Contents

**Preface**

**1 Introduction**

**Part A: Fundamentals and Models**

**2 Structure of Rock**

2.1 Introduction

2.2 Rock Groups

2.3 Intact Rock

2.4 Faults

2.5 Folds

2.6 Discontinuities

2.7 Rock Mass

**3 Stress-Strain Behavior of Jointed Rock**

3.1 Introduction

3.2 Intact Rock

3.3 Discontinuities

3.4 Rock Mass

**4 Squeezing Rock**

4.1 Introduction

4.2 Phenomenon

4.3 Yielding Support

4.4 Convergence-Confinement Method

4.5 Example of a Tunnel in Squeezing Rock

**5 Rock Salt**

5.1 Introduction

5.2 Stress-Strain Behavior

5.3 Validation of Model

**6 Permeability and Seepage Flow**

6.1 Introduction

6.2 Porous Intact Rock

6.3 Discontinuities

6.4 Rock Mass

**7 Coupling of Stress-Strain Behavior and Seepage Flow**

7.1 Introduction

7.2 Permeability of a Discontinuity as a Function of Stress

7.3 Rock Mass

**8 Swelling Rock**

8.1 Introduction

8.2 Swelling Mechanisms

8.3 Water Uptake

8.4 Swelling

8.5 Coupled Model

8.6 Characteristic Parameters of the Gypsum Keuper

8.7 Gypsum Keuper in its Natural Condition

8.8 Calibration of Model

**9 Rock Mass In-Situ Stress**

9.1 Introduction

9.2 Stresses due to Gravity

9.3 Tectonic Stresses

9.4 Stresses due to Pre-loading and Subsequent Unloading

9.5 Residual Stresses

9.6 Effect of Rock Mass Inhomogeneity

**Part B: Analysis and Design Methods**

**10 Finite Element Method (FEM)**

10.1 Introduction

10.2 The Principle of FEM

10.3 Element Types

10.4 Computation Section

10.5 Stress-Strain Analyses

10.6 Seepage Flow Analyses

10.7 FEM Program Systems and Related Modules Provided by WBI

**11 Stability of Rock Wedges and Excavation Surfaces**

11.1 Introduction

11.2 Potential Failure Modes of Rock Wedges

11.3 Stability of Rock Wedges against Sliding

11.4 Stability of Rock Wedges against Rotation

11.5 Stability of Multiple Rock Blocks

11.6 Stability of Rock Columns and Layers against Buckling

**12 Design Methods**

12.1 Introduction

12.2 Design Based on Rock Mechanical Models

12.3 Design Methods Based on the Assessment of the Rock Mass Behavior

12.4 Design Based on Classification Systems

12.5 Flaws and Deficiencies of Classification Systems

12.6 Case History Road Tunnel Österfeld

12.7 Conclusions

**Part C: Exploration, Testing and Monitoring**

**13 Site Investigation**

13.1 Introduction

13.2 Evaluation of Documents

13.3 Rock Exposures

13.4 Test Pits

13.5 Boreholes

13.6 Exploration Adits and Shafts

13.7 Test Excavations

13.8 Exploration During Construction

13.9 Mapping of Rock Surfaces

13.10 Evaluation of a Structural Model

**14 Laboratory Tests**

14.1 Introduction

14.2 Petrographic Investigations

14.3 Water Content, Density, Porosity and Related Properties

14.4 Deformability and Strength of Intact Rock

14.5 Shear Strength of Discontinuities

14.6 Swelling

14.7 Slake Durability and Disintegration Resistance

14.8 Abrasiveness

**15 Field Tests**

15.1 Introduction

15.2 Borehole Expansion Tests

15.3 Plate Loading Tests

15.4 Flat Jack Tests

15.5 Triaxial Tests

15.6 Gallery Tests

15.7 Direct Shear Tests

15.8 Permeability Tests

**16 Stress Measurements**

16.1 Introduction

16.2 Stress Relief

16.3 Stiff Inclusion

16.4 Compensation Method

16.5 Hydraulic Methods

16.6 Methods of Large-Scale In-situ Stress Determination

16.7 Case Studies

16.8 The World Stress Map Project – Results of Stress Measurements

**17 Monitoring**

17.1 Introduction

17.2 Geodetic Measurements

17.3 Monitoring of Vertical Displacements on the Ground Surface

17.4 Monitoring of Rock Displacements along Boreholes

17.5 Monitoring of Relative Displacements between Rock Surfaces

17.6 Pressure Monitoring

17.7 Anchor Force Measurements

17.8 Monitoring of Water Level and Water Pressure

17.9 Automatic Data Acquisition

17.10 Examples

**18 Evaluation of Rock Mechanical Parameters**

18.1 General Procedure

18.2 Examples

**19 Examples of Testing and Monitoring Programs**

19.1 Introduction

19.2 Urban Railway Stuttgart, Hasenberg Tunnel, Construction Lot 15, Exploration Shaft and Adits

19.3 Urban Railway Stuttgart, Construction Lot 11, Exploration Shaft and Adit

**Part D: Applications and Case Histories**

**20 NATM Tunneling**

20.1 Introduction

20.2 Fundamentals of the NATM

20.3 Tunneling under Stuttgart Airport Runway

**21 TBM Tunneling**

21.1 Fundamentals

21.2 Stability of the Temporary Face and Shield Design, Example

21.3 Shield Design, Example

21.4 Stability and Permeability Changes of the Rock Mass in the Machine Area, Example

21.5 Design of the Segmental Lining, Examples

**22 Powerhouse Cavern Estangento-Sallente**

22.1 Project

22.2 Site Investigation and Testing Prior to Construction

22.3 Location of the Powerhouse Cavern

22.4 Rock Mechanical Model

22.5 Stability Analyses

22.6 Monitoring Program

22.7 Mapping and Monitoring Results during Excavation of Vault

22.8 Support of the Cavern Walls

22.9 Mapping and Monitoring Results during Excavation of Benches

22.10 Back Analyses

22.11 Conclusions

**23 Tunneling in Swelling Rock**

23.1 Introduction

23.2 Influence of the Elevation of the Anhydrite Surface on Swelling Pressure and Heaving

23.3 Urban Railway Tunnel in Stuttgart, Construction Lot 12

**24 Rehabilitation of Urft Dam**

24.1 Introduction

24.2 Project

24.3 Rehabilitation Concept

24.4 Site Investigation and Testing

24.5 Rehabilitation Works

24.6 Monitoring

24.7 Back Analyses

24.8 Stability Proof

24.9 Conclusions

**25 Stabilization of a Rock Mass Slide**

25.1 Original Design

25.2 Revised Design

25.3 Back Analysis of Monitoring Results

25.4 Installation of Additional Tendons

25.5 Long-Term Monitoring

**References**

**Index**

Walter Wittke

WBI GmbH

Im Technologiepark 3

69469 Weinheim

Germany

**Library of Congress Card No.:**

applied for

**British Library Cataloguing-in-Publication Data**

A catalogue record for this book is available from the British Library.

**Bibliographic information published by**

**the Deutsche Nationalbibliothek**

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;

detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.

© 2014 Wilhelm Ernst & Sohn, Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Rotherstraße 21,

10245 Berlin, Germany

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by

photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission

from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be

considered unprotected by law.

**Print ISBN:** 978-3-433-03079-0

**ePDF ISBN:** 978-3-433-60431-1

**ePub ISBN:** 978-3-433-60429-8

**eMobi ISBN:** 978-3-433-60428-1

**oBook ISBN:** 978-3-433-60430-4

Preface

In 1984, my book on Rock Mechanics was published. Some years later, it was translated into English and Chinese and to a larger part also into Russian.

Since then, together with my co-workers, I have extended our anisotropic jointed rock model (AJRM) and the corresponding analysis methods to a wider spectrum of rock types. Furthermore, our design approach has been applied to many projects in tunneling, dam and slope design. Monitoring and back analyses have helped us to gain a far better understanding of rock mass behavior and to assess the corresponding properties.

Therefore, I decided to publish a new book with the title

Rock Mechanics

based on an

Anisotropic Jointed Rock Model

AJRM

I hope that many colleagues will follow this design method in order to achieve safer and more economic solutions.

In Part A, the fundamentals of our modeling concept are outlined and in part B our means of analyzing structures in and on jointed rock are presented. Part C is devoted to exploration, testing and monitoring and in Part D applications and case studies are presented.

I would like to thank Dr.-Ing. Dipl.-Phys. Johannes Kiehl for his valuable contribution to this book. He has carefully studied and summarized the recent literature related to a number of chapters of parts A and C, and he has compiled the material gathered in WBI Worldwide Engineering over many years.

Furthermore, I would like to thank Dipl.-Ing. Christa Mühlen-Senz for contributing the excellent figures to this book and Mrs. Ute Kratz for compiling the whole manuscript.

My thanks also go to my former and present colleagues in the WBI Company for their contributions to the development of models and computer programs and to the design of the related engineering projects.

In this context, I would specially mention Dr.-Ing. Bernd Pierau, Dr.-Ing. Claus Erichsen, Dr.-Ing. Bettina Wittke-Schmitt, Dr.-Ing. Patricia Wittke-Gattermann, Dr.-Ing. Martin Wittke, as well as Dipl.-Ing. Dieter Schmitt and Dipl.-Ing. Meinolf Tegelkamp.

Finally, I want to thank my wife Lilian for the many years of personal support and understanding.

*Walter Wittke*

This book focuses on the fundamentals of rock mechanics as a basis for the safe and economical design and construction of tunnels, dam foundations and slopes in jointed and anisotropic rock. It is divided into four main parts:

A Fundamentals and models

B Analysis and design methods

C Exploration, testing and monitoring

D Applications and case histories.

The rock mechanical models presented account for the influence of discontinuities on the stress-strain behavior and the permeability of jointed rock masses. They are implemented in three-dimensional finite element analysis methods, which enable a realistic simulation of the load-carrying behavior of structures in rock. Also, advanced methods to describe squeezing, swelling and creeping rock masses are implemented. The corresponding computer programs developed and applied by the author and his colleagues will soon be available for sale.

In comparison to the rock mechanics book first published by the author in German in 1984 (Wittke 1984), this book has been completely revised and is now based on more than 40 years of experience in the design and construction of tunnels, dams and slopes in many kinds of rock.

The book is aimed at students taking advanced courses in geotechnics and rock mechanics, at postgraduate students (PhD) and at researchers as an introduction, as well as at civil engineers working as consultants, planners and designers, or in the construction industry and within public authorities. The required academic background for readers is a graduate degree (BEng or BSc) in civil engineering. However, inter-disciplinary working professionals coming from geology, mining and natural sciences can also benefit from this book.

The practical examples and case histories presented in this book should facilitate the consideration of rock mechanics in the reader’s own work.

Fundamentals and Models

Rock mass is a compound material including the intact rock and discontinuities such as joints, bedding-parallel discontinuities, faults and other planes of weakness forming a more or less complex structure. The objective of this chapter is to show how different types of rock masses can be characterized by means of structural models. These structural models form the basis for the description of the mechanical and hydraulic behavior of rock masses which will be considered in the subsequent chapters.

It is convenient to classify the rocks in the earth’s crust into three different groups according to their geological origin: igneous, sedimentary and metamorphic rocks (Fig. 2.1).

Igneous rocks, also referred to as “magmatic rocks”, are formed from the solidification of the magma. They are divided into plutonic, dyke and volcanic rocks, depending on the depth and rate of their cooling (Fig. 2.1 and 2.2). Volcanic and dyke rocks are formed by the cooling of extruded magma either in voids and cracks of other rocks in various depths (dyke rocks) or at the earth’s surface (volcanic rocks). Plutonic rocks are formed at greater depth. It is also convenient to subdivide igneous rocks into acid, intermediate and basic rocks depending on the amount of silica in their composition (Farmer 1968).

Sedimentary rocks are largely formed by the sedimentation and subsequent cementing of mechanically or chemically degraded rocks from all three main groups. They are divided into three subgroups according to their formation: clastic, chemical and organic sedimentary rocks. Clastic sediments may be further subdivided into pyroclastic sediments of volcanic origin and detrital sediments, and they are classified according to the grain size of their components as in the usual classification of granular soil. Important subgroups of the detrital sediments are arenaceous rocks such as sandstone, and argillaceous rocks such as mudstone and shale. Chemical sediments are deposits from water-based solutions in the sea or inland waters. Important representatives are salt rocks and calcareous rocks such as dolomite and limestone (Fig. 2.2).

Metamorphic rocks may be either igneous or sedimentary rocks which have been altered physically by the application of intense heat and/or pressure at some time in their geological history. Various types of rock metamorphosis caused by different processes and conditions can be differentiated (Murawski 1983). Examples of metamorphic rocks are listed in Fig. 2.2.

Between the main groups, various transitions exist so that naturally occurring rocks cannot always clearly be classified into one of the three groups.

The earth’s crust is estimated to be composed of 95% of igneous rocks, 5% of sedimentary rocks and an insignificant proportion of metamorphic rocks. The sedimentary rocks however make up approximately 75% of the earth’s surface and are thus of considerable importance in rock engineering (Leitmeier 1950, Wagenbreth 1970).

The main rock-forming minerals of igneous rocks are feldspar, quartz, amphibole, pyroxene and mica. The major mineral constituents of sedimentary rocks are quartz, calcite, clay minerals and locally also salt minerals, gypsum and anhydrite. The metamorphic rocks contain further minerals such as chlorite, garnet and epidote. A total of 90% of the earth’s crust is formed by the minerals feldspar, quartz, amphibole, pyroxene and mica. Thus, only a few minerals form the principal components of rock.

In the following the aggregate of mineral particles, of which a rock is basically composed, will be denoted as “intact rock”. The minerals that represent the basic rock structure normally take on the form of crystals but may exist also as amorphous molecule aggregates such as silica. The fine-grained mineral fraction in which larger grains or crystals are embedded is referred to as “rock matrix”. The mineral particles are cemented together by the rock matrix or by mechanical bonding at contact interfaces between the grains forming the “grain skeleton”.

When describing the structure of intact rock, the term “texture” is used for the description of appearance, shape, size and size distribution of the individual grains and aggregates of mineral particles. The grain size, as in soil mechanics, is described by the terms fine-grained, medium-grained and coarse-grained. With regard to rock mechanical properties such as deformability, strength and permeability these attributes, however, are less important.

The spatial orientation of grains in the grain skeleton is described by the “grain structure”, also referred to as “grain fabric”. This term also covers imperfections in the structure such as pores, cracks, inclusions and grain boundaries.

Sedimentary and igneous rocks frequently exhibit a random grain structure. Such structures are characterized by a statistically uniform distribution of particles. The homogeneous sandstone shown in Fig. 2.3 represents an example for such an intact rock. A further example of an intact rock with random grain structure is the rock salt, illustrated in Fig. 2.4.

The most important structural feature of sedimentary rocks is their bedding as a consequence of the sedimentation process during rock formation. When for example platy particles are deposited with their longitudinal axes predominantly horizontal then a planar structure arises.

The structure of metamorphic rocks is frequently based on the primary grain structure of the original rock. Depending on the degree of metamorphosis, the primary structure such as the bedding may be partially or completely maintained while at the same time new secondary structures may come into being. It also occurs, however, that the primary structure will be completely replaced by a secondary structure. Such a secondary structure can be a “schistosity” or “foliation”, which also reveals a planar grain structure. Examples of intact rocks with a planar grain structure are shown in Figs. 2.5 to 2.7.

If two planar structures with different orientations are present, the lines of intersection of both structures may form a linear grain structure, also referred to as “lineation”.

From the rock mechanics point of view intact rocks can be classified with respect to their grain structure. Most intact rocks can be related to a random and a planar grain structure as represented by the models shown in Fig. 2.8. Intact rocks with a random grain structure are most likely characterized by an isotropic behavior with regard to deformability and strength. In contrast, intact rocks with a planar grain structure can exhibit a marked anisotropy with respect to deformability and strength (Section 3.2).

With regard to rock engineering all intact rocks can be regarded as homogeneous because the grains are very small compared to the dimension of an engineering structure.

Faults are discontinuities on which relative shear displacements of the neighboring rock mass take place or have taken place (Metz 1967). Also shear zones, which are bands of material in which local shear failure of the rock has previously taken place due to stress relief in an otherwise unaltered rock (Brady & Brown 2006), may be categorized in the broadest sense as faults.

Most faults can be classified into three types based on the sense of relative movement or slip as illustrated in Fig. 2.9. A fault where the relative movement is approximately vertical is known as a “dip-slip fault”. Starting from the initial undisturbed state of the rock mass, downward and upward relative movements lead to normal and reverse dip-slip faults, respectively, or simply normal and reverse faults (Fig. 2.9, top). A downward displaced block between two normal faults dipping towards each other is called a “graben”. An upward displaced block between two normal faults dipping away from each other is referred to as a “horst”. Where the slip is approximately horizontal, the fault is known as a “strike-slip” fault (Fig. 2.9, lower left). An “oblique-slip fault” has non-zero vertical and horizontal slip components (Fig. 2.9, lower right).

Faults may be large, pervasive features of several meters in thickness or may be of relatively limited local extent of millimeters in thickness. Displacements up to several kilometers may arise.

The formation of faults is frequently associated with high stresses of tectonic origin (Section 9.3). As a consequence, the intact rock may be decomposed into small pieces. Thick fillings of pulverized rock material such as mylonites are therefore often to be found within a fault.

Folding of rock mass is a consequence of regional deformation due to tectonic forces. A systematic overview of folds, including a genetic interpretation of their origin is given by Ashgirei (1963).

As an example, in Fig. 2.10 a regularly folded rock mass with two fold axes is shown (Wittke 1990). A fold with downward diverging limbs is known as an “anticline” and a fold whose limbs diverge upward is denoted as “syncline” (Murawski 1983).

The shear and tensile stresses initiated in the rock mass during folding, in most cases lead to fractures, i.e. to the forming of joints. In the example represented in Fig. 2.10, these are denoted with respect to the position of the fold axes as diagonal, transversal and longitudinal joints and are oriented perpendicular to the folded rock layers. Thus, homogeneous areas with closely spaced parallel joints appear. The joints are frequently interrupted by discontinuities that are parallel to the bedding or the schistosity (Fig. 2.10). However, in highly ductile rocks such as rock salt, folding widely takes place without fracturing and jointing.

Folds may be large structures (Fig. 2.11) or may be of a smaller local scale (Fig. 2.12). They appear predominantly in metamorphic and sedimentary rocks.

The term “discontinuity” in rock mechanics is used as a collective term for all planes of weakness along which the coherence of intact rock is interrupted. In recent literature the term discontinuity is often replaced by the term “fracture”.

A classification of discontinuities can be based on the magnitude of shear displacement that the surfaces of the discontinuity have suffered. Discontinuities are called “joints” if the shear displacement is zero or too small to be visible. Faults are discontinuities on which larger shear displacements have taken place. Another classification of discontinuities with regard to their extent was introduced by Müller (1963). Many further attempts to classify discontinuities, particularly joints, according to certain attributes can be found in literature.

The terms “tension joint” or “extension joint” and “shear joint” are used to qualify the origin of joints (Stini 1929). Joints may be formed by different processes such as the cracking of initially latent bedding planes of already consolidated sediments due to shrinkage during drying (tension joints) and by tectonic processes (tension or shear joints). The term “contraction joint” is used for joints that are a result of tensile stresses initiated in igneous rocks by shrinkage due to the cooling of magma. A “master joint”, also known as a “main joint” or “major joint”, is a persistent joint plane of great extent, generally constituting the dominant jointing of an area.

The walls of a discontinuity are frequently slickensided and may be coated with minerals of low shear resistance, such as graphite and chlorite. Slickensides consist of thin films of finely pulverized rock and are smooth in appearance (Fig. 2.13). The surfaces therefore exhibit very low shear strength. The direction of relative displacement is often visible from the slickenside’s striae.

Figure 2.14 shows a granite in which the jointing in preferred directions is clearly visible.

The sandstone represented in Fig. 2.15 exhibits horizontal, persistent and closed bedding-parallel discontinuities, as well as vertical joints that frequently terminate at the bedding-parallel discontinuities. Locally, open joints appear.

Fig. 2.16 shows a closely bedded claystone with vertical joints. After drying and subsequent contact with water, such a rock may become a mud. A rock mass with such a behavior is called “slaking”.

A sedimentary rock mass often consists of an alternating sequence of different intact rocks. In the example illustrated in Fig. 2.17, between sandstone and siltstone layers bedding-parallel shear zones appear.

The clay slate represented in Fig. 2.18 is an example of a rock mass with an orthogonal system of vertical discontinuities (D1 and D2) and horizontal schistosity-parallel discontinuities (Sch), which in this particular case have the same orientation as the bedding.

Fig. 2.19 illustrates a tuff, as an example for a pyroclastic sediment. Such a rock mass often exhibits practically no discontinuities. The same is true for the rock salt represented in Fig. 2.20 as an example for a chemical sediment.

The water solubility of chemical sediments can lead to large openings in the ground. Figure 2.21 shows a schematic section through the White Jurassic formation at the Swabian Alb. In the banked limestone, which has been subjected to the Rhenanian karst formation, various karst structures appear, such as karstified master joints (Fig. 2.22) and large karst channels (Fig. 2.23). In the overlying massive limestones of the Danubian karst formation major karst structures such as horizontal and vertical caves, holes and larger cavities can be found (Fig. 2.21).

Also in rock salt water solubility can lead to major karst structures known as “saliniferous karst”. The existence of such structures may be indicated by dolines and sinkholes which can be observed, for example, around the Dead Sea (Fig. 2.24).

Discontinuities usually occur as sets or families with more or less parallel orientation. The whole assemblage of discontinuities present in a rock mass is called a “discontinuity system”. The individual discontinuities of a set are characterized by parameters describing orientation, appearance and persistence.

The three-dimensional orientation of a discontinuity is well-defined by two angles (Fig. 2.25). The so-called “angle of strike” (strike angle) α is measured from the north in a clockwise direction until the intersection of the discontinuity and the horizontal (contour line). The angle β between the horizontal and the line of dip of the discontinuity is referred to as the “angle of dip” (Fig. 2.25). According to this definition, the angular range of α and β is as follows:

(2.1)

(2.2)

Alternatively, instead of α the “dip direction” α_{d} can be used, which describes the angle between north and the projection of the line of dip on the horizontal being defined (Fig. 2.25) as

(2.3)

With the aid of the equal-area projection of the lower hemisphere, the orientation of discontinuities as subsequently outlined can be represented in a two-dimensional diagram. This method was first developed as a tool for use in structural geology and then extended to rock engineering applications. A detailed description of this method is given in Müller (1963), Adler et al. (1969), Hoek & Bray (1977), Goodman (1980), Hoek & Brown (1980b), Priest (1985), Wittke (1990) and in various geological and rock mechanical textbooks.

The lower hemisphere of a unit sphere is denoted as the “lower reference hemisphere”. The representation of the discontinuity’s orientation into the lower reference hemisphere is shown in Fig. 2.26. A discontinuity forming a tangent plane on the lower reference hemisphere can be characterized by its contact point referred to as a “pole”. This is illustrated in Fig. 2.26 (upper) by means of a vertical discontinuity (A), a horizontal discontinuity (B) and an oblique discontinuity (C). Thus, the orientation of a discontinuity is well-defined by the location of its pole. The intersection of a discontinuity running through the center of the reference hemisphere and the surface of the lower reference hemisphere is a great circle, and the normal of the discontinuity intersects the surface of the lower reference hemisphere at its pole (Fig. 2.26, lower).

To obtain a two-dimensional representation of poles the stereographic projection of the lower reference hemisphere into a horizontal plane is carried out in a way that the projection plane has the same area as the surface of the lower reference hemisphere. The construction of this so-called “equal-area projection”, also referred to as the “Schmidt projection” or “Lambert projection”, is illustrated in Fig. 2.27. In this projection, the great circles are mapped as straight lines and the circles of latitude, also referred to as “small circles”, are mapped as circles onto a circular plane. As a result, a polar diagram known as a “polar equal-area net” is obtained.

Fig. 2.28 shows the lower reference hemisphere that is, in another way, subdivided into great and small circles. The connecting line of the center of the reference hemisphere to a point of a vertical small circle encloses the angle η with the horizontal. The great circles are inclined by an angle β to the horizontal. An equal-area projection of the lower reference hemisphere subdivided in this way, analogous to that of Fig. 2.27, leads to the so-called “equatorial equal-area net” illustrated in Fig. 2.28. The equatorial equal-area net enables us to map a great circle of the lower reference hemisphere representing a discontinuity onto a polar equal-area net. As an example, Fig. 2.29 shows the projection of the great circle and the pole P of a discontinuity with a strike angle of α = 115° and a dip angle of β = 60°.

The coordinates α_{P} and β_{P} of a discontinuity’s pole are related to its angles α_{d} and β as follows (Fig. 2.29):

(2.4)

(2.5)

The measured orientation of discontinuity sets in homogeneous rock units are subject to scatter and must therefore be evaluated statistically. For this purpose, the angles of strike and dip of the discontinuities measured during mapping are entered in a polar equal-area net leading to a pole plot (Fig. 2.30, upper left). According to the so-called “Schmidt contouring method”, subsequently the poles that are located within 1% of the area of the polar equal area net are counted (Fig. 2.30, upper right) to determine the areal density of the poles or discontinuities, respectively, denoted as the “pole density”. The pole density D is defined as the counted number of discontinuities n divided by the total number of discontinuities m:

(2.6)

Counting of poles can be carried out by means of computer software or simply by using a square grid placed over the net with side lengths of a tenth of the polar equal-area net’s diameter. With the aid of a counting circle, also having a diameter of a tenth of the net, the poles and discontinuities, respectively, can be counted by centering the circle over each node and on each field of the grid. To enable continuous counting over the boundary of the net a counting bar can be used. As a result, a plot of pole densities is obtained (Fig. 2.30, center left). On this basis the pole densities are contoured selecting several pole density intervals (Fig. 2.30, center right). Thus, the areas of the most frequent orientations can be identified and the discontinuities can be grouped into discontinuity sets by means of a so-called “Schmidt contour diagram” (Fig. 2.30, lower).

When grouping the discontinuities into sets it is important that only discontinuities of same appearance and rock mechanical relevance are accounted for.

The orientation limits for each set in most cases are identified by eye. This subjective approach has the advantage of allowing a rating of the particular site with no further mathematical treatment. Objective probabilistic methods and algorithms for the automatic grouping of discontinuities into sets are reported in literature. For further treatment of this subject see Section 13.9.2.

Stereographic projection is also used for the determination and representation of angles between discontinuities and between line elements or vectors such as lines of dip, dip vectors and intersection lines of discontinuities (Goodman 1980). Another possible application of the stereographic projection technique is the representation of measured principal normal stresses (Section 16.2.1, Fig. 16.6).

The appearance of discontinuities is significantly governed by the surface characteristics, which are particularly important with regard to the estimation of shear strength. Surface profiles of discontinuities are qualitatively described as “stepped”, “undulating” and “planar” and, at a smaller scale, classified into “rough”, “smooth” and “slickensided” (ISRM 1978e). At the large scale stepped and undulating profiles are often designated as “un even”. Discontinuities are referred to as slickensided when their surfaces are very smooth in at least one direction because of a relative shear displacement (Fig. 2.31).

Furthermore, discontinuities can be open or closed and may contain coatings and veins of minerals such as quartz or calcite or may have fillings (Fig. 2.31). Aperture is an important parameter of open discontinuities since it has an essential influence on the permeability of a rock mass. Apertures, however, are difficult to estimate and to determine (Chapters 6 and 13).

The spacing s defines the smallest distance between neighboring discontinuities of a set. Where the direction of measurement does not run perpendicular to discontinuities of the corresponding set the spacing of two discontinuities may be calculated as follows (Wittke 1990):

(2.7)

in which d is the measured distance of both discontinuities, α^{*} is the angle between the strike of the discontinuities and the direction of measurement and β is the dip angle of the discontinuities (Fig. 2.32, left).

The reciprocal value of the discontinuity spacing s was introduced by Müller (1950) as “discontinuity density” or “intensity number” k of a discontinuity set.

Discontinuities are often interrupted by rock bridges, or terminate against other discontinuities (cf. Fig. 2.15). Such discontinuities, in literature, are also referred to as “impersistent” or “intermittent” discontinuities, in the following are denoted as “non-persistent” discontinuities.

A measure of the degree to which discontinuities persist is the so-called “planar degree of separation”. According to Pacher (1959), the planar degree of separation κ_{p} is defined as the sum of the separated rock areas A_{i} divided by a reference area A (Fig. 2.32, right):

(2.8)

κ_{p} is an important parameter for the estimation of shear strength of non-persistent discontinuities (Section 3.3.3).

In a similar way Müller (1974) defined the “linear degree of separation” κ_{ℓ} as a measure of persistence of discontinuities on a two-dimensional rock exposure:

(2.9)

where ℓ is a reference length along the trace of a discontinuity and ℓ_{i} are the separated sections of l, also denoted as “trace lengths”.

With the aid of κ_{p} and s or κ_{p} and k, respectively, the spatial degree of separation κ_{s} as a measure of persistence of a discontinuity set relating to a reference volume V = A ⋅ s may be described as follows (Pacher 1959):

(2.10)

Models of the grain structure of the intact rock and the discontinuity system must be superimposed to obtain a structural model of a rock mass. Fig. 2.33 shows examples of structural models as a result of such a superposition. These examples show that, to a certain extent, structural models allow us to estimate whether the deformability of a rock mass is isotropic or anisotropic (Chapter 3).

Fig. 2.34 shows a discontinuity system with five sets of a gneiss that is encountered in the section Bodio of the Gotthard basetunnel which was driven using a TBM. The joints of the sets KI and KII, and KIII and KV are steeply dipping and strike parallel to and perpendicularly to the tunnel axis, respectively. The joints of the set KIV are also steeply dipping and strike diagonally to the tunnel axis. Because the latter are mechanically less important they are not represented in Fig. 2.34. The discontinuities of the set KS, which are parallel to the schistosity, are more or less horizontal. Fig. 2.35 (upper) shows an idealized structural model of this rock mass in which sets KI and KII as well as KIII and KV are each combined into one vertical set. These sets can form rock wedges which may fall into the tunnel (Fig. 2.35, lower).

As another example, Fig. 2.36 shows the structural model of the clay slate, which was illustrated in Fig. 2.18 and encountered in the construction pit for the powerhouse at the dam site of the Selingue project in Mali. This rock mass is separated by the discontinuity sets Sch, D1 and D2. The intact rock has a planar grain structure caused by the schistosity. The schistosity-parallel discontinuities, Sch, have a mean orientation of α_{Sch} = 160° and β_{Sch} = 90° and exhibit rusty coatings and often also fillings of fine-grained soil. They are closely spaced with a mean spacing of less than 1 m. The average trace length exceeds 10 m. The joints of D1 have a mean orientation of α_{D1} = 70° and β_{D1} = 90° as well as an average spacing of 2.5 m. These discontinuities also often have rusty coatings, but their extent in general is significantly less than that of the schistosity-parallel discontinuities. The joints of set D2 are more or less horizontal. They frequently exhibit fillings of cohesive soil. Their average spacing and extent amount to 1 m and 4 m, respectively. In addition, various joints with the same orientation but extending more than 20 m were encountered. Since some of them are filled with cohesive soil they exhibit low shear strength and therefore lead to a large reduction of the powerhouse block’s safety against sliding. Because of their large extent, in the order of magnitude of the size of the powerhouse blocks, they need to be considered as individual elements (Wittke 1990). The structural model and the most important rock mechanical parameters, also specified in Fig. 2.36, lead to the corresponding rock mechanical model.

Fig. 2.37 shows the structural model of another clay slate encountered at the dam site for a planned drinking water reservoir near Wiesbaden in Germany (Wittke & Schetelig 1978, Wittke 1990). As in the case of the other clay slate, the intact rock has a planar grain structure caused by the schistosity, leading to a pronounced anisotropic deformability and strength. As a result of exploration the schistosity-parallel discontinuities Sch and the joint sets D1 and D2 strike perpendicular and parallel to the valley and form an orthogonal discontinuity system. The mean orientations as well as the most important rock mechanical parameters of these sets leading to the rock mechanical model are also specified in Fig. 2.37.

The majority of rock masses, within a certain homogeneous area, can be described as a solid material (intact rock) separated by one or more sets of approximately planar and parallel discontinuities. The intact rock may have a random or planar grain structure. Major structural features such as master joints, faults and shear zones generally occur as individual elements (Wittke 1990). However, in some cases rock masses without discontinuities exist such as the tuff and the rock salt shown in Figs. 2.19 and 2.20.

For both deep and shallow underground openings the rock mass normally represents the main load-carrying structure. Consequently, the stress-strain behavior of the rock mass is of great importance for the stability and thus the design and construction of tunnels, caverns and other underground openings in rock. Similarly when concentrated loads are applied to a rock mass, as in the case of concrete dam foundations and abutments, the rock represents a critical part of the overall structure. When slopes are constructed, the rock mass, in combination with potential retaining structures, also has the task of carrying loads due to self-weight and other impacts.

An investigation of the stability of rock engineering structures therefore requires an adequate description of the stress-strain behavior of the rock mass, taking into account the influence of the structure and particularly the influence of discontinuities.

The recommended model for the stress-strain behavior, which is subsequently formulated, is based on structural models as described in Section 2.7.3. It was developed in the 1970s and, since 1980, successfully applied by WBI, in many cases in combination with numerical analysis methods, to solve practical problems in rock engineering (Wittke 1990, Wittke 2000b, Wittke et al. 2002, Wittke et al. 2006).

The elastic behavior of intact rock with random grain structure can be considered as isotropic. Two elastic constants, Young’s modulus E and Poisson’s ratio ν, defined by means of an applied uniaxial normal stress σ_{z} and normal strains ε_{x} and ε_{z} perpendicular and parallel to σ_{z}, respectively, are then sufficient to describe the linear elastic stress-strain behavior (Fig. 3.1). The shear modulus G results from E and ν as a dependent elastic constant:

(3.1)

In an arbitrarily oriented Cartesian coordinate system (x,y,z) normal stresses σ and normal strains ε as well as shear stresses τ and shear strains γ are related to each other by Hooke’s law:

(3.2)

Equation (3.2) can also be expressed in a more compact matrix form:

(3.3)

In (3.3)

and

are the stress and strain vectors in which σ_{x}, σ_{y}, σ_{z}, τ_{xy}, τ_{yz}, τ_{zx} and are the components of the symmetric stress and strain tensors, respectively, and

is called the “elasticity matrix”.

As an example, Fig. 3.2 shows the axial and lateral strains ε_{x} and ε_{z}, respectively, of a granite specimen during loading and unloading, measured under uniaxial normal stress σ_{z} (Walsh 1965c). The stress-strain curves are both more or less linear and reversible up to high stress levels. Deviations from linearity at low stress levels can be attributed to the closing of cracks that are pre-existing in the specimen. The sliding between contacting crack surfaces at the beginning of unloading creates a slight hysteresis loop. From a practical point of view, however, these deviations from a linear elastic stress-strain behavior in most rocks can be neglected.

For intact rocks with planar grain structure, the assumption of isotropic elastic behavior usually represents an inadmissible simplification. From experience, intact rocks with planar grain structure such as schist, slate and some argillaceous rocks often exhibit a significantly lower Young’s modulus perpendicular to the structure planes than parallel to them (Figs. 2.5 – 2.7 and 2.16 – 2.18). The elastic behavior of such rocks is therefore normally anisotropic and can be described by transverse isotropy. Such an elastic behavior is specified by five independent elastic constants. Two Young’s moduli E_{1} and E_{2} characterize the deformability parallel and perpendicular to the structure planes. Parallel to a structure plane the deformability is assumed to be isotropic. A structure plane is therefore called an “isotropic plane”. The shear modulus G_{2} describes the deformability for a shear loading parallel to the isotropic plane. Furthermore, two Poisson’s ratios ν_{1} and ν_{2} are needed.

In Fig. 3.3 the elastic constants of a transversely isotropic rock are defined by means of stresses and strains applied to a cube-shaped specimen using a Cartesian coordinate system (x',y',z') that is related to the isotropic plane. The z' axis coincides with the direction perpendicular to the isotropic plane and the x' and y' axes lie in the isotropic plane.

Figure 3.3 defines, in addition to E_{1}, E_{2}, G_{2}, ν_{1} and ν_{2} the elastic constants G_{1} and ν_{3}, which are dependent on E_{1}, E_{2}, ν_{1} and ν_{2}.

In the coordinate system (x',y',z') the stress-strain relationship for transverse isotropy can be expressed as:

(3.4)

with

where n = E_{1}/E_{2} and m = 1 - ν_{1} - 2⋅n⋅ν_{2}^{2}.

The inverse relation of Equation (3.4) is

(3.5)

in which

is denoted as the “compliance matrix”.

For stability analyses, a global coordinate system (x,y,z) is needed that is related to the engineering structure’s overall geometry. This usually does not coincide with the coordinate system (x',y',z') introduced above. Linking of the two coordinate systems can be achieved by using two angles α and β describing the direction of the contour line and the inclination of the line of dip of the isotropic plane in relation to the global coordinate system (Fig. 3.4). These angles were introduced in Section 2.7.2 (Fig. 2.25).

Stress and strain vectors {σ'} and {ε'} can be computed from {σ} and {ε} with the aid of transformation matrices [T] and [T^{*}] respectively:

(3.6)

(3.7)

The dependency of [T] and [T^{*}] from the angles α and β is given by the following equations:

(3.8)

(3.9)

in which

ℓ_{1} = sinα, |
m_{1} = cosα, |
n_{1} = 0, |

ℓ_{2} = cosβ cosα, |
m_{2} = -cosβ sinα, |
n_{2} = -sinβ, |

ℓ_{3} = -sinβ cosα, |
m_{3} = sinβ sinα, |
n_{3} = -cosβ. |

Insertion of (3.6) and (3.7) into (3.4) and multiplication by [T]^{-1} yields

The calculation of [T]^{-1} leads to the relationship

(3.10)

Thus, (3.4) can be replaced by

(3.11)

with

Equation (3.11) describes the relation between stresses and strains for a transversely isotropic rock in the global coordinate system.

The inverse relation of (3.11)

(3.12)

is obtained by inverting the elasticity matrix [D], which can be accomplished in consideration of (3.10) as follows:

From the thermodynamic constraint of positive definite elastic strain energy the theory of elasticity provides lower and upper bounds of the elastic constants for isotropic and transversely isotropic bodies. The following relations are valid for an isotropic elastic body (Love 1927):

(3.13)

and

(3.14)

The corresponding restrictions for a transversely isotropic elastic body are (Pickering 1970):

(3.15)

and

(3.16)

Relations (3.14) and (3.16) imply that, in principle, negative values of Poisson’s ratios may also be possible. However, with a few exceptions regarding highly anisotropic rocks only, Poisson’s ratios between 0 and 0.5 are reported in the related literature (Vutukuri et al. 1974, Hatheway & Kiersch 1986, Gercek 2007).

Further restrictions for Poisson’s ratios ν_{1} and ν_{2} of transversely isotropic rocks were found by Knops & Payne (1971)

(3.17)

and Amadei (1996)

(3.18)

Both inequalities were later derived by Exadaktylos (2001) assuming plane strain conditions.

Not all intact rocks can be described by isotropic or transversely isotropic elastic behavior. The general anisotropic body with no symmetry at all, exhibits 21 elastic constants (Lekhnitskii 1963). However, the effort needed to evaluate more than five elastic constants is unjustifiably high and therefore carried out only in exceptional cases.

Experience has shown that intact rocks with random grain structure may be considered as isotropic with regard to their strength. The shear strength of those rocks can be described by an approximation using the Mohr-Coulomb failure criterion:

(3.19)

_{IR}_{IR}_{IR}_{2}_{1}_{3}Fig. 3.5