Modelling in Geomechanics: Russian Translations Series 107 9789054102199, 9780203746851, 9054102195

Table of contents :
Cover
Half Title
Title Page
Copyright Page
Preface
Table of Contents
Part I Theory, Method and Technique of Modelling
1. Theoretical Principles of the Method
1.1 General Observations
1.2 Conditions of Similarity
1.3 Scale of Time
1.4 General Systematic Procedures
1.5 Verification of Reliability of Experimental Results
2. Construction and Testing of Models
2.1 General Assumptions
2.2 Computation of Height of Two-Dimensional Model for Longwall Faces
2.3 Simulation of Large Depths on Two-Dimensional Models
2.4 Reproduction of Properties of Rock Mass
2.5 Test of Models
3. Equivalent Materials
3.1 Requirements Imposed on Equivalent Materials
3.2 Theoretical Principles of Preparing Composite Materials
3.3 Basic Materials for Preparing Equivalent Materials
3.4 Technology for Preparing Equivalent Materials
3.5 Physical and Mechanical Properties of Equivalent Materials
4. Measurements on Models
4.1 General Concepts
4.2 Dynamometer Base and Mounting
4.3 Self-contained Microdynamometers
4.4 Construction of Microdynamometers (Example)
4.5 Measurement of Pressures on Mine Supports
4.6 Measurement of Deformation and Displacement
5. Planning Experimental Investigations
5.1 General Concepts
5.2 Selection of Initial Factors and Geomechanical Parameters
5.3 Procedure for Formulating Plan of Experiments
5.4 Analysis of Results of Investigations
5.5 Verification of Results and Determination of their Region of Applicability
5.6 Illustrative Example of a Problem from Group I
Part II Forecasting Geomechanical Processes during Mining
6. Forecasting Influence of Geological Factors on Stability of Mine Workings
6.1 Influence of Size of Cross-section on Convergence in a Drift
6.2 Influence of Depth of Mining on Stress Distribution in Vicinity of a Mine Working
6.3 Influence of Angle of Seam Dip on Displacement of Contour of Mine Working
6.4 Influence of Location of Longitudinal Axis of Mine Working with Respect to Strike Lines of Weak Planes
6.5 Influence of Residual Tectonic Stresses on State of Mine Workings in Fissured Rocks
6.6 Influence of Mechanical Properties of Surroundings on Development of Rock Pressure
6.7 Investigation of Stability of Drifts Depending on Type of Anchor Supports [Roof Bolts]
7. Forecasting Conditions of Interaction in 'Lateral Rocks support' Systems in Isolated Mine Workings
7.1 General Remarks
7.2 Interaction between Rocks and Yielding Supports in a Mine Working
7.3 Forecasting Effective Parameters for Supports of Mine Workings
7.4 Influence of Rigidity of Support on Interaction Process in 'Support-lateral Rocks' System
7.5 Comparison of Results of Observations in Mines and Laboratory Investigations
7.6 Approximate Overall Estimation of Working Conditions of 'Rock Massmine Working' System
8. Forecasting Manifestation of Rock Pressure in Preparatory Mine Workings
8.1 General Observations
8.2 Investigation of Stability of Development Workings Driven in Stratified Rocks
8.3 Investigation of Methods for Protecting Development Workings Driven in Stratified Seams
8.4 Forecasting Appearance of Rock Pressure during Over- and Undercutting of Seams in Development Workings
9. Forecasting Manifestation of Rock Pressure in Longwall Faces
9.1 General Observations
9.2 Determination of Variation of Stresses in Rocks due to Influence of Mining Operations
9.3 Forecasting Anticipated Load on a Support during First Cave-in of Main Roof
9.4 Investigation of Mechanism of Displacement of Roof Blocks and their Interaction with the Support
9.5 Influence of Seam Thickness on Nature of Stress Distribution Near Working Face
10. New Mechanisms of Manifestation of Rock Pressure
10.1 General Remarks
10.2 Working of Supports during Sudden Collapse of Roofs
10.3 Rapid Variation of Stresses in Abutments of Seams
10.4 Investigation of Deformation Properties of Fissured Rocks
10.5 Failure of Rocks Surrounding Mine Working at Large Depths
11. Principal Trends in Further Development of Investigations on Models
11.1 Geomechanical Problems
11.2 Extraction of Single Seams
11.3 Extraction of Multiple Seams
11.4 Investigation of Interaction between Supports and Lateral Rock Walls
11.5 Local Problems of Geomechanics Related to Conduction of Underground Mining Operations
References
Appendix

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RUSSIAN TRANSLATIONS SERIES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

26. 27. 28. 29. 30. 31. 32. 33.

34. 35. 36. 37. 38. 39. 40. 41.

42. 43. 44. 45. 46. 47. 48. 49. 50. 51.

K.Ya. Kondrat'ev et al. (editors): USSRiUSA Bering Sea Experiment D.V. Nalivkin: Hurricanes. Storms and Tornadoes V.M. Novikov (editor): Handbook of Fishery Technology, Vol. F.G. Martyshev: Pond Fisheries R.N. Burukovskii: Key to Shrimps and Lobsters V.M. Novikov (editor): Handbook of Fishery Technology, Vol. 4 V.P. Bykov (editor): Marine Fishes N.N. Tsvelev: Grasses of the Soviet Union L.V. Metlitskii et al.: Controlled Atmosphere Storage of Fruits M.A. G1azovskaya: Soils of the World (2 volumes) V.G. Kort & V.S. Samoilenko: Atlantic Hydrophysical Polygon-70 M.A. Mardzhanishvili: Seismic Design of Frame-panel Buildings and Their Structural Members E'.A. Sokolenko (editor): Water and Salt Regimes of Soils: Modeling and Management A.P. Bocharov: A Description of Devices Used in the Study of Wind Erosion of Soils E.S. Artsybashev: Forest Fires and Their Control R.Kh. Makasheva: The Pea N.G. Kondrashova: Shipboard Refrigeration and Fish Processing Equipment S.M. Uspenskii: Ufe in High lAtitudes A.V. Rozova: Biostratigraphic Zoning and Trilobites of the Upper Cambrian and Lower Ordovician of the Northwestern Siberian Platform N.!. Barlcov: Ice Shelves of Antarctica V.P. Averldev: Shipboard Fish Scouting and Electronavigational Equipment D.F. Petrov (Editor-in-Chief): Apomixis and Its Role in Evolution and Breeding G.A. Mchedlidze: General Features of the Paleobiological Evolution of Cetacea M.G. Ravich et al.: Geological Structure of Mac. Robertson Land (&st Antarctica) L.A. Timokhov (editor): Dynamics of Ice Cover K.Ya. Kondrat'ev: Changes in Global Climate P.S. Nartov: Disk Soil-Working Implements V.L. Kontrirnavichus (Editor-in-Chief): Beringia in the Cenozoic Era S.V. Nerpin & A.F. Chudnovskii: Heat and Mass Transfer in the Plant-Soil-Air System T.V. Alekseeva et al.: Highway Machines N.!. K1enin et al.: Agricultural Machines V.K. Rudnev: Digging of Soils by Earthmovers with Powered Parts A.N. Zelenin et al.: Machines for Moving the Earth Systematics. Breeding and Seed Production of Potatoes D.S. Orlov: Humus Acids of Soils M.M. Sevemev (editor): Wear of Agricultural Machine Parts Kh.A. Khachatryan: Operation of Soil-working Implements in Hilly Regions L.V. Gyachev: Theory of Surfaces of Plow Bottoms S.V. Kardashevskii et al.: Testing of Agricultural Technological Processes M.A. Sadovskii (editor): Physics of the Earthquake Focus I.M. Dolgin: Climate of Antarctica V.V. Egorov et al.: Classification and Diagnostics of Soils of the USSR V.A. Moshkin: Castor E'.1. Sarukhanyan: Structure and Variability of the Antarctic Circumpolar Current V.A. Shapa (Chief Editor): Biological Plant Protection A.1. Zakharova: Estimation of Seismicity Parameters Using a Computer M.A. Mardzhanishvili & L.M. Mardzhanishvili: Theoretical and Experimental Analysis of Members of Earthquake-proof Frame-panel Buildings S.G. Shul'rnan: Seismic Pressure of Water on Hydraulic Structures Yu.A. lbad-zade: Movement of Sediments in Open Channels I.S. Popushoi (Chief Editor): Biological and Chemical Methods of Plant Protection K.V. Novozhilov (Chief Editor): Microbiological Methods for Biological Control of Pests of Agricultural Crops (continued)

RUSSIAN TRANSLATIONS SERIES 52. K.l. Rossinskii (editor): Dynamics and Thermal Regimes of Rivers K.V. Gnedin: Operating Conditions and Hydraulics of Horizontal Settling Tanks G.A. Zakladnoi & V.F. Ratanova: Stored-grain Pests and Their Control Ts.E. Mirtskhulava: Reliability of Hydro-reclamation Installations la.S. Ageikin: Off-the-road Mobility of Automobiles 57. A.A. Kmito & Yu.A. Sklyarov: Pyrheliometry 58. N.S. Motsonelidze: Stability and Seismic Resistance of Buttress Dams 59. la.S. Ageikin: Off-the-road Wheeled and Combined Traction Devices 60. Iu.N. Fadeev & K.V. Novozhilov: Integrated Plant Protection 61. N.A. Izyumova: Parasitic Fauna of Reservoir Fishes of the USSR and Its Evolution 62. O.A. Skarlato (Editor-in-Chief): Investigation of Monogeneans in the USSR 63. A.1. Ivanov: Alfalfa 64. Z.S. Bronshtein: Fresh-water Ostracoda 65. M.G. Chukhrii: An At/as of the Ultrastructure of Viruses of Lepidopteran Pests of Plants 66. E.A. Bosoi et al.: Theory, Construction and Calculations of Agricultural Machines, Vol. 1 67. G.A. Avsyuk (Editor-in-Chief): Data of Glaciological Studies 68. G.A. Mchedlidze: Fossil Cetacea of the Caucasus 69. A.M. Akramkhodzhaev: Geology and Exploration of Oil- and Gas-bearing Ancient Deltas 70. N.M. Berezina & D.A. Kaushanskii: Presowing Irradiation of Plant Seeds 71. G.V. Lindberg & Z.V. Krasyukova: Fishes of the Sea of Japan and the Adjacent Areas

53. 54. 55. 56.

of the Sea of Okhotsk and the Yellow Sea

72. N.1. Plotnikov & I.I. Roginets: Hydrogeology of Ore Deposits 73. A.V. Balushkin: Morphological Bases of the Systematics and Phylogeny of the Nototheniid

Fishes 74. 75. 76. 77.

E.Z. Pozin et al.: Coal Cutting by Winning Machines S.S. Shul'rnan: Myxosporidia of the USSR G.N. Gogonenkov: Seismic Prospecting for Sedimentary Formations I.M. Batugina & I.M. Petukhov: Geodynamic Zoning of Mineral Deposits for Planning

and Exploitation of Mines 1.1. Abrarnovich & I.G. KIushin: Geodynamics and Metallogeny of Folded Belts M.V. Mina: Microevolution of Fishes K.V. Konyaev : Spectral Analysis of Physical Oceanographic Data A.1. Tseitlin & A.A . Kusainov: Role of Internal Friction in Dynamic Analysis of Structures E.A. Kozlov: Migration in Seismic Prospecting E.S. Bosoi et al.: Theory, Construction and Calculations of Agricultural Machines, Vol. 2 84. B.B. Kudl)'ashov and A.M. Yakovlev: Drilling in the Permafrost 85. T.T. KIubova: Clayey Reservoirs of Oil and Gas 86. G.1. Arnurskii et al.: Remote-sensing Methods in Studying Tectonic Fractures in Oi/- and

78. 79. 80. 81. 82. 83.

87. 88. 89. 90. 91. 92. 93. 94.

95. 96. 97.

Gas-bearing Formations A.V. Razvalyaev: Continental Rift Formation and Its Prehistory V.A. Ivovich and L.N. Pokrovskii: Dynamic Analysis of Suspended Roof Systems N.P. Kozlov (Technical Editor): Earth's Nature from Space M.M. Grachevskii and A.S. Kravchuk: Hydrocarbon Potential of Oceanic Reefs of the World K.V. Mikhailov it al.: Polymer Concretes and Their Structural Uses D.S. Orlov: Soil Chemistry L.S. Belousova & J...V. Denisova: Rare Plants of the World T.1. Frolova et al.: Magmatism and Transformation of Active Areas of the Earth's Crust Z.G. Ter-Martirosyan: Rheological Parameters of Soils and Design of Foundations S.N. Alekseev et al.: Durability of Reinforced Concrete in Aggressive Media F.P. Glushikhin et al.: Modelling in Geomechanics

MODELLING

IN

GEOMECHANICS

GEOMECHANICS IN GEOMECHANICS

F.P. Glushikhin, G.N. Kuznetsov, M.F. Shklyarskii, V.N. Pavlov and M.S. Zlotnikov

RUSSIAN TRANSLATIONS SERIES 97

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A BALKEMA BOOK

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P.O. Box 447, 2300 AK Leiden, The Netherlands e-mail: [email protected] www.crcpress.com - www.taylorandfrancis.com © 1993 Copyright reserved CRC PresslBalkema is an imprint ofthe Taylor & Francis Group, an informa business No claim to original U.S. Govermnent works ISBN 13: 978-90-5410-219-9 (hbk) DOT: 10.1201/9780203746851 This book contains infonnation obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectiry in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For pennission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfmncis.com and the CRC P,'ess Web site at http://www.crcpress.com

Translation of: Modelirovanie v geomekhanika, Nedra, Moscow, 1991 Distributed in USA and Canada by: A,A. Balkema Publishers

Preface

During mining for mineral resources and the erection of subterranean con­ structions one has to deal with a medium that is extremely complex in composi­ tion and has diverse mechanical properties and laws of deformation. To achieve these objectives one is compelled to utilise a large set of technological schemes and types of mine workings. In spite of the vast scale on which mining work and related scientific investigations are carried out, the development of mineral resources is such a complex problem that no unique laws have been defined to date to describe the behaviour of rock massifs which might be disturbed dur­ ing mining operations or when they interact with engineering structures. These processes are being studied throughout the world by various methods-from full-scale observations to theoretical interpretations and analytical descriptions. Amongst them physical modelling on equivalent materials (EM) holds a particu­ larly significant place. This method, proposed and developed by G.N. Kuznetsov in 1936, received considerable impetus in the post-war years. Essentially it con­ sists of preparing a model from a portion of the rock mass obtained from mine workings to investigate the processes described earlier. The models are devel­ oped from such equivalent materials as satisfy similarity criteria in reflecting the mechanical properties of rocks under natural conditions. The method of equivalent materials has received worldwide recognition and is extensively utilised in many areas of rock mechanics. Many a complex geomechanical problem has been resolved by its utilisation. In recent years physical modelling has been carried out at the level of multiple-factor predictive complex investigation in view of improvements in the methods of investigation, development of new equivalent materials and new techniques of measurement. These have helped in establishing new laws of de­ formation of rock masses and prediction of possible consequences (phenomena) since the values of principal influencing factors vary widely. The need for predictive evaluation is dictated not only by the problems un­ der investigation, but also by those circumstances which result in the idle time of highly mechanised modem face machinery or its inefficient operation, leading to a considerably greater loss to the national economy compared to that of 15 to 20 years ago. Predictions are based on results of physical modelling utilising equivalent materials and their thorough verification under natural conditions. Such verifications are carried out on the basis of a multiplicity of data accu­ mulated during investigation in mines over a wide range of variations of many specific observations. In some cases physical modelling enables better in-depth

vi investigation of geomechanical processes and improves the existing theoretical description of these processes or proposes new ones. The All-Union Scientific Research Institute for Mine Surveying (VNIMI) in its last report devoted to this topic and published in 1968 [21], brought out the theoretical principles of modelling with a description of the systematic pro­ cedure to be followed to achieve it together with solutions to several specific problems in mining. The present work is devoted to correlation of the results of investigations carried out at the VNIMI on models during the last twenty years. The book consists of two parts. Part I briefly reviews the theoretical principles of the method and new developments in procedures and techniques which help to extend it to a wider range of problems in geomechanics that could be solved through modelling. Part 11 is devoted to a presentation of the predictive investi­ gations conducted by the VNIMI on the problems of geomechanics encountered during underground mining for coal or the erection of subterranean construc­ tions, particularly those which are difficult to solve through theoretical methods and are practically impossible to resolve through direct observations in mines. The concluding chapter discusses the problems faced in geomechanics and the extent to which these have been investigated and suggests the methods to be adopted for modelling them with equivalent materials. The Preface and various conclusions drawn throughout the book were written by F.P. Glushikhin; Chapter 1 by G.N. Kuznetsov in collabora­ tion with F.P. Glushikhin; Chapter 2 by F.P. Glushikhin, M.F. Shklyarskii and G.N. Kuznetsov; Chapter 3 by M.S. Zlotnikov, F.P. Glushikhin and G.N. Kuznetsov; Chapter 4 by F.P. Glushikhin, V.N. Pavlov and M.F. Shklyar­ skii; Chapter 5 by M.F. Shklyarskii, V.N. Pavlov and F.P. Glushikhin; Chap­ ters 6, 7, 8 and 10 by F.P. Glushikhin and M.F. Shklyarskii; Chap­ ter 9 by F.P. Glushikhin, V.N. Pavlov and M.F. Shklyarskii; Chapter 11 by F.P. Glushikhin, M.F. Shklyarskii, G.N. Kuznetsov and V.N. Pavlov. The authors would be grateful for constructive criticism of this work.

Contents

Preface

v

PART I

THEORY, METHOD AND TECHNIQUE OF MODELLING

1. Theoretical Principles of the Method 1.1 General Observations 1.2 Conditions of Similarity 1.3 Scale of Time 1.4 General Systematic Procedures 1.5 Verification of Reliability of Experimental Results 2. Construction and Testing of Models 2.1 General Assumptions 2.2 Computation of Height of Two-Dimensional Model for

Longwall «aces 2.3 Simulation of Large Depths on Two-Dimensional Models 2.4 Reproduction of Properties of Rock Mass 2.5 Test of Models 3. Equivalent Materials 3.1 Requirements Imposed on Equivalent Materials 3.2 Theoretical Principles of Preparing Composite Materials 3.3 Basic Materials for Preparing Equivalent Materials 3.4 Technology for Preparing Equivalent Materials 3.5 Physical and Mechanical Properties of Equivalent Materials 4. Measurements on Models 4.1 General Concepts 4.2 Dynamometer Base and Mounting 4.3 Self-contained Microdynamometers 4.4 Construction of Microdynamometers (Example) 4.5 Measurement of Pressures on Mine Supports 4.6 Measurement of Deformation and Displacement 5. Planning Experimental Investigations 5.1 General Concepts 5.2 Selection of Initial Factors and Geomechanical Parameters 5.3 Procedure for Formulating Plan of Experiments

1

3

3

4

9

14

19

22

22

30

36

44

47

51

51

54

58

65

73

80

80

82

89

95

99

102

106

106

109

111

viii 5.4 Analysis of Results of Investigations 5.5 Verification of Results and Determination of their Region of

Applicability 5.6 Illustrative Example of a Problem from Group I

113

116

118

PART II

FORECASTING GEOMECHANICAL PROCESSES DURING MINING

6. Forecasting Influence of Geological Factors on Stability of

Mine Workings 6.1 Influence of Size of Cross-section on Convergence in a Drift 6.2 Influence of Depth of Mining on Stress Distribution in

Vicinity of a Mine Working 6.3 Influence of Angle of Seam Dip on Displacement of Contour

of Mine Working 6.4 Influence of Location of Longitudinal Axis of Mine Working

with Respect to Strike Lines of Weak Planes 6.5 Influence of Residual Tectonic Stresses on State of Mine

Workings in Fissured Rocks 6.6 Influence of Mechanical Properties of Surroundings on

Development of Rock Pressure 6.7 Investigation of Stability of Drifts Depending on Type of

Anchor Supports [Roof Bolts] 7. Forecasting Conditions of Interaction in 'Lateral Rocks-

support' Systems in Isolated Mine Workings 7.1 General Remarks 7.2 Interaction between Rocks and Yielding Supports in a Mine

Working 7.3 Forecasting Effective Parameters for Supports of Mine

Workings 7.4 Influence of Rigidity of Support on Interaction Process in

'Support-lateral Rocks' System 7.5 Comparison of Results of Observations in Mines and

Laboratory Investigations 7.6 Approximate Overall Estimation of Working Conditions of

'Rock Massmine Working' System 8. Forecasting Manifestation of Rock Pressure in Preparatory

Mine Workings 8.1 General Observations 8.2 Investigation of Stability of Development Workings Driven

in Stratified Rocks

123

125

125

126

127

130

132

135

137

142

142

143

147

148

151

156

160

160

161

ix 8.3 Investigation of Methods for Protecting Development

Workings Driven in Stratified Seams 8.4 Forecasting Appearance of Rock Pressure during Over- and

Undercutting of Seams in Development Workings

9. Forecasting Manifestation of Rock Pressure in Longwall Faces 9.1 General Observations 9.2 Determination of Variation of Stresses in Rocks due to

Influence of Mining Operations 9.3 Forecasting Anticipated Load on a Support during First

Cave-in of Main Roof 9.4 Investigation of Mechanism of Displacement of Roof Blocks

and their Interaction with the Support 9.5 Influence of Seam Thickness on Nature of Stress Distribution

Near Working Face

10. New Mechanisms of Manifestation of Rock Pressure 10.1 General Remarks 10.2 Working of Supports during Sudden Collapse of Roofs 10.3 Rapid Variation of Stresses in Abutments of Seams 10.4 Investigation of Deformation Properties of Fissured Rocks 10.5 Failure of Rocks Surrounding Mine Working at Large Depths

11. Principal Trends in Further Development of Investigations on

Models 11.1 Geomechanical Problems 11.2 Extraction of Single Seams 11.3 Extraction of Multiple Seams 11.4 Investigation of Interaction between Supports and Lateral

Rock Walls 11.5 Local Problems of Geomechanics Related to Conduction of

Underground Mining Operations

165

171

178

178

179

186

194

202

206

206

206

210

217

221

227

227

229

230

233

236

References

239

Appendix

242

PART I

THEORY, METHOD AND TECHNIQUE

OF MODELLING

1

Theoretical Principles of the Method 1.1 General Observations Modelling by the method of equivalent materials is based on the substitu­ tion of natural rocks by such artificial materials whose physical and mechanical properties have specific correlations with the corresponding properties of the nat­ ural rocks. These correlations, governed by the general principles of mechanical similarity, are helpful in achieving a close analogy between the geomechanical processes in nature and in the models under the action of gravitational forces. The method of equivalent materials helps to reproduce in a model the var­ ious structures of rock masses and minerals and to conduct in them all the principal mining operations carried out during extraction of minerals and sup­ porting of mine workings with a fair degree of approximation to such operations under natural conditions. Modelling materials that satisfy similarity conditions in terms of strength and deformation properties, development of structural formations analogous to the natural rock mass being modelled as also the use of instruments which mimic the working of supports in the model, enable us to study a wide variety of processes of deformation and failure of a rock mass as well as their interactiQn with the support during the development and operation of mine workings using models made from equivalent materials. Modelling through the method of equivalent materials forms an indispens­ able part of the various techniques used in the investigations of mining geo­ mechanics. Results obtained from modelling facilitate better interpretation of observations from mines help to more precisely organise and carry out such work with less expense. If the mechanism of the process under investigation is known, studies on models help to establish in most cases a more accurate and reliable computational method for resolving the corresponding problem in the field. A rational combination of modelling and analytical methods consid­ erably simplifies these problems and brings the computational techniques close to reality. Investigation of problems in mining geomechanics may also be carried out using other physical modelling methods such as photoelastic, centrifugal and structural. The underlying principles of these methods and the areas of their

DOI: 10.1201/9780203746851-1

4 applicability in resolving problems of rock mechanics are very distinct. Each has certain advantages for studying some specific aspect of a given process. The method ofphotoelasticity helps to obtain a detailed picture of the stress field in the mechanical system under consideration. The centrifugal method, based ~n the substitution of gravitational forces by the centrifugal, ensures mechanical similarity during investigations on models made from natural rock species. However, when such a model is used it becomes quite difficult to represent through a model rock masses of large size and the advance of mine workings. The method of structural modelling embraces all modes of construction of models reproducing the principal elements of the general shape and structure of the mechanical system being investigated and enables one to study the effect of geometry of the system on its mechanical state and nature of interaction between the structural elements constituting the given system. In such a case the mechan­ ical properties and the structural elements of the system under investigation may differ from those obtained by the theory of similarity. When such models are constructed from an existing set of structural ele­ ments, which may be repeatedly used in different models, the tedious task of building and testing them is significantly reduced, thereby making it possible to carry out tests on a mass scale. The use of structural models composed of equivalent materials which pre­ serve similarity of mechanical properties of the rock mass, enable investigation of the influence of the characteristics of the rocks on the mechanical properties of the massif. Research ,work involving modelling with equivalent materials can only be of high quality when carried out by qualified personnel who have the requisite theoretical background and experience in working in this field and when the laboratories in which such work is carried out are equipped with special test facilities and measuring instruments. 1.2 Conditions of Similarity Modelling of any physical phenomenon is based on the theory of similarity. A phenomenon which occurs under geometrically similar conditions is said to be similar if the ratio of analogous quantities remains constant at all compatible points. These ratios, designated as similarity constants (or multiplying factors), are not assigned arbitrarily since they are specifically governed by the laws of nature. In most cases such a relationship may be expressed mathematically in terms of an equation. For phenomena which are similar such equations must be of the same form. While modelling any mechanical process the researcher is free to choose the values (dimensions) of three principal quantities: length, time and mass. Thereafter the dimensions and numerical values of all other characteristics of

5 the process under consideration (velocity, acceleration, internal stresses in the elements of the system etc.) are defined. Accordingly, mechanical similarity is dependent upon the specification of the multiplying factors (scales) for length (geometrical similarity), time (kinematic similarity) and mass (dynamic similar­ ity). Elements representing the original full-scale system (prototype) are indi­ cated by the subscript 'n' and those of the model by the subscript 'm'. The mUltiplying factors for the corresponding elements of the system are designated by the Greek letter a with the subscript of that element which is described by the given multiplier. The criteria for geometric similarity are expressed in terms of ratios of linear dimensions of the full-scale and model elements In/lm = a

· .. (1.1)

l

while the criteria for kinematic similarity are expressed in terms of ratios of time ... (1.2) tn/t m = at· Dynamic similarity, when it is required to have a constant ratio between the masses of any two similar elements, for a given ratio of their linear dimensions a l and consequently also their volumes a~, is determined by the ratio of their densities · .. (1.3) Pn/Pm = ap'

Using these three multiplying factors (l.l), 0.2) and (1.3), it is possible to establish a relationship between any other elements (factors) of a mechanical system in conformity with their dimensions. Since force P is equal to the product of mass Tn and acceleration a, one may express the ratio of forces acting on similar elements of the system as

ap

= Pn/Pm = Tnnan/(mmam)'

· .. (104)

Expressing acceleration in terms of its dimensions and mass of elements in terms of their density and volume we have Pn m

Pnl~ n

Pml~ -t­2 m

· .. (1.5)

_ Pmt~ Pm 14m

= NI

_ idem.

· .. (1.6)

ap=p=T or

Pnt~

7 Pn n ­

The set of expressions given in equation (1.6) are dimensionless; their iden­ tity characterises the similarity of the mechanical systems under consideration and they are known as similarity criteria. Equation (1.6) is known as Newton's criterion and it defines the general conditions for dynamic similarity of mechanical systems.

6 The condition of equality of similarity criteria for the prototype and model may be likewise expressed as a ratio equal to unity:

P

t2

Pn

14

~ n

P

t2

~

P

14

=

m m

a

P

=

a 2a - 1a - 4 I t pi·

· .. (1.7)

Equation (1.7) is known as an indicator of similarity of the process. When several different forces of a physical nature (gravitational force, inter­ nal frictional force etc.) act concomitantly, the similarity conditions of mechan­ ical systems are governed by several criteria or indicators of similarity, each of which corresponds to the given type of force. If the process under investigation can be expressed in terms of an equa­ tion, it is easy to determine from it magnitudes of the corresponding criteria and indicators of similarity. Then the equation has to be transformed into a non-dimensional form, dividing all its terms by one of them. After substituting each of the exponential units by the corresponding multiplying factors a and subsequent reduction, we obtain the desired value of the similarity indicator. In the case of differential equations, the relationship for determining sim­ ilarity indicators may be obtained by dropping all signs of differentiation and dividing all terms by one of them. The resultant non-dimensional sets are the similarity criteria. The corresponding similarity indicators are obtained by tak­ ing a ratio of the criteria for prototypes and models and substituting the ratio of elements of identical dimensions by the corresponding multiplying factors a. To illustrate the foregoing let us consider an elementary example of a rigidly fixed cantilever beam of rectangular cross-section subjected to its own self­ weight lh and a uniformly distributed external load q. If it is assumed for this beam that the width of cross-section is I, height is h, length is I, specific weight is 1, modulus of elasticity is E, the magnitude of the uniformly distributed load is q, then the differential equation for the curved axis of the beam with the origin of the co-ordinate axes located at the free end may be written as d2y

dx 2

(lh + q)x 2 /2 Eh 3 /12

· .. (1.8)

Ignoring the sign of differentiation in the above equation and dividing all its terms by y/x2 the following non-dimensional sets are obtained: III =

6'lhx 2 . x 2 3

Eh .y

and Il2 =

6qx 2 . x 2 3

Eh-·y

.

· .. (1.9)

Let us obtain the ratio of each non-dimensional set for the prototype and model:

X4)

( 61 Eh 2 y

a..,ai

X4)

n

( 61 Eh 2 y

m

aEa 2 a l -l

a '"jalaE = I.

... (1.10)

7 q 4 ( 6 X ) Eh 3 y

n

q 4 ( 6 X ) Eh 3 y

aqa~ CX CX 3CX E I I

m

=

CXqCX

EI =

I.

· .. (l.ll)

The similarity indicator (l.l 0) expresses the similarity condition for the action of gravitational force due to the weight of the beam while the indicator (1.11) expresses the similarity condition for the action of the external load q. While modelling the flexure of the beam considered in the above example, it is necessary to satisfy both the similarity indicators. If the complexity of the phenomenon under investigation is due to multi­ plicity of factors influencing it whose interdependence is not clearly delineated, then it is not possible to formulate a well-substantiated physical relationship even in a differential fonn. In such cases the general Newton law of dynamic similarity given by equations 0.6) and (1.7) is used together with the method of dimensional analysis. It is well known that the processes of deformation, rupture and displace­ ment of rock masses due to the influence of progressive working of mines are caused by the gravitational force and the forces arising out of internal stresses in the rocks. Assuming that these forces together with similarity of geometric characteristics of the system, its initial conditions and similarity of boundary conditions uniquely determine the state of the system, we obtain the similarity criteria corresponding to the simultaneous action of these forces as N N _n_ = _m_ = idem ' 1n ln 1m lm

· .. (l.l2)

or the corresponding similarity indicator may be obtained as -I

-I

cxNcx'"] cx l

=

I

,

... (1.13)

which are identical to those obtained earlier (1.10). The quantity N in equation (1.12) corresponds to various characteristic states of the prototype and model expressed in terms of ratios of forces and areas (for instance elastic limits, modulus of elasticity or modulus of deformation, external loading per unit area etc.). For similar systems the scale of all forces acting on them must be identical. The similarity criteria for external forces (for instance reaction at the supports)

p

p

_n___m_ = idem

1nl~ 1ml~

· .. (1.14)

and the corresponding similarity indicator is CX p CX::- I CX- 3 1

=

I.

... (l.lS)

Considering equations (1.12) to (1.15) together with the necessity for numer­ ical equality of all non-dimensional characteristics of the prototype and model, we obtain the following equations for the set of characteristics of the model when the characteristics of the prototype and the ratios of model and prototype

8 characteristics (disregarding the influence of time) are specified:

(RJm = (RJnQ:;-IQ\1 (Re)m

= (Re)nQ:;-IQ\1

principal similarity conditions for failure (breakage) processes;

(Ri)m = (RJnQ::;-IQ\1 C m = C nQ-,-I QI- I

tan 'Pm - tan '-Pn

P:n

} }

= Pn Q ,.- I Q I-3

Em = EnQ"-I 0: 1-I 1Jm

=

Vn

similarity condition for external forces; elastic deformation processes or deformations lying within proportion­ ality limits,

where Rc, Re and Ri are the corresponding limits of compression, elongation and flexures; C is the bonding force, 'P is the angle of internal friction, 1J is Poisson's ratio, P is the external force and E is the modulus of elasticity or proportionality. The foregoing basic similarity conditions of the process of rupture envisage determination of the simplest experimental strength characteristics. In such a case the similarity condition is expressed in terms of geometric similarity of the schedule of specifications of the material used for the model and the corre­ sponding rock mass represented as envelopes of the limiting Mohr Circles. Such strength specifications must reflect not only the ultimate state of the monolithic rock, but also the weak surfaces existing in it. The multiplying factor connecting the corresponding stresses on the stress diagrams of prototype and model must satisfy the following equality Q

iT

=

Q

T

=

Q"QI·

... (1.16)

Agreement between all the foregoing conditions of similarity for the me­ chanical characteristics of materials of the models is achieved with the help of well-known approximations. If it is possible to identify a factor which has a predominant influence on the process of breakage, then the problem of selecting an equivalent material may be simplified by comparing the strength character­ istics of the materials of the model and prototype in terms of some arbitrary functional characteristic of the strength of the material corresponding to the

type of failure. In numerical terms such functional characteristics take into con­ sideration not only the physical and mechanical properties of the material but also the influence of several significant factors related to the shape of the failed element and conditions governing its stressed state. In most cases when the re­ quirements of similarity are satisfied even through one functional characteristic, it is possible to obtain an adequate correspondence between the prototype and

9 the model for the process under investigation. Such functional characteristics may be utilised to describe limiting spans of caved-in stratified roofing, depth to which a support jack sinks into a yielding floor of mine working etc. The basic similarity conditions described in the foregoing for the mechanical processes that occur in the pro-totype and model due to the action of gravitational force and internal stresses, hold good not only for static but also for dynamic loads. For instance when an elastic body falls from a height h due to its own weight and collides with another stationary body of a much greater mass and rigidity, the maximum stress developed in the falling body at the instant of impact is ...(1.17) G dmax = J6E l h. The relationship between stresses acting at the corresponding points in the prototype and the model is determined by O'"d

=

G dn

En lnhn Emlmhm

~

dm

but 0E = O'"QI' from which,

=

VO'E 0'"0'1

... (1.18) ... (1.19)

O'crd

= 0'" ° I'

... (1.20)

i.e., the relationship between dynamic stresses would be exactly the same as that between stresses caused by loading under static conditions. It should be remem­ bered that the conditions specified by equation (1.19) have to be necessarily satisfied to obtain this relationship. For static loading it is not strictly necessary to satisfy the condition 0.19); although small differences in the scales of O'E and Q" give rise to some variation in the amount of strain, these do not affect the nature of stress distribution in the model.

1.3 Scale of Time The problem of prolonging the action of geomechanical processes in models has been and is of interest to many researchers. The diversity of states of a rock mass and the variety of geomechanical processes occurring in it give rise to significant variations in the scale of time even for a single process during various stages of its development. Geomechanical processes occurring during extraction of minerals and con­ struction of various structures in a rock massif may be arbitrarily divided into several groups distinguished from each other in the degree of complexity of the mechanical systems involved in these processes. The first group consists of processes whose development with passage of time may be specified taking into consideration the properties of the surroundings. The second group constitutes all processes governed by highly complex systems, whose displacement or de­ formation with time is known only for individual elements. In such cases an

10

experimental-cum-analytical approach is utilised to describe the development of the process. The third group consists of complex processes that occur in a multielement system whose deformation with time may not be amenable to analytical description. The scale of time for such processes is established on the basis of comparison of results of experimental investigations conducted on prototypes and models for the specific phenomenon of each process. The fourth group comprises processes whose duration has no effect on the end result. Free fall of a body, the motion of individual elements on an inclined plane and other quick-acting processes are encompassed in the first group. For these the scale of time is determined by the geometric scale [21]:

at=A,

.. . (1.21)

where a l is the linear scale of the model. When elastic waves pass through the material of a model (a process be­ longing to the second group) the rate of propagation is a quantity which is determined by the scale of time. For this it is necessary to compare the speed of propagation of elastic waves in rock masses c n and the equivalent material c of the model cm' i.e., a c = ~ and then the scale of time is obtained as cm -I

at = alae'

... (1.22)

However, there are processes which occur primarily due to the action of gravitational force but are complicated by other factors whose influence has to be accounted for by introducing correction coefficients. The scale of time for modelling processes that occur during the movement of loose material on an inclined plane is an example. In this case the movement of a particle is influenced by the angle of inclination e of the surface over which displacement occurs, the coefficient of friction f between the loose material and the surface and a coefficient K representing some properties of the material. In general, the scale of time for this process may be represented as at = va~a;;-2abine - fm cos e)/(sin e - fn cose),

... (1.23)

where a K is the scale of coefficients K, characterising the process of displace­ ment of the loose material of different grain sizes over the inclined plane and aa is the scale of acceleration. The coefficients K for the materials of prototype and model are determined from the equation Kj

=vj/'vh

gl(sinEl - fcasEl).

where Vi is the rate of displacement of the material determined experimentally, g is the gravitational acceleration and I is the distance through which the material is displaced. Adhering to the similarity conditions in which angle of inclination of the surfaces and the coefficients of friction for the prototype and model are the

11

same, the expression for determining the scale of time simplifies to -I

~

... (1.24)

at = aKaa val'

The characteristic process for resolving the problem of providing supports in mine workings driven in weak rocks is that of deformation of the rock mass with time. Considering the properties of the equivalent materials, the scale of time is obtained from the condition of equality of the relative deformations of the prototype and model for identical loading of the rock, i.e., when the following conditions are satisfied:

(m = (n; (G/R)m = (G/R)n' If deformation of the material of the model is described by the linear theory of successive creep, then the scale of time for the case of a constant load may be obtained from the expression at

=~

[1 -- am tn 8m

(E~ E o.m)]

1 (1-",)

. .. (\.25)

Gm

where Q m and 8m are the parameters (of the nucleus) of creep of the model material; (t,n is the relative deformation of the prototype material during time tn; EO,m is the modulus of initial deformation of the model material and Gm is the stress developed in the model. When equivalent materials whose deformation with time is non-linear are used, the scale of time is determined by taking into consideration this non­ linearity for the same conditions of equality of relative deformations and stresses: 0t

= -

I exp {I - In

tm

nm

[~(A()m(~nm -

Qla:Gn)] }

AomGn o 1 0'1

,

... (1.26)

where nm , mm and Aom are coefficients determined from results of tests on equivalent materials, a'j is the scale of specific weight of the model and G n is the stress developed under natural conditions. While determining the endurance limit of supporting structures of the system (pillars, fringe areas) and walls of the mine workings, the characteristic influence is exerted by the ability of the rock mass to collapse with the passage of time. It has been established through special investigations that the process of failure in the form of growth of microfissures begins long before loss of continuity is evident and its duration depends on the size of the specimen under test and its strength characteristics [7, 14]. A similar principle was found to be applicable during failure of equivalent materials. In such a case the relative co-ordinates of characteristic points on the failure curves plotted on the basis of various physical methods of investigating the state of the material (as also direct measurements) almost coincide while the duration for failure varies from fractions of a second to several years (earth's crust). Naturally, the data points have a considerable

12

scatter (up to 30%) but for such species of rocks the mechanical properties of even a single layer may vary over a wide range. The relationship established between the duration of failure and the endurance limit from these data and the size of the specimen made it possible to obtain the following general expression for the experimental results: log tIT = log N" + M" log Ra; } log tl = log NI + MI log I,

... (1.27)

where t" and tl are the durations of failure when strength and dimensions are changed; Ra is the ultimate strength of the rock in uniaxial compression; I is the linear dimension of the specimen subjected to failure and N", NI' M" and MI are coefficients. Taking antilogs for equation (1.27) the expression for determining the du­ ration of failure processes is obtained as tIT = N" R~"'} tl = NII M ,.

... (1.28)

It should be remembered here that the stresses in the rocks exceed the endurance limit while the value of t is obtained by measuring it from the instant when microfailure commences. Naturally the actual stress level has a strong influence on the duration of failure but the scale of time is determined from the condition of equality of relative stresses by a simple comparison of the results from calculations for the model and the prototype for strength of the rock mass and dimensions of the specimen respectively. These relationships may be expressed in a generalised form as follows: a a

t(T

t,1

=

". } a M'ry a Ma' (RM '" - M") I l n M' fM' M") a a '1\ ,- I tol I , a

=

too

I

... (1.29)

where a to ." and a to ., are the ratios of coefficients in equation (1.27) for the model and prototype respectively; M'" and M'I are the coefficients for the model while M"" and M"I are those for the prototype. If the kinetic theory of dgidity is true and failure of all rock species including equivalent materials as well follow one single law, then the values of the desired coefficients must be identical. Then the scale of time for failure of the rock masses, considering their strength and dimensions of the specimen, is determined from the expression at

,cr.

I

= aMcr~.(M"+M,) ~

~l

... (1.30)

Results of investigations of models have enabled approximate determination of the magnitudes of the coefficients: N" = 1.4; NI = 2512; M" = 2; and MI = 0.7. Utilising these for determining the scale of time during failure (adhering strictly to all other similarity conditions) we have ru

'-'t

=a 2-. a 2I . 7

. .. (1.31)

13 In practice, it is extremely difficult to obtain a strict correspondence between the properties of equivalent materials and the rock masses and hence the actual scale of time would be somewhat different from that theoretically calculated from equation (1.31). For this let us utilise equation (1.29): (Xt.".!

= (Xt ,,(Xt I ()

I)

(x~~lx(M~+M~IRIM~-M~II(M:-M:'1 !

I n n

... (1.32)

This expression takes into account nearly all the characteristic features of fail­ ure with passage of time both for natural and equivalent materials. Results of tests on rock masses have shown that the duration in which the processes leading to failure accelerate, i.e., a sharp increase in the rate at which fissures develop preceding final failure, constitutes 7.3 to 20% of the total time in which load is imposed. Similar figures have been given by other researchers. For in­ stance, according to the data of S.N. Zhurkov this quantity lies between IO.S and 19.1%. While measuring the stresses developed in models, it was established that the nature of their variation with time is similar. The ratio of the time duration in which the failure process accelerates to the overall duration of loading varied between 4 and 7.1 %. This shows that the overall nature of failure in equivalent materials and natural rock masses is the same. In the case of geomechanical processes of the third group in which the nature of interaction between several mechanical systems is quite complex, it is almost impossible to account for the characteristics of the entire set of parameters of the process. For such problems it is acceptable to determine the scale of time in terms of the duration of one or the other process based on results of experiments carried out in the field or under laboratory conditions (for specific points). In such a case one should compare those parameters which reflect the interaction between the entire set of factors influencing the development of the process. For instance, the influence of successive cycles of working a seam on the intensity of convergence of the rock mass in the winning face is dependent upon many factors (rate at which the shearer advances, strength of supports and their properties, strength of rocks in the floor and roof, depth of mine working etc.) whose combined influence is extremely difficult to evaluate analytically. The foregoing is related to the drivage of a pair of workings and construction of shaft stations etc. Naturally, such comparisons must be carried out by adhering strictly to the similarity conditions during modelling. While determining geomechanical parameters on which time has little or no effect (processes falling under the fourth group), it may not be neces­ sary to determine the scale of time, e.g., investigation of successive caving of strata, nature of interaction between blocks of rock masses, direction of forces acting in the system, determining the zone of influence of mine workings etc.

14

1.4 General Systematic Procedures The quality of experiments and the reliability of the data derived from them are evaluated on the basis of the following: I) Reproducing in the model the principal factors governing the course of the processes under investigation and selection of such parameters whose mea­ surement facilitates identification of the law governing the process, its nature and characteristic features as also the desired characteristics with reference to mining; 2) Providing a fairly complete and reliable identification of the desired laws or specific mining parameters of the process under investigation by a rational selection of the type, number and scale of the models being constructed; 3) Ensuring correspondence between boundary conditions of the model and the section of the rock mass; 4) Ensuring correspondence (based on the requirements of the theory of similarity) between physicomechanical characteristics of the materials used for models and the real rock massif being modelled, taking into account its structural features; 5) Ensuring agreement between force and kinematic characteristics of de­ vices simulating the action of props in the models, the real indices of these characteristics and range of their variation under natural conditions; 6) Ensuring correspondence between the time schedule of mining operations with respect to advancement of mine working in the models, the time scale of the process of deformation and displacement of the rock mass surrounding the mine workings; 7) Providing suitable instrumentation for taking observations and measure­ ments during experiments on models for investigating a given process (recording all possible variations together with their accuracy), taking into account the spe­ cific features of the problem under investigation; 8) Ensuring agreement between the methods used for processing the data and analysis of the experimental results of the problem under investigation. Here it is important to note that the desired degree of correspondence of the foregoing conditions may vary depending on the problem but the first condition will always be the fundamental one. When it is violated to a significant extent, the experiment is no longer representative in spite of other conditions being satisfied. Geomechanical problems are extremely diverse but in most cases they may be classified under one of the following types: I) Establishing the basic principles governing the process under investiga­ tion, culminating in verification and identification of various factors affecting the course of the process and in determining the interaction between these factors and the parameters of the said process; 2) Comparison of experimental and analytical parameters of the process, i.e., practical verification of the applicability of analytical methods to the solution of a said problem;

15

3) Detennination of such mining indices or parameters of mine workings whose specific and ultimate magnitudes must be known for the solution of a real practical problem; 4) Target -oriented forecasting of geomechanical processes, taking into ac­ count the simultaneous action of many factors over a wide range of measuring their numerical values. Problems of the first type are the most important and widespread in model­ ling practice. At the same time, in most cases, considerable reduction and sim­ plification in the evaluation requirements can be effected while solving such problems depending on their scope and purpose. This is because such investiga­ tions require considerable planning and simplification of actual field conditions with selective elimination of the influence of specific factors in different sets of tests. Hence while satisfying the procedural requirements, evaluation of the representativeness of the experimental results largely depends on the accuracy of the numerical factors, general nature of the laws governing the process and reli­ ability of detennining the range of natural conditions wherein the experimental results could be utilised. The problems of the second type can also be greatly simplified in tenns of their requirements. This is because the analytical methods being verified usually portray the natural conditions to a significant extent. Hence in the first stage of investigations it suffices to model with significant accuracy only those factors and parameters of the process which are required in the given calculation procedure while in the second stage the factors which bring the conditions in the model closer to those under natural conditions and enable one to verify whether the given analytical method is applicable to a real problem are modelled. Problems of the third type are extremely difficult from the point of view of satisfying all the desired conditions in the model since, in such models, it becomes necessary to ensure the maximum possible correspondence with those under natural conditions for a particular case. Since the quantitative factors which largely influence the natural conditions are not well known and may vary between some limits, the method adopted for developing the models must envis­ age the ability to reproduce these factors in them within similar corresponding limits. Problems of the fourth type are extremely specific since they cover almost all real problems in mining and geology. They are fairly difficult to solve because they require a large number of models for which the similarity conditions are already prescribed according to some specific plan and it is important to adhere to them strictly since the experiment deals with a vast amount of infonnation. Mechanical processes that occur in rock masses around mine workings and the interaction of such rocks with the supports in all cases depend upon the following set of factors: the properties and structure of the surroundings (rock masses) wherein the processes under investigation originate; shape and si7e of cavities fonned by mining in these surroundings; mining methods adopted

16

and the progress of mining operations with time and characteristics of artificial structures or mechanisms used for supporting mine workings. For each specific problem in mining engineering the role of individual factors constituting the system and their magnitudes are distinct. Hence for successfully solving the problem, one must isolate those factors which are of major importance for a given case. The set of parameters selected for the process must be sufficient not only to identify the general laws governing the process, but also to verify and evaluate the role of some of the more important influencing factors. It would be desirable to constitute this set in such a way that it enables one to verify that the condition of balance of forces acting on the model is satisfied in terms of the measured values of parameters representing the forces. The type of scale and the number of models required in a given set of experiments should be determined by taking into consideration the overall me­ chanical arrangement of the process, the number of alternatives and the range of variation of the influencing factors in accordance with the scope of the problem and the method adopted for its investigation. The possibility of conducting the requisite experiments on two- or three-dimensional models (or their judicious combination) is ascertained from the specific nature of the problem with due regard to satisfaction of the desired boundary conditions at the outer faces of the model. Usually two-dimensional models are used to solve problems in which it is required to determine pressure, deformation and displacement that vary signif­ icantly only in two directions. On the other hand, when these vary in a three­ dimensional space, it becomes necessary to use a three-dimensional model and in some cases a combination of three- and two-dimensional models. While any method may be used to study a process, the calculations and practical recommendations from investigations on models must be based on statistical data and results of a systematic set of different types of experiments. Adhering to the requirements of boundary conditions in models is of great importance for obtaining reliable results from the experiments. At the same time, it often becomes extremely difficult and sometimes even impossible to satisfy these conditions. Hence in all cases it is necessary to estimate the extent of error that may be permissible in achieving them. First of all it should be made clear whether the domain of variation of the initial state of the rock mass brought about by the process of mining in the model reaches its boundaries or not. In the latter case, before commencing the experiments it is enough to specify the boundary conditions corresponding to the virgin rock mass in natural conditions and to hold them constant during experimentation. In the very first of the above-mentioned cases, holding the boundary co_n­ ditions constant would introduce distortion in the development of the process under consideration and hence they should be altered while conducting exper­ iments in conformity with the special features of the process. Such a situation

17

mostly occurs during tests on large-scale models with changing contour of mine workings which do not permit representation of a large portion of the rock mass on the experimental stand. If it is not required at the very outset to establish the law governing variation of boundary conditions at the outside surfaces of the model, conforming to the nature of deformation, displacement and failure processes occurring in the model during tests, then the experimental programme must envisage preliminary tests on a small-scale model wherein the boundary conditions at the outside surfaces are kept constant during the course of the tests. The nature of deformation, pressure and displacement at various sections of the model corresponding to the conditions at the exterior contour of the subsequent large-scale models must be studied in sufficient detail on the small-scale model. The results obtained from such experiments must provide the corresponding boundary conditions and the nature of their variation at the surface of the large-scale model during tests conducted on it. For those faces of the model which remain unexposed during tests, the desired boundary conditions are established by means of various instruments assigning either a load or a deformation (in some cases this is achieved by sim­ ply inhibiting deformation). In the case of two-dimensional models, often the boundary conditions at the exposed front and rear surfaces have to be established in such a way that they ensure a state of two-dimensional deformation in the model. In certain cases, because of two-dimensional modelling and the nature of the loads acting on them, the deformations, failures and displacements occurring in them are not directed towards the exposed front and rear surfaces. Violation of boundary conditions at these surfaces may therefore be neglected as they do not give rise to any significant distortion in the development of the processes of deformation and failure of the rock mass surrounding the mine workings. In the rest of the cases, especially while modelling a rock layer at great depth, one should try to provide the conditions corresponding to two-dimensional deforma­ tion of the model during experimentation. Deviation of the boundary conditions between 15 to 20% from those spec­ ified may be considered permissible since the characteristics defining the me­ chanical properties and structure of the rock mass in the model also vary between the same limits. While selecting the materials for the models (based on what is required from the theory of similarity including the scale factor), it should be ensured that those characteristics of physicomechanical properties and structure of the rock mass which determine the course of the process under investigation are reproduced in the model. The rest of the characteristics may, in this case, deviate to some extent from those required by similarity conditions.

It is necessary that the scales of deformation moduli of specific portions of the rock mass in the models be identical. Such an assumption simplifies the selection of equivalent materials since it permits the use of materials with a

18

lower modulus of deformation in the models and simplifies the recording of small deformations in the elements of the model. As long as the absolute magnitudes of deformations in the model remain small and do not lead to distortion of the stress field, the low moduli of deforma­ tion of the materials modelling the rock mass would have no adverse effect on the nature of stress distribution and hence would faithfully reproduce the state of the rock mass. Naturally use of materials of low modulus of deformation must be taken into consideration while selecting the devices used to simulate the supports or fillings in the models. When this condition is satisfied, a low modulus of deformation does not introduce distortions in the force field when the supports or the fillings interact with the adjacent rock masses and the continuity of the force field is not violated. If the programme for preparation of models and their testing envisages a representation of the process with significant simplification of the actual natural conditions, there is no need to select materials with physicomechanical properties and structural formation exactly corresponding to the specific natural rock mass. In such cases a series of models are always used in which the numerical values of the influencing factors are developed from a regular variable series. The values of parameters may be arbitrarily selected but restricted over such a range as would be possible under natural conditions. Such an approach makes it possible to establish the general laws governing the process and to elucidate the role of individual factors in the development of this process. Devices which simulate the action of supports in models must provide similarity of geometric, kinematic and force conditions on the one hand, and reproduction of all operations in the model for installation, removal and shifting of supports closely corresponding to the conditions in the field in the required sequence and at the corresponding places in the mine workings on the other hand. Apart from mechanical similarity in terms of load-bearing capacity of the supports and their flexibility, the devices must be amenable to the measurement of all force and kinematic characteristics of the supports with sufficient accuracy and simple and rapid computations, in order to synchronise the recording of all instruments installed in the experimental set-up. It is desirable to carry out automatic continuous recording of the instrument readings characterising the strength and flexibility of the supports. The completeness of the data recorded and their accuracy should be specified, taking into account the specific features of the problem. The requisite frequency and accuracy of the parameters recorded must sat­ isfy both the overall range of their values as well as the nature of their variation. For a monotonous and well-defined variation of a parameter over its entire range the number of observations may be limited to just a few. When a significant non-linearity is found to exist in the relationship being studied, the number of observations must be increased since those areas in which a sharp variation in

19

the process occurs must be investigated with great care. If the process under in­ vestigation has extrema over the range of experimentation, it becomes necessary to reduce the intervals between observations at the place where an extremum is expected to occur, in order to accurately determine its location. The pro­ cess parameters under study should be determined with such an accuracy that it becomes possible to establish a reliable expression for the desired relationship. The experimental data obtained from models are processed and analysed in conformity with the purpose of investigations. In the most general case the purpose of an investigation is to establish the mechanism of a process and the basic laws governing it. These laws are usually brought to light by comparing the results from tests on a large number of discrete models or a single model wherein data are recorded at different stages of investigation when the process is one of a periodic nature. It is better to compare the interdependence of the experimental data in rel­ ative (non-dimensional) terms. These non-dimensional terms must represent the physical nature of the process with sufficient reliability. There are no ready-made rules for the selection of such non-dimensional factors. However, in order to facilitate their selection one may use dimensional analysis, in particular, Buck­ ingham's theorem (Pi theorem). When all the (n) variables influencing a given process and the (k) funda­ mental physical units appearing in these variables are known (in mechanical processes they are usually the three quantities, namely mass, length and time) then it is possible to establish (n - k) non-dimensional combinations of the variables which are sufficient to describe the given process. Often the possible number of non-dimensional combinations of the variables exceeds (n - k). In such cases one may select from among them those combinations which represent the physical nature of the process to the best possible extent. The selection of such a combination of variables (they may also be designated as non-dimensional functional characteristics) would be deemed to be correct if the relationship es­ tablished between them is found to be applicable over a wide range of the basic parameters of the process deduced in the experiment. If the purpose of the experiment is to obtain real values of the parameters, the data obtained from experiments are processed and evaluated for its reliability through the usual methods of statistical analysis. All final relationships or numerical factors obtained as a result of processing and analysis of the experimental data must be accompanied by an error analysis and the range in which such errors may vary. 1.5 V erification of Reliability of Experimental Results

The first stage of verification must consist of a careful and objective eval­ uation to ensure whether the experiments satisfy the eight basic conditions for conducting them (as given in section 1.4) together with comments on those

20 aspects of the above conditions which were not realised in the given experiment. A summary of all such elements which do not conform to the desired conditions is helpful in making a qualitative assessment of the reliability of experimental results. The following methods and procedures may be utilised to estimate the error and reliability of each of the experimental results: Comparison of the ex­ perimental data with field data; qualitative comparison of the external features revealing the process development and comparison of similar quantitative pa­ rameters (displacement, deformation, pressure etc.) in terms of their absolute or relative values. When a disparity between the actual and the desired conditions or errors are detected, control experiments are conducted on the models to ascertain the possible deviation in the values derived from the parameters of the process or to verify (if necessary) whether the procedure followed for the basic experiments was per se correct. Evaluation of the accuracy and reliability of the quantita­ tive parameters of the process derived from experimental results is carried out by the methods of theory of probability and statistical analysis on the basis of the probabilistic nature of all initial data (according to the well-known analyt­ ical relationship between these parameters and the initial data or according to empirical relationships based on experiments). I The extent to which the laws established from experiments may be generalised is evaluated on the basis of the proximity of experimental points obtained over a wide range of conditions. The selection of a given method or set of methods for analysing and verify­ ing the experimental results is influenced not only by the scope of the problem under consideration, but also by the overall magnitude and type of information obtained from the experiments. If the information is insufficient, it becomes necessary, in most cases, to restrict the verification to qualitative methods and to conduct control experiments. An objective assessment of the reliability of experimental results together with comments on their shortcomings and recommendations regarding the pos­ sible directions and methods through which they can be improved and made more accurate must follow from the results of the foregoing verification. Such an evaluation must be provided by the researcher himself since it is he who knows all the details and conditions under which the experiments were conducted. This objective assessment at the end of the final report is of great importance to other researchers duplicating the same experimental conditions and range of variation in parameters as a means of increasing the effectiveness

and quality of such investigations. In the case of prognostic investigations it becomes difficult to compare the multi variable relationships obtained from experiments with field data mostly 1 A detailed treatment of the corresponding methodical procedures is given in 'Reliability of Modelling in Building Construction' by V.!. Mostachenko. Stroizdat. Moscow. 1974.

21 because of the absence of proper expressions. In such cases reliable results from field investigations are viewed vis-a-vis all the necessary laboratory information and from the relationships thus derived the geomechanical parameters under investigation are determined for comparison.

2

Construction and Testing of Models 2.1 General Assumptions Construction of any model from equivalent materials to study the ground pressure developed in rock masses due to mining must be based on concepts dealing with the laws governing the processes that occur in them. These laws are usually established on the basis of results obtained from in-the-mine and laboratory investigations as well as the accumulated field data. The development of models may also be based on theoretical investigations, which are checked to ensure that the desired conditions are reproduced as closely as possible in the models. To satisfy the desired conditions described in section lA, the investigator must group the initial data for setting up the model. The composition of the initial data depends on the type of problem to be investigated. When the problem is related to general aspects of geomechanics and ground pressure, the initial data may be specified by the researcher himself or he may utilise data gathered from mining in similar regions. The structure of information for the physical and mechanical properties of the surrounding rock mass and the desired mineral is determined by the actual problem under study. Apart from the necessary information on density and compressive and tensile strength of the materials, the initial data may also include information on the deformation and rheological characteristics, bending and crushing strength, scale effect etc. (depending on the principal governing factors of the process to be studied). The process of investigation using models is divided into several indepen­ dent stages which, when executed, help to satisfy the following requirements: a) formulation of the problem and development of the model; b) construction of the models; c) preparation of the models for testing; d) testing of the models and conduction of observations; e) processing the observed data and presentation of the investigatory results. The reliability of the results depends upon the quality and accuracy with which each of these stages of investigation is executed. Experience has shown that it is advisable to use small-scale models to study the general problem of

DOI: 10.1201/9780203746851-2

23 geomechanics whereas in the case of real problems (applicable to specific regions or methods of mining) large-scale models should be used. Since the character­ istics of the natural rock mass reproduced in the model are often based on an extremely restricted amount of data (for some of the observation points some­ times these data simply do not exist), for a more comprehensive investigation of the process, several models are prepared wherein the parameters investigated are varied over certain limits. The number of models is governed by the number and magnitude of factors that vary during investigations as well as the qual­ ity of construction of the models. The number of models may be increased or decreased during the course of an investigation. In designing an experiment one important problem is selection of the type of model. In conformity with the general systematic procedures (see section 1.4), two-dimensional models are often used in those cases in which the problem can be reduced to a two-dimensional one, i.e., when the absence of rock mass at the sides of the model does not affect the nature of the processes occurring in it. Three-dimensional models are far more complex and used whenever the rock mass exerts an appreciable influence. The dimensions of the model are determined on the basis of the problem to be investigated. If the processes to be studied are periodic or of long duration (stability of roof, secondary settlements, displacement of the rock mass), the di­ mensions selected for the model should be such that the phenomena investigated are reproduced to a scale not less than three times. Models are generally made and tested on special stands which allow con­ struction of the model in the required size and reproduction of the geomining and technical situations of the given problem. All experimental stands must be convenient for maintenance and conduction of the necessary measurements. The stands are made sufficiently rigid to resist deformation due to the weight of the model and the stress concentration caused thereby not to give rise to any harmful effect on the stressed state of the rock mass of the model. In view of the diverse types of problems investigated through models, ex­ perimental stands with the maximum possible range of applications are often constructed. All experimental stands can be subdivided into two-dimensional (horizontal or swivelling type) and three-dimensional (likewise horizontal and swivelling type). Sometimes it becomes necessary to construct auxiliary portable stands, mostly used for developing and selecting materials, testing of measuring instruments etc. during the process of construction of the models. The effective height of a horizontal stand is greater than that of a swivel stand when they are placed in the same room (this is necessitated by the arrangement of the apparatus). Because of restrictions on the height of the room and technical difficulties, large depths are often reproduced in the model by simulating that

portion of the rock mass lying adjacent to the portion under investigation. The influence of that part of the rock mass not reaching the roof is compensated by an artificial load.

24

Parameters of the compensating load (force, point of its application, its dis­ tance from the portion under investigation etc.) are selected based on the actual problem but subject to the condition that the compensating load exert no signif­ icant effect on the parameters of the problem. The variety of problems investigated on EM models stipulate a differential approach towards determination of the boundary conditions. Such an approach may be based on an analysis and study of the zone of influence of the mine working and the mechanism of deformation and failure met with in a real prob­ lem. Insofar as the mining industry is concemed, all the problems may be broadly classified into the following groups: 1) Investigation of the process of strengthening and supporting individual mine workings such as drifts, trunk roadways, chambers etc. 2) Investigation of the nature of deformation of the rock mass in the vicinity of the working place of long faces, determination of parameters of the initial openings, development of methods to protect the preparatory mine workings from the influence of stoping etc. 3) Determining the zone of bearing pressure, search for a rational mutual disposition of workings in formations of contiguous seams, study of the influence of pillars and fringe portions abandoned in the converging strata etc. 4) Selection and computation of the load-bearing capacity of supports, deter­ mining the stress distribution in the elements of supporting elements, transverse loading on the pre-face area of a long face, study of the kinematics of mecha­ nised supports etc. 5) Investigation of the laws goveming displacement of rocks and subsidence of surface; 6) Investigation of the stability of banks in open-pit mines; 7) Study of the properties of rock masses under different types of loading conditions; 8) Investigation of the stability of mine workings of large cross-sectional area. All these problems can be solved with the help of EM models, utilising vari­ ous systematic methods of approach to ensure similarity of the stress deformation conditions. In general, the farther the distance from the mine working at which an arti­ ficial load is imposed, the smaller the extent to which it will affect the process of defonnation of the material close to the workings. In practice, because of the

restricted dimensions of the chambers, the test apparatus and other equipment must determine the smallest permissible dimensions of the model. The minimum size of the model for problems of the first group may be established from the dimensions of the zone of influence of the mine working which, according to experimental data, is found to be S = 3.5 D, where D is the maximum dimension of the mine working.

25 The length of a two-dimensional model with the mine working located at the centre is A = 2S

+B =

7D + B,

... (2.1)

where B is the width of the two-dimensional model. In the case of three-dimensional models, the influence of the walls of the experimental stand (equipment) is reduced by redistribution of the stresses over the contour of the model by means of special spacers made of flexible materials (modulus of rigidity between 0.241 and 0.588 MPa) placed between the body of the model and the walls carrying the load. The zone of influence of winning faces is quite significant. Hence in this case the dimensions of the model also increase. When working a longwall, it is extremely important that the superincumbent load not affect the parameters of the first cave-in until the first roof collapse occurs. The span of the first cave-in (problems of the second group) is influenced by a number of factors, of which the major ones are depth of working H, the ultimate compressive strength of the rock mass Rc and the thickness of the rock layer ha measured from the stratum. For problems of the third group, when the zone of influence of the working face reaches the earth's surface, the minimum height of the model should be determined from the similarity of stress-deformation condition of the seam and the surrounding rock mass. This similarity criterion may be taken to be that of stress distribution. During investigation of specific constructions of mechanised support in mine drifts the effect of their interaction with the surrounding rocks, as in other cases, is utilised in models whose dimensions are dictated by the zone of interaction. Considering the results of observations under natural conditions and experiments on models during investigation of problems of the fourth group, the height hm of the model for working faces must be not less than 10 rno in thin and medium layers of rock mass and not less than 20 rno in thick rock layers (rno is the thickness of the layer being extracted) and in the case of drifts it should not be less than its diameter D. The height of a model for investigating the displacement of rocks and the earth's surface (problems of the fifth group) must correspond to the depth of mining while its length must at least be 3.5 times its height. Investigation of the stability of banks of open cast mines and benches requires complete reproduction of the conditions leading to development of rockslides. In such cases the height of the model is determined by the number of benches and the thickness of the underlying rock-bed. The width of the model must not be less than 5 times its height when limiting devices are not employed and not less than the model height h m when such devices are utilised.

Problems of the seventh group are usually investigated on special set-ups wherein one can impose various combinations of stresses. The dimensions of the models, excluding mine workings, used for this purpose are determined by

26

the structural parameters of the rock mass being modelled, consisting of not less than 8 to 10 structural elements (strata, blocks etc.). Investigations for the stability of workings of large cross-sectional areas (rooms in salt deposits, underground chambers of hydroelectric stations etc.) most often requires the construction of three-dimensional models whose linear dimensions must be not less than 5 times the size of the working in the corre­ sponding directions, except in those cases wherein the problem may be reduced to a two-dimensional one. The problems considered in the foregoing refer to typical conditions. How­ ever, there exist such other problems whose solution requires models of di­ mensions significantly different from those described above. These dimensions are selected either on the basis of theoretical considerations or are determined experimentally by conducting a series of special tests. In order to obtain a state of plane-deformation in a two-dimensional model, deformations in the direction normal to the plane of the model have to be prevented. Experience in model construction has shown that it is possible to obtain entirely satisfactory results by utilising special tie bars in the form of wires with bearing surfaces at their ends, which are installed in the model during its construction. The lateral deformations depend on the extent to which the lateral surface of the model is overlapped by these bearing surfaces, which is defined by the overlap coefficient Ko = So/Srn,

... (2.2)

where So is the total overlapped area and Srn is the frontal surface area of the model. The relationship between Ko and load lH/Rc was plotted from the results of special investigations on a model of 20 cm width with bearing surface of tie bars equal to 2.72 cm2 . It may be observed from Fig. 2.1 that at 1 H/Re = 2 the entire lateral surface of the model has to be covered by the bearing surfaces (of the tie bars). Thus the use of this method is restricted by the depth of mining. For models of other widths this relationship has to be established once again. Another method of ensuring similarity of stress-deformation conditions in a two-dimensional model consists of altering the width of the models. The greater the width of the model, the more the amount of material in a spatial stressed state and the less the area to be covered by tie bars in the model. Apart from ensuring two-dimensional deformations of the model, it is also important to ensure stability of its two-dimensional state, especially when it is used at large

depths. The load supported by a narrow and tall model may be obtained as a critical compressive load in the longitudinal direction of the tie wires from the formula

R=

TI 2 E/A 2 ,'

... (2.3)

27

Fig. 2.1. Relationship between overlap coefficient and extent of loading.

where E is the modulus of deformation of the material of the model, MPa; >-- = I-Lh'm

V{12 b is the non-dimensional stability criterion; ~L is a coefficient; h~,

is the height of the model, m and b is the width of the model, m. A graphic solution to equation (2.3), taking into account the depth of mining under natural conditions and scale of model , is shown in Fig. 2.2. The path followed for obtaining a solution to the problem is shown in the figure by a broken line with arrows indicating the direction. The width of the model which satisfies the specified initial stability of the model under a given load is thus determined. For a two-dimensional model of relatively large dimensions, its width is one of the most important factors which ensures authenticity of modelling. If the width of the model is insufficient (such models may be chosen for reducing time and labour), it may warp and the results obtained may be distorted. Two-dimensional models are widely used since they are simpler to construct, easy to visualise and allow reproduction of the engineering process with relative ease. Apart from this the measurement of pressure and deformation becomes considerably simpler in two-dimensional models, permitting the use of extremely simple measuring devices and instruments. At the same time, when two-dimensional models are used, in many cases it becomes necessary to restrain deformation of the material in the direction normal to the plane of the model. The restraining devices used are transparent enclosures (organic glass), metallic tie bars, elastic tapes of high coefficient of friction which are laid on the rock mass of the model etc. Use of plane models without restraining devices is permissible only when the equivalent materials are sturdy , the loads are small and in those cases in which the transverse deformation of the model does not affect the course of the processes under investigation, for instance, while determining the limiting span and length of an individual overhang.

28

Fig. 2.2. Nomogram for determining the minimum width of a model depending on the influencing factors.

Rigid enclosing walls may be pennitted when modelling weak rocks and when the width of the model is not less than 40 cm. In this case the friction force appearing at the walls exerts no significant effect on the behaviour of the rock at the midportion of the model. Tape liners of high friction coefficient (for instance, emery paper) are also suitable for weak equivalent materials (weak natural rocks or small-scale models). They enable the use of models of small width (15 to 20 cm) without introducing any significant disturbance in the stress field of the model. Metal ties whose length equals the width of the model are used in the case of equivalent materials of medium strength in the fonn of double-ended anchors.

29

It is preferable to use three-dimensional models since they more accurately reflect the conditions during mining of mineral deposits and the processes of deformation and failure of the surrounding rocks. However, their complexity, the laborious nature of their construction and use, difficulties involved in carrying out measurements, poor clarity etc. restrict their field of application. Three-dimensional models are often used for investigating the deformation of rocks and their interaction with the supports (in shafts, preparatory and major mine workings, at the junctions of shafts to study their mutual influence etc.). They are also frequently used to study strata with a sharp dip. Due to the complexity of simulating the mining process, three-dimensional models vary widely in construction. This is because each specific problem re­ quires a new approach towards the construction of its model to obtain a solution. For instance, it is necessary to provide for visual observations while investigat­ ing the stability of roofs and pillars in mines. In such cases the type of floor rocks is not simulated while the bottom of the stand is made transparent. In the case of investigations applicable to long winning faces it is necessary to provide for the possibility of leaving contour pillars, retrieval of measuring devices from the model or leads for recording electric signals. Studying problems associated with working series of seams becomes ex­ tremely difficult due to the complications involved in mining and carrying out the necessary measurements. Special experiments on three-dimensional models have shown that the height of the models must not exceed their width since in the case of a large height for the model, the friction at the walls of the experimental stand causes signif­ icant distortion of stress distribution in the models. For this reason the three­ dimensional models are usually constructed on a small scale-I: 100 to 1: 150. Construction of large-scale models requires extremely large experimental stands, which are beset with several engineering problems. During construction of three-dimensional models it is necessary to ensure that the time required for laying and rolling the bed layers is minimal since preparation of large volumes of material requires a considerably greater amount of time. It is very important to ensure that the volumetric capacity of the mixing plant corresponds to that of the bed being laid. If the capacity of the mixer is insufficient to lay the bed in one operation, rolling is carried out in two rounds, with only part of the bed being rolled at one time. Simultaneous use of two mixers for laying the bed as a single layer is permitted only when all parameters of the mixers and the volumes of the mix prepared in them are identical. Considerable technical difficulties are encountered in simulating the winning of a seam and shifting of supports during investigations of the interaction be­ tween the supports and the rock mass in long faces. The seam may be worked from outside the model by cutting it out with a special type of saw, includ­ ing wire saws, or through a hole drilled in the bottom of the stand. A model

30 of mechanised supports may be self-propelled or may be manipulated through a hole in the bottom of the stand. In many cases, when the deformation and strength properties of the strata do not influence the solution of the problem under investigation, the strata made from equivalent materials may be replaced by retractable planks whose successive retraction from the model simulates the process of winning. Models of various types of supports may be attached to the ends of these planks. To simplify the winning of a seam and manipulation of the supports in three­ dimensional models, a metal frame with flanges of specific width is placed on the seam made from equivalent material and is fixed to the bed of the experimental stand. An equivalent material which simulates the roof is placed on the frame. After the model is constructed the casing is removed from the bed, which now lies uncovered. During this period the roof rock masses are placed on the flanges of the frame and the central portion of the bed. When the latter is excavated the rock masses cave in through the opening of the frame while the material resting on the flanges of the frame simulates the pillars at the boundaries of the working face. The foregoing qlethod is applicable in those cases wherein deformation and collapse of pillars or the fringe portions of the rock-bed exert no significant effect on the process under consideration. The interaction between supports and the floor rocks may be investigated in this manner. In such a case the floor is also artificially constituted. 2.2 Computation of Height of Two-Dimensional Model for Longwall Faces The theory of modelling with equivalent materials enables us to replace part of the rock mass with an artificial compensating load, provided this substitution does not distort the stress distribution and deformation in the rock mass in the vicinity of the mine working or in the zone under investigation. This gives rise to the problem of selecting the correct dimensions for a model that would be suitable for a specific problem and specific conditions. It is well known that the processes of deformation of rocks due to extraction of minerals depends on many factors. Similarity of the processes in the model and the field must be achieved by ensuring similarity of all parameters governing these processes. However, since the processes occurring in nature are often defined qualitatively, i.e., the parameters governing the stress-deformation state of the rocks are not clear, it would be better to ensure similarity of the state of stress at nodal points, conserving similarity of all other criteria, for instance, at the points of maximum stress in load-bearing zones at appropriate distances from the edge of the strata. Thus the first similarity criterion for the stress­ deformation state may be taken to be that of identity of the stress concentration coefficients at the load-bearing zones. The second similarity criterion is the identity (at the scale of the model) of

31 additional forces applied at the top edge of the model in the region affected by mining out. If similarity of all mechanical characteristics of the seam and the adjoining rock mass are conserved in the model, it may be assumed that the nature of ground-pressure distribution in the seam and the adjoining rocks would be iden­ tical subject to the above criteria being identical. Obviously, while determining the concentration coefficients it would be necessary to take into account both the strength and deformation properties of the seam and the rock mass. The es­ sential condition in this case is taken to be that of identity of the concentration coefficients under natural conditions (calculated on the basis of the theory of ultimate equilibrium) and in the model (determined on the basis of the theory of beams on elastic foundations with a variable coefficient of bedding): kmax = k~ax' The magnitude of k max may be determined from Prof. K.A. Ardashev ' s equation, which takes into account the strength characteristics of a coal-bed: No 2'-!,x ma> 25. The volumetric compression of the material of low deformation modulus at a pressure of I MPa, disregarding dissolved air as well as air inclusions is given by ~ V 2 = ex V2(j

=

o(jt2Trd~j4 = 1.925 x 10- 12 m 1 .

. .. (4.15)

The total change in volume below the outer diaphragm (4) at a pressure of MPa is obtained from the expression ~Vy = V3

+ ~V2 = 7.205

x 1O- 12 m 3.

. .. (4.16)

97

Fig. 4.7. Characteristics of microdynamometer MDG·2. I and 2-----{;alibration curves obtained in oil and solid equivalent material: 3-variation of deformation of material as a function of true stress in the medium.

The effective deflection of the outer diaphragm is given by fa = V y /(1\r2 x 0.333) = 0.275 microns.

. .. (4.17)

The effective modulus of deformation of the transducer under the same conditions is given by a

ata

E

fo

I

x 1.85 X 10- 3 2.756 X 10-7 = 6712 MPa.

... (4,18)

Thus the modulus of deformation of the transducer obtained from theoret­ ical calculations is found to be sufficiently high (compared to the modulus of deformation of the equivalent materials used extensively for modelling), which satisfies the requirements specified earlier. Experimental determination of the effective modulus of deformation of transducers on the test stand IZV -I showed that it depends on the load. At the commencement of loading it is much smaller than the calculated value but with increase of load the modulus of deformation rapidly increases and at a load of I MPa attains 50 to 70% of the theoretically predicted value. To determine the effective region of utilisation of these transducers, they were placed in specimens of d!fferent types of equivalent materials and were tested with identical loading and unloading conditions. The readings of the instruments were recorded and the deformations of the specimens were measured in the zone where the dynamometers were placed. The variation of deformation as a function of load was plotted from all these observations on a single graph. The calibration curve of the corresponding instrument, obtained from tests in a fluid medium, was plotted on the same graph (Fig. 4.7). Comparing the readings of the transducer placed in an equivalent material with that in a liquid medium (caHbration curve), the ratio of (arnea/atrue) was determined for each level of loading while the modulus of deformation of the material Em was determined based on the measured deformation of the speci­

98

Fig. 4.H. Determination of effective region of utilisation of microdynamometers in terms of: (a) the ratio of moduli of deformation of the instrument and the material and (b) reading of the dynamometer whose principle of operation is based on elongation (strain) of the sensing element. I-at end of loading: 2--during operation as a dynamometer.

mens. Comparing the latter with the modulus of deformation of the instrument all conjugate points were plotted with co-ordinate axes of (Err/Em) and (Gmea!Grrue) (Fig. 4.8, a). The scatter of measured values on the graph in most cases is explained by the inaccuracy of measurements, differences in the con­ ditions of contact between the instrument and the medium as well as random effects. This graph is helpful in determining the effective region of application of the transducer in terms of the ratio of modulus of deformation of the in­ strument and that of the medium. Assuming that a 10% error in measurement of (omea/Glrue) is permissible, it can be seen that the given construction of the instrument is suitable for measurements in the limits of ratio of moduli of defor­ mations Err/Em of I to 7. The actual deviations were from +9% to -6%. For a reliability index of P = 0.9 statistical analysis of the data over the given region resulted in a correction coefficient 11 = 1.025 ± 0.014. Use of this transducer for values of (Err/Em) < 0.5 is ineffective since the errors may be substantial. Figure 4.8, a shows the theoretical curve computed from the constructional data of the dynamometer. A comparison of this curve with the actual data shows that the basic theoretical and operational requirements have been taken into account fairly well in the construction of the instrument. Figure 4.8, b shows a similar graph for a dynamometer designed on the principle of elongation of the sensing element. As can be seen from a comparison of the graphs (see Fig. 4.8, a and b) the sensitivity of this instrument is far worse than that of microdynamometer MDG-2. Tests conducted on microdynamometer MGD-2 under various operating conditions showed that for parameters characterising dynamic processes occur­ ring in both nature and the model, the measurement error does not exceed E rT ,

99

± 10% with a probability P = 0.9 (pennissible for characteristics of the stress­ defonnation state of the model). Tests were not conducted to verify the appli­ cability of microdynamometers for dynamic parameters during rockbursts.

4.5 Measurement of Pressures on Mine Supports The experience accumulated in the field of constructing models of mine supports enabled us to identify two directions in the development of modelling methods. The first suggests a solution to the principal problems of rock me­ chanics on small-scale models in which a support functions as just an element resisting the displacement of rocks in mine workings. In such a case it is im­ portant to know the resistance of the support and its nature. Construction of models of supports for solving such problems may differ fundamentally from their actual construction in natural conditions. The second direction presupposes a study of the interaction between the supports and the rocks at the walls for the purpose of verifying and developing design parameters for the supports or for ascertaining the influence of the latter on the state of the lateral walls and the nature of their displacement and collapse. Hence models of supports, apart from kinematic parameters, must also take into account all constructional features of the prototype in natural conditions. Such investigations are often carried out on large-scale models, which considerably simplifies the construction of models of actual supports. In confonnity with these directions, the requirements specified for construc­ tion of models of supports are slightly different. Developing an efficient model of supports to solve problems of the first and partially of the second group is not difficult. Such can be designed in a simplified fonn. Construction of models of supports to solve most of the problems under the second group is associated with many difficulties since it is often necessary to make the model contemporary, mechanised and self-propelled through hydraulics. The task is further compli­ cated by the fact that the model has to be equipped with measuring instruments, preferably of the automatic strip-chart recorder type. Since, in practice, it is dif­ ficult to satisfy all the requirements for modelling supports in a specific problem, certain simplifications are pennitted in the construction of such models. Models of supports are constructed based on general principles of force­ measuring equipment, taking into account the requirements of the theory of similitude and specific features of such equipment when used in models. The basic elements utilised in such modelling are: rigid (nominally strong), yield­ ing, with an increasing resistance (to compression), and pliable elements with constant and variable compressive resistance characteristics. Blocks, struts, pivots etc. (during construction of rigid supports) are used as rigid elements. Yielding elements with an increasing resistance characteristic are simulated by helical springs, cantilever or simply supported beams, friction elements, brackets or a combination of these elements. Constant or almost con­

100 stant compressive resistance may be provided by lever arrangements, hydraulic struts and piezometers, friction posts etc. The reactions of models of supports and their flexibility are measured em­ ploying well-known principles. Wherever possible systems for measurement of force and displacement should be incorporated. For modelling supports on a small-scale, use is made of lever arrangements with constant or increasing re­ sistance as well as hydraulic piezometers since, because of their small size, it is extremely difficult to use any other system. For large-scale models of supports measurements are conducted with the help of resistance strain gauges cemented to the elements of the model subject to deformations. In the case of hydraulic models of supports, measurements may be carried out with the help of high sensitivity hydraulic or strain-gauge manometers. Before conducting the investigations it is desirable to provide measuring de­ vices in all sections of the support. Otherwise the dynamometer sections should be located in the central portion of the model and exactly similar models of the support but without measuring devices should be installed along its sides. Just as in the case of plane models, the mine workings in three-dimensional models must be anchored over the entire length. Of late, investigations on models often include a study of the specific nature of interaction between various types of supvorts and the adjacent rock mass. For this models of mechanised supports are constructed in which the specific features of each type of construction of supports are reproduced while retaining the functions of all primary elements. The resultant model of supports may not be a universal one, however. Since there are many types of mechanised supports, the cost of constructing models for them and the preparatory time for experimental investigations are sharply increased. One way to reduce these expenditures might be to construct universal elements of models of mechanised supports that could be effectively utilised in assemblying various types of supports with different parameters (modular models of supports). During development of hydraulic models of mechanised supports, it is im­ portant to provide for successive translatory motion and conformity with the nature of interaction between the support and the adjacent rocks. It is possible to construct such models along the same principles used for construction of full-scale supports with minimum pressure drop in the system for overcoming friction and hydraulic resistance (use of graduated glass cylinders, special types of fluids, the principle of interconnected vessels etc.). Constant resistance of a support may be ensured through a hydraulic ac­ cumulator filled with oil delivered by a pump, while loading of the support is carried out with weights or individual loading devices for sections or struts. In such cases the pressure drop in the line from the hydraulic accumulator to the plunger of the support is experimentally calibrated. Displacement of the roof is monitored by transducers. Sometimes specially selected helical springs are used in place of the hydraulic struts in simplified models.

101

_ Models of supports for major and preparatory mine workings do not differ from each other in principle. They are arbitrarily subdivided into rigid and yielding supports. If the study pertains to the ultimate load-bearing capacity of a support or the influence of external factors on the nature of its failure, the model of support is designed in such a way that during the process of experimentation it undergoes failure. For this it is either made from artificial (equivalent) materials (rigid and flexible concrete supports) or from metal, while the moment of resistance is determined from conditions of similarity. Models of anchor supports used in investigations must also be provided with measuring instruments for determining the load and the limiting pull-out force. The functioning of anchor supports is often 'itudied either on large-scale models or on separate blocks of equivalent materials. All the mining conditions in studies of models are reproduced to explore the feasibility of using anchors, the effect of their technical specifications on mining conditions etc. The extent to which rocks are joined together, the force required to pull them apart etc., are determined on blocks. The basic parameters of rock bolts are determined by taking into account the scale of modelling and the method of their installation while their anchoring is determined experimentally. Models of supports for mine workings used in three-dimensional models differ from those for major mine operations only in the type of measuring instruments used, which are usually governed by the actual problem. In all cases where several models of supports of identical construction are placed in a mining operation, it becomes necessary to provide them with identical strength and maximum resistance irrespective of whether they carry measuring devices or otherwise. The contact pressure of the supports is also measured in models, i.e., stress distribution at the contact surface between two media with different mechanical properties (between two layers in a model, between the model and the walls of the experimental stand, at the interface of the medium and supports in a mine working). This implies relatively constant conditions of contact between the measuring element and the medium and compliance with the requirements for their rigidity ratio. Since most constructions of supports used in mines are designed to 'avoid' heavy loads, their modulus of deformation is selected in such a way that they are not subjected to all the stresses of the massif. Consequently, the transducers placed at the top of the supports do not sense the stresses in the rock massif of the model but merely the pressure exerted by it on the support. Furthermore, distortion in the interaction between the material and the sup­ port gives rise to random contact conditions. Often the 'mine working' is driven into the massif of the model after its construction and to ensure constant con­ ditions of contact between the supports placed in the mine working and the

102 surrounding rocks becomes almost impossible. Hence the pressure sensed by the individual transducers placed on the surface of the supports must perforce be random in nature. In such cases, for a more detailed measurement the size of each measuring device must be as small as possible while the dimensions of the contacting elements in the model should be similar to their dimensions in natural conditions (transducer, projections etc.). Thus the development of a reliable device for measuring the contact pres­ sure on a support is beset with many technical difficulties. An approach which holds promise is one in which the entire surface of the support is covered by dynamometric devices. In such a case the modulus of deformation of the mea­ suring devices, taking into consideration the contact between the support and the measuring elements, must exceed the modulus of deformation of the support. Based on measurements of contact pressure, graphs are plotted for pressure distribution in terms of averaged parameters of individual instruments which indicate the loading condition of supports under a given type of deformation of the rock mass.

4.6 Measurement of Deformation and Displacement Deformation and displacement in plane models is usually measured on the open surface over the edge portion of the model assuming that a plane model deforms uniformly over its entire width. Hence it is important to monitor the state of the model as a whole. When the edge portions break down, the readings of the measuring instrument could be erroneous. The points at which measurements are to be made in the model are specified before commencement of the experimental work. Their number and location are fixed in such a way as to obtain a fairly complete representation of both the nature and extent of deformation and displacement of the rock mass of the model and the contour of the mine working cross-section. The method and the Instruments used for measurements should be selected while taking into account the spacing of the points and their number as well as the characteristics and dimensions of the measuring devices. Instruments with a maximum possible range of measurements and arranged in a closely spaced network should be installed at those places of the model where large displacements are expected to occur. The frequency of measurements depends on the rate at which the process occurs and is determined in conformity with the actual problem under investi­ gation. When the expected rate of displacement exceeds 0.5 mm/minute, it is desirable to use an automatic strip-chart recorder in order to obtain a continuous or digital record of the instrument readings. All primary measuring devices (transducers) for measurement of deforma­ tion and displacement may be subdivided into three groups: mechanical, opti­ comechanical and electromechanical.

103 The first group consists of the widely used strain-gauge pick-ups which are based on the transformation (amplification) of the measured quantity: flexible and modular strain gauges, co-ordinating devices, reference gauges (including depth gauges), mechanical recorders etc. Measurements are usually conducted either by means of special scales or microscopes. The second group consists of optical and mirror type strain gauges. Devices in which the measured quantity is converted into electric signals, constitute the third group. These consist of wire strain gauges, thermal strain gauges, deformation gauges with wire resistance strain-gauge transducers, inductive deformation pick-ups etc. Methods of photofixation and photogrammetry, based on photographing the entire model or part of it fitted with special markers (points), are also used for these measurements. The measuring devices are installed on the surface of the model using lugs in the shape of needles (for light-weight devices) or pins (for large-scale mod­ els). Careful and proper installation of the device is of considerable importance. Misalignments, poor fixture in the model and other errors may lead to mis­ representation of results and detachment of the device from its setting. All the measuring devices and transducers must be brought to full working condition and be calibrated before installation. Relative displacement measurements of the rock mass of the model are carried out by means of devices which have a measuring base of 20 to 50 mm. The absolute displacement of individual points is usually measured relative to the rigid element of the stand or the fixed scales provided on plane models. Determination of deformation and displacement in three-dimensional mod­ els becomes extremely difficult since it depends on the placement of measuring devices within the model. In this case the working of the devices is complicated by the distortion of the stress distribution near the device and interaction of the latter with the surrounding medium. All these factors must be taken into con­ sideration both during the design-development stage of the model and during selection of the type of measuring devices. Because of the difficulties involved in the construction of three-dimensional models, it is desirable to utilise each with maximum effectiveness to derive maximum information. Hence one should concomitantly install deformometers (strain gauges) for measuring deformation of the material above supports and measuring devices for determining large displacements above goafs. At the same time the model must not be overequipped with measuring devices. Hence the number of measurement points is selected on the basis of the problem under investigation, scale of the model and size of the measuring devices. Deformations inside the model are measured with the help of deformometers whose principle of operation is based on the deformation of elastic plates, beams, pins and other elements with wire transducers. Detailed investigations on their performance in models have shown that they usually do not reflect the true

104 defonnation of the material. Hence it is advisable to calibrate them in separate specimens of the material of the model. Experience has revealed that the accuracy of these devices is best when Err/Em lies between 0.01 to 0.05. In fact the magnitude of this ratio depends on the construction of the device, method of its installation, type of material etc. (see Fig. 4.8, b). The current trend in measurement of displacement inside a model is to use devices which cause no elastic interaction with the surrounding medium. Often the relative displacement of the contour of the cross-section of a mine working is measured in models. The absolute displacements of individual points of the contour are measured for solving specific problems, for instance during a study of the effect of location of a mine working in relatively weak rocks and shear zones, mining in a suite of strata, selecting the method of supporting a mine working driven in rocks which are non-homogeneous both in structure and strength etc. In plane models the measuring instruments should be installed in the central part of the model so as to avoid edge effects and accidental breakdown of the devices. In three-dimensional models the points at which measuring devices are installed are selected in such a way that they provide complete and reliable infonnation. For instance, during a study of the influence of a winning face on a development working situated in a stratum and protected by pillars, the measuring equipment (instrument) should be so installed that the effect of mining on the development working is fully realised (both before and after approaching the measurement point). The measuring instruments used are: mechanical dial-type indicating gauges and resistance and inductive-type transducers. In many cases the instruments used are capable of simultaneously measuring displacement and reactions at the supports of the mine working. While measuring displacement it is important to ensure that there is a reliable contact between the measuring device and the equivalent material. Supporting platfonns, contour reference points etc. are used for this purpose. It is extremely important to provide minimum rigidity for the measuring equipment so that it has no significant effect on displacement. From this point of view use of various types of inductive pick-ups for plane and three-dimensional models and method of photofixation for plane models is found to be the most acceptable. In the latter case markers (reference points) are cemented on the model close to the contour of the mine working. The instruments used for measurement of defonnation and displacement of rocks close to the mine working are the same as those used for any dis­ placement of the massif. In large-scale three-dimensional models depth-wise reference points, installed at various depths from the mine working, may be used. The absolute displacement of individual points on the contour of the mine cross-section is measured by special attachments whereby the immobility of the

105

reference element is ensured by fixing it outside the model. For measurement of dynamic displacements the instruments used should be inertia-free with rigidity close to zero.

5

Planning Experimental Investigations 5.1 General Concepts One of the advantages of the method of EM modelling is that investigations can be conducted under any desired state of the material right up to the ultimate limits, i.e., approaching real conditions. Increase in the depth of mine workings is accompanied by changes in the influence of various factors, increase in the cost of drivage and supporting of mine workings as also the equipment used for mining. This requires that the results of complex investigations take into account the overall effect of all phenomena occurring in the rock mass, in the mine working and under laboratory conditions be highly reliable. However, experience has shown that when experiments are planned for actual mining conditions it is not possible to take into account all the influencing factors and their range of variations since the numerical values of such factors are not always known. Investigations on EM models enable us to approach the problem from the other side-to vary the numerical values of the factors under investigation over a specific range and thus establish functional relationships. However, the multi­ plicity of influencing factors and the wide range of variation in their numerical values (for instance, strength of the rocks, structural properties of the rock mass, depth at which mining is carried out etc.) make it necessary to formulate a large number of experiments (sometimes as many as a thousand) which is practically impossible to achieve in a short period of time. Under these circumstances it is advisable to plan the experiments on a scientific basis not only regarding their required number, but also the actual conditions under which they are to be conducted (determining the combination of factors to be investigated and their numerical values for each of the experiments). This eliminates any arbitrary assumptions in assigning experimental conditions, making it possible to obtain general relationships between the desired parameters and the aggregate effect of the factors under study. Analysis has shown that it is possible to adopt the method of systematic planning of experiments based on the use of combination squares (Latin squares). This method [33] is a particular solution to statistical planning of experiments. It is very clear and very helpful in significantly reducing the number of experiments. The method of Latin squares

DOI: 10.1201/9780203746851-5

107

was first employed by M.M. Protod'yakonov Uunior) during investigation of strength properties of rocks under laboratory conditions. The method of Latin squares was initially verified by the results of a large number of investigations conducted earlier in the modelling laboratory of the VNIMI and was used thereafter during formulation of experimental investi­ gations to solve a number of problems. Based on experience the method of systematic planning of experiments can be recommended for extensive use in all laboratories where investigations are conducted on EM models. At the same time certain specific features of its application directly related to investigations on equivalent materials were identified during the process of its adaptation. The method of systematic planning of experiments assumes not only the influence of factors in a real set, but also correlating functions between factors when their numerical values vary over a wide range, i.e., empirical correlations. The task of systematic planning of experiments during investigations on models consists of a study of the simultaneous effect of several geomining and technical factors on geomechanical parameters for a minimal number of experiments. The principle of systematic planning of experiments using the Latin square method consists of the following. The region of planning is bounded by a square drawn on the XY plane-the so-called A-type basic Latin square (Fig. 5.1). At the top and left sides of the A-type basic Latin square the factors under investigation are indicated as a, b, c, d, e and f together with their numerical values. Depending on the number of factors to be investigated (preferably even number and not less than two) the basic square (field) is divided into much smaller squares. The extent of subdivision is determined from the formula

A = K/2

+ I,

... (5.1)

where K is the number of influencing factors. The number of elementary squares m in a planning field is governed by the number of actual numerical values of the factors n: m

nk.

. .. (5.2)

Before setting up the Latin squares, the numerical values of the factors to be investigated are codified in accordance with the rules of systematic planning of experiments. Codification consists of the following operations: several nu­ merical values are assigned to each factor with the smallest value being unity. All subsequent values are expressed by numbers greater than unity. Here it is desirable that the numerical values of these factors be varied according to some single law (arithmetic or geometric progression or a more complex law). While the overall value of each of them is greater than the number of factors by one but not less than three, it should be remembered that with an increase in the number of numerical values of factors the accuracy of results improves. The experience gained in systematic planning of experiments has shown that depending on the actual problem, the number of numerical values (levels) of

108

Fig. 5.1. Planning field of Latin squares.

factors, taking into account the volume of experimental work, may differ. When preliminary experiments are carried out or when the experimental material is inadequate, it may be assumed that n = 3. In this case it is possible to estimate only the extent of influence of a factor or its trend. To establish a relationship it is necessary to take n between 4 and 7. If a more accurate relationship is desired, then the experiments should be planned with n between 9 and 11. However, in this case the total number of experiments sharply increases. Increasing n up to 13 adds practically little to the accuracy of the derived relationships but significantly increases the labour involved in computations. For the present investigations it suffices to assume n between the limits of 4 and 7 while this number for a given Latin square must be identical and an odd number for all the factors under investigation. The location of any of the smallest squares is determined by specific group­ ing of the numerical values of the factors investigated. Planning of experiments with such a grouping of elementary squares with a specific set of conditions in the combination field of type-A squares consists of conducting the experiment (grouping of numerical values of factors) so that the investigation under all conditions can be achieved with the least number of experiments. This is done by satisfying the following condition: only one ele­ mentary square should be selected in each row and each column. Its location in the combination field is determined by the specific initial data of the experiment. With such planning the total number of experiments reduces by n 2 times. The position of each proposed experiment in the Latin square (elementary squares) is selected in such a way that during model testing the numerical values of the factors under investigation change simultaneously, i.e., each proposed experiment must consist of a new set of numerical values of these factors. It is

109 desirable that the grouping of elementary squares in the field of the basic A-type squares is not monotonous. After selecting the elementary squares with the conditions for conducting all the experiments and their verification, experiments are commenced adhering strictly to the corresponding conditions. Each experiment, planned in confor­ mity with the stipulated requirements, must be carefully thought out and must strictly comply with the similarity conditions of all geomining and technical factors. As the experiments are completed, their results are entered in that ele­ mentary square of the combination field under whose conditions the experiments were conducted. After filling all the selected elementary squares with results of investigations one proceeds towards processing these experimental data. Determining the influence of geomining and technical factors on the process under investigation makes it possible to discard those factors which exert a negligible influence on the process in further investigations and to concentrate on the principal factors among them or to proceed to the region of planned investigations of a new set of factors not yet studied. Although the method of systematic planning of experiments has several ad­ vantages compared to the element-by-element (classical) approach, it also has disadvantages: (l) it is possible to obtain the desired relationships only after completion of the entire cycle of experiments and (2) the results of tests con­ ducted on models, as in the case of other experimental investigations, are valid only over the range of numerical values of factors assumed in the experiments.

5.2 Selection of Initial Factors and Geomechanical Parameters It is well known that the processes investigated on models depend on a large number of geomining and technical factors. Identifying the set of neces­ sary basic factors mostly depends on the qualifications and intuitive skills of the researcher. Before commencing investigations the first task is always to select the set of factors which are of interest and the limits of variation of their nu­ merical values. To resolve this problem one must select the necessary laboratory equipment (stands, equivalent materials, instruments) and simultaneously pro­ vide for satisfying the conditions of similarity between the model and natural conditions. When the range of variation of the geomining and technical factors is determined, one should take into account the technical feasibility of conducting the experiments (depth of mine workings may be simulated only over specific limits: similarly the strength of the rocks is limited by the scale of modelling etc.). The geomining and technomining factors (a, b, c, d, e, f) whose effect on a given process is investigated on models are the independent variables from the point of view of mathematical analysis. These are: depth of mining, strength of rocks, thickness of the coal seam, characteristics of the supports, stratification, fissures and fractures, rate of advance of the face etc. Apart from this, during

110

investigations it becomes necessary to specify definite numerical values (in an ascending order) for each of the factors. The number of factors and their values are generally determined by the type of process under investigation. In the first place one should select those factors which have the most significant effect. The range of variation of each factor must include those which are the most representative for the problem investigated. For instance, when the study concerns stress distribution in the support region of longwalls over a coal seams of different thickness and varying strength, it is necessary to take into consideration the depth at which the strata are located, structure and strength of the rocks, nature of fissures in the coal seams etc. If the same study relates to drift mining, it is necessary to consider additional factors such as size of the mine working and characteristics of the supports. Special attention needs to be given to the selection of the geomechanical characteristic since it forms the primary aim of all investigations on models. A vast amount of information and a whole range of output data are obtained after completion of the planned cycle of experiments. These data may be classified according to their significance in the process under study as follows: -Principal geomechanical characteristics-these are of great interest to the researcher (for instance, displacement of roofs in faces or drifts, parameters in the bearing pressure zone etc.); -Subsidiary geomechanical characteristics which are measured simultane­ ous with the basic characteristics and which are interrelated with them but are not of much interest to the researcher during experimentation. They may be cal­ culated during processing of data and analysis of results (for instance, resistance of supports, which is measured along with displacement of the roof or contour of the mine working, velocity, acceleration etc.); -Unexpected geomechanical characteristics which may be obtained as a result of subsequent analysis of results of investigations (for instance, extent of fissures and fractures in the rocks, stratification etc.). This classification of the geomechanical characteristics into three types is entirely arbitrary since a given characteristic may be principal in one investiga­ tion but subsidiary or entirely unexpected in another. In general the geomechanical characteristic or parameter is the unknown which is a function of many independent variable factors. The aim of investiga­ tions on models is to study the dependence of one geomechanical characteristic on several factors or the relationship between several geomechanical parameters and the same set of factors:

y, = f,(a,b,c,d, ... ); }

Y2 = f 2 (a,b,c,d, ... );

Y3 = f3 (a,b,c,d, ... );

;~ '~'i~(~:b:~:d:::

:).

... (5.3)

III

Fig. 5.2. Combination field of Latin square.

5.3 Procedure for Formulating Plan of Experiments Before setting out to plan experiments the researchers must precisely formu­ late the problem and select the appropriate experimental stands and measuring instruments for its solution. The next step is to select that Latin square which is the most suitable for the specific problem. The elementary squares (ES) are arranged in the combination field by the trial and error method (Fig. 5.2). The central square, which is divided into ele­ mentary squares, is taken as the base. The ES are numbered from left to right and line by line. In the present case n = 5 and the number of ES is 11/ 25. After this, squares similar to the central are placed on each of its sides and also divided into elementary squares. Numbers corresponding to the ES in the cental square are assigned to those in the portions of these squares adjoining the central square. This is done in such a way that one obtains an inclined square with numbered ES. Five parallel broken lines are now drawn diagonally and successively num­ bered 1, 2, 3,4 and 5 from top to bottom. These diagonal parallel lines intersect an identical number of ES and correspond to the number of numerical values of the factors.

112

Fig. 5.3. Latin square with plan of experiments.

Thereafter, the difference between the ES of the combination field and the basic square is highlighted. A complete square is set out for this purpose (Fig. 5.3) in which the numbered ES are accommodated. The numbers of ES lying along the diagonal in Fig. 5.2 are placed in the basic square along a horizontal (broken) line. The location of an ES in Fig. 5.3 must correspond to its position in the squares of Fig. 5.2. Thus the ES numbered consecutively in Fig. 5.3 are spread over the entire field of the basic square with only one ES located in each central square. With such a construction the numerical values of the factors along the central line and in the central column change consecutively. If the plan formulated in this way does not satisfy the researcher, it may be altered. The initial plan is altered in various ways as follows: by turning the diagonal lines through 90° with subsequent distribution of the ES; transposition of ES in the combination field of the basic square; turning the initial plan through 90, 180 or 270° relative to the codified values of the factors. It is possible to transpose the ES in a combination field in several ways. The combination field represented in Fig. 5.2 may be presented in tabular form: 14

20

10 II 17

7 13

23

19

22

18

3 9

24 5

10 II 17

7 13

IS

6

23

19

22

18

14

3 9 IS

24

20

21 5 6 2 4 25 16 12 4 25 16 12 8 If the numbers in the first column are carried to the end of each row then we obtain a new combination plan. One may simultaneously transpose two, three or even more number of columns. Each time we obtain a new arrangement for the location of the ES in the combination field, i.e., a new plan of experiments is formulated. For this purpose the rows in the basic Latin square may be trans­ posed. All the variants described above for planning experiments are equivalent 21 2 8

113

but one should select that which is the most suitable from the point of view of the conditions under which actual experimentation is to be conducted. This problem has to be resolved each time by the researcher himself. When a new series of experiments is planned to study all or part of the factors affecting the course of the geomechanical process, the plan should be constituted in such a way that it occupies the entire field while the initial data specified in this case should be such that it is possible to construct a model of the rock mass. When an additional series of experiments is planned, the plan is constituted in such a way that it is possible to utilise all the results derived from the previous experiments. It should be remembered here that some portion of these may not be used in developing the plan. However, they may be useful in verifying the laws established on the basis of experiments conducted according to the given plan. Devising a plan is also extremely useful for processing any statistical data regarding multifactor systems over a wide range of their variations. Experience has shown that it is advisable to consider several plans, varying them by one of the methods described earlier. The type of Latin square selected after preliminary analysis, with the ele­ mentary squares spread over different parts of the primary Latin square, forms the basic Latin square and no change in the planned initial data for the proposed experiments during the process of investigations is permissible.

5.4 Analysis of Results of Investigations Planning of experiments on models significantly reduces their number. How­ ever, because of the simultaneous influence of the many factors considered here the initial data become very large. These data have to be analysed carefully according to a specific programme and in the appropriate order. At the end of each test the results of investigation (numerical values of one or more unknowns of the geomechanical process) are inscribed near (or within) each corresponding elementary square. After all the ES are filled (completion of the entire series of experiments), tables are set up with numerical values of the geomechanical characteristics and two of the unknowns located in each. Codified numerical values of a single factor are placed horizontally (in the upper portion of the table), the second in the extreme left column (vertically). While preparing these tables it is necessary to adhere to the following rules: -The number of cells in the table must be equal to the number of squares in which the numerical values of the unknown factors vary;

-The values of the unknown geomechanical parameters are entered into those cells of the table whose co-ordinates are identical with the corresponding co-ordinates of the elementary square in the Latin square field.

114

After the table is filled, the corresponding values of the geomechanical characteristics of the process are added row-wise and column-wise. Their mean value is determined and the relationship between the unknown geomechanical characteristic and each of the factors being investigated is plotted with the help of these values. A generalised empirical relationship is developed on the basis of the graphic functional relationships of each of these influencing factors. Each of these re­ lationships is valid for mean numerical values of all other factors. The results are further processed by taking into account the nature of these relationships. If the relationship is linear or nearly so, the general expression is deduced on the basis of the following equation: F = Yay

+ f1 (Aj

~ Aay)

+ f2(Bj

~ Bay)

+ f3(C j ~ Cay) + ... + 6,

... (5.4)

where F is the geomechanical parameter; Yay is the mean value of the geome­ chanical parameter for all the observations; f1 (A), f2 (B), ... are the functional relationships of each of the factors and 6 is the mean square error of measure­ ments. To determine the value of the function with values of the influencing factor obtained from natural conditions, the coefficients in the linear relationships must be calculated as follows: ... (5.5) Kn = KcNc/Nn, where Kn and Kc are the coefficients for the decoded and coded values of respec­ tive factors; Ne and Nn are the coded and decoded values of the corresponding factors. When the functional relationships are far more complex, the empirical ex­ pressions are deduced as follows: i) The results are grouped according to the importance of each factor, often simultaneously for two factors; ii) The effect of all other factors is neutralised by averaging their action; iii) The influence of each of the factors in the overall result is brought out utilising graphic representation; iv) In the case of a factor which has a strong influence, a particular empir­ ical expression is obtained which takes into consideration its influence on the geomechanical parameters with all other conditions kept constant at their mean values; v) The initial data is re-estimated based on an averaged or a single value of the strongly influencing factor which neutralises its effect;

vi) The calculations are repeated for the next most influencing factor etc. (method of successive approximations); vii) If recalculation does not provide a conclusive result, one of the most reliable intermediate results should be used for a fresh calculation; viii) Particular empirical expressions are combined together to form an over­ all expression;

115

Fig. 5.4. Illustrative example of relationship between geomechanicai parameter (GMP) and unknown factor.

a-increasing trend of geomechanical parameter, band c--decreasing trend of geomechanicai

parameter.

ix) The calculated values are compared with the initial data and their stan­ dard deviation is obtained, When the data are analysed in this manner, extremely complicated rela­ tionships may result for the geomechanical parameters as functions of unknown geomining and technical factors. During tests on models the numerical values of factors are varied over a given range (generally the lower limit is greater than zero). This must be borne in mind while selecting the actual mathematical relationship. If the researcher is interested in expanding the range of applicability of the deduced expression by piece-wise variation of the factors from 0 to 1 (it is presumed here that the factors are in the coded form), then further analysis of the experimental results is carried out and the value of the geomechanical parameters is obtained corresponding to zero value of the factor, assuming, of course, that this analysis is based on logical arguments and the physical nature of the geomechanical characteristics, Thus the relationship derived from an analysis of experiments for an increas­ ing trend in the geomechanical parameters with respect to the unknown factor may be represented over the portion 1-3 of the variable factor a l (Fig. 5.4, a) by a straight line or, after further analysis (assuming zero value for the factor), by a higher order curve, for example, y =

KvaJ·

... (5.6)

which is shown by a broken line in Fig. 5.4, a. Over the region 1-3 the straight line intercept and the higher order curve almost coincide.

116

When the general trend of the geomechanical parameter is of a decreasing nature (Fig. 5.4, b), the variation of the relationship over the region 1-3 of a2 may be approximated by a straight line or (considering a 2 from 0-3) by a curve of the type y =

K/Ja2.

If the geomechanical parameter has a specific value at zero value of the factor a3 (Fig. 5.4, c), the curve approximating the relationship may be described by . .. (5.7) y = K exp( -va3). After the equation for each of the relationships has been selected, it must be transformed in such a way as to obtain straight lines on the graph. For this the procedure is to take logarithms, group the factors on the co-ordinate axes etc. When the relationships are transformed to obtain straight lines, new expressions are developed in terms of the parameters on the two co-ordinate axes. Further operations follow the same procedure as for linear relationships. It is almost impossible to advance any readymade functional relationships for approximating the processes. In each case the researcher has to select the necessary function.

5.5 Verification of Results and Determination of their Region of Applicability Results of investigations are verified in two ways depending on the problem under consideration. If the tests were planned on the basic Latin square field and experiments were conducted for the assumed multifactor system, then in order to verify the validity of the deduced relationships several additional tests should be conducted. The location of the proposed additional tests (ES) in the basic Latin square field should preferably be at the corners such that the experimental conditions of each of the tests in this new series differs significantly. Results of the additional tests are compared with the functional relationships developed earlier. If they agree with the general relationship, it may be assumed that the experiments were conducted properly and the general functional rela­ tionship is applicable over the entire range of the unknown factors. If the results of the additional experiments differ significantly from the deduced relationships and there is no doubt regarding their validity, it may

be assumed that analysis of the data from the original series of experiments was incorrect and needs to be modified by selecting more precise functional relationships. If the data obtained from experiments conducted in conformity with the generally accepted methods and processed according to the stipulated method results in a general mathematical expression or nomogram, it is then verified by

117

computing the geomechanical parameters for those conditions of the experiments which do not appear in the plan but are present in the region of variation of the factors considered during planning of the experiments. If the divergence between the experimental data and results of analysis by the stipulated method does not exceed the value specified earlier, one may stop at this stage and assume that the experimental results have been analysed correctly. If they differ significantly from the deduced relationship, the reliability of all the experiments should then be verified and a new series of experiments conducted if necessary. Experiments conducted in conformity with the method of systematic plan­ ning of experiments and analysis of their results require in-depth knowledge of the subject of investigations and considerable experience in selecting the functional relationships. The recommendations given in the following are based on the theory of systematic planning as well as experience accumulated in its application for the conditions of modelling with equivalent materials. The problems to which the method of systematic planning of experiments may be applied can be divided into several groups. Group I: Preliminary planning of experiments aimed at establishing a quan­ titative relationship between the unknown geomechanical parameter and several simultaneously acting influential factors. For such planning it is necessary to know the overall qualitative laws governing the processes occurring in the rock masses as also the principal factors which affect them. The quantitative infor­ mation must contain information on the limits of numerical values of the factors under natural conditions and those which may be deemed to be reasonable. Group Il: Analysis of the available quantitative information regarding some of the processes which may have been obtained experimentally, gathered at the existing sites etc. Often the method of systematic planning of experiments is found to be extremely useful in those cases where all attempts to establish a multifactor relationship are unsuccessful in spite of apparent availability of abundant information. Group Ill: Exploring conditions for conducting additional experiments, for instance in the case of insufficient information regarding processes when, instead of the complete cycle of experiments, it becomes necessary to conduct certain specific experiments but under unknown conditions. The method of systematic planning of experiments may also be utilised for evaluating the results of data gathered from mine working investigations, wherein the test conditions were not selected earlier and most often were inad­ vertent. Expanding the region of application of the method of systematic planning of experiments to embrace investigations under natural conditions may be achieved by:

118 i) Detennining the set of factors and geomechanical parameters which may be investigated at present with the help of modem facilities; ii) Development of maps-plans for equipping observation posts for each aspect of the investigations; iii) Developing a computer program for preparing the plan (Latin square) of experiments.

5.6 Illustrative Example of a Problem from Group I It is required to detennine the relationship between displacement of the contour of the preparatory mine working of a seam and the depth H at which it is located. The thickness of the seam is mo, the compressive strength of the sur­ rounding rock mass is Rc and the rigidity of supports with increasing resistance is n. The scale of modelling is 1 : 100 and the model is three-dimensional. Let us consider a five-factor Latin square with five numerical values for each factor and a single law for the variation of the unknown factors. The depth at which the seam is located varies from 200-1000 m. The compressive strength of rocks varies from 15-75 MPa. The rigidity of the supports varies from 3-15 kPa/cm. The fractional thickness of the seam maID (D is the diameter of the shaft) varies from 0.5-2.5 m. The ratio of compressive strength of the coal seam and the surrounding rocks RalR = 0.5 for the entire series of experiments. Let us introduce all the assumed values of the factors into their corresponding places in the rows and columns of the Latin square graph (Fig. 5.5). Planning of experiments consists of selecting 25 of the elementary squares from among the 625 with specific combinations of the numerical values of the factors. The selected ES are shown in black in Fig. 5.5. Combination of numerical values of factors for these ES is decided from the conditions under which each specific experiment is conducted. The displacement obtained from experiments on models is shown against each of the blackened squares. Based on the experimental results inscribed in the Latin square field, tables are constituted for developing expressions for their relationships. The coded values of the factor maID (from 1 to 5) are placed at the head of Table 5.1. The observed values of the geomechanical parameters (conver­ gence ~DID), their sum and average are entered in the respective rows and the last two rows of the table as percentages. The coded values of the other factor-compressive strength Rc-are entered in the first column. The sum and the averaged values of displacements of the mine working (geomechanical pa­ rameter) ~DID are entered in the last two columns on the right side of the table. The average values of the geomechanical parameter for this case vary between 4.72 and 9.38. The values of the geomechanical parameter (relative convergence ~DID) are entered in Table 5.2 for the two other factors, i.e., rigidity of the supports

119

--

Fig. 5S Results of experiments in the Latin square field .

Si and depth of mine working H-factors for which the coded value is varied within each central square from I to 5, repeating the same values 5 times over the field of the larger A-type Latin square. Completing the entries in Table 5.2 is somewhat more difficult compared to Table 5.1. The average values of the geomechanical parameter obtained from exper­ iments with these co-ordinates are entered in the cell H = I and Si = I. The elementary square with these co-ordinates is located in the middle with Rc = 2 and 1110/0 = 4 and the value of the geomechanical parameter is 4.5 . In other words, to fill each of the cells of the table with H and Si co-ordinates, one has to search over the Latin square field for values with the same co-ordinates. The value of the geomechanical parameter in the cell with H = I and Si = 2 in Table 5.2 is obtained from Rc = I and 1110/0 = I of the Latin square (where

120 Table 5.1. Table of values from Latin squares

Rc 2

maID 3

4

5

~

I 2 3 4 5

4.8 4.3 3.5 2.7 6.3

7.7 7.2 5.0 5.5 3.6

11.2 6.0 2.5 6.0 5.2

13.2 4.5 8.6 5.5 4.5

10.0 14.0 10.5 3.6 4.0

~GMP

21.6

29.0

30.9

36.3

42.1

GMP. v

4.32

5.80

6.18

7.26

(llDID)

46.9 36.0 30.1 23.3 23.6

8.42

Average (llD(D) 9.38 7.20 6.02 4.66 4.72

6.39

Table 5.2. Table of values from Latin square with respect to H and SI co-ordinates SI

H

~GMP

2

3

4

2 3 4 5

4.5 7.7 6.0 10.5 6.3

4.8 5.5 5.0 5.2 14.0

3.6 3.5 4.5 7.2 11.2

2.5 4.0 4.3 13.2 5.5

3.6 6.0 10.0 2.7 8.6

~GMP

35.0

34.5

30.0

29.5

30.9

GMP.v

7.0

1

6.90

6.0

5.90

GMP. v

5

6.18

19.0 26.7 29.8 38.8 45.6

3.8 5.34 5.96 7.76 9.12

6.39

the corresponding elementary square is located) where the value of ~DID is found to be 4.8. For the cell H = 1 and n = 3 the value of the geomechanical parameter is taken from the cell with the co-ordinates Rc = 4 and maID = 5, where ~DID = 3.6. For H = 1 and n = 4 the value of the geomechanical parameter is obtained as 2.5 from the elementary square with the co-ordinates Rc = 3 and maID = 3. For the last cell in the first row of the table, i.e., H = 1 and n = 5, the value of the geomechanical parameter is obtained as 3.6 from the ES with the corresponding co-ordinates for which Rc and maID = 2. The rest of the rows in the table are completed in the same manner. After completing Table 5.2 the values were totalled and their average value determined both row-wise and column-wise. The average values of the geome­ chanical parameter obtained in this way for the corresponding coded values of maID, Rc, S1 and H were plotted in Fig. 5.6. The coded values of the fac­ tors were measured along the abscissa and the corresponding average values of the geomechanical parameter measured along the ordinate. Thus Fig. 5.6 shows four relationships between the geomechanical parameter and all the unknown factors. All these four relationships, within the limits of their variation, may be approximated by straight-line intercepts.

121

Fig. 5.6. Relationship between convergence in a mine working and various factors (indicated on corresponding lines).

Let us analyse these graphs. For the case under consideration all the four unknown factors have a significant influence on convergence. Two of them--depth of location and thickness of seam-tend to increase it while the other two-strength and rigidity of the surrounding rocks-tend to decrease it. The depth at which the mine working is located has the strongest influence on the process under investigation. An empirical expression is fitted for each of the graphs. These hold good with all other factors at their average value. Since all the relationships happen to be linear, the relative convergence in the mine working considering all the factors is determined on the basis of equation (5.4):

.6.0/0 = 0.01[6.39

+ 1.29(Hj -

3)

+ 0.68(m o/O j -

- 0.33(nj - 3)] ± 0.016.

3) - 0.08(R c . j - 3)­ . .. (5.8)

The number 6.39 in this expression is the average value of the geomechanical parameter determined from all the measurements; 3 refers to the average coded value of each factor and ± 0.016 represents the square error. The values of the factors are likewise substituted in the equation in their coded form. The coefficients preceding the parentheses represent the extent of influence of each of factor. After decoding and substituting the numerical values for the factors, equation (5.8) transforms to:

.6.0/0

= 0.01[6.39 + 0.00645(Hj - 600) + 1.36 (maiD - 1.5)­ - 0.053(Rj - 45) - II(nj - 9] ± 0.016, ... (5.9)

where the average values of the factors are subtracted in the parentheses, i.e., av­ erage value of depth of mine working Hay = 600 m, average ratio of seam thick­ ness to diameter of mine working (maID)ay = 1.5, average compressive strength of the rocks Rc,ay = 45 MPa and average rigidity of the supports n = 9 kPa/cm.

122 The relationship obtained above is the unknown and is valid within the limits of variation of the numerical values of all the factors. Equation (5.9) is verified by computing the relative displacement for all the experimental points. It is found from analysis of the data that the discrep­ ancy between the computed values and the experimental data essentially varies between I to 26%. Results of experimental investigations of problems falling under groups 11 and III are processed similarly. The only difference is that the basic Latin square field is filled on the basis of actual conditions, taking into account the geomin­ ing and technical factors of interest and the total number of experiments. The greater the number of experimental results, the more accurate and complete is the filling up of the basic Latin square field. The functional relationships ob­ tained in this way will largely correspond to the physical nature of the processes under investigation.

PART 11

FORECASTING GEOMECHANICAL

PROCESSES DURING MINING

6

Forecasting Influence of Geological Factors on Stability of Mine Workings 6.1 Influence of Size of Cross-section on Convergence in a Drift It is well known from practical experience that the greater the cross-section of a mine working, the more the perceptible displacement of the rock mass and the greater the difficulty in providing roof supports. But the representative data available to date do not suffice for establishing any sort of relationship to determine the influence of size of mine working on displacement of rocks. Investigations were conducted on plane (two-dimensional) models, simulating a continuous medium with brittle and ductile properties to resolve this problem. In order to ensure clarity of experiments, drifts of all dimensions were located in one single model. The model was loaded with the pneumatic loading device described in Part I. The mine drifts were of circular cross-section and were unsupported. Scaled to natural conditions, the experimental conditions were as follows: Dimensions of the portion of rock mass, m ..... 200 x 200 x 200 Compressive strength of rock mass, MPa ....... 12 to 15 Dimensions of drift, m ........................ 3, 5, 9 I Relative stress in massif l'H/R c .......•.•...... The convergence in the drift was selected as the parameter representing the geomechanical process occurring in it. It was determined by means of a special type of automatic measuring device with a facility for multipoint recording from several electronic bridges. Results of investigations are shown in the form of a functional relationship between the relative convergence in the drift and its diameter at the instant when the deformation of the rock mass stabilised (Fig. 6.1, a). These results show that the relationship is non-linear; the increase in displacement with increasing size of the drift is more pronounced in the case of brittle materials than for ductile ones. This is explained by the fact that convergence in small diameter mine workings differs significantly from that in large diameter ones. It is obvious that

for excessive loading and uniform strength of the rock mass in the vicinity of a large diameter drift the onset of failure occurs at several regions of the rocks and

DOI: 10.1201/9780203746851-6

126

Fig. 6.1. Relationship between relative convergence in a drift to its diameter (a) and area of cross­ section (b) for brittle materials (I) and for materials with CTJlRc = I (2).

becomes quasi-ductile, i.e., in tenns of defonnation properties it almost behaves like a ductile material. Several research documents consider the area of cross-section as the dimen­ sion of the mine working. It can be seen from Fig. 6.1, b that the relationship between 'area of cross-section of drift and convergence' becomes linear both for brittle as well as ductile massifs. Consequently, for mine workings of cir­ cular cross-section in a continuous medium the basic relationship between the convergence ~d and the area of cross-section for stress ratio 0' lIRe = I in the rock mass may be expressed as ~d = 0.785 a d 2 •

where a is a coefficient which takes into account the defonnation properties of the rocks (a fraction) and d is the diameter of the mine working, m. Results of investigations showed that the dimension of a drift has a signifi­ cant effect on the relative convergence and must be considered during analysis of results of any investigations. It should be remembered that the relationships given here are applicable only for the experimental conditions described above.

6.2 Influence of Depth of Mining on Stress Distribution in Vicinity of a Mine Working It is well known that stresses are usually redistributed near the mine work­ ing. This process is influenced by several factors including the depth at which the mine is located. With increase in depth of mine working the maximum value of stress ratio does not increase whereas stability of the drift deteriorates. _ The stressed state of the rocks surrounding a single drift of circular cross­ section of radius R was detennined on a plane model. The dimensions of the

127

Fig. 6.2.

PI/Ro

Relationship between bearing pressure parameters and depth of mine working for (I)

= 0.9 and (2) PI/R,· = 1.8.

model were 63 x 63 x 20 cm. A homogeneous rock mass was simulated in the model with a compressive strength Rc = 0.07 MPa (Paraffin-sand 'mix­ ture). The experiments were conducted in two stages. In the first stage the external pressure was increased up to PI = 0.9 Rc and up to 1.8 Rc in the second. In both cases the pressure was measured by dynamometers, working on the friction principle, located within the model. The maximum stress concentration coefficient in the first stage of loading was 2.1 and was observed at a distance of about 0.5 R from the periphery of the drift (Fig. 6.2). In the second stage, for an external load of 1.8 Rc, the maximum stress concentration coefficient was registered at a distance of about 1.6 R from the periphery of the mine working. Results of investigations showed that with increase in depth at which the drift is located, with all other conditions remaining unchanged and stresses exceeding the compressive strength of rocks, both qualitative as well as quan­ titative, changes occur in the stress distribution near the drift. The stress con­ centration coefficient decreases while the distance at which maximum stresses occur as also the dimensions of all influencing zones increase. Consequently, at stresses exceeding the strength of the rocks during uniaxial compression, the influence of depth of mining becomes extremely significant compared to that at small depths. This influence needs to be investigated over a wide range of variations of geological conditions.

6.3 Influence of Angle of Seam Dip on Displacement of Contour of Mine Working Changes in the stability of a drift driven in a homogeneous stratified rock mass with horizontal, sloping and steeply sloping bed were investigated on models. The general parameters of these tests transformed to natural conditions

128

Fig. 6.3. Arrangement of test stand: (a) front elevation; (b) plan.

are given below: Portion of the rock mass under test, m ...... . ..... 54 x 54 x 54 Scale of modelling .............................. I : 50 Compressive strength of the material of model, MPa 25 Thickness of layers, m ......................... . 5 Diameter of drift, m ............................. In order to obtain a valid comparison of the experimental results, a large­ size model was first prepared (300 x 120 x 20 cm) from which each successive small-size model was cut. While preparing small-size models, rigid frames which could be dismantled were used to facilitate their installation at the desired angle with respect to the horizontal plane. After the models were prepared, each together with its frames was placed one after the other in the volumetric test chamber developed by G.H. Kuznetsov (Fig. 6.3). The test chamber (1), 63 x 63 x 63 cm in size, consists of two rigid side plates (3), rear plate (5), and two horizontal rigid plates consisting of one upper plate (2) and one lower plate (4). The front side of the chamber is covered by a thick organoglass plate (7) in which there is a circular hole in the centre for driving the mine working and installing devices for measuring displacement of the contour of the drift and simulating the working of supports. The side, top and bottom edges of the model are loaded by jackscrews (8) operated by electric motors. The force measuring device consists of a steel ring with resistance strain gauges (6) fixed on it. This test stand is capable of operating under specified con­ ditions of loading and defonnation rates. A constant load on the specimen is en­ sured by the set of Belleville spring washers. All force characteristics of loading of the model are recorded on the strip chart of the automatic electronic recorder. The test stand is designed for a maximum vertical load of 0.6 MN and a maximum lateral load of up to 0.3 MN. After the model was installed in the test

129

Fig. 6.4. Arrangement of device simulating working of a support with increasing resistance. I-rigid hollow body of device; 2-flat steel springs; 3-walls of mine working; 4-resistance strain gauges.

chamber, a drift of a circular cross-section was driven in it. Devices simulating the working of supports with increasing resistance and low rigidity were inserted in the drift (Fig. 6.4). Tests were conducted on the models with increase of load through steps of 0.04 MPa and a lateral thrust ratio of 0.3. With increase in vertical load on the top surface of the models, the front uncovered surface was photographed. Figure 6.5 shows the relationship between the relative convergence of the contour of the drift from the roof and floor with respect to the dip angle of the seams at various values of the vertical stresses. The tendency of monotonous increase of relative convergence with increasing stressed state of the rock mass of the model is clearly evident. It follows from an analysis of these curves that for small vertical stresses or with increase in depth of mining, the relative convergence from the roof and floor of the mine working is almost independent of the dip angle of the layers. With increase in the load applied at the top and increase of dip angle from 30 up to 60°, convergence decreases at the same stress values a I' Thus, for instance, for a maximum stress a I = 0.24 MPa in the experiment, the relative convergence for the model with dip angles of seams at 30, 60 and 45° was 12.2%, 5.4% and 8.1 % respectively. Based on the same data it is possible to develop a relationship between relative convergence and the stressed state (aIRc) of the rock mass for various dip angles of the seams (Fig. 6.6). Points AI' A2 and A3 represent the physical condition of the rock mass of the model at the instance when cracks begin to develop in it at the contour of the mine working. The location of these points on the curves becomes displaced towards larger values of (a lIRe) with an increase in the dip angle. This indicates that, all other conditions remaining the same, with an increase in dip angle of the layers of rock mass the stability of the mine working improves. The sequence of appearance of cracks in the rock mass surrounding the drift in models is of specific interest to researchers. To illustrate this Fig. 6.7

130

Fig. 6.5. Relationship between convergence in a mine working and dip angle of the rock layers. I to 6--at depth of mining corresponding to 256.512.768. 1024. 1280 and 1536 m.

Fig. 6.6. Relationship between convergence in a mine working and stress conditions (