Strength Analysis in Geomechanics Part 1 ppsx

Foundations of Engineering Mechanics Series Editors: V.I. Babitsky, J. Wittenburg Foundations of Engineering Mechanics Series Editors: Vladimir I. Babitsky, Loughborough University, UK Further volumes of this series can be found on our homepage: springer.com Wojciech, S. Aleynikov, S.M. Spatial Contact Problems in Geotechnics, 2006 ISBN 3-540-25138-3 Skubov, D.Y., Khodzhaev, K.S. Non-Linear Electromechanics, 2006 ISBN 3-540-25139-1 Feodosiev, V.I., Advanced Stress and Stability Analysis Worked Examples, 2005 ISBN 3-540-23935-9 Lurie, A.I. Theory of Elasticity, 2005 ISBN 3-540-24556-1 Sosnovskiy, L.A., TRIBO-FATIGUE · Wear-Fatigue Damage and its Prediction, 2005 ISBN 3-540-23153-6 Andrianov, I.V., Awrejcewicz, J., Manevitch, L.I. (Eds.) Asymptotical Mechanics of Thin-Walled Structures, 2004 ISBN 3-540-40876-2 Ginevsky, A.S., Vlasov, Y.V., Karavosov, R.K. Acoustic Control of Turbulent Jets, 2004 ISBN 3-540-20143-2, Kolpakov, A.G. Stressed Composite Structures, 2004 ISBN 3-540-40790-1 Shorr, B.F. The Wave Finite Element Method, 2004 ISBN 3-540-41638-2 Svetlitsky, V.A. Engineering Vibration Analysis - Worked Problems 1, 2004 ISBN 3-540-20658-2 Babitsky, V.I., Shipilov, A. Resonant Robotic Systems, 2003 ISBN 3-540-00334-7 Le xuan Anh, Dynamics of Mechanical Systems with Coulomb Friction, 2003 ISBN 3-540-00654-0 Nagaev, R.F. Dynamics of Synchronising Systems, 2003 ISBN 3-540-44195-6 Neimark, J.I. Engineering, 2003 ISBN 3-540-43680-4 Perelmuter, A.V., Slivker, V.I. Numerical Structural Analysis, 2003 ISBN 3-540-00628-1 Lurie, A.I., Analytical Mechanics, 2002 ISBN 3-540-42982-4 Mechanics of Periodically Heterogeneous Structures, 2002 ISBN 3-540-41630-7 Babitsky, V.I., Krupenin, V.L. Systems, 2001 ISBN 3-540-41447-9 Landa, P.S. Regular and Chaotic Oscillations, 2001 ISBN 3-540-41001-5 Wittbrodt, E., Adamiec-Wojcik, I., ISBN 3-540-32351-1 Jens Wittenburg, Karlsruhe University, Germany ISBN 978-3-540-44718-4 ISBN 978-3-540-37052-9 ISBN 978-3-540-37261-8 (Continued after index) Elsoufiev, S.A. Strength Analysis in Geomechanics, 2007 2007 Vibration of Strongly Nonlinear Discontinuous Awrejcewicz, J., Krysko, V.A., . Thermo-Dynamics of Plates and Shells, Mechanics of Structural Elements, 2007 Mathematical Models in Natural Science and Manevitch, L.I., Andrianov, I.V., Oshmyan, V.G. Dynamics of Flexible Multibody Systems, 2006 Krysko, A.V. Slivker, V.I. Strength Analysis in Geomechanics Serguey A. Elsoufiev With 158 Figures and 11 Tables Series Editors: f¨ur Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the mate- rial is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper ISSN print edition: 1612-1384 springer.com V.I. Babitsky Department of Mechanical Engineering Loughborough University Germany Loughborough LE11 3TU, Leicestershire United Kingdom J. Wittenburg Technische MechanikInstitut Kaiserstraße 12 Universit¨at Karlsruhe (TH) 76128 Karlsruhe package Cover-Design: deblik, Berlin A E LT X © Springer-Verlag Berlin Heidelberg 2007 Typesetting:Data conversion by the author\and SPiusing Springer Author: ISBN-10: 3-540-37052-8 ISBN-13: 978-3-540-37052-9 Library of Congress Control Number: 2006931488 SPIN: 11744382 62/3100/SPi-543210 Serguey A. Elsoufiev 44807 Bochum, Germany Vierhausstr. 27 Foreword It is hardly possible to find a single rheological law for all the soils. However, they have mechanical properties (elasticity, plasticity, creep, damage, etc.) that are met in some special sciences, and basic equations of these disciplines can be applied to earth structures. This way is taken in this book. It represents the results that can be used as a base for computations in many fields of the Geomechanics in its wide sense. Deformation and fracture of many objects include a row of important effects that must be taken into account. Some of them can be considered in the rheological law that, however, must be simple enough to solve the problems for real objects. On the base of experiments and some theoretical investigations the con- stitutive equations that take into account large strains, a non-linear unsteady creep, an influence of a stress state type, an initial anisotropy and a damage are introduced. The test results show that they can be used first of all to finding ultimate state of structures – for a wide variety of monotonous load- ings when equivalent strain does not diminish, and include some interrupted, step-wise and even cycling changes of stresses. When the influence of time is negligible the basic expressions become the constitutive equations of the plasticity theory generalized here. At limit values of the exponent of a hard- ening law the last ones give the Hooke’s and the Prandtl’s diagrams. Together with the basic relations of continuum mechanics they are used to describe the deformation of many objects. Any of its stage can be taken as maximum allowable one but it is more convenient to predict a failure according to the criterion of infinite strains rate at the beginning of unstable deformation. The method reveals the influence of the form and dimensions of the structure on its ultimate state that are not considered by classical approaches. Certainly it is hardly possible to solve any real problem without some assumptions of geometrical type. Here the tasks are distinguished as anti- plane (longitudinal shear), plane and axisymmetric problems. This allows to consider a fracture of many real structures. The results are represented by relations that can be applied directly and a computer is used (if neces- sary) on a final stage of calculations. The method can be realized not only in VI Foreword Geomechanics but also in other branches of industry and science. The whole approach takes into account five types of non-linearity (three physical and two geometrical) and contains some new ideas, for example, the consideration of the fracture as a process, the difference between the body and the element of a material which only deforms and fails because it is in a structure, the simplicity of some non-linear computations against linear ones (ideal plastic- ity versus the Hooke’s law, unsteady creep instead of a steady one, etc.), the independence of maximum critical strain for brittle materials on the types of structure and stress state, an advantage of deformation theories before flow ones and others. All this does not deny the classical methods that are also used in the book which is addressed to students, scientists and engineers who are busy with strength problems. Preface The solution of complex problems of strength in many branches of industry and science is impossible without a knowledge of fracture processes. Last 50 years demonstrated a great interest to these problems that was stimulated by their immense practical importance. Exact methods of solution aimed at finding fields of stresses and strains based on theories of elasticity, plastic- ity, creep, etc. and a rough appreciation of strength provide different results and this discrepancy can be explained by the fact that the fracture is a com- plex problem at the intersection of physics of solids, mechanics of media and material sciences. Real materials contain many defects of different form and dimensions beginning from submicroscopic ones to big pores and main cracks. Because of that the use of physical theories for a quantitative appreciation of real structures can be considered by us as of little perspective. For technical applications the concept of fracture in terms of methods of continuum me- chanics plays an important role. We shall distinguish between the strength of a material (considered as an element of it – a cube, for example) and that of structures, which include also samples (of a material) of a different kind. We shall also distinguish between various types of fracture: ductile (plastic at big residual strains), brittle (at small changes of a bodies’ dimensions) and due to a development of main cracks (splits). Here we will not use the usual approach to strength computation when the distribution of stresses are found by methods of continuum mechanics and then hypotheses of strength are applied to the most dangerous points. Instead, we consider the fracture as a process developing in time according to constitutive equations taking into account large strains of unsteady creep and damage (development of internal defects). Any stage of the structures’ deformation can be supposed as a dangerous one and hence the condition of maximum allowable strains can be used. But more convenient is the application of a criterion of an infinite strain rate at the moment of beginning of unstable deformation. This approach gives critical strains and the time in a natural way. When the influence of the latter is small, ultimate loads may be also VIII Preface found. Now we show how this idea is applied to structures made of different materials, mainly soils. The first (introductory) chapter begins with a description of the role of engineering geological investigations. It is underlined that foundations should not be considered separately from structures. Then the components of geo- mechanics are listed as well as the main tasks of the Soil Mechanics. Its short history is given. The description of soil properties by methods of mechanics is represented. The idea is introduced that the failure of a structure is a process, the study of which can describe its final stage. Among the examples of this are: stability of a ring under external pressure and of a bar under compression and torsion (they are represented as particular cases of the common approach to the stability of bars); elementary theory of crack propagation; the ultimate state of structures made of ideal plastic materials; the simplest theory of retaining wall; a long-time strength according to a criterion of infinite elonga- tions and that of their rate. The properties of introduced non-linear equations for unsteady creep with damage as well as a method of determination of creep and fracture parameters from tests in tension, compression and bending are given (as particular cases of an eccentric compression of a bar). In order to apply the methods of Chap. 1 to real objects we must intro- duce main equations for a complex stress state that is made in Chap. 2. The stresses and stress tensor are introduced. They are linked by three equilib- rium equations and hence the problem is statically indeterminate. To solve the task, displacements and strains are introduced. The latters are linked by compatibility equations. The consideration of rheological laws begins with the Hooke’s equations and their generalization for non-linear steady and unsteady creep is given. The last option includes a damage parameter. Then basic ex- pressions for anisotropic materials are considered. The case of transversally isotropic plate is described in detail. It is shown that the great influence of anisotropy on rheology of the body in three options of isotropy and loading planes interposition takes place. Since the problem for general case cannot be solved even for simple bodies some geometrical hypotheses are introduced. For anti-plane deformation we have five equations for five unknowns and the task can be solved easily. The transition to polar coordinates is given. For a plane problem we have eight equations for eight unknowns. A very useful and unknown from the literature combination of static equations is received. The basic expressions for axi-symmetric problem are given. For spherical coordi- nates a useful combination of equilibrium laws is also derived. It is not possible to give all the elastic solutions of geo-technical problems. They are widely represented in the literature. But some of them are included in Chap. 3 for an understanding of further non-linear results. We begin with longitudinal shear which, due to the use of complex variables, opens the way to solution of similar plane tasks. The convenience of the approach is based on the opportunity to apply a conformal transformation when the results for simple figures (circle or semi-plane) can be applied to compound sections. The displacement of a strip, deformation of a massif with a circular hole and Preface IX a brittle rupture of a body with a crack are considered. The plane deformation of a wedge under an one-sided load, concentrated force in its apex and pressed by inclined plates is also studied. The use of complex variables is demonstrated on the task of compression of a massif with a circular hole. General relations for a semi-plane under a vertical load are applied to the cases of the crack in tension and a constant displacement under a punch. In a similar way rela- tions for transversal shear are used, and critical stresses are found. Among the axi-symmetric problems a sphere, cylinder and cone under internal and exter- nal pressures are investigated. The generalization of the Boussinesq’s problem includes determination of stresses and displacements under loads uniformly distributed in a circle and rectangle. Some approximate approaches for a com- putation of the settling are also considered. Among them the layer-by-layer summation and with the help of the so-called equivalent layer. Short infor- mation on bending of thin plates and their ultimate state is described. As a conclusion, relations for displacements and stresses caused by a circular crack in tension are given. Many materials demonstrate at loading a yielding part of the stress-strain diagram and their ultimate state can be found according to the Prandtl’s and the Coulomb’s laws which are considered in Chap. 4, devoted to the ultimate state of elastic-plastic structures. The investigations and natural observations show that the method can be also applied to brittle fracture. This approach is simpler than the consequent elastic one, and many problems can be solved on the basis of static equations and the yielding condition, for example, the tor- sion problem, which is used for the determination of a shear strength of many materials including soils. The rigorous solutions for the problems of cracks and plastic zones near punch edges at longitudinal shear are given. Elastic-plastic deformation and failure of a slope under vertical loads are studied among the plane problems. The rigorous solution of a massif compressed by inclined plates for particular cases of soil pressure on a retaining wall and flow of the earth between two foundations is given. Engineering relations for wedge pen- etration and a load-bearing capacity of a piles sheet are also presented. The introduced theory of slip lines opens the way to finding the ultimate state of structures by a construction of plastic fields. The investigated penetration of the wedge gives in a particular case the ultimate load for punch pressure in a medium and that with a crack in tension. A similar procedure for soils is re- duced to ultimate state of a slope and the second critical load on foundation. Interaction of a soil with a retaining wall, stability of footings and different methods of slope stability appreciation are also given. The ultimate state of thick-walled structures under internal and external pressures and compression of a cylinder by rough plates are considered among axi-symmetric problems. A solution to a problem of flow of a material within a cone, its penetration in a soil and load-bearing capacity of a circular pile are of a high practical value. Many materials demonstrate a non-linear stress-strain behaviour from the beginning of a loading, which is accompanied as a rule by creep and damage. This case is studied in Chap. 5 devoted to the ultimate state of structures [...]... Spherical Coordinates 1 1 1 2 4 4 6 11 13 13 14 17 21 24 26 31 31 33 34 34 35 36 40 40 40 42 44 45 XII 3 4 Contents Some Elastic Solutions 3 .1 Longitudinal Shear 3 .1. 1 General Considerations 3 .1. 2 Longitudinal Displacement of Strip 3 .1. 3 Deformation... 11 1 4.2 .10 Some Methods of Appreciation of Slopes Stability 11 3 4.3 Axisymmetric Problem 11 7 4.3 .1 Elastic-Plastic and Ultimate States of Thick-Walled Elements Under Internal and External Pressure 11 7 4.3.2 Compression of Cylinder by Rough Plates 11 9 4.3.3 Flow of Material Within Cone 12 1 4.3.4 Penetration... 15 1 5.3.3 Cone Penetration and Load-Bearing Capacity of Circular Pile 15 4 5.3.4 Fracture of Thick-Walled Elements Due to Damage 15 6 6 Ultimate State of Structures at Finite Strains 16 1 6 .1 Use of Hoff’s Method 16 1 6 .1. 1 Tension of Elements Under Hydrostatic Pressure 16 1 6 .1. 2 Fracture Time... area in a surrounding of the investigated point and dP – the resultant of forces acting in it 1. 4 Main Properties of Soils 5 P n M l dP dA Fig 1. 1 Stresses in massif of soil P A a a p Fig 1. 2 Stresses in compressed bar The value and the direction of stress p depend not only on a meaning of external forces and the position of the point but also on the direction of a cross-section If vector p is inclined... P (Fig 1. 1) in a point M a part of the body under cross-section nMl is in an equilibrium with internal stresses p which are distributed non-uniformly in the part of the massif If they are constant in a cross-section the relation for their determination (Fig 1. 2) is p = P/A (1. 1) where A is an area of cross-section aa The non-uniformly distributed stresses can be found according to expression (1. 2) p... of important results in the geomechanics and other disciplines The works of Soviet scientists V Sokolovski, L Kachanov and A.Il’ushin in this field are also well-known The special interest have their investigations in the Theory of Plasticity of a hardening material which describes the real behaviour of a continuum and includes as 4 1 Introduction: Main Ideas particular cases the linear elasticity and... Main Ideas 1. 1 Role of Engineering Geological Investigations 1. 2 Scope and Aim of the Subject: Short History of Soil Mechanics 1. 3 Use of the Continuum Mechanics Methods 1. 4 Main Properties of Soils 1. 4 .1 Stresses in Soil 1. 4.2... The theoretical investigations that are actual now were contained in the works of French scientists (Prony, 18 10; Coulomb, 17 78; Belidor, 17 29; Poncelet, 18 30 and others) who were solving the problem of soil pressure on a retaining wall, and Russian academician Fussa (the end of the eighteenth century) who received a computational method for a beam on elastic foundation Until the beginning of the twentieths... thanks to the famous principle of Saint-Venant In calculations according to the scheme of the Elasticity Theory the main task is the determination of stress and strain fields An estimation of a strength has as a rule an auxiliary character since a destruction in one point or in a group of them does not lead to a failure of a structure The Galilei’s idea of an appreciation of the strength of the whole... 13 9 5.2.5 Wedge Under Bending Moment in its Apex 14 0 5.2.6 Load-Bearing Capacity of Sliding Supports 14 4 5.2.7 Propagation of Cracks and Plastic Zones near Punch Edges 14 6 5.3 Axisymmetric Problem 14 9 5.3 .1 Generalization of Boussinesq’s Solution 14 9 5.3.2 Flow of Material within Cone . nature”. Contents 1 Introduction: Main Ideas 1 1 .1 Role of Engineering Geological Investigations . . . . . . . . . . . . . . . 1 1.2 Scope and Aim of the Subject: Short History ofSoilMechanics 1 1.3 UseoftheContinuumMechanicsMethods. 18 7 Appendix A 19 1 Appendix B 19 3 Appendix C 19 5 Appendix D 19 7 Appendix E 19 9 Appendix F 2 01 Appendix G 203 Appendix H 207 Appendix I 209 Appendix J 211 Appendix K 213 Appendix L 215 Appendix M 217 Appendix. general use. Printed on acid-free paper ISSN print edition: 16 12 -13 84 springer.com V.I. Babitsky Department of Mechanical Engineering Loughborough University Germany Loughborough LE 11 3TU, Leicestershire United