MOVING LOADS ON ICE PLATES

SOLID MECHANICS AND ITS APPLICATIONS Volume 45 Series Editor:

G.M.L. GLADWELL Solid Mechanics Division, Faculty of Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3Gl

Aims and Scope of the Series

The fundamental questions arising in mechanics are: Why? How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For a list of related mechanics titles, see final pages.

Moving Loads on Ice Plates by

VERNON A. SQUIRE Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand

ROGER J. HOSKING Department of Mathematics and Statistics. James Cook University. Townsville. Australia

ARNOLD D. KERR Department of Civil Engineering. University of Delaware, Newark. U.S.A.

and

PATRICIA J. LANGHORNE Department of Physics, University of Otago, Dunedin. New Zealand

KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-13: 978-94-010-7238-0 DOl: 10.1007/978-94-009-1649-4

e-ISBN-13: 978-94-009-1649-4

Published by Kluwer Academic Publishers, P.O. Box 17,3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

Printed on acid-free paper

All Rights Reserved © 1996 Kluwer Academic Publishers Softcover reprint of the hardcover 1st edition 1996 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

The Emperor of China on the move, hauled by a team of eight. (Lorch, 1977; © Walter Lorch 1977; reproduced here by kind permission of the author; source Schweizer Turn und Sportmuseum.)

To our parents

All royalties from the sale of this book will be donated to the 'Save the Children Fund'.

Ice sledge on the Tegernsee, Bavaria, Germany. In the background a man crosses over on a unique ice scooter. Eighteenth century engraving. (Lorch, 1977; © Walter Lorch 1977; reproduced here by kind permission of the author; source Deutsches Museum, Miinchen.)

Table of Contents

PREFACE

xi

1 PREAMBLE

1

2 STRUCTURE AND PROPERTIES OF ICE PLATES 2.1 Introduction... 2.2 Atomic Structure .. 2.3 Sea Ice . . . . . . . . 2.3.1 Introduction 2.3.2 Interface Stability and Microstructure 2.3.3 Structure of a Sea Ice Sheet . 2.3.4 Salinity and Desalination . . 2.3.5 The Brine Content of Sea Ice 2.4 Lake and River Ice . 2.4.1 Introduction . . . . . . . . . 2.4.2 Impurities........... 2.4.3 Structure of a Nonsaline Ice Sheet 2.5 Some Mechanical Properties of Ice . . . . 2.5.1 Introduction . . . . . . . . . . . . 2.5.2 The Rheology of Ice and its Constitutive Laws

9 9

3

CONTINUUM MECHANICS 3.1 Introduction........ 3.2 Notation and Definitions. . . 3.2.1 The Stress Tensor .. 3.2.2 Infinitesimal Strain and Strain Rate 3.2.3 Momentum Balance . . 3.3 Linear Elasticity . . . . . . . . . . . 3.3.1 Stress-Strain Relations. . . . 3.3.2 Deviatoric Stress and Strain. 3.3.3 Plane Stress and Plane Strain. 3.4 The Thin Elastic Plate Equation . . . 3.4.1 Approximate Expressions for Stresses and Strains. vii

9

10 10 15 20 26 28 29 29 30 30 33 33 34

47 47 47 48 49 50 50 50 51 52 52 52

4

3.4.2 Momentum Equations 3.4.3 The Plate Equation 3.5 Inhomogeneity . . 3.6 Viscoelasticity. .. .. 3.7 Elastic Plate of Finite Thickness 3.8 The Fluid Foundation .. .. 3.8.1 Hydrostatic Buoyancy and Fluid Inertia 3.8.2 The Continuity Equation " 3.8.3 The Newtonian Constitutive Equation 3.8.4 The Navier-Stokes, Euler and Bernoulli Equations 3.9 Surface Gravity Waves at a Free Surface . .. .. 3.9.1 Free Surface and Bottom Boundary Conditions 3.9.2 The Dispersion Equation 3.10 The Floating Ice Plate . 3.10.1 Elastic Model. 3.10.2 Viscoelastic Model 3.10.3 Finite Thickness

54 55 56 59 60 61 61 62 63 63 64 65 65 66 66 67 68

HISTORICAL PERSPECTIVES 4.1 Precursor 4.2 Beam on a Winkler Base. 4.2.1 No Damping .. 4.2.2 With Damping .. 4.2.3 The Effect of Axial Forces. 4.2.4 Timoshenko Beam 4.3 Plate on a Winkler Base . 4.4 Pioneering Analyses of a Plate on a Fluid Base 4.4.1 Stationary Loads. .. .. 4.4.2 Moving Loads. . . 4.5 Liquid under Gravity with Surface Tension 4.6 Concluding Remarks ... .

69 69

5 THEORETICAL ADVANCES 5.1 5.2

5.3

Introduction... .. Dispersion Relation and Critical Speed. 5.2.1 Dispersion Relation for Plane Waves 5.2.2 Moving Loads. .. .. 5.2.3 Energy Accumulation and Critical Speed Steady State Wave Generation .. 5.3.1 Formal Solution for the Deflexion . . . 5.3.2 Asymptotic Plate Deflexion Formulae 5.3.3 Steady State Wave Patterns .

viii

70 70 80 84 86 88

90 90 93 99 101 105 105 106 107 109

110 111 111 113 114

5.3.4 A Complementary Study . . . . . . . . . . . . . . . 117 5.4 Impulsively-Started Line Load . . . . . . . . . . . . . . . . 121 5.4.1 Formal Time-Dependent Solution for the Deftexion. 121 5.4.2 Evolution of the Response in the Vicinity of the Load 124 5.4.3 Spatial Development of the Wave System 127 5.5 Bukatov-Zharkov Theory 130 5.5.1 The Model .. 131 5.5.2 Results . . . . . . 133 5.6 Viscoelastic Response .. 133 135 5.6.1 Formal Solution for the Deftexion . 137 5.6.2 The Deftexion in the Line of Motion 5.6.3 Asymptotic Deftexion Formulae and Steady State Wave Patterns . . . . . . . . . . . . . . . . 141 5.7 Water Stratification . . . . . . . . . . . . . 145 5.7.1 Dispersion Relation for Plane Waves 147 150 5.8 Plate of Finite Thickness 154 5.9 Vibrating Loads .. 5.10 Concluding Remarks. . . 157

6

EXPERIMENTS INVOLVING MOVING LOADS 6.1 Introduction . . . . . . . . . . . 6.2 Pioneering Experimental Work 6.2.1 Kubo (1980) . . . . 6.2.2 Wilson (1955, 1958) .. 6.2.3 Anderson (1958a) . . . . 6.2.4 Sunberg-Falkenmark (1963) 6.2.5 Eyre (1977) . . . . . . . . . 6.2.6 Goodman and Holdsworth (1978) . 6.2.7 Beltaos (1981) . . . . . . . 6.3 Recent Experimental Work . . . . 6.4 The Field Programme of Takizawa 6.4.1 Takizawa (1978) . . . . . . 6.4.2 Takizawa's Second Experiment 6.5 Laboratory Tests . . . . . . . . . . . . 6.6 The Field Experiments of Squire and Colleagues 6.6.1 A Bipolar Effort . . . . . . . 6.6.2 McMurdo Sound, Antarctica 6.7 Conclusion . . . . . . . . . . . . . .

IX

159 159 159 159 160 162 163 163 164 165 165 166 166 167 176 178 178 179 196

7 IMPLICATIONS AND CONCLUSIONS 7.1 Introduction. . . . . . . . . . 7.2 Critical Speed Vcrit = Cmin .. 7.2.1 Kheysin's Expression. 7.2.2 Poisson's ratio 7.2.3 Water Density 7.2.4 Ice Density .. 7.2.5 Water Depth 7.2.6 In-plane Forces 7.2.7 Viscosity ... 7.2.8 Synopsis . . . . 7.2.9 Repercussions. 7.3 Peak Deflexions and Strains 7.3.1 Temperature .... 7.3.2 Estimates for the Amplification Factor. 7.4 Waves at Distance 7.5 Aircraft 7.6 Finale ......

199

199 200 200 201 202 202 202 203 203 203 204 206 207 207 209 211 211

BIBLIOGRAPHY

213

INDEX

223

x

PREFACE

The goal of this book is to present a succinct, contemporary account of research on the topic of moving loads on floating sheets of freshwater ice or sea ice, drawing where necessary on historical developments which have led to our present understanding and on related information about the character of the ice. Moving loads might include vehicles of various kinds travelling upon the surface of the ice, but also landing aeroplanes. The salient feature is that the pressure exerted by the load on the ice moves across its surface, rather than being immobile. The subject embraces both experimental and theoretical work, so each is included. It is a fascinating area because the ingenious theories that have evolved over the last few decades have steadily become more polished, and consequently they have become very able to match observational data. In many cases the fit has been quite remarkable. But what does this mean to the agency that wishes to operate vehicles or land aircraft in the Arctic or Antarctic? It means that we understand the problem well, and accordingly, that we can make reliable forecasts about what will occur. Despite an abundance of first class research papers and reports on loads travelling across ice sheets, there has been no attempt to synthesize the many diverse threads into a common theme until this book. This is surprising as in many ways it is a perfect topic to narrate because it is at an advanced stage of development, and has the capacity to lead to sound and far-reaching conclusions. It is the authors' aim to fill this gap by making sense of the many disjoint studies that have taken place and by binding them into a coherent story with consistent nomenclature. Certainly, aspiring and practising cold-regions engineers should find our treatise useful, but we hope also to capture the interest of the various organizations that operate routinely and solve practical, down-to-earth problems on the ice in harsh polar climes. Graduate students in ice physics, ice engineering, or polar marine hydrodynamics, or indeed their mentors, may find the book timely. The trend towards increased exploration and exploitation of the Arctic and Antarctic demands more ambitious and creative solutions to solve apparently impossible problems. Cost-effective answers may lie herein. The sophistication of the theory necessitates some applied mathematics.

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We make no apology for this as the physics involved can be intriguing but complicated, yet we recognize also that readers may not be quite as enthused as we are about complex variable theory, asymptotic analysis, and Fourier transforms. The mathematics can approach, and at times transcend, graduate level, despite the authors having tried to write the text in a way which allows lesser mathematicians to move forward and to extract useful ideas and results. This book was completed using some of the facilities at the Scott Polar Research Institute, University of Cambridge, and the Department of Civil and Environmental Engineering, Clarkson University, while the principal author and Dr Patricia Langhorne were on Sabbatical Leave from the University of Otago, New Zealand. Because of the geographic heterogeneity of its authors the book could not have been completed without ethernet and the internet, employing Fetch and Telnet where necessary. It was typeset in UTEX on a Macintosh 540c using the unrivalled application Textures 1. 7 by Blue Sky Research. Other applications used herein are MATLAB, Spyglass Plot, MacDraw Pro, and MacDraft. Many colleagues and friends have helped directly and indirectly in the preparation of this book, and sadly they cannot all be named in person. We extend our sincere thanks to these unidentified souls. We mention specifically a few with whom we have collaborated, who have hosted Study Leaves, who have helped beyond the call of duty, or who have provided as yet unpublished data, photographs, or manuscripts to make the book as upto-date as possible. They are Professor George Batchelor, Mr Mark Borrie, Dr David Cole, Professor Anthony G. Collins, Mr John Davys, Dr John Dougherty, Dr Tim Haskell, Dr John Heap, Mr Arnold Heine, Dr Fausto Milinazzo, Mr William Mills, Mr Peter Monypenny, Mr Glen Rayner, Dr Bill Robinson, Ms Shirley Sawtell, Dr Ruben Schulkes, Professor Hayley Shen, Dr Nirmal Sinha, Associate Professor Alfred Sneyd, Professor Gil Strang, Professor Alar Toomre, Dr Peter Wadhams, and Dr Darryn Waugh. V.A.S. and P.J.L. are grateful to the University of Otago and the New Zealand Foundation for Research, Science and Technology for their support, and to the staff at the New Zealand Antarctic Programme for prolonged assistance. R.J.H. acknowledges the support of the Special Study Program at James Cook University, Australia. Lastly, the authors are indebted to their respective families who stoically tolerated the hummocky path to publication with an enigmatic smile. V.A.S., R.J.H., A.D.K., P.J.L. Clarkson University, N.Y., U.S.A. November 1995

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