Foundations of Engineering Mechanics

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Elsoufiev, S.A.Strength Analysis in Geomechanics, 2007

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Strength Analysis inGeomechanics

Serguey A. Elsoufiev

With 158 Figures and 11 Tables

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V.I. BabitskyDepartment of Mechanical EngineeringLoughborough University

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Loughborough LE11 3TU, LeicestershireUnited Kingdom

J. WittenburgTechnische MechanikInstitut

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Foreword

It is hardly possible to find a single rheological law for all the soils. However,they have mechanical properties (elasticity, plasticity, creep, damage, etc.)that are met in some special sciences, and basic equations of these disciplinescan be applied to earth structures. This way is taken in this book. It representsthe results that can be used as a base for computations in many fields of theGeomechanics in its wide sense. Deformation and fracture of many objectsinclude a row of important effects that must be taken into account. Some ofthem can be considered in the rheological law that, however, must be simpleenough to solve the problems for real objects.

On the base of experiments and some theoretical investigations the con-stitutive equations that take into account large strains, a non-linear unsteadycreep, an influence of a stress state type, an initial anisotropy and a damageare introduced. The test results show that they can be used first of all tofinding ultimate state of structures – for a wide variety of monotonous load-ings when equivalent strain does not diminish, and include some interrupted,step-wise and even cycling changes of stresses. When the influence of timeis negligible the basic expressions become the constitutive equations of theplasticity theory generalized here. At limit values of the exponent of a hard-ening law the last ones give the Hooke’s and the Prandtl’s diagrams. Togetherwith the basic relations of continuum mechanics they are used to describe thedeformation of many objects. Any of its stage can be taken as maximumallowable one but it is more convenient to predict a failure according to thecriterion of infinite strains rate at the beginning of unstable deformation. Themethod reveals the influence of the form and dimensions of the structure onits ultimate state that are not considered by classical approaches.

Certainly it is hardly possible to solve any real problem without someassumptions of geometrical type. Here the tasks are distinguished as anti-plane (longitudinal shear), plane and axisymmetric problems. This allowsto consider a fracture of many real structures. The results are representedby relations that can be applied directly and a computer is used (if neces-sary) on a final stage of calculations. The method can be realized not only in

VI Foreword

Geomechanics but also in other branches of industry and science. The wholeapproach takes into account five types of non-linearity (three physical andtwo geometrical) and contains some new ideas, for example, the considerationof the fracture as a process, the difference between the body and the elementof a material which only deforms and fails because it is in a structure, thesimplicity of some non-linear computations against linear ones (ideal plastic-ity versus the Hooke’s law, unsteady creep instead of a steady one, etc.), theindependence of maximum critical strain for brittle materials on the types ofstructure and stress state, an advantage of deformation theories before flowones and others.

All this does not deny the classical methods that are also used in the bookwhich is addressed to students, scientists and engineers who are busy withstrength problems.

Preface

The solution of complex problems of strength in many branches of industryand science is impossible without a knowledge of fracture processes. Last 50years demonstrated a great interest to these problems that was stimulatedby their immense practical importance. Exact methods of solution aimed atfinding fields of stresses and strains based on theories of elasticity, plastic-ity, creep, etc. and a rough appreciation of strength provide different resultsand this discrepancy can be explained by the fact that the fracture is a com-plex problem at the intersection of physics of solids, mechanics of media andmaterial sciences. Real materials contain many defects of different form anddimensions beginning from submicroscopic ones to big pores and main cracks.Because of that the use of physical theories for a quantitative appreciation ofreal structures can be considered by us as of little perspective. For technicalapplications the concept of fracture in terms of methods of continuum me-chanics plays an important role. We shall distinguish between the strength ofa material (considered as an element of it – a cube, for example) and that ofstructures, which include also samples (of a material) of a different kind. Weshall also distinguish between various types of fracture: ductile (plastic at bigresidual strains), brittle (at small changes of a bodies’ dimensions) and dueto a development of main cracks (splits).

Here we will not use the usual approach to strength computation when thedistribution of stresses are found by methods of continuum mechanics and thenhypotheses of strength are applied to the most dangerous points. Instead, weconsider the fracture as a process developing in time according to constitutiveequations taking into account large strains of unsteady creep and damage(development of internal defects). Any stage of the structures’ deformationcan be supposed as a dangerous one and hence the condition of maximumallowable strains can be used. But more convenient is the application of acriterion of an infinite strain rate at the moment of beginning of unstabledeformation. This approach gives critical strains and the time in a naturalway. When the influence of the latter is small, ultimate loads may be also

VIII Preface

found. Now we show how this idea is applied to structures made of differentmaterials, mainly soils.

The first (introductory) chapter begins with a description of the role ofengineering geological investigations. It is underlined that foundations shouldnot be considered separately from structures. Then the components of geo-mechanics are listed as well as the main tasks of the Soil Mechanics. Its shorthistory is given. The description of soil properties by methods of mechanics isrepresented. The idea is introduced that the failure of a structure is a process,the study of which can describe its final stage. Among the examples of thisare: stability of a ring under external pressure and of a bar under compressionand torsion (they are represented as particular cases of the common approachto the stability of bars); elementary theory of crack propagation; the ultimatestate of structures made of ideal plastic materials; the simplest theory ofretaining wall; a long-time strength according to a criterion of infinite elonga-tions and that of their rate. The properties of introduced non-linear equationsfor unsteady creep with damage as well as a method of determination of creepand fracture parameters from tests in tension, compression and bending aregiven (as particular cases of an eccentric compression of a bar).

In order to apply the methods of Chap. 1 to real objects we must intro-duce main equations for a complex stress state that is made in Chap. 2. Thestresses and stress tensor are introduced. They are linked by three equilib-rium equations and hence the problem is statically indeterminate. To solvethe task, displacements and strains are introduced. The latters are linked bycompatibility equations. The consideration of rheological laws begins with theHooke’s equations and their generalization for non-linear steady and unsteadycreep is given. The last option includes a damage parameter. Then basic ex-pressions for anisotropic materials are considered. The case of transversallyisotropic plate is described in detail. It is shown that the great influence ofanisotropy on rheology of the body in three options of isotropy and loadingplanes interposition takes place. Since the problem for general case cannotbe solved even for simple bodies some geometrical hypotheses are introduced.For anti-plane deformation we have five equations for five unknowns and thetask can be solved easily. The transition to polar coordinates is given. For aplane problem we have eight equations for eight unknowns. A very useful andunknown from the literature combination of static equations is received. Thebasic expressions for axi-symmetric problem are given. For spherical coordi-nates a useful combination of equilibrium laws is also derived.

It is not possible to give all the elastic solutions of geo-technical problems.They are widely represented in the literature. But some of them are includedin Chap. 3 for an understanding of further non-linear results. We begin withlongitudinal shear which, due to the use of complex variables, opens the wayto solution of similar plane tasks. The convenience of the approach is basedon the opportunity to apply a conformal transformation when the results forsimple figures (circle or semi-plane) can be applied to compound sections.The displacement of a strip, deformation of a massif with a circular hole and

Preface IX

a brittle rupture of a body with a crack are considered. The plane deformationof a wedge under an one-sided load, concentrated force in its apex and pressedby inclined plates is also studied. The use of complex variables is demonstratedon the task of compression of a massif with a circular hole. General relationsfor a semi-plane under a vertical load are applied to the cases of the crackin tension and a constant displacement under a punch. In a similar way rela-tions for transversal shear are used, and critical stresses are found. Among theaxi-symmetric problems a sphere, cylinder and cone under internal and exter-nal pressures are investigated. The generalization of the Boussinesq’s problemincludes determination of stresses and displacements under loads uniformlydistributed in a circle and rectangle. Some approximate approaches for a com-putation of the settling are also considered. Among them the layer-by-layersummation and with the help of the so-called equivalent layer. Short infor-mation on bending of thin plates and their ultimate state is described. As aconclusion, relations for displacements and stresses caused by a circular crackin tension are given.

Many materials demonstrate at loading a yielding part of the stress-straindiagram and their ultimate state can be found according to the Prandtl’s andthe Coulomb’s laws which are considered in Chap. 4, devoted to the ultimatestate of elastic-plastic structures. The investigations and natural observationsshow that the method can be also applied to brittle fracture. This approach issimpler than the consequent elastic one, and many problems can be solved onthe basis of static equations and the yielding condition, for example, the tor-sion problem, which is used for the determination of a shear strength of manymaterials including soils. The rigorous solutions for the problems of cracks andplastic zones near punch edges at longitudinal shear are given. Elastic-plasticdeformation and failure of a slope under vertical loads are studied amongthe plane problems. The rigorous solution of a massif compressed by inclinedplates for particular cases of soil pressure on a retaining wall and flow of theearth between two foundations is given. Engineering relations for wedge pen-etration and a load-bearing capacity of a piles sheet are also presented. Theintroduced theory of slip lines opens the way to finding the ultimate state ofstructures by a construction of plastic fields. The investigated penetration ofthe wedge gives in a particular case the ultimate load for punch pressure in amedium and that with a crack in tension. A similar procedure for soils is re-duced to ultimate state of a slope and the second critical load on foundation.Interaction of a soil with a retaining wall, stability of footings and differentmethods of slope stability appreciation are also given. The ultimate state ofthick-walled structures under internal and external pressures and compressionof a cylinder by rough plates are considered among axi-symmetric problems.A solution to a problem of flow of a material within a cone, its penetration ina soil and load-bearing capacity of a circular pile are of a high practical value.

Many materials demonstrate a non-linear stress-strain behaviour from thebeginning of a loading, which is accompanied as a rule by creep and damage.This case is studied in Chap. 5 devoted to the ultimate state of structures

X Preface

at small non-linear strains. The rigorous solution for propagation of a crackand plastic zones near punch edges at anti-plane deformation is given. Thegeneralization of the Flamant’s results and the analysis of them are presented.Deformation and fracture of a slope under vertical loads are considered interms of simple engineering relations. The problem of a wedge pressed byinclined plates and a flow of a material between them as well as penetrationof a wedge and load-bearing capacity of piles sheet are also discussed. Theproblem of the propagation of a crack and plastic zones near punch edges attension and compression as well as at transversal shear are also studied. Aload-bearing capacity of sliding supports is investigated. A generalization ofthe Boussinesq’s problem and its practical analysis are fulfilled. The flow of thematerial within a cone, its penetration in a massif and the load-bearing capa-city of a circular pile are studied. As a conclusion the fracture of thick-walledelements (an axi-symmetrically stretched plate with a hole, sphere, cylinderand cone under internal and external pressures) are investigated. The resultsof these solutions can be used to predict failure of the voids of different formand dimensions in soil.

In the first part of Chap. 6, devoted to the ultimate state of structuresat finite strains, the Hoff’s method of infinite elongations at the moment offracture is used. A plate and a bar at tension under hydrostatic pressure areconsidered. Thick-walled elements (axi-symmetrically stretched plate with ahole, sphere, cylinder and cone under internal and external pressure) are stud-ied in the same way. The reference to other structures is made. The secondpart of the chapter is devoted to mixed fracture at unsteady creep. The sameproblems from its first part are investigated and the comparison with theresults by the Hoff’s method is made. The ultimate state of shells (a cylinderand a torus of revolution) under internal pressure as well as different mem-branes under hydrostatic loading is studied. The comparison with test data isgiven. The same is made for a short bar in tension and compression. In con-clusion the fracture of an anisotropic plate in biaxial tension is investigated.The results are important not only for similar structures but also for a find-ing the theoretical ultimate state of a material element (a cube), which areformulated according to the strength hypotheses. The found independence ofcritical maximum strain for brittle materials on the form of a structure andthe stress state type can be formulated as a “law of nature”.

Contents

1 Introduction: Main Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Role of Engineering Geological Investigations . . . . . . . . . . . . . . . 11.2 Scope and Aim of the Subject: Short History

of Soil Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Use of the Continuum Mechanics Methods . . . . . . . . . . . . . . . . . . 21.4 Main Properties of Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 Stresses in Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.2 Settling of Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.3 Computation of Settling Changing in Time . . . . . . . . . . . 11

1.5 Description of Properties of Soils and Other Materialsby Methods of Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5.2 The Use of the Elasticity Theory . . . . . . . . . . . . . . . . . . . . 141.5.3 The Bases of Ultimate Plastic State Theory . . . . . . . . . . 171.5.4 Simplest Theories of Retaining Walls . . . . . . . . . . . . . . . . 211.5.5 Long-Time Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5.6 Eccentric Compression and Determination of Creep

Parameters from Bending Tests . . . . . . . . . . . . . . . . . . . . . 26

2 Main Equations in Media Mechanics . . . . . . . . . . . . . . . . . . . . . . . 312.1 Stresses in Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 Displacements and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3 Rheological Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.1 Generalised Hooke’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.2 Non-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3.3 Constitutive Equations for Anisotropic Materials . . . . . . 36

2.4 Solution Methods of Mechanical Problems . . . . . . . . . . . . . . . . . . 402.4.1 Common Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4.2 Basic Equations for Anti-Plane Deformation . . . . . . . . . . 402.4.3 Plane Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.4.4 Axisymmetric Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.4.5 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

XII Contents

3 Some Elastic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.1 Longitudinal Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.1.2 Longitudinal Displacement of Strip . . . . . . . . . . . . . . . . . . 483.1.3 Deformation of Massif with Circular Hole

of Unit Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.1.4 Brittle Rupture of Body with Crack . . . . . . . . . . . . . . . . . 503.1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Plane Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2.1 Wedge Under One-Sided Load . . . . . . . . . . . . . . . . . . . . . . 523.2.2 Wedge Pressed by Inclined Plates . . . . . . . . . . . . . . . . . . . 533.2.3 Wedge Under Concentrated Force in its Apex.

Some Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.2.4 Beams on Elastic Foundation . . . . . . . . . . . . . . . . . . . . . . . 613.2.5 Use of Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2.6 General Relations for a Semi-Plane Under

Vertical Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.2.7 Crack in Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2.8 Critical Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.2.9 Stresses and Displacements Under Plane Punch . . . . . . . 683.2.10 General Relations for Transversal Shear . . . . . . . . . . . . . . 693.2.11 Rupture Due to Crack in Transversal Shear . . . . . . . . . . . 693.2.12 Constant Displacement at Transversal Shear . . . . . . . . . . 703.2.13 Inclined Crack in Tension . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3 Axisymmetric Problem and its Generalization . . . . . . . . . . . . . . . 723.3.1 Sphere, Cylinder and Cone Under External

and Internal Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.3.2 Boussinesq’s Problem and its Generalization . . . . . . . . . . 743.3.3 Short Information on Bending of Thin Plates . . . . . . . . . 793.3.4 Circular Crack in Tension . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4 Elastic-Plastic and Ultimate State of PerfectPlastic Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.1 Anti-Plane Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.1.1 Ultimate State at Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . 854.1.2 Plastic Zones near Crack and Punch Ends . . . . . . . . . . . . 86

4.2 Plane Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.2.1 Elastic-Plastic Deformation and Failure of Slope . . . . . . . 884.2.2 Compression of Massif by Inclined Rigid Plates . . . . . . . 904.2.3 Penetration of Wedge and Load-Bearing Capacity

of Piles Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.2.4 Theory of Slip Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2.5 Ultimate State of Some Plastic Bodies . . . . . . . . . . . . . . . 994.2.6 Ultimate State of Some Soil Structures . . . . . . . . . . . . . . . 1034.2.7 Pressure of Soils on Retaining Walls . . . . . . . . . . . . . . . . . 108

Contents XIII

4.2.8 Stability of Footings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.2.9 Elementary Tasks of Slope Stability . . . . . . . . . . . . . . . . . . 1114.2.10 Some Methods of Appreciation of Slopes Stability . . . . . 113

4.3 Axisymmetric Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.3.1 Elastic-Plastic and Ultimate States of Thick-Walled

Elements Under Internal and External Pressure . . . . . . . 1174.3.2 Compression of Cylinder by Rough Plates . . . . . . . . . . . . 1194.3.3 Flow of Material Within Cone . . . . . . . . . . . . . . . . . . . . . . 1214.3.4 Penetration of Rigid Cone and Load-Bearing Capacity

of Circular Pile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.4 Intermediary Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5 Ultimate State of Structures at Small Non-Linear Strains . . 1255.1 Fracture Near Edges of Cracks and Punch

at Anti-Plane Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.1.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.1.2 Case of Crack Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 1265.1.3 Plastic Zones near Punch Edges . . . . . . . . . . . . . . . . . . . . . 127

5.2 Plane Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.2.1 Generalization of Flamant’s Problem. . . . . . . . . . . . . . . . . 1285.2.2 Slope Under One-Sided Load . . . . . . . . . . . . . . . . . . . . . . . 1315.2.3 Wedge Pressed by Inclined Rigid Plates . . . . . . . . . . . . . . 1345.2.4 Penetration of Wedge and Load-Bearing Capacity

of Piles Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.2.5 Wedge Under Bending Moment in its Apex . . . . . . . . . . . 1405.2.6 Load-Bearing Capacity of Sliding Supports . . . . . . . . . . . 1445.2.7 Propagation of Cracks and Plastic Zones

near Punch Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.3 Axisymmetric Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.3.1 Generalization of Boussinesq’s Solution . . . . . . . . . . . . . . . 1495.3.2 Flow of Material within Cone . . . . . . . . . . . . . . . . . . . . . . . 1515.3.3 Cone Penetration and Load-Bearing Capacity

of Circular Pile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545.3.4 Fracture of Thick-Walled Elements Due to Damage . . . . 156

6 Ultimate State of Structures at Finite Strains . . . . . . . . . . . . . 1616.1 Use of Hoff’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.1.1 Tension of Elements Under Hydrostatic Pressure . . . . . . 1616.1.2 Fracture Time of Axisymmetrically Stretched Plate . . . . 1626.1.3 Thick-Walled Elements Under Internal

and External Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1646.1.4 Final Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.2 Mixed Fracture at Unsteady Creep . . . . . . . . . . . . . . . . . . . . . . . . 1666.2.1 Tension Under Hydrostatic Pressure . . . . . . . . . . . . . . . . . 166

XIV Contents

6.2.2 Axisymmetric Tension of Variable Thickness Platewith Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.2.3 Thick-Walled Elements Under Internaland External Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.2.4 Deformation and Fracture of Thin-Walled ShellsUnder Internal Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.2.5 Thin-Walled Membranes Under Hydrostatic Pressure . . 1766.2.6 Two Other Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1786.2.7 Ultimate State of Anisotropic Plate

in Biaxial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Appendix E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Appendix F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Appendix G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Appendix H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Appendix I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Appendix J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Appendix K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

Appendix L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Appendix M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Appendix N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229