This book is an introduction to numerical analysis in geomechanics and is intended for advanced undergraduate and beginning graduate study of the mechanics of porous, jointed rocks and soils. Although familiarity with the concepts of stress, strain and so on is assumed, a review of the fundamentals of solid mechanics including concepts of physical laws, kinematics and material laws is presented in an appendix. Emphasis is on the popular finite element method but brief explanations of the boundary element method, the distinct element method (also known as the discrete element method) and discontinuous deformation analysis are included. Familiarity with a computer programming language such as Fortran, C++ or Python is not required, although programming excerpts in Fortran are presented at the end of some chapters. This work begins with an intuitive approach to interpolation over a triangular element and thus avoids making the simple complex by not doing energy minimization via a calculus of variations approach so often found in reference books on the finite element method. The presentation then proceeds to a principal of virtual work via the well-known divergence theorem to obtain element equilibrium and then global equilibrium, both expressed as stiffness equations relating force to displacement. Solution methods for the finite element approach including elimination and iteration methods are discussed. Hydro-mechanical coupling is described and extension of the finite element method to accommodate fluid flow in porous geological media is made. Example problems illustrate important concepts throughout the text. Additional problems for a 15-week course of study are presented in an appendix; solutions are given in another appendix.

This book is about geoplasticity, solid mechanics of rock, jointed rock and soil beyond the domain of a purely elastic deformation. Plastic deformation is irreversible and begins at the limit to elasticity with any attempt at further loading. Stress at the limit to elasticity is "strength" which is described by a functional relationship amongst stresses, that is, by a yield function or failure criterion. Mohr-Coulomb, Drucker-Prager and Hoek-Brown criteria are well-known examples in geomechanics. Beyond the elastic limit, but still within the realm of small strain increments, a total strain increment is the sum of an elastic increment and a plastic increment. The elastic increment is computed through an incremental form of Hooke's law, isotropic or anisotropic as the case may be. Computation of the plastic part is at the core of any plasticity theory and is approached through the concept of a plastic potential. The plastic potential is a function of stresses and perhaps other material parameters such as plastic strain and temperature. Derivatives of the plastic potential with respect to stress lead to the plastic part of the total strain increment. If the yield criterion and plastic potential are the same, then the plastic stress-strain relationships are "associated rules of flow" and follow a "normality" principle. Normality is in reference to a graphical portrayal in principal stress space where the plastic strain increment is perpendicular to the yield surface. If the plastic potential and yield criterion are different, as is often the case in geoplasticity, then the rules of flow are "non-associated". Drucker's famous stability postulate implies normality at a smooth point on the yield surface, convexity of the yield function and other important features of plasticity theory in geomechanics. However, there is no point to proceeding to theoretical analyses without physical justification. Hence, the physical foundations for application of plasticity theory to rock, jointed rock and soil are examined in Chapter 2 of this book. A brief review of continuum mechanics principles is given in Chapter 3. Chapter 4 focuses on plane plastic strain and "sliplines". The technical literature is replete with numerous diagrams of sliplines, especially in discussions of foundations on soils, but the relevant mathematics is often lacking and with it genuine understanding. Examples illustrate application of theory to traditional geomechanics problems such as computation of retaining wall forces in soils, foundation bearing capacity of soil and rock, wedge penetration of rock under confining pressure and others. Brief discussions of anisotropy, visco-plasticity and poro-plasticity are presented in Chapters 6, 7 and 8. This book will be of interest to civil, geological and mining engineers, particularly those involved in reliable design of excavations and foundations beyond elasticity, especially in jointed rock.

This book is an introduction to numerical analysis in geomechanics and is intended for advanced undergraduate and beginning graduate study of the mechanics of porous, jointed rocks and soils. Although familiarity with the concepts of stress, strain and so on is assumed, a review of the fundamentals of solid mechanics including concepts of physical laws, kinematics and material laws is presented in an appendix. Emphasis is on the popular finite element method but brief explanations of the boundary element method, the distinct element method (also known as the discrete element method) and discontinuous deformation analysis are included. Familiarity with a computer programming language such as Fortran, C++ or Python is not required, although programming excerpts in Fortran are presented at the end of some chapters. This work begins with an intuitive approach to interpolation over a triangular element and thus avoids making the simple complex by not doing energy minimization via a calculus of variations approach so often found in reference books on the finite element method. The presentation then proceeds to a principal of virtual work via the well-known divergence theorem to obtain element equilibrium and then global equilibrium, both expressed as stiffness equations relating force to displacement. Solution methods for the finite element approach including elimination and iteration methods are discussed. Hydro-mechanical coupling is described and extension of the finite element method to accommodate fluid flow in porous geological media is made. Example problems illustrate important concepts throughout the text. Additional problems for a 15-week course of study are presented in an appendix; solutions are given in another appendix.

This comprehensive introduction to rock mechanics treats the basics of rock mechanics in a clear and straightforward manner and discusses important design problems in terms of the mechanics of materials. This extended third edition includes an additional chapter on Foundations on Jointed Rock. Developed for a complete class in rock engineering, this volume uniquely combines the design of surface and underground rock excavations and addresses: * rock slope stability in surface excavations, from planar block and wedge slides to rotational and toppling failures * shaft and tunnel stability, ranging from naturally-supported openings to analysis and design of artificial support and reinforcement systems * entries and pillars in stratified ground * three-dimensional caverns, with emphasis on cable bolting and backfill * geometry and forces of chimney caving, combination support and trough subsidence * rock bursts and bumps in underground excavations, with focus on dynamic phenomena and on fast and sometimes catastrophic failures. The numerous exercises and examples familiarize the reader with solving basic practical problems in rock mechanics through various design analysis techniques and their applications. Supporting the main text, appendices provide supplementary information about rock, joint, and composite properties, rock mass classification schemes, useful formulas, and an extensive literature list. The large selection of problems at the end of each chapter can be used for home assignment. A solutions manual is available to course instructors. Explanatory and illustrative in character, this volume is suited for courses in rock mechanics, rock engineering and geological engineering design for undergraduate and first year graduate students in mining, civil engineering and applied earth sciences. Moreover, it will form a good introduction to the subject of rock mechanics for earth scientists and engineers from other disciplines.

This book is about geoplasticity, solid mechanics of rock, jointed rock and soil beyond the domain of a purely elastic deformation. Plastic deformation is irreversible and begins at the limit to elasticity with any attempt at further loading. Stress at the limit to elasticity is "strength" which is described by a functional relationship amongst stresses, that is, by a yield function or failure criterion. Mohr-Coulomb, Drucker-Prager and Hoek-Brown criteria are well-known examples in geomechanics. Beyond the elastic limit, but still within the realm of small strain increments, a total strain increment is the sum of an elastic increment and a plastic increment. The elastic increment is computed through an incremental form of Hooke's law, isotropic or anisotropic as the case may be. Computation of the plastic part is at the core of any plasticity theory and is approached through the concept of a plastic potential. The plastic potential is a function of stresses and perhaps other material parameters such as plastic strain and temperature. Derivatives of the plastic potential with respect to stress lead to the plastic part of the total strain increment. If the yield criterion and plastic potential are the same, then the plastic stress-strain relationships are "associated rules of flow" and follow a "normality" principle. Normality is in reference to a graphical portrayal in principal stress space where the plastic strain increment is perpendicular to the yield surface. If the plastic potential and yield criterion are different, as is often the case in geoplasticity, then the rules of flow are "non-associated". Drucker's famous stability postulate implies normality at a smooth point on the yield surface, convexity of the yield function and other important features of plasticity theory in geomechanics. However, there is no point to proceeding to theoretical analyses without physical justification. Hence, the physical foundations for application of plasticity theory to rock, jointed rock and soil are examined in Chapter 2 of this book. A brief review of continuum mechanics principles is given in Chapter 3. Chapter 4 focuses on plane plastic strain and "sliplines". The technical literature is replete with numerous diagrams of sliplines, especially in discussions of foundations on soils, but the relevant mathematics is often lacking and with it genuine understanding. Examples illustrate application of theory to traditional geomechanics problems such as computation of retaining wall forces in soils, foundation bearing capacity of soil and rock, wedge penetration of rock under confining pressure and others. Brief discussions of anisotropy, visco-plasticity and poro-plasticity are presented in Chapters 6, 7 and 8. This book will be of interest to civil, geological and mining engineers, particularly those involved in reliable design of excavations and foundations beyond elasticity, especially in jointed rock.