows: 1) Direct compression of the ship hull 2) Direct shear of the ship hull 3) Bending of the hull in the horizontal plane (but not in the vertical plane) 4) Twisting of the hull 5) Local buckling of the plating 6) Local external impacts and pressures from the breaking and broken ice 7) Distortion of the hull by heat radiation from the sun and uneven internal heating of the ship. The strain gage rosettes used had three legs radiating from a point: C Cy B VERTICAL A, a 45~ / X X AA HOR IZONTAL

ENGINEERING RESEARCH INSTITUTE Bc-2 UNIVERSITY OF MICHIGAN The horizontal leg was designated A, the inclined leg was designated B, and the vertical leg was designated C. The strain in the horizontal leg A is Ex, the strain in the inclined leg is E and the strain in the vertical leg is Ey. These strains are related to the shear strain, it, as follows: E = E _+ Ey + 'X Ycos 2a + sin 2a 2 2 2 For a = 45~, G + l 2 2 The average stresses under the strain gage (assuming no buckling to occur) are: E Orx - 2 x + VEy) (tension or compression) ~'Y - 1(Cy E+ Ax) (tension or compression) E (shear stress) xy 2(1 + ) The signals recorded were proportional to changes (from zeroes established with the ship floating in still water) in 6x, &y, and E. For most of the gages, only leg A was connected to the oscillograph. Only gage No. 29 was used consistently as a rosette. When leg A is used alone the signal excludes the effects of direct shear and twisting of the hull, and the effect of bending of the hull is made negligible because the neutral plane is near the gage level. Several of the plating gages show distinct signals which are associated with the bending vibration of the ship. These are most probably associated with slight buckling of the plating. That this is possible is indicated

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN 3 by the following facts. The plating at the strain gage locations is practically flat or only slightly curved. For gage locations 8, 11, 20, 29, 33, and. 43 the ship's lines indicate the plating to be slightly convex inward, which would lead us to expect tension changes for external impacts against the ice, Gage 55 is located at a point where there is a slight double curvature at right angles. This would lead us to expect external impacts to give either tension or compression depending on the exact location relative to the gage of the loading. Gage locations 67, 79, and 86 are on plating slightly concave inward. For these gages external impacts should show compressive stresses. The actual records of impact strains show the following effects: Gage 8 always compression Gage 11 always tension Gage 20 always tension Gage 29 A always compression Gage 29 B always tension Gage 29 C always compression Gage 33 mostly compression, a few tension Gage 43 always tension Gage 55 about half tension and half compression Gage 67 always tension Gage 79 always compression Gage 86 always tension The local curvature of the plating created in the construction of the ship dominates the sign of the signal since tension and compression responses to external impacts are not determined by location according to the ship's lines. The strains with bending frequencies are not analyzed. No average strains are analyzed because there was uncontrolled and unrecorded shifting of

ENGINEERING RESEARCH INSTITUTE Bc-4 l UNIVERSITY OF MICHIGAN zero lines to space the traces evenly after the run was started. The external impacts of the ice on the plating give rise to transient signals which carry their own zeroes. These will be analyzed. Before analyzing the records of ice impacts, the general nature of the stresses in the plating will be analyzed for a few idealized and simplified conditions. It would have been better to have had only four gages, all used as complete rosettes, located with even spacing between the bow and the midship section, than the arrangement we used. The rosette analysis indicates that a single leg - such as leg A - may give an erroneous indication of the magnitude of the maximum stress actually occurring at the gage location. 1. Analysis of Simple Plate Loading Situations To determine correctly the thickness of the hull plating on an icebreaker, it is necessary to make an adequate stress analysis. This analysis is complicated by two factors, namely: (1) the complex nature of the ship's hull and its supporting structure, and (2) the rather vague and random nature of the loading due to the ice field. It is possible, however, to obtain indicative results by idealizing these two phases of the problem. In the analyses to follow, the ship plating is analyzed for three different idealized structural plans: a. A simply supported infinitely long strip of one frame spacing width b. A fixed edged infinitely long strip of one frame spacing width c. A plate continuous over three spans, each span one frame spacing in width. It is further assumed that the relative deflection of the frames, decks, and bulkheads is small and therefore may be neglected. These assumptions appear

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Bc-5 reasonable in view of the apparent stiffness of these structures. The assumption of an infinite strip vertically between frames is conservative. The analyses show that the nearness of the deck to the point of application of the load is only important for deck-to-load distances less than the frame spacing. For loadings nearer to the deck than this, the stresses in the plate are diminished, and vanish completely if the load is applied at the deck level and the deck is rigidly attached to the ship plating at this point. The simply supported strip and the fixed edged strip represent an upper and a lower limit. The actual situation at the support is more nearly represented by the threespan plate. The loading is taken as a concentrated load at the center of the plate, which is the worst condition. A comparison with the line load is made for the plate of three spans showing the reduction in stresses resulting from a distribution of this load. Other distributions of the load will give corresponding reductions. The method of analysis is approximate. Exact solutions by means of trigonometric series were attempted for the plates of three spans, but the convergence was so poor as to render these solutions worthless. For the sake of uniformity of procedure, the approximate methods were also applied to the plates of a single span. This latter step made it possible to check the accuracy of the approximate method with exact series solutions which do converge sufficiently rapidly to permit evaluation. The error in the deflections obtained. by the approximate method is of the order of three percent. The method. used consists of assuming the general form of the solution and then evaluating the coefficients by equating the work done by the external loads to the gain in elastic energy of the plate. The assumed form of the solution was selected so that all boundary conditions were automatically satisfied. Such a procedure

Bc-6 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN is direct and lends itself well to the problem at hand. 2. Simply Supported Strip with a Concentrated Load The nature of this plate and its loading is shown in Figure 1. The solution for the deflections was assumed to be of the form w = w [3(x/a) - 4(x/a)3] (1 + Ay/a) e-Y/a 0 - x 4 a/2 O z y < o where w = deflection wo = deflection under the load P a = plate span and the coordinate axes are as shown in Figure 1. The value of wo as determined by energy considerations is wo = 0.1796 Pa2/Eh3 The exact solution for this special example has also been evaluated, and it yields wo = 0.1849 Pa2/Eh3 which shows the error to be less than three percent. The variation of the deflection along the two axes of symmetry is shown in Figure 1. The moments and stresses are obtained from the usual expressions = Eh3 2w + J

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Bc-8 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN MY l~2) (2 + x2 12(1 - 2) 2) andx 2 Y h2 6 _ Eh _ 2w xy 2(1l+) ax y where MxMy = moments per unit length of plate E = Young's Modulus i = Poisson's Ratio h = plate thickness ~TxJ y = bending stresses in the plate. The stress distribution along the two axes of the plate is shown in Figure 2. 35. Fixed Edged Strip with a Concentrated Load The plate and its loading for this example are shown in Figure 3. The equation of the deflection surface was obtained in the same manner as in (2.) above. w = 4 wo (x/a)2 (3 - 4x/a) (1 + ty/a) e-nY/a 0 _ x _ a/2 O _ y c Qo where wo = 0.06289 Pa2/Eh3 is the deflection under the load P. The deflections and the stresses 6x and ay along the two axes of symmetry are shown in Figures 3 and 4.

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Bc - 12 CI A. UNIVERSITY OF MICHIGAN 4. Plate Continuous over Three Spans with a Concentrated Load at the Center The plate and its loading are shown in Figure 5. This example was chosen so as to evaluate the effect of side span flexibility. The deflection surface has a different expression for each span. The x-axis is the horizontal axis of symmetry, and the y-axis is the left edge of- the span in question. The deflection surface is expressed by Side Span w 112w wO(x/a)3 - (x/a) (1 + ay/a) e-2(j 0 x _ a O - y 4 0 Center Span w = 14 Wo 6(x/a) + 9(x/a)2 - 20(x/a)3] (1 + y/a) e-:KY/a 0 _ x - a/2 0 y The deflection under the load (wo) is given by wo = 0.1096 Pa2/Eh3 The deflections and stresses in the side spans and the central span are illustrated in Figures 5 and 6. 5. Plate Continuous over Three Spans with a Line Load on the Center Span Thi example was analyzed to obtain ae inodication on the effect of | a vr.riation in the load distribution on the plate. The load was assumed to be of intensity q per unit length and distributed along a line on the center

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ENGINEERING RESEARCH INSTITUTE Bc-15 UNIVERSITY OF MICHIGAN | span as shown in Figure 7. The axes are located in the same manner as for the previous example. The deflection surfaces have the equations Left Span w 16 wo (x/a) [(x/a)2 1 (1 y/a) e -Y/a 0 z x z a 0 L y ( o Center Span 16 Wo(X/a) [2 + /a) lO(x/a)2 + 5(x/a) (1 + iy/a)e-jy/a 0xa 0 ~ y where the deflection under the load is (wo) wo = 0.06632 qa3/Eh3 The nature of the deflections and the stresses is shown in Figures 7 and 8. 6. Summary The deflections in the plate are seen to be of the form w = a Pa2/Eh3 and the stresses are of the form 0. = B p/h2 where a and p are functions of the load location and distribution. It is significant that the maximum stresses are independent of the magnitude of the span a, as long as P remains constant. The deflections, however, increase as the square of a.

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CX)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~" " ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 'UL w 'T1 ~ ~ ~ ~ ~ ~~~~~~~~~~~1 G)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1F 'm~~~~~~~~~~~~~~~~~~~~1IIA

Bc-18 | ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN The maximum stresses at the center of the span as shown in the figures are: 1) Simply supported strip with a concentrated load at mid-span: max. = 1.476 P/h2; ax = 1.329 P/h2 2) Fixed edged strip with a concentrated load at mid-span: aCS = 0.932 P/h2; ma = 0.590 p/h2 x max. y max. 3) Plate continuous over three spans, concentrated load,center of midspan: max. =.o098 P/h2; 6 ax = o.870 p/h2 (rx max. y max. 4) Plate continuous over three spans, line load across center span: max = 0.50 qa/h2; max. = 0.47 qa/h2 Table I indicates the location, frequency, magnitude, and sign of the signals which are assumed to be the result of external, local impact between the ship plating and the ice. Table II shows the frequency of occurrence of impact signals. 1) The most remarkable fact which leaps out of Table II is that the impact signals are almost completely absent from the high speed runs (designated by the lower case letter c) under almost all draft conditions and propeller combinations. In draft condition A, where the shell plating gages were at ice level, only gage 67 shows impacts when the stern propellers alone are operating. Under draft condition C, even keel, 19-foot draft, no impacts are registered at high speed. However, under condition B, even keel, 17-foot draft, all the high speed runs show some impacts. The speed itself may be connected with the observed fact. At high speeds the ice, when ploughed

Bc-19 TABLE I IMPACT M4EASUREMENTS Run No. of Maximum Average Average No. Gage Impacts Type Amplitude Amplitude Duration 4-A-l-a 8 14 C* 43.8 x 10-6 19.2 x 10-6.67 sec 11 14 T** 22.8 x 10-6 9.1 x 10-6.75 sec 20 6 T 14.5 x 10-6 10.0x 10-6 1.0 sec 33 9 C 42.5 x 10-6 18.0 x 10-6.6 sec 43 6 T 11.4 x 10-6 6.o0 x 10-6.7 sec 55 4 C 8.0 x 10-6 6.0 x 10-6 1.0 sec 55 1 T 7.5 x 10-6 7.5 x 10i-6 1.8 sec 67 8 T 10.2 x 10-6 5.0 x 10-6.7 sec 79A 5 C 8.0 x 10-6 5.0 x 1ic6 1.0 sec 29B 5 T 20.8 x 10-6 10.4 x 10-6.7 sec 4-A-l-b 8 59 C 49.3 x 10-6 16.5 x 10-6.1 sec 11 46 T 38.2 x 10-6 9.1 x 1O-6.1 sec 20 36 T 25.6 x 10-6 10.7 x 10 6.25 sec 33 2 T 18.7 x 10-6 15.0 x 10-6.5 sec 33 29 C 37.0 x 10-6 12.0 x 10-6.2 sec 43 32 T 28.6 x 10-6 14.0 x 10-6.3 sec 55A 10 T 19.5 x lOi6 8.1 x lO-6.2 sec 55A 12 C 21.6 x 10-6 8.7 x 10-6.2 sec 67 19 T 30.8 x 10-6 10.3 x 10-6.2 sec 79A 17 C 17.2 x 10-6 8.0 x 10-6.5 sec 86 Most of the vibrations look like bending. 29B 15 T 52.0 x 10 18. x 10-6.3 sec 4-A-l-c 8,11,29B, 20,33,43, Bending frequencies only. 55A,79A,86 67 10 T 30.8 x 10-6 10.9 x 10-6.2 sec 4-A-2-a 8 7 C 48.o x o6 30.0 x 10-6 10.7 sec 11 5 T 26.0 x 10-6 8.0 x 10-6.6 sec 20 6 T 11.0 x 10-6 7.0 x 10-6.5 sec 33 5 C 37.0 x 10-6 12.0 x 106.7 sec 143 6 T 14.5 x 10-6 5.0 x 10-6.7 sec 55A 6 C 8.0 x 10-6 5.0 x 10-6.5sec 67 7 T 18.8 x 10-6 8.6 1.0 sec Impact signals directed toward the tnime signals are compression strains. Impact signals directed away from the time signals are tension strains.

Bc -20 TABLE I, continued Run No. of Maximum Average Average No. Gage Impacts Type Amplitude Amplitude Duration 4-A-2-a 79A 6 C 34.3 x 10-6 20.0 x 10-6 1.0 sec 86 5 T 29.4 x 10-6 10.0 x 10-6 1.3 sec 29A 3 C 30.7 x 10-6 25.0 x 10-6.9 sec 29B 3 T 7.8 x 10o6 6.0 x 1o-6.9 sec 29C 3 C 10.0 x 10-6 8.0 x 10-6.9 sec 4-A-2-b 8 6 C 54.8 x 10-6 30.0 x 10-6.4 sec 11 12 T 11.0 x 10-6 8.0 x 10-6.25 sec 20 8 T 21.0 x 10-6 8.0 x 10-6.25 sec 33 12 C 16.0 x 10-6 8.o0 x 10-6.25 sec 43 9 T 25.0 x 10-6 19.0 x 10-6.25 sec 55A 7 T 20.0 x 10-6 13.0 x 10-6.25 sec 67 11 T 53.0 x 10-6 25.0 x 10-6.25 sec 79A 5 C 54.0 x 10-6 36.0 x 10-6.25 sec 86 3 T 62.0 x 10-6 32.0 x 10-6.25 sec 29A 15 C 45.0 x 10-6 32.0 x 10-6.2 sec 29B 15 T 14.0 x 10-6 10.0 x 10-6.2 sec 29C 15 C 17.0 x 10-6 9.0 x 10-6.2 sec 4-A-2-c 8,11,20 all 33, gage43 Bending predominates - impacts not discernible. 55A.167 4-A-3-a 8 10 C 84.0 x 1o-6 61.3 x 10-6.4 sec 11 7 T 70.9 x 1O-6 35.4 x o-6.3 sec 20 12 T 38.0 x 10o 24.0 x 10-6.4 sec 33 10 C 37.4 x lo6 34.6 x lo0-6.3 sec 43 8 T 40.0 x 10-6 20.0 x 10-6.5 sec 55A 5 C 38.0 x 10-6 14.o x 10-6.5 sec 67 8 T 69.0 x 10-6 38.0 x 10-6.6 sec 79A 10 C 54.0 x 10-6 34.o x 10-6.6 sec 86 5 T 30.0 x 106 25.0 x 10-6,5 sec 29A 9 C 130.0 x 10-6 100.0 x 10i6.5 sec 29B 9 T 42.0 x 10-6 27.0 x 106.5 sec 29C 9 C 38.0 x 10-6 26.0 x 10-6.5 sec 4-A-3-b 8 9 C 84.0 x 10-6 61.3 x 10-6.3 sec 11 12 T 70.0 x 10o6 35.0 x lo-.25 sec 20 10 T 38.0 x 10-6 24.0 x o10-6.25 sec 35 6 C 57.0 x l0-6 34.0 x l0-6.25 sec 43 8 T 40.0x 10-6 20.0 x 10-6.3 sec 55A 10 T 38.ox i0o6 13.0 x 10-6.5 sec 67 6 T 69.0 x 10-6 38.o x 10-6.75 sec 79A 3 C 54.0x 1i-6 35.0 x 10-6.5 sec 86 1 T 30.6 x 10-6 1.0 sec 29A 4 C 183.0 x 10-6 99.0 x a0-6 29B 4 T 61.0 x 10-6 31.0 x 10-6 29C 4 C 36.0 x i0-6 26.0 x 10-6

Bc-21 TABLE I, continued Run No. of Maximum Average Average No. Gage Impacts Type Amplitude Amplitude Duration 4-A-3-c all Bending frequencies predominate; no impacts readily gages discernible. 4-B-l-a 8 6 C 5.76 x 10-6 3.7 x 0-6 1.0 sec 11 3 T 4.2 x 10-6 3 5 x 10-6 1.0 sec 20 4 T 3.8 x 10-6 2.5 x 10-6 1.0 sec 33 8 C 24.0 x 10-6 12.0 x 10-6 1.0 sec 43 4 T 2.7 x 10-6 5.4 x 10-6 1.0 sec 55A 7 C 27.0 x 10-6 10.0 x 10-6 1.0 sec 67 8 T 7.2 x 10-6 1.0 xse 79A 5 C 12.0 x 10-6 8.o x 10-6 1.0 sec 86 2 T 6.2 x i0r6 5.2 x 10-6 1.0 sec 29A 8 C 36.0 x 10-6 26.0 x 10-6.75 sec 29B 8 T 18.0 x 10-6 11.0 x 10-6.75 sec 29C 8 C 16.0 x 10-6 8.o x 10-6.75 sec 4-B-I-b 8 11 Bending only. 20 33 4 T 11.5 x 106 6.2 x lO-6.5 sec 43 6 T 17.6 x 10-6 13.6 x 10c6.5 sec 55A 7 T 43.0 x 10-6 16.0 x 10-6.5 sec 67 4 T 24.0x i-6 14.o x l-c6.5 sec 79A 4 C 4o.6 x 10-6 24.0 x 10-6.5 sec 86 4 T 37.0 x 10-6 23.0 x 10-6.5 se 29A 2 C 58.o x 10-6 36.0 x 10-6.5 sec 29B 2 T 19.0 x 10-6 14.0 x lo-6.5 sec 29C 2 C 24.0 x 10-6 16.0 x 10-6.5 sec 4-B-1-c 8 4 C 24.0 x 10-6 17.0 x iO-6.25 sec 11 3 T 21.0 x 10- 15.0 x 10-6.25 sec 20 5 T 53.0 x 10-6 25.0 x 10-6.25 sec 33.4 C 43.0 x 10-6 27.0 x 10-6.25 sec 43 9 T 51.0 x 10-6 22.0 x 10-6.25 sec 55A 7 T 50.0 x lO-6 21.0 x 0 x 6.25 sec 67 8 T 68.0 x 10-6 34.0 x 106.25 sec 79A 5 C 64.o x 10-6 39.0 x 10-6.25 sec 86 7 T 34.0 x 10-6 24.0 x 10-6.25 sec 29A 6 C 71.0 x 10-6 56.0 x 10-6.25 sec 29B 6 T 29.0 x 10-6 24.0 x 10-6.25 sec 29C 6 C 18.0 x 10-6 8.0 x 10-6.25 sec 4-B-2-a 8,11,20 No record. 6 55A 7 T 20.5 x iO6 10.0 x O-60.5 sec 67 6 T 34.0 x 106 14.0 x 10-6.5 sec 79A 7 C 21.0x 10-6 13.0 x 10-6.5 sec 86 3T 1 4.0 x 1 0-6 10.0 x 10-6.05 sec

Bc-22 TABLE I, continued Run No. of Maximum Average Average No. Gage Impacts Type Amplitude Amplitude Duration 4-B-2-a 29A 3 C 22.0 x 10-6 10.Ox 10-6. sec 29B 3 T 16.0 x 10-6 8.0 x 10-6.5 sec 29C 3 C 18.0 x 10-6 8.0 x 10-6.5 sec 4-B-2-b 8 1 C 23.3 x 10-6.5 sec 11 1 T 18.8 x 10-6.5 sec 20 2 T 24.3 x 10-6 15.0 1065 sec 33 2 C 26.0 x 10-6 18.0 x 10-6.5 sec 43 4 T 47.0x 106 32.0x 10-6.0 sec 55A 3 C 30.0 x 10-6 17.0 x10-6.5 sec 67 4 T 33.0 x l 16.0 x 10-6.5 sec 79A 6 C 36.o x 10-6 23.0 x 10-6.5 sec 86 3 T 36.0 x 10-6 24.0 x 10-6.5 sec 29A 1 C 25.0 x 10-6.5 sec 29B 1 T 22.0 x 10-6.5 sec 29C 1 C 20.0 x 10-6.5 sec 4-B-2-c 8 2 C 25.0 x 1010-.25 sec 11 1 T 22.0 x 10-6.25 sec 20 Bending predominates; no impacts discernible. 33 43 4 T 47.0 x 10-6 36.0 x 10-6.5 sec 55A 3 C 16.0 x 10-6 14.0 x 10-6.25 sec 67 Bending seems to predominate. 6 79A 2 C 44.0 x 10- 34.0 x 10-6.2 sec 86 3 T 44.0 x lo-6 22.0 x 10-6.25 sec 29A 29B Bending seems to predominate. 29C 4-B-3-a 8 4 C 19.4 x lO-6 9.0 x 10-6.25 sec 11 3 T 8. x 10-6 5.0 x 6.25 sec 20 2 T 10.0 x 10-6 6.0 x 10-6.2 sec 33 8 C 17.0 x 10-6 12.0 x 10-6.5 sec 43 10 T 14.0 x 10-6 8.0 x 10-6.5 sec 55A 6 T Half tension and half compression. 55A 4 C 67 7 T 16.5 x 10-6 11.0 x 10-6.33 sec 79A 8 C 21.6 x 10-6 13.0 x 10-6.5 sec 86 5 T 18.0 x 10-6 10.0 x 10-6.5 sec 29A 5 C 23.0 x 10-6 13.0 x 10-6.25 sec 29B 5 T 22.0 x 10-6 11.0 x 10-6.25 sec 29C 5 C 21.0 x 10-6 9.0 x 10-6.25 sec

Bc-23 TABLE I, continued Run No. of Maximum Average Average No. Gage Impacts Type Amplitude Amplitude Duration 4-B-3-b 8 Bending seems to dominate 11 9 T 14.0 x 10- 11.0 x 0-6.1 sec 20 7 T 22.0 x 10-6 14.0 x 10-6.1 sec ~33 3 C 11.0 x 10-6 10.0 x 10-6.1 sec 43 12 10-6.- 6 43 1I4 T 15.0 x 0-6 12.0 x 106.2 sec 55A 7 C 17.0 x 10- 9.0 x 10-6.2 sec 67 6 T 40.0 x 190 x 0-6.13 sec 79A 6 C 33.0 x 10-6 24.0 x 1O-6.2 sec 86 5 T 17.0 x l0-6 12.0 x 10-6.2 sec 29A 29B Bending seems to predominate. 29C 4-B-3-c 8 Bending seems to predominate. 11 2 T 13.0 x 10 10.0 x 106.13 sec 20 5 T 50.0 x 1o-6 34.0 x 10-6.13 sec 33 4 C 24.0 x 10-6. 18.0 x 10-6.13 sec 43 6 T 27.0 x 10-6 18.0 x l0-6.13 sec 55A Both tension and compression. 67 4 T 25.0 x 10- 16.0 x i0-6.1 sec 79A 8 C 41.0 x 10-6 33.0 x l0-6.2 sec 86 2 T 45.o 0x 106 30.0 x 0 6.1 sec 29A 3 C 29.0 x 10-6 15.0 x l0-6.1 sec 29B 3 T 20.0 x 10-6 14.o x 10-6.1 sec 29C 3 C 13.0 x 10-6 9.0 x 10-6.1 sec 4-C-l-a- No impacts show up. 4-C-l-b 8 3 C 14.0 x 10-6 11.0 x 10-6.2 sec 11 2 T 8.0 x 10-6 7.0 x 10-6.33 sec 20 None apparent. 33 43 Gage drifts all over the lot; no good? 55A 3 C 11.0 x 10-6 7.0 x 10-6.33 sec 679 Do not show any definite impacts; 7A86 bending is quite evident. 86 29A 29B No true impacts. 29C 4-C-l-c It would be difficult to determine impacts on this test, for the average amplitude of vibration is large and there are no bunches which.definitely look like impacts.

Bc-24 TABLE I, concluded Run No. of Maximum Average Average No. Gage Impacts Type Amplitude Amplitude Duration 4-C-2-a 8 2 C 6.0 x 0o-6 4.0 x 10-6.2 sec 11 1 T 3.0 x 10-6.33 sec 20 1 T 2.0 x 10-6 1.0 sec 33 4 C 8.0 x 10-6 5.0 x 10-6 1.0 sec 43 1 T 6.0 x 10-6.5 sec 55A 6 "7 None. 79A 4 C 8.0 x 10-6 3.0 x 10-6.75 sec 86 3 T 7.0 x 10-6 4.0 x 10-6.5 sec 29A 4 C 2.0 x 10-6 2.0 x 10-6.5 sec 29B 4 C 6.0o x 10-6 4.0 x 10-6.5 sec 29C 4 T 4.0 x 10-6 2.0 x 10-6.5 sec 4-.C-2-b 8 3 C 18.0 x l0-6 10.0 x 10-6.25 sec 11 None discernible. 20 2 T 100 x 0-o6 6.0o x 10-6.25 sec 33 1 C 28.0 x 10-6.2 sec 43 1 T 13.0 x ]_06.2 sec 55A 2 T 13 0 x 0 10.0 x 10-6.2 sec 67 3 T 19.0 x 10-6 11.0 x 10-6.2 sec 79A 3 C 34.o x ]0-6 18.0 x 10-6.2 sec 86 2 T 17.0 x 1~ 14o0 x 10-6.2 sec 29A 4 C 32.0 x 10- 12.0 x 10-.2 sec 29B 4 T 31.0 x 10-6 10r0 x 10-6.2 sec 29C 4 C 24.0 x 10-6 60o x 10-6.2 sec 4-C-2-c No definite impact shocks on any of these gages. 4-C-3-a There are no definite impacts noticeable on these gages save for 79A 3 C 7.6 x 10-6 5.0 x 10-6 1.0 sec 86 2 T 9.0 x 10-6 6.0 x 10-6 1.0 sec 4-C-3-b 8 None. 11 2 T 10.0 x 10-6 7.0 x 10-6.2 sec 20 4 T 53.6 x 10-6 14.0 x 10-6.2 sec 33 5 C 21.4 x 1o-6 10.0 x 10-6.2 sec 43 Gage wanders; no good. 55A 6 C 18.0 x 10- 6.25 sec 67 2 T 12.0 x 10-6 8.0o x 10-6.25 sec 79A 2 23.0 x ]0o-6 14.25 sec 86 None discernible. 29A 5 C 15.0 x.0-6 7.0 x 1o-6.2 sec 29B 5 T 13.0 x 10-6 9.0 x.0-6.2 sec 29C 5 C 8.0 x o10-6 5.0 x o10-6.2 sec 4-C-3-c No impacts are discernible on the gages; amplitlude of vibration is pretty large.

Bc-25 TABLE II NUVIBER OF IMPACTS FOR EACH GAGE DURING EACH RUN Run Gage Number No. 8 11 20 33 43 55 67 79 86 29 4A I a 14 14 6 9 6 5 8 5 — 5 b 59 46 36 31 32 22 19 17 -- 15 c -.- -- -10 - - -- 4A II a 7 5 6 5 6 6 7 6 5 3 b 6 12 8 12 9 7 11 5 3 -- 4A IIIa 10 7 '12 10 8 5 8 10 5 9 b 9 12 10 6 8 10 6 3 1 4 4B I a 6 3 4 8 4 7 8 5 2 8 b -- - - 4 6 7 4 4 4 6 c 4 3 5 4 9 7 8 5 7 6 4B II a -- -- -- -- -- 7 6 7 3 3 b 1 1 2 2 4 3 4 6 3 1 c 2 1 -- -- 4 3 -- 2 3 4B IIIa 4 3 2 8 10? 7 8 5 5 b -- 9 7 3 14 7 6 5 -- c -- 2 5 4 6? 4 8 2 3 4C I a -- -- -- -- -- -- -- -- b 3 2 -- -- -- 3... c - -- - -- - - -- 4C II a 2 1 1 4 1 -- -- 4 3 4 b 3 -- 2 1 1 2 3 3 2 4 c -- -.. - - - -. _ _.- _-_- - _.4C IIIa - -- --. — _ 3 2 -- b -- 2 4 -- 6 2 2 -- 5 C _- _- _- _- _- _- _- _- _- _ ____

Bc - 26 | ENGINEERING RESEARCH INSTITUTE |cI2 6UNIVERSITY OF MICHIGAN under, is given a higher downward velocity, perhaps enough to overcome the flotation force long enough to establish a water shield between the hull and the submerging ice. 2) The frequency of impacts on each gage is plotted in Figure 9. The distribution along the hull is too irregular to draw any conclusions except that gage No. 8 near the bow does receive more impacts than gage No. 86 located near the midship section. Even this is true only for the overall total and for draft condition A (15 feet 3 inches forward, 18 feet 9 inches aft). The effect of draft is more noticeable. Draft Submergence of Gages Total Number Fwd. Aft below Water Line of Impacts A 15'3" 18'9" 1/2' to 1-1/2' 655 B 17'0" 17'2" 2-1/2' 363 C 18'8 19'0" 3-1/2' to 4' 80 Under draft condition A the gages were very near to the ice level (7-inch to 18-inch thick ice). Those impacts which did occur were easily registered. The gages are very sensitive to the location of the impact. A point load located at a vertical distance of + 16 inches above or below the gage will register only 5 percent as much signal as the same load would give right at the gage. In draft condition C there probably were just as heavy impacts as in draft condition A, but they were too far above the gage to induce signals. It can be surmised that no impacts occur at all at a distance of 4-1/2 to 5 feet below the water line with ice no thicker than 18 inches. At that depth the traffic pattern of the ice blocks has been arranged parallel to the

Bc-27 200 FIG. 9 150 TOTAL OF ALL RUNS A, B,C. Co o 100 A z 5 50 8 II 20 29 33 43 55 67 79 86 B.c GAGE NUMBER

~0 ro TABLE III Strains, microinches/inch, on each gage leg xCrx ma un, ~ 29A 29B 29C Run E ~~~s ~lbs/in2 lbs/in2 lbs/in2 lbs/in2 4AII a - 30.7 + 7.8 - 10 - 1120 - 626 660 - 1578 4A IIIa -130.0 + 42.0 - 38 - 4650 -2540 2810 - 6595 4BI a - 36.0 +18.0 - 16 - 1350 -891 1016 - 2160 4BI b - 58.0 +19.0 - 24 - 2140 -1350 1385 - 3185 4B I c - 71.0 + 29.0 - 18 - 2510 -1288 1700 - 3709 4BII a - 22.0 +16.0 - 18 * - 890 - 825 831 - 1690 4BIIb - 25.0 +22.0 - 20 * - 1023 - 925 1027 - 1999 4B IIIa - 23.0 + 22.0 - 21 *- - 925 1016 - 1956 4B IIIc - 29.0 + 20.0 - 13 - o1088 - 726 948 - 1874 4CII a 2.0 - 6.o + 4?? - 1287 -1122 1312 - 2522 4cIIb - 32.0 +31.0 -24 * - 495 - 396 543 - 992 4A IIIb -183.0 + 61.0 - 36 - 64oo -3000 -1119 - 6735

ENGINEERING RESEARCH INSTITUTE Bc-29 -UNIVERSITY OF MICHIGAN ship' s hull. 3) The duration of the impact signals ran from.1 second to 1.8 seconds, with no apparent correlation with any of the set conditions. These durations depended on the size of the ice block encountered; the very long durations might have been caused by the ice being jambed in the surrounding ice instead of free. 4) An analysis of the rosette gage No. 29 is of interest. The highest impact strain to occur in the whole test period happened to be on this gage. In Table III most of the loads were not centered on the gage location. Those records marked by the * come closest to this condition. All these stresses are low. The largest stress encountered is - 6735 lbs/in2. If leg A had been read alone, it would have indicated a maximum stress of - 6400 lbs/in2, only 95 percent of the true maximum stress. The actual maximum stress for Run 4AIIIb was probably greater than - 6755 lbs/in2 Probably the largest impact stress encountered by the plating was less than 10,000 lbs/in2.

APPENDIX C, SECTION a PHYSICAL PROPERTIES OF LAKE ICE WITH PARTICUIAR REFERENCE TO STRENGTH BY JAMES T. WIISON JOEN MI. HORETH June, 1949

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX C, SECTION a PHYSICAL PROPERTIES OF IAKE ICE WITH PARTICULAR REFERENCE TO STREN(GTH INTRODUCTION This report details the experimental procedures followed and the results obtained for a number of tests made on natural and "artificial" lake ice during the fall and winter of 1947-48. The work was done under the auspices of the Department of Engineering Research, University of Michigan, as part of a project conducted for the United States Coast Guard. As the main project dealt with the problems of ice-breaking, the ice properties studied and reported here were those thought most likely to be important in this connection. Primary emphasis was placed on tests to determine the tensile strengt and the shear strength of lake ice and to determine the possible effects of temperature and crystallinity on these properties. A search of the existing literature showed that while much was known about the elastic properties of single ice crystals and natural ice sheets, less information was available on the strength of natural lake and river ice. (See references 1, 2 and 3) Furthermore, there was little available information on the crystal structure of such natural ice. It was decided to undertake tests on lake ice or on artificial ice so frozen as to closely approximate natural lake ice, and at the same time, search all the

ENGINEERING RESEARCH INSTITUTE Ca-2 UNIVERSITY OF MICHIGAN available literature and compile a detailed bibliography and summary of information for lake ice. The bibliography and summaries are presented in a separate report. The results of the various tests are given here. TfPES OF TESTS Tests were made to determine the tensile strength and shear strength of lake ice. No tests of compressive or crushing strength were made. A considerable number of such tests have been reported in the literature, and furthermore, information on crushing strength did not seem to be so necessary for the problem on hand. Some tests were made to determine the coefficient of friction between steel and ice. These tests were rather crude and gave only approximate results; however, they are in good agreement with some extensive Russian data that came to our attention late in the course of the project. Values of tensile strength were determined by breaking bars of ice in pure bending. Shear strengths were determined by subjecting short beams to pure shear. It is well known that ice crystallizes in the hexagonal system and that the ice sheets which form on natural water bodies are usually composed of crystals with their optic axes vertical, or nearly so. In the bending tests, the beams were arranged so that the crystals were vertical, transverse to the long axis in the beam and parallel to the applied forces. In the shearing tests the applied forces were also parallel to the crystal axes. If 'the axes of the crystals were horizontal transverse to the axis of the beam,

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ca-3 and at right angles to the applied forces, the strength would be expected to be the same, and what work had been done previously by other investigators indicated this to be the case. (See references 1 and 2.) If the crystals were arranged with their axes parallel to the length of the beam, the results would be expected to be quite different. No tests were made for this orientation because it did not seem that the results would be of interest to the main project. Little information was available on the size of the crystals to be expected in lake ice. Such information was quite essential as it was desirable that the test specimens should be several times larger than the crystal dimensions. Careful attention was paid to the size as well as the orientation of the component crystals. A number of samples were studied in polarized light, and the crystal dimensions measured. The results of these studies are presented in Appendix A, of this Appendix C, Section a. TEST APPARATUS AND PROCEIXRE Most of the tests were performed in a refrigerator room in the East Engineering Building on the-campus of the University of Michigan. The special specimens of artificial ice were also frozen in this room. The temperatures in the room are quite constant, and the fluctuations amounting to a few degrees and resulting from the presence of the experimenters and the attendant opening and shutting of the door, were not great enough to affect any of the results. A series of tests were conducted January 27-30, 1948, on the U. S. Coast Guard Cutter Mackinaw. Tests on the Mackinaw were carried out on the deck where the temperatures never rose above freezing and were usually around 20~F.

ENGINEERING RESEARCH INSTITUTE Ca-4 UNIVERSITY OF MICHIGAN Most of the testing done at Ann Arbor was done at either -9~F or at 32~F. A few shear specimens were tested at 2~F. Throughout most of the tests the refrigerator room was maintained constantly at -9~F. The testing equipment was left in the refrigerator room so that it would always be at the same temperature as the specimens tested. It was inconvenient to raise the temperature of the cold room to 32~F. In order to test specimens at this temperature, specimens were taken out of the cold room and allowed to warm up until the specimen began to show slight effects of melting. This usually took 15 to 30 minutes. Thermometers inserted into some of the specimens showed that even the larger specimens were at a temperature of 30 to 320F by the time the outer surface began to melt noticeably. For testing, the warmed specimens were taken back into the refrigerator room and tested immediately. Usually less than three minutes elapsed between the time the specimens were taken into the room and the final failure of the specimens. In the section on results, these specimens are correlated with a temperature of 320F. It might be more precise to correlate them with a temperature of about 300?, but the temperature effects are so small that this would make no change in any of the results. Bending Tests The special testing machine, built for these experiments, is shown in the photograph in Plate 1. Two 4-inch I-beams, approximately three feet in length, were held one above the other by four adjustable tie rods. The The tie rods were 5/8 inch in diameter and about three feet long. They were threaded for a considerable part of their length to allow for adjustment. This made a much stronger structure than was needed; however, at the time it

Ca-5 PLATE 1 TESTING MACHINE FOR ICE SPECIMENS

ENGINEERING RESEARCH INSTITUTE Ca-6 UNIVERSITY OF MICHIGAN was constructed it was thought that some of the tests might involve measuring the deflection of the test beam. A considerably lighter construction would have been particularly desirable for the tests that were conducted on the Mackinaw. Plate 1 shows the equipment set up for bending tests. Two hardwood blocks with rounded lower surfaces were attached to the lower surface of the upper I-beam. A hydraulic Jack was attached to the upper surface of the lower I-beam. Mounted on the Jack was a crosshead carrying two wooden blocks with rounded upper surfaces. The test beams were rested on the blocks on the crosshead, and then forced upward against the two blocks on the upper I-beam. In the setup shown in Plate 1, the upper blocks are spaced 26 inches apart, and the blocks on the crosshead are spaced 10 inches apart. These spacings were used for tests on beams about 30 inches in length. Smaller spacings of about 14 inches for the upper blocks and 5 inches for the lower blocks were used for a number of tests on smaller beams. The test beams were of rectangular cross section, and except for the special tests conducted on the Mackinaw, the height and width were approximately the same. The beams used with the wider spacing were from 2-1/2 to 4 inches wide, from 3 to 6 inches high, and about 30 inches long. Beams used with smaller spacing were usually about 2-1/2 inches square and about 17 inches in length. In the tests performed on the Mackinaw, it was thought desirable to try to obtain representative values for the full thickness of the ice sheet, and consequently, some of the beams were about one foot high. The preparation of the specimens is described in more detail below.

ENGINEERING RESEARCH INSTITUTE C. UNIVERSITY OF MICHIGAN Ca-7 The force applied to the beams was measured by means of a gauge which measured the oil pressure on the ram of the hydraulic Jack. This method seemed precise enough, considering that the individual ice specimens varied so much in strength. The Jack and gauge were calibrated against a standard test machine. Calibrations were made at 70'F and at -9~F. It was feared that the low temperatures might so80 increase the viscosity of the oil in the hydraulic jack as to make unreliable the measurements at low temperatures and to avoid this, the oil in the Jack was thinned with gasoline. The experience gained during the tests indicated that this was probably an unnecessary precaution. It is well known that ice behaves plastically for slowly-applied stresses, and it is known also that at low temperatures ice becomes extremely brittle. Consequently, it was realized that the strength values obtained might depend upon the rate of stress application. However, work done by Brown (see reference 1) indicated that over quite a wide range, the rupture modulus did not seem to be a function of the rate of application of stress. Some work done during the course of the experiment described here indicated the same result. In the interest of uniformity, an attempt was made to carry each bend test to rupture in from one to three minutes. It was thought that this rate was slow enough so that there would be no danger of snapping the ice due to a sudden application of stress, and at the same time was slow enough so that there would not be any great amount of plastic deformation before rupture. Several bending tests were performed, in which the speciment was loaded at a slow rate. These specimens were loaded with increments of force equal to about one-quarter of the estimated force necessary

ENGINEERING RESEARCH INSTITUTE Ca-8 UNIVERSITY OF MICHIGAN for rupture. A ten- to fifteen-minute wait was made between each loading. The results for these tests did not differ significantly from the results obtained when the specimens were loaded to failure in from one to three minutes. The specimens usually broke with a clean tension break somewhere between the two blocks on the crosshead. Plate 2 shows a photograph of the broken surfaces of one of the specimens. Occasionally, the break occurred directly above one of the blocks on the crosshead. In no case did a specimen fail by crushing at a support or in the compressed side of the beam. Each beam was carefully measured before it was broken and the broken cross section was usually remeasured afterward. The measurements were made to the nearest.05 inch. Considering the irregularities described in the section on the preparation of specimens, greater accuracy of measurements did not seem necessary. The experiments were usually performed with two operators. While one operated the hydraulic Jack, the other observed the gauge and read the pressure at the instant of failure. The rupture modulus (tensile strength) was calculated for each specimen by the formula Tensile strength (psi) = 2 F l 2 abZ where F is the force exerted by the Jack, 1 is the spacing of the upper blocks, s is the spacing of the lower blocks, a is the width of the beam, and b is the height of the beam. The bearing blocks were so spaced in proportion to the dimensions of the test beam that the tensile stress in the span between the two central bearing blocks should be exceeded before any part of the beam was subjected to a shear stress equalling its shear strength. This condition will be fulfilled if

Ca-9 PLATE 2 BROKEN SURFACES OF TEST BLOCKS

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ca-10 k< b b where k is the ratio of the tensile strength to the shear strength. Shear Tests For the shear tests a special shearing tool was constructed. It was used in the testing machine described above. Essentially, it consisted of three hardwood blocks faced with plates of 1/4-inch brass. Two of the blocks were fastened in a box-like structure and separated by about three inches. The third block was Just wide enough to slide freely between them. The two fixed blocks, in their supporting structure, could be attached to either the upper or lower I-beam, and the third block attached to the Jack. Short test specimens of rectangular cross section were rested on the fixed blocks and broken by forcing the movable block up against them. A drawing of the shearing tool is shown in Figure 1. The test specimens were usually 2 to 2-1/2 inches square, and 5 or 6 inches long. The same general procedure was followed for the shear tests as for the bending tests. The Jack was worked by one operator while another read the gauge. Each test was carried to completion in from one to three minutes. In many cases, the specimens broke cleanly, with two parallel breaks —the central section being "punched" out. In some cases, the central section was badly shattered. The shear strengths were calculated by the formula where F s1 the force applied by the Jack, and A is the cross section of the specimen.

o00o; foIvaES I I III I 3A u. Z I~~~~~~~~T8

ENGINEERING RESEARCH INSTITUTE Ca_-.2 UNIVERSITY OF MICHIGAN Preparation of Specimens It was necessary to start the tests before specimens of natural lake ice were available. "Artificial" lake ice was prepared by freezing ice in special containers. Various attempts to prepare specimens by freezing ice in wood trays and molds met with failure because of the expansion on freezing. The ice would have an irregular upper surface, usually with one or more open cracks, and an irregular crystal orientation. Two special containers were constructed and used with complete success. The larger container consisted of a vertical-sided wooden tray about 2 feet by 4 feet by 7 inches deep. The inner walls of this tray were padded with about three inches of excelsior. An inner container of thin sheet rubber rested against the bottom of the wooden tray and against the excelsior-padded sides. The expansion was taken up by the excelsior padding so that the specimen frozen was uncracked and undistorted. Plate 3 shows a photograph of the larger tray. The handles shown frozen in the ice were to facilitate removal of the ice cake from the tray. Another tray about half as large and lined with sponge rubber gave equally good results. All of the specimens of artificial ice were frozen at a temperature of about -90F. The water was undisturbed and the air in the room quiet during the period of freezing. The resulting ice cake consisted of long crystals about 1/2 inch in diameter and extending from the top to the bottom of the cake. The optic axes of the crystals were vertical. It was originally thought that considerable experimentation might have to be done in order to obtain specimens that had crystals of the proper size; however, the conditions described produced specimens that had crystals about the sage size as those in the ice obtained from Lake Michigar during the expedition on the Mackinaw. I

Ca-13 PLATE 3 TRAY FOR FREEZING SPECIMENS

ENGINEERING RESEARCH INSTITUTE Ca-14 UNIVERSITY OF MICHIGAN The shear test specimens were prepared from the artifical cake by sawing and shaping with ordinary carpenter tools. Most of the specimen preparation was done in a temperature of -9~F. A somewhat higher temperature would have been more desirable as ice is much less brittle near its freezing point, but it was not desirable to raise the temperature of the test room because of other experiments that were in progress. No particular difficulty was encountered in preparing the specimens. Some of the first specimens were prepared with a saw that had 10 teeth to the inch, but most;of them were cut with a saw having 5-1/2 teeth to the inch. The finer toothed saw gave a somewhat better surface but because of the slow rate of sawing, and perhaps because the teeth clogged more easily, it was more difficult to produce a plane surface. In most cases, the sawn surfaces were not further smoothed, the irregularities being so much smaller than the crystal size that it seemed unlikely they would weaken the specimen. The larger ice beams for bending tests were usually sawed out so that the upper and lower surfaces of the original ice cake were surfaces of the finished beam. The smaller beams and the shear test specimens were sawed out in special miter boxes. In earlier work done by Brown and Finlayson (references 1 and 2) the specimens were apparently very carefully smoothed by such operations as planing, sanding and milling. If such operations are carried on at temperatures around 30'F, very smooth surfaces can be produced. However, it seems likely that there is considerable danger of causing a number of small cracks in the specimen by such operations. The results they obtained on such smooth specimens scatter at least as widely as those reported here for smoothly sawn specimens. It seems likely that the desirable effects of polishing or smoothing y be offset by interal cracks set up during the smoothing. It

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ca-15 was thought preferable to work on the specimens as little as possible in order not to damage them. Specimens of ice from the Mackinac Straits region were obtained January 27 through 30, during a cruise on the U. S. Coast Guard Cutter Mackinaw. These specimens were sawed out of large blocks broken out as the ship made a lane through the ice. These large blocks were hoisted on deck for the preparation of the specimens. The original blocks and the final specimens were examined very carefully for any cracks or flaws that had been produced as a result of the action of the ice breaker. Strength tests were made on many of these specimens while still on the Mackinaw and a number of blocks from these specimens were brought back to Ann Arbor for examination. Table I is a summary of the information obtained from the three most studied specimens. TEST RESULTS Bending Tests The results obtained from the bending tests are suimarized in Table II and Figure 2. The only previous work thought to be directly comparable to the bending tests reported here was done by Brown (see reference 1). The data obtained by Brown are summarized in Table III and are shown, together with the data reported here, in Figure 2. Brown's results were from ice cut from the St. Lawrence River near Montreal. His tests differed from those reported here only in that he used several different rates of loading. However, he found no systematic effect of loading rate on the strength. He also tested beams in which the crystals were horizontal and transverse to both the axis of the beam and the applied force. None of his data obtained from such beams are presented here; however, the results obtained were in close agreement

TABLE I LAKE MICHIGAN ICE TESTED ON USOGO MACKINAW Strength Specimen Ice Thickness Tensile Shear Observed Structure in Polarized Light 1 12 including 181 77 Upper third: Cloudy and with many air holes. Grades down1 of cloudy (1 test) (3 tests) ward to clear ice with vertical prismatic ice on top crystals with diameters up to 0.3 inch. Middle third: Vertical prismatic crystals. Largest crystals 3/4 inch in diameter. Average diameter about 1/2 inch. A "sheared" zone shows in one section. Bottom third: Vertical prismatic crystals up to about 1 inch in diameter. (See Plate 4a) 2 12 including 167 71 Upper third: Cloudy with many air holes. Grades downard to 2 of cloudy (2 tests) (6 teats) ice that is cloudy without macroscopic air ice on top holes. (See Plate Ib) Middle third: As upper third, but grades downward into ice with vertical prismatic crystals up to 1 inch in diameter. (See Plate 4c) Bottom third: Vertical prismatic crystals up to 2 inches in diameter. 10-1/4 in- 279 79 Bottom half: Vertical prismatic crystals-with average diacluding 3/4 (4 tests) (4 tests) meter increasing from about 1/2 inch in upper of cloudy ice part to about 1-1/2 inches in the lower part. on top (See Plate- 4d)

Ca-17 TABLE II BENDING TESTS (TENSILE STRENGTH) "Artificial" Lake Ice Frozen and Tested Ann Arbor 32.F -9 F psi psi 126 226 133 226 142 231 154 248 167 256 168 261 190 273 191 286 204 296 240 266 Average 180 256 Lake Michigan Ice Collected from and Tested on the U.S.C.G.C. Mackinaw, Temperature 120F-20~F Description Tensile Strength (See Table I) psi Top 7" of Spec. 1 - - - - - - - - - - - - 181 Cloudy ice up. Top 7" of Spec. 2 - ----------- 151 Cloudy ice up Center section of Spec. 2 - - - - - - - - - - - - 183 Almost full section of Spec. 3. Cloudy ice up - - - - - - - - - - - - - 260 Top half of Spec. 3 - - - - - - - 241 Bottom half of Spec. 3 - - - - - - - - - - - - - - 270 Top third of Spec. 3 - - - - - - - - - - - - - - - 344 Cloudy ice down Average 233

300 o?. 0 250 — 5 1 T I I g 1 2~~~~~~~t40 Co A 200 A CT 150 Co RESULTS OF BENDING TESTS c 00 0) Legend oAnn Arbor 50,St. Lawrence oLake Michigan 0 -320 3~0 250 200 150 100 50 0" _50 Temperature- OF FIGURE 2

Ca-19 TABLE III BENDING TESTS (TENSILE STRENGTH), ST. LAWRENCE RIVER ICE (After Brown, see reference 1) 29?F 15 OF psi psi 126 177 133 178 1311~ 191 143 201 158 202 177 203 193 205 197 209 200 222 210 230 229 237 306 311 Average 184 214

ENGINEERING RESEARCH INSTITUTE Ca-20 UNIVERSITY OF MICHIGAN with those which are presented. As can be seen in Tables IIand IIIand Figure 2, there is a large variation from specimen to specimen. This variation is less at low temperature. This is perhaps due to a better bond between the crystals when the specimen is at a low temperature. The values obtained for ice in Lake Michigan are in good agreement with the values obtained for the ice frozen at Ann Arbor and the ice from the St. Lawrence River. It is to be noted that there is no more scatter of the Lalre Michigan values than for the carefully prepared artificial ice. A smooth curve was fitted through the average values at each temperature. The Lake Michigan ice was excluded. This curve indicates the following relationship between tensile strength and temperature: Tensile Strength (psi) = 240 - 1.7 x ~F - 0.01 x (OF)2 No great significance should be attached to the coefficient of the temperature squared. However, an increase in tensile strength of about 3/4 of a per cent per degree Fahrenheit decrease in temperature seems quite real. This is in qualitative agreement with previous-recorded results for compressive strength (see Bibliographic Report). As the average temperature of a sheet of lake ice will rarely be below 20~F, 200 psi is indicated as a good average value to take for the tensile strength of such ice. Most of the ice specimens tested during the cruise on the U. S. Coast Guard Cutter Mackinaw contained one or more bands of cloudy or white ice. There was almost always such a band 1/2 to 2 inches thick in the upper part of the ice sheet, and occasionally there were bands within the sheet. These bands did not seem to have any appreciable effect on the tensile strength of the ice. This may be explainable in terms of the crystal structure. When the ice fails in tension, the failure seems more

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ca-21 likely to occur between the individual ice crystals. The nature of the broken surfaces indicates this to be the case. A typical specimen of the Lake Michigan ice consisted of some 10 inches of solid, blue ice representing the bottom part of the ice sheet. This blue ice was made up of vertical, prismatic ice crystals with an average diameter of about 1/2 inch. Many of the crystals extended vertically for the full 10 inches of the blue ice thickness. On top of the blue ice there was 1/2 to 2 inches of cloudy white ice. This cloudy white ice was composed of very small crystals randomly oriented. The cloudy appearance was due, at least in part, to tiny air bubbles included in the ice and to the smaller randomly oriented crystals. It is thought that the random orientation may have in part made up for the strength deficiencies due to the air inclusion. Several tests were made in which the cloudy ice was in the upper, or tension, side of the test specimen. These specimens seemed to be as strong as those made up completely of the blue ice. As will be shown later, this cloudy ice was considerably weaker in shear. Shear Tests The results of the shear tests are presented in Table IV and Figure 3. The results of some comparable tests reported by Finlayson (see reference 2) are included in Figure 3 and summarized in Table V. Various other similar tests are summarized in the Bibliographic Report, but those by Finlayson seemed the most comparable, in method and scope, to the work reported here. As can be seen in Table IV, variations from specimen to specimen are even greater than for the bending tests. Finlayson reported only averages and states that each value "represents the average of many tests." He

Ca-22 TABLE IV SEEAR TESTS "Artificial" Lake Ice 32~F 22F -10~F psi psi psi 66 58 65 85 59 70 88 70 80 89 78 82 100 88 113 102 91 122 103 104 135 115 112 153 126 116 160 161 120 173 Average 94 90 115 Lake Michigan Ice Collected from and Tested on the U.S.C.G.C. Mackinaw, Temperature 12@F - 20@F psi 641 66e From Spec. No. 1 (See Table I) 92_ 39' Upper, Middle, and Lower Sections 65 * of Ice Sheet (See Spec. No. 2, 88J Table r) 491 Upper, Middle, and Lower Sections 103 of Ice Sheet (See Spec. No. 2, 86j Table I) 821 Upper, Middle (2 spec.) and Lower 81 Sections of Ice Sheet (See Spec. 74 No. 3, Table I) 80o Average 75

Ca-23 TABLE V AVERAGE SEEAR STRENGTH OF RIVER ICE (After Finlayson, see reference 2) Temperature Shear Strength eF ppsi 29 101 28 102 26 98 8 99 5 91 4 94 -3 101

FIGURE 3 150 I I I I 125 0. "75 LO00 50 RESULTS OF SHE A R TESTS Legend 2 Ann Arbor 25 a Manitoba ~Lake Michigan 320 300 250 200 150 100 50 0 - 50 -I0 Temperature-0F

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ca-25 indicates that he found considerable variation in the mean values, with a minimum of 68 psi and a maximum of 353 psi. To facilitate the inclusion of Finlayson's data, Figure 3 shows only averages. The averages for Lake Michigan ice are averages not only for the temperature indicated, but also for representative sections taken from the top, middle, and bottom of the ice sheet. As can be seen from the data, there is no apparent effect of the temperature. The Lake Michigan ice should be excluded from any consideration in this regard. The low values obtained for it can be explained on other grounds. It seems likely that the shear strength of ice must depend on temperature in somewhat the same fashion as the tensile strength. Possibly its temperature coefficient is considerably less and neither our experiments nor Finlayson's were sensitive enough to determine the coefficient. It seems likely that the temperature coefficient of the shear strength might be considerably less. The tensile strength is probably determined largely by the bond between the individual crystals, while the shear strength is probably determined partly by this bond and partly by the way in which the crystals are wedged together. In all the data reported here, the long direction of the prismatic crystals was parallel to the applied force. The crystals are only crudely prismatic, and their faces may depart considerably from parallelism. As a result, there would be a tendency for the crystals to wedge together under the influence of shearing stresses. The shear specimens of Lake Michigan ice that were tested on the Mackinaw were tested in sets, with each set so chosen as to be representative of the ice sheet. The make-up of these sets is shown in Table IV. There was a definite tendency for the cloudy ice to be considerably weaker in shear..._

ENGINEERING RESEARCH INSTITUTE Ca-26 UNIVERSITY OF MICHIGAN No compressive tests were made of the cloudy ice, so it is not certain that this weakness was not the result of crushing; however, most of the cloudy specimens broke with what seemed to be clean, shear fractures with the central portion of the specimen punched out. The results obtained indicate that lake ice has a shear strength of about 100 pounds psi, decreasing to as low as 50 pounds psi if there is a considerable content of cloudy ice. Friction Tests: Two sets of friction tests were made. Both were limited to a consideration of steel on ice. The first set was carried out during the winter of 1947-48 and was designed to obtain values of the coefficient of kinetic friction between steel and ice. The second set was made during the winter of 1948-49 and was designed to determine the coefficient of static friction between steel and ice. A search of the literature (See bibliographic report) furnished a number of values for comparison. In general these values are about 1/10 for kinetic friction and between 1/5 and 1/2 for static friction. There are at least three factors that must be considered in designing friction tests on ice or in evaluating the results. (1) The low melting point of ice may lead to "friction melting" which will furnish a lubricating water film. (2) Stress concentration may cause "pressure melting" and consequent lubrication. (3) The low melting point of ice may allow "welding" to take place between the surfaces under test. These factors may explain some of the inconsistancy in published values, as well as the observations on the increase in the coefficient of kinetic friction with decreasing temperature or decreasing load. Values of kinetic friction were determined by sliding steel blocks down inclined plates of ice. The inclination of the ice surface would be adjusted until a steel block would slide down the surface without acceleration

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ca-27 when once started. The tangent of the angle of inclination is then the coefficient of kinetic friction. A number of tests at -9~F gave values between 0.08 and 0.10 for the coefficient. The inclination method can be used to determine the coefficient of static friction by slowly increasing the inclination of the ice surface with a steel block resting on it and measuring the inclination at the time the block begins to slide. The tangent of this angle of inclination is the coefficient of static friction. This method was tried but consistent values could not be obtained. Slight Jarring of the equipment while the inclination was being increased would cause the block to start prematurely. Trouble was also encountered with the block "freezing down" to the ice. The most satisfactory tests of static friction were carried out by sliding steel plates on the smooth ice of a frozen pond. The ratio of the force necessary to start the plate sliding to the weight of the plate is then the coefficient of static friction. Three sets of values were obtained from these tests: (1) a series of low values of about 1/10, (2) a consistent series indicating a value of very close to 1/4, (3) a series of high values which were as high as 1-1/2. These three sets of values are interpreted as follows. The low values are due to a film of water on the ice from either an earlier test or from the plate having become warm enough in the sun to melt some of the ice. The values of about 1/4 represent the true value of the coefficient of static friction for steel on ice. The high values are the result of the steel plate freezing to the ice surface. For all of the tests the steel was unpainted and unpolished. If frictional melting is of importance for kinetic friction painted steel might show lower values because of the relatively low thermal conductivity of the paint.

ENGINEERING RESEARCH INSTITUTE Ca-28 UNIVERSITY OF MICHIGAN In evaluating the effects of friction on the operation of ice breaking ships it must be kept in mind that the hull of most ships is far from smooth and consequently there must be a considerable amount of "machining" of the ice which will exert a frictional drag. This is a difficult factor to evaluate. The coefficient of friction for metal on snow is greater than for metal on ice. It is difficult to evaluate how much of this effect may be due to deformation of the snow surface.. A thick snow cover is known to impede the forward progress of an ice breaking ship. Much of this effect can probably be attributed to the fact that snow piles up around the bow of the ship and greatly increases the area of contact. This pile of snow must be continually sheared as the ship advances. In summary, it was found that the coefficient of kinetic friction etween smooth but unpolished and unpainted steel plates and ice is about 0.1 t temperatures well below 32@F and that for the same conditions the cofficient of static friction is about 0.25. These values seem to be in substantial agreement with previously determined values.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ca-29 REFERENCES Particular reference should be made to the bibliographic report prepared as part of this project (Bibliography on Ice, Richard Strong). General References 1. Barnes, H. T., Ice Engineering, Renouf Publishing Company, Montreal (1928) 2. Dorsey, N. E. Properties of Ordinary Water Substance, Reinhold Publishing Corporation, New York (1940) Specific References 1. Brown, E. W., Experiments on the Strength of Ice, Report of the Joint Board of Engineers on the St. Lawrence Waterway Project, App. F, pp. 423-453 (1926) 2. Finlayson, J. N., Tests on the Shearing Strength of Ice, Can. Eng., Vol. 53, pp. 101-103 (1927) 3. Weinberg, B., Mechanical Properties of Ice, Assoc. Int. d'Hydrol., Sci. Bull. No. 23, pp. 509-535 (1938)

ENGINEERING RESEARCH INSTITUTE Ca-30 UNIVERSITY OF MICHIGAN APPENDIX A STRUCTURE OF LAKE ICE It is well known that the ice which forms on the surface of natural water bodies is usually composed of prismatic ice crystals oriented with their long axes vertical and at right angles to the surface of refrigeration. Ice crystallizes in the hexagonal system, and the long axes of the prismatic crystals correspond to the optic, or "c", axis. Little information could be found in ice literature about the cross-sectional size or shape of these crystals in lake and river ice. The shape of the cross section might well influence the strength. Also, for the tests described above, it seemed quite desirable that the test specimens be made large enough so that they would include several crystals within the cross section of the test specimens. There are several methods by which the crystal outlines may be made evident. Ice exposed to radiation usually melts somewhat more rapidly along the crystal interfaces, and consequently, the structure of the ice may be made evident by this etching process. Some attempts were made to study the structure of ice in this manner, but some other methods were found more useful. The artificial ice frozen at Ann Arbor was usually at a temperature of about -9~F. When these specimens were removed from the refrigerator room to warmer surroundings, the frost pattern which formed on them made the crystal structure quite evident. This pattern is quite clear if viewed in light that strikes the surface near grazing incidence, but is evident in almost any aspect. However, the most satisfactory method for observing the

Light Box Polaroids Ice Specimen. 'I FIGUTE 4 H ARRAEEIM FOR OBSERVIES CRYSTALIITY

ENGINEERING RESEARCH INSTITUTE Ca-32 UNIVERSITY OF MICHIGAN crystals is to view a section of the ice placed between sheets of polaroid which have their transmission directions crossed at right angles. Figure 4 is a schematic drawing of the apparatus that was used to examine ice specimens in polarized light. Two ten-inch polaroid disks were placed about six inches apart and a holder for ice sections placed between them. A light box containing several small mazda bulbs behind a sheet of opal glass was used to illuminate the polaroid-ice system. Ice sections were examined visually and then photographed with a 35mm camera placed fourteen inches from the ice section. All of the ice sections examined were approximately three-eighths of an inch thick, and were usually about four inches square. All were cut parallel to the surface of the ice sheet. Ice crystallizes in the hexagonal system (optically uniaxial) and consequently, if the optic axis of the crystal is directed towards the observer, the crystal section will appear black when viewed between crossed polaroids. As the optic axis is turned from parallelism with the line of sight, the color of the crystal cross section will shift from black through grey to white when the optic axis makes an angle of about 8 degrees (for ice sections about three-eighths of an inch thick) with the line of sight. By the time the optic axis is off by 11 or 12 degrees, the section will be a reddish orange. This makes it possible to estimate quite accurately the direction of the.optic axes of the crystals. Sections of the "artificial" lake ice frozen at Ann Arbor were examined in polarized light and it was found that in them the component crystals were prismatic with their long axes (optic axes) approximately perpendicular to the surface of refrigeration. The average crystal diameter was

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Caapproximately one-half inch. Only rarely were any of the crystal axes more than ten degrees off from perpendicularity with the refrigeration surface. Five specimens of the Lake Michigan ice encountered during the test on the Mackinaw were brought back to Ann Arbor for study. These specimens were vertical prisms cut out of the ice sheet. These prisms were sectioned parallel to the surface of the ice sheet and each section examined and photographed in polarized light. The observations for those specimens from blocks on which strength tests were made are summarized in Table I. Several of the photographs are reproduced in Plate 4. All of the specimens showed a thin layer of cloudy white ice in the upper section. When examined in polarized light, this zone of cloudy ice was shown to contain a number of air holes and to be made of small crystals that had a random orientation. This zone of random orientation seems to extend below the cloudy ice that is seen when the specimen is examined in ordinary light. Below this zone the Lake Michigan ice was composed of vertical prismatic crystals with an average crystal diameter of approximately three-quarters of an inch. The larger crystals that extended for almost the full thickness of the ice sheet enlarge downward, so that near the bottom of the ice sheet, the average crystal diameter was an inch and one-half or more. The crystals in the natural ice were somewhat more irregular than those in the artificial ice, and in the natural ice there were occasionally "sheared" zones in which the crystals were randomly oriented. It will be noted that specimen No. 3 listed in Table I gave a considerably higher value of tensile strength than specimens No. 1 and 2. Specimens No. 2 and 3 were picked up from the same general ice area, and the difference is probably due to the small number of tests.

Ca-34 a b c d PLATE 4 ICE SECT IONS BETWEEN CROSSED POLAROID x 3/7

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN A number of questions are left unanswered by the observations. All of the ice examined came from a quite uniform ice sheet that had formed rapidly and was made up largely of clear blue ice. It showed very little effect of having grown by the piling of one ice sheet on top of another. In only a few cases were any internal bands of cloudy ice observed. It is not thought that these observations on structure can be generalized to thicker ice sheets, to ice formed more slowly, or to ice sheets that have been in existence fora number of weeks and subjected to the various disturbances caused by winds and currents. No crystals with diameters larger than about two inches were observed; however, it is understood that much larger crystals have been observed in Lake Michigan ice. These larger crystals may form where the ice freezes over a longer period of time, or it is possible that they develop from the smaller crystals later in the history of the ice sheet.

ENGINEERING RESEARCH INSTITUTE Ca-36 UNIVERSITY OF MICHIGAN APPENDIX B SUGGESTIONS FOR FFURTHIER WORK Strength Tests The results to be found in the literature and those reported here indicate quite satisfactorily the range of tensile and shear strength to be expected for lake ice. However, no test has been developed that can be performed easily and quickly to furnish an approximate value of the strength of a particular ice sheet. The tests described here, as well as those outlined in the Bibliographic Report, all require the preparation of special specimens. Experiments with simple tests, such as penetration tests, might show a good correlation with the strength measured by a more precise and orthodox method. Such penetration tests have yielded good results on packed snow in Switzerland. Of the tests reported here, the simplest and quickest is the shear test as it requires a relatively small specimen. A relatively simple apparatus could be constructed for obtaining shear strength under shipboard conditions. From an academic viewpoint, it would be desirable to perform some more precise tests to try to determine whether or not there is a temperature coefficient of the shear strength. This would be necessary if measurements of shear strength were to be used in an estimation of the tensile strength of a particular ice sheet. Friction Tests While there is a considerable amount of Russian literature that reports various friction measurements done in connection with ice breaking tests,

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ca-37 there is still some doubt as to the effective values of the coefficients of static and kinetic friction between steel plates and ice under practical ice breaking conditions. Considerable further work might be done on this phase of the problem. Ice Structure An amazingly small amount of information is available on the structure of lake ice. It would seem desirable to obtain a great deal of information on this subject. Ice specimens could be collected from a number of selected stations at regular intervals throughout an ice season in order to determine changes that probably take place in the ice structure with the passage of time and with increasing ice thickness. Special studies should continue into the period of ice breakup. Such data seem quite desirable if the results presented in this report are to be extrapolated to thicker and older ice sheets. Such structure studies might be compared profitably with observations or aerial photographs made from a low flying airplane or helicopter. Such observations might indicate a relationship between ice structure as seen in small specimens, and the general.structure of the ice field.,! [,, [ [ [.... J, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I

APPENDIX C, SECTION b BIBLIOGRAPHY ON LAKE ICE BY RICHARD STRONG January, 1948

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ABSTRACT A bibliography on lake ice is compiled on the basis of a comprehensive search of the literature. The subject matter of the paper is presented in two parts: first, a complete alphabetical bibliography, and second, a series of individual bibliographies with descriptions and tabulations of data for the following subjects: General Formation and structure of lake ice Strength of ice Ice friction Hardness of ice Elastic constants Viscosity and plasticity of ice Ice breakers and ice breaking Sources of bibliographic material

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN BIBLIOGRAPHY ON LAKE ICE BY: Richard Strong INTRODUCTION The compilation of this bibliography was undertaken at the request of the United States Coast Guard and was financed by them. An intensive search of all pertinent literature was made using as guides the comprehensive bibliographies of Barnes (1928) and Dorsey (1940). Few titles are added to theirs and those which have been added were omitted by Barnes and Dorsey only because of later dates of publication. On the other hand, many of their references are omitted from this bibliography because they do not pertain to lake ice. In general, it is found that very few papers deal directly with lake ice. In many cases it is found difficult to draw a line between articles indirectly concerning lake ice and those concerning borderline subjects such as river, glacier, and sea ice. Often, the exclusion or inclusion of a title is purely arbitrary, although the objective throughout is to include any work which concerns itself directly or indirectly with lake ice in any aspect. Thus, much of H. T. Barnes' work on river ice is omitted. The early controversies of Tyndall, Forbes, Faraday, J. Thomson and W. Thomson (Lord Kelvin) concerning the structure and movement of glaciers are largely ignored. Similarly, later work on glacier ice is included only where it deals with such problems as viscosity and plasticity of ice in general. The problems of sea ice are completely omitted. It is hoped, however, that where borderline fields are somewhat slighted, sufficient references are included to serve as guides to more comprehensive bibliographies. In the organization of this paper, the bibliographic treatment is emphasized. The text serves only to relate and evaluate the references and to present the more important data. Not all references were thought important enough to mention by name. However, these are included in the alphabetical bibliography for the sake of completeness. The writer wishes to acknowledge an indebtedness to Dr. James T. Wilson of the Department of Geology, University of Michigan,for his assistance and encouragement during the preparation of this manuscript.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN BIBLIOGRAPHY ON LAKE ICE AIPHABETICAL BIBLIOGRAPHY (Trans.) American Geophysical Union. Andreev, S. M. (1934?), Experiments of V. N. Pineghin in 1924 on the resistance of river ice (ready for printing, 1934). Probably in Russian. Not examined. Andreev, S. M., and Arnold-Alabieff, V. I. (1937), Tests on the strength of the ice of Neva River and Gulf of Finland during 1930-36 made at the Scientific Meteorological Ice Station of the Hydrometeorological Service of the USSR (computed in 1937). Probably in Russian. Not examined. Andrews, T. (1886), Observations on pure ice and snow: Proc. Roy. Soc. London, vol. 40, pp. 544-549. Arnold-Alabieff, V. I. (1938), Experiments on external friction of ice: Jour. Appl. Physics, vol. 7, pp. 873-878. This is apparently the Russian publication and was not examined. Arnold-Alabieff, V. I. (1938), The external friction of ice: Assoc. Int. d'Hydrol. Sci., Bull. N 23, pp. 563-570. Arnold-Alabieff, V. I. (1937), The visible structure of atmospheric ice and of the ice cover of the water: Meteorologia i Hydrologia, vol. 3, pp. 103-106. Probably in Russian. Not examined. Barnes, H. T. (1906), Ice formation, John Wiley and Sons, Inc., New York. Barnes, H. T. (1910), The orientation of crystals in a flux of heat: Nature, vol. 83, p. 276. Barnes, H. T. (1914), The crushing strength of ice: Roy. Soc. Can. Proc. and Trans., 3d ser., vol. 8, part III, p. 19. Barnes, H. T. (1928), Ice engineering, Renouf Pub. Co. Montreal. Barnes, H. T. (1929), The science of ice engineering: Sci. Monthly, vol. 29, pp. 289-297.

ENGINEERING RESEARCH INSTITUTE Cb -2~ l UNIVERSITY OF MICHIGAN Barnes, H. T., Hayward, J. W., and McLeod, N. M. (1914), The expansive force of ice: Roy. Soc. Can. Proc. and Trans., 3d ser., vol. 8, part III, p. 29. Barnes, W. H. (1929), The crystal structure of ice between O0C and -1830C: Proc. Roy. Soc. London, ser. A, vol. 125, pp. 670-693. Bartlett, R. A. (1928), Ice navigation, in Am. Geog. Soc. Sp. Pub. no. 7 (Problems of Polar research), pp. 427-444. Bassin, M. M. (1934?), Tests of the ice cover of the Svirj River on resistance to compression, shearing, and bending during thaw in spring of 1934 (MS in Inst. Hydotech.) Probably in Russian. Not examined. Bentley, W. A. (1907), Ice and snow: Monthly Wea. Rev., vol. 35, pp. 348, 397, 439, 512, 584, pl. I-XXXI. See also U. S. Wea. Bur. Ann. Summary, vol. 30, pp. 607-616, pl. I-XXII, 1902. Bell, G. C. (1911), Results of experiments on the strength of ice: Proc. Maine Soc. Civil Eng., vol. 1, pp. 41-46. Belokonj, P. N. (1938), On the coefficient of friction in the ice cover: Meteorologia i Hydrologia, vol. 4, pp. 116-131. Bevan, Benjamin (i826), Account of an experiment on the elasticity of ice: Philos. Trans., vol. 116, part III, p. 304. Bianconi, J. J. (1876), Nouvelles experiences sur la flexibilite de la glace: Compt. Rend., vol. 82, p. 1193. Blackwelder, E. (1940), The hardness of ice: Am. Jour. Sci., vol. 238 (whole ser.), pp. 61-62. Bowden, F. P., and Hughes, T. P. (1939), The mechanism of sliding on snow and ice: Proc. Roy. Soc. London, ser. A, vol. 172, pp. 280-298. Boyle, R. W., and Sproule, D. C. (1931), Velocity of longitudinal vibrations in solid rods (ultrasonic method) with special reference to the elasticity of ice: Can. Jour. Research, vol. 5, p. 601. Bregman, G. R. (?), The Atlantic influence on the processes of icebreaking and freezing of rivers: Bull. Inst. Hydrotech., vol. 4, pp. 176-183. Probably in Russian. Not examined. Brewster, 0. (1834), Krystallform des Eises: Ann. d. Physik u. Chemie (Pogg.), vol. 32, p. 399. Bridgman, P. W. (1912), Water in the liquid and five solid forms under pressure: Proc. Am. Acad., vol. 47, pp. 439-458.

ENGINEERING RESEARCH INSTITUTE Cb-3 UNIVERSITY OF MICHIGAN. Bridgman, P. W. (1912), Thermodynamic properties of liquid water to 80~ and 12,000 kgm.: Proc. Am. Acad., vol. 48, pp. 309-362. Brockamp, B., and Mothes, H. S. (1930), Seismische Untersuchungen am Pasterze-Gletscher: Zeit. f. Geophysik, vol. 6, pp. 428-500. Brown, E. W. (1926), Experiments on the strength of ice, Rpt. of the Joint Bd. of Eng. on the St. Lawrence Waterway Project, Appendix F, pp. 423 -453. Buchanan, J. Y. (1887), On ice and ice brines: Nature, vol. 35, pp. 608-611, and vol. 36, pp. 9-12. Bydin, F. J. (1932), A study of ice formation under natural conditions: Bull. Inst. Hydrotech., vol. 4, pp. 176-183. Probably in Russian. Not examined. Bydin, F. J., and Petrunichev (1932), Investigations of the formula of growth of surface ice proposed by M. P. Porovkin: Bull. Inst. Hydrotech., vol. 4, pp. 184-195. Probably in Russian. Not examined. Davydov, V. V. (1938), Theoretical investigations of the impact of a ship on ice: Problemy Arktiki (Problems of the Arctic), no. 5/6, pp. 103 -124. Probably in Russian. Not examined. Deeley, R. M. (1908), The viscosity of ice: Proc. Roy. Soc. London, ser. A, vol. 81, p. 250. Deeley, R. M., and Parr, P. H. (1913), The viscosity of glacier ice: Phil. Mag., 6th ser., vol. 26, pp. 85-111. Devik, Olaf (1944), Ice formation in lakes and rivers: Geog. Jour., vol. 103, p. 193. Dieke, J. C. (1864),,,Ueber Eisbildung und Entstehung der Scrinde und Spalten in den Eisdecken der Susswasserseen: Ann. d. Physik u. Chemie (Pogg.), vol. 121, p. 165. Directorate of engineering development, Dept. of Nat'l. Defence, Report on ice and snow, Ottawa, April 1944. Dobrowolski, A. B. (1923), HistorJa Naturalna Lodu (Natural History of ice), Warsaw. Dorsey, N. E., Properties of ordinary water substance, Reinhold Publishing Corp., New York. Ewing, M., Crary, A. P., and Thorne, A. M., Jr. (1934), Propagation of elastic waves in ice, Part I: Physics, vol. 5, p. 165, and Part II, p. 181.

Cb-4 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Fabian, 0. (1887), Uber die Dehnbarkeit und Elastizitat des Eises: Repert. Exp. Phys. (Carlo), vol. 13, pp. 447-457. Faraday, M. (1860), Note on regelation: Proc. Roy. Soc. London, vol. 10, pp. 440-450. Finlayson, J. N. (1927), Tests on the shearing strength of ice: Can. Eng., vol. 53, pp. 101-103. (Journal) of Glaciology, published by the British Glaciological Society. Golovkov, M. P. (1936), On the structure and morphological singularities of ice crystals: Jour. Gen. Chem., vol. 7, pp. 335-340. This is apparently a Russian publication. The article is probably in Russian and was not examined. Gulston, A., Ice breakers and their services: Jour. Soc. Arts, vol. 52, p. 215. Hargis, C. D. (1922), The viscosity and rigidity of ice: Phys. Rev., 2d ser., vol. 19, pp. 526-527. Hawkes, L. (1930), Some notes on the structure and flow of ice: Geog. Mag., vol. 67, pp. 111-123. Hess, Hans (1902), Elastizitat und innere Reibung des Eises: Ann. d. Physik (Drudes), vol. 8, pp. 405-431. Hess, Hans (19041), Die Gletscher. Hugi, F. J. (1830-1), Edin. New. Phil. Jour., vol. 10, pp. 337-338. Hydrographic Office of the Navy Department, Pilot Chart of the North Atlantic for July 1946: Physical properties of sea ice (on rear of chart); reprinted from Report of the Arctic, Desert and Tropic Information Center of the Army Air Forces. (U. S.) Hydrographic Office Chart No. 2601, S. P. (reverse side of), Notes on navigation in ice, March, April, and May 1947. (Reprint of 'Notes on navigation on ice', Hydrographic Publication H. D. 372, Hydrographic Department, London. H. C. 6536/42, N. I. D. 1484/42, 1942.) Keefer, T. C. (1898), Ice floods and winter navigation in the lower St. Lawrence: Roy. Soc. Can. Proc. and Trans., 2d ser., vol. 4, part III, p.3. Kerry, J. G. G. (1947), Ice blockade of Canadian ports: The Dock and Harbor Authority, March 1947. Klocke, F. (1879), Ueber die optische Structur des Rises: Neues Jahrb., pp. 272-285.

ENGINEERING RESEARCH INSTITUTE Cb-} UNIVERSITY OF MICHIGAN Knoop, Frederick, Peters, C. G., and Emerson, W. B. (1939), A sensitive pyramidal-diamond tool for indentation measurements: Jour. Res. of the Nat. Bur. Standards, Research Paper 1220, vol. 23. Koch, K. R. (1885), Beitrage zur Kentniss der Elastizitat des Eises: Ann. d. Physik u. Chemie (Wied), vol. 25, pp. 438-450. Koch, K. R. (1913), Uber die Plastizitat des Eises: Ann. d. Physik (Drudes), vol. 41, pp. 709-727. Koch, K. R. (1914), Uber die Elastizitat des Eises: Ann. d. Physik (Drudes), vol. 45, pp. 237-258. Koechlin, Rene (1944), Les glaciers et leur m6canisme, F. Rouge and Cie S. A., Lausanne. Kohler, R. (1929), Beobachtungen an Profilen auf See-Eis: Zeitschr. Geophysik, vol. 5, pp. 314-316. Komarovskij, A. N. (1932), The structure and physical properties of the ice cover of fresh waters, Moscow-Leningrad, 51 pp. In Russian. Not examined. London, L., and Seban, A. (1943), Rate of ice formation: Trans. ASME, vol. 65, pp. 771-778. Mallet, R. (1845), On the brittleness and non-plasticity of glacier ice: Phil. Mag., 3d ser., vol. 26, pp. 586-593. Mariutin, T. P. (1936), On the strength of sea ice: Meteorologia i Hydrologia, vol. 1, pp. 70-73. In Russian. Not examined. Matsuyama, Motonori (1920), On some physical properties of ice: Jour. Geol., vol. 28, pp. 607-631. Maxwell, J. C. (1868), On the dynamical theory of gases: Phil. Mag., 4th ser., vol. 35, pp. 129-145, 185-217. McConnel, J. C., and Kidd, D. A. (1888), On the plasticity of glacier and other ice: Proc. Roy. Soc. London, vol. 44, pp. 331-367. McConnel, J. C. (1889), On the plasticity of glacier and other ice: Nature, vol. 39, p. 203. McConnel, J. C. (1891), On the plasticity of an ice crystal: Proc. Roy. Soc. London, vol. 49, pp. 323-343. Megaw, H. D. (1934), Cell dimensions of ordinary and 'heavy' ice: Nature, vol. 134, pp. 900-901. Morphy, H. (1913), The influence of pressure on surface friction: Phil. Mag., 6th ser., vol. 25, p. 133.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Moseley, H. (Canon of Bristol) (1870), On the mechanical properties of ice: Phil. Mag., 4th ser., vol. 39, p. 1. 'Moseley, H. (1871), On the mechanical impossibility of the descent of glaciers by their weight only: Phil. Mag., 4th ser., vol. 42, pp.4 188-149. Palache, Charles, Berman, H., and Frondel, C. (1944), The system of mineralogy of J. D. and E. S. Dina, vol. I, John Wiley and Sons, Inc., New York. Parsons, W. J. Jr. (1940), Ice in the northern streams of the United States: Trans. Am. Geophys. Union, part III, p. 170. Pfaff, F. (1875), Versuche uber die Plastizitat des Eises: Ann. d. Physik u. Chemie, (Pogg.), vol. 155, pp. 169-174. Physics of the Earth (1942), Vol. 9. (Hydrology), McGraw-Hill Book Co., ch. 9. Pineghin, N. V. (1927), On the changes of the modulus of elasticity and of Poisson's ratio by river ice at compression: Nauka i Tekhnika (Science and Technics), vol. 5, pp. 1-6. In Russian. Not examined. Plyler, E. K. (1925), Some properties of an ice crystal: Jour. Elisha Mitchell Sci. Soc., vol. 41, p. 18. Plyler, E. K. (1926), The growth of ice crystals: Jour. Geol., vol. 34, p. 367. Polar Record: printed in Great Britain for the Scott Polar Research Institute, Cambridge. Reich, M., and Stierstadt, 0. (1931), Messung der Schallgeschwindigkeit von Stoffen in festen und geschmolzenen Zustand: Physics, vol. 32, pp. 124-130. Reusch, E. (1864), Beitrage zur Lehre vom Eis: Ann. d. Physik u. Chemie (Pogg.), vol..121, pp. 573-578. Rogers, A. F. (1937), Introduction to the study of minerals, 3d ed., Mc Graw-Hill Book Co., Inc., New York. Romanowicz, H., and Honigman, E. J. M. (1932), Forsch. Gebiete Ingenieurw., vol. 3, p. 99. Root, C. J. (1944), Great Ltkes frozen across: Bull. Am. Meteorological Soc., vol. 25, p. 203. Seliakov, N. J. (1936), The structure of river ice: Compt. Rend. Acad. Sci. URSS, vol. 1, pp. 280-287. In Russian with abstract in English.... * _ _.~~~~~~~~~~~~~~~~

ENGINEERING RESEARCH INSTITUTE Cb-7 UNIVERSITY OF MICHIGAN SelJakov, N. (same as above?) (1936A), To what class of symmetry does ordinary ice belong?: Compt. Rend. Acad. Sci. URSS, vol. 10, pp. 293-294. Seljakov, N. (1936B), Some remarks on a and B ice: Ccmpt. Rend. Acad. Sci. URSS, vol. 11, p. 227. SelJakov, N. (1937), The nature of ordinary ice: Compt. Rend. Acad. Sci. URSS, vol. 14, pp. 181-186. Schwedoff, T. (1889), Jour. d. Phys., 2d ser., vol. 8, pp. 341-359. Schwedoff, T. (1890), Jour. d. Phys., 2d ser., vol. 9, pp. 34-46. Sharp, R. P. (1947), Suitability of ice for aircraft landings: Trans. Am. Geophys. Union, vol. 28, p. 111. ShepelevskiJ, A. (?), On the velocity of growth of the ice cover at a given temperature of the upper surface of the ice. (MS in the editorial section of the Arctic Institute). Probably in Russian. Not examined. ShouleJkin, V. V. (1932), The temperature curve in the ice cover of the seas: Trudy TaimyrskoJ Hydrografichesko Ekspedicii (Transactions of the Taimyr Hydrographic Expedition), vol. 2, pp. 49-68. Probably in Russian. Not examined. Silliman, Benjamin (1821), Circumstances connected with the formation of ice on still water and with the continued action of cold on the fluid underneath: Am. Jour. Sci., vol. 3, (whole ser. ), p. 179. Sokolov, J. A. (1926), Young's modulus for a natural ice crystal: Jour. Appl. Phys., vol. 3, pp. 275-277. In Russian. Sukhorukov, V. V. (1938), The types of ice breakers and their shapes: Trudy Leningradskogo OtdeleniJa VsesoJuznogo Nauchnogo i IngenieroTekhnicheskogo Obshchestva Vodnogo Transporta (Transactions of the Leningrad Section of the All-Union Society of Science and Engineering Technology), no. 2/3 (vol. 2/3?), pp. 117-148. In Russian. Not examined. Swift, H. W. (1926), Determination of the modulus of elasticity by dynamical methods: Philos. Mag., 7th ser., vol. 2, pp. 351-368. Tammann, G. (1902), Ueber die Ausflussgeschwindigkeit Krystallisirter Stoffe: Die Ausflussgeschwindigkeit des Eises und seine Schmelzcurve: Ann. d. Physik (Drudes), vol. 7, pp. 198-224. Tammann, G.,and Dreyer, K. L. (1934), Die Eisbildung auf GewAssern und die Bildung von Kunsteis: Naturwissenschaften, vol. 22, pp. 613-614. Tamura, S. T. (1905), The mathematical theory of ice formation: Monthly Wea. Rev., vol. 33, p. 55.

Cb-8 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Tarr, R. S., and Rich, J. L. (1912), The properties of ice: experimental studies: Zeitschr. Gletscherkunde, vol. 6, pp. 226-249. See also Nature, vol. 91, p. 307, 1913. Tarr, R. S., and Von Engeln, O. D. (1915), Experimental studies of ice with reference to glacier structure and motion: Zeitschr. Gletscherkunde: vol. 9, p. 81. Teichert, C. (1939), Corrosion by wind-blown snow in polar regions: Am. Jour. Sci., vol. 237 (whole ser.), pp. 146-148. Thomson, J. (1857), On the plasticity of ice, as manifested in glaciers: Proc. Roy. Soc. London, vol. 8, pp. 455-457. Thornton, W. M. (1919), The thermal conductivity of solid insulators: Philos. Hag., 6th ser., vol. 38, pp. 705-707. Timonoff, V. E. (1938), On the establishment of a working hypothesis of ice phenomena in lakes and rivers: Assoc. Int. d'Hydrol. Sci. Bull. N 23. Trouton, F. T. (1899), Arrangement of the crystals of certain substances in freezing: Proc. Roy. Dub. Soc., new ser., vol. 8, p. 691. Trowbridge, J., and McRae, A. L. (1885), Elasticity of ice: Am. Jour. Sci., vol. 129 (whole ser.), pp. 349-355. Tyndall, J., and Huxley, T..H. (1858), On the structure and motion of glaciers: Philos. Mag., 4th ser., vol. 15, p. 365. Tyndall, John (1858A), Proc. Roy. Soc. London, vol. 9, pp. 76-80. Tyndall, J. (1858B), On some physical properties of ice: Philos. Mag., 4th ser., vol. 16, pp. 333-354. Vedel, P. (1895), The growth and sustaining power of ice: Jour. Franklin Inst., vol. 140, p. 355. Vitman, F. F., and Shandrikov, P. P. (1938), Some investigations of the mechanical properties of ice: Trans. Arctic Inst., vol. 110, pp. 83-100. In Russian with abstract in English. Von Engeln, 0. D. (1915), Experimental studies and observations on ice structure: Am. Jour. Sci., vol. 190 (whole ser.), pp. 449-473. Weinberg, B. (1905), Uber die innere Reibung des Eises: Ann. d. Physik (Drudes), vol. 18, p. 81. Weinberg, B. (1907A), Uber den Koeffizienten der innere Reibung des Gletschereis und seine Bedeutung fur die Theorien der Gletscherbewegung: Zeitschr. Gletscherkunde, vol. 1, pp. 321-347.

UNIVERSITY OF MICHIGAN Weinberg, Boris (1907B), Uber die innere Reibung des Eises: Ann. d. Physik, 4th ser., vol. 22, pp. 321-332. Weinberg, B. (1938), Mechanical properties of ice: Assoc. Int. d'Hydrol. Sci. Bull. N 23, pp. 509-535. Weinberg, B. (1940), List of latest publications of USSR on ice and snow: Trans. Am. Geophys. Union, part III, Appendix A, p. 757. Weinberg, B. P. (Boris?) (1929), The influence of temperature on the mechanical resistance of river ice: Bull. Central Geophys. Obs., vol. 2, pp. 22-33. In Russian with an English abstract. Not examined. Williams, F. M., and Williams, F. P. (1926), Report on ice breaking: Int. Cong. of Nav. Wilson, J. T., and Horeth, J. M. (1948), Bending and shear tests on lake ice: To be published in Trans. Am. Geophys. Union. Winchell, Horace (1945), The Knoop micro-hardness tester as a mineralogical tool: Am. Mineralogist, vol. 30, pp. 583-595. Yoshida, U., and Tsuboi, S. (1929), Examination of ice crystals by x-rays: Mem. Coll. Sci., Kyoto, ser. A, vol. 12, pp. 203-207. Zubov, N. N. (1940), The drift of the ice-breaker Sedov: Nature, vol. 145, pp. 533-539. Addendum Johnson, H. F. (Rear Adm., USCG, Ret.) (1946), Development of icebreaking vessels for the U. S. Coast Guard: Trans. Soc. Naval. Arch. and Marine Eng., vol. 54, pp. 112-151. Runeberg, Robert (1889), On steamers for winter navigation and icebreaking: Proc. Inst. Civil Eng., vol. 47, paper 2371, pp. 277-239. Runeberg, Robert (1900), Steamers for winter navigation and icebreaking: Proc. Inst. Civil Eng., vol. 57, paper 3191, pp. 109-129. Simonson, D. R. (1936), Bow characteristics for ice-breaking: Jour. Amer. Soc. Naval Eng., vol. 48, no. 2.

Cb-10 | ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN -EAL RFERENCES Barnes, H. T. (1906), Ice formation with special reference to anchor ice and frazil, John Wiley and Sons, New York: 250 pp., 40 figs. This book contains no bibliography but a list of authors to whom reference is made is appended. Contents Chapter I: Physical laws governing the transfer of heat: Radiation. Surface emissivity and solar radiation. Conduction. Convection. Chapter II: Physical constants of ice: Density. Maximum density of water. Heat of fusion or latent heat. Specific heat of ice. Specific heat of water and the thermal unit. Thermal conductivity of ice and snow. Thermal conductivity of water. Coefficient of expansion. Relative hardness or penetrability. Plasticity. Elasticity. Viscosity. Vapor pressure of ice and supercooled water. Electric properties. Chapter III: Formation and structure of ice: Crystalline structure. Snow crystals. Structure of solid ice. Quincke's theory. Regelation. Spontaneous crystallization. Glacier motion. Modifications of ice. Supercooling of water. Chapter IV: Sheet, frazil and anchor ice: Sheet ice. Rate of growth of surface ice. Frazil-ice. Artificial production. Anchor-ice. Historical. Distinguished from frazil. Radiation as the cause Of anchor-ice formation. Ice floods in the St. Lawrence. Chapter V: Precise temperature measurements: The platinum resistance thermometer. Mercury thermometer compared. Method of measurement. Refinement of apparatus. Sample results showing accuracy attained. Chapter VI: River temperatures: Instruments used in investigations of river temperatures. First series of results under surface ice. Second series of tests at the Lachine Rapids. Details of the observations. General considerations. Freezing point diagram. Conditions which govern the formation of frazil and anchor-ice. Chapter VII: Theories to account for frazil and anchor-ice: Early theories of Arago, Eisdale, and Farquharson. Later theories of Francis, Bell, Hunt, Henshaw and Lord Kelvin. Discussion of Henshaw's paper. Views of Keefer. Extract from article on anchor-ice from the Montreal Flood Commission Report. Depth of formation of anchor-ice in fresh and salt waters.

ENGINEERING RESEARCH INSTITUTE Cb-ll UNIVERSITY OF MICHIGAN Chapter VIII: Methods of overcoming the ice problem in engineering work: Construction and situation of power works. When a surface sheet is valuable and when not. Vulnerable spots in a power house. Artificial heating and steam injection. Electric heating of racks. Effect of head on water. Effect of rapid fall in temperature conditions. Volume of ice formed by radiation. Erosive velocity of water and its probable effect on anchor-ice formation.

Cb-12 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN| Barnes, H. T. (1928), Ice engineering, Renouf Publishing Co., Montreal: 353 PP., 71 figs. This book is a revision of Barnes' earlier work. It contains a comprehensive bibliography, arranged by subjects, of about three-hundred seventy-five entries. Contents Chapter I: Equilibrium of the ice-water system. Colloidal ice. The crystal structure of ice. Chapter II: Physical constants of ice: density, specific heat, hardness, elastic constants, heat of fusion, evaporation, sublimation, color and bacteriology. Chapter III: Rate of growth and melting of surface-formed ice, infra-red absorption of ice, various kinds of ice. Chapter IV: Theories of formation of anchor-ice, effect of light and heat on its disintegration. Chapter V: Frazil ice. Winter ice floods. Chapter VI: Ice remedial work. Use of steam, thermit, hot water, calcium chloride and other chemicals. Chapter VII: Ice pressure and expansion. Chapter VIII: Ice navigation and ice breaking. Chapter IX: Conservation of heat in lakes and rivers for ice prevention, evaporation, power houses and power canals. Chapter X: Glacier ice and icebergs. Bibliography.

ENGINEERING RESEARCH INSTITUTE Cb-13 UNIVERSITY OF MICHIGAN. Dobrovolski, A. B. (1923), HistorJa Naturalna Lodu (Natural history of ice). Warsaw: 940 pp., 340 figs. Edition of the Kasa dla osob pracujacych na polu naukowem imienia D-ra Mianowskiego. In Polish. Due to the comparative obscurity of this comprehensive book, it seems advisable to include here the introduction and table of contents. The author appended a translation in French of the introduction and table of contents which was translated into English by the present writer. INTRODUCTION Ice is certainly one of the most important solid constituents of the earth's surface. Glaciers alone occupy an area of more than fifteen million square kilometers, which is three per cent of the total surface of the earth and ten per cent of the continental area. An entire continent, Antarctica, is covered with this rock-like substance which is both crystalline and sedimentary in nature. The frigid zones are covered with snow during the winter and snow forms a sort of ephemeral rock in the temperate zones and in the extensive polar regions which are not covered by glaciers. The polar expanses, on the other hand, remain frozen to a certain depth during the entire year, forming a conglomerate cemented by ice. Another sort of ice rock, comparable to the igneous rocks, forms the polar ice pack and chokes the rivers and lakes of the continents. The rivers of the temperate zone and especially those of the subpolar regions which remain unfrozen in winter due to turbulence and rapids transport detached masses of sandyappearing ice and form spongy and often large deposits of anchor ice. Large expanses of white frost appear on the plains due to radiational cooling. Under certain meteorological conditions, ob3ects may be coated with glaze or rime ice. Rime ice is developed in mountainous regions in large, rocklike masses which cover the sharp peaks and sometimes form small glaciers as the result of avalanches. Finally the atmosphere, within the limits of the troposphere, is always filled with finely powdered ice or with ice condensed in the form of crystalline fogs or in clouds of snow, sleet and hail. It can be seen, then, that ice forms a sort of envelope around the earth, a cryosphere, which is very homogeneous as to composition but extremely varied in its aspect. This envelope is intimately related to the lithosphere, the hydrosphere and the atmosphere. It has upper and lower limits, the latter being very high over tropical regions but lowering gradually toward the polar regions where it penetrates below the surface. It varies in general with the seasons. In contrast to the other solid constituents of the earth, ice is a separate entity, having exceptional thermodynamic and other properties. First, ice has an exceptionally low melting point which results in a variability in state unknown in the other solid constituents of the globe; it

Cb-l14 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN appears and disappears, continually increasing or diminishing in amount by absorbing energy or by producing it, by assimilating or producing water vapor or liquid water. Furthermore, in the change from the solid to the liquid state (or vice versa) the volume change is in reverse of the ordinary. Consequently, an increase in pressure lowers the melting point instead of raising it. For these two reasons - wide distribution and singularity of behavior, ice plays a very important role in the economy of nature. It is the basis of a group of physical, mechanical and morphological phenomena, as well as the cause of a whole chain of variations, both megascopic and microscopic, in the atmosphere, in the water and in the soil. In my opinion, therefore, the study of the cryosphere could and ought to forms a separate science, a special branch of physical geography, to be known as cryograph, of which glaciology would constitute a special phase concerned with the geology and history of the globe. The varied mode of existence of ice; the influence that it exerts on meteorological, geological, oceanographical, hydrographical and biological phenomena; finally its specific properties as a rock and as a mineral have long since lead to studies of this substance by the most diverse specialists. Work has been done by mathematicians and astronomers, physicists and chemists, crystallographers and mineralogists, geographers and geologists, meteorologists and climatologists, biologists and'also engineers and.explorers. The results have been published in the reviews dedicated to the various specialties - in the publications of the academies or in reports on expeditions. Articles on ice are found in publications of all sorts. This makes it difficult to orient oneself, not only in the general field of ice research but also in regard to specific problems. It is often easy to ignore the general accomplishments and problems in the field of ice, and it is also easy to ignore the history of a specific problem. This aggravating state of affairs is disappearing as the study of ice continues and the number of published works increases. In my opinion, two things would be of great usefulness to the various specialists concerned with ice and the phenomena which depend on it. First would be a monograph, as complete as possible, presenting a view of the whole subject by means of a detailed and systematic discussion of all problems relating to ice, with regard to their evolution as well as to their present state of development, and containing a bibliography and a complete iconography. Second would be an international publication which would group (systematically) the workers and their works, and by that means would give to the whole of the research a rational direction, and to the investigators, the possibility of reading the works of others and of being read by others.* *For this purpose, the Review of Glaciology, the publication of the International commission for the study of Glaciers, could be revived in expanded form.

ENGINEERING RESEARCH INSTITUTE Cb-15 UNIVERSITY OF MICHIGAN _ The HistorJa Naturalna Lodu is an attempt to consolidate the research done in the field and in the laboratory on ice of all types and of all origins. It is a sort of index to the problems of the important yet little known subject of ice. It includes a history of each of the problems, the results of investigation, and the problems which are not yet solved or are questionably solved. The book is divided into fifteen chapters. The first three discuss the conditions of formation and growth of ice; Chapter IV studies the crystallography of ice (symmetry, combinations, principal aspects, twinning, geometric constants, polymorphism); Chapters V, VI, and VII, atmospheric ice; Chapters VIII and IX, the freezing of calm water (with the formation of an ice sheet) and running water (anchor ice); Chapter X, ice in the soil; Chapters XI and XII, the snow cover; Chapters XIII and XIV, the glaciers; finally Chapter XV, halo phenomena. The thermodynamic constants are treated in Chapters I and II; the electrical and magnetic properties as well as the acoustical peculiarities are contained in Chapter XI; the optical properties in Chapters XI and XV; and the mechanical properties in Chapter XIV. I have not treated all questions with the same importance. I have not emphasized those which depend only in an indirect manner on the specific properties of ice nor those which already have been treated in excellent detailed and well-known monographs. Therefore, in trying to discuss completely the conditions of formation of atmospheric ice (Ch. III), I have restricted the discussion of hail to the essential points of the problem, to methods of study and above all to the most recent theories, especially that of A. Wegener; I have substituted as complete a bibliography and iconography as possible in place of a discussion of the more obsolete theories. I have also cited the works containing an historical review of the problem. Similarly, in the discussion of the snow cover, I have treated the physical and morphological properties in a very detailed fashion, but I have not dwelt at all on the times of accumulation and melting of snow since the study of these phenomena falls within the domain of the climatologist; I have treated similarly and for analogous reasons the layer of ice covering the water; I have omitted the special subject of icebergs, which is very important to the polar scientist but which is not of particular interest in regard to the properties of ice. Furthermore, (a discussion of) the interesting properties of icebergs will be found in various chapters of the -book in the form of examples and illustrations pertinent to the general discussion. Concerning glaciers, I have ignored the climatological and geological problems by omitting the glacial epochs and by limiting myself to a citation of the classical works in which the student will find complete discussions of these problems together with their bibliographies; on the other hand, I have considered at length the structure and movement of glaciers since these phenomena are a direct expression, on a large scale, of the mechanical, physical and structural properties of ice in general and of glacier ice in particular. In this discussion I have tried to analyze the basic premises of the various theories of recrystallization and movement of glaciers. I have excluded from Chapter XV treating the halo phenomena, the fundamental mathematical deductions as being merely applications of the general formulae of reflection of light to the special problem of' ice crystals.

Cb-1j6 | ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN This theoretical background is discussed in several excellent monographs. On the other hand, I have tried to give a detailed statement and an exhaustive discussion of the physical bases of the halo theory as it is related to the physical properties of ice and to the form and structure of atmospheric crystals. Finally, I have mentioned only occasionally the morphology of clouds since this depends very little on the specific properties of ice. Nearly all the questions relative to ice are treated in conjunction with the more general problems of which they constitute particular cases. For example, the subject of the supercooling of water is preceded by a resume of the problem of supercooling in general (as investigated by Tammann); the theory of formation of dunes par obstacle is accompanied by a resume of modern hydrodynamical theory regarding an obstacle in a current (Karman, Prandtl); etc. These various additions have strongly augmented the scope of the present work, and in my opinion have added to its value: considered in the light of the more general problems, questions relating to ice have acquired the space and importance which is due them and thus have been clarified. During the writing of HistorJa Naturalna Lodu, I have carried on some original research, the results of which appear in the present book. The problems investigated were: (1) the origin of the structure of 'heavy' and 'light' sleet (Frostgraupeln and Reifgraupeln of A. Wegener) Chapters III and VII; (2) the influence of a surface of refrigeration on the orientation of ice crystals forming on that surface, Chapters VII and VIII; (3) the general crystallography of ice (symmetry, crystal habit, twinning), Chapter IV; (4) certain questions of the special crystallography (unequal growth in the direction of the principal axis, systems of internal cavities, radial aggregations of crystals in the form of rods and plates), Chapters V and VII; (5) the classification of ice rocks, Chapter XI; (6) the recrystallization of a layer of ice formed on a water surface, Chapter VIII; (7) the mechanical theories of dunes par obstacle, Chapter XII; (8) the fundamental bases of the various theories of the glacier grain. Chapter XIII; (9) a critical review of modern theories of glacier movement, Chapter XIV; (10) the physical basis of the halo theories, Chapter XV. In order that the present work, which is to be published in Polish, be of value to foreign students, I have appended a table of contents in French which, in my opinion, is sufficiently detailed to serve. The important thing is the understanding of the different problems which relate to ice and a knowledge of the authors and their works. The table of contents comprises a detailed index to these problems. With the aid of this table, it is easy to find the pages on which the problems of particular interest are treated. Printed there in bold-faced type are the names of the authors who have investigated each special problem and whose works appear in the bibliography at the end of each chapter. These bibliographies include, in general, only works on ice and on associated phenomena. Works cited in the text having only an indirect connection with ice are mentioned on the spot in the form of a reference or in parentheses following the name of the author. In those cases where it would have been necessary to cite a large number of such works, I have inserted them in a special

ENGINEERING RESEARCH INSTITUTE cb-17 UNIVERSITY OF MICHIGAN Dorsey, N. E. (1940), Properties of ordinary water-substance in all its phases: water vapor, water and all the ices, Reinhold Publishing Corp., New York: 673 pp., 13 figs., 289 tables. Over four-hundred references are given as footnotes. The following abstract of the table of contents includes only those sections of the book dealing with ice. Contents Section IIC. Ice. 56. Foreword. 57. Types of ice. 58. Appearance of ice-I. 59. Forms and formation of ice. Crystallographic structure. Structure of ice in bulk. Internal melting. Flowers of ice. Formation of frazil or needle ice. Formation of an ice sheet. Growth and orientation of ice crystals. Recrystallization. Regelation. Purity of ice. Production of homogeneous ice. Monocrystals of ice. Freezing of supercooled water. Icicles. Hail. Snow and frost. Glaciers. Sea-ice. 60. Molecular data for ice. Association of molecules in ice. Structure of the molecule of ice. 61. Interaction of ice and corpuscular radiation. 62. Adhesiveness of ice. 63. Sliding friction of ice. 64. Deformability of ice. Descriptive treatment. Linear compression. Extension. Flexure. Punching. Penetration. Flowing. Recovery from stress. Brittleness.

ENGINEERING RESEARCH INSTITUTE Cb-18 lUNIVERSITY OF MICHIGAN section of the bibliography of the chapter. I have done the same for those problems which possess a very important literature. Hence the bibliography added to Chapter III contains two separate sections; Grad (hail) and Pieczary lodowe (ice grottoes), and that of Chapter IV is divided into two sections: BiblJografJa dolyczaca lodu (bibliography relating to ice) and BiblJografja bezposrednia lodu nie dotyczaca (bibliography not relating to ice). On the other hand, for the two chapters on. glaciers (XIII and XIV), I have given only a single bibliography which is placed at the end of Chapter XIV. The manuscript of the present work was completed in 1916 but could not be published until May 1923. In the intervening period I have been occupied with other work and was able to proofread and revise the text and bibliography during the printing of the book, which was completed in May 1923. For this reason certain works appearing after 1916 have been omitted. Table of Contents Chapter I. Freezing of water II. Sublimation of water vapor III. Growth of atmospheric ice IV. Crystallography of ice V. Crystallization of water vapor in the free air (clouds and crystalline fogs - snow) VI. Crystallization of water vapor on the surface of a solid body (white frost, some varities of white frost and of arborescence on windows) VII. Freezing of water droplets VIII. Freezing of calm water (ice cover) IX. Freezing of running water X. Freezing of water in the soil XI. Bed of snow - physical properties XII. Snow cover - description XIII. Glaciers, structure XIV. Glaciers, movement XV. The crystals of atmospheric ice and the halo phenomena.

ENGINEERING RESEARCH INSTITUTE ' UNIVERSITY OF MICHIGAN Cb-19 64. Deformability of ice (continued). Quantitative treatment. Young's modulus. Poisson's ratio. Rigidity. Tensile strength. Strength of linear compression. Shearing strength. Hardness. Plasticity and viscosity. Sustaining power of an ice sheet. 65. Acoustic and other vibrational data for ice. Velocity of transmission. Reflectivity. 67. Pressure-volume-temperature associations for ice. Density of snow. Density of ice-I at one atmosphere. Densities of the ices not at their melting points. Thermal expansion of ice (cubical). Compressibility of ice. Ice-I. Ice-VI. Ice-VII. 68. Coefficient of linear expansion of ice. 69. Thermal energy of ice-I. Specific heat of ice. Entropy of ice. 70. Thermal conductivity of ice and snow. Single crystals. Ice in bulk. Snow. 71. Refractivity of ice. 72. Reflectivity of ice and snow. Ice. Snow. 73. Luminescence of ice. Fluorescence of ice. Rayleigh scattering by ice. 1Raman scattering by ice. Effect of temperature. 74. Diffraction of x-rays by ice. 75. Absorption and transmission of radiation by ice and by snow. Ice. Snow. Glaciers and neves. 76. Emissivity of ice and of snow. 77. Photoelectric emission by ice. 78. Absorption spectrum of ice. 79. Optical rotation by ice.

UNIVERSITY OF MICHIGAN 80. Dielectric properties of ice. Dielectric constant of ice. Dielectric absorption of ice. Dielectric strength of ice. 81. Electrical conductivity of ice. 82. Miscellaneous electrical data for ice. Pyroelectric effect. 83. Magnetic susceptibility of ice. Section III. Multiple-phase systems. 92. Pressure temperature associations for equilibrium between ice and another phase. Triple points. Ice and water vapor. Vapor-pressure of ice-I. Density and specific volume of vapor saturated with respect to ice. Ice and water. Melting point of ice. Ice-I: normal melting point and triple point. Effect of a solute. Ice and ice. 93. Phase diagram for water and the ices. 94. Surface charges on water and on ice. Section IV. Phase transition. 97. Freezing and melting. Ice needles. Supercooling of water. Superheating of ice. Rate of freezing and melting. Rate of melting: effect of tension. Crystalloluminescence. 98. Transition of ice to ice.

DEPARTMENT OF ENGINEERING RESEARCH Cb-21 UNIVERSITY OF MICHIGAN Dorsey, N. B. (1940), The properties of ordinary water substance, Reinhold Pub. Corp. New York. Faraday, M. (1860), Note on regelaticon: Proc. Roy. Soc. London, vol. 10, pp. 440-450. Hawkes, L. (1930), Some notes on the structure and flow of ice: Oeo~g. Nag., vol. 67. pp. 111-123. Hess, Hans (1904), Die Gletscher. Hugl, F. J. (1830-1), Edin. New Phil. Jour., vol. l0, pp. 37-338. Kerry, J. G. G. (1945), Ice blockade of Canadian ports: The dock and harbor authority, March, 1947. [locke, F. (1879), Ueber die optische Structur des Rises: Neues Jahrb., pp. 272-285. Koch, K. R. (1913), Ueber die Plastizitgt des Eieew: Ann. d. Physik u. Chemie (Drudes), vol. 41, pp. 709-727. Megaw, H. D. (1934), Cell dimensions of ordinary and 'heavy' ice: Nature, vol. 134, pp. 900-901. McConnel, J. C., and Kidd, D. A. (1888), On the plasticity of glacier and other ice: Proc. Roy. Soc. London, vol. 44, pp. 331-367. McConnel, J. C. (1889), On the plasticity of glacier and other ice: Nature, vol. 39, p. 367. Parsons,W. J., Jr. (1940), Ice in the northern streams of the United States: Trans. Am. Geophys. Union, part III, p. 170. Plyler, E. K. (1925), Some properties of an ice crystal: Jour. Elisha Mitchell Sci. Soc., vol. 41, p. 18. Plyler, E. K. (1926), The growth of ice crystals: Jour. Geol., vol. 34., p. 367. Root, C. J. (1944), Great Lakes frozen across: Bull. Am. Meteorological Soc., vol. 25, p. 203. SelJakov, N. (1936A), To what class of symmetry does ordinary ice belong?: C. Rde l'Acad. Sci. de I'URSS, vol. 10, pp. 293-294. SelJakov, N. (1936B), Some remarks on a and B ice: C. R. de l'Acad. Sci.de 1'URSS, vol. 11, p. 227. SelJakov, N. (1937), The nature of ordinary ice: C. R. de l'Acad. Sci. de 1'URSS, vol. 14, pp. 181-186.

Cb-22 DEPARTMENT OF ENGINEERING RESEARCH |i UNIVERSITY OF MICHIGAN F01MATION AND STRUCTUIr CF LAKE ICE Bibliography Barnes, H. T. (1906), Ice formation, John Wiley and Sons, Inc., New York. Barnes, H. T. (1910), The orientation of crystals 'in a flux of heat: Nature, vol. 83, p. 276. Barnes, H. T. (1928), Ice engineering, Renouf Pub. Co., Montreal. Barnes, H. T. (1929), The science of ice engineering: Sci. Monthly, vol. 29, pp. 289-297. Barnes, W. H. (1929), The crystal structure of ice between O*C and -183'C: Proc. Roy. Soc. London, ser. A, vol. 125, pp. 670-693. Bentley, W. A. (1907), Ice and snow: Monthly Wea. Rev., vol. 35, pp. 348, 397, 439, 512, 584, pl. I-XXXI. See also U.S. Wea. Bur. Annual Sunmary, vol. 30, pp. 607-616, pl. I-XXII, 1902. Brewster, 0. (1834), Krystallform des Eises: Ann. d. Physik u. Chemie (Pogg.), vol. 32, p. 399. Bridgman, P. W. (1912), Water in the liquid and five solid forms under pressure: Proc. Am. Acad., vol. 47, pp. 439-458. Bridgman, P. W. (1912), Thermodynamic properties of liquid water to 80'C and 12,000 kgm: Proc. Am. Acad., vol. 48, pp. 309-362. Buchanan, J. Y. (1887), On ice and ice brines: Nature, vol. 35, pp. 608-611 and vol. 36, pp. 9-12. Devik, Olaf (1944), Ice formation in lakes and rivers: Geog. Jour., vol. 103, p. 193. Dieke, J. C. (1864), Ueber Eisbildung und BntsiLehung der Scrunde und Spalten in den Eisdecken der Susswasserseen: Ann. d. Physik u. Chemie (Pogg.), vol. 121, p. 165. Dobrowdski, A. B. (1923), HistorJa Naturalna Lodu (Natural History of Ice), Warsaw, p. 353.

DEPARTMENT OF ENGINEERING RESEARCH Cb-23 UNIVERSITY OF MICHIGAN Silliman, BenJamin (1821), Circumstances connected with the formation of ice on still water and with continued action of cold on the fluid underneath: Am. Jour. Sci., vol. 3 (whole ser.), p. 179. Tammarnn, G., and Dreyer, K. L. (1934), Die Eisbildung auf Gewlssern und die Bildung von Kunsteis: Naturwissenschaften, vol. 22, pp. 613-614. Tamura, S. T. (1905), The mathematical theory of ice formation: Monthly Wea. Rev., vol. 33, p. 55. Tarr, R. S., and Rich, J. L. (1912), The properties of iCe: experimental studies: Zeitschr. Gletscherkunde, vol. 6, pp. 226-249. See also Nature, vol. 91, p. 307. Trouton, F. T. (1899), Arrangement of the crystals of certain substances in freezing: Proc. Roy. Dublin Soc., new ser., vol. 8, p. 691-. Tyndall, John (1858A), Proc. Roy. Soc. London, vol. 9, pp. 76-80. Tyndall, John (1858B), On some physical properties of ice: Phil. Mag., 4th ser., vol. 16, pp. 333-554. Vedel, P. (1895), The growth and sustaining power of ice: Jour. Franklin Inst., vol. 140, p. 355. Vitman, F. F., and Shandrikov, P. P. (1938), Some investigations of the mechanical properties of ice: Trans. Arctic Inst., vol. 110, pp. 83-100 In Russian with abstract in English. Wilson, J. T., and Horeth, J. M. (1948), Bending and shear tests on lake ice: To be published in Trans. Am,.Geophys. Union. Yoshida, U., and Tsuboi, S. (1929), Examination of ice crystals by x-rays: Mem. Coll. Sci., Kyoto, ser. A, vo1. 12, pp. 203-207.

Cb-24 DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN Forms of Ice light forms of ice have been described, including the most common, ice-I, a low-temperature vitreous form (see Dorsey, 1940: p. 395) and six other varieties whieh have been reported by Bridgman (1912). Dorsey describes ice-I as crystalline, having a conchoidal fracture and vitreous luster. Barnes (1928) states that the color of natural ice is a vivid blue, the depth of blueness being- a measure of the purity. The blue color is prow bably caused by scattering by ice molecules which are large with respect to the wavelength of blue light. Crystallographic Data Ice is coamonly referred to the hexagonal system (Hess, 1904; Tarr and Rich, 1912; Barnes, 1928: etc.) However, many different interpretatibns have been placed on the results of x-ray analyses,of ice. SelJakov (1936A, 1936B, 1937) has reported two forms of ice-I, which may resolve soa of the differences observed. He describes a ice as a hexagonal form which is stable from 0' to -183'C (see W. H. Barnes, 1929) and $ ice as a rhambohedral form which develops from supercooled water. W. H. Barnes (1929) has obtained dimensions for the unit cell of ice-I which he refers to the dihexagonal b$pyramidal class or the ditrigonal bipyraidal class (probably the former): a = 4.53,J; c a 7.4lA; c/a a 1.634; cell contents - I4 20 molecules. Dorsey states tiat these dimensions are believed to be correct within a few parts in a thousand. Optical Data Ordinary ice is uniaxial and positive. The indices of refraction for sodium light (5893 A) arew = 1.3090 and6 = 1.3104. (See Dorsey, 1940, pp. 484-485.) The Structure of Iake Ice Lake ice caomonly exhibits a banded or stratified structure in crosssection, with alternate bands of clear glassy ice and of more or less opaque ice containing xaerous air bubbles. (See Silliman, 1821; Faraday, 1860; Tyndall, 18583; Bentley, 1907; Barnes, 1906; Taumann and Dreyer, 1934; etc.) The stratification is expressed in various ways and is thought by Bentley (1907) to be caused by variations in the rate of freezing. He describes the bubbles as being tubular and giving a milky white appearance to the ice. According to Bentley the ice which freezes most rapidly contains the greatest number of air tubes and the layers. containing the greatest concentration of tubes usually lie in the upper half of a given section of ice. Dieke (1864) and Koch (1913) state that the air tubes are always normal to the freezing surface. Silliman (11l) describes a 15 inch layer of pond ice which contained

DEPARTMENT OF ENGINEERING RESEARCH Cb-25 UNIVERSITY OF MICHIGAN 21 layers of decreasing thickness downward. He states that the air bubbles were concentrated at the top of each layer. His explonation is that nocturnal cooling causes dissolved air to be expelled from the water and that the resulting air bubbles accumulate at the bottom of the previous layer. The water then freezes, forming a more or less clear layer below the air bubbles. The pond ice used by Tarr and Rich (1912) in their experiments was 30 cm thick with the top 10 cm composed of finely granular ice, the next 15 cm composed of coarsely crystalline ice with prismatic crystals standing perpendicular to the water surface, and the bottom 5 cm composed of finely granular ice with diversely oriented cyrstals. Wilson and Horeth (1948) state that ice from Lake Michigan usually had a cloudy zone as much as 2 inches thick at the top of a layer 1 foot thick. Koch (1913) describes lake ice used by him in Labrador for elasticity experiments. One piece of ice was 76 cm thick. The upper 36 cm was composed of alternate thin layers of ice with air bubbles and tubes, and of clear ice. The lower 40 cm was completely air-free. Koch also describes ice from Lago della Crocetta which was 85 cm thick and of which only the lower 10 cm was free of air bubbles. Vitman and Shandrikov (1938) imply in the manner of tabulation of their data that the lower layer of the ice used by them was usually porous, white and opaque and that the upper layer was devoid of air bubbles. Except for the statements of Silliman and Bentley, few investigators have attempted to explain the stratification of lake ice. The association of high rates of freezing with concentration of air bubbles in the uppermost layers is consistent with the theory, to be described in more detail, that freezing proceeds more rapidly below a thin layer of ice than below a thick layer. The writer suggests that there exists a critical diurnal range of temperature (in the case of the stratified ice) such that the flow of heat outward from the water (or ice) surface occurs only at night or during alternate time periods of any (fairly short) length so that freezing occurs intermittently. Under such conditions, if Sillimans idea concerning the expulsion of dissolved water is correct, a laminated structure could result. In any case, it is difficult to explain the laminated structure on the basis of continuous freezing. The presence of air bubbles in ice probably affects its physical properties but no systematic investigation of this effect was found in the literature. The orientation of ice crystals in lake ice, or in any ice formed on unagitated water, is easily determined. There is almost complete agreement on a uniformly vertical orientation of ice crystals in sheet ice. (See Brewster, 1834; Trouton, 1899; Barnes, 1906, 1910, 1928; McConnel, 1889; Yoshida und Tsuboi, 1929; Megaw, 1934; Tammann and Dreyer, 1934; Dorsey, 1940; and many others.) The orientation can be determined by means of a petrographic microscope or by means of a pair of polarizing prisms. A phenomenon first described by Tyndall (1858A) and named "Tyndall's flowers of ice" after him is also readily employed for the same purpose. Tyndall's flowers are described by Barnes (1906, 1928) and Dorsey (1940). When a beam of light is passed through a piece of ice, water bubbles appear which have the form of flowers with six coplanar petals. Their appearance is accompanied by

UNIVERSITY OF MICHIGAN a distinct clicking noise. The petals lie in a plane which is normal to the optic axis of the associated crystal and each of the petals corresponds to a secondary hexagonal axis. Hawkes (1930) suggested that the nucleus of each flower is a dust grain which is able to absorb heat and melt the surrounding ice when the beam of light is passed through the ice. The clicking noise was explained by Tyndall on the basis of the sudden contraction of the water as the ice melts. In a given sample of lake ice, the flowers lie in parallel horizontal planes, thus indicating the uniformly vertical orientation of the Crystals. The unique orientation of ice crystals described above is ascribed by Barnes (1910) and Trouton (1899) to the fact that the thermal conductivity of ice is greater along the major axis than in directions perpendicular to it. According to Trouton the ratio of the conductivities parallel to and normal to the major axis is 22 to 21. This is apparently sufficient so that the crystals orient themselves with the long axis parallel to the direction of flow of heat. Yoshida and Tsuboi (1929) conclude their paper by saying that "all directions parallel to the basal pinacoid of hexagonal ice crystals are equally suited for growth, which occurs more readily in these directions than perpendicular to the basal pinacoid. Ice formed on the surface of calm water exposed to cold air on a clear night is usually composed of monocrystals of considerable size with basal planes nearly parallel to the surface." Thus the direction of greatest thermal conductivity appears not to be related to the directions of most rapid growth. Bentley (1907) describes six types of ice crystals. His classification is based largely on shape. The types are the lanceolate, the discoidal, the solid hexagonal, the flower-like, the spandrelliform, and the coralline. The needle-like lanceolate crystals are invariably the first to form when water begins to freeze. All six forms can coexist and most forms pass through scallop and branch-like stages before assuming their characteristic shapes. In thick masses, ice seldom exhibits perfectly hexagonal crystals. Instead, the crystals are rounded or of irregular shape. The size of the component crystals has not been investigated thoroughly. Tarr and Rich (1912) describe pond ice crystals which are from 1 to 1-1/2 am in diameter and from 5 to 15 cm in length. Wilson and Horeth (1948) noted few crystals larger than 1 inch in diameter in Lake Michigan ice but the crystal diameter was always largest near the bottom of the layer. McConnel and Kidd (1888) describe ice from St. Moritz lake which was built up of vertical columns. The colu!mns were a centimeter or less in diameter and a foot or more in length (equal to the thickness of the ice). They state that each column was a single crystal and that the optic axis was horizontal. The latter condition seems unlikely. Ice crystal diameters can easily be measured under polarized light. Further information concerning crystal size can be obtained by means of an etching process. Ice surfaces can be etched by application of:infra-red (or ordinary) light so that the crystals are outlined and stand out from the surface in low relief. The phenomenon was first described by Hugi (1830-i) and was subsequently recognized and explained by Buchanan (1877 and quoted by Hawkes, 1930) on the basis of differential absorption of heat by

DEPARTMENT OF ENGINEERING RESEARCH Cb-27 UNIVERSITY OF MICHIGAN intercrystalline liquid brine. The presence of this intercrystallic material has been demonstrated for sea ice and also for ice formed from very dilute salt solutions. Data are shown for water originally containing 7 parts per million of C1 (as NaCl). Under these conditions, liquid brine is still present between the ice crystals at temperatures as low as -19'C. The ultimate freezing point of such brine is -21.72'C. Tarr and Rich (1912) state that eutectic proportions of 23.5 percent NaCl and 76.5 percent H 0 are reached at approximately -22'C (the cryohydric point). The remaining liquid crystallies in the eutectic proportions and the resulting mixture of salt and ice crystals is termed the cryohydrate. The ice which forms initially is formed from pure water. Hence, the residual solution is enriched in dissolved material. This intercrystallic brine becomes more and more concentrated until the eutectic proportions are reached. By these arguments, ice near the freezing point must contain some liquid brine in equilibrium with the ice. Buchanan explains the etching of ice crystals on the basis of the reater absorptive power of the brine, which results in melting at the crystal boundaries. Plyler (1925-1926) subsequently found that the intercrystalline layers have strong absorption bands in the infra-red spectrum, which substantiates Buchanan's theory. In these same arguments also lies the explanation for the formation of "rotten" ice in the spring. Rotten or "candled" ice is observed on any body of water covered with ice as soon as the ice is exposed to the sun at temperatures near the freezing point. Such ice appears solid but actually is not because the intercrystalline eutectic mixture, which serves as cement, is melted. Formation of Lake Ice Growth of an ice sheet:A quiet body of water, such as a lake or pond, vhen exposed to cold air will be cooled gradually at its surface, and convection currents will distribute the chilled water throughout the lake or pond until a uniform temperature of 4'C is reached. At this temperature water has its maximum density; further cooling is not accompanied by convection. Dorsey (1940) describes a convective cell in the atmosphere directly above the water surface, which is the mechanism for the transport of heat away from the water. The heat conductivity of soil and rock is so small that it is a minor or even negligible factor in the cooling process, except perhaps in very shallow water. Hence, loss of heat is more or less restricted to conduction and radiation to the overlying atmosphere. The importance of atmospheric convection is emphasized by Dorsey (1940) who states that air moving across a body of water gains heat and tends to rise. Thus a convective cell is initiated and a continual supply of cool air moves across the water. The temperature gradient and therefore the flow of heat are at all times directed upward. However, the normal air masses present during the time when freezing occurs are relatively dry. Water vapor is necessary for the absorption of radiant energy. It would seem, therefore, that convection currents would have to be maintained by conductiont from the water surface to the air. The

UNIVERSITY OF MICHIGAN insulating properties of air are well known and the writer doubts that conduction plays a major role in the cooling process. Devik (1944) has shown the relative importance of the various cooling processes. Assuming an air temperature of -100C and a water temperature of O0C, the heat loss b3 infra-red radiation is 13.8 cal/cm2/hour with clear sky and 4.9 cal/cm /hour with cloudy sky. Under the same conditions, convection (caused by conduction from water to air) will result in a heat loss from the water of 2.8 cal/cm /hour with no wind, and 11.5 cal/cm2/hour with a wind of 5 m/sec. Consideration of the heat loss due to evaporation assumes that the latent heat of vaporization is furnished entirely by the water. This is approximately true since the specific heat of water is approximately 4 times that of air and the thermal conductivity of water is approximately 20 times that of air. Devik states that the loss of heat due to evaporation, if the vapor pressure is 2.6 mm Hg and other conditions are as above, is 1.7 cal/cm2/hour if the air is calm and 7.7 cal/cm2/hour if the wind speed is 5 m/sec. The formation of an ice sheet is described by Barnes (1928), Hess (1904), Tammann and Dreyer (1934), Dorsey (1940), Parsons (1940), Devik (1944) and others. The shallow water at the edge of a lake is cooled to the temperature of maximum density before the deeper water near the center of the lake. Therefore, ice formation begins at the shore. Needles of ice (lanceolate form) shoot out from the shore and gradually enlarge until the surface is covered. From this time on, growth is accomplished by thickening. The extent of the initial ice sheet is controlled by the amount of water surface which has been cooled. If the entire surface of the lake is supercooled more or less simultaneously, the sheet of ice extends rapidly over the surface. In large bodies of water, such as the Great Lakes, it is doubted that supercooling reaches the center of the surface during a normal winter. The only record of complete freezing of one of the Great Lakes was given by Root (1944). However, no records are kept of the extent of the ice cover of the Great Lakes from year to year so that complete information on this subject is lacking. (See Kerry, 1947.) The orientation of the initial ice needles is in question. If, as Yoshida and Tsuboi (1929) suggest, directions parallel to the basal pinacoid are most favorable for growth, detached crystals will become platelike and will float with their optic axes vertical. However, Klocke (1879) has observed that with rapid and severe chilling the first needles are formed with optic axes horizontal. Subsequent growth occurs in the normal manner with the formation of crystals with vertical optic axes. When the first thin layer of ice becomes continuous, further growth is in the nature of thickening, either by freezing of water directly below the ice or by agglomeration of ice crystals on the under surface (Barnes, 1929). Accumulations of snow on the top of the ice also accomplish an increase in thickness. The rate of growth of lake ice:The rate of thickening of a sheet of ice is controlled by the air temperature, the effectiveness of radiation and conduction in removing heat, the presence of a snow cover on the ice, evaporation from the ice surface,

DEPARTMENT OF ENGINEERING RESEARCH Cb-29 UNIVERSITY OF MICHIGAN and other factors. Barnes (1928) states that radiation ceases when the ice forms. However, the thermal conductivity of ice is high (0.0057) and it is thought that radiation not from the underlying water surface but from the surface of the ice is an effective cooling process. However, it is apparent that the cooling of the water takes place largely by means of the conduction of heat through the ice, regardless of the manner in which the heat is disposed of from the ice surface. Various empirical relationships have been established, from which the rate of thickening can be determined. The following notation will be used in presenting such formulas: L = heat of fusion = 80 cal/gm Jo= density of ice = 0.9166 gm/cm3 K = thermal conductivity of ice = 0.0057 cal/sec.cm.-C 0 = temperature of the ice surface in 'C E = ice thickness. in cm t = time in seconds C = an experimentally determined constant. Vedel (1895) derived the following formula: where C is an experimentally determined constant and is of the order of 10 for cgs units. Tamura (1905) established a formula which tales into account more definitely the physical properties of ice: Barnes (1928) derived a useful formula for calculation, on the basis of analysis of observed data: Eu- _1 1 tKQ 'Lp Barnes' and Tamura's formulas can be compared on the basis of the time necessary for the formation of a layer of ice E cm thick: tBarnes - tTeamura = O E 0.000155 E The difference in results is largest for large thickness and for temperature near OC. The rate of thickening as derived from Tamura's formula is: dE Kg

Cb-30 DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN Therefore, the rate of thickening is greatest for low temperature and for thin,ide, as might be expected. Devik (1944) has shown that when a sheet of ice is thin, the temperature gradient through it is steep. The heat loss is great and the thickness increases rapidly. When the ice has attained considerable thickness, the temperature gradient is much smaller and the rate of thickening is decreased. If the ice is covered by a snoaw layer, the temperature gradient in the ice is a minimum, since the snow acts as an insulating layer. Barnes (1928) also gives a more refined development of formulas for the rate of thickening which attempts to take into account the effect of convection currents in the water, evaporation from the ice, and the effect of radiation. Table I was taken from Dorsey (1940) and was computed by means of Barnes' formula. TABLE I The Rate of Thickening of an Ice-8beet (From Dorsey) The body of the table comprises values of t, the time necessary to form a layer of ice 1 cm thick at a surface temperature of Q. 0 -5 -10 -20 -30 -40 1 64 mm 32 in 16 min 11 mn 8.0 min 2 2.9 hr 1.4 hr.43 min 29 min 21 min 10 1.79 da 21.4 hr 10.7 hr 7.1 hr 5.4 hr 15 3.80 da 1.90 d 22.8 hr 15.2 hr 11.4 hr 20 6.55 da 3.28 da 1.64 da 26.2 hr 19.7 hr 30 14.29 da 7.15 da 3.57 da 2.38 da 1.79 da 60 55.4 da 27.69 da 13.85 da 9,23 da 6.92 da 90 123.3 da 61.6 da 30.8 da 20.6 da 15.4 da

DEPARTMENT OF ENGINEERING RESEARCH Cb-3 UNIVERSITY OF MICHIGAN Metmorphism of an ice layer:The term "metamorphism" is used loosely in this connection. The development of rotten ice during the spring is well known, but apparently there are earlier changes in the physical structure of ice. Dobrowolski (1923) includes a sub-section on the metamorphism of an ice layer in chapter VIII. It was not possible to have this translated so correlation of his statements with those made here must remain a subject for future investigat ion. Barnes (1929) has stated that surface ice becomes coarser with age, the larger crystals growing at the expense of the smaller ones. Bentley (1907) discusses the structure of old ice more thoroughly. He states, "When closely examined, old ice, as a result of slight internal melting, or of changes of structure due to its being repeatedly subjected to cold and changes of temperature, often reveals traces of its former open crystalline 'pre-solid' character. Such old ice presents faint evidence of a cellular or honey-comb-like structure, the cell walls being mainly normal to the surface of the ice." Bentley attributes the formation of long slender air tubes to the aging process also. He reports that the air tubes "are arranged perpendicularly to its (the ice's) surface, but ofttimes parallel to and at the lines of intersection of two or more of the faintly outlined cell walls....the main cause, in most cases, must be attributed to the fact that such ice sheets undergo lateral expansion and contraction during and subsequent to solidifying. Such internal stresses tend to squeeze the air into the ice along the lines of fracture and of least resistance, i.e., into the so-called cell walls, or into their points of intersection. " This is the only detailed statement concerning changes in layer ice subsequent to formation which was found in the literature (with the exception of Dobrowolski).

Cb-32 DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN STRENGTH OF ICE Bibliography Crushing strength (strength in cor ression):Barnes, H. T. (1914), The crushing strength of ice: Roy. Soc. Can. Proc. and Trans., 3d ser., vol. 8, part III, p. 19. Barnes, H. T., Hayward, J. W., and McLeod, N. M. (1914), The expansiv force of ice: Roy. Soc. Can. Proc. and Trans., 3d ser., vol. 8, part III, p. 29. Barnes, H. T., (1928), Ice engineering, Renouf Pub. Co., Montreal, P. 54 Bell, G. C. (1911), Results of experiments on the strength of ice: Proc. Maine Socl Civil 5ng., vol. 1, pp. 41-46. Brown, E. W. (1926), Experiments of the strength of ice, Rpt. of the Joint Bd. of Eng. on the St. Lawrence Waterway Project, Appendix F, pp. 423 -453. Dorsey, N. E. (1940), The properties of ordinary water-substance, Reinhold Publ. Corp., New York, pp. 448-449. Finlayson, J. N. (1927), Tests on the shearing strength of ice: Can. Eng., vol. 53, pp. 101-103. K[oechlin, Rne' (1944), Lee glaciers et leur mecanisme, F. Rouge and Cie. S. A., Lausanne, p. 18. Hess, Hans (1904), Die Gletscher, pp. 22-23. Moseley, Henry (Canon of Bristol) (1870), On the mechanical properties of ice: Phil. Nag., vol. 39, 4th ser., p. 1. Roanowicz,., and Honilgmn, E. J. M. (1932): Forsch. Gebiete Ingenieurw., vol. 3, p. 99. Sharp, R. P. (1947), Suitability of ice for aircraft landings: Trans. Am. Geophys. Union, vol. 28, p. 111. Tarr, R. S. and Rich, J. L. (1912), The properties of ice: experimental studies: Zeitschr. Oletscherkunde, vol. 6, pp. 226-249. See also Nature, vol. 91, p. 307, 1913.

DEPARTMENT OF ENGINEERING RESEARCH Cb-33 UNIVERSITY OF MICHIGAN Tarr, R. S., and Von Engeln, O. D. (1915), Experimental studies of ice with reference to glacier structure and motion: Zeitschr. Gletscherkunde, vol. 9, p. 81. Von Engeln, O. D. (1915), Experimental studies and observations on ice structure: Am. Jour. Sci., vol. 190 (whole ser.), pp. 449-473. Vitman, F. F., and Shandrikov, N. P. (1938), Some investigations of the mechanical strength of ice: Trans. Arctic Inst., vol. 110, pp. 83-100. In Russian. Weinberg, Boris, (1936), Mechanical properties of ice: Assoc. Int. d'Hydrol. Sc., Bull. N23, p. 509. Weinberg, B. P. (Boris?) (1929), The influence of the temperature on the mechanical resistance of river-ice: Bulletin of the Central Geophys. Obs., vol. 2, pp. 22-53. In Russian with English abstract. Tension: - Barnes, H. T., Hayward, J. W., and McLeod, N. M. (1914), The expansive force of ice: Boy. Soc. Can. Proc. and Trans., 3d ser., vol. 8., part III, p. 29. Barnes, H. T. (1928), Ice engineering, Renouf Pub. Co., Montreal, pp. 214-217. Dorsey, N. E. (1940), The properties of ordinary water-substance, Reinhold Pub. Corp., New York, pp. 447-449. Finlayson, J. N. (1927), Tests on the shearing strength of ice: Can. Eng., vol. 53, pp. 101-103. Koechlin, Rene (1944), Les glaciers et leur mecanisme, F. Rouge and Cie S. A., Lausanne, p. 18. Moseley, Henry (Canon of Bristol) (1870), On the mechanical properties of ice: Phil. Mag., 4th ser., vol. 39, p. 1 Reusch, E. (1864), Beitrage zur Lehre vom Eis: Ann. d. Physik u. Chemie (Pogg.), vol. 121, pp. 573-578. Romanowicz, H., and Honigman, E. J. M. (1932): Forsch. Gebiete Ingenieurw., vol. 3, p. 99. Weinberg, Boris (1936), Mechanical properties of ice: Assoc. Int. d'Iydrol. Sc., Bull. N23, p. 509.

Cb-34 I DEPARTMENT OF ENGINEERING RESEARCH _b I7 4 UNIVERSITY OF MICHIGAN Shear strength: - Barnes, H. T. (1928), Ice engineering, Renouf Pub. Co., Montreal, pp. 217. Finlayson, J. N. (1927), Tests on the shearing strength of ice: Can. Eng., vol. 53, pp. 101-103. Dorsey, N. E. (1940), The properties of ordinary water-substance, Reinhold Pub. Corp., New York, pp. 448-450. Matsuyama, Motonori (1920), On some physical properties of ice: Jour. Geol., vol. 28, pp. 607-631. Moseley, Henry (Canon of Bristol) (1870), On the mechanical properties of ice: Phil. Mag., 4th ser., vol. 39, P. 1 Weinberg, Boris (1936), Mechanical properties of ice: Assoc. Int. d'Hydrol. Sc., Bull. N23, p. 509. Wilson, J. T., and Horeth, J. M. (1948), Bending and shear tests on lake ice: To be published in Trans. Am. Geophys. Union. Bending (determination of tensile strength): - Brown, E. W. (1926), Experiments on the strength of ice: Rpt. of the Joint Bd. of Eng. on the St. Lawrence Waterway Project, Appendix F, pp. 423-453. Hess, Hans (1902), Elastizitat und innere Reibung des Eises: Ann. d. Physik (Drudes), vol. 8, pp. 405-431. Tarr, R. S., and Rich, J. L. (1912), The properties of ice: experimental studies: Zeitschr. Gletscherkunde, vol. 6, pp. 226-249. See also Nature, vol. 9, P. 307, 1913. Weinberg, Boris (1936), Mechanical properties of ice: Assoc. Int. d'Hydrol. Sc., Bull. N23, p. 509. Wilson, J. T., and Horeth, J. M. (1948), Bending and shear tests on lake ice: To be published in Trans. Am. Geophys. Union.

DEPARTMENT OF ENGINEERING RESEARCH 3 UNIVERSITY OF MICHIGAN Additional references on the strength of ice (unexamined):Andreev, S. M. (1934?), Experiments of V. N. Pineghin in 1924 on the resistance of river ice. (No publication listed). In Russian. Andreev, S. M., and Arnold-Alabieff, W. J. (Computed in 1937), Tests of the strength of the ice of the Neva River and Gulf of Finland during 1930-36 made at the scientific meteorological ice station of the Hydrometeorological Service of the USSR. (No publication listed) In Russ ian. Bassin, M. M. (1934?), Test on the ice cover of the Svirj River on resistance to compression, shearing and bending during thaw in spring of 1934,, MS in Inst. Hydrotech. In Russian. KomarovskiJ, A. N. (1932), The structure and physical properties of the ice cover of fresh waters, Moscow-Leningrad. Mariutin, T. P. (1936), On the strength of sea ice: Meteorologia i Hydrologia, vol. 1, No. 2, pp. 70-73.

DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN Description and Tabulation of Experimental ]esults Crushing strength (strength in ccMpression):Determinations of the strength of ice in linear compression are swumarized in Table II. It is seen that complete descriptions of the conditions under which the determinations were made are usually lacking. The average of all the determinations listed in this table is 625 psi for those with the load applied parallel to the crystallographic axis but without regard for other sources of variation, and 607 psi for those made with the load applied perpendicular. This 1.5 percent variation is probably smaller thn the error introduced by ignoring the possibility of other controlling factors than the direction of application of the load. Brown (1926) states that the rate of application of the load and the temperature both affect the magnitude of the crushing strength. His measurements were made as nearly as possible at a constant rate of loading (1000 pounds per 2 seconds). His results show a significant variation with temperature. Similarly the results of Vitman and Shandrikov (1938) indicate increasing strength with decreasing temperature. The rate of loading is not stated and their results differ materially frcm those of Brown. (See figure 1.) Barnes (1914) reports that "the only effect of varying the position of the axis of the ice with respect to the direction of the pressure appeared to be the way the block burst." There is, however, some consistency in the relationship between the average values, with those measured parallel to the crystallographic axis being slightly higher (Barnes, 1914) and Bell, 1911). This variation is small and deviations from the normal are large for individual specimens. With regard to the effect of temperature, the results of Brown (1926) and Vitman and Shandrikov (1938) have been cited previously. Tarr and Von Engeln (1915) state in their conclusions that "variations of temperature within limits between 10~ and 25~F do not seem to exert a notable effect on the crushing strength of the ice." It is possible that the slow and variable rate of loading employed in their measurements obscured the effect of temperature. Barnes (1928) stresses the relationship between temperature and crushing strength although he points out that the increase in hardness observed by many may be only a reflection of the increase in viscosity with decrease in temperature. He comes to no conclusion and states only that further experimentation is desirable. The article by Weinberg (1929), which was not examined, would undoubtedly be pertinent in this connection. As indicated by Vitman and Shandrikov (1938), the structure and temperature history of the ice exert a dominant influence on crushing strength, with clear young ice being stronger than old porous ice. Weinberg (1936) gives a summary of determinations, mostly by Russian investigators. These values have been corrected to a temperature

DEPARTMENT OF ENGINEERING RESEARCH Cb-37 UNIVERSITY OF MICHIGAN of -3'C according to Weinberg's method (Weinberg, 1929). The data are reproduced in Table III. The sources of these data, which for the most part have not been examined, are listed in the original reference. Sharp (1947), in discussing thd suitability of ice for aircraft landings, uses an average value for crushing strength in theoretical calculations of the thickness of ice necessary to support a given load. Although there may be some question as to the applicability of crushing strength in this sense, his results for river, lake and sea ice should stand in co~rect relation to each other. He finds that the required thickness of young sea Ice for a given load is about three times that of river ice, with lake ice being intermediate.

Cb-38 DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN TABLE II Summary of Measurements of Compressive Strength of Ice Crushine Temperature Direction of Source of Data Type of Ice Strength F Force with psi respect to....... optic axis 356 32 normal Barnes (1914)1 River ice 370 32 parallel 590 25 to 40 normal Bell (1911)2 Artificial ice 659 25 to 40 parallel 363 32? Barnes (and McKay) River ice (1928) 300 28 normal Brown (1926) River ice 693.14 normal. 811 2 no rmal 400?? Barnes, Hayward, Assumed to be and McLeod (1914) an average value. 568 18(?)? Romanowicz and Artificial ice Honigman (1932)3 611 626 t tt 198 32? Tarr an. Rich Pond ice (1912) 1000 18 to 20 parallel Tarr and Von Engeln Pond ice (1915) and Von Engeln (1915) 3550... normal 312?? Koechlin (1944)? 308 above? Moseley (1870)? freezing

DEPARTMENT OF ENGINEERING RESEARCH Cb-39 UNIVERSITY OF MICHIGAN TABLE II (continued) Crushing Temperature Direction of ISource of Data. Type 6f ice Strength oF force with psi respect to I opt!ic axis 355? | Hess (1904)? 1800 11 parallel Finlayson (1927) River ice 1050 11 noal 700? parallel Vitman and River ice. (?)5 Shandrikov (1938) 640? normal River ice (T) 6 2.44? "prallel River ice 314? parallel " River ice (?) 3JA parallel " River ice (?)8 227? parallel River ice (?)9 473? parallel " River ice (?)l0 318 7 parallel River ice () 11 650 7 parallel. River ice (?)12 298? parallel | River ice (?)13 559? parallel " River ice (?)1)4 Remarks: - ~locks were heard to crack at a pressure approximately one half the ultimate crushing force. 2One of Bell's values was high enough (1128 psi) so that he considered it to be unrepresentative and ignored it in computing the average. 37-pm cubes were loaded at the rate of 43 psi/sec. 4I3se data xe considered unreliable by the authors due to the conditions under which the measurements were made. 5Average of 14 values. 6Average of 9 values. Young clear ice with numerous air bubbles.

UNIVERSITY OF MICHIGAN TABLE II (continued) 7Average of 13 values. Samples taken from lower, extremely porous layer. 8Average of 11 valuaes. Samples from upper, less porous layer. The ice was dark-colored but almost transparent. 9Average of 12 values. Samples from lower, porous layer. Light-colored opaque ice. Average of 11 values. Samples from upper, less porous layer. Average of 11 values. Lower, porous layer. 12 Average of 10 values. Upper layer, absolutely devoid of cavities. 13Average of 10 values. Lower, very porous layer. 14 -Average of 10 values. Middle layer, entirely non-porous, coarsely crystalline.

TABLE III Compressive Strength of Ice (after Weinberg) Upper layers Lower layers 0 parallel no parallel normal Authors n median mean n median mean n median mean n median mean Bezsonov (1923) 2 - 735 1 - 219 11 907 907 7 282 286 Pineghin (1923) (11) - 371 (11) - 307 (22) - 459 (22) - 381 Pineghin (1924) 16 296 482 11 333 340 25 472 508 22 310 411 Sergheev (1928) 2 - 523 2 - 351 4 476 481 4 228 234 Arnold-Alabieff - 11 539 550 - 32 430 442 (1929) Ice Station 11 478 430 11 407 376 28 566 587 22 275 272 (1930-36) Sergheev (1928) - 4 232 259 8 185 448 - - dir. of axis unknown 0 is the orientatin of the force with respect to the crystallographic axis. Values of crushing strength given in pounds per square inch. n is the number of observations (values in parentheses are assumed).

Cb-42 TABLE III (continued) Authors n median mean n median mean Vasenko (1899) 3 367 316 1 465 (natural ice) Makarov (1911) 8 533 566 (artificial ice) Bell (1911) 6 670 674 2 - 626 Barnes (and 7 280 311 9 306 300 xCI,~y) (1928) Krayger (1922) 7 867 2091 8 597 1070 Finlayson (1927) 7 - 1920 1 112 n median mean Moseley (1870) 1 309 asenkoo (1899) 13 375 326 (artificial ice) Gzovsky (Barnes (10) 209 1928) Mees (Barnes, 1928) (1) - 401 Frahling (Vasenko, (10) - 148 1899) Bell (1911) 3 - 882 Von Ingeln (1915) 3 - 933 Katanskij (1917) 2 - 1004 Brown (1926) (15) 483 Ludlow (Barnes, (20) 398 1928)

1000 800 a l I ZC 400 Z 200 0~ -10~ — 200 -30~ -40~ -50~ -60~ — 70~ TEMPERATURE-~C Figure I - Relation between crushing strength and temperature. ( - Brown (1926). -Vitman and Shandrlkov (1938)

Cb-14 DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN Tensile strength:Measurements of the tensile strength of ice are rare and recorded experimentation has been somewhat haphazard. Dorsey (1940) makes the following observations concerning tensile strength: "The tensile strength of ice may be expected to depend upon the structure of the ice and the direction of the line of stress with reference to the optic axis of the c ystal, or crystals, of which the specimen is composed." Reusch (1864) gives 967 psi as the maximum tension at the moment of breaking of a bar supported at the ends and loaded at the middle. The temperature at the time was a few degrees above freezing. Moseley (1870) made a series of measurements at temperatures of 70- to 757F. The average of eight determinations is 99.3 psi. Hess (1904) and Koechlin (1944), who is probably quoting from Hess, give 100 to 114 psi as the tensile strength. The conditions ft the determination are not given. Barnes, Hayward, and McLeod (114) state that the tensile strength of ice is probably less than 200 psi. Finlayson (1927) tested samples of artificial ice frozen in cement testing briquette molds and obtained the values 103 psi for water ice and 136 psi for ice formed from a mixture of snow and water. The observed variation from the average value was small. Barnes (1928) quotes the American Civil Engineers Handbook as giving 142 to 223 psi for tensile strength. Romanowicz and Honigman (1932), working at -8'C, obtained the three man values 229, 260 and 251 psi. The rate of loading was 1.42 psi/sec. (See Dorsey, 1940, p.448.) These results are summarized in Table IV. Weinberg (1936) summarizes Russian work on tensileastrength and also cites values by xoseley, Finlayson, and McConnel and Lidd. These data are reproduced in Table V. However, McConnel and Kidd were investigating plasticity and apparently did not break specimens for the purpose of determining tensile strength. Since Weinberg does not attempt to Justify the inclusion of a value based on McCanel and Kidd's work, this value has been omitted. All values in Table V have been corrected to -3-C by Weinberg's method. (Bee bibliography on crushing strength: Weinberg, 1929).

DEPARTMENT OF ENGINEERING RESEARCH CbD-45 UNIVERSITY OF MICHIGAN TABL 1IV Sunary of Tensile Strength Measurements Tensile Strength Author Remarks psi 967 Reusch (1864) Temp. a few degrees above freez 99.3 Moseley (1870) Temp. 70' to 757S 100-114 Hess (1904) and Koechlin (1944) less than 200 Barnes, Hayward, and McLeod (1914) 103 Finlayson (1927) Water ice 136 Ice formed from mixture of snow and water 144-223 Barnes (1928) quoting Am. Civ. 3ng. Handbook 229 (mean) Romanowicz and Honigman Temp. -8'C. Rate of loading: (1932) 1.42 psi/sec. 260 (mean) 251 (mean) 353 (max) 210 (m) n)

UNIVERSITY OF MICHIGAN TABLE V Russian Determinations of Tensile Strength (After Weinberg) Tensile Strength Author Remarks psi 182 Pineghin (1924) Upper layer, force parallel to axis. 102 Upper layer, force normal to axis. 172 Lower layer, force parallel to axis. 125 Lower layer, force normal to axis. 100-113 Schumacher, Moritz Values quoted by Hess (1904) and Apparently quoted Koechlin (1944) may also be from frcm Mousson (1858) Mousson. 100 Moseley (1870) Average of 8 specimens. 34 Fabian (1877) 179 Fruhling; quoted from AVerage of 9 specimens. Vasenko (1899) 90 + 46 Vasenko (1899) Average of two specimens. 193 + 20 Finlayson Average of 10 specimens. All values are corrected to -3'C according to Weinberg's method.

DEPARTMENT OF ENGINEERING RESEARCH Cb-47 UNIVERSITY OF MICHIGAN Shear strength:The shear strength of ice usually has been determined by the bending of ice bars or by punching or cutting ice bars in such a way that they rupture in shear. It can be shown in the case of an ice beam h cm high lying on two supports 1 cm apart and loaded at two points s cm apart (s 1) that if 21 - s/h is less than K where K is the ratio of tensile strength to shear strength (2 to 4), the beam will break in shear; if 21 - s/h is greater than K, the beam will break in tension. Moseley (1870) broke 13 specimens in shear and determined the average shear strength to be 111 psi. The temperature varied from 70e to 75"F. Using the same experimental procedure, Moseley (1869 and 1871) had previously found 74.5 psi to be the average shear strength of two specimens of ice formed by tamping small pieces of ice, a few at a time, into the test apparatus. The later (1870) value seems more applicable although it also was in part based on specimens formed by tamping. Finlayson (1927) measured the shear strength of 3 by 3 inch bars of river ice in a 30,000 pound Riehle testing machine. Loading was done at a slow rate but was otherwise unspecified. Specimens were tested both parallel and perpendicular -to the crystallographic axis and at various temperatures. He found 98 psi and 114 psi as average shear strengths parallel and perpendicular to the crystallographic axis respectively. Finlayson also found that the shear strength of artificial ice was about 80 percent that of river ice. Weinberg (1936) lists values of shear strength determined by Pineghin (also by Moseley and Finlayson). The average of Pineghin's values, corrected to -3eC, for all orientations and types of ice ea on the basis of the assumed (by Weinberg) number of observations is 106 psi. Wilson and Horeth (1948) tested the shear strength of artificial ice frozen to simulate lake ice and of Lake Michigan ice parallel to the crystallographic axis. The average of their results for 30 specimens of artificial ice was 99.7 psi. Thirteen specimens of lake ice had an average shear strength of 75 psi. These determinations were made at various temperatures (see Table VI). Table VI is a ccmpilation of the results cited above. Barnes (1928) and Dorsey (1940) also give brief summaries of shear strength determinations. The dependence of shear strength on temperature is not clearly established. The data of Wilson and Horeth (1948) and of Finlayson (1927) show a random relationship. On the other hand, Weinberg has apparently assumed, erroneously or not, that such a relationship exists, as indicated by his correction of PineghiA's data to -3'C. Matsuyama (1920) has noted a significant dependence of the rigidity modulus on temperature, which should be reflected in shear strength. He has expressed this relationship as follows: n = (Q.18 - 0.095t - 0.0020t2)x 106

Cb-48 DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF' MICHIGAN where n is the modulus of rigidity, not in cgs units as Matsuyama indicates, but in gms/cm2, and t is temperature in *C. It is to be expected that careful attention to the details of temperature, orientation, structure, absence of stresses other than shear, and rate of application of the load may eventually reveal a relationship between temperature and shear strength.

DEPARTMENT OF ENGINEERING RESEARCH Cb-49 UNIVERSITY OF MICHIGAN I TABLEI VI Shear Strength of Ice Shear Strengt TWap. Author Remarks 75 70-75 Moseley (1869) Specimeno formed by tamping and regelation. 111 70-75 Moseley (1870) Some specimens of solid ice, others formed by tamping and regelation. 88 corr. Pineghin to (Weinberg, 1938) Upper layer. Shear parallel to axis. -3'C Crystal axis. 102 27F Upper layer. Shear normal to axis. 107 Lower layer. Shear parallel to axis. 117 Lower layer. Shear normal to axis. 101 -3 Finlayson (1927) 102 14 98 5 99 | 8 Iiver ice. Shear parallel to axis. 115 10 Wilson and Horet (1948) ft 2Averages of 10 specimens each of arti i90 2 cial ice. Shear grallel to axis. 94 32 75 12-20 n Average of 13 samples of Lake Michigan ice. Shear parallel to crystal axis.

Cb-i DEPART5XlMENT OF ENGINEERING RESEARCHJ..... UNIVERSITY OF MICHIGAN TABLE VI (continued) Shear Strength Temp. Author Remarks Psi F. 131 -10 Finlayson 111 -9 108 n" 109 -3 110' 2 103 |4 River ice. Shear normal to crystal axis. 112 8 120 11 914 23 132 26 114 28 110 30

DEPARTMENT OF ENGINEERING RESEARCH C-51 UNIVERSITY OF MICHIGAN Bending (determination of tensile strength):The determination of tensile strength by the bending of ice bars has been undertaken by few investigators. As indicated in the section on shear strength, if a bar of ice h cm high lies on supports 1 cm apart and is loaded in two places s cm apart (s C 1), then if 21 - s/h is greater than K where K is the ratio of the tensile strength to the shear strength (2 to 4), the bar will break in tension. Compressive stress is present but the crushing strength is so high, comparatively, that the specimen breaks in tension only. The dependence of tensile strength as determined by bending on temperature seems clearly defined. The results of bending tests by Brown (1926) and Wilson and Horeth (1948) are consistent and are represented approximately by the relation: tensile strength (psi) 240 - 1.7t - 0.01t2 where t is the temperature in "F. (See Wilsonand Horeth, 1948). The results of Brown (1926) and Wilson and Horeth (1948) are given in Table VII. Weinberg (1936) has included a 8smry table marked "Mean values of the limit of plasticity of ice by bending" which can probably be interpreted as tensile strength determinations, All data as apparently corrected to -3'C by Weinberg. (See Weinberg, 1929; bibliography on crushing strength.) In rechecking Weinberg's sumary, the writer was unable to find certain values cited by Weinberg in the original references. This was true of values credited to Hess, Koch, Pfaff, Reusch, and Brown. The value credited to Tarr and Rich (1912) was found to agree with data in the original reference. New computations were made using the data given by Hess (1902). The average of these values is 502 psi for samples composed of single ice crystals. Only the values credited by Weinberg to Russian authors are given in Table VIII. The references cited were not examd and the data, therefore,ate not verified.

DEPARTMENT OF ENGINEERING RESEARCH |Cb -2 UNIVERSITY OF MICHIGAN TABIE VII Tasuile Strength of Ice as Determined by Bending Tenile.. Strength Temp. Author Remarks psi _F 155 28-30 Brown (1926) 9 specimens of St. Lawrence River ice. Crystals horizontal. 1814 28-30 12 specimens of St. Lawrence River ice. Cryetals vertical. 239 14-16 n 12 specimens of St. Lawrence River ice. Crystals horizontal. 214 14-16 12 specimens of St. Lawrence River ice. Crystals vertical. 180 32 Wilso nd Ioreth: 11 specimens of artificial ice. (1948) Crystals vertical. 214 12-20 " 6 specimens of Lake Michigan ice. Crystals vertical. 256 _-9 9 specimens of artificial ice. Crystals vertical. 158? Tarr and Rich 3 specimens of pond ice. Fourth specimen (1912) was apparently overloaded and breaking utress was too high.

DEPARTMENT OF ENGINEERING RESEARCH Cb-53 UNIVERSITY OF MICHIGAN TARIZ VIII Sunmary of Determzinations of Tensile Strength by Bending by Russian Investigators (After Weinberg) Layer Orientation No. of Obs. Mean Values (psi) axis II force 93 208 Upper axis lforce 25 230 199 unknown 31 151 axis II force 260 216 axis Lforce 62 185 Lower 228 229 axis Illength 30 458 unknown 79 179 axis I force 5 242 Not axis force 2 492 271 Indicated axis II length 4 8148 unknown 110 247 ~~~~- ~~~~~~~-~~~~~-~ ~

ENGINEERING RESEARCH INSTITUTE Cb-54 UNIVERSITY OF MICHIGAN ICE FRICTION Bibliography Arnold-Alabieff, V. I. (1938), The external friction of ice: Assoc. Int. d'Hydrol. Sci., Bull. N 23, pp. 563-570. Bowden, F. P., and Hughes, T. P. (1939), The mechanism of sliding on ice and snow: Proc. Roy. Soc. London, ser. A, vol. 172, pp. 280-298. Deeley, R. M. (1908), The viscosity of ice: Proc. Roy. Soc. London, ser. A, vol. 81, p. 250. McConnel, J. C. (1891), On the plasticity of ice crystal: Proc. Roy. Soc. London, vol. 49, pp. 323-343. Morphy, H. (1913), The influence of pressure on surface friction: Phil. Mag., 6th ser., vol. 25, p. 133. The following references were not examined. Arnold-Alabieff, V. I. (1937), Experiments on the external friction of ice: J. of Applied Physics, vol. 7, pp. 873'878. This is apparently a Russian Journal and the article is probably in Russian. Arnold-Alabieff, V. I. (1933), Ice service and ice stations of the Leningrad Hydrcoeteorological Direction, Report of the IV Hydrological Conference of the Baltic states, Leningrad, September, 1933. Bull. Arctic Inst., No. 5/6, 1935, Leningrad. The previous two references are taken from Arnold-Alabieff (1938) and apparently deal with the experimental procedures on which the data in the article cited are based. BelokonJ, P. N. (1938), On the coefficient of friction in the ice cover: Meteorologia i Hydrologia, vol. 4, pp. 116-131. Probably in Russian.

ENGINEERING RESEARCH INSTITUTE Cb-55 UNIVERSITY OF MICHIGAN Description and Tabulation of Experimental Data The power required to drive an ice-breaking ship onto a layer of ice is partly a function of the coefficient of friction between the plates of the ice-breaker and the ice. The question of ice friction has been investigated largely with reference to the process of ice skating. Pressure melting due to concentration of load on a small area of contact is thought to be effective in forming a film of water which serves as a lubricant between the ice and the blade of the skate. The friction between a flat, painted, rusted or otherwise roughened plate of metal (such as the plates of a ship) has been studied only recently and by few investigators. Of the papers cited, the three which were examined treated aspects of the latter problem. Morphy's (1913) treatment is the least applicable to the problem of ice-breaking. In order to investigate the static friction of ice, he placed a small aluminum sled on an inclined ice surface and, under varying loads, determined the angle at which sliding occurred. He found that the tangent of the angle at which sliding occurred was 0.36+0.01 if the load were less than 14.3 grams weight. If the load were greater than 15 grams weight, the tangent of the corresponding angle was 0.17+0.01. In each range the angle was found to be independent of the load. Bowden and Hughes (1939) investigated the friction of miniature ski of various materials moving in contact with ice. Their results demonstrate. the independence of the coefficient of kinetic friction from load, apparent area of contact, and speed of sliding. Temperature of the ice, however, affected the friction markedly, the friction rising as the temperature of the ice fell. Variation in friction of the ski of different materials indicated the dependence of friction on thermal conductivity of the ski material. This suggests frictional melting as an important factor in the formation of the water film during sliding. This theory, together with the non-dependence of friction on load and area of contact, minimizes greatly the effect of pressure melting in sliding under conditions comparable to those of the experiments. The authors agree qualitatively with Morphy by stating, "The F/W curve was at first linear showing that /- s was independent of the load, but at heavier loads the curve became concave to the load axis, showing that, becomes less at heavier loads." Tables IX to XI, all from Bowden and Hughes, show the relationship between friction and various quantities: Table IX shows the influence of area of contact on friction in the case of ice sliding on ice; Table X shows the effect of temperature on static friction for ice on ice; Table XI shows the effect of increasing the thermal conductivity and heat capacity of a hollow ebonite ski whose bottom was covered with a thin sheet of copper. This is done by filling the hollow ski above the copper sheet with mercury; Table XI shows further the effect of variable thermal conductivity; Table XII shows the influence of wax on the ski bottom; Table XIII shows the effect of temperature on the coefficient of kinetic friction between a waxed hickory ski and ice. The figures are all from Bowden and Hughes.

~Cb,6 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Table IX Relation between Friction and Area of Contact Apparent area Mean of crntact temperature Exp. no. cm. C / k 1 o.6 -1.4 0.019 2.3 -2.0 0.019 2 o.6 -3.0 0.017 2.5 -3.0 0.019 3 0.02 -1 to -10 0.016 3.1 -3.0 0.021 Table X Relation between Friction and Temperature Temperature, 0C 0 -12 -71 -82 -110 -, s 0.05-0.15 0.3 0.5 0.5 0.5 Table XI Relation between Friction and Thermal Conductivity of Ski A. k Metal (no mercury) (mercury on ski) kAg/fk Copper 0.022 0.031 1.4 0.027 0.034 1.3 0,032 0.043 1.3 0.032 0.054 1.7 Constantan 0.021 0.020 1.0 B. Load 200 g. Load 1000 g. Brass ski 0.010 0.005 Ebonite ski 0.025 0.010

ENGINEERING RESEARCH INSTITUTE Cb-57 UNIVERSITY OF MICHIGAN Table XII Influence of Waxed Ski on Friction Load 200 g. Nature of ski surface Temp. -30C Temp. -70C Unwaxed hickory 0.08 Waxed hickory 0.03 0.04 Unwaxed brass 0.030 0.05 Waxed brass 0.025 0.045 Table XIII Influence of Temperature on Kinetic Friction of Waxed Ski Temperature, C... 0 -3 ca.-10 -40 k 0.04 0.o9 0.18 0.4 30 (B) 20.. "QE A I / 0 ~ 0 200 400 600 w (GRAMS) Figure 2. The relation between the frictional force (F) and the load (W) for ice sliding on ice. Curve (a) mean surface temperature -3.3~C; curve (b) mean surface temperature -27.5~; curve (c) mean surface temperature 00.

.......Cb-58 | ENGINEERING RESEARCH INSTITUTE Cb-58 TUNIVERSITY OF MICHIGAN 0.12 320 GRAMS 650 GRAMS 0.04 0oC -40 - 80 -1200 I 608 MEAN TEMPERATURE Figure 3. The influence of temperature on the kinetic friction between ice surfaces. 0.12 0.0 0.04 0 000 -200 -400 -600 -800 MEAN TEMPERATURE Figure 4. The influence of temperature on the friction of brass, ebonite and ice sliding on ice.

ENGINEERING RESEARCH INSTITUTE Cb-59 UNIVERSITY OF MICHIGAN 6 ______..I 0 0 25 50 75 1X0 W (KILOS) Figure 5. The F/W relation for real ski on snow. Arnold-Alabieff (1938) studied ice friction according to the follaowing plan: Kind of friction: Friction of rest (static friction) Friction of motion (kinetic friction) Character of friction: Dry friction Fluid friction Friction with self lubrication Type of ice: fresh water, transparent, turbid, sea, polar, etc. Material: ship steel, smooth, new,old, rusty, painted; concrete, cement covering, etc. The following statements are of interest in a qualitative sense: "From the practice of navigation in ice it is known e.g. that ice mixed with snow and covered with it produces greater friction than that without snow." "Also crystalline ice should give different coefficients depending on the direction of the surface of friction relative to the axis of the crystals. This suggestion follows from the statements of McConnel (1891) and Deeley (1908) according to whom ice has a different viscosity dependent on direction to the axis of the crystals. For the same reason granular ice of crystalline structure ought to have a different coefficient of friction than that of monocrystalline ice with identical directions of the axes of separate crystals."

Cb-6.0 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN The results of experiments by Arnold-Alabieff are shown in Tables XIV and XV. Table XIV Coefficients of Friction Kind of Friction Coefficient of Friction Friction at rest.........30 - 0.50 Friction in motion........ 0.03 - 0.05

Table XV Coefficients of Friction for Various Surfaces and Types of Ice Neva ice Baltic Sea ice Kara Sea ice Kind of ice: Fresh water ice De-salted ice Polar salt-water ice Kind of Smooth Painted Smooth Painted Smooth Painted Friction metal metal metal metal metal metal Friction of rest 0.15-0.25 0.35-0.40 0.15-0.20 --- 0.15-0.25 0.30-0.35 Friction of motion 0.10-0.15 --- 0.10-0.15 --- 0.10-0.20 0.20!n

Cb-62 ENGINEERING -RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN HIARDNESS OF ICE Bibliography Andrews, T. (1886), Observations on pure ice and snow: Proc. Roy. Soc., London, Vol. 40, pp. 54-5149. Barnes, HI. T. (1928), Ice engineering, Renouf Publishing Co., Montreal, pp. 46-47. Blackwelder, B. (1940), The hardness of ice: Am. Jour. Sci., Vol. 238 (whole ser.), pp. 61-62. Dorsey, N. E. (1940), Properties of ordinary water substance, Reinhold Publishing Corp., pp. 450-451. Knoop, Frederick, Peters, C. G., and Emerson, W. B. (1939), A sensitive pyramidal-diamond tool for indentation measurements: Jour. Res. of the Nat. Bur. Standards, Research Paper 1220, Vol. 23. Palache, Charles, Berman, H., and Frondel, C. (1944), The system of mineralogy of J. D. and E. S. Dana, Vol. I, John Wiley and Sons, Inc., Nev York, p. 494. Rogers, A. F. (1937), Introduction to the study of minerals, 3rd Ed., McGraw-Hill Book Co., Inc., New York, p. 326. Teichert, Curt (1939), Corrosion by wind-blown snow in polar regions: Am. Jour. Sci., Vol. 237 (whole ser.), pp. 146-148. Winchell, Horace (1945), The Kloop micro-hardness tester as a mineralogical tool: Am. Mineralogist, Vol. 30, pp. 583-595.

ENGINEERING RESEARCH INSTITUTE cb-63 UNIVERSITY OF MICHIGAN Description and Tabulation of Experimental Data Hardness may be expressed as resistance to abrasion or as resistance to indentation. The two properties are not necessarily related in a given substance. Resistance to Abrasion: Moha' scale is ccmaonly used to express hardness in terms of resistance to abrasion. This scale is applied almost exclusively to the determination of the relative hardness of minerals. Textbooks on mineralogy (Rogers, 1937 and Palache and others, 194) give 1-1/2 as the relative hardness of ice on Mobs' scale (between talc and gypsum). Later work by Teichert (1939) and Blackwelder (1940) in connection with the corrasive power of wind-blown snow shows an increase in relative hardness with a decrease in temperature. Table IVI gives a sueary of the values of the hardnesses of ice for different temperatures. Temperature Relative Hardness Equivalent Source of Data *,,C ~-e Mohs' Scale Mineral? 1-1/2 Talc to (Roers (1937) vpsu (Palache ad others (194) -15 2-3 NypsMu to Teichert (1939) calcite -30 3-4 Calcite to Teichert (1939) fluorite -40 approx. 4 Fluorite Teichert (1939) -44 exactly 4 Fluorite Teichert (1939) -78.5 approx. 6 Orthoclase Blackwelder (1940) Table XVI. Variation in relative hardness of ice (Mobs' scale) with temperature. Resistance to Indentation: Hardness is expressed in terms of resistance to indentation as Brinell hardness, Rockwell hardness or Shore scleroscope hardness. Various other tests have been suggested recently, some of which are merely refinements of these. In the Brinell hardness testing machine an impression is left in the sample being tested by a steel ball under a given load applied for a given length of time. The Brinell hardness number is the ratio of the load in kilograms teo the area of ithe indentation tin square millimeters. The Rockwell tester uses either a steel ball or a 120 -

Cb-64 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN diamond cone under minor and major loads applied successively. The increase in the depth of the impression formed by the minor load to that formed by the major load determines the hardness. The value of hardness is read directly from the machine. The Shore scleroscope, which in a strict sense does not measure resistance to indentation employs a diamond-tipped hammer which is allowed to fall on a polished surface of the test sample. The height of rebound measured on an arbitrary linear scale is the hardness. Dorsey (1940) states that no hardness data determined by the above methods have been found but that no difficulty should be anticipated in determining the Shore scleroscope.hardness of ice. As for the Brinell test, he states: "The Brinell hardness number for ice would have no significance except at temperatures so low that the rate at which ice yields progressively under the action of a constant load applied to a small area of its surface is negligible; say, at temperatures below -30~C." A more recent method of testing hardness has been developed by Knoop and others (1939). This method employs a diamond-shaped indenter, the ratio of whose diagonals is one to seven. It is stated that this method of determining hardness is applicable to brittle materials and to those with low elastic limits since recovery of the impression takes place only along the minor diagonal. This enables the unrecovered value of the major diagonal and hence the unrecovered area of the indentation to be determined. Instruments manufactured for applying this test employ impact loading. Thus, in testing ice, the plastic deformation which occurs upon slow application of a load would be minimized, lending greater significance to the results than is the case for Brinell and Rockwell hardness values. No measurements on ice were reported in the paper cited. The hardness of the minerals of Mohs' scale as determined by the Knoop method is shown in Table XVII for purposes of comparison with Table XVI. Mineral Enoop Hardness Mohst Hardness Gypsum 32 2 Calcite 135 3 Fluorite 163 4 Apatite 360-430 5 Orthoclase 563 6 Quartz 710-790 7 Topaz 1250 8 Diamond 8000-8500 10 Table XVII. Relation between Knoop hardness and Mobs' scale.

ENGINEERING RESEARCH INSTITUTE-6 UNIVERSITY OF MICHIGAN Further discussion of the applicability of this method of testing is given by Winchell (1945). Measurements of the penetrability of ice by Andrews (1886) were interpreted by him as representing the relative hardness of ice. Barnes (1928) refers to these determinations in his discussion of the plasticity of ice. Dorsey (1940) points out the evident connection between penetrability and progressive deformation under a constant load. Andrews' measurements were made by allowing a 0.292 by 16 inch blunt steel rod to penetrate ice under a constant load of 181-1/2 pounds. Observations of the rate of penetration were made at temperatures from -35"F to 32'F. His observations show that ice is very resistant to penetration from -40oC to about 6~C. From 60C to the melting point, the resistance rapidly decreases.

Cb-66 I UNIVERSITY OF MICHIGAN ELASTIC CONSTANTS Bibliography Bevan, Benjamin (1826), Account of an experiment on the elasticity of ice: Phil. Trans., ser. B, vol. 116, p. 304. Boyle, R. W., and Sproule, D. C. (1931), Velocity of longitudinal vibrations in solid rods (ultrasonic) method with special reference to the elasticity of ice: Can. Jour. Research, vol. 5, p. 601. Brockamp, B., and Mothes, H. S. (1930), Seismische Untersuchungen am Pasterze-Gletscher: Zeit. f., Geophysik, vol. 6, pp. 428-500. Brown, E. W. (1926), Experiments on the strength of ice, Rpt. of the Joint Bd.. of Eng. on the St. Lawrence River Waterway Project, Appendix F, pp. 423-453 Dorsey, N. E. (1940), Properties of ordinary water-substance, Reinhold Publishing Co., New Tork, pp. 450-451. Ewing, M., Crary, A. P., and Thorne, A. M. (1934), Propagation of elastic waves in ice, Part I: Physics, vol. 5, p. 165; Part II, p. 181. Fabian, 0. (1877), Uber die Dehnbarkeit und Elastizitat des Eises: Repert. Exp. Phys. (Carl), vol. 13, pp. 447-457. Hargis, C. D. (1922), The viscosity and rigidity of ice: Phys. Rev., 2nd ser., vol. 19, pp. 526-527. Hess, Hans (1902), Elastizitat und innere Reibung des Eises: Ann d. Physik (Drudes), vol. 8, pp. 405-431. Hess, Hans (1904), Die Gletscher, pp. 21-22. Koch, K. R. (1885), Beitrage zur Kenntniss der Elastizitat des Eises: Ann. d. Physik, u. Chemie (wied), vol. 25, pp. 438-450. Koch, K. R. (1913), Iber die Elastizitat des Eises: Ann. d. Physik, 4th ser., vol. 41, pp. 709-727. Koch, K. R. (1914), tber die El~stizitat des Eises: Ann. d. Physik, 4th ser., vol. 45, pp. 237-258. Kohler, R. (1929), Beobachtungen an Profilen auf See-Eis: Zeit. f. Geophysik, vol. 5, pp. 314-316.

DEPARTMENT OF ENGINEERING RESEARCH Cb-67 UNIVERSITY OF MICHIGAN Matsuyama, Motonori (1920), On some physical properties of ice: Jour. Geol., vol. 28, pp. 607-631. Moseley, Henry (Canon of Bristol) (1870), On the mechanical properties of ice: Phil. Mag., 4th ser., vol. 39, p. 1. Moseley, Henry (1871), On the mechanical impossibility of the descent of glaciers by their weight only: Phil. Mag., 4th ser., vol. 42, pp. 138-149. Pineghin, N. V. (1927), On the changes of the modulus of elasticity and of Poisson's ratio by river ice at compression: Nauka i Tekhnika (Science and Technics), vol. 5, no. 3-4 (11), pp. 1-6. In Russian. Reich, M. and Stierstadt, 0. (1931), Messung der Schallgeschwindigkeit von Stoffen in festen und geschmolzenen Zustand: Physik. Zeit., vol. 32,. pp. 124-130. Reusch, E. (1864), Beitrbge zur Lehre vom Eis: Ann. d. Physik u. Chemis (pogg.), vol. 121, pp. 573-578. Reusch, E. (1880), Elastizittt des Eiases: Ann d. Physik u. Chemie (Wied.), vol. 9, pp. 329-334. Sokolov, J. A. (1926), Young's modulus for a natural ice crystal: Jour. Appl. Physics, vol. 3, pp. 275-277. In Russian. Not examined. Swift, H. W. (1926), Determination of the modulus of elasticity by dynamical methods: Phil Hag., 7th ser., vol. 2, pp. 351-368. Tammann, G. (1902), Ueber die Ausflusn geschwindigkeit krystallisirter Stoffe. Die Ausflussgeswindigkeit des Eises und seine Schmelzcurve: Ann. d. Physik, 4th ser., vol. 7, pp. 198-224. Thornton, W. M. (1919), The thermal conductivity of solid insulators: Phil. Mag., 6th ser., vol. 38, pp. 705-707. Trowbridge, John, and McRae, A. L. (1885), Elasticity of ice: Am. Jour. Sci., vol. 129 (whole ser.), pp. 349-355. Weinberg, B. (1907A), ifber den Koeffizienten der innere Reibung des Gletschereis und seine Bedeutung fur die Theorien der Gletscherbevegung: Zeit. f. Gletecherk., vol. 1, pp. 321-347. Weinberg, B. (1907B), Uber die innere Reibung des Eises, II: Ann d. Physik, 4th ser., vol. 22, pp. 321-332. Weinberg, Boris (1936), Mechanical properties of ice: Assoc. Int. d'Hydrol. Sci. Bull. N. 23, pp. 509-535.

cb-68 DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN Description and Tabulation of Experimental Data Data for the computation of Young's modulus and the modulus of rigidity have been derived from the bending of bars of ice, from bars of ice mounted as torsion pendulums, from the longitudinal extension of ice samples under a load, by determining the pitch of the tone given off by a bar of ice vibrating at its resonant frequency, by compression, and by determining the velocity of transmission of longitudinal and transverse waves through ice. The dynamic methods which make use of longitudinal and transverse vibrations have thus far given much more consistent results than the static methods involving bending, compression, and tension. Weinberg (1936) states that "it will be safer to assume that (on the basis of) our present knowledge on the moduli of elasticity of ice we can only say that Young's modulus is of the order 700 to 800 kg/mm2 and the modulus of rigidity of the order 250 to 300 kg/mm2." However, his compilation of values does not include the important dynamic determinations of Boyle and Sproule (1931) or of Ewing, Crary, and Thorne (1934) which Dorsey (1940) states are the most accurate available. The values determined by Boyle and Sproule for Young's modulus vary from 918 to 1070 kg/mm2. This variation, according to them, is much smaller than that observed in the results of static tests but is too large to be assigned to instrumental error. They list the possible sources of error in applications of their ultrasonic method as follows: "(1) Error in measurement of the length of the ice rods; (2) ErrOr due to possibly mistaking the identity of one mode or type of vibration for a totally different mode or type, e.g., assuming that the fifth mode of transverse vibration is the first mode of longitudinal vibration; (3) Error in measurement of frequency." (1) and (3) together were not over one per cent. An error in (2) would result in a value of velocity which was in error by some multiple of the true velocity; in practice this type of error is easily avoided. Weinberg (1936) summarizes the possible causes of variation in experimental results. Most tables of values are grouped so as to take into account the first three, but, according to Weinberg, most authors ignore the last five possible causes of variation in reporting their results. The causes are: 1. Type of ice. 2. Relation between the direction of application of the force and the direction of the crystallographic axis. 3. The temperature at which the experiments are done. 4. The conditions of formation of the ice. 5. The temperature history of the ice from the time of formation to the time of the test. 6. The rate of application of the load. 7. The size of the sample and the precision of its preparation. 8. The character of the pressing surface. Presumably this refers to the method of application of the load. To this list might be added such factors as salinity in the case of brackish and sea ice, and the elastic history of the sample.

DEPARTMENT OF ENGINEERING RESEARCH Cb-69 UNIVERSITY OF MICHIGAN Most of these factors have been taken into account by one or the other of the investigators on ice but in none of the sets of experimental results examined by the writer were even a majority of the factors considered by a single author. For instance, Brown (1926) has investigated the effect of the rate of application of the load in static determinations of Young's modulus (figures 1 to 5) and has carefully noted the sizes of the samples used in his tests. At the sam time he fails to consider the fact that in most cases his apparatus has exceeded the elastic limit of ice and hence is not measuring a true value of Young's modulus. On the other hand, most of the investigators who have employed dynamic methods involving vibrations have not accounted for the possible effect of variation in frequency of vibration, i.e., the rate of application of the load, on the values of the elastic constants. By applying Maxwell and Schvedoff's (see Viscosity and Plasticity of Ice, Maxwell (1868) and Schwedoff (1890)) theory of relaxation to ice Weinber computes the elastic limit of sheet ice (Neva River) to be 0.57 kg/cmt and of glacier ice to be 0.09 kg/cm2. On the basis of this determination he proposes a valid objection to most determinations of the moduli of elasticity. He says, "almost all authors have been investigating ice beyond its limits of elasticit and.....this one circumstance does not permit us to consider their results on the moduli with full confidence." Most authors have not measured Young's modulus of an ice sample under different loads. However the results of Brown (1926) show decreasing values of Young's modulus with increasing load. It appears that the elastic limit has been exceeded in this case. The above objection may apply equally well to dynamic determinations of elasticity. The tables of values of the various elastic constants contain the more important data on this subject. Tables XVII and III (after Dorsey) include the results of static and dynamic determinations of Young's modulus. Table XX lists static and dynamic determinations of the modulus of rigidity. The single value which was determined by dynamic methods (Ewing, Crary, and Thorne, 1934) is as yet (1940) unsupported but, according to Dorsey, is to be preferred over the static values. Table XXI lists values of Poisson's ratio. Weinberg (1936) has erroneously quoted Koch (1914) as having given 0.314 +0.007 and 0.248 +0.007 for Poisson's ratio. The values given by Koch refer instead to the ratio of the modulus of rigidity to Toung's modulus in a given sample of ice. Figures 6, 7 and 8 (from Brown, 1926) show the relationship between Young's modulus and the rate of application of the load in compression tests. Figures 9 and 10 show a similar relationship for bending tests. These tests were carried on under various temperature conditions as indicated in the figures.

Cb-70 TABZLE XVII Young's Modulus - Dynuamic Determation (after Dorsey) I. Longitudinal vibrations. Unite of I - kg wt/ma2. O = angle between optic axis and length of sample. t a temperature *C. 3 0 t Source Type of ice 9k7 0 -26 Boyle and Sproule(1931) River ice 967 0 -26 n 1|040 0 -26 n n 1110 0 -26 970 0 -26 " n 900 145 -26 n I I 990 90 -26 f 915.90 -26 " 990 90 -26 f " - 945 90 -26 n " " 877 90 -5 Trowbridge and McRae(1934) Fresh pond ice 960 90 -4 Reich and Stierstadt(1931) Artificial ice II. Transverse vibrations 622 t -13(?) Trowbridge and McRae(19314) Artificial ic 884 90 -7 Koch (1885) Lake ice 236 90 0(?) Reusch (1880) Artificial ic 710 90 Kohler (1929) Lae ice -

Cb-71 TABLE XIX Young's Modulus of Ice - Static Method Bending of Beams (after Dorsey) Unit of E = kg/mm2, temperature = t-C. E, optic axis parallel to t Author Type of Ice len.th width depth 182 383 0 to -1 Hess (1902) Single crystal 59 -2 to -5 254 418 -1 to -5 67 194 336 1 185 60 92 -3.5 Matsuyama (1920)? E, orientation of optic axis with respect to length: parallel normal 642 -5.4 Koch (1885) Lake ice 860 -5 to 7 Trowbridge and ice McRae (1885) 609 622 -6.5 to -7.8 Koch (1913) Clear lake ice 656 -6.5 to -7.8 Koch (1913) 696 696 -9 Koch (1885) Lake ice 1120 958 (?K) - Koch (1914) Lake ice 950 (?) Moseley (1870) (?) 500o (?) Bevan (1826) (?)

Cb-72 ABLE XX Rigidity of Ice (after Dorsey) N = modulus of rigidity in kmegadynes/cm2 radian - 1019 kg/cm2 radian. t - temperature in *C. A. Dynamic Method. N 91.7 = 0.5; t = -5 to -15; Ewing, Crary, and Thorne (1934); artificial ice. B. Static Method., orientation of ~ptic aB with ~t Author Type of ice espect to length: arallel normal 27.2 29.4? Koch (1914) Lake ice 10 0 Weinberg (1907B)? 17 -5 " 1.6 -7.5 Matsuyama(1920) Artificial ice 1.8 -6 orientation unknovn 28.2 t Hargis (1922) Artificial ice 27? Kohler (1929) Lake ice glacier ice 8 0 Weinberg (1907B)? 34 -5 7 I - ~~~~~~ _ _ g I _ i~~~~~~~~~~

Cb-73 TABLE XXI Poisson's Ratio for Ice Poisson's Ratio Author Remarks 0.365 + O.007 lwing, Crary, Computed from the velocity of and Thorne (1934) longitudinal waves in the range -5'C to -15'C. 0.30 Kohler (1929) Computed from horizontal velocity of waves in an "isotropic" ice sheet. 0.38 + o.49 Weinberg (1907A) Computed from static observations. Direction of extension not indicated. 0.326 Pineghin (1927) Computed from two different series of observations. Static 0.358 + 0.047 Pineghin (1927) method-either bending or longitudinal compression. 0.361 Brockamp and Computed from velocities of Mothes (1930) longitudinal and transverse waves in glacier ice. Weinberg thinks it applies to river ice also.

I~. d RANGE OF LOAD I. 0 I' N Ne I0 VALU ESI OF E FO ICEI AT ROF O 3-F. ST. LAWRENCE WATERWAY --. sDIAGRAM SHOWING I8 I_ RESJLTS OF EXPERIMENTS ON PHYSICAL PROPERTIES OF ICE W 7 -S...TO ACCOMPANY REPORT OF JOINT BOARD -3 7 OF ENGINEERS, DATED NOV. 16, 1926 t r 4h 7th 8 cr ist 2d 3 rd 4 th 5th 6,th 7 th 8/h INCREMENT OF 1000 LBS. INITIAL LOAD 250 LBS. Fig. 6. Relationship betwen Young's modulus (E) and rate of application of load. Temperature 27 to 3~F. Compression tests. Fro Brown (1926).

I I I. I.RANGE OF LOAD...-J0.- J m m.. maP ~ 0,,, o ' ' 9 3 I1 0 0 go o6 o o o o Ic D\, 0 0 _ "i9 ' - ST. LAWRENCE WATERWAY ~~~~~z \\ x: O~~~~~~~DIAGRAM SHOWING cico 8 5 nRESUTS OF EXPERIMENTS ON 7 IPHYSICAL PROPERTIES OF ICE W_ 7 TO ACCOMPANY REPORT OF JOINT BOARD <" I I I I I QHVSIjAL PROPEF~TI~S " - sL l ]\ N i \ v P F ENGINEERS. DATED NOV. 16. j9 IsSt nd rd th 5th th 7th 8 th 9th INCREMENT OF 1000 LBS. INITIAL LOAD 250 LBS. Fig. 7. Relationship between TYon's duus ) ad rt appliti of loa ITeperature 1r to li'?. Compression tests. r (1926). I t ~~~cs Tempeaaa~~~aaa~~rature Ik~~F to i~"F 0ompreemton teltin. Fro Brovn (1~'26).

I '-RANGE OF LOAD I. I* ar r 9 5T 0 0 00 tD0o 2 c~~ cW cm o 0. ~~ ~~ ~~~0 0 0 0 in to I VAWES OF "E" FOR ICE AT 280F TO 30'F 2 b 3 _ n 1\ I I ST LAWRENCE WATERWAY 0 5 DIAGRAM SHOWING W.J2 RESULTS OF EXPERIMENTS ON 3 A PHYSICAL PROPERTIES OF ICE TO ACCOMPANY REPORT OF JOINT BOARD _______ OF ENGINEERS, DATED NOV. 16,1926 33 J2 E-. 1st 2nd 3 rd 4 th 5th 6 th 7th INCREMENT OF 1000 LBS.-INITIAL LOAD 250 LBS. Fig. 8. Relationship between Youmg's modulus (3) and rate of application of load. Temperature 28&F to 301F. Comnpression tests. From Brown (1926).

o ~ CRYSTALS- HORIZONTAL Co BEAM TESTS ---14F TO 16~F OCRYSTALS-VERTICAL o:: I,5 HH FIRST STAGE - -- SECOND STAGE -I Lo a Ilt 5 10 O20 40 LOADING RATE IN SECONDS Fig. 9. Relationship between Young's modulus (E) and rate of application of load. Temperature 14'F to 167~. Bending tests. Frau Brown (1926). = | | | | | | | | | | ~ CRYSTALS- HORIZONTAL o- - - | | | - | BEAM TESTS --- 28 TO 30F. ' o CRYSTALS-VERTICAL ~l~-" 0 ~~~~~~ ~. --- SECOND STAGE 7 2 I - 19 0 5 I4 20 0 LOADING RATE IN SECONDS Fig. 10. Relationship between Young's modulus (E) and rate of application of load. Temperature 28~F to 30'F. Bending tests. From Brown (1926).

UNIVERSITY OF MICHIGAN VISCOSITY AND PLASTICITY OF ICE Bibliography Andrews, T. (1886), Observations on pure ice and snow: Proc. Roy. Soc. London, Vol. 40, pp. 544-549. Bianconi, J. J. (1876), Nouvelles experiences sur la flexibilite de la glace: Compt. Rend., Vol. 82, p. 1193, Deeley, R. M. (1908), The viscosity of ice: Proc. Roy. Soc. London, ser. A, Vol. 81, p. 250. Deeley, R. M., and Parr, P. H. (1913), The viscosity of glacier ice: Phil. Mag., 6th ser., Vol. 26, pp. 85-111. Dorsey, N. E. (1940), Properties of ordinary water-substance, Reinhold Pub. Corp., New York, pp. 451-457. Hargis, C. D. (1922), The viscosity and rigidity of ice: Phys. Rev., 2nd ser., Vol. 19, pp. 526-527. Hess, Hans (1902), Elastizitit und innere Reibung des Eises: Ann. d. Physik (Drudes), Vol. 8, pp. 405-431. Hess, Hans (1904), Die Gletscher, pp. 14-21. Mallet, R. (1845), On the brittleness and non-plasticity of glacier ice: Phil. MNag., 3rd ser., Vol. 26, pp. 586-593. Maxwell, J. C. (1868), On the dynamical theory of gases: Phil. Nag., 4th ser., Vol. 35, pp. 129-145, 185-217. McConnel, J. C., and Kidd, D. A. (1888), On the plasticity of glacier and other ice: Proc. Roy. Soc. London, Vol. 44, pp. 331-367. McConnel, J. C. (1891), On the plasticity of an ice crystal: Proc. Roy. Soc. London, Vol. 49, pp. 323-343. Pfaff, F. (1875), Versuche uber die plastizitaXt des Eises: Ann. d. Physik. u. Chemie (Pogg.), Vol. 155, pp. 169-174. Reusch, E. (1864), Beitrige zur Lehre vom Eis: Ann. d. Physik. u. Chemie (Pogg.), Vol. 121, pp. 573-578. Schwedoff, T. (1889), Jour. d. Phys., 2nd ser., Vol. 8, pp. 341-359.

ENGINEERING RESEARCH INSTITUTE | UNIVERSITY OF MICHIGAN Schwedoff, T. (1890), Jour. d. Phys., 2nd ser., Vol. 9, PP. 34-46. Thomson, J. (1857), On the plasticity of ice as manifested in glaciers: Proc. Roy. Soc. London, Vol. 8, pp. 455-457. Tyndall, John, and Huxley, T. H. (1858), On the structure and motion of glaciers: Phil. Mag., 4th ser., Vol. 15, p. 365. Weinberg, Boris (1905), Uber die innere Reibung des Eises: Ann. d. Physik (Drudes), Vol. 18, p. 81. Weinberg, Boris (1907A), Uber die innere Reibung des Eises: Ann. d. Physik, 4th ser., Vol. 22, pp. 321-332. Weinberg, Boris (1907B), IUber den Koeffizienten der innere Reibung des Gletschereis und seine Bedeutung for die Theorien der Gletscherbewegung: Zeit. f. Gletecherk., Vol. 1, pp. 321-347.

Cb-80 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Description and Tabulation of Experimental Data Ice is a plastic solid in that it possesses an elastic limit for small forces but is continuously deformed by stresses exceeding the elastic limit. Dorsey (1940) follows Maxwell's (1868) treatment as extended by Schwedoff (1889-90) in his discussion of the viscosity and plasticity of ice. He considers a plastic solid, ice, bounded by two parallel plates, one of which is moving relative to the other. A velocity gradient dV/dx is established in the solid and each plate experiences a tangential drag of P units per unit area. The elastic limit is taken as p units per unit area. Equation (1) is a statement of the proportionality of the tangential drag to the velocity gradient: P - p dV (1) dx The constant of proportionality,/A, is the viscosity and is a characteristic property of the solid, being independent of the other variables. Equation (1) may be taken as a definition of a plastic solid. Unless P exceeds p, dV/dx is zero. Dorsey further chooses to denote the usual expression Pdx/dV as apparent visco sit, e. Equation (2) relates viscosity and apparent viscosity: e =/A P dx (2) dV Weinberg (1905, 1907A) was the first to investigate quantitatively the inverse relationship between the apparent viscosity and the velocity gradient. He expressed his experimental data by means of an equation of the form of (3): "ie = A'o((aL ) t(3) t where t is the temperature in ~C, YW is the rate of shear in radians per second, and4,o is the value of viscosity at 0~C and I/v=o. Dorsey cites several irregularities in application of this formula and has recomputed the values which appear in Table XXII. Weinberg (1907A, 1907B) also observed that when the stress is in the nature of a torsion, the elastic limit of ice is not a function of temperature. J. Thomson (1857), Pfaff (1875), Bianconi (1876) and Andrews (1886) maintain, on the other hand, that ice is a perfectly viscous solid (elastic limit equal to zero). Mallet (1845) has also attempted to demonstrate the non-plastic nature of ice. From experimental data on the bending of rectangular bars, Hess (1902, 1904) deduced that for moderate loads,+ e increases with time, the rate being essentially constant after five minutes. However, under loads approaching the breaking strength,45 showed a decrease with time. Hess'

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN. values of apparent viscosity were on the order of 1/100 as great as those of Weinberg (Table XXIV). McConnel (1891) made similar observations and Deeley (1908) computed from his data the viscosity and found it to be of the same order as Hess' results. (Dorsey was unable to reproduce these computations.) On the basis of these data, Dorsey has estimated that the apparent viscosity measured parallel to the optic axis is perhaps a hundred times that measured normal to the optic axis. Deeley and Parr (1913) have computed viscosities from data by McConnel and Kidd (1888) which illustrate dependence on temperature, the structure of the ice andthe direction of the shear (Table XXIII). Hargis (1922) has computed the viscosity from data obtained by mounting a cylinder of ice as a torsion pendulum. The values varied from 3.8 x 109 to 6.33 x 109 poises. Dorsey summarizes the available data on viscosity of ice as follows: 1. None of the available data for the plasticity or for the viscosity of ice are entirely satisfactory. 2. Values ofAe derived from the bending of bars are of the order of 1010 poises, those for axial torsion and those from the longitudinal stretching of bars are of the order of 1012 poises. 3. Although McConnel's and McConnel and Kidd's data indicate thatA4 for shear parallel to the optic axis is about 100 times as great as lor shear perpendicular to that axis, Hess' data indicate that the difference is slight. 4. The value of_,e increases as the rate of shear decreases (Table I). 5. When the stress is kept constant,Ae increases with the time the stress has been applied. Whether this involves other phenomena than those pertaining to the variation with the rate of shear cannot be determined from the data now available. 6. The value ofA increases very rapidly as the temperature decreases, a decrease of 10~C being accompanied by a five-fold increase inc for river ice, and a 26-fold increase for glacier ice. This increase in ( tcauses a marked, but in general a smaller, increase.in/-. 7. An attempt to fit the data of Hess, of M~Connel, and of McConnel and Kidd to Weinberg's equation has been unsuccessful. 8. Owing to the absence of important data, to significant variations in the procedures followed, and to variations in the structure and the purity of the ice used, it is impossible to correlate satisfactorily the data obtained by different observers.

ENGINEERING RESEARCH INSTITUTE Cb-ce UNIVERSITY OF MICHIGAN Table XXII Viscosity of River and Glacier Ice (Froma Dorsey, 1940, p. 454) 'M =AO(a - b/t)'t - c/9o; temperature = t-C; rate of shear = radians/sec., corresponding to a difference of v meters/year in the velocities of two planes of slipping that are 100 mekers apgt. In his paper of 1905, Weinberg gives/t = ( 2.44 - 4.02t - O.277t ) x 10e poises when the mean value of 9 is about 10 radians/sec.; this formula is probably Ot so good as the other. Caputations by the copiler. Unit ofA = lO2 poises, of f and of v as already indicated; temp. t=C. I. River ice. Planes of slipping age perpendicular to optic axis. 9.5, a = 1.12, b = 0.54~C, c = 5 x 10' poise-radian/sec. y, 104-< (? 5 10-S} 100 10,7- 5x10 — v 31.6(?) 15.8 31.6 316 631 0 12.4 110o 60 14.5 10.5 9.5 -0.1 12.8 112 62 16.5 12.5 11.5 -0.5 14.5 114 64 19.1 15.1 14.1 -1.0 16.7 116 66 20.8 16.8 15.8 -2.0 21.6 118 68 23.3 19.3 18.3 _-7.0 27.0 121 71 25.9 21.9 20.9 -4.0 33;0 124 74 28.6 24.6 23.6 -5.o 39.5 126 76 31.5 27.5 26.5 -7.5 58.2 135 85 40.3 36.3 35.3 -10.0 80.3 147 97 52.3 48.3 47.3 -12.5 93.0 163 113 67.3 63.9 62.9 -15.0 135.0 184 134 88.6 84.6 83.6 II. Glacier ice. -40 = 3.8, a - 1.32, b - 0.65'C, c * 8 x 104 poiseradian/sec. y 10-9 510-9 10-. 10-7.5x10-7 v 316 15.8 31.6 4 316 6 31 0 83.8 19.8 11.8 4.6 4.o 3.8 -0.1 84.7 20.7 12.7 5.5 4.8 4.7 -0.5 86.1 22.1 14.1 6.9 6.3 6.1 -1.0 87.5 23.5 15.5 8.3 7.6 7-5 -2.0 90.3 26.3 18.3 11.1 10.5 10.3 -3.0 93.8 29.8 21.9 14.6 14.0 13.8 ~ -4.30 98.4 34.4 26.4 19. 2 I8.5 18.4 -5.0 104.4 40.4 32.4 25.2 24.5 24.4 -7.5 129.0 65.0 57.0 49.8 49.2 49.0 -10.0 178.0 114.4 106.4 99.2 98.6 98.4 -2.5 278.0 214.0 206.0 198.0 198.0 198.0 -15.o 475.0 411.0 I403.0 396.0 395.0 395.0

ENGINEERING RESEARCH INSTITUTE cb-83 UNIVERSITY OF MICHIGAN Table HIII Viscosity of Glacier Ice Adapted from R. M. Deeley and P. H. Parr (From Dorsey, 1940, p. 456) Unit of A 1012 poises Obsorver C onputer, Dr. Main 1888 R. M. Deeley 1912 6.0 McConnel and Kidd 1888 R. M. Deeley 1912 84.5 B. Weinberg 1907 B. Weinberg 1907 8.o Tyndall and others 1907 R. M. Deeley 1908 78.9 Blimncke and Hess 1907 B. Weinberg 1906 17.4 Blumcke andess 1910 B. Weinberg 1910 17.5 Bl'mcke and Hess 1910 Deeley and Parr 1913 147.7a Bliumcke and Hess 1910 Deeley and Parr 1913 125.0a a. From motion of glaciers in winter.

UNIVERSITY OF MICHIGAN Table XXIV Viscosity of River Ice (From Dorsey, 1940, p. 456) Computed from Hess' (1902, 1904) data Values were derived from the bending of horizontal rectangular bars supported at the ends and loaded at the middle. P = load; M = bending moment per unit of cross sectional area = P1/4ab;-. = value of the apparent viscosity as computed from the rate of shear 7sec. after the load was applied; 1 = length between supports; a = vertical thickness; b = horizontal breadth; vertical and horizontal refer to position of the bar when loaded for test. All three bars were cut from the same sheet of ice. Unit of P = 1 gram force, of N = 1 gram force/cm., ofA4 = 1012 poises. Axis Parallel to 1 Parallel to a P 2000 5000 6000 1000 1500oo 2000 n 1350 3400 4000 2350 2350 3100 15 o.os 0.105 o.oo055 o.o75 o.loo o.0o8 60 0.175 0.115 0.0360 0.075 0.110o 0.070 120 O.1 0.130 0.0365 0.075 0.090 0.00 o. 300 0.110 0.160 0.0350 0.080 0.120 0.120 1200 0.120 Optic axis parallel to b P 1000 1500 2000 3000 M 1500 2250 3000 4450 15 0.037 0.037 0.024 0.110 60 0.080 0.110 0.060 0.ogo90 120 0.120 0.100 0.100o 300 0.210 0.190 0.170

ENGINEERING RESEARCH INSTITUTE Cb-85 UNIVERSITY OF MICHIGAN ICE BREAKERS AND ICE B]EAKING The articles examined were mostly non-technical. Of the Russian articles, which were not available for examination, Davydov's is the only one of a technical nature, Judging from the titles. Barnes, H. T. (1928), Ice engineering, Renouf Publishing Company, Montreal, chapter 8, pp. 229-299. Following are the paragraph headings and a brief statement of the contents of each: Work at the Northumberland Straits: a description of the ice encountered, meteorological conditions, and methods of breaking ice. Ice breaking on the St. Lawrence: similar discussion. Note on the formation of ice over Lake St. Peter: similar discussion. Measurement of water temperatures by microthermometer: a description of the electrical resistance thermometer used. Effect of sun on general ice conditions: absorption of the sun's rays in the water. Influence of ice on the temperature of the water. Effect of convection: A limit to the thickness to which ice can grow is postulated due to the fact that the water under the ice is usually slightly warmer than the freezing point. Atmospheric humidity: Even when the relative humidity is 100 per cent, evaporation may take place both from ice and from water since they are generally warmer than the air. The temperature of the layer of air next to the ice or water is raised by conduction, thus lowering the relative humidity and permitting evaporation. Effect of radiation: This effect is difficult to evaluate but is probably small. Ice-breakers and their services: a review of Gulston's article. Ice-breaking tugs: a discussion of existing ice-breaking tugs and of the effect of the lines of the ship and of the bow propellor on efficiency as ice-breakers.

Cb-86 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Bartlett, R. A. (1928), Ice navigation, Am. Geog. Soc. Sp. Pub. No. 7 (Problems of Polar research), pp. 427-444. This article includes non-technical discussions of the following subjects with special reference to polar sea-ice: Polar ships and their construction The training grounds for ice navigation Ice navigating with Peary Varying ice conditions from year to year Wooden ships versus steel ships Ice movements in Baffin Bay and Labrador waters Bregman, G. R. (?), The Atlantic influence on the processes of ice breaking and freezing of rivers (To be printed in the Transactions of the Hydrological Institute). Probably in Russian. Not examined. The title of this article may be a mis-translation of the Russian and may actually refer to the melting and freezing of ice in rivers. Davydov, V. V. (1938), Theoretical investigations of the impact of a ship on ice, Problemy Arktiki (Problems of the Arctic): No. 5/6, pp. 103 -124. In Russian. Not examined. Gulston, A. (1904), Ice-breakers and.their services: Jour. Soc. Arts, vol. 52, p. 215. This article is a history of the development of ice-breakers to the date of the paper and includes descriptions of most of the ice breakers which had been in use previous to that time. (U. S.) Hydrographic Office Chart No. 2601, S. P., March, April and May, 1947 (reverse side of), Notes on navigation in ice. (Reprint of 'Notes on navigation in ice', Hydrographic publication H. D. 372, Hydrographic Department, London. H. C. 6536/42, N. I. D. 1484/42, 1942. Not examined. Johnson, H. F. (Rear Adm., USCG, Ret.) (1946), Development of icebreaking vessels for the U. S. Coast Guard: Trans. Soc. Naval. Arch. and Marine Eng., vol. 54, 'pp. 112-151. Includes table of characteristics of various ice-breakers built during the period 1890 to 1945 and table of relative strengths of certain ice-breakers. Seven pages of discussion follow the main paper. Paragraph headings are as follows: Data available on ice conditions Trends of foreign designs Coast Guard ice-breaker design developments Detail requirements of ice-breaker designs Selection of machinery Propellors and shafting Operating difficulties

ENGINEERING RESEARCH INSTITUTE Cb 87 UNIVERSITY OF MICHIGAN Subdivision Heeling and trimming arrangements Steering gear and rudders Towing arrangements Topside icing and insulation References cited include Runeberg (1889, 1900) (see below) and: Simonson, D. R. (1936), Bow characteristics for ice-breaking: Jour. Amer. Soc. Naval Eng., vol. 48, No. 2. Keefer, T. C. (1898), Ice floods and winter navigation on the lower St. Lawrence: Roy. Soc. Can. Proc. and Trans., 2d ser., vol. 4, part III, p. 3. Not examined. Runeberg, Robert (1889), On steamers for winter navigation and icebreaking: Proc. Inst. Civil Eng., vol. 47, paper 2371, pp. 277-239. Includes theoretical discussion of the ice-breaking process and descriptions of existing ice breakers (as of 1889). Paragraph headings are as follows: (Introduction) Ice breaking by a continually progressing steamer Ice breaking power of a steamer when charging Two forces of subordinate influence Effect produced by the continued working of the engines Frictional resistance caused by change of motion Displacement of metacenter Details of construction Particulars of some ice breaking steamers Runeberg, Robert (1900), Steamers for winter navigation and icebreaking: Proc. Inst. Civil Eng., vol. 57, paper 3191, pp. 109-129. A continuation of the earlier paper. Includes further discussion of designs and descriptions of existing ice breakers. Six pages of correspondence on this subject follow the paper. Williams, F. M., and Williams, F. P. (1926), Report on ice-breaking, International Congress of Navigation. Barnes quotes from this paper in his chapter on ice-breaking. Sukhorukov, V. V. (1938), The types of ice breakers and their shapes: Trudy Leningradskogo Otdelenija Vsesojuznogo Nauchnogo i Ingeniero-Technicheskogo Obshchestva Vodnogo Transporta (Transactions of the Leningrad Section of the All-Union Society of Science and Engineering Technology), vol. 2/3, pp. 117-148. In Russian. Not available for examination. Zubov, N. N. (1940), The drift of the ice-breaker Sedov: Nature, vol. 145, pp. 533-539.

Cb-88 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN SOURCES OF BIBLIOGRAPEIC MATERIAL ON ICE The following references contain bibliographies on various aspects of ice which may be considered more or less complete at the time of publicatio Barnes, H. T. (1906), Ice formation, John Wiley and Sons, Inc., New York. Barnes, H. T. (1928), Ice engineering, Renouf Publishing Company, Montreal. Dobrowolski, A. B. (1923), HistorJa Naturalna Lodu (Natural history of ice), Warsaw. Dorsey, N. E. (1940), The properties of ordinary water-substance, Reinhold Pub. Corp., JNew York. Physics of the Earth (1942), vol. 9, (Hydrology), 0. E. Meinzer, ed., MlcGraw-Hill Book Co., Inc., ch. 9 (bibliography). Weinberg, Boris (1940), List of the latest publications of the USSR on ice and snow: Trans. Am. Geophys. Union, part III, Appendix A, p. 757. The following are sources of current bibliographic material on ice: Transactions of the American Geophysical Union: in addendum to reports of committee on snow. Headings in bibliography as follows: General Sea and River Ice Glaciers Underground Ice Polar Physics of Snaw and Ice The Journal of Glaciology: published semi-annually by the British Glaciological Society, London, Volume 1, number 1, dated January 1947. Recent publications listed under the general heading of glaciological literature. Polar Record: published semi-annually in Great Britain for the Scott Polar Research Institute, Camibridge. Volume 1, number 1, dated January 1931. Recent publications Usted under the heading of "Recent Advances in Polar Research."

APPENDIX C, SECTION c STRESS IN A PLANE SEEET OF ICE LOADING IN THE PLANE OF THE ICE SHEET BY P. F. CHENEA August, 1949

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX C, SECTION c STRESS IN A PLANE SHEET OF ICE LOADING IN THE PLANE OF THE ICE SHEET 1. Introduction: The nature of the action of an icebreaker in breaking ice is two-fold. The ship forms a strip of open water in moving through the ice. The end of this strip is shaped in a form similar to the bow plan at the waterline. As the ship moves into this wedge, it exerts a force in the plane of the ice sheet and, in addition, a force normal to the ice sheet as the ice is pushed under the ship. The nature of the stresses arising from these two sets of forces is different. The forces in the plane of the ice cause a state of plane stress to be established in the ice which ultimately produces failure in the form of long radial cracks emanating from the bow of the ship. The second set of forces, normal to the ice field, produces bending in the ice sheet which ultimately causes failure along lines roughly at right angles to the radial cracks. The details of the icebreaking process as well as the overall effects depend upon which type of failure occurs first. For the relatively thin ice that was encountered

C c - 2 | ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN in the ice-breaking tests, there is considerable photographic evidence to show that the radial cracks precede those due to bending. There is reason to believe, however, that this may not be the case for ice of greater thickness. To illustrate the general nature of the stresses which occur in an ice field during the passage of an icebreaker, three sets of solutions have been obtained. These solutions are those for: a. Plane stresses in a continuous ice field. b. Bending stresses in a continuous ice field. c. Bending stresses in a radially cracked ice field. 2. Plane Stresses in a Continuous Ice Field The nature of the plane stresses in a continuous ice sheet may be studied by assuming that the ice has the form of an infinite sheet with a sector missing as shown in Figure 1. A typical loading along one side of the wedge is also shown. The actual loading is symmetrical and may involve tangential forces as well as normal forces. A general solution for this problem exists, and it may be stated as follows: let r r +r r -r rr r 9 - then = a logr + br2 + c r2logr + d r22 0 a0 0 0 + a' + a1 r sin9 - c1 rG cosG

cc-3 rrG x CXc FI. C.c. FIG. I

ENGINEERING RESEARCH INSTITUTE Cc-4 | UNIVERSITY OF MICHIGAN 1 1 (blr3 + alr-1 + blr log r) cos 0 + (dlr3 + cjlr1 + dir log r) sin G + E (anrn + bnrn+ + a-n+2) cos nG n=2 +; (Cnrn + dnrn+2 + cnrn + dnr-n+2) sin n G n=2 The constants a, b, c and d are to be determined by the nature of the loading along the free boundaries. Certain simplifications may be made by assuming the general characteristics of the load. For example, if the loading is symmetrical and involves negligible shear forces, then it can be shown that the normal stresses along the boundary may have the form: ((ye'=i (2bo + 3c0) + 2co log r + 6(a3 cos 53+ bl cos jr + 12 (a4 Cos08 4 + b2 cos 2( )r2 + 20(a5 cos 59+b3 cos 3)r3 1 1 1 + b* cos~ - + blr Co-r - ao 7a + 2(al cos0 + bI cos 3 ) r3 + 6(al cos 2R+ bl cos 4() + Where all other constants are to be zero. It is seen that any loading along the boundary that may be represented by power series in r can be solved by this method. As a simple example of the above solutions, we may consider the case where all constants except ao vanish. Then: = ao log r ao r

ENGINEERING RESEARCH INSTITUTE I UNIVERSITY OF MICHIGAN c-c This solution approximates the type of loading that probably exists on the ice sheet,as shown in Figure 2. It is zero at r -- * and very high near the bow (in fact infinite if the bow has a sharp point). Many other solutions can be obtained from the above general analyses. All solutions show that the nature of the tip of the bow curvature to be the governing f'actor, suggesting that this type of failure may be controlled by adjusting the bow sharpness. For example, if it is desired to crack the ice radially and then by bending, this sequence may be obtained by making the bow sufficiently sharp. The reverse sequence may be obtained by blunting the bow.

cc-6 x G EDGE OF ICE C, c, IG. 2

APPENDIX C Section d - Part I BENDING STRESSES IN A CONTINUOUS SEMI-INFINITE ICE FIELD BY ROBERT W. PEACH September, 1949

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX C Section d Part I BENDING STRESSES IN A CONTINUOUS SENI-INFINITE ICE FIELD TEE PROBLEM The problem that was attacked with some success was the plate on an elastic foundation with an applied concentrated load. The basic equation is, 2 - 2 + 2 -Kw = -q ax2 axay a in which the term Kw represents the reaction of the elastic support which is seen to be proportional to the deflections, as it should be for a sheet of ice floating on water. The constant K is the buoyancy force of the water. If the ice is pushed down one foot, then each square foot of the ice sheet will have the buoyancy force of one cubic foot of water; thus the constant K is in terms of buoyancy force in pounds per square foot per foot of displacement. Also in the above expression we have q, the intensity of the loading on the plate. This we shall discuss in more detail later. In the following development D = Eh3/12(1_ -2)

CdI-2 | ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Finally for Mx, My, and Mxy we have Mx = -D f(2w 2w 2w K D ( ) x2 a2 2 M = -D + w 2=w In these, / is Poisson's ratio. On substituting these expressions into the plate equation we have. D — x- 2 2D(1 -A) D -D - -A -q _q k 22 2 x -- 2 D - %2 2Kw or, W 26'w "kw Kw + 2. + D+ Kw which may also be written V4w + Kw = D D When this is applied to an infinite plate with a concentrated load we have the Hertz' solution, which is, without development, wl = _-2 / Ke io (r See Figure 1 for a plot of the Keio function. This solution is very nice theoretically but lacks the usual features of so many theoretical solutions,

-I______________ KELVINS BESSEL FUNCTION OF ZERO ORDER (KEI0o) _e 1 2 3 4 5 6 7 8 C.d FIG. I

Cd.I-4 | ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN namely, it doesn't fit our particular problem, which is breaking ice by the application of a concentrated load on the free edge of the elastically supported plate. Below we shall develop a solution, which is approximate, for the first case of the icebreaker approaching a sheet of ice which may be considered a semi-infinite plate with a concentrated load applied to the free edge. If this can be broken, then the ship rill proceed to the second case of the problem, which shall be briefly outlined after we demonstrate the solution to the first case. This second problem is one of an infinite plate with a cut of approximately the width of the ship and approaching the bow of the ship in some irregular curve. THE FINITE DIFFERENCE APPROACH In order to indicate the quality of the wool we are going to pull over your eyes, we shall begin with a little introduction to finite differences. It will be recalled from the very elementary calculus that the fundamental definition was f'(x) = f(x + a) - f(x) a in which we usually let a approach zero and thus obtain the infinitisimal calculus. If we do not let a approach zero, we have the finite difference rather than the derivative. Furthermore, for the solution of our problem we shall define the derivative or finite difference as f'(x) = ax = f(x + a/2) - f(x -a/2) a In our problem we have partials rather than total derivatives and adopting finite difference notation gives

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN - =D A x2 A2 and D x2y Thus we have the expressions: \wo = Wo+a/2 o-a/2 and = otb/2 o-/2 a: o b for the approximation to the first derivatives in the x and y directions. Notice that we have generalized and used different intervals in the x and ydirections, namely, a and b respectively. The approximations to the second derivatives are, a 0 2 a and, / w = W o+b - 2wo + Wo-b - b2 In a similar manner we have for the approximations to the higher derivatives, iA2\3 = ot+3a/2 - 3Wo+a/2 + 3Wo-a/2 - wo-3a/2 0'

ENGINEERING RESEARCH INSTITUTE CdI-6 UNIVERSITY OF MICHIGAN /4Wo = Wo+3b/2 3Wo+b/2 + 3Wo-b/2 Wo-b/2 O b b3 1 - 4w + 6w - 4w + w ) -4 (o+wo+ao.o-a o-2a z 4 = -To2 4w + 6w - 4w + o-2b) and finally, Z\22wo - 1 (Wo+a+b = 2Wo+a-b - 2o+b + o -2ob + Wo-a+b 2Wo-a + Wo-ab) If we take b equal to a and number the points as in Figure 2, y 16 8 1' 4 12 17 3 _0 I_ 15 L__ 2 10 9 -6 14 C.d. Figure 2

ENGINEERING RESEARCH INSTITUTE CdI-7 UNIVERSITY OF MICHIGAN I these become, w= 3 -^w = 4 - W2 2a 2a A%2 w3 2wo +wl A w4 2Wo +w2 2 Wo2 AQwo= a4 ( w7 - 4 w3 + 6 w - 4 wl + w5 ) A Wo 4 ( w8 2 4 4 + 6wo 4 w2 + w6 ) aend, wo= a ( wg - 2 w3 + w 2 4 + 4 Wo -2 w2 + w2 -2 Wl+ 9 ) After obtaining these results,to make some use of them we naturally substitute them into the plate equation. This operation results in, ~- [w5 - 4 wl + 6 wo - 4 w3 + w7 w+ 2 w9 - 4 w4 + 8 wo 42 + 2 4 w3 + 2 10 + 8 - 4 w4 + 6 4 w2 + w6] K0o q D D or,,

I[I-8 | UNIVERSITY OF MICHIGAN [20wo - 8 (wl +w2 +w3 + ) +w5 +W6 +w7 +W8 + 2( w9 + W1o + wl + Wl12 + - = q D D If we substitute our particular desired values of load and ice thickness we will naturally obtain the solution for this problem. However, in order to make our results more general we shall put our equation into a non-dimensional form so that any load or ice thickness may be inserted and the appropriate values of deflections, moments, and stresses computed. As we have a concentrated load, we shall use two equations. In way of the load we have, V4w +:. =.(1) and we assume the concentrated load is distributed over half a grid space to either side and into the plate. Away from the load we will have, V4 + D — = o (2) D This is a much closer approach to the problem than a true concentrated load or a distributed load. For the first step (using R. V. Southwell's method) let, x = Lx' and y = LY'; thus, V4w = 1 V'4

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Then our first equation becomes, 7'4w + KL W = D D In this it will be seen that KI4/D is non-dimensional and we may let it be represented by X. Finally if we let w = D we will be able to remove all dimensional units and have, 4wt + xw' = 1 (1) V,4w' + w' = 0 (2) for our two equations. In finite differences these are, (20+a4)w~ - 8(Wl+W2+W3+W4) + w5 +.w6 + 7 + w + 2(w9+Wlo+wll+w12) = a4 (1) (20+a4X)w - 8(Wl+W2+w3+w4) + 5 6 w7+ 2(w+w+ll+2) + (2) Whereas this appears non-dimensional, it cannot be solved without using some value for a and X. Now,it is not possible to select X and still keep the problem non-dimensional. This will be discussed in more detail be low when we treat the actual problem. Furthermore, whereas a is the number of grid lengths in the given length L, it is a variable depending on the L selected.

ENGINEERING RESEARCH INSTITUTE CdI-10 UNIVERSITY OF MICHIGAN TOE SEMI-INFITE PLATE PRQBiM The main difficulty with the Hertz' solution for our particular problem is the fact that it does not have the free edge. We need to develop boundary conditions for the free edge that must be satisfied by the basic equation. The first condition is that in the plate far from the load the deflections will be zero. As the deflections obtained by finite differences tend to become smaller away from the load we shall not worry about this condition. The main worry, and it is a big one, is to satisfy two conditions along the free edge, the EKirchhoff-Kelvin boundary condition concerning shears, and the zero moment condition across the free edge. Let us take our axes as in Figure 3. y / _ p FREE EDGE Figure 3 Then the second boundary condition becomes, or,

ENGINEERING RESEARCH INSTITUTE CdI-11 UNIVERSITY OF MICHIGAN = 0.2 +/-?~2 In finite differences this is, w4 - 2wo + w2 / (1 -2wo + w3) +, = 0 a2 a4 or,,2 + w4 + 4(W1,+w3) - 2(1 +r)w = o The Kirchhoff-Kelvin condition is, (QY _ _ -y o or, + (2 - ) a 2 - 0 y-3 o Now in finite differences, aw 2 Wo+a+b/2 - 2wo+b/2 + wo-a+b/2 a2b and a,3= nt 5b3 (Wo+3b/2 - 3Wo+b/2 + 3Wo-b/2 - Wo-3b/2)

ENGINEERING RESEARCH INSTITUTE CdI-.12 |UNIVERSITY OF MICHIGAN As we are not using half-points we must change these forms to ones more readily usable. With increased inaccuracy, we may double the interval in the direction necessary in order to use whole rather than half-points. Thus (see Figure 2), 3 3w A2 l ( w12 2 w4 + Wl - wg + W2 - wo) and, y5w 1 3 8a3( 16 - 3W4 + 3W2 - 'V4 ) Then this boundary condition becomes, 0= wl6 3 w4 + 3 w 2 - w 4 w 2w + v w9 + 2 o 3 + (2 -j) 2-a or, 0 = ( 19- 8At) ( w2 - w4 ) - 4( 2 -,)) (wg + 0-o - W 1- w2) - W14+W16 As can be seen at a glance the boundary conditions are not affected by any dimensional changes. They are merely statements of ratios of adjacent deflections to each other. At this point it is necessary to consider what grid size we shall use. At first we are confronted with the obvious difficulty that though the plate is infinite the grid size cannot depend on this infinite dimension; that is, we must select some definite length for a basis. On page 9 above we found that, A=

ENGINEERING RESEARCH INSTITUTE CdI-13 UNIVERSITY OF MICHIGAN and it was remarked that whereas this was a non-dimensional parameter it actually depends on the dimensions selected. The thickness enters in this parameter through the stiffness factor and. the grid size through the grid length L. At once it is seen that L cannot be taken as infinite, as our equation would become indeterminant. For the sake of obtaining an answer, let us assume the following values: K = 62.4 lbs/ft3/ft = 1/3 E = 500,000 lbs/in2 h = 2 ft P = 1,000,000 lbs. Then, D =26.48 x 108 in-lbs, or 5.4 x 107 ft-lbs. 12(1 -/ ) Now this gives, for the assumed thickness, >X = 62.4 L = 11.56 x 10-7 L4 5.4 x l07 and depending on the value chosen for L we may make this parameter X as large or small as we wish. Before going on with this discussion it is necessary to speak of residuals. If we transpose the left side of Equation 1, page 9, to the right we have, 0 = a4 - (20 + a4X)wo +8(wI + w2 + w3 + w4) - 5- w 6 - w7 - w8 - 2(w9 + w0 +wl +w12))

ENGINEERING RESEARCH INSTITUTE CdI-14 I UNIVERSITY OF MICHIGAN This should be zero if the equation is satisfied; however, it usually is not as the values of w are not exact. This discrepancy is called the residual. We could have transposed the a4 rather than the other factors, but by doing it as above we will remove positive residuals by positive increments of displacement. Usually we change w in the appropriate magnitude to make the residual at wo become as small as desired. Let us assume that the sum of the residuals over the whole plate is zero} that there are some positive and some negative. If a4% were zero then any change in the residual at wo would change the surrounding points both positively and negatively, but would not change the sum of the residuals which would remain zero. There would always be the chance that with enough effort the residuals could all be made zero. With the factor a4X in the equation this chance disappears and only by making it as small as possible will we be able to reduce the residuals to reasonable values. To return to our discussion of X we find that by making L small enough we may make X very small compared to the factor 20 in the wo term. For convenience we shall always take a as 1, which means that our characteri tic length L is the distance between grid points. At this point it is unwise to rush headlong into selecting L the smallest value possible as that means we shall need more points in order to cover the area. This naturally entails a tremendous amount of work, especially if the assumed initial values of w are far from their correct values. If we start with an L of 40 feet we will have only 36 points to cover an area 320 feet from the load along the edge and 120 feet into the plate. This gives x ' 2.96, so our equations become, 1 - 22.96 wo + 8(wl+w2+wi+w4) - w5-w6-w7-w~ - 2(w9+wlO+wll+w12) = 0 (1)

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN and - 22.96 wo + 8(wl + w2 + w34) - W5 -w-w7- W8 - 2(wg + Wlo'+ w + W12)= o (2) This assumes P is distributed over a rectangle L (40 ft. ) in length along the edge of the plate and L/2 (20 ft.) into the plate. Thus, - P _ 2P q L —. and, a L4, 2P L4'W 2P L2w' D w L2D D| If we use R. V. Southwell's practice of multialying our equation through by approiate powers of ten to make our answer in whole numbers accurate to the power of ten used, we have on using 1000, 2P L2wt 1000 D This will give our answer correct to three figures. Next let us consider a little interpolation. On closer examination of the boundary conditions it will be observed that from the boundary conditions we- may find w2 and w14 but not w6 if w is the deflection along the boundary. These values are fictitious in that the plate does not exist at these points but they are necessary to satisfy the plate equation for points on and near the boundary and also the boundary conditions. Thus it is necessary to interpolate to find the value of w6. Lagrange's interpolation

CI-1I 6ENGINEERING RESEARCH INSTITUTE CdI-16 | UNIVERSITY OF MICHIGAN formula: wc(x-a)(x-b) wb(x-a)(x-c) wa(x-b)(x-c) WX +........,. --— +- _ + (c-a)(c-b) (b-a)(b-c) (a-c)(a-b) With equal intervals and points numbered as in Figure 4, this reduces to W-C W-b W-0 W o -C -b -a o Figure 4 w w - c + w _ -b 3 -a 3 See Figure 6 for the general scheme of numbering points. After finding the values of w on points on the first, coarse net, it will be necessary to interpolate to obtain values for the second, fine net. For this two sets of interpolation formulas were developed from Lagrange's for mula using four points. In Figure 5 are indicated the various relations. Wx -c I wx x d c x b a d x c b a Figure 5

ENGINEERING RESEARCH INSTITUTE cdI-17 UNIVERSITY OF MICHIGAN In general I was used; however, in order to cut down on possible errors from the boundaries II was used near the boundaries. As on the finer net it was decided to use an L of 10 ft., it was necessary to find w at x x = L/4, L/2, and 3L/4. The formulas are: x L = 1 ( -7 wa + 105 wb + 35 wc 5 wd) 1 1 x = 2L 16 = ( _wa + 9wb + 9wc wd) and, x = L, Wx =8 ( -5 wa + 35 wb + 105 wc - 7 wd) II 1 1 x = L, = w= (5 vW - 27 vb + 135 wc + 15 wd) 1 1 =2 L, wx = (a - 5 b + 15 wc + 5 wd) and, x L, wx:L3wb+(Wc+ To obtain a starting value any method may be used; some other solution which is a little similar to the problem, a uniform distribution from an assumed maximum under the load to zero at any desired distance, or all deflections zero. Using the latter method the initial deflection under the load was found to be 43.6. After much strenuous labor the values given in Figure 6 were obtained. The values to the left of each point are displacements w' and the values to the right are the residuals. All residuals are below 22.96)meaninn that to make any zero would change the

WR t0 0-Il 0:-9 0.-l 0..0 0,,0 0-0 010 00 Ic COARSE NET DISPLACEMENTS AND RESIDUALS 5 1.6 2-1.9 01- 0 -10 0-3 0 0 olo FIGURE 6 3 3-15.5 17 -10.6 5 -1.5 I -1.3 0-1.3 0 -2.7 00 0 0 0 0 111.6 -.1 54 114.9 19 -10o. 6-182 I 4.6 0 1.6 0 -. 0 0 0olo FREE EDGE 228.6 8 3.1 25.7 9. 0.31 -0.3 l ~ 959.2 -55.7, -33.3, 5. 8, -16.2. - 2.6 0.70 o, 2303.5 -362.4 - 157.9 -4.41 -48.4 -6.9 21 0 ol 4 I1 2 3 4 5 6 7 8 C.d.

ENGINEERING RESEARCH INSTITUTEdI-19 UNIVERSITY OF MICHIGAN displacements by less than one whole number and the results are accurate to the whole number or to three places of accuracy. Before going into the fine net it is well to consider some peculiarities of the problem knowledge of which was necessary to obtain a result in reasonable time. These might have helped in the solution of the coarse net but are almost essential to the fine net solution. The first concerns the boundary conditions. It is essential to note that no matter what method, including sheer intuition, is used,the equation of the plate must be satisfied at all points and the boundary conditions must also be satisfied by the final results. Thus if by observing the behavior of the boundary conditions we can obtain better values of w it will shorten the labor of the problem. Taking, as 1/3 in the boundary conditions gives, 1 8 w2 +w4+3+ (wl + w3) - o and, 3 (W2 - w4) - (w9 + w10 - wl - wl2) - w4 + w16 = 0 3 3 The deflection wo is on the free edge of the plate. From the first one we would find in the conventional manner the point w2 as the others being inside the plate are all known. Now suppose that wo is in error of only 1 unit in the last place. This would result in residuals for a wo as an interior point of less than the accuracy being considered. However, on the boundary this introduces an error in w2 of about 3. Now if this value is used in the second equation the error becomes of the order of 50 for wp4.

ENGINEERING RESEARCH INSTITUTE CdI-20 UNIVERSITY OF MICHIGAN This large error will be reduced to about 20 when the value of w6 is calculated. But this is large enough to change the residual at wo considerably. In trying to solve the problem one feels that he has a bull by the tail and each little twist of the tail throws him all over the lot. It is possible to have these small errors over-correct each other to such an extent that the residuals will oscillate with increasing amplitude. In order to throw the bull we shall first tie down his tail and then go to work on the rest of him. That is, we assume some value of w14 which appears from a few trys to be near the correct value. From this we find w2 and then wo. Next we compute the residuals at all points affected by the boundary and improve those that depend on the values of w on the free edge but not across the boundary. To change the values on the edge would merely be untying the tail. With the residuals known for the edge points, it is a simple matter to estimate how to change the values at wb and w.c so that the residuals will be reduced. Then wa and wo may be recomputed and the residuals checked again. It will be found to be a slow process but one that converges rather than diverges as in the case of the conventional attack. Except in the vicinity of the load it was found that interpolation seemed to increase the accuracy of the results. For example, in the coarse net point 0, 3 had a displacement of 6 and a residual of -18.2. In the fine net the point becomes 0, 12 with a displacement of 6.o and a residual of 0.1. The reason for this is probably two-fold; first the coarse net covers more space and thus will introduce greater errors from distant points. Second, the interpolation puts a smooth curve through adjacent points thus more nearly fitting the deflection surface than the arbitrary initial values found from the coarse net.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN CdI 21 If they are needed very much, block operators can be very useful. In the coarse net they are not necessary as only single points need be changed. In the fine net the chances of using block operators is somewhat greater but only about three or four were necessary. In Figure 7 we have the basic operator in which wo is changed by +1. The figures by each point indicate the magnitude of the change in the residuals at the point. A few of the operators developed are indicated in Figure 8. In Figure 8a there are three unit displacements in a row. Next in Figure 8b we have a square operaor with a unit displacement at each of the four corners. Finally in Figure 8c we have a three displacement Lshaped operator. For the fine net we shall use L of 10 ft. and keep a as 1. In this finer grid the only factor in our equations that is changed is a5\ which becomes a = 104 x 11.56 x 10-7 = 0.01156 and our equation becomes, 1-20.01 wo + 8(wl+w2+w53+wU) -5-w6-w7-w8 -2(9 +w+w1ll+12) = 0 and for the desired accuracy we shall multiply all the displacements by 1000. At this point it is desirable to estimate the effect of different thicknesses of ice on the value of ai. Suppose the ice is only half as thick as assumed or 1 foot; then as h3 is proportional to D and D is in - versley proportional to X this change means that a4i will be eight times as large but this changes the coefficient of wo from 20.01 to 20.08 oyr less

CdII -2ENGINEERING RESEARCH INSTITUTE CdI-22 | UNIVERSITY OF MICHIGAN 8 - II -2 4 -8 12-2 A:1I 71- 1 +8 o -(20ta4X) 1 +8 5,10 -2 2 +8 9 -2 6 -I1 Figure 7 -i -I -I, -I -I =-2+5 +5 -2 -2 -t6 it4 -6 -2 - 1 5 A6A-6 4 5 - -I t7 -13 A'4 A k1 317 11 I 5 A6A-6 +65 -1 +62 L6 6 -2 -2 +5 +5 -2 -I -I -I -I -I a. b. -I - -2 -6 -5 -2 -I 1+7 A-14 A-4 +5 -I -3 4 +6 -1 -3 +7 -2 C.

ENGINEERING RESEARCH INSTITUTE CdI-23 UNIVERSITY OF MICHIGAN than one half a per cent. Likewise it is seen that increasing the thickness to say) 10, changes the coefficient from 20.01 to 20.00+. Thus, for the thicker ice the effect of the elastic support becomes smaller, or the ice itself supports the load with little aid from the elastic foundation. The foundation always contributes something to the support but with a thicker plate better able to take a loadthe foundation disappears as a factor to the accuracy with which our answers are obtained. The finer net was obtained with a coefficient of wo of 20.01. The displacements found on the fine net are shown in Figure 9. Having found the displacements on this fine net, we might proceed to a finer net but life is too short. Rather let us proceed to the determination of the first and second differences. If the first differences are used to find the second differences large errors will appear in the results, as this method takes in a span of w values twice that defined by the second differences on which our equations are based. (See pages 4 to 11.) Now, M= -D(&2 w + 2 w) My = D(1 -)) &w and, = -D( 2 w + A22 w) In these, 2 Wo = wl +w3 -2 wo a2

0 ~ ~ ~~~~~~~ - ~~~~~~~~~~~~.Q.2~ ~~~~~- 4 - - ]. 0.2 0.1.. 2.. 0. 0.2 0..2 -0.2 -i 7J. 2. 0.0.. 1.709 1.2 0.8 07 0.7 0.4 0.6 0.1 O -0.310.6-0.31 0.2 -0.30.2 -0.40.3 -O. d0. -0. 0.1-0.,..- -O. -. 0.2 O 1 0, -1.2-0.1 -0.7-0.2 -.4 0.71 0.2 1.-0.1 1.4 0 1OO.2. 0.9. 0.8b.4 0.2"O O.1'O.1 0.1 0.4 0 0.3 -.1 -0.b 0.1 0.1 O.2 -0.1 - - i 1. 7,, 0 2.1, O., -h 6n -. — -1.". '06I L 6'f - Z-1.8' i.4A W-3 1. 3 1 3 01O-2 0.0 0 '-0.:-0. 0.18T 4~% 0~ 8.1,. 8.6 1.6 8.1-.5 7.0 1.4 5.61.1 4.270.9 310.8 2.2 -0.5 1.30.3 0.610.3 0.310.3 0.1, 0.2 0,0.2 00.2.1 0.0.1 0. - Z 0.3 -1.4'-O.6 -1.0 1.4 -0.6 2.2-03 2.b 2.3 0.3 2. 0.2 1.7 0 1.60.2 1.4 0.4 0.9 O. 0 0.1 'O.O 0.1 0.3 0.1 0.1 -0.1 02 0 0.2 -0.1 f31 132 140A.1.4 411.61.9 9.4 1.5 7.3 J.3 5. 2 0.9 4.1 0.7 2.705 1.6 10.4 1.010.2 0.6,0.2 -0.30. 0.2 OdO.1 0.1,0.2 0.',0 0.2.. 0,, -35 0.6 -2.1 -0.8 -1.6 2.1 -0.8 35 -0.6 36T0.1 3.200.4 3.0-0.2 2.60.1 2.17 0.3 1.7 0.5 1.2.2 0.8 -0. 1 0.7 0.2 0.6 0 0.470.1 0.2 -0.1 0.2.0 e ~ 3~jjQ~.0.. 18..142 I2. LT 9.O.Q 6..&.1 4. 0.L.. 0. 10.TI.4 0.8 0.3 0.5 0.2 0..2 0.20.2 QW.Q.Q -0.g 8 -30 -5 0 -2.2 3.0 1.5 5.3 -0.8 5.5 0 4.7103 4.600.4 3.3i0.2 2.8,0.5 2.3 0.5 i. 0.5 1.2 0.1. 0.2 0.70.1 0.4 0.1 0.4 0 0.3-0.1O d2 3.. 38 3.3 31.- 3.1 27.4..8 22.2 -2.6 176 2.1 137 1.8 10.4 1.6 7.5.3 5.1 1.1 35.0.9 24 1.7 1.6 0.6 1.0.6 0.7 0.4 0.. 0.3 0.2 0.4 -6.7 1.3 -3 -2.4 40-3.0 3.9 -2.4 6.7 -1.3 75 0.6, 6.0.7 4.9 0.6 4.1 0.4 3.7-05 3.060.8 2.,0 5 1.9 0.3 1.360.2 1.0 0.3 0.7 0.1 0.b -0.1 0.5-0. I 3 45. 47.'3.1 45028 39.51 24 32.3 2.0 25.6J.5 20.2-0.9.157-0.6 11.730.4 8.30.4 6.0-0.3 4.2"0.4 3.0-0.3 2.1.0.2 1.5 10.2 1.1 0.1 O.7 0 0.32-0.1 -e 18.1, 2 0".718 -8.0 1.7 -5.1 3.4 42 5.1 -3.4 8.0 -1.7 9.1 0.5 8391.3 6.30.9 5.1 0.5 44.0.6 3.5 0. 1. 2.8,0.5 2.250.6 1.6203 1.3.0.3 1. I0.2 08,0 0.8 1 0 -.1 61.5 64.5-1.9 611.5-8 54.0`1.5 44.4'1.0 35.150.8 27610.5 21.6"0.2 16.3"0.2 11.90 8.8 -.0.1 6.4 -0.1 4.7 -0.1 3.40 2.510 1.840.1 1.240 0.50.1 -9.62.1 - 4.5 O-6.0 6.2 -4.5 9.6-2.1 10.703 9.9 1.8 78 1.5 5.9 0.7 4.80.9 3.9 "1.3 2.9 0.7 2.2 0.7 1.7 0.6 1.3 0.4 1.1 0.2 1.0-0.1 1.00. 1I ' a 79.8, 83.8 40.8 79. 0.5 70.0,0 57.50 454.03 3550.3 27.7-0 21.16,0.2 15.5, 0.1 11.50 8.5,0 6.300.1 4.710 3.510 2.610 1.7 0. I0..0.1 -9.6 2.7 -5.8'5.8 -8.0 5.8'-5.8 9.6 2.7 11.1 0.4 10.4"2.2 8.7 2.1 6.7 1.2 5.1 1.0 4.0 1.6 2.9 1.0 2.1 0.8 1.6 0.6 1.30.4 1.0 0.3 0.9 0 1,1 0 0f6.0, 976 02.3Q 3.1 97,6A2.4 86.0.J 7.90.1 56.0:07 43.70.9 8 71-0.5 2. 19.0:-0.6 14.2,;Q.4 10.6,:0.3 8.(,0.2 6,'0.2 4.5.1 3.40 2.3 1.0 0.2 FREE -11.1 -35 -0.7 -6.9 0 -9.4 0.7 -6.9 11.1 -3.5 12.6"0.2 11.4'2.6 8.8 2.4 6.8 1.8 5.3 1.4 3.7'1.9 2.5'1.2 1.8 1.0 1.5 0.6 1.30.5 08' 0.4 0.70 1.4 -0.2 EDGE 0:1,, 1!7,8,;.123.9, 17.8, 103.1 84.4, 65.9, 51.0, 39.4, 29.8, 21.9, 16.5, 12.4, 9.5 7.1, 54, 4.2,, 2.9, 1.0 12.6, 144.5, 154.9, 144.5 123.6 98.9 73.1 56.0 43.3, 32.7d 23.8, 18.1i 13.7, 10.5,, 8.2, 6.4, 4.8 31 1.0 147.4, 177.7, 1954, 177.7, 147.4- 114.4, 77.7,, 58.7, 45.5, 34.5, 24.5, 18.4, 13.9, 10.9L 8.9, 7.5, 5.6 5.7, 0. 10 FOOT SPACING KEY: Kwa22W FIGURE 9 Wa W a?' wa2'w DISPLACEMENTS -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ~C.d I__ I I C~d. I

ENGINEERING RESEARCH INSTITUTE CdI-25 UNIVERSITY OF MICHIGAN /A2 =o w2+. - 2 wo and, z Xwo = -wg + w10 - wU + W12 a2 The values of M1, My, and M.y found from these are given in Figure 9. Eowever, these are not true values, as they were bated on w' which needs to be multiplied by certain factors to bring it into line with the true w. Thusif we call these values of moments f(Mx), f(M), and f(Mxy) then the true values will be, f= () D 32 pL2 M, + a2 1000 D Substituting in the values for a and L gives, Mx = - 3.2 Pf(mx) and, = - 3.2 Pf(My) For Mxy using /= 1/3 we have, = y (Y) a - 32 PL2 XY a2 1000 D or, Mxy Pf (M )

CdI-26 ENGINEERING RESEARCH INSTITUTE' ~I-2~ lUNIVERSITY OF MICHIGAN Values of 3.2f(Mx), 53.2f (M) and 6.4/3 f(Mlqy) were plotted in Figures 10, 11, and 12 respectively. These curves present a rather confusing appearance as they represent cross-plots in order to make sure the values were fair. Most values fell close to the final curves as cross.plotted to the large scale used. As the stresses and moments are related by, 6Mx These same curves may be used for finding the stresses, If we let the values from the curve be Nx, Ny and Nxy, then, 6yP My = -P Ny and, Mxy = P Nxy In fairness to any prospective users of the curves the large squares (1/2 inch) represent units and the smallest squares are tenths. Furthermore, each curve is plotted with its line as a zero base line; that is, curve c uses the line c for a zero base and curve 7 uses the line 7 as a base, It was originally intended to use the values on the coarse net at a considerable distance from the load in the final results, but the valus

, -II I I 1 I, M\ 1 N a51 i - I iI I < I Ir I 2 I ~I/, A iFIGURE 10 04 / /j 0 I <o 2 3 4 5 6 7 8 4 < < I 7 I 9 1 I 1 12 SCALE N IS VALUE FROM CURVE P IS LOAD APPLIED h IS THICKNESS -~!- M~~x - PNx h2 x EXAMPLE FOR POINT d,11 Nx= 1.20 FEET/FT. THEN Mx 1.20 P FT -LBS./FT (IF P IS I N LBS.) O-x-7.20.LBS/FT2 (h IN FT.) C. d.I

AD! -PN' -~ o II I V / / d./ I I 1I /8 1 13 1L FIGURE II MO y AND Y6? I I I I! I 7 N~ IS VALUE FROM CURVE P IS LOAD APPLIED h IS THICKNESS My -PN EXAMPLE FOR POINT d,9 N - 3.28 FEET/FT THEN My - 328 FT-LBS/FT (P IN LBS) Iy - '19.68 P/h2 LBS/FT2( h IN FT) C.d. I

('Se-1 NI Si A d) Ib/SB' J d S S - t, =4 S. AN Ol'O 3-IdVYVX3 8 O.A Nd VYW C]31-ld dV CIVO Si d 3Aano YYOH-4~ 3n -iV Si N ZI 38nou)I Axvy ~~I ~~~~I ~~~~I I 01 6 8 9 3 1 oi- 6- $ I 0 1 ~ 'To ~~~s " ' I~~~~~~~~~f7o/ J1I 1 I Y 9~~~~~~~~~~~1 I M I I ~~~~~~~~~Y I I V I I I PI I I R III 1Y lg~~~~~~~~~~~~~~~~~~~~-.00 ZD I I ai igl I i_/ I I I 81 I i ug ~ ~ 17 IX~~~!/ I~~ r~~~n u.I\ n I I n r i u I i r I v ~~~~~~~~~~.......L.... I r/i i i/i i 1 1~~2

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN CdI-31 obtained from the fine net tended towards zero so the values from the coarse net were not used. Let us now consider the accuracy of the solution. At this moment it is appropriate to discuss the amount of work entailed with various degrees of accuracy. Our results were assumed to be accurate to three significant figures. Actually the temptation was overly strong so that the values were worked to an accuracy in almost the entire plate of four significant figures. This is shown by the fact that the residua;ls are 2 or less at most points on the fine net. For three-figure accuracy the residuals should be less than 20. To obtain four- significant-figure accuracy would probably have required 1/3 more of the time spent. This was with the values as they now stand. To obtain five-figure accuracy would probably require twice as long as four-figure accuracy or about two to three months full time for one person with the 200-odd points used in the fine net. Except adjacent to the load, the accuracy probably could be obtained more rapidly by using a finer net. The difficulties near the load would remain, however. Besides the discussion of the accuracy to which the solution was worked, we have a comparison with the Hertz' solution. Using the constants on page 13 we find the Hertz' solution to be, -W- P Keio i r where 4 Keio = r at r - O0 and, = P 106 8I' 8J~ - i7W4x54707

ENGINEERING RESEARCH INSTITUTE CdI-32 UNIVERSITY OF MICHIGAN Or, =1 - 2.16 ft. at P On the 10-foot grid of the finite difference solution we have, w- 32PL2w' = 32x 106 x 102 x102.3 1000 D 5.4 x 107 x 103 or, w = 6.o6 ft. at P This is an expected difference from the infinite to the semi-infinite plate. In Table I are the data used to draw up Figure 13. It will be noticed that the deflection under the load is almost three times as large for the semi-infinite plate as for the infinite plate. All values of deflections approach 0 at 100 to 160 feet from the load. Furthermore, the deflections along the free edge remain large for a longer distance than those into the plate. The latter starts out with the same deflection as the edge curve but rapidly approaches the Hertz' solution and almost becomes tangent at about 80 feet and then runs almost tangent from there out. All of this is to be expected and lends confidence to the finite difference solution. The deflections have not been plotted. They would be desirable but they are not essential.

CdI-33 TABLE I Hertz F. Diff. F. Diff. r, x, yx WI 0 2.16 6.o6 6.o6 10 1.98 5.78 4.^97 20 1.65 5.10 3.82 30 1.38 4.20 2.79 40 1*10 3.32 1.94 50 0.88 2.59 1.29 60 0.66 2.00 o.83 70 0.50 1.52 0.51 80 0.33 1.13 0.28 90 0.19 o.84 0.14 100 o.o8 o063 0.05 110 0 047 - 120 -0.03 0.36 130 -oo6 0.27 14o -o.o6 0.20 150 -0.03 0.14 160o 0.01 IN FEET

COMPARISON OF HERTZ AND 6 FINITE DIFFERENCE SOLUTIONS P= 1000,0001ooooo E=500,000*/ IN2, ~:'/3 ~ K: 62,4/ FT3 5\......__ h=2FT. Iw w _tJ Z- 4 o 03 LL 2 CL)~ ~~~A 0 -0 20 40 60 80 100 120 140 160 80 C.d. I DISTANCE FROM LOAD-FEET FIG.1 DISTANCE FROM LOAD- ~FEET

ENGINEERING RESEARCH INSTITUTE I UNIVERSITY OF MICHIGAN CdI-35 TEE SHIP StADPE As to be expected there are some similarities between the semi-infinite plate and the ship shape plate. The boundary conditions are of the same general form and by suitable manipulation of the shape of the boundary they can be made identical. The same plate equation must be satisfied. Horever, at points near the boundary it will have a modified pattern. Assume the shape and grid is as shown in Figure 14. _ IY_ lb a (B) C Figure 14i

ENGINEERING RESEARCH INSTITUTE CdI-36 I UNIVERSITY OF MICHIGAN We may assume the shape at (A) which would simplify calculations, and the approach used in the semi-infinite plate could be used as all edges are normal to the grid. The boundary conditions would not have to be altered. If the more nearly true shape as at (B) is used,then the varying grid lengths as b and c have to be incorporated in the pattern to obtain the deflections along the free edge. Another alternative is to assume straight line variation from the edge of the plate to points (1) and (2)j which is better) as several points are needed across the boundary and it is desirable to keep them on the grid. This method does not lend itself to the boundary conditions we used in the previous problem. First,these conditions were for normals to the free edgejthusthe points obtained from the boundary conditions will not fall on the grid points in general. There are two possibilities of handling the boundary conditions. First they may be transformed to lines along the grid lines, but this will necessitate a different boundary condition at each point. Possibly a better way would be to interpolate values inside the plate at points parallel to the edge and use these as basis for satisfying the boundary conditions, This is illustrated at (C). It will be necessary to compute a great number of points by interpolation in this method, and without finding the altered boundary conditions it is difficult to determine which is more involved. Another suggestion is to use the semi-infinite solution transformed by conformal mapping to a slot approaching the ship shape.

APPENDIX C Section d - Part II BENDING STRESSES UNDER A CONCENTRATED LOAD ON, THE EDGE OF A SEMI-INFINITE ICE SHEET BY JESSE ORMONDROYD September, 1949

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX C Section d - Part II BENDING STRESSES UNDER A CONCENTRATED LOAD ON THE EDGE OF A SEMI-INFINITE ICE SHEET From the results of the finite difference calculations made by Peach, the stress on the bottom of the ice sheet right under the load is ' r = 164.4 P If rx = 200 lb/in2 (ultimate strength of ice) the force necessary to crack.the ice is P 200 h2 1.215 h2 h in inches P in pounds h in. h2in.2 P = 1.215 h21b 12 144 175 24 576 700 36 1296 1575 48 2304 2850 60 3600 4380

CdII-2 ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN It seems obvious that the calculated forces to crack an ice sheet of thickness h are too small. Calculated stresses just under the load are evidently not too accurate with the finite difference method. A more rough and ready approach is offered by a formula given on page 250 of "Plates and Shells" by Timoshenko. The formula holds for a semi-infinite slab supported on an elastic foundation and carrying loads P pounds spaced at intervals of length a inches along the free edge. When a is made very large, the maximum stress on the under surface of the slab just under the load is x (max) = 0.529 (1 + o.54) [1o0 0.71 where J = Poisson's ratio for ice P = Total load in lbs. h = Thickness of ice,inches E = Modulus of elasticity for ice, lbs/in2 K = Spring constant of elastic foundation lb/in.Ainch deflection. c = Radius of the circle of area over which the load P is distributed. b =.6c + h2 - 0.675 h when c< 1.724 h b = c when c > 1.724 h If c = o P is a concentrated load and b = 0.325 h 4 = 0.011157 h

ENGINEERING RESEARCH INSTITUTE CdII-3 UNIVERSITY OF MICHIGAN When the plate is supported in water K = 62..4 - 0.0361 lb 1728 in. E = 500,000 lb/in.2 for ice Eh3 1.242 x 109 K=bL h Solving for P rx h2 0.529 (1 + 0.54) ) [log1-() - 0.711 If - = 200 lb/in.2 andi =.3 x P = 325.362 h2 Eh3 [log ().71] h /Eh 3 Eh3 h3. 2 h2 ih () log(Th ) log o 0.71 h bs 11~~ Eh 3\ (I~[og0 0 71 Elbs. 10 124.2 x 106 8.0934 7.3834 100 13.52 4,400 20 62.1 x 106 7.7931 7.0831 400 56.50 18,390 30 41.4 x 106 7.6170 6.9070 900 130.00 42,300 50 24.85x 106 7.3953 6.6853 2,500 374.00 374 121,500 100 12.42x 106 7.0941 6.3841 10,000 1,565.00 508,400 200 6.21x 106 6.7931 6.0831 40,000 6,590.00 2,142,000 300 4.14x 106 6.6170 5.9070 90,000 15,240.00 4,960,000

~cdrI-4 | ENGINEERING RESEARCH INSTITUTE I UNIVERSITY OF MICHIGAN Figure 1 shows the relationship between the concentrated force P and h the thickness of ice which it can crack at the bottom of the ice sheet just below the load. The maximum static force that the U.S.C.G.C. Mackinaw can exert resting on a ledge of ice at a point just forward of the skeg is about 10 percent of its displacement tonnage. At 5,000 tons displacement, it could exert a vertical force P = 1,120,000 pounds. At 4,000 tons displacement, it could exert a vertical force, P = 896,000 pounds. If the ship ran up on the edge of the ice at full speed, it might exert about twice as much vertical force, 2,240,000 pounds at 5,000 tons displacement and 1,792,000 pounds at 4,000 tons displacement. From Figure 1 under static conditions, the Mackinaw could crack a solid sheet of ice 146 inches (12 feet 2 inches) thick at 5,000 tons displacement and a sheet 129 inches (10 feet 9 inches) thick at 4,000 tons displacement. Dynamically the Mackinaw might defeat a sheet of ice 205 inches (17 feet 1 inch) thick at 5,000 tons displacement and 183 inches (15 feet 3 inches) thick at 4,000 tons displacement. Ice probably never gets over 4 feet thick in the Great Lakes. This thickness can be classified as "thin" for the Mackinaw. A rough rule for ice thicknesses to be met in the Great Lakes is P = 46 h This would hold for the ship resting statically on the straight edge of an unbroken ice sheet. In a field of ice already penetrated and cracked, the maximum vertical force at the bow is probably much less than half of the amount given in the approximate formula - perhaps nearer to P = 1 2 or even less.

CdII-5 140 - 0 D 100 0 60- - o 60 --- - 02E 0 4 8 12 16 20 24 28 32 36 40 44 48 h, INCHES, ICE THICKNESS _ 0 9.P DYNAMIC A= 5000 TONS.2 P DYNAMIC =4000 TONS. 0 m33 P AT 5000 TONS DISPLACEMENT P AT 4000 TONS DISPLACEMENT 0 50 100 150 200 250 300 h,INCHES, ICE THICKNESS C.d.2 FIG. I

APPENDIX C, SECTION e STRESS ANALYSIS OF A CRACKED ICE SHEET BY P. F. CHENEA P. NAGEDI August, 1949

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX C, SECTION e STRESS ANALYSIS OF A CRACKED ICE SHEET LEGEND Normal stress on planes perpendicular to x-axis. E Modulus of elasticity in tension and compression. M Bending moment in a beam. Q Shearing force in a beam. I Moment of inertia of a plane figure. E bh3 D Flexural rigidity of a plate = 1 2 i2 Poisson's ratio. k Modulus of foundation, pounds/square foot. h Thickness of plate. w Deflection in a beam. Specific weight (fresh water). b Width of plate. STRESSES IN AN ICE SHEET DUE TO LOADING ON THE EDGE 1. The Equation of Bending Moment We consider a semi-infinite plate of constant width b on an elastic foundation under the load Qo as shown in Figure 1. We have the following relationsl: 1Timoshenko,"Strength of Materials", Part II.

6 so (RECTANGULAR) LOAD Qo APPLIED ATO 0 x CRACK LINE h THICKNESS C.e. FIG. I

ENGINEERING RESEARCH INSTITUTE Ce-3 UNIVERSITY OF MICHIGAN M d2w D 2 D dx D dx3 kw d4w dx where D is the flexural rigidity of the plate and k is the modulus of the foundation. The last equation can be written as: d4w + 4;w = 0 dX where w4 = k 12 k (1 2) k = 5 4 E h3 b The solution of this differential equation is w = eX(A cos Xx + B sin Xx) + e-Xx(G cos Xx + H sin Xx) (1) where A, B, G, and H are arbitrary constants to be determined by the boundary conditions. The boundary conditions are: a) = Q d b) M = 0D w = c) w -_- O as x --- o

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN The last condition yields at once that A = B = 0 so that (1) reduces to w = e X(G cos Xx + H sin Xx) (2) Now dwx ke-X (G cos Xx + H sin Xx) + e-Xx(-GX sin Xx + Hk cos Xx) dx = kXe-x [(G - H)cos Xx + (G + H)sin x] d = 2e2Xx [G-I)cos Xx + (G+H)sin x- 2eXx E(G'-H)sin kx + (G+H)cosx] = 22e X(G sin x - H cos Xx) The second boundary condition yields: (2) -= 2 2 H = 0 So H = 0 and thus (2) reduces to w = G ex cosXx (3) Then dw _ - G X e (cos Xx + sin x) dw = G _2 e2x sinXx dx2

ENGINEERING RESEARCH INSTITUTE Ce-5 UNIVERSITY OF MICHIGAN dx3 = 2 G X3 e-Xx (sin Xx - cos x) Thus from the first boundary condition we have: (d3w) - 2 G = GQ/ So G - Qo/23D and we have for the deflection w = Qo e-Xx cos Xx 2X3D Thus we have for the bending moment M = D d2 = QO-xsinXx dx2 x This solution is valid for a plate of constant width. 2. The Point of Maximum Stress An approximation to the results on plates of varying width can be found by letting the width b of the plate have the form b = bo(X/h)P where bo is constant and P varies from 0 to 1 (O K K L 1); d may be regarded as the"shape factor"; that is, for the case of rectangle P = 0, for parabola 5 = 1/2, and for the wedge 5 = 1. Using the usual beam formula, the stress is given by: e = M(h/2)_

UNIVERSITY OF MICHIGAN where b33 I _ -b h = b (x1/h) h 12 ~ 12 So we have X o~ (eX sin 7Xx X b h2-' x3 If we differentiate the above and set it equal to zero, we get dO -e- sin Xx + ee-X cos x} - Xe-Xe Sin Xx xl Xboh Lb x2 or (ox + A) sin Xx = Xx cos Xx or tan x = X Xx + X where x is the distance where the maximum stress occurs. Now the load per linear foot is given by Qo % r Xh2 (x/h) 0b - 6 eXx sin Xx 3. Determination of the Distance x and the Load (Q /b,) for Various Thicknesses of the Ice Sheet In Figures 2 and 3, x and Qo/bo are plotted vs. the shape factor, for various thicknesses h (from 1 to 10 feet), using the following data: E = 500,000 psi = 62.4 lbs/ft3 (k = bd) ) = 1/3 max = 200 lbs/in2 (Modulus of rupture)

DISTANCE x FOR MAXIMUM STRESS VS. SHAPE FACTOR A (FOR VARIOUS THICKNESS h-FEET) I00 u. 60 iL I20 I 0II I I I 3I9 I9 O 3 9 C.e. 8 4 2 4 10 FIG.2 PARABOLA RECTANGLE WEDGE SHAPE FACTOR -'

ce-8 50 BREAKING LOAD ( Q bo SHAPE FACTOR / (FOR VARIOUS THICKNESS h - FEET) 45 40 35 30 25 15,0 h4 REOTANG~I PARABOLA (SHAPE FACTOR ) WEDGE

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Ce-9 X = k/i7iD = o.o3988 h-3/4 per foot or 1/k = 25.7 h3/4 feet 4. Differential Equation of the Deflection of a Wedge The analysis presented in Section 1 pertains to the case of the rectangular ice sheet, i.e., when 3 = 0. Let us now consider the extreme case of 1 = 1, namely that of a wedge as shown. In accordance with Section 2, the width b of the ice sheet in this case is given by b = bo(x/h) I and the corresponding flexural rigidity of the plate is D = EI = Ebh3 12(1- _2) Eb h2 = 0 x = ix 12(1 _ 2) GC.e. FIG. 4 Thus the bending moment and shear at a section x are M = x d 2 dx dx 2 d2] = [d dx3w and

Cse -10 | ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN -tb w = a2 [ d2wl d4w dawl - b w = --- x = x + 2 $ dx2 L xdx4 d dx3 This last equation can be written as d4w 2 d 3w 4 + x 3 + Xw = (4 dx dx where k4 = bo/S h. The fourth-order linear differential equation (4) is a self adjoint equation which may be solved, by iteration methods. The series solution in this case appears to be tedious and impractical. 5. Evaluation of the Results The analysis of an ice sheet of rectangular plan form (1 = 0) and the corresponding points of Figure 2, where distance x for maximum stress is plotted vs. shape factor 1, is exact. Thus for values of 1 close to zero the error is small. In order to investigate the error for larger values of 1, we shall consider the extreme case of PB = 1, i.e., the wedge. We thus proceed to determine an error function w* by substituting the solution (3) into Equation (4) and calculating the error w* that must be added to the solution (3). The procedure is as follows: -Xx w = G e cos Xx (3) where G = Qo/2X3D. Substituting of (3) into the left-hand member of (4) leads to L(w) - d4w+ 2 w + 4w 4 (1 - tan xx dx6 x dx3, kx

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN The error function w* is now computed from (4) by setting its right-hand member equal to this residual L(w). Thus we have d4w* 2 d _w 4 4 + G% 3 +jC GX, (1 tan Xx)3 dx-4 x dx3 Gt where G = Qo/23D. Since the term X4w* is numerically very small, we proceed to integrate the above equation neglecting this term. x2 d3w] G4 [3x2 4 (1 - tan Xx is an equivalent form of the previous equation. Direct integration yields. = GX [x2 4_ dxThus the error M* of the bending moment is M* = D GDk x dx We have from Section 1 d2w' Qo -Xx M = D d w 2 ex sin Xx = 2 2 G D e-kX sin Xx dx2 x The error involved in the stress may be determined from the ratio 6*/0-or simply from M*/M. Thus 2 2 _ 2x] -Xx 2 e sin Xx The value of this error for the stress near the load is

ENGINEERING RESEARCH INSTITUTE Ce-12 UNIVERSITY OF MICHIGAN X2 _ x 2 lim (M*/M) = lim = -1 x-O x-PO 2 e-Xx sin Xx This in general indicates the results for 1 = 1 are 100 percent in error.

Ce-13 -r FOR h18 0 =80 #/IN = 960 #/FT z300l C. FIG.5 0 00 0 10 20 30 40 50 60 h,INCHES, THICKNESS OF ICE G.e FIG.5

APPENDIX C, SECTION f BEPORT ON ICE CRACK PHOTOGBRAPH U. S. C. G. CUTTER MACKINAW WILLIAM W. HAGERTY JAMES R. PACEARD August 15, 1949

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN APPENDIX C, SECTION f REPORT ON ICE CRACK PHOTOGRAPHY U. S. C. G. CUTTER MACKINAW This report includes the following material: (1) a statement of the factual data obtained by the ice crack photography party; (2) an exhibition and discussion of a representative set of the photographs; (3) recommendations for the area of future study to supplement the existing data; and (4) comments on the experimental method used and recommendations concerning the practice to be followed in future work. 1. FACTUAL DATA The principal data obtained were about two hundred feet of black-andwhite 16-mm. film of good quality and fifty of colored 16-mm. film of good quality. In addition, about three hundred feet of black-and-white film of poor quality were taken, but are not suitable for study. The film records the speed of ice crack propagation and the crack patterns developed due to the motion of the ship through the ice. The record includes the ice-breaking action at many speeds of the ship, starting from barely perceptible motion to about twelve knots. Exact information on speeds at the instant of exposure is lacking.

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Valuable experience was gained in taking such pictures. Ice, snow, and light conditions varied widely during the taking of these photographic records, so that considerable judgment had to be exercised by the camera personnel. A summary of this knowledge will be included in Section 4 of this report. Exposures were taken from two positions: (a) One position was with the camera mounted on a framework attached to the bow of the ship at a position about six feet above the ice or water level and about twelve feet forward of the point where the bow of the ship contacts the ice. The reason for taking photographs from this position was to show the close-up action of the ice-breaking process at the bow. (b) The second position for taking pictures was the weather deck, about thirty feet above the level of the ice. From this position, the ice-breaking action could be observed over a wider area and the total range of the different kinds of ice cracks that developed could be recorded. 2. EXHIBITION AND DISCUSSION OF A REPRESENTATIVE SET OF PHOTOGRAPHS All photographs shown on the following pages are enlargements of single frames taken from the 16-mm. film record.

of ship Figure 1 o This picture shows some of the details of the camera and the framework used for taking pictures down near the ice at the bow of the ship. The legs of the frame were made from standard 3/4-inch pipe and were supported from ears welded to the ship itself. Another view of the frame and supports is shown by Figure 4. A flat piece of steel plate served as a table upon which the camera could be mounted. The outer end of the frame was supported by the chain and cable shown running to the upper righthand corner of the picture, The camera is shown being put into the aluminum box which protected it. The box was then mounted on a fixture attached to the light-colored pipe which is shown bolted to the top of the frame. Another view of the pipe and fixture is shown by Figure 5. A chain was attached to the aluminum box and a release was designed so that a sharp pull on the chain attached to the four corners of the aluminum box would release the camera and box from the supporting pipe. In this way, the camera could be rescued if it appeared that the framework was about to be damaged by windrows of ice. The shiny round plate on the front of the box is a lucite disc which covers the lens of the camera. The disc was rotated at high speeds by the small motor shown and in this means the camera lens was kept free from flying particles of ice and snow. The round object in the lower righthand corner of the table is a counterweight to balance the camera.

Cf.4 ~ENGINEERING RESEARCIH INSTITUTE ~~l4 U NIvERSITY OF MI}.IGAN___NE YOMCG of ship Figure 2 This shows a view of a patch of clear ice as taken from the weather deck about thirty feet above the ice. The orthogonal patterns formed by the new cracks are clearly shown. The diagonal crack is of older origin. The white areas are patches of snow. The vertical crack running directly forward of the ship was frequently visible for 75 to 100 feet ahead. The thrust or spoke-like cracks nearly always formed first, with the curved or bending cracks appearing shortly after, The pattern shown might be called the primary crack pattern, since in the area immediately forward of the ship more numerous and random cracks wrould appear. Approximte speed of the ship was four to six knots,

ENGINEERING RESEARCH INSTITUTE ' UNIVERSITY OF MICHIGAN Cf5 ir of s130lp Figure 3 In this picture taken from the weather deck, the primary crack hich always ran straight forward of the bow is shown along the right side of the picture. Many older cracks are also included in this frame. Of special interest is the consistent series of bending cracks which are shown crossig the primary crack, These bending cracks have appeared in rapid succession, and this particular frame is taken fsrom the film ediately after the appearance of these bending cracks. Approximate speed of the ship was two to three knots.

ENGINEERING RESEARCH INSTITUTE' ' Cf.-6 UNIVERSITY OF MICHIGAN 3 Fire 4 W.':............... i~~~~~~~~~~~~~~~~~~....................................................................~i;:;~~~~~~~~ La~................. laft nor"Aa r in Lk+,nA +U^,i,^w~g i~a~efo "- -.C&44. — 1A. 1 ---3 1 — ~e~ge

t~~~~~~~~~~~~~~~~............!_ - -....W.... |ID 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~............................... -.....a......... -a a s -- g — - i-.: a a s - S l- - i..................................... iEE-E -E iE-E............ Ss~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. g.....-.-.; aaaaa~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...-..........: C,',,,.,-'''' ''''''"''"''''''""''"'"'""'''"""'"'""''''''''"''''~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.......... W""'S-;'-""'""""''""';''"";'"' ';;;';';"~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..................:: --........... aa -- g a a - -g...... g 0 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~............. tlmt~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~................*.}#... ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~....:..: 7{...................... In this case the sh~~~tp ts at rest................................. relatitely 1arge area of au3>port ts shown It ts twortaE to te the.............................. pr~~eponderame of thrast crac~~~ks a-nd t~~w:re~lat~~tvely few h........................ sh arolmd the oupport area* This picture wae alsof taR-en fr the..................................................-...... d~~~~~~~~~~~~~eek~ ~ ~ ~~~~~~ ~...............

Figure 6 This picture was taken with the camera moulnted on the framework and down near the bow of the ship. In this case the ship was moving alowly, prob ably aboutl three or four knots. Slow speeds tusually peimitted best developmuent of the bending cracks. The manner in which the cracks develo ped throtuh the ice is also shown in this picture. From this and many other pictures it appears that; in the case of ice about fitoeen inches thick, a vertical crack will develop thbrouh the top half of the ice and diagonal spreadout cracks will deo velop in the bottom hlf, as8 shown in the auxiliary sketch. The thrust criack in the Top of Ice ower right corner appeared shortly after.......... the appearance of the bending crack shown........ running upward to the right. As a retult,..... 'its direction ch ar:ed slightly when cros\ %:.sing the bending c rack. A porti.on o the............. camera f wrameork and of the bow of the ship Bottom of Ice is visible in the u per left corner of the frame, In cross section Imany cracks appeared approximately as in the sketch. This explains why some cracks appear so wide in the photographs.......... 1~~~~~~~~~~~~~~~~~~~~......:I - I:::::::: -:': --— ~-~~~~~~~~~~~~~~~~~~~~~~~~~......

l ENGINEERING RESEARCH INSTITUTE Cf-9 UNIVERSITY OF MICHIGAN f / of ship Figure 7 This picture was also taken from the framework mounted on the bow. In this case the ship was moving at an estimated rate of eight knots. The 32-frame per second rate was barely sufficient to stop the motion. The thickness could probably be determined by suitable calibration of such pictures as these, Attention is called to the large block of ice shown near the upper right portion of the frame. It is about to fold under the curved portion of the ship, The manner in which the ice seems to fold under the ship is clearly shown in many of the films taken, and is undoubtedly related to the efficiency of the ice-breaking process.

ENGINEERING RESEARCH INSTITUTE Cf -10 UNIVERSITY OF MICHIGAN 3. RECOMMENDATIONS FOR TEE AREA OF FUTURE STUDY The photographic sequences obtained demonstrate that it is possible to take clear pictures of the ice-breaking process. With the experience gained, future pictures should be still better. The orthogonal patterns formed indicate that the clear ice is homogeneous as far as the ice-breaking action is concerned. It appears, however, that the crack patterns developed are definitely related to the speed of the ship. For example, the thrust cracks develop much earlier than the bending cracks, a process which becomes more and more evident as the speed of the ship is increased. In view of the fact that the thrust cracks predominate relative to the bending cracks, it might be well to determine the reason for this predominance and to establish whether it is beneficial to efficient ice-breaking. A number of specific points can be considered. (a) The question arises as to the extent to which this action is related to the angle between the ship's centerline and the ice at the bow. On the Mackinaw it is approximately thirty degrees. Tests could be made to determine the fracture patterns caused by loading at different angles. This could be done rather easily under static conditions. (b) Tests should be made to determine whether ice will break more readily when the breaking is effected by an excess of radial cracks, or whether it is more desirable to develop an excess of bending cracks. The effect of ice thickness should also be studied. In other words, should ice-breaking be purely a thrust action or purely a bending action, or which combination of the two should be used for maximum efficiency?

ENGINEERING RESEARCH INSTITUTE Cf-ll UNIVERSITY OF MICHIGAN (c) If possible, data should be obtained when the Mackinaw is breaking ice of about three or four feet thickness. It is probably impractical to break any quantity of such thick ice in a laboratory. 4. COMMIETS ON THE EXPERIMENITAL METHOD USED AND RECOMMENDATIONS CONCERNING THE PRACTICE TO BE FOLLOWED IN FUTURE WORK This discussion will be divided into four parts: (a) the effect of weather and ice conditions on the taking of good photographs; (b) mechanical features of the apparatus used; (c) film material; and (d) camera techniques. a. The Effect of Weather and Ice Conditions 1. The cold weather experienced slows down moving parts. It is advisable to have all outside equipment completely cleaned and serviced for cold weather operation. 2. In general, only small patches of clear ice were found on this trip. This was generally unsatisfactory, especially for picture sequences taken from the weather deck. 3. The dark, clear, smooth ice permits more readable pictures than the cloudy or gray ice. 4. The camera platform should always be elevated when approaching windrows, since their height cannot always be accurately estimated from the weather deck. b. Mechanical Features of the Apparatus Used 1. The camera used was an Eastman Eastman Cine-Kodak Special, 16 mm., equipped with Kodak-Anastigmat f.2.7, 15-mm. lens.

Cf -12 ENGINEERING RESEARCH INSTITUTE...... UNIVERSITY OF MICHIGAN 2. An electrical drive should be provided for the camera to eliminate the time required to stop the ship and send a man down to wind the camera. 3. For the same reason of eliminating delay, it would be preferable to use a 200-foot capacity magazine rather than the 100-foot capacity magazine used in this program. A 200-foot magazine would give about four minutes at 32 frames per second with 16-mm. film. An extra magazine should be provided to permit rapid film changes. 4. The Plexiglas spinning lens shield was very successful in that flying snow and ice particles were prevented from obscuring the lens. The shield did not seem to interfere optically to any noticeable extent. A preliminary difficulty experience when using the Plexiglas shield was that the motor was not perfectly free from oil. When cold wind blew on it, the shield was slowed down to such an extent that it did not throw the snow off. When the system was cleaned and lubricated with kerosene, the results were very good. 5. The aluminum box provided adequate protection for the camera and also provided a fine means of supporting it. The box was constructed from 3/16-inch aluminum sheets and the inner base was provided with a fitting which mated with that of the camera. 6. The camera platform should be attached to the ship about twelve to fifteen feet above the ice line rather than about five feet as in January, 1948. This would not only protect the platform from windrows, but in addition would remove the framework from the field of view of the camera. 7. A windlass or some other mechanical means should be provided to permit one operator to elevate the camera platform rapidly. 8. The camera box mount should be adjustable to permit shifting the camera from side to side in order to follow the sun as the ship changes course.

ENGINEERING RESEARCH INSTITUTE Cf-13 UNIVERSITY OF MICHIGAN In January, 1948, it was necessary to unbolt the mounting pipe and reverse its position. It would have been better to provide dual mountings, one on either end of the pipe. The principle of the safety chain used to rescue the camera was very satisfactory on several occasions when the platform ran into windrows and ice blocks before it could be elevated. In no case was the camera damaged. c. Film Material 1. Negative Super XX was used to provide the necessary speed, color sensitivity and latitude, as well as to permit a maximum of manipulation and ease of duplication. 2. Kodachrome could be used at the bow position only in the bestlight conditions, due to the slow speed of this film. The pictures taken with Kodachrome from the deck gave a very readable rendition. d. Camera Techniques 1. In order to stop the action when shooting from the bow, the camera speed should be 32 frames per second at low speeds below six knots, 64 frames per second at intermediate speeds up to ten or twelve knots, and probably 128 frames per second at higher speeds. From the deck, 32 frames per second is satisfactory. 2. Wide-angle lenses appear to be necessary due to the limitations of distances involved. 3. Reddish-colored filters appear to give good results with blackand-white pictures. These filters tone down the dark blue of the ice and increase the contrast of the white cracks. A Kodachrome artificial light to daylight type of filter was used to keep the filter factor down under poorer light conditions and appears to have been effective. The best filter to use with a particular color or shade of ice should be determined by experiment.

c-14 I ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN 4. The angle of shooting the ice-breaking action at the bow was about thirty degrees. This seems to have been a good compromise between visibility and stopping the action. 5. When shooting from the weather deck, it was necessary to select a clear patch of ice ahead of the ship and to follow it in to the bow. A support should be provided so that operator will not be required to hold the camera in his hand and lean over the side to take pictures from the deck. 6. There is a great difference between the reflecting power of the clear ice and the snow-covered areas. Exposures to show the cracks must be carefully determined by readings taken from the clear ice. Using the W6ston Master II Lightmeter, readings of 1600 were observed from the snow, while at the same time the clear ice showed only from 100 to 200. 7. All shots should be identified at beginning and end by the rise of a slate. The slate should give run number and camera and film data. It is not possible to identify runs separately after the test has been completed without such data. 8. A scale should be included in all shots, both those taken at the bow and those taken from the weather deck. Ship speeds could then be computed directly from the pictures. 9. A level should be mounted on the camera box to give the sighting angle. This should be put on the slate of Item 7.