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1.4: Stereographic Projection - Geosciences

1.4: Stereographic Projection - Geosciences


Introduction

Stereographic projection is a powerful method for solving geometric problems in structural geology. Whereas crystallographers use an upper hemisphere projection, structural geologists always use the lower hemisphere. The on-line visualization tool at https://app.visiblegeology.com/stereonet.html may also help.

Stereogram basics

There are two parts to any stereographic projection. The projection itself, or stereogram, is usually drawn on tracing paper, and represents a bowl-shaped surface embedded in the Earth. The stereographic net or stereonet is the 3-D equivalent of a protractor. It is used to measure angles on the projection. To measure angles, we need to rotate the net relative to the tracing paper. For practical reasons we usually turn the tracing paper and keep the net fixed, but it is important to remember that in reality, the projection has a fixed orientation and the net should be rotated to make measurements.

To construct a stereogram, take a sheet of tracing paper and draw a circle, with the same radius as an available stereonet. This circle is known as the primitive. Mark the centre with a cross, and mark a north arrow on the primitive at the top of the page. Mark E, S and W (or 090, 180 and 270) points at 90° intervals around the primitive. It is sometimes helpful to reinforce the centre with adhesive tape on the back of the tracing paper.

The stereonet may be reinforced with card to extend its life. It is convenient to place an old-fashioned thumb tack through the centre of the net. Protect yourself and others from the thumb tack by keeping it embedded in an eraser while not in use. There are several varieties of stereonet available. We will start with a Wulff net, which is used for the construction of the true, or equal-angle stereographic projection. In later labs we will use a Schmidt net, which constructs an equal-area projection.

Principle of stereographic projection

For stereographic projection, a line or a plane is imagined to be surrounded by a projection sphere (Fig. 1a). A plane intersects the sphere in a trace that is a great circle that bisects the sphere precisely. A line intersects the sphere in a point. To image features on a sheet of paper, these traces and points are projected from a point at the summit or zenith of the sphere onto the equatorial plane. This is clearer in a diagram (Fig. 2b), which shows the method for stereographic projection of a dipping plane. A family of planes dipping at various increments is shown in Fig. 3a. Planes project as curves that are actually perfectly circular arcs called cyclographic traces or just great circles. Lines project as points also known as poles.

As a general principle, planes that dip at low angles are represented by great circles having significant curvature and lying closer to the primitive, whereas steeply dipping planes are characterized by straighter great circles passing close to the centre of the plot. All vertical planes will project as straight lines passing through the centre of the stereogram.

Sometimes we represent a plane by its pole. The pole to a plane is the plot of a line perpendicular to the plane. For a horizontal plane the pole is in the centre of the net. Gently dipping planes have poles near the centre; steeply dipping planes have poles near the edge. The pole is always in the opposite quadrant to the great circle. Poles are used when plotting numerous great circles would make the plot cluttered and confusing.

Features of the net

The stereographic net assists in the construction of great circles and points. It contains a family of great circles intersecting at the top and bottom points of the net (Fig. 3a). Every fifth cyclographic trace is bolder so that ten degree increments can be easily counted. By rotating the net, a great circle can be maneuvered into any desired strike and dip orientation. The net also contains small circles (Fig. 3b) that are helpful in solving rotation problems, and act as a scale of pitch angles along each great circle.

One great circle on the net corresponds to a vertical plane and is straight. It runs from top to bottom of the net. One trace in the family of small circles is also straight, from left to right on the net; it is actually a great circle too. These two intersecting lines form four straight radii which are crucial for counting angles of dip and plunge.

When the plot is located so that its north arrow coincides with the top of the net, it is said to be in the reference position.

Two applications of the stereographic projection

True and apparent dip

In lab 2 you encountered the terms ‘true dip’ and ‘apparent dip’. Refer back to lab 2 if you need to remind yourself of the difference. In principle, it’s possible to make conversions between true and apparent dip by trigonometry. However, it’s generally much easier to make the conversion using the stereographic projection. The construction is shown in Fig. 5.

True and apparent thickness

A common problem in stratigraphy is to determine the true thickness of a formation. True thickness is measured perpendicular to the plane of bedding. Often, in the field or in a subsurface well, an apparent thickness is measured – one that is oblique to bedding, and therefore overestimates the true thickness (Fig. 6). There are many trigonometric methods for calculating true thickness, depending on the exact circumstances of measurement. However, one method using the stereographic projection works every time: multiply the apparent thickness by the cosine of the angle (θ) between the pole to bedding and the line of measurement (the trend and plunge of the traverse, tape measure, or whatever measuring device was used).

True thickness = Measured thickness × cos(θ)


Truncated tesseract

Schlegel diagram
(tetrahedron cells visible)
Type Uniform 4-polytope
Schläfli symbol t
Coxeter diagrams
Cells 24 8 3.8.8
16 3.3.3
Faces 88 64
24
Edges 128
Vertices 64
Vertex figure
( )v
Dual Tetrakis 16-cell
Symmetry group B4, [4,3,3], order 384
Properties convex
Uniform index 12 13 14

The truncated tesseract is bounded by 24 cells: 8 truncated cubes, and 16 tetrahedra.

Alternate names Edit

  • Truncated tesseract (Norman W. Johnson)
  • Truncated tesseract (Acronym tat) (George Olshevsky, and Jonathan Bowers) [1]

Construction Edit

The Cartesian coordinates of the vertices of a truncated tesseract having edge length 2 is given by all permutations of:

Projections Edit

In the truncated cube first parallel projection of the truncated tesseract into 3-dimensional space, the image is laid out as follows:

  • The projection envelope is a cube.
  • Two of the truncated cube cells project onto a truncated cube inscribed in the cubical envelope.
  • The other 6 truncated cubes project onto the square faces of the envelope.
  • The 8 tetrahedral volumes between the envelope and the triangular faces of the central truncated cube are the images of the 16 tetrahedra, a pair of cells to each image.

Images Edit

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph
Dihedral symmetry [12/3] [4]

A polyhedral net

Truncated tesseract
projected onto the 3-sphere
with a stereographic projection
into 3-space.

Related polytopes Edit

The truncated tesseract, is third in a sequence of truncated hypercubes:

Truncated hypercubes
Image .
Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
Coxeter diagram
Vertex figure ( )v( )
( )v

( )v

( )v
( )v ( )v ( )v

Bitruncated tesseract

Two Schlegel diagrams, centered on truncated tetrahedral or truncated octahedral cells, with alternate cell types hidden.
Type Uniform 4-polytope
Schläfli symbol 2t<4,3,3>
2t<3,3 1,1 >
h2,3
Coxeter diagrams

=
Cells 24 8 4.6.6
16 3.6.6
Faces 120 32
24
64
Edges 192
Vertices 96
Vertex figure
Digonal disphenoid
Symmetry group B4, [3,3,4], order 384
D4, [3 1,1,1 ], order 192
Properties convex, vertex-transitive
Uniform index 15 16 17

The bitruncated tesseract, bitruncated 16-cell, or tesseractihexadecachoron is constructed by a bitruncation operation applied to the tesseract. It can also be called a runcicantic tesseract with half the vertices of a runcicantellated tesseract with a construction.

Alternate names Edit

  • Bitruncated tesseract/Runcicantic tesseract (Norman W. Johnson)
  • Bitruncated tesseract (Acronym tah) (George Olshevsky, and Jonathan Bowers) [2]

Construction Edit

A tesseract is bitruncated by truncating its cells beyond their midpoints, turning the eight cubes into eight truncated octahedra. These still share their square faces, but the hexagonal faces form truncated tetrahedra which share their triangular faces with each other.

The Cartesian coordinates of the vertices of a bitruncated tesseract having edge length 2 is given by all permutations of:

Structure Edit

The truncated octahedra are connected to each other via their square faces, and to the truncated tetrahedra via their hexagonal faces. The truncated tetrahedra are connected to each other via their triangular faces.

Projections Edit

Stereographic projections Edit

The truncated-octahedron-first projection of the bitruncated tesseract into 3D space has a truncated cubical envelope. Two of the truncated octahedral cells project onto a truncated octahedron inscribed in this envelope, with the square faces touching the centers of the octahedral faces. The 6 octahedral faces are the images of the remaining 6 truncated octahedral cells. The remaining gap between the inscribed truncated octahedron and the envelope are filled by 8 flattened truncated tetrahedra, each of which is the image of a pair of truncated tetrahedral cells.

Stereographic projections

Colored transparently with pink triangles, blue squares, and gray hexagons

Related polytopes Edit

The bitruncated tesseract is second in a sequence of bitruncated hypercubes:

Bitruncated hypercubes
Image .
Name Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
Coxeter
Vertex figure
( )v

< >v

< >v

< >v
< >v < >v

Truncated 16-cell
Cantic tesseract

Schlegel diagram
(octahedron cells visible)
Type Uniform 4-polytope
Schläfli symbol t<4,3,3>
t<3,3 1,1 >
h2
Coxeter diagrams

=
Cells 24 8 3.3.3.3
16 3.6.6
Faces 96 64
32
Edges 120
Vertices 48
Vertex figure
square pyramid
Dual Hexakis tesseract
Coxeter groups B4 [3,3,4], order 384
D4 [3 1,1,1 ], order 192
Properties convex
Uniform index 16 17 18

The truncated 16-cell, truncated hexadecachoron, cantic tesseract which is bounded by 24 cells: 8 regular octahedra, and 16 truncated tetrahedra. It has half the vertices of a cantellated tesseract with construction .

It is related to, but not to be confused with, the 24-cell, which is a regular 4-polytope bounded by 24 regular octahedra.

Alternate names Edit

  • Truncated 16-cell/Cantic tesseract (Norman W. Johnson)
  • Truncated hexadecachoron (Acronym thex) (George Olshevsky, and Jonathan Bowers) [3]

Construction Edit

The truncated 16-cell may be constructed from the 16-cell by truncating its vertices at 1/3 of the edge length. This results in the 16 truncated tetrahedral cells, and introduces the 8 octahedra (vertex figures).

(Truncating a 16-cell at 1/2 of the edge length results in the 24-cell, which has a greater degree of symmetry because the truncated cells become identical with the vertex figures.)

The Cartesian coordinates of the vertices of a truncated 16-cell having edge length 2√2 are given by all permutations, and sign combinations:

An alternate construction begins with a demitesseract with vertex coordinates (±3,±3,±3,±3), having an even number of each sign, and truncates it to obtain the permutations of

(1,1,3,3), with an even number of each sign.

Structure Edit

The truncated tetrahedra are joined to each other via their hexagonal faces. The octahedra are joined to the truncated tetrahedra via their triangular faces.

Projections Edit

Centered on octahedron Edit

The octahedron-first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure:

  • The projection envelope is a truncated octahedron.
  • The 6 square faces of the envelope are the images of 6 of the octahedral cells.
  • An octahedron lies at the center of the envelope, joined to the center of the 6 square faces by 6 edges. This is the image of the other 2 octahedral cells.
  • The remaining space between the envelope and the central octahedron is filled by 8 truncated tetrahedra (distorted by projection). These are the images of the 16 truncated tetrahedral cells, a pair of cells to each image.

This layout of cells in projection is analogous to the layout of faces in the projection of the truncated octahedron into 2-dimensional space. Hence, the truncated 16-cell may be thought of as the 4-dimensional analogue of the truncated octahedron.

Centered on truncated tetrahedron Edit

The truncated tetrahedron first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure:

  • The projection envelope is a truncated cube.
  • The nearest truncated tetrahedron to the 4D viewpoint projects to the center of the envelope, with its triangular faces joined to 4 octahedral volumes that connect it to 4 of the triangular faces of the envelope.
  • The remaining space in the envelope is filled by 4 other truncated tetrahedra.
  • These volumes are the images of the cells lying on the near side of the truncated 16-cell the other cells project onto the same layout except in the dual configuration.
  • The six octagonal faces of the projection envelope are the images of the remaining 6 truncated tetrahedral cells.

Images Edit

Related polytopes Edit

A truncated 16-cell, as a cantic 4-cube, is related to the dimensional family of cantic n-cubes:


Complex Analysis

This is a clever, concise, concrete, and classical complex analysis book, aimed at undergraduates with no background beyond single-variable calculus. The book has an eclectic flavor rather than develop any general theories, the authors work toward a number of classical results, and usually take the shortest path to get there.

The book generally takes an analytic rather than a geometric approach the Cauchy-Riemann equations are central. It works up to analytic functions by going through polynomials and entire functions, and only then considers functions analytic on a disk and then analytic on a region. The elementary functions are developed as extensions of those functions on the reals, rather than as power series. Similarly, the book starts with polygonal paths with only horizontal or vertical segments, and works up to general curves. There are no Riemann surfaces, and multi-valued functions such as the logarithm are sidestepped by explicitly defining a useful branch and showing that it has the desired properties.

The book has a modest number of applications, including some discussion of fluid flow and the Riemann mapping theorem. Most of the applications are to other branches of mathematics rather than to other sciences, and cover fields such as combinatorics and evaluation of definite integrals and infinite series. There is even has a complete proof of the prime number theorem.

There are many exercises, and they cover a wide range of difficulty, from routine applications of techniques in the text through quite challenging problems. Answers to all exercises are given in the back of the book, although usually they are sketches of the answer in a couple of sentences rather than a detailed answer.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

1. The Complex Numbers
Introduction
1.1. The Field of Complex Numbers
1.2. The Complex Plane
1.3. Topological Aspects of the Complex Plane
1.4. Stereographic Projection The Point at Infinity
Exercises

2. Functions of the Complex Variable z
Introduction
2.1. Analytic Polynomials
2.2. Power Series
2.3. Differentiability and Uniqueness of Power Series
Exercises

3. Analytic Functions
3.1. Analyticity and the Cauchy-Riemann Equations
3.2. The Functions e z , sin z, cos z
Exercises

4. Line Integrals and Entire Functions
Introduction
4.1. Properties of the Line Integral
4.2. The Closed Curve Theorem for Entire Functions
Exercises

5. Properties of Entire Functions
5.1. The Cauchy Integral Formula and Taylor Expansion for Entire Functions
5.2. Liouville Theorems and the Fundamental Theorem of Algebra
Exercises

6. Properties of Analytic Functions
Introduction
6.1. The Power Series Representation for Functions Analytic in a Disc
6.2. Analyticity in an Arbitrary Open Set
6.3. The Uniqueness, Mean-Value, and Maximum-Modulus Theorems
Exercises

7. Further Properties of Analytic Functions
7.1. The Open Mapping Theorem Schwarz’ Lemma
7.2. The Converse of Cauchy’s Theorem: Morera’s Theorem The Schwarz Reflection Principle
Exercises

8. Simply Connected Domains
8.1. The General Cauchy Closed Curve Theorem
8.2. The Analytic Function Log z
Exercises

9. Isolated Singularities of an Analytic Function
9.1. Classification of Isolated Singularities Riemann’s Principle and the Casorati-Weierstrass Theorem
9.2. Laurent Expansions
Exercises

10. The Residue Theorem
lO.1. Winding Numbers and the Cauchy Residue Theorem
lO.2. Applications of the Residue Theorem
Exercises

11. Applications of The Residue Theorem to the Evaluation of Integrals and Sums
Introduction
11.1. Evaluation of Definite Integrals by Contour Integral Techniques
11.2. Application of Contour Integral Methods to Evaluation and Estimation of Sums
Exercises

12. Further Contour Integral Techniques
12.1. Shifting the Contour of Integration
12.2. An Entire Function Bounded in Every Direction
Exercises

13. Introduction to Conformal Mapping
13.1. Conformal Equivalence
13.2. Special Mappings
Exercises

14. The Riemann Mapping Theorem
14.1. Conformal Mapping and Hydrodynamics
14.2. The Riemann Mapping Theorem
Exercises

15. Maximum-Modulus Theorems for Unbounded Domains
15.1. A General Maximum-Modulus Theorem
15.2. The Phragmén-Lindelöf Theorem
Exercises

16. Harmonic Functions
16.1. Poisson Formulae and the Dirichlet Problem
16.2. Liouville Theorems for Re f Zeroes of Entire Functions of Finite Order
Exercises

17. Different Forms of Analytic Functions
Introduction
17.1. Infinite Products
17.2. Analytic Functions Defined by Definite Integrals
Exercises

18. Analytic Continuation The Gamma and Zeta Functions
Introduction
18.1. Power Series
18.2. The Gamma and Zeta Functions
Exercises

19. Applications to Other Areas of Mathematics
Introduction
19.1. A Partition Problem
19.2. An Infinite System of Equations
19.3. A Variation Problem
19.4. The Fourier Uniqueness Theorem
19.5. The Prime-Number Theorem
Exercises

Appendices
1. A Note on Simply Connected Regions
II. Circulation and Flux as Contour Integrals
III. Steady-State Temperatures The Heat Equation
IV. Tchebychev Estimates


Book Description

Drilled shafts in rock are widely used as foundations of heavy structures such as highway bridges and tall buildings. Although much has been learned about the analysis and design of drilled shafts in rock, all the major findings are published in the form of reports and articles in technical journals and conference proceedings. This book is the first to present and summarize the latest information in one volume, highlighting for the reader the principle differences between foundations in soil, and foundations in rock masses containing discontinuities.

This book presents methods for characterizing discontinuities in jointed rock masses, and considering their effects on the behaviour of drilled shafts. A valuable tool for practitioners in geological engineering, rock mechanics and foundation engineering.


Table of Contents

Preface
1 Introduction to Basic
1.1 The BASIC Approach
1.2 The Elements of BASIC
1.3 Checking Programs
1.4 Different Computers and Variants of BASIC
1.5 Graphics Commands
1.6 Special Commands
Bibliography
2 Structure of Materials
Essential Theory
2.1 Introduction
2.2 Atoms on Crystal Planes
2.3 Dislocations
2.4 Stereographic Projection
2.5 Atomic and Weight Percent
2.6 Hume-Rothery Primary Solid Solubility Rules
2.7 Binary Eutectic Equilibrium Diagram
2.8 Coring in a Binary Alloy
2.9 Quantitative Metallography
Programs
2.1 Atoms on a Crystal Plane
2.2 Dislocations
2.3 Stereographic Projection
2.4 Atomic and Weight Percent Compositions
2.5 Hume-Rothery Primary Solid Solubility
2.6 Binary Eutectic Equilibrium Diagram
2.7 Coring
2.8 Quantitative Metallography
Problems
References
Bibliography
3 Thermodynamics and Kinetics of Solids
Essential Theory
3.1 Introduction
3.2 Thermodynamic Relationships
3.3 Rates of Reaction - The Arrhenius Equation
3.4 Diffusion
3.5 Analysis of Resistivity Data
3.6 Shear Transformation (Martensitic Reactions)
3.7 Corrosion
Programs
3.1 Calculation of Activation Energy-Diffusion of Cu in Cuo
3.2 Calculation of Reaction Times from Activation Energy
3.3 Calculation of Diffusion Profiles - Fick's Second Law
3.4 Analysis of Resistivity Data: Johnson-Mehl
3.5 Determination of Hardenability
3.6 Corrosion - Calculation of Cell Voltages
Problems
References
Bibliography
4 Mechanical Properties of Polymers
Essential Theory
4.1 Introduction
4.2 Degree of Polymerization
4.3 Average Molecular Weights
4.4 Elastic Strain and Elastic Energy Stored
4.5 Work Done In Deformation
4.6 Stress or Strain Relaxation
4.7 Viscoelastic Modulus
4.8 Elastomer Stress-Strain Curve
4.9 Molecular Weight and Strength of Polystyrene
Programs
4.1 Degree of Polymerization
4.2 Weight and Number Average Molecular Weights
4.3 Elastic Strain and Energy
4.4 Work Done In Deformation
4.5 Anelastic Relaxation Time
4.6 Viscoelastic Modulus
4.7 Elastomer Stress-Strain
4.8 Molecular Weight and Tensile Compact Strength
Problems
Reference
Bibliography
5 Deformation and Strength of Crystalline Materials
Essential Theory
5.1 Introduction
5.2 Theoretical Strength
5.3 Deformation of Single Crystals-Critical Resolved Shear Stress
5.4 Tensile Deformation of Polycrystalline Materials
5.5 Three-Point Bend Testing
5.6 Hardness
Programs
5.1 Critical Resolved Shear Stress
5.2 Tensile Analysis
5.3 Yield Point Phenomena
5.4 Neutral Axis and Inter-Laminar Shear Stress (ILSS)
5.5 Hardness Vickers-Ocular to Hw Conversion
Problems
References
Bibliography
6 Materials Properties Comparisons
Essential Theory
6.1 Introduction
6.2 Order of Merit Classification
6.3 The Data
Program
Problems
Reference
Bibliography
Index


1.4: Stereographic Projection - Geosciences

Stability assessment around a railway tunnel using terrestrial laser scanner data and finite element analysis

* Universidad Católica de la Santísima Concepción - Concepción, CHILE

** Vale S.A. - Minas Gerais, BRASIL

*** Universidade de São Paulo - São Paulo, BRASIL

Geotechnical analysis of tunnels in complex geo-structural environments requires an advanced understanding of the inter-block structure effect on rock mass behavior, such as joints and fractures systems, bedding and foliation planes, among other discontinuity types. The conventional approach for preliminary geotechnical analysis of tunnels is based on a continuous-equivalent system representation of rock mass, i.e. without explicit consideration of systematic discontinuity systems. However, to obtain a closer to reality results of the rock mass expected behavior, geo-structural data should be included from the initial stage of geotechnical analysis. A case study is used to analyze the implications of the discontinuity systems inclusion on the rock mass stability around tunnel. Two-dimensional finite element numerical models were developed using three different models to the generation of rock discontinuity systems. The obtained results show that fracture intensity parameter help to generated more realistic two-dimensional DFNs.

Keywords: Tunnel TLS (terrestrial laser scanning) FEM (finite element method) DFN (discrete fracture networks)

The construction of geologically realistic discontinuity system networks to use in the geomechanical evaluation of underground excavations has gained ground over the conventional techniques of considering rock mass as a continuous equivalent material and the geomechanical classification systems based on empirical data. Currently the discrete fracture networks are the technique that is mainly used. It offers the possibility of maximizing the use of geo-structural data collected via manual geotechnical mapping or remote sensing techniques, like digital photogrammetry and laser scanning ( Elmo et al., 2014 ).

( Cacciari and Futai, 2017 ) have presented a methodology for tridimensional numerical simulations of tunnels excavated in discontinuous rock masses, based on the terrestrial laser scanner technique (TLS) and the generation of discrete fracture networks (DFNs). These authors have discussed several aspects related with the mapping of discontinuities using TLS, statistical analysis of discontinuities using window sampling methods and tridimensional numerical modeling using DFNs.

The construction of underground excavation projects is often made in rock masses with discontinuity systems such as failures, stratification planes and foliation, seams and fractures, among others. These discontinuity systems induce the formation and instability of rock blocks and wedges during the execution and operation stages of underground works. The discontinuity systems usually occur in packages, which can be geometrically described by their orientation, trace length, persistence and spacing. In addition, the low and sometimes non-existent resistance to the cross-section of the discontinuity systems, high levels of tension in situ, as well as the confinement loss conditions during the execution of the excavations, induce different modes of instability and the failures of rock blocks formed by the intersection of two or more discontinuity systems, such as sliding, toppling and falls or collapses within the excavation.

To study the effect of the inclusion of geo-structural data on the stability of underground excavations built in discontinuous rock masses, a 10-m long section of the Monte Seco tunnel, located in the Southeast of Brazil, has been used as a case study. For this, two-dimensional numerical simulations considering a semi-discontinuous elasto-plastic means have been made using the FEM-RS2 finite element commercial software ( Rocscience, 2015 ).

In the numerical simulations which explicitly consider rock mass discontinuity systems, three discontinuity network generation methodologies have been evaluated, namely: deterministic parallel, statistical parallel and the Baecher empirical model ( Rocscience, 2015 ) ( Baecher et al., 1977 ). This was done by directly importing the entry parameters, which in this case, correspond to TLS measurements of the discontinuity system's geometrical parameters. In this way, discontinuity planes are created, defined by their specific geometric characteristics, which, at the same time, generate two-dimensional networks of the discontinuity systems coupled to the finite elements model.

The results obtained show and highlight the significant effect of the explicit modeling of discontinuity systems over the stress-displacement patterns around the studied tunnel, being able to quantitatively visualize the effects on the concentration and relaxation of stresses, shear zones, the formation of blocks and the displacement trajectories. In addition, the over-excavation profiles obtained using the numerical modeling have been compared with TLS measurements, through which a good fit with the onsite observations has been obtained.

Numerical simulations of rock tunnels must consider the complexities related to the interaction of three-dimensional tunnel geometry and true geometrical discontinuity system (i.e., orientation, trace length, persistence and spacing). Discontinuity systems are usually represented deterministically without accounting for uncertainty and spatial variability which represent inherent characteristics of rock mechanics problems ( Einstein and Baecher, 1983 ). Terrestrial remote sensing techniques (e.g., digital photogrammetry and laser scanning) now provide a convenient additional tool to help reduce these problems. The remote sensing data provides key geotechnical information such as discontinuity orientation and length as well as the location of each discontinuity measurement.

The volume of data can be significantly greater both in terms of magnitude and the areal extent mapped compared to traditional geotechnical data mapping ( Fekete and Diederichs, 2013 ). The acquired discontinuity data can be used to develop stochastic DFNs for a more realistic representation of discontinuity systems ( Havaej et al., 2016 ). The application of remote sensing methods for discontinuity mapping has increased significantly in the last decade. There are several applications of remote sensing techniques in rock engineering practice, such as rock mass characterization ( Tonon and Kottenstette, 2006 ) ( Ferrero et al., 2009 ) ( Lato et al., 2009 ), ( Lato et al., 2010 ) ( Gigli and Casagli, 2011 ) ( Lato et al., 2013 ) ( Otoo et al., 2013 ) ( Deliormanli et al., 2014 ) ( Lai et al., 2014 ), rock slope stability ( Strouth et al., 2006 ) ( Ghirotti and Genevois, 2007 ) ( Sturzenegger and Stead, 2009 ) ( Lato and Vöge, 2011 ) ( Lato et al., 2012 ) ( Lato et al., 2015 ) ( Tuckey and Stead, 2016 ) and underground excavations (Fekete et al., 2010) ( Styles et al., 2010 ) ( Fekete and Diederichs, 2013 ) ( Lato and Diederichs, 2014 ) ( Preston et al., 2014 ) ( Walton et al., 2014 ) ( Cacciari and Futai, 2017 ) ( Delaloye et al., 2015 ) ( Villalobos et al., 2017 ).

DFNs have been used in a wide range of geomechanical problems (e.g., open pits, tunneling, block caving, reservoir geomechanics, etc.). In order to define a fracture network to represent a natural discontinuity system, ( Elmo, 2006 ) proposed that at least three sets of parameters are required: fracture size distribution, fracture orientation distribution and fracture density. ( Dershowitz et al., 2014 ) defining two different parameters, the areal intensity P21 and volumetric intensity P32, to represent the degree of fracturing in rock masses. P21 and P32 are defined as the cumulative length of fractures per unit area and the cumulative area of fractures per unit volume, respectively ( Havaej et al., 2016 ).

In this study, we use a two-dimensional finite element code to investigate the rock mass stability around the Monte Seco tunnel, located in Minas Gerais, Brazil. (Figure 1) illustrates the methodology adopted. TLS is performed at the site in order to characterize the rock mass. The point cloud derived from the TLS is used to reproduce a realistic rock mass geometry for a 10-m length representative section of the tunnel which is then incorporated into the numerical simulations using finite elements. Discontinuity mapping is also performed using the TLS data, which allows developing realistic DFNs. TLS data is also used to reproduce overbreak profiles of tunnel sections which are then compared with two-dimensional numerical simulation results.

Figure 1 Methodology adopted for rock mass characterization and subsequent numerical simulations

3. Case study: Monte Seco Tunnel

In Brazil, there are several old tunnels of the highway and railroad system that date from the 1950s. They were built in rock masses with a very good geotechnical quality but without any reinforcement or support system. Today, some of these tunnels have seen localized problems with the formation and falls of rock blocks, mainly associated to the distribution of discontinuity systems and one-off degradation processes of the associated geomechanical parameters.

The Monte Seco Tunnel is an old linear underground project built for the Vitoria-Minas railroad in the State of Espirito Santo in the southeast of Brazil see (Figure 2). It belongs to the mining company, VALE S.A. This tunnel has required a series of geological and geotechnical investigations to provide geomechanical parameters and its stability assessments.

Considering this problem, a joint project was started between the Polytechnic School of the Universidade de São Paulo and the Mining Company, VALE S.A., to propose a study methodology of the tunnel's current state. Thus, the Monte Seco tunnel has been transformed into a valuable source of geomechanical information to study the stability of underground projects built in discontinuous rock masses. Below, the main characteristics and properties considered for this work, are presented.

Figure 2 Location of the Monte Seco tunnel and geotechnical investigations performed at the tunnel site ( Cacciari and Futai, 2017 )

3.1 Geo-structural data collection

The Monte Seco tunnel is located in the Province of Mantiqueira, built in a rock mass formed by a Gneiss (metamorphism of sedimentary rock), with mylonitic texture, comprising felsic (with a predominance of quartz and feldspar) and mafic bands (with a predominance of biotite and amphibolite), with pronounced foliation, mainly due to the orientation of mica. In addition, in several portions within the tunnel, as well as in the outcrop of external rocks, pockets of Granite inserted in the Gneiss were seen with diameters of between 1.0 and 3.0-m, with abrupt contacts, without foliation and with a pegmatitic texture.

( Cacciari and Futai, 2017 ) have geotechnically characterized the discontinuity systems using the remote techniques known as TLS to generate the point clouds shown in (Figure 3a). The methodology used can be summarized in the following three steps: (i) complete mapping of discontinuities in the TLS point clouds, measuring all the positions, trace lengths and orientations of each set of discontinuities (ii) discontinuity analysis to determine the probability density functions of the diameters and orientations of each set of discontinuities and (iii) calculation of the volumetric intensity parameters, P32 and P21 for each set of discontinuities.

The mapping of the discontinuity systems in the TLS points cloud basically consists in the interpretation of the discontinuities present in the tunnel's rock surface (i.e., walls and roof) and make their respective measurements. The orientation is measured through the selection of coplanar points to the exposed areas of the discontinuities and extracting the normal vector to the adjusted plane for these points. Below, the normal vectors measured are converted to the geo-structural notation (i.e., Dip and DipDir). The traces are the intersection between the discontinuities and the exposed rock surface within the tunnel. Finally, the trace lengths are measured taking the distance between the final points of the adjusted polylines in these traces. (Figures 3b) and (Figure 3c) show examples of the trace length measurements and orientation in the points cloud.

Figure 3 a) Faro Focus 3D laser scanner model and an example of the Monte Seco Tunnel's TLS image (point cloud) generated by this instrument, b) measurement of trace lengths and c) measurement of orientation

3.2 Geo-structural data analysis

The Monte Seco Tunnel was built in a Gneiss rock mass, where four discontinuity systems were identified and characterized during the in-situ inspection. Two fracture systems along cut F1 and F2, a laminar jointing system F3, and a foliation system Sn. These discontinuity systems were mapped in detail with the TLS images. (Figure 4a) shows the orientation measurements in the TLS images with the identification of each one of these discontinuity systems and the tunnel direction. (Figure 4b) preliminarily shows the formation and instability of the blocks which tend to slide and fall inside the tunnel excavation.

In most cases found in technical literature, trace lengths are described using log-normal, gamma or exponential distributions ( McMahon, 1974 ) ( Call et al., 1976 ) ( Baecher et al., 1977 ) ( Priest and Hudson, 1981 ) ( Kulatilake and Wu, 1984 ) ( Villaescusa and Brown, 1992 ) ( Zhang and Einstein, 2000 ). All the trace lengths of the F1, F2 and F3 systems mapped in the tunnel's TLS images were analyzed by statistical tests to find the distribution form that they best fit. (Figure 5) shows the distribution that best fits for each discontinuity system, with this being log-normal distribution in all cases. The foliation system is considered as persistent in the tunnel's scale therefore, its traces were not statistically analyzed. Particularly for the 10-m long tunnel section evaluated, the F3 system was not found. The geo-structural data considered for the numerical simulations are summarized in (Table 1).

Figure 4 Stereographic plot of the discontinuity orientation measured in TLS images: (a) pole concentrations, (b) representative planes without F3 system

Figure 5 Log-normal distributions fitted to F1, F2 and F3 trace length data

3.3 Intact rock and discontinuity parameters

Following the recommendations indicated by the International Society for Rock Mechanics ( Bieniawski and Bernede, 1979 ) ( Ulusay, 2015 ), uniaxial compression tests were made considering three different orientations of the foliation system. In addition, the measurement of the axial and radial deformations has been made using strain gauges installed on the test specimens. With this, the resistance to the uniaxial compression and the elastic parameters were obtained, i.e. the elasticity modulus and Poisson coefficient of the intact rock ( Ito, 2016 ). These parameters have been used in the numerical simulations.

(Table 2) presents a summary of the results obtained for the uniaxial compression tests. From the results, a substantial reduction in the resistance was seen, close to 50%, mainly in the test specimens which have altered minerals and cracks.

In addition, a series of different tests, such as direct and indirect traction, tilt test and sclerometer have been made to determine the geomechanical parameters of the discontinuity systems ( Barrios, 2014 ) ( Ito, 2016 ) ( Monticelli, 2014 ). For the three discontinuity systems (F1, F2 and Sn), the Barton- Bandis failure criteria has been considered (Bandis et al., 1981 ) ( Barton et al., 1985 ). (Table 3) presents a summary of the corresponding shear strength parameters for each discontinuity system.

The normal and shear stiffness of discontinuity systems were estimated from rock mass modulus, intact rock modulus and joint spacing. It is assumed that the deformability of a rock mass is due to the deformability of the intact rock and the deformability of the discontinuities in the rock mass, with K N = 3.5 GPa/m for F1 and F2, and K N = 2.8 GPa/m for Sn. In this paper, the shear stiffness of the discontinuities was estimated using the ratio K N /K S = 10.

A very important and influential aspect within this work is the degradation the different discontinuity planes have experienced as time has gone by. ( Monticelli, 2014 ) made the characterization of the alteration processes of the Monte Seco Tunnel's rock mass, concluding that the type of weathering is a moderate to strong intensity chemical type, presenting a strong structural control related to the presence of fractures and foliation. The foliation intensifies the alteration process of the rock matrix of the blocks formed around the tunnel, facilitating the percolation of water through the inter and trans mineral fissures formed in the altered planes. Both in the rock matrix and in the discontinuity planes, the different degrees of alteration have a micro-morphological nature (fissures and pores) related to expansion-contraction processes of the secondary mineralogy comprising Pyrite, Chlorite and Smectite aggregates, the latter confirmed in the X-ray diffractometry tests, where it was also seen that the fracture planes have alteration processes controlled by the inter-mineral fissures connected throughout the foliation, and that the trans-mineral fissures are quite expressive and parallel to the fracture planes occurring with and without Iron Chlorite Oxide fills.

Table 1 Geo-structural data used for two-dimensional DFNs generation

Table 2 Intact rock strength and elastic parameters

Table 3 Barton-Bandis joint strength parameters

4. FEM numerical modelling

Based on the geological and rock mechanics information described above, plane-strain semi- discontinuous elasto-plastic numerical models were completed using a commercial software.

4.1 Boundary conditions and in-situ stresses

Because the main objective of this work was to observe the rock mass stability behavior around the tunnel excavation, the edges of the finite elements model have been restricted to the horizontal and vertical directions. Triangular finite elements with three nodes were used in the analysis, considering a higher density finite elements mesh near the excavation see (Figure 6a). The tunnel shape is defined as a 6.5-m high and 6.0-m wide horseshoe see (Figure 6b). The in-situ vertical stress state was estimated considering the lithology column on the tunnel's roof, and with a horizontal/vertical stress ratio K0 = 1,35. The shallow tunnel is located at a depth ranging from 35-m to 45-m. A vertical stress of 1.0 MPa was applied to the model to simulate the gravitational loading of the overburden rock mass strata. This is based on an average overburden depth of 40-m and an overburden material density of 25 kN/m 3 .

Figure 6 (a) Geometry and boundary conditions of finite elements model, (b) dimensions of horseshoe tunnel section, and (c) random rock joint system generation ( Villalobos et al., 2017 )

The rock mass material model selected was the generalized Hoek-Brown (Hoek et al., 2002 ). It has been adapted in accordance with the methodology described in ( Diederichs, 2007 ), whereby peak and residual strength parameters are selected so that strain-softening behavior occurs close to the excavation perimeter whilst under increasing confinement (i.e., further from the excavation perimeter, strain-hardening occurs). Three clearly defined rock discontinuity systems (fracture planes: F1, F2 and foliation plane: Sn) were considered to follow the Barton- Bandis failure criteria (Bandis et al., 1981 ) ( Barton et al., 1985 ).

4.3 Generation of rock discontinuity systems

The numerical modelling has incorporated the geo-structural data to evaluate the formation and stability of rock blocks around the tunnel excavation. The rock mass was modelled as elastic-plastic material type intersected by geological discontinuities, in this case the network of rock discontinuity systems presented in (Table 1). Considering the mechanical properties of intact rock and discontinuities, one can predict major failures controlled by discontinuities in the semi-discontinuous model. It was considered as an intact rock material with good mechanical properties, as a result the discontinuities are generally much weaker mechanical properties than the intact rock blocks. Therefore, the semi-discontinuous numerical modelling provides a valuable analysis tool because the displacement of unstable blocks along rock discontinuity systems is allowed. For the semi-discontinuous numerical modeling, which explicitly considered the rock mass' discontinuity systems, three discontinuity mesh generation methodologies have been evaluated. These are described below.

4.3.1 Parallel deterministic model

The parallel deterministic model was developed for rock discontinuities that define a network of parallel discontinuities with a fixed spacing and orientation. In this case, deterministic refers to the fact that the spacing, length and persistence of the joints is assumed to be constant (i.e., known precisely with no statistical variation). However, the parallel deterministic model does allow randomness of the joint location. The orientation defined by the trace plane in the discontinuity network is simply the cross-sectional plane of the model. The spacing is the perpendicular distance between the parallel discontinuity planes. For the parallel deterministic model, the spacing is a constant value.

4.3.2 Parallel Statistical Model

The parallel statistical model allows defining a network of parallel discontinuities with defined statistical distributions, log-normal in this case, for the spacing, length and persistence of discontinuities. The orientation parameters for the parallel statistical discontinuity network model have the same definition as the parallel deterministic model. The spacing, trace length and persistence can be defined as a random variable by selecting a statistical distribution, and entering the mean, standard deviation and relative minimum and maximum values. For the parallel statistical model, if both the trace length and persistence are defined as random variables, then the length of each discontinuity segment, and the gap of intact material between adjacent discontinuity segments, will be variable, according to the statistical distributions defined.

The major feature of the Baecher model is the assumption of circular discontinuity shape. The following geometric parameters are required to generate the discrete discontinuity network: density of the discontinuities (number of discontinuities per unit area), the orientation distribution of these discontinuities, the size and shape of the discontinuities. Discontinuity centers are located uniformly in space, using a Poisson process and the discontinuities are generated as discs with a given radius and orientation. As a result of the fracture location, shape and size process of the model, discontinuities terminate in intact rock and intersect each other. Any combination of discontinuity size, location and orientation assumptions is possible ( Grenon et al., 2017 ).

5. Outcomes and Evaluation

An explicit network of rock discontinuity systems was generated within a limited area to reduce computation time while maintaining a high accuracy in the immediate vicinity of the tunnel excavation. The generation of rock discontinuity systems was used to provide more realistic representations of the jointing patterns based on the geo-structural data.

Below, the results of the two-dimensional analysis are presented, explicitly considering the geo-structural data. It is important to indicate that this sensitivity analysis has the purpose of evaluating, in a simple manner, the stability of the rock mass around the tunnel. From this data, it is possible to verify the failure patterns defined by the intersection of two or more discontinuity systems, and the influence of the geo-structural data on the numerical model.

5.1 Displacements and failure patterns

The variation of the displacement boundary and the extension of the shear zone on the tunnel's perimeter for the three methods used to generate the two-dimensional discontinuity networks, is shown in (Figure 7). In this work, the analysis focus considers the shear of the discontinuity systems, which induce the formation and instability of rock blocks around the excavation.

The zone disturbed around the tunnel is the region where the original state of in situ stresses has been affected due to the excavation works. This is the zone where normally the rock blocks have important displacements and the tangential stresse have the highest increase. In this way, the displacements and stresses are the factors which control the tunnel's stability. In the studied case, the stress field in situ induces the concentration of stresses mainly in the tunnel walls.

The results obtained with the deterministic parallel and statistical parallel methods (Figures 7a) and (Figure 7c) have total displacements of between 3 to 5 mm, giving a good representation of the displacement pattern and failure experimented locally on the tunnel's West wall, i.e., block sizes close to 1.35-m however, there was not a good fit with what was seen on the East wall. The results of the Baecher model, shown in (Figure 7e) and (Figure 7f), having maximum displacements of the tunnel area of between 3 to 4 mm, indicating greater shear zones with average lengths of close to 0.75-m, both on the East and West walls of the tunnel. This model provides results that are closer to what has been seen and measured in situ, as it assumes much more realistic spatial distribution of discontinuity planes, using the Poisson process. It is worth stating that the formation of unstable blocks in the tunnel's outline is very sensitive to the geometric parameters of the discontinuities (orientation, spacing and persistence), as well as the shear resistance parameters and their stiffness.

5.2 Overbreak measurements

The results obtained with the Baecher model, with a shear zone length of between 0.5 to 1.0-m, are similar to the onsite observations. For this scenario, block slippage is seen with maximum apexes that are equal to 1.0-m on the East wall right wall in all (Figure 7) and (Figure 8), which fits very well with the over excavation measurements made using TLS in the tunnel. These results are similar to that seen in the tunnel's overexcavation profiles, which were measured with TLS. The overexcavation measurements made, show that the type of failure that commonly occurred inside the tunnel, has been block falls with apexes below 1.5-m on the East wall, while failure mechanisms have also been seen on the West wall with apexes between 0.85 to 1.25-m.

Figure 7 Displacement contours and yielded zone for different rock discontinuity generation methods: (a) and (b) Parallel deterministic, (c) and (d) Parallel statistical, (e) and (f) Baecher model

Figure 8 Overbreak profiles measured using TLS data

In this study, statistical characterization of rock discontinuity systems and a two-dimensional finite element analysis commercial software was used to demonstrate and highlight the importance of explicit modelling rock discontinuity systems on the displacements and stability patterns around tunnels excavated in discontinuous rock masses. With the proposed methodology, using TLS data and numerical simulation through finite elements, the formation and instability of rock blocks around a tunnel excavated in discontinuity rock mass could be evaluated simply and quickly.

Due to the existence of rock discontinuity systems, a plastic zone is formed around the excavation. Some discontinuity plastic zones also appear far away from the tunnel but they have little influence on the stability of the excavation. The results from semi-discontinuous numerical models show an anisotropic behavior on the displacement patterns.

The terrestrial laser scanner technique is a practical and powerful tool for discontinuity mapping in tunnels because it overcomes difficulties associated with the traditional hand-made geological mapping.

The results of the deterministic parallel and statistical parallel models do not show a good fit with the over-excavation profiles of the tunnel's outline measured via TLS. This is because the two-dimensional generation of the DFNs systems using these models, does not consider the facture's intensity parameter.

The results of the Baecher model, where shear zones were obtained, both for the discontinuity systems and the rock mass of between 0.75 to 1.50-m around the tunnel, showed a good fit with the over excavation profiles of the tunnel's outline measured using TLS. This is because this model considers the fracture intensity parameter for the P21 area, which is derived from the fracture intensity parameter by volume P32. Therefore, it makes the systems generated from two-dimensional DFNs more realistic.

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